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Deflection of plates with nonlinear boundary supports using generalized differential quadrature
Pergamon
0045-7949(94)EO198-B
Computers & Srrucrures Vol. 53. No. 4. pp. 993-999. I994 Copyright C:: 1994 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0045.7949/w $7.00 + 0.00
DEFLECTION OF PLATES WITH NONLINEAR BOUNDARY SUPPORTS USING GENERALIZED DIFFERENTIAL QUADRATURE R. M. Lin, M. K. Lim and H. Du School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263, Republic of Singapore (Received 20 April 1993) Abstract-A global method of generalized differential quadrature has been presented in this paper to solve the problems of the dektions of plates with genera1 nonlinear elastic boundary supports. The proposed method has been found to be computationally very efficient and numerically very accurate, especially in the case where nonlinear problems are to be solved. The weighting coefficientsfor the approximation of the derivatives required in differential quadrature formulation are calculated in a very simple way yet without any restriction on the choice of grid points. Numerical implementations of the method is strai~tfo~ard and different boundary conditions can be easily incorporate. Numerical results demonstrate the great potential of the method for nonlinear analysis of structural components.
1. INTRODUCTION Physical problems are often described by partial differential equations, either linear or nonlinear and in most cases, their closed form solutions are extremely difficult (if they exist at all) to establish. As a result, approximate numerical methods have been widely used to solve partial differential equations which arise in almost all engineering disciplines. The most commonly used numerical methods for such applications are the finite element method and the finite difference method and nowadays, most engineering problems can be solved by finite element or finite difference methods to satisfactory accuracy if a proper and sufficient number of grid points are used. However, in a large number of practical applications where only reasonably accurate solutions at few specified physical coordinates are of interest, the finite element method or the finite difference method becomes inapprop~ate since they still require a large number of grid points and so large a computer capacity, especially in the cases of nonlinear problems where iteration becomes inevitable. More recently, spectra1 and pseudospectral methods have provided attractive techniques for the solutions of smooth engineering problems, using much fewer grid points. Among the family of these methods, the Chebyshev pseudospectral method [l] is commonly used to solve fluid problems. In seeking a more efficient numerical method which requires fewer grid points yet achieves acceptable accuracy, the method of differential quadrature, which is based on the assumptions that the partial derivatives of a function in one direction can be expressed as a linear combination of the function values at all mesh points 993
along that direction, was introduced by Bellman et al. [2,3]. Since then, applications of the differential quadrature method to various engineering problems have been investigated and their successes have demonstrated the potential of the method as an attractive numerical analysis technique [4-l I]. However, there do exist some major difficulties in the application of the original method of differential quadrature proposed by Bellman et ai. [2,3]. Obviously, the key issue regarding the application of the method of differential quadrature is to determine the weighting coefficients for any order partial derivatives. For the weighting coefficients of the first order derivatives, two techniques were suggested by Bellman et al. [2,3]. One is to solve a set of linear algebraic equations. Unfortunately however, when the number of grid points becomes reasonably large, the coefficient matrix of the linear algebraic equation system tends to be ill-conditioned and its accurate inverse becomes difficult to obtain. This restricts the range of practical applicability of the method of differential quadrature. The other technique is to compute the weighting coefficients by a simple algebraic’ formulation, but with the necessity that the coordinates of the grid points have to be chosen as the roots of shifted Legendre polynomials. This means that if the number of grid points is specified, the distributions of grid points are the same for different physical problems and different boundary conditions. In order to overcome these difficulties, a method of generalized differential quadrature was developed by Shu and Richards and has been applied to solve some problems in fluid dynamics [12, 131. Preliminary results have demonstrated the effectiveness and efficiency of the method.
994
R. M. Lin er OL
This paper presents the generalized differential quadrature method and investigates its applications to the solutions of nontinear structural problems. The specific problem to be solved in this paper is the deflection of plates with general nonlinear elastic boundary supports. Such problems arise in engineering practice where plates or plate-like structures rest on elastic supports whose stiffness characteristics are nonlinear. By applying the proposed generalized differential quadrature method to such nonlinear problems, the difficulties encountered in the original quadrature have been overcome and the weighting coeffCents are given by simple numerical expressions, both for the first order and higher order partial derivatives. It will be shown that the method is very easy COimplement numerically and is computationally very efficient. Numerical examples demonstrate the practical applicability of the proposed method.
