Analysis of eccentrically stiffened plates with mixed boundary conditions using differential quadrature method

Applied Mathematical Modelling 22 (1998) 251±275

Analysis of eccentrically sti€ened plates with mixed boundary conditions using di€erential quadrature method Zahid A. Siddiqi a, Anant R. Kukreti

b,*

a

b

Civil Engineering Department, University of Engineering Technology, Lahore, Pakistan School of Civil Engineering and Environmental Sciences, University of Oklahoma, 202 West Boyd Street, Room 334, Norman, OK 73019, USA Received 4 July 1995; received in revised form 10 February 1998; accepted 3 March 1998

Abstract Di€erential quadrature solution for the ¯exural analysis of eccentrically sti€ened plates subjected to transverse uniform loads is presented. In-plane forces in the plate are considered to take into account the axial sti€ness of the plate and the interaction between the beams and the plate due to the eccentricity. Torsional and shear sti€nesses of the beams are also considered. The analysis procedure presented can be used for: point loads applied at the corners of the plate segments; roller point supports at the corners of the plate segments; and outer edges having di€erent combinations of boundary conditions, which includes, free, simply supported, clamped, or resting on beams. The method gives the same accuracy for the moments and shears as that for the de¯ections and is computationally ecient and simple to program. The results for single panels with complicated boundary conditions are compared with the available exact results. Two examples, one with central eccentric sti€ener, and the other with two central mutually perpendicular sti€eners are analyzed and compared with the available results. An example of a plate with no sti€eners but with mixed boundary conditions is also analyzed and compared with the ®nite element results. All the results are close to the published results. Ó 1998 Published by Elsevier Science Inc. All rights reserved.

1. Introduction Structural plate systems sti€ened by ribs in one or two directions are widely used in buildings, bridges, ships, aircraft, and machines. Sti€ening of the plate is used to increase its load carrying capacity and to prevent buckling, especially in case of in-plane loads. When the ribs are closely and evenly spaced, the analysis may be carried out by smearing-out the sti€ness properties of the ribs to get an equivalent orthotropic homogeneous slab of constant thickness [29]. The general bending of the ribbed plate system is computed using any such conventional method, and the results are superimposed to those resulting from the local bending of the sti€eners and the unsti€ened slab-panels. When the ribs are few and far apart, the structure acts like plate panels resting on beams. There is a need to develop methods to analyze the general system covering these two extreme cases and also the intermediate cases. The beams and plate-segments should be considered separately but proper interaction should be provided between the two components so the *

Corresponding author. Tel.: +1 405 325 5911; fax: +1 405 325 4217.

0307-904X/98/$19.00 Ó 1998 Published by Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 9 8 ) 1 0 0 0 5 - 7

252

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

loads are resisted collectively. One way to accomplish this is to ®nd the independent solutions of the plate-panels and the beams separately subjected to unknown interactive forces between the structural components besides the direct loads [1,12]. The unknown interactive forces are then iteratively calculated to satisfy the compatibility between the two structural components. The e€ect of torsion and shear transfer from the beams to the plate are generally neglected in these methods. In addition, the working formulas presented [1,12] are very lengthy for any design calculations. Another approach to solve sti€ened plate systems is the ®nite element method, the use of which is very limited for such problems, because of the requirement for signi®cant computer storage and time to model the whole plate-beam systems accurately and the unavailability of suitable commercial software that can be used. The analysis of the sti€ened plate system becomes more complicated when the sti€eners are placed at some eccentricity from the middle surface of the slab (the more practical case), and for which the in-plane forces in the plate-segments become signi®cant to a€ect the overall system behavior. Some literature [4,10,21±27] is available for static analysis of ribbed plates considering the interaction between the beams and the plate for the case when beams are present at some eccentricity with respect to the plate middle surface. Torsional and bending strength of the beams are considered in all of these references, but shear strength is included in only a few [10]. McBean [21] documented the mathematical derivations of the governing di€erential equations and the associated boundary conditions using the principle of minimum potential energy. A sti€ener element compatible with the quadrilateral plate elements is derived. Any solution based on this combination of elements provides a lower bound on the strain energy. Thirty-two degrees of freedom are associated with each plate element, 10 with each sti€ener element. Chang [4] advanced the developments of the di€erential equations and the boundary conditions for such sti€ened plate systems. A Levy type solution for the displacements was assumed to separate the variables and to convert the partial di€erential equations to ordinary di€erential equations. Laplace and Fourier transformation method were applied to some limited cases. Navier type solutions for rectangular sti€ened plates with both discrete and smeared idealizations were also presented. Rossow and Ibrahimkhail [26] have presented a constraint approach in the ®nite element method to solve such problems. Deb and Booton [10] derived linear ®nite element models based on Mindlin's shear distortion theory for bending of eccentrically sti€ened plates under transverse loading. Peng-Cheung et al. [25] used B-Spline functions for the static, vibration and stability analysis of sti€ened plates. Kukreti and Rajapaksa [17] analyzed ribbed and grid plate systems used for bridge decks using an energy-based approach, where polynomial functions are used to model de¯ections and moments in the slab. The eccentricity of the sti€eners was neglected in their analysis. The order of the polynomial to be used for the each ®eld variable to be approximated may easily be changed. Sheikh and Mukhopadhay [27] proposed a general spline ®nite strip method for the analysis of sti€ened plates. Mukhopadhay [22] has presented a semi-analytical ®nite di€erence method to solve the di€erential equations of Chang [4]. Using a certain transformation, the resulting equations are reduced to ordinary di€erential equations with constant coecients. Harik and Guo [13] have presented a ®nite element analysis of eccentrically sti€ened plates in free vibration. In this paper a di€erential quadrature method is presented to predict the ¯exural behavior of eccentrically sti€ened plates. The interaction between the eccentrically placed beams is accounted for by considering the in-plane sti€ness and displacements of the plate. The method accounts for the major bending strength of the beams as well as their torsional and shear strengths. Lateral bending sti€ness of the beams may easily be added, if required. The method is developed for a regular grid with di€erent size beams in each direction. However, it can be extended to systems with di€erent beam spacings and sizes. Point loads can be applied at the intersection of the beams. Each outer edge between two beams can have di€erent edge conditions, which includes free,

