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Post-buckling of cylindrically orthotropic linearly tapered circular plates by finite element method
Computers & Structures Printedin GreatBritain.
Vol. 21,
No. 5. pp. %9-972.
1985 0
0045-7949185 $3.00 + .lN 1985 Pergamon PressLtd.
POST-BUCKLING OF CYLINDRICALLY ORTHOTROPIC LINEARLY TAPERED CIRCULAR PLATES BY FINITE ELEMENT METHOD K. KANAKA Aerospace
RAW and G. VENKATESWARA RAO Division, Vikram Sarabhai Space Center, Trivandrum-69.5022, India
Structures
(Received 2 November 1983) Abstract-The post-buckling behavior of tapered circular plates with cylindrically orthotropic material properties is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and orthotropy parameter. Both simply supposed and clamped boundary conditions are considered.
NOTATION a radius of the circuIar plate c central deflection
Doi2fl
Wd -
we)
E,, Ea Young’s moduli in r and 6 directions [g], [G ] element and assembled geometric stiffness matrices thickness at the center of the plate thickness at the edge of the plate element and assembled elastic stiffness matrices radial coordinate inner and outer radii of the ring elements radial displacement transverse displacement taper parameter generalized
(=
y)
strength to weight ratio. One of the essential requirements is to predict the buckling and post-buckling behavior of such structures to assess the reliability of their design. While analyzing the global characteristics like stability, it is sufficient if these structural components are treated with equivalent orthotropic material properties. In this paper, the study is confined to the buckling and post-buckling behavior of cylindrically orthotropic tapered circular plates, through the finite element formulation developed by the authors recently 11, 21. Linear buckling load parameters and empirical formulas giving radial load ratios in terms of the central deflection to the thickness ratios in the postbuckling range for various taper and orthotropy parameter values are presented in the tables comprehensively for the first time in the literature.
coordinates FINITE ELEMENT FORMULATION
E0
E , orthotropy *
parameter
The circular plate considered ness which is of the form
-
Nr N,.”
h = hfJl - ar), where OLis the taper parameter
$$
is of tapering thick-
(1)
given by
linear buckling load parameter
01 = (h, - h,)/h,,.
curvatures eigenvector circumferential coordinate Poisson’s ratio in r and 8 directions
Cases of positive and negative cy, indicating tapering down from and towards the center, respectively, are considered. The plate with an external compressive load N,. per unit length at the boundary is discretized into N number of ring finite elements whose strain energy is given by
Superscripts - I inverse of the matrix T transpose of the matrix Subscripts L linear NL nonlinear
+ Dl,yt + &x$ + L),~x,.x~}Ydr de.
INTRODUCTION Layered composite in modern aerospace
structures structures
are of frequent use owing to their high
(2)
131
The work done by the external load N, per unit length at the boundary of the ring element is given 969
K. KANAKARAW and G. VENKATESWARA RAO
970
Table 1. Convergence
study of post-buckling results of linearly tapered circular plates
Simply supported -0.2 P
tG*
-
Clamped 0.2
-0.2
0.2
4
8
4
8
4
8
4
8
5.6878 0.2396 0.0036 7.9438 0.2186 0.0030 10.2442 0.2013 0.0025
5.6879 0.2396 0.0036 7.943 1 0.2188 0.0030 10.2428 0.2016 0.0026
2.9779 0.3038 0.0061 3.9993 0.2874 0.005 I 5.0158 0.2720 0.0044
2.9778 0.3036 0.0061 3.9986 0.2875 0.0052 5.0146 0.2720 0.0046
20.7007 0.1806 0.0048 26.3036 0.1738 0.0042 3 I .6948 0.1663 0.0037
20.6916 0.1809 0.0050 26.2811 0.1745 0.0044 3 I .6608 0.1670 0.0039
9.8297 0.2300 0.0068 12.3221 0.2266 0.006 I 14.6906 0.2208 0.0055
9.8246 0.2301 0.0069 12.31 I2 0.2270 0.0063 14.6737 0.221 I 0.0057
* N, number of elements.
by
are the assembled elastic stiffness and geometric stiffness matrices, respectively, i indicating the element number. A is the eigenvalue and (6) is the eigenvector. Using any standard algorithm to extract eigenvalues and eigenvectors, eqn (6) can be solved with appropriate boundary conditions incorporated. The nonlinear terms occuring in the stiffness matrix are determined through a numerical method described in Appendix A.
where RF is the radial
load dist~bution per unit length within the element. Full details of strain-displacement relations and elastic constants are included in Appendix A. Assuming displacement distributions of the form u = al + cq
+ a~?
