THEORETICAL AND EXPERIMENTAL STUDY OF CORRUGATED PLATE STRUCTURES K.N. Saha, D. Misra, S. Ghosal, P.K. De and G. Pohit Department of Mechanical Engineering Jadavpur University, Kolkata 700 032 Abstract In the present study, corrugated plate has been modelled as a plate with stiffeners. The original corrugated plate is taken as a series of stiffeners and another backing plate has been considered as the main plate. The corrugated plate is simulated by making the thickness of this backing plate approaching zero such that the backing plate becomes wafer thin and the model describes the corrugated plate only. The static problem of plate bending has been solved by using a variational method based on assumed transverse and in-plane displacement fields of the plate. The displacement fields are described by products of orthogonal beam bending functions. These functions are generated by satisfying the specific boundary conditions of the plate. The differential equation governing the system involves geometric as well as material non-linearity. The geometric non-linearity only has been addressed in this study and the solution is obtained by using an iterative technique. The model has been validated by comparing the results with those of other researchers. Experimental verification of the same has also been carried out. Good agreement has been observed in both the cases. Displacement and stress fields for different load conditions, boundary conditions and geometries have been obtained in the study. Keywords: Stiffened plate, non-linear, large deflection, box container, corrugations.

Nomenclature a, b, h B, H D di E,ν I Q q u, v, w U,V φ, ϕ, α , β nsx, nsy

plate dimensions along x-, y- and z- direction breadth and height of the stiffener flexural rigidity of the plate, D = Eh 3 / 12(1 − ν 2 ) unknown co-efficient in the assumed displacement fields elastic modulus and poison’s ratio second M.O.I of the stiffener first M.O.I of the stiffener uniform load on the plate displacements in x-, y- and zdirections respectively strain and potential energy functions used to describe displacement fields number of stiffeners along x- and y- directions respectively

1. INTRODUCTION Rectangular plate structures are used for various applications in many branches of engineering. The load carrying capacity of such structures is greatly enhanced if the plate is made corrugated and such structures are widely used in ship building, freight container construction and in many other applications. However, a mathematical model of such a structure is very difficult to develop as the mid-plane of the plate has a very complex distribution in space, and hence their analysis is too complicated and CPU time demanding. The primary objective of this paper is to present the mathematical modelling of corrugated plate with geometric non-linearity. A few results of the analysis are also reported in this paper. Thin walled structures, in general, undergo large deflection and their static as well as dynamic response are needed to the designer. Among the recent research work on large amplitude vibration, Kobayashi and Leissa [1] observed the behaviour of rectangular shallow shell supported on shear diaphragms by using Galerkin principle. Wang et al. [2] used a numerical technique based on boundary element method to solve the static large deflection problems of thin elastic plates. Saha et al. [3, 4] studied vibration and dynamic stability of plates with elastic restraint along edges and plates on nonhomogeneous Winkler foundation. Benamar et al. [5] examined, theoretically and experimentally, the dynamic behaviour of fully clamped rectangular plates under large amplitudes of vibration. All the above work is focussed on the non-linear aspects of plate without stiffener or corrugation. An extensive review work is carried out by Mukherjee and Mukhopadhyay [6] on stiffened plates. Various methods have been reported in the textbook by Troitsky [7], in connection with various types of stiffened plate problems and out of them Huber's method is the most widely used. Huber modelled this structure as an equivalent plate with a modified flexural rigidity depending on the geometry of the corrugation. However, Huber's method is limited for only the particular geometries mentioned by him. Lam and Hung [8] used a partitioning method to study the vibration characteristics of plates with stiffened openings. A numerical technique using super finite element has been used by Koko and Olson [9] for large deflection elasto-plastic analysis of stiffened plates. A host of work, covering a numerous aspect of stiffened plate structure, is carried out by Bedair [10-11]. Bedair [12,13] employed sequential quadratic programming technique to determine the fundamental frequency of stiffened plates. Effect of geometric non-linearity on stiffened plate structure, is carried out by Chattopadhyay et al. [14] by using spline finite elements. Recently, in Ref. [15], Sheikh and Mukhopadhyay investigated geometric non-linear analysis of stiffened plates based on Von-Karman’s plate theory. It is observed that research work on stiffened plates is carried out in connection with different types of problems. One of the earlier studies on vibration problem was taken up by Kirk [16]. Among the later work, which involves various types and degrees of complications, Nair and Rao [17] investigated the problem of varying stiffener length. Following the method of Kantorovich, buckling mode localization in rib-stiffened plates has been investigated in several papers by Xie [18-19]. Sapountzakis and Katsikadelis [20], in a recent work, observed the influence of in-plane loading as well as in-plane

