- Email: [email protected]
Post-buckling analysis of moderately thick cylindrically orthotropic circular plates
Computers& Smctwes Vol. 29. No. 4. pp. 125-127, 1988 Priated is Great Britain.
0045-7949/88 53.00 + 0.00 0 1988 Pergamon Press plc
TECHNICAL NOTE POST-BUCKLING ANALYSIS OF MODERATELY THICK CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATES K. KANAKA RAJU and G. VENKATFFWARARAO Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India
(Receiued
17 April 1987)
Abstract-A simple finite element formulation is presented to aid study of the post-buckling behaviour of moderately thick circular plates with cylindrically orthotropic material properties. Linear critical loads and ratios of nonlinear radial loads to linear critical loads are given for various values of the parameters involved.
1.
INTRODUCTION
Post-buckling behaviour of isotropic circular plates has been studied using continuum and finite element methods [l, 21. Using the simple finite element formulation developed by Rao and Raju [3], post-buckling behaviour of isotropic thick circular plates [4] and cylindrically orthotropic circular plates [5] has been studied. Dumir [6] employed the collocation method of solving the associated differential equations and presented results for the post-buckling behaviour of cylindrically orthotropic, moderately thick annular plates. His [6] results for the specific case of moderately thick isotropic circular plates are in excellent agreement with those of the present authors [S]. However, a simple finite element formulation for the title problem is not available in the literature. In this note, the earlier finite element formulation of[3] has been further generalized to consider the effects of cylindrical orthotropy and transverse shear on the post-buckling behaviour of circular plates.
elements and the strain energy LI of a typical element bounded by radii I, and r2 is given by
+D,,xS+~D,~X,X~+D~X;
+$ Ghe;}r where
C,, =A
Eh
l-v,v,
Eeh c,, = l-v,v,
c,*= vg c,, =
v,
c,
(3)
and
2. FINITE ELEMENT FORMULATION
E,h’
The nonlinear strain-displacement relations of a moderately thick circular plate, for the axisymmetric case, are du c,=-&+j
(2)
dr de,
D,, = 12( 1 -
1 dw * z 0
D,=
v, ve)
Esh’ 12(1 -v, vs)
D,, = veD,, = v, D,,
fg = !! r
(1)
.
(4)
In eqns (2)-(4), h is the thickness of the plate, G is the shear modulus, E, and Es are Young’s moduli in the r and 0 directions respectively and v, and v0 are the corresponding Poisson’s ratios. The work done, W, by the external compressive radial load, N,, per unit length applied at the outer edge is given by
where t,, eDare plane strains, 1,. ~0 are curvatures, e,z is the transverse shear strain, r and 0 are radial and circumferential coordinates respectively and c is the shear rotation. The circular plate is discretized into a set of annular plate
where lo, is the radial load distribution in the plate computed by performing prebuckling stress analysis. 725
Technical Note
126
Assuming cubic displacement polynomials for u, w and c as
Table 1. Values of I.‘ and li for simply supported clamped circular plates B
hia
1
0.001 0.05 0.1 0.15
4.0690 4.0580 4.0254 3.7920
0.2848 0.2855 0.2878 0.2917
Clamped z 1, 14.6894 0.2087 14.5453 0.2107 14.1321 0.2167 13.4952 0.227 1
3
0.001 0.05 0.1 0.15
9.8513 9.7974 9.6405 9.3904
0.2963 0.2975 0.3015 0.3083
32.8033 32.2974 30.8778 28.7714
0.2227 0.2253 0.2331 0.2460
5
0.001 0.05 0.1 0.15
15.2893 15.1666 14.8137 14.2618
0.2942 0.2959 0.3014 0.3107
49.4308 48.4202 45.6212 41.5865
0.2223 0.2254 0.2348 0.2495
lo
0.001 0.05 0.1 0.15
28.4008 27.9975 26.8764 25.1982
0.2892 0.2918 0.3004 0.3150
88.8117 85.9663 78.3547 68.0826
0.2167 0.2211 0.2339 0.2568
u=a,+qr+a,rz+a,r3 w = a, + a,r + a$ 4 =cr,+a,,r
+ agr3
+a,,r’+a,,r’,
(6)
where a,, a*. . . are generalized coordinates, and fohowing standard procedures, the element stiffness and geometric stiffness matrices can be obtained as
1 1
[S]r [!?][S] r dr dtJ [T-l]
[S]‘@][S] r dr d@ [T-l].
