Rapid Repair and Replacement Techniques For Transportation Infrastructure Damaged from Natural and Man-Made Disasters WisDOT Project ID: 1009-04-08 CMSC: WO 1.9
Final Report
September, 2009
Submitted to the Wisconsin Department of Transportation
Michael G. Oliva Lawrence Bank Robert Sivak
Construction and Materials Support Center University of Wisconsin – Madison Department of Civil and Environmental Engineering
Rapid Repair and Replacement Techniques for Transportation Infrastructure Damaged from Natural and Man-made Disasters
Project Report for Wisconsin Department of Transportation
Professor M. G. Oliva, Professor L. Bank, Robert Sivak University of Wisconsin Department of Civil and Environmental Engineering
September 2009
Report Summary The objective of the report summarized here was to investigate methods for reducing the impact that infrastructure damage, from serious manmade or natural incidents, would have on the transportation network in Wisconsin. To that end the following results and information are provided to the Wisconsin Department of Transportation. Section 2 summarizes an investigation into rapid repair/construction techniques that have been used by other States in responding to disasters and provides an overall view of possible bridge systems that can be used for rapid construction. Based on the experience of other agencies it is recommended that the State, possibly in co-operation with adjoining states, consider purchasing prefabricated temporary bridging components (such as an Acrow bridge) that can be used to immediately replace an essential transportation structure in an emergency. Section 3 examines alternate contracting procedures that could be used in an emergency to quickly get a design and contractor on the job to replace a damaged structure. The current process used by WisDOT in procuring emergency services meets the desired goal. It is recommended, however, that added emphasis be placed on the crucial time aspect. In the interest of the traveling public and freight movement the DOT should consider the use of incentives and disincentives in emergency contracts which would provide monetary bonuses for expedited repair work and penalties for slow response. An essential component in achieving a rapid response in an emergency situation is understanding that an emergency exists. Section 5 examines the factors that are involved in judging whether a disaster situation should receive emergency response. A decision making framework was developed that can be used by first responders to evaluate the structural situation and decide whether an emergency response is needed. This framework and a guide for its use are provided in a special section following this summary.
From examining State data it appears that one of the most common situations of damage in bridges involves vehicle impacts with bridge girders. Section 6 proposes a new method that could be used for very rapid replacement of precast concrete highway bridge girders that have been damaged by impact. Girder segments in 25 ft. and 10 ft. lengths would be precast and stored by the State. In an emergency these components could be quickly assembled, through post-tensioning, to create girders in the needed span length within a day or two. This would prepare a rapid substitute for the current method of having a precaster form, cast and cure a special girder. Designs have been developed for precast girder segments, and the needed post-tensioning to connect them into the desired length girder, to meet a variety of emergency girder replacement needs. Plan sheets for the girder segments are in a special section following this summary. Recommendations: 1. The WisDOT should consider purchasing temporary emergency bridge components that could be used to quickly bypass a damaged bridge. 2. When contracting for emergency construction services to replace critical structural components, the WisDOT should consider time based monetary incentives and dis-incentives in the contracts. 3. WisDOT should examine the proposed decision framework for designating emergency situations with regard to bridge/structural incidents, modify it as appropriate and make it available to district maintenance engineers for their use. 4. An alternate system of replacing damaged bridge girders, due to vehicle impact, should be considered. The proposed modular precast concrete girder system should be considered for implementation.
Highway Bridge Incident Emergency Decision Framework
A proposed decision making framework for examining a disaster situation and making a determination of whether it should be ranked as an “emergency” for the purpose of repair or reconstruction is provided on the following sheets. Instructions: Each of the 12 factors listed in the left hand column should be considered for the structure being examined. For each of the factors, one of the ranking values described with the factor should be assigned to the structure and entered into the value column. The value should then be multiplied by the weighting factor listed in the weight column and the resulting total value should be entered in the next column. All of the total values should be summed up and at the bottom of the sheet the sum should be entered. The maximum value (greatest emergency) the structure could receive is 224.7. The WisDOT should set a threshold value at which an emergency status would be applied. This should be done by applying the framework to previous bridge disasters, developing a rating factor, and correlating the ratings with past practice in deciding whether an emergency existed. A threshold value of 140-160 may be appropriate for establishing an emergency condition for a repair/reconstruction contract.
Factor
possible value
A. Feature on the Bridge Interstate highway or other freeway
weight 8.2
3.0
Non-freeway U.S. highway
2.5
Non-freeway state highway or arterial roadway County highway or collector roadway
2.0
Other local road
1.0
1.5
B. Feature under the Bridge
6.8
Interstate highway or other freeway
3.0
Railroad
2.5
Non-freeway U.S. highway
2.5
Non-freeway state highway or arterial roadway County highway or collector roadway
2.0
Buildings or parking lots
1.0
1.5
Other local road
1.0
Natural feature (river, ravine, etc.)
1.0
C. Location Urban Area (by population)
value
5.9 3.0
Suburban Area (by population)
1.5
Rural Area (by population)
0.5
6.1
D. AADT (per lane) after any lane closures > 20,000 /lane
3.0
10,000 – 20,000 / lane
2.5
5,000 – 10,000 /lane
2.0
3,000 – 5,000 /lane
1.5
< 3,000 /lane
1.0
Unknown
2.0
E. Truck Traffic
4.8
> 30 % trucks
3.0
20% – 30%
2.0
10% – 20%
1.0
< 10%
0.5
Unknown
1.5
F. Detour Length
4.7
> 40 miles
3.0
20 – 40 miles
2.0
10 – 20 miles
1.0
0 - 10 miles
0.5
G. Effect on Local Traffic
6.1
Major access route for hospital, fire, or police dept. Major access route for local school
3.0
All else
0.5
2.0
total
notes
H. Affect on local business Bridge serves major industry or shipping area All else
4.4 3.0 0.5
9.3
I. Amount of Damage Post-incident condition rating ≤ 3
3.0
Post-incident condition rating = 4
2.0
Post-incident condition rating = 5
1.0
Post-incident condition rating ≥ 5
0.5
J. Effect on Traffic Under Bridge (after damage) Entire roadway must be closed
3.0
Partial lane closures required
1.5
No lane closures needed
0.5
Building beneath must be vacated
3.0
Parking lot underneath must be closed off Water traffic underneath seriously affected
1.5 2.0
4.4
K. Condition of Bridge Before Incident Inventory rating (Item 66) < 045 (4.5 metric tons) Inventory rating of 045 - 136
3.0
Inventory rating of 136 - 272
1.0
Inventory rating > 272
0.5
2.0
L. Structural Complexity Components require significant time to fabricate Special equipment needed for construction Location poses difficulties in placement of equipment or materials All else
9.0
5.2 3.0 3.0 3.0 0
Total value (max possible of 224.7) = high value = emergency
Proposed Modular Precast Girder System for Rapid Girder Replacement One of the most common incidents occurring with highway bridges is vehicle impact with overhead bridge girders. The use of prefabricated modular precast girders, modules which could be rapidly assembled to form a girder in a desired length, could provide an optimal solution for quick replacement of damaged girders. The modules would include 25ft long end segments with draped prestressing strands and 10ft long interior segments with straight strands. The 10ft segments could be cut to any desired length to meet full girder length needs. The prefabricated modules are connected together to form a full length girder by using full length post-tensioning. The modules are combined to form a spliced girder. The precast modules would be built and held in a storage yard until and incident occurs. When needed, the modules could be post-tensioned together in less than a day and could be available for bridge erection within 2 days of an incident occurrence. The post-tensioned required for an array of span situations and required girder moment capacities has been calculated and is listed in the Appendix of the full report. Plan sheets for the precast girder modules (45” AASHTO I girder) are following.
77
78
i
Table of Contents Section 1: Introduction.................................................................................................................... 1 1.1 Problem Description ............................................................................................................. 1 1.2 Project Description................................................................................................................ 2 1.2.1 Desired Outcomes........................................................................................................... 2 1.2.2 Scope of Work ................................................................................................................ 3 1.3 Organization of the Report.................................................................................................... 4 Section 2: Rapid Construction Techniques and Materials.............................................................. 5 2.1 Introduction........................................................................................................................... 5 2.2 Concrete Superstructure Elements........................................................................................ 6 2.3 Steel Superstructure Elements .............................................................................................. 9 2.4 Substructure Systems .......................................................................................................... 11 2.5 Other Techniques or Methods............................................................................................. 11 2.6 Recommendations............................................................................................................... 14 Section 3: Emergency Contracting Procedures............................................................................. 16 3.1 Introduction......................................................................................................................... 16 3.2 Contracting Methods........................................................................................................... 16 3.3 Current WisDOT Policy ..................................................................................................... 24 3.4 Recommendations…………………………………………………………………………26 Section 4: Definition of Rapid Construction ................................................................................ 27 4.1 Introduction......................................................................................................................... 27 4.2 Definition of Rapid Construction........................................................................................ 27 Section 5: Definition of Emergency Situation .............................................................................. 29 5.1 Introduction......................................................................................................................... 29 5.2 Process ................................................................................................................................ 29 5.3 Survey Results .................................................................................................................... 31 5.4 A Formula to Assess Bridge Incidents ............................................................................... 35 5.5 Application of the Formula to an Actual Bridge Incident in Wisconsin ............................ 42 Section 6: Segmental Girder Design............................................................................................. 44
ii 6.1 Introduction......................................................................................................................... 44 6.2 Practicality of System ......................................................................................................... 44 6.3 Development of System...................................................................................................... 45 6.3.1 Dimensions (choosing the girder size and segment lengths)........................................ 45 6.3.2 Analysis and Design ..................................................................................................... 47 6.3.3 Final Product................................................................................................................. 55 Section 7: Summary and Recommendations ................................................................................ 56 7.1 Summary ............................................................................................................................. 56 7.2 Recommendations............................................................................................................... 57 Section 8: References.................................................................................................................... 59 Appendices……………………………………………………………………………………….62
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Section 1: Introduction 1.1 Problem Description A properly operating transportation infrastructure that is able to safely and efficiently move people and goods is critical to the State of Wisconsin. Serious incidents that damage infrastructure and cause highway routes to be out of service can have a serious effect on the State, in terms of both the economy and the way of life of the residents. Freight haulers using an impacted route may have to travel at a slower speed through the affected area or follow a lengthy detour to avoid the damage. The additional transit time incurred can lead to an unwanted increase in costs and shipment delays. The local population’s way of life can also be affected in much the same way: closed routes and/or detours cause inconveniences and time delays that the general public is forced to deal with. In addition, emergency services can be seriously affected, as emergency vehicles may be forced to find alternate routes to emergency calls or to hospitals. It is for these reasons that long term facility closures need to be reduced to days or weeks. Other state highway agencies have tried various rapid repair or replacement techniques in response to damage caused by natural or man-made disasters; however, these methods have not been cataloged or summarized, and documentation of their use exists mainly as project reports with limited distribution. Prefabricated elements and systems, on-site prefabrication, off-site construction, rapid curing materials, and innovate construction techniques are some examples of rapid response methods used in previous instances. These techniques and materials need to be cataloged and evaluated for use in Wisconsin with the most promising techniques summarized for use by the Wisconsin Department of Transportation (WisDOT).
2 Construction productivity is not the only factor that affects the speed of the repair or replacement of a damaged structure. The contracting and project delivery method used can have a significant effect on the effectiveness of the response to an incident. The use of the right contracting method for a project can ensure a successful response. Various alternative methods have been used by other agencies, and these methods should also be identified and compared to current WisDOT practices, so that WisDOT can take steps to allow for their use in emergency situations.
1.2 Project Description 1.2.1 Desired Outcomes Through this project, the project team aimed to accomplish the following goals: 1. Provide WisDOT with a current list of the various rapid repair or replacement techniques that have been used by other agencies. This list would provide a description of the various methods as well as contact information related to their use (when available). A recommendation would also be provided for the most promising techniques to be used by WisDOT. 2. Investigate the various contracting methods that could be used in an emergency response as well as the current practice followed by WisDOT. These methods would be evaluated, and a recommendation provided to WisDOT as to which method might be best able to provide a quick and efficient response for a damaged structure. 3. Further enable WisDOT to respond quickly and efficiently to situations involving damaged infrastructure. This would be accomplished through the development of a decision making framework for WisDOT to follow during their response to a situation.
3 4. Design and development ideas for an innovative method of replacing precast girders, which are often hit in collisions with overpasses.This new system would use prefabricated pre-cast post-tensioned concrete girders to allow a very rapid response to a girder damage situation. It is believed that through the accomplishment of these goals, WisDOT will be better able to respond to situations where a structure has been seriously damaged.
1.2.2 Scope of Work To achieve the above goals and ensure that WisDOT is able to respond quickly to infrastructure incidents, the following work and research was performed: •
A literature review of rapid bridge replacement and repair techniques, as well as various contracting methods used in the repair or replacement of damaged highway structures was completed.
•
The identified construction and contracting methods and their potential use in Wisconsin were evaluated.
•
A decision making framework for WisDOT to follow when responding to damaged infrastructure components was prepared. Creation of this frame work included: o Development of a definition of what is considered as “rapid” construction for the sake of this project and future actions of WisDOT; o Development of a definition of what is or isn’t a situation that requires an emergency response by WisDOT; This process includes a survey of relevant WisDOT personnel to determine what factors are considered when evaluating a situation, as well as their perceived importance when making a decision;
4 •
Development and design of a post-tensioned, pre-cast concrete girder system to be used by WisDOT in situations requiring very rapid girder replacement; and
•
Preparation of this final report with recommendations.
1.3 Organization of this Report The rest of this report is organized into three general parts. Part 1 deals with the review and evaluation of the existing techniques used by other state transportation agencies in rapid repair projects and their contracting methods. This includes a literature review of existing techniques, as well as their evaluation and a recommendation of the most promising techniques. Sections 2&3 of the report address these issues. Part 2 of this report introduces new methods that WisDOT could utilize in their response to damaged highway structures. This part includes a definition of rapid construction, a decision framework strategy for evaluating an emergency situation, as well as the development of a posttensioned, pre-cast concrete girder system for very rapid response to damaged girder incidents. Sections 4-6 include this material. The final part of this report, Section 7, is comprised of a summary and conclusions including recommendations for construction and contracting methods to be used by WisDOT in future situations.
5
Section 2: Rapid Construction Techniques and Materials 2.1 Introduction Various techniques and materials have been used in bridge projects to accelerate the construction process.
To identify and catalog the use of these materials for WisDOT, a
comprehensive literature review was conducted. Included in the review were such items as technical reports on completed construction projects, proceedings from bridge construction related conferences, news articles, publications released by the Federal Highway Administration (FHWA), American Association of State Highway and Transportation Officials (AASHTO) documents, reports from the Transportation Research Board (TRB), and technical reports prepared by consultants for state DOTs. During the literature review, various techniques and methods were identified that could minimize the effect of a damaged structure by allowing the route to open sooner than it would via ordinary construction methods, or even remain open during construction. Examples of these methods include pre-fabricated sections, off-site prefabrication, and temporary bridging. The various techniques and materials are described below.
When applicable and available,
information has been provided on past examples of their use, as well as contact information for the individuals or agencies involved. Please be aware that the contact information provided may only be as current as the source where it was found.
6 2.2 Concrete Superstructure Elements Pre-fabricated Deck Elements Placement of the concrete deck of a bridge is perhaps one of the most time-consuming phases of bridge construction. Formwork must first be erected between the girders, and then the reinforcement must be laid out, and then the concrete placed and cured. Construction crews must wait until the concrete has cured long enough to reach the desired strength before the formwork can be removed. There are pre-cast systems that can be used to minimize this process. Some of the headaches of this process can be avoided through the use of pre-cast concrete deck panels. Concrete deck panels can either be full-depth or partial-depth. Full depth panels would be fabricated to the desired thickness of the bridge deck, and could eliminate the need for any cast in place concrete. They typically do not require a riding surface; however, in some instances, a thin concrete or asphalt wearing surface may be applied (Hieber, Wacker, Eberhard, & Stanton, 2005). Partial-depth panels are much the same, but require an additional asphalt or cast-in-place concrete overlay to provide a proper wearing surface and/or additional strenght, although the panel itself can serve as stay-in-place formwork (Fouad, Rizk, Stafford, & Hamby, 2006). Panels are often connected to each other via grouted shear keys, and can be posttensioned together to provide continuity in the deck. To connect the panels to the girders, “shear pockets” or voids are cast into the panels at intervals dictated by the design. The pockets line up with mechanical connectors that are installed on the girders. The shear pockets are filled with grout to complete the connection and obtain composite action between the deck and girders (Fouad, Rizk, Stafford, & Hamby, 2006). WisDOT has used full-depth pre-cast deck panels for structure B-13-161 on I-90/94 as part of the Innovative Bridge Research and Construction (IBRC) Program (Markowski, 2005; Ehmke, 2006). The system developed in that project was
7 aimed at allowing rapid night replacement of portions of deck with full traffic on again during the following day. Another type of pre-fabricated deck system is the Exodermic™ deck system, which consists of a reinforced concrete slab composite with an unfilled steel grid. The concrete portion of this system can be pre-cast at the fabrication plant, or can be cast at the site, with the steel grid serving as a stay-in-place form (The D. S. Brown Company). The Exodermic™ system was used to successfully replace the deck of the Tappan Zee Bridge in New York during night time closures while allowing all lanes of traffic to be open during the day (Federal Highway Administration, 2008). Contact the Thruway Authority Public Affairs Office of the New York State DOT at (518) 436-2700 for more information on the use of this system.
Pre-fabricated Beam Systems The use of pre-cast concrete girders is obviously a familiar subject to WisDOT. It is difficult to drive anywhere in the state without driving on or under a bridge that utilizes this technology. There are other pre-cast elements, however, that can be used to expedite the repair or construction of a bridge. Bulb-tee girders, double-tee beams, channel beams, and box beams are examples of pre-cast elements that can be placed adjacent to one another to serve as both the beams and the deck. Or, the elements may be used as a stay-in-place formwork if a wearing surface is desired. Grouted keyways or welded plates may be used to connect adjacent elements, and transverse post-tensioning can be used as well, or as a way to create a composite structures (Hieber, Wacker, Eberhard, & Stanton, 2005; Fouad, Rizk, Stafford, & Hamby, 2006).
Pre-fabricated Slab Systems
8 A unique system for slab structures is the Poutre Dalle system. This system consists of inverted precast concrete tee-beams which are laid adjacent to each other, and then made composite by placing concrete between the webs and over the tops of the beams (Prefabricated Bridge Elements and Systems in Japan and Europe, Summary Report, 2004). Minnesota has developed a pre-cast slab system similar to the Poutre Dalle, but with some modifications (Bell II, French, & Shield, 2006). The Minnesota DOT should be contacted for more information on the use of this system. For slab bridges without beams, slab beams could be used to rapidly construct the superstructure of the bridge. These would also be placed adjacent to each other and connected by grouted shear keys, mechanical connections, or post-tensioning (Fouad, Rizk, Stafford, & Hamby, 2006). This type of system was used successfully with box beams in Colorado during the construction of the Mitchell Gulch Bridge, which was constructed completely of pre-cast components and constructed over one weekend. Contact Pam Hutton with the Colorado DOT at (303) 757-9118 (AASHTO Techology Implementation Group and Federal Highway Administration, 2004).
Total Superstructure Systems As another method to minimize construction time, a completely pre-fabricated concrete superstructure could be used. To accomplish this, a complete concrete deck is cast onto pre-cast concrete girders, either in a pre-cast concrete plant, or at or near the construction site. Once the concrete has cured and reached the necessary strength, the entire unit can be moved into place on the bridge using self propelled modular transporters or launching techniques (Fouad, Rizk, Stafford, & Hamby, 2006). This type of system was used on the I-10/Lake Pontchartrain Bridge
9 in Louisiana in 2002 and on the Richmond-San Rafael Bridge in California in 2004-2005. Contact Lynn Marsalone with the Louisiana DOT, at (504) 278-7457, for further information on the I-10 Bridge, and Usen Inyang with CalTrans, at (510) 231-7828, for information on the Richmond-San Rafael Bridge (AASHTO Techology Implementation Group and Federal Highway Administration, 2004).
2.3 Steel Superstructure Elements Decked Steel Girders Similar to the concrete beam systems mentioned above, decked steel girders could be used to minimize or eliminate the hassle of using a cast-in-place concrete deck. This system consists of traditional steel girders with a pre-cast concrete top flange. There are two types of decked girders available: those with a full depth flange, and those with a partial depth flange. The two types are much like the two types of pre-cast concrete deck panels. Full depth flange girders provide a full pre-cast concrete deck, while the partial depth flange girders provide a formwork for the required cast-in-place concrete. Adjacent beams can be connected through welded connections, shear keys, or post-tensioning, much like concrete deck panels or adjacent concrete beam systems (Cisneros, Fulginiti, Krause, Medlock, & Wandzilak, 2008). Although no information could be found on any projects utilizing this exact system, more information about this can be obtained by contacting Robert A. Cisneros of High Steel Structures, Inc.
He can be reached by phone at (717) 293-4086, or by e-mail at
[email protected].
Total Superstructure Units
10 Analogous to the pre-fabricated concrete superstructure systems described previously, the steel version consists of a complete concrete deck cast onto steel girders, either at the project site, or at a fabrication plant. Once properly cured, the unit is then placed into its final position on the bridge as one unit (SDR Engineering Consultants, Inc., 2005). This type of system was used to replace 45 of the 50 span of the I-95 Bridge over the James River in Richmond, VA. For this project, crews were able to replace one three-lane-wide span of the bridge with work that occurred only between 7:00 PM and 6:00 AM. For more information on this project, contact Dina
Kukreja
with
the
Virginia
DOT
at
(804)
786-5172
or
by
e-mail
at
[email protected] (Federal Highway Administration, 2008). A variation on this type of system is the Inverset type system. This system is fabricated by inverting steel girders and supporting them at their ends. Formwork is then suspended from the “bottom flanges” of the inverted beams (which are actually the top flanges) and concrete is poured into the forms. Once the concrete has cured, the system is flipped over such that the concrete slab is on top. The effect of this inverted casting is similar to pre-stressing, in that the deck is in a crack-resistant compressive state (Fouad, Rizk, Stafford, & Hamby, 2006). The Inverset type system was used to replace fire-damaged spans on the New York Thruway (I-87) Bridge over Central Park Avenue after a tanker truck exploded underneath the bridge in 1997 (Bai & Burkett, 2005). With the use of this system, crews were able to open one of the spans to traffic approximately one week after construction began. The New York State DOT can be contacted for more information.
11 2.4 Substructure Systems In some bridge incidents, it may not be the superstructure that is damaged, but the substructure of the bridge, such as the columns or piers, pier caps, or even the abutments. In instances like this, these bridge elements may need to be repaired or replaced quickly. This can be achieved through the use of pre-cast members. Bridge piers can be pre-cast, either as one single element, or as segments that are connected together on site by reinforcement splices or post-tensioning. The columns can be topped with a pre-cast or cast-in-place pier cap (Hieber, Wacker, Eberhard, & Stanton, 2005). The University of Wisconsin has conducted research into pre-fabricated bridge subsystems for WisDOT (Okumus, 2008), and a bridge has been constructed using precast abutments in Baldwin (B-55-217). Another Wisconsin bridge, B-06157, will be constructed with all precast pier caps in 2010. The Texas DOT has had experience with several projects that have incorporated pre-cast substructure elements (Federal Highway Administration, 2008). David Hohmann is the current director of the Bridge Division of the Texas DOT and may be able to provide additional information. He can be reached at (512) 4162183. Mary Lou Ralls, the former Texas State Bridge Engineer may also be able to provide additional input on the use of this technology in Texas (
[email protected]).
2.5 Other Techniques or Methods There are several other materials, methods, or systems that could be used to accelerate the repair or replacement of a structure, or alleviate the impact on the travelling public, that are not necessarily classified by the above categories. These methods are described below.
Temporary Bridges
12 Temporary bridges could be used to keep a route open to the public while work is completed on the damaged structure. The use of temporary bridges could help minimize the overall impact of a bridge incident to the travelling public by avoiding lengthy detours. In addition, temporary bridges can be assembled using only a few people (5-7 people) plus minimal necessary equipment (crane or excavator) (Smith, 2008). The temporary bridge could be placed over the site to allow work to continue on the structure underneath, as was done in New York in 1997 when a tanker truck exploded underneath the New York State Thruway (I-87) while travelling on the Central Park Avenue. For this project, two two-lane temporary bridges were erected at the site to allow traffic to flow on the Thruway while the new structure was designed and constructed (Bai & Burkett, 2005). This type of bridge does not have to be used for only temporary bridges; they can be used as permanent bridges as well. In St. John’s County, FL, an Acrow bridge system was used as a permanent replacement for a bridge over Tributary to Deep Creek. The total bridge replacement was able to be completed in 30 days because of the use of the Acrow Bridge (Leonard, 2008). For more information on this method, contact the New York State
DOT,
Michael
Leonard
of
HDR
Engineering
(904-598-8906
or
[email protected]), or one of the companies that fabricates temporary bridges, such as
Mabey
Bridge
&
Shore,
Inc.
(www.mabey.com),
Bailey
Bridges,
Inc.
(www.baileybridge.com), or Acrow Corporation of America (www.acrowusa.com).
Stay-In-Place Formwork As mentioned previously, a lot of time can be spent on the concrete casting process of bridge construction. But this time could be diminished slightly through the use of stay-in-place formwork. By not spending time removing formwork from the bridge, construction could be completed sooner. Materials that could be used include fiber reinforced polymer forms, sheet
13 metal or cardboard tubes (Spottiswoode, 2007). WisDOT has constructed several bridges using stay-in-place FRP formwork (Ringelstetter, 2006; Bae, 2008).
Although the stay-in-place
formwork was implemented in these projects as parts of other systems, they provide evidence that the use of stay-in-place formwork is possible in Wisconsin.
Staged Construction In situations where part of the damaged structure is still useable, a staged construction process could be employed to keep the route open while still allowing construction crews to complete work on the damaged portion. Then, when that work is completed, traffic can be diverted to the newly constructed portion, and if needed, the remainder of the bridge can be repaired. This method of construction could work well with the use of temporary bridging. This technique has been used to minimize traffic disruption during normal bridge or road construction projects in Wisconsin, but could be used successfully as an emergency repair technique.
Off-Site or Near-Site Prefabrication Instead of constructing a replacement structure in place and having the work disrupt traffic patterns, the replacement span(s) could be pre-fabricated away from its final location, either in a temporary staging yard near the site, or at a fabrication plant. The State of Utah has utilized this technique with the replacement of two spans of an I-215 overpass with one single span. The new span was constructed on temporary abutments near the site. The existing spans were removed using self-propelled modular transporters (SPMTs), and the new span was installed using the same method. The use of pre-fabrication and SPMTs allowed the entire process to be completed during a single weekend-long closure (Cho, 2007). CalTrans used a similar method to replace a portion of the roadway connecting the San Francisco-Oakland Bay
14 Bridge with Yerba Buena Island (Aragon, 2007). The respective DOTs for each project can be contacted for more information. The City of Phoenix AZ, erected an entire pedestrian bridge over I-17 at Maryland Avenue that was prefabricated in a fabrication plant 90 miles from the site. The entire bridge superstructure was trucked to the site from the plant, and lifted into place during a weekend-long closure of the freeway. The concrete deck was then poured during another weekend closure a week later (Cannon, Peterson, & Bales, 2007). Jerry Cannon of TranSystems Corporation’s Tucson office (520-792-2200) should be contacted for further information on this project. 2.6 Recommendations Through a comprehensive literature review of existing rapid construction and repair techniques, various systems and techniques that may be used to accelerate WisDOT’s response to damaged highway structures have been cited. Of those identified, there are a few methods that should be further investigated and possibly implemented by WisDOT.
WisDOT should
seriously consider the use of temporary bridge systems in specific appropriate incidents. The use of temporary bridging would allow for a route to stay open to the travelling public while work continues on the damaged structure, or while planning is underway for a replacement. This would allow WisDOT to use the best possible method to repair or replace the structure, while minimizing disruption to local traffic. WisDOT should contact adjoining states to examine whether it might be reasonable to cooperatively purchase and have temporary bridge components from one of the manufacturers stored and available for rapid use. Whenever possible, WisDOT should utilize pre-fabricated elements when repairing or replacing a damaged structure, including pre-fabricated beams and deck systems. Pre-fabricated elements can provide a higher quality project, and can minimize the amount of time spent on site.
15 WisDOT should also investigate the possibility of purchasing some of these pre-fabricated elements beforehand and stockpiling them for emergency use. It is important to note that it is not just the materials or construction techniques used that make a rapid response successful. In the response to the collapse of a portion of the MacArthur Maze in Oakland in 2007, CalTrans was able to reopen the damaged structure to traffic 25 days after the original incident, without using any special materials or construction techniques. Instead, the project was able to be completed so quickly because of a commitment to expedite the project from everyone involved: CalTrans personnel were available 24 hours a day; final plans for the replacement structure and contract documents were completed within 3 days; and meetings were held every day to avoid problems that could lead to delays (Franklin, Wahbeh, Speer, & Pohll, 2007). The key lesson to learn from this example is that proper planning and dedication is also necessary for a successful response. WisDOT should prepare a plan for dealing with emergencies, including communication amongst personnel, deadlines for the completion of project documents, and project meetings.
16
Section 3: Emergency Contracting Procedures
3.1 Introduction The materials and construction methods used on a project are not the only factors in a successful rapid response. First it is essential to be able to identify when an emergency exists and then a rapid personnel response is needed. The type of contract that is used in securing consultant and contractor services is also a critical component of the process. Therefore, it is important to identify the various contracting methods that might be used to accelerate WisDOT’s response to damaged structures. As part of this study various contracting methods for use in rapid construction are identified, and the strengths and weaknesses in their use are identified. Additionally, the current WisDOT policy for emergency contracting is reviewed, to determine what changes, if any, should be made. Lastly, recommendations are provided on further actions WisDOT can take to allow for an effective response to emergency situations.
3.2 Contracting Methods There are several contracting methods that could be used in the procurement of emergency repairs for a damaged structure. The various methods are described below. All of these methods and their pros and cons are summarized in Table 3-1, along with examples of projects that employed each method. The information provided in this section is garnered from personal knowledge, as well as discussions with Gary Whited (Program Manager, Construction and Materials Support Center, University of Wisconsin - Madison), and review of AASHTO’s
17 Primer on Contracting For the Twenty-first Century (Fifth Edition, 2006), and the Minnesota DOT’s Innovative Contracting Guidelines (December, 2005).
A+B Bidding In this method of contracting, potential contractors or consultants submit bids for the total cost of the project (“A”), as well as a time estimate (“B”). The time estimate is multiplied by a cost per unit of time, and this value is added to the project cost for an “adjusted cost.” The lowest adjusted cost is awarded the contract. The bids could also include a “C” factor, such as total road or lane closure time, and a cost rate could be applied to this number, similar to the total project time estimate, that would cause the total cost of the project to increase with an increase in the road or lane closure time. This method can be useful because it can account for the total time of the project, which in emergency repair is the most important consideration.
However, this method does not
necessarily assure that the bid with the shortest project time wins the contract.
Best-Value Contracting Similar to A+B contracting, this method involves the contractor or consultant providing an estimate of project cost and time. The difference is that bidders must also include their qualifications for the project. The contracting agency then provides a value to each of the selection criteria to attain a “technical score.” The bidder with the best technical score wins the contract. Time becomes an issue with this method, as project teams may require a longer time to prepare their submittals. There may also be more time needed for the evaluation of the bids, as
18 Description A+B Bidding - Potential contractors submit bids for cost as well as a time estimate - Could include a "C" - road closure time - A cost per day rate is applied to the time estimate, and that amount is added to the project cost bid for an "adjusted" cost bid - Lowest "adjusted cost" wins project Best Value Contracting - Contractor gives cost, time (similar to A+B), and qualifications or proposed plans - Qualifications/plans are scored, as well as cost and time estimate - Best ratio between cost/time and qualifications/plans scores is awarded contract Design-Build - Contractor/consultant (DB Team) bid on design and construction of project - Construction begins as final design is still being developed - Contracting agency typ. Designs project to 30% level
Pros
Cons
Projects
- Accounts for the length of construction - Puts some emphasis on time instead of cost alone - Could allow for shortest time to win project -Lowest "Total Cost" wins
- Still requires bidding process, which wastes time - Doesn't ensure fastest construction time
- I-35W Bridge; Minneapolis; 2007 - MacArthur Maze; Oakland, CA; 2007
- Allows DOT to put emphasis on construction speed
- Requires longer submittal process - Technical merits have to be scored before project can be bid - Can be subject to legal challenges by unselected bidders
- I-35W Bridge; Minneapolis; 2007
- Project time is reduced since construction begins before design is finished - Only one contract to bid - Selection criteria could emphasize speed of construction
- Still requires bidding or RFP process - Does not ensure fastest construction time
- I-40 Washout; North Carolina - I-5 Expansion; Washington - I-15 expansion; Salt Lake City
19
Design-Build-Finance-Maintain, could also be P3 (Public-Private Partnerships) - One contract is issued for all bridges - One contract covers all bridges, Would most likely include "Project Teams" or for a long time- Takes "Consortium"- Contractor is responsible for responsibility off DOT's hands design, construction and maintenance of all structures over period of contract- DOT reimburses contractor by a constant amount over the length of the contract Emergency Contractor/Consultant; Preferred Contractor; On-Call Contractor - Contract is issued with one consultant or - Moves bidding period from after contractor, making them responsible for any the incident to a time when it emergency repair work that may be needed doesn't affect project - Exact conditions of contract may vary from - Gets design/construction started situation to situation earlier - "Emergency Services Contract" - Still negotiate how it works when incident happens -Boiler plate and contract language done ahead of time - Can task contractor to do repairs/replacement (task order/ work order) - DOT could do entire design and contractor does work
- May stretch out single contractor- Doesn't necessarily cover emergency constructionNo mention of construction timePrivitization of bridges?- DOT loses control of who does work May not be most qualified contractor (if teams)
- Missouri DOT (Proposed, but not utilized)
- Is contractor expected to drop everything for emergency project? (-Master agreement states how soon they have to mobilize on project) - May take longer for projects due to resources available at time of incident
- Chesapeake Bay Bridge Railings; Maryland
20
Cost-Plus Contracting - Contractor is reimbursed for costs incurred, plus an agreed upon profit - Could be costs plus an hourly rate or so
- Can provide incentive to speed up construction - Contractor receives higher payment with faster construction
Negotiated Contract -Agency picks someone to get a bid, and if it's - Time can be saved with no reasonable, they negotiate contract bidding and expedited negotiationsAllows DOT to select properly qualified contractors Incentives/Disincentives - DOT awards or penalizes contractor based on project finish compared to a previously set date - Contractor gets additional payment if finished early, loses money if finishing late - Could also be inclusion of lane rental fees (paying a per day or hour fee for each lane that has to be closed during construction) - Possibly a part of A+B? - Typ. used with low-bid, but could be applied to all.
- Puts emphasis on construction time - Motivates contractor to work harder and finish early - Can lead to low bids (see MacArthur Maze)
- Requires bidding process - Provides no incentive to limit total costs
- I-40 Bridge over Arkansas River; OK
- Cost could be high- How do you ensure accelerated construction?
- Can increase monetary cost of project - How do you decide what the rates are? - Can take resources from other projects - In terms of contractor; would take men and machines from other projects for use on projects with higher incentives
Incentives: - MacArthur Maze; Oakland, CA; 2007 - I-70 Corridor; Indiana Lane Rental Fees: - I-35/40 Interchange, OK; - Denver Interchange, CO; - I-295 Bridge Deck Replacement, Portland, ME
21 the contracting agency has to review all of the qualifications as well. This method could also be subject to legal challenges by the unselected bidders.
Design-Build Instead of bidding the design and construction of a project separately, they can be bid on as a total package. Teams of contractors and consultants provide bids on the cost of the design and construction of the project. Typically, construction begins on the project before the final design is completed, but the contracting agency has the project designed to the 30% level of completion. The design-build method of contracting can save time by combining two bidding process into one. While time is saved in this part of the process, there is typically no assurance that the overall project is completed quickly, and some other measures must be taken to accomplish that end.
Design-Build-Finance-Maintain/Public-Private Partnerships Much like design-build, this contracting method incorporates the design and construction of a structure into one contract. This method, however, also includes the maintenance of the structure, and covers many structures instead of one. In this method, project teams compete for the responsibility of designing, constructing, and maintaining many bridges on the state’s highway network. The winning team would be responsible for project costs, but would be reimbursed by the state on a yearly or monthly basis. This contracting method could be helpful for emergency incidents, because the responsibility for the maintenance of the bridge is already assigned, and therefore, additional contract negotiations would not need to take place. The length of the project could be an issue,
22 since there is no limit for construction time, although this could incorporated into the contract through special fee structures or penalties. There could be situations where the contractor working on the repair or replacement is not the best qualified firm for the project, which may lead to quality issues.
Emergency Contractor/Consultant Agencies could also enter into an “Emergency Services Contract,” where a firm serves as an emergency or on-call contractor, and is responsible for emergency repairs or replacements. In this method of contracting, the contractor/consultant agrees to provide design or construction services for damaged structures in emergency situations.
General contract terms can be
negotiated before any incident occurs, and specific details can be worked out once an incident occurs. The use of an emergency contractor/consultant can be advantageous because it moves much of the contracting process from after the incident happens to before, when time is not much of a factor. This method could, however, lead to issues in terms of resources available to the contractor or consultant. If the incident occurs during a busy period, the contractor/consultant may be spread too thin across different projects, and may not be able to complete the project in a timely manner.
There may also be issues with how quickly the contractor/consultant can
mobilize to the site.
Cost-Plus Contracting This method of contracting entails the contractor or consultant being reimbursed for the costs incurred during the project, plus an agreed upon profit, such as a certain percentage of total cost. Cost-plus contracting could easily include incentives to speed up the project, such as
23 basing the amount of payment on total project time or on the completion date. However, this method still requires a bidding process, and does not necessarily limit the total cost of the project, although cost should not be a major factor in emergency repairs.
Negotiated Contracts Perhaps the simplest of all of the contracting methods mentioned here, negotiated contracts allow the DOT to simply contact a contractor or consultant and negotiate a contract to complete the needed repairs or replacements. A significant amount of time can be saved by using this method, since the entire bidding process is skipped. This method could also allow the most qualified firm to complete the project, because the DOT could select the firm from a preferred list, or based on previous experience or evaluations of their work. Costs for a project with this type of contract could be high, since a low bid is not sought. Incentives could easily be included to motivate the contractor or consultant to accelerate the project.
Incentives/Disincentives To provide motivation for project teams to complete their work early or with minimal interruption, monetary incentives (bonuses) or disincentives (penalties) could be included in the contracts described previously. Possible incentives could include a certain bonus for each day (or other division of time) that the project is completed ahead of schedule or just a lump sum bonus for finishing early. Disincentives could include a cost per day or even per hour penalty for how far past the target date the project is completed. Along these same lines, contractors could also be charged a “lane rental fee.” For every hour that a lane must be closed to complete the work, the contractor is charged a certain amount. This type of incentive could be very useful in situations where a damaged bridge spans a very busy roadway, such as an Interstate Highway.
24 The use of incentives and disincentives is a good way to promote accelerated construction. The prospect of additional income can motivate contractors to work at a faster pace, and could possibly lead to lower bids, because of the additional income from incentives. Minor problems could arise, however, such as determining the actual amounts for the incentives and disincentives, as well as determining the target date for completion. Also, the use of incentives could lead to contractors focusing their efforts on the more incentive-laden projects.
3.3 Current WisDOT Policy Currently, Wisconsin Statute 84.07(1b), “Emergency Repair and Protection of State Trunk Highways” allows WisDOT to enter into negotiated contracts with private contractors or agencies to handle the repair and replacement of damaged structures on state trunk highways. A section of the new Maintenance Manual (Kinar,2009) expands on this and provides the process followed to gain approval for these types of agreements. This process is generalized as follows: 1. The damaged structure is inspected and regional WisDOT personnel determine whether emergency repairs need to be completed. 2. The appropriate party submits justification for emergency repairs to the State Highway Maintenance Engineer and the Proposal Management Chief, including a description of the damage and any specialized repair techniques that may be needed, as well as cost and time estimates of the repair. 3. The State Highway Maintenance Engineer and Proposal Management Chief quickly review the submittal and decide on the method of repair, either on an emergency basis or as part of a letting.
25 4. After being informed that the repairs will be handled as an emergency, regional personnel contact three contractors qualified to perform the repairs. In a small number of situations, only one contractor need be contacted. On very large projects, up to six contractors are contacted. The contractors are contacted to determine their availability to do the work, as well as their ability to meet the required deadline for the work to be finished. 5. Following the initial contact, WisDOT personnel provide contractors with written information on the incident and the proposed repairs. 6. While the contractors prepare their proposals, a draft emergency repair agreement is prepared. 7. While the region is getting proposals from contractors, the BHO director sends the justification to the DTSD Administrator. If they agree, the justification is passed on to the WisDOT Secretary’s Office. If they also agree, it is passed on to the Governor’s Office for verbal approval. 8. Once verbal approval is received, the region is notified and is allowed to work with the chosen contractor to finalize the agreement. 9. The final agreement for the repairs is completed and sent to the contractor for signature. Work may begin once the contractor has signed and returned the agreement. 10. The two signed copies of the agreement are sent, along with the necessary paperwork, to the State Highway Maintenance Engineer, who then forwards it on to the Proposal Management Section for further processing. 11. Once the proper paperwork is completed, one copy is returned to the contractor, and one copy is filed away.
