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Post-buckling behaviour of stepped circular plates under uniform compression
Pergamon
00457949(94)00343-2
TECHNICAL
Compurrrs & Srrurrures Vol. 54. No. 4. pp. 175-777. 1995 Copyright ‘0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 $9.50 + 0.00
NOTE
POST-BUCKLING BEHAVIOUR OF STEPPED CIRCULAR PLATES UNDER UNIFORM COMPRESSION G. Venkateswara Structural
Design and Analysis
Rao, N. Rajasekhara Naidu and K. Kanaka Raju Division, Structural Engineering Group, Trivandrum-695 022, India (Received
22 August
Vikram
Sarabhai
Space Centre,
1993)
Abstract-A simple, direct finite element formulation is used to study the post-buckling behaviour of stepped circular plates subjected to a uniform compressive load. Both simply supported and clamped boundary conditions are considered. Numerical results are obtained for different step thicknesses and step radii and are presented in the form of tables. The results indicate the sensitiveness of the buckling and post-buckling behaviour of stepped circular plates to the step thickness and step radius.
INTRODUCTION
A simple, direct finite element formulation proposed to study the post-buckling behaviour of elastic isotropic circular plates [1], has been found to yield very accurate results when compared with the continuum solution [2]. This simple, direct formulation has made the consideration of the complicating effects such as shear deformation, orthotropy, taper, elastic foundation, elastic edge restraints, etc., easy and these effects have been studied in a series of papers [3-S]. In the present paper the post-buckling behaviour of a stepped circular plate (Fig. 1) is studied using the proposed finite element formulation. The following sections contain a brief description of the finite element formulation, numerical results and discussion of the problem considered.
FINITE ELEMENT
where i7, is the radial load distribution, obtained from the stress analysis, in the element. Following the general procedures as presented in [9] the element matrices are derived assuming cubic displacement distributions for u and w as
FORMULATION
The strain-displacement relations axisymmetric case, are given by
of a circular
plate for
(1) where L, and t0 are the strains, 1(1,and tiO are the curvatures, u and w are the radial and lateral displacements and r and 0 are the radial and circumferential coordinates, respectively. The circular plate of radius a is idealized with a number of annular ring finite elements. The strain energy U of a typical element bounded by the radii r, and rz is given by cl=;
E is the Young’s modulus, h is the thickness of the plate, v is the Poisson’s ratio (taken as 0.3 in the present study). It is to be noted here that as the present study is concerned with the stepped plate, an appropriate value of thickness h has to be used in eqn (2). The work done W by the external compressive load N, is given by
2n ‘2 [C(c;+c;+2vL,LQ) ssII ,I + D ($3 + JI: +
u = a, + a2r + a,r2 + a4r3
(4)
w = a, + fx6r + u,r* + tlgr3,
(5)
and
where &,-a, are the generalized coordinates, with U, du/dr, w and dw/dr as degrees of freedom per node. The final matrix equation governing the post-buckling behaviour of the plate is given by [K](6)
+L[cl{6}
=o.
(6)
In eqn (6), [K] is the assembled elastic stiffness matrix and [C] is the assembled geometric stiffness matrix, (6) is the eigenvector and I is the eigenvalue. An iterative numerical method, which is briefly described in the next section is used to solve eqn (6) to obtain
2v$,Wlrd rd 0, (2)
where
and c=_
Eh 1 - Y2’
DC_
Eh’ 12(1 - VZ)’
1,,=!g,
(8)
Technical
Note per unit length. The results (j.L and I,v,) are obtained for values of h,/h equal to 1.4, 1.2. 1.0, 0.8, 0.6 and a,/~ equal to 0.25. 0.5 and 0.75 for both simply supported and clamped plates. The values of I, obtained with an eight-element idealization of the plate agree very well with those given in [IO], after suitably redefining the non-dimensional parameter, for the parameters h, /h and u, /a considered in that reference. Hence all the numerical results presented in this paper are obtained using an eight-element idealization of the plate. i.,, values are presented in the form of the load ratio y (defined as y = I,v,/i.,). for which an empirical formula is obtained through a least square curve fitting technique as
Nr per unit length
\
Nr*Nr t--a----I Fig. 1. A stepped
circular
plate
where I, is the critical load parameter, i,v, is the post-buckling load parameter and N,,, is the critical externally applied radial load per unit length.