boundary condition on deflection profile of ribbed plates. Schubak et al. [21] investigated the problem through a rigid–plastic modelling of blast loaded stiffened plate and conducted extensive experiment for validation. A study of the behavior of concentrically and eccentrically stiffened laminated plates based on finite element discretisation is presented in Ref. [22]. In the present paper, a thin isotropic stiffened plate, incorporating the effect of geometric non-linearity, is studied by developing a specific numerical methodology. The problem is formulated by using a variational method. The assumed deflection field, required for the analysis, is constituted through linear combinations of beam functions corresponding to the specific boundary conditions of the plate. The numerical results in the study are validated with those of other researchers and experimental verification of the same has also been carried out [23, 24]. The agreement found is quite good, thus strongly establishing the mathematical model. 2. ANALYSIS The governing differential equation for the problem is formulated through an energy method, the underlying principle being the extremisation of total energy of the system in its equilibrium state. A clamped rectangular stiffened plate (a x b x h) when subjected to a uniformly distributed transverse load (q) produces deflection (w). For this deformed configuration the strain energies stored in the system comes from, (i) the strain energy stored in the plate due to bending (Ub), (ii) the strain energy of the plate due to stretching of its mid surface (Um) and (iii) the strain energy stored in the stiffeners (Us). The expressions of the energies are given below, and the different notations used are described in the nomenclature. 2    2  2 D a b  ∂ 2 w ∂ 2 w  ∂ 2 w ∂ 2 w   ∂ w  U b = ∫ ∫  − 2 + 2(1 − υ) + dxdy ,  ∂x∂y  2 0 0  ∂x ∂y  ∂x ∂y 2       

Um

2 2 2 2 2 2  ∂u  ∂w   ∂v  ∂v  ∂w   1  ∂w   ∂w   Eh a b  ∂u            = + + + + +       ∫∫  ∂x  ∂x   ∂y  ∂y  ∂y   4  ∂x   ∂y   2(1 − υ) 0 0  ∂x       ∂u ∂v 1 ∂v  ∂w  2 1 ∂u  ∂w  2    + + 2υ  +    2 ∂x  ∂x    ∂x ∂y 2 ∂y  ∂y    2 2  1 − υ  ∂u  ∂u ∂v  ∂v  ∂u ∂w ∂w ∂v ∂w ∂w     +  dxdy + +  +2 +2 ∂y ∂x  ∂x  ∂y ∂x ∂y ∂y ∂x ∂y   2  ∂y   

2

2 2 4 2  2   d2w   d2v  d u 1  d 2 w   du  dw   U s = ∑ ∫ I yp  2  + I yzp  2  + A yp  2  +  2  +     dx  4  dx   dx  dx   i =1 2 0   dx    dx    2 2 2    du  d 2 w   d 2 w  dw  2   nsx b     d w d u  − Q yzp 2  2  +  2    dx + ∑ ∫ I xq  2  +I xzq  2  dy i =1 0    dx  dx   dx  dx      dy    4 2  2    dv  d 2 w   d 2 w  dw  2   d v 1  d 2 w   dv  dw    − Q yzp 2  2  +  2    dy + A xq  2  +  2  +    dy  4  dy   dy  dy     dy  dy   dy  dy        nsy E a  

Thus the total strain energy in the stiffened plate is U = U p + U s = U b + U m + U s , where Up represents the energy of the plate and Us is the sum of the energies stored in the stiffeners along x- and y- axis. The work function due to external forces is a b W = −V = ∫ ∫ (qw)dxdy , where V is the total potential energy. The energies are expressed 0 0 through the non-dimensional quantities ξ = x / a, η = y / b , while deflection and load retain their dimensions. The principle of conservation of total energy of the system δ (U +V ) = 0 yields the governing equation of the equivalent corrugated plate system.  11  1 ∂4w 2 ∂4w ∂4w ∂4w  1 ∂4w 2(1−υ)  ∂4w + + − + −2 Dab∫ ∫  dξdηδw 4 ∂ξ4 a2b2 ∂ξ2∂η2 b4 ∂η4 a2b2  ∂ξ2∂η2 ∂η2∂ξ2 ∂ξ∂η∂ξ∂η   a  00     +

