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Parametric instability characteristics of rectangular plates subjected to localized edge loading (compression or tension)
Cmpurm
00457949(94)E0277-9
& Srrumres Vol. 54. No. I, pp. 73-82. 1995 Copynght t 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved mm-7949195 $9.50 + 0.00
PARAMETRIC INSTABILITY CHARACTERISTICS OF RECTANGULAR PLATES SUBJECTED TO LOCALIZED EDGE LOADING (COMPRESSION OR TENSION) P. J. Deolasi and P. K. Datta Department
of Aerospace
Engineering,
Indian (Received
Institute
of Technology,
Kharagpur
721302,
India
1 June 1993)
Abstract-The parametric instability behaviour of a plate subjected to localized in-plane compressive or tensile periodic edge loading is studied, considering the effects of transverse shear deformation and rotary inertia. Different edge loading cases have been considered. Compression and tension buckling results are discussed. The vibration analysis for this problem shows that for certain parameters of tensile loading, the frequency of the plate initially rises with the load, but then begins to decrease with increasing tension, showing the onset of tension buckling. Dynamic instability behaviour under compressive periodic edge loading shows that the instability regions are influenced by the load band width and its location on the edge. Parametric instability behaviour under tensile periodic edge loading shows that the instability regions appear early for positions of the load near the corner of the edge. Instability behaviour due to different load parameters is discussed.
The situation in which loads are applied over only part of the edges, causing non-uniform initial stresses, has been studied [l-5] for static stability behaviour. The free vibration analysis of a rectangular plate subjected to a pair of oppositely directed in-plane concentrated forces was obtained by Leissa and Ayoub [6]. Structural elements subjected to in-plane periodic forces may induce transverse vibrations, which may be resonant for certain combinations of natural frequency of transverse vibration, the frequency of the in-plane forcing function and magnitude of the in-plane load. The spectrum of values of parameters causing unstable motion is referred to as the regions of dynamic instability or parametric resonance. Bolotin, in his text on dynamic stability [7], gives a thorough review of the problems involving parametric excitation of structural elements. Hutt and Salam [8] used a thin plate finite element model to study the dynamic instability of plates subjected to uniform periodic edge loading. The study of the parametric instability behaviour of plate subjected to localized edge loading is new. Recently, Prabhakara and Datta [9] have studied the effects of localized regions of damage in a plate, causing non-uniform stress distribution, on the dynamic instability behaviour of the plate. The problems discussed so far were concerned with the behaviour of plates subjected to compressive loading. However, plates can also buckle when subjected to forces that are only tensile. If a thin sheet is subjected to eccentric tensile loading on the opposite edges, one observes that the sheet wrinkles at several places away from the application of the load.
NOTATION plate lengths in the x- and y-directions respectively load band width plate flexural rigidity Young’s modulus plate thickness shear correction factor stiffness matrix elastic and geometric stiffness matrices respectively mass matrix edge load critical buckling load nodal displacement vector time deflection of mid-plane of the plate Cartesian coordinates static and dynamic load factors respectively non-dimensional load non-dimensional buckling load non-dimensional frequency Poisson’s ratio rotations of the normal about the K- and y-axes respectively mass density of the plate material initial in-plane stresses angular frequency of in-plane forcing function angular frequency of transverse vibration fundamental frequency of transverse vibration
1. INTRODUCTION
The study of static and dynamic behaviour of plate elements subjected to non-uniform in-plane stresses is of considerable importance to mechanical, aerospace and structural engineers. Large numbers of references are found in the published literature which deal with the buckling and vibration behaviour of rectangular plates subjected to in-plane uniform initial stresses. 73
14
P. J. Deolasi
Such wrinkling (i.e. buckling) can occur whenever the solution to the plane elasticity problem yields internal stresses that are compressive in any direction at any point. Tension buckling of rectangular sheets due to concentrated forces has recently been studied by Leissa and Ayoub [lo]. A plate subjected to periodic localized tensile edge loading will also exhibit dynamic instability phenomena due to the presence of compressive stresses in the plate causing static instability. The present paper deals with the problems of static and dynamic stability behaviour of rectangular plates subjected to partial edge loading. Both the compressive and tensile loading cases have been considered. Finite element formulation is applied for obtaining initial non-uniform stress distribution in the plate and also to solve the buckling and vibration problems. These results are then utilized for determination of the dynamic instability zones, for different loading conditions. 2. ANALYSIS
and P. K. Datta
components, W, 0, and 0,. are specified at each node. Mindlin’s hypothesis is applied to include the effects of transverse shear deformation and rotary inertia. The expressions for the total potential energy and the kinetic energy of the plate are as follows. Total potential energy:
11 dy dx
+jJob {;[~y(~>‘+~fg) +2r,,
( I aw aw
--
ax ay
The problem considered here consists of a rectangular plate (a x b) subjected to in-plane localized edge loading, as shown in Fig. 1, The loading shown is compressive in nature. Reversing the direction of loading will give the corresponding tensile loading problems. The boundaries of the plate are simply supported along all the edges. An eight-noded quadratic isoparametric element is used in the present analysis. Three displacement
(4
HI
dy dx
(1)
(b) P
P
.c-l c
I--
b
a
P
(4
Fig.
P/2
PI2
P/2
P/2
I. Description of the plate problems. (a) Partial edge loading at one end. (b) Concentrated loading. (c) Partial edge loading at the centre. (d) Loading at both ends.
edge
Jnstability of rectangular plates
75
mn, where m denotes the number of approximate
and kinetic energy:
half waves in the x-direction and n denotes the number of approximate half waves in the y-direction. 2.1. Dynamic stability problem For the problems described in Fig. 1, the edge load P is periodic and can be expressed in the form
where D is the plate flexural rigidity and D = Eh’/12 (1 - v2); 6,) ?v and zlr are the initial, in-plane stresses; k is the shear correction factor which accounts for the fact that transverse shear strain is not constant over the plate thickness. (k = 5/6 is used in the present analysis.) Upon assuming suitable interpolation functions for w, 0, and 0,. in eqns (1) and (2) in terms of nodal displacements, the total potential energy and kinetic energy can be written in matrix form as
P(t) = P, + P, cos Rt,
(6)
where P, is the static portion of P, P, is the amplitude of the dynamic portion of P and Q is the frequency of the dynamic portion of P. The stress distribution in the plate will be non-uniform and periodic and any element will be subjected to in-plane stresses of the form a, = a,, + r?,, cos Rt = Lip, + 6P, cos nt a! = fJT,.+ fJrrcos Rt = CP, + JP, cos nt
. (7)
T._ = 5,,, + t,, cos Rt = f?P, +fp, cos Rt i where {q} is the vector of generalized co-ordinates, [K] is the stiffness matrix and [M] is the consistent mass matrix. The stiffness matrix can be split into two parts,
Let P,$= rPe,
[Kl= [KC1- P[S], where [K,] is the elastic stiffness matrix, [S] is the stress stiffness matrix, also called the geometric stiffness matrix, which is obtained from the in-plane stress distribution cry, a!, r’, , and P is magnitude of the edge load. The equation of equilibrium for a plate subjected to in-plane loads can be written as
WI Gi) + Kl +I) - fvl blj = 0.
