Thin-Walled Structures 32 (1998) 275–287
Nonlinear vibration of thin plates with initial stress by spline finite strip method Y.K. Cheunga,*, D.S. Zhub, V.P. Iuc a
Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong, PR China b Department of Mechanics, Huazhong University of Science and Technology, Wuhan, PR China c Department of Civil Engineering, University of East Asia, Macau, Macau
Abstract The spline finite strip method and the incremental time–space finite element procedure are used to analyse large amplitude vibration of plates with initial stresses. Two improvements for the procedure are presented. The free vibration and the internal resonance of plates with initial stress as well as the forced vibration of plates with damping and initial stress are computed. The results compared favourably with those available in other publications. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear vibration of plates; Internal resonance; Spline finite element; Nonlinear frequency– response behavior
1. General It is well known that when the vibration amplitude becomes comparable to or larger than the plate thickness, the nonlinear interaction between the bending and stretching of the plate becomes significant, leading to a variety of complex responses, which cannot be analysed by the linear theory of plates. The problem should then be treated with the nonlinear theory of plates, as that the effect of large amplitudes can be considered. With the increasing use of thin plate members in modern light weight structures, it will be of great technical importance to clarify the whole aspect of the nonlinear responses of the plate undergoing large amplitude flexural vibrations.
* Corresponding author. 0263-8231/98/$ - see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 2 2 - 6
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The basic equations for the nonlinear vibration of plates are the dynamic analogs of the von Karman equations modified by the transverse inertia term. Based on these equations, numerous studies have been conducted on the nonlinear vibration of circular, rectangular and other shaped plates including anisotropic and laminated plates under various boundary conditions. An extensive bibliography can be found in Refs. [1,2]. The finite element method has gained wide acceptance and application in engineering practice during the past 30 years. It can be said that this method is the only one suitable for the analysis of various complicated engineering structures up to now. Therefore, the study on the finite element method used for the nonlinear vibration of structures is a very important topic for the analysis of engineering practical problems and the application of nonlinear vibration theory. Many researchers have devoted a lot of effort to the study on this topic, and the bibliography on their work can be found in Refs. [1,2]. However, all these methods, including the direct integration methods [3], have some difficulty in obtaining the complex responses [1], such as internal resonance, superharmonic resonance and so on. Cheung and colleagues [4,5] developed an increment time–space finite element procedure for nonlinear structural vibration. This procedure is capable of treating highly nonlinear problems with complex responses because the in-plane displacement and inertia are taken into account and an increment-iteration procedure is adopted. The nonlinear free vibration, forced vibration and internal resonance responses of the plates have been computed by this method and the modified DKT plate element [5,6]. This method has also been applied to the nonlinear vibration of laminated plates [7]. Nonlinear vibration of plates under various initial conditions is encountered often in engineering practice. Some problems with this type of vibrations have been analysed by several researchers using analytical methods. For example, Crawford and Atluri [8] and Harari [9] analysed the nonlinear free vibration of plates with initial stress, Chen and Doong [10] and Bhimaraddi [11] analysed the large amplitude free vibration of moderately thick plates and laminated plates with initial stress, respectively. Yamaki and colleagues [12,13] performed the analysis and experiment for the nonlinear free and forced vibrations of a clamped plate with initial deflection and initial edge displacement. The finite strip method developed by Cheung [14] in the late 1960s has gained wide acceptance and has been applied to the nonlinear analysis of structures [15,16]. In the past few years, Cheung and colleagues [17] introduced the spline finite strip/element method, which has been considered very successful in linear analyses of structures [18,19]. More recently, the method has been extended to large deflection analysis of arbitrary shaped thin plates [20] and the postbuckling analysis of shells [21–23]. This has greatly enlarged the range of application of the finite strip method. Due to the fact that spline functions have good interpolation ability for sharply varying deformations, this method is particularly suitable for the analysis of large and complex deformation problems. The present study concerns the first application of the spline finite strip/element method to nonlinear vibration of plates using the incremental time–space finite element procedure with some improvement. The free vibration and internal resonance
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277
of the plates with the initial stress as well as the forced vibration with damping and initial stress of plates are computed. The results are in reasonable agreement with the ones in the available references.