 2 2  Ehab 11  1 ∂w ∂2u 1 ∂w ∂2v  1 ∂v ∂2w 1 ∂u ∂2w 1−υ  1 ∂w ∂ u 1 ∂u ∂ w + + + + ∫∫   +υ  (1−υ2) 00 a3 ∂ξ ∂ξ2 b3 ∂η ∂η2  a2b ∂η ∂ξ2 a2b ∂ξ ∂η2  2 ab2 ∂ξ ∂η2 ab2 ∂η ∂η∂ξ 1 ∂w ∂ 2v 1 ∂v ∂ 2w  + + a 2b ∂ξ ∂ξ2 a 2b2 ∂ξ ∂ξ∂η  2  2 2 2 1 ∂ 2w  ∂w   1 1  ∂w  ∂ 2w 1  ∂w  ∂ 2w 1 ∂ 2w  ∂w  (δw) +    +   + + 2 a 4  ∂ξ  ∂ξ2 b4  ∂η  ∂η2 a 2b2 ∂ξ2  ∂η  a 2b2 ∂η2  ∂ξ   



  1 ∂2u 1 ∂w ∂2u  1 ∂2v 1 ∂w ∂2w 1−υ  1 ∂2u 1 ∂2v 1 ∂w ∂2w  + + υ + + + +      (δu) a2 ∂ξ2 a3 ∂ξ ∂ξ2  ab 2 ∂η ∂ξ∂η 2 b2 ∂η2 ab ∂ξ∂η ab2 ∂η ∂ξ∂η  ∂ ξ ∂ η a b         1 ∂2v 1 ∂w ∂2v  1 ∂2u 1 ∂w ∂2w 1−υ  1 ∂2w 1 ∂2u 1 ∂w ∂2w       +  + + + + (δv) dξdη +υ + b2 ∂η2 b3 ∂η ∂η2  ab ∂ξ∂η ab2 ∂ξ ∂ξ∂η 2 a2 ∂ξ2 ab ∂ξ∂η ab2 ∂ξ ∂ξ∂η         2  3  d2u dw d2w du E  nsy 1 1 d4w dwd2w d w dw  d2w   −4Q  +3Ayp + +2Aypa + ∑ ∫ 2I yzp 3 dξ +  2  δw 2 p =1a3 0 yp dξ4 dξ dξ2 dξ2 dξ dξ2 dξ  dξ  dξ         2 2 3 4 4 2 1     d v  nsx 1  dwd w dw d w dwd w du − Qyzp2a 3 δu + Iyzp 4 δvdξ+ ∑ 3 ∫ 2Ixq 4 +3Axq + 2Ayp a2 2 +2Ayp a 2 dη dη2 dξ dξ dη dξ  dξ   dξ   q=1b 0  2  3 d2v dw d2w dv  d3u d3v d w dw  d2w  +2Axq b 2 + 2  − 4Qxzq 3 + +2bQxzq 3  δw + 2Ixzq 3  δu 2    dη dη  dη   dη  dη  dη dη dη dη    11  d3w   dwd2w d2v 2 bQ + 2b2Axq 2 + 2bAxq − = 0 δvdη − q∫ ∫ δwdξdη yzp 2 3 dη dη dη    dη  00

+

Displacements u, v and w are approximated by w( ξ ,η )= ∑inw =1 d i φi ψ i , u( ξ ,η )= ∑i =1 d i + nwα iψ i nu

and v( ξ ,η )= ∑i =1 d i + nw+ nu φ i β i respectively, where φ (ξ ), ψ (η ), α (ξ ) and β (η ) are sets of nv

orthogonal polynomial functions. The starting functions of these orthogonal sets satisfy the corresponding boundary conditions of the plate. Substituting the above series and replacing δ by ∂ / ∂d i , the governing equation can be written in the matrix form as [K ]{d } = q{R} , where [K ] is a square matrix of the order (nw+nu+nv), the matrix elements are given in the appendix.