(3)
Equation (3) can be reduced to the governing equations for buckling and vibration problems as follows:
GlI = 0.
and
P, = ljPCr,
(8)
where CI and b are the percentages of the static buckling load PC,, which is obtained from the solution of eqn (4); a and p are termed as static and dynamic load factors, respectively. Usually, P,, is taken as the reference load. In the present analysis, for compressive loading, the buckling load for a fully loaded plate (c/a = 1.0) is taken as the reference load. Whereas for tensile loading, the tensile buckling load corresponding to concentrated loading at one end of the edge is taken as the reference load. Substituting eqns (7) and (8) in eqn (3), the following equation in matrix form is obtained:
PflGiI + [Kl - ap,.,PJ
(1) Buckling problem, {4} = 0:
KI +I) - pm
In eqn (7), a,, = CP,, CT,,= FP,, and z,,, = FP, are the static portions of o,~, oYand z,,., respec&vely, whereas the other terms are time dependant; G, b; E, I?, C and f are constant multipliers.
(4)
(2) Free vibration problem: assuming that the plate vibrates harmonically with angular frequency o, eqn (3) reduces to
- /I?PCr[s,]cos Qr] {q} = 0. (9) In eqn (9), the matrices [&I and [S,] are given by
(5) Both eqns (4) and (5) represent the eigenvalue problems. The eigenvalues are the critical buckling loads, PC,for eqn (4) or squares of natural frequencies, co2 for a known load P for eqn (5). The eigenvectors correspond to the mode shapes of buckling or vibration. A mode shape is identified by its mode number
in which IS,], [S,.] and [S,,.] are the stability matrices associated with or, uJ. and 7,,., respectively, for an element. Equation (9) represents a system of second order differential equations with periodic coefficients of the Mathieu-Hill type. The boundaries of the dynamic
P. J. Deolasi and P. K. Datta
76 instability regions are formed by solutions of period T and 2T, where T boundaries of the primary instability period 2T) are determined [7,8] from
the periodic = 2lIjQ. The regions (with the equation
Equation (10) represents an eigenvalue problem for known values of CI, /I and P,,. The two conditions under a plus or minus sign correspond to the two boundaries of the dynamic instability region. The eigenvalues are R, which give the boundary frequencies of the instability regions for given values of x and /I. 2.2. Finite element formulation An eight-noded quadratic isoparametric element is present analysis. Three displacement components, W, 0, and 0, are specified at each node. The element matrices can be expressed as: used in the
Elastic stiffness
matrix.
h[B]r[D][B] Geometric
stiffness
[S], =
dx dy
matrix.
Table 1. Comparison of non-dimensional buckling loads, y,, for a plate subjected to concentrated loading at the two opposite edges alb
Present
Leissa and Ayoub [6]
Yamaki [I]
1.0 2.0 0.5
25.720 29.852 28.851
25.814 30.061 26.523
25.7 29.7
into a two dimensional array of rectangular elements. Element elastic stiffness matrices are obtained with 2 x 2 Gauss sampling points to avoid possible shear locking. Element mass matrices are also obtained with 2 x 2 sampling points. The geometric stiffness matrix [S] is essentially a function of the in-plane stress distribution in the element due to applied edge loading. Since the stress field is non-uniform, plane stress analysis is carried out using finite element technique to determine the stresses at Gauss sampling points. Element geometric stiffness matrices are obtained with 3 x 3 Gauss sampling points. Element matrices are assembled into global matrices using skyline technique. Subspace iteration method is adopted throughout to solve the eigenvalue problems (4), (5) and (10). The non-dimensional loads, 7, and non-dimensional frequencies, i, are defined as:
oh ss
Non-dimensional
[BG]r[6][BG] dx d?.
load, ‘; = g
0 0
Consistent
Non-dimensional
mass matrix,
[N]‘m[N]
dx dy.
Further details of the formulation of the element matrices are given in Appendix A. The overall matrices [K,], [S] and [M] in eqn (3) are obtained by assemblies of corresponding element matrices and after elimination of the rows and columns corresponding to zero displacements. The vector of generalized coordinates consists of only active nodal displacements. The advantages of the symmetries of the plate geometry and the in-plane stress distribution are exploited by grouping the buckling and vibration modes into symmetric and antisymmetric modes about the axis or axes of symmetry. Thus only one side of the axis of symmetry needs to be considered. For loading cases shown in Fig. I(a) and (b), only the upper half, and for loading cases shown in Fig. I(c) and (d) only the upper left quarter of the plate are considered. This greatly reduces the computational efforts and memory space requirements.
E. = wb’JphlD.
To check the validity of the program, the buckling and vibration results are compared with those available in the literature. The comparisons are shown in Tables 1 and 2. All the results are obtained with a 10 x 10 mesh for a full plate. 3. RESULTS AND
DISCUSSION
Results have been obtained for the problems described in Fig. 1 for two plate aspect ratios, a/b = 1 and a/b = 2. Figure 2 shows a typical plot of nondimensional compressive buckling load vs c/a for the loading case shown in Fig. I(a) for the fundamental mode. It is observed that for small values of c/a, the Table 2. Comparison of non-dimensional frequencies, I without in-plane load. (Figures within the brackets indicate the results of Leissa and Ayoub [6]) i
alb 1.0
2.3. Computation A computer program has been written to perform all the necessary computations. The plate is divided
frequency,
2.0
Mode 11
Mode 12
Mode 21
Mode 22
19.729 (19.739)
49.148 (49.348)
49.148 (49.348)
78.432 (78.957)
Mode I I
Mode 21
Mode 31
Mode 12
12.282 (12.337)
19.582 (19.739)
31.362 (32.076)
39.398 (41.946)
Instability of rectangular plates 100
r
71
I20
-
*-
c/0=0.0
- - . c/r=0.2 c/r=O.B 80
60
oL---0
0.2
0.4
0.6
0.8
1.0
-0
400
800
Non-dimensional
cla
1200 tensile
load,
1600 y
2. Non-dimensional buckling load vs c/a for partial edge loading at one end.
Fig. 4. Non-dimensional fundamental frequency vs non-dimensional tensile load for partial edge loading at one end (a/b = 1).
buckling load is usually higher because of the edge restraint of the edge x = 0. As c/a increases, the buckling load decreases. However, as c/a approaches unity (corresponding to full load over the edges) the buckling load again increases due to the restraining effect of the other edge, x = a. Figure 3 shows the variation of non-dimensional frequencies with a non-dimensional compressive load for the loading case shown in Fig. l(a), for c/a = 0.2 and c/u = 0.8 and for u/b = 2. It is observed that as the load increases, the frequencies decrease and the fundamental frequencies become zero at the respective values of the buckling load. There is considerable variation in the load-frequency curves as c/u changes. For the same loading case as shown in Fig. l(a), with tensile loading, it is observed that as c/a changes
from 0 to 1, the tensile buckling load increases monotonically and rapidly from minimum to infinity; this is because of the fact that the strength and area of the compressive zone produced due to partial tensile loading reduce as c/u increases. Figure 4 shows the plots of non-dimensional fundamental frequencies vs a non-dimensional tensile load for different values of c/u and for u/b = 1. It is observed that for small values of c/u (c/a = 0.0 and c/u = 0.2) the fundamental frequency initially rises with the tensile load. As the load further increases, the frequency starts falling down and becomes zero at the respective value of the tensile buckling load; this is due to the presence of a substantial compressive zone within the plate under tensile edge loading of small band width. For
c/8=0.8 - - * c/1=0.2
.
higher values of c/u (c/a = 0.8) the fundamental frequencies keep on increasing over a considerable range. For other loading cases described in Fig. 1, the buckling and vibration behaviour have been studied with c/u as a parameter and the results of similar behaviour have been obtained. Figure 5 shows the general buckling and vibration mode shapes for compressive and tensile edge loading problems. It follows from the buckling and vibration analysis that the dynamic instability behaviour of a plate under localized edge loading will be of considerable interest. The dynamic instability behaviour under localized loading is discussed in the following sections. 3.1. Dynamic instability behuviour under compressive loading
0
I
0
20
I\1 40
Non-dimensional
!I 60
80
I
I
I
too
120
140
compressive
load,
y
Fig. 3. Non-dimensional frequencies vs non-dimensional compressive load for partial edge loading at one end (a/b = 2).