(7) (8)
The elements of the matrices [E] and [g] above are given in the Appendix and the matrices [S] (connecting the quantities u, duldr, dw/dr , d2w/dq2, [, d[/dr to the generalized coordinates a,-a,r) and T (connecting the nodal degrees of freedom, defined later, to the generalized coordinates al-e& are the usual transformation matrices that can be easily derived, The nodal degrees of freedom considered for the present ring element are u, du/dr, w, dw/dr, [ and dc/dr at each node, with two nodes per element. The final matrix equation governing the post-buckling behaviour of the plates is obtained as ]Kl{S) +A[Gl@) =O,
(9)
where [K] and [G] are the assembled elastic stiffness (nonlinear) and geometric stiffness matrices, ,I is the radial load parameter (N,a’/D,,) and {a} is the eigenvector, a being the radius of the plate. Equation (9) is the standard eigenvalue problem except that [K], being a nonlinear matrix, has elements containing unknown values of a, du/dr and dw/dr. A numerical iterative scheme, described in the next section, is employed to evaluate the nonlinear terms and, in conjunction with a standard eigenvalue extraction scheme, enables (9) to be solved. 3.
ITERATIVE
SOLUTION
SCHEME
The
matrix eqn (9) above is solved by an iterative numerical method with the following steps. (i) The stiffness matrix [I(] is obtained in the first step neglecting all the nonlinear terms, yielding the linear stiffness matrix [KL]. Using [KL] and [Cl, the linear critical load parameter ,IL and the linear eigenvector {S,} can be obtained from eqn (9) by any standard eigenvalue extraction algorithm. (ii) For a specified maximum deflection c/h, at the centre of the plate, the linear eigenvector is scaled up by c/h times, so that the resultant vector will have a displacement c/h at the maximum deflection point. (iii) Using the scaled up eigenvector, the nonlinear terms in the stiffness matrix [K] are obtained. (iv) Using the new [K] and [G] and treating the problem as a linear eigenvalue problem, the radial load parameter ,I, and the nonlinear eigenvector {S,) are obtained. (v) Steps (ixiv) are repeated by replacing (&I by {S, 3 in step (ii) and iterations continued till a converged radial load parameter I,, is obtained to a prescribed accuracy (in the present study, the accuracy is of order 10e4). (vi) Steps (i)-(v) are repeated for various values of c/h. 4. NUMERICAL RESULTS DISCUSSION The
AND
iterative scheme given above is used to obtain the
Simply supported L? 1,
and
linear stability parameter I, (N,_aq/D,,, where N,_ is the critical load) and the radial load parameter L,, ( = N,aq/D,, ) for cylindrically orthotropic circular plates with the values of the orthotropy parameter /I( =&/E,) taken as 1,3,5 and 10. As in the study of Dumir [6], the larger Poisson’s ratio is taken as 0.25 and G/E,,, as 0.4. The thickness to radius ratio (h/u ) of the plates is considered to vary as 0.001 (thin plate), 0.05, 0.1 and 0.15. L, values are obtained for Central deflection to thickness ratio (c/h) varying between 0.0 and 1.0 in steps of 0.2, and based on these values an empirical formula is obtained for l,.,,/,IL through a least squares fit as
(10) It has been observed that bc
(iv) As /I increases from 1 to 10, rl increases initially and then decreases for h/a values of less than or equal to 0.1, however a values increase monotonically for h/a = 0.15. It can be further seen that the percentage increase is greater in the case of a simply supported plate than in that of a clamped plate as far as I, variation with /I increasing is considered. However, all the other variations given in observations (ii)above are found to be more pronounced in the case of clamped plates than in the case of simply supported plates. REFERENCES
1. J. M. T. Thompson and G. W. Hunt, A General Theury of Elastic Stabifity. John Wiley, London (1973).
Technical Note L. C. Wellford and G. M. Dib, Finite element methods for nonlinear eigenvalue problems and the postbuckling behaviour of elastic plates. Compur. Struct. 6, 413-418 (1976). G. V. Rao and K. K. Raju, A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation. Comput. Sfruct. 17, 233-236 (1983). K. K. Raju and G. V. Rao, Finite element analysis of post-buckling behaviour of cylindrically orthotropic circular plates. Fibre Sci. Technol. 19, 145-154 (1983). K. K. Raju and G. V. Rao, Post-buckling analysis of moderately thick circular plates. J. appl. Mech. 50, 468470 (1983). 6. P. C. Dumir, Axisymmetric post-buckling of orthotropic tapered thick annular plates. J. appl. Mech. 52, 725-727
(1985).
