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Thermally induced stability and vibration of initially stressed laminated composite plates.

1. Introduction

For the past decades, the composite materials are widely used in spacecraft and engineering industries because of their higher tensile strength and lower weight. Composite plate structures are often applied at elevated temperature environments. In such thermal circumstances, the thermal induced compressive stresses will be developed in the composite plates and consequently lead to the change in mechanical behaviors. The thermally induced behavior of composite plate plays an important role in the design of structural components in thermal environments. Thus, the studies on thermal vibration and buckling of composite plates are increasing considerably in recent years.

Many investigations on thermally induced behaviors of composite plates are concerned with the thermal stability and vibration. The critical buckling temperatures of laminated plates based on a finite strip method were studied by Dawe and Ge [1]. In the pre-buckling stage, an in-plane thermal stress analysis was conducted first, and a buckling analysis was followed using the determined in-plane stress distribution. Wang et al. [2] presented the local thermal buckling of laminated plate using the delaminated buckling model. The analytical predictions for the critical temperature yielding the local delamination buckling are shown to correlate well with experimental results. Shian and Kuo [3] developed a thermal buckling analysis method for composite sandwich plates. The results show that the buckling mode of sandwich plate depends on the fiber orientation in the faces and the aspect ratio of the plate. Thermal buckling analysis of cross-ply laminated hybrid composite plates with a hole subjected to a uniform temperature rise was investigated by Avci et al. [4]. The effects of hole size, lay-up sequences and boundary conditions on the thermal buckling temperatures were investigated. The equivalent mechanical loading concept was used to study various thermal buckling problems of simple laminated plate configurations by Jones [5]. The results were given in the form of buckling temperature change from the stress-free temperature against plate aspect ratio curves. Matsunaga [6] presented the thermal buckling of laminated plates using the principle of virtual work. Several sets of truncated mth order approximate theories were applied to solve the eigenvalue problems. Modal transverse shear and normal stresses could be calculated by integrating the equilibrium equations.

The governing equations for determining thermal buckling of imperfect sandwich plates were developed by Zakeri and Alinia [7]. The buckling thermal stress remains unchanged for aspect ratios greater than five. The structural optimization of a laminated plate subjected to thermal and shear loading was considered by Teters [8]. The optimization criteria depend on two variable design parameters of composite properties and temperature. Thermal buckling analysis of composite laminated plates under uniform temperature rise was investigated by Shariyat [9]. A numerical scheme and a modified instability criterion are used to determine the buckling temperature in a computerized solution. A thermal buckling response of symmetric laminated plates subjected to a uniformly distributed temperature load was presented by Kabir et al. [10]. The numerical results were presented for various significant effects such as length-to-thickness ratio, plate aspect ratio and modulus ratio. Thermal buckling behavior of imperfect laminated plates based on first order plate theory was studied by Pradeep and Ganesan [11]. A decoupled thermo-mechanical analysis is used to deal with the thermal buckling and vibration behavior of sandwich plates. The variation of natural frequency and loss factor with temperature was studied by Owhadi and Shariat [12]. The plate was assumed to be under the longitudinal temperature rise. The effects of initial imperfections on buckling loads were discussed. A perturbation technique was used by Verma and Singh [13] to find the buckling temperature of laminated composite plates subjected to a uniform temperature rise. It was found that small variations in material and geometric properties of the composite plate significantly affect the buckling temperature of the laminated composite plate. Wu [14] investigated the stresses and deflections of a laminated plate under thermal vibration using the moving least squares differential quadrature method. The method provides rapidly convergent and accurate solutions for calculating the stresses and deflections.

The thermal buckling behavior of the laminated plates subjected to uniform and/or non-uniform temperature fields was studied by Ghomshei [15]. The influence parameters of plate aspect ratio, cross-ply ratio and stiffness ratio on the critical temperature were presented. Rath [16] the free behavior of laminated plates subjected to varying temperature and moisture. A simple laminated plate model is developed for the vibration of composite plates subjected to hygrothermal loading. The results showed the effects of geometry, material and lamination parameters of woven fiber laminate on the vibration of composite plates for different temperature. Ghugal [17] presented the flexural response of cross-ply laminated plates subjected to thermo-mechanical loads. The shear deformation theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. Thermal stresses for three-layer symmetric cross-ply laminated plates subjected to uniform linear and nonlinear and thermo-mechanical loads are obtained. The governing equations for laminated beams subjected to uniform temperature rise are derived by Fu [18]. The effects of the transverse shear effects and boundary conditions on the thermal buckling and post-buckling of the beams are discussed. A differential quadrature method is applied to obtain the maximum buckling temperature of laminated composite by Malekzadeh [19]. The direct iterative method in conjunction with genetic algorithms is used to determine the optimum fiber orientation for the maximum buckling temperature.