26 3.4 Recommendations Current WisDOT policy, outlined above, meets the desired goals of an emergency response to a damaged structure. Time is a major factor in emergency responses, and the current WisDOT process addresses that, though indirectly. The contracting process is expedited to allow the needed work to begin as quickly as possible. Through the negotiated agreement, WisDOT is able to set a timeline for the completion of the repairs but does not set incentives/penalties. The slow bidding process is avoided. Looking at the negative aspects of the various methods above, the current WisDOT practice avoids most of the weaknesses. The current process can account for the length of time for the project, and can avoid stretching a contractor too thin in resources by finding the firm that is most available to do the work. The cost may be a little higher by this method, but the time savings make up for the increase in cost. It can be concluded that the current emergency contracting process is adequate for an emergency response. Some suggested changes could make the current process work better. Extra emphasis should be placed on the time aspect of the whole process. Time is the critical aspect in an emergency situation. In the interest of the traveling public and the freight economy, the DOT should include the use of incentives and disincentives in the negotiated contract. The use of monetary bonuses and penalties could aide in expediting repair work and getting the system back in service quickly.
27
Section 4: Definition of Rapid Construction
4.1 Introduction While the literature review identified many useful materials, construction techniques, and contracting methods that can be used to rapidly respond to a damaged structure, there were still aspects that seemed to be missing. One of these was a definition of what, exactly, is considered rapid construction. To address this, a definition of rapid construction was defined for the purpose of this report and future actions by WisDOT. It is important to remember that in this report, the term “rapid construction” does not refer to just the construction aspect of a project, as is usually done in other projects. For this report, “rapid construction” refers to all factors including: problem identification, planning, design, and construction.
4.2 Definition of Rapid Construction For the majority of bridges “rapid construction” is a process that can be divided into two categories: emergency response and very rapid response. An emergency response will take several days to several weeks. These are typically necessary when a structure absolutely needs to be reopened as soon as possible. A very rapid response would last from a week to a couple months. This type of response would be used when it is not necessary for the replacement or repair of the structure to be done immediately. Although there are two separate types of response, it is possible that both may be used for the same bridge. An example of this would be an emergency response resulting in a temporary bridge being placed to open the route,
28 while a very rapid response is used to construct a permanent replacement for the damaged structure. It is not possible for these types of responses to be used on all structures addressed in this report. Some structures may have unique or difficult structural systems or have accessibility problems that would require additional attention and time. For these structures, this report focuses on a response process that may last longer than several months.
29
Section 5: Definition of Emergency Situation
5.1 Introduction Throughout the literature review, several reports about bridge incidents where an emergency was declared were encountered. While these reports explained what this designation meant in terms of procurement and contracting abilities, there was never an explanation of when, exactly, a bridge incident becomes an emergency situation, and what factors are considered to determine this. For this reason, it appeared to be beneficial to develop a definition of what exactly an emergency situation in Wisconsin would be, i.e. what factors are considered, and at what threshold the situation becomes an emergency. With this definition, it is hoped that WisDOT would be able to quickly determine when an emergency response should be used, and when a more conventional response is acceptable. This section describes this definition and its development.
5.2 Process The purpose of this section is to provide a standard definition of what constitutes an emergency situation when a damaged highway structure is involved. To prepare this definition, the Bridge Maintenance Engineers in each of the eight districts of Wisconsin were contacted. The Bridge Maintenance Engineers listed in Table 5-1 were selected since they were the individuals responsible for deciding whether or not an incident involving damaged infrastructure
30 is an emergency. Bruce Karow, Chief Structure Maintenance Engineer, was also contacted for his input. Individuals were contacted via electronic mail (e-mail) and by phone. Table 5-1 - Contacted Bridge Maintenance Engineers
Name
Position
Bruce Karow
Chief Structure Maintenance Engineer
Matthew Murphy
District 1 Bridge Maintenance Engineer
John Bolka
District 2 Bridge Maintenance Engineer
Dale Weber
District 3 Bridge Maintenance Engineer
Tom Hardinger
District 4 Bridge Maintenance Engineer
Dave Bohnsack
District 5 Bridge Maintenance Engineer
Greg Haig
District 6 Bridge Maintenance Engineer
Brock Gehrig
District 7 Bridge Maintenance Engineer
Allan Bjorklund
District 8 Bridge Maintenance Engineer
In the first phase of contact, the selected experts were surveyed on what factors they take into account when assessing a damaged structure. Once all people had been contacted, the factors were compiled into a list, which was then sent back to everyone for the second phase of contact. In this phase, they were asked to rank the factors in terms of how important each factor is when assessing a damaged structure. The factors were ranked on a scale of 1 to 10, with 10 meaning very important and 1 meaning not important at all. Individuals were allowed to use the same ranking more than once. These rankings were used to create a formula that can be used to determine whether or not an emergency exists. The results of the survey and rankings are discussed in the following section.
31
5.3 Survey Results Factors Considered After the survey of the Bridge Maintenance Engineers, the following list of factors (and explanations) was compiled: •
Feature on the Bridge o What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc)
•
Feature under the Bridge o What the bridge passes over (Another highway, railroad, terrain feature, etc.)
•
Location o Where the bridge is located. Is it in a rural, suburban, or urban area?
•
Average Annual Daily Traffic o The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable.
•
Truck Traffic o The percentage of the daily traffic that is composed of trucks.
•
Length of Detour and possible detour routes o How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (countryside, residential neighborhoods, school zones)?
•
Effect on local traffic
32 o Would closing the bridge significantly affect fire and medical response or school bussing? •
Effect on local businesses o Are the roadways involved providing primary access to a major business or shipping area? Would closing the bridge cause hardships for local businesses?
•
Amount of damage o Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open?
•
Effect on Traffic Under Bridge o Is there any danger of debris or material falling off of the structure and causing damage or injury? If over a roadway, can all lanes of traffic be left open under the bridge?
•
Condition of Bridge before Accident o Was it scheduled to be replaced in the near future? Would it be acceptable to load post the bridge until its scheduled replacement takes place, and at what load is the bridge posted?
•
Structural Complexity o Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or equipment? Would difficult construction techniques be needed to repair the bridge? Does the location pose difficulties in terms of getting materials or equipment into place?
33 Ranking of Factors The list was then sent out to the Bridge Maintenance Engineers for ranking. rankings given by each person are summarized in Table 5-2 below: Table 5-2: Rankings of Factors by Maintenance Engineers (10 = important)
Reviewers Factor 1
2
3
4
5
6
7
8
9
Feature On
10
10
8
6
7
8
7
8
10
Feature Under
7
8
5
6
5
8
7
8
7
Location
8
7
7
3
4
5
5
6
8
AADT
8
7
3
2
8
8
3
7
9
Truck Traffic
6
5
2
2
8
6
3
5
6
8
4
2
2
3
7
4
6
6
8
9
5
2
6
7
5
5
8
Effect on Business
6
3
3
2
2
6
5
5
8
Amount of Damage
7
10
7
10
10
10
10
10
10
8
6
8
10
9
10
10
10
10
8
5
4
8
1
4
2
7
1
7
3
5
8
3
7
7
6
1
Detour Length and Route Effect on Local Traffic
Effect on Traffic Under Condition Before Structural Complexity
The
34 The averages and other statistics are summarized in Table 5-3 below. The important factors with high averages included the feature on the bridge, amount of damage and effect on traffic under. Table 5-3: Statistics of Factor Rankings (Values rounded to the nearest tenth) high numbers = very important
Statistics Factor High
Low
Avg
Var
St Dev
Feature On (A)
10
6
8.2
2.2
1.5
Feature Under (B)
8
5
6.8
1.4
1.2
Location (C)
8
3
5.9
3.1
1.8
AADT (D)
9
2
6.1
7.1
2.7
Truck Traffic (E)
8
2
4.8
4.2
2.1
8
2
4.7
4.8
2.2
Effect on Local Traffic (G)
9
2
6.1
4.6
2.2
Effect on Business (H)
8
2
4.4
4.3
2.1
Amount of Damage (I)
10
7
9.3
1.8
1.3
10
6
9.0
2.0
1.4
Condition Before (K)
8
1
4.4
7.8
2.8
Structural Complexity (L)
8
1
5.2
5.7
2.4
Detour Length and Route (F)
Effect on Traffic Under (J)
35 5.4 A Formula to Assess Bridge Incidents A simple formula that involves the averages of the rankings in Table 5-3 was created for use in determining whether an emergency exists. To account for the various levels of importance for each of the factors, a weighting multiplier equal to the average ranking in Table 3 is applied. The highest ranking (worst emergency situation) that could be attained would have a value of 224.7.
Where:
A = value for the feature on the bridge B = Value for the feature under the bridge C = Value for the location of the bridge D= Value for average annual daily traffic And so on, as listed in Table 3 above.
Suggested values for each of the variables are provided in Table 5-4; however, these are only suggestions. Explanations of unusual values are provided in the text following the table. For each variable, a value should be assigned that corresponds to a certain element or level for that factor, and to avoid confusion the range of allowable values should be the same for all variables (i.e. between zero and three).
36
Table 5-4: Suggested Values for Variables
A. Feature on the Bridge
Interstate highway or other freeway Non-freeway U.S. highway Non-freeway state highway or arterial roadway County highway or collector roadway Other local road B. Feature under the Bridge Interstate highway or other freeway Railroad Non-freeway U.S. highway Non-freeway state highway or arterial roadway County highway or collector roadway Buildings or parking lots Other local road Natural feature (river, ravine, etc.) C. Location Urban Area (by population) Suburban Area (by population) Rural Area (by population) D. AADT > 69,000 veh/day (A4) 44,000 – 69,000 veh/day (A4) 8,700 – 44,000 veh/day (A3) 3,500 – 8,700 veh/day (A2) < 3,500 veh/day (A1) Unknown E. Truck Traffic > 30 % trucks 20% – 30% 10% – 20% < 10% Unknown F. Detour Length > 60 minutes 30 – 60 minutes 15 – 30 minutes 0 - 15 minutes As an alternative: > 40 miles 20 – 40 miles 10 – 20 miles
3.0 2.5 2.0 1.5 1.0 3.0 2.5 2.5 2.0 1.5 1.0 1.0 1.0 3.0 1.5 0.5 3.0 2.5 2.0 1.5 1.0 2.0 3.0 2.0 1.0 0.5 1.5 3.0 2.0 1.0 0.5 3.0 2.0 1.0
37 0 - 10 miles
G. Effect on Local Traffic Major access route for hospital, fire, or police dept. Major access route for local school All else H. Affect on local business Bridge serves major industry or shipping area All else I. Amount of Damage Bridge in danger of collapse or has collapsed Unusable, could collapse if hit again Bridge can be left partially open Bridge can remain in full operation J. Effect on Traffic Under Bridge Entire roadway must be closed Partial lane closures required No lane closures needed Building beneath must be vacated Parking lot underneath must be closed off Water traffic underneath seriously affected K. Condition of Bridge After Incident Bridge should be posted at less than 5 tons Bridge should be posted at 5-15 tons Bridge should be posted at 15-30 tons Bridge should be posted at over 30 tons L. Structural Complexity Components require significant time to fabricate Special equipment needed for construction Location poses difficulties in placement of equipment or materials All else
0.5
3.0 2.0 0.5 3.0 0.5 3.0 2.5 1.5 0.5 3.0 1.5 0.5 3.0 1.5 2.0 3.0 2.0 1.0 0.5 3.0 3.0 3.0 0
The following are some notes on the values for the variables: •
Values for the AADT were taken from the Facilities Development Manual, and correspond to the different classifications of state highways.
38 •
Both the AADT and Truck Traffic variables provide a value to use if the value for the roadways is not known.
•
AADT, Amount of Truck Traffic, Affect on Local Businesses, and Detour Length should be evaluated for both the roadway over the bridge and the roadway under the bridge when applicable. The worse (higher) of the two values should be used.
•
Detour Length can be evaluated in terms of time and distance. Time was chosen as the measurement because people may relate everything in terms of time (such as Milwaukee is about an hour east of Madison, instead of saying Milwaukee is about 70 miles east of Madison). Time also often has a direct relation to cost when measuring impacts on business travel or freight transportation. Detour length as a distance can be used when the speed of the detour is unknown. To keep the two alternatives equivalent to each other, a detour speed of 40 miles per hour was assumed.
•
The Effect on Traffic Under the Bridge should include lane closures required to perform all necessary work to stabilize the bridge, and any effect such work has on the traffic (such as temporary shoring) through construction of the replacement bridge.
•
The values for the Condition of the Bridge Before Incident were relatively difficult to define, the suggested load posting after the incident is a better measure though the authors suggest that this factor may not necessarily be relevant.
•
For Structural Complexity, the bridge elements referred to would be unique elements that need to be fabricated specially and may need to include a bidding process.
The Sufficiency Rating
39 In 1995, the Federal Highway Administration (FHWA) introduced Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s Bridges (“The Guide”) (Report No. FHWA-PD-96-001). “The Guide” details how different elements and features of bridges should be recorded and coded for their inclusion in the National Bridge Inventory. “The Guide” also presents the “Sufficiency Rating,” a formula that takes several features and attributes of a bridge into account to determine whether that structure is sufficient to remain in service. Typically, a bridge with a sufficiency rating less than 80 is eligible to have work done to bring it back to its proper level of service. The sufficiency rating formula provides “benchmarks” for certain factors and a value assigned to each level for use in the formula. This is very similar to the emergency factor formula developed above.
The factors above and their rating values were developed
independently of the FHWA’s sufficiency rating; however, our “bridge score” could also be calculated using the ranges of values for applicable factors in the sufficiency rating. Table 5-5, below, presents revised emergency factors based on data used in sufficiency ratings, with the altered factors in bold.
40 Table 5-5 – Values for Formula Variables Using Sufficiency Rating Values
A. Feature on the Bridge
Interstate highway or other freeway Non-freeway U.S. highway Non-freeway state highway or arterial roadway County highway or collector roadway Other local road B. Feature under the Bridge Interstate highway or other freeway Railroad Non-freeway U.S. highway Non-freeway state highway or arterial roadway County highway or collector roadway Buildings or parking lots Other local road Natural feature (river, ravine, etc.) C. Location Urban Area (by population) Suburban Area (by population) Rural Area (by population) D. AADT (per lane) after any lane closures > 20,000 /lane 10,000 – 20,000 / lane 5,000 – 10,000 /lane 3,000 – 5,000 /lane < 3,000 /lane Unknown E. Truck Traffic > 30 % trucks 20% – 30% 10% – 20% < 10% Unknown F. Detour Length > 60 minutes 30 – 60 minutes 15 – 30 minutes 0 - 15 minutes As an alternative: > 40 miles 20 – 40 miles 10 – 20 miles 0 - 10 miles
3.0 2.5 2.0 1.5 1.0 3.0 2.5 2.5 2.0 1.5 1.0 1.0 1.0 3.0 1.5 0.5 3.0 2.5 2.0 1.5 1.0 2.0 3.0 2.0 1.0 0.5 1.5 3.0 2.0 1.0 0.5 3.0 2.0 1.0 0.5
41 G. Effect on Local Traffic Major access route for hospital, fire, or police dept. Major access route for local school All else H. Affect on local business Bridge serves major industry or shipping area All else I. Amount of Damage Post-incident condition rating ≤ 3 Post-incident condition rating = 4 Post-incident condition rating = 5 Post-incident condition rating ≥ 5 J. Effect on Traffic Under Bridge (after damage) Entire roadway must be closed Partial lane closures required No lane closures needed Building beneath must be vacated Parking lot underneath must be closed off Water traffic underneath seriously affected K. Condition of Bridge Before Incident Inventory rating (Item 66) < 045 (4.5 metric tons) Inventory rating of 045 - 136 Inventory rating of 136 - 272 Inventory rating > 272 L. Structural Complexity Components require significant time to fabricate Special equipment needed for construction Location poses difficulties in placement of equipment or materials All else
3.0 2.0 0.5 3.0 0.5 3.0 2.0 1.0 0.5 3.0 1.5 0.5 3.0 1.5 2.0 3.0 2.0 1.0 0.5 3.0 3.0 3.0 0
42 5.5 Application of the Formula to an Actual Bridge Incident in Wisconsin This section will demonstrate the use of the formula to determine whether a certain incident requires an emergency response.
The bridge incident examined in this example
occurred with bridge B-13-089, which carries River Road over I-39/90/94 in the Town of Windsor. This structure was damaged April 30, 2008, when the boom arm on a truck was not in the fully lowered position as it passed under the bridge. Bridge girders over I-39/90/94 were damaged. Evaluation of the factors used in this example was done based on the inspection report for the structure, e-mail correspondence with DOT employees, internet-based mapping services, and personal knowledge of the area. Application of the formula is shown below in table format (Table 5-6). Applications of the formula to this and other bridge incidents from the Bridge Incident Response Database (Ghorbanpoor & Dudek, 2007; http://www.uwm.edu/CEAS/bird/) are provided in Appendix A.2. The incident involving structure B-13-089 received a score of 114.6 out of 224.7, the worst possible score (see Table 5-6). This score would be compared to an emergency response threshold (to be selected by WisDOT) to determine whether an emergency response is required for the repair/replacement process. If all factors were given a value of 3.0, 2.0, or 1.0, the incident would receive scores of 224.7, 149.8, and 74.9, respectively.
43
Table 5-6: Application of Emergency Formula to Structure B-13-089
Factor
Value
Weight
Total
Feature On
1.0
8.2
8.2
River Rd. – Local road
Feature Under
3.0
6.8
20.4
I-39/90/94 – Interstate highway
Location
0.5
5.9
2.95
Rural area
AADT
3.0
6.1
18.3
790 on, 65840 under. Larger value used.
Truck Traffic
1.5
4.8
7.2
Unknown
Detour Length
0.5
4.7
2.35
Approximately 10 minutes*
Effect on Local Traffic
0.5
6.1
3.05
No schools or emergency facilities nearby
Effect on Business
0.5
4.4
2.2
No major businesses nearby
Amount of Damage
2.5
9.3
23.25
In danger of collapse if hit again
Effect on Traffic Under
1.5
9
13.5
Lane closures needed to remove loose debris
Condition Before
3.0
4.4
13.2
Unknown
Structural Complexity
0.0
5.2
0.0
Standard concrete girder bridge
INCIDENT SCORE:
Notes
114.6
*Detour time was taken as the time needed to get from one side of the bridge to the other following the provided detour.
44
Section 6: Segmental Girder Design
6.1 Introduction As an additional method to help increase the speed of response for damaged structures, the development of a pre-cast, post-tensioned segmental concrete girder system to be used in the replacement of damaged concrete bridge girders was investigated. The system would consist of concrete bridge girders cast in segments, which would be stored at some location until their use in an emergency. One or more states adjacent to Wisconsin might join in storing these critical components and they could be shared in a disaster situation. If a precast concrete bridge girder was damaged by a vehicle strike below, the pre-cast segments could quickly be post-tensioned together to form one single girder (or multiple girders, if necessary) that would be transported to and installed in place of the damaged girder(s). A system such as this could reduce the amount of time that a damaged structure is out of service by 1 to 10 days, thus minimizing the impact of the closure on the local and state economy.
6.2 Practicality of System To determine if such a system would be practical for use in Wisconsin, engineers at Spancrete and County Materials, the two main suppliers of pre-cast concrete girders for bridges in Wisconsin, were contacted for information on their emergency production of pre-cast concrete girders. From these discussions, it was determined that approximately one week to one month would be needed to deliver a pre-cast girder to the site of a damaged bridge depending on other
45 demands existing at a plant (Holien, 2008; Kirchner, 2008). Based on this information, the team considered the use of the segmental girder system to be quite practical, as it would allow the needed girders to be delivered in only a couple days instead of a week or more.
6.3 Development of System 6.3.1 Dimensions (choosing the girder size and segment lengths) The first step in the development of the segmental girder system was to determine the size of the girder and the lengths of the segments. The first thought was to choose the size of the girder based on the availability and complexity of the formwork (i.e. if modifications need to be made to other formwork in order to use it for a different girder type) for each size and type of girder. However, in discussions with the producers, it was clear that each of the girder types and sizes have their own formwork. In addition, it was difficult to say which form size would be most readily available in an emergency, because that would change as the pre-casters work on different projects (Holien, 2008; Kirchner, 2008). As a second method of deciding on a girder size, the team looked at the number of bridges in the State using each girder size.
Data on girder usage was obtained from the
WisDOT’s Highway Structures Information System, with help from Travis McDaniel at WisDOT. The number of State bridges using each size of girder can be seen in Figure 6-1 below. The “UNKNOWN” category refers to slab bridges or bridges where data on the girders is not known. Based on this data, the 45 inch deep concrete girder was chosen to provide an example design of a segmental girder system because it is most commonly used and thus likely to suffer damage in the future.
46
Figure 6-1 Distribution of precast concrete girder sizes in Wisconsin
Data on the various span lengths of the bridges utilizing 45 inch deep girders was also obtained in the same manner, and can be seen in Figure 6-2 below. These numbers were used to determine the lengths of the individual segments, as well as the span lengths that the system was designed for. Using this data, the team decided to analyze three possible span lengths: 60 feet, 80 feet, and 100 feet. These span lengths represented the upper and lower limits on span length for the 45-in girder (based on Table 19.1a of the November 1999 edition of the Wisconsin Bridge Manual), as well as a length that fell within that range. The replacement girder would be built up from a series of segments to achieve the desired total length. The standard segment lengths chosen were 25 feet for the end segments, to provide room for strand drape, and 10 feet for the interior segments. To allow the system to be used for spans lengths that are not multiples of 10 feet, one of the 10 foot sections could be cut to the needed length.
47
Figure 6-2 - Distribution of span lengths for 45 inch precast I-girder in Wisconsin
6.3.2 Analysis and Design
Assumptions The segmental girder system was designed according to the provisions of the Wisconsin DOT’s Bridge Manual (2008), and AASHTO’s LRFD Bridge Design Specifications (2004). As mentioned above, three different bridge lengths were analyzed to determine required capacity: 60 feet, 80 feet, and 100 feet. All three bridges were designed as single spans. An exact bridge width was not assumed, however, the bridges were assumed to have 5 girders for load distribution. The design process focused on interior girders. Concrete deck thickness was assumed to be 8 inches, with no future wearing course. It was also assumed that the precast girder segments would be in storage for at least one year, for loss calculations, before being put into use.
48 The girder was designed as having three post-tensioning ducts, two of which would contain a maximum of 12 strands, and one with a maximum of 7 strands. The actual number of strands required in a particular bridge replacement depends on the bridge length and loading. In the middle portion of the girder, these ducts are straight and stacked on top of each other, with the top-most duct being the 7-strand duct. The numbering convention for the ducts is as follows: duct one is the lowest duct, duct two is the middle duct, and duct three is the top duct. In the end segments the ducts are raised up in the girder section to provide the required spacing between end anchorages, and to control the concrete stresses in this portion of the beam. The post-tensioning tendons were assumed to be composed of 0.6 inch diameter, 270 ksi low-relaxation steel strand. The duct sizes were selected based on the thickness of the girder web, the required concrete cover that would be needed, and the diameters of the ducts themselves. The duct sizes were based on technical data for VSL’s post-tensioning ducts (http://www.vsl.net/). The posttensioning strands will be anchored at the end of the girder using an anchorage system similar to what is shown in Figure 6-3 below. The system was designed based on a Type E Stressing Anchorage from VSL. Other types or brands may be used, but the designer will need to check that spacing requirements are met between the separate anchorages and between the anchorages and the edge of the concrete.
49
Figure 6-3 - An example of a post-tensioning anchorage (Source: VSL)
Analysis The dead loads applied in the analysis model included the weight of the girders, concrete deck, steel braces between girders, and concrete parapet walls. The live loads applied to the structure were a 640 pound per foot “lane load” and the HL-93 truck and tandem loadings. The moments and shear forces resulting from the loads were determined using the PCBRIDGE™ computer program, which calculated the maximum moments at specific intervals over the length as the loads were moved across the span. The data from PCBRIDGE™ was brought into spreadsheets, where load and distribution factors were applied, and loads were added together to form the proper load cases. The factored loads from the spreadsheets were used to design the girders. Maximum moments and shear forces for the two controlling LRFD limit states and span length are shown in Table 6-1.
Table 6-1 Maximum girder moments (ft-kips) and shear forces (kips) for each load case and span length
60’
80’
100’
Moment
Shear
Moment
Shear
Moment
Shear
Service I
1549.47
122.64
1917.13
118.87
1969.87
101.05
Service III
1261.93
99.59
1573.49
97.09
1638.09
83.34
Strength I
3661.17
278.13
4791.68
279.98
5178.02
246.13
50
Moment Design Using the results of the analyses, girders for the three span lengths were designed. Since the system is post-tensioned, a large part of the design involved the calculation of the number of strands needed, and the amount of stress loss in the strands. There were two types of pre-stress losses: instantaneous losses, which include friction loss, elastic shortening loss, and anchorage slip; and long term losses, which include losses due to creep and shrinkage of the concrete, and relaxation of the strands themselves. An initial loss estimate of 7.9% was assumed to obtain a preliminary number of strands to use and perform initial stress checks. This estimate is lower than what may typically be used as a preliminary estimate for normal post-tensioning. Since the girder segments would be in storage for a significant length of time, the effects from concrete creep and shrinkage would be significantly reduced. Once an arrangement of strands was found that provided the needed capacity and kept the concrete stresses within the allowable limits (.268 ksi in tension, 4.8 ksi in compression), the actual losses in the strands were calculated. These calculations followed the refined method in the AASHTO LRFD specification. All losses were calculated for each duct, and the values given are per individual strand. The long term losses were calculated twice, once for the time period between the stressing of the strands and deck placement (assumed to be 10 days), and once more for the time from deck placement to the end of the structure’s useful life (assumed to be 50 years), due to the change in geometry from the deck. Descriptions of the types of losses and the results of the loss calculations are summarized below:
Instantaneous Losses
51 Anchorage Slip: It was assumed that there would be no slip at the anchorages, due to the jack type that would be used. Therefore, there would be no loss in the pre-stress as a result. If a jacking process is used that cannot provide zero anchorage slip, this loss would need to be calculated.
Friction: Friction loss results from the strands rubbing against the sides of the ducts as they are stressed, particularly in curved ducts. The amount of stress loss per strand through the end segments is approximately 3 ksi in duct one, 7 ksi in duct two, and 11 ksi in duct three. In the interior segments, the loss can be taken as .04 ksi per foot of length in all three ducts.
Elastic Shortening: As some strands are stressed, they can cause a change in stress in strands that have already been seated. No elastic shortening occurs from a strands own stressing, since the concrete girder is exposed to the resulting stresses before the strands are seated. The amount of elastic shortening loss varies between ducts, and depends heavily on the number of strands used in the design, so this loss should be calculated for each specific case. However, if the prescribed jacking pattern is used (duct two, duct one, duct three), there should never be any elastic shortening loss in duct three. Duct one would experience a maximum loss per strand of 9.2 ksi, and duct two would have a maximum loss of 19.1 ksi, if all three ducts contained the maximum amount of strands.
Long Term Losses, Between Stressing and Deck Placement Shrinkage: As concrete ages, it loses water, which causes a change in volume. This then causes the post-tensioning strands to shorten, which reduces the stress in the strand. The actual amount
52 of loss per strand depends on the number of strands used; however, the loss should be less than .015 ksi for the aged segments and short time interval between stressing and decking.
Creep: When a sustained load (such as the post-tensioning) is applied to concrete, the concrete never really stops deforming. This change in the concrete length causes a shortening of the tendons also, and thus, a loss of stress. Creep loss is dependant on the concrete stiffness and the number of strands used or amount of prestress, as more prestress would mean a higher creep. The maximum creep loss between stressing and deck placement is 3.7 ksi.
Relaxation: As the tendons remain at a sustained elongation the strands lose stress. The use of low-relaxation strands was assumed for this design, and, per the AASHTO design specifications, our relaxation loss is 1.2 ksi.
Long Term Losses, After Deck Placement Shrinkage: Once again, the magnitude of the loss due to added girder shrinkage depends on the number of strands; however, a maximum additional loss of .5 ksi can be expected.
Creep: The addition loss due to creep is also dependant on the number of strands, but a maximum additional loss of 7.6 ksi can be assumed.
Relaxation: The additional relaxation loss can again be taken as 1.2 ksi.
53 Shrinkage of Deck Concrete: Shrinkage of the concrete in the deck can actually cause a gain in the tendon stresses, due to the eccentricity of the deck and the tendons. A maximum gain of 8.1 ksi can be assumed
After calculation of these losses, concrete stresses at various points along the girder were checked to ensure that they still fell within the desired limits. Once it was determined that these limits were met, the ultimate moment capacity was calculated for the three spans and their strand arrangement. The moment capacities for selected strand arrangements are presented below in Table 6-2. The moment capacities for all strand arrangements are provided in Appendix B.5. Note that there are two values for each moment capacity: one for the 60 and 80 foot spans, and one for the 100 foot span. The reason for this is because of the difference in the effective flange width provided by the bridge deck used in the calculations. Table 6-2 - Selected Strength-1 Moment Capacities
Number of strands in each duct
Moment Capacity (ft-kip)
Duct 1 (Bottom)
Duct 2 (Middle)
Duct 3 (Top)
60/80 ft. span
100 ft span
1
0
0
251.80
251.54
0
1
0
234.32
234.06
0
0
1
218.58
218.31
12
0
0
2934.99
2898.37
0
12
0
2725.38
2688.88
0
0
7
1502.31
1489.64
12
12
7
6773.74
6548.63
54 Shear Design Shear capacity and required reinforcement was calculated for each of the three span lengths. As can be seen in Table 6-1 above, the 80 foot span had the highest girder shear force in the Strength 1 limit state. It would be expected that the longer span would have the higher shear forces, due to the higher DL and longer span. This is true for the total lane load shear. Because the girders are spaced closer on the longer spans, the individual girder shear decreases, causing the shear force to be lower than for shorter spans with wide girder spacing. For all three span lengths, the post-tension tendons contributed to the shear capacity significantly where draped, and the required spacing for shear stirrups was large. Instead the design spacing was at the maximum allowed spacing for shear reinforcement, which was 12 inches. This stirrup spacing was checked to insure it provided adequate interaction with the concrete deck to provide composite action. The 12 inch spacing proved adequate for all three span lengths.
Anchorage Zone Design In post-tensioned systems, special attention needs to be paid to the area near the anchorages, due to bursting forces that can result from the high compressive stresses in this area. Additional reinforcement must be provided in this area to handle these forces. This area will also need a larger cross-section to accommodate the anchorage hardware. In this case, the anchorage zone cross-section is a 22 inch by 45 inch extending up to 4 feet in from the end of the girder. At this point, there is a 1 foot long “transition section” where the cross-section tapers back to the normal girder shape. The anchorage zone is treated as two different areas: the local zone and the general zone. Reinforcement in the local zone, which is the area near the anchorages, is detailed by the post-
55 tensioning system provider, as it was in this case. In the example local zone, the additional reinforcement is provided by spiral reinforcement around the embedded portion of the anchorages, as detailed by VSL. Reinforcement needs for the general zone, which entails everything else in the anchorage zone, was determined with a strut-and-tie model. In the strut-and-tie model, the “struts” were compression members that were positioned to distribute compressive forces from the anchorages across the entire girder cross section. The “ties” were tension members that ran perpendicular to the struts, and the forces in these members represented the bursting forces that could occur. The ties were provided by reinforcement in the form of stirrups, similar to that used to resist shear forces. As a result, in the anchorage zone, the stirrups are spaced 3 inches apart as needed for ties instead of 12 inches. 6.3.3 Final Product The final result of this design is a post-tensioned, segmental girder system, with two types of segments. The interior segments are 10 feet in length (or shorter), and 45 inches tall. The cross-sections of these sections are the same as the cross-sections of a typical 45 inch concrete I-girder. Each segment has three straight post-tensioning ducts. Shear reinforcement in the form of stirrups in the webs are spaced 12 inches apart. Two longitudinal reinforcement bars run along the bottom of each segment to reinforce the segment during transportation. The end segments are the same as the interior segments, except the post-tensioning ducts are draped for a portion of these segments. In addition, the last 4 feet of the end segments have a larger cross-section and a closer stirrup spacing to account for the anchorage zone. Exact dimensions are given in the plans in Appendix B.4.
56
Section 7: Summary and Recommendations
7.1 Summary When a bridge is damaged by a man-made or natural incident, it is important that the structure be repaired and the routes quickly reopened to traffic. If this is not done, the effects on the travelling public and the state economy could be significant. This report is intended to provide WisDOT with tools needed to respond to bridge incidents and open the roadways to traffic quickly. A literature review was conducted to identify the different elements, systems, and construction techniques that have been used to accelerate the bridge construction process. Information was provided about each of these, as well as some examples of their use, and contact information (when available) for the agencies or individuals involved, in case WisDOT would like additional information. Through this same literature review, certain contracting methods that could expedite construction projects were identified. The positives and negatives of each contracting method were also identified, and these methods were compared to the current contracting policy followed by WisDOT. To further provide WisDOT with tools for reacting to incidents, a decision making framework for the agency to follow when determining emergency response procedures was developed. This framework allows the responding persons to inspect the damaged structure and determine relatively quickly whether or not an emergency response is needed. From that point, WisDOT could utilize their normal approach or the various techniques and materials identified
57 through the literature review to repair or replace the damaged structure and open the routes to traffic. As an additional tool for WisDOT (and possibly adjoining states) to use in a disaster response, a post-tensioned pre-cast segmental concrete replacement girder system was developed. This system could be produced and stored ahead of time, and then quickly assembled to make girders of various lengths that can be transported to wherever they are needed and installed within a matter of days. With this report the Wisconsin Department of Transportation has added information and tools to respond to bridge incidents and reopen damaged structures quickly and efficiently.
7.2 Recommendations WisDOT should examine the techniques suggested in this report for use in response to future bridge/structure incidents. Specifically, the agency should consider the purchase of one or more temporary bridge systems. The use of these systems could allow the routes affected by a damaged structure to reopen very quickly and remain open during most, if not all, of the repair or replacement process. WisDOT is also strongly recommended to utilize prefabricated elements for disaster response whenever possible, because the use of these elements can minimize route closures and the amount of time that construction crews are on site. The current WisDOT policy for emergency contracting is adequate. A list of alternate contracting methods that may be used in emergency response are provided and compared these to the current WisDOT contracting policy.
WisDOT should examine alternate contracting
techniques for emergencies and particularly place a high emphasis on the time aspect of the
58 responser. WisDOT should consider the use of incentives and/or disincentives in their contracts to further motivate the contractors or consultants to complete rebuilding projects fast. A decision making framework has been developed and is described in this report. WisDOT should apply the procedure described to previous disaster situations and develop their own calibrated definition for a “threshold score” out of the 224.7 available to define when a situation should be designated an emergency. With this framework, WisDOT would be able to quickly determine the type of response that should be initiated for a bridge incident, either an emergency or an accelerated standard response, and thus begin the proper actions sooner, leading to the damaged structure being put back into use earlier. WisDOT should modify this framework to fit their needs, and distribute it to the necessary personnel to begin implementing it as soon as possible. A newly developed post-tensioned, pre-cast segmental girder system that could be assembled and installed in only a few days to replace damaged precast bridge girders has been presented.
It is recommended that WisDOT review this system, modify aspects of it as
necessary, and judge whether this system could be of benefit to Wisconsin or in partnership with adjoining states. This system could significantly assist WisDOT in their response to frequently damaged girders due to vehicle collisions.
59
Section 8: References AASHTO Techology Implementation Group and Federal Highway Administration. (2004). Prefabricated Bridges 2004. Retrieved October 21, 2007, from http://www.fhwa.dot.gov/bridge/prefab/2004best.pdf American Association of State Highway and Transportation Officials. (2004). LRFD Bridge Design Specifications. American Association of State Highway and Transportation Officials. (2006). Primer on Contracting for the Twenty-first Century. Retrieved April 20, 2009, from http://www.transportation.org/sites/construction/docs/Primer%20on%20Contracting%202006.pd f Aragon, G. (2007, September 10). Bay Bridge Rehab Gets a Lift With 6,500-Ton Deck Section. Engineering News-Record , 259 (9), p. 14. Bae, H.-U. (2008). Design of Reinforcement-Free Bridge Decks with Wide Flange Prestressed Precast Girders. PhD Dissertation, Department of Civil and Environmental Engineering, University of Wisconsin - Madison. Bai, Y., & Burkett, W. R. (2005). Rapid Bridge Replacement After an Extreme Event. TRB 2005 Annual Meeting CD-ROM. Washington, D.C. Bell II, C. M., French, C. E., & Shield, C. K. (2006, September). Application of Precast Decks and Other Elements to Bridge Structures. Retrieved November 1, 2007, from http://www.lrrb.org/pdf/200637.pdf Cannon, J. A., Peterson, D., & Bales, J. (2007). Symposium Paper on the Maryland Avenue Pedestrian & Bicycle Bridge. 2007 World Steel Bridge Symposium Proceedings (CD-ROM). New Orleans, LA: National Steel Bridge Alliance. Cho, A. (2007, November 5). Utah Embraces Accelerated Construction Method. Engineering News-Record , 259 (16), p. 15. Cisneros, R. A., Fulginiti, L. H., Krause, S. L., Medlock, R. D., & Wandzilak, T. J. (2008). Decked Girders for Accelerated Bridge Construction. In A. Azizinamini (Ed.), Proceedings: 2008 Accelerated Bridge Construction - Highway for Life Conference (pp. 157-162). Baltimore, MD: Federal Highway Administration. Ehmke, F. G. (2006). Analysis of a Bridge Deck Built on Interstate Highway 39/90 with FullDepth, Precast, Prestressed Concrete Deck Panels. MS Thesis, Department of Civil and Environmental Engineering, University of Wisconsin - Madison.
60 Federal Highway Administration. (2008, May 12). PBES Innovative Projects. Retrieved October 21, 2007, from Prefabricated Bridge Elements and Systems: http://www.fhwa.dot.gov/bridge/prefab/projects.cfm Federal Highway Administration. (1995). Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s Bridges. Report No. FHWA-PD-96-001, Washington, D.C. Fouad, F. H., Rizk, T., Stafford, E. L., & Hamby, D. (2006). A Prefabricated Precast Concrete Bridge System for the State of Alabama. Retrieved October 10, 2007, from http://utca.eng.ua.edu/projects/final_reports/05215fnl.pdf Franklin, A., Wahbeh, M., Speer, D., & Pohll, M. (2007). The Collapse and Reconstruction of the MacArthur Maze. 2007 World Steel Bridge Symposium Proceedings (CD-ROM). New Orleans, LA: National Steel Bridge Alliance. Ghorbanpoor, A., & Dudek, J. (2007). Bridge Integrated Analysis and Decision Support. Final Report for Wisconsin Highway Research Project 0092-04-15, University of Wisconsin Milwaukee. Hieber, D. G., Wacker, J. M., Eberhard, M. O., & Stanton, J. F. (2005). State-of-the-Art Report on Precast Concrete Systems for Rapid Construction of Bridges. Retrieved October 10, 2007, from http://www.wsdot.wa.gov/research/reports/fullreports/594.1.pdf Holien, T. (2008, October 30). Phone Conversation. (R. Sivak, Interviewer) Kinar, J. (2009) Private communication (WisDOT) Kirchner, C. (2008, November 5). E-mail communication. (R. Sivak, Interviewer) Leonard, M. (2008). Accelerated Bridge Replacement in a Rural Environment. In A. Azizinamini (Ed.), Proceedings: 2008 Accelerated Bridge Construction - Highway for Life Conference (pp. 175-177). Baltimore, MD: Federal Highway Administration. Mabey Bridge & Shore, Inc. (n.d.). Retrieved from http://mabey.com/ Malla, A. P. (2007). Development of a Specification for Thin Stay-In-Place Forms For Bridge Deck Construction. MS Thesis, Department of Civil and Environmental Engineering, University of Wisconsin - Madison. Markowski, S. M. (2005). Experimental and Analytical Study of Full-Depth Precast/Prestressed Concrete Deck Panels for Highway Bridges. Department of Civil and Environmental Engineering, University of Wisconsin - Madison.