2. Spline finite strip/element method In the context of the spline finite strip method, the structure is discretized by using n strips along the x-axis and m sections along the y-axis, so that there are n ⫻ m subdomains and (m ⫹ 3) ⫻ (n ⫹ 1) nodes for computation. The interpolation functions for displacement u, v, w in the rectangular strip are expressed as a product of B3 spline functions ⌽1(y) ⫺ ⌽8(y) with m ⫹ 3 nodes in the y direction and a conventional linear interpolation N1(x) ⫺ N2(x) for u, v and beam functions N3(x) ⫺ N6(x) for w in the other direction. For more complicated problems, it is no longer efficient to adopt the linear interpolations N1, N2 for displacements u, v. In the present study the beam functions are also used for u, v in the x direction, in which case,
冦冧 冤 u
N3 N4 0
0
v ⫽
0
0 N3 N4 0
w
0
0
0
0
0 N5 N6 0 0
0
0 N3 N4 0
0
0 N5 N6 0 0
0
0
冥
0 •
0 N5 N6
⌽1
0
ui ⌽2
ui,x
⌽3
vi
⌽4
vi,x
0 ⌽5
wi
⌽6
•
⌽7 ⌽8
vi ⫹ 1
⌽10
vi,x⫹ 1
⌽11
ui ⫹ 1 ui,x⫹ 1
⌽9
0
wi,x
wi ⫹ 1
⌽12 wi,x⫹ 1
where ui, vi, wi are displacement parameter vectors associated with the displacements of nodal line i, and ui,x ⫽
冉 冊
i
∂u ∂x
and so on. For the detailed theory and formulation
of the spline finite strip method, please see Refs. [17,24,25]. The spline finite strip method not only inherits the advantages and the applicability
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of the classical finite strip method but also possesses some outstanding characteristics. For example, the adopted B3 spline is made up of simple piecewise cubic polynomials with C2 continuity everywhere. It reduces the risk of unstable calculations and improves accuracy, and is therefore suitable for the analysis of problems with complex deformations. 3. Time–space finite element or finite strip procedure By the total Lagrangian formulation, the virtual work equation with respect to displacement increments for a dynamic problem can be written as
冕冕冕
(ij ⫹ ⌬ij )␦(⌬⑀ij )dV(0) ⫹
v
冕冕冕
∂2(ui ⫹ ⌬ui) ␦(⌬ui)dV(0) ⫽ W ∂t2
(1)
v
i,j ⫽ 1,2,3 where ⑀ij is Green’s strain tensor, ij is the second Piona–Kirchhoff stress tensor, and dV(0) are the elementary volume in the undeformed state ⍀(0), denotes the density of the material, t stands for time, ui is the displacement at ⍀(N) state and W is the external virtual work expression at the ⍀(N+1) state. ij, ⑀ij are taken in the ⍀(N) state and referred to the undeformed configuration. All increments are taken from the ⍀(N) state to the ⍀(N+1) state. The present study is only concerned with the periodic motion of the elastic system. Introducing a dimensionless time ⫽ ( ⫹ ⌬)t, where is the frequency of vibration at ⍀(N) state and ⌬ is its increment from the ⍀(N) state to the ⍀(N+1) state, and invoking Hamilton’s principle, one can obtain
冕 再冕冕冕 冕冕
2
0
(ij ⫹ ⌬ij )␦(⌬⑀ij )dv ⫹ ( ⫹ ⌬)2
v
∂2(ui ⫹ ⌬ui) ␦(⌬ui)dv ∂2
(2)
v
( ⫹ ⌬)Pi␦(⌬ui)ds其d ⫽ 0,
⫺
冕冕冕
i,j ⫽ 1,2,3
s
where is a load factor on proportional loading; ⌬ is its increment and Pi is a generalized total surface load vector for plates. The superscripts (0) on V and s have been ignored. Based on von Karman’s large deflection plate theory, the strain–displacement relationship is ∂u1 1 ∂u3 2 ∂2u3 ⫹ ⫺ x3 2 ∂x1 2 ∂x1 ∂x1
⑀⫽
冉 冊 冉 冊
∂u2 1 ∂u3 ⫹ ∂x2 2 ∂x2
2
⫺ x3
∂2u3 ∂x22
.