+ ffd3,
w = ff5 + a6r + a,r’
and through
the standard
+ ad,
principles
~MERICAL RESULTS AND DI~USSION (5)
[3] the element
stiffness [kl and element geometric stiffness [gl matrices are obtained (see Appendix A). After the usual assembly procedure, the matrix equation governing the ~st-buckling behavior of the plate is obtained as
[Kl@I + h[Gl@I = 0,
(6)
where
Employing the above formulation the post-buckling behavior of linearly tapered circular plates, with various tapers and orthotropic properties, is evaluated both for simply supported and clamped boundary conditions. The results are presented in the form of linear buckling load parameter A,(= N,.a*/lAJ and the coefficients Z, 5 of an empirical formula for the radial load ratio y (= h,&~) evaluated through a least square method (from the y values obtained for various c/ho ratios by the numerical method given in the Appendix) in the form y = 1 + iifc/ho)* + T;(c/h,j4.
i= 1
\
Table 1 gives the results of a convergence study made with four- and eight-element idealizations of the plates for the simply supported and clamped plates for three values of orthotropy parameter 6 (1 .O, 1.4, and I .8) and two values of taper parameter a (-0.2 and 0.2). The convergence of the results
/
and
Table 2. Post-buckling i3
a
(71
results of simply supported linearly tapered circular plates (eight-element
solution).
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
6.5447 0.2257 0.0032 9.2017 0.2048 0.0026 Il.9221 0.1880 0.0023
5.6879 0.2396 0.0036 7.943 1 0.2188 0.0030 10.2428 0.2016 0.0026
4.9070 0.2544 0.0041 6.7996 0.2340 0.003s 8.7203 0.2167 0.0029
4.1978 0.2702 0.0047 5.7650 0.2506 0.0040 7.3468 0.2334 0.0034
3.5562 0.2866 0.0053 4.8334 0.2684 0.0046 6.1142 0.2517 0.0039
2.9778 0.3036 0.0061 3.9986 0.2875 0.0052 5.0146 0.2720 0.0046
2.4582 0.3209 0.0071 3.2546 0.3078 0.0060 4.0400 0.2942 0.0054
Post-buckling Table 3. Post-buckling
of cylindrically orthotropic
971
plates
results of clamped linearly tapered circular plates (eight-element
solution)
0.0
0.1
0.2
0.3
17.5382
14.6826
12.1149
9.8246
7.8011
0.1809 0.0050 26.2811
0.0054 0.1913 22.2152
0.0058 0.2028 18.5409
0.0063 0.2157 15.2444
0.0069 0.2301 12.3112
0.2462 0.0076 9.7267
0.0041 0.1648 37.1180
0.1745 0.0044 31.6608
0.1854 0.0048 26.7054
0.1976 0.0052 22.2345
0.0057 0.2114 18.2300
0.0063 0.2270 14.6737
0.0071 0.2448 1I .5472
0.0036 0.1571
0.0039 0.1670
0.0042 0.1781
0.1906 0.0046
0.0051 0.2048
0.0057 0.2211
0.2398 0.0064
P
o
-0.3
-0.2
I.0
AL
24.1528
20.6916
1.4
-a6 AL
0.0047 0.1716 30.7524
1.8
-a6 AL -a&
-0.1
can be seen to be very good and all further results are presented for an eight-element idealization. Table 2 gives the results for simply supported tapered circular plates (eight-element solutions) for three values of the orthotropy parameter p (1 .O, 1.4, and 1.8) and for various values of the taper parameter a ranging between -0.3 and 0.3. It can be seen from this table that for each l3 as the taper value increases from - 0.3 to 0.3, the linear buckling load decreases, but the radial load ratio increases (as Z increases). The effect of nonlinearity thus is more pronounced when large values of (Yare considered. Also as l3 increases, the radial load ratio decreases. The nonlinear effects can be seen to be more pronounced when small values of (3 and large values of (Yare considered. In Table 3, the results for clampled plates are presented and they show the same tendencies as the simply supported plates. In this case, as l3 increases, though the linear buckling load increases, the radial load ratio decreases. The effect of nonlinearity, as expected, is less in the case of clamped plates than the simply supported plates. However, the linear buckling loads are larger for clamped plates. In all the cases when the taper is negative, the linear buckling load is more, but the effect of nonlinearity on the load ratios is less. The results for p = 1.0 and (Y = 0.0 corresponding to isotropic circular plates of uniform thickness can be seen to be in very good agreement with the continuum solutions [4]. The present results are further accurate up to terms of (cDz~)~. For all other p values and a. = 0.0 (cylindrically orthotropic circular plates of uniform thickness), the present results are the same as those presented in an earlier paper PI.