3. SOLUTION METHODOLOGY The unknown coefficients {d} can be obtained from {d } = q[K ]−1 {R} . As the coefficient matrix [K] is a function of the unknown coefficients {d}, the solution is obtained through an iterative scheme. In each iterative step, the matrix [K] is computed using old values {d}old and the solution vector is updated using the relaxation parameter λ as {d } = λ{d }new + (1 − λ ){d }old . During each step of iteration, the error vector {ε } = q[K ]−1 {R} − {d } is computed. If error is not within the permitted value of tolerance, the process is repeated with new {d }old = {d} until {ε } becomes less than the tolerance. A tolerance value of 0.5% has been used in the computational work. 4. RESULTS AND DISCUSSIONS Both numerical and experimental results are obtained in the present work. Numerical results pertain to stiffened and corrugated plates while experiment is carried out for stiffened plate only. Results are obtained with twenty-five plate displacement functions (u, v and w) along each coordinate axis, the particular number of functions being arrived at after carrying out the necessary convergence test. It is observed that the convergence of the numerical iteration scheme depends on the tolerance value of the error {ε} as well. The results of the analysis are compared with those of Koko and Olson [6] and Sheikh and Mukhopadhyay [8] with appropriate configuration and parameter values and are furnished in Fig. 1 (a, b). From the figures, it is observed that the agreement is quite good. The detail of the geometry and the dimensions of the stiffened plate considered are shown in Fig. 2. The validation is carried out through experiment also [24], a photograph of the set up is shown in Fig. 3. 20

25 Central deflection at point A

Deflection at point B

Present study

Present study

Rao ( taken from [8] )

Rao ( taken from [8] )

20

Sheikh and Mukhopadhyay [8]

Sheikh and Mukhopadhyay [8]

Super element, Koko and Olson [6]

Super element, Koko and Olson [6]

Finite strip, Koko and Olson [6]

Finite strip, Koko and Olson [6]

Deflection (mm)

Deflection (mm)

15

10

15

10

5 5

0

0

0.0

0.2

0.4

Load (MPa)

(a)

0.6

0.8

1.0

0.0

0.2

0.4

Load (MPa)

0.6

0.8

1.0

(b)

Fig. 1: Comparative study of deflections at selective points of a stiffened plate

Fig. 2: Geometry of stiffened plate considered for comparative study

Fig. 3: A photograph of the experimental set up Results are further generated for a 400 mm square steel plate with one centrally placed rectangular stiffener. Combinations of three different cross sectional dimensions of the stiffeners and three different thickness of plate are used for the purpose. Material properties are taken as E = 2.1 × 1011 Pa, ρ = 7850 Kg/m3 and ν = 0.3. The numerical results are shown in Fig. 4a-4c as deflection vs. pressure plot. Each of the figures corresponds to the plate of a particular thickness viz. 1.25 mm, 1.6 mm and 2.5 mm. Results for un-stiffened plate are also included in the figures for the purpose of a better understanding of the problem. With increase in stiffener section plate deflection reduces, as expected, but it is observed that the slope of the curve becomes constant for stiffened plates.

0.75

0.75 SQUARE PLATE: 400 x 400 x 1.6

UNSTIFFENED

UNSTIFFENED

STF: (3 X 24.4)

STF: (3 X 24.4)

STF: (6.3 x 25)

STF: (6.3 x 25)

STF: (9.3 x 25)

Normalised central deflection (w/h)

Normalised central deflection (w/h)

SQUARE PLATE: 400 x 400 x 1.25

0.50

0.25

0.00

STF: (9.3 x 25)

0.50

0.25

0.00 0

100

200

300 Normalised pressure (qa4/Eh4 )

400

500

0

100

200

300

Normalised pressure (qa4/Eh4

(a)

400

500

)

(b)

0.75 SQUARE PLATE: 400 x 400 x 2.5

UNSTIFFENED STF: (3 X 24.4)

Normalised central deflection (w/h)

STF: (6.3 x 25) STF: (9.3 x 25)