∂u1 ∂u2 ∂u3 ∂u3 ∂2 u 3 ⫹ ⫹ ⫺ 2x3 ∂x1 ∂x1 ∂x2 ∂x1∂x2 ∂x2
(3)
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279
When one gives the displacement a set of increments, the strain can be written in the following form
⑀ ⫽ ⑀0 ⫹ ⑀1 ⫹ ⑀2
(4)
where ⑀0, ⑀1 and ⑀2 represent, respectively, the parts containing 0, 1, 2 order terms of the displacement increment and all are vectors. Substituting the linear elastic constitutive relationship and Eq. (4) into Eq. (2), and ignoring the third order and fourth order terms of the displacement increment and the frequency increment in the results, the following formula can be obtained
冕 再冕冕冕 冕冕冕冋 冕冕
2
[(␦⑀1)TD⑀1 ⫹ (␦⑀2)TD⑀0 ⫹ (␦⑀1)TD⑀0]dV
0
v
⫹
2
册
∂2ui ∂2ui ∂2⌬ui 2 ␦(⌬ui)dV 2 ⫹ 2⌬ 2 ⫹ ∂ ∂ ∂2
v
( ⫹ ⌬)Pi␦(⌬ui)ds其d ⫽ 0,
⫺
(5)
i ⫽ 1,2,3
s
in which D is the usual linear elastic constitutive relationship matrix of the plate. Substituting the displacement interpolation formula ui ⫽ Si nan and ⌬ui ⫽ Si n⌬an,
(6)
where Sin is the shape function and an is the nodal displacement with n ⫽ 1,…,N and N is the number of the nodal degrees of freedom, into Eq. (5), then a corresponding finite element equation can be obtained easily. When one considers only the analysis of periodic vibration the vectors an and ⌬an are periodic functions of with a period of 2. They are, therefore, suitable for expanding into Fourier series, while a reduced basis consisting of linear eigenvectors of the same problem is used in order to achieve higher efficiency, i.e. a1
T 11 $ T 1 $ T M 1
·
·
·
·
·
·
·
·
·
·
an ⫽ T
· 1 n
· n
$ T
$ T
M n
·
·
·
·
·
·
·
·
·
·
·
·
aN
T
1 N
$ T
N
冘
1 d ⯗
1 d ⯗
d cosi ⫹ 冘 d sinj , ⯗ ⯗
$ TM N
fc
i⫽0
fs
j⫽1
d M i
d M j
(7)
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or an ⫽ T n d ␣F␣ and ⌬an ⫽ T n ⌬d ␣F␣,
(8)
where T stands for the nth element of the th mode vector in the reduced basis, and F␣ stands for a trigonometric function in the ␣th term of Fourier series. After the Fourier series expansion and the reduced basis treatment the number of unknowns related to the displacement becomes ⫹ ␣, and d ␣ is a component of the unknown vector. A superscript denotes a mode number from 1 to M, and a subscript ␣ denotes a term number of the Fourier series from 1 to f, where M is the number of the chosen reduced basis and f ⫽ 1 ⫹ fc ⫹ fs is the total number of terms of the Fourier series chosen. After substituting Eqs. (6) and (8) into Eq. (5), the multiplication of 2 苲 4 trigonometric functions and their integration with respect to will exist in each element of the matrixes and vectors obtained. Their calculations can be performed using the following general formulae n
cosi · cosj ⫽
1 [cos(i ⫺ j) ⫹ cos(i ⫺ j)], 2
sini · sinj ⫽
1 [cos(i ⫺ j) ⫺ cos(i ⫺ j)], 2
sini · cosj ⫽
1 [sin(i ⫺ j) ⫹ sin(i ⫺ j)] 2
and
冦
冕
cosi · cosj d ⫽
冕
cosi · cosj d ⫽
冕
cosi · cosj d ⬅ 0.
2
0
2
0
0
i⫽j
i ⫽ j⫽0 ,
2 i ⫽ j ⫽ 0
再
0 i⫽j or i ⫽ j ⫽ 0 , i ⫽ j⫽0
2
0
It is easy to see that their results are equal to 0 or or 2 only. In the present work, only viscous damping is considered. When such a damping C is introduced, starting from the very beginning, an extra integral associated with the virtual work done by the damping forces
冕冕冕
C
v
∂ui ␦(⌬ui)dV(0) ∂t
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281
is added to Eq. (1). In the case of modal damping covered in this study, the global damping matrix exhibits a diagonal form with the th diagonal coefficient equal to 2() (), where () is the damping ratio of the th chosen mode of the reduced basis and is the linear frequency of the corresponding mode. Depending on the type of problems to be tackled, the choice of the Fourier expansion may vary. In general, with the presence of viscous damping, both sine and cosine terms must be included to cater for the phase lag of /2 between the velocity and the displacement. When a plate with initial in-plane stress is considered, the following integral should be added to the left side of Eq. (5)
冕 冕冕冕
2
[(␦⑀1)TS 0/h ⫹ (␦⑀2)TS 0/h]dV d,
0
v
where S ⫽ [S 0x S 0y S 0xy] is an initial in-plane force vector distributed uniformly and h is the thickness of the plate. It is notable that when S0 are expanded into Fourier series, only the coefficient of the cos(0) term is not zero because S0 are constants with respect to . There is a more detailed representation on basic formulation of the procedure in Refs. [5,6]. Two improvements are presented in our study: (1) The reduced basis is adopted for the in-plane and transverse nodal displacements instead of only for transverse displacement as in Refs. [5,6]. The feasibility of the improvement is due to the adoption of the spline finite strip, which can calculate accurately the higher order in-plane modes and so the interaction between in-plane modes and transverse displacement modes can be included, and this has been confirmed by the computational examples. This improvement reduces the computer’s memory needed and heightens computational efficiency further. (2) The supposition that the resultant force of the in-plane stress within any element is approximately a constant, or alternatively the adoption of an average value in the calculation, is eliminated, so that the complicated deformation state can be described accurately by the procedure together with the spline function. 0
