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Free vibration of plates with stepped variations in thickness on non-homogeneous elastic foundations
Journal of Sound and Vibration (1995) 183(3), 533–545
FREE VIBRATION OF PLATES WITH STEPPED VARIATIONS IN THICKNESS ON NON-HOMOGENEOUS ELASTIC FOUNDATIONS F. J, H. P. L K. H. L Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 (Received 22 November 1993, and in final form 22 April 1994) A finite element model is presented for the analysis of the free vibration of plates with multiple stepped variations in thickness resting on non-homogeneous elastic foundations. Based on Mindlin plate theory, the model includes transverse shear deformation as well as bending–extension coupling in cases of plates with stepped sections eccentrically located with respect to the mid-planes. The section of elastic foundation under a plate element is treated as a separate foundation element. The transverse deformation of these foundation elements is made to be consistent with the deflection of plate elements being supported, resulting in a consistent stiffness matrix for the elastic foundation. Numerical results are in good agreement with the available reported results. The effect of eccentricity of the locations of stepped sections on the natural frequencies is found to be not negligible. The elastic foundations are found to have a significant effect on the fundamental natural frequencies of both uniform and stepped plates. Natural frequencies and mode shapes of rectangular plates and circular plates with multiple eccentrically stepped sections resting on non-homogeneous elastic foundations are presented.
1. INTRODUCTION
The free vibration of plates with stepped variations in thickness, commonly known as stepped plates, and plates resting on elastic foundations has been studied extensively due to their practical applications in engineering structures. Cortinez and Laura [1] computed the natural frequencies of stepped rectangular plates by means of the extended Kantorovich method. Bambill et al. [2] obtained the fundamental frequencies of stepped simply supported rectangular plates by using the Rayleigh–Ritz method. The free vibration of rectangular plates with multiple stepped sections using the analytical strip method was investigated by Harik et al. [3]. The free vibrations of stepped sector plates and stepped circular plates were studied by Molaghasemi and Harik [4] using a similar analytical strip method, by Mizusawa [5] using a spline element method and by Laura et al. [6] using the Ritz method. However, all of these studies were restricted to symmetrically stepped plates with coplanar mid-planes for all the constituent stepped sections. In practical applications, plates may be eccentrically stepped due to structural considerations or space constraints. An example of such a plate is that with multiple stepped sections and a common flat bottom surface. In this case, there is coupling between extension and bending due to the offset between the mid-planes of adjacent stepped sections of the plate. The related problem of free vibration of beams resting on elastic foundations was investigated by Kukla [7], Lee and Ke [8], Nielsen [9], Wang [10] and Lai et al. [11]. In contrast, there were relatively few studies on the vibration of plates resting on elastic 533 0022–460X/95/230533 + 13 $08.00/0
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foundations. Laura and Gutierrez [12, 13] studied the free vibration of uniform rectangular plates supported on a non-homogeneous elastic foundation using the Rayleigh–Ritz method. Horenberg and Kerstens [14] also studied the same problem by the modal constraint method. An analytical solution in terms of Bessel functions was presented by Harik and Molaghasemi [15] for the free vibration of annular sector plates resting on homogeneous elastic foundations. The free vibration of plates on elastic foundations was also analyzed by Bowles [16] using the finite element method, with the elastic foundations modelled as linear springs attached to the nodes of the elements. The same procedure was used by the computer software ANSYS [17] for the analysis of such problems. However, the convergence of the spring support model is dependent on the number and the length of the elements, as pointed out by Chilton and Wekezer [18]. They presented a new model for homogeneous elastic foundations by formulating a consistent foundation stiffness matrix based on the same shape functions for rectangular plate elements. However, the plate elements were non-conforming elements based on the Kirchhoff assumption with neglect of the transverse shear deformation. The objective of this study is to present a finite element model for the analysis of the free vibration of plates with multiple stepped sections resting on non-homogeneous elastic foundations. Based on Mindlin plate theory, this model includes the transverse shear deformation as well as bending–extension coupling in cases of plates with eccentrically stepped sections. The main task of the finite element formulation is to provide a direct way for the bending–extension coupling to be enforced, given the amount of eccentricity of the step. Essentially, this requires the in-plane translation of the mid-planes of the stepped sections to be related by the eccentricity of the step and the rotations of the normals to the mid-planes. There are many commerical finite element software codes which provide Mindlin plate elements, but none provide explicit eccentrically stepped plate elements. The only way to solve such eccentrically stepped problems with codes such as ANSYS [17] and ABAQUS [19] is to make use of the multi-point constraints with which the user specifies the relationship between the in-plane translations and the rotations at the nodes on the steps. However, this can be tedious and time-consuming in cases in which there are many nodes along steps. The elastic foundation is assumed to be of Winkler type. The section of elastic foundation under a plate element is treated as a separate foundation element, and the transverse deformation of a foundation element is made compatible with the transverse deflection of the plate element being supported. The subspace method is used to compute the modal parameters. Natural frequencies are compared with the available published results. The effects of eccentricity of the locations of the stepped sections and elastic foundations on the natural frequencies and mode shapes are studied. Natural frequencies and mode shapes of a rectangular plate and a circular plate with multiple stepped sections resting on non-homogeneous elastic foundations are presented to illustrate the use of the present method.
Figure 1. A typical plate with multiple stepped sections on elastic foundations.
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Figure 2. An eight-node serendipity plate element.
2. THEORY AND FORMULATIONS
A typical plate with multiple stepped sections, resting on an elastic foundation, is shown in Figure 1. Three types of elements are required for modelling the plate; namely, the general plate element, the connecting plate element and the elastic foundation element. The general plate element is based on the Mindlin plate theory and a full description of its formulations can be found in references [20] and [21]. In the present work, the eight-node serendipity plate elements, shown in Figure 2, are adopted. 2.1. Consider two elements (element n and element m) at a connecting boundary of two adjacent stepped sections, as shown in Figure 3. The local co-ordinates for element n and element m are x, y, z and x1 , y1 , z1 with the x–y plane and x1–y1 plane as the mid-planes of element n and element m, respectively. Nodes 5, 6 and 7 are at the connecting boundary. The element stiffness and element mass matrices and element nodal displacement vectors are denoted by [k], [m] and {d} for element n, and [k]1 [m]1 and {d}1 for element m. The vector {d} is defined by {d} = [d1 d2
d3
d4
d5
d6
d7
d8 ]T,
(1)
with {d}1 being defined in a similar manner. Here, {di } is used to denote the five degrees of freedom at node i as follows: {d} = [u0i
v0i w0i 8xi
8yi ]T,
i = 1, . . . , 8,
(2)
where u0i , v0i and w0i are nodal translations of the mid-plane of the plate, and 8xi and 8yi are nodal rotations of the plane normal to the mid-plane of the plate in the x–z and y–z planes, respectively. In order to satisfy the displacement continuity conditions at the connecting boundary, let w0j = w0j1 ,
8xj = 8xj1 ,
8yj = 8yj1 ,
j = 5, 6, 7.