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

253

simply supported, clamped, or resting on beams. The beams are considered as line-elements and their width is neglected. Results for single panels with complicated boundary conditions are compared with the exact solutions. Two examples of plate panels, one with one central sti€ener and the other with two mutually perpendicular central sti€eners are analyzed and compared with the available ®nite element results [4,21]. Finally, a plate with no sti€eners but with mixed boundary conditions is analyzed and the results are compared with the ®nite element results. It is noted that the di€erential quadraturte results, when compared with the ®nite element results, give nearly the same level of accuracy for the moments and shears as that for the de¯ections, which is very helpful for most of the design problems where accurate prediction of internal stress resultants is more important then de¯ections. The method itself does not impose any size limitation on the problem, just like the ®nite element method. However, computer speed and its memory along with the round-o€ error in the calculations determine the maximum size of the problem which can be analyzed. 2. Description of the quadrature method An approximation to the value of a function integral or derivative at a given discrete point as a weighted linear sum of function values at discrete points of the independent variable is called an integral or a di€erential quadrature [6], respectively. A quadrature approximation at the ith discrete point is given by Lff …x†gi 

N X

Wij f …xj †

…1†

jˆ1

in which L is any di€erential, integral, or combined operator applied to a smooth and analytic function f(x), where x is the independent spatial variable; x1 ; x2 ; . . . ; xN are the sampling points considered in the domain; f(xj ) are the function values at these points; and wij are the weights attached to these function values. In the start, f(xj ) and the coecients wij are treated as the unknown coecients in this method. To determine the coecients wij , the following power polynomial is used to approximate the function f(x) f …x† ˆ

N N X X akÿ1 fk …x† ˆ akÿ1 xkÿ1 : kˆ1

…2†

kˆ1

Substituting each term, fk (x), of this polynomial, called a test function [6], into Eq. (1) leads to: Lfxkÿ1 g ˆ

N X wij xjkÿ1 for i and k ˆ 1; 2; . . . ; N :

…3†

jˆ1

In Eq. (3), the left-hand side can be solved for the chosen operator L. For a set of selected values of xj , Eq. (3) is written for all values of k ˆ 1; 2; . . . ; N to complete the polynomial up to (N ) 1) order. This represents a set of N linear algebraic equations which are solved for wij . This set of equations gives a unique solution, since the matrix of elements xjkÿ1 is a vandermonde matrix. Thus, the resulting system can be solved analytically, as described by Bj orck and Pereyra [3]. The weighting coecients, wij , so obtained are then substituted into Eq. (3) to express the derivatives and integrals of a function at a discrete point in terms of values of the function at all discrete points. This solution is then substituted into the governing di€erential equation for the problem for each i including the boundary conditions at some of the points. The resulting system of equations is now solved for the unknown function values.

254

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Civan [5] has shown that by choosing a few discrete points in the quadrature formulation, one may derive the well-proven ®nite di€erence di€erential and integral formulae. The quadrature method has been observed to provide very rapid convergence if the expected solution is smooth and continuous [2,5±9,11,15,16,18±20]. The ®rst main advantage of using quadrature method is that it provides higher order polynomial ®tting. The second advantage is that the same formulation may be used for di€erential terms of any order, integral terms, integro-di€erential terms, and a linear combination of all these types [5,6]. Di€erential quadrature method has been successfully used to solve governing di€erential equations for various boundary value and eigen value problems in structural mechanics. Through these applications, it has been shown that the method is easy to implement numerically, is computationally very ecient, and produces results with a similar degree of accuracy as other acceptable methods (e.g., ®nite element method, ®nite di€erence method, Galerkin's method, Ritz method, etc.). Some of the early work includes that of Jang et al. [15], which reports application of di€erential quadrature method to ®nd de¯ections in various structural elements (beams, membrane, and thin circular and rectangular plates), and buckling behavior of columns and plates with di€erent boundary conditions. Sherbourne and Pandey [28] extended the application of the method to buckling analysis of beams and plates, which result in solution of di€erential equations with variable coecients. The applications chosen by them included a laminated composite plate under variable compression, a variable thickness plate under constant uniaxial compression, and the lateral-torsional buckling of an I-beam. Lin et al. [19] demonstrated the application of the di€erential quadrature method to solve the problems of the de¯ections of plates with general nonlinear elastic boundary constraints. They investigated the application of the Lagrange interpolated polynomial as the test functions in order to avoid ill-conditioning of the resulting equations when the number of grids needed is reasonably large. This selective literature review illustrates the evolution of the application of the di€erential quadrature method in the area of structural mechanics. Using the experiences gained regarding selection of grid points, spacing between grid lines, modeling of forced and natural boundary conditions, procedures to avoid illconditioning of resulting equations, and procedures to ensure fast convergence, researchers extended the application of the method in progression to solve more complex structural mechanics problems [2,11,15,16,19,20,28]. But, in all these studies the method was applied to problems in which the same governing di€erential equations described the behavior in the total system domain. There are numerous problems in which the total structural system consists of sub-structures, and the behavior of the di€erent sub-structures is described by di€erent unique governing di€erential equations. Also, the interaction of forces and compatibility of deformations between adjoining sub-structures must be accounted in the analysis. This paper further extends the application of the di€erential quadrature method to such a problem, which includes ¯exural analysis of a thin rectangular plate with sti€ener beams spanning in both directions. To our knowledge, this is the ®rst time the method has been applied to study this boundary value problem. 3. Assumptions The following assumptions are made for the problem studied: 1. Transverse shear deformations are neglected for the plate. 2. The material of the sti€ener (beam) is elastic, homogeneous and isotropic. 3. Deformations due to bending vary linearly for both the plate and the beams. 4. The sti€ener is prismatic and has a symmetric cross-section.