From the literature reviewed, researches on the vibration and buckling of initially stressed laminate plates under thermal environmental condition seem to be lacking. The vibration and stability behaviors of initially-stressed laminate plates have been investigated by Chen et al. [20-21] in recent years. The studies revealed that the initial stress in structures may significantly influence the behaviors of laminated plates. Therefore, while studying the thermal buckling and vibration behavior of laminate plates, the effect of initial stress should be taken into account. In this paper, the equilibrium equations for a laminated plate subjected to the arbitrary initial stress and thermal condition are established by using variation method. The temperature field is assumed to be uniform plus linearly distributed through the plate thickness. The effects of various parameters on the critical temperature, natural frequencies and buckling loads in thermal environments are presented.

2. Equilibrium equations

Following a similar technique described by Brunelle and Robertson [22] and Chen et al. [20-21], Hamilton's principle is applied to derive nonlinear equations of the composite plate including the effects of rotary inertia and transverse shear. For an initially stressed body which is in equilibrium and subjected to a time-varying incremental deformation, the Hamilton's principle can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where U, K, [W.sub.e] and [W.sub.i] are the strain energy, kinetic energy, work of external forces and internal forces, respectively; [[sigma].sub.ij] and [[epsilon].sub.ij] are the stresses and strains; [v.sub.i] are the displacements referred to the spatial frame; [rho] is the density; [X.sub.i] is the body force per unit initial volume and [p.sub.i] is the external force per unit initial surface area. The application of the minimum total energy principle leads to the general equations and boundary conditions. Assume that the stresses and applied forces are constant, and substitute Eq. (2) into Eq. (1). Then taking the variation and integrating the kinetic energy term by parts with respected to time, Eq. (1) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If a rectangular plate is considered, the equations can be rephrased in xy coordinates. The incremental displacements are assumed to be of the following forms:

[v.sub.x] (x, y, z, t) = [u.sub.x] (x, y, t) + z[[phi].sub.x] (x, y, t); [v.sub.y] (x, y, z, t) = [u.sub.y] (x, y, t) + z[[phi].sub.y] (x, y, t); [v.sub.z] (x, y, z, t) = w (x, y, t), (4)

where [u.sub.x], [u.sub.y] and w are the displacements of the middle surface in the x, y and z direction, respectively; [[phi].sub.x] and [[phi].sub.y] denotes the rotation angle about y and x axis, respectively. The two edges of a rectangular plate are set along x and y axes, respectively. The stress-strain relations are taken to be those of uncoupled linear thermal elasticity. Hence, the constitutive relations for the kth lamina including the thermal effect can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [C.sub.ij] are the elastic constants of lamina; [[alpha].sub.ij] are thermal expansion coefficients and [DELTA]T is the temperature rise. The stress-displacement relations are found to be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

Substitute Eqs. (4)-(6) into Eq. (3), perform all necessary partial integrations and group the terms by the five independent displacement variations, [delta][u.sub.x], [delta][u.sub.y], [delta]w, [delta][[phi].sub.x] and [delta][[phi].sub.y], to yield the following five governing equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (11)

where [f.sub.x], [f.sub.y], [f.sub.z], [m.sub.x] and [m.sub.y] are the lateral loadings. The arbitrary initial stresses are included in the stress resultants [N.sub.ij], [M.sub.ij] and [M..sup.*.sub.IJ] x [N.sup.T.sub.ij], [M.sup.T.sub.ij] and [M.sup.T*.sub.jj] are thermal stress resultants. The coefficients associated with material parameters, initial stress, thermal stress resultants and rotary inertia are defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where all the integrals are integrated through the thickness h of the plate from -h/2 to h/2.