3.1.1. Partial edge loading at one end. Figure 6(a) and (b) shows the boundaries of the first four dynamic instability regions for c( = 0.2, s( = 0.6, for c/a = 0.2, c/u = 0.8 and for a/b = 1. It is observed that for small values of c/u (c/a = 0.2), the widths of the
dynamic
instability
regions are usually smaller in
P. J. Deolasi and P. K. Datta
78
(4
(b) Fig. 5. Buckling
and vibration
mode shapes;
(a) compressive
comparison to those for c/a close to unity (c/a = 0.8); this is because of the fact that the restraint of the edge x = 0 has a stabilizing effect on the dynamic instability behaviour of the plate for a small load band width.
edge loading,
(b) tensile edge loading.
As the static load factor, tl increases, approaching the value corresponding to the buckling load of the plate, the widths of the instability regions increase and the regions shift inwards on the frequency ratio 2.4
c/a=O.8 - - * cla=O.2
2.0 c
ta) c/a=o.s - - . c/.¶=O.l
I 0 2.4
r
2 @I -
4
6
8
I
10
0
cls=O.8
2 @I -
2.4 2,0
4
6
8
6
8
10
c/a=03
- - * c/a=O.l
1.6
p
1.2 0.8
I 0
2
4
6
8
10
R/o,
Fig. 6. Regions of dynamic instability for partial compressive edge loading at one end (a/b = 1); (a) a = 0.2, (b) a = 0.6.
0.4
0
-I 0
2
4
10
n/o,
Fig. 7. Regions of dynamic instability for concentrated compressive edge loading (u/b = 1); (a) a = 0.2, (b) r = 0.6.
Instability
of rectangular
axis. The rate of shift is lower for a small value of c/a (c/a = 0.2) than that for c/a close to unity (c/a = 0.8); this shows that for small values of c/a, the plate is less susceptible to dynamic instability. The dynamic instability behaviour for a/b = 2 shows that the instability behaviour is similar in nature to that for a/b = 1, but the load parameter c/a has a more pronounced effect on the behaviour in this case. 3.1.2. Concentrated edge loading. Figure 7(a) and (b) shows the boundaries of the first four dynamic instability regions for c( = 0.2, c1= 0.6, for c/a = 0.1, c/a = 0.5 and for a/b = I. It is observed that for small values of c/a (c/a = 0.1) the widths of the instability regions are usually smaller because of the edge restraint of the edge x = 0. As the point load moves towards the centre of the edge, the sizes of the instability zones increase, attaining the maximum value for c/a = 0.5, indicating the maximum instability effect. As Y increases, the widths of the instability regions increase and the regions shift inwards on the frequency ratio axis, as in the previous case. For a/b = 2, the instability regions are similar in nature to those of problem 1. 3.1.3. Purtial edge loading at the centre. Figure 8(a) and (b) shows the boundaries of the first four dynamic instability regions for CY= 0.2, cz = 0.6, for c/a = 0.2,
plates
cJa = 0.8 and for a/b = I. For small values of c/a (c/a = 0.2), the widths of the instability regions are usually larger than those for c/a close to unity (c/a = 0.8); this means that the plate is more susceptible to dynamic instability for band loading at the centre of the edges. This is because, for loading near the centre, the restraints of the edges are not much effective. As c( increases, approaching the value corresponding to the buckling load, the widths of the instability regions increase and the regions shift inwards on the frequency ratio axis. For a/b = 2, the dynamic instability behaviour is similar to that for a/b = 1, but the load band width c has a less pronounced effect on the behaviour when a/b = 2. 3. I .4. Partial edge loading at both ends. Figure 9(a) and (b) shows the boundaries of the first four dynamic instability regions for c( = 0.2, c1= 0.6, for c/a = 0.2, c/a = 0.8 and for a/b = 1. It is observed that for small values of c/a (c/a = 0.2) the widths of the instability zones are smaller as compared to those for c/a close to unity (c/a =0.8); this means that the plate is less susceptible to dynamic instability for band loading near the ends of the edges. This is because of the edge restraining effects (edges x = 0 and x = a). The effect of the dynamic load factor, cz on the instability regions is similar to the previous loading cases.
c/a=0.8 - - * c/0=0.2
- -
“0
(b) - -
2.4 2,0
*
79
4
2 (b) - -
2.4
c/a=0.8 c/a=0.2
cla=O.S
* c/a=o.2
2.0
2
4
6
8
10
Fig. 8. Regions of dynamic instability for partial compressive edge loading at the centre (a/b = 1); (a) G(=0.2, (b) r = 0.6.
0
8
10
I 6
8
10
c/a=O.8
* c/a=o.2
I 0
6
2
4
I
Fig. 9. Regions of dynamic instability for partial compressive edge loading at both ends (a/b = I); (a) r = 0.2, (b) CL= 0.6.
80
P. J. Deolasi
I .6 1.4
f
(a) clr~O.8 - - * c/p=O.2
and P. K. Datta
1.6 1.4
F
(a) c/a=o.s - - . c/a=O.l
1.0 p 0.8 0.6 0.4 0.2
I 0 I .6 1.4
F
2
4
8
6
10
12
n
14
0
1.6
(b) c/a=0.8 - - . &CO.2
2
4
6
8
10
12
14
I
I
8
10
12
14
(b) c/a=03 - - . c,a=O.l
1.4 1.2
I 0
2
4
8
6
10
12
0
14
n/w,
2
4
6
n/o,
Fig. 10. Regions of dynamic instability for partial tensile edge loading at one end (a/b = 1); (a) G(= 0.2, (b) a = 0.5.
Fig. 11. Regions of dynamic instability for concentrated tensile edge loading (a/b = 1); (a) a = 0.2, (b) u = 0.5.
