Baseline Determination for Tall Buildings Using the S-MR Method Ramsés Rodríguez and J. Alberto Escobar Instituto de Ingeniería, UNAM, Cd. Universitaria, CP. 04510, México.

Nomenclature

K λ M φ ω ki mi

System stiffness matrix Eigenvalue System mass matrix Mode shape Angular frequency

n

Number of degrees of freedom Mode shape information matrix Unknown stiffness-mass ratios vector Eigenvalue information vector Transpose Indicates damage Damage indicator Number of modes Geometric information of the system’s stiffness matrix

A u b T *

β

NM

Cj NE

z β σβ α

Lateral stiffness of the i-th storey Translational mass of the i-th floor

Number of elements Normalized damage indicator Mean of damage indicators Standard deviation of damage indicators Severity of damage indicator

ABSTRACT The Stiffness-Mass Ratios Method, S-MRM [1] is evaluated for baseline modal parameters determination of output-only tall buildings as applied to a three-dimensional 15-storey building using simulated data. Once the baseline state of the structure has been identified a damage detection methodology is applied in conjunction with modal parameters from the damaged state. This paper will contribute with a non-iterative approach for undamaged state determination of tall buildings. 1. Introduction Vibrational damage detection methods are a useful tool since they give information of the health of the structure without destructing or affecting the facility when damage is hidden within the structure behind a slab, below water or ground surface and when it is impossible to be located just through visual inspection. These vibrational damage detection methods use a certain indicator (parameter) that changes along the structure and that compares information from the undamaged (baseline state) and the damaged state of the structure. This indicator may also be able to identify presence, location and severity of damage.

Researchers have investigated various different indicators for damage identification. Initial studies for damage detection focused on the use of natural frequencies and/or mode shapes as the vibration signature parameters. In 1994, Zimmerman and Kaouk [2] published a paper whose indicator of damage is based on changes in stiffnesses. The method called Changes in Flexibility Matrix, was used to detect damage [3]. Zhang and Aktan developed the Changes in Uniform Load Surface Curvature [4]. The Damage Index Method developed by Stubbs [5] was studied by Farrar and Jauregui [6] in comparison with these other methods of damage identification, and it was the one that performed the best in terms of accuracy for damage detection in the 140 Bridge experiment. In addition, the Damage Index Method offers the following advantages: 1) no need to unit-mass normalization of the mode shapes, 2) good results for structures in which only few first modes are available (3 or less), and 3) application can be extended up to determine the structural reliability level using some statistical approach to predict consequences in terms of reliability of life of the structure. The main problem in the use of vibrational damage detection procedures is the need for a baseline, which can be a set of modal parameters from the undamaged state of the structure, that in most of the cases does not exist or is not available. Consequently, in this paper, structural engineers will be offered a non-iterative method for system identification of the undamaged state of the structure. This method is based on ratios between stiffness and mass values from the eigenvalue problem that just makes use of the information from the damaged structure. In this study the methodology is applied to a three-dimensional 15-storey building using simulated data. Once the baseline state of the structure has been identified the Damage Index Method is applied in conjunction with modal parameters from the damaged state to identify presence, location and severity of damage. 2. Stiffnesss-Mass Ratios Method, S-MRM The S-MR method [1] is a non-iterative approach for identification of baseline modal parameters that only utilizes damaged response data. The mass mi at each floor level is lumped together. Similarly, the lateral stiffness of the i-th storey is given by k i . Then the system mass and stiffness matrices for an n degree-of-freedom system can be indicated as unknowns. From the matrix eigenvalue problem, whose solution gives the natural frequencies and mode shapes of a system, we have:

[K − λM ]φ = 0

(1)

Substituting the first extracted eigenvalue, λ1 and the first mode shape, φ1 , from the damaged structure into equation (1) and leaving k i and mi as unknowns, the following system of linear equations can be obtained:

( k1 + k2 − λ1m1 ) φ11 − k2φ21 −k2φ11 + ( k2 + k3 − λ1m2 ) φ 21 − k3φ31 M −kiφi −1,1 + ( ki + ki +1 − λ1mi ) φi1 − ki +1φi +1,1 M −knφn −1,1 + ( kn − λ1mn ) φn1

= = = = = =

0 0 M 0 M 0

(2)

i = 1,2...n Dividing each equation i from the set of equations (2) by mi and φ i 1 and grouping the terms with the same stiffness coefficient together, the equivalent system is now in terms of the unknown ratios of stiffnesses and masses, ki mi . In matrix form and factoring the unknown ratios, this system can be expressed by:

A1u = b1

(3)

where A1 is the matrix containing coefficients in terms of just mode shapes, u is a vector of the unknown ratios, and b1 is the vector containing the eigenvalues, λ1 , as shown in Equations (4), (5) and (6). A1 is a non-square

matrix, n x 2n − 1 and u is a vector of length 2n − 1 .