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

255

5. Warping is considered negligible for the beams. As in the elementary theory for bending and torsion, the relations for pure bending and St. Venant torsion are adopted herein, because the distributions of shearing force and torque are relatively continuous and the sti€ener is long relative to its depth. Warping discontinuities due to torsion will occur on the boundary between sti€eners, but this incompatibility is presumed to have a negligible e€ect on the strain bounds. Thus, the present study is restricted to thin-walled sti€eners made-up from open cross-sections that are symmetrical about a vertical axis where the torsion and bending e€ects may be uncoupled. More uncertainty will be involved when this method is used for torsionally sti€ (closed) cross-sections. 6. All loads are applied normal to the middle-surface of the plate. 7. The sti€eners are torsionally weak and either weak in shear or rigid in shear. When the shear e€ect is included, the de¯ection of the sti€ener is increased (overall rigidity reduces), changing the de¯ected behavior of the slab segments. For smaller span/depth ratios, the e€ect of shear increases. For I-beams the e€ects are similar to those for rectangular beams, except that the relative magnitude of the shear de¯ection is usually two or three times greater. For beams of sandwich construction, the increase in de¯ection due to shear may be as high as 50%. 8. Sti€eners are located only along the boundaries of the plate segments. Sti€eners are assumed as line elements with no width. 9. Concentrated loads are only applied at the corners of the plate segments. 10. The common normal to the plate and the sti€ener before bending remains straight after bending. 11. Bending of sti€eners in the plane of the plate is negligible. However, lateral bending of these sti€eners may easily be incorporated, if required. 12. The second order e€ects, in the plate segments, corresponding to the membrane forces, is neglected and it is assumed that local buckling does not occur in the plate. 4. The governing equations A thin plate having thickness h, ¯exural rigidity Df , and axial rigidity Da built monolithically with concentric or eccentric beams is analyzed by using the di€erential quadrature method. The beams/sti€eners have ¯exural rigidity Bfx or Bfy , axial rigidity Bax or Bay , torsional rigidity Btx or Bty , and shear rigidity Bsx or Bsy , where the subscript x denotes sti€eners spanning in the x cartesian direction and y denotes sti€eners spanning in the y cartesian direction. The eccentricity from the mid-plane of the plate to the centroid of the individual beam is denoted by ex and ey along the x and y directions, respectively. The transverse loading and de¯ections are assumed to be in the z direction. The plate deformations in the x, y, and z directions are denoted by u, v, and w, respectively. The plate moment acting on the x-face about the y-axis is denoted by Mxx and the plate moment acting on the y-face about the x-axis is denoted by Myy . The plate twisting moment acting on x-face about the x-axis or the plate twisting moment acting on the y-face about the y-axis is denoted by Mxy . The plate shear force acting on the x-face is denoted by Qx , and that on the y-face is denoted by Qy . The axial forces in the x and y directions are denoted by Nx and Ny , respectively, whereas the in-plane shear force is denoted by Nxy . All these stress resultants are forces and moments per unit length of the respective edge of the plate and are graphically illustrated in Fig. 1. The governing di€erential equations for the transversely loaded isotropic thin plate free to deform along the three directions, based on all the aforementioned nomenclature and assumptions, may be written as:

256

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Fig. 1. Moments, shear forces, and in-plane forces acting on a thin plate segment.

o4 w o4 w o4 w q ‡ 2m ‡ ˆÿ ; 4 2 2 4 ox ox oy oy Df

…4†

o2 u 1 o2 v 1 o2 u ‡ ˆ 0; …1 ‡ m† ‡ …1 ÿ m† ox2 2 oxoy 2 oy 2

…5†

o2 v 1 o2 u 1 o2 v ‡ ˆ 0: …1 ‡ m† ‡ …1 ÿ m† oy 2 2 oxoy 2 ox2

…6†

Fig. 2 shows the interaction between the plate and the beam running in the y direction at x ˆ a. The shear at the plate edge acts as a distributed load on the beam, giving the following di€erential equation for bending of the beam:   o4 w Vx Df o 3 w o3 w ; …7a† ˆ ˆ ‡ …2 ÿ m† oy 4 Bfy Bfy ox3 oxoy 2   o2 w Df o3 w o3 w : …7b† ˆÿ ‡ …2 ÿ m† Bsy ox3 oy 2 oxoy 2 The di€erential equation for the bending caused by the moment of the in-plane plate forces acting at an eccentricity from the beam centroid, Nxy ey , as shown in Fig. 2, is as follows.

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

257

Fig. 2. Interacting forces at a beam±slab junction.

o2 w Nxy ey ˆÿ : …8† 2 Bfy oy For plane sections passing through the composite plate and beam to remain plane after bending, the de¯ection, vb , of any point lying on the centroid of the beam in y direction is related to the de¯ection, v, of the plate in y direction by the following expression. vb ˆ v ÿ ey

ow : oy

…9†

258

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Axial force at the centroid of the beam can be calculated by di€erentiating Eq. (9) with respect to y and multiplying with the beam axial sti€ness. This force must be equal and opposite to the inplane force at the edge of the slab panel in the y direction, Nxy , giving,   ov o2 w …10† Nxy ˆ ÿBay ÿ ey 2 : oy oy Substituting the value of Nxy from Eq. (10) into Eq. (8) gives   o2 w Bay ov o2 w ˆ ey ÿ ey 2 : Bfy oy oy 2 oy

…11†

Using continuous and di€erentiable functions for the three displacements, u, v, and w, Eqs. (7b) and (11) may be di€erentiated twice with respect to y and combined with Eq. (7a), giving the following di€erential equation for bending of the beam including the interaction with the plate and the shear deformation in the beam:      5  o4 w Df o3 w o3 w Bay o3 v o4 w Df ow o5 w ‡ ey : ˆ ‡ …2 ÿ m† ÿ ey 4 ÿ ‡ …2 ÿ m† Bfy oy 3 Bsy ox3 oy 2 oy 4 Bfy ox3 oxoy 2 oy oxoy 4 …12† Observing that Nx is assumed zero for the edge beams, if the lateral bending is neglected, and its equal and opposite values cancel for the intermediate beams, the di€erential equation for twisting of the beam may be obtained, as shown by Timoshenko [29],   o3 w Df o2 w o2 w …13† ˆÿ ‡m 2 : Bty ox2 oxoy 2 oy The axial force in beam produced per unit length must equate the axial force in the slab edge per unit length (Nxy ), giving the following governing di€erential equation for axial displacement of the beam.  2    ov o3 w Eh ou ov ÿ ey 3 ˆ ‡ : …14† oy 2 oy 2…1 ‡ m†Bay oy ox In addition to the above three di€erential equations for the beam, the slopes of the two plate segments meeting at an intermediate beam (sti€ener) must be equal and the two axial forces Nx must be equal and opposite.