61 Minnesota Department of Transportation. (2005, December). Innovative Contracting Guidelines. Retrieved April 20, 2009, from http://www.dot.state.mn.us/const/tools/documents/Guidelines.pdf Okumus, P. (2008). Rapid Bridge Construction Technology: Precast Elements for Substructures. MS Thesis, Department of Civil and Environmental Engineering, University of Wisconsin Madison. Prefabricated Bridge Elements and Systems in Japan and Europe, Summary Report. (2004, May). Retrieved November 12, 2007, from http://www.fhwa.dot.gov/bridge/prefab/pbesscan.pdf Ringelstetter, T. E. (2006). Investigation of Modular FRP Grid Reinforcing Systems with Integral Stay-In-Place Form for Concrete Bridge Decks. MS Thesis, Department of Civil and Environmental Engineering, University of Wisconsin - Madison. SDR Engineering Consultants, Inc. (2005). Prefabricated Steel Bridge Systems. Retrieved November 13, 2007, from http://www.fhwa.dot.gov/bridge/prefab/psbsreport.pdf Shahawy, M. A. (2003). NCHRP Synthesis 324: Prefabricated Bridge Elements and Systems to Limit Traffic Disruption During Construction. Washington, D.C.: Transportation Research Board. Smith, R. (2008). Using Pre-Fabricated Temporary Bridges to Expedite Bridge Repair/Replacement. In A. Azizinamini (Ed.), Proceedings: 2008 Accelerated Bridge Construction - Highway for Life Conference (pp. 185-189). Baltimore, MD: Federal Highway Administration. Spottiswoode, A. J. (2007). An Investigation into the use of Paperboard Tube Segments for Bridge Deck Formwork. MS Thesis, Department of Civil and Environmental Engineering, University of Wisconsin - Madison. The D. S. Brown Company. (n.d.). Exodermic Bridge Deck. Retrieved November 13 2007, from http://www.exodermic.com/ University of Wisconsin - Milwaukee. (2007). Bridge Incident Response Data (BIRD). (UWM College of Engineering & Applied Science) Retrieved February 14, 2008, from http://www.uwm.edu/CEAS/bird/ VSL. (n.d.). Post-Tensioning and Specialty Reinforcement Systems. Retrieved from http://vsl.net/ Wisconsin Department of Transportation. (2009). Bridge Maintenance Manual. Wisconsin Department of Transportation. (2008). Wisconsin Bridge Manual.
62
Appendix A: Definition of Emergency Situation A.1: Factor Ranking Responses
63
Survey Response 1 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: _10__ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) 7___ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) 8___ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? 8___ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. 6___ Truck Traffic What percentage of the daily traffic is trucks. 8___ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? 8___ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? 6___ Effect on local businesses Would closing the bridge cause hardships for local businesses? 7___ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? 8___ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? 8___ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? 7___ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 2 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: 10__ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) 8___ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) 7___ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? 7___ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. 5___ Truck Traffic What percentage of the daily traffic is trucks. 4___ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? 9___ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? 3___ Effect on local businesses Would closing the bridge cause hardships for local businesses? 10___ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? 6___ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? 5___ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? 3___ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 3 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: __8_ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) __5_ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) ___7 Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? _3__ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. __2_ Truck Traffic What percentage of the daily traffic is trucks. __2_ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? __5_ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? __3_ Effect on local businesses Would closing the bridge cause hardships for local businesses? _7__ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? __8_ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? _4__ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? __5 Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 4 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: 6 __ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) 6___ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) 3___ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? 2___ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. 2___ Truck Traffic What percentage of the daily traffic is trucks. 2___ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? 2___ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? 2___ Effect on local businesses Would closing the bridge cause hardships for local businesses? 10__ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? 10__ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? 8___ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? 8___ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 5 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: __7_ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) __5_ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) __4_ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? __8_ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. __8_ Truck Traffic What percentage of the daily traffic is trucks. _3__ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? __6_ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? _2__ Effect on local businesses Would closing the bridge cause hardships for local businesses? __10_ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? _9__ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? __1_ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? __3_ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 6 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: _8__ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) _8__ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) _5__ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? __8_ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. __6_ Truck Traffic What percentage of the daily traffic is trucks. __7_ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? _7__ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? _6__ Effect on local businesses Would closing the bridge cause hardships for local businesses? _10__ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? _10__ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? _4__ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? _7__ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
69
Survey Response 7 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: __7_ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) __7_ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) __5_ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? _3__ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. __3_ Truck Traffic What percentage of the daily traffic is trucks. __4_ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? __5_ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? __5_ Effect on local businesses Would closing the bridge cause hardships for local businesses? _10_ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? _10_ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? __2_ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? _7__ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 8 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: _8__ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) _8__ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) __6_ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? __7_ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. __5_ Truck Traffic What percentage of the daily traffic is trucks. __6_ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? __5_ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? __5_ Effect on local businesses Would closing the bridge cause hardships for local businesses? _10__ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? __10_ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? __7_ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? __6_ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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Survey Response 9 Using a scale of 1-10, with 10 being the highest, please rate each of the following factors in terms of how important they are when determining whether a bridge incident is an emergency situation: 10__ Feature on What roadway the bridge carries (Interstate Highway, U.S. Highway, State Highway, etc) 7__ Feature under What the bridge passes over (Another highway, local roads, river, lake, valley, etc) 8__ Location Where the bridge is located. Is it in a rural or urban area? Is it located near schools or residential areas? 9__ AADT The average annual daily traffic carried by the roadway on the bridge. Could possibly consider the AADT of the roadway under the bridge, if applicable. 6__ Truck Traffic What percentage of the daily traffic is trucks. 6__ Length of Detour and possible detour routes How long of a detour would vehicles need to follow, and on what classification of roads would the route use? What kinds of areas would the detour pass through (farm fields, residential neighborhoods, school zones)? 8__ Effect on local traffic Would closing of bridge significantly affect fire and medical response? School bussing? 8__ Effect on local businesses Would closing the bridge cause hardships for local businesses? 10_ Amount of damage Is the bridge in danger of collapse without immediate action? Would the bridge collapse if it were hit again while in its current condition? Is the bridge still able to carry traffic? Can temporary repairs be utilized to keep it open? 10_ Safety of Traffic under Bridge Is there any danger to of debris or material falling off of the structure and causing damage or injury? Can all lanes of traffic be left open under the bridge? 1__ Condition of Bridge before Accident Was it scheduled to be replaced in the near future? Is it a new or old bridge? Was it structurally or functionally obsolete? 1__ Structural Complexity Is the bridge a complex structure, such as an arch or suspension bridge? Would repair or replacement of the structure require any special materials or parts?
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A.2: Additional Examples of Emergency Formula Use
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Table A-1: Application of Emergency Formula to Structure B-13-0089 (Copy of table in text), 2008
Factor
Value Weight
Total
Notes
Feature On
1.0
8.2
8.2
River Rd. – Local road
Feature Under
3.0
6.8
20.4
I-39/90/94 – Interstate highway
Location
0.5
5.9
2.95
Rural area
AADT
2.5
6.1
18.3
790 on, 65840 under. Larger value used.
Truck Traffic
1.5
4.8
7.2
Unknown
Detour Length
0.5
4.7
2.35
Approximately 10 minutes
0.5
6.1
3.05
Effect on Business
0.5
4.4
2.2
Amount of Damage
2.5
9.3
23.25
1.5
9
13.5
1.0
4.4
13.2
0.0
5.2
0.0
Effect on Local Traffic
Effect on Traffic Under Condition Before Structural Complexity
No schools or emergency facilities nearby No major businesses nearby In danger of collapse if hit again Lane closures needed to remove loose debris Inv rating of HS11 Standard concrete girder bridge
INCIDENT SCORE: 102.75 Additional Notes: -Bridge was struck by oversized truck on 4/30/2008 -Detour length calculated as time to get to other side of bridge following the provided detour
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Table A-2: Application of Emergency Formula to Structure B-13-0264 (Seminole Hwy. Over Madison Beltline), 2006
Factor
Value Weight
Total
Notes
Feature On
2.0
8.2
8.2
Seminole Hwy - minor arterial
Feature Under
3.0
6.8
20.4
USH 12/14/18/151 freeway
Location
3.0
5.9
2.95
Located in Madison
AADT
3.0
6.1
18.3
13,700 on (2006), 105,406 under (2003), largest used
Truck Traffic
0.5
4.8
7.2
3% Truck Traffic Under
Detour Length
0.5
4.7
2.35
Detour length of 3 miles
0.5
6.1
3.05
No major business nearby
Effect on Business
0.5
4.4
2.2
No schools or emergency facilities nearby
Amount of Damage
1.5
9.3
23.25
Only southbound lanes were closed
1.5
9
13.5
Lane closures required during clean-up & repair
0.5
4.4
13.2
Inv. Rating of HS19
0.0
5.2
0.0
Standard Concrete Girder Bridge
Effect on Local Traffic
Effect on Traffic Under Condition Before Structural Complexity
INCIDENT SCORE:
112.45
Additional Notes: - Bridge was struck by oversized load on 1/13/2006
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Table A-3: Application of Emergency Formula to Structure B-40-0377 (S. 116th St. over I-43 in Milwaukee), 2003
Factor
Value Weight
Total
Notes
Feature On
1.0
8.2
8.2
S. 116th St. - Local Road
Feature Under
3.0
6.8
20.4
I-43 - Interstate Hwy
Location
3.0
5.9
2.95
In Milwaukee Metro Area
2.5
6.1
18.3
3,100 on (2005), 46,300 (2007); Highest value used
0.5
4.8
7.2
0% on, 1% under
0.5
4.7
2.35
2 miles for 116th St; 2 miles for I-43 closures
2.0
6.1
3.05
Whitnall HS is just south of the bridge
0.5
4.4
2.2
No major businesses nearby
1.5
9.3
23.25
Shoulder of southbound lane of 116th St. closed
3.0
9
13.5
Entire NB lanes of I-43 closed for girder repair
0.5
4.4
13.2
Inv. Rating of HS21
0.0
5.2
0.0
Standard Concrete Girder Bridge
AADT Truck Traffic Detour Length Effect on Local Traffic Effect on Business Amount of Damage Effect on Traffic Under Condition Before Structural Complexity
INCIDENT SCORE: 123.85 Additional Notes: - Bridge was struck by oversized load on 10/12/2003 - Most recent traffic data is used, because traffic data at time of incident was unavailable - Detour length for I-43 based on route described in case history background (B.I.R.D.)
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Appendix B: Segmental Girder System B.1: Girder Plan Drawings
77
78
79
B.2: Moment Capacity by Strand Layout
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Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
0.217 0.434 0.651 0.868 1.085 1.302 1.519 0.217 0.434 0.651 0.868 1.085 1.302 1.519 1.736 0.434 0.651 0.868 1.085 1.302 1.519 1.736 1.953 0.651 0.868 1.085 1.302 1.519 1.736 1.953 2.17 0.868 1.085 1.302 1.519 1.736
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 10.675 10.675 10.675 10.675 10.675 10.675 10.675 13.900 12.288 11.750 11.481 11.320 11.213 11.136 11.078 13.900 12.825 12.288 11.965 11.750 11.596 11.481 11.392 13.900 13.094 12.610 12.288 12.057 11.884 11.750 11.643 13.900 13.255 12.825 12.518 12.288
218.58 435.82 651.74 866.34 1079.63 1291.62 1502.30 234.32 451.57 667.49 882.09 1095.37 1307.36 1518.05 1727.45 467.32 683.23 897.83 1111.12 1323.10 1533.79 1743.19 1951.30 698.98 913.58 1126.86 1338.84 1549.53 1758.93 1967.04 2173.88 929.32 1142.61 1354.59 1565.27 1774.67
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Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
1.953 2.17 2.387 1.085 1.302 1.519 1.736 1.953 2.17 2.387 2.604 1.302 1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387 2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 12.108 11.965 11.848 13.900 13.363 12.979 12.691 12.467 12.288 12.141 12.019 13.900 13.439 13.094 12.825 12.610 12.434 12.288 12.163 13.900 13.497 13.183 12.933 12.727 12.556 12.412 12.288 13.900 13.542 13.255 13.020 12.825 12.660 12.518 12.395 13.900
1982.78 2189.62 2395.19 1158.35 1370.33 1581.01 1790.41 1998.52 2205.36 2410.93 2615.24 1386.07 1596.76 1806.15 2014.26 2221.10 2426.67 2630.98 2834.03 1612.50 1821.89 2030.00 2236.83 2442.40 2646.71 2849.76 3051.57 1837.63 2045.74 2252.57 2458.14 2662.44 2865.50 3067.30 3267.87 2061.48
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Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 0.217 0.434 0.651 0.868 1.085
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.578 13.314 13.094 12.908 12.748 12.610 12.489 13.900 13.607 13.363 13.156 12.979 12.825 12.691 12.572 13.900 13.631 13.404 13.209 13.040 12.892 12.762 12.646 13.900 13.652 13.439 13.255 13.094 12.951 12.825 12.712 17.480 14.078 12.943 12.376 12.036
2268.31 2473.87 2678.18 2881.23 3083.03 3283.60 3482.93 2284.05 2489.61 2693.91 2896.96 3098.76 3299.33 3498.66 3696.77 2505.35 2709.65 2912.69 3114.49 3315.05 3514.38 3712.49 3909.38 2725.38 2928.43 3130.22 3330.78 3530.11 3728.21 3925.10 4120.77 251.80 469.05 684.97 899.56 1112.85
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Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
1.302 1.519 1.736 0.434 0.651 0.868 1.085 1.302 1.519 1.736 1.953 0.651 0.868 1.085 1.302 1.519 1.736 1.953 2.17 0.868 1.085 1.302 1.519 1.736 1.953 2.17 2.387 1.085 1.302 1.519 1.736 1.953 2.17 2.387 2.604 1.302
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 11.809 11.647 11.526 15.690 14.018 13.183 12.681 12.347 12.108 11.929 11.789 15.093 13.989 13.326 12.884 12.569 12.332 12.148 12.001 14.795 13.971 13.422 13.029 12.735 12.506 12.323 12.173 14.616 13.959 13.490 13.138 12.864 12.646 12.466 12.317 14.497
1324.83 1535.52 1744.92 484.79 700.71 915.31 1128.59 1340.58 1551.26 1760.66 1968.77 716.46 931.05 1144.34 1356.32 1567.00 1776.40 1984.51 2191.35 946.80 1160.08 1372.06 1582.75 1792.14 2000.25 2207.09 2412.66 1175.83 1387.81 1598.49 1807.88 2015.99 2222.83 2428.40 2632.71 1403.55
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Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387 2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953 2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.951 13.541 13.223 12.968 12.760 12.586 12.439 14.411 13.944 13.581 13.291 13.053 12.855 12.687 12.543 14.348 13.939 13.613 13.346 13.123 12.935 12.774 12.634 14.298 13.936 13.639 13.392 13.183 13.004 12.849 12.713 14.258 13.932 13.661 13.431 13.234
1614.23 1823.62 2031.73 2238.57 2444.13 2648.44 2851.50 1629.97 1839.36 2047.47 2254.30 2459.87 2664.18 2867.23 3069.03 1855.10 2063.21 2270.04 2475.61 2679.91 2882.96 3084.76 3285.33 2078.95 2285.78 2491.34 2695.64 2898.69 3100.49 3301.06 3500.39 2301.52 2507.08 2711.38 2914.42 3116.22
85
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 0.434 0.651 0.868 1.085 1.302 1.519 1.736 1.953 0.651
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.064 12.914 12.783 14.225 13.930 13.679 13.465 13.279 13.116 12.972 12.845 14.198 13.927 13.695 13.494 13.318 13.162 13.024 12.900 14.175 13.925 13.709 13.519 13.352 13.203 13.070 12.950 17.480 15.212 14.078 13.397 12.943 12.619 12.376 12.187 16.287
3316.79 3516.12 3714.22 2522.82 2727.11 2930.16 3131.96 3332.51 3531.84 3729.94 3926.83 2742.85 2945.89 3147.69 3348.24 3547.57 3745.67 3942.55 4138.22 2961.62 3163.42 3363.97 3563.29 3761.39 3958.27 4153.93 4348.40 502.27 718.19 932.79 1146.07 1358.05 1568.74 1778.13 1986.25 733.94
86
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
0.868 1.085 1.302 1.519 1.736 1.953 2.17 0.868 1.085 1.302 1.519 1.736 1.953 2.17 2.387 1.085 1.302 1.519 1.736 1.953 2.17 2.387 2.604 1.302 1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.884 14.042 13.481 13.080 12.779 12.546 12.359 15.690 14.687 14.018 13.541 13.183 12.904 12.681 12.499 15.332 14.556 14.001 13.586 13.262 13.004 12.792 12.615 15.093 14.462 13.989 13.621 13.326 13.085 12.884 12.714 14.923 14.392 13.979 13.649 13.378
948.53 1161.82 1373.80 1584.48 1793.87 2001.99 2208.82 964.28 1177.56 1389.54 1600.22 1809.61 2017.72 2224.56 2430.13 1193.30 1405.28 1615.96 1825.35 2033.46 2240.30 2445.87 2650.17 1421.03 1631.71 1841.10 2049.20 2256.04 2461.60 2665.91 2868.96 1647.45 1856.84 2064.94 2271.77 2477.34
87
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953 2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.153 12.962 12.799 14.795 14.337 13.971 13.671 13.422 13.210 13.029 12.872 14.696 14.294 13.965 13.690 13.458 13.260 13.087 12.937 14.616 14.258 13.959 13.707 13.490 13.302 13.138 12.993 14.551 14.228 13.955 13.720 13.517 13.340 13.183 13.044 14.497
2681.64 2884.69 3086.50 1872.58 2080.68 2287.51 2493.07 2697.38 2900.42 3102.23 3302.79 2096.42 2303.25 2508.81 2713.11 2916.16 3117.96 3318.52 3517.85 2318.99 2524.55 2728.85 2931.89 3133.69 3334.25 3533.57 3731.67 2540.28 2744.58 2947.62 3149.42 3349.97 3549.30 3747.40 3944.28 2760.32
88
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3
10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 0.651 0.868 1.085 1.302 1.519 1.736 1.953 2.17 0.868 1.085 1.302 1.519 1.736
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.203 13.951 13.732 13.541 13.373 13.223 13.089 14.451 14.181 13.947 13.743 13.562 13.402 13.258 13.129 14.411 14.162 13.944 13.752 13.581 13.428 13.291 13.166 17.480 15.779 14.758 14.078 13.591 13.227 12.943 12.717 16.585 15.403 14.615 14.052 13.630
2963.35 3165.15 3365.70 3565.02 3763.12 3960.00 4155.66 2979.09 3180.88 3381.43 3580.75 3778.84 3975.72 4171.38 4365.84 3196.61 3397.16 3596.48 3794.57 3991.44 4187.10 4381.56 4574.81 751.41 966.01 1179.29 1391.27 1601.95 1811.35 2019.46 2226.29 981.75 1195.04 1407.02 1617.70 1827.09
89
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
1.953 2.17 2.387 1.085 1.302 1.519 1.736 1.953 2.17 2.387 2.604 1.302 1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387 2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.302 13.039 12.824 16.048 15.153 14.513 14.033 13.660 13.362 13.117 12.914 15.690 14.974 14.436 14.018 13.684 13.410 13.183 12.990 15.434 14.839 14.377 14.007 13.704 13.451 13.238 13.055 15.243 14.735 14.329 13.997 13.720 13.486 13.285 13.111 15.093
2035.20 2242.03 2447.60 1210.78 1422.76 1633.44 1842.83 2050.94 2257.77 2463.33 2667.64 1438.50 1649.18 1858.57 2066.68 2273.51 2479.07 2683.37 2886.42 1664.92 1874.31 2082.42 2289.25 2494.81 2699.11 2902.16 3103.96 1890.05 2098.16 2304.98 2510.54 2714.84 2917.89 3119.69 3320.25 2113.90
90
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.652 14.290 13.989 13.734 13.515 13.326 13.160 14.974 14.583 14.258 13.982 13.746 13.541 13.362 13.204 14.876 14.526 14.230 13.976 13.756 13.563 13.394 13.243 14.795 14.478 14.206 13.971 13.765 13.583 13.422 13.277 14.726 14.437 14.186 13.967 13.773
2320.72 2526.28 2730.58 2933.62 3135.42 3335.98 3535.30 2336.46 2542.02 2746.31 2949.35 3151.15 3351.70 3551.03 3749.13 2557.75 2762.05 2965.09 3166.88 3367.43 3566.75 3764.85 3961.73 2777.78 2980.82 3182.61 3383.16 3582.48 3780.57 3977.45 4173.11 2996.55 3198.34 3398.89 3598.21 3796.30
91
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 0.868 1.085 1.302 1.519 1.736 1.953 2.17 2.387 1.085 1.302 1.519 1.736 1.953 2.17 2.387 2.604 1.302
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.601 13.447 13.308 14.667 14.401 14.168 13.963 13.780 13.617 13.470 13.336 14.616 14.370 14.152 13.959 13.786 13.631 13.490 13.362 17.480 16.119 15.212 14.564 14.078 13.699 13.397 13.150 16.764 15.749 15.024 14.481 14.058 13.720 13.443 13.212 16.287
3993.17 4188.83 4383.29 3214.07 3414.62 3613.93 3812.02 4008.89 4204.55 4399.00 4592.25 3430.35 3629.66 3827.74 4024.61 4220.26 4414.71 4607.97 4800.03 999.23 1212.51 1424.49 1635.17 1844.56 2052.67 2259.50 2465.07 1228.26 1440.24 1650.91 1860.30 2068.41 2275.24 2480.80 2685.11 1455.98
92
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387 2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953 2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.485 14.884 14.416 14.042 13.736 13.481 13.265 15.946 15.287 14.774 14.365 14.029 13.750 13.513 13.310 15.690 15.133 14.687 14.322 14.018 13.761 13.541 13.350 15.491 15.010 14.615 14.287 14.009 13.771 13.565 13.384 15.332 14.909 14.556 14.257 14.001
1666.66 1876.04 2084.15 2290.98 2496.54 2700.84 2903.89 1682.40 1891.78 2099.89 2306.72 2512.28 2716.57 2919.62 3121.42 1907.53 2115.63 2322.45 2528.01 2732.31 2935.35 3137.15 3337.71 2131.37 2338.19 2543.75 2748.04 2951.09 3152.88 3353.44 3552.76 2353.93 2559.49 2763.78 2966.82 3168.61
93
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.780 13.586 13.414 15.202 14.825 14.505 14.232 13.995 13.787 13.604 13.441 15.093 14.753 14.462 14.210 13.989 13.794 13.621 13.466 15.002 14.693 14.425 14.190 13.984 13.800 13.635 13.487 14.923 14.640 14.392 14.173 13.979 13.805 13.649 13.507 14.855
3369.16 3568.49 3766.58 2575.22 2779.51 2982.55 3184.34 3384.89 3584.21 3782.30 3979.18 2795.25 2998.28 3200.07 3400.62 3599.94 3798.03 3994.90 4190.56 3014.02 3215.80 3416.35 3615.66 3813.75 4010.62 4206.28 4400.73 3231.54 3432.08 3631.39 3829.48 4026.34 4222.00 4416.44 4609.70 3447.81
94
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34 4.557 4.774 4.991 1.085 1.302 1.519 1.736 1.953 2.17 2.387 2.604 1.302 1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.593 14.363 14.158 13.975 13.810 13.660 13.525 14.795 14.553 14.337 14.144 13.971 13.814 13.671 13.541 17.480 16.346 15.536 14.928 14.456 14.078 13.768 13.510 16.883 15.996 15.331 14.814 14.400 14.061 13.779 13.540 16.457 15.734 15.172 14.723 14.355
3647.12 3845.20 4042.06 4237.71 4432.16 4625.41 4817.46 3662.84 3860.92 4057.78 4253.43 4447.87 4641.12 4833.17 5024.04 1245.74 1457.71 1668.39 1877.78 2085.88 2292.71 2498.27 2702.57 1473.45 1684.13 1893.52 2101.62 2308.45 2514.01 2718.31 2921.35 1699.87 1909.26 2117.36 2324.19 2529.74
95
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953 2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.048 13.788 13.566 16.138 15.531 15.045 14.648 14.317 14.037 13.796 13.588 15.889 15.368 14.941 14.585 14.285 14.027 13.803 13.608 15.690 15.234 14.854 14.533 14.257 14.018 13.809 13.625 15.527 15.123 14.781 14.488 14.233 14.011 13.815 13.640 15.392
2734.04 2937.08 3138.88 1925.00 2133.10 2339.93 2545.48 2749.78 2952.82 3154.61 3355.17 2148.84 2355.66 2561.22 2765.51 2968.55 3170.34 3370.90 3570.22 2371.40 2576.96 2781.25 2984.28 3186.07 3386.62 3585.94 3784.03 2592.69 2796.98 3000.02 3201.80 3402.35 3601.67 3799.76 3996.63 2812.72
96
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.029 14.718 14.448 14.213 14.004 13.819 13.654 15.277 14.948 14.663 14.414 14.194 13.999 13.824 13.666 15.179 14.878 14.616 14.384 14.178 13.993 13.828 13.677 15.093 14.817 14.574 14.357 14.163 13.989 13.831 13.688 15.019 14.763 14.536 14.333 14.150
3015.75 3217.54 3418.08 3617.39 3815.48 4012.35 4208.01 3031.48 3233.27 3433.81 3633.12 3831.21 4028.07 4223.73 4418.17 3249.00 3449.54 3648.85 3846.93 4043.79 4239.44 4433.89 4627.14 3465.27 3664.57 3862.65 4059.51 4255.16 4449.60 4642.85 4834.90 3680.30 3878.38 4075.24 4270.88 4465.32
97
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 1.302 1.519 1.736 1.953 2.17 2.387 2.604 2.821 1.519 1.736 1.953 2.17 2.387 2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.985 13.834 13.697 14.953 14.715 14.503 14.311 14.138 13.981 13.837 13.705 17.480 16.508 15.779 15.212 14.758 14.387 14.078 13.816 16.969 16.182 15.570 15.081 14.680 14.346 14.064 13.822 16.585 15.928 15.403 14.973 14.615 14.312 14.052 13.827 16.287
4658.56 4850.61 5041.48 3894.10 4090.96 4286.60 4481.04 4674.27 4866.32 5057.18 5246.86 1490.93 1701.61 1910.99 2119.09 2325.92 2531.48 2735.77 2938.82 1717.35 1926.73 2134.83 2341.66 2547.21 2751.51 2954.55 3156.34 1942.47 2150.57 2357.40 2562.95 2767.24 2970.28 3172.07 3372.63 2166.31
98
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.726 15.266 14.884 14.560 14.283 14.042 13.832 16.048 15.560 15.153 14.808 14.513 14.257 14.033 13.836 15.853 15.421 15.056 14.743 14.472 14.235 14.025 13.839 15.690 15.304 14.974 14.687 14.436 14.215 14.018 13.842 15.552 15.204 14.902 14.638 14.405
2373.13 2578.69 2782.98 2986.02 3187.81 3388.36 3587.67 2388.87 2594.42 2798.71 3001.75 3203.54 3404.08 3603.40 3801.49 2610.16 2814.45 3017.48 3219.27 3419.81 3619.13 3817.21 4014.08 2830.18 3033.22 3235.00 3435.54 3634.85 3832.94 4029.80 4225.46 3048.95 3250.73 3451.27 3650.58 3848.66
99
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34 4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.198 14.012 13.845 15.434 15.117 14.839 14.594 14.377 14.182 14.007 13.848 15.332 15.041 14.784 14.556 14.352 14.168 14.001 13.850 15.243 14.974 14.735 14.521 14.329 14.155 13.997 13.852 15.164 14.914 14.691 14.490 14.309 14.143 13.993 13.854 15.093
4045.52 4241.17 4435.62 3266.46 3467.00 3666.31 3864.38 4061.24 4256.89 4451.33 4644.58 3482.73 3682.03 3880.11 4076.97 4272.61 4467.05 4660.29 4852.34 3697.76 3895.83 4092.69 4288.33 4482.77 4676.00 4868.05 5058.91 3911.56 4108.41 4304.05 4498.48 4691.72 4883.76 5074.62 5264.30 4124.13
100
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.123 4.34 4.557 4.774 4.991 5.208 5.425 1.519 1.736 1.953 2.17 2.387 2.604 2.821 3.038 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953 2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.861 14.652 14.462 14.290 14.133 13.989 13.856 17.480 16.629 15.968 15.439 15.005 14.645 14.339 14.078 17.033 16.326 15.761 15.299 14.913 14.587 14.308 14.066 16.684 16.084 15.592 15.182 14.835 14.538 14.281 14.055 16.406 15.885 15.451 15.083 14.769
4319.77 4514.20 4707.43 4899.47 5090.32 5280.00 5468.50 1734.82 1944.21 2152.31 2359.13 2564.68 2768.98 2972.01 3173.81 1959.95 2168.05 2374.87 2580.42 2784.71 2987.75 3189.54 3390.09 2183.79 2390.61 2596.16 2800.45 3003.48 3205.27 3405.82 3605.13 2406.34 2611.89 2816.18 3019.21 3221.00
101
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.496 14.257 14.046 16.178 15.720 15.332 14.999 14.711 14.458 14.236 14.038 15.988 15.580 15.229 14.926 14.660 14.426 14.217 14.031 15.828 15.460 15.141 14.862 14.615 14.396 14.201 14.024 15.690 15.356 15.063 14.805 14.576 14.370 14.186 14.018 15.571
3421.54 3620.86 3818.94 2627.63 2831.92 3034.95 3236.73 3437.27 3636.58 3834.67 4031.53 2847.65 3050.68 3252.46 3453.00 3652.31 3850.39 4047.25 4242.90 3066.41 3268.19 3468.73 3668.04 3866.12 4062.98 4258.62 4453.06 3283.93 3484.46 3683.76 3881.84 4078.70 4274.34 4468.78 4662.02 3500.19
102
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34 4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906 4.123 4.34 4.557 4.774 4.991 5.208 5.425 4.123 4.34 4.557 4.774 4.991
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.265 14.995 14.755 14.540 14.347 14.172 14.013 15.466 15.184 14.934 14.710 14.508 14.325 14.160 14.008 15.374 15.113 14.879 14.669 14.479 14.306 14.148 14.004 15.292 15.049 14.831 14.633 14.453 14.288 14.138 13.999 15.219 14.992 14.786 14.599 14.429
3699.49 3897.56 4094.42 4290.06 4484.50 4677.73 4869.78 3715.22 3913.29 4110.14 4305.78 4500.21 4693.45 4885.49 5076.35 3929.01 4125.86 4321.50 4515.93 4709.16 4901.20 5092.05 5281.73 4141.59 4337.22 4531.64 4724.87 4916.91 5107.76 5297.43 5485.93 4352.94 4547.36 4740.59 4932.62 5123.46
103
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
5.208 5.425 5.642 1.736 1.953 2.17 2.387 2.604 2.821 3.038 3.255 1.953 2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.272 14.128 13.996 17.480 16.724 16.119 15.624 15.212 14.863 14.564 14.304 17.082 16.442 15.917 15.480 15.111 14.794 14.519 14.279 16.764 16.210 15.749 15.359 15.024 14.734 14.481 14.257 16.504 16.018 15.607 15.255 14.949 14.682 14.446 14.237 16.287
5313.13 5501.63 5688.96 1977.42 2185.52 2392.34 2597.89 2802.18 3005.21 3207.00 3407.55 2201.26 2408.08 2613.63 2817.91 3020.95 3222.73 3423.28 3622.59 2423.82 2629.36 2833.65 3036.68 3238.46 3439.01 3638.31 3836.40 2645.10 2849.38 3052.41 3254.19 3454.73 3654.04 3852.12 4048.98 2865.12
104
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.855 15.485 15.164 14.884 14.636 14.416 14.219 16.103 15.715 15.379 15.085 14.826 14.595 14.389 14.203 15.946 15.594 15.287 15.016 14.774 14.559 14.365 14.189 15.809 15.488 15.205 14.954 14.728 14.526 14.342 14.176 15.690 15.395 15.133 14.898 14.687
3068.15 3269.93 3470.46 3669.77 3867.85 4064.71 4260.35 3083.88 3285.66 3486.19 3685.50 3883.57 4080.43 4276.07 4470.51 3301.39 3501.92 3701.22 3899.30 4096.15 4291.79 4486.23 4679.46 3517.65 3716.95 3915.02 4111.87 4307.51 4501.94 4695.18 4887.22 3732.68 3930.75 4127.59 4323.23 4517.66
105
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9
8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906 4.123 4.34 4.557 4.774 4.991 5.208 5.425 4.123 4.34 4.557 4.774 4.991 5.208 5.425 5.642 4.34 4.557 4.774 4.991 5.208 5.425 5.642 5.859 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.496 14.322 14.164 15.585 15.312 15.068 14.848 14.650 14.469 14.304 14.153 15.491 15.238 15.010 14.803 14.615 14.444 14.287 14.143 15.407 15.171 14.957 14.762 14.584 14.421 14.272 14.133 15.332 15.110 14.909 14.725 14.556 14.401 14.257 14.125 17.480
4710.89 4902.93 5093.78 3946.47 4143.32 4338.95 4533.37 4726.60 4918.64 5109.49 5299.16 4159.04 4354.67 4549.09 4742.32 4934.35 5125.19 5314.86 5503.36 4370.39 4564.81 4758.03 4950.06 5140.90 5330.56 5519.06 5706.38 4580.52 4773.74 4965.77 5156.61 5346.27 5534.75 5722.08 5908.24 2218.73
106
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.17 2.387 2.604 2.821 3.038 3.255 3.472 2.17 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 16.800 16.243 15.779 15.386 15.050 14.758 14.503 17.122 16.536 16.048 15.634 15.280 14.973 14.704 14.467 16.829 16.316 15.882 15.510 15.188 14.906 14.657 14.436 16.585 16.130 15.741 15.403 15.108 14.847 14.615 14.408 16.378 15.971 15.618 15.309 15.036
2425.55 2631.10 2835.38 3038.41 3240.19 3440.74 3640.05 2441.29 2646.83 2851.12 3054.14 3255.93 3456.47 3655.77 3853.85 2662.57 2866.85 3069.88 3271.66 3472.20 3671.50 3869.58 4066.44 2882.59 3085.61 3287.39 3487.92 3687.23 3885.30 4082.16 4277.80 3101.35 3303.12 3503.65 3702.95 3901.03
107
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34 4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.794 14.577 14.382 16.201 15.833 15.511 15.226 14.973 14.747 14.544 14.359 16.048 15.712 15.416 15.153 14.917 14.705 14.513 14.338 15.914 15.606 15.332 15.087 14.866 14.666 14.485 14.319 15.795 15.511 15.256 15.027 14.820 14.632 14.460 14.302 15.690
4097.88 4293.52 4487.96 3318.85 3519.38 3718.68 3916.75 4113.60 4309.24 4503.67 4696.91 3535.11 3734.41 3932.48 4129.32 4324.96 4519.39 4712.62 4904.66 3750.14 3948.20 4145.05 4340.68 4535.10 4728.33 4920.37 5111.22 3963.93 4160.77 4356.40 4550.82 4744.04 4936.08 5126.92 5316.59 4176.49
108
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10
9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.123 4.34 4.557 4.774 4.991 5.208 5.425 4.123 4.34 4.557 4.774 4.991 5.208 5.425 5.642 4.34 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557 4.774 4.991 5.208 5.425 5.642 5.859 6.076 2.17 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.426 15.189 14.974 14.778 14.600 14.436 14.286 15.596 15.350 15.127 14.925 14.740 14.571 14.415 14.271 15.511 15.281 15.071 14.880 14.705 14.544 14.395 14.257 15.434 15.218 15.020 14.839 14.673 14.519 14.377 14.244 17.480 16.861 16.346 15.910 15.536
4372.12 4566.54 4759.76 4951.79 5142.63 5332.29 5520.78 4387.84 4582.25 4775.47 4967.50 5158.34 5348.00 5536.48 5723.80 4597.97 4791.19 4983.21 5174.04 5363.70 5552.18 5739.50 5925.66 4806.90 4998.92 5189.75 5379.40 5567.88 5755.20 5941.35 6126.35 2458.76 2664.30 2868.58 3071.61 3273.39
109
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.212 14.928 14.678 17.155 16.615 16.158 15.766 15.427 15.130 14.868 14.635 16.883 16.406 15.996 15.642 15.331 15.057 14.814 14.596 16.654 16.227 15.857 15.533 15.247 14.993 14.766 14.561 16.457 16.072 15.734 15.437 15.172 14.936 14.723 14.530 16.287
3473.93 3673.23 3871.31 2680.04 2884.32 3087.34 3289.12 3489.66 3688.96 3887.03 4083.89 2900.06 3103.08 3304.85 3505.39 3704.68 3902.76 4099.61 4295.25 3118.81 3320.58 3521.12 3720.41 3918.48 4115.33 4310.97 4505.40 3336.32 3536.85 3736.14 3934.21 4131.05 4326.69 4521.12 4714.35 3552.58
110
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34 4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906 4.123 4.34 4.557 4.774 4.991 5.208 5.425 4.123 4.34 4.557 4.774 4.991
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.936 15.626 15.351 15.105 14.884 14.683 14.501 16.138 15.816 15.531 15.275 15.045 14.837 14.648 14.475 16.006 15.710 15.445 15.206 14.990 14.794 14.615 14.451 15.889 15.614 15.368 15.144 14.941 14.755 14.585 14.429 15.784 15.529 15.298 15.088 14.896
3751.87 3949.93 4146.78 4342.41 4536.83 4730.06 4922.10 3767.60 3965.66 4162.50 4358.13 4552.55 4745.77 4937.81 5128.65 3981.38 4178.22 4373.85 4568.27 4761.49 4953.52 5144.36 5334.02 4193.95 4389.57 4583.98 4777.20 4969.23 5160.06 5349.72 5538.21 4405.29 4599.70 4792.92 4984.94 5175.77
111
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11
9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
5.208 5.425 5.642 4.34 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557 4.774 4.991 5.208 5.425 5.642 5.859 6.076 4.774 4.991 5.208 5.425 5.642 5.859 6.076 6.293 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.720 14.558 14.409 15.690 15.451 15.234 15.036 14.854 14.687 14.533 14.390 15.605 15.381 15.176 14.989 14.816 14.657 14.509 14.372 15.527 15.316 15.123 14.945 14.781 14.629 14.488 14.356 17.480 16.913 16.433 16.022 15.665 15.353 15.078 14.834 17.182
5365.43 5553.91 5741.23 4615.42 4808.63 5000.65 5191.48 5381.13 5569.61 5756.92 5943.08 4824.34 5016.36 5207.19 5396.84 5585.31 5772.62 5958.77 6143.77 5032.07 5222.89 5412.54 5601.01 5788.32 5974.46 6159.46 6343.30 2697.51 2901.79 3104.81 3306.58 3507.12 3706.42 3904.49 4101.34 2917.52
