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Thermal deformation and stress analysis of composite laminated plates by finite element method
oc!45-7949/m s3.M) + 0.00 I’er~mon Press pk
THERMAL DEFORMATION AND STRESS ANALYSIS OF COMPOSITE LAMINATED PLATES BY FINITE ELEMENT METHOD L.-W.-N
and L.-Y.
CHEN
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, Republic of China (Received 23 May 1989) Abstract-Thermal deformations and stresses, induced by a non-uniform temperature field in composite laminated plates, are studied by the fmite element method. The stiffness matrix and load vector are derived based on the minimum potential energy. All three displacements of the middle surface are expressed as products of one-dimensional first-order Hermite interpolation polynomials. Numerical results are presented as nondimensional curves. The effects of boundary conditions, temperature ~st~butions, aspect ratios, and other parameters on the thermal deformations and stresses are investigated.
It is not surprising that many papers and reports have been written on this subject. Maulbetsch [I] presents solutions of the problem of thermal stresses in thin plates with simply supported edges when the temperature varies along the thickness of the plates. A differential equation governing the deflection of an elastic plate subjected to non-uniform temperature distribution is derived by Sokolnikoff and Sokolnikoff [2]. Aleck [3] obtains an approximate solution for the stresses induced by a uniform change in temperature in a thin plate clamped along an edge. Thermal stresses induced in a flat, rectangular plate by non-uniform heating are determined both experimentally and theoretically by Heldenfels and Roberts 141. Boley [5] presents an analytical successive-approximation method for the solution of twodimensional heat and thermal stress problems. Explicit formulas for the calculations of stresses and deflections are given. Thermal bending of a rectangular plate with two parallel edges simply supported and the other two edges supported in any manner is analyzed by Das and Navaratna 161. Thermal and shrinkage stresses in restrained plates subjected to uniform heating are investigated by Sundara Raja Iyengar and Chandrashekhara [7]. Lee [S] obtains a three-dimensional series solution for thick plates subjected to general temperature distribution. Thermal bending of a moderately thick plate subjected to a temperature distribution which is antisymmetric about the middle plane of the plate is studied by Das and Rath [9]. Alvarez and Cooper [lo] determine the stress distribution for a thin plate with temperature as a function of thickness. They generalize to permit the thermal expansion coefficient, Young’s modulus, and Poisson’s ratio to be functions of temperature. An analysis with a shear deformation capability for the thermal bending of thick rectangular plates is presented by Bapu Rao 1111.
NOTATION
width of the plate in x and y directions extensional, coupling, and bending stiffness matrix Young’s modulus in the fiber and tranverse to the fiber direction shear modulus in x-y plane thickness of plate stiffness matrix of the element load vector of the element normalized moment resultants normalized thermal moment resultants normalized force resultants normalized thermal force resultants reduced stiffness matrix temperature distribution transpose of a matrix middle surface displacement of the plate in x, y, and z directions normalized middle surface displacement of the plate in x, y, and z directions thermal expansion coefficients membrane strain curvature Poisson’s ratio
As the technology of composite material becomes more highly developed, it is now possible to use composite material in high-temperature situations. Consequently, the thermal deformations and stresses which are induced by non-unifo~ temperature in composite structures grow to be an important consideration in structural design; and the determination of such quantities in composite laminated plates is of practical interest. 41
42
L.-W. CHEN and L.-Y. CKEN
All the analyses mentioned above are restricted to isotropic plates, Therefore, the plane stress and bending problems are solved inde~ndentIy. However, thermal stretching-ending coupling could be signi%ant when Iaminated anisotropic plates are considered. A general thermoelastic theory for a thin heterogeneous anisotropic plate has been established first by Stavsky [12]. It is shown that thermal coup&g terms appear. Thermal defo~a~ans and stresses in symmetric laminates and ~t~s~etr~~ cross-ply and angle-piy laminates are investigated by Wu and Tauchert[l3,14]. The method of Levy and the double Fourier series are used to obtain solutions. The coupled problem of bending and stretching of composite sandwich plate due to a thermal field and rn~ha~~a~ foad is treated by We~nste~~ er of. [15] using finite element method. Recent& the therma stress anafysis of a skew plate with mixed in-plane boundary conditions using the finite element approach has also been investigated [ 161. Xn the present paper, an analysis of thermal deformations and stresses for laminated composite plates by finite element method is presented. The stiffness matrix is derived based on the principle of minimum potential energy. Transverse shear strain energy is neglected. Expressing the assumed displacement state over the middle surface of the plate element as products uf one-dimensional, first-order Hermite poIynomjals, it is possible to ensure that the displacement state for the assembled set of such elements is geometrically admissible 1173. After comparing present solutions with closed-form solutions for some problems available in the literature, thermal defa~a~ons and stresses are investigated.