5. Boundary conditions Chang [4] has derived the boundary conditions at the edge x ˆ a as the following sets of kinetic and kinematic boundary conditions. For a particular type of boundary, only one condition is to be picked out of each set, either force-related (kinetic) or displacement-related (kinematic).   ou o2 w Nx ‡ Bax …15† ÿ ex 2 d…y ÿ yi † ˆ 0 or u ˆ 0; ox ox Nxy ˆ 0 or

v ˆ 0;

…16†

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

 Df

259

    2  o2 w o2 w ou Bfx Bty o3 w 2 o w d…y ÿ yi † ˆ 0 ‡ m 2 ‡ Bax ÿ ex ‡ ‡ ex ‡ Bax ox2 oy ox ox2 Bax oxoy 2

or

ow ˆ0 ox …17†

 Df

    3  o3 w o3 w o2 u Bfx 2 o w ‡ Bax ÿ ex 2 ‡ d…y ÿ yi † ˆ 0; ‡ …2 ÿ m† ‡ ex Bax ox3 oxoy 2 ox ox3

or w ˆ 0; …18†

where

(

d…y ÿ yi † ˆ

0

where there is no beam;

1

if there is beam:

…19†

In addition, at the corners of the plate o2 w ˆ0 oxoy

or w ˆ 0:

…20†

6. Di€erential quadrature equivalents of governing di€erential equations and boundary conditions The variables are normalized using the following nomenclature: X ˆ x=A;

…21†

Y ˆ y=B;

…22†

b ˆ A=B;

…23†

qA4 ; Df

…24†

aˆ

R1 ˆ 1 ‡

e2y Bay ; Bfy

…25†

R2 ˆ

Df B ; Bfy b3

…26†

R3 ˆ

Df B …2 ÿ m†; Bfy b

…27†

R3a ˆ ey Bay B=Bfy ;

…28†

R4 ˆ

…2 ÿ m†Df ; Bsy Bb

…29†

R5 ˆ

Df ; Bsy Bb3

…30†

R9 ˆ

Df B ; bBty

…31†

260

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

R10 ˆ Df Bmb=Bty ;

…32†

R14 ˆ

ey ; B

…33†

R15 ˆ

EhB ; 2…1 ‡ m†Bay

…34†

R16 ˆ

EhB ; 2…1 ‡ m†bBay

…35†

R22 ˆ ex ABax =Df ; ÿ
R23 ˆ Bfx ‡ Bax e2x =Df ;

…36† …37†

R30 ˆ Bax =Da ;

…38†

R31 ˆ Bax ex =Da A;

…39†

R32 ˆ Bty b=Df B:

…40† 2

2

3

3

4

4

Also, denoting the weighting coecients of the operators o=oX ; o =oX ; o =oX ; o =oX ; respectively, by the symbols A1ij ; A2ij ; A3ij ; and A4ij . The same weighting coecients will be valid for the operators o=oY ; o2 =oY 2 ; o3 =oY 3 ; o4 =oY 4 , respectively. The governing di€erential equations, Eqs. (4)±(6), can be expressed in di€erential quadrature form, respectively, as follows: N N X N N X X X 2 4 A4ik Wkj ‡ 2mb A2ik A2jm Wkm ‡ b A4jm Wim ˆ ÿ1 kˆ1

kˆ1 mˆ1

mˆ1

for i; j ˆ 1; 2; . . . ; N …except at boundary points†;

…41†

N N X N N X X X b b2 A2ik Ukj ‡ …1 ‡ m† A1ik A1jm Vkm ‡ …1 ÿ m† A2jm Uim ˆ 0 2 2 kˆ1 kˆ1 mˆ1 mˆ1

for i; j ˆ 1; 2; . . . ; N …except at boundary points†; b2

…42†

N N X N N X X b …1 ÿ m† X A2im Vmj ‡ …1 ‡ m† A1ik A1jm Ukm ‡ A2jk Vik ˆ 0 2 2 mˆ1 kˆ1 mˆ1 kˆ1

for i; j ˆ 1; 2; . . . ; N …except at boundary points†:

…43†

The governing di€erential equations for the sti€eners, Eqs. (12)±(14), which also serve as boundary conditions for the plate segments, become in di€erential quadrature form, respectively, as follows: R1

N N N X N N X X X X A4im Wmj ÿ R2 A3ik Wkj ÿ R3 A1ik A2jm Wkm ‡ R3a A3im Vmj mˆ1

kˆ1

‡ R5

N X N X

kˆ1 mˆ1

N X N X A3ik A2jm Wkm ‡ R4 A1ik A4jm Wkm ˆ 0

kˆ1 mˆ1 N X N X kˆ1 mˆ1

A1ik A2jm Wkm ‡ R9

mˆ1

for i ˆ 1; N ; j ˆ 1; 2; . . . ; N ;

…44†

kˆ1 mˆ1 N X kˆ1

A2ik Wkj ‡ R10

N X A2im Wmj ˆ 0 for i ˆ 1; N; j ˆ 1; 2; . . . ; N ; …45† mˆ1

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

261

N N N N X X X X A2im Vmj ÿ R14 A3im Wmj ÿ R15 A1im Umj ÿ R16 A1ik Vkj ˆ 0 mˆ1

mˆ1

mˆ1

kˆ1

for i ˆ 1; N ; j ˆ 1; 2; . . . ; N :

…46†

The outer sti€ened plate boundary conditions, given by Eqs. (15)±(20), become in di€erential quadrature form, respectively, as follows: " # N N N N X X X X A1ik Ukj ‡ mb A1im Vmj ‡ R30 A1ik Ukj ÿ R31 A2ik Wkj d…y ÿ yi † ˆ 0 kˆ1

mˆ1

kˆ1

kˆ1

or Uij ˆ 0 for i ˆ 1; N ; j ˆ 1; 2; . . . ; N ; N N X 1X A1im Umj ‡ A1ik Vkj ˆ 0 b kˆ1 mˆ1

or Vij ˆ 0

…47† for i ˆ 1; N ; j ˆ 1; 2; . . . ; N

…48†

" N N N N X X X X A2ik Wkj ‡ mb2 A2im Wmj ‡ ÿR22 A1ik Ukj ‡ R23 A2ik Wkj kˆ1 mˆ1 kˆ1 kˆ1 # N X N X A1ik A2jm Wkm d…y ÿ yi † ˆ 0 ‡R32 kˆ1 mˆ1

or

N X

A1ik Wkj ˆ 0 for i ˆ 1; N ; j ˆ 1; 2; . . . ; N ;