NUMERICAL
METHOD
The iterative method followed to solve eqn (6) is briefly described in the following steps: 1. In the first step the stiffness matrix [K] is obtained by neglecting all the non-linear terms to yield the linear stiffness matrix [KL]. Using [KL] and [G] the linear critical load parameter 1, and linear eigenvector {S,} are obtained by solving eqn (6). 2. The onset of post-buckling occurs when the plate deflects laterally for a given central deflection c. i.,, corresponding to the actual deflection c is computed as follows: for a specified c, the linear eigenvector (6,) is scaled up by c times so that the resulting vector will have a lateral displacement c at the maximum deflection point. 3. Using the scaled up eigenvector, the non-linear terms in the stiffness matrix [K] are obtained through numerical integration. 4. With the new [K] and [G ] and treating the problem as a linear eigenvalue problem, the post-buckling load parameter I,“, and the non-linear eigenvector {S,,,} are obtained. 5. Steps l-4 are repeated by replacing (6,) with {dhL} in step 1 to obtain a converged A,, to the prescribed accuracy. say 10d4. 6. Steps l-5 are repeated for different values of c.
NUMERICAL
RESULTS
AND DISCUSSION
Using the above formulation and the numerical method discussed in the previous section the values of the i., and E.,v, of a stepped circular plate are evaluated for various values of the central deflection c. The geometry of the stepped plate is given in Fig. 1. The circular plate is of radius a and the stepped portion extends from the centre of the plate to a radius a,. The plate thickness where there is no step is h and the thickness of the plate at the stepped region is h,. The plate is subjected to a uniform compression of intensity N,
from the values of y obtained for different values of c/h, ranging from 0.0 to 1.0 in steps of 0.2. It is observed that fi
Table
1. Values of 1, and r7 of a simply supported circular
stepped
plate
4th al/a
1.4
1.2
1.0
0.8
0.6
0.25
i., a
4.7017 0.2463
4.4960 0.2536
4.1978 0.2702
3.8096 0.3030
3.4060 0.3502
0.50
1, d
6.4026 0.1574
5.3551 0.1963
4.1978 0.2702
3.1147 0.3933
2.2967 0.5565
0.75
i LL 9.3659 a 0.1136
6.4761 0.1723
4.1978 0.2702
2.5617 0.4392
1.5141 0.7306
stepped
circular
Table 2. Values of i,, and a of a clamped plate
a/u
1.4
1.2
1.o
0.8
0.6
0.25
iL d
16.9950 0.1694
16.0356 0.1794
14.6825 0.2028
13.0626 0.2457
11.6054 0.2977
0.50
E., 22.2682 c? 0.1082
18.3054 0.1470
14.6825 0.2028
II.7079 0.2830
8.9556 0.4945
0.75
i., u
19.6091 0.1418
14.6825 0.2028
9.9702 0.3392
5.3968 0.6967
25.3264 0.1068
Technical The trend in the values of /I, and d can be explained by considering the relative stiffness of the steps (in terms of a, /a and h, /h ) of the stepped circular plates. CONCLUDING
REMARKS
The post-buckling behaviour of stepped, elastic circular plates subjected to a uniform compressive load is studied in this note. It is observed that the size of the step and the extent of the step have a major significance both on the buckling loads and the post-buckling behaviour of stepped circular plates.
REFERENCES
G. Venkateswara Rao and K. Kanaka Raju, A reinvestigation of postbuckling behaviour of elastic circular plates using a simple finite element formulation. Comput. Struct. 17, 233-236 (1983). J. M. T. Thompson and G. W. Hunt, A General Theor) of Elastic Stability. John Wiley, London (1973). K. Kanaka Raju and G. Venkateswara Rao, Postbuckling analysis of moderately thick elastic circular plates. ASME J. Appl. Mech. 50, 468-470 (1983).
Note
171
Raju and G. Venkateswara Rao. Finite 4. K. Kanaka element analysis of postbuckling behaviour of cylindrically orthotropic circular plates. Fibre Sci. Technol. 19, 145-154 (1983). Raju and G. Venkateswara Rao, Post5. K. Kanaka buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation. Comput. Strucf. 18, 118331188 (1984). Rao, Postbuck6. K. Kanaka Raju and G. Venkateswara ling of cylindrically orthotropic linearly tapered circular plates by finite element method. Comput. Struct. 21, 969 -972 (1985). Rao, Postbuck7. K. Kanaka Raju and G. Venkateswara ling of thick circular plates with edges restrained against rotation. AIAA J. 24, 1882-1884 (1986). Naidu, K. Kanaka Raju and G. 8. N. Rajasekhara Venkateswara Rao, Postbuckling behaviour of circular plates on an axisymmetric elastic partial foundation under uniform compressive loads. Compuf. Strucr. 46, 187-190 (1993). 9. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science. McGraw-Hill, New York (1971). 10. Column Research Committee of Japan (Ed.), Handbook of Structural Stability. Corona, Tokyo (1971).