4. Numerical examples Based on the formulation stated in Section 3, problems of the nonlinear stationary periodic vibration can be solved by using the increment-iteration procedure. In all the following examples, four finite strips and four sections of the spline are used to discretize one quarter of the plate due to symmetry. Ten modes at the most are used in the reduced basis with three flexural modes, while the others are extensional modes. The Fourier expansion of the displacements for undamped systems includes up to the third harmonic terms and the transverse motions of plates are described as
冘 3
w(x,y,t)/h ⫽
Aicosit,
0
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where w is the transverse displacement of the plate and Ai is termed the ith harmonic dimensionless amplitude, which is the sum of the contributions of the mode shapes in the reduced basis multiplied by the corresponding generalized coordinates. 4.1. Nonlinear free vibration of square plates with initial stress The nonlinear free vibration of a square plate with initial stress is computed. The plate parameters are assumed to be 1 h √D/(h) ⫽ 1, ⫽ 1, ⫽ 0.3, a2 a where a is the edge length of the square plate, D is the bending stiffness and is Poisson’s ratio. The boundary conditions are w ⫽ u ⫽ v ⫽ 0 at x ⫽ 0,a and y ⫽ 0,a. The plate is subject to an initial internal force with an intensity equal to S 0i ⫽ ␣i
2D , a2
where ␣i is a dimensionless constant. This initial internal force is distributed uniformly along the y-axis direction or both the x-axis and the y-axis directions. The solution process starts from an initial solution near the corresponding known linear solution with a sufficiently small amplitude. For the nonlinear fundamental frequency, the initial solution can be taken as the fundamental linear frequency for the problem with the initial stress, and the generalized coordinate corresponding to the cost term and the first reduced basis are small values with all the others being zero. It is to be noted that starting from any specified frequency, the computation will automatically converge to some nearby linear frequency of the system. This characteristic is helpful in the analysis of a system in which the linear vibration frequency is not known in advance. The backbone curves with various initial internal forces in the x-axis direction are shown in Fig. 1, where A1c represents the first harmonic dimensionless amplitudes at the centre point c. It can be seen that the curve obtained without the initial stress is in agreement with Ref. [6] and the initial compressive force produces a softening effect on the nonlinear frequency but the initial tensile force produces a hardening effect. The curves obtained by Crawford and Atluri [8] using a perturbation technique are also shown in the same figure for comparison. They themselves have pointed out that the upper part of the curves obtained by them are not consistent with experiment. However, the experimental results were not given. The backbone curves of a square plate with the initial internal force in both axis directions are also calculated and the results shown also in Fig. 1. It can be seen that the results for ␣x ⫽ ␣y ⫽ ⫾ 1 are identical to those of ␣x ⫽ ⫾ 2 while ␣y ⫽ 0.
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 275–287 S0x = αx
π2D , a2
S0y = αy
283
π2D a2
A1c Ref. [6] and present method present method 2.0
αx = αy = 1
Ref. [8]
αx = αy = 0 αx = αy = −1 αx = −2, αy = 0
αx = 2, αy = 0
1.0
0.0 1.0
2.0
ωω
/ 11
Fig. 1. Backbone curve of square plate with initial stress.
4.2. Forced vibration of plates with damping and initial stress To check further the correctness of the computational procedure adopted, the forced vibration of a square plate without damping and without initial stresses is computed first. The parameters and the boundary conditions of the plate are the same as the square plate in Section 4.1. The plate is excited by a uniformly distributed force with an intensity equal to q ⫽ 1.5
Dh cost. a4
The computational procedure is divided into two steps. Firstly the response at a certain fixed frequency is computed by increasing the exciting force incrementally to its prescribed value. After that, the amplitude–frequency response is computed with the forcing term remaining constant. The result computed for the centre of the plate is shown in Fig. 2. It can be seen that there are two separate branches of solution under resonance, which correspond to the in-phase and the out-of-phase responses to the exciting force, respectively. The amplitudes of the left and the right side solutions are denoted by A1c, A3c and A⬘1c, A⬘3c, respectively, for clarity. It can be seen from the figure that the result computed is in agreement with Ref. [6]. Based on the previous computation, the forced vibrations with damping and different initial stresses for the same plate are computed. Due to the presence of the damping force that is related to the velocities of the vibrating system, a cosine har-
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1.5 Ref. [6]
+A1c’ +A3c× 10, −A′1c’ −A′3c× 10
present method A1c A′1c
1.0
0.5
A3c A′3c
0.0 0.6
0.8
1.0
Fig. 2.