(3a)
For in-plane translations, the following relations can be established: u0j1 = u0j − e8xj ,
voj1 = v0j − e8yj ,
j = 5, 6. 7.
Figure 3. Elements in the connecting boundary of two adjacent stepped sections.
(3b)
. .
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Here e is the magnitude of the eccentricity between the x–y plane and x1–y1 plane, and is positive when the right section is lower than the left section. Therefore, {dj }1 = [l]{dj },
j = 5, 6, 7,
(4)
L G −eG 0 G. 0G G 1l
(4a)
where 1 K G G0 [l] = G0 G0 G k0
0
0
−e
1
0
0
0
1
0
0 0
0 0
1 0
0
By using equation (4), the transformation of the element displacement vector for element m can be established as {d}1 = [T]{d }1,
(5)
where {d }1 = [d11
d21 d31 d41 d5
d6
d7
d81 ]T,
[T] = Diag [I
I
I
I
l
l
l
I ]. (5a, b)
The matrix [I] is a 5 × 5 unit matrix, and [l] is given by equation (4a). Also, the transformations for the element stiffness and element mass matrices of element m can be expressed as [k ]1 = [T]T[k]1[T],
[m¯ ]1 = [T]T[m]1[T].
(6)
The displacement continuity conditions have been incorporated into [k ]1 and [m¯ ]1, and the 1 nodal displacements {dj } are replaced by {dj } (j = 5, 6, 7). [k ] and [m¯ ]1 are to be used to assemble the global stiffness and mass matrices. A similar treatment can be carried out for cases in which the connecting boundaries of adjacent stepped sections are at the other edges of the elements. 2.2. An eight-node serendipity plate element and the supporting Winkler-type are shown in Figure 4. The elastic foundation is assumed to deflect in a manner consistent with the plate element being supported. The total potential energy of the foundation element is U = 12
g
c(x, y)w 2(x, y) dx dy,
A
Figure 4. An elastic foundation element.
(7)
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Figure 5. Stepped square plates with all the edges simply supported. (a) Plate I, symmetrically stepped; (b) plate II, eccentrically stepped.
where c(x, y) is the elastic coefficient of the foundation, w(x, y) is the deflection of the plate element, and A is the area of the bottom surface of the element. Upon introducing 8
w(x, y) = s Ni (j, h)w0i ,
(8)
i=1
where Ni (j, h), i = 1, . . . , 8, written as U = 12 {dc }T
are the shape functions, equation (7) can be re-
gg 1
1
−1
−1
c(j, h)[N]T[N] det =J= dj dh{dc },
(9)
where [N] = [N1
N2
N3
{dc } = [w01 w02 w03
N4 w04
N5 w05
N6 w06
N7 w07
N8 ],
(10a) T
w08 ] .
(10b)
The stiffness matrix of the foundation element is therefore given by
gg 1
[kc ] =
−1
1
c(j, h)[N]T[N] det =J= dj dh.
−1
T 1 Fundamental frequency coefficients (bi ) of stepped square plates; vi = (bi /4a 2 )zD1 /rh1 , D1 = Eh13 /12(1 − n 2 ) h2 /h1
Present
Bambill et al. [2]
0·70 0·75 0·80 0·85 0·90 0·95
15·586 16·394 17·012 17·665 18·307 18·964
15·619 16·412 17·121 17·789 18·507 19·102
(11)
0·3 0·5 0·8 1·0 1·2 1·5 2·0 2·5 3·0
3·3 4·0 0·7 0·0 2·4 5·7 12·4 15·6 13·4
7·30 11·97 17·01 19·50 22·08 26·21 33·85 42·21 52·87
h2 /h1 7·55 12·47 17·14 19·50 22·62 27·79 38·54 49·92 61·07
First mode ZXXXXCXXXXV Plate I Plate II D (%) 17·03 18·16 42·23 49·11 56·29 65·33 79·98 94·72 109·4
17·23 18·41 42·35 49·11 56·43 66·11 82·47 99·05 115·2
1·2 0·9 0·3 0·0 0·2 1·2 3·0 4·4 5·1
Second mode ZXXXXXCXXXXV Plate I Plate II D (%) 45·77 53·68 70·87 79·13 95·26 113·4 143·1 164·4 168·5
49·13 55·14 71·07 79·13 95·49 114·7 147·0 165·1 169·0
6·8 2·6 0·3 0·0 0·2 1·1 2·6 0·4 1·3
Third mode ZXXXXXCXXXXV Plate I Plate II D (%)
58·49 67·10 89·07 100·3 118·2 135·1 155·1 172·2 200·2
66·01 71·01 90·23 100·3 119·1 137·4 155·8 178·9 210·2
11·3 5·5 1·0 0·0 0·7 1·5 0·6 3·9 4·8
Fourth mode ZXXXXXCXXXXV Plate I Plate II D (%)
T 2 The effect of step eccentricity on the frequency coefficients (bi ); vi = (bi /4a 2 )zD1 /rh1 , D1 = Eh13 /12(1 − n 2 )
538 . .
539
2.3. By assembling all the element stiffness matrices and element mass matrices, the eigenvalue equation is obtained as ([K] − v 2[M]){D} = {0},
(12)
where [K] and [M] are the global stiffness matrix and the global mass matrix, v is the natural frequency and {D} is the corresponding mode shape. Note that the thickness:span ratios of plates in the present study are assumed to be large enough for shear locking to be avoided. Otherwise, the reduced and selective integration methods or other methods suggested by Huang [22] should be adopted to overcome this problem. 3. NUMERICAL RESULTS AND DISCUSSION
In order to validate the above analysis and formulation, the natural frequencies and mode shapes have been computed for several examples. Some of the results are compared with available reported results. 3.1. Two stepped square plates wtih all the edges simply supported are shown in Figures 5(a) and (b). The dimensions and material properties of the two plates are assumed to be the same except that plate I is symmetrically stepped and plate II is eccentrically stepped with a flat bottom surface. The fundamental frequencies of plate I for varying thickness ratios, together with the results obtained by Bambill et al. [2] using the Rayleigh–Ritz method, are given in Table 1. It can be seen that, for symmetrically stepped plates, the present results are in good agreement with those of Bambill et al. [2]. The effects of step eccentricity in the location of the stepped sections on the natural frequencies can be seen from Table 2 by comparing the results for plates I and II. The results show the following. 1. Due to the bending–extension coupling induced by step eccentricity, the natural frequencies of the eccentrically stepped plate are higher than those of the symmetrically stepped plate. 2. When the ratio of h2 to h1 is close to unity, the effect of eccentricity on the natural frequencies is negligible, as expected.