APPENDIX The non-zero elements of the symmetric matrices [I;] and [g], which are of the order 6 x 6, are as follows:
E,, =
v,Qg 2r i
721
0045-7949/88 53.00 + 0.00 0 1988 Pergamon Press plc
TECHNICAL NOTE POST-BUCKLING ANALYSIS OF MODERATELY THICK CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATES K. KANAKA RAJU and G. VENKATFFWARARAO Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India
(Receiued
17 April 1987)
Abstract-A simple finite element formulation is presented to aid study of the post-buckling behaviour of moderately thick circular plates with cylindrically orthotropic material properties. Linear critical loads and ratios of nonlinear radial loads to linear critical loads are given for various values of the parameters involved.
1.
INTRODUCTION
Post-buckling behaviour of isotropic circular plates has been studied using continuum and finite element methods [l, 21. Using the simple finite element formulation developed by Rao and Raju [3], post-buckling behaviour of isotropic thick circular plates [4] and cylindrically orthotropic circular plates [5] has been studied. Dumir [6] employed the collocation method of solving the associated differential equations and presented results for the post-buckling behaviour of cylindrically orthotropic, moderately thick annular plates. His [6] results for the specific case of moderately thick isotropic circular plates are in excellent agreement with those of the present authors [S]. However, a simple finite element formulation for the title problem is not available in the literature. In this note, the earlier finite element formulation of[3] has been further generalized to consider the effects of cylindrical orthotropy and transverse shear on the post-buckling behaviour of circular plates.
elements and the strain energy LI of a typical element bounded by radii I, and r2 is given by
+D,,xS+~D,~X,X~+D~X;
+$ Ghe;}r where
C,, =A
Eh
l-v,v,
Eeh c,, = l-v,v,
c,*= vg c,, =
v,
c,
(3)
and
2. FINITE ELEMENT FORMULATION
E,h’
The nonlinear strain-displacement relations of a moderately thick circular plate, for the axisymmetric case, are du c,=-&+j
(2)
dr de,
D,, = 12( 1 -
1 dw * z 0
D,=
v, ve)
Esh’ 12(1 -v, vs)
D,, = veD,, = v, D,,
fg = !! r
(1)
.
(4)
In eqns (2)-(4), h is the thickness of the plate, G is the shear modulus, E, and Es are Young’s moduli in the r and 0 directions respectively and v, and v0 are the corresponding Poisson’s ratios. The work done, W, by the external compressive radial load, N,, per unit length applied at the outer edge is given by
where t,, eDare plane strains, 1,. ~0 are curvatures, e,z is the transverse shear strain, r and 0 are radial and circumferential coordinates respectively and c is the shear rotation. The circular plate is discretized into a set of annular plate
where lo, is the radial load distribution in the plate computed by performing prebuckling stress analysis. 725
Technical Note
126
Assuming cubic displacement polynomials for u, w and c as
Table 1. Values of I.‘ and li for simply supported clamped circular plates B
hia
1
0.001 0.05 0.1 0.15
4.0690 4.0580 4.0254 3.7920
0.2848 0.2855 0.2878 0.2917
Clamped z 1, 14.6894 0.2087 14.5453 0.2107 14.1321 0.2167 13.4952 0.227 1
3
0.001 0.05 0.1 0.15
9.8513 9.7974 9.6405 9.3904
0.2963 0.2975 0.3015 0.3083
32.8033 32.2974 30.8778 28.7714
0.2227 0.2253 0.2331 0.2460
5
0.001 0.05 0.1 0.15
15.2893 15.1666 14.8137 14.2618
0.2942 0.2959 0.3014 0.3107
49.4308 48.4202 45.6212 41.5865
0.2223 0.2254 0.2348 0.2495
lo
0.001 0.05 0.1 0.15
28.4008 27.9975 26.8764 25.1982
0.2892 0.2918 0.3004 0.3150
88.8117 85.9663 78.3547 68.0826
0.2167 0.2211 0.2339 0.2568
u=a,+qr+a,rz+a,r3 w = a, + a,r + a$ 4 =cr,+a,,r
+ agr3
+a,,r’+a,,r’,
(6)
where a,, a*. . . are generalized coordinates, and fohowing standard procedures, the element stiffness and geometric stiffness matrices can be obtained as
1 1
[S]r [!?][S] r dr dtJ [T-l]
[S]‘@][S] r dr d@ [T-l].
