# Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

Authors: D. V. Bambill; C. A. Rossit
Journal: Journal of Engineering
 (2013)

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## Abstract

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.


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## Body

## 1. Introduction

The transverse-free vibrations and buckling of plates which are subjected to edge loads acting in their middle planes are areas of research which have received a great deal of attention in the past century.As it was stated experimentally by Hearmon [1] for the case of a beam, bifurcation buckling may be regarded as a special case of the vibration problem, that is, determining the in-plane stresses which cause vibration frequencies to reduce to zero.Most of the work has dealt with rectangular plates having uniformly distributed in-plane edge loads. In that case, the governing differential equations of motion and equilibrium have constant coefficients, yielding exact solutions for frequencies and buckling loads straightforwardly when two opposite edges of the plates are simply supported.Many researchers have analyzed both the buckling and vibration of rectangular plates subjected to in-plane stress field. Among them, one can mention Kang and Leissa [2]; Leissa and Kang [3]; Bassily and Dickinson [4]; Dickinson [5]; Kielb and Han [6]; Kaldas and Dickinson [7].For the linearly varying loading, the governing differential equations have variable coefficients.Leissa and Kang [3] found exact solutions for the free vibration and buckling problems of the SS-C-SS-C isotropic plate loaded at its simply supported edges by linearly varying in-plane stresses.They also found the exact solution [8] for the buckling of rectangular plates having linearly varying in-plane loading on two opposite simply supported edges, with different boundary conditions at the other opposite edges.Within the realm of the classical theory of plates, the case of buckling and vibrations problems for all the possibilities of boundary conditions and linearly varying in-plane forces offers considerable difficulty. This is the reason why it is quite common to make use of the Ritz variational method.


## 2. Approximate Analytical Solution

In the case of a transversely vibrating, thin, isotropic plate subjected to in-plane forcesNx, Ny, and Nxy, (Figure 1 and (5)), the maximum value of the potential energy due to bending deformation is
(1)Umax=12D∫A[(∂2W∂x-2+∂2W∂y-2)2+2(1-ν)×(∂2W∂x-2∂2W∂y-2-(∂2W∂x-∂y-)2)]dx-dy-,
where W=W(x,y) is the deflection amplitude of the middle plane of the plate, D is the well known flexural rigidity D=Eh3/12(1-ν2), E is the Young modulus, and ν is the Poisson coefficient.Figure 1
Rectangular vibrating plate subjected to in-plane loads:N-x, N-y, and N-xy.While the maximum of the kinetic energy is(2)Tmax=12ρω2∫AhW2dx-dy-,
where ρ is the density of the plate material, ω is the circular frequency, and h is the thickness of the plate.And the maximum potential energy of the internal stresses caused by the in-plane loading is(3)τN=12∫AP(N-x(∂W∂x-)2+N-y(∂W∂y-)2+2N-xy∂W∂x-∂W∂y-)dx-dy-.The lengths of the sides of the rectangular plate area in the x direction and b in the y direction. The coordinates are written in the dimensionless form as follows:
(4)x=x-a,y=y-b.And the in-plane forces are expressed as (Bambill et al. [9]):
(5)Nx=N-xb2D,Ny=N-yb2D,Nxy=N-xyb2D.Then, the governing functional of the system is(6)J[W]=12D[ba3∫ANx(∂W∂x)2dxdy+1ab∫ANy(∂W∂y)2dxdy+2a2∫ANxy∂W∂x∂W∂ydxdy(∂W∂x)]+12D[ba3∫A(∂2W∂x2)2dxdy+2νab∫A∂2W∂x2∂2W∂y2dxdy+ab3∫A(∂2W∂y2)2dxdy+2(1-ν)ab∫A(∂2W∂x∂y)2dxdy]-12ρabω2∫AhW2dxdy.Equation (6) satisfies, if W is the exact solution, the condition:
(7)δJ[W]=0.Following the Ritz method, the expression of the deflection of the plate is approximated in the form of a truncated series:(8)W≅Wa(x,y)=∑m=1M∑n=1NAmnXm(x)Yn(y),
where Xm(x) and Yn(y) are the characteristic functions for the normal modes of vibration of beams with end conditions nominally similar to those of the opposite edges of the plate in each coordinate direction [10]. Consequently, they satisfy the essential boundary conditions, as the method requires.The variational equation (7) is replaced by the homogeneous linear system of equations:
(9)∂J[Wa]∂Aql=0,q={1,2,…M};l={1,2,…N},
and using nondimensional variables, becomes
(10)abD∂J[Wa]∂Aql=0.Finally, one obtains a homogeneous linear system of equations in terms of theAql’s parameters.The nontriviality condition of the system (10) requires the determinant to be zero:
(11)|S·[τqlmn]+[uqlmn]-Ω2·[tqlmn]|=0,
where Ωmn=ωmna2ρh/D are the frequency coefficients.The elements of the matrices involved in (11) are given by
(12)τqlmn=1λ2∫AnNx*(dXqdxdXmdx)(YlYn)dxdy+∫AnNy*(XqXm)(dYldydYndy)dxdy×1λ∫AnNxy*(dXqdxXqdXmdxdYldyYn+dXqdxXmYldYndy)dxdy,
where λ=a/b is the aspect ratio
(13)Nx=S·Nx*,Ny=S·Ny*,Nxy=S·Nxy*,
where S is a factor that indicates the magnitude of the in-plane loading system, regarding the relative value of the forces. Consider
(14)uqlmn=λ-2∫A(d2Xqdx2d2Xmdx2)(YlYn)dxdy+ν∫A[(Xqd2Xmdx2)(d2Yldy2Yn)+(d2Xqdx2Xm)(Yld2Yndy2)]dxdy+λ2∫A(XqXm)(d2Yldy2d2Yndy2)dxdy+2(1-ν)∫A(dXqdxdXmdx)(dYldydYndy)dxdy,tqlmn=λ-2∫A(XqXm)(YlYn)dxdy.As it is known, the conditionΩ=0 in (11) yields the critical value of the in-plane loading.