3. GENERALIZED DIFFERENTIAL QUADRATL’RE
The method of differential quadrature is developed based on the assumption that the partial derivative of a function with respect to a space variable at a given discrete point can be expressed as a weighted linear sum of the function values at all discrete points in the domain of that variable. To illustrate the concept. let us consider the first derivative of a one dimensional function f(x). Suppose _yi (i = 1, N) are the grid
points obtained by subdividing the x-variable into N discrete values, then the first derivative ZJ(x)/& at x =xi can be writlen as,
where ~1:~ are the weighting coefficients for first derivatives. The key issue regarding the applications of bifferential quadrature is how to determine accurately and efficiently the weighting c&Ficients required. In the original formulation of differential quadrature [2, 31, twu approaches were proposed. One assumes the test fUnCtiftnSgk(x) to begA = n’(k = I, N - I), lad& ing to a set of linear algebraic equations from which the weighting co&Gents can be determined. However, it has ken Tuuud that the linear algebraic equation system thus formulated becomes ill-conditioned when N is large. The other approach asfumes the test function to he an Nth order Legendre polynomial9 leading to simple algebraic expression for the weighting coefficients. However, it requires that X, (i = 1, N) have lo he chosen to be the roots of shifted Legendre polynomial. This means that once the number of grid points N is specified, the roots of the shifted Legendre polynomial are given, thus the distriblltion of the grid pointi is fixed regardless the physical problems being considered .
Ln order to find simple algebraic expressions for the coefficients yet without restricting the choice of grid meshes, a generaked differestial weighting
quadrature method was developed [IS, 131. In a generalized differential quadrature, the test functions are assumed to be the Lagrange interpolated polynomial a5
where
M(x)
=
fl (x -A-,) ,=I
and
Upon substituting (2) into (l), following mlatiunships can be established [l2. 131, M’“(x,)
cil),
‘-’
(s, - x J )M”‘(X)J
forifj,i=l,N c(J ‘I
-
and
j=l,N
M’2’(x,)for i=j,i = l,N.
2M”)(X,)
(3)
(4)
it can be seen that (3) and (4) are very simple algebraic expression for ctj’, yet there is no restriction in the choice of grid coordinates. It is worth mentioning here however that the determination of c!,‘) requires the availability of the second order derivative of M(x} which is more difficult to obtain. instead of using (4) to calculate cC’, a more convenient relationship can lx established to obtain cl,?. It can be shown frum a Taylor series expansion relationship holds for c(l) 7:9
that
the following
Thus, the c.oeticitnts ~1:’ can be calculated as:
Similarly, the weighting coefficients for second and higher order derivatives can be computed. Again, assume the mth order derivative can be expressed as
for
i = 1, N,
for j = I, 2, . . . , N,
(7)
995
Deflection of plates with ncntlinear boundary supports then an amazing recurrence relationship can be established for the mth order weighting coefficients c{y) when Lagrange interpolated polynomials are used as test functions, Cii (m) _
m
C!y-
(
,,-‘),$)
for
xi - xj
i#j,
The determination following,
i=l,N
1 and
j=l,N.
(8)
of ciy) can be done from the
cp) = -jCi,+ic$)
(10)
Y(x) = ,J(x,)p.(X)?
1)
-A--.-
are obtained, it is very easy to determine the function values in the overall domain in terms of polynomial approximation,
for i = 1, N.
(9)
To summarize, the recurrence relationships (8) and (9) together with the formulations for the coefficients of first derivatives (3) and (4) constitute a complete formula for the determination of the weighting coefficients from the first to as high as the (N - 1)th order derivatives. There is no need to solve a set of linear algebraic equations which can be ill-conditioned when the number of grids is reasonably large and there are no restrictions on the coordinates of the chosen grid points. Further, the expressions for the determination of the weighting coefficients are so simple and easy to be implemented numerically. Such features makes the generalized differential quadrature method the most favourable for solving practical problems in structural analysis, especially for nonlinear structural problems. Also, it should be pointed out that the generalized differential quadrature method can be applied to solve both ordinary and partial differential equations. The application of this method to static problems leads to a set of algebraic equations with the function values at grid points as unknowns, while its application to time dependent dynamic problems results in a set of ordinary differential equations with time dependent function values at grid points as unknowns which can then be solved by the existing integration scheme. Finally, once the function values at all grids
where g,(x) are Lagrange interpolated polynomials as expressed in (2). For a multi-dimensional case, each direction can be treated in the same way as in a one-dimensional case and so the above arguments are also applicable. Here, the results for a two-dimensional case are given since they are needed in later analyses. Suppose there are N grid points in the x-direction and M grid points in the y-direction, then we have
i=l,N
and
j=l,M
(11)
i=l,N
and
j=l,M
(12)
i=l,N
and
j=l,M,
(13)
where c$‘)and Er) are the weighting coefficients which are determined in the same way as for the one-dimensional case. 3. PLATES WITH GENERAL NONLINEAR ELASTIC SUPPORTS
The governing equation for a rectangular shown in Fig. 1 under load q(x, y) is,
a% KG@
+d4”=&,
Fig. I. Rectangular plate with general nonlinear elastic support.