Figure 6. A uniform plate on non-homogeneous elastic foundations.
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T 3 The history of convergence for natural frequencies (Hz) Element mesh 2×2 3×3 4×4 5×5 6×6 7×7
First mode
Second mode
Third mode
Fourth mode
Fifth mode
115·63 115·73 115·75 115·76 115·76 115·78
172·25 133·79 132·31 131·93 131·81 131·72
339·73 176·79 161·92 158·13 157·81 157·33
348·00 215·42 184·85 180·21 179·84 179·67
717·66 422·29 234·22 217·69 214·64 213·85
3. When h2 is very much larger than h1 , the effect of eccentricity on the fundamental frequency is significant. For instance, when h2 /h1 is larger than 2·0 in this specific example, the difference in the fundamental frequencies is above 10%. However, the natural frequencies for the higher modes do not change significantly as h2 /h1 varies. 4. When h2 is very much smaller than h1 the step eccentricity has a noticeable effect on the higher natural frequencies. As shown in the table, when h2 /h1 = 0·3, the relative differences of the third and the fourth natural frequencies are 6·8% and 11·3%, respectively. However, in this case, the fundamental frequency is not significantly affected. From the above results, one can conclude that the eccentricity of the steps may have a significant effect on both the fundamental frequency and the higher natural frequencies. This effect is greatly dependent on the ratios of the heights of the steps, the positions of the steps, the modes under consideration and, of course, the boundary conditions. 3.2. As pointed out in the Introduction, the spring support model for the elastic foundation may have convergence problems when applied to the free vibration of beams on elastic foundations. In view of this, the convergence of the present elastic foundation element model has been first checked by computing the natural frequencies of a simply supported uniform plate on the elastic foundation shown in Figure 6 for k2 = k1 . It can be seen from Table 3 that the convergence of the present model is excellent. T 4 Fundamental frequency coefficients (bi ) of square plates on non-homogeneous elastic foundations; vi = (bi /4a 2 )zD/rh , D = Eh 3/12(1 − n 2 ), k1 = k'1 (D/a 4 ), k2 = k'2 (D/a 4 )
k'1
k'2
0 0 0 20 50 100 20 20 50 100
20 50 100 0 0 0 50 100 20 20
CCCC SSSS ZXXXXXXXCXXXXXXV ZXXXXXXXCXXXXXXV Laura & Gutierrez Laura & Gutierrez Present [13] Present [13] 39·26 44·51 52·40 35·66 36·15 36·93 44·75 52·33 39·98 40·74
39·89 45·10 52·66 36·33 36·84 37·66 45·37 52·89 40·65 41·40
CCCC, four edges clamped; SSSS, four edges simply supported.
25·04 31·80 40·52 20·68 22·63 25·41 32·78 41·35 27·84 30·21
25·48 32·23 41·09 21·21 23·25 26·29 33·15 41·81 28·29 30·83
541
T 5 The effect of an elastic foundation on the natural frequency coefficients (bi ) of square plates; vi = (bi /4a 2 )zD/rh, D = Eh 3/12(1 − n 2 ), k1 = k2 = k'(D/a 4 ) k' 0 200 400 600 800 1000
First mode
Second mode
Third mode
Fourth mode
Fifth mode
Sixth mode
19·50 24·19 28·11 31·52 34·63 37·47
49·11 51·15 53·12 54·99 56·83 58·60
79·13 80·41 81·67 82·91 84·13 85·37
100·3 101·2 102·2 103·4 104·2 105·1
132·6 133·4 134·1 134·9 135·7 136·5
179·5 179·9 180·4 181·0 181·5 182·4
A uniform plate resting on a non-homogeneous elastic foundation is shown in Figure 6. The elastic coefficient of the foundation for the inner square section is k2 and that for the surrounding outer section is k1 . The dependence of the fundamental frequencies on the elastic coefficients of this non-homogeneous elastic foundation is illustrated in Table 4. The results are in excellent agreement with the Rayleigh–Ritz solutions presented by Laura and Gutierrez [13]. It can be concluded that the present foundation element is adequate for the analysis of the free vibration of plates on nonhomogeneous elastic foundations. In Table 5 are shown the first six natural frequencies of a square plate of uniform thickness for a range of values of the elastic coefficient of foundation. It is found that the effect of the elastic foundation on the fundamental natural frequency is significant, but the natural frequencies for the higher modes are not greatly affected by the elastic foundation. 3.3. - In order to illustrate the use of the present method for determining the effects of multiple stepped sections and elastic foundations on the modal parameters, the natural frequencies and mode shapes of clamped rectangular and circular plates with several geometrical configurations and supporting foundations have been computed. A stepped rectangular plate resting on non-homogeneous elastic foundations with four edges clamped is shown in Figure 7. The thickness ratios of the stepped sections are h1 :h2 :h3 = 1·0:0·8:0·5.
(13)
The elastic coefficients of the non-homogeneous elastic foundations are k1 = 20k0 ,
k1 = 50k0 ,
k1 = 100k0 ,
(14)
Figure 7. A rectangular plate with multiple stepped sections resting on non-homogeneous elastic foundations.
542
. .
Figure 8. Natural frequency coefficients (bi ) and mode shapes for a rectangular plate with multiple stepped sections resting on non-homogeneous foundations (a without elastic foundation (b).
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Figure 9. A circular plate with multiple stepped sections resting on non-homogeneous elastic foundations.
where k0 = (1/a 4 )Eh13 /12(1 − n 2 ).
(14a)
The natural frequencies and mode shapes of the plate are shown in Figure 8(a). In Figure 8(b) are shown the natural frequencies and mode shapes of a corresponding stepped rectangular plate with the same geometrical configuration but without the supporting elastic foundation. It is found that the fundamental frequency of the plate is affected more by the presence of the elastic foundation. The effect of the elastic foundations on the first mode shape is also more significant than the effect on the higher mode shapes. A stepped circular plate resting on nonhomogeneous elastic foundations (denoted as plate IV) is shown in Figure 9. The thickness ratios of the stepped sections and the elastic coefficients of foundation are the same as in the foregoing example for the rectangular plate except that k0 is redefined as k0 = {1/(3a)4 }Eh13 /12(1 − n 2 ).