(7) (8)
The elements of the matrices [E] and [g] above are given in the Appendix and the matrices [S] (connecting the quantities u, duldr, dw/dr , d2w/dq2, [, d[/dr to the generalized coordinates a,-a,r) and T (connecting the nodal degrees of freedom, defined later, to the generalized coordinates al-e& are the usual transformation matrices that can be easily derived, The nodal degrees of freedom considered for the present ring element are u, du/dr, w, dw/dr, [ and dc/dr at each node, with two nodes per element. The final matrix equation governing the post-buckling behaviour of the plates is obtained as ]Kl{S) +A[Gl@) =O,
(9)
where [K] and [G] are the assembled elastic stiffness (nonlinear) and geometric stiffness matrices, ,I is the radial load parameter (N,a’/D,,) and {a} is the eigenvector, a being the radius of the plate. Equation (9) is the standard eigenvalue problem except that [K], being a nonlinear matrix, has elements containing unknown values of a, du/dr and dw/dr. A numerical iterative scheme, described in the next section, is employed to evaluate the nonlinear terms and, in conjunction with a standard eigenvalue extraction scheme, enables (9) to be solved. 3.
ITERATIVE
SOLUTION
SCHEME
The
matrix eqn (9) above is solved by an iterative numerical method with the following steps. (i) The stiffness matrix [I(] is obtained in the first step neglecting all the nonlinear terms, yielding the linear stiffness matrix [KL]. Using [KL] and [Cl, the linear critical load parameter ,IL and the linear eigenvector {S,} can be obtained from eqn (9) by any standard eigenvalue extraction algorithm. (ii) For a specified maximum deflection c/h, at the centre of the plate, the linear eigenvector is scaled up by c/h times, so that the resultant vector will have a displacement c/h at the maximum deflection point. (iii) Using the scaled up eigenvector, the nonlinear terms in the stiffness matrix [K] are obtained. (iv) Using the new [K] and [G] and treating the problem as a linear eigenvalue problem, the radial load parameter ,I, and the nonlinear eigenvector {S,) are obtained. (v) Steps (ixiv) are repeated by replacing (&I by {S, 3 in step (ii) and iterations continued till a converged radial load parameter I,, is obtained to a prescribed accuracy (in the present study, the accuracy is of order 10e4). (vi) Steps (i)-(v) are repeated for various values of c/h. 4. NUMERICAL RESULTS DISCUSSION The
AND
iterative scheme given above is used to obtain the
Simply supported L? 1,
and
linear stability parameter I, (N,_aq/D,,, where N,_ is the critical load) and the radial load parameter L,, ( = N,aq/D,, ) for cylindrically orthotropic circular plates with the values of the orthotropy parameter /I( =&/E,) taken as 1,3,5 and 10. As in the study of Dumir [6], the larger Poisson’s ratio is taken as 0.25 and G/E,,, as 0.4. The thickness to radius ratio (h/u ) of the plates is considered to vary as 0.001 (thin plate), 0.05, 0.1 and 0.15. L, values are obtained for Central deflection to thickness ratio (c/h) varying between 0.0 and 1.0 in steps of 0.2, and based on these values an empirical formula is obtained for l,.,,/,IL through a least squares fit as
(10) It has been observed that bc
(iv) As /I increases from 1 to 10, rl increases initially and then decreases for h/a values of less than or equal to 0.1, however a values increase monotonically for h/a = 0.15. It can be further seen that the percentage increase is greater in the case of a simply supported plate than in that of a clamped plate as far as I, variation with /I increasing is considered. However, all the other variations given in observations (ii)above are found to be more pronounced in the case of clamped plates than in the case of simply supported plates. REFERENCES
1. J. M. T. Thompson and G. W. Hunt, A General Theury of Elastic Stabifity. John Wiley, London (1973).
Technical Note L. C. Wellford and G. M. Dib, Finite element methods for nonlinear eigenvalue problems and the postbuckling behaviour of elastic plates. Compur. Struct. 6, 413-418 (1976). G. V. Rao and K. K. Raju, A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation. Comput. Sfruct. 17, 233-236 (1983). K. K. Raju and G. V. Rao, Finite element analysis of post-buckling behaviour of cylindrically orthotropic circular plates. Fibre Sci. Technol. 19, 145-154 (1983). K. K. Raju and G. V. Rao, Post-buckling analysis of moderately thick circular plates. J. appl. Mech. 50, 468470 (1983). 6. P. C. Dumir, Axisymmetric post-buckling of orthotropic tapered thick annular plates. J. appl. Mech. 52, 725-727
(1985).
APPENDIX The non-zero elements of the symmetric matrices [I;] and [g], which are of the order 6 x 6, are as follows:
E,, =
v,Qg 2r i
721