ay4
D
y),
plate
(14)
996
R. M. Lin ef al.
where D is the flexural rigidity. Consider general nonlinear elastic supports, then the boundary conditions can be expressed as,
while plates with all edges clamped correspond
to
K:=K;=K;=K:=cc and
I
D($+p$)=K:(y,$$
x =o,
D&
K,=K,=K,=K,=co.
(20)
(15) $+(2-&j
4. APPLICATION
= -K,(y,w)w,
)
where K:( y, w) and K, (y, w) are the nonlinear rotational and translational stiffness along the edge x = 0. Similarly, for other three edges, we have,
D~;+P$)=-K:(Y$)~(16)
OF GENERALIZED QUADRATURE
DIFFERENTIAL
Before applying the generalized differential quadrature, it is necessary to normalize the equations to be solved. Let X =x/a, Y = y/b, W = W//I, p = a/b, where a is the length of the plate, b is the width and h is the thickness, then the governing equation (14) and boundary condition (15) become
x = a, $+(2-p)$
>
=K,(y,w)w
+B4gy=&x,
Y)
(21)
a2w -@+pP’$R: Yg
x = 0,
( >fg
a2w --c(2-p)/32$ ; aX ax2 a
(
= -R,(Y,
W)W,
where
D&+p$)=K;(x,$)t
(17)
y =O, )
I
Dr$+p$)=
Y =b,
7
(22)
= -4(x,
c(
w)w
-K:(x,$)$
and R,(Y, W)=a’K,(y,w)/D.
(18)
)
Y,!$=aKfy.g)/D
=K4(x,w)w.
Applying the generalized differential quadrature, (21)
By properly choosing the values for the rotational and translational stiffnesses, it is obvious that different classical plate support conditions can be simulated. Plates with all edges simply supported correspond to following setting of stiffness values,
becomes,
+fl” 3 CL’!., c$‘W, = a4qijlDh k=l
and K, = K, = K, = K4 = co
(19)
fori=i,N,
and
j=l,N)
(23)
Deflection of plates with nonlinear boundary supports
997
Table 1. Central deflections of C-C-C-C plate at different fl values ~1, =aira- b,r PDQI 0.00127qa’/D
bla 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
wIx-a/29-b/21151
Difference (%)
0.00126qa4/D 0.00150qa4/D
0.78 0.66 0.58 0.00 0.00 0.00 0.00 0.00 0.41 0.00 0.39
0.00151 qa4/D 0.00173 qa4/D 0.00191 qa4/D 0.00207 qa”/D 0.00220 qa4/D 0.00230 qa4/D 0.00238 qa4/D 0.00244 qa4/D 0.00249 qa4/D 0.00253 qa4/D
0.00172 qa4/D 0.00191 qa4/D 0.00207 qa4/D 0.00220 qa4/D 0.00230 qa4/D 0.00238 qa4/D 0.00245 qa4/D 0.00249 qa4/D 0.00254 qa4/D
Table 2. Convergence test of generalized differential quadrature method (/3 = I .O) N
7
9
11
13
15
17
w,(904/D)
1.23438E- 3
1.26649E - 3
1.26537E- 3
1.26531E- 3
1.26533E- 3
1.26533E - 3
and (22) becomes,
Y,,,
z
>zc\yw,
C#Wkj
k=l
k=l
(24) 2 d#W,+(2-/‘)p’
5 c\‘k’2 k=l
?$Wk,,,= -K,(Y,j,
W,j)W,j
In=,
forj=l,N,.