(14b)
In order to make comparisons, three other corresponding circular plates (denoted as plates I, II and III) are also considered. The natural frequency coefficients of the four circular plates are given in Table 6. The mode shapes of the stepped circular plate on the
T 6 A comparison of natural frequency coefficients (bi ) for different circular plates; vi = {bi /(3a)2 }zD1 /rh1 , D1 = Eh13 /12(1 − n 2 ) Plate Plate Plate Plate Plate
I II III IV
First mode
Second mode
Third mode
Fourth mode
Fifth mode
Six mode
10·26 13·67 6·56 10·95
21·24 22·42 13·54 15·67
34·83 36·61 24·40 25·07
39·78 40·38 27·34 28·53
60·81 61·67 39·80 40·44
88·42 89·59 42·00 42·74
Plate I: h1 = h2 = h3 , k1 = k2 = k3 = 0; plate II: h1 = h2 = h3 , k1 $ k2 $ k3 $ 0; plate III: h1 $ h2 $ h3 , k1 = k2 = k3 = 0; plate IV: h1 $ h2 $ h3 , k1 $ k2 $ k3 $ 0.
544
. .
Figure 10. Mode shapes of a circular plate with multiple stepped sections resting on non-homogeneous elastic foundations (plate IV).
nonhomogeneous elastic foundations (plate IV) are shown in Figure 10. It is found that the effect of multiply stepped sections on natural frequencies, both for plates with and without supporting elastic foundations, is significant. The higher the mode order, the greater this effect is on the natural frequencies. It is also found that the increases in the fundamental natural frequencies of uniform and stepped circular plates due to the elastic foundations are 33% and 67% of those of corresponding circular plates without elastic foundation, respectively. However, the increases in the natural frequencies of mode 6 are only 1% and 2%, respectively. It can be concluded that the effect of the elastic foundation on the fundamental natural frequency of both uniform and stepped circular plates is more important than that on the natural frequencies for the higher modes.
4. CONCLUSIONS
A finite element model for the analysis of the free vibration of plates with multiple stepped sections resting on nonhomogeneous elastic foundations has been presented. Three types of elements have been used in modelling of the plate. These are the general plate element, the connecting plate element and the elastic foundation element. Both plate elements are based on Mindlin plate theory, with the connecting element being used to relate the inplane translations of the mid-planes of the eccentrically stepped sections. This provides explicitly the bending–extension coupling which is present in such problems. The
545
formulation of the elastic foundation element is made consistent with that of the plate element to which it is attached. The numerical results from the analysis of test problems are in good agreement with available reported results. It is found that the effect of eccentricity of the locations of the stepped sections on the natural frequencies is not negligible, and the effect of the elastic foundation on the fundamental natural frequencies of both uniform and stepped plates is much more significant than on the higher natural frequencies. Natural frequencies and mode shapes of stepped rectangular plates and circular plates resting on non-homogeneous elastic foundations have also been presented for various combinations of stepped sections and elastic foundations.
REFERENCES 1. V. H. C and P. A. A. L 1990 Journal of Sound and Vibration 147, 457–461. Analysis of vibrating rectangular plates of discontinuously varying thickness by means of the Kantorovich extended method. 2. D. V. B, P. A. A. L, A. B and R. C 1991 Journal of Sound and Vibration 150, 167–169. Fundamental frequency of transverse vibration of symmetrically stepped simply supported rectangular plates. 3. I. E. H, X. L and N. B 1992 Journal of Sound and Vibration 153, 51–62. Analytical solution to the free vibration of rectangular plates. 4. R. M and I. E. H 1991 Journal of Sound and Vibration 149, 323–326. Natural frequencies of vibrating sector plates with stepped thickness. 5. T. M 1991 Computers and Structures 41, 377–383. Vibration of stepped sector plates by the spline element method. 6. P. A. A. L, D. A and H. L 1991 Journal of Sound and Vibration 146, 533–537. Numerical experiments on vibrating circular plates with stepped thickness and with edges elastically restrained against rotation and translation. 7. S. K 1991 Journal of Sound and Vibration 149, 259–265. Free vibration of a beam supported on a stepped elastic foundation. 8. S. Y. L and H. Y. K 1990 Computers and Structures 34, 421–429. Free vibration of non-uniform beams resting on non-uniform elastic foundation with general elastic end restraints. 9. J. C. O. N 1991 Journal of Sound and Vibration 145, 279–287. Eigenfrequencies and eigenmodes of beam structures on an elastic foundation. 10. J. W 1991 Journal of Sound and Vibration 149, 315–322. Vibration of stepped beams on elastic foundations. 11. Y. C. L, B. Y. T, W. S. L and B. R. B 1992 Journal of Structural Engineering 118, 853–858. Dynamic response of beams on elastic foundation. 12. P. A. A. L and R. H. G 1984 Journal of Sound and Vibration 95, 537–552. Vibration of plates elastically supported on a nonhomogeneous foundation. 13. P. A. A. L and R. H. G 1985 Journal of Sound and Vibration 101, 307–315. Transverse vibrations of rectangular plates on inhomogeneous foundation, part I: Rayleigh–Ritz method. 14. A. G. H and J. G. M. K 1985 Journal of Sound and Vibration 101, 315–324. Transverse vibrations of rectangular plates on inhomogeneous foundation, part II: modal constraint method. 15. I. E. H and H. R. M 1990 Journal of Sound and Vibration 138, 524–528. Vibration of sector plates on elastic foundations. 16. J. E. B 1977 Foundation Analysis and Design. New York: McGraw-Hill; second edition. 17. A 1992 Engineering Analysis System Version 5.0, User’s Manual. Houston, Texas: Swanson Analysis Systems, Inc. 18. D. S. C and J. W. W 1990 Journal of Structural Engineering 116, 3236–3241. Plates on elastic foundation. 19. A 1992 Version 5.2 User’s Manual. Pawtucket: Hibbitt, Karlsson & Sorensen, Inc. 20. R. D. C, D. S. M and M. E. P 1989 Concepts and Applications of Finite Element Analysis. New York: John Wiley. 21. T. J. R. H 1987 The Finite Element Method. Englewood Cliffs, New Jersey: Prentice-Hall. 22. H. C. H 1989 Static and Dynamic Analysis of Plates and Shells. Berlin: Springer-Verlag.