…49†

kˆ1

" # N N X N N N X X X X 2 A3ik Wkj ‡ …2 ÿ m†b A1ik A2jm Wkm ‡ ÿR22 A2ik Ukj ‡R23 A3ik Wkj d…y ÿ yi † ˆ 0 kˆ1

kˆ1 mˆ1

kˆ1

or Wij ˆ 0 for i ˆ 1; N ; j ˆ 1; 2; . . . ; N ; N X N X

A1ik A1jm Wkm ˆ 0 or

Wij ˆ 0

for i; j ˆ 1; N :

kˆ1

…50† …51†

kˆ1 mˆ1

Eqs. (41)±(43) for all plate segments between intermediate sti€eners can be grouped together as a single matrix equation representing the whole system. Di€erential quadrature equations for the beam bending, twisting, and axial deformation given by Eqs. (12)±(14) serve the purpose of boundary conditions for the plate segments at the junction with the inner beams and the other plate segments. These conditions along with the slope and transverse axial force compatibility of the two plate elements meeting at inner beams, are substituted in place of the di€erential quadrature equivalent of the governing di€erential equations at the outer two lines of the quadrature points parallel to the beams for each plate segment [5±7,20]. These equations are to be solved for the unknown horizontal and vertical displacements, u, v, and w, for all the plate segments and beam elements at the chosen discrete points so as to satisfy the outer boundary conditions of the system as a whole, which are given by Eqs. (15)±(18). These conditions are satis®ed by replacing the di€erential quadrature equivalents of the governing di€erential equations for the two lines of the quadrature points lying parallel to the boundary of the system as a whole with the above equations. Wherever there is a discontinuity in the type of the boundary, the displacement-related boundary conditions are given preference and only those are incorporated in the ®nal set-up for the solution. The resulting system of equations is solved for the unknown normalized displacements at the discrete points. All the response parameters are now calculated using their quadrature expressions, corresponding weighting coecients, and the normalized displacements at the discrete points.

262

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

7. Results and discussion 7.1. Example 1 A rectangular plate of dimensions A ´ B, without sti€eners and with following boundary conditions is analyzed to check the accuracy and convergence of the method: one of the edge of dimension A is assumed to be free while the opposite edge is assumed to be built-in (®xed); and both the edges of dimension B are assumed to be simply supported. For this example, the aspect ratio, b ˆ A/B, is varied from 10 to 0.1 to see its e€ect on the convergence. Zeros of Chebyshev polynomials, zeros of Legendre polynomials, equally spaced points, and Chebyshev±Gauss±Lobbato points were tried as the quadrature points, and it was observed that using all these types of points, the solution converges to the same value. However, zeros of Chebyshev polynomials provided more rapid convergence in this example and the results are reported for these only. Depending upon the aspect ratio, the number of quadrature points in one direction were varied from 7 to 15 to achieve a practical accuracy (de®ned as a solution when the results did not vary by more than 3%) and from 11 to 21 for the re®ned solutions (de®ned as a solution when the results did not vary by more than 1%) to see the convergence. The results obtained by the quadrature method are shown in Table 1 and compared with the results reported by Timoshenko [29], which are termed as exact. In this table, low-order refers to the solutions with practical accuracy and re®ned refers to the high-order re®ned solutions obtained to see the convergence. As shown in Table 1, eleven number of quadrature points gives results varying only in the last digits, except for very small aspect ratio where 15±21 quadrature points are sucient for convergence. The results with 7±15 quadrature points are also shown, which are reasonably close to the exact results showing very fast convergence. It is further observed that the de¯ections and moments have nearly the same percentage di€erence when compared to the exact solution which is an added advantage of this method over the ®nite element method, which would usually require more elements to be considered to obtain the same percentage accuracy for moments and shears as that for de¯ections. 7.2. Example 2 The second example used to validate the method is the analysis of a uniformly loaded square plate supported on identical edge beams and with a point roller support on four corners. The effect of gradually reducing the beam bending sti€ness on the plate response is studied. The beams are concentric and are taken to have negligible torsional sti€ness but in®nitely large shear sti€ness. The ratio of beam sti€ness to the plate sti€ness normalized along the x direction of the plate, …EI†=…ADf †, is varied from 1020 to 0 representing simple support and free edge with no beams, respectively, as the two extremes. Zeros of Chebyshev polynomials, zeros of Legendre polynomials, equally spaced points, and Chebyshev±Gauss±Lobbato points were tried as the quadrature points, and it was observed that using all these types of points, the solution converges to the same value. However, zeros of Chebyshev polynomials provided more rapid convergence in this example and the results are reported for these only. Seven quadrature points in one direction were found to give reasonable accuracy (de®ned as a solution when the results at the center of the plate did not vary by more than 3%) and eleven quadrature points were found to give a re®ned solution (de®ned as a solution when the results at the center of the plate did not vary by more than 0.5%) and to investigate the convergence trend, except for no-beam case where eleven and twenty one quadrature points were considered, respectively. The results obtained by the quadrature method are shown in Table 2 and compared with the results reported by Timoshenko [29], which are

De¯ections and bending moments for a uniformly loaded rectangular plate with one edge of dimension A free, other edge of dimension A built-in, and the remaining two edges of dimension B simply supported Exact a wmax (qA4 /Df )

Low-order wmax (qA4 /Df )

Re®ned wmax (qA4 /Df )

Mxx at center of free edge (qA2 )

(1)

(2)

(3)

(4)

(5)

1 10 3 2 1.5 1 2/3 1/2 1/3 0.1 0

0.125 ´ 10ÿ4

b

a b

0.00116 0.00364 0.00662 0.0113 0.0141 0.0150 0.0152 0.0152

0.125 ´ 10ÿ4 (7) 0.00114 (7) 0.00357 (7) 0.00646 (7) 0.0115 (7) 0.0141 (9) 0.0154 (9) 0.0155 (11) 0.0148 (15)

b

0.125 ´ 10ÿ4 (11) 0.0116 (11) 0.00364 (11) 0.00663 (11) 0.0112 (11) 0.0142 (11) 0.0150 (11) 0.0152 (15) 0.0152 (21)

Exact

b

Ref. [29]. Numbers in brackets are the number of quadrature points.

a

0 0.0078 0.0293 0.0558 0.0972 0.123 0.131 0.133 0.133

Myy at center of built-in edge (qA2 )

Corresponding to Column (3) (6)

Corresponding to Column (4) (7)

0.35 ´ 10ÿ5 0.0080 0.0293 0.0557 0.1008 0.126 0.135 0.135 0.131

0.20 ´ 10ÿ5 0.0078 0.0293 0.0559 0.0972 0.124 0.131 0.133 0.134

Exact

a

(8) )0.0050 )0.0476 )0.0798 )0.1009 )0.119 )0.124 )0.125 )0.125 )0.125

Corresponding to Column (3) (9)

Corresponding to Column (4) (10)

)0.00481 )0.0470 )0.0786 )0.0994 )0.119 )0.124 )0.123 )0.124 )0.107

)0.0050 )0.0476 )0.0798 )0.1009 )0.118 )0.124 )0.125 )0.125 )0.126

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Table 1

263

264

De¯ections and bending moments for a uniformly loaded square plate supported on identical edge beams resting at four corners (m ˆ 0.25) (EI)/(ADf )