For a/b = 2, the dynamic similar to that of a/b = 1.
a = 0.5, for c/a = 0.1, c/a = 0.5 and for a/b = 1. It is observed that for positions of the concentrated load near the corner of the edges (c/a = 0.1). the first and third instability regions appear at a lower frequency ratio in comparison to the position of the load near the centre of the edges (c/a = 0.5). However, opposite is the case for the second and fourth instability regions. The effect of the static load factor, r is similar to the previous partial edge loading problem.
instability
behaviour
is
3.2. Dynamic instability behaviour under tensile loading As discussed earlier, the tensile loading acting on the plate, producing compressive stresses within the plate, will cause parametric instability behaviour of the plate. The dynamic instability behaviour for different loading cases is discussed as follows. 3.2.1. Partial edge loading at one end. Figure IO(a) and (b) shows the regions of dynamic instability for CL=0.2, u =0.5, for c/a =0.2, c/a = 0.8 and for a/b = 1. It is observed that for small values of c/a (c/a = 0.2), the instability regions shift inward on the frequency ratio axis, indicating the onset of instability much earlier in comparison to larger values of c/a (c/a = 0.8); this is due to the fact that for small values of c/a, the compressive zone on the plate is much wider, causing the instability effects to appear earlier. However, the widths of the instability regions are found to be larger for higher values of c/a. As c( increases, the widths of the instability regions decrease and the regions shift outwards on the frequency ratio axis, indicating less susceptibility of the plate to the dynamic instability behaviour due to the static load parameter; this is just the opposite case to that of the compressive edge loading. 3.2.2. Concentrated edge loading. Figure 1 l(a) and (b) shows the instability regions for a = 0.2,
3.2.3. Partialedge loading at the centre. Figure 12(a) and (b) shows the boundaries of instability regions for a =0.2, a = 0.5, c/a = 0.2, c/a = 0.8 and for a/b = 1. For a small load band width (c/a = 0.2) near the centre of the edges, the first and third instability regions appear at higher frequency ratios, whereas second and fourth instability regions appear at lower frequency ratios in comparison to those for a higher load band width (c/a = 0.8). However, the shift in the onset of instability regions on the frequency ratio axis is comparatively much lower than that of the previous cases. Also, the widths of the instability regions are not much affected by the load band width. The effect of static load factor, a, on the instability regions is similar to that of the previous loading cases, as discussed earlier. The instability behaviour due to partial edge loading at both ends is similar in nature to that of partial edge loading at one end.
Instability 1.6 1.4
F 1.2 1
of rectangular
(a) c/so.8 - - . c/a=o.2
1.0 p 0.8 0.6 0.4 0.2 0 0 1.6 1.4
F 1.2 b
I 2
4
6
8
10
12
I 14
tb) c/a=O.8 - - . &=0.2
1.0 p 0.8 0.6 -
plates
81
ratio axis, thus resulting in instabilities at lower excitation frequency. However, for the band loading at the centre, the effect of load band width is the reverse. The effect of static load factor shows that it has a destabilizing effect on the dynamic instability behaviour of the plate. (4) Parametric instability behaviour under tensile periodic edge loading shows that the instability regions start at lower values of frequency ratios for the position of the load near the end of the edge. However, the width of the instability region increases as the load band width increases and as the load approaches the centre of the edge, but the instability zones move outwards on the frequency ratio axis with increasing load band width, thus resulting in instabilities at higher frequency ratios. The effect of an increase of the static load factor shows that it has a stabilizing effect on the dynamic instability behaviour of the plate.
0.4 0.2 O0
REFERENCES 2
4
6
8
10
12
14
Fig. 12. Regions of dynamic instability for partial tensile edge loading at the centre (a/b = 1); (a) a = 0.2, (b) c( = 0.5.
4. CONCLUSION
The results from a study of the static and dynamic instability behaviour of a plate subjected to partial compressive or tensile periodic in-plane edge loading can be summarized as follows. (1) Compression buckling analysis shows that the buckling load is usually higher for application of the loading near the ends of the edges; this is because of the restraining effects of the supported edges which undergo small out of plane deformation. Natural frequencies decrease as the compressive load increases and the fundamental frequency becomes zero at the respective value of the buckling load. (2) Tension buckling occurs due to compressive stresses produced in the plate due to localized tensile edge loading. The value of the tensile buckling load depends on the strength and extent of the compressive zone. Natural frequencies under tensile loading initially rise with the load, but begin to fall down beyond a certain value of the load and gradually approach zero as the load approaches the tensile buckling load. (3) Dynamic instability behaviour under compressive periodic edge loading shows that the width of the instability region increases as the load band width increases and as the load approaches the centre of the edge. The instability regions for the above cases appear to move inward on the frequency
Buckling of a rectangular plate under I. N. Yamaki, locally distributed forces applied on two opposite edges. Report of the Institute of High Speed Mechanics, Tohoku University, Japan, 34-55-71 (i954). and J. M. Gere, Theory of 2. S. P. Timoshenko Elastic Stabilify (2nd Edn). McGraw-Hill, New York (1961). 3. K. C. Rockey and D. K. Bagchi, Buckling of plate girder webs under partial edge loading, Znt. J. Mech. Sci
12, 61 (1970). 4. M. Z. Khan and A. C. Walker, Buckling of plates subjected to localized edge loading. The Structural Engineer 50, 225-232, (1972). Plate buckling 5. H. H. Spencer and H. Surjanhata, under paitial edge loading. deu. Mech. (Proc. of the 19th Midwestern Mech. Conf.) 13. 83-84 (1985). 6. A. W. Leissa and E. F. Ayoub; V&ration and buckling of a simply supported redtangular plate subjected to a nair of in-plane concentrated forces. J. Sound Vibr. 127. i55S171 (i988). 7 V. V. Bolotin, The Dynamic Stability qf Elastic Systems. Holden-Day, San Francisco (1964). 8 J. M. Hutt and A. E. Salam, Dynamic stability of plates bv finite element method. J. Enanp Mech. Dio. A.S.C.E. 67, 879-899 (1971). and P. K. Datta, Parametric instabil9 D. L. Prabhakara ity characteristics of rectangular plates with localized damage subjected to in-plane periodic load. Comput. Sfruct. 49, 825-836 (1994). 10 A. W. Leissa and E. F. Ayoub, Tension buckling of rectangular sheets due to concentrated forces. J. Engng Mech. A.S.C.E. 115, 2749-2762 (1989). I
APPENDIX
Finite
element
1. Element
formulation
Y
A
of element
elastic stiffness
matrices
matrix:
[B17[DI[B]dy dr, where [B] = [[B,], [&I.
, [BJ,
P. J. Deolasi
82
and P. K. Datta
?N,
O Ox 0 -~ 3N, 0
[&I=
113 i2 [HI
0
6 r?N, dN, -_ _ ; , = 1,2,...,8;
au
a>
0
N,
-N,
0
dN, -
ax
;;
[H] =
dN,
dJ Nf are the shape
h[Hl
functions
geometric
1
corresponding
[Sl,= where
stiffness
to eight nodes
3. Element
[&I= [[BGIl, [&I. dN,
ax
dN,
-0 dJ
PaIT= O OzI$-O
[NJ, ,[N,ll ;
N,
i=l,2
,,_.,
8.
N,
[iI= 0
0
I’ dN, 6N,
[Nl= KN,l,
d.r dx,
, [&II, 0
1
mass matrix:
iNsI=
matrix:
h ‘I [B,]r[b][B,] ss0 0
;;
i
where
2. Element
;,:
[HI
Oi i = I. 2,.
, 8;
The actual integration for element matrices are carried out by transformation of physical coordinates (x,J) into natural coordinates (I, s). Then the integrals are converted into forms convenient for numerical evaluation by Gaussquadrature integration formula.