FREE VIBRATION OF PLATES WITH STEPPED VARIATIONS IN THICKNESS ON NON-HOMOGENEOUS ELASTIC FOUNDATIONS F. J, H. P. L K. H. L Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 (Received 22 November 1993, and in final form 22 April 1994) A finite element model is presented for the analysis of the free vibration of plates with multiple stepped variations in thickness resting on non-homogeneous elastic foundations. Based on Mindlin plate theory, the model includes transverse shear deformation as well as bending–extension coupling in cases of plates with stepped sections eccentrically located with respect to the mid-planes. The section of elastic foundation under a plate element is treated as a separate foundation element. The transverse deformation of these foundation elements is made to be consistent with the deflection of plate elements being supported, resulting in a consistent stiffness matrix for the elastic foundation. Numerical results are in good agreement with the available reported results. The effect of eccentricity of the locations of stepped sections on the natural frequencies is found to be not negligible. The elastic foundations are found to have a significant effect on the fundamental natural frequencies of both uniform and stepped plates. Natural frequencies and mode shapes of rectangular plates and circular plates with multiple eccentrically stepped sections resting on non-homogeneous elastic foundations are presented.
1. INTRODUCTION
The free vibration of plates with stepped variations in thickness, commonly known as stepped plates, and plates resting on elastic foundations has been studied extensively due to their practical applications in engineering structures. Cortinez and Laura [1] computed the natural frequencies of stepped rectangular plates by means of the extended Kantorovich method. Bambill et al. [2] obtained the fundamental frequencies of stepped simply supported rectangular plates by using the Rayleigh–Ritz method. The free vibration of rectangular plates with multiple stepped sections using the analytical strip method was investigated by Harik et al. [3]. The free vibrations of stepped sector plates and stepped circular plates were studied by Molaghasemi and Harik [4] using a similar analytical strip method, by Mizusawa [5] using a spline element method and by Laura et al. [6] using the Ritz method. However, all of these studies were restricted to symmetrically stepped plates with coplanar mid-planes for all the constituent stepped sections. In practical applications, plates may be eccentrically stepped due to structural considerations or space constraints. An example of such a plate is that with multiple stepped sections and a common flat bottom surface. In this case, there is coupling between extension and bending due to the offset between the mid-planes of adjacent stepped sections of the plate. The related problem of free vibration of beams resting on elastic foundations was investigated by Kukla [7], Lee and Ke [8], Nielsen [9], Wang [10] and Lai et al. [11]. In contrast, there were relatively few studies on the vibration of plates resting on elastic 533 0022–460X/95/230533 + 13 $08.00/0
7 1995 Academic Press Limited
534
. .
foundations. Laura and Gutierrez [12, 13] studied the free vibration of uniform rectangular plates supported on a non-homogeneous elastic foundation using the Rayleigh–Ritz method. Horenberg and Kerstens [14] also studied the same problem by the modal constraint method. An analytical solution in terms of Bessel functions was presented by Harik and Molaghasemi [15] for the free vibration of annular sector plates resting on homogeneous elastic foundations. The free vibration of plates on elastic foundations was also analyzed by Bowles [16] using the finite element method, with the elastic foundations modelled as linear springs attached to the nodes of the elements. The same procedure was used by the computer software ANSYS [17] for the analysis of such problems. However, the convergence of the spring support model is dependent on the number and the length of the elements, as pointed out by Chilton and Wekezer [18]. They presented a new model for homogeneous elastic foundations by formulating a consistent foundation stiffness matrix based on the same shape functions for rectangular plate elements. However, the plate elements were non-conforming elements based on the Kirchhoff assumption with neglect of the transverse shear deformation. The objective of this study is to present a finite element model for the analysis of the free vibration of plates with multiple stepped sections resting on non-homogeneous elastic foundations. Based on Mindlin plate theory, this model includes the transverse shear deformation as well as bending–extension coupling in cases of plates with eccentrically stepped sections. The main task of the finite element formulation is to provide a direct way for the bending–extension coupling to be enforced, given the amount of eccentricity of the step. Essentially, this requires the in-plane translation of the mid-planes of the stepped sections to be related by the eccentricity of the step and the rotations of the normals to the mid-planes. There are many commerical finite element software codes which provide Mindlin plate elements, but none provide explicit eccentrically stepped plate elements. The only way to solve such eccentrically stepped problems with codes such as ANSYS [17] and ABAQUS [19] is to make use of the multi-point constraints with which the user specifies the relationship between the in-plane translations and the rotations at the nodes on the steps. However, this can be tedious and time-consuming in cases in which there are many nodes along steps. The elastic foundation is assumed to be of Winkler type. The section of elastic foundation under a plate element is treated as a separate foundation element, and the transverse deformation of a foundation element is made compatible with the transverse deflection of the plate element being supported. The subspace method is used to compute the modal parameters. Natural frequencies are compared with the available published results. The effects of eccentricity of the locations of the stepped sections and elastic foundations on the natural frequencies and mode shapes are studied. Natural frequencies and mode shapes of a rectangular plate and a circular plate with multiple stepped sections resting on non-homogeneous elastic foundations are presented to illustrate the use of the present method.
Figure 1. A typical plate with multiple stepped sections on elastic foundations.
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Figure 2. An eight-node serendipity plate element.
2. THEORY AND FORMULATIONS
A typical plate with multiple stepped sections, resting on an elastic foundation, is shown in Figure 1. Three types of elements are required for modelling the plate; namely, the general plate element, the connecting plate element and the elastic foundation element. The general plate element is based on the Mindlin plate theory and a full description of its formulations can be found in references [20] and [21]. In the present work, the eight-node serendipity plate elements, shown in Figure 2, are adopted. 2.1. Consider two elements (element n and element m) at a connecting boundary of two adjacent stepped sections, as shown in Figure 3. The local co-ordinates for element n and element m are x, y, z and x1 , y1 , z1 with the x–y plane and x1–y1 plane as the mid-planes of element n and element m, respectively. Nodes 5, 6 and 7 are at the connecting boundary. The element stiffness and element mass matrices and element nodal displacement vectors are denoted by [k], [m] and {d} for element n, and [k]1 [m]1 and {d}1 for element m. The vector {d} is defined by {d} = [d1 d2
d3
d4
d5
d6
d7
d8 ]T,
(1)
with {d}1 being defined in a similar manner. Here, {di } is used to denote the five degrees of freedom at node i as follows: {d} = [u0i
v0i w0i 8xi
8yi ]T,
i = 1, . . . , 8,
(2)
where u0i , v0i and w0i are nodal translations of the mid-plane of the plate, and 8xi and 8yi are nodal rotations of the plane normal to the mid-plane of the plate in the x–z and y–z planes, respectively. In order to satisfy the displacement continuity conditions at the connecting boundary, let w0j = w0j1 ,
8xj = 8xj1 ,
8yj = 8yj1 ,
j = 5, 6, 7.