3. Solution of the governing equations

Because the stability and vibration behaviors of the investigated initially stressed laminate composite plate are affected by various parameters, it would be difficult to present results for all cases. Thus, only the initially-stressed simply supported cross-ply laminate plate under the combined uniform and linear thermal loading is investigated. The lateral loads and body forces [f.sub.x], [f.sub.y], [f.sub.z], [m.sub.x] and my are taken to be zero. The only nonzero initial stress is assumed to be (Fig. 1)

[[sigma].sub.xx] = [[sigma].sub.n] + 2z[[sigma].sub.m]/h, (13)

which comprises of the constant uniaxial stress [[sigma].sub.n] and bending stress [[sigma].sub.m]. Hence, the nonzero axial stress resultants are [N.sub.xx] = h[[sigma].sub.n], [M.sub.xx] = [Sh.sup.2][[sigma].sub.n]/ 6 and [M.sup.*.sub.xx] = [h.sup.3] [[sigma].sub.n] / 12. The factor S = [[sigma].sub.m] / [[sigma].sub.n] denotes the ratio of a bending stress to a normal stress. For the cross-ply plate, the stiffness coefficients [C.sub.16], [C.sub.26] and [C.sub.45] will be equal to zero in Eqs. (6) and (7).

[FIGURE 1 OMITTED]

The combined uniform and linear temperature distribution is of the form as

[DELTA]T = [T.sub.o] + 2z[T.sub.g], (14)

where [T.sub.o] is the uniform temperature rise and [T.sub.g] is the temperature gradient. The nonzero thermal stress resultants are [N.sup.T.sub.ij] = -[[alpha].sub.ij][C.sub.ij] [T.sub.o] h, [M.sup.T.sub.ij] = -[[alpha].sub.ij] [C.sub.ij][T.sub.g] [h.sup.2] / 6 and [M.sup.T*.sub.ij] = -[[alpha].sub.ij] [C.sub.ij] [T.sub.o][h.sup.3] / 12.

For a simply supported laminated plate, the boundary conditions along the x-constant edges are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (15, a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (15, b)

and along the y-constant edges are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (16, a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16, b)

For the simply supported plate, the displacement fields satisfying the geometric boundary conditions are given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

All summations are summed up from m, n = 1 to [infinity]. For a buckling problem, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is neglected in Eq. (17). Substituting the initial stress (13), temperature distribution (14) and displacement fields (17) into the governing Eqs. (8)-(12), and collecting the coefficients for any fixed values of m and n leads to the following eigenvalue equation:

([C] - [lambda][G]){[??]} = {0}; {[??]} = [[U.sub.mn], [V.sub.mn], [W.sub.mn], [[PSI].sub.xmn], [[PSI].sub.ymn]].sup.T], (18)

in which parameter [lambda] refers to the corresponding frequency or buckling coefficient. For the vibration problems, the coefficients of the symmetric matrix [C] and [G] are expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the thermal buckling problem, the coefficients of matrix [C] are given by neglecting thermal induce stresses resultant terms in the stiffness matrix in Eq. (18) and the coefficients of matrix [G] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As to the buckling load problems, the coefficients of the symmetric matrix [C] are given by neglecting the initial stress resultant terms of the matrix [C]. The coefficients of the matrix [G] are:

[G.sub.1,1] = [G.sub.2,2] = [G.sub.3,3] = [[alpha].sup.2]; [G.sub.1,4] = [G.sub.2,5] = S[[alpha].sup.2] / 6h; [G.sub.4,4] = [G.sub.5,5] = [[alpha].sup.2] / 12[h.sup.2].

4. Results and discussion

For verifying the present computer program, the close agreements between the present results and those in Matsunaga [23], Liu and Huang [24] for cross-ply plates as shown in Tables 1-2 demonstrate the accuracy and effectiveness of the present method. Parametric studies are carried out to examine the effects of various variables on the vibration and stability response of laminate plates under thermal environments. The following non- dimensional natural frequency ([OMEGA] = [omega][b.sup.2] [square root of [rho] / [h.sup.2] [E.sub.y]), buckling coefficient ([K.sub.f] = [b.sup.2 [N.sub.xx] / [E.sub.y]) and thermal buckling coefficient (T = [??]T [[alpha].sub.yy] [10.sup.4]) are defined and used throughout the vibration and stability study. If the stress is tensile, then the buckling coefficient [K.sub.f] is positive. There is no initial stress when [K.sub.f] = 0 and S = 0. The critical buckling temperature is denoted by [T.sub.cr].