00457949(94)E0277-9
& Srrumres Vol. 54. No. I, pp. 73-82. 1995 Copynght t 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved mm-7949195 $9.50 + 0.00
PARAMETRIC INSTABILITY CHARACTERISTICS OF RECTANGULAR PLATES SUBJECTED TO LOCALIZED EDGE LOADING (COMPRESSION OR TENSION) P. J. Deolasi and P. K. Datta Department
of Aerospace
Engineering,
Indian (Received
Institute
of Technology,
Kharagpur
721302,
India
1 June 1993)
Abstract-The parametric instability behaviour of a plate subjected to localized in-plane compressive or tensile periodic edge loading is studied, considering the effects of transverse shear deformation and rotary inertia. Different edge loading cases have been considered. Compression and tension buckling results are discussed. The vibration analysis for this problem shows that for certain parameters of tensile loading, the frequency of the plate initially rises with the load, but then begins to decrease with increasing tension, showing the onset of tension buckling. Dynamic instability behaviour under compressive periodic edge loading shows that the instability regions are influenced by the load band width and its location on the edge. Parametric instability behaviour under tensile periodic edge loading shows that the instability regions appear early for positions of the load near the corner of the edge. Instability behaviour due to different load parameters is discussed.
The situation in which loads are applied over only part of the edges, causing non-uniform initial stresses, has been studied [l-5] for static stability behaviour. The free vibration analysis of a rectangular plate subjected to a pair of oppositely directed in-plane concentrated forces was obtained by Leissa and Ayoub [6]. Structural elements subjected to in-plane periodic forces may induce transverse vibrations, which may be resonant for certain combinations of natural frequency of transverse vibration, the frequency of the in-plane forcing function and magnitude of the in-plane load. The spectrum of values of parameters causing unstable motion is referred to as the regions of dynamic instability or parametric resonance. Bolotin, in his text on dynamic stability [7], gives a thorough review of the problems involving parametric excitation of structural elements. Hutt and Salam [8] used a thin plate finite element model to study the dynamic instability of plates subjected to uniform periodic edge loading. The study of the parametric instability behaviour of plate subjected to localized edge loading is new. Recently, Prabhakara and Datta [9] have studied the effects of localized regions of damage in a plate, causing non-uniform stress distribution, on the dynamic instability behaviour of the plate. The problems discussed so far were concerned with the behaviour of plates subjected to compressive loading. However, plates can also buckle when subjected to forces that are only tensile. If a thin sheet is subjected to eccentric tensile loading on the opposite edges, one observes that the sheet wrinkles at several places away from the application of the load.
NOTATION plate lengths in the x- and y-directions respectively load band width plate flexural rigidity Young’s modulus plate thickness shear correction factor stiffness matrix elastic and geometric stiffness matrices respectively mass matrix edge load critical buckling load nodal displacement vector time deflection of mid-plane of the plate Cartesian coordinates static and dynamic load factors respectively non-dimensional load non-dimensional buckling load non-dimensional frequency Poisson’s ratio rotations of the normal about the K- and y-axes respectively mass density of the plate material initial in-plane stresses angular frequency of in-plane forcing function angular frequency of transverse vibration fundamental frequency of transverse vibration
1. INTRODUCTION
The study of static and dynamic behaviour of plate elements subjected to non-uniform in-plane stresses is of considerable importance to mechanical, aerospace and structural engineers. Large numbers of references are found in the published literature which deal with the buckling and vibration behaviour of rectangular plates subjected to in-plane uniform initial stresses. 73
14
P. J. Deolasi
Such wrinkling (i.e. buckling) can occur whenever the solution to the plane elasticity problem yields internal stresses that are compressive in any direction at any point. Tension buckling of rectangular sheets due to concentrated forces has recently been studied by Leissa and Ayoub [lo]. A plate subjected to periodic localized tensile edge loading will also exhibit dynamic instability phenomena due to the presence of compressive stresses in the plate causing static instability. The present paper deals with the problems of static and dynamic stability behaviour of rectangular plates subjected to partial edge loading. Both the compressive and tensile loading cases have been considered. Finite element formulation is applied for obtaining initial non-uniform stress distribution in the plate and also to solve the buckling and vibration problems. These results are then utilized for determination of the dynamic instability zones, for different loading conditions. 2. ANALYSIS
and P. K. Datta
components, W, 0, and 0,. are specified at each node. Mindlin’s hypothesis is applied to include the effects of transverse shear deformation and rotary inertia. The expressions for the total potential energy and the kinetic energy of the plate are as follows. Total potential energy:
11 dy dx
+jJob {;[~y(~>‘+~fg) +2r,,
( I aw aw
--
ax ay
The problem considered here consists of a rectangular plate (a x b) subjected to in-plane localized edge loading, as shown in Fig. 1, The loading shown is compressive in nature. Reversing the direction of loading will give the corresponding tensile loading problems. The boundaries of the plate are simply supported along all the edges. An eight-noded quadratic isoparametric element is used in the present analysis. Three displacement
(4
HI
dy dx
(1)
(b) P
P
.c-l c
I--
b
a
P
(4
Fig.
P/2
PI2
P/2
P/2
I. Description of the plate problems. (a) Partial edge loading at one end. (b) Concentrated loading. (c) Partial edge loading at the centre. (d) Loading at both ends.
edge
Jnstability of rectangular plates
75
mn, where m denotes the number of approximate
and kinetic energy:
half waves in the x-direction and n denotes the number of approximate half waves in the y-direction. 2.1. Dynamic stability problem For the problems described in Fig. 1, the edge load P is periodic and can be expressed in the form
where D is the plate flexural rigidity and D = Eh’/12 (1 - v2); 6,) ?v and zlr are the initial, in-plane stresses; k is the shear correction factor which accounts for the fact that transverse shear strain is not constant over the plate thickness. (k = 5/6 is used in the present analysis.) Upon assuming suitable interpolation functions for w, 0, and 0,. in eqns (1) and (2) in terms of nodal displacements, the total potential energy and kinetic energy can be written in matrix form as
P(t) = P, + P, cos Rt,
(6)
where P, is the static portion of P, P, is the amplitude of the dynamic portion of P and Q is the frequency of the dynamic portion of P. The stress distribution in the plate will be non-uniform and periodic and any element will be subjected to in-plane stresses of the form a, = a,, + r?,, cos Rt = Lip, + 6P, cos nt a! = fJT,.+ fJrrcos Rt = CP, + JP, cos nt
. (7)
T._ = 5,,, + t,, cos Rt = f?P, +fp, cos Rt i where {q} is the vector of generalized co-ordinates, [K] is the stiffness matrix and [M] is the consistent mass matrix. The stiffness matrix can be split into two parts,
Let P,$= rPe,
[Kl= [KC1- P[S], where [K,] is the elastic stiffness matrix, [S] is the stress stiffness matrix, also called the geometric stiffness matrix, which is obtained from the in-plane stress distribution cry, a!, r’, , and P is magnitude of the edge load. The equation of equilibrium for a plate subjected to in-plane loads can be written as
WI Gi) + Kl +I) - fvl blj = 0.
