Thermally induced stability and vibration of initially stressed laminated composite plates.
Chen, Chun-Sheng ; Chen, Wei-Ren ; Lin, Hung-Wei 等
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.
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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