[FIGURE 2 OMITTED]

Fig. 2 presents the effect of modulus ratio on the thermal buckling temperature of plates with different stack layers. The critical temperature increases monotonically as the modulus ratio or/and layer number increase. The critical temperatures of eight-layer plates under different temperature gradient [T.sub.g] are given in Table 3. The increasing temperature gradient reduces the thermal buckling temperature, and its influence on the critical temperature is less than the layer number. The effects of modulus ratio and span ratio on critical temperature parameters are shown in Fig. 3. The buckling temperature of plate with a smaller span ratio is always higher than that with a larger span ratio, especially for the plate with a higher modulus ratio. Thus, with a higher modulus, higher stacking number of layer, lower span ratio and lower gradient temperature, the laminated plate has a higher thermal buckling temperature.

[FIGURE 3 OMITTED]

The effect of buckling coefficient on the natural frequency of plates under various uniform temperature rises can be observed in Fig. 4. The natural frequency decreases with the increasing initial compressive stress and temperature rise. The buckling load can be obtained when the natural frequency approaches zero. Meanwhile, the plate under a lower temperature rise has a greater buckling coefficient. Fig. 5 shows the effect of modulus ratio on the natural frequency of laminated plates. The laminate plate with higher modulus ratio has a larger vibration frequency and higher buckling load.

The buckling load and natural frequency of laminate plate with different layer numbers and modulus ratios under uniform temperature rise are shown in Tables 4-5. The plate with larger stack layer number or/and higher modulus ratio has a higher critical buckling load and natural frequency. It can also be observed that the buckling load and natural frequency decreases steadily with the increasing uniform temperature rise. Thus, the two-layered plate with smallest modulus ratio and under higher temperature rise will possess the smallest buckling load and natural frequency.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The effect of different temperature gradient on buckling load and natural frequency of plates is presented in Tables 6-7. When the linear gradient temperature increases the buckling load and natural frequency coefficient slightly decrease. The laminated plate with lower modulus ratio and under higher uniform temperature and temperature gradient has a smaller critical buckling and vibration frequency. The influence of temperature gradient on the buckling load and natural frequency for laminate plates is less apparent than that of uniform temperature.

Variations of critical temperature and natural frequency with the linear temperature change for initially stressed laminate plates are shown in Tables 8-9. It is evident that the compressive initial stress ([K.sub.f] < 0) produces a decreasing effect on the critical temperature and natural frequency, and the tensile initial stress has a reverse effect. Likewise, the initially stressed laminate plate with higher modulus ratio has a larger critical temperature than the one with lower modulus ratio.

The effect of bending stress ratio on the critical buckling coefficient for initially stressed plates under uniform temperature is given in Table 10. As can be seen, the increasing bending stress ratio decreases the critical buckling load. The influence of bending stress on the natural frequency of initially stressed plates under a fixed uniform temperature is presented in Table 11. The vibration frequency decreases with the increase in bending stress. However, the natural frequency is not affected by the increasing bending stress when the plate is subject to the pure bending stress only. The lowest natural frequency can be observed for the plate with a lower modulus ratio and under a higher bending stress.

5. Conclusions

The vibration and buckling behaviors of initially stressed and thermally stressed laminate plates have been described and discussed in this paper. The results demonstrate the influence of the modulus ratio, number of layer, initial stress and thermal stress on the vibration and buckling behaviors of laminate plates. Following the above discussions, the preliminary results are summarized as follows:

1. The modulus ratio, number of layer and uniform temperature has an apparent influence on natural frequency and buckling load. They are slightly affected by the temperature gradient rise and bending stress.

2. With the increasing modulus ratio and number of layer, the critical temperature, buckling load and natural frequency increase. The uniform temperature has a reverse effect.

3. The compressive stress significantly reduces the natural frequency and critical temperature but the tensile stress produces an opposite effect.

Received November 13, 2014

Accepted January 06, 2016

Acknowledgements

This research was supported by the Ministry of Science and Technology through the grant NSC98-2221-E-262 -009 -MY3.

References

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(19.) Malekzadeh, P.; Vosoughi, A.R.; Sadeghpour, M. 2014. Thermal buckling optimization of temperature-dependent laminated composite skew plates, J. Aero. Engng. 27: 64-75. http://dx.doi.org/10.1061/CASCEjAS.1943-5525.00002 20.

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Chun-Sheng Chen, Wei-Ren Chen, Hung-Wei Lin

THERMALLY INDUCED STABILITY AND VIBRATION OF INITIALLY STRESSED LAMINATED COMPOSITE PLATES

Chun-Sheng Chen *, Wei-Ren Chen **, Hung-Wei Lin ***

* Department of Mechanical Engineering, Lunghwa University of Science and Technology, Guishan Shiang 33306, Taiwan, E-mail: cschen@mail.lhu.edu.tw.