(3)
Equation (3) can be reduced to the governing equations for buckling and vibration problems as follows:
GlI = 0.
and
P, = ljPCr,
(8)
where CI and b are the percentages of the static buckling load PC,, which is obtained from the solution of eqn (4); a and p are termed as static and dynamic load factors, respectively. Usually, P,, is taken as the reference load. In the present analysis, for compressive loading, the buckling load for a fully loaded plate (c/a = 1.0) is taken as the reference load. Whereas for tensile loading, the tensile buckling load corresponding to concentrated loading at one end of the edge is taken as the reference load. Substituting eqns (7) and (8) in eqn (3), the following equation in matrix form is obtained:
PflGiI + [Kl - ap,.,PJ
(1) Buckling problem, {4} = 0:
KI +I) - pm
In eqn (7), a,, = CP,, CT,,= FP,, and z,,, = FP, are the static portions of o,~, oYand z,,., respec&vely, whereas the other terms are time dependant; G, b; E, I?, C and f are constant multipliers.
(4)
(2) Free vibration problem: assuming that the plate vibrates harmonically with angular frequency o, eqn (3) reduces to
- /I?PCr[s,]cos Qr] {q} = 0. (9) In eqn (9), the matrices [&I and [S,] are given by
(5) Both eqns (4) and (5) represent the eigenvalue problems. The eigenvalues are the critical buckling loads, PC,for eqn (4) or squares of natural frequencies, co2 for a known load P for eqn (5). The eigenvectors correspond to the mode shapes of buckling or vibration. A mode shape is identified by its mode number
in which IS,], [S,.] and [S,,.] are the stability matrices associated with or, uJ. and 7,,., respectively, for an element. Equation (9) represents a system of second order differential equations with periodic coefficients of the Mathieu-Hill type. The boundaries of the dynamic
P. J. Deolasi and P. K. Datta
76 instability regions are formed by solutions of period T and 2T, where T boundaries of the primary instability period 2T) are determined [7,8] from
the periodic = 2lIjQ. The regions (with the equation
Equation (10) represents an eigenvalue problem for known values of CI, /I and P,,. The two conditions under a plus or minus sign correspond to the two boundaries of the dynamic instability region. The eigenvalues are R, which give the boundary frequencies of the instability regions for given values of x and /I. 2.2. Finite element formulation An eight-noded quadratic isoparametric element is present analysis. Three displacement components, W, 0, and 0, are specified at each node. The element matrices can be expressed as: used in the
Elastic stiffness
matrix.
h[B]r[D][B] Geometric
stiffness
[S], =
dx dy
matrix.
Table 1. Comparison of non-dimensional buckling loads, y,, for a plate subjected to concentrated loading at the two opposite edges alb
Present
Leissa and Ayoub [6]
Yamaki [I]
1.0 2.0 0.5
25.720 29.852 28.851
25.814 30.061 26.523
25.7 29.7
into a two dimensional array of rectangular elements. Element elastic stiffness matrices are obtained with 2 x 2 Gauss sampling points to avoid possible shear locking. Element mass matrices are also obtained with 2 x 2 sampling points. The geometric stiffness matrix [S] is essentially a function of the in-plane stress distribution in the element due to applied edge loading. Since the stress field is non-uniform, plane stress analysis is carried out using finite element technique to determine the stresses at Gauss sampling points. Element geometric stiffness matrices are obtained with 3 x 3 Gauss sampling points. Element matrices are assembled into global matrices using skyline technique. Subspace iteration method is adopted throughout to solve the eigenvalue problems (4), (5) and (10). The non-dimensional loads, 7, and non-dimensional frequencies, i, are defined as:
oh ss
Non-dimensional
[BG]r[6][BG] dx d?.
load, ‘; = g
0 0
Consistent
Non-dimensional
mass matrix,
[N]‘m[N]
dx dy.
Further details of the formulation of the element matrices are given in Appendix A. The overall matrices [K,], [S] and [M] in eqn (3) are obtained by assemblies of corresponding element matrices and after elimination of the rows and columns corresponding to zero displacements. The vector of generalized coordinates consists of only active nodal displacements. The advantages of the symmetries of the plate geometry and the in-plane stress distribution are exploited by grouping the buckling and vibration modes into symmetric and antisymmetric modes about the axis or axes of symmetry. Thus only one side of the axis of symmetry needs to be considered. For loading cases shown in Fig. I(a) and (b), only the upper half, and for loading cases shown in Fig. I(c) and (d) only the upper left quarter of the plate are considered. This greatly reduces the computational efforts and memory space requirements.
E. = wb’JphlD.
To check the validity of the program, the buckling and vibration results are compared with those available in the literature. The comparisons are shown in Tables 1 and 2. All the results are obtained with a 10 x 10 mesh for a full plate. 3. RESULTS AND
DISCUSSION
Results have been obtained for the problems described in Fig. 1 for two plate aspect ratios, a/b = 1 and a/b = 2. Figure 2 shows a typical plot of nondimensional compressive buckling load vs c/a for the loading case shown in Fig. I(a) for the fundamental mode. It is observed that for small values of c/a, the Table 2. Comparison of non-dimensional frequencies, I without in-plane load. (Figures within the brackets indicate the results of Leissa and Ayoub [6]) i
alb 1.0
2.3. Computation A computer program has been written to perform all the necessary computations. The plate is divided
frequency,
2.0
Mode 11
Mode 12
Mode 21
Mode 22
19.729 (19.739)
49.148 (49.348)
49.148 (49.348)
78.432 (78.957)
Mode I I
Mode 21
Mode 31
Mode 12
12.282 (12.337)
19.582 (19.739)
31.362 (32.076)
39.398 (41.946)
Instability of rectangular plates 100
r
71
I20
-
*-
c/0=0.0
- - . c/r=0.2 c/r=O.B 80
60
oL---0
0.2
0.4
0.6
0.8
1.0
-0
400
800
Non-dimensional
cla
1200 tensile
load,
1600 y
2. Non-dimensional buckling load vs c/a for partial edge loading at one end.
Fig. 4. Non-dimensional fundamental frequency vs non-dimensional tensile load for partial edge loading at one end (a/b = 1).
buckling load is usually higher because of the edge restraint of the edge x = 0. As c/a increases, the buckling load decreases. However, as c/a approaches unity (corresponding to full load over the edges) the buckling load again increases due to the restraining effect of the other edge, x = a. Figure 3 shows the variation of non-dimensional frequencies with a non-dimensional compressive load for the loading case shown in Fig. l(a), for c/a = 0.2 and c/u = 0.8 and for u/b = 2. It is observed that as the load increases, the frequencies decrease and the fundamental frequencies become zero at the respective values of the buckling load. There is considerable variation in the load-frequency curves as c/u changes. For the same loading case as shown in Fig. l(a), with tensile loading, it is observed that as c/a changes
from 0 to 1, the tensile buckling load increases monotonically and rapidly from minimum to infinity; this is because of the fact that the strength and area of the compressive zone produced due to partial tensile loading reduce as c/u increases. Figure 4 shows the plots of non-dimensional fundamental frequencies vs a non-dimensional tensile load for different values of c/u and for u/b = 1. It is observed that for small values of c/u (c/a = 0.0 and c/u = 0.2) the fundamental frequency initially rises with the tensile load. As the load further increases, the frequency starts falling down and becomes zero at the respective value of the tensile buckling load; this is due to the presence of a substantial compressive zone within the plate under tensile edge loading of small band width. For
c/8=0.8 - - * c/1=0.2
.
higher values of c/u (c/a = 0.8) the fundamental frequencies keep on increasing over a considerable range. For other loading cases described in Fig. 1, the buckling and vibration behaviour have been studied with c/u as a parameter and the results of similar behaviour have been obtained. Figure 5 shows the general buckling and vibration mode shapes for compressive and tensile edge loading problems. It follows from the buckling and vibration analysis that the dynamic instability behaviour of a plate under localized edge loading will be of considerable interest. The dynamic instability behaviour under localized loading is discussed in the following sections. 3.1. Dynamic instability behuviour under compressive loading
0
I
0
20
I\1 40
Non-dimensional
!I 60
80
I
I
I
too
120
140
compressive
load,
y
Fig. 3. Non-dimensional frequencies vs non-dimensional compressive load for partial edge loading at one end (a/b = 2).