 1   0  0 A1 =  0  M   0 

 φ 21  1 −  φ 11  

 k  u =  1   m1  b1 = [λ1,1

0 0

0

0

 φ11  1 −   φ 21  0

 φ31  1 −  0  φ 21  0 O

0

0

0

M

M

M

0

0

0

 k2     m1 

 k2   m2

λ1, 2 K λ1,i

0

k    L  i    mi 

0

0

L

0

0

L

 φi −1,1   L 1 −  φ i ,1  

L

 φi +1,1  1 −  L   φ i ,1   O

0

L

 k i +1   k    L  n   mi   m n 

0

L

    0   0    0   M   φ n −1,1  1 −    φ n , 1   0

(4)

T

(5)

K λ1,n ]

T

(6)

Repeating the process for all ω i and φ i , an overdetermined system of linear equations is obtained, whose number of rows is equal to the square of the number of degrees of freedom in the system. This system is expressed and solved as shown in equation (7).

Au = b

(7)

u = ( AT A) −1 AT b

(8)

where A is the matrix containing individual A1 matrices for each mode, and similarly b is the vector containing T

the individual bi vectors containing the eigenvalues. It is necessary to mention that the product ( A A) in equation T

(7) is always invertible, since for dynamic problems using the shear beam model the rank (r) of ( A A) is always T

T

T

greater or at least equal to the column space of ( A A) . Physically r ( A A)
3. Damage Index Method Stubbs et al. [5] proposed a method that uses and indicator based on the relationship between the material stiffness properties of the undamaged and the damaged element of the structure. This damage indicator for the j-th element, β j , is obtained as: NE  *T  * *T * φ C φ φi Ckφi K i  +  ∑ ∑ j i  i i =1  k =1   βj = NM NE  T  * T ∑  φ i C j φ i + ∑ φ i C k φ i  K i  i =1  k =1   NM

(9)

j = 1,2...NE These indicators are then normalized to provide more robust statistical criteria for damage localization. The normalized damage indicator for the j-th element, z j , is given by:

zj =

(β

j

−β)

(10)

σβ

In this study positive z j values give indication of potential damage location. According to this method, severity of damage is expressed as the fractional change in stiffness of an element:

αj =

1 −1 βj

(11)

No damage is present when α j = 0 ; if damage is present then α j < 0 . Note also that if α j = −1 all stiffness capacity has been lost for that element. 4. Damage Assessment of a Three-Dimensional 15-Storey Building A 15-storey reinforced concrete building [7] designed based on the Mexico City’s Federal District code (MCFDC) was studied. Figure 1 shows the geometry and member properties of the undamaged structure. SAP2000 [8] was utilized to model the building. The center of mass (CM) at each floor was located at a distance from the center of stiffness (CS) equal to 20% of plan dimension. Location of CM is shown in Figure 1. The mass values at each floor are presented on Table 1. Table 1. Translational mass (kg) and mass moment of inertia (kg m2 ) values at each floor of the studied 15-storey concrete building. i-th floor mi Ioi

1 2 3 4, 5 6 7, 8 345450 339276 334768 330358 326144 322028 25801440 25248720 24846920 24446100 24069780 23694440

i-th Floor mi Ioi

9 10, 11 12 13, 14 15 318108 314188 310660 307034 286258 23344580 22994720 22671320 22346940 20481020

15-th floor 0.75m x 0.35m all beams cols. 0.60m x 0.60m

Y

cols. 0.65m x 0.65m 14 @ 3m

cols. 0.70m x 0.70m

3.6m

cols. 0.75m x 0.75m

CM

cols. 0.80m x 0.80m

3.5m

18 m

18 m

3.6m

CS

X Elevation

Typical floor

Figure 1. Studied 15-storey concrete building. Three damage patterns were studied: 1) 90% stiffness reduction of 12-th storey columns, 2) 70% stiffness reduction of 3-rd and 8-th storey columns, and 3) 40% stiffness reduction of 2 columns at 2-nd and 14-th storeys. Modal parameters of the analytical model were computed for the undamaged state and the three damage patterns. Table 2 shows the first mode-computed frequencies. Table 2. First mode frequencies of the 15-storey concrete building (Hz). Undamaged state 0.629

Damage pattern 1 0.614

Damage pattern 2 0.610

Damage pattern 3 0.629

Note from Table 2 that even for damage patterns 1 and 2 changes on frequencies are almost neglectable indicating global modest changes in modal parameters. In this investigation, baseline modal parameters in the X direction using the S-MRM were identified for two different cases: 1) using all computed damaged mode shapes and 2) using only the first three computed damaged mode shapes. Last case was studied to evaluate the S-MRM for the case when only a few number of modes are available. Computed damaged mode shapes for damage pattern 1 were substituted into equation (4) as well as the computed damaged frequencies into Equation (6). Equation (8) was computed to solve for baseline modal parameters according with the S-MRM. In order to locate damage, equation (9) was computed for every damage pattern using the identified baseline *

mode shapes from the S-MRM and the computed damaged mode shapes. Both K i and K i in equation (9) were taken as equal to the identified K i by S-MRM. It is very important to mention since indicates that only changes in modal parameters are contributing to identify damage.