Similarly, generalized differential quadrature can be applied to boundary conditions (16), (17) and (18), leading to a set of overdetermined nonlinear algebraic equations with W, as unknowns which can be solved using existing nonlinear algorithms such as the Newton-Raphson method [14]. In the actual solution process, only the equations in (23) with i = 3, N, - 2 and j = 3, N, - 2 are used and the remainder are from the boundary conditions. To check the accuracy and convergence of the generalized differential quadrature method, a rectangular plate with all edges clamped which corresponds to boundary condition (20) has been studied. When uniform loading q(x,y) = q. and uniform mesh N, = NJ = N = 9 are considered, the central deflections w L =o,2,rEb,2are tabulated in Table 1 with different /l values. Comparison of the results with those from the existing literature shows the accuracy of the generalized differential quadrature method, even when a very coarse mesh (N = 9) is used. To illustrate the convergence, a different number of grid points N are used and the results are shown in Table 2 from which it can be seen that convergence is in general very fast (6 digit accuracy when N = 15). For the general case of a nonlinear elastic support, a square plate (p = 1.0) is investigated. A uniform grid is used with N, = N, = N = 9. All the translational stiffnesses on the four edges are assumed to be the same xi = x2 = k, = & = K and the cubic hard-
ening stiffness is assumed with K to be expressed as, K(W)=&(l+yWZ)
(25)
whose forcedisplacement relationship is shown in Fig. 2. Also, all the rotational stiffnesses on the four edges are assumed to be the same, c=z= G=c= KT with KT expressed as Kr( W’) = Ki(l + yr IV”), where IV’ is the rotational
Fig. 2. Force-displacement._relationship of cubic hardening stmness.
998
R. M. Lin et al
Linear Deflection
Nonlinear Deflection
0.0
1.0
2.0
3.0
4.0
5.0
Uniform Load
6.0
7.0
8.0
9.0
10.0
qa4/Dh
Fig. 3. Nonlinear deflection of a square plate under different loading.
displacement. The nonlinear deflections were solved corresponding to parameter settings K, = 10.0, y = 75.0, Kz = 5.0 and or = 75.0 and the results for the central deflection w lx=a,2J=b,2are shown in Fig. 3. It can be seen that due to the hardening effect of the nonlinear stiffnesses, the deflection becomes much smaller than that in the linear stiffness case, especially when higher loading is considered. To see the deflection pattern of the plate, a case of loading qa4/Dh = 10.0 was considered and the results are shown in Fig. 4. 5. CONCLUDING
REMARKS
In this paper, an improved and generalized differential quadrature method, which has overcome the drawbacks which exist in the originally proposed differential quadrature, has been introduced and applied to solve nonlinear structural problems. The proposed method gives very simple algebraic for-
mulae for the determination of weighting coefficients which are required by differential quadrature approximation yet without restricting the choice of grid mesh. Accurate results can be obtained using the proposed method even when relatively few grid points are considered. As compared with other numerical methods such as the finite element or the finite difference methods, the proposed method requires much less memory storage and computational time. Also, in the formulation of the present method, boundary conditions can be very easily incorporated. For nonlinear problems, convergence has been found to be very fast and only few iterations are usually required for most practical applications. The numerical accuracy and computational efficiency of the proposed method have demonstrated its great potential for wide applications to structural problems, especially to nonlinear structural problems. REFERENCES
1. U. Ehrenstein and R. Peyret. A Chebyshev collocation method for the Navier-Stokes equations with applications to double-diffusive convection. Inf. J. Numer. Meth.
Fluids 9, 421452
(1989).
2. R. E. Bellman and J. Casti. Differential quadrature and long-term integration. J. Math. Anal. Appl. 34, 235-238 (1971).
3. R. Bellman, B. G. Kashef and J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Computat. Phys. 10, 40-52 (1972).
4. R. Bellman and B. G. Kashef. Solution of the partial differential equation of the Hodgkins-Huxley model using differential quadrature. Math. Biosci. 19, l-8 (1974).
5. J. 0. Mingle, The method of differential quadrature for transient nonlinear diffusion, J. Math. Anal. Applic. 71, 403411
Fig. 4. Deflection pattern of a square plate resting on
nonlinear support.
(1979).
6. G. Naadimuthu, R. E. Bellman, K. M. Wang and E. S. Lee, Differential quadrature and partial differential equations: some numerical results. J. Mafh. Anal. Applic. 98, 220-235 (1984).