3.1.1. Partial edge loading at one end. Figure 6(a) and (b) shows the boundaries of the first four dynamic instability regions for c( = 0.2, s( = 0.6, for c/a = 0.2, c/u = 0.8 and for a/b = 1. It is observed that for small values of c/u (c/a = 0.2), the widths of the
dynamic
instability
regions are usually smaller in
P. J. Deolasi and P. K. Datta
78
(4
(b) Fig. 5. Buckling
and vibration
mode shapes;
(a) compressive
comparison to those for c/a close to unity (c/a = 0.8); this is because of the fact that the restraint of the edge x = 0 has a stabilizing effect on the dynamic instability behaviour of the plate for a small load band width.
edge loading,
(b) tensile edge loading.
As the static load factor, tl increases, approaching the value corresponding to the buckling load of the plate, the widths of the instability regions increase and the regions shift inwards on the frequency ratio 2.4
c/a=O.8 - - * cla=O.2
2.0 c
ta) c/a=o.s - - . c/.¶=O.l
I 0 2.4
r
2 @I -
4
6
8
I
10
0
cls=O.8
2 @I -
2.4 2,0
4
6
8
6
8
10
c/a=03
- - * c/a=O.l
1.6
p
1.2 0.8
I 0
2
4
6
8
10
R/o,
Fig. 6. Regions of dynamic instability for partial compressive edge loading at one end (a/b = 1); (a) a = 0.2, (b) a = 0.6.
0.4
0
-I 0
2
4
10
n/o,
Fig. 7. Regions of dynamic instability for concentrated compressive edge loading (u/b = 1); (a) a = 0.2, (b) r = 0.6.
Instability
of rectangular
axis. The rate of shift is lower for a small value of c/a (c/a = 0.2) than that for c/a close to unity (c/a = 0.8); this shows that for small values of c/a, the plate is less susceptible to dynamic instability. The dynamic instability behaviour for a/b = 2 shows that the instability behaviour is similar in nature to that for a/b = 1, but the load parameter c/a has a more pronounced effect on the behaviour in this case. 3.1.2. Concentrated edge loading. Figure 7(a) and (b) shows the boundaries of the first four dynamic instability regions for c( = 0.2, c1= 0.6, for c/a = 0.1, c/a = 0.5 and for a/b = I. It is observed that for small values of c/a (c/a = 0.1) the widths of the instability regions are usually smaller because of the edge restraint of the edge x = 0. As the point load moves towards the centre of the edge, the sizes of the instability zones increase, attaining the maximum value for c/a = 0.5, indicating the maximum instability effect. As Y increases, the widths of the instability regions increase and the regions shift inwards on the frequency ratio axis, as in the previous case. For a/b = 2, the instability regions are similar in nature to those of problem 1. 3.1.3. Purtial edge loading at the centre. Figure 8(a) and (b) shows the boundaries of the first four dynamic instability regions for CY= 0.2, cz = 0.6, for c/a = 0.2,
plates
cJa = 0.8 and for a/b = I. For small values of c/a (c/a = 0.2), the widths of the instability regions are usually larger than those for c/a close to unity (c/a = 0.8); this means that the plate is more susceptible to dynamic instability for band loading at the centre of the edges. This is because, for loading near the centre, the restraints of the edges are not much effective. As c( increases, approaching the value corresponding to the buckling load, the widths of the instability regions increase and the regions shift inwards on the frequency ratio axis. For a/b = 2, the dynamic instability behaviour is similar to that for a/b = 1, but the load band width c has a less pronounced effect on the behaviour when a/b = 2. 3. I .4. Partial edge loading at both ends. Figure 9(a) and (b) shows the boundaries of the first four dynamic instability regions for c( = 0.2, c1= 0.6, for c/a = 0.2, c/a = 0.8 and for a/b = 1. It is observed that for small values of c/a (c/a = 0.2) the widths of the instability zones are smaller as compared to those for c/a close to unity (c/a =0.8); this means that the plate is less susceptible to dynamic instability for band loading near the ends of the edges. This is because of the edge restraining effects (edges x = 0 and x = a). The effect of the dynamic load factor, cz on the instability regions is similar to the previous loading cases.
c/a=0.8 - - * c/0=0.2
- -
“0
(b) - -
2.4 2,0
*
79
4
2 (b) - -
2.4
c/a=0.8 c/a=0.2
cla=O.S
* c/a=o.2
2.0
2
4
6
8
10
Fig. 8. Regions of dynamic instability for partial compressive edge loading at the centre (a/b = 1); (a) G(=0.2, (b) r = 0.6.
0
8
10
I 6
8
10
c/a=O.8
* c/a=o.2
I 0
6
2
4
I
Fig. 9. Regions of dynamic instability for partial compressive edge loading at both ends (a/b = I); (a) r = 0.2, (b) CL= 0.6.
80
P. J. Deolasi
I .6 1.4
f
(a) clr~O.8 - - * c/p=O.2
and P. K. Datta
1.6 1.4
F
(a) c/a=o.s - - . c/a=O.l
1.0 p 0.8 0.6 0.4 0.2
I 0 I .6 1.4
F
2
4
8
6
10
12
n
14
0
1.6
(b) c/a=0.8 - - . &CO.2
2
4
6
8
10
12
14
I
I
8
10
12
14
(b) c/a=03 - - . c,a=O.l
1.4 1.2
I 0
2
4
8
6
10
12
0
14
n/w,
2
4
6
n/o,
Fig. 10. Regions of dynamic instability for partial tensile edge loading at one end (a/b = 1); (a) G(= 0.2, (b) a = 0.5.
Fig. 11. Regions of dynamic instability for concentrated tensile edge loading (a/b = 1); (a) a = 0.2, (b) u = 0.5.