0.50

0.25

0.00 0

100

200 Normalised pressure (qa4/Eh4

300

400

)

(c)

Fig. 4: Load-deflection behaviour of stiffened plates (numerical results) 5. EXPERIMENTAL OBSERVATION An experiment has been carried out wherein a box structure with square plates at the top is subjected to internal pneumatic pressure and the results are shown in Fig. 5 (a-c). Rectangular-section stiffeners are fixed to the plate separately through end-clamps and oriented parallel to the plate edges. The plate deflections, corresponding to various pressure loading, are measured with the help of dial gauges placed at the center. There are provisions for fixing stiffeners with different combinations of sectional dimensions, number and positions. Experiments are carried out for a single centrally placed stiffener with three different dimensions. In the experimental results, the nature of the curves is same but the actual deflections seem to be much higher. When the plate and stiffener thicknesses are at par (Fig. 5c, 2.5

mm plate, 3 mm stiffener), stiffener effect is found to be negligible and beyond a certain value of non-dimensional load (300) the behaviour of the stiffened plate becomes identical to that of the un-stiffened one. This may be attributed to the fact that beyond that particular load a plastic hinge develops in the beam stiffener and it ceases to contribute against plate deflection. 0.75

1.00

0.75

Unstiffened

Normalised central deflection (w/h)

Normalised central deflection (w/h)

Square plate - 400 x 400 x 1.25

Stf: (3 x 24.4)

0.50

Stf: (6.3 x 25) Stf: (9.3 x 25)

0.25

Square plate - 400 x 400 x 1.6 Unstiffened Stf: (3 x 24.4) Stf: (6.3 x 25) Stf: (9.3 x 25)

0.50

0.25

0.00

0.00 0

100

200

300

Normalised pressure (qa4/Eh4

400

0

100

200

Normalised pressure (qa4/Eh4

)

(a)

300

400

)

(b) 1.75

Normalised central deflection (w/h)

1.50

1.25

1.00

0.75

0.50

Square plate - 400 x 400 x 2.5 Unstiffened Stf: (3 x 24.4) Stf: (6.3 x 25) Stf: (9.3 x 25)

0.25

0.00 0

100

200

Normalised pressure (qa4/Eh4

300

400

)

(c)

Fig. 5: Load-deflection behaviour of stiffened plates (experimental results) With this background, it is felt that corrugated plate may be modelled as a plate with stiffeners. The original corrugated plate may be considered as a continuous series of stiffeners having the corrugation geometry with a backing plate serving as the main plate. The simulation is achieved by making the thickness of this backing plate approaching zero such that the model describes the corrugated plate only. It can be pointed out that the increase in weight due to corrugation is nominal but the reduction of deflection is substantial when compared to the case of a conventionally stiffened plate.

6. CONCLUDING REMARKS In the present work, a numerical method is proposed to study the large deflection behaviour of clamped stiffened rectangular plates and excellent agreement is observed with earlier theoretical works. The effect of stiffener geometry on the load-deflection behaviour is investigated. An experimental study is also carried out. It is observed that mathematical model predicts the non-linear behaviour of stiffened plate quite well. The experimental deflections for the stiffened plates are found to be higher than the theoretical results. This can be attributed to the insufficiency in modelling the actual boundary conditions of the plate-stiffener system.