112
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 16.681 16.252 15.880 15.555 15.268 15.013 14.784 16.929 16.483 16.095 15.757 15.458 15.192 14.954 14.740 16.713 16.310 15.958 15.647 15.371 15.124 14.902 14.700 16.525 16.160 15.837 15.550 15.294 15.063 14.854 14.664 16.361 16.027 15.729 15.463 15.224
3120.54 3322.32 3522.85 3722.14 3920.21 4117.06 4312.70 3136.28 3338.05 3538.58 3737.87 3935.94 4132.79 4328.42 4522.85 3353.78 3554.31 3753.60 3951.66 4148.51 4344.14 4538.56 4731.79 3570.04 3769.33 3967.39 4164.23 4359.86 4554.28 4747.50 4939.53 3785.05 3983.11 4179.95 4375.58 4570.00
113
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906 4.123 4.34 4.557 4.774 4.991 5.208 5.425 4.123 4.34 4.557 4.774 4.991 5.208 5.425 5.642 4.34 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.007 14.810 14.631 16.216 15.909 15.633 15.385 15.161 14.957 14.771 14.600 16.088 15.803 15.547 15.315 15.104 14.911 14.735 14.572 15.973 15.708 15.468 15.250 15.051 14.869 14.701 14.546 15.869 15.622 15.397 15.192 15.003 14.830 14.670 14.522 15.775
4763.22 4955.25 5146.09 3998.84 4195.68 4391.30 4585.72 4778.93 4970.96 5161.79 5351.45 4211.40 4407.02 4601.43 4794.65 4986.67 5177.50 5367.16 5555.64 4422.74 4617.15 4810.36 5002.38 5193.21 5382.86 5571.34 5758.65 4632.87 4826.07 5018.09 5208.92 5398.56 5587.04 5774.35 5960.50 4841.79
114
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12
10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.774 4.991 5.208 5.425 5.642 5.859 6.076 4.774 4.991 5.208 5.425 5.642 5.859 6.076 6.293 4.991 5.208 5.425 5.642 5.859 6.076 6.293 6.51 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.543 15.332 15.138 14.959 14.794 14.642 14.500 15.690 15.472 15.272 15.088 14.918 14.761 14.615 14.479 15.612 15.406 15.217 15.043 14.881 14.731 14.591 14.460 17.480 16.957 16.508 16.119 15.779 15.479 15.212 14.973 17.205 16.738 16.334 15.980 15.668
5033.80 5224.62 5414.27 5602.74 5790.05 5976.19 6161.18 5049.51 5240.33 5429.97 5618.44 5805.74 5991.88 6176.87 6360.72 5256.04 5445.68 5634.14 5821.44 6007.58 6192.56 6376.40 6559.10 2934.99 3138.01 3339.78 3540.31 3739.60 3937.67 4134.52 4330.15 3153.75 3355.51 3556.04 3755.33 3953.39
115
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.34 3.038 3.255 3.472 3.689 3.906 4.123 4.34 4.557 3.255 3.472 3.689 3.906 4.123 4.34 4.557 4.774 3.472 3.689 3.906 4.123 4.34 4.557 4.774 4.991 3.689 3.906 4.123 4.34 4.557 4.774 4.991 5.208 3.906
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.391 15.143 14.919 16.969 16.549 16.182 15.858 15.570 15.312 15.081 14.871 16.764 16.383 16.048 15.749 15.482 15.242 15.024 14.827 16.585 16.237 15.928 15.652 15.403 15.178 14.973 14.786 16.427 16.108 15.822 15.564 15.331 15.120 14.927 14.749 16.287
4150.24 4345.87 4540.29 3371.24 3571.77 3771.06 3969.12 4165.96 4361.59 4556.01 4749.23 3587.50 3786.79 3984.84 4181.68 4377.31 4571.73 4764.95 4956.97 3802.51 4000.57 4197.41 4393.03 4587.45 4780.66 4972.69 5163.52 4016.30 4213.13 4408.75 4603.16 4796.38 4988.40 5179.23 5368.88 4228.85
116
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.123 4.34 4.557 4.774 4.991 5.208 5.425 4.123 4.34 4.557 4.774 4.991 5.208 5.425 5.642 4.34 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557 4.774 4.991 5.208 5.425 5.642 5.859 6.076 4.774 4.991 5.208 5.425 5.642
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.991 15.726 15.485 15.266 15.067 14.884 14.715 16.161 15.887 15.639 15.413 15.207 15.018 14.844 14.684 16.048 15.792 15.560 15.347 15.153 14.973 14.808 14.655 15.946 15.706 15.487 15.287 15.102 14.932 14.774 14.628 15.853 15.628 15.421 15.231 15.056
4424.47 4618.88 4812.09 5004.11 5194.94 5384.59 5573.07 4440.19 4634.60 4827.80 5019.82 5210.65 5400.29 5588.77 5776.08 4650.32 4843.52 5035.53 5226.35 5416.00 5604.47 5791.77 5977.92 4859.23 5051.24 5242.06 5431.70 5620.17 5807.47 5993.61 6178.60 5066.95 5257.77 5447.41 5635.87 5823.17
117
Table B.1: Moment Capacities by Strand Layout, 60 ft. and 80 ft. Spans (104 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
5.859 6.076 6.293 4.991 5.208 5.425 5.642 5.859 6.076 6.293 6.51 5.208 5.425 5.642 5.859 6.076 6.293 6.51 6.727
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.894 14.743 14.603 15.768 15.556 15.360 15.180 15.013 14.858 14.714 14.580 15.690 15.489 15.304 15.133 14.974 14.825 14.687 14.558
6009.31 6194.29 6378.13 5273.48 5463.11 5651.57 5838.87 6025.00 6209.98 6393.81 6576.51 5478.82 5667.27 5854.56 6040.69 6225.67 6409.50 6592.19 6773.74
118
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
0.217 0.434 0.651 0.868 1.085 1.302 1.519 0.217 0.434 0.651 0.868 1.085 1.302 1.519 1.736 0.434 0.651 0.868 1.085 1.302 1.519 1.736 1.953 0.651 0.868 1.085 1.302 1.519 1.736 1.953 2.170 0.868 1.085 1.302 1.519 1.736
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 10.675 10.675 10.675 10.675 10.675 10.675 10.675 13.900 12.288 11.750 11.481 11.320 11.213 11.136 11.078 13.900 12.825 12.288 11.965 11.750 11.596 11.481 11.392 13.900 13.094 12.610 12.288 12.057 11.884 11.750 11.643 13.900 13.255 12.825 12.518 12.288
218.31 434.77 649.37 862.15 1073.11 1282.27 1489.64 234.06 450.51 665.12 877.89 1088.85 1298.01 1505.38 1710.98 466.26 680.86 893.64 1104.59 1313.75 1521.12 1726.71 1930.55 696.61 909.38 1120.34 1329.49 1536.86 1742.45 1946.28 2148.37 925.12 1136.08 1345.23 1552.59 1758.18
119
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
1.953 2.170 2.387 1.085 1.302 1.519 1.736 1.953 2.170 2.387 2.604 1.302 1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387 2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 12.108 11.965 11.848 13.900 13.363 12.979 12.691 12.467 12.288 12.141 12.019 13.900 13.439 13.094 12.825 12.610 12.434 12.288 12.163 13.900 13.497 13.183 12.933 12.727 12.556 12.412 12.288 13.900 13.542 13.255 13.020 12.825 12.660 12.518 12.395 13.900
1962.02 2164.10 2364.46 1151.82 1360.97 1568.33 1773.92 1977.75 2179.83 2380.18 2578.82 1376.71 1584.07 1789.66 1993.48 2195.56 2395.91 2594.54 2791.47 1599.81 1805.39 2009.21 2211.29 2411.64 2610.27 2807.19 3002.42 1821.13 2024.95 2227.02 2427.36 2625.99 2822.91 3018.14 3211.69 2040.68
120
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 0.217 0.434 0.651 0.868 1.085
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.578 13.314 13.094 12.908 12.748 12.610 12.489 13.900 13.607 13.363 13.156 12.979 12.825 12.691 12.572 13.900 13.631 13.404 13.209 13.040 12.892 12.762 12.646 13.900 13.652 13.439 13.255 13.094 12.951 12.825 12.712 17.480 14.078 12.943 12.376 12.036
2242.75 2443.09 2641.71 2838.63 3033.85 3227.40 3419.28 2258.48 2458.82 2657.44 2854.35 3049.57 3243.11 3434.99 3625.22 2474.55 2673.16 2870.07 3065.28 3258.82 3450.69 3640.92 3829.50 2688.88 2885.79 3081.00 3274.53 3466.40 3656.62 3845.20 4032.15 251.54 467.99 682.59 895.37 1106.33
121
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
1.302 1.519 1.736 0.434 0.651 0.868 1.085 1.302 1.519 1.736 1.953 0.651 0.868 1.085 1.302 1.519 1.736 1.953 2.170 0.868 1.085 1.302 1.519 1.736 1.953 2.170 2.387 1.085 1.302 1.519 1.736 1.953 2.170 2.387 2.604 1.302
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 11.809 11.647 11.526 15.690 14.018 13.183 12.681 12.347 12.108 11.929 11.789 15.093 13.989 13.326 12.884 12.569 12.332 12.148 12.001 14.795 13.971 13.422 13.029 12.735 12.506 12.323 12.173 14.616 13.959 13.490 13.138 12.864 12.646 12.466 12.317 14.497
1315.48 1522.85 1728.45 483.74 698.34 911.11 1122.07 1331.22 1538.59 1744.18 1948.02 714.08 926.86 1137.81 1346.96 1554.33 1759.92 1963.75 2165.83 942.60 1153.55 1362.70 1570.06 1775.65 1979.48 2181.56 2381.92 1169.29 1378.44 1585.80 1791.39 1995.21 2197.29 2397.64 2596.27 1394.18
122
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387 2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953 2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.951 13.541 13.223 12.968 12.760 12.586 12.439 14.411 13.944 13.581 13.291 13.053 12.855 12.687 12.543 14.348 13.939 13.613 13.346 13.123 12.935 12.774 12.634 14.298 13.936 13.639 13.392 13.183 13.004 12.849 12.713 14.258 13.932 13.661 13.431 13.234
1601.54 1807.12 2010.95 2213.02 2413.37 2612.00 2808.92 1617.28 1822.86 2026.68 2228.75 2429.10 2627.72 2824.64 3019.87 1838.60 2042.41 2244.48 2444.82 2643.44 2840.36 3035.58 3229.13 2058.15 2260.22 2460.55 2659.17 2856.08 3051.30 3244.84 3436.72 2275.95 2476.28 2674.89 2871.80 3067.01
123
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 0.434 0.651 0.868 1.085 1.302 1.519 1.736 1.953 0.651
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.064 12.914 12.783 14.225 13.930 13.679 13.465 13.279 13.116 12.972 12.845 14.198 13.927 13.695 13.494 13.318 13.162 13.024 12.900 14.175 13.925 13.709 13.519 13.352 13.203 13.070 12.950 17.480 15.212 14.078 13.397 12.943 12.619 12.376 12.187 16.287
3260.55 3452.42 3642.64 2492.01 2690.61 2887.52 3082.73 3276.26 3468.13 3658.35 3846.92 2706.34 2903.24 3098.45 3291.97 3483.84 3674.05 3862.62 4049.57 2918.96 3114.16 3307.69 3499.54 3689.75 3878.32 4065.26 4250.59 501.21 715.82 928.59 1139.54 1348.69 1556.06 1761.65 1965.48 731.56
124
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
0.868 1.085 1.302 1.519 1.736 1.953 2.170 0.868 1.085 1.302 1.519 1.736 1.953 2.170 2.387 1.085 1.302 1.519 1.736 1.953 2.170 2.387 2.604 1.302 1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.884 14.042 13.481 13.080 12.779 12.546 12.359 15.690 14.687 14.018 13.541 13.183 12.904 12.681 12.499 15.332 14.556 14.001 13.586 13.262 13.004 12.792 12.615 15.093 14.462 13.989 13.621 13.326 13.085 12.884 12.714 14.923 14.392 13.979 13.649 13.378
944.33 1155.29 1364.43 1571.80 1777.38 1981.21 2183.30 960.08 1171.03 1380.18 1587.53 1793.12 1996.95 2199.03 2399.37 1186.77 1395.92 1603.27 1808.86 2012.68 2214.76 2415.10 2613.73 1411.66 1619.01 1824.59 2028.41 2230.49 2430.83 2629.45 2826.37 1634.75 1840.33 2044.15 2246.22 2446.55
125
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953 2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.153 12.962 12.799 14.795 14.337 13.971 13.671 13.422 13.210 13.029 12.872 14.696 14.294 13.965 13.690 13.458 13.260 13.087 12.937 14.616 14.258 13.959 13.707 13.490 13.302 13.138 12.993 14.551 14.228 13.955 13.720 13.517 13.340 13.183 13.044 14.497
2645.17 2842.09 3037.31 1856.06 2059.88 2261.95 2462.28 2660.90 2857.81 3053.03 3246.57 2075.61 2277.68 2478.01 2676.62 2873.53 3068.74 3262.28 3454.15 2293.41 2493.74 2692.35 2889.25 3084.46 3277.99 3469.86 3660.07 2509.46 2708.07 2904.97 3100.18 3293.70 3485.57 3675.78 3864.35 2723.79
126
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3
10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 0.651 0.868 1.085 1.302 1.519 1.736 1.953 2.170 0.868 1.085 1.302 1.519 1.736
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.203 13.951 13.732 13.541 13.373 13.223 13.089 14.451 14.181 13.947 13.743 13.562 13.402 13.258 13.129 14.411 14.162 13.944 13.752 13.581 13.428 13.291 13.166 17.480 15.779 14.758 14.078 13.591 13.227 12.943 12.717 16.585 15.403 14.615 14.052 13.630
2920.69 3115.89 3309.41 3501.27 3691.48 3880.04 4066.98 2936.41 3131.61 3325.13 3516.98 3707.18 3895.74 4082.68 4268.00 3147.33 3340.84 3532.69 3722.88 3911.44 4098.37 4283.68 4467.40 749.04 961.81 1172.76 1381.91 1589.27 1794.85 1998.68 2200.76 977.55 1188.50 1397.65 1605.01 1810.59
127
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
1.953 2.170 2.387 1.085 1.302 1.519 1.736 1.953 2.170 2.387 2.604 1.302 1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387 2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.302 13.039 12.824 16.048 15.153 14.513 14.033 13.660 13.362 13.117 12.914 15.690 14.974 14.436 14.018 13.684 13.410 13.183 12.990 15.434 14.839 14.377 14.007 13.704 13.451 13.238 13.055 15.243 14.735 14.329 13.997 13.720 13.486 13.285 13.111 15.093
2014.41 2216.49 2416.83 1204.24 1413.39 1620.74 1826.32 2030.14 2232.22 2432.56 2631.18 1429.13 1636.48 1842.06 2045.88 2247.95 2448.29 2646.90 2843.82 1652.22 1857.80 2061.61 2263.68 2464.01 2662.63 2859.54 3054.76 1873.53 2077.35 2279.41 2479.74 2678.35 2875.26 3070.47 3264.01 2093.08
128
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.652 14.290 13.989 13.734 13.515 13.326 13.160 14.974 14.583 14.258 13.982 13.746 13.541 13.362 13.204 14.876 14.526 14.230 13.976 13.756 13.563 13.394 13.243 14.795 14.478 14.206 13.971 13.765 13.583 13.422 13.277 14.726 14.437 14.186 13.967 13.773
2295.14 2495.47 2694.08 2890.98 3086.19 3279.72 3471.59 2310.87 2511.20 2709.80 2906.70 3101.91 3295.43 3487.29 3677.50 2526.92 2725.53 2922.42 3117.62 3311.14 3503.00 3693.21 3881.77 2741.25 2938.14 3133.34 3326.86 3518.71 3708.91 3897.47 4084.40 2953.86 3149.06 3342.57 3534.42 3724.61
129
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 0.868 1.085 1.302 1.519 1.736 1.953 2.170 2.387 1.085 1.302 1.519 1.736 1.953 2.170 2.387 2.604 1.302
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.601 13.447 13.308 14.667 14.401 14.168 13.963 13.780 13.617 13.470 13.336 14.616 14.370 14.152 13.959 13.786 13.631 13.490 13.362 17.480 16.119 15.212 14.564 14.078 13.699 13.397 13.150 16.764 15.749 15.024 14.481 14.058 13.720 13.443 13.212 16.287
3913.17 4100.10 4285.41 3164.77 3358.28 3550.13 3740.31 3928.86 4115.79 4301.10 4484.80 3374.00 3565.83 3756.02 3944.56 4131.48 4316.78 4500.49 4682.60 995.03 1205.98 1415.12 1622.48 1828.06 2031.88 2233.95 2434.29 1221.72 1430.86 1638.22 1843.79 2047.61 2249.68 2450.02 2648.64 1446.60
130
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387 2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953 2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.485 14.884 14.416 14.042 13.736 13.481 13.265 15.946 15.287 14.774 14.365 14.029 13.750 13.513 13.310 15.690 15.133 14.687 14.322 14.018 13.761 13.541 13.350 15.491 15.010 14.615 14.287 14.009 13.771 13.565 13.384 15.332 14.909 14.556 14.257 14.001
1653.95 1859.53 2063.34 2265.41 2465.74 2664.36 2861.27 1669.69 1875.27 2079.08 2281.14 2481.47 2680.08 2876.99 3072.20 1891.00 2094.81 2296.87 2497.20 2695.81 2892.71 3087.92 3281.45 2110.55 2312.60 2512.93 2711.53 2908.43 3103.64 3297.16 3489.02 2328.33 2528.66 2727.26 2924.15 3119.35
131
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.780 13.586 13.414 15.202 14.825 14.505 14.232 13.995 13.787 13.604 13.441 15.093 14.753 14.462 14.210 13.989 13.794 13.621 13.466 15.002 14.693 14.425 14.190 13.984 13.800 13.635 13.487 14.923 14.640 14.392 14.173 13.979 13.805 13.649 13.507 14.855
3312.87 3504.73 3694.93 2544.38 2742.98 2939.87 3135.07 3328.59 3520.44 3710.64 3899.20 2758.71 2955.59 3150.79 3344.30 3536.15 3726.34 3914.89 4101.82 2971.31 3166.50 3360.01 3551.85 3742.04 3930.59 4117.51 4302.82 3182.22 3375.73 3567.56 3757.75 3946.29 4133.21 4318.51 4502.21 3391.44
132
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340 4.557 4.774 4.991 1.085 1.302 1.519 1.736 1.953 2.170 2.387 2.604 1.302 1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.593 14.363 14.158 13.975 13.810 13.660 13.525 14.795 14.553 14.337 14.144 13.971 13.814 13.671 13.541 17.480 16.346 15.536 14.928 14.456 14.078 13.768 13.510 16.883 15.996 15.331 14.814 14.400 14.061 13.779 13.540 16.457 15.734 15.172 14.723 14.355
3583.27 3773.45 3961.99 4148.90 4334.20 4517.89 4700.00 3598.98 3789.15 3977.69 4164.59 4349.88 4533.57 4715.67 4896.20 1239.20 1448.34 1655.69 1861.26 2065.08 2267.14 2467.48 2666.09 1464.08 1671.43 1877.00 2080.81 2282.87 2483.20 2681.81 2878.72 1687.16 1892.74 2096.54 2298.60 2498.93
133
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953 2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.048 13.788 13.566 16.138 15.531 15.045 14.648 14.317 14.037 13.796 13.588 15.889 15.368 14.941 14.585 14.285 14.027 13.803 13.608 15.690 15.234 14.854 14.533 14.257 14.018 13.809 13.625 15.527 15.123 14.781 14.488 14.233 14.011 13.815 13.640 15.392
2697.54 2894.44 3089.65 1908.47 2112.28 2314.34 2514.66 2713.26 2910.16 3105.37 3298.89 2128.01 2330.07 2530.39 2728.99 2925.88 3121.08 3314.60 3506.46 2345.80 2546.12 2744.71 2941.60 3136.80 3330.32 3522.17 3712.37 2561.84 2760.44 2957.32 3152.52 3346.03 3537.88 3728.07 3916.62 2776.16
134
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.029 14.718 14.448 14.213 14.004 13.819 13.654 15.277 14.948 14.663 14.414 14.194 13.999 13.824 13.666 15.179 14.878 14.616 14.384 14.178 13.993 13.828 13.677 15.093 14.817 14.574 14.357 14.163 13.989 13.831 13.688 15.019 14.763 14.536 14.333 14.150
2973.04 3168.23 3361.74 3553.58 3743.77 3932.32 4119.24 2988.77 3183.95 3377.45 3569.29 3759.48 3948.02 4134.93 4320.24 3199.67 3393.17 3585.00 3775.18 3963.72 4150.63 4335.92 4519.62 3408.88 3600.71 3790.88 3979.42 4166.32 4351.61 4535.30 4717.40 3616.42 3806.59 3995.12 4182.01 4367.30
135
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 1.302 1.519 1.736 1.953 2.170 2.387 2.604 2.821 1.519 1.736 1.953 2.170 2.387 2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 13.985 13.834 13.697 14.953 14.715 14.503 14.311 14.138 13.981 13.837 13.705 17.480 16.508 15.779 15.212 14.758 14.387 14.078 13.816 16.969 16.182 15.570 15.081 14.680 14.346 14.064 13.822 16.585 15.928 15.403 14.973 14.615 14.312 14.052 13.827 16.287
4550.98 4733.07 4913.59 3822.29 4010.81 4197.71 4382.99 4566.66 4748.75 4929.26 5108.20 1481.55 1688.90 1894.47 2098.28 2300.34 2500.66 2699.27 2896.17 1704.64 1910.20 2114.01 2316.07 2516.39 2714.99 2911.89 3107.10 1925.94 2129.74 2331.80 2532.12 2730.72 2927.61 3122.81 3316.33 2145.48
136
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.726 15.266 14.884 14.560 14.283 14.042 13.832 16.048 15.560 15.153 14.808 14.513 14.257 14.033 13.836 15.853 15.421 15.056 14.743 14.472 14.235 14.025 13.839 15.690 15.304 14.974 14.687 14.436 14.215 14.018 13.842 15.552 15.204 14.902 14.638 14.405
2347.53 2547.85 2746.44 2943.33 3138.53 3332.05 3523.90 2363.26 2563.58 2762.17 2959.05 3154.25 3347.76 3539.60 3729.80 2579.30 2777.89 2974.78 3169.96 3363.47 3555.31 3745.50 3934.05 2793.62 2990.50 3185.68 3379.18 3571.02 3761.20 3949.75 4136.66 3006.22 3201.40 3394.90 3586.73 3776.91
137
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340 4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.198 14.012 13.845 15.434 15.117 14.839 14.594 14.377 14.182 14.007 13.848 15.332 15.041 14.784 14.556 14.352 14.168 14.001 13.850 15.243 14.974 14.735 14.521 14.329 14.155 13.997 13.852 15.164 14.914 14.691 14.490 14.309 14.143 13.993 13.854 15.093
3965.45 4152.36 4337.65 3217.12 3410.61 3602.44 3792.61 3981.14 4168.05 4353.34 4537.03 3426.33 3618.15 3808.32 3996.84 4183.74 4369.03 4552.71 4734.80 3633.86 3824.02 4012.54 4199.44 4384.71 4568.39 4750.48 4930.98 3839.73 4028.24 4215.13 4400.40 4584.07 4766.15 4946.65 5125.59 4043.94
138
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.123 4.340 4.557 4.774 4.991 5.208 5.425 1.519 1.736 1.953 2.170 2.387 2.604 2.821 3.038 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953 2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.861 14.652 14.462 14.290 14.133 13.989 13.856 17.480 16.629 15.968 15.439 15.005 14.645 14.339 14.078 17.033 16.326 15.761 15.299 14.913 14.587 14.308 14.066 16.684 16.084 15.592 15.182 14.835 14.538 14.281 14.055 16.406 15.885 15.451 15.083 14.769
4230.83 4416.09 4599.76 4781.83 4962.32 5141.25 5318.63 1722.11 1927.67 2131.48 2333.53 2533.85 2732.45 2929.34 3124.54 1943.41 2147.21 2349.26 2549.58 2748.17 2945.06 3140.26 3333.78 2162.95 2364.99 2565.31 2763.90 2960.78 3155.98 3349.49 3541.33 2380.73 2581.04 2779.62 2976.51 3171.69
139
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.496 14.257 14.046 16.178 15.720 15.332 14.999 14.711 14.458 14.236 14.038 15.988 15.580 15.229 14.926 14.660 14.426 14.217 14.031 15.828 15.460 15.141 14.862 14.615 14.396 14.201 14.024 15.690 15.356 15.063 14.805 14.576 14.370 14.186 14.018 15.571
3365.20 3557.04 3747.23 2596.77 2795.35 2992.23 3187.41 3380.91 3572.75 3762.93 3951.47 2811.08 3007.95 3203.13 3396.63 3588.46 3778.64 3967.17 4154.08 3023.67 3218.85 3412.34 3604.17 3794.34 3982.87 4169.78 4355.07 3234.57 3428.06 3619.88 3810.05 3998.57 4185.47 4370.75 4554.43 3443.77
140
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340 4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906 4.123 4.340 4.557 4.774 4.991 5.208 5.425 4.123 4.340 4.557 4.774 4.991
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.265 14.995 14.755 14.540 14.347 14.172 14.013 15.466 15.184 14.934 14.710 14.508 14.325 14.160 14.008 15.374 15.113 14.879 14.669 14.479 14.306 14.148 14.004 15.292 15.049 14.831 14.633 14.453 14.288 14.138 13.999 15.219 14.992 14.786 14.599 14.429
3635.59 3825.75 4014.27 4201.16 4386.44 4570.12 4752.20 3651.30 3841.46 4029.97 4216.86 4402.13 4585.80 4767.88 4948.38 3857.16 4045.67 4232.55 4417.82 4601.48 4783.55 4964.05 5142.98 4061.37 4248.25 4433.51 4617.17 4799.23 4979.72 5158.64 5336.01 4263.94 4449.20 4632.85 4814.91 4995.39
141
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
5.208 5.425 5.642 1.736 1.953 2.170 2.387 2.604 2.821 3.038 3.255 1.953 2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.272 14.128 13.996 17.480 16.724 16.119 15.624 15.212 14.863 14.564 14.304 17.082 16.442 15.917 15.480 15.111 14.794 14.519 14.279 16.764 16.210 15.749 15.359 15.024 14.734 14.481 14.257 16.504 16.018 15.607 15.255 14.949 14.682 14.446 14.237 16.287
5174.31 5351.67 5527.49 1960.88 2164.68 2366.73 2567.04 2765.63 2962.51 3157.71 3351.22 2180.41 2382.46 2582.77 2781.36 2978.24 3173.42 3366.93 3558.77 2398.19 2598.50 2797.08 2993.96 3189.14 3382.64 3574.48 3764.66 2614.23 2812.81 3009.68 3204.86 3398.36 3590.19 3780.37 3968.90 2828.53
142
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.855 15.485 15.164 14.884 14.636 14.416 14.219 16.103 15.715 15.379 15.085 14.826 14.595 14.389 14.203 15.946 15.594 15.287 15.016 14.774 14.559 14.365 14.189 15.809 15.488 15.205 14.954 14.728 14.526 14.342 14.176 15.690 15.395 15.133 14.898 14.687
3025.40 3220.58 3414.07 3605.90 3796.07 3984.60 4171.50 3041.13 3236.30 3429.79 3621.61 3811.77 4000.30 4187.20 4372.48 3252.01 3445.50 3637.32 3827.48 4016.00 4202.89 4388.17 4571.84 3461.21 3653.03 3843.18 4031.70 4218.59 4403.86 4587.53 4769.60 3668.74 3858.89 4047.40 4234.28 4419.55
143
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9
8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906 4.123 4.340 4.557 4.774 4.991 5.208 5.425 4.123 4.340 4.557 4.774 4.991 5.208 5.425 5.642 4.340 4.557 4.774 4.991 5.208 5.425 5.642 5.859 1.953
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.496 14.322 14.164 15.585 15.312 15.068 14.848 14.650 14.469 14.304 14.153 15.491 15.238 15.010 14.803 14.615 14.444 14.287 14.143 15.407 15.171 14.957 14.762 14.584 14.421 14.272 14.133 15.332 15.110 14.909 14.725 14.556 14.401 14.257 14.125 17.480
4603.21 4785.28 4965.77 3874.60 4063.10 4249.98 4435.24 4618.89 4800.96 4981.44 5160.37 4078.80 4265.67 4450.93 4634.58 4816.63 4997.12 5176.03 5353.39 4281.37 4466.62 4650.26 4832.31 5012.79 5191.70 5369.05 5544.86 4482.31 4665.94 4847.99 5028.46 5207.36 5384.71 5560.52 5734.79 2197.88
144
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.170 2.387 2.604 2.821 3.038 3.255 3.472 2.170 2.387 2.604 2.821 3.038 3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 16.800 16.243 15.779 15.386 15.050 14.758 14.503 17.122 16.536 16.048 15.634 15.280 14.973 14.704 14.467 16.829 16.316 15.882 15.510 15.188 14.906 14.657 14.436 16.585 16.130 15.741 15.403 15.108 14.847 14.615 14.408 16.378 15.971 15.618 15.309 15.036
2399.92 2600.23 2798.81 2995.69 3190.87 3384.37 3576.21 2415.65 2615.96 2814.54 3011.41 3206.59 3400.09 3591.92 3782.09 2631.69 2830.26 3027.13 3222.31 3415.80 3607.63 3797.80 3986.33 2845.99 3042.86 3238.03 3431.52 3623.34 3813.50 4002.03 4188.92 3058.58 3253.75 3447.23 3639.05 3829.21
145
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340 4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.794 14.577 14.382 16.201 15.833 15.511 15.226 14.973 14.747 14.544 14.359 16.048 15.712 15.416 15.153 14.917 14.705 14.513 14.338 15.914 15.606 15.332 15.087 14.866 14.666 14.485 14.319 15.795 15.511 15.256 15.027 14.820 14.632 14.460 14.302 15.690
4017.73 4204.62 4389.90 3269.46 3462.94 3654.76 3844.91 4033.43 4220.31 4405.58 4589.25 3478.66 3670.47 3860.62 4049.13 4236.01 4421.27 4604.93 4787.01 3686.18 3876.32 4064.83 4251.70 4436.96 4620.62 4802.68 4983.17 3892.03 4080.53 4267.40 4452.65 4636.30 4818.36 4998.84 5177.76 4096.23
146
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10
9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.123 4.340 4.557 4.774 4.991 5.208 5.425 4.123 4.340 4.557 4.774 4.991 5.208 5.425 5.642 4.340 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557 4.774 4.991 5.208 5.425 5.642 5.859 6.076 2.170 2.387 2.604 2.821 3.038
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.426 15.189 14.974 14.778 14.600 14.436 14.286 15.596 15.350 15.127 14.925 14.740 14.571 14.415 14.271 15.511 15.281 15.071 14.880 14.705 14.544 14.395 14.257 15.434 15.218 15.020 14.839 14.673 14.519 14.377 14.244 17.480 16.861 16.346 15.910 15.536
4283.10 4468.34 4651.99 4834.04 5014.51 5193.42 5370.77 4298.79 4484.03 4667.67 4849.72 5030.18 5209.08 5386.43 5562.24 4499.72 4683.36 4865.40 5045.86 5224.75 5402.09 5577.89 5752.16 4699.04 4881.07 5061.53 5240.42 5417.75 5593.54 5767.80 5940.55 2433.12 2633.42 2832.00 3028.86 3224.04
147
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.255 3.472 3.689 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.212 14.928 14.678 17.155 16.615 16.158 15.766 15.427 15.130 14.868 14.635 16.883 16.406 15.996 15.642 15.331 15.057 14.814 14.596 16.654 16.227 15.857 15.533 15.247 14.993 14.766 14.561 16.457 16.072 15.734 15.437 15.172 14.936 14.723 14.530 16.287
3417.53 3609.36 3799.53 2649.15 2847.72 3044.59 3239.76 3433.25 3625.07 3815.23 4003.76 2863.45 3060.31 3255.48 3448.96 3640.78 3830.94 4019.46 4206.35 3076.03 3271.19 3464.67 3656.49 3846.64 4035.16 4222.04 4407.31 3286.91 3480.39 3672.20 3862.35 4050.86 4237.74 4423.00 4606.66 3496.10
148
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340 4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906 4.123 4.340 4.557 4.774 4.991 5.208 5.425 4.123 4.340 4.557 4.774 4.991
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.936 15.626 15.351 15.105 14.884 14.683 14.501 16.138 15.816 15.531 15.275 15.045 14.837 14.648 14.475 16.006 15.710 15.445 15.206 14.990 14.794 14.615 14.451 15.889 15.614 15.368 15.144 14.941 14.755 14.585 14.429 15.784 15.529 15.298 15.088 14.896
3687.91 3878.05 4066.56 4253.43 4438.69 4622.34 4804.41 3703.62 3893.76 4082.26 4269.13 4454.38 4638.03 4820.09 5000.57 3909.47 4097.96 4284.82 4470.07 4653.71 4835.76 5016.24 5195.14 4113.66 4300.52 4485.76 4669.40 4851.44 5031.91 5210.81 5388.16 4316.22 4501.45 4685.08 4867.12 5047.58
149
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11
9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
5.208 5.425 5.642 4.340 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557 4.774 4.991 5.208 5.425 5.642 5.859 6.076 4.774 4.991 5.208 5.425 5.642 5.859 6.076 6.293 2.387 2.604 2.821 3.038 3.255 3.472 3.689 3.906 2.604
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.720 14.558 14.409 15.690 15.451 15.234 15.036 14.854 14.687 14.533 14.390 15.605 15.381 15.176 14.989 14.816 14.657 14.509 14.372 15.527 15.316 15.123 14.945 14.781 14.629 14.488 14.356 17.480 16.913 16.433 16.022 15.665 15.353 15.078 14.834 17.182
5226.47 5403.81 5579.61 4517.14 4700.77 4882.80 5063.25 5242.14 5419.47 5595.26 5769.52 4716.45 4898.48 5078.93 5257.81 5435.13 5610.92 5785.17 5957.90 4914.16 5094.60 5273.47 5450.79 5626.57 5800.81 5973.54 6144.76 2666.61 2865.18 3062.04 3257.21 3450.69 3642.50 3832.67 4021.18 2880.91
150
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689 3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 16.681 16.252 15.880 15.555 15.268 15.013 14.784 16.929 16.483 16.095 15.757 15.458 15.192 14.954 14.740 16.713 16.310 15.958 15.647 15.371 15.124 14.902 14.700 16.525 16.160 15.837 15.550 15.294 15.063 14.854 14.664 16.361 16.027 15.729 15.463 15.224
3077.76 3272.92 3466.40 3658.22 3848.37 4036.88 4223.77 3093.49 3288.64 3482.12 3673.93 3864.08 4052.58 4239.46 4424.73 3304.36 3497.83 3689.64 3879.78 4068.29 4255.16 4440.42 4624.07 3513.55 3705.35 3895.49 4083.99 4270.85 4456.11 4639.75 4821.81 3721.06 3911.19 4099.69 4286.55 4471.80
151
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11
5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906 4.123 4.340 4.557 4.774 4.991 5.208 5.425 4.123 4.340 4.557 4.774 4.991 5.208 5.425 5.642 4.340 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.007 14.810 14.631 16.216 15.909 15.633 15.385 15.161 14.957 14.771 14.600 16.088 15.803 15.547 15.315 15.104 14.911 14.735 14.572 15.973 15.708 15.468 15.250 15.051 14.869 14.701 14.546 15.869 15.622 15.397 15.192 15.003 14.830 14.670 14.522 15.775
4655.44 4837.49 5017.96 3926.90 4115.39 4302.25 4487.49 4671.12 4853.17 5033.63 5212.53 4131.09 4317.94 4503.18 4686.81 4868.85 5049.31 5228.20 5405.54 4333.64 4518.87 4702.49 4884.53 5064.98 5243.86 5421.20 5596.99 4534.56 4718.18 4900.21 5080.65 5259.53 5436.86 5612.64 5786.89 4733.87
152
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12
10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 0 0 0 0 0 0 0 0 1 1 1 1 1
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.774 4.991 5.208 5.425 5.642 5.859 6.076 4.774 4.991 5.208 5.425 5.642 5.859 6.076 6.293 4.991 5.208 5.425 5.642 5.859 6.076 6.293 6.510 2.604 2.821 3.038 3.255 3.472 3.689 3.906 4.123 2.821 3.038 3.255 3.472 3.689
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.543 15.332 15.138 14.959 14.794 14.642 14.500 15.690 15.472 15.272 15.088 14.918 14.761 14.615 14.479 15.612 15.406 15.217 15.043 14.881 14.731 14.591 14.460 17.480 16.957 16.508 16.119 15.779 15.479 15.212 14.973 17.205 16.738 16.334 15.980 15.668
4915.89 5096.33 5275.20 5452.52 5628.29 5802.54 5975.26 4931.57 5112.00 5290.87 5468.18 5643.94 5818.18 5990.90 6162.11 5127.67 5306.53 5483.84 5659.60 5833.83 6006.54 6177.74 6347.45 2898.36 3095.22 3290.37 3483.85 3675.65 3865.80 4054.31 4241.19 3110.94 3306.09 3499.56 3691.37 3881.51
153
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0
3.906 4.123 4.340 3.038 3.255 3.472 3.689 3.906 4.123 4.340 4.557 3.255 3.472 3.689 3.906 4.123 4.340 4.557 4.774 3.472 3.689 3.906 4.123 4.340 4.557 4.774 4.991 3.689 3.906 4.123 4.340 4.557 4.774 4.991 5.208 3.906
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.391 15.143 14.919 16.969 16.549 16.182 15.858 15.570 15.312 15.081 14.871 16.764 16.383 16.048 15.749 15.482 15.242 15.024 14.827 16.585 16.237 15.928 15.652 15.403 15.178 14.973 14.786 16.427 16.108 15.822 15.564 15.331 15.120 14.927 14.749 16.287
4070.01 4256.89 4442.14 3321.81 3515.28 3707.08 3897.22 4085.71 4272.58 4457.83 4641.48 3531.00 3722.79 3912.92 4101.42 4288.28 4473.52 4657.17 4839.22 3738.50 3928.63 4117.12 4303.97 4489.21 4672.85 4854.89 5035.36 3944.34 4132.82 4319.67 4504.91 4688.54 4870.57 5051.03 5229.92 4148.52
154
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10
1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4
4.123 4.340 4.557 4.774 4.991 5.208 5.425 4.123 4.340 4.557 4.774 4.991 5.208 5.425 5.642 4.340 4.557 4.774 4.991 5.208 5.425 5.642 5.859 4.557 4.774 4.991 5.208 5.425 5.642 5.859 6.076 4.774 4.991 5.208 5.425 5.642
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 15.991 15.726 15.485 15.266 15.067 14.884 14.715 16.161 15.887 15.639 15.413 15.207 15.018 14.844 14.684 16.048 15.792 15.560 15.347 15.153 14.973 14.808 14.655 15.946 15.706 15.487 15.287 15.102 14.932 14.774 14.628 15.853 15.628 15.421 15.231 15.056
4335.37 4520.60 4704.22 4886.25 5066.70 5245.59 5422.92 4351.07 4536.29 4719.91 4901.93 5082.38 5261.26 5438.58 5614.36 4551.98 4735.59 4917.61 5098.05 5276.92 5454.24 5630.01 5804.26 4751.28 4933.29 5113.72 5292.59 5469.90 5645.67 5819.90 5992.62 4948.97 5129.40 5308.26 5485.56 5661.32
155
Table B.2: Moment Capacities by Strand Layout, 100 ft. Span (74.4 in. effective flange width)
Strands in Each Duct 1 (Bottom)
2 (Middle)
3 (Top)
Total Area of PT Cables (in.2)
12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12
5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
5.859 6.076 6.293 4.991 5.208 5.425 5.642 5.859 6.076 6.293 6.510 5.208 5.425 5.642 5.859 6.076 6.293 6.510 6.727
Eccentricity of PT Nominal Moment Cables Capacity (ft-kip) (in. below cgc) 14.894 14.743 14.603 15.768 15.556 15.360 15.180 15.013 14.858 14.714 14.580 15.690 15.489 15.304 15.133 14.974 14.825 14.687 14.558
5835.55 6008.26 6179.46 5145.07 5323.93 5501.22 5676.97 5851.20 6023.90 6195.10 6364.80 5339.59 5516.88 5692.63 5866.84 6039.54 6210.73 6380.42 6548.63
156
B.3 Girder Design Calculations
157
Segmental Girder Design Rapid Repair and Replacement Project 60 ft span, AASTHO 45-in I-Girder
ASSUMPTIONS: Single Span Bridge Span Length = 60 ft. 8000 psi concrete (full strength @ transfer) 270 ksi low relax strand (prestress)
Design Units: 1 in := ft 12 Girder Spacing:
k := 1000lbf
Parapet walls @ 338 lb/ft of wall 8"deck thickness fcg := 8ksi fsu := 270ksi
ksi := 1000
lbf 2
fcgi := 8ksi d strand := 0.6in
fcd := 4ksi
β := 0.85
in
45 in I-girder, for 60 ft span, use same spacing as 65' span: From WBM:
spacing := 12ft
Deck thickness: Assume 8" thick deck, after loss of wearing surface we will have a 7.5" deck. t := 8in
t eff := 7.5in
Span Lengths: Span length is 60 ft With 12" bearing pads at either end, (the effective span (for simply supported beams) is 99 ft) assume L is 100 ft. Lspan := 60ft Bridge Width: Assume 50 ft width
158
Loading: We will want to look at the bridge in the following limit states: Strength I, Service I, Service III, and Fatigue. Load factors from LRFD T3.4.1-1:
Strength I Service I Service III Fatigue
DL 1.25 1.0 1.0 N/A
LL 1.75 1.0 0.8 0.75
Our dynamic load allowances, according to LRFD T3.6.2.1-1: 15% for Fatigue Limit State 33% for all other Limit States Loads applied to each girders: Girder self weight.................... ..583 k/ft (given in WBM) 8' Deck slab........................... 1.2 k/ft (150 pcf x thickness (8") x tributary area width(12') ) No future wearing surface Loads distributed among all girders: Sidewalk, median................... .472 k/ft* Diaphragms........................... .512 k on each girder Parapet wall.......................... .774 k/ft k wgirder := .583 ft
k wslab := 1.2 ft
k ws.m := .472 ft
k wdiaph := .512k wparapet := .774 ft
Total Girder Loads: On simple span girders: wdiaph WDCsimp := wgirder + wslab + Lspan
k WDCsimp = 1.792 ⋅ ft
On composite girder: WDCcont :=
ws.m + wparapet 5
k WDCcont = 0.249 ⋅ ft
Live Loads: Use HL93 truck (either design truck (below left) + lane or design tandem (below right) + lane) for +M Use special fatigue truck for fatigue limit state (HL93 truck with rear axles @ 30')
159
Load Distribution: This bridge meets the simple dist. parameters (const. width, assumed >4 beams, parallel equal beams, road overhang <= 3 ft, standard cross-section, etc), so we can use the AASHTO simple distribution guidelines. According to table 4.6.2.2.1-1, we have a type "k" bridge. According to T4.6.2.2.2b-1 our equations are:
S 0.06 + ⎛⎜ ⎞⎟ 14 ⎝ ⎠
One Design Lane Loaded:
Two or more lanes loaded:
S ⎞ 0.075 + ⎛⎜ ⎟ 9.5 ⎝ ⎠
0.4
S ⋅ ⎛⎜ ⎞⎟ ⎝ L⎠
0.6
S ⋅ ⎛⎜ ⎞⎟ ⎝ L⎠
0.3
0.2
⎛ Kg ⎞ ⎟ ⋅⎜ ⎜ 12.0⋅ L⋅ t 3 ⎟ s ⎠ ⎝
⎛ Kg ⎞ ⎟ ⋅⎜ ⎜ 12.0⋅ L⋅ t 3 ⎟ s ⎠ ⎝
0.1
0.1
The longitudinal stiffness parameter, K g , is: Kg := n gd⋅ ⎡Ig + ⎛ Ag ⋅ eg ⎣ ⎝
2⎞⎤
⎠⎦
Calculation of "n": Eg := 33000 ⋅ .150
Ed := 33000 ⋅ .150
1.5
⋅ 8 ⋅ ksi
Eg = 5422.453⋅
1.5
⋅ 4 ⋅ ksi
n gd :=
Ed = 3834.254⋅
Eg
k 2
in
k 2
in
n gd = 1.414
Ed
Girder Properties from WBM: 4
Ig := 125390in 2
Ag := 560in
3
y T := 24.73in
ST := 5070in
y B := −20.27 in
SB := −6186in
3
Distance between center of gravity of girder and center of gravity of deck (with 2" haunch), eg : t eg := y T + 2in + 2
eg = 30.73 ⋅ in
Longitudinal stiffness parameter: Kg := n gd⋅ ⎡Ig + ⎛ Ag ⋅ eg ⎣ ⎝
2⎞⎤
5
The relative long stiffness is: Krel :=
4
Kg = 9.252 × 10 ⋅ in
⎠⎦ ⎛⎜ Kg ⎞⎟ ⎜ 12⋅ L⋅ t3 ⎟ ⎝ ⎠
160
For single span: One lane loaded:
Two or more lanes loaded:
spacing ⎞ gi1 := 0.06 + ⎛⎜ ⎟ ⎝ 14ft ⎠
0.4
spacing ⎞ gi2 := 0.075 + ⎛⎜ ⎟ ⎝ 9.5ft ⎠
spacing ⎞ ⋅ ⎛⎜ ⎟ ⎝ Lspan ⎠
0.6
0.3
spacing ⎞ ⋅ ⎛⎜ ⎟ ⎝ Lspan ⎠
⎛ Kg ⋅ 12 ⎞ ⎟ ⋅⎜ 3 ⎜ 12.0⋅ L ⎟ span⋅ t ⎠ ⎝
0.2
0.1
⎛ Kg⋅ 12 ⎞ ⎟ ⋅⎜ 3⎟ ⎜ 12.0⋅ L t ⋅ span ⎠ ⎝
gi1 = 0.696
0.1
gi2 = 0.989
2 lane distribution factor controls: gi := max( gi2 , gi1 ) = 0.989 Load Cases a.) DL on initial simple span girders b.) unfactored post DL on composite girders c.) factored post DL on composite girders d.) HL-93 truck with lane load e.) HL-93 tandem with lane load f.) Fatigue truck NO NEGATIVE MOMENT TO DESIGN FOR - SIMPLE SPAN!!! Load combinations for Service-1 moments (all include post-DL): sv1-1) truck + lane LL ..... for +M in span sv1-2) tandem + lane LL ..... for +M in span Load combinations for Service-3 moments (include post DL): sv3-1) truck + lane LL ..... for +M in span sv3-2) tandem + lane LL ..... for +M in span Load combinations for Strength-1 moments (include all loads): [ same sets as Service 1, but with Strength 1 load factors ] Load combinations for Stength-1 shear: str1-sh1) truck + lane LL str1-sh2) tandem + lane LL Used PCBRIDGE to find the forces from the load cases, then used EXCEL to combine and factor them.