(3a)
For in-plane translations, the following relations can be established: u0j1 = u0j − e8xj ,
voj1 = v0j − e8yj ,
j = 5, 6. 7.
Figure 3. Elements in the connecting boundary of two adjacent stepped sections.
(3b)
. .
536
Here e is the magnitude of the eccentricity between the x–y plane and x1–y1 plane, and is positive when the right section is lower than the left section. Therefore, {dj }1 = [l]{dj },
j = 5, 6, 7,
(4)
L G −eG 0 G. 0G G 1l
(4a)
where 1 K G G0 [l] = G0 G0 G k0
0
0
−e
1
0
0
0
1
0
0 0
0 0
1 0
0
By using equation (4), the transformation of the element displacement vector for element m can be established as {d}1 = [T]{d }1,
(5)
where {d }1 = [d11
d21 d31 d41 d5
d6
d7
d81 ]T,
[T] = Diag [I
I
I
I
l
l
l
I ]. (5a, b)
The matrix [I] is a 5 × 5 unit matrix, and [l] is given by equation (4a). Also, the transformations for the element stiffness and element mass matrices of element m can be expressed as [k ]1 = [T]T[k]1[T],
[m¯ ]1 = [T]T[m]1[T].
(6)
The displacement continuity conditions have been incorporated into [k ]1 and [m¯ ]1, and the 1 nodal displacements {dj } are replaced by {dj } (j = 5, 6, 7). [k ] and [m¯ ]1 are to be used to assemble the global stiffness and mass matrices. A similar treatment can be carried out for cases in which the connecting boundaries of adjacent stepped sections are at the other edges of the elements. 2.2. An eight-node serendipity plate element and the supporting Winkler-type are shown in Figure 4. The elastic foundation is assumed to deflect in a manner consistent with the plate element being supported. The total potential energy of the foundation element is U = 12
g
c(x, y)w 2(x, y) dx dy,
A
Figure 4. An elastic foundation element.
(7)
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Figure 5. Stepped square plates with all the edges simply supported. (a) Plate I, symmetrically stepped; (b) plate II, eccentrically stepped.
where c(x, y) is the elastic coefficient of the foundation, w(x, y) is the deflection of the plate element, and A is the area of the bottom surface of the element. Upon introducing 8
w(x, y) = s Ni (j, h)w0i ,
(8)
i=1
where Ni (j, h), i = 1, . . . , 8, written as U = 12 {dc }T
are the shape functions, equation (7) can be re-
gg 1
1
−1
−1
c(j, h)[N]T[N] det =J= dj dh{dc },
(9)
where [N] = [N1
N2
N3
{dc } = [w01 w02 w03
N4 w04
N5 w05
N6 w06
N7 w07
N8 ],
(10a) T
w08 ] .
(10b)
The stiffness matrix of the foundation element is therefore given by
gg 1
[kc ] =
−1
1
c(j, h)[N]T[N] det =J= dj dh.
−1
T 1 Fundamental frequency coefficients (bi ) of stepped square plates; vi = (bi /4a 2 )zD1 /rh1 , D1 = Eh13 /12(1 − n 2 ) h2 /h1
Present
Bambill et al. [2]
0·70 0·75 0·80 0·85 0·90 0·95
15·586 16·394 17·012 17·665 18·307 18·964
15·619 16·412 17·121 17·789 18·507 19·102
(11)
0·3 0·5 0·8 1·0 1·2 1·5 2·0 2·5 3·0
3·3 4·0 0·7 0·0 2·4 5·7 12·4 15·6 13·4
7·30 11·97 17·01 19·50 22·08 26·21 33·85 42·21 52·87
h2 /h1 7·55 12·47 17·14 19·50 22·62 27·79 38·54 49·92 61·07
First mode ZXXXXCXXXXV Plate I Plate II D (%) 17·03 18·16 42·23 49·11 56·29 65·33 79·98 94·72 109·4
17·23 18·41 42·35 49·11 56·43 66·11 82·47 99·05 115·2
1·2 0·9 0·3 0·0 0·2 1·2 3·0 4·4 5·1
Second mode ZXXXXXCXXXXV Plate I Plate II D (%) 45·77 53·68 70·87 79·13 95·26 113·4 143·1 164·4 168·5
49·13 55·14 71·07 79·13 95·49 114·7 147·0 165·1 169·0
6·8 2·6 0·3 0·0 0·2 1·1 2·6 0·4 1·3
Third mode ZXXXXXCXXXXV Plate I Plate II D (%)
58·49 67·10 89·07 100·3 118·2 135·1 155·1 172·2 200·2
66·01 71·01 90·23 100·3 119·1 137·4 155·8 178·9 210·2
11·3 5·5 1·0 0·0 0·7 1·5 0·6 3·9 4·8
Fourth mode ZXXXXXCXXXXV Plate I Plate II D (%)
T 2 The effect of step eccentricity on the frequency coefficients (bi ); vi = (bi /4a 2 )zD1 /rh1 , D1 = Eh13 /12(1 − n 2 )
538 . .
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2.3. By assembling all the element stiffness matrices and element mass matrices, the eigenvalue equation is obtained as ([K] − v 2[M]){D} = {0},
(12)
where [K] and [M] are the global stiffness matrix and the global mass matrix, v is the natural frequency and {D} is the corresponding mode shape. Note that the thickness:span ratios of plates in the present study are assumed to be large enough for shear locking to be avoided. Otherwise, the reduced and selective integration methods or other methods suggested by Huang [22] should be adopted to overcome this problem. 3. NUMERICAL RESULTS AND DISCUSSION
In order to validate the above analysis and formulation, the natural frequencies and mode shapes have been computed for several examples. Some of the results are compared with available reported results. 3.1. Two stepped square plates wtih all the edges simply supported are shown in Figures 5(a) and (b). The dimensions and material properties of the two plates are assumed to be the same except that plate I is symmetrically stepped and plate II is eccentrically stepped with a flat bottom surface. The fundamental frequencies of plate I for varying thickness ratios, together with the results obtained by Bambill et al. [2] using the Rayleigh–Ritz method, are given in Table 1. It can be seen that, for symmetrically stepped plates, the present results are in good agreement with those of Bambill et al. [2]. The effects of step eccentricity in the location of the stepped sections on the natural frequencies can be seen from Table 2 by comparing the results for plates I and II. The results show the following. 1. Due to the bending–extension coupling induced by step eccentricity, the natural frequencies of the eccentrically stepped plate are higher than those of the symmetrically stepped plate. 2. When the ratio of h2 to h1 is close to unity, the effect of eccentricity on the natural frequencies is negligible, as expected.