1.2
1.4
1.6
ωω
/ 11
Fundamental harmonic resonance of a square plate.
monic term in the Fourier expansion of the displacements produces a sine harmonic term. For consistency, it is necessary to include the sine harmonic terms in the expansion. Hence, the expression for the out-of-plane motion is rewritten as
冘 冘 3
w(x,y,t)/h ⫽
i⫽0
冘 3
Acicosit ⫹
Asj sinjt
j⫽1
3
or ⫽ A0 ⫹
Aicos(it ⫹ i).
i⫽1
The two branches of the undamped responses are connected in this case through a gradual change of the phase difference of each harmonic amplitude. The computation is carried out for a constant damping ratio ⫽ 0.05. The amplitude response curves with different initial stresses are illustrated in Fig. 3. It can be seen from the figure that the initial tensile stress reduces the amplitude of the vibration with a translation of the backbone curve to the direction of increasing frequency, but the initial compressive stress produces an opposite effect. This trend is in agreement with the conclusion that initial tensile stress increases the stiffness of plates but initial compressive stress reduces it. 4.3. Internal resonance of a rectangular plate with initial stress The nonlinear free vibration of a rectangular plate with initial stress as shown in Fig. 4 is also studied. For such a plate without initial stress, initial resonance will
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 275–287
S0x = α x
Amplitude at center / thickness
1.0
π2D α2
ξ = 0.05
αx = −2 αx = 0 αx = 2 0.5
0.0 0.6
0.8
Fig. 3.
S0x = αx y
α
1.0
1.2
ωω
/ 11
1.4
Damped vibration of a square plate with initial stress.
2α
2.0
π2D α2
αx = β
S0y = 0
A1c
αx = 0
β = 0.78125
x Amplitude / thickness
S0y = 0
285
αx = β
αx = −β
αx = 0 αx = −β
present method Ref. [8] 1.0
A3c
αx = 0 αx = β αx = −β 0.0 10
Fig. 4.
20
30
40
ω
Backbone curve of rectangular plate with initial stress.
not occur in the studied frequency region. However, it can be seen from the computational result that the internal resonance will appear in the plate with an initial stress. This is due to the fact that the linear frequency 13 of the rectangular plate under the initial compressive stress is slightly larger than three times the fundamental frequency 11. Therefore, for large amplitudes, internal resonance occurs due to the cos3t term being excited. The response curves computed are shown in Fig. 4. It can be seen that such internal resonance appears only in a narrow frequency range
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Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 275–287
( is from 10.9 to 11.1) with the transfer of energy from A1 to A3. The internal resonance curve under initial compressive stress is enlarged and plotted in Fig. 5. This curve is similar to the internal resonance curve for the rectangular plate with a length/width ratio 1.5 and without initial stresses in Ref. [6]. It can also be seen from Fig. 4 that all A3c curves increase suddenly with the transfer of energy from A1 to A3 when is close to 13. The curves obtained using the perturbation technique in Ref. [8] are also plotted in Fig. 4. Similar to the square plate problem mentioned previously, the curves are inaccurate for amplitude ratios larger than one as pointed out by the authors of this reference, and no reference was made about the internal resonance phenomenon obtained in our study.
5. Conclusion 1. The spline finite strip method has been successfully extended to the nonlinear vibration field. The free vibration and the internal resonance of plates with initial stress as well as the force vibration of plates with damping and initial stress are analysed. All examples demonstrated the capability and versatility of the method. 2. Two improvements for the incremental time–space finite element procedure have been incorporated due to the excellent interpolation ability of the spline function. The improvements have increased computational efficiency and capability to analyse the complex nonlinear dynamic response of the procedure. 3. Internal resonance will appear in a plate under certain initial stress but does not occur in the same plate without initial stress. This phenomenon is confirmed by the computation and is significant for the vibration analysis of engineering structures.
A1c, A3c
π2D
2a S0x = −0.78125
α
y
a2
S0y = 0 x
A1c 0.5
A1c
A3c
A3c
0.0 8.5
9
Fig. 5.
10
11
12
Free internal resonance of rectangular plate with initial stress.
ω
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287
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