REFERENCES
G. Venkateswara Rao and K. Kanaka Raju, A reinvestigation of post-buckling behavior of elastic circular plates using a simple finite element formulation. Comput. Srruct. 17, 233-236 (1983). K. Kanaka Raju and G. Venkateswara Rao, Finite element analysis of post-buckling behavior of cylindrically orthotropic circular plates. Fiber Sci. Technol. 19,145154 (1983). 0. C. Zienkiewicz, Finite Element Method in Engineering Science, McGraw-Hill, New York (1971). J. M. T. Thompson and G. W. Hunt, A General Theory ofElastic Stability. Wiley, New York (1973). APPENDIX A
The strain-displacement relations of a circular plate for axisymmetric conditions are given by
le
= ulr,
xr=
-2,
(AlI
a%
1 aw xe=--- r ar ’ Along with eqns (l)-(5) of the text and eqn (Al), following standard procedures, the element stiffness [k] and the geometric stiffness [g], matrices can be obtained as
WI = lT_‘IT [ 12n 6:’ S*DSr dr dg] [T-t]
(A2)
and
[g] = [Z’-‘lT
[l*” J]:’S%Sr
dr d6] [T-l].
(A3)
The matrices S, D, 5, and T are given below.
Dl = CONCLUDINGREMARKS
A finite element formulation for the post-buckling analysis of linearly tapered circular plates with orthotropic material properties is presented. Nonlinear effects are found to be more pronounced in the case of plates with large positive taper parameter values and small values of orthotropy parameter under simply supported boundary conditions. The formulation presented is quite general and covers a wide ranee of structural elements.
Cl2 -
Cl
C,zw’
C,w’
r
2r2 ()
C,W’ -
( 0
0
2
9+~w.2+3+~u r2
012 r
>
DI r Dl
(A4)
972
K. KANAKARAJU and G. VENKATESWARA RAO at each node and the elements in the matrix D which contain the degrees of freedom explicitly are the nonlinear terms, which are evaluated through a numerical method given below.
where E,h c, = 1 - Y,Ye’
D,
c* = - Eeh
D2 =
1 -
lJ,Y@ ’
c,* = uec,=
=
D,z
I&,
E,h3
12(l - u,ve) ’ Eeh3 12(1 -
= VRD,
Y,V@) ’ = v,Dz,
NUMERICALMETHOD LW
where h varies as given in eqn (I) of the text:
(‘46)
1 2r
3?
0
0
0
0
[Sl =
(‘47) lrltj 0 1
[Tl
I+
00
0
0
34 0
0
0
0
0
00
2rl 0
00
0
0
0
0
0
0
(A@
= 0 00
1
2rZ 3r: 0
0
The degrees of freedom considered are u, u’, W, and W’
The matrix equation (6) of the text is solved by an iterative method with the following steps: (i) The stiffness matrix IKl is obtained in the first steu neglecting all the nonlinea; terms, yielding the linear stiffness matrix [&,I. Using [&I and [G] the linear critical load parameter ALand linear eigenvector {S,} are obtained from eqn (8) by any standard eigenvalue extraction method. (ii) Now, for a specified maximum deflection, c/ho at the center of the plate, the linear eigenvector {&} is scaled up by c/ho times, so that the resultant vector will have a displacement c/ho at the maximum deflection point. (iii) Using the scaled up eigenvector, the nonlinear terms in the stiffness matrix [K] are obtained through numerical integration. (iv) Using the new [K] and [G] and treating the problem as a linear eigenvalue problem, the radial load parameter ANLand the nonlinear eigenvector {&L} are obtained. (v) Steps (ii)are repeated by replacing {&} by {~NL} in step (ii) to obtain a converged radial load parameter to the prescribed accuracy (in the present study, 10m4). (vi) Steps (i)-(v) are repeated for various values of c/h,,
.