Deflection of plates with nonlinear boundary supports 7. F. Civan and C. M. Sliepcevich, Application of differential quadrature to transport processes. J. Muth. Anal. Applic. 93, 206221 (1983). 8. F. Civan and C. M. Sliepcevich, Solution of the Poisson equation by differential quadrature. Int. J. Numer. Meth. Engng 19, 711-724 (1983). 9. C. W. Bert, S. K. Jang and A. G. Striz, New methods for analyzing vibration of structural components. AIAA Jnl 26, 612-618 (1988). 10. A. G. Striz, S. K. Jang and C. W. Bert, Nonlinear bending analysis of thin circular plates by differential quadrature. Thin- Walled Struct. 6, 5142 (1988). 11. S. K. Jang, C. W. Bert and A. G. Striz, Application of differentialquadraturetostaticanalysisofstructuralcomponents. Int. J. Numer. Meth. Engng28,561-577 (1989).
999
12. C. Shu and B. E. Richards, High resolution of natural convection in a square cavity by generalized differential quadrature. Proc. 3rd Int. Conf. on Advances in Numer. Methods in Engineering: Theory and Applications, Vol. II, pp. 978-985. Swansea (1990). 13. C. Shu and B. E. Richards, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. ht. J. Numer. Meth. Fluids 15, 791-798 (1992). 14. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes. Cambridge University Press, Cambridge (1986). 15. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd Edn. McGraw-Hill, New York (1959).
0045-7949(94)EO198-B
Computers & Srrucrures Vol. 53. No. 4. pp. 993-999. I994 Copyright C:: 1994 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0045.7949/w $7.00 + 0.00
DEFLECTION OF PLATES WITH NONLINEAR BOUNDARY SUPPORTS USING GENERALIZED DIFFERENTIAL QUADRATURE R. M. Lin, M. K. Lim and H. Du School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263, Republic of Singapore (Received 20 April 1993) Abstract-A global method of generalized differential quadrature has been presented in this paper to solve the problems of the dektions of plates with genera1 nonlinear elastic boundary supports. The proposed method has been found to be computationally very efficient and numerically very accurate, especially in the case where nonlinear problems are to be solved. The weighting coefficientsfor the approximation of the derivatives required in differential quadrature formulation are calculated in a very simple way yet without any restriction on the choice of grid points. Numerical implementations of the method is strai~tfo~ard and different boundary conditions can be easily incorporate. Numerical results demonstrate the great potential of the method for nonlinear analysis of structural components.
1. INTRODUCTION Physical problems are often described by partial differential equations, either linear or nonlinear and in most cases, their closed form solutions are extremely difficult (if they exist at all) to establish. As a result, approximate numerical methods have been widely used to solve partial differential equations which arise in almost all engineering disciplines. The most commonly used numerical methods for such applications are the finite element method and the finite difference method and nowadays, most engineering problems can be solved by finite element or finite difference methods to satisfactory accuracy if a proper and sufficient number of grid points are used. However, in a large number of practical applications where only reasonably accurate solutions at few specified physical coordinates are of interest, the finite element method or the finite difference method becomes inapprop~ate since they still require a large number of grid points and so large a computer capacity, especially in the cases of nonlinear problems where iteration becomes inevitable. More recently, spectra1 and pseudospectral methods have provided attractive techniques for the solutions of smooth engineering problems, using much fewer grid points. Among the family of these methods, the Chebyshev pseudospectral method [l] is commonly used to solve fluid problems. In seeking a more efficient numerical method which requires fewer grid points yet achieves acceptable accuracy, the method of differential quadrature, which is based on the assumptions that the partial derivatives of a function in one direction can be expressed as a linear combination of the function values at all mesh points 993
along that direction, was introduced by Bellman et al. [2,3]. Since then, applications of the differential quadrature method to various engineering problems have been investigated and their successes have demonstrated the potential of the method as an attractive numerical analysis technique [4-l I]. However, there do exist some major difficulties in the application of the original method of differential quadrature proposed by Bellman et ai. [2,3]. Obviously, the key issue regarding the application of the method of differential quadrature is to determine the weighting coefficients for any order partial derivatives. For the weighting coefficients of the first order derivatives, two techniques were suggested by Bellman et al. [2,3]. One is to solve a set of linear algebraic equations. Unfortunately however, when the number of grid points becomes reasonably large, the coefficient matrix of the linear algebraic equation system tends to be ill-conditioned and its accurate inverse becomes difficult to obtain. This restricts the range of practical applicability of the method of differential quadrature. The other technique is to compute the weighting coefficients by a simple algebraic’ formulation, but with the necessity that the coordinates of the grid points have to be chosen as the roots of shifted Legendre polynomials. This means that if the number of grid points is specified, the distributions of grid points are the same for different physical problems and different boundary conditions. In order to overcome these difficulties, a method of generalized differential quadrature was developed by Shu and Richards and has been applied to solve some problems in fluid dynamics [12, 131. Preliminary results have demonstrated the effectiveness and efficiency of the method.