Figure 6. A uniform plate on non-homogeneous elastic foundations.
. .
540
T 3 The history of convergence for natural frequencies (Hz) Element mesh 2×2 3×3 4×4 5×5 6×6 7×7
First mode
Second mode
Third mode
Fourth mode
Fifth mode
115·63 115·73 115·75 115·76 115·76 115·78
172·25 133·79 132·31 131·93 131·81 131·72
339·73 176·79 161·92 158·13 157·81 157·33
348·00 215·42 184·85 180·21 179·84 179·67
717·66 422·29 234·22 217·69 214·64 213·85
3. When h2 is very much larger than h1 , the effect of eccentricity on the fundamental frequency is significant. For instance, when h2 /h1 is larger than 2·0 in this specific example, the difference in the fundamental frequencies is above 10%. However, the natural frequencies for the higher modes do not change significantly as h2 /h1 varies. 4. When h2 is very much smaller than h1 the step eccentricity has a noticeable effect on the higher natural frequencies. As shown in the table, when h2 /h1 = 0·3, the relative differences of the third and the fourth natural frequencies are 6·8% and 11·3%, respectively. However, in this case, the fundamental frequency is not significantly affected. From the above results, one can conclude that the eccentricity of the steps may have a significant effect on both the fundamental frequency and the higher natural frequencies. This effect is greatly dependent on the ratios of the heights of the steps, the positions of the steps, the modes under consideration and, of course, the boundary conditions. 3.2. As pointed out in the Introduction, the spring support model for the elastic foundation may have convergence problems when applied to the free vibration of beams on elastic foundations. In view of this, the convergence of the present elastic foundation element model has been first checked by computing the natural frequencies of a simply supported uniform plate on the elastic foundation shown in Figure 6 for k2 = k1 . It can be seen from Table 3 that the convergence of the present model is excellent. T 4 Fundamental frequency coefficients (bi ) of square plates on non-homogeneous elastic foundations; vi = (bi /4a 2 )zD/rh , D = Eh 3/12(1 − n 2 ), k1 = k'1 (D/a 4 ), k2 = k'2 (D/a 4 )
k'1
k'2
0 0 0 20 50 100 20 20 50 100
20 50 100 0 0 0 50 100 20 20
CCCC SSSS ZXXXXXXXCXXXXXXV ZXXXXXXXCXXXXXXV Laura & Gutierrez Laura & Gutierrez Present [13] Present [13] 39·26 44·51 52·40 35·66 36·15 36·93 44·75 52·33 39·98 40·74
39·89 45·10 52·66 36·33 36·84 37·66 45·37 52·89 40·65 41·40
CCCC, four edges clamped; SSSS, four edges simply supported.
25·04 31·80 40·52 20·68 22·63 25·41 32·78 41·35 27·84 30·21
25·48 32·23 41·09 21·21 23·25 26·29 33·15 41·81 28·29 30·83
541
T 5 The effect of an elastic foundation on the natural frequency coefficients (bi ) of square plates; vi = (bi /4a 2 )zD/rh, D = Eh 3/12(1 − n 2 ), k1 = k2 = k'(D/a 4 ) k' 0 200 400 600 800 1000
First mode
Second mode
Third mode
Fourth mode
Fifth mode
Sixth mode
19·50 24·19 28·11 31·52 34·63 37·47
49·11 51·15 53·12 54·99 56·83 58·60
79·13 80·41 81·67 82·91 84·13 85·37
100·3 101·2 102·2 103·4 104·2 105·1
132·6 133·4 134·1 134·9 135·7 136·5
179·5 179·9 180·4 181·0 181·5 182·4
A uniform plate resting on a non-homogeneous elastic foundation is shown in Figure 6. The elastic coefficient of the foundation for the inner square section is k2 and that for the surrounding outer section is k1 . The dependence of the fundamental frequencies on the elastic coefficients of this non-homogeneous elastic foundation is illustrated in Table 4. The results are in excellent agreement with the Rayleigh–Ritz solutions presented by Laura and Gutierrez [13]. It can be concluded that the present foundation element is adequate for the analysis of the free vibration of plates on nonhomogeneous elastic foundations. In Table 5 are shown the first six natural frequencies of a square plate of uniform thickness for a range of values of the elastic coefficient of foundation. It is found that the effect of the elastic foundation on the fundamental natural frequency is significant, but the natural frequencies for the higher modes are not greatly affected by the elastic foundation. 3.3. - In order to illustrate the use of the present method for determining the effects of multiple stepped sections and elastic foundations on the modal parameters, the natural frequencies and mode shapes of clamped rectangular and circular plates with several geometrical configurations and supporting foundations have been computed. A stepped rectangular plate resting on non-homogeneous elastic foundations with four edges clamped is shown in Figure 7. The thickness ratios of the stepped sections are h1 :h2 :h3 = 1·0:0·8:0·5.
(13)
The elastic coefficients of the non-homogeneous elastic foundations are k1 = 20k0 ,
k1 = 50k0 ,
k1 = 100k0 ,
(14)
Figure 7. A rectangular plate with multiple stepped sections resting on non-homogeneous elastic foundations.
542
. .
Figure 8. Natural frequency coefficients (bi ) and mode shapes for a rectangular plate with multiple stepped sections resting on non-homogeneous foundations (a without elastic foundation (b).
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Figure 9. A circular plate with multiple stepped sections resting on non-homogeneous elastic foundations.
where k0 = (1/a 4 )Eh13 /12(1 − n 2 ).
(14a)
The natural frequencies and mode shapes of the plate are shown in Figure 8(a). In Figure 8(b) are shown the natural frequencies and mode shapes of a corresponding stepped rectangular plate with the same geometrical configuration but without the supporting elastic foundation. It is found that the fundamental frequency of the plate is affected more by the presence of the elastic foundation. The effect of the elastic foundations on the first mode shape is also more significant than the effect on the higher mode shapes. A stepped circular plate resting on nonhomogeneous elastic foundations (denoted as plate IV) is shown in Figure 9. The thickness ratios of the stepped sections and the elastic coefficients of foundation are the same as in the foregoing example for the rectangular plate except that k0 is redefined as k0 = {1/(3a)4 }Eh13 /12(1 − n 2 ).