Vol. 21,
No. 5. pp. %9-972.
1985 0
0045-7949185 $3.00 + .lN 1985 Pergamon PressLtd.
POST-BUCKLING OF CYLINDRICALLY ORTHOTROPIC LINEARLY TAPERED CIRCULAR PLATES BY FINITE ELEMENT METHOD K. KANAKA Aerospace
RAW and G. VENKATESWARA RAO Division, Vikram Sarabhai Space Center, Trivandrum-69.5022, India
Structures
(Received 2 November 1983) Abstract-The post-buckling behavior of tapered circular plates with cylindrically orthotropic material properties is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and orthotropy parameter. Both simply supposed and clamped boundary conditions are considered.
NOTATION a radius of the circuIar plate c central deflection
Doi2fl
Wd -
we)
E,, Ea Young’s moduli in r and 6 directions [g], [G ] element and assembled geometric stiffness matrices thickness at the center of the plate thickness at the edge of the plate element and assembled elastic stiffness matrices radial coordinate inner and outer radii of the ring elements radial displacement transverse displacement taper parameter generalized
(=
y)
strength to weight ratio. One of the essential requirements is to predict the buckling and post-buckling behavior of such structures to assess the reliability of their design. While analyzing the global characteristics like stability, it is sufficient if these structural components are treated with equivalent orthotropic material properties. In this paper, the study is confined to the buckling and post-buckling behavior of cylindrically orthotropic tapered circular plates, through the finite element formulation developed by the authors recently 11, 21. Linear buckling load parameters and empirical formulas giving radial load ratios in terms of the central deflection to the thickness ratios in the postbuckling range for various taper and orthotropy parameter values are presented in the tables comprehensively for the first time in the literature.
coordinates FINITE ELEMENT FORMULATION
E0
E , orthotropy *
parameter
The circular plate considered ness which is of the form
-
Nr N,.”
h = hfJl - ar), where OLis the taper parameter
$$
is of tapering thick-
(1)
given by
linear buckling load parameter
01 = (h, - h,)/h,,.
curvatures eigenvector circumferential coordinate Poisson’s ratio in r and 8 directions
Cases of positive and negative cy, indicating tapering down from and towards the center, respectively, are considered. The plate with an external compressive load N,. per unit length at the boundary is discretized into N number of ring finite elements whose strain energy is given by
Superscripts - I inverse of the matrix T transpose of the matrix Subscripts L linear NL nonlinear
+ Dl,yt + &x$ + L),~x,.x~}Ydr de.
INTRODUCTION Layered composite in modern aerospace
structures structures
are of frequent use owing to their high
(2)
131
The work done by the external load N, per unit length at the boundary of the ring element is given 969
K. KANAKARAW and G. VENKATESWARA RAO
970
Table 1. Convergence
study of post-buckling results of linearly tapered circular plates
Simply supported -0.2 P
tG*
-
Clamped 0.2
-0.2
0.2
4
8
4
8
4
8
4
8
5.6878 0.2396 0.0036 7.9438 0.2186 0.0030 10.2442 0.2013 0.0025
5.6879 0.2396 0.0036 7.943 1 0.2188 0.0030 10.2428 0.2016 0.0026
2.9779 0.3038 0.0061 3.9993 0.2874 0.005 I 5.0158 0.2720 0.0044
2.9778 0.3036 0.0061 3.9986 0.2875 0.0052 5.0146 0.2720 0.0046
20.7007 0.1806 0.0048 26.3036 0.1738 0.0042 3 I .6948 0.1663 0.0037
20.6916 0.1809 0.0050 26.2811 0.1745 0.0044 3 I .6608 0.1670 0.0039
9.8297 0.2300 0.0068 12.3221 0.2266 0.006 I 14.6906 0.2208 0.0055
9.8246 0.2301 0.0069 12.31 I2 0.2270 0.0063 14.6737 0.221 I 0.0057
* N, number of elements.
by
are the assembled elastic stiffness and geometric stiffness matrices, respectively, i indicating the element number. A is the eigenvalue and (6) is the eigenvector. Using any standard algorithm to extract eigenvalues and eigenvectors, eqn (6) can be solved with appropriate boundary conditions incorporated. The nonlinear terms occuring in the stiffness matrix are determined through a numerical method described in Appendix A.
where RF is the radial
load dist~bution per unit length within the element. Full details of strain-displacement relations and elastic constants are included in Appendix A. Assuming displacement distributions of the form u = al + cq
+ a~?