994
R. M. Lin er OL
This paper presents the generalized differential quadrature method and investigates its applications to the solutions of nontinear structural problems. The specific problem to be solved in this paper is the deflection of plates with general nonlinear elastic boundary supports. Such problems arise in engineering practice where plates or plate-like structures rest on elastic supports whose stiffness characteristics are nonlinear. By applying the proposed generalized differential quadrature method to such nonlinear problems, the difficulties encountered in the original quadrature have been overcome and the weighting coeffCents are given by simple numerical expressions, both for the first order and higher order partial derivatives. It will be shown that the method is very easy COimplement numerically and is computationally very efficient. Numerical examples demonstrate the practical applicability of the proposed method.
3. GENERALIZED DIFFERENTIAL QUADRATL’RE
The method of differential quadrature is developed based on the assumption that the partial derivative of a function with respect to a space variable at a given discrete point can be expressed as a weighted linear sum of the function values at all discrete points in the domain of that variable. To illustrate the concept. let us consider the first derivative of a one dimensional function f(x). Suppose _yi (i = 1, N) are the grid
points obtained by subdividing the x-variable into N discrete values, then the first derivative ZJ(x)/& at x =xi can be writlen as,
where ~1:~ are the weighting coefficients for first derivatives. The key issue regarding the applications of bifferential quadrature is how to determine accurately and efficiently the weighting c&Ficients required. In the original formulation of differential quadrature [2, 31, twu approaches were proposed. One assumes the test fUnCtiftnSgk(x) to begA = n’(k = I, N - I), lad& ing to a set of linear algebraic equations from which the weighting co&Gents can be determined. However, it has ken Tuuud that the linear algebraic equation system thus formulated becomes ill-conditioned when N is large. The other approach asfumes the test function to he an Nth order Legendre polynomial9 leading to simple algebraic expression for the weighting coefficients. However, it requires that X, (i = 1, N) have lo he chosen to be the roots of shifted Legendre polynomial. This means that once the number of grid points N is specified, the roots of the shifted Legendre polynomial are given, thus the distriblltion of the grid pointi is fixed regardless the physical problems being considered .
Ln order to find simple algebraic expressions for the coefficients yet without restricting the choice of grid meshes, a generaked differestial weighting
quadrature method was developed [IS, 131. In a generalized differential quadrature, the test functions are assumed to be the Lagrange interpolated polynomial a5
where
M(x)
=
fl (x -A-,) ,=I
and
Upon substituting (2) into (l), following mlatiunships can be established [l2. 131, M’“(x,)
cil),
‘-’
(s, - x J )M”‘(X)J
forifj,i=l,N c(J ‘I
-
and
j=l,N
M’2’(x,)for i=j,i = l,N.
2M”)(X,)
(3)
(4)
it can be seen that (3) and (4) are very simple algebraic expression for ctj’, yet there is no restriction in the choice of grid coordinates. It is worth mentioning here however that the determination of c!,‘) requires the availability of the second order derivative of M(x} which is more difficult to obtain. instead of using (4) to calculate cC’, a more convenient relationship can lx established to obtain cl,?. It can be shown frum a Taylor series expansion relationship holds for c(l) 7:9
that
the following
Thus, the c.oeticitnts ~1:’ can be calculated as:
Similarly, the weighting coefficients for second and higher order derivatives can be computed. Again, assume the mth order derivative can be expressed as
for
i = 1, N,
for j = I, 2, . . . , N,
(7)
995
Deflection of plates with ncntlinear boundary supports then an amazing recurrence relationship can be established for the mth order weighting coefficients c{y) when Lagrange interpolated polynomials are used as test functions, Cii (m) _
m
C!y-
(
,,-‘),$)
for
xi - xj
i#j,
The determination following,
i=l,N
1 and
j=l,N.
(8)
of ciy) can be done from the
cp) = -jCi,+ic$)
(10)
Y(x) = ,J(x,)p.(X)?