Another important comment is that only the first mode shape was utilized to compute damage indicators using equation (9). Figures 2a, 2c and 2e show the normalized damage indicators using equation (10) for case 1 (all damaged modes utilized by S-MRM to obtain baseline modal parameters) and Figures 2b, 2d and 2f for case 2 (the first three damaged modes were used by the S-MRM). Positive normalized damage indicator on Figure 2a indicates damage location on the 12-th storey for damage pattern 1, which matches with the actual damage location. It can also been observed a false damage location on the 15-th storey. Figure 2b is showing that the methodology is correctly identifying damage 1 location for the case when only the first three damaged mode shapes were used to identify baseline modal parameters however some other false damage location were identified primarily on the 2-nd storey. For damage pattern 2, damage location was more accurately identified for case 2 than for case 1. This can be observed on Figure 2d which identified damage location on the 3-rd and 8-th storey corresponds to actual damage location however a false damage location was present on the 2-nd storey similarly for damage pattern 1. The fact that case 2 performed more adequately than case 1 is convenient for common practices when a few number of modes are available. It can be observed on Figure 2e that the actual damage location on the 14-th storey for pattern 3 was identified for case 1 and not for case 2 (Figure 2f). As it can be expected, for moderate damage cases on superior stories the more number of modes used by the S-MRM the more accurate the damage localization will be. For case 2 on Figure 2f, damage location on the 2-nd storey was accurately identified as well as some other false damage locations on the neighborhood. In general, the three modes utilized on case 2 were enough so that the S-MRM performed adequately for damage location. Table 3 presents severity of damage values obtained with equation (11) for the accurately identified damage locations. According to the Damage Index Method damage is present when severity of damage is negative. Table 3 shows two positive severity of damage indicators indicating that damage is not present which is false. This is another indication that the S-MRM performed more adequately for case 2. However it can be observed that the severity of damage value for case 1 pattern 1 is very close to the actual severity of damage value and case 2 is underestimating severity. Case 2 is also underestimating severity of damage for damage pattern 2 and overestimating for pattern 3. Last results show that for severe damage cases the use of great number of damaged modes to obtain baseline modal parameters is preferred. It can be also mentioned that when a few number of modes are used to obtain baseline modal parameters using the S-MRM analyst must consider that severity of damage values might be underestimated. Table 3. Severity of damage. Damage Storey Case 1 Case 2 pattern 1 12 -0.92 -0.61 3 ---0.59 2 8 0.03 -0.34 2 ---0.67 3 14 0.13 ---

15

Storey

1

-1

0

1

2

3

-1

a) Damage pattern 1 case 1.

0

1

2

3

z

b) Damage pattern 1 case 2.

15

Storey

1

-1

0

1

2

3

-1

c) Damage pattern 2 case 1.

0

1

2

3

z

d) Damage pattern 2 case 2.

15

Storey

1

-1

0

1

2

3

-1

e) Damage pattern 3 case 1.

0

2

3

f) Damage pattern 3 case 2.

Figure 2. Normalized damage indicator. Note: Actual location of damage is indicated by the symbol

1

.

z

Conclusions In this investigation the non-iterative S-MRM demonstrated its capacity for baseline modal parameters identification of tall buildings using simulated data without knowing actual undamaged state of the structure. Solely damaged modal information was used and the method identified adequately damage location for the case when only the first three modes were utilized. However potential analysts must consider that severity of damage values were underestimated. It has also been shown that the Damage Index Method provided good damage identification results utilizing only the first mode. References [1]

Barroso, L., and Rodríguez, R. “Damage Detection Utilizing The Damage Index Method To A Benchmark Structure”, Journal Of Engineering Mechanics. Vol. 130, No. 2, pp. 142-151, 2004.

[2]

Zimmerman, D. C., and Kauk, M. "Structural Damage Detection Using a Minimum Rank Update Theory", Journal of Sound and Vibration, Vol. 116, No. 2, pp. 221-231, 1994.

[3]

Pandey, A. K., and Biswas, M. "Damage Detection in Structures Using Changes in Flexibility" Journal of Sound and Vibration, Vol. 169, No. 1, pp. 3-17, 1991.

[4]

Zhang, Z., and Aktan, H. M. "The Damage Indices for Constructed Facilities", 13-th International Modal Analysis Conference, Tennessee, pp. 1520–1529 , 1995.

[5]

Stubbs, N., and Kim J. “Damage Localization in Structures Without Baseline Modal Parameters”, American Institute of Aeronautics and Astronautics Journal. Vol. 34, No. 8, pp. 1644-1649, 1996.

[6]

Farrar, C. R., and Jauregui, D. A. "Comparative Study of Damage Identification Algorithms Applied to a Bridge: 1. Experiment", Smart Materials and Structures, Vol. 7, pp. 704-719. 1998.

[7]

Stark, R., and Lira, J. “Respuesta de Estructuras de Concreto Diseñadas de Acuerdo con el RDF-87 Ante Sismos en la Ciudad de México”, Seminario Internacional Evaluación de Estructuras de Concreto, México D.F., 1991.

[8]

Wilson, E., and Habibullah A. “SAP2000, Structural Analysis Manual User’s”, Computers and Structures Inc., Berkeley California, 2000.