Center of the panel

Mxx at x ˆ 0 face

Exact a w (qA4 /Df )

Low-order w (qA4 /Df )

Re®ned w (qA4 /Df )

Mxx ˆ Myy Exact a (qA2 )

(1)

(2)

(3)

(4)

1020 (1) 100 50 25 10 5 4 3 2 1 0.5 0

0.00406 0.00412 0.00418 0.00429 0.00464 0.00519 0.00546 0.00588 0.00668 0.00873 0.01174 0.0257

0.00406 (7) 0.00412 (7) 0.00418 (7) 0.00430 (7) 0.00466 (7) 0.00523 (7) 0.00550 (7) 0.00594 (7) 0.00676 (7) 0.00885 (7) 0.01185 (7) 0.0250 (11)

a b

b

0.00406 (11) 0.00412 (11) 0.00418 (11) 0.00430 (11) 0.00465 (11) 0.00520 (11) 0.00547 (11) 0.00589 (11) 0.00670 (11) 0.00876 (11) 0.01176 (11) 0.0256 (21)

b

Mxx ˆ Myy Corresponding to Column (4) (qA2 ) (7)

Exact

(5)

Mxx ˆ Myy Corresponding to Column (3) (qA2 ) (6)

0.0460 0.0462 0.0463 0.0467 0.0477 0.0494 0.0502 0.0515 0.0539 0.0601 0.0691 0.1109

0.0460 0.0462 0.0464 0.0468 0.0478 0.0495 0.0504 0.0517 0.0542 0.0604 0.0694 0.1087

0.0461 0.0462 0.0464 0.0468 0.0478 0.0495 0.0503 0.0516 0.0540 0.0601 0.0692 0.1104

Data from Ref. [29], considering ®rst four odd terms of the assumed series for the de¯ection function. Numbers in brackets are the numbers of quadrature points.

a

(qA2 )

Corresponding to Column (3) (qA2 )

Corresponding to Column (4) (qA2 )

(8)

(9)

(10)

0 ) ) 0.0002 0.0024 0.0065 0.0085 0.0117 0.0177 0.0332 0.0559 0.1527

0 0.0005 0.0009 0.0018 0.0044 0.0086 0.0107 0.0139 0.0199 0.0352 0.0571 0.1487

0 0.0004 0.0009 0.0018 0.0043 0.0058 0.0104 0.0136 0.0195 0.0346 0.0566 0.1516

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Table 2

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

265

labeled as the exact solutions. The Timoshenko results are for ®rst four odd terms in the assumed series type de¯ection function and may have relatively more error in moments. In this table, loworder refers to the solutions with reasonable accuracy and re®ned refers to the high-order re®ned solutions obtained to see the convergence. In this table, the results for the central de¯ection and moments are shown. It can be seen that these results are very close to the exact results, and the results rapidly converge as the number of quadrature points are increased. However, the plate bending moments at the center of the edges obtained by the quadrature method vary signi®cantly from the exact result, but the moments at these points is usually very small. The di€erential quadrature results obtained at these points converged to the re®ned results given in the table using all types of quadrature points and by increasing the number of these points. 7.3. Example 3 A simply supported square plate with one sti€ener subjected to uniformly distributed load is considered, which has also been solved by McBean [21]. The dimensions and the material properties are shown in Fig. 3. The beam hangs down from the slab but is monolithically built. Shear deformations of the beam are neglected. Chebyshev±Gauss±Lobatto points are used for the quadrature points and seven quadrature points in one direction provided reasonable accuracy. The results for the eccentric case and the case where beams are considered as concentric, along with the ®nite element results reported by McBean, are shown in Fig. 4. Both the results are practically the same, the maximum di€erence in the central de¯ection for the eccentric case is 2.8%. The central de¯ection of the plate converges as the number of quadrature points are increased, as shown in Fig. 5, for the eccentric case, using the zeros of the Chebyshev polynomials. Maximum di€erence of the converged central de¯ection using various types of quadrature points from the McBean's results is lesser than 6.5%. For the ®nite element results, McBean used 16 elements, each having 32 degrees of freedom, to model one quarter of the plate. In addition, matching number of sti€ener elements were used, each having ten degrees of freedom. The di€erential quadrature results shown in Fig. 4 only required 273 equations to solve the complete problem. Fig. 6 shows the variation of central de¯ection of the McBean's plate as the shear strength of the beam is varied from its original value for the rectangular cross-section. The change in de¯ection is small except for very small

Fig. 3. Simply supported square plate with one sti€ener.

266

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Fig. 4. De¯ected shape of McBean's plate.

Fig. 5. Convergence of central de¯ection of McBean's plate.

shear strength and it almost decreases linearly when shear strength is increased beyond 40% of it's original value. But as the shear strength is reduced beyond 40% of it's original value, the de¯ection increases exponentially. For the results shown in Fig. 6, zeros of Chebyshev polynomials were used, and seven quadrature points in one direction were considered. 7.4. Example 4 A simply supported rectangular plate with one sti€ener in each direction subjected to a uniformly distributed load and a concentrated load at the center is analyzed, which has also been solved by Chang [4]. The dimensions and the material properties are shown in Fig. 7. The beams hang down from the plate, but are monolithically built. The problem is analyzed both for the eccentric case and the case where the two beams are considered concentric. Shear deformations of

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

267

Fig. 6. Variation of central de¯ection of McBean's plate due to reduction in shear strength of the beam.

Fig. 7. Simply supported rectangular plate with one sti€ener in each direction.

the beams are neglected. Chebyshev points are used and the total number of quadrature points is eleven, except for the eccentric case subjected to uniform load where 11 Legendre points are used. The results along the total system center-line and the plate segment center-line (quarter-line of the system) for both the concentric and the eccentric cases subjected to a uniform load, along with the ®nite element results of Chang, are shown in Figs. 8±10. There is a perfect match of the displacements. Fig. 9 shows the plate moment Mxx along half of the shorter beam and along the shorter center-line of one plate-segment. The moment along the beam is up to 45% higher than the values obtained by the ®nite element method. However this pattern is consistent with the results reported by Rossow and Ibrahimkhail [26] and many other researchers [10,22,24]. The moment at the center of one plate quarter is at most 8.5% lower than the ®nite element values. Again this is within the range of the variation of the results published for the moments [10,22,24]. Fig. 10 shows the plate moment Myy along half of the longer beam and along the longer center-line of one

268

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Fig. 8. De¯ection of plate with two sti€eners under uniform load.