1)
-A--.-
are obtained, it is very easy to determine the function values in the overall domain in terms of polynomial approximation,
for i = 1, N.
(9)
To summarize, the recurrence relationships (8) and (9) together with the formulations for the coefficients of first derivatives (3) and (4) constitute a complete formula for the determination of the weighting coefficients from the first to as high as the (N - 1)th order derivatives. There is no need to solve a set of linear algebraic equations which can be ill-conditioned when the number of grids is reasonably large and there are no restrictions on the coordinates of the chosen grid points. Further, the expressions for the determination of the weighting coefficients are so simple and easy to be implemented numerically. Such features makes the generalized differential quadrature method the most favourable for solving practical problems in structural analysis, especially for nonlinear structural problems. Also, it should be pointed out that the generalized differential quadrature method can be applied to solve both ordinary and partial differential equations. The application of this method to static problems leads to a set of algebraic equations with the function values at grid points as unknowns, while its application to time dependent dynamic problems results in a set of ordinary differential equations with time dependent function values at grid points as unknowns which can then be solved by the existing integration scheme. Finally, once the function values at all grids
where g,(x) are Lagrange interpolated polynomials as expressed in (2). For a multi-dimensional case, each direction can be treated in the same way as in a one-dimensional case and so the above arguments are also applicable. Here, the results for a two-dimensional case are given since they are needed in later analyses. Suppose there are N grid points in the x-direction and M grid points in the y-direction, then we have
i=l,N
and
j=l,M
(11)
i=l,N
and
j=l,M
(12)
i=l,N
and
j=l,M,
(13)
where c$‘)and Er) are the weighting coefficients which are determined in the same way as for the one-dimensional case. 3. PLATES WITH GENERAL NONLINEAR ELASTIC SUPPORTS
The governing equation for a rectangular shown in Fig. 1 under load q(x, y) is,
a% KG@
+d4”=&,
Fig. I. Rectangular plate with general nonlinear elastic support.
ay4
D
y),
plate
(14)
996
R. M. Lin ef al.
where D is the flexural rigidity. Consider general nonlinear elastic supports, then the boundary conditions can be expressed as,
while plates with all edges clamped correspond
to
K:=K;=K;=K:=cc and
I
D($+p$)=K:(y,$$
x =o,
D&
K,=K,=K,=K,=co.
(20)
(15) $+(2-&j
4. APPLICATION
= -K,(y,w)w,
)
where K:( y, w) and K, (y, w) are the nonlinear rotational and translational stiffness along the edge x = 0. Similarly, for other three edges, we have,
D~;+P$)=-K:(Y$)~(16)
OF GENERALIZED QUADRATURE
DIFFERENTIAL
Before applying the generalized differential quadrature, it is necessary to normalize the equations to be solved. Let X =x/a, Y = y/b, W = W//I, p = a/b, where a is the length of the plate, b is the width and h is the thickness, then the governing equation (14) and boundary condition (15) become
x = a, $+(2-p)$
>
=K,(y,w)w
+B4gy=&x,
Y)
(21)
a2w -@+pP’$R: Yg
x = 0,
( >fg
a2w --c(2-p)/32$ ; aX ax2 a
(
= -R,(Y,
W)W,
where
D&+p$)=K;(x,$)t
(17)
y =O, )
I
Dr$+p$)=
Y =b,
7
(22)
= -4(x,
c(
w)w
-K:(x,$)$
and R,(Y, W)=a’K,(y,w)/D.
(18)
)
Y,!$=aKfy.g)/D
=K4(x,w)w.