on the middle surface of a flat plate as shown in Fig. I. Components of displacement along the coordinate axes x, Y, and z are denoted by 19, tP, and woI resp~tively. For a laminated anisotropic thin plate, the strain-displacement relations are [ 12)
where
and
(3)
The rectangular coordinates X, y, and z are taken
(a) si mply
supported
(b)
clamped
The constitutive fotm fl2]
relations
NX i
&’
1
Denoting in-plane stresses by crjr then 112 (A$, &fj) = H/,EI. b&l, 2) d& s -l/Z whore ET represents the transverse modulus of a unidirectional ply, and H = a/h. Aij, Bij, and I);j are extensional, coupling, and bending stiffnesses of the laminates defined in terms of reduced stiffness coefficients Qij by El7]. IiZ Q,(l, 3, P)dF. fs) (A,,, Bjj, Dji) = l/E; s -112 The normalized thermal force and moment resultants are defined as follows:
be obtained from the corresponding matrix Qij and [al ) u2I 0] for principal material directions, The normalized strain energy of the laminated plate subjected to thermal loading is given by:
Expressions for displacements ii and B are similar. Since each of the three displacements of the middle surface are expressed as products of ore-Dimensions E&mite ~nte~~atjo~ ~ol~nomiaIs, this leads to an element with a total of 48 degrees of freedom with 12 degrees of freedom per node. The resulting displacement vector is
L.-W. CHENand L.-Y. CHEN
44
be written in the form 1 I lJ= l/2 [41’wJ’[~lPJ Is-1 -I
B
+ [wPl[~J
0
+ [S,lW’l) IJ I d5 dv,
Y
(14)
where
L
(15)
[xl= [&lIql;
[q] is the resulting displacement vector of the element which is a 48 x 1 vector; [S,] and [S,] are 3 x 48 matrices which relate the membrane strain and curvature of the element to the vector [q], respectively; and ( .I 1 is the determinant of the Jacobian matrix. The stiffness matrix and load vector are derived based on the principle of minimum potential energy and are obtained by numerical integration using a 3 x 3 Gaussian quadrature formula. [K’]=
’ ss-I
1
-I
wJ[~l[&l+
The frontal method [18] is used to assemble finite element stiffness and load vectors into a global stiffness matrix and load vector and then solve for unknowns. RESULTS
Fig. 2. Finite element modelings.
(4 Temperature
distribution Ci = To = 1.0, Co = r, =O.O, v =0.3, GI= 1.0 x 10-6, a/b = 1.0, a/h = 100.0, and E = 1.0. Co = 1.O, C, = 0.0, @I Temperature distribution To = 95.31, T, = 150.0, v = 0.33, CI= 12.7 x 10-6, a/b = 1.5, a/h = 144.0, and E = 1.0. The results of (a) and (b) for the various modelings shown in Fig. 2 are given in Table 1 and Fig. 3,
PJ’[~lw,1
+ [U’WTl) IJ I d5dv. (17)
NUMERICAL
AND DISCUSSION
In the following, the laminated plate is analyzed under the following temperature distribution without any external loads. T(X, 7, I) = (Co + C,Z)T,(Z,j$
(18)
respectively. The numerical value of non-dimensional displacement R at the center of the plate with a simply supported boundary condition is compared with the results of those models that are successive refinements of model A. From Table 1, it can be noted that monotonic convergence is exhibited. In Fig. 3, the variation of the thermal force resultants, N,, N,,, and N,,, of the plate along the y direction for various element meshes is shown, together with the corresponding closed-form solutions of Heldenfels and Roberts [4]. A satisfactory agreement is seen. For the sake of simplicity and without loss of generality, only two types of temperature distribution are considered in the following cases, namely T(ji, 7) and T(T), where T&p):
where r,(x,p)=
o
T,+2T,y r,-2T,(y-
1)
1/2
1.