** Department of Mechanical Engineering, Chinese Culture University, Taipei 11114, Taiwan, E-mail: wrchen@faculty.pccu.edu.tw.

*** Department of Electrical Engineering, Lee Ming Institute of Technology, Taishan 24305, Taiwan, E-mail: hwlin@mail.lit.edu. tw.

cross ref http://dx.doi.org/10.5755/j01.mech.22.1.8682
Table 1
Comparison of minimum critical temperatures of
three-layer cross-ply laminated composite plates
[0[degrees]/90[degrees] /0[degrees]]

a / h    Matsunaga [23]   Present

20/10        0.3334       0.3438
20/6         0.2465       0.2554
20/5         0.2184       0.2216
20/4         0.1763       0.1802
20/3         0.1294       0.1299
20/2         0.0746       0.0731
20           0.0230       0.0219

Table 2
Comparison of vibration frequencies of a [0/90]s square
plate in thermal environment

                        [alpha].sub.xx]/[[alpha].sub.yy]

[T.sub.o]    Source    -0.05     0.1      0.2      0.3

-50         Liu [24]   15.149   15.247   15.320   15.394
            Present    15.165   15.277   15.351   15.425

0           Liu [24]   15.150   15.150   15.150   15.150
            Present    15.179   15.179   15.179   15.179

Table 3
Effect of gradient temperature on critical temperature
parameter of plates with different modulus ratio
(a / b = 1; a / h = 10; n = 8; [K.sub.f] = 0; S = 0)

                                [E.sub.x] / [E.sub.y]

[T.sub.g] /     5        10       20       30        40        50
[T.sub.o]

0             4.4621   6.3727   9.1758   11.0261   2.2410    13.0226
5             4.4545   6.3634   9.1653   11.0161   12.2320   13.0147
10            4.4321   6.3355   9.1340   10.9861   12.2050   12.9911
20            4.3467   6.2285   9.0125   10.8688   12.0990   12.8981
40            4.0580   5.8585   8.5775   10.4380   11.7016   12.5443

Table 4
Effect of layer number and modulus ratio on critical
buckling coefficient of plates under various uniform
temperature rise (a / b = 1, a / h = 10, S = 0,
[T.sub.g] / [T.sub.o] = 0)

                   [E.sub.x] / [E.sub.y]

n   [T.sub.o] /     5        10        20
    [T.sub.cr]

2        0        5.1392   6.1869    8.0733
       0.25       3.8544   4.6402    6.0549
        0.5       2.5696   3.0935    4.0366
       0.75       1.2848   1.5468    2.0183

4        0        6.4432   9.4152    14.6917
       0.25       4.8324   7.0614    11.0188
        0.5       3.2216   4.7076    7.3459
       0.75       1.6108   2.3538    3.6729

6        0        6.6812   9.9874    15.7990
       0.25       5.0109   7.4905    11.8492
        0.5       3.3406   4.9937    7.8995
       0.75       1.6703   2.4968    3.9497

8        0        6.7642   10.1858   16.1786
       0.25       5.0731   7.6394    12.1340
        0.5       3.3821   5.0929    8.0893
       0.75       1.6910   2.5465    4.0446

                    [E.sub.x] / [E.sub.y]

n   [T.sub.o] /     30        40        50
    [T.sub.cr]

2        0        9.8419    11.5230   13.1266
       0.25       7.3815    8.6422    9.8450
        0.5       4.9210    5.7615    6.5633
       0.75       2.4606    2.8808    3.2817

4        0        19.2579   23.2504   26.7707
       0.25       14.4434   17.4378   20.0780
        0.5       9.6290    11.6252   13.3853
       0.75       4.8145    5.8126    6.6926

6        0        20.7515   25.0224   28.7425
       0.25       15.5636   18.7668   21.5569
        0.5       10.3757   12.5112   14.3712
       0.75       5.1878    6.2556    7.1855

8        0        21.2585   25.6186   29.4008
       0.25       15.9438   19.2139   22.0506
        0.5       10.6292   12.8092   14.7003
       0.75       5.3146    6.4046    7.3501