00457949(94)00343-2
TECHNICAL
Compurrrs & Srrurrures Vol. 54. No. 4. pp. 175-777. 1995 Copyright ‘0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 $9.50 + 0.00
NOTE
POST-BUCKLING BEHAVIOUR OF STEPPED CIRCULAR PLATES UNDER UNIFORM COMPRESSION G. Venkateswara Structural
Design and Analysis
Rao, N. Rajasekhara Naidu and K. Kanaka Raju Division, Structural Engineering Group, Trivandrum-695 022, India (Received
22 August
Vikram
Sarabhai
Space Centre,
1993)
Abstract-A simple, direct finite element formulation is used to study the post-buckling behaviour of stepped circular plates subjected to a uniform compressive load. Both simply supported and clamped boundary conditions are considered. Numerical results are obtained for different step thicknesses and step radii and are presented in the form of tables. The results indicate the sensitiveness of the buckling and post-buckling behaviour of stepped circular plates to the step thickness and step radius.
INTRODUCTION
A simple, direct finite element formulation proposed to study the post-buckling behaviour of elastic isotropic circular plates [1], has been found to yield very accurate results when compared with the continuum solution [2]. This simple, direct formulation has made the consideration of the complicating effects such as shear deformation, orthotropy, taper, elastic foundation, elastic edge restraints, etc., easy and these effects have been studied in a series of papers [3-S]. In the present paper the post-buckling behaviour of a stepped circular plate (Fig. 1) is studied using the proposed finite element formulation. The following sections contain a brief description of the finite element formulation, numerical results and discussion of the problem considered.
FINITE ELEMENT
where i7, is the radial load distribution, obtained from the stress analysis, in the element. Following the general procedures as presented in [9] the element matrices are derived assuming cubic displacement distributions for u and w as
FORMULATION
The strain-displacement relations axisymmetric case, are given by
of a circular
plate for
(1) where L, and t0 are the strains, 1(1,and tiO are the curvatures, u and w are the radial and lateral displacements and r and 0 are the radial and circumferential coordinates, respectively. The circular plate of radius a is idealized with a number of annular ring finite elements. The strain energy U of a typical element bounded by the radii r, and rz is given by cl=;
E is the Young’s modulus, h is the thickness of the plate, v is the Poisson’s ratio (taken as 0.3 in the present study). It is to be noted here that as the present study is concerned with the stepped plate, an appropriate value of thickness h has to be used in eqn (2). The work done W by the external compressive load N, is given by
2n ‘2 [C(c;+c;+2vL,LQ) ssII ,I + D ($3 + JI: +
u = a, + a2r + a,r2 + a4r3
(4)
w = a, + fx6r + u,r* + tlgr3,
(5)
and
where &,-a, are the generalized coordinates, with U, du/dr, w and dw/dr as degrees of freedom per node. The final matrix equation governing the post-buckling behaviour of the plate is given by [K](6)
+L[cl{6}
=o.
(6)
In eqn (6), [K] is the assembled elastic stiffness matrix and [C] is the assembled geometric stiffness matrix, (6) is the eigenvector and I is the eigenvalue. An iterative numerical method, which is briefly described in the next section is used to solve eqn (6) to obtain
2v$,Wlrd rd 0, (2)
where
and c=_
Eh 1 - Y2’
DC_
Eh’ 12(1 - VZ)’
1,,=!g,
(8)
Technical
Note per unit length. The results (j.L and I,v,) are obtained for values of h,/h equal to 1.4, 1.2. 1.0, 0.8, 0.6 and a,/~ equal to 0.25. 0.5 and 0.75 for both simply supported and clamped plates. The values of I, obtained with an eight-element idealization of the plate agree very well with those given in [IO], after suitably redefining the non-dimensional parameter, for the parameters h, /h and u, /a considered in that reference. Hence all the numerical results presented in this paper are obtained using an eight-element idealization of the plate. i.,, values are presented in the form of the load ratio y (defined as y = I,v,/i.,). for which an empirical formula is obtained through a least square curve fitting technique as
Nr per unit length
\
Nr*Nr t--a----I Fig. 1. A stepped
circular
plate
where I, is the critical load parameter, i,v, is the post-buckling load parameter and N,,, is the critical externally applied radial load per unit length.