161
Factored Design Moments: The results from PCBRIDGE were multiplied by DLA, load factors, and distribution factor in Excel. Service I and Service III loads do not include DL on non-composite span. BUT Strength I does include this. Moments from the initial non composite DL: V = 53.76 k
M ss := 806.4ft⋅ k
Moments from composite DL, unfactored: M = 112.05 ft-k
M dc := 112.05ft⋅ k
M ss = 806.4 ⋅ ft⋅ k
Girder plus deck, simple span, unfactored
M s1c := 1549.47ft⋅ k
Service I Max Moment
(Truck and lane, plus post-comp DL)
M s3c := 1261.93ft⋅ k
Service III Max Moment
(Truck and lane, plus post-comp DL)
M str1 := 3661.17ft⋅ k
Strength I Max Moment
(Truck and lane, plus FULL DL)
M g45 := 262.35ft⋅ k
Girder alone on SS
162
Preliminary Design: Control of PT design: Full PT will be transferred well after concrete has cured - T is maximum and concrete strength is maximum. According to LRFD, our allowable stresses under the service loads: Initial compression before losses (5.9.4.1.1) fcia := .6⋅ fcgi
fcia = 4.8⋅ ksi
Compression after losses (5.9.4.2.1) fca := .45⋅ fcg
fca = 3.6⋅ ksi
Effective flange width on composite girder: Top flange width of girder:
b := 16in
Long span effective length:
Lelong := Lspan
Effective flange width:
(Distance between inflection points on DL moment diagram - full span because simple supported)
fle.long.a := 0.25⋅ Lelong = 180 ⋅ in fle.long.b := 12⋅ t + 0.5⋅ b = 104 ⋅ in fle.long.c := spacing = 144 ⋅ in
(
)
flelong := min fle.long.a , fle.long.b , fle.long.c
flelong = 104 ⋅ in
fle := flelong Transforming the slab into equivalent girder concrete: fle flet := flet = 73.539⋅ in n gd Composite Section Properties (Ignore haunch): 2
Ag = 560 ⋅ in
2
Aeffslab := t eff ⋅ flet
Acg := Ag + Aeffslab
Aeffslab = 551.543 ⋅ in 2
Acg = 1111.543⋅ in
Calculating location of center of gravity for composite girder: Ag Aeffslab 7.5in ⎞ dist := −y B⋅ + ⋅ ⎛⎜ 45in + 2in + ⎟ 2 ⎠ Acg Acg ⎝ dist = 35.394⋅ in The center of gravity for the composite section is 35.394 in from the bottom of the girder.
163
Using the dimensions given in the example for a composite beam, the dimension calculated above, and the parallel axis theorem, I can find the I-value for the composite girder as a whole: Ieffslab :=
flet⋅ t eff
3
4
Ieffslab = 10341.437 ⋅ in
3
height comp := 45in + 2in + 7.5in
height comp = 54.5⋅ in
2 ⎡ 7.5in ⎞ ⎤ ⎡ ⎛ ⎢ ⎥ + ⎣Ig + Ag⋅ ( dist + y B) 2⎤⎦ Icg := Ieffslab + Aeffslab⋅ ⎜ height comp − dist − ⎟ 2 ⎠⎦ ⎣ ⎝ 5
4
Icg = 3.939 × 10 ⋅ in Using the same dimensions I can also find the section moduli to the top the deck, the bottom of the girder, and the top of the girder flange: Icg 3 (To top of deck) Scgd := Scgd = 20615.616 ⋅ in height comp − dist Icg Scgb := 4 3 (To bottom of girder) Scgb = −1.113 × 10 ⋅ in −dist Icg Scgt := 45in − dist y cgd := height comp − dist
3
Scgt = 41003.856 ⋅ in y cgd = 19.106⋅ in
(To top of girder flange) y cgb := −dist
y cgb = −35.394⋅ in
Estimating PT Loss: To get an early estimate of how much loss we will have in the strands, we can use T5.9.5.3-1 (from 2004 edition) and calculate the average loss:
⎛ ⎝
loss := 33⋅ ⎜ 1.0ksi − 0.15⋅
fcg − 6ksi ⎞ 6
⎟ + 6ksi ⎠
(PPR = 1.0)
loss = 37.35 ⋅ ksi Since we are using low relaxation strands: loss := loss − 6ksi loss = 31.35 ⋅ ksi This seems high, so instead we will try: loss := 15ksi Strands will be tensioned up to a maximum of 218.7 ksi (.90f py) temporarily to offset friction and seating losses. Immediately after anchor set, the stress limit is 200 ksi (.74f pu = 199.8, rounded to 200). So the strands will be tensioned to 200 ksi and anchored. (Per T5.9.3-1) fpt := 200ksi Assume elastic shortening loss to be: ES := 10ksi Total assumed lossed: Δf := ES + loss
Δf = 25⋅ ksi
164
Effective prestress after losses: fe := 200ksi − ES − loss
fe = 175 ⋅ ksi
And our percent loss: pl :=
loss
pl = 7.895 ⋅ %
fpt − ES
Design amount of prestress to prevent tension at bottom of beam under full load (at center span) after 50 years Calcuating stress at the bottom of the beam due to non-composite and composite loading (Service 3 limit state): Components of moment @ center span: (NOT NECESSARILY THE MAXIMUM VALUE!) M ss = ⋅ ft⋅ k
Girder/deck/steel diaphragms, non-composite, unfactored
M s1c = 1549.47 ⋅ ft⋅ k
LL and composite DL, factored
M s3c = 1261.93 ⋅ ft⋅ k
LL and composite DL, factored
M g45 = 262.35⋅ ft⋅ k
Girder alone on SS
These all contribute to the full Service 1 and Service 3 moments. Total moments @ mid-span for Service 1 and 3 cases: M s1 := M ss + M s1c
M s3 := M ss + M s3c
M s1 = 2355.87 ⋅ ft⋅ k
M s3 = 2068.33 ⋅ ft⋅ k
The simple span moment will occur on a non-composite section, while the other moments will occur on a composite section. So the Service 3 tension stress at the bottom of the beam will be: M s3c M ss fbs3 := + fbs3 = −2.925 ⋅ ksi (Negative value means tension.) Scgb SB According to T5.9.4.2.2-1 in the LRFD, the limit on tension stress in service 3 condition is: fcg ftall := −.0948 ksi⋅ f = −0.268 ⋅ ksi ksi tall So we want a final prestress of: fpreq := fbs3 − ftall
fpreq = −2.657 ⋅ ksi
Using the prestress loss we previously calculated, we know that the initial compression from the prestress needs to be: fpreq fpireq := fpireq = −2.885 ⋅ ksi pl = 0.079 1 − pl Try 3 ducts, .6" diamater strands, two 12 strand ducts, one 7 strand duct. From VSL data for Polypropelene Plastic Duct: OD12 := 3.58in OD7 := 2.87in
Includes ribs
165
Assume they are bundled together in the middle portion of the span (one of top of the other), and have 1" of cover between bottom of girder and bottom duct (ignoring ribs on the bottom side of the duct). Even though ducts are bundled, they will still be .2 in apart because of the ribs. We are assuming the ribs will not hit each other. Numbering convention: Lowest duct is #1, middle duct is #2, top most duct is #3. cover := 1in OD12 educt1 := y B + cover + = −17.48 ⋅ in 2 OD12
educt2 := y B + cover + OD12 +
2
= −13.9⋅ in
educt3 := y B + cover + OD12 + OD12 +
OD7 2
= −10.675⋅ in
Guess a number of strands to use: ns 3 := 3
2
ns := 16
As := 0.217in
ns 2 := 6 ns 1 := 7
⎛ ns1 ⎞ ⎛ ns2 ⎞ ⎛ ns3 ⎞ estrands := educt1⋅ ⎜ ⎟ + educt2⋅ ⎜ ⎟ + educt3⋅ ⎜ ⎟ = −14.862⋅ in ⎝ ns ⎠ ⎝ ns ⎠ ⎝ ns ⎠ y_over_rsq := −.09053
in 2
in
term2 := 1 + estrands⋅ y_over_rsq = 2.345 term3 :=
(
Ag
2
term2
= 238.763 ⋅ in
)
Apt Apt := ns ⋅ A⋅ fspt fbi := = 2.908 ⋅ ksi term3 Forces in tendons: fpt = 200 ⋅ ksi
Calculations From WisDOT Design Data for 45 in I-girder
OK!
To := ns ⋅ fpt⋅ As
To.1int := ns 1 ⋅ fpt⋅ As = 303.8 ⋅ kip
To = 694.4 ⋅ kip
To.2int := ns 2 ⋅ fpt⋅ As = 260.4 ⋅ kip
To.3int := ns 3 ⋅ fpt⋅ As = 130.2 ⋅ kip
166
Check stresses in beam at mid-span and at the end under girder weight to avoid premature failure. Check using full PT load (conservative): At mid-span: Bottom: To.1int⋅ educt1 + To.2int⋅ educt2 + To.3int⋅ educt3 + M g45 To.1int + To.2int + To.3int fbint := + Ag SB
(
) (
fbint = 2.399 ⋅ ksi fcia = 4.8⋅ ksi
Allowed:
)
OK
At top: ftint :=
(To.1int + To.2int + To.3int) (To.1int⋅ educt1 + To.2int⋅ educt2 + To.3int⋅ educt3) + Mg45 +
Ag
Allowed:
ftint = −0.175 ⋅ ksi ftall = −0.268 ⋅ ksi fcia = 4.8⋅ ksi
ST
OK
GIRDER IS OK AT MIDSPAN IN THE INITIAL CONDITION! End of beam: Because of spacing needs for the anchorages, the ducts CANNOT be the same height at the end as they are in the middle. First figure out where the strands would be at the end of the girder using VSL Type E Stressing Anchorage, we need: - an edge spacing of 8.185" for the 12-strand ducts and 6.51" for the 7-strand duct - an anchorage spacing of 14.37" for the 12-strand ducts and 11.02" for the 7-strand duct These spacings are from the center of the anchorage. Figuring out the eccentricities of the ducts at the end of the beam: spacings are rounded for easier fabrication
ed1.end := y B + 8.2in = −12.07 ⋅ in ed2.end := ed1.end + 14.4in = 2.33⋅ in
larger spacing used because of larger duct
ed3.end := ed2.end + 14.4in = 16.73 ⋅ in Check edge spacing at top: y T − ed3.end = 8 ⋅ in
> 6.51" OK
Now check stresses at end of girder using these eccentricities: To.1int⋅ ed1.end + To.2int⋅ ed2.end + To.3int⋅ ed3.end To.1int + To.2int + To.3int fbeint := + Ag SB
(
) (
Allowed:
fbeint = 1.383 ⋅ ksi OK fcia = 4.8⋅ ksi
)
167
fteint :=
(To.1int + To.2int + To.3int) + (To.1int⋅ ed1.end + To.2int⋅ ed2.end + To.3int⋅ ed3.end) Ag
ST
fteint = 1.066 ⋅ ksi Allowed: ftall = −0.268 ⋅ ksi fcia = 4.8⋅ ksi
OK
To determine point where draped strands become straight (hold down point), pick distance along girder where all strands are straight, and determine parabolic equation to match the ducts' eccentricities at that point, and their eccentricities at the end of the girder. All three ducts become straight at 20 feet from girder end. The eccentricities for each duct are: educt1 = −17.48 ⋅ in
ed1.end = −12.07 ⋅ in
educt2 = −13.9⋅ in
ed2.end = 2.33⋅ in
('+' means above cgc of girder) educt3 = −10.675⋅ in ed3.end = 16.73 ⋅ in The distance above the bottom of the girder for the center of each duct, as a function of x (0
drape1 ( x ) := 9.3924⋅ 10
The eccentricity of each duct as a function of x:
− 5) 2 ( ⋅ x − .045x + −12.07 −4 2 ecc2( x ) := ( 2.8177⋅ 10 ) ⋅ x − .13525x + 2.33 −4 2 ecc3( x ) := ( 4.7578⋅ 10 ) ⋅ x − .2284x + 16.73
ecc1( x ) := 9.3924⋅ 10
The slopes of the ducts at any point x (up to x = 240): − 4) ( ⋅ x − 0.045 −4 slope2( x ) := ( 5.6354⋅ 10 ) ⋅ x − 0.13525 −4 slope3( x ) := ( 9.5156⋅ 10 ) ⋅ x − .2284
slope1( x ) := 1.8785⋅ 10
168
Instantaneous Losses: Anchorage Set: Assume that there is no slip at the anchorages, plus, tendons are pulled to higher stress to account for this if it were to happen: ΔfpA := 0ksi Friction Loss: Using AutoCAD, the drape of the strands was approximated with a circle, and the angle change of the strands was found. fpj := fpt K := .0002 μ := .23 K and μ taken from T5.9.5.2.2b-1 Duct 1: From end of beam to end of drape:
⎡
ΔfpF1d := fpj⋅ ⎣1 − e
(
)
x 1d := 20 α1d := 2.577deg = 0.045
− K⋅ x1d+ μ⋅ α1d ⎤
⎦ = 2.848 ⋅ ksi
From end of drape to center of beam: x 1s := 10
(
)
⎡
− K⋅ x1s + μ⋅ α1s ⎤
⎡
− K⋅ x2d+ μ⋅ α2d ⎤
α1s := 0
ΔfpF1s := fpj⋅ ⎣1 − e ⎦ = 0.4⋅ ksi Duct 2: From end of beam to end of drape: x 2d := 20 α2d := 7.688deg = 0.134 ΔfpF2d := fpj⋅ ⎣1 − e
(
)
⎦ = 6.852 ⋅ ksi
From end of drape to center of beam: x 2s := 10 α2s := 0
⎡
ΔfpF2s := fpj⋅ ⎣1 − e
(
)
− K⋅ x2s + μ⋅ α2s ⎤
⎦ = 0.4⋅ ksi
Duct 3: From end of beam to end of drape:
⎡
ΔfpF3d := fpj⋅ ⎣1 − e
(
)
x 3d := 20 α3d := 12.844deg = 0.224
− K⋅ x3d+ μ⋅ α3d ⎤
⎦ = 10.809⋅ ksi
From end of drape to center of beam: x 3s := 10 α3s := 0
⎡
ΔfpF3s := fpj⋅ ⎣1 − e
(
)
− K⋅ x3s + μ⋅ α3s ⎤
⎦ = 0.4⋅ ksi
Friction Loss Summary: To1int.end := fpt⋅ ns 1 ⋅ As = 303.8 ⋅ kip To2int.end := fpt⋅ ns 2 ⋅ As = 260.4 ⋅ kip To3int.end := fpt⋅ ns 3 ⋅ As = 130.2 ⋅ kip
( ( (
) ) )
To1int.mid := fpt − ΔfpF1d − ΔfpF1s ⋅ ns 1 ⋅ As = 298.866 ⋅ kip To2int.mid := fpt − ΔfpF2d − ΔfpF2s ⋅ ns 2 ⋅ As = 250.958 ⋅ kip To3int.mid := fpt − ΔfpF3d − ΔfpF3s ⋅ ns 3 ⋅ As = 122.903 ⋅ kip
169
Use an average in each duct for other loss calcs: To1int.end + To1int.mid To1.int := = 301.333 ⋅ kip 2 To2.int := To3.int :=
To2int.end + To2int.mid 2 To3int.end + To3int.mid 2
= 255.679 ⋅ kip = 126.552 ⋅ kip
Elastic Shortening Losses: To re-estimate elastic shortening loss, we will need a jacking pattern, since any elastic shortening that occurs will be caused by the jacking of the different ducts. Jacking pattern: Duct 2, Duct 1, Duct 3. fcgi Es Eci := 1820⋅ ⋅ ksi Es := 28500ksi n := = 5.536 ksi Eci AT MIDSPAN: Loss in duct 2 (from pulling ducts 1 and 3): fc.2mid :=
To.1int Ag
+
To.3int Ag
+
(To.1int⋅ educt1)⋅ educt2 + (To.3int⋅ educt3)⋅ educt2 Ig
Ig
fs.2mid := n ⋅ fc.2mid = 8.403 ⋅ ksi ΔT2mid := fs.2mid⋅ ns 2 ⋅ As = 10.941⋅ kip Loss in duct 1 (from pulling duct 3): To.3int To.3int⋅ educt3 ⋅ educt1 fc.1mid := + Ag Ig
(
)
fs.1mid := n ⋅ fc.1mid = 2.36⋅ ksi ΔT1mid := fs.1mid⋅ ns 1 ⋅ As = 3.585 ⋅ kip Loss in duct 3:
NONE
ΔT3 := 0kip
AT END: Loss in duct 2 (from pulling ducts 1 and 3): fc.2end :=
To.1int Ag
+
To.3int Ag
+
(To.1int⋅ ed1.end)⋅ ed2.end + (To.3int⋅ ed3.end)⋅ ed2.end Ig
fs.2end := n ⋅ fc.2end = 4.138 ⋅ ksi ΔT2end := fs.2end ⋅ ns 2 ⋅ As = 5.387 ⋅ kip
Ig
170
Loss in duct 1 (from pulling duct 3): To.3int⋅ ed3.end ⋅ ed1.end To.3int fc.1end := + Ig Ag
(
)
fs.1end := n ⋅ fc.1end = 0.126 ⋅ ksi ΔT1end := fs.1end ⋅ ns 1 ⋅ As = 0.192 ⋅ kip Loss in duct 3:
NONE
Elastic Shortening Losses Summary: Forces in ducts at mid-span: To.1mid := ns 1 ⋅ fpt⋅ As − ΔT1mid = 300.215 ⋅ kip To.2mid := ns 2 ⋅ fpt⋅ As − ΔT2mid = 249.459 ⋅ kip To.3mid := ns 3 ⋅ fpt⋅ As − ΔT3 = 130.2 ⋅ kip Forces in ducts at end: To.1end := ns 1 ⋅ fpt⋅ As − ΔT1end = 303.608 ⋅ kip To.2end := ns 2 ⋅ fpt⋅ As − ΔT2end = 255.013 ⋅ kip To.3end := ns 3 ⋅ fpt⋅ As − ΔT3 = 130.2 ⋅ kip
Time Dependent Losses: TIME DEPENDANT LOSSES, BETWEEN TRANSFER AND DECK PLACEMENT Shrinkage Loss, between transfer and deck placement: According to AASHTO Eq 5.9.5.4.2a-1, the prestress loss due to shrinkage of grider concrete between time of transfer and deck placement is: ΔfpSR := ε bid⋅ Ep ⋅ Kid t f := 50 × 365 = 18250 days
Final age in days (50 years)
t i := 365 days
Age at transfer, days (1 year)
Ep := Es ε bid := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 H for Wisconsin = 72
Eq 5.4.2.3.3-1
conc. shrnk. strain from transfer to deck placement
H := 72
k hs := 2.00 − 0.014 ⋅ H = 0.992
humidity factor for shrinkage
k vs := 1.45 − 0.13⋅ VS
factor for effect of volume-to-surface ratio
VS :=
Ag ( 16 + 2 ⋅ 7 + 2 ⋅ 19 + 2 ⋅ 7 + 22 + 2 ⋅ 6.36 + 2 ⋅ 10.61 )in
= 4.06⋅ in
volume/surface ratio
171
k vs := 1.45 − 0.13⋅
k f := k td :=
VS
5ksi 1ksi + fcgi
= 0.922 but >=1.0
in
k vs := 1.0
factor for effect of conc. strength
= 0.556
t
time development factor
61 − 4 ⋅ fcgi + t
t1 := 373
t1 = time between end of cure and deck t2 = time between end of cure and transfer
t2 := 363 t1
k td1 := 61 − 4 ⋅
fcgi ksi
t2
= 0.928 k td2 := + t1
61 − 4 ⋅
fcgi ksi
= 0.926 + t2
k td := k td1 − k td2 = 0.002 ε bid := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 −7
ε bid = −4.868 × 10 Kid :=
1 Ag ⋅ estrands Es Aps ⎡⎢ ⋅ ⋅ 1+ 1+ Ig Eci Ag ⎢ ⎣
(
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti
) ⎤⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥( 2
⎦
creep coefficient
− 0.118
transformed section coefficient
psi factor for 1 year, 1 year + 10 days, and 50 years
humidity factor for creep
k hc := 1.56 − 0.008 ⋅ H = 0.984 2
Aps := ns ⋅ As = 3.472 ⋅ in
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti Kid.SR :=
− 0.118
= 0.001
1
⎡A ⋅ e Es Aps ⎡⎢ ⎣ g strands ⋅ ⋅ 1+ 1+ ⎢ Ig Eci Ag ⎣
(
ΔfpSR := −ε bid ⋅ Ep ⋅ Kid.SR = 0.013 ⋅ ksi
2⎤ ⎤
) ⎦ ⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥(
= 0.936
⎦
loss from shrinkage btwn transfer and deck
172
Creep Loss, between transfer and deck placement: Uses most of above values... Ep ΔfpCR := ⋅f ⋅ψ ⋅K Eci cgp b.td.ti id t := 10 t is 10 days now, since it is from time of transfer to deck t k td := = 0.256 fcgi 61 − 4 ⋅ +t ksi ψb.td.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i Kid.CR :=
comp := mom :=
− 0.118
− 0.118
= 0.133 = 0.133
To adjust these values for t=10 days ti is still 365 days
1 2⎤ ⎤ ⎡A ⋅ ( e Es Aps ⎡⎢ g strands) ⎦ ⎥ ⎣ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Ig Eci Ag ⎢ ⎣ ⎦
= 0.931
(To.1mid + To.2mid + To.3mid) (
(These terms are split up so they fit on the page) Ag To.1mid⋅ educt1 + To.2mid⋅ educt2 + To.3mid⋅ educt3 + M g45 ⋅ estrands
)
fcgp := comp + mom
fcgp = 2.039 ⋅ ksi
Ig
Ep loss from creep btwn transfer and deck ΔfpCR := ⋅f ⋅ψ ⋅K = 1.394 ⋅ ksi Eci cgp b.td.ti id.CR Relaxation Loss, between transfer and deck placement: ΔfpR1 := 1.2ksi Since we are using low-relaxation strand, we can assume a relaxation loss of 1.2 ksi TIME DEPENDANT LOSSES, AFTER DECK PLACEMENT Shrinkage ΔfpSD := ε bdf ⋅ Ep ⋅ Kdf ε bdf := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 k td :=
t 61 − 4 ⋅ fcgi + t
t1 := 18248
Eq 5.4.2.3.3-1
conc. shrnk. strain from transfer to deck placement
t2 := 373 t1 = time from end of cure to 50 years t2 = time from end of cure to deck
173
t1
k td1 := 61 − 4 ⋅
= 0.998
fcgi
61 − 4 ⋅
+ t1
ksi
t2
k td2 :=
fcgi
k td := k td1 − k td2 = 0.071
ksi
ε bdf := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 = −1.866 × 10
= 0.928 + t2
−5
− 0.118
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i = 0.037 epc := ⎡estrands − y B − y cgb ⎤ = −29.986⋅ in
(
⎣
Kdf.SD :=
)⎦
Below center of gravity of composite section
1 2 Acg⋅ ( epc) ⎤⎥ Ep Aps ⎡⎢ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Icg Eci Acg ⎢ ⎣ ⎦
ΔfpSD := −ε bdf ⋅ Ep ⋅ Kdf.SD = 0.501 ⋅ ksi
= 0.941
loss from shrinkage btwn deck and 50 yrs
Creep Loss: Ep Ep ΔfpCD := ⋅ fcgp⋅ ψb.tf.ti − ψb.td.ti ⋅ Kdf + ⋅ Δfcd⋅ ψb.tf.td⋅ Kdf Eci Eci
(
)
50 years - (1 yr + 10 days)
t := 17875 t
k td := 61 − 4 ⋅
fcgi ksi
= 0.998 +t
ψb.td.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i Kdf.CD :=
(
− 0.118
= 0.517 = 0.517
1 2 Acg⋅ ( epc) ⎤⎥ Ep Aps ⎡⎢ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Icg Eci Acg ⎢ ⎣ ⎦
)
Δfcd := M ss ⋅ t d := 375
− 0.118
(−estrands) Ig
= 0.923
Eci + ΔfpCR + ΔfpSR + ΔfpR1 ⋅ = 1.618 ⋅ ksi Ep
(
)
the age of the girder at the time the deck is placed.
ψb.tf.td := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t d
− 0.118
= 0.515
Ep Ep ΔfpCD := ⋅ fcgp⋅ ψb.tf.ti − ψb.td.ti ⋅ Kdf.CD + ⋅ Δfcd⋅ ψb.tf.td⋅ Kdf.CD = 4.261 ⋅ ksi Eci Eci
(
)
174
Relaxation Loss: According to 5.9.5.4.3c, ΔfpR2 := ΔfpR1
Shrinkage of Deck Concrete: Loss/gain resulting from shrinkage of deck concrete, per 5.9.5.4.3d: Ep ΔfpSS := ⋅ Δfcdf ⋅ Kdf ⋅ 1 + 0.7⋅ ψb.tf.td Eci epc ⋅ ed⎤ ε ddf ⋅ Ad ⋅ Ecd ⎡ 1 Δfcdf := ⋅⎢ + ⎥ Icg 1 + 0.7⋅ ψd.tf.td Acg ⎣ ⎦
(
)
( )
ε ddf := −k vsd⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 k f := 1
different concrete strength (now 4ksi).
Ad := 7.5in⋅ ( 3ft + 3ft + 4 ⋅ spacing) VSd :=
2 ⋅ 7.5in + 2 ⋅ ( 3ft + 3ft + 4 ⋅ spacing) VSd in
= 3.707 ⋅ in
61 − 4 ⋅
fcgi ksi
= 0.928 +t
= 0.968 But must be at least 1.0 k vsd := 1.0 k f and k hs stay the same
ε ddf := −k vsd⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 Ecd := 3640ksi
t
k td :=
Ad
k vsd := 1.45 − 0.13⋅
t := 373
MOE for 4 ksi concrete
in ed := y cgd + −7.5 = 15.356⋅ in 2 ψd.tf.td := 1.9⋅ k vsd⋅ k hc⋅ k f ⋅ k td⋅ t d
− 0.118
= 0.862
−epc⋅ −ed ⎞ ε ddf ⋅ Ad ⋅ Ecd ⎛ 1 Δfcdf := ⋅⎜ + ⎟ = 1.313 ⋅ ksi Icg 1 + 0.7⋅ ψd.tf.td Acg ⎝ ⎠ Ep ΔfpSS := ⋅ Δfcdf ⋅ Kdf.CD⋅ 1 + 0.7⋅ ψb.tf.td = 9.131 ⋅ ksi Eci This value is positive, so it means we have a gain resulting from the shrinkage of the deck.
(
)
Summary of losses (at mid-span): Instantaneous losses: ΔfpF1 := ΔfpF1d + ΔfpF1s = 3.248 ⋅ ksi ΔfpF2 := ΔfpF2d + ΔfpF2s = 7.252 ⋅ ksi
Friction Losses
175
ΔfpF3 := ΔfpF3d + ΔfpF3s = 11.208⋅ ksi fs.1mid = 2.36⋅ ksi
Elastic Shortening Losses
fs.2mid = 8.403 ⋅ ksi fs.3mid := 0ksi
Anchorage Losses
ΔfpA = 0
Time dependant losses, between PT transfer and deck placement: ΔfpSR = 0.013 ⋅ ksi
ΔfpCR = 1.394 ⋅ ksi
ΔfpR1 = 1.2⋅ ksi
Time dependant losses, between deck placement and 50 years: ΔfpSD = 0.501 ⋅ ksi
ΔfpCD = 4.261 ⋅ ksi
ΔfpR2 = 1.2⋅ ksi
ΔfpSS = 9.131 ⋅ ksi
TOTAL TIME DEPENDANT LOSSES: ΔfpLT := ΔfpSR + ΔfpCR + ΔfpR1 + ΔfpSD + ΔfpCD + ΔfpR2 − ΔfpSS = −0.563 ⋅ ksi This is used as an average value for all ducts. Total losses for each duct: Δfpe1 := ΔfpLT + fs.1mid + ΔfpF1 = 5.045 ⋅ ksi Δfpe2 := ΔfpLT + fs.2mid + ΔfpF2 = 15.092⋅ ksi Δfpe3 := ΔfpLT + fs.3mid + ΔfpF3 = 10.646⋅ ksi % loss: ploss1 :=
Δfpe1 fpt
⋅ 100 = 2.523
ploss2 :=
Δfpe2 fpt
⋅ 100 = 7.546 ploss3 :=
loss after elastic shortening: ΔfpLT = −0.563 ⋅ ksi as a % ΔfpLTper :=
ΔfpLT fpt
⋅ 100 = −0.281
a gain after transfer!
Initial stress after jacking: fi1 := fpt − fs.1mid − ΔfpF1 = 194.392 ⋅ ksi fi2 := fpt − fs.2mid − ΔfpF2 = 184.345 ⋅ ksi fi3 := fpt − fs.3mid − ΔfpF3 = 188.792 ⋅ ksi Effective stress after all loss: fe1 := fpt − Δfpe1 = 194.955 ⋅ ksi fe2 := fe3 :=
fpt − Δfpe2 = 184.908 ⋅ ksi fpt − Δfpe3 = 189.354 ⋅ ksi Total effective prestress force: T1 := As⋅ ns 1 ⋅ fe1 = 296.136 ⋅ k
Δfpe3 fpt
⋅ 100 = 5.323
176
T2 := As⋅ ns 2 ⋅ fe2 = 240.75⋅ k T3 := As⋅ ns 3 ⋅ fe3 = 123.27⋅ k To1 := As⋅ ns 1 ⋅ fi1 = 295.281 ⋅ k To2 := As⋅ ns 2 ⋅ fi2 = 240.018 ⋅ k To3 := As⋅ ns 3 ⋅ fi3 = 122.903 ⋅ k
Stress Check @ mid-span: Here we will check at the midspan at the time of transfer and at 50 years down the road. Data needed: To1 = 295.281 ⋅ k
To2 = 240.018 ⋅ k
To3 = 122.903 ⋅ k
T1 = 296.136 ⋅ k
T2 = 240.75⋅ k
T3 = 123.27⋅ k
educt1 = −17.48 ⋅ in
educt2 = −13.9⋅ in
educt3 = −10.675⋅ in
M ss = 806.4 ⋅ ft⋅ k
Non-composite DL, unfactored
M s3c = 1261.93 ⋅ ft⋅ k
Service 3 composite load
M s1c = 1549.47 ⋅ ft⋅ k
Service 1 composite load
Initial condition after jacking: Top of girder: fti :=
(To1 + To2 + To3) + (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg45 ST
Ag
fti = −0.139 ⋅ ksi
Allowable: fcia = 4.8⋅ ksi
OK!
ftall = −0.268 ⋅ ksi Bottom of girder: To1 + To2 + To3 To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3 + M g45 fbi := + SB Ag
(
) (
fbi = 2.252 ⋅ ksi
Allowable: fcia = 4.8⋅ ksi
)
OK!
ftall = −0.268 ⋅ ksi Final Condition (50 years): Top of girder (under Service 1 loading): ft1 :=
T1 + T2 + T3 Ag
ft1 = 1.6⋅ ksi
+
T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss ST Allowable: fcia = 4.8⋅ ksi
OK!
+
M s1c Scgt
177
Top of girder (under Service 3 loading): ft1 :=
T1 + T2 + T3 Ag
+
T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss ST Allowable: fcia = 4.8⋅ ksi
ft1 = 1.516 ⋅ ksi
+
M s3c Scgt
OK!
Bottom of girder (under Service 1 loading): M s1c T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss T1 + T2 + T3 fb3 := + + Scgb SB Ag Allowable: ftall = −0.268 ⋅ ksi HIGH!
fb3 = −0.466 ⋅ ksi
Bottom of girder (under Service 3 loading): M s3c T1 + T2 + T3 T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss fb3 := + + Scgb Ag SB Allowable: ftall = −0.268 ⋅ ksi OK!
fb3 = −0.156 ⋅ ksi
Top of deck slab (under Service 1 loading): M s1c ft.deck := ft.deck = 0.902 ⋅ ksi Scgd
Allowable: fcia = 4.8⋅ ksi
OK!
Check stress at hold down point (20 ft) at jacking: wgirder M g20 := ⋅ Lspan − 20ft ⋅ 20ft = 233.2 ⋅ kip ⋅ ft 2
(
fti.20 :=
(To1 + To2 + To3) (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg20 Ag
fti.20 = −0.208 ⋅ ksi fbi.20 :=
)
+
Allowable:
ST ftall = −0.268 ⋅ ksi
OK!
(To1 + To2 + To3) (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg20 Ag
fbi.20 = 2.309 ⋅ ksi
+
Allowable:
SB fcia = 4.8⋅ ksi
OK!
Ultimate Strength - Moment Capacity: Check flexural strength capacity @ midspan: Aps := ns ⋅ As 2
Aps = 3.472 ⋅ in
fe = 175 ⋅ ksi
fpe := fe
fpu := fsu
0.5⋅ fsu = 135 ⋅ ksi
fpe > 0.5⋅ fpu
OK!!!
At failure, when the concrete is crushing, we can assume the tendon stress to be:
178
c fps := fpu⋅ ⎛⎜ 1 − k f ⋅ ⎞⎟ dp ⎝ ⎠
k f := 0.28
Where "c" is defined as the neutral axis location. We will calculate it below:
b = 16⋅ in
girder flange width
t f := 7in
flange thickness at outer edge
t fi := 7in
flange thickness at start of radius
b w := 7in
girder web width
hau := 2in
haunch thickness
We will assume that the compression block is in the deck, and calculate capacity as if it is a rectangular section, and use the compression strength of the deck concrete Aps ⋅ fpu
c :=
⎛ fcg − 4ksi ⎞ ⎟ = 0.65 ⎝ 1ksi ⎠
fpu
.85⋅ fcd⋅ β⋅ b + k f ⋅ Aps ⋅ d
β1 := .85 − .05⋅ ⎜
p
dp is the distance from the compression face to the tendon centroid:
d p := y T − estrands + hau + teff d p = 49.092⋅ in
fle = 104 ⋅ in
Calculate c, taking width b as the effective flange width Aps ⋅ fpu
c :=
fpu
.85⋅ fcd⋅ β⋅ fle + k f ⋅ Aps ⋅ d
= 3.064 ⋅ in
p Neutral Axis is in deck, good
β⋅ c = 2.605 ⋅ in c fps := fpu⋅ ⎛⎜ 1 − k f ⋅ ⎞⎟ dp ⎝ ⎠
fps = 265.281 ⋅ ksi
Nominal moment cpacity of the composite section: a := β⋅ c
a = 2.605 ⋅ in
2
Aps = 3.472 ⋅ in
a M n := Aps ⋅ fps ⋅ ⎛⎜ d p − ⎞⎟ 2⎠ ⎝ M n = 3668.037⋅ ft⋅ k Capacity: Mr = φ*Mn , and in prestressed concrete, φ=1.00, so:
179
M r := 1.0⋅ M n M r = 3668.037⋅ ft⋅ k Required capacity, from strength I state: M str1 = 3661.17 ⋅ ft⋅ k
Moment capacity is OK!!!