Table 5
Effect of layer number and modulus ratio on the natural
frequency of plates under various uniform temperature rise
(a / b=1; a/ h=10; [K.sub.f] = 0; S = 0; [T.sub.g]/[T.sub.o] = 0)

                     [E.sub.x] / [E.sub.y]

n   [T.sub.o] /      5         10        20
    [T.sub.cr]

2        0         7.1219    7.8142    8.9264
       0.25        6.1678    6.7674    7.7305
        0.5        5.0360    5.5256    6.3119
       0.75        3.5610    3.9073    4.4632

4        0         7.9745    9.6397    12.0417
       0.25        6.9061    8.3483    10.4284
        0.5        5.6388    6.8164    8.5148
       0.75        3.9873    4.8200    6.0209

6        0         8.1204    9.9284    12.4872
       0.25        7.0325    8.5982    10.8143
        0.5        5.7420    7.0204    8.8298
       0.75        4.0602    4.9642    6.2436

8        0         8.1707    10.0265   12.6364
       0.25        7.0760    8.6832    10.9434
        0.5        5.7776    7.0899    8.9353
       0.75        4.0853    5.0133    6.3182

                    [E.sub.x] / [E.sub.y]

n   [T.sub.o] /     30        40        50
    [T.sub.cr]

2        0        9.8558    10.6643   11.3822
       0.25       8.5354    9.2356    9.8573
        0.5       6.9692    7.5409    8.0485
       0.75       4.9280    5.3323    5.6912

4        0        13.7866   15.1484   16.2548
       0.25       11.9395   13.1189   14.0771
        0.5       9.7486    10.7116   11.4939
       0.75       6.8934    7.5743    8.1274

6        0        14.3112   15.7151   16.8428
       0.25       12.3939   13.6097   14.5863
        0.5       10.1196   11.1123   11.9097
       0.75       7.1556    7.8576    8.4214

8        0        14.4850   15.9012   17.0346
       0.25       12.5444   13.7708   14.7524
        0.5       10.2424   11.2438   12.0453
       0.75       7.2425    7.9506    8.5173

Table 6
Effect of linear temperature rise on the critical buckling
coefficient (a / b = 1; a / h = 10; n = 8; S = 0)

                                [T.sub.g] / [T.sub.o]

[E.sub.x] /   [T.sub.o] /      0          5        10
[E.sub.y]     [T.sub.cr]

10                 0        10.1858    10.1858   10.1858
                 0.25        7.6394    7.6385    7.6356
                  0.5        5.0929    5.0892    5.0779
                 0.75        2.5465    2.5380    2.5126

40                 0        25.6186    25.6186   25.6186
                 0.25       19.2139    19.2127   19.2092
                  0.5       12.8092    12.8045   12.7903
                 0.75        6.4046    6.3939    6.3620

                             [T.sub.g] / [T.sub.o]

[E.sub.x] /   [T.sub.o] /       20           40
[E.sub.y]     [T.sub.cr]

10                 0         10.1858      10.1858
                 0.25         7.6243       7.5792
                  0.5         5.0327       4.8515
                 0.75         2.4109       2.0013

40                 0         25.6186      25.6186
                 0.25        19.1950      19.1381
                  0.5        12.7335      12.5049
                 0.75         6.2338       5.7151

Table 7
Effect of linear temperature rise on the natural frequency
(a / b = 1; a / h = 10; n = 8; [K.sub.f] = 0; S = 0)

                                [T.sub.g] / [T.sub.o]

[E.sub.x] /   [T.sub.o] /
[E.sub.y]     [T.sub.cr]       0          5        10

10                 0        10.0265    10.0265   10.0265
                 0.25        8.6832    8.6827    8.6811
                  0.5        7.0899    7.0872    7.0794
                 0.75        5.0133    5.0050    4.9799

40                 0        15.9012    15.9012   15.9012
                 0.25       13.7708    13.7704   13.7691
                  0.5       11.2438    11.2418   11.2355
                 0.75        7.9506    7.9440    7.9241

                             [T.sub.g] / [T.sub.o]

[E.sub.x] /   [T.sub.o] /
[E.sub.y]     [T.sub.cr]        20           40

10                 0         10.0265      10.0265
                 0.25         8.6747       8.6490
                  0.5         7.0478       6.9198
                 0.75         4.8780       4.4444

40                 0         15.9012      15.9012
                 0.25        13.7640      13.7436
                  0.5        11.2105      11.1095
                 0.75         7.8439       7.5105