## 3. Numerical Evaluations

Hearmon [1] has experimented on a fixed-free strip. Admittedly, the problem is analytically simpler in the case of one-dimensional domains. As an example, let us try with a pinned-pinned transversely vibrating beam, subjected to an axial compressive force P. The expression of the frequency coefficient is
(15)Ωn=ρA0EIL2ωn=(nπ)2[1-PL2EI(nπ)2];withn=1,2,3,4…,
where ρ is the density of the material, A0 is the cross-section, L is the length, and EI the flexural rigidity of the beam.All the Euler buckling loads are determined making zero expression (15). For n=1, the critical buckling load of the beam, Pcrit, is obtained.Plotting the valuesΩn=(nπ)2(1-(P/n2Pcrit)) of the first three frequency coefficients depending upon the ratio P/Pcrit yield regular curves as is shown in Figure 2. The presence of the compressive axial load P does not alter the order of the modal shapes of the beam.Figure 2
Curves of the frequency coefficients of transverse vibration for a pinned-pinned beam under axial compressive loadP.In the case of a plate, in general, and due to the bidimensional behavior induced by the torsional rigidity, the compressive in-plane load may alter both the order and shape of the modal shapes associated to each natural frequency.This situation has an important technological signification from the point of view of vibration control.Certainly, the modal shape of a natural resonant frequency must be known in order to suppress it. In the case of in-plane loading, this shape can be different from the expected one.Due to the quantity and variability of the parameters involved in the description of the behaviour of these kinds of structures, just a few representative cases will be considered to demonstrate the convenience of the procedure and the importance of the situation.All the values are determined takingM=N=15 in (8).Table1 shows the values of the first natural frequency coefficients for a CCFF plate subjected to a general in-plane loading: linear load in x direction (α=2)—bending moment, constant load in y direction (β=0), and constant shear force Nxy=N1=N2=N).Table 1
The first six natural frequency coefficientsΩi for a C-C-F-F plate under general in-plane loading.
N
/
N
crit
0
0.25
0.5
0.75
1
λ
=
a
/
b
=
0.75; Ncrit=5.95168
Ω
1
5.39789
4.75442
3.95041
2.84503
0
Ω
2
15.7547
15.2037
14.5835
13.8967
13.1443
Ω
3
23.8207
23.9071
24.0071
24.1145
24.2241
Ω
4
36.3999
36.2826
36.1113
35.7654
35.1888
Ω
5
38.1446
37.5193
36.9334
36.5072
36.2961
Ω
6
59.7605
59.5098
59.2367
58.9439
58.6327
λ
=
a
/
b
=
1; Ncrit=3.69137
Ω
1
6.93254
6.06065
4.99441
3.56413
0
Ω
2
23.9780
23.9778
23.6766
23.0314
22.1751
Ω
3
26.6265
26.0033
25.6318
25.5511
25.6238
Ω
4
47.7179
47.4465
47.1607
46.8601
46.5440
Ω
5
62.8830
62.6774
62.3443
61.9069
61.3970
Ω
6
65.6818
65.4396
65.3185
65.2954
65.3384
λ
=
a
/
b
=
2; Ncrit=1.85222
Ω
1
17.1515
14.9296
12.2458
8.69504
0
Ω
2
36.4410
36.0647
35.6415
35.1668
34.6357
Ω
3
73.5943
73.8555
74.0906
74.2962
74.4680
Ω
4
91.1002
89.5702
88.0081
86.4144
84.7901
Ω
5
115.363
114.262
113.147
112.021
110.886
Ω
6
131.962
132.267
132.566
132.855
133.130In order to show the influence of the in-plane loading, the next two examples are presented.Table2 shows the natural frequency coefficients for a C-C-SS-SS square plate under uniform compression in the x direction (Nx=N, α=0, and Ny=Nxy=0).Table 2