For a/b = 2, the dynamic similar to that of a/b = 1.
a = 0.5, for c/a = 0.1, c/a = 0.5 and for a/b = 1. It is observed that for positions of the concentrated load near the corner of the edges (c/a = 0.1). the first and third instability regions appear at a lower frequency ratio in comparison to the position of the load near the centre of the edges (c/a = 0.5). However, opposite is the case for the second and fourth instability regions. The effect of the static load factor, r is similar to the previous partial edge loading problem.
instability
behaviour
is
3.2. Dynamic instability behaviour under tensile loading As discussed earlier, the tensile loading acting on the plate, producing compressive stresses within the plate, will cause parametric instability behaviour of the plate. The dynamic instability behaviour for different loading cases is discussed as follows. 3.2.1. Partial edge loading at one end. Figure IO(a) and (b) shows the regions of dynamic instability for CL=0.2, u =0.5, for c/a =0.2, c/a = 0.8 and for a/b = 1. It is observed that for small values of c/a (c/a = 0.2), the instability regions shift inward on the frequency ratio axis, indicating the onset of instability much earlier in comparison to larger values of c/a (c/a = 0.8); this is due to the fact that for small values of c/a, the compressive zone on the plate is much wider, causing the instability effects to appear earlier. However, the widths of the instability regions are found to be larger for higher values of c/a. As c( increases, the widths of the instability regions decrease and the regions shift outwards on the frequency ratio axis, indicating less susceptibility of the plate to the dynamic instability behaviour due to the static load parameter; this is just the opposite case to that of the compressive edge loading. 3.2.2. Concentrated edge loading. Figure 1 l(a) and (b) shows the instability regions for a = 0.2,
3.2.3. Partialedge loading at the centre. Figure 12(a) and (b) shows the boundaries of instability regions for a =0.2, a = 0.5, c/a = 0.2, c/a = 0.8 and for a/b = 1. For a small load band width (c/a = 0.2) near the centre of the edges, the first and third instability regions appear at higher frequency ratios, whereas second and fourth instability regions appear at lower frequency ratios in comparison to those for a higher load band width (c/a = 0.8). However, the shift in the onset of instability regions on the frequency ratio axis is comparatively much lower than that of the previous cases. Also, the widths of the instability regions are not much affected by the load band width. The effect of static load factor, a, on the instability regions is similar to that of the previous loading cases, as discussed earlier. The instability behaviour due to partial edge loading at both ends is similar in nature to that of partial edge loading at one end.
Instability 1.6 1.4
F 1.2 1
of rectangular
(a) c/so.8 - - . c/a=o.2
1.0 p 0.8 0.6 0.4 0.2 0 0 1.6 1.4
F 1.2 b
I 2
4
6
8
10
12
I 14
tb) c/a=O.8 - - . &=0.2
1.0 p 0.8 0.6 -
plates
81
ratio axis, thus resulting in instabilities at lower excitation frequency. However, for the band loading at the centre, the effect of load band width is the reverse. The effect of static load factor shows that it has a destabilizing effect on the dynamic instability behaviour of the plate. (4) Parametric instability behaviour under tensile periodic edge loading shows that the instability regions start at lower values of frequency ratios for the position of the load near the end of the edge. However, the width of the instability region increases as the load band width increases and as the load approaches the centre of the edge, but the instability zones move outwards on the frequency ratio axis with increasing load band width, thus resulting in instabilities at higher frequency ratios. The effect of an increase of the static load factor shows that it has a stabilizing effect on the dynamic instability behaviour of the plate.
0.4 0.2 O0
REFERENCES 2
4
6
8
10
12
14
Fig. 12. Regions of dynamic instability for partial tensile edge loading at the centre (a/b = 1); (a) a = 0.2, (b) c( = 0.5.
4. CONCLUSION
The results from a study of the static and dynamic instability behaviour of a plate subjected to partial compressive or tensile periodic in-plane edge loading can be summarized as follows. (1) Compression buckling analysis shows that the buckling load is usually higher for application of the loading near the ends of the edges; this is because of the restraining effects of the supported edges which undergo small out of plane deformation. Natural frequencies decrease as the compressive load increases and the fundamental frequency becomes zero at the respective value of the buckling load. (2) Tension buckling occurs due to compressive stresses produced in the plate due to localized tensile edge loading. The value of the tensile buckling load depends on the strength and extent of the compressive zone. Natural frequencies under tensile loading initially rise with the load, but begin to fall down beyond a certain value of the load and gradually approach zero as the load approaches the tensile buckling load. (3) Dynamic instability behaviour under compressive periodic edge loading shows that the width of the instability region increases as the load band width increases and as the load approaches the centre of the edge. The instability regions for the above cases appear to move inward on the frequency
Buckling of a rectangular plate under I. N. Yamaki, locally distributed forces applied on two opposite edges. Report of the Institute of High Speed Mechanics, Tohoku University, Japan, 34-55-71 (i954). and J. M. Gere, Theory of 2. S. P. Timoshenko Elastic Stabilify (2nd Edn). McGraw-Hill, New York (1961). 3. K. C. Rockey and D. K. Bagchi, Buckling of plate girder webs under partial edge loading, Znt. J. Mech. Sci
12, 61 (1970). 4. M. Z. Khan and A. C. Walker, Buckling of plates subjected to localized edge loading. The Structural Engineer 50, 225-232, (1972). Plate buckling 5. H. H. Spencer and H. Surjanhata, under paitial edge loading. deu. Mech. (Proc. of the 19th Midwestern Mech. Conf.) 13. 83-84 (1985). 6. A. W. Leissa and E. F. Ayoub; V&ration and buckling of a simply supported redtangular plate subjected to a nair of in-plane concentrated forces. J. Sound Vibr. 127. i55S171 (i988). 7 V. V. Bolotin, The Dynamic Stability qf Elastic Systems. Holden-Day, San Francisco (1964). 8 J. M. Hutt and A. E. Salam, Dynamic stability of plates bv finite element method. J. Enanp Mech. Dio. A.S.C.E. 67, 879-899 (1971). and P. K. Datta, Parametric instabil9 D. L. Prabhakara ity characteristics of rectangular plates with localized damage subjected to in-plane periodic load. Comput. Sfruct. 49, 825-836 (1994). 10 A. W. Leissa and E. F. Ayoub, Tension buckling of rectangular sheets due to concentrated forces. J. Engng Mech. A.S.C.E. 115, 2749-2762 (1989). I
APPENDIX
Finite
element
1. Element
formulation
Y
A
of element
elastic stiffness
matrices
matrix:
[B17[DI[B]dy dr, where [B] = [[B,], [&I.
, [BJ,
P. J. Deolasi
82
and P. K. Datta
?N,
O Ox 0 -~ 3N, 0
[&I=
113 i2 [HI
0
6 r?N, dN, -_ _ ; , = 1,2,...,8;
au
a>
0
N,
-N,
0
dN, -
ax
;;
[H] =
dN,
dJ Nf are the shape
h[Hl
functions
geometric
1
corresponding
[Sl,= where
stiffness
to eight nodes
3. Element
[&I= [[BGIl, [&I. dN,
ax
dN,
-0 dJ
PaIT= O OzI$-O
[NJ, ,[N,ll ;
N,
i=l,2
,,_.,
8.
N,
[iI= 0
0
I’ dN, 6N,
[Nl= KN,l,
d.r dx,
, [&II, 0
1
mass matrix:
iNsI=
matrix:
h ‘I [B,]r[b][B,] ss0 0
;;
i
where
2. Element
;,:
[HI
Oi i = I. 2,.
, 8;
The actual integration for element matrices are carried out by transformation of physical coordinates (x,J) into natural coordinates (I, s). Then the integrals are converted into forms convenient for numerical evaluation by Gaussquadrature integration formula.