7. REFERENCES [1]. Y. Kobayashi and A. W. Leissa, “Large amplitude free vibration of thick shallow shells supported by shear diaphragms,” International Journal of Non-Linear Mechanics, 30(1), pp. 57-66, 1995. [2]. W. Wang, X. Ji and M. Tanaka, “A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates,” Computational Mechanics, 26, pp. 58-65, 2000. [3]. K. N. Saha, R. C. Kar and P.K. Dutta, “Free Vibration Analysis of Rectangular Midline Plates with Elastic Restraints Uniformly Distributed along the Edges,” Journal of Sound and Vibration, 192(4), pp. 885-904, 1996. [4]. K. N. Saha, R. C. Kar and P. K. Dutta, “Dynamic Stability of a Rectangular Plate on Non-homogenous Winkler Foundations,” Computers and Structures, 63(6), pp. 1213-1222, 1997. [5]. R. Benamar, M. M. K. Bennouna and R. G. White, “The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structure, Part II: Fully clamped rectangular isotropic plates,” Journal of Sound and Vibration, 164(2), pp. 295-316, 1993. [6]. A. Mukherjee and M. Mukhopadhyay, “A review of dynamic behavior of stiffened plates”, The Shock and Vibration Digest, 18(6), pp. 3-8, 1986. [7]. M. S. Troitsky, Stiffened plates: Bending, Stability and Vibration, Elsevier Scientific Publishing Co., Amsterdam, 1976. [8]. K. Y. Lam and K. C. Hung, “Vibration study on plates with stiffened openings using orthogonal polynomials and partitioning method”, Computers and Structures, 37(3), pp. 295-304, 1990. [9]. T. S. Koko and M. D. Olson, “Non-linear analysis of stiffened plates using super elements”, International Journal of Numerical Methods in Engineering, 31(2), pp. 319-343, 1991. [10]. O. K. Bedair, “Fundamental frequency determination of stiffened plates using sequential quadratic programming”, Journal of Sound and Vibration, 199(1), pp. 87-106, 1997. [11]. O.K. Bedair, “A Contribution to the stability of stiffened plates under uniform compression”, Composite Structures, 66(5), pp. 535-570, 1998.

[12]. Bedair, O.K., “Influence of stiffener location on the stability of stiffened plates under compression”, International Journal of Mechanical Sciences, 39(1), pp. 3349, 1997. [13]. O.K. Bedair, “Analysis of stiffened plates under lateral loading using sequential quadratic program”, Computers and Structures, 62(1), pp. 63-80, 1997. [14]. B. Chattopadhyay, P.K. Sinha and M. Mukhopadhyay, “Geometrically nonlinear analysis of composite stiffened plates using finite elements”, Composite Structures, 31(2), pp. 107-118, 1995. [15]. A. H. Sheikh and M. Mukhopadhyay, “Geometric non-linear analysis of stiffened plates by spline finite strip method”, Computers and Structures, 76(3), pp. 765785, 2000. [16]. C. L. Kirk, “Natural frequencies of stiffened rectangular plates”, Journal of Sound and Vibration, 13(4), pp. 375-388, 1970. [17]. P. S. Nair and M. S. Rao, “On vibration of plates with varying stiffener length”, Journal of Sound and Vibration, 199(1), pp. 87-106, 1984. [18]. W. C. Xie, “Buckling mode localization in randomly disordered multispan continuous beams”, AIAA Journal, 33(6), pp. 1142-1149, 1995. [19]. W.C. Xie, “Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners” Computers & Structures, 67(1-3), pp. 175-189, 1998. [20]. E. J. Sapountzakis, and J. T. Katsikadelis, “Elastic deformation of ribbed plates under static, transverse and inplane loading”, Computers & Structures, 74, pp. 571-581, 2000. [21]. R. B. Schubak, M. D. Olson and D. L. Anderson, “Rigid-plastic modeling of blastloaded stiffened plates – Part I: One-way stiffened plates”, International Journal of Mechanical Sciences, 35(3/4), pp. 289-306, 1993. [22]. E.A. Sadek and S.A. Tawfik, “A finite element model for the analysis of stiffened laminated plates”, Computers & Structures, 75(4), pp. 369-383, 2000. [23]. A. P. Kumar, “On large deflection analysis of rectangular stiffened plates”, P. G. Thesis, Department of Mechanical Engineering, Jadavpur University, India, 2000. [24]. A. P. Kumar, K.N. Saha, D. Misra, P.K. De and S. Ghosal, “Large deflection behaviour of plates under uniform Transverse Loading”, Proceedings of National Symposium on “Recent Advances in Experimental Mechanics”, IIT, Kanpur, pp. 418-429, 2000.