+ ffd3,
w = ff5 + a6r + a,r’
and through
the standard
+ ad,
principles
~MERICAL RESULTS AND DI~USSION (5)
[3] the element
stiffness [kl and element geometric stiffness [gl matrices are obtained (see Appendix A). After the usual assembly procedure, the matrix equation governing the ~st-buckling behavior of the plate is obtained as
[Kl@I + h[Gl@I = 0,
(6)
where
Employing the above formulation the post-buckling behavior of linearly tapered circular plates, with various tapers and orthotropic properties, is evaluated both for simply supported and clamped boundary conditions. The results are presented in the form of linear buckling load parameter A,(= N,.a*/lAJ and the coefficients Z, 5 of an empirical formula for the radial load ratio y (= h,&~) evaluated through a least square method (from the y values obtained for various c/ho ratios by the numerical method given in the Appendix) in the form y = 1 + iifc/ho)* + T;(c/h,j4.
i= 1
\
Table 1 gives the results of a convergence study made with four- and eight-element idealizations of the plates for the simply supported and clamped plates for three values of orthotropy parameter 6 (1 .O, 1.4, and I .8) and two values of taper parameter a (-0.2 and 0.2). The convergence of the results
/
and
Table 2. Post-buckling i3
a
(71
results of simply supported linearly tapered circular plates (eight-element
solution).
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
6.5447 0.2257 0.0032 9.2017 0.2048 0.0026 Il.9221 0.1880 0.0023
5.6879 0.2396 0.0036 7.943 1 0.2188 0.0030 10.2428 0.2016 0.0026
4.9070 0.2544 0.0041 6.7996 0.2340 0.003s 8.7203 0.2167 0.0029
4.1978 0.2702 0.0047 5.7650 0.2506 0.0040 7.3468 0.2334 0.0034
3.5562 0.2866 0.0053 4.8334 0.2684 0.0046 6.1142 0.2517 0.0039
2.9778 0.3036 0.0061 3.9986 0.2875 0.0052 5.0146 0.2720 0.0046
2.4582 0.3209 0.0071 3.2546 0.3078 0.0060 4.0400 0.2942 0.0054
Post-buckling Table 3. Post-buckling
of cylindrically orthotropic
971
plates
results of clamped linearly tapered circular plates (eight-element
solution)
0.0
0.1
0.2
0.3
17.5382
14.6826
12.1149
9.8246
7.8011
0.1809 0.0050 26.2811
0.0054 0.1913 22.2152
0.0058 0.2028 18.5409
0.0063 0.2157 15.2444
0.0069 0.2301 12.3112
0.2462 0.0076 9.7267
0.0041 0.1648 37.1180
0.1745 0.0044 31.6608
0.1854 0.0048 26.7054
0.1976 0.0052 22.2345
0.0057 0.2114 18.2300
0.0063 0.2270 14.6737
0.0071 0.2448 1I .5472
0.0036 0.1571
0.0039 0.1670
0.0042 0.1781
0.1906 0.0046
0.0051 0.2048
0.0057 0.2211
0.2398 0.0064
P
o
-0.3
-0.2
I.0
AL
24.1528
20.6916
1.4
-a6 AL
0.0047 0.1716 30.7524
1.8
-a6 AL -a&
-0.1
can be seen to be very good and all further results are presented for an eight-element idealization. Table 2 gives the results for simply supported tapered circular plates (eight-element solutions) for three values of the orthotropy parameter p (1 .O, 1.4, and 1.8) and for various values of the taper parameter a ranging between -0.3 and 0.3. It can be seen from this table that for each l3 as the taper value increases from - 0.3 to 0.3, the linear buckling load decreases, but the radial load ratio increases (as Z increases). The effect of nonlinearity thus is more pronounced when large values of (Yare considered. Also as l3 increases, the radial load ratio decreases. The nonlinear effects can be seen to be more pronounced when small values of (3 and large values of (Yare considered. In Table 3, the results for clampled plates are presented and they show the same tendencies as the simply supported plates. In this case, as l3 increases, though the linear buckling load increases, the radial load ratio decreases. The effect of nonlinearity, as expected, is less in the case of clamped plates than the simply supported plates. However, the linear buckling loads are larger for clamped plates. In all the cases when the taper is negative, the linear buckling load is more, but the effect of nonlinearity on the load ratios is less. The results for p = 1.0 and (Y = 0.0 corresponding to isotropic circular plates of uniform thickness can be seen to be in very good agreement with the continuum solutions [4]. The present results are further accurate up to terms of (cDz~)~. For all other p values and a. = 0.0 (cylindrically orthotropic circular plates of uniform thickness), the present results are the same as those presented in an earlier paper PI.