The first six natural frequency coefficientsΩi for a C-C-SS-SS plate under uniform Nx loading (α=0).
λ
=
a
/
b
=
1 Ncrit=61.4151
N
/
N
crit
0
0.1
0.25
0.5
0.75
1
Ω
1
27.0542
25.7489
23.6368
19.5326
14.0587
0
Ω
2
60.5387
58.4465
54.9733
48.6153
41.2865
32.4139
Ω
3
60.7863
60.1238
59.2827
57.8385
56.3769
54.8441
Ω
4
92.8371
91.4269
89.2672
85.5386
81.6286
77.5081
Ω
5
114.557
112.082
108.152
101.257
93.8564
85.8259
Ω
6
114.704
114.350
113.923
113.207
112.485
111.755Figure3 shows that a minimal presence of in-plane loading (10% of the critical value) dramatically modifies the mode shapes, while changes in the values of frequencies may not be noticed (0.3% in the sixth frequency). It is important to point out that the small load can be originated by thermal variations and restrictions on plane displacements imposed by the external supports.Figure 3
Normal modes of vibration of a square C-C-SS-SS plate under uniformNx loading (λ=a/b=1, Ncrit=61.4151).Finally, Table3 shows the results for a rectangular C-C-SS-SS plate subjected to shear in-plane forces.Table 3
The first six natural frequency coefficientsΩi for a C-C-SS-SS plate under uniform shear Nxy loading.
λ
=
a
/
b
=
1.5 Ncrit=89.8137
N
/
N
crit
0
0.1
0.25
0.4
0.5
0.75
0.9
1
Ω
1
44.8904
44.7005
43.7991
42.4070
40.2822
32.4323
22.3996
0
Ω
2
76.5451
76.1864
74.3165
70.7817
67.4142
54.4376
41.3590
27.7956
Ω
3
122.319
122.349
122.548
121.465
117.487
104.882
95.5807
88.7020
Ω
4
129.393
128.871
126.129
122.871
123.101
123.415
123.108
122.473
Ω
5
152.529
152.707
153.680
155.033
155.912
157.429
157.633
157.273
Ω
6
202.615
200.101
194.748
188.611
184.103
171.463
163.073
214.018Figure4 shows that the third and fourth natural frequencies interchange their normal modes as Nxy increases. This situation is noticeable from Figure 5.Figure 4
Normal modes of vibration of a rectangular C-C-SS-SS plate under uniformNxy loading (λ=a/b=1.5, Ncrit=89.8137).Figure 5
The first six natural frequency coefficients Ωi for a C-C-SS-SS plate under uniform shear Nxy loading (λ=a/b=1.5, Ncrit=89.8137).This means that for a given value ofNxy, between 0.25 and 0.4 of the critical value, there are two normal modes for the same natural frequency (repeated frequency). This is an important point in vibration control, since when repeated frequencies arise in a system, the related vibration mode shape cannot be uniquely determined. Any linear combination of the modes is still valid for the repeated frequency.In order to evaluate the accuracy of the expounded procedure, comparison is made with the results obtained in [3] for a SS-C-SS-C plate loaded at its simply supported edges by linearly varying in-plane stresses (Tables 4 and 5).Table 4
Comparison of nondimensional critical buckling loadsNcrit=N-critb2/D for a SS-C-SS-C plate.
Load(α)
Solution
λ
=
a
/
b
0.4
0.5
0.6
0.7
0.8
0.9
1.0
[A]M = 5
93.3059
75.9452
69.6553
69.1116
72.0966
77.5543
75.9452
[A]  M = 10
93.2555
75.9146
69.6351
69.0972
72.0859
77.5460
75.9146
0
[A]  M = 15
93.2477
75.9105
69.6323
69.0954
72.0846
77.5450
75.9105
[A]  M = 20
93.2476
75.9101
69.6323
69.09531
72.0844
77.5449
75.9101
[B]
93.247
75.910
69.632
69.095
72.084