Applying the generalized differential quadrature, (21)
By properly choosing the values for the rotational and translational stiffnesses, it is obvious that different classical plate support conditions can be simulated. Plates with all edges simply supported correspond to following setting of stiffness values,
becomes,
+fl” 3 CL’!., c$‘W, = a4qijlDh k=l
and K, = K, = K, = K4 = co
(19)
fori=i,N,
and
j=l,N)
(23)
Deflection of plates with nonlinear boundary supports
997
Table 1. Central deflections of C-C-C-C plate at different fl values ~1, =aira- b,r PDQI 0.00127qa’/D
bla 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
wIx-a/29-b/21151
Difference (%)
0.00126qa4/D 0.00150qa4/D
0.78 0.66 0.58 0.00 0.00 0.00 0.00 0.00 0.41 0.00 0.39
0.00151 qa4/D 0.00173 qa4/D 0.00191 qa4/D 0.00207 qa”/D 0.00220 qa4/D 0.00230 qa4/D 0.00238 qa4/D 0.00244 qa4/D 0.00249 qa4/D 0.00253 qa4/D
0.00172 qa4/D 0.00191 qa4/D 0.00207 qa4/D 0.00220 qa4/D 0.00230 qa4/D 0.00238 qa4/D 0.00245 qa4/D 0.00249 qa4/D 0.00254 qa4/D
Table 2. Convergence test of generalized differential quadrature method (/3 = I .O) N
7
9
11
13
15
17
w,(904/D)
1.23438E- 3
1.26649E - 3
1.26537E- 3
1.26531E- 3
1.26533E- 3
1.26533E - 3
and (22) becomes,
Y,,,
z
>zc\yw,
C#Wkj
k=l
k=l
(24) 2 d#W,+(2-/‘)p’
5 c\‘k’2 k=l
?$Wk,,,= -K,(Y,j,
W,j)W,j
In=,
forj=l,N,.
Similarly, generalized differential quadrature can be applied to boundary conditions (16), (17) and (18), leading to a set of overdetermined nonlinear algebraic equations with W, as unknowns which can be solved using existing nonlinear algorithms such as the Newton-Raphson method [14]. In the actual solution process, only the equations in (23) with i = 3, N, - 2 and j = 3, N, - 2 are used and the remainder are from the boundary conditions. To check the accuracy and convergence of the generalized differential quadrature method, a rectangular plate with all edges clamped which corresponds to boundary condition (20) has been studied. When uniform loading q(x,y) = q. and uniform mesh N, = NJ = N = 9 are considered, the central deflections w L =o,2,rEb,2are tabulated in Table 1 with different /l values. Comparison of the results with those from the existing literature shows the accuracy of the generalized differential quadrature method, even when a very coarse mesh (N = 9) is used. To illustrate the convergence, a different number of grid points N are used and the results are shown in Table 2 from which it can be seen that convergence is in general very fast (6 digit accuracy when N = 15). For the general case of a nonlinear elastic support, a square plate (p = 1.0) is investigated. A uniform grid is used with N, = N, = N = 9. All the translational stiffnesses on the four edges are assumed to be the same xi = x2 = k, = & = K and the cubic hard-
ening stiffness is assumed with K to be expressed as, K(W)=&(l+yWZ)
(25)
whose forcedisplacement relationship is shown in Fig. 2. Also, all the rotational stiffnesses on the four edges are assumed to be the same, c=z= G=c= KT with KT expressed as Kr( W’) = Ki(l + yr IV”), where IV’ is the rotational
Fig. 2. Force-displacement._relationship of cubic hardening stmness.
998
R. M. Lin et al
Linear Deflection
Nonlinear Deflection
0.0
1.0
2.0
3.0
4.0
5.0
Uniform Load
6.0
7.0
8.0
9.0
10.0
qa4/Dh
Fig. 3. Nonlinear deflection of a square plate under different loading.
displacement. The nonlinear deflections were solved corresponding to parameter settings K, = 10.0, y = 75.0, Kz = 5.0 and or = 75.0 and the results for the central deflection w lx=a,2J=b,2are shown in Fig. 3. It can be seen that due to the hardening effect of the nonlinear stiffnesses, the deflection becomes much smaller than that in the linear stiffness case, especially when higher loading is considered. To see the deflection pattern of the plate, a case of loading qa4/Dh = 10.0 was considered and the results are shown in Fig. 4. 5. CONCLUDING
REMARKS
In this paper, an improved and generalized differential quadrature method, which has overcome the drawbacks which exist in the originally proposed differential quadrature, has been introduced and applied to solve nonlinear structural problems. The proposed method gives very simple algebraic for-
mulae for the determination of weighting coefficients which are required by differential quadrature approximation yet without restricting the choice of grid mesh. Accurate results can be obtained using the proposed method even when relatively few grid points are considered. As compared with other numerical methods such as the finite element or the finite difference methods, the proposed method requires much less memory storage and computational time. Also, in the formulation of the present method, boundary conditions can be very easily incorporated. For nonlinear problems, convergence has been found to be very fast and only few iterations are usually required for most practical applications. The numerical accuracy and computational efficiency of the proposed method have demonstrated its great potential for wide applications to structural problems, especially to nonlinear structural problems. REFERENCES
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Fig. 4. Deflection pattern of a square plate resting on
nonlinear support.
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