Co= 1.0, C, =O.O, T,=O.O, and T, = 150.0;
T(Z): Co = 0.0, Ci = 1.0, To= 150.0, and T, =O.O.
l/2 (19)
Co, C, , To, Tl are constants. To demonstrate the performance of the present finite element formulation in solving the problems of thermal stress, two examples, for which closed-form solutions are available, have been considered.
Initially, laminated plates subjected to temperature distribution T(Z, 7) are studied. Numerical results have been given for various aspect ratios a/b. It may be seen from Fig. 4a that the displacement E along the y direction of the plate increases with the coordinate 7, and is antisymmetric about the center line of the plate for a/b > 1, while in the case of a/b < 1, the
Table 1. Convergence test
Central disnlacement.
8’
x
Element model A
Element model B
Element model C
Ref. [9]
0.190799 x 10-d
0.190807 x 1o-4
0.190810 x 1o-4
0.1914 x 1o-4
Thermal deformation and stress analysis of plates
45
d Ref.4
0
Ref. 4 Modeling A
B
A
Modeling
B
C
n
Modeling
C
s
0
Modeling
n
Modeling
Cl Modeling
GA
A
0.040 0.020 0.000 -0.020 -0.040 -0.080 -0.080 -0.
T;=O.9436
too p
-O-b2po
0.60
0.70
0.80
0.90
1.
0.60
0.70
0.80
0.90
I.
Y Fig.
3a. Stress resultant N,v.
Fig. 3b. Stress resultant N,,.
0.000 I
-0.00
I
o A f7
I
I
Ref.4 Modeling Modeling Modeling
A B C
-0.01 N ‘I -0.01
-0.02
-0.02
-0.03
Fig. 3c. Stress resultant NxY.
displacement ij shows an opposite trend. The variations of the force resultant NYand moment resultant MXYas a function of coordinate J are shown in Fig. 4b and c, respectively. Both NYand h4_, reach an extreme value at the center line of the plate. It may also be observed that NY and MXYinduced in a plate for aspect ratio a Jb = 1 are always higher than for other aspect ratios. In a plate which is heated nonuniformly, the various elements tend to expand with the temperature but are partically prevented from doing so by neighboring elements. Such a phenomenon is particularly significant in a rectangular plate. In Fig. 4a, it is shown that the displacement for the aspect ratio a/f, = 1 is the smallest of all three
aspect ratios, so that the stress and moment resultant
NY and MXYinduced in the square plate become the highest. Figure 5 presents the effects of the boundary conditions on the deformation, force resultant, and moment resultant. Since the clamped plate is completely fixed along the edges, and is more difficult to expand, it is not surprising that the force and moment resultants of the clamped plate are higher than those of the simply supported plate. Plots of shear force and moment resultant vs coordinate J are shown for various lamination angles in Fig. 6. For the lamination angle 8 between 0 and 45”, it is noticed that the larger the value of the lamination angle, the
L.-w.
46
&EN and L.-Y. C&N
Y
(a) Displacement
%
(b)
Force
resultant
N,
7 (c) Moment resultant
M.,
Fig. 4. Distribution of thermal deformation, force and moment resultants (EL/& = 40.0; GUI& = 0.5; ~~~~0.25; a/h = 100.0; G(,= CL~ = 1.0 x 10-6; [-45”/45”],; simply supported).