Fig. 9. Moment Mxx of plate with two sti€eners under uniform load.

Fig. 10. Moment Myy of plate with two sti€eners under uniform load.

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

269

Fig. 11. De¯ection of plate with two sti€eners under concentrated load.

plate-quarter. The moment along the beam is up to 12.5% higher than the values obtained by the ®nite element method. The moment at the center of one plate quarter is at most 5.5% lower than the ®nite element values. The moment at the quarter length of the long beam has a di€erence of about 51%. All these di€erences are within the range of the variation of the results published for the moments [10,22,24]. For the ®nite element results, 16 plate elements and 8 sti€ener elements were required to model one quarter of the plate. On the other hand, the di€erential quadrature method required 1323 equations for the whole system with the added advantage of possible inclusion of shear strength of the beams and obtaining direct solution at more points. The plate of Fig. 7 is also analyzed for a concentrated central load of thousand pounds acting at the intersection of the two beams. The de¯ection variations along half of the longer side at middle and quarter of the shorter dimension are shown in Fig. 11 for both the concentric and the eccentric cases. The corresponding moments Mxx and Myy are shown in Figs. 12 and 13, respectively. In all these ®gures comparison is made with the ®nite element results reported by Chang [4]. A perfect agreement was found for the de¯ections as well as for the moments. Only seven Chebyshev quadrature points in each direction per plate segment are sucient to get a converged solution, giving a total of 507 simultaneous equations for the whole system. 7.5. Example 5 A six panel concrete slab with mixed boundary conditions but without sti€eners was analyzed using the di€erential quadrature method and ®nite element computer software I-DEAS [14]. The details of the slab including the material properties and loading are shown in Fig. 14. Besides the regular supports, one roller point support is provided at one exterior corner and one at the interior junction of the panels. It is to be noted that this example considers all the possible boundary conditions and complexities. Wherever two di€erent boundary conditions meet at a single point, preference is given to the kinematic boundary conditions and kinetic boundary conditions are discarded at that point. Fig. 15 shows the results for the de¯ection along the line y ˆ 150 in. obtained using both the methods. Only di€erence between the two results is that the ®nite element results failed to provide the high bending sti€ness of the slab at the intermediate roller support probably because the software I-DEAS uses non-conforming thin shell bending elements, which poorly

270

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Fig. 12. Moment Mxx of plate with two sti€eners under concentrated load.

Fig. 13. Moment Myy of plate with two sti€eners under concentrated load.

computed moment Mxy . This is indicated by a relatively sharp change of slope around the point support in the results obtained by the ®nite element, thus predicting higher de¯ections between the supports. Variation of bending moment Mxx along y ˆ 150 in. is shown in Fig. 16 for both the methods. Di€erential quadrature method predicted high concentrations of bending moments at the built-in and the intermediate roller supports showing local instability of the solution. The results within 10 in. on each side of these supports is not shown in Fig. 16. The left span moments, not considering the values very close to the supports, are in close agreement with the ®nite element

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

271

Fig. 14. Six panel concrete slab with mixed boundary conditions.

Fig. 15. De¯ection of six panel slab along y ˆ 150 in.

method, whereas there is di€erence in the bending moments for the right panel. However, the differential quadrature results with portions showing sagging moments on both sides and zero moment (at the singular right support) appears to be more reasonable. Four noded thin shell elements were used for the ®nite element analysis requiring 3692 equations after condensation. Seventeen Chebyshev quadrature points in each direction per panel were used for the di€erential quadrature solution requiring 1617 equations. 8. Conclusions A di€erential quadrature solution for the ¯exural analysis of eccentrically sti€ened plates subjected to transverse uniform loads is presented. In-plane forces in the plate are considered to take

272

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

Fig. 16. Moment Mxx of six panel slab along y ˆ 150 in.

into account the axial sti€ness of the plate and the interaction between the beams and the plate due to the eccentricity. Torsional and shear sti€nesses of the beams are also considered. Point loads may be applied at the corners of the plate segments. Roller point supports may also be provided at these corners of the plate segments. The outer edges between the beams may have di€erent boundary conditions out of the following four types: free, simply supported, clamped, or resting on beams. The method gives nearly same accuracy for the moments and shears as that for the de¯ections, in contrast to the ®nite element method. The results for single panels with complicated boundary conditions are compared with the exact results. Four types of quadrature points, zeros of Chebyshev polynomials, zeros of Legendre polynomials, equally spaced points, and Chebyshev±Gauss±Lobbato points are tried. It is found that using all these types of points, the solution converges to nearly the same values for all the examples, however, the Chebyshev quadrature points provide more rapid convergence for all the problems studied. Once the displacements are known, the matrix expressions for any required force may then be easily developed using the appropriate coecients for the di€erential quadrature analogs of the standard equations relating the plate internal forces to the respective displacement functions and the already calculated de¯ections. It is found that the di€erential quadrature results are equal in accuracy to the ®nite super-elements and better than the lower-order ®nite elements. Notation a A Aij B Bax Bay Bfx Bfy Bsx Bsy Btx Bty Da Df

dimension along x-axis panel length in x-direction weighting coecients for di€erent operators panel length in y-direction axial rigidity of the x-sti€ener axial rigidity of the y-sti€ener ¯exural rigidity of the x-sti€ener ¯exural rigidity of the y-sti€ener shear rigidity of the x-sti€ener shear rigidity of the y-sti€ener torsional rigidity of the x-sti€ener torsional rigidity of the y-sti€ener axial rigidity of the plate (Eh=…1 ÿ m2 †) ¯exural rigidity of the plate (Eh3 =‰12…1 ÿ m2 †Š)

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

ex ey E f(x) h Mxx Mxy Myy N Nx Nxy Ny q Qx Qy R1 R2 R3 R3a R4 R5 R9 R10 R14 R15 R16 R22 R23 R30 R31 R32 u U v V wij w W x X y Y a b d L