Vertical Shear Capacity: Design vertical shear reinforcing at exterior end of span. Shear force has a different distribution factor than moment, and that has been accounted for in the load cases above. g shear1 := 0.36 + g shear2 := 0.2 +
spacing 25.0ft
spacing 12ft
= 0.84 2
spacing ⎞ − ⎛⎜ ⎟ = 1.082 ⎝ 35ft ⎠
Vu := 278.13k
ϕv := 0.9
Controls
Max shear from Strength 1 case
Critical section is taken at a distance of dv from the face of the support. dv is the distance between resultants of the tensile and compressive forces due to flexure, and no less than .9*de or .72*h. .72⋅ height comp = 39.24 ⋅ in .9⋅ .5⋅ spacing = 64.8⋅ in
Use this as value of dv
d v := −estrands + y T + hau + teff −
a 2
= 47.789⋅ in
d v := 65in But, there are draped strands in this area of the beam, so the e and a values will be different. Assuming standard bearing bad width of 8 inches, the distance to the critical section from the end of the girder: 8in Lcrit := ⎛⎜ + d v⎞⎟ + .5ft = 6.25 ft ⎝ 2 ⎠ Eccentricity of the strands at critical section: Lcrit x := = 75 in ns 2 ns 3 ⎛ ns1 ⎞ es_crit := ⎜ ⋅ ecc1( x ) + ⋅ ecc2( x ) + ⋅ ecc3( x ) ⎟ ⋅ in = −8.435 ⋅ in ns ns ⎝ ns ⎠ Stress block at critical section: d p_crit := −es_crit + y T + hau + teff = 42.665⋅ in
180
Aps ⋅ fpu
c :=
.85⋅ fcd⋅ β⋅ fle + k f ⋅ Aps ⋅ d
fpu
= 3.056 ⋅ in
p_crit
acrit := β⋅ c = 2.598 ⋅ in
In the deck which is good. acrit d v_crit := −estrands + y T + hau + teff − = 47.793⋅ in 2 Use:
d v := 65in
x ⋅ in = 6.25 ft
The nominal shear resistance is calculated as:
(
Vn := min Vc + Vs + Vp , 0.25⋅ fcg⋅ b v ⋅ d v + Vp
)
Vp is taken as zero in this calculation, but not in the calculation of V cw The following values are needed for the shear calculations values taken at L = 6, to be conservative: Vd := 46kip Vi := 173.8kip
shear force at section due to unfactored DL (sum of DL shear, no shear dist factor because DL is not distributed by factor factored shear at section due to LL (Str 1 env - DL shear)
M dnc := 290.3kip⋅ ft
Total unfactored dead load moment acting on the non-composite section (moment from non-comp DL)
M max := 979kip ⋅ ft
Max factored moment at section due to externally applied loads (Str 1 env - DL moments)
M max = 11748 ⋅ kip ⋅ in
Needs to be in kip-in
Modulus of rupture: fr := −.20ksi⋅
⎛ fcg ⎞ ⎜ ⎟ = −0.566 ⋅ ksi ⎝ ksi ⎠
Compressive stress in concrete due to effective prestress only, after all losses, at extreme tensile fiber of the section where stress is caused by externally applied loads: fcpe :=
(T1 + T2 + T3) + (T1⋅ ecc1(x)⋅ in + T2⋅ ecc2(x)⋅ in + T3⋅ ecc3(x)⋅ in) = 2.09⋅ ksi
Ag SB Moment causing flexural cracking due to externally applied loads
⎛
M cre := Scgb⋅ ⎜ fr + −fcpe −
⎝
M dnc ⎞ SB
⎟ ⎠
= 23286.739 ⋅ kip ⋅ in
THE MULTIPLYING OF Mdnc BY 12 IS NOT DONE HERE BECAUSE MATHCAD IS ABLE TO CONVERT IT TO KIP-IN
181
Minimum web width within depth
b v := 7in
AUTOMATICALLY
fcg Vci1 := .06ksi⋅ ⋅ b ⋅ d = 77.216⋅ kip ksi v v fcg Vi⋅ M cre Vci2 := .02ksi⋅ ⋅ b v ⋅ d v + Vd + = 416.243 ⋅ kip ksi M max
(
)
Vci := max Vci1 , Vci2 = 416.243 ⋅ kip
ft :=
fb :=
(T1 + T2 + T3) (T1⋅ ecc1(x)⋅ in + T2⋅ ecc2(x)⋅ in + T3⋅ ecc3(x)⋅ in) (Mdnc) +
Ag
+
ST
ST
(T1 + T2 + T3) (T1⋅ ecc1( x) ⋅ in + T2⋅ ecc2( x) ⋅ in + T3⋅ ecc3( x) ⋅ in) (Mdnc) +
Ag
+
SB
ft − fb fpc := fb − y cgb⋅ = 0.919 ⋅ ksi 45in The contribution of the PT tendons: At critical section: Slope of each duct at: x = 75 −slope1( x ) = 0.031
−slope2( x ) = 0.093
−slope3( x ) = 0.157
Shear contribution of each duct: Vpx1 := Vpx2 :=
fe1⋅ ns 1 ⋅ As⋅ −slope1( x ) = 9.154 ⋅ k fe2⋅ ns 2 ⋅ As⋅ −slope2( x ) = 22.386⋅ k
Vpx3 := fe3⋅ ns 3 ⋅ As⋅ −slope3( x ) = 19.357⋅ k Vpx := Vpx1 + Vpx2 + Vpx3 = 50.897⋅ kip
fcg ⎛ ⎞ Vcw := ⎜ .06ksi⋅ + .3⋅ fpc⎟ ⋅ b v ⋅ d v + Vpx = 253.58⋅ kip ksi ⎝ ⎠
(
)
Vc := min Vcw , Vci = 253.58⋅ kip Shear resistance: φv := 0.9 Vu = 278.13⋅ kip
SB
= 0.754 ⋅ ksi
= 1.527 ⋅ ksi
182
Vu Vn := = 309.033 ⋅ kip φv Required steel capacity: Vp := 0kip
Because it is already accounted for
Vs := Vn − Vc − Vp = 55.453⋅ kip
2
Av := .40in d v = 65⋅ in cotθ :=
fy := 60ksi
1 if Vci < Vcw fpc ⎛ ⎜ ksi min⎜ 1.8 , 1.0 + 3 ⋅ ⎜ fcg ⎜ ksi ⎝
⎞ ⎟ ⎟ otherwise ⎟ ⎟ ⎠
cotθ = 1.8 cotθ Vs := Av ⋅ fy ⋅ d v ⋅ s cotθ s := Av ⋅ fy ⋅ d v ⋅ = 50.637⋅ in Vs Checking maximum spacing: v u :=
Vu φv ⋅ b v ⋅ d v
smax1 :=
= 0.679 ⋅ ksi
(
)
fcg if v u < .125ksi⋅ ksi
(
)
fcg if v u ≥ .125ksi⋅ ksi
min .8⋅ d v , 24in min .4⋅ d v , 12in
smax1 = 12⋅ in Check minimum reinforcing (LRFD5.8.2.5):
183
Av ⋅ fy
smax2 :=
.0316ksi⋅ b v ⋅
(
fcg
= 38.36 ⋅ in
ksi
)
smax := min smax1 , smax2 = 12⋅ in Therefore, use
s := 12in
cotθ Vs := Av ⋅ fy ⋅ d v ⋅ = 234 ⋅ kip s
Check Vn requirements: Vn1 := Vc + Vs + Vp = 487.58⋅ kip Vn2 := .25⋅ fcg⋅ b v ⋅ d v + Vp = 910 ⋅ kip
(
)
Vn := min Vn1 , Vn2 = 487.58⋅ kip Vr := φv ⋅ Vn = 438.822 ⋅ kip
>
Vu = 278.13⋅ kip
OK!
Would normally check if web reinforcing is needed over entire span by checking the above calcs at various points. However, for simplicity of design, and to aid in retraint of PT ducts, use 12 in spacing along entire span.
Composite Action/Interface shear design: b vi := 16in v ui :=
Vu b vi⋅ d v
width of top flange available to bond to the deck
= 0.267 ⋅ ksi
kip Vui := v ui⋅ b vi = 51.347⋅ ft
(
Vn := c⋅ Acv + μ⋅ Avf ⋅ fy + Pc
)
Nominal shear resistance Vn shall not be greater than the lesser of: Vn1 := K1 ⋅ fcd⋅ Acv c := .28ksi K1 := 0.3
Vn2 := K2 ⋅ Acv μ := 1.0
184
Units are added to this term to make units work out later
K2 := 1.8ksi 2
in Acv := b vi = 192 ⋅ ft
Area of concrete considered to be engaged in shear transfer
For an exterior girder, Pc is the weight of the deck, haunch, parapet, and wearing surface (but there is no wearing surface in this design) wc := .150
kip
Assume an overhang of 3 ft:
3
ft wc⋅ teff
(
Pcd := ⋅ spacing + soh 2 ⋅ spacing
)2 = 0.879 ⋅
soh := 3ft
kip ft
kip Pch := hau⋅ 16in⋅ wc = 0.033 ⋅ ft kip Pcp := wparapet = 0.774 ⋅ ft kip Pc := Pcd + Pch + Pcp = 1.686 ⋅ ft Stirrup spacing is:
s = 12⋅ in 2
Av
in Avf := = 0.4⋅ ft s
kip Vn := c⋅ Acv + μ⋅ Avf ⋅ fy + Pc = 79.446⋅ ft kip Vn1 := K1 ⋅ fcd⋅ Acv = 230.4 ⋅ ft kip Vn2 := K2 ⋅ Acv = 345.6 ⋅ ft
(
)
kip Vn := min Vn , Vn1 , Vn2 = 79.446⋅ ft
(
)
kip Vr := φv ⋅ Vn = 71.502⋅ ft
>
kip Vui = 51.347⋅ ft
Stirrup spacing is adequate!
Tensile steel to support individual segments before girder is tensioned together Maxium moment in 25 ft end segment and 10 ft interior segment: M 25 := 1.5wgirder⋅
( 25ft) 8
2
= 68.32 ⋅ kip ⋅ ft
M 10 := 1.5wgirder⋅
( 10ft) 8
2
= 10.931⋅ kip ⋅ ft
185
Assume that the segments are simply supported when they are moved, and that the concrete strength at moving is only 6500 psi. fci := 6.5ksi For 6500 psi concrete:
⎛ fci − 4ksi ⎞ ⎟ = 0.725 ⎝ 1ksi ⎠
β1 := .85 − .05⋅ ⎜
Assume single layer of #4 bars at 1.5 in. cover to exterior edge of steel fy = 60000 ⋅ psi
2
As4 := .2in
d := 45in − 1.5in − .25in = 43.25 ⋅ in
Assume compression block is contained entirely within upper flange. c := 7in a := β1 ⋅ c = 5.075 ⋅ in Moment capacity: a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ 2⎠ ⎝
( )
Solve for needed area of steel. M 25 2 As.req := = 0.336 ⋅ in a fy ⋅ ⎛⎜ d − ⎞⎟ 2⎠ ⎝ But, minimum reinforcement needed, per ACI 10.5.1: 3psi⋅ As.min :=
fci psi
2
⋅ 7 in⋅ d = 1.22⋅ in
fy
Try 6 #4 bars: 2
As := 6 ⋅ As4 = 1.2⋅ in
a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ = 244.275 ⋅ kip⋅ ft 2⎠ ⎝
( )
Check actual value of a: As⋅ fy a := = 0.814 ⋅ in .85⋅ fci⋅ b
186
M 25
As.req :=
2
a⎞
fy ⋅ ⎛⎜ d − ⎟ 2⎠ ⎝
= 0.319 ⋅ in
6 #4 bars still works.
Check tensile strain when concrete crushes: c := .003 c .003⋅
a β1
= 1.123 ⋅ in
= ⋅
x d−c
d−c c
= 0.112
Steel strain is greater than .005, so tensile controlled, therefore ϕ = 0.9 ϕ := 0.9
Actual moment capacity: a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ = 257.057 ⋅ kip⋅ ft 2⎠ ⎝
( )
ϕ⋅ M n = 231.351 ⋅ kip ⋅ ft
>
M 25 = 68.32 ⋅ kip ⋅ ft
OK!
Place 6 #4 bars at bottom of each segment, three on each side of the PT ducts.
187
Segmental Girder Design Rapid Repair and Replacement Project 80 ft span, AASTHO 45-in I-Girder
ASSUMPTIONS: Single Span Bridge Span Length = 80 ft. 8000 psi concrete (full strength @ transfer) 270 ksi low relax strand (prestress)
Design Units: 1 in := ft 12 Girder Spacing:
k := 1000lbf
Parapet walls @ 338 lb/ft of wall 8"deck thickness fcg := 8ksi fsu := 270ksi
ksi := 1000
lbf 2
fcgi := 8ksi d strand := 0.6in
fcd := 4ksi
β := 0.85
in 45 in I-girder, for 80 ft span:
From Table 19-1 of WBM: spacing := 9.5ft
Deck thickness: Assume 8" thick deck, after loss of wearing surface we will have a 7.5" deck. Span Lengths:
t := 8in
t eff := 7.5in
Span length is 80 ft With 12" bearing pads at either end, (the effective span (for simply supported beams) is 79 ft) assume L is 80 ft. Lspan := 80ft Bridge Width: Assume 50 ft width??? (affects amount of dead load to each girder)
188
Loading: We will want to look at the bridge in the following limit states: Strength I, Service I, Service III, and Fatigue. Load factors from LRFD T3.4.1-1:
Strength I Service I Service III Fatigue
DL 1.25 1.0 1.0 N/A
LL 1.75 1.0 0.8 0.75
Our dynamic load allowances, according to LRFD T3.6.2.1-1: 15% for Fatigue Limit State 33% for all other Limit States Loads applied to each girders: Girder self weight.................... ..583 k/ft (given in WBM) 8' Deck slab........................... .950 k/ft (150 pcf x thickness (8") x tributary area width(9.5') ) No future wearing surface Loads distributed among all girders: Sidewalk, median................... .472 k/ft* Diaphragms........................... .406 k on each beam Parapet wall.......................... .774 k/ft (2 walls, one on each side) k wgirder := .583 ft
k wslab := .950 ft
k ws.m := .472 ft
k wdiaph := .406k wparapet := .774 ft
Total Girder Loads: On simple span girders: wdiaph WDCsimp := wgirder + wslab + Lspan
k WDCsimp = 1.538 ⋅ ft
On composite girder: ws.m + wparapet WDCcont := 5
k WDCcont = 0.249 ⋅ ft
Live Loads: Use HL93 truck (either design truck (below left) + lane or design tandem (below right) + lane) for +M Use special fatigue truck for fatigue limit state (HL93 truck with rear axles @ 30')
189
Load Distribution: This bridge meets the simple dist. parameters (const. width, assumed >4 beams, parallel equal beams, road overhang <= 3 ft, standard cross-section, etc), so we can use the AASHTO simple distribution guidelines. According to table 4.6.2.2.1-1, we have a type "k" bridge. According to T4.6.2.2.2b-1 our equations are: 0.1 0.4 0.3 ⎛ Kg ⎞ S⎞ S⎞ ⎛ ⎛ One Design Lane Loaded: ⎟ 0.06 + ⎜ ⎟ ⋅ ⎜ ⎟ ⋅ ⎜ ⎝ 14 ⎠ ⎝ L ⎠ ⎜ 12.0⋅ L⋅ t 3 ⎟ s
⎝
Two or more lanes loaded:
S ⎞ 0.075 + ⎛⎜ ⎟ ⎝ 9.5 ⎠
0.6
S ⋅ ⎛⎜ ⎞⎟ ⎝ L⎠
0.2
⎛ Kg ⎞ ⎟ ⋅⎜ ⎜ 12.0⋅ L⋅ t 3 ⎟ s ⎠ ⎝
⎠
0.1
The longitudinal stiffness parameter, K g , is: Kg := n gd⋅ ⎡Ig + ⎛ Ag ⋅ eg ⎣ ⎝
2⎞⎤
⎠⎦
Calculation of "n": Eg := 33000 ⋅ .150
Ed := 33000 ⋅ .150
1.5
⋅ 8 ⋅ ksi
Eg = 5422.453⋅
1.5
⋅ 4 ⋅ ksi
n gd :=
Ed = 3834.254⋅
Eg
k 2
in
k 2
in
n gd = 1.414
Ed
Girder Properties from WBM: 4
Ig := 125390in 2
Ag := 560in
3
y T := 24.73in
ST := 5070in
y B := −20.27 in
SB := −6186in
3
Distance between center of gravity of girder and center of gravity of deck (with 2" haunch), eg : t eg := y T + 2in + 2
eg = 30.73 ⋅ in
Longitudinal stiffness parameter: Kg := n gd⋅ ⎡Ig + ⎛ Ag ⋅ eg ⎣ ⎝
2⎞⎤
⎠⎦
5
The relative long stiffness is: Krel :=
4
Kg = 9.252 × 10 ⋅ in
⎛⎜ Kg ⎞⎟ ⎜ 12⋅ L⋅ t3 ⎟ ⎝ ⎠
190
For single span: One lane loaded:
Two or more lanes loaded:
spacing ⎞ gi1 := 0.06 + ⎛⎜ ⎟ ⎝ 14ft ⎠
0.4
spacing ⎞ gi2 := 0.075 + ⎛⎜ ⎟ ⎝ 9.5ft ⎠
spacing ⎞ ⋅ ⎛⎜ ⎟ ⎝ Lspan ⎠
0.6
0.3
spacing ⎞ ⋅ ⎛⎜ ⎟ ⎝ Lspan ⎠
⎛ Kg ⋅ 12 ⎞ ⎟ ⋅⎜ 3⎟ ⎜ 12.0⋅ L span⋅ t ⎠ ⎝
0.2
0.1
⎛ Kg⋅ 12 ⎞ ⎟ ⋅⎜ 3⎟ ⎜ 12.0⋅ L span⋅ t ⎠ ⎝
gi1 = 0.541
0.1
gi2 = 0.771
2 lane distribution factor controls: gi := max( gi2 , gi1 ) = 0.771 Load Cases a.) DL on initial simple span girders b.) unfactored post DL on composite girders c.) factored post DL on composite girders d.) HL-93 truck with lane load e.) HL-93 tandem with lane load f.) Fatigue truck NO NEGATIVE MOMENT TO DESIGN FOR - SIMPLE SPAN!!! Load combinations for Service-1 moments (all include post-DL): sv1-1) truck + lane LL ..... for +M in span sv1-2) tandem + lane LL ..... for +M in span Load combinations for Service-3 moments (include post DL): sv3-1) truck + lane LL ..... for +M in span sv3-2) tandem + lane LL ..... for +M in span Load combinations for Strength-1 moments (include all loads): [ same sets as Service 1, but with Strength 1 load factors ] Load combinations for Stength-1 shear: str1-sh1) truck + lane LL str1-sh2) tandem + lane LL Used PCBRIDGE to find the forces from the load cases, then used EXCEL to combine and factor them.
191
Factored Design Moments: The results from PCBRIDGE were multiplied by DLA, load factors, and distribution factor in Excel. Service I and Service III loads do not include DL on non-composite span. BUT Strength I does include this. Moments from the initial non composite DL: V = 61.52 k
M ss := 1230.4ft⋅ k
Moments from composite DL, unfactored: M = 199.2 ft-k
M dc := 199.2ft⋅ k
M ss = 1230.4⋅ ft⋅ k
Girder plus deck, simple span, unfactored
M s1c := 1917.13ft⋅ k
Service I Max Moment Truck and lane plus Post DL
M s3c := 1573.49ft⋅ k
Service III Max Moment Truck and lane plus Post DL
M str1 := 4791.68ft⋅ k
Strength I Max Moment Truck and lane plus FULL DL
M g45 := 466.4ft⋅ k
Girder alone on SS
192
Preliminary Design: Control of PT design: Full PT will be transferred well after concrete has cured - T is maximum and concrete strength is maximum. According to LRFD, our allowable stresses under the service loads: Initial compression before losses (5.9.4.1.1) fcia := .6⋅ fcgi
fcia = 4.8⋅ ksi
Compression after losses (5.9.4.2.1) fca := .45⋅ fcg
fca = 3.6⋅ ksi
Effective flange width on composite girder: Top flange width of girder:
b := 16in
Long span effective length:
Lelong := Lspan
Effective flange width:
(Distance between inflection points on DL moment diagram - full span because simple supported)
fle.long.a := 0.25⋅ Lelong = 240 ⋅ in fle.long.b := 12⋅ t + 0.5⋅ b = 104 ⋅ in fle.long.c := spacing = 114 ⋅ in
(
)
flelong := min fle.long.a , fle.long.b , fle.long.c
flelong = 104 ⋅ in
fle := flelong Transforming the slab into equivalent girder concrete: fle flet := flet = 73.539⋅ in n gd Composite Section Properties (Ignore haunch): 2
Ag = 560 ⋅ in
2
Aeffslab := t eff ⋅ flet
Acg := Ag + Aeffslab
Aeffslab = 551.543 ⋅ in 2
Acg = 1111.543⋅ in
Calculating location of center of gravity for composite girder: Ag Aeffslab 7.5in ⎞ dist := −y B⋅ + ⋅ ⎛⎜ 45in + 2in + ⎟ Acg Acg ⎝ 2 ⎠ dist = 35.394⋅ in The center of gravity for the composite section is 35.394 in from the bottom of the girder.
193
Using the dimensions given in the example for a composite beam, the dimension calculated above, and the parallel axis theorem, I can find the I-value for the composite girder as a whole: Ieffslab :=
flet⋅ t eff
3
4
Ieffslab = 10341.437 ⋅ in
3
height comp := 45in + 2in + 7.5in
⎡
Icg := ⎢Ieffslab + Aeffslab⋅ ⎛⎜ height comp − dist −
⎣
7.5in ⎞
⎝
height comp = 54.5⋅ in
2⎤
2 ⎟ ⎥ + ⎡⎣Ig + Ag⋅ ( dist + y B) ⎤⎦ ⎠⎦
2
5
4
Icg = 3.939 × 10 ⋅ in Using the same dimensions I can also find the section moduli to the top the deck, the bottom of the girder, and the top of the girder flange: Icg 3 (To top of deck) Scgd := Scgd = 20615.616 ⋅ in height comp − dist Icg Scgb := 4 3 (To bottom of girder) Scgb = −1.113 × 10 ⋅ in −dist Icg Scgt := 45in − dist y cgd := height comp − dist
3
Scgt = 41003.856 ⋅ in y cgd = 19.106⋅ in
(To top of girder flange) y cgb := −dist
y cgb = −35.394⋅ in
Estimating PT Loss: To get an early estimate of how much loss we will have in the strands, we can use T5.9.5.3-1 (from 2004 edition) and calculate the average loss:
⎛ ⎝
loss := 33⋅ ⎜ 1.0ksi − 0.15⋅
fcg − 6ksi ⎞ 6
⎟ + 6ksi ⎠
(PPR = 1.0)
loss = 37.35 ⋅ ksi Since we are using low relaxation strands: loss := loss − 6ksi loss = 31.35 ⋅ ksi This seems high, so instead we will try: loss := 15ksi Strands will be tensioned up to a maximum of 218.7 ksi (.90f py) temporarily to offset friction and seating losses. Immediately after anchor set, the stress limit is 200 ksi (.74f pu = 199.8, rounded to 200). So the strands will be tensioned to 200 ksi and anchored. (Per T5.9.3-1) fpt := 200ksi Assume elastic shortening loss to be: ES := 10ksi Total assumed lossed: Δf := ES + loss
Δf = 25⋅ ksi
194
Effective prestress after losses: fe := 200ksi − ES − loss
fe = 175 ⋅ ksi
And our percent loss: pl :=
loss
pl = 7.895 ⋅ %
fpt − ES
Design amount of prestress to prevent tension at bottom of beam under full load (at center span) after 50 years Calcuating stress at the bottom of the beam due to non-composite and composite loading (Service 3 limit state): Components of moment @ center span: (NOT NECESSARILY THE MAXIMUM VALUE!) M ss = 1230.4⋅ ft⋅ k
Girder/deck/steel diaphragms, non-composite, unfactored
M s1c = 1917.13 ⋅ ft⋅ k
LL and composite DL, factored
M s3c = 1573.49 ⋅ ft⋅ k
LL and composite DL, factored
M g45 = 466.4 ⋅ ft⋅ k
Girder alone on SS
These all contribute to the full Service 1 and Service 3 moments. Total moments @ mid-span for Service 1 and 3 cases: M s1 := M ss + M s1c
M s3 := M ss + M s3c
M s1 = 3147.53 ⋅ ft⋅ k
M s3 = 2803.89 ⋅ ft⋅ k
The simple span moment will occur on a non-composite section, while the other moments will occur on a composite section. So the Service 3 tension stress at the bottom of the beam will be: M s3c M ss fbs3 := + fbs3 = −4.084 ⋅ ksi (Negative value means tension.) Scgb SB According to T5.9.4.2.2-1 in the LRFD, the limit on tension stress in service 3 condition is: fcg ftall := −.0948 ksi⋅ f = −0.268 ⋅ ksi ksi tall So we want a final prestress of: fpreq := fbs3 − ftall
fpreq = −3.815 ⋅ ksi
Using the prestress loss we previously calculated, we know that the initial compression from the prestress needs to be: fpreq fpireq := fpireq = −4.142 ⋅ ksi pl = 0.079 1 − pl Try 3 ducts, .6" diamater strands, two 12 strand ducts, one 7 strand duct. From VSL data for Polypropelene Plastic Duct: OD12 := 3.58in OD7 := 2.87in
Includes ribs
195
Assume they are bundled together in the middle portion of the span (one of top of the other), and have 1" of cover between bottom of girder and bottom duct (ignoring ribs on the bottom side of the duct). Even though ducts are bundled, they will still be .2 in apart because of the ribs. We are assuming the ribs will not hit each other. Numbering convention: Lowest duct is #1, middle duct is #2, top most duct is #3. cover := 1in OD12 educt1 := y B + cover + = −17.48 ⋅ in 2 OD12
educt2 := y B + cover + OD12 +
2
= −13.9⋅ in
educt3 := y B + cover + OD12 + OD12 +
OD7 2
= −10.675⋅ in
Guess a number of strands to use: ns := 24
ns 3 := 6
2
As := 0.217in
ns 2 := 9 ns 1 := 9
⎛ ns1 ⎞ ⎛ ns2 ⎞ ⎛ ns3 ⎞ estrands := educt1⋅ ⎜ ⎟ + educt2⋅ ⎜ ⎟ + educt3⋅ ⎜ ⎟ = −14.436⋅ in ⎝ ns ⎠ ⎝ ns ⎠ ⎝ ns ⎠ y_over_rsq := −.09053
in 2
in
term2 := 1 + estrands⋅ y_over_rsq = 2.307 term3 :=
Ag term2
2
= 242.749 ⋅ in
Calculations From WisDOT Design Data for 45 in I-Girder
Apt := ns ⋅ As fbi :=
(Apt⋅ fpt) term3
= 4.291 ⋅ ksi
Forces in tendons: fpt = 200 ⋅ ksi
OK!
To := ns ⋅ fpt⋅ As
To.1int := ns 1 ⋅ fpt⋅ As = 390.6 ⋅ kip
To = 1041.6⋅ kip
To.2int := ns 2 ⋅ fpt⋅ As = 390.6 ⋅ kip
To.3int := ns 3 ⋅ fpt⋅ As = 260.4 ⋅ kip
196
Check stresses in beam at mid-span and at the end under girder weight to avoid premature failure. Check using full PT load (conservative): At mid-span: Bottom: To.1int⋅ educt1 + To.2int⋅ educt2 + To.3int⋅ educt3 + M g45 To.1int + To.2int + To.3int fbint := + Ag SB
(
) (
fbint = 3.386 ⋅ ksi fcia = 4.8⋅ ksi
Allowed:
)
OK
At top: ftint :=
(To.1int + To.2int + To.3int) (To.1int⋅ educt1 + To.2int⋅ educt2 + To.3int⋅ educt3) + Mg45 +
Ag
Allowed:
ftint = −0.002 ⋅ ksi ftall = −0.268 ⋅ ksi fcia = 4.8⋅ ksi
ST
OK
GIRDER IS OK AT MIDSPAN IN THE INITIAL CONDITION! End of beam: Because of spacing needs for the anchorages, the ducts CANNOT be the same height at the end as they are in the middle. First figure out where the strands would be at the end of the girder using VSL Type E Stressing Anchorage, we need: - an edge spacing of 8.185" for the 12-strand ducts and 6.51" for the 7-strand duct - an anchorage spacing of 14.37" for the 12-strand ducts and 11.02" for the 7-strand duct These spacings are from the center of the anchorage. Figuring out the eccentricities of the ducts at the end of the beam: spacings are rounded for easier fabrication
ed1.end := y B + 8.2in = −12.07 ⋅ in ed2.end := ed1.end + 14.4in = 2.33⋅ in
larger spacing used because of larger duct
ed3.end := ed2.end + 14.4in = 16.73 ⋅ in Check edge spacing at top: y T − ed3.end = 8 ⋅ in
> 6.51" OK
Now check stresses at end of girder using these eccentricities: To.1int⋅ ed1.end + To.2int⋅ ed2.end + To.3int⋅ ed3.end To.1int + To.2int + To.3int fbeint := + Ag SB
(
) (
Allowed:
fbeint = 1.771 ⋅ ksi OK fcia = 4.8⋅ ksi
)
197
fteint :=
(To.1int + To.2int + To.3int) + (To.1int⋅ ed1.end + To.2int⋅ ed2.end + To.3int⋅ ed3.end) Ag
Allowed:
ST
fteint = 1.969 ⋅ ksi ftall = −0.268 ⋅ ksi fcia = 4.8⋅ ksi
OK
To determine point where draped strands become straight (hold down point), pick distance along girder where all strands are straight, and determine parabolic equation to match the ducts' eccentricities at that point, and their eccentricities at the end of the girder. All three ducts become straight at 20 feet from girder end. The eccentricities for each duct are: educt1 = −17.48 ⋅ in
ed1.end = −12.07 ⋅ in
educt2 = −13.9⋅ in
ed2.end = 2.33⋅ in
('+' means above cgc of girder) educt3 = −10.675⋅ in ed3.end = 16.73 ⋅ in The distance above the bottom of the girder for the center of each duct, as a function of x (0
drape1 ( x ) := 9.3924⋅ 10
The eccentricity of each duct as a function of x:
− 5) 2 ( ⋅ x − .045x + −12.07 −4 2 ecc2( x ) := ( 2.8177⋅ 10 ) ⋅ x − .13525x + 2.33 −4 2 ecc3( x ) := ( 4.7578⋅ 10 ) ⋅ x − .2284x + 16.73
ecc1( x ) := 9.3924⋅ 10
The slopes of the ducts at any point x (up to x = 240): − 4) ( ⋅ x − 0.045 −4 slope2( x ) := ( 5.6354⋅ 10 ) ⋅ x − 0.13525 −4 slope3( x ) := ( 9.5156⋅ 10 ) ⋅ x − .2284
slope1( x ) := 1.8785⋅ 10
198
Instantaneous Losses: Anchorage Set: Assume that there is no slip at the anchorages, plus, tendons are pulled to higher stress to account for this if it were to happen: ΔfpA := 0ksi Friction Loss: Using AutoCAD, the drape of the strands was approximated with a circle, and the angle change of the strands was found. fpj := fpt K := .0002 μ := .23 K and μ taken from T5.9.5.2.2b-1 Duct 1: From end of beam to end of drape:
⎡
ΔfpF1d := fpj⋅ ⎣1 − e
(
)
x 1d := 20 α1d := 2.577deg = 0.045
− K⋅ x1d+ μ⋅ α1d ⎤
⎦ = 2.848 ⋅ ksi
From end of drape to center of beam: x 1s := 20
(
)
⎡
− K⋅ x1s + μ⋅ α1s ⎤
⎡
− K⋅ x2d+ μ⋅ α2d ⎤
α1s := 0
ΔfpF1s := fpj⋅ ⎣1 − e ⎦ = 0.798 ⋅ ksi Duct 2: From end of beam to end of drape: x 2d := 20 α2d := 7.688deg = 0.134 ΔfpF2d := fpj⋅ ⎣1 − e
(
)
⎦ = 6.852 ⋅ ksi
From end of drape to center of beam: x 2s := 20 α2s := 0
⎡
ΔfpF2s := fpj⋅ ⎣1 − e
(
)
− K⋅ x2s + μ⋅ α2s ⎤
⎦ = 0.798 ⋅ ksi
Duct 3: From end of beam to end of drape:
⎡
ΔfpF3d := fpj⋅ ⎣1 − e
(
)
x 3d := 20 α3d := 12.844deg = 0.224
− K⋅ x3d+ μ⋅ α3d ⎤
⎦ = 10.809⋅ ksi
From end of drape to center of beam: x 3s := 20 α3s := 0
⎡
ΔfpF3s := fpj⋅ ⎣1 − e
(
)
− K⋅ x3s + μ⋅ α3s ⎤
⎦ = 0.798 ⋅ ksi
Friction Loss Summary: To1int.end := fpt⋅ ns 1 ⋅ As = 390.6 ⋅ kip To2int.end := fpt⋅ ns 2 ⋅ As = 390.6 ⋅ kip To3int.end := fpt⋅ ns 3 ⋅ As = 260.4 ⋅ kip
( ( (
) ) )
To1int.mid := fpt − ΔfpF1d − ΔfpF1s ⋅ ns 1 ⋅ As = 383.478 ⋅ kip To2int.mid := fpt − ΔfpF2d − ΔfpF2s ⋅ ns 2 ⋅ As = 375.658 ⋅ kip To3int.mid := fpt − ΔfpF3d − ΔfpF3s ⋅ ns 3 ⋅ As = 245.287 ⋅ kip
199
Use an average in each duct for other loss calcs: To1.int := To2.int := To3.int :=
To1int.end + To1int.mid 2 To2int.end + To2int.mid 2 To3int.end + To3int.mid 2
= 387.039 ⋅ kip = 383.129 ⋅ kip = 252.844 ⋅ kip
Elastic Shortening Losses: To re-estimate elastic shortening loss, we will need a jacking pattern, since any elastic shortening that occurs will be caused by the jacking of the different ducts. Jacking pattern: Duct 2, Duct 1, Duct 3. fcgi Es Eci := 1820⋅ ⋅ ksi Es := 28500ksi n := = 5.536 ksi Eci AT MIDSPAN: Loss in duct 2 (from pulling ducts 1 and 3): fc.2mid :=
To.1int Ag
+
To.3int Ag
+
(To.1int⋅ educt1)⋅ educt2 + (To.3int⋅ educt3)⋅ educt2 Ig
Ig
fs.2mid := n ⋅ fc.2mid = 12.333⋅ ksi ΔT2mid := fs.2mid⋅ ns 2 ⋅ As = 24.085⋅ kip Loss in duct 1 (from pulling duct 3): To.3int To.3int⋅ educt3 ⋅ educt1 fc.1mid := + Ag Ig
(
)
fs.1mid := n ⋅ fc.1mid = 4.72⋅ ksi ΔT1mid := fs.1mid⋅ ns 1 ⋅ As = 9.218 ⋅ kip Loss in duct 3:
NONE
ΔT3 := 0kip
AT END: Loss in duct 2 (from pulling ducts 1 and 3): fc.2end :=
To.1int Ag
+
To.3int Ag
+
(To.1int⋅ ed1.end)⋅ ed2.end + (To.3int⋅ ed3.end)⋅ ed2.end Ig
fs.2end := n ⋅ fc.2end = 6.399 ⋅ ksi ΔT2end := fs.2end ⋅ ns 2 ⋅ As = 12.498⋅ kip
Ig
200
Loss in duct 1 (from pulling duct 3): To.3int⋅ ed3.end ⋅ ed1.end To.3int fc.1end := + Ig Ag
(
)
fs.1end := n ⋅ fc.1end = 0.253 ⋅ ksi ΔT1end := fs.1end ⋅ ns 1 ⋅ As = 0.494 ⋅ kip Loss in duct 3:
NONE
Elastic Shortening Losses Summary: Forces in ducts at mid-span: To.1mid := ns 1 ⋅ fpt⋅ As − ΔT1mid = 381.382 ⋅ kip To.2mid := ns 2 ⋅ fpt⋅ As − ΔT2mid = 366.515 ⋅ kip To.3mid := ns 3 ⋅ fpt⋅ As − ΔT3 = 260.4 ⋅ kip Forces in ducts at end: To.1end := ns 1 ⋅ fpt⋅ As − ΔT1end = 390.106 ⋅ kip To.2end := ns 2 ⋅ fpt⋅ As − ΔT2end = 378.102 ⋅ kip To.3end := ns 3 ⋅ fpt⋅ As − ΔT3 = 260.4 ⋅ kip
Time Dependent Losses: TIME DEPENDANT LOSSES, BETWEEN TRANSFER AND DECK PLACEMENT Shrinkage Loss, between transfer and deck placement: According to AASHTO Eq 5.9.5.4.2a-1, the prestress loss due to shrinkage of grider concrete between time of transfer and deck placement is: ΔfpSR := ε bid⋅ Ep ⋅ Kid t f := 50 × 365 = 18250 days
Final age in days (50 years)
t i := 365 days
Age at transfer, days (1 year)
Ep := Es ε bid := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 H for Wisconsin = 72
Eq 5.4.2.3.3-1
conc. shrnk. strain from transfer to deck placement
H := 72
k hs := 2.00 − 0.014 ⋅ H = 0.992
humidity factor for shrinkage
k vs := 1.45 − 0.13⋅ VS
factor for effect of volume-to-surface ratio
VS :=
Ag ( 16 + 2 ⋅ 7 + 2 ⋅ 19 + 2 ⋅ 7 + 22 + 2 ⋅ 6.36 + 2 ⋅ 10.61 )in
= 4.06⋅ in
volume/surface ratio
201
k vs := 1.45 − 0.13⋅
k f := k td :=
VS
5ksi 1ksi + fcgi
= 0.922 but >=1.0 k vs := 1.0
in
factor for effect of conc. strength
= 0.556
t
time development factor
61 − 4 ⋅ fcgi + t
t1 := 373
t1 = time between end of cure and deck t2 = time between end of cure and transfer
t2 := 363 t1
k td1 := 61 − 4 ⋅
fcgi ksi
t2
= 0.928 k td2 := + t1
61 − 4 ⋅
fcgi ksi
= 0.926 + t2
k td := k td1 − k td2 = 0.002 ε bid := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 −7
ε bid = −4.868 × 10 Kid :=
1 Ag ⋅ estrands Es Aps ⎡⎢ ⋅ ⋅ 1+ 1+ Ig Eci Ag ⎢ ⎣
(
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti
) ⎤⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥( 2
transformed section coefficient
⎦
creep coefficient
− 0.118
humidity factor for creep
k hc := 1.56 − 0.008 ⋅ H = 0.984 2
Aps := ns ⋅ As = 5.208 ⋅ in
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti Kid.SR :=
− 0.118
= 0.001
1
⎡A ⋅ e Es Aps ⎡⎢ ⎣ g strands ⋅ ⋅ 1+ 1+ ⎢ Ig Eci Ag ⎣
(
ΔfpSR := −ε bid ⋅ Ep ⋅ Kid.SR = 0.013 ⋅ ksi
2⎤ ⎤
) ⎦ ⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥(
= 0.91
⎦
loss from shrinkage btwn transfer and deck
202
Creep Loss, between transfer and deck placement: Uses most of above values... Ep ΔfpCR := ⋅f ⋅ψ ⋅K Eci cgp b.td.ti id t := 10 t is 10 days now, since it is from time of transfer to deck t k td := = 0.256 fcgi 61 − 4 ⋅ +t ksi ψb.td.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i Kid.CR :=
comp :=
mom :=
− 0.118
− 0.118
= 0.133 = 0.133
To adjust these values for t=10 days ti is still 365 days
1 2⎤ ⎤ ⎡A ⋅ ( e Es Aps ⎡⎢ g strands) ⎦ ⎥ ⎣ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Ig Eci Ag ⎢ ⎣ ⎦
(To.1mid + To.2mid + To.3mid)
= 0.902
(These terms are split up so they fit on the page)
Ag
(To.1mid⋅ educt1 + To.2mid⋅ educt2 + To.3mid⋅ educt3 + Mg45)⋅ estrands
fcgp := comp + mom fcgp = 2.83⋅ ksi
Ig
Ep ΔfpCR := ⋅f ⋅ψ ⋅K = 1.876 ⋅ ksi Eci cgp b.td.ti id.CR
loss from creep btwn transfer and deck
Relaxation Loss, between transfer and deck placement: Since we are using low-relaxation strand, we can assume a relaxation loss of 1.2 ksi ΔfpR1 := 1.2ksi TIME DEPENDANT LOSSES, AFTER DECK PLACEMENT Shrinkage ΔfpSD := ε bdf ⋅ Ep ⋅ Kdf
203
Eq 5.4.2.3.3-1 conc. shrnk. strain from transfer to ε bdf := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 deck placement t k td := 61 − 4 ⋅ fcgi + t t1 := 18248 t2 := 373 t1 t2 k td1 := = 0.998 k td2 := = 0.928 fcgi fcgi 61 − 4 ⋅ 61 − 4 ⋅ + t1 + t2 ksi ksi k td := k td1 − k td2 = 0.071 ε bdf := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 = −1.866 × 10
−5
− 0.118
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i = 0.037 epc := ⎡estrands − y B − y cgb ⎤ = −29.56 ⋅ in
(
⎣
Kdf.SD :=
)⎦
Below center of gravity of composite section
1 Acg⋅ epc Ep Aps ⎡⎢ ⋅ ⋅ 1+ 1+ Icg Eci Acg ⎢ ⎣
2⎤
( ) ⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥(
= 0.916
⎦
loss from shrinkage btwn deck and 50 yrs
ΔfpSD := −ε bdf ⋅ Ep ⋅ Kdf.SD = 0.487 ⋅ ksi Creep Loss:
Ep Ep ΔfpCD := ⋅ fcgp⋅ ψb.tf.ti − ψb.td.ti ⋅ Kdf + ⋅ Δfcd⋅ ψb.tf.td⋅ Kdf Eci Eci
(
)
50 years - (1 yr + 10 days)
t := 17875 t
k td := 61 − 4 ⋅
fcgi ksi
= 0.998 +t
ψb.td.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i Kdf.CD :=
(
− 0.118
= 0.517 = 0.517
1 Acg⋅ epc Ep Aps ⎡⎢ ⋅ ⋅ 1+ 1+ Icg Eci Acg ⎢ ⎣
2⎤
( ) ⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥(
)
Δfcd := M ss ⋅ t d := 375
− 0.118
(−estrands) Ig
= 0.891
⎦
Eci + ΔfpCR + ΔfpSR + ΔfpR1 ⋅ = 2.258 ⋅ ksi Ep
(
)
the age of the girder at the time the deck is placed.
204
ψb.tf.td := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t d
− 0.118
= 0.515
Ep Ep ΔfpCD := ⋅ fcgp⋅ ψb.tf.ti − ψb.td.ti ⋅ Kdf.CD + ⋅ Δfcd⋅ ψb.tf.td⋅ Kdf.CD = 5.738 ⋅ ksi Eci Eci
(
)
Relaxation Loss: According to 5.9.5.4.3c, ΔfpR2 := ΔfpR1
Shrinkage of Deck Concrete: Loss/gain resulting from shrinkage of deck concrete, per 5.9.5.4.3d: Ep ΔfpSS := ⋅ Δfcdf ⋅ Kdf ⋅ 1 + 0.7⋅ ψb.tf.td Eci ε ddf ⋅ Ad ⋅ Ecd ⎡ 1 epc ⋅ ed⎤ Δfcdf := ⋅⎢ + ⎥ 1 + 0.7⋅ ψd.tf.td Acg Icg ⎣ ⎦
(
)
( )
ε ddf := −k vsd⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 k f := 1
t := 373
Ad := 7.5in⋅ ( 3ft + 3ft + 4 ⋅ spacing) VSd :=
Ad 2 ⋅ 7.5in + 2 ⋅ ( 3ft + 3ft + 4 ⋅ spacing)
k vsd := 1.45 − 0.13⋅
VSd in
= 3.697 ⋅ in
61 − 4 ⋅
fcgi ksi
= 0.928 +t
= 0.969 But must be at least 1.0 k vsd := 1.0
ε ddf := −k vsd⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 Ecd := 3640ksi
t
k td :=
k f and k hs stay the same
MOE for 4 ksi concrete
in ed := y cgd + −7.5 = 15.356⋅ in 2 ψd.tf.td := 1.9⋅ k vsd⋅ k hc⋅ k f ⋅ k td⋅ t d
− 0.118
= 0.862
ε ddf ⋅ Ad ⋅ Ecd ⎛ 1 −epc⋅ −ed ⎞ Δfcdf := ⋅⎜ + ⎟ = 1.004 ⋅ ksi 1 + 0.7⋅ ψd.tf.td Acg Icg ⎝ ⎠ Ep ΔfpSS := ⋅ Δfcdf ⋅ Kdf.CD⋅ 1 + 0.7⋅ ψb.tf.td = 6.739 ⋅ ksi Eci This value is positive, so it means we have a gain resulting from the shrinkage of the deck.