higher the shear stress and moment resultant become. Figure 7 shows the variation of force and moment resultants against the coordinate y for four different 2 positions. It is clear that the force and moment resultant distributions are symmetric about the center lines of the plate, so that the analysis can be confined to the first quadrant only. However, in further study on thermal buckling, the entire domain of the plate is investigated. Secondly, the laminated plate is analyzed under the temperature distribution T(z). The displacement ii, for various aspect ratios a/b is presented in Fig. 8a. It is observed that the displacement tifdecreases as the
aspect ratio a/b increases and always reaches a minimum value at the center line of the plate. The variation of the moment resultant Mxy as a function of coordinate 7 for various X positions is shown in Fig. 8b. It may be seen that the distribution of Mx, is antisymmetric about the center lines of the plate. Finally, comparisons have been made for the force and moment resultants N, and M, under temperature distributions T(j;_,7) and T(Z) which are symmetric and antisymmetric about the center line and middle surface of the plate, respectively. Figure 9 shows that the symmetric temperature distribution corresponds to the symmetric force and moment resultant distri-
Thermal deformation and stress analysis of plates
simply
0.006
47
bution, while the antisymmetric temperature distribution corresponds to the antisymmetric force and moment resultant distribution.
supported
CONCLUSION
A finite element formulation for laminated composite plates subjected to non-uniform thermal loading is presented. The displacement state for the assembied set of Hermite elements is geometricatly admissible. Monotonic convergence is also ensured. After comparing present solutions with closedform solution for some problems available in the literature. thermal deformations and stresses are in-
~~~~~
;=oT 0.20
.
,
*
*
0.40
0.60
0.80
0.120 1.
I
0.090
-
0.080
-
7 (a)
Displacement
7
0.100 x=0.5564
simply
supported -0.060-
-0.300x=0.25 -0.
00’ b .oo
0.20
0.40
0.80
0.60
1.00
-0.12ol
0.00
u (b)
Force
0.20
0.40
_
0.60
0.80
I
1.00
Y
resultant
N.
(a)
0.009
Force
resultant
N,,
-
simply supported
:~, 1
0.20
0.40
-
0.60
0.80
1,
Y
(c) Moment Fig.
5.
Distribution
resultant
M.,
of thermal deformations, force and
ii=O.,%
*
,
*
0.20
0.40
0.80
0.80
F
(b) Moment resultant
1
hL,
Fig. 6. Distribution of thermal force and moment resultants
moment resultants (EL/E, = 40.0; G,,/E,=OS;v,,=O.25; (EJET=40.0; G,,/E,=OS;v,,= 0.25;a/h = lOO.O;a/b = a/h= 100.0; a/b= 1.0; at =a*= 1.0x 10e6; [-30"/30"],).1.0; a,=a*= 1.01IO-&; [-ff"/P],; simply supported). CM 35/1-D
L.-W. CHENand L.-Y. CHEN
48
r
0.008 0.006
0,004
0.002 M" 0.000
f
-o*oo2/ -0.004
-0.006
-0.t:6
00
Force
0.40
0.80
0.80
1
0
Y
Y
(a)
0.20
(b) Moment
resultant N,
vestigated. It is shown that thermal stresses induced in the clamped plate are higher than those in the simply supported plate. The temperature distribution has a significant infiuence on the dist~bution of thermal stress. Thermal defo~ations induced in a
resultant
I&
square rectangular plate are always smaller than those in a rectangular plate which is not square. Thermal buckling of laminated plates subjected to God-unifo~ temperature ~st~butio~ is a further topic to be investigated.
0.0018
0.0015
0.0012 K 0.0009
9.0008
a.0003
0.0000 0.00
0.20
0.40 ;; 0.80
0.80
0.80
1.00 (b) Moment
Y
result ant a,
Fig. 8. Distribution of thermal deformation and moment resultant (EL&= 40.0; CAT/&= 0.3 tt&+=0.25; n/h = 100.0; a/b = 1.0; DL,= a2 = I.0 x IO-*; [-30”/3#“],; simply supported).
1. 0
Thermal deformation and stress analysis of plates
49
REIXRENCRS 1. J.
0.020
-
N.
-0.040-
-0.090x=0.75 -0.
oeol 0.00
0.20
0.40
0.80
0.90
1. 00
Y
(a) Force resultant
N.