77.545
75.910
[A]M = 5
174.533
145.286
134.809
134.624
140.981
152.024
145.286
[A]  M = 10
174.395
145.215
134.765
134.593
140.958
152.007
145.215
1
[A]  M = 15
174.379
145.207
134.760
134.590
140.956
152.005
145.207
[A]  M = 20
174.377
145.206
134.760
134.590
140.956
152.005
145.206
[B]
174.4
145.2
134.8
134.6
141.0
152.0
145.2
[A]M = 5
401.518
392.147
412.162
424.140
401.518
392.143
392.147
[A]  M = 10
400.478
391.589
411.812
422.594
400.478
391.398
391.589
2
[A]  M = 15
400.410
391.548
411.790
422.490
400.410
391.351
391.548
[A]  M = 20
400.399
391.548
411.787
422.472
400.399
391.343
391.547
[B]
400.4
391.5
411.8
422.5
400.4
—
391.5
[A]: present approach with different M=N and [B]: [3].Table 5
Comparison of the first six natural frequency coefficients for a SS-C-SS-C plate under bending moment inx-direction (α=2).
λ
=
a
/
b
N
/
N
crit
Solution
Ω
1
Ω
2
Ω
3
Ω
4
Ω
5
Ω
6
0
[A]
13.6858
23.6465
38.6942
42.5871
51.6767
58.647
[B]
13.69
23.65
—
42.59
51.67
—
0.5
0.5
[A]
11.4936
24.152
37.9343
38.8757
52.632
58.7048
[B]
11.49
24.15
37.93
—
52.63
—
0.95
[A]
3.92178
24.906
27.6496
39.3337
52.6657
58.8581
[B]
3.926
24.91
27.65
—
52.66
—
0
[A]
28.9509
54.7433
69.3271
94.5862
102.217
129.096
[B]
28.95
54.74
69.33
94.59
102.2
—
1
0.5
[A]
27.4647
45.9744
69.6407
87.2215
96.6079
129.164
[B]
27.47
45.97
69.65
87.22
96.61
—
0.95
[A]
15.6871
23.3557
46.9793
70.3963
99.624
110.599
[B]
15.70
23.37
46.96
70.41
99.63
110.6
0
[A]
95.2625
115.803
156.357
218.973
254.138
277.308
[B]
95.26
115.8
156.4
—
254.1
277.3
2
0.5
[A]
94.7438
109.859
137.29
183.897
254.214
254.254
[B]
94.76
109.9
137.3
183.9
254.2
—
0.95
[A]
62.7485
77.444
93.4166
93.4229
101.133
187.917
[B]
62.82
77.50
93.46
93.47
—
—
[A]: present approach and [B]: [3].In Table4, values of Ncrit are compared for three different cases of the x direction load: constant (α=0), linear with null value at one extreme (α=1), and bending moment (α=2), and different aspect radii of the plate. A convergence study is also made. As it can be seen, taking M=N=15 provides an excellent accuracy from an engineering viewpoint.


## 4. Conclusions

The classical, variational method of Ritz has been successfully used in the present study to obtain an approximate, yet quite accurate, solution to a difficult elastodynamics problem.Natural frequencies and mode shapes of transverse vibration are obtained for a meaningful combination of the boundary conditions of a thin plate subjected to general in-plane loads. The critical values of the in-plane forces which cause instability of the plates are also obtained.The obtained values are the outcome of an algorithm, relatively simple to implement, [11] which allows studying these with only the assistance of a PC.Additional complexities like orthotropic material characteristics can be taken into account [12].The agreement with results available in the literature is excellent. Nevertheless, it is also possible to increase the number of terms in the summation on (8) to increase the accuracy.No claim of originality is made, but it is hoped that the present work draws the attention to the effect that the presence of plane stress state may have on the effectiveness of vibration control on plates.



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