(
)
205
Summary of losses (at mid-span): Instantaneous losses: ΔfpF1 := ΔfpF1d + ΔfpF1s = 3.647 ⋅ ksi ΔfpF2 := ΔfpF2d + ΔfpF2s = 7.651 ⋅ ksi
Friction Losses
ΔfpF3 := ΔfpF3d + ΔfpF3s = 11.607⋅ ksi fs.1mid = 4.72⋅ ksi
Elastic Shortening Losses
fs.2mid = 12.333⋅ ksi fs.3mid := 0ksi
Anchorage Losses
ΔfpA = 0
Time dependant losses, between PT transfer and deck placement: ΔfpSR = 0.013 ⋅ ksi
ΔfpCR = 1.876 ⋅ ksi
ΔfpR1 = 1.2⋅ ksi
Time dependant losses, between deck placement and 50 years: ΔfpSD = 0.487 ⋅ ksi
ΔfpCD = 5.738 ⋅ ksi
ΔfpR2 = 1.2⋅ ksi
ΔfpSS = 6.739 ⋅ ksi
TOTAL TIME DEPENDANT LOSSES: ΔfpLT := ΔfpSR + ΔfpCR + ΔfpR1 + ΔfpSD + ΔfpCD + ΔfpR2 − ΔfpSS = 3.776 ⋅ ksi This is used as an average value for all ducts. Total losses for each duct: Δfpe1 := ΔfpLT + fs.1mid + ΔfpF1 = 12.142⋅ ksi Δfpe2 := ΔfpLT + fs.2mid + ΔfpF2 = 23.759⋅ ksi Δfpe3 := ΔfpLT + fs.3mid + ΔfpF3 = 15.383⋅ ksi % loss: ploss1 :=
Δfpe1 fpt
⋅ 100 = 6.071
ploss2 :=
Δfpe2
loss after elastic shortening: ΔfpLT = 3.776 ⋅ ksi as a % ΔfpLTper :=
ΔfpLT fpt
⋅ 100 = 1.888
Initial stress after jacking: fi1 := fpt − fs.1mid − ΔfpF1 = 191.633 ⋅ ksi fi2 := fpt − fs.2mid − ΔfpF2 = 180.017 ⋅ ksi fi3 := fpt − fs.3mid − ΔfpF3 = 188.393 ⋅ ksi Effective stress after all loss: fe1 := fpt − Δfpe1 = 187.858 ⋅ ksi
fpt
⋅ 100 = 11.879 ploss3 :=
Δfpe3 fpt
⋅ 100 = 7.691
206
fe2 := fpt − Δfpe2 = 176.241 ⋅ ksi fe3 := fpt − Δfpe3 = 184.617 ⋅ ksi Total effective prestress force: T1 := As⋅ ns 1 ⋅ fe1 = 366.886 ⋅ k T2 := As⋅ ns 2 ⋅ fe2 = 344.199 ⋅ k T3 := As⋅ ns 3 ⋅ fe3 = 240.372 ⋅ k To1 := As⋅ ns 1 ⋅ fi1 = 374.26⋅ k To2 := As⋅ ns 2 ⋅ fi2 = 351.573 ⋅ k To3 := As⋅ ns 3 ⋅ fi3 = 245.287 ⋅ k
Stress Check @ mid-span: Here we will check at the midspan at the time of transfer and at 50 years down the road. Data needed: To1 = 374.26⋅ k
To2 = 351.573 ⋅ k
To3 = 245.287 ⋅ k
T1 = 366.886 ⋅ k
T2 = 344.199 ⋅ k
T3 = 240.372 ⋅ k
educt1 = −17.48 ⋅ in
educt2 = −13.9⋅ in
educt3 = −10.675⋅ in
M ss = 1230.4⋅ ft⋅ k
Non-composite DL, unfactored
M s3c = 1573.49 ⋅ ft⋅ k
Service 3 composite load
M s1c = 1917.13 ⋅ ft⋅ k
Service 1 composite load
Initial condition after jacking: Top of girder: fti :=
(To1 + To2 + To3) + (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg45 ST
Ag
fti = 0.067 ⋅ ksi
Allowable: fcia = 4.8⋅ ksi
OK!
ftall = −0.268 ⋅ ksi Bottom of girder: To1 + To2 + To3 To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3 + M g45 fbi := + SB Ag
(
fbi = 3.1⋅ ksi
) (
Allowable: fcia = 4.8⋅ ksi ftall = −0.268 ⋅ ksi
Final Condition (50 years): Top of girder (under Service 1 loading):
)
OK!
207
ft1 :=
T1 + T2 + T3 Ag
+
ft1 = 2.458 ⋅ ksi
T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss ST Allowable: fcia = 4.8⋅ ksi
+
M s1c Scgt
OK!
Top of girder (under Service 3 loading): ft3 :=
T1 + T2 + T3 Ag
ft3 = 2.357 ⋅ ksi
+
T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss ST Allowable: fcia = 4.8⋅ ksi
+
M s3c Scgt
OK!
Bottom of girder (under Service 1 loading): T1 + T2 + T3 M s1c T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss fb1 := + + Ag Scgb SB fb1 = −0.53⋅ ksi
Allowable: ftall = −0.268 ⋅ ksi HIGH!
Bottom of girder (under Service 3 loading): T1 + T2 + T3 M s3c T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss fb3 := + + Ag Scgb SB fb3 = −0.16⋅ ksi
Allowable: ftall = −0.268 ⋅ ksi OK!
Top of deck slab (under Service 1 loading): M s1c ft.deck := ft.deck = 1.116 ⋅ ksi Scgd
Allowable: fcia = 4.8⋅ ksi
OK!
Check stress at hold down point (20 ft) at jacking: wgirder M g20 := ⋅ Lspan − 20ft ⋅ 20ft = 349.8 ⋅ kip ⋅ ft 2
(
fti.20 :=
(To1 + To2 + To3) + (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg20 ST
Ag
fti.20 = −0.209 ⋅ ksi fbi.20 :=
)
Allowable:
ftall = −0.268 ⋅ ksi
OK!
(To1 + To2 + To3) + (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg20 SB
Ag
fbi.20 = 3.326 ⋅ ksi
Allowable:
fcia = 4.8⋅ ksi
Ultimate Strength - Moment Capacity: Check flexural strength capacity @ midspan:
OK!
208
Aps := ns ⋅ As 2
Aps = 5.208 ⋅ in
fe = 175 ⋅ ksi
fpe := fe
fpu := fsu
0.5⋅ fsu = 135 ⋅ ksi
fpe > 0.5⋅ fpu
OK!!!
At failure, when the concrete is crushing, we can assume the tendon stress to be: c fps := fpu⋅ ⎛⎜ 1 − k f ⋅ ⎞⎟ dp ⎝ ⎠
k f := 0.28
Where "c" is defined as the neutral axis location. We will calculate it below:
b = 16⋅ in
girder flange width
t f := 3in
flange thickness at outer edge
t fi := 7in
flange thickness at start of radius
b w := 7in
girder web width
hau := 2in
haunch thickness
We will assume that the compression block is in the deck, and calculate capacity as if it is a rectangular section, and use the compression strength of the deck concrete
c :=
Aps ⋅ fpu
β = 0.85
fpu .85⋅ fcd⋅ β⋅ b + k f ⋅ Aps ⋅ dp
d p := y T − estrands + hau + teff
dp is the distance from the compression face to the tendon centroid:
d p = 48.666⋅ in
fle = 104 ⋅ in
Calculate c, taking width b as the effective flange width Aps ⋅ fpu
c :=
fpu
.85⋅ fcd⋅ β⋅ fle + k f ⋅ Aps ⋅ d β⋅ c = 3.872 ⋅ in
= 4.556 ⋅ in
p Neutral Axis is in deck, good
c fps := fpu⋅ ⎛⎜ 1 − k f ⋅ ⎞⎟ dp ⎝ ⎠
fps = 262.923 ⋅ ksi
Nominal moment cpacity of the composite section: a := β⋅ c
a = 3.872 ⋅ in
2
Aps = 5.208 ⋅ in
209
a M n := Aps ⋅ fps ⋅ ⎛⎜ d p − ⎞⎟ 2⎠ ⎝ M n = 5332.292⋅ ft⋅ k Capacity: Mr = φ*Mn , and in prestressed concrete, φ=1.00, so: M r := 1.0⋅ M n M r = 5332.292⋅ ft⋅ k Required capacity, from strength I state: M str1 = 4791.68 ⋅ ft⋅ k
Moment capacity is OK!!!
Vertical Shear Capacity: Design vertical shear reinforcing at exterior end of span. Shear force has a different distribution factor than moment, and that has been accounted for in the load cases above. spacing g shear1 := 0.36 + = 0.74 25.0ft g shear2 := 0.2 +
spacing 12ft
2
−
⎛ spacing ⎞ = 0.918 ⎜ 35ft ⎟ ⎝ ⎠
Vu := 280k
ϕv := 0.9
Controls
Max shear from Strength 1 case
Critical section is taken at a distance of dv from the face of the support. dv is the distance between resultants of the tensile and compressive forces due to flexure, and no less than .9*de or .72*h. .72⋅ height comp = 39.24 ⋅ in Use this as value of dv a d v := −estrands + y T + hau + teff − = 46.73 ⋅ in 2 .9⋅ .5⋅ spacing = 51.3⋅ in
d v := 51.3in But, there are draped strands in this area of the beam, so the e and a values will be different. Assuming standard bearing bad width of 8 inches, the distance to the critical section from the end of the girder: Lcrit :=
⎛ 8in + d ⎞ + .5ft = 5.108 ft ⎜ 2 v⎟ ⎝ ⎠
Eccentricity of the strands at critical section: Lcrit x := = 61.3 in ns 2 ns 3 ⎛ ns1 ⎞ es_crit := ⎜ ⋅ ecc1( x ) + ⋅ ecc2( x ) + ⋅ ecc3( x ) ⎟ ⋅ in = −6.137 ⋅ in ns ns ⎝ ns ⎠
210
Stress block at critical section: d p_crit := −es_crit + y T + hau + teff = 40.367⋅ in Aps ⋅ fpu
c :=
.85⋅ fcd⋅ β⋅ fle + k f ⋅ Aps ⋅ d
= 4.531 ⋅ in
fpu p_crit
In the deck which is good.
acrit := β⋅ c = 3.852 ⋅ in d v_crit := −estrands + y T + hau + teff − Use:
d v = 51.3⋅ in
acrit 2
= 46.74 ⋅ in
x ⋅ in = 5.108 ft
The nominal shear resistance is calculated as:
(
Vn := min Vc + Vs + Vp , 0.25⋅ fcg⋅ b v ⋅ d v + Vp
)
Vp is taken as zero in this calculation, but not in the calculation of V cw The following values are needed for the shear calculations values taken at L = 5, to be conservative: Vd := 62kip Vi := 174.1kip
shear force at section due to unfactored DL (sum of DL shear, no shear dist factor because DL is not distributed by factor factored shear at section due to LL (Str 1 env - DL shear)
M dnc := 288.4kip⋅ ft
Total unfactored dead load moment acting on the non-composite section (moment from non-comp DL)
M max := 745.6kip ⋅ ft
Max factored moment at section due to externally applied loads (Str 1 env - DL moments)
M max = 8947.2⋅ kip⋅ in
Needs to be in kip-in
Modulus of rupture: fr := −.20ksi⋅
⎛ fcg ⎞ ⎜ ⎟ = −0.566 ⋅ ksi ⎝ ksi ⎠
Compressive stress in concrete due to effective prestress only, after all losses, at extreme tensile fiber of the section where stress is caused by externally applied loads: fcpe :=
(T1 + T2 + T3) (T1⋅ ecc1(x)⋅ in + T2⋅ ecc2(x)⋅ in + T3⋅ ecc3(x)⋅ in)
+ Ag SB Moment causing flexural cracking due to externally applied loads
= 2.655 ⋅ ksi
211
THE MULTIPLYING OF Mdnc BY M dnc ⎞ ⎛ M cre := Scgb⋅ ⎜ fr + −fcpe − ⎟ = 29613.171 ⋅ kip⋅ in 12 IS NOT DONE HERE SB BECAUSE MATHCAD IS ABLE TO ⎝ ⎠ Minimum web width within depth
b v := 7in
CONVERT IT TO KIP-IN AUTOMATICALLY
fcg Vci1 := .06ksi⋅ ⋅ b ⋅ d = 60.941⋅ kip ksi v v fcg Vi⋅ M cre Vci2 := .02ksi⋅ ⋅ b v ⋅ d v + Vd + = 658.545 ⋅ kip ksi M max
(
)
Vci := max Vci1 , Vci2 = 658.545 ⋅ kip
ft :=
fb :=
(T1 + T2 + T3) (T1⋅ ecc1(x)⋅ in + T2⋅ ecc2(x)⋅ in + T3⋅ ecc3(x)⋅ in) (Mdnc) +
Ag
+
ST
ST
(T1 + T2 + T3) (T1⋅ ecc1( x) ⋅ in + T2⋅ ecc2( x) ⋅ in + T3⋅ ecc3( x) ⋅ in) (Mdnc) +
Ag
+
SB
ft − fb fpc := fb − y cgb⋅ = 1.403 ⋅ ksi 45in The contribution of the PT tendons: At critical section: Slope of each duct at: x = 61.3 −slope1( x ) = 0.033
−slope2( x ) = 0.101
−slope3( x ) = 0.17
Shear contribution of each duct: Vpx1 := Vpx2 :=
fe1⋅ ns 1 ⋅ As⋅ −slope1( x ) = 12.285⋅ k fe2⋅ ns 2 ⋅ As⋅ −slope2( x ) = 34.663⋅ k
Vpx3 := fe3⋅ ns 3 ⋅ As⋅ −slope3( x ) = 40.88 ⋅ k Vpx := Vpx1 + Vpx2 + Vpx3 = 87.828⋅ kip
fcg ⎛ ⎞ Vcw := ⎜ .06ksi⋅ + .3⋅ fpc⎟ ⋅ b v ⋅ d v + Vpx = 299.946 ⋅ kip ksi ⎝ ⎠
(
)
Vc := min Vcw , Vci = 299.946 ⋅ kip Shear resistance:
SB
= 1.215 ⋅ ksi
= 2.095 ⋅ ksi
212
φv := 0.9 Vu = 280 ⋅ kip Vu Vn := = 311.111 ⋅ kip φv Required steel capacity: Vp := 0kip
Because it is already accounted for
Vs := Vn − Vc − Vp = 11.165⋅ kip
2
Av := .40in d v = 51.3⋅ in cotθ :=
fy := 60ksi
1 if Vci < Vcw fpc ⎛ ⎜ ksi min⎜ 1.8 , 1.0 + 3 ⋅ ⎜ fcg ⎜ ksi ⎝
⎞ ⎟ ⎟ otherwise ⎟ ⎟ ⎠
cotθ = 1.8 cotθ Vs := Av ⋅ fy ⋅ d v ⋅ s cotθ s := Av ⋅ fy ⋅ d v ⋅ = 198.495 ⋅ in Vs Checking maximum spacing: v u :=
Vu φv ⋅ b v ⋅ d v
smax1 :=
= 0.866 ⋅ ksi fcg
(
)
if v u < .125ksi⋅
(
)
fcg if v u ≥ .125ksi⋅ ksi
min .8⋅ d v , 24in min .4⋅ d v , 12in
ksi
213
smax1 = 12⋅ in Check minimum reinforcing (LRFD5.8.2.5): Av ⋅ fy
smax2 :=
.0316ksi⋅ b v ⋅
(
fcg
= 38.36 ⋅ in
ksi
)
smax := min smax1 , smax2 = 12⋅ in Therefore, use
s := 12in
cotθ Vs := Av ⋅ fy ⋅ d v ⋅ = 184.68⋅ kip s
Check Vn requirements: Vn1 := Vc + Vs + Vp = 484.626 ⋅ kip Vn2 := .25⋅ fcg⋅ b v ⋅ d v + Vp = 718.2 ⋅ kip
(
)
Vn := min Vn1 , Vn2 = 484.626 ⋅ kip Vr := φv ⋅ Vn = 436.164 ⋅ kip
>
Vu = 280 ⋅ kip
OK!
Would normally check if web reinforcing is needed over entire span by checking the above calcs at various points. However, for simplicity of design, and to aid in retraint of PT ducts, use 12 in spacing along entire span.
Composite Action/Interface shear design: b vi := 16in v ui :=
Vu b vi⋅ d v
width of top flange available to bond to the deck
= 0.341 ⋅ ksi
kip Vui := v ui⋅ b vi = 65.497⋅ ft
(
Vn := c⋅ Acv + μ⋅ Avf ⋅ fy + Pc
)
Nominal shear resistance Vn shall not be greater than the lesser of: Vn1 := K1 ⋅ fcd⋅ Acv
Vn2 := K2 ⋅ Acv
214
c := .28ksi
μ := 1.0
K1 := 0.3 K2 := 1.8ksi
Units are added to this term to make units work out later 2
in Acv := b vi = 192 ⋅ ft
Area of concrete considered to be engaged in shear transfer
For an exterior girder, Pc is the weight of the deck, haunch, parapet, and wearing surface (but there is no wearing surface in this design) wc := .150
kip ft
Assume an overhang of 3 ft:
3
soh := 3ft
wc⋅ teff kip 2 Pcd := ⋅ spacing + soh = 0.771 ⋅ 2 ⋅ spacing ft
(
)
kip Pch := hau⋅ 16in⋅ wc = 0.033 ⋅ ft kip Pcp := wparapet = 0.774 ⋅ ft kip Pc := Pcd + Pch + Pcp = 1.578 ⋅ ft s = 12⋅ in
Stirrup spacing is:
2
Av
in
Avf := = 0.4⋅ s ft kip Vn := c⋅ Acv + μ⋅ Avf ⋅ fy + Pc = 79.338⋅ ft kip Vn1 := K1 ⋅ fcd⋅ Acv = 230.4 ⋅ ft kip Vn2 := K2 ⋅ Acv = 345.6 ⋅ ft
(
)
kip Vn := min Vn , Vn1 , Vn2 = 79.338⋅ ft
(
)
kip Vr := φv ⋅ Vn = 71.404⋅ ft
>
kip Vui = 65.497⋅ ft
Stirrup spacing is adequate!
Tensile steel to support individual segments before girder is tensioned together Maxium moment in 25 ft end segment and 10 ft interior segment:
215
M 25 := 1.5wgirder⋅
( 25ft)
2
8
= 68.32 ⋅ kip ⋅ ft
M 10 := 1.5wgirder⋅
( 10ft) 8
2
= 10.931⋅ kip ⋅ ft
Assume that the segments are simply supported when they are moved, and that the concrete strength at moving is only 6500 psi. fci := 6.5ksi For 6500 psi concrete:
⎛ fci − 4ksi ⎞ ⎟ = 0.725 ⎝ 1ksi ⎠
β1 := .85 − .05⋅ ⎜
Assume single layer of #4 bars at 1.5 in. cover to exterior edge of steel fy = 60000 ⋅ psi
2
As4 := .2in
d := 45in − 1.5in − .25in = 43.25 ⋅ in
Assume compression block is contained entirely within upper flange. c := 7in a := β1 ⋅ c = 5.075 ⋅ in Moment capacity: a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ 2⎠ ⎝
( )
Solve for needed area of steel. M 25 2 As.req := = 0.336 ⋅ in a fy ⋅ ⎛⎜ d − ⎞⎟ 2⎠ ⎝ But, minimum reinforcement needed, per ACI 10.5.1: 3psi⋅ As.min :=
fci psi
2
⋅ 7 in⋅ d = 1.22⋅ in
fy
Try 6 #4 bars: 2
As := 6 ⋅ As4 = 1.2⋅ in
a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ = 244.275 ⋅ kip⋅ ft 2⎠ ⎝
( )
Check actual value of a: As⋅ fy a := = 0.814 ⋅ in .85⋅ fci⋅ b
216
M 25
As.req :=
2
a⎞
fy ⋅ ⎛⎜ d − ⎟ 2⎠ ⎝
= 0.319 ⋅ in
6 #4 bars still works.
Check tensile strain when concrete crushes: c := .003 c .003⋅
a β1
= 1.123 ⋅ in
= ⋅
x d−c
d−c c
= 0.112
Steel strain is greater than .005, so tensile controlled, therefore ϕ = 0.9 ϕ := 0.9
Actual moment capacity: a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ = 257.057 ⋅ kip⋅ ft 2⎠ ⎝
( )
ϕ⋅ M n = 231.351 ⋅ kip ⋅ ft
>
M 25 = 68.32 ⋅ kip ⋅ ft
OK!
Place 6 #4 bars at bottom of each segment, three on each side of the PT ducts.
217
Segmental Girder Design Rapid Repair and Replacement Project 100 ft span, AASTHO 45-in I-Girder
ASSUMPTIONS: Single Span Bridge Span Length = 100 ft. 8000 psi concrete (full strength @ transfer) 270 ksi low relax strand (prestress)
Parapet walls @ 338 lb/ft of wall 8"deck thickness fcg := 8ksi fcgi := 8ksi fcd := 4ksi fsu := 270ksi d strand := 0.6in fpy := .9⋅ fsu = 243 ⋅ ksi
Design Units: 1 in := ft 12
k := 1000lbf
Girder Spacing:
ksi := 1000
lbf 2
β := 0.85
in 45 in I-girder, for 100 ft span:
From Table 19-1 of WBM, spacing equals 6 ft for 102' span, 6'6" for 97' span. Interpolate: spacing := 6ft − 2ft⋅
6.5ft − 6ft 97ft − 102ft
spacing = 6.2 ft
Deck thickness: Assume 8" thick deck, after loss of wearing surface we will have a 7.5" deck. t := 8in
t eff := 7.5in
Span Lengths: Span length is 100 ft With 8" bearing pads at either end, (the effective span (for simply supported beams) is 99 ft) assume L is 100 ft. Lspan := 100ft Bridge Width: Assume 50 ft width
218
Loading: We will want to look at the bridge in the following limit states: Strength I, Service I, Service III, and Fatigue. Load factors from LRFD DL LL T3.4.1-1: Strength I 1.25 1.75 Service I 1.0 1.0 Service III 1.0 0.8 Fatigue N/A 0.75 Our dynamic load allowances, according to LRFD T3.6.2.1-1: 15% for Fatigue Limit State 33% for all other Limit States Loads applied to each girders: Girder self weight.................... ..583 k/ft (given in WBM) 8' Deck slab........................... .620 k/ft (150 pcf x thickness (8") x tributary area width(6.2') ) No future wearing surface Loads distributed among all girders: Sidewalk, median................... .472 k/ft Diaphragms........................... .529 k on each beam Parapet wall.......................... .774 k/ft (2 walls, one on each side) k wgirder := .583 ft
k wslab := .620 ft
k ws.m := .472 ft
k wdiaph := .529k wparapet := .774 ft
Total Girder Loads: On simple span girders: wdiaph WDCsimp := wgirder + wslab + Lspan
k WDCsimp = 1.208 ⋅ ft
On composite girder: WDCcont :=
ws.m + wparapet 5
k WDCcont = 0.249 ⋅ ft
Live Loads: Use HL93 truck (either design truck (below left) + lane or design tandem (below right) + lane) for +M Use special fatigue truck for fatigue limit state (HL93 truck with rear axles @ 30')
219
Load Distribution: This bridge meets the simple dist. parameters (const. width, >4 beams, parallel equal beams, road overhang <= 3 ft, standard cross-section, etc), so we can use the AASHTO simple distribution guidelines. According to table 4.6.2.2.1-1, we have a type "k" bridge. According to T4.6.2.2.2b-1 our equations are: S 0.06 + ⎛⎜ ⎞⎟ 14 ⎝ ⎠
One Design Lane Loaded:
Two or more lanes loaded:
0.4
S ⎞ 0.075 + ⎛⎜ ⎟ 9.5 ⎝ ⎠
S ⋅ ⎛⎜ ⎞⎟ ⎝ L⎠
0.3
0.6
S ⋅ ⎛⎜ ⎞⎟ ⎝ L⎠
⎛ Kg ⎞ ⎟ ⋅⎜ ⎜ 12.0⋅ L⋅ t 3 ⎟ s ⎠ ⎝
0.2
The longitudinal stiffness parameter, K g , is: Kg := n gd⋅ ⎡Ig + ⎛ Ag ⋅ eg ⎣ ⎝
0.1
⎛ Kg ⎞ ⎟ ⋅⎜ ⎜ 12.0⋅ L⋅ t 3 ⎟ s ⎠ ⎝
0.1
2⎞⎤
⎠⎦
Calculation of "n": Eg := 33000 ⋅ .150
Ed := 33000 ⋅ .150
1.5
⋅ 8 ⋅ ksi
1.5
⋅ 4 ⋅ ksi
n gd :=
Eg Ed
Eg = 5422.453⋅
Ed = 3834.254⋅
k 2
in
k 2
in
n gd = 1.414
Girder Properties from WBM: 4
Ig := 125390in 2
Ag := 560in
3
y T := 24.73in
ST := 5070in
y B := −20.27 in
SB := −6186in
3
Distance between center of gravity of girder and center of gravity of deck (with 2" haunch), eg : t eg := y T + 2in + 2
eg = 30.73 ⋅ in
Longitudinal stiffness parameter: Kg := n gd⋅ ⎡Ig + ⎛ Ag ⋅ eg ⎣ ⎝
2⎞⎤
⎠⎦
The relative long. stiffness is: Krel :=
5
4
Kg = 9.252 × 10 ⋅ in Kg ⎞ ⎛ ⎜ ⎟ 3⎟ ⎜ 12⋅ L ⋅ t span ⎠ ⎝
Krel = 0.125
220
For single span: One lane loaded:
Two or more lanes loaded:
spacing ⎞ gi1 := 0.06 + ⎛⎜ ⎟ ⎝ 14ft ⎠
0.4
spacing ⎞ gi2 := 0.075 + ⎛⎜ ⎟ ⎝ 9.5ft ⎠
spacing ⎞ ⋅ ⎛⎜ ⎟ ⎝ Lspan ⎠
0.6
0.3
spacing ⎞ ⋅ ⎛⎜ ⎟ ⎝ Lspan ⎠
⎛ Kg ⋅ 12 ⎞ ⎟ ⋅⎜ 3⎟ ⎜ 12.0⋅ L span⋅ t ⎠ ⎝
0.2
0.1
⎛ Kg⋅ 12 ⎞ ⎟ ⋅⎜ 3⎟ ⎜ 12.0⋅ L span⋅ t ⎠ ⎝
gi1 = 0.387
0.1
gi2 = 0.537
2 lane distribution factor controls: gi := max( gi2 , gi1 ) = 0.537 Load Cases a.) girder, deck, and diaphragm DL on initial simple span girders b.) sidewalk, median, parapet DL on composite girders c.) HL-93 truck with lane load d.) HL-93 tandem with lane load e.) Fatigue truck NO NEGATIVE MOMENT TO DESIGN FOR!!! - SIMPLE SPAN Load combinations for Service-1 moments (all include post-DL): sv1-1) truck + lane LL ..... for +M in span sv1-2) tandem + lane LL ..... for +M in span Load combinations for Service-3 moments (include post DL): sv3-1) truck + lane LL ..... for +M in span sv3-2) tandem + lane LL ..... for +M in span Load combinations for Strength-1 moments (include all loads): [ same sets as Service 1, but with Strength 1 load factors ] Load combinations for Stength-1 shear: str1-sh1) truck + lane LL str1-sh2) tandem + lane LL Used PCBRIDGE to find the forces from the load cases, then used EXCEL to combine and factor them.
221
Factored Design Moments: The results from PCBRIDGE were multiplied by DLA, load factors, and distribution factor in Excel. Service I and Service III loads do not include DL on non-composite span. BUT Strength I does include this. Moments from the initial non composite DL on SS, unfactored: M ss := 1510ft⋅ k
V = 60.4 k Moments from composite DL, unfactored: M = 311.25 ft-k
M dc := 311.25ft⋅ k
M ss = 1510⋅ ft⋅ k
Girder plus deck, simple span, unfactored
M s1c := 1969.87ft⋅ k
Service I Max Moment Truck and lane plus Post DL
M s3c := 1638.09ft⋅ k
Service III Max Moment Truck and lane plus Post DL
M str1 := 5178.02ft⋅ k
Strength I Max Moment Truck and lane plus FULL DL
M g45 := 728.75ft⋅ k
Girder alone on SS
222
Preliminary Design: Control of PT design: Full PT will be transferred well after concrete has cured - T is maximum and concrete strength is maximum. According to LRFD, our allowable stresses under the service loads: Initial compression before losses (5.9.4.1.1) fcia := .6⋅ fcgi
fcia = 4.8⋅ ksi
Compression after losses (5.9.4.2.1) fca := .45⋅ fcg
fca = 3.6⋅ ksi
Effective flange width on composite girder: Top flange width of girder:
b := 16in
Long span effective length:
Lelong := Lspan
Effective flange width:
(Distance between inflection points on DL moment diagram - full span because simple supported)
fle.long.a := 0.25⋅ Lelong = 300 ⋅ in fle.long.b := 12⋅ t + 0.5⋅ b = 104 ⋅ in fle.long.c := spacing = 74.4⋅ in
(
)
flelong := min fle.long.a , fle.long.b , fle.long.c
flelong = 74.4⋅ in
fle := flelong Transforming the slab into equivalent girder concrete: fle flet := flet = 52.609⋅ in n gd Composite Section Properties (Ignore haunch): 2
Ag = 560 ⋅ in
Aeffslab := t eff ⋅ flet
2
Aeffslab = 394.566 ⋅ in 2
Acg := Ag + Aeffslab Acg = 954.566 ⋅ in Calculating location of center of gravity for composite girder: Ag Aeffslab 7.5in ⎞ dist := −y B⋅ + ⋅ ⎛⎜ 45in + 2in + ⎟ Acg Acg ⎝ 2 ⎠ dist = 32.869⋅ in
The center of gravity for the composite section is 32.869 in from the bottom of the girder.
223
Using the dimensions given in the example for a composite beam, the dimension calculated above, and the parallel axis theorem, I can find the I-value for the composite girder as a whole: Ieffslab :=
flet⋅ t eff
3
4
Ieffslab = 7398.105⋅ in
3
height comp := 45in + 2in + 7.5in
height comp = 54.5⋅ in
2 ⎡ 7.5in ⎞ ⎤ ⎡ ⎛ ⎢ ⎥ + ⎣Ig + Ag⋅ ( dist + y B) 2⎤⎦ Icg := Ieffslab + Aeffslab⋅ ⎜ height comp − dist − ⎟ 2 ⎠⎦ ⎣ ⎝ 5
4
Icg = 3.478 × 10 ⋅ in Using the same dimensions I can also find the section moduli to the top the deck, the bottom of the girder, and the top of the girder flange: Icg 3 (To top of deck) Scgd := Scgd = 16080.187 ⋅ in height comp − dist Icg Scgb := 4 3 (To bottom of girder) Scgb = −1.058 × 10 ⋅ in −dist Icg Scgt := 45in − dist y cgd := height comp − dist
3
Scgt = 28672.634 ⋅ in y cgd = 21.631⋅ in
(To top of girder flange) y cgb := −dist
y cgb = −32.869⋅ in
Estimating PT Loss: To get an early estimate of how much loss we will have in the strands, we can use T5.9.5.3-1 (from 2004 edition) and calculate the average loss: fcg − 6ksi ⎞ ⎛ (PPR = 1.0) loss := 33⋅ ⎜ 1.0ksi − 0.15⋅ ⎟ + 6ksi 6 ⎝ ⎠ loss = 37.35 ⋅ ksi Since we are using low relaxation strands: loss := loss − 6ksi loss = 31.35 ⋅ ksi This seems high, so instead we will try: loss := 15ksi Strands will be tensioned up to a maximum of 218.7 ksi (.90f py) temporarily to offset friction and seating losses. Immediately after anchor set, the stress limit is 200 ksi (.74f pu = 199.8, rounded to 200). So the strands will be tensioned to 200 ksi and anchored. (Per T5.9.3-1) fpt := 200ksi Assume elastic shortening loss to be: ES := 10ksi Total assumed lossed: Δf := ES + loss
Δf = 25⋅ ksi
224
Effective prestress after losses: fe := 200ksi − ES − loss
fe = 175 ⋅ ksi
And our percent loss: pl :=
loss
pl = 7.895 ⋅ %
fpt − ES
Design amount of prestress to prevent tension at bottom of beam under full load (at center span) after 50 years Calcuating stress at the bottom of the beam due to non-composite and composite loading (Service 3 limit state): Components of moment @ center span: (NOT NECESSARILY THE MAXIMUM VALUE!) M ss = 1510⋅ ft⋅ k
Girder/deck/steel diaphragms, non-composite, unfactored
M s1c = 1969.87 ⋅ ft⋅ k
LL and composite DL, factored
M s3c = 1638.09 ⋅ ft⋅ k
LL and composite DL, factored
M g45 = 728.75⋅ ft⋅ k
Girder alone on SS
These all contribute to the full Service 1 and Service 3 moments. Total moments @ mid-span for Service 1 and 3 cases: M s1 := M ss + M s1c
M s3 := M ss + M s3c
M s1 = 3479.87 ⋅ ft⋅ k
M s3 = 3148.09 ⋅ ft⋅ k
The simple span moment will occur on a non-composite section, while the other moments will occur on a composite section. So the Service 3 tension stress at the bottom of the beam will be: M s3c M ss fbs3 := + fbs3 = −4.787 ⋅ ksi (Negative value means tension.) Scgb SB According to T5.9.4.2.2-1 in the LRFD, the limit on tension stress in service 3 condition is: fcg ftall := −.0948 ksi⋅ f = −0.268 ⋅ ksi ksi tall So we want a final prestress of: fpreq := fbs3 − ftall
fpreq = −4.519 ⋅ ksi
Using the prestress loss we previously calculated, we know that the initial compression from the prestress needs to be: fpreq fpireq := fpireq = −4.906 ⋅ ksi pl = 0.079 1 − pl Try 3 ducts, .6" diamater strands, two 12 strand ducts, one 7 strand duct. From VSL data for Polypropelene Plastic Duct: OD12 := 3.58in OD7 := 2.87in
Includes ribs
225
Assume they are bundled together in the middle portion of the span (one of top of the other), and have 1" of cover between bottom of girder and bottom duct (ignoring ribs on the bottom side of the duct). Even though ducts are bundled, they will still be .2 in apart because of the ribs. We are assuming the ribs will not hit each other. Numbering convention: Lowest duct is #1, middle duct is #2, top most duct is #3. cover := 1in OD12 educt1 := y B + cover + = −17.48 ⋅ in 2 educt2 := y B + cover + OD12 +
OD12 2
= −13.9⋅ in
educt3 := y B + cover + OD12 + OD12 +
OD7 2
= −10.675⋅ in
Guess a number of strands to use: ns 3 := 4
2
ns := 28
As := 0.217in
ns 2 := 12 ns 1 := 12
⎛ ns1 ⎞ ⎛ ns2 ⎞ ⎛ ns3 ⎞ estrands := educt1⋅ ⎜ ⎟ + educt2⋅ ⎜ ⎟ + educt3⋅ ⎜ ⎟ = −14.974⋅ in ⎝ ns ⎠ ⎝ ns ⎠ ⎝ ns ⎠ y_over_rsq := −.09053
in 2
in
term2 := 1 + estrands⋅ y_over_rsq = 2.356 term3 :=
(
Ag
2
term2
= 237.736 ⋅ in
Calculations From WisDOT Design Data for 45 in I-girder
)
Apt Apt := ns ⋅ A⋅ fspt fbi := = 5.112 ⋅ ksi term3 Forces in tendons: fpt = 200 ⋅ ksi
OK!