0.009
0.000
0.003 M., 0.000
Lo.o03-
T(q -o.oos-
%=0.75 ,3
-0.009-
T
‘“‘8’.%0
0.20)
0.40
0.60
0.80
I. 00
F
Moment resultant h& Fig. 9. Distribution of thermal force and moment resuitant (E‘ jar = 40.0; G,,/Er = 0.5; vLT= 0.25; a/h = 100.0; a/b = 1.0; LX,= o+= 1.0 x 10m6; [-30”/30”],; simply supported). (b)
L. Maulbetsch, Thermal stresses in plates. J. appl. Mech., ASME 2, Al41-Al46 (1935). 2. I. S. Sokolnikoff and E. S. Sokolnikoff, Thermal stresses in elastic plates. Am. Math. Sot. Trans. 45, 235-255 (1939). 3. B. J. Aleck, Thermal stresses in a rectangular plate clamped along an edge. J. appl. Mech., ASME 16, 11%i22 (1949). 4. R. R. Heldenfels and W. M. Roberts, Experimental and theo~ti~ly dete~ination of thermal stresses in a flat plate. NACA TN 2769 (1952). 5. 8. A. Boley, The determination of temperature, stresses, and deflection in two-dimensional thermoelastic problems. J. Aeronaut. Sci. 23, 67-75 (1962). 6. Y. C. Das and D. R. Navaratna, Thermal bending of rectangular plate. J. Aerospace Sci. 29, 1397-1399 (1962). 7. K. T. Sundara Raja Iyengar and K. Chandrashe~ra, Thermal stresses in restrained rectangular plates. J. Engng Mech. Div., ASCE 94, I-21 (1968). 8. C. W. Lee, Thermal stresses in a thick plate. Innt. J. Solids Struct. 6, 6055615 (1970). 9. Y. C. Das and B. K. Rath, Thermal bending of moderately thick rectangular plate. AIAA Jnl 10, 1349-1352 (1972). 10. I. S. Alvarez and A. R. Cooper Jr, Thermal stress distribution for a thin plate with temperature variation through the thickness. J. appi. mech., ASME 42, 513-514 (1975). 11. M. N. Bapu Rao, Thermal bending of thick rectangular plates. Nucl. Engng Des. 54, 115-l 18 (1979). 12, Y. Stavsky, Thermoelasticity of heterogeneous aeolotropic plates. J. Engng Mech. Div., ASCE SP, 89-105 (1963). 13. C. H. Wu and T. R. Tauchert, Thermoelastic analysis of laminated plates. 1. Symmetric specially orthotropic laminates. J. therm. Stresses 3, 247-259 (1980). 14. C. H. Wu and T. R. Tauchert, Thermoelastic analysis of laminated plates. 2. Antisymmetric cross-ply and angle-ply laminates. J. therm. Srresses 3, 365-378 (1980). 15. F. Weinstein, S. Putter and Y. Stavslcy, Thermoelastic stress analysis of anisotropic composite sandwich plates by finite element method. Compur. Srruct. 17, 31-36 (1983). 16. N. Ganesan and M. S. Dhotarad, Thermal stress analysis of skew plates by finite element method. Compur. Strucr. 21, 1013-1023 (1985). 17. F. K. Bogner, R. L. Fox and L. A. Schmit, A cylindri~i shell discete element. AIAA J&5, 745-750 (1967). 18. E. Hinton and D. R. J. Owen, Finire EZement Programming. Academic Press, London (1979)
THERMAL DEFORMATION AND STRESS ANALYSIS OF COMPOSITE LAMINATED PLATES BY FINITE ELEMENT METHOD L.-W.-N
and L.-Y.
CHEN
Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, Republic of China (Received 23 May 1989) Abstract-Thermal deformations and stresses, induced by a non-uniform temperature field in composite laminated plates, are studied by the fmite element method. The stiffness matrix and load vector are derived based on the minimum potential energy. All three displacements of the middle surface are expressed as products of one-dimensional first-order Hermite interpolation polynomials. Numerical results are presented as nondimensional curves. The effects of boundary conditions, temperature ~st~butions, aspect ratios, and other parameters on the thermal deformations and stresses are investigated.