(14b)
In order to make comparisons, three other corresponding circular plates (denoted as plates I, II and III) are also considered. The natural frequency coefficients of the four circular plates are given in Table 6. The mode shapes of the stepped circular plate on the
T 6 A comparison of natural frequency coefficients (bi ) for different circular plates; vi = {bi /(3a)2 }zD1 /rh1 , D1 = Eh13 /12(1 − n 2 ) Plate Plate Plate Plate Plate
I II III IV
First mode
Second mode
Third mode
Fourth mode
Fifth mode
Six mode
10·26 13·67 6·56 10·95
21·24 22·42 13·54 15·67
34·83 36·61 24·40 25·07
39·78 40·38 27·34 28·53
60·81 61·67 39·80 40·44
88·42 89·59 42·00 42·74
Plate I: h1 = h2 = h3 , k1 = k2 = k3 = 0; plate II: h1 = h2 = h3 , k1 $ k2 $ k3 $ 0; plate III: h1 $ h2 $ h3 , k1 = k2 = k3 = 0; plate IV: h1 $ h2 $ h3 , k1 $ k2 $ k3 $ 0.
544
. .
Figure 10. Mode shapes of a circular plate with multiple stepped sections resting on non-homogeneous elastic foundations (plate IV).
nonhomogeneous elastic foundations (plate IV) are shown in Figure 10. It is found that the effect of multiply stepped sections on natural frequencies, both for plates with and without supporting elastic foundations, is significant. The higher the mode order, the greater this effect is on the natural frequencies. It is also found that the increases in the fundamental natural frequencies of uniform and stepped circular plates due to the elastic foundations are 33% and 67% of those of corresponding circular plates without elastic foundation, respectively. However, the increases in the natural frequencies of mode 6 are only 1% and 2%, respectively. It can be concluded that the effect of the elastic foundation on the fundamental natural frequency of both uniform and stepped circular plates is more important than that on the natural frequencies for the higher modes.
4. CONCLUSIONS
A finite element model for the analysis of the free vibration of plates with multiple stepped sections resting on nonhomogeneous elastic foundations has been presented. Three types of elements have been used in modelling of the plate. These are the general plate element, the connecting plate element and the elastic foundation element. Both plate elements are based on Mindlin plate theory, with the connecting element being used to relate the inplane translations of the mid-planes of the eccentrically stepped sections. This provides explicitly the bending–extension coupling which is present in such problems. The
545
formulation of the elastic foundation element is made consistent with that of the plate element to which it is attached. The numerical results from the analysis of test problems are in good agreement with available reported results. It is found that the effect of eccentricity of the locations of the stepped sections on the natural frequencies is not negligible, and the effect of the elastic foundation on the fundamental natural frequencies of both uniform and stepped plates is much more significant than on the higher natural frequencies. Natural frequencies and mode shapes of stepped rectangular plates and circular plates resting on non-homogeneous elastic foundations have also been presented for various combinations of stepped sections and elastic foundations.
REFERENCES 1. V. H. C and P. A. A. L 1990 Journal of Sound and Vibration 147, 457–461. Analysis of vibrating rectangular plates of discontinuously varying thickness by means of the Kantorovich extended method. 2. D. V. B, P. A. A. L, A. B and R. C 1991 Journal of Sound and Vibration 150, 167–169. Fundamental frequency of transverse vibration of symmetrically stepped simply supported rectangular plates. 3. I. E. H, X. L and N. B 1992 Journal of Sound and Vibration 153, 51–62. Analytical solution to the free vibration of rectangular plates. 4. R. M and I. E. H 1991 Journal of Sound and Vibration 149, 323–326. Natural frequencies of vibrating sector plates with stepped thickness. 5. T. M 1991 Computers and Structures 41, 377–383. Vibration of stepped sector plates by the spline element method. 6. P. A. A. L, D. A and H. L 1991 Journal of Sound and Vibration 146, 533–537. Numerical experiments on vibrating circular plates with stepped thickness and with edges elastically restrained against rotation and translation. 7. S. K 1991 Journal of Sound and Vibration 149, 259–265. Free vibration of a beam supported on a stepped elastic foundation. 8. S. Y. L and H. Y. K 1990 Computers and Structures 34, 421–429. Free vibration of non-uniform beams resting on non-uniform elastic foundation with general elastic end restraints. 9. J. C. O. N 1991 Journal of Sound and Vibration 145, 279–287. Eigenfrequencies and eigenmodes of beam structures on an elastic foundation. 10. J. W 1991 Journal of Sound and Vibration 149, 315–322. Vibration of stepped beams on elastic foundations. 11. Y. C. L, B. Y. T, W. S. L and B. R. B 1992 Journal of Structural Engineering 118, 853–858. Dynamic response of beams on elastic foundation. 12. P. A. A. L and R. H. G 1984 Journal of Sound and Vibration 95, 537–552. Vibration of plates elastically supported on a nonhomogeneous foundation. 13. P. A. A. L and R. H. G 1985 Journal of Sound and Vibration 101, 307–315. Transverse vibrations of rectangular plates on inhomogeneous foundation, part I: Rayleigh–Ritz method. 14. A. G. H and J. G. M. K 1985 Journal of Sound and Vibration 101, 315–324. Transverse vibrations of rectangular plates on inhomogeneous foundation, part II: modal constraint method. 15. I. E. H and H. R. M 1990 Journal of Sound and Vibration 138, 524–528. Vibration of sector plates on elastic foundations. 16. J. E. B 1977 Foundation Analysis and Design. New York: McGraw-Hill; second edition. 17. A 1992 Engineering Analysis System Version 5.0, User’s Manual. Houston, Texas: Swanson Analysis Systems, Inc. 18. D. S. C and J. W. W 1990 Journal of Structural Engineering 116, 3236–3241. Plates on elastic foundation. 19. A 1992 Version 5.2 User’s Manual. Pawtucket: Hibbitt, Karlsson & Sorensen, Inc. 20. R. D. C, D. S. M and M. E. P 1989 Concepts and Applications of Finite Element Analysis. New York: John Wiley. 21. T. J. R. H 1987 The Finite Element Method. Englewood Cliffs, New Jersey: Prentice-Hall. 22. H. C. H 1989 Static and Dynamic Analysis of Plates and Shells. Berlin: Springer-Verlag.