To := ns ⋅ fpt⋅ As
To.1int := ns 1 ⋅ fpt⋅ As = 520.8 ⋅ kip
To = 1215.2⋅ kip
To.2int := ns 2 ⋅ fpt⋅ As = 520.8 ⋅ kip
To.3int := ns 3 ⋅ fpt⋅ As = 173.6 ⋅ kip
226
Check stresses in beam at mid-span and at the end under girder weight to avoid premature failure. Check using full PT load (conservative): At mid-span: Bottom: To.1int⋅ educt1 + To.2int⋅ educt2 + To.3int⋅ educt3 + M g45 To.1int + To.2int + To.3int fbint := + Ag SB
(
) (
fbint = 3.698 ⋅ ksi fcia = 4.8⋅ ksi
Allowed:
)
OK
At top: ftint :=
(To.1int + To.2int + To.3int) (To.1int⋅ educt1 + To.2int⋅ educt2 + To.3int⋅ educt3) + Mg45 +
Ag
Allowed:
ftint = 0.306 ⋅ ksi ftall = −0.268 ⋅ ksi fcia = 4.8⋅ ksi
ST
OK
GIRDER IS OK AT MIDSPAN IN THE INITIAL CONDITION! End of beam: Because of spacing needs for the anchorages, the ducts CANNOT be the same height at the end as they are in the middle. First figure out where the strands would be at the end of the girder using VSL Type E Stressing Anchorage, we need: - an edge spacing of 8.185" for the 12-strand ducts and 6.51" for the 7-strand duct - an anchorage spacing of 14.37" for the 12-strand ducts and 11.02" for the 7-strand duct These spacings are from the center of the anchorage. Figuring out the eccentricities of the ducts at the end of the beam: spacings are rounded for easier fabrication
ed1.end := y B + 8.2in = −12.07 ⋅ in ed2.end := ed1.end + 14.4in = 2.33⋅ in
larger spacing used because of larger duct
ed3.end := ed2.end + 14.4in = 16.73 ⋅ in Check edge spacing at top: y T − ed3.end = 8 ⋅ in
> 6.51" OK
Now check stresses at end of girder using these eccentricities: To.1int⋅ ed1.end + To.2int⋅ ed2.end + To.3int⋅ ed3.end To.1int + To.2int + To.3int fbeint := + Ag SB
(
) (
Allowed:
fbeint = 2.521 ⋅ ksi OK fcia = 4.8⋅ ksi
)
227
fteint :=
(To.1int + To.2int + To.3int) + (To.1int⋅ ed1.end + To.2int⋅ ed2.end + To.3int⋅ ed3.end) Ag
Allowed:
ST
fteint = 1.742 ⋅ ksi ftall = −0.268 ⋅ ksi fcia = 4.8⋅ ksi
OK
To determine point where draped strands become straight (hold down point), pick distance along girder where all strands are straight, and determine parabolic equation to match the ducts' eccentricities at that point, and their eccentricities at the end of the girder. All three ducts become straight at 20 feet from girder end. The eccentricities for each duct are: educt1 = −17.48 ⋅ in
ed1.end = −12.07 ⋅ in
educt2 = −13.9⋅ in
ed2.end = 2.33⋅ in
('+' means above cgc of girder) educt3 = −10.675⋅ in ed3.end = 16.73 ⋅ in The distance above the bottom of the girder for the center of each duct, as a function of x (0
drape1 ( x ) := 9.3924⋅ 10
The eccentricity of each duct as a function of x:
− 5) 2 ( ⋅ x − .045x + −12.07 −4 2 ecc2( x ) := ( 2.8177⋅ 10 ) ⋅ x − .13525x + 2.33 −4 2 ecc3( x ) := ( 4.7578⋅ 10 ) ⋅ x − .2284x + 16.73
ecc1( x ) := 9.3924⋅ 10
The slopes of the ducts at any point x (up to x = 240): − 4) ( ⋅ x − 0.045 −4 slope2( x ) := ( 5.6354⋅ 10 ) ⋅ x − 0.13525 −4 slope3( x ) := ( 9.5156⋅ 10 ) ⋅ x − .2284
slope1( x ) := 1.8785⋅ 10
228
Instantaneous Losses: Anchorage Set: Assume that there is no slip at the anchorages, plus, tendons are pulled to higher stress to account for this if it were to happen: ΔfpA := 0ksi Friction Loss: Using AutoCAD, the drape of the strands was approximated with a circle, and the angle change of the strands was found. fpj := fpt K := .0002 μ := .23 K and μ taken from T5.9.5.2.2b-1 Duct 1: From end of beam to end of drape:
⎡
ΔfpF1d := fpj⋅ ⎣1 − e
(
)
x 1d := 20 α1d := 2.577deg = 0.045
− K⋅ x1d+ μ⋅ α1d ⎤
⎦ = 2.848 ⋅ ksi
From end of drape to center of beam: x 1s := 30
(
)
⎡
− K⋅ x1s + μ⋅ α1s ⎤
⎡
− K⋅ x2d+ μ⋅ α2d ⎤
α1s := 0
ΔfpF1s := fpj⋅ ⎣1 − e ⎦ = 1.196 ⋅ ksi Duct 2: From end of beam to end of drape: x 2d := 20 α2d := 7.688deg = 0.134 ΔfpF2d := fpj⋅ ⎣1 − e
(
)
⎦ = 6.852 ⋅ ksi
From end of drape to center of beam: x 2s := 30 α2s := 0
⎡
ΔfpF2s := fpj⋅ ⎣1 − e
(
)
− K⋅ x2s + μ⋅ α2s ⎤
⎦ = 1.196 ⋅ ksi
Duct 3: From end of beam to end of drape:
⎡
ΔfpF3d := fpj⋅ ⎣1 − e
(
)
x 3d := 20 α3d := 12.844deg = 0.224
− K⋅ x3d+ μ⋅ α3d ⎤
⎦ = 10.809⋅ ksi
From end of drape to center of beam: x 3s := 30 α3s := 0
⎡
ΔfpF3s := fpj⋅ ⎣1 − e
(
)
− K⋅ x3s + μ⋅ α3s ⎤
⎦ = 1.196 ⋅ ksi
Friction Loss Summary: To1int.end := fpt⋅ ns 1 ⋅ As = 520.8 ⋅ kip To2int.end := fpt⋅ ns 2 ⋅ As = 520.8 ⋅ kip To3int.end := fpt⋅ ns 3 ⋅ As = 173.6 ⋅ kip
( ( (
) ) )
To1int.mid := fpt − ΔfpF1d − ΔfpF1s ⋅ ns 1 ⋅ As = 510.267 ⋅ kip To2int.mid := fpt − ΔfpF2d − ΔfpF2s ⋅ ns 2 ⋅ As = 499.841 ⋅ kip To3int.mid := fpt − ΔfpF3d − ΔfpF3s ⋅ ns 3 ⋅ As = 163.179 ⋅ kip
229
Use an average in each duct for other loss calcs: To1int.end + To1int.mid To1.int := = 515.534 ⋅ kip 2 To2.int := To3.int :=
To2int.end + To2int.mid 2 To3int.end + To3int.mid 2
= 510.321 ⋅ kip = 168.39⋅ kip
Elastic Shortening Losses: To re-estimate elastic shortening loss, we will need a jacking pattern, since any elastic shortening that occurs will be caused by the jacking of the different ducts. Jacking pattern: Duct 2, Duct 1, Duct 3. fcgi Es Eci := 1820⋅ ⋅ ksi Es := 28500ksi n := = 5.536 ksi Eci AT MIDSPAN: Loss in duct 2 (from pulling ducts 1 and 3): fc.2mid :=
To.1int Ag
+
To.3int Ag
+
(To.1int⋅ educt1)⋅ educt2 + (To.3int⋅ educt3)⋅ educt2 Ig
Ig
fs.2mid := n ⋅ fc.2mid = 13.59 ⋅ ksi ΔT2mid := fs.2mid⋅ ns 2 ⋅ As = 35.388⋅ kip Loss in duct 1 (from pulling duct 3): To.3int To.3int⋅ educt3 ⋅ educt1 fc.1mid := + Ag Ig
(
)
fs.1mid := n ⋅ fc.1mid = 3.147 ⋅ ksi ΔT1mid := fs.1mid⋅ ns 1 ⋅ As = 8.194 ⋅ kip Loss in duct 3:
NONE
ΔT3 := 0kip
AT END: Loss in duct 2 (from pulling ducts 1 and 3): fc.2end :=
To.1int Ag
+
To.3int Ag
+
(To.1int⋅ ed1.end)⋅ ed2.end + (To.3int⋅ ed3.end)⋅ ed2.end Ig
fs.2end := n ⋅ fc.2end = 6.517 ⋅ ksi ΔT2end := fs.2end ⋅ ns 2 ⋅ As = 16.971⋅ kip
Ig
230
Loss in duct 1 (from pulling duct 3): To.3int⋅ ed3.end ⋅ ed1.end To.3int fc.1end := + Ig Ag
(
)
fs.1end := n ⋅ fc.1end = 0.168 ⋅ ksi ΔT1end := fs.1end ⋅ ns 1 ⋅ As = 0.439 ⋅ kip Loss in duct 3:
NONE
Elastic Shortening Losses Summary: Forces in ducts at mid-span: To.1mid := ns 1 ⋅ fpt⋅ As − ΔT1mid = 512.606 ⋅ kip To.2mid := ns 2 ⋅ fpt⋅ As − ΔT2mid = 485.412 ⋅ kip To.3mid := ns 3 ⋅ fpt⋅ As − ΔT3 = 173.6 ⋅ kip Forces in ducts at end: To.1end := ns 1 ⋅ fpt⋅ As − ΔT1end = 520.361 ⋅ kip To.2end := ns 2 ⋅ fpt⋅ As − ΔT2end = 503.829 ⋅ kip To.3end := ns 3 ⋅ fpt⋅ As − ΔT3 = 173.6 ⋅ kip
Time Dependent Losses: TIME DEPENDANT LOSSES, BETWEEN TRANSFER AND DECK PLACEMENT Shrinkage Loss, between transfer and deck placement: According to AASHTO Eq 5.9.5.4.2a-1, the prestress loss due to shrinkage of grider concrete between time of transfer and deck placement is: ΔfpSR := ε bid⋅ Ep ⋅ Kid t f := 50 × 365 = 18250 days
Final age in days (50 years)
t i := 365 days
Age at transfer, days (1 year)
Ep := Es ε bid := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 H for Wisconsin = 72
Eq 5.4.2.3.3-1
conc. shrnk. strain from transfer to deck placement
H := 72
k hs := 2.00 − 0.014 ⋅ H = 0.992
humidity factor for shrinkage
k vs := 1.45 − 0.13⋅ VS
factor for effect of volume-to-surface ratio
VS :=
Ag ( 16 + 2 ⋅ 7 + 2 ⋅ 19 + 2 ⋅ 7 + 22 + 2 ⋅ 6.36 + 2 ⋅ 10.61 )in
= 4.06⋅ in
volume/surface ratio
231
k vs := 1.45 − 0.13⋅
k f := k td :=
5ksi 1ksi + fcgi
VS in
= 0.922 but >=1.0 k vs := 1.0 factor for effect of conc. strength
= 0.556
t
time development factor
61 − 4 ⋅ fcgi + t
t1 := 373
t1 = time between end of cure and deck t2 = time between end of cure and transfer
t2 := 363 t1
k td1 := 61 − 4 ⋅
fcgi ksi
t2
= 0.928 k td2 := + t1
61 − 4 ⋅
fcgi ksi
= 0.926 + t2
k td := k td1 − k td2 = 0.002 ε bid := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 −7
ε bid = −4.868 × 10 Kid :=
1 Ag ⋅ estrands Es Aps ⎡⎢ ⋅ ⋅ 1+ 1+ Ig Eci Ag ⎢ ⎣
(
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti
) ⎤⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥( 2
transformed section coefficient
⎦
creep coefficient
− 0.118
psi factor for 1 year, 1 year + 10 days, and 50 years
humidity factor for creep
k hc := 1.56 − 0.008 ⋅ H = 0.984 2
Aps := ns ⋅ As = 6.076 ⋅ in
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti Kid.SR :=
− 0.118
= 0.001
1
⎡A ⋅ e Es Aps ⎡⎢ ⎣ g strands ⋅ ⋅ 1+ 1+ ⎢ Ig Eci Ag ⎣
(
ΔfpSR := −ε bid ⋅ Ep ⋅ Kid.SR = 0.012 ⋅ ksi
2⎤ ⎤
) ⎦ ⎥ ⋅ 1 + 0.7⋅ ψ b.tf.ti) ⎥(
= 0.893
⎦
loss from shrinkage btwn transfer and deck
232
Creep Loss, between transfer and deck placement: Uses most of above values... Ep ΔfpCR := ⋅f ⋅ψ ⋅K Eci cgp b.td.ti id t := 10 t is 10 days now, since it is from time of transfer to deck t k td := = 0.256 fcgi 61 − 4 ⋅ +t ksi ψb.td.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ ti ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i Kid.CR :=
comp := mom :=
− 0.118
− 0.118
= 0.133 = 0.133
To adjust these values for t=10 days ti is still 365 days
1 2⎤ ⎤ ⎡A ⋅ ( e Es Aps ⎡⎢ g strands) ⎦ ⎥ ⎣ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Ig Eci Ag ⎢ ⎣ ⎦
= 0.884
(To.1mid + To.2mid + To.3mid) (
(These terms are split up so they fit on the page) Ag To.1mid⋅ educt1 + To.2mid⋅ educt2 + To.3mid⋅ educt3 + M g45 ⋅ estrands
fcgp := comp + mom
)
fcgp = 3.145 ⋅ ksi
Ig
Ep ΔfpCR := ⋅f ⋅ψ ⋅K = 2.043 ⋅ ksi Eci cgp b.td.ti id.CR
loss from creep btwn transfer and deck
Relaxation Loss, between transfer and deck placement: Since we are using low-relaxation strand, we can assume a relaxation loss of 1.2 ksi ΔfpR1 := 1.2ksi TIME DEPENDANT LOSSES, AFTER DECK PLACEMENT Shrinkage ΔfpSD := ε bdf ⋅ Ep ⋅ Kdf ε bdf := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 t k td := 61 − 4 ⋅ fcgi + t t1 := 18248
t2 := 373
Eq 5.4.2.3.3-1
conc. shrnk. strain from transfer to deck placement
t1 = time from end of cure to 50 years t2 = time from end of cure to deck
233
t1
k td1 := 61 − 4 ⋅
= 0.998
fcgi
61 − 4 ⋅
+ t1
ksi
t2
k td2 :=
fcgi
k td := k td1 − k td2 = 0.071
ksi
ε bdf := −k vs ⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 = −1.866 × 10
= 0.928 + t2
−5
− 0.118
ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i = 0.037 epc := ⎡estrands − y B − y cgb ⎤ = −27.572⋅ in
(
⎣
Kdf.SD :=
)⎦
Below center of gravity of composite section
1 2 Acg⋅ ( epc) ⎤⎥ Ep Aps ⎡⎢ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Icg Eci Acg ⎢ ⎣ ⎦
ΔfpSD := −ε bdf ⋅ Ep ⋅ Kdf.SD = 0.479 ⋅ ksi
= 0.9
loss from shrinkage btwn deck and 50 yrs
Creep Loss: Ep Ep ΔfpCD := ⋅ fcgp⋅ ψb.tf.ti − ψb.td.ti ⋅ Kdf + ⋅ Δfcd⋅ ψb.tf.td⋅ Kdf Eci Eci
(
)
50 years - (1 yr + 10 days)
t := 17875 t
k td := 61 − 4 ⋅
fcgi ksi
= 0.998 +t
ψb.td.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i ψb.tf.ti := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t i Kdf.CD :=
(
− 0.118
= 0.517 = 0.517
1 2 Acg⋅ ( epc) ⎤⎥ Ep Aps ⎡⎢ ⋅ ⋅ 1+ 1+ ⎥ ⋅ ( 1 + 0.7⋅ ψb.tf.ti) Icg Eci Acg ⎢ ⎣ ⎦
)
Δfcd := M ss ⋅ t d := 375
− 0.118
(−estrands) Ig
= 0.871
Eci + ΔfpCR + ΔfpSR + ΔfpR1 ⋅ = 2.752 ⋅ ksi Ep
(
)
the age of the girder at the time the deck is placed.
ψb.tf.td := 1.9⋅ k vs ⋅ k hc⋅ k f ⋅ k td⋅ t d
− 0.118
= 0.515
Ep Ep ΔfpCD := ⋅ fcgp⋅ ψb.tf.ti − ψb.td.ti ⋅ Kdf.CD + ⋅ Δfcd⋅ ψb.tf.td⋅ Kdf.CD = 6.838 ⋅ ksi Eci Eci
(
)
234
Relaxation Loss: According to 5.9.5.4.3c, ΔfpR2 := ΔfpR1
Shrinkage of Deck Concrete: Loss/gain resulting from shrinkage of deck concrete, per 5.9.5.4.3d: Ep ΔfpSS := ⋅ Δfcdf ⋅ Kdf ⋅ 1 + 0.7⋅ ψb.tf.td Eci epc ⋅ ed⎤ ε ddf ⋅ Ad ⋅ Ecd ⎡ 1 Δfcdf := ⋅⎢ + ⎥ Icg 1 + 0.7⋅ ψd.tf.td Acg ⎣ ⎦
(
)
( )
ε ddf := −k vsd⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 k f := 1
different concrete strength (now 4ksi).
Ad := 7.5in⋅ ( 3ft + 3ft + 4 ⋅ spacing) VSd :=
2 ⋅ 7.5in + 2 ⋅ ( 3ft + 3ft + 4 ⋅ spacing) VSd in
= 3.675 ⋅ in
61 − 4 ⋅
fcgi ksi
= 0.928 +t
= 0.972 But must be at least 1.0 k vsd := 1.0 k f and k hs stay the same
ε ddf := −k vsd⋅ k hs ⋅ k f ⋅ k td⋅ 0.00048 Ecd := 3640ksi
1 yr + 10 days t
k td :=
Ad
k vsd := 1.45 − 0.13⋅
t := 373
MOE for 4 ksi concrete
in ed := y cgd + −7.5 = 17.881⋅ in 2 ψd.tf.td := 1.9⋅ k vsd⋅ k hc⋅ k f ⋅ k td⋅ t d
− 0.118
= 0.862
−epc⋅ −ed ⎞ ε ddf ⋅ Ad ⋅ Ecd ⎛ 1 Δfcdf := ⋅⎜ + ⎟ = 1.028 ⋅ ksi Icg 1 + 0.7⋅ ψd.tf.td Acg ⎝ ⎠ Ep ΔfpSS := ⋅ Δfcdf ⋅ Kdf.CD⋅ 1 + 0.7⋅ ψb.tf.td = 6.747 ⋅ ksi Eci This value is positive, so it means we have a gain resulting from the shrinkage of the deck.
(
)
Summary of losses (at mid-span): Instantaneous losses: ΔfpF1 := ΔfpF1d + ΔfpF1s = 4.045 ⋅ ksi ΔfpF2 := ΔfpF2d + ΔfpF2s = 8.049 ⋅ ksi
Friction Losses
235
ΔfpF3 := ΔfpF3d + ΔfpF3s = 12.005⋅ ksi fs.1mid = 3.147 ⋅ ksi
Elastic Shortening Losses
fs.2mid = 13.59 ⋅ ksi fs.3mid := 0ksi
Anchorage Losses
ΔfpA = 0
Time dependant losses, between PT transfer and deck placement: ΔfpSR = 0.012 ⋅ ksi
ΔfpCR = 2.043 ⋅ ksi
ΔfpR1 = 1.2⋅ ksi
Time dependant losses, between deck placement and 50 years: ΔfpSD = 0.479 ⋅ ksi
ΔfpCD = 6.838 ⋅ ksi
ΔfpR2 = 1.2⋅ ksi
ΔfpSS = 6.747 ⋅ ksi
TOTAL TIME DEPENDANT LOSSES: ΔfpLT := ΔfpSR + ΔfpCR + ΔfpR1 + ΔfpSD + ΔfpCD + ΔfpR2 − ΔfpSS = 5.025 ⋅ ksi This is used as an average value for all ducts. Total losses for each duct: Δfpe1 := ΔfpLT + fs.1mid + ΔfpF1 = 12.216⋅ ksi Δfpe2 := ΔfpLT + fs.2mid + ΔfpF2 = 26.663⋅ ksi Δfpe3 := ΔfpLT + fs.3mid + ΔfpF3 = 17.03 ⋅ ksi % loss: ploss1 :=
Δfpe1 fpt
⋅ 100 = 6.108
ploss2 :=
Δfpe2
loss after elastic shortening: ΔfpLT = 5.025 ⋅ ksi as a % ΔfpLTper :=
ΔfpLT fpt
⋅ 100 = 2.512
Initial stress after jacking: fi1 := fpt − fs.1mid − ΔfpF1 = 192.809 ⋅ ksi fi2 := fpt − fs.2mid − ΔfpF2 = 178.362 ⋅ ksi fi3 := fpt − fs.3mid − ΔfpF3 = 187.995 ⋅ ksi Effective stress after all loss: fe1 := fpt − Δfpe1 = 187.784 ⋅ ksi fe2 := fe3 :=
fpt − Δfpe2 = 173.337 ⋅ ksi fpt − Δfpe3 = 182.97⋅ ksi Total effective prestress force: T1 := As⋅ ns 1 ⋅ fe1 = 488.989 ⋅ k
fpt
⋅ 100 = 13.332
ploss3 :=
Δfpe3 fpt
⋅ 100 = 8.515
236
T2 := As⋅ ns 2 ⋅ fe2 = 451.369 ⋅ k T3 := As⋅ ns 3 ⋅ fe3 = 158.818 ⋅ k To1 := As⋅ ns 1 ⋅ fi1 = 502.073 ⋅ k To2 := As⋅ ns 2 ⋅ fi2 = 464.454 ⋅ k To3 := As⋅ ns 3 ⋅ fi3 = 163.179 ⋅ k
Stress Check @ mid-span: Here we will check at the midspan at the time of transfer and at 50 years down the road. Data needed: To1 = 502.073 ⋅ k
To2 = 464.454 ⋅ k
To3 = 163.179 ⋅ k
T1 = 488.989 ⋅ k
T2 = 451.369 ⋅ k
T3 = 158.818 ⋅ k
educt1 = −17.48 ⋅ in
educt2 = −13.9⋅ in
educt3 = −10.675⋅ in
M ss = 1510⋅ ft⋅ k
Non-composite DL, unfactored
M s3c = 1638.09 ⋅ ft⋅ k
Service 3 composite load
M s1c = 1969.87 ⋅ ft⋅ k
Service 1 composite load
Initial condition after jacking: Top of girder: fti :=
(To1 + To2 + To3) + (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg45 ST
Ag
fti = 0.394 ⋅ ksi
Allowable: fcia = 4.8⋅ ksi
OK!
ftall = −0.268 ⋅ ksi Bottom of girder: To1 + To2 + To3 To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3 + M g45 fbi := + SB Ag
(
) (
fbi = 3.348 ⋅ ksi
Allowable: fcia = 4.8⋅ ksi
)
OK!
ftall = −0.268 ⋅ ksi Final Condition (50 years): Top of girder (under Service 1 loading): ft1 :=
T1 + T2 + T3 Ag
ft1 = 3.103 ⋅ ksi
+
T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss ST Allowable: fcia = 4.8⋅ ksi
OK!
+
M s1c Scgt
237
Top of girder (under Service 3 loading): ft3 :=
T1 + T2 + T3 Ag
+
T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss ST Allowable: fcia = 4.8⋅ ksi
ft3 = 2.965 ⋅ ksi
+
M s3c Scgt
OK!
Bottom of girder (under Service 1 loading): M s1c T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss T1 + T2 + T3 fb1 := + + Scgb SB Ag Allowable: ftall = −0.268 ⋅ ksi HIGH!
fb1 = −0.53⋅ ksi
Bottom of girder (under Service 3 loading): M s3c T1 ⋅ educt1 + T2 ⋅ educt2 + T3 ⋅ educt3 + M ss T1 + T2 + T3 fb3 := + + Scgb SB Ag Allowable: ftall = −0.268 ⋅ ksi OK!
fb3 = −0.154 ⋅ ksi
Top of deck slab (under Service 1 loading): M s1c ft.deck := ft.deck = 1.47⋅ ksi Scgd
Allowable: fcia = 4.8⋅ ksi
OK!
Check stress at hold down point (20 ft) at jacking: wgirder M g20 := ⋅ Lspan − 20ft ⋅ 20ft = 466.4 ⋅ kip ⋅ ft 2
(
fti.20 :=
(To1 + To2 + To3) (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg20 Ag
fti.20 = −0.227 ⋅ ksi fbi.20 :=
)
+
Allowable:
ST OK!
ftall = −0.268 ⋅ ksi
(To1 + To2 + To3) (To1⋅ educt1 + To2⋅ educt2 + To3⋅ educt3) + Mg20 Ag
fbi.20 = 3.857 ⋅ ksi
+
Allowable:
SB fcia = 4.8⋅ ksi
OK!
Ultimate Strength - Moment Capacity: Check flexural strength capacity @ midspan: Aps := ns ⋅ As 2
Aps = 6.076 ⋅ in
fe = 175 ⋅ ksi
fpe := fe
fpu := fsu
0.5⋅ fsu = 135 ⋅ ksi
fpe > 0.5⋅ fpu
OK!!!
At failure, when the concrete is crushing, we can assume the tendon stress to be:
238
c fps := fpu⋅ ⎛⎜ 1 − k f ⋅ ⎞⎟ dp ⎝ ⎠
k f := 0.28
Where "c" is defined as the neutral axis location. We will calculate it below:
b = 16⋅ in
girder flange width
t f := 7in
flange thickness at outer edge
t fi := 7in
flange thickness at start of radius
b w := 7in
girder web width
hau := 2in
haunch thickness
We will assume that the compression block is in the deck, and calculate capacity as if it is a rectangular section, and use the compression strength of the deck concrete
c :=
Aps ⋅ fpu
β = 0.85
fpu .85⋅ fcd⋅ β⋅ b + k f ⋅ Aps ⋅ dp
dp is the distance from the compression face to the tendon centroid:
d p := y T − estrands + hau + teff d p = 49.204⋅ in
fle = 74.4⋅ in
Calculate c, taking width b as the effective flange width Aps ⋅ fpu
c :=
fpu
.85⋅ fcd⋅ β⋅ fle + k f ⋅ Aps ⋅ d
Neutral Axis is in deck, good
= 7.312 ⋅ in
p
β⋅ c = 6.215 ⋅ in c fps := fpu⋅ ⎛⎜ 1 − k f ⋅ ⎞⎟ dp ⎝ ⎠
fps = 258.765 ⋅ ksi
Nominal moment cpacity of the composite section: a := β⋅ c
a = 6.215 ⋅ in
2
Aps = 6.076 ⋅ in
a M n := Aps ⋅ fps ⋅ ⎛⎜ d p − ⎞⎟ 2⎠ ⎝ M n = 6039.538⋅ ft⋅ k Capacity: Mr = φ*Mn , and in prestressed concrete, φ=1.00, so:
239
M r := 1.0⋅ M n M r = 6039.538⋅ ft⋅ k Required capacity, from strength I state: M str1 = 5178.02 ⋅ ft⋅ k
Moment capacity is OK!!!
Vertical Shear Capacity: Design vertical shear reinforcing at exterior end of span. Shear force has a different distribution factor than moment, and that has been accounted for in the load cases above. spacing g shear1 := 0.36 + = 0.608 25.0ft g shear2 := 0.2 +
spacing 12ft
2
−
Vu := 246k
⎛ spacing ⎞ = 0.685 ⎜ 35ft ⎟ ⎝ ⎠ ϕv := 0.9
Controls
Max shear from Strength 1 case
Critical section is taken at a distance of dv from the face of the support. dv is the distance between resultants of the tensile and compressive forces due to flexure, and no less than .9*de or .72*h. .72⋅ height comp = 39.24 ⋅ in .9⋅ .5⋅ spacing = 33.48 ⋅ in a d v := −estrands + y T + hau + teff − = 46.096⋅ in 2
Use this as value of dv
But, there are draped strands in this area of the beam, so the e and a values will be different. Assuming standard bearing bad width of 8 inches, the distance to the critical section from the end of the girder: 8in Lcrit := ⎛⎜ + d v⎞⎟ + .5ft = 4.675 ft 2
⎝
⎠
Eccentricity of the strands at critical section: Lcrit x := = 56.096 in ns 2 ns 3 ⎛ ns1 ⎞ es_crit := ⎜ ⋅ ecc1( x ) + ⋅ ecc2( x ) + ⋅ ecc3( x ) ⎟ ⋅ in = −7.227 ⋅ in ns ns ⎝ ns ⎠ Stress block at critical section: d p_crit := −es_crit + y T + hau + teff = 41.457⋅ in
240
Aps ⋅ fpu
c :=
.85⋅ fcd⋅ β⋅ fle + k f ⋅ Aps ⋅ d
fpu
= 7.256 ⋅ in
p_crit
acrit := β⋅ c = 6.167 ⋅ in
In the deck which is good.
d v_crit := −estrands + y T + hau + teff −
acrit 2
= 46.12 ⋅ in
This is the same as the value of dv we calculated above, so OK!
The nominal shear resistance is calculated as:
(
Vn := min Vc + Vs + Vp , 0.25⋅ fcg⋅ b v ⋅ d v + Vp
)
Vp is taken as zero in this calculation, but not in the calculation of V cw The following values are needed for the shear calculations values taken at L = 4.5, to be conservative: Vd := 66.294kip Vi := 145.4kip
shear force at section due to unfactored DL (sum of DL shear, no shear dist factor because DL is not distributed by factor factored shear at section due to LL (Str 1 env - DL shear)
M dnc := 259.6kip⋅ ft
Total unfactored dead load moment acting on the non-composite section (moment from non-comp DL)
M max := 520.82kip ⋅ ft
Max factored moment at section due to externally applied loads (Str 1 env - DL moments)
M max = 6249.84 ⋅ kip ⋅ in
Needs to be in kip-in
Modulus of rupture: fr := −.20ksi⋅
⎛ fcg ⎞ ⎜ ⎟ = −0.566 ⋅ ksi ⎝ ksi ⎠
Compressive stress in concrete due to effective prestress only, after all losses, at extreme tensile fiber of the section where stress is caused by externally applied loads: fcpe :=
(T1 + T2 + T3) + (T1⋅ ecc1(x)⋅ in + T2⋅ ecc2(x)⋅ in + T3⋅ ecc3(x)⋅ in)
Ag SB Moment causing flexural cracking due to externally applied loads
= 3.273 ⋅ ksi
241
M dnc ⎞ ⎛ M cre := Scgb⋅ ⎜ fr + −fcpe − ⎟ = 35293.282 ⋅ kip⋅ in SB ⎝ ⎠ Minimum web width within depth
b v := 7in
fcg Vci1 := .06ksi⋅ ⋅ b ⋅ d = 54.759⋅ kip ksi v v
THE MULTIPLYING OF Mdnc BY 12 IS NOT DONE HERE BECAUSE MATHCAD IS ABLE TO CONVERT IT TO KIP-IN AUTOMATICALLY
fcg Vi⋅ M cre Vci2 := .02ksi⋅ ⋅ b v ⋅ d v + Vd + = 905.631 ⋅ kip ksi M max
(
)
Vci := max Vci1 , Vci2 = 905.631 ⋅ kip
ft :=
fb :=
(T1 + T2 + T3) (T1⋅ ecc1(x)⋅ in + T2⋅ ecc2(x)⋅ in + T3⋅ ecc3(x)⋅ in) (Mdnc) +
Ag
+
ST
ST
(T1 + T2 + T3) (T1⋅ ecc1( x) ⋅ in + T2⋅ ecc2( x) ⋅ in + T3⋅ ecc3( x) ⋅ in) (Mdnc) +
Ag
+
SB
ft − fb fpc := fb − y cgb⋅ = 1.461 ⋅ ksi 45in The contribution of the PT tendons: At critical section: Slope of each duct at: x = 56.096 −slope1( x ) = 0.034
−slope2( x ) = 0.104
−slope3( x ) = 0.175
Shear contribution of each duct: Vpx1 := Vpx2 :=
fe1⋅ ns 1 ⋅ As⋅ −slope1( x ) = 16.852⋅ k fe2⋅ ns 2 ⋅ As⋅ −slope2( x ) = 46.779⋅ k
Vpx3 := fe3⋅ ns 3 ⋅ As⋅ −slope3( x ) = 27.797⋅ k Vpx := Vpx1 + Vpx2 + Vpx3 = 91.427⋅ kip
fcg ⎛ ⎞ Vcw := ⎜ .06ksi⋅ + .3⋅ fpc⎟ ⋅ b v ⋅ d v + Vpx = 287.656 ⋅ kip ksi ⎝ ⎠
(
)
Vc := min Vcw , Vci = 287.656 ⋅ kip Shear resistance:
SB
= 0.979 ⋅ ksi
= 2.769 ⋅ ksi
242
φv := 0.9 Vu = 246 ⋅ kip Vu Vn := = 273.333 ⋅ kip φv Required steel capacity: Vp := 0kip
Because it is already accounted for
Vs := Vn − Vc − Vp = −14.323⋅ kip Since this is negative, we technically do not need any shear reinforcement beyond the minimum spacing, but we will need to include the minimum reinf.
However, for later calcs, still calculate the following: 2
Av := .40in d v = 46.096⋅ in cotθ :=
fy := 60ksi
1 if Vci < Vcw fpc ⎛ ⎜ ksi min⎜ 1.8 , 1.0 + 3 ⋅ ⎜ fcg ⎜ ksi ⎝
⎞ ⎟ ⎟ otherwise ⎟ ⎟ ⎠
cotθ = 1.8 cotθ Vs := Av ⋅ fy ⋅ d v ⋅ s cotθ s := Av ⋅ fy ⋅ d v ⋅ = −139.033 ⋅ in Vs Checking maximum spacing: v u :=
Vu φv ⋅ b v ⋅ d v
smax1 :=
= 0.847 ⋅ ksi fcg
(
)
if v u < .125ksi⋅
(
)
fcg if v u ≥ .125ksi⋅ ksi
min .8⋅ d v , 24in min .4⋅ d v , 12in
ksi
243
smax1 = 12⋅ in Check minimum reinforcing (LRFD5.8.2.5): Av ⋅ fy
smax2 :=
.0316ksi⋅ b v ⋅
(
fcg
= 38.36 ⋅ in
ksi
)
smax := min smax1 , smax2 = 12⋅ in Therefore, use
s := 12in
cotθ Vs := Av ⋅ fy ⋅ d v ⋅ = 165.945 ⋅ kip s
Check Vn requirements: Vn1 := Vc + Vs + Vp = 453.601 ⋅ kip Vn2 := .25⋅ fcg⋅ b v ⋅ d v + Vp = 645.342 ⋅ kip
(
)
Vn := min Vn1 , Vn2 = 453.601 ⋅ kip Vr := φv ⋅ Vn = 408.241 ⋅ kip
>
Vu = 246 ⋅ kip
OK!
Would normally check if web reinforcing is needed over entire span by checking the above calcs at various points. However, for simplicity of design, and to aid in retraint of PT ducts, use 12 in spacing along entire span.
Composite Action/Interface shear design: b vi := 16in v ui :=
Vu b vi⋅ d v
width of top flange available to bond to the deck
= 0.334 ⋅ ksi
kip Vui := v ui⋅ b vi = 64.04 ⋅ ft
(
Vn := c⋅ Acv + μ⋅ Avf ⋅ fy + Pc
)
Nominal shear resistance Vn shall not be greater than the lesser of: Vn1 := K1 ⋅ fcd⋅ Acv
Vn2 := K2 ⋅ Acv
244
c := .28ksi
μ := 1.0
K1 := 0.3 K2 := 1.8ksi
Units are added to this term to make units work out later 2
in Acv := b vi = 192 ⋅ ft
Area of concrete considered to be engaged in shear transfer
For an exterior girder, Pc is the weight of the deck, haunch, parapet, and wearing surface (but there is no wearing surface in this design) wc := .150
kip ft
Assume an overhang of 3 ft:
3
soh := 3ft
wc⋅ teff kip 2 Pcd := ⋅ spacing + soh = 0.64⋅ 2 ⋅ spacing ft
(
)
kip Pch := hau⋅ 16in⋅ wc = 0.033 ⋅ ft kip Pcp := wparapet = 0.774 ⋅ ft kip Pc := Pcd + Pch + Pcp = 1.447 ⋅ ft s = 12⋅ in
Stirrup spacing is:
2 Av in Avf := = 0.4⋅ s ft
kip Vn := c⋅ Acv + μ⋅ Avf ⋅ fy + Pc = 79.207⋅ ft kip Vn1 := K1 ⋅ fcd⋅ Acv = 230.4 ⋅ ft kip Vn2 := K2 ⋅ Acv = 345.6 ⋅ ft
(
)
kip Vn := min Vn , Vn1 , Vn2 = 79.207⋅ ft
(
)
kip Vr := φv ⋅ Vn = 71.287⋅ ft
>
kip Vui = 64.04 ⋅ ft
Stirrup spacing is adequate!
Tensile steel to support individual segments before girder is tensioned together Maxium moment in 25 ft end segment and 10 ft interior segment:
245
M 25 := 1.5wgirder⋅
( 25ft)
2
8
= 68.32 ⋅ kip ⋅ ft
M 10 := 1.5wgirder⋅
( 10ft) 8
2
= 10.931⋅ kip ⋅ ft
Assume that the segments are simply supported when they are moved, and that the concrete strength at moving is only 6500 psi. fci := 6.5ksi For 6500 psi concrete:
⎛ fci − 4ksi ⎞ ⎟ = 0.725 ⎝ 1ksi ⎠
β1 := .85 − .05⋅ ⎜
Assume single layer of #4 bars at 1.5 in. cover to exterior edge of steel fy = 60000 ⋅ psi
2
As4 := .2in
d := 45in − 1.5in − .25in = 43.25 ⋅ in Assume compression block is contained entirely within upper flange. c := 7in a := β1 ⋅ c = 5.075 ⋅ in Moment capacity: a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ 2⎠ ⎝
( )
Solve for needed area of steel. M 25 2 As.req := = 0.336 ⋅ in a fy ⋅ ⎛⎜ d − ⎞⎟ 2⎠ ⎝ But, minimum reinforcement needed, per ACI 10.5.1: 3psi⋅ As.min :=
fci psi
2
⋅ 7 in⋅ d = 1.22⋅ in
fy
Try 2 #4 bars: 2
As := 2 ⋅ As4 = 0.4⋅ in
a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ = 81.425⋅ kip ⋅ ft 2⎠ ⎝
( )
Check actual value of a: As⋅ fy a := = 0.271 ⋅ in .85⋅ fci⋅ b
246
M 25
As.req :=
2
a⎞
fy ⋅ ⎛⎜ d − ⎟ 2⎠ ⎝
= 0.317 ⋅ in
Works
Check tensile strain when concrete crushes: a
c := .003 c .003⋅
β1
= 0.374 ⋅ in
= ⋅
x d−c
d−c c
= 0.343
Steel strain is greater than .005, so tensile controlled, therefore ϕ = 0.9 ϕ := 0.9
Actual moment capacity: a M n := As ⋅ fy ⋅ ⎛⎜ d − ⎞⎟ = 86.229⋅ kip ⋅ ft 2⎠ ⎝
( )
ϕ⋅ M n = 77.606⋅ kip⋅ ft
>
M 25 = 68.32 ⋅ kip ⋅ ft
OK!
Place 2 #4 bars at bottom of each segment, three on each side of the PT ducts.
247
Anchorage zone design: The end sections will be a rectangular section for a distance of 48 in. The dimensions of this section will be 45in deep by 22in wide (width of bottom flange) Design force for PT anchorage zone and tendons: fanc := 1.2⋅ fpj⋅ 12⋅ As = 1152⋅ kip In-plane force effects (on concrete above ducts): Ignore, significant cover above ducts Out of plane force effects (blowing out the side of the ducts): fanc kip Fu.out := = 4.4⋅ ft π⋅ 1000in d c.out :=
7in − OD12 2
+
OD12 2
Designing for curvature of the 7-strand duct, since it is lowest. Designing for a 12 strand duct, since it is the widest
fcg kip Vr.out := φv ⋅ .125⋅ d c.out⋅ ksi⋅ = 13.364⋅ ksi ft Strut and Tie Model The force at each anchorage, broken up into horizontal and vertical components: 2
As := 0.217in P3 := fpj⋅ 7 ⋅ As = 303.8 ⋅ kip P2 := fpj⋅ 12⋅ As = 520.8 ⋅ kip P1 := fpj⋅ 12⋅ As = 520.8 ⋅ kip x := 0 angle3 := −atan( slope3( x ) ) = 12.866⋅ deg angle2 := −atan( slope2( x ) ) = 7.703 ⋅ deg angle1 := −atan( slope1( x ) ) = 2.577 ⋅ deg Horizontal Components P3h := P3⋅ cos( angle3) = 296.173 ⋅ kip P2h := P2⋅ cos( angle2) = 516.101 ⋅ kip P1h := P1⋅ cos( angle1) = 520.273 ⋅ kip Vertical Components P3v := P3⋅ sin( angle3) = 67.646⋅ kip P2v := P2⋅ sin( angle2) = 69.803⋅ kip P1v := P1⋅ sin( angle1) = 23.412⋅ kip
drape3 ( x ) = 37 drape2 ( x ) = 22.6 drape1 ( x ) = 8.2
248
Determine the moment at the end of the anchorage zone to determine the stresses on the cross-section:
(
)
M 48 := − P1v + P2v + P3v ⋅ 48in + P1h ⋅ ed1.end + P2h ⋅ ed2.end + P3h ⋅ ed3.end = −7843.533⋅ kip ⋅ in Find the stress at various points on the cross-section: Top of girder: M 48 P1h + P2h + P3h ft48 := + = 0.833 ⋅ ksi ST Ag Bottom of top flange: fbtf48 := Top of web:
P1h + P2h + P3h Ag P1h + P2h + P3h
ftw48 :=
fbw48 :=
P1h + P2h + P3h Ag
(M48)⋅ (yT − 7in) Ig
= 1.27⋅ ksi
(M48)⋅ (yT − 11.5in)
+
Ag
Bottom of web:
+
Ig
+
(M48)⋅ (yB + 14.5in) Ig
= 1.552 ⋅ ksi
= 2.74⋅ ksi
Top of bottom flange: ftbf48 := Bottom of girder: fb48 :=
P1h + P2h + P3h Ag
P1h + P2h + P3h Ag
+
+
(M48)⋅ (yB + 7in) Ig
M 48 SB
= 3.21⋅ ksi
= 3.647 ⋅ ksi
Using these values, I found the resultants of the following areas: -Rectangular portion of top flange -Transition area from top flange to web -Web -Transition area from web to bottom flange -Rectangular area of bottom flange ⎛ ft48 + fbtf48 ⎞ Ptf := 16in⋅ 7 in⋅ ⎜ ⎟ = 117.767 ⋅ kip 2
⎝ ⎠ ⎛ ftw48 + fbw48 ⎞ Pw := 7in⋅ 19in⋅ ⎜ ⎟ = 285.448 ⋅ kip 2 ⎝ ⎠ ⎛ ftbf48 + fb48 ⎞ Pbf := 22in⋅ 7 in⋅ ⎜ ⎟ = 527.999 ⋅ kip 2 ⎝ ⎠
The two transition areas will be simplified - average the stresses at the top and bottom of the
249
areas, and multiply that by the area of the section: ⎛ fbtf48 + ftw48 ⎞ 16in + 7in Ptt := ⎜ ⎟⋅ ⋅ 4.5in = 73.031⋅ kip 2 2
⎝ ⎠ ⎛ ftbf48 + fbw48 ⎞ 22in + 7in Pbt := ⎜ ⎟⋅ ⋅ 7.5in = 323.537 ⋅ kip 2 2 ⎝ ⎠
These forces will be applied at the centroids of each area. Strut and Tie model results: We need to figure out the stirrup spacing to handle these loads. For this area, there will only be #4 ducts around the outside of the anchorage zone, because confinement is not needed for the ducts because of all the additional concrete. At 12 in. from end: 23.3 kips, so from the end of the girder to 18 in. from the end, we need: As.0_18 :=
23.3kip
2
= 0.388 ⋅ in
fy
spacing0_18 :=
18in
= 18.541⋅ in
A s.0_18 Av
Our stirrup spacing of 12 in. provides enough strength. At 24 in. from the end: 76.5 kips, so from 18 in. to 30 in: As.18_30 :=
76.5kip fy
2
= 1.275 ⋅ in
spacing18_30 :=
12in As.18_30
= 3.765 ⋅ in
Av
Stirrups need to be spaced at 3.5 inches in this area. At 36 in. from the end: 94.5 kips, so from 30 in. to 42 in: As.30_42 :=
94.5kip fy
2
= 1.575 ⋅ in
spacing30_42 :=
12in As.30_42
= 3.048 ⋅ in
Av
Stirrups need to be spaced at 3.0 inches in this area. At 48 in. from the end: 19.9 kips, so from 42 in. to 48 in: As.42_48 :=
19.9kip fy
2
= 0.332 ⋅ in
spacing42_48 :=
6in As.42_48 Av
= 7.236 ⋅ in
Stirrups need to be spaced at 7.0 inches in this area. For simplicity in fabrication, space stirrups at 3.0 inches in the anchorage zone, with one bar located right at 48 in mark. Stirrups will be spaced at 12 in in the rest of the girder.