It is not surprising that many papers and reports have been written on this subject. Maulbetsch [I] presents solutions of the problem of thermal stresses in thin plates with simply supported edges when the temperature varies along the thickness of the plates. A differential equation governing the deflection of an elastic plate subjected to non-uniform temperature distribution is derived by Sokolnikoff and Sokolnikoff [2]. Aleck [3] obtains an approximate solution for the stresses induced by a uniform change in temperature in a thin plate clamped along an edge. Thermal stresses induced in a flat, rectangular plate by non-uniform heating are determined both experimentally and theoretically by Heldenfels and Roberts 141. Boley [5] presents an analytical successive-approximation method for the solution of twodimensional heat and thermal stress problems. Explicit formulas for the calculations of stresses and deflections are given. Thermal bending of a rectangular plate with two parallel edges simply supported and the other two edges supported in any manner is analyzed by Das and Navaratna 161. Thermal and shrinkage stresses in restrained plates subjected to uniform heating are investigated by Sundara Raja Iyengar and Chandrashekhara [7]. Lee [S] obtains a three-dimensional series solution for thick plates subjected to general temperature distribution. Thermal bending of a moderately thick plate subjected to a temperature distribution which is antisymmetric about the middle plane of the plate is studied by Das and Rath [9]. Alvarez and Cooper [lo] determine the stress distribution for a thin plate with temperature as a function of thickness. They generalize to permit the thermal expansion coefficient, Young’s modulus, and Poisson’s ratio to be functions of temperature. An analysis with a shear deformation capability for the thermal bending of thick rectangular plates is presented by Bapu Rao 1111.
NOTATION
width of the plate in x and y directions extensional, coupling, and bending stiffness matrix Young’s modulus in the fiber and tranverse to the fiber direction shear modulus in x-y plane thickness of plate stiffness matrix of the element load vector of the element normalized moment resultants normalized thermal moment resultants normalized force resultants normalized thermal force resultants reduced stiffness matrix temperature distribution transpose of a matrix middle surface displacement of the plate in x, y, and z directions normalized middle surface displacement of the plate in x, y, and z directions thermal expansion coefficients membrane strain curvature Poisson’s ratio
As the technology of composite material becomes more highly developed, it is now possible to use composite material in high-temperature situations. Consequently, the thermal deformations and stresses which are induced by non-unifo~ temperature in composite structures grow to be an important consideration in structural design; and the determination of such quantities in composite laminated plates is of practical interest. 41
42
L.-W. CHEN and L.-Y. CKEN
All the analyses mentioned above are restricted to isotropic plates, Therefore, the plane stress and bending problems are solved inde~ndentIy. However, thermal stretching-ending coupling could be signi%ant when Iaminated anisotropic plates are considered. A general thermoelastic theory for a thin heterogeneous anisotropic plate has been established first by Stavsky [12]. It is shown that thermal coup&g terms appear. Thermal defo~a~ans and stresses in symmetric laminates and ~t~s~etr~~ cross-ply and angle-piy laminates are investigated by Wu and Tauchert[l3,14]. The method of Levy and the double Fourier series are used to obtain solutions. The coupled problem of bending and stretching of composite sandwich plate due to a thermal field and rn~ha~~a~ foad is treated by We~nste~~ er of. [15] using finite element method. Recent& the therma stress anafysis of a skew plate with mixed in-plane boundary conditions using the finite element approach has also been investigated [ 161. Xn the present paper, an analysis of thermal deformations and stresses for laminated composite plates by finite element method is presented. The stiffness matrix is derived based on the principle of minimum potential energy. Transverse shear strain energy is neglected. Expressing the assumed displacement state over the middle surface of the plate element as products uf one-dimensional, first-order Hermite poIynomjals, it is possible to ensure that the displacement state for the assembled set of such elements is geometrically admissible 1173. After comparing present solutions with closed-form solutions for some problems available in the literature, thermal defa~a~ons and stresses are investigated.