NUMERICAL
METHOD
The iterative method followed to solve eqn (6) is briefly described in the following steps: 1. In the first step the stiffness matrix [K] is obtained by neglecting all the non-linear terms to yield the linear stiffness matrix [KL]. Using [KL] and [G] the linear critical load parameter 1, and linear eigenvector {S,} are obtained by solving eqn (6). 2. The onset of post-buckling occurs when the plate deflects laterally for a given central deflection c. i.,, corresponding to the actual deflection c is computed as follows: for a specified c, the linear eigenvector (6,) is scaled up by c times so that the resulting vector will have a lateral displacement c at the maximum deflection point. 3. Using the scaled up eigenvector, the non-linear terms in the stiffness matrix [K] are obtained through numerical integration. 4. With the new [K] and [G ] and treating the problem as a linear eigenvalue problem, the post-buckling load parameter I,“, and the non-linear eigenvector {S,,,} are obtained. 5. Steps l-4 are repeated by replacing (6,) with {dhL} in step 1 to obtain a converged A,, to the prescribed accuracy. say 10d4. 6. Steps l-5 are repeated for different values of c.
NUMERICAL
RESULTS
AND DISCUSSION
Using the above formulation and the numerical method discussed in the previous section the values of the i., and E.,v, of a stepped circular plate are evaluated for various values of the central deflection c. The geometry of the stepped plate is given in Fig. 1. The circular plate is of radius a and the stepped portion extends from the centre of the plate to a radius a,. The plate thickness where there is no step is h and the thickness of the plate at the stepped region is h,. The plate is subjected to a uniform compression of intensity N,
from the values of y obtained for different values of c/h, ranging from 0.0 to 1.0 in steps of 0.2. It is observed that fi
Table
1. Values of 1, and r7 of a simply supported circular
stepped
plate
4th al/a
1.4
1.2
1.0
0.8
0.6
0.25
i., a
4.7017 0.2463
4.4960 0.2536
4.1978 0.2702
3.8096 0.3030
3.4060 0.3502
0.50
1, d
6.4026 0.1574
5.3551 0.1963
4.1978 0.2702
3.1147 0.3933
2.2967 0.5565
0.75
i LL 9.3659 a 0.1136
6.4761 0.1723
4.1978 0.2702
2.5617 0.4392
1.5141 0.7306
stepped
circular
Table 2. Values of i,, and a of a clamped plate
a/u
1.4
1.2
1.o
0.8
0.6
0.25
iL d
16.9950 0.1694
16.0356 0.1794
14.6825 0.2028
13.0626 0.2457
11.6054 0.2977
0.50
E., 22.2682 c? 0.1082
18.3054 0.1470
14.6825 0.2028
II.7079 0.2830
8.9556 0.4945
0.75
i., u
19.6091 0.1418
14.6825 0.2028
9.9702 0.3392
5.3968 0.6967
25.3264 0.1068
Technical The trend in the values of /I, and d can be explained by considering the relative stiffness of the steps (in terms of a, /a and h, /h ) of the stepped circular plates. CONCLUDING
REMARKS
The post-buckling behaviour of stepped, elastic circular plates subjected to a uniform compressive load is studied in this note. It is observed that the size of the step and the extent of the step have a major significance both on the buckling loads and the post-buckling behaviour of stepped circular plates.
REFERENCES
G. Venkateswara Rao and K. Kanaka Raju, A reinvestigation of postbuckling behaviour of elastic circular plates using a simple finite element formulation. Comput. Struct. 17, 233-236 (1983). J. M. T. Thompson and G. W. Hunt, A General Theor) of Elastic Stability. John Wiley, London (1973). K. Kanaka Raju and G. Venkateswara Rao, Postbuckling analysis of moderately thick elastic circular plates. ASME J. Appl. Mech. 50, 468-470 (1983).
Note
171
Raju and G. Venkateswara Rao. Finite 4. K. Kanaka element analysis of postbuckling behaviour of cylindrically orthotropic circular plates. Fibre Sci. Technol. 19, 145-154 (1983). Raju and G. Venkateswara Rao, Post5. K. Kanaka buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation. Comput. Strucf. 18, 118331188 (1984). Rao, Postbuck6. K. Kanaka Raju and G. Venkateswara ling of cylindrically orthotropic linearly tapered circular plates by finite element method. Comput. Struct. 21, 969 -972 (1985). Rao, Postbuck7. K. Kanaka Raju and G. Venkateswara ling of thick circular plates with edges restrained against rotation. AIAA J. 24, 1882-1884 (1986). Naidu, K. Kanaka Raju and G. 8. N. Rajasekhara Venkateswara Rao, Postbuckling behaviour of circular plates on an axisymmetric elastic partial foundation under uniform compressive loads. Compuf. Strucr. 46, 187-190 (1993). 9. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science. McGraw-Hill, New York (1971). 10. Column Research Committee of Japan (Ed.), Handbook of Structural Stability. Corona, Tokyo (1971).