273

eccentricity of the sti€ener in x-direction from the plate centroid eccentricity of the sti€ener in y-direction from the plate centroid Young's modulus of elasticity of the plate material function values at the quadrature points thickness of the plate moment per unit length along x-face around y-axis moment per unit length along x-face around x-axis or moment per unit length along y-face around y-axis moment per unit length along y-face around x-axis number of quadrature points per panel in one direction in-plane force per unit length along x-face in x-direction in-plane force per unit length along x-face in y-direction or in-plane force per unit length along y-face in x-direction in-plane force per unit length along y-face in y-direction uniformly distributed load on the plate shear force per unit length acting on x-face shear force per unit length acting on y-face 1 ‡ e2y Bay =Bfy Df B=…Bfy b3 † Df B…2 ÿ m†=…Bfy b† ey Bay B=Bfy …2 ÿ m†Df =…Bsy Bb† Df =…Bsy Bb3 † Df B=…bBty † Df Bmb=Bty ey =B EhB=‰2…1 ‡ m†Bay Š EhB=‰2…1 ‡ m†Bay Š ex ABax =Df …Bfx ‡ Bax e2x †=Df Bax =Da Bax ex =…Da A† Bty =…Df B† displacement of the plate in x-direction u normalized by a displacement of the plate in y-direction v normalized by a weights attached to the function values f(xj ) displacement of the plate in z-direction w normalized by a variable distance along x-direction x/A variable distance along y-direction y/B qA4 /Df aspect ratio of the plate panel (A/B) delta unit function de®ned in the text any di€erential, integral, or combined operator on f(x)

274

o m

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

partial derivative Poisson's ratio of the material

References [1] R.A. Adel Malek, Design formulas for bridge decks, Master's Thesis presented to North Carolina State University at Raleigh, North Carolina (1969). [2] C.W. Bert, S.K. Jang, A.G. Striz, Nonlinear bending analysis of orthotropic rectangular plates by the method of di€erential quadrature, Computational Mechanics 5 (2/3) (1989) 217±226. [3] A. Bj orck, V. Pereyra, Solution of the Vandermonde system of equations, Mathematics of Computation 24 (1970) 893±903. [4] S.P. Chang, Analysis of eccentrically sti€ened plates, Ph.D. Thesis presented to University of Missouri, Columbia (1973). [5] F. Civan, Quadrature and cubature methods for numerical solution of integro-di€erential equations, in: A.H. Sheikh (eds.), Integral Methods in Science and Engineering ± 90, Series in Computational and Physical Processes in Mechanics and Thermal Sciences, Hemisphere, New York, 1991, pp. 282±297. [6] F. Civan, Solving multivariable mathematical models by the quadrature and cubature methods, Numerical Methods for Partial Di€erential Equations 10 (1994) 545±567. [7] F. Civan, Numerical simulation by the quadrature and cubature methods, in: SPE Paper No. 28703, Proceedings of the SPE International Petroleum Conference and Exhibition of Mexico, Veracruz, Mexico, 10±13 October 1994, pp. 353±363. [8] F. Civan, C.M. Sliepcevich, Di€erential quadrature for multidimensional problems, Journal of Mathematical Analysis and Applications 101 (1984) 423±443. [9] F. Civan, C.M. Sliepcevich, Solving integro-di€erential equations by the quadrature methods, in: F.R. Payne, C.C. Corduneanu, A.H. Sheikh, T. Huang (Eds.), Integral Methods in Science and Engineering, Hemisphere, New York, 1986, pp. 106±113. [10] A. Deb, M. Booton, Finite element models for sti€ened plates under transverse loading, Computers and Structures 28 (3) (1988) 361±372. [11] J. Farsa, A.R. Kukreti, C.W. Bert, Fundamental frequency analysis of single spacially orthotropic, generally orthotropic and anisotropic rectangular layered plates by di€erential quadrature method, Computers and Structures 46 (3) (1993) 465±477. [12] H.V.S. Gangarao, A.A. Elmeged, V.K. Chaudhary, Macroapproach for ribbed and grid plate systems, ASCE Journal of Engineering Mechanics 101 (1) (1975) 25±39. [13] I.E. Harik, M. Guo, Finite element analysis of eccentrically sti€ened plates in free vibration, Computers and Structures 49 (6) (1993) 1007±1015. [14] I-DEAS Master Series 1.3c. Structural Dynamics Research Corporation, Milford, Ohio. [15] S.K. Jang, C.W. Bert, A.G. Striz, Application of di€erential quadrature to static analysis of structural components, International Journal for Numerical Methods in Engineering 28 (3) (1989) 561±577. [16] A.R. Kukreti, J. Farsa, C.W. Bert, Fundamental frequency of tapered plates by di€erential quadrature, Journal of Engineering Mechanics 118 (6) (1992) 1221±1238. [17] A.R. Kukreti, Y. Rajapaksa, Analysis procedure for ribbed and grid plate systems used for bridge decks, ASCE Journal of Structural Engineering 116 (2) (1990) 372±391. [18] A.R. Kukreti, Z.A. Siddiqi, Analysis of ¯uid storage tanks including foundation-superstructure interaction using di€erential quadrature method, Journal of Applied Mathematical Modeling 21 (1997) 193±205. [19] R.M. Lin, M.K. Lim, H. Du, De¯ection of plates with nonlinear boundary supports using generalized di€erential quadrature, Computers and Structures 53 (4) (1994) 993±999. [20] M. Malik, Di€erential quadrature method in computational mechanics: New developments and applications, Ph.D. Dissertation, University of Oklahoma, Norman, USA (1994). [21] M.D. McBean, Analysis of sti€ened plates, Ph.D. Thesis presented to Stanford University, California (1968)n. [22] M. Mukhopadhay, Sti€ened plates in bending, Computers and Structures 50 (4) (1994) 541±548. [23] M. Mukhopadhay, Sti€ened plate plane stress elements for the analysis of ships' structures, Computers and Structures 13 (1981) 563±573. [24] G.S. Palani, N.R. Iyer, T.V.S.R. Appa Rao, An ecient ®nite element model for static and vibration analysis of eccentrically sti€ened plates/shells, Computers and Structures 43 (4) (1992) 651±661. [25] S. Peng-Cheng, H. Dade, W. Zongmu, Static, vibration and stability analysis of sti€ened plates using B spline functions, Computers and Structures 27 (1) (1993) 73±78.

Z.A. Siddiqi, A.R. Kukreti / Appl. Math. Modelling 22 (1998) 251±275

275

[26] M.P. Rossow, A.K. Ibrahimkhail, Constraint method analysis of sti€ened plates, Computers and Structures 8 (1978) 51±60. [27] A.H. Sheikh, M. Mukhopadhay, Analysis of sti€ened plate with arbitrary planform by the general spline ®nite strip method, Computers and Structures 42 (1) (1992) 53±67. [28] A.N. Sherbourne, M.D. Pandey, Di€erential quadrature method in the buckling analysis of beams and composite plates, Computers and Structures 40 (1991) 903±913. [29] S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, 1959.