Appendix : The sub-matrix elements of [K] (=[Kp]+[Ks]) are given by

[ ][ ][ ] [K ] [ ] [ ] [ ] [ ][ ][ ]

 K p11 K p12 K p13    =  K p 21 K p 22 K p 23  and  K p31 K p32 K p33   

p

nw nw 

[K ] = ∑ ∑ d ∫ ∫ Dab a1

[ ][ ][ ] [K ] [ ] [ ] [ ] [ ][ ][ ] s

 K s11 K s12 K s13    =  K s 21 K s 22 K s 23  , where  K s 31 K s 32 K s33   

  1 dφ d 4 ψ i  Ehab φ ψ d i ∫ ∫  3 i ψ i dξdη + j j 4 4 2 2 2 2 4 4  2 dξ a b dξ dη b dξ  1− υ i =1 j =1  a dξ    00 2 2 2 2 nu nv  1 d φi  d β m  dβ 1 d ψ i nu d αm d αk 1 dψ i nv ψ k + 3 φi + υ 2 ψ i ∑ d k φ k k + 2 φi ψm  d ∑ dk ∑ d mφm 2 2  2 2 ∑ m dη m =1  a b dξ  dη  dη ab dη m =1 dξ dξ b k =1 k =1 2 2  (1 − υ)  1 dφi ψ nu d α d βk + 1 dφi dψ i nu d α d ψ m + 1 φ d ψ i nv d d φm β + ∑ m m ∑ m  m i i ∑ k k 2  ab 2 dξ k =1 dη a 2b dη m =1 dη 2 ab 2 dξ dη i m =1 dξ 2  2 2  1 dφi dψ i nv d φ k  1  1 d 2 φi  nw dφii 1 d 2 ψi  nw dψ ii      + 2 β + ψ ψ + φ φ d d d ∑ k  4 k i  ∑ ii ii  i  ∑ ii ii dη  2 a b dξ dη i k =1 dξ b 4 d η2  ii =1  ii =1 dξ    2  a dξ 2 2   nw   1 d dψ ii  1 d 2 ψ i  nw dφii  dξdη  + 2 2 φi   + 2 2 2 ψ i  ∑ d ii φii φ ψ ψ d ∑  ii ii j j     dη  a b dξ a b dη2  ii =1 dξ  ii =1    p 11

11

i



d 4 φi

ψi +

2

d 2 φi d 2 ψ i

+

1

φi

(

)

 nsy

2   1 1 d 4φi d 2φi  nw d 2φii  ψ + ψ ψ 2 I 3 A c ∫  yp 4 i 11 yp i ∑ ii ii  2 p =1 a 3 0  dξ dξ2  ii =1 dξ2 i =1 j=1      dφ  d3φi  nw dφii   nu d 2α k  d 2φi  nu d 2α k     + 2a A yp  i ψ i  ∑ a k − ψ i  ∑ cii + ψ ψ ψ ψii  a 4 Q  ∑ k i k k yzp  2 2 2 3   dξ dξ dξ  dξ  k =1  ii =1 dξ   k =1   dξ 2 nsx 1 1   nw d 2φii  d 2φ d 4ψ i d 2ψi  nw dψ ii    c φ  + 2i ψ i  ∑ cii ψ ii  φ jψ j dξ + ∑ 3 ∫ 2 I xq φi + 3 A xqφi 2 4 2  ∑ ii ii dη  ξ η η dξ d b d d q =1  0  ii =1  ii =1   dψ  nv d 2βm  d 2ψ i  nv dβm   i    b m φm b + φ φ + 2A xq b φi ∑ ∑ i m m dη   dη2  dη2  m =1  dη  m =1   d 3ψ  nw dψ ii  d 2ψ i  nw d 2ψ ii   i    − 4Q zxq φi φ + φ φ c c d φ ψ η  ∑ ∑  ii ii i ii ii j j   3 dη   dη2  ii =1 dη2    dη  ii =1 

[Ks ] = [K

E  ∑ 11 ] + ∑ ∑ d i   nw nw

(

)

(

[K ] = ∑ ∑ d p

nw nu

12

i =1 j =1

nw υ d φ d ψ ii d 2 φ ii d ψ i nw  Ehab 1 1  1 d φ i ψ ψ ii + φ d ∑ ∑ d ii ii  ∫ ∫ i ii 1 − υ 2  a 3 dξ 2 2 i dξ dη d η ii =1 dξ ab ii =1  00 nw  d φ i ii d ψ ii  1− υ dψ i + φ ∑ d ii  α jψ j d ξ d η d  2 i η ξ η d d d 2 ab ii =1  

i + nw

(

[K ] = ∑ s 12

nw

i =1

nu

∑ d i + nw

j=1

)