REFERENCES
G. Venkateswara Rao and K. Kanaka Raju, A reinvestigation of post-buckling behavior of elastic circular plates using a simple finite element formulation. Comput. Srruct. 17, 233-236 (1983). K. Kanaka Raju and G. Venkateswara Rao, Finite element analysis of post-buckling behavior of cylindrically orthotropic circular plates. Fiber Sci. Technol. 19,145154 (1983). 0. C. Zienkiewicz, Finite Element Method in Engineering Science, McGraw-Hill, New York (1971). J. M. T. Thompson and G. W. Hunt, A General Theory ofElastic Stability. Wiley, New York (1973). APPENDIX A
The strain-displacement relations of a circular plate for axisymmetric conditions are given by
le
= ulr,
xr=
-2,
(AlI
a%
1 aw xe=--- r ar ’ Along with eqns (l)-(5) of the text and eqn (Al), following standard procedures, the element stiffness [k] and the geometric stiffness [g], matrices can be obtained as
WI = lT_‘IT [ 12n 6:’ S*DSr dr dg] [T-t]
(A2)
and
[g] = [Z’-‘lT
[l*” J]:’S%Sr
dr d6] [T-l].
(A3)
The matrices S, D, 5, and T are given below.
Dl = CONCLUDINGREMARKS
A finite element formulation for the post-buckling analysis of linearly tapered circular plates with orthotropic material properties is presented. Nonlinear effects are found to be more pronounced in the case of plates with large positive taper parameter values and small values of orthotropy parameter under simply supported boundary conditions. The formulation presented is quite general and covers a wide ranee of structural elements.
Cl2 -
Cl
C,zw’
C,w’
r
2r2 ()
C,W’ -
( 0
0
2
9+~w.2+3+~u r2
012 r
>
DI r Dl
(A4)
972
K. KANAKARAJU and G. VENKATESWARA RAO at each node and the elements in the matrix D which contain the degrees of freedom explicitly are the nonlinear terms, which are evaluated through a numerical method given below.
where E,h c, = 1 - Y,Ye’
D,
c* = - Eeh
D2 =
1 -
lJ,Y@ ’
c,* = uec,=
=
D,z
I&,
E,h3
12(l - u,ve) ’ Eeh3 12(1 -
= VRD,
Y,V@) ’ = v,Dz,
NUMERICALMETHOD LW
where h varies as given in eqn (I) of the text:
(‘46)
1 2r
3?
0
0
0
0
[Sl =
(‘47) lrltj 0 1
[Tl
I+
00
0
0
34 0
0
0
0
0
00
2rl 0
00
0
0
0
0
0
0
(A@
= 0 00
1
2rZ 3r: 0
0
The degrees of freedom considered are u, u’, W, and W’
The matrix equation (6) of the text is solved by an iterative method with the following steps: (i) The stiffness matrix IKl is obtained in the first steu neglecting all the nonlinea; terms, yielding the linear stiffness matrix [&,I. Using [&I and [G] the linear critical load parameter ALand linear eigenvector {S,} are obtained from eqn (8) by any standard eigenvalue extraction method. (ii) Now, for a specified maximum deflection, c/ho at the center of the plate, the linear eigenvector {&} is scaled up by c/ho times, so that the resultant vector will have a displacement c/ho at the maximum deflection point. (iii) Using the scaled up eigenvector, the nonlinear terms in the stiffness matrix [K] are obtained through numerical integration. (iv) Using the new [K] and [G] and treating the problem as a linear eigenvalue problem, the radial load parameter ANLand the nonlinear eigenvector {&L} are obtained. (v) Steps (ii)are repeated by replacing {&} by {~NL} in step (ii) to obtain a converged radial load parameter to the prescribed accuracy (in the present study, 10m4). (vi) Steps (i)-(v) are repeated for various values of c/h,,
.