Table 8
Effect of initial stresses on the critical temperature of
plates under linear temperature rise
(a / b = 1; a / h = 10; n = 8; S = 0)

                              [T.sub.g] / [T.sub.o]

[E.sub.x] /
[E.sub.y]     [K.sub.f]      0         5        10

10                4       8.8753    8.8571    8.8034
                  0       6.3727    6.3634    6.3355
                 -4       3.8702    3.8667    3.8564

40                4       14.1523   14.1402   14.1042
                  0       12.2410   12.2320   12.2050
                 -4       10.3297   10.3233   10.3041

                           [T.sub.g] / [T.sub.o]

[E.sub.x] /
[E.sub.y]     [K.sub.f]       20           40

10                4         8.5996       7.9259
                  0         6.2285       5.8585
                 -4         3.8161       3.6697

40                4        13.9628      13.4365
                  0        12.0990      11.7016
                 -4        10.2284       9.9425

Table 9
Effect of initial stresses on the natural frequency of plates
under linear temperature rise
(a / b = 1, a / h = 10, n = 8, S = 0, [T.sub.o] / [T.sub.cr] = 0.5)

                              [T.sub.g] / [T.sub.o]

[E.sub.x] /
[E.sub.y]     [K.sub.f]      0         5        10

10                4       9.4734    9.4714    9.4655
                  0       7.0899    7.0872    7.0794
                 -4       3.2844    3.2788    3.2617

40                4       12.8803   12.8785   12.8731
                  0       11.2438   11.2418   11.2355
                 -4       9.3244    9.3219    9.3144

                           [T.sub.g] / [T.sub.o]

[E.sub.x] /
[E.sub.y]     [K.sub.f]       20           40

10                4         9.4419       9.3468
                  0         7.0478       6.9198
                 -4         3.1927       2.8991

40                4        12.8512      12.7632
                  0        11.2105      11.1095
                 -4         9.2842       9.1620

Table 10
Effect of bending ratios on critical buckling coefficient of
plates under different uniform temperature rise
(a / b = 1, a / h = 10, n = 8, [T.sub.g] / [T.sub.o] = 0)

                                         S

[E.sub.x] /   [T.sub.o] /
[E.sub.y]     [T.sub.cr]       0        10        20

10                 0        10.1858   10.1263   9.9553
                 0.25       7.6394    7.6058    7.5084
                  0.5       5.0929    5.0780    5.0341
                 0.75       2.5465    2.5427    2.5316

40                 0        25.6186   25.5432   25.3213
                 0.25       19.2139   19.1715   19.0461
                  0.5       12.8092   12.7904   12.7343
                 0.75       6.4046    6.3998    6.3857

                                         S

[E.sub.x] /   [T.sub.o] /
[E.sub.y]     [T.sub.cr]      30        40        50

10                 0        9.6925    9.3639    8.9952
                 0.25       7.3560    7.1608    6.9358
                  0.5       4.9641    4.8721    4.7627
                 0.75       2.5135    2.4890    2.4587

40                 0        24.9643   24.4897   23.9190
                 0.25       18.8430   18.5704   18.2383
                  0.5       12.6429   12.5186   12.3650
                 0.75       6.3625    6.3305    6.2903

Table 11
Effect of initial bending stress on the natural frequency of
plates under uniform temperature rise
(a / b = 1, a / h = 10, n = 8, [T.sub.o] / [T.sub.cr] = 0.5,
[T.sub.g] / [T.sub.o] = 0)

                                       S

[E.sub.x] /   [K.sub.f]      0        10        20
[E.sub.y]

10                4       9.4734    9.4685    9.4540
                  0       7.0899    7.0899    7.0899
                 -4       3.2844    3.2704    3.2281

40                4       12.8803   12.8796   12.8775
                  0       11.2438   11.2438   11.2438
                 -4       9.3244    9.3235    9.3205

                                       S

[E.sub.x] /   [K.sub.f]     30        40        50
[E.sub.y]

10                4       9.4297    9.3956    9.3514
                  0       7.0899    7.0899    7.0899
                 -4       3.1563    3.0528    2.9140

40                4       12.8739   12.8690   12.8626
                  0       11.2438   11.2438   11.2438
                 -4       9.3156    9.3088    9.3000
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Author:Chen, Chun-Sheng; Chen, Wei-Ren; Lin, Hung-Wei
Publication:Mechanika
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2016
Words:5958
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