E  nsy 2 ∑ 2 p =1a 2 

1



∫ A yp

0 

)

3    dφi  nw d 2φii  − Q yzp d φi ψ i  α jψ j dξ ψ i ∑ cii ψ ii  dξ  ii =1 dξ 2 dξ3   

(

)

[K ] = ∑ ∑ d p 13

 Ehab 1 1  1 dψ i nw υ d φi nw dφii dψ ii d 2ψ ii φ φ + 2 ψ i ∑ d ii d ∑ ii ii i + nw + nu  2 2 ∫∫ 3 i dη ii =1 dξ dη dξ 2a b dξ ii =1 i =1 j=1  1 − υ 0 0  b  1 − υ dφi nw dφii dψ ii  + 2 ψ i ∑ d ii  φ jβ j dξdη 2a b dξ ii =1 dξ dη   nw nv

(

)

( )

[K ] = ∑ ∑ d s 13

i =1 j=1

s

nu nu

22

i =1 j=1

[K ]= ∑ ∑ d s

i =1 j=1

i + nw

nu nv

23

i =1 j=1

[K ] = 0 p

nv nu

32

i =1i =1 nv nv

33

i =1 j=1

[K ]= ∑ ∑ d s

i =1 j=1

3

0

yzp

 d 3αi  d φ ψ ξ ψ   i j j dξ3  

(

)

(

)

)

( )

 Ehab

(

)

[

]

  υ 1 − υ  dα i dψ i  s +   φ jβ j dξdη K 23 = 0  1 − υ 0 0  ab 2ab  dξ dη  

i + nw + nu 

(

2

)

11

nv

( )

∫ ∫ 

nw

31

i =1 j =1

i

E  nsx 1 ∑ 2 q = 1 b 3 

1



∫ 2b 0



Q xzq φ i

 d 3 β i  φ ψ dη  3  j j  dξ 

(

)

  υ 1 − υ  dφ dβ  +   α jψ j dξdη   1 − υ 0 0  ab 2ab  dξ dη   Ehab

i + nw 

(

2

i + nw + nu

11

)

(

∫ ∫ 

 Ehab

)

[K ] = 0 s

32

2 2   1 d βi 1 − υ d φi  β φ β dξdη + φ i 2 2 e2 i  j j 2 2  dη 2a dξ  1 − υ 0 0  b 

i + nw + nu 

nv nv

33

1

1  nsx  E  nsy 1 1  2 dα d 4ψ    ∑ 3 ∫ 2a A yp 2i ψ i  φψ j dξ + ∑ 13 ∫ I xzq α i 4  α jψ j dη 2  p =1a 0  dξ dη  q =1 b 0    

s

[K ] = ∑ ∑ d p

p =1

[K ] = ∑ ∑ d

31

[K ] = ∑ ∑ d p

i

i =1 j=1

(

[K ] = ∑ ∑ d p

nu nw

21

)

 Ehab 1 1  1 d 2α  1 − υ d 2ψ i  i ψ + αi α ψ dξdη   ∫ ∫ i + nw i 2 2 2  j j 2 2 2b dη   1 − υ 0 0  a dξ 

nu nu

22

 nsy

[K ] = ∑ ∑ d E2  ∑ a1 ∫ 2a Q

21

[K ] = ∑ ∑ d p

(

i + nw + nu

[K ] = 0 ; p

 E  nsx 2 1  dψ  nw d 2ψ ii  d 3ψ i   Q d − φ α β η ∑ 2 ∫ A xq φi i  ∑ cii φii  xzq i j j 2 q =1b 0  dη  ii =1 dη2  dξ3   

nw nv

(

)

11

( )

∫∫

 nsx 1 1  E  nsy 1 1  d 4φ  d 4β    ∑ 3 ∫ I yzp 4i βi  φ jβ j dξ + ∑ 3 ∫ 2A xq φi 4i  φ jβ j dη 2 p =1a 0   dξ dξ  q =1 b 0   

( )

( )

The right hand vector is given by {R1} {R} = {R 2 } where {R 3 }

nw

11

i =1

00

{R1} = ∑ ab ∫ ∫ (φiψ i )dξdη ,

{R 2 } = 0 {R 3} = 0