on the middle surface of a flat plate as shown in Fig. I. Components of displacement along the coordinate axes x, Y, and z are denoted by 19, tP, and woI resp~tively. For a laminated anisotropic thin plate, the strain-displacement relations are [ 12)
where
and
(3)
The rectangular coordinates X, y, and z are taken
(a) si mply
supported
(b)
clamped
The constitutive fotm fl2]
relations
NX i
&’
1
Denoting in-plane stresses by crjr then 112 (A$, &fj) = H/,EI. b&l, 2) d& s -l/Z whore ET represents the transverse modulus of a unidirectional ply, and H = a/h. Aij, Bij, and I);j are extensional, coupling, and bending stiffnesses of the laminates defined in terms of reduced stiffness coefficients Qij by El7]. IiZ Q,(l, 3, P)dF. fs) (A,,, Bjj, Dji) = l/E; s -112 The normalized thermal force and moment resultants are defined as follows:
be obtained from the corresponding matrix Qij and [al ) u2I 0] for principal material directions, The normalized strain energy of the laminated plate subjected to thermal loading is given by:
Expressions for displacements ii and B are similar. Since each of the three displacements of the middle surface are expressed as products of ore-Dimensions E&mite ~nte~~atjo~ ~ol~nomiaIs, this leads to an element with a total of 48 degrees of freedom with 12 degrees of freedom per node. The resulting displacement vector is
L.-W. CHENand L.-Y. CHEN
44
be written in the form 1 I lJ= l/2 [41’wJ’[~lPJ Is-1 -I
B
+ [wPl[~J
0
+ [S,lW’l) IJ I d5 dv,
Y
(14)
where
L
(15)
[xl= [&lIql;
[q] is the resulting displacement vector of the element which is a 48 x 1 vector; [S,] and [S,] are 3 x 48 matrices which relate the membrane strain and curvature of the element to the vector [q], respectively; and ( .I 1 is the determinant of the Jacobian matrix. The stiffness matrix and load vector are derived based on the principle of minimum potential energy and are obtained by numerical integration using a 3 x 3 Gaussian quadrature formula. [K’]=
’ ss-I
1
-I
wJ[~l[&l+
The frontal method [18] is used to assemble finite element stiffness and load vectors into a global stiffness matrix and load vector and then solve for unknowns. RESULTS
Fig. 2. Finite element modelings.
(4 Temperature
distribution Ci = To = 1.0, Co = r, =O.O, v =0.3, GI= 1.0 x 10-6, a/b = 1.0, a/h = 100.0, and E = 1.0. Co = 1.O, C, = 0.0, @I Temperature distribution To = 95.31, T, = 150.0, v = 0.33, CI= 12.7 x 10-6, a/b = 1.5, a/h = 144.0, and E = 1.0. The results of (a) and (b) for the various modelings shown in Fig. 2 are given in Table 1 and Fig. 3,
PJ’[~lw,1
+ [U’WTl) IJ I d5dv. (17)
NUMERICAL
AND DISCUSSION
In the following, the laminated plate is analyzed under the following temperature distribution without any external loads. T(X, 7, I) = (Co + C,Z)T,(Z,j$
(18)
respectively. The numerical value of non-dimensional displacement R at the center of the plate with a simply supported boundary condition is compared with the results of those models that are successive refinements of model A. From Table 1, it can be noted that monotonic convergence is exhibited. In Fig. 3, the variation of the thermal force resultants, N,, N,,, and N,,, of the plate along the y direction for various element meshes is shown, together with the corresponding closed-form solutions of Heldenfels and Roberts [4]. A satisfactory agreement is seen. For the sake of simplicity and without loss of generality, only two types of temperature distribution are considered in the following cases, namely T(ji, 7) and T(T), where T&p):
where r,(x,p)=
o
T,+2T,y r,-2T,(y-
1)
1/2
1.
Co= 1.0, C, =O.O, T,=O.O, and T, = 150.0;
T(Z): Co = 0.0, Ci = 1.0, To= 150.0, and T, =O.O.
l/2 (19)
Co, C, , To, Tl are constants. To demonstrate the performance of the present finite element formulation in solving the problems of thermal stress, two examples, for which closed-form solutions are available, have been considered.
Initially, laminated plates subjected to temperature distribution T(Z, 7) are studied. Numerical results have been given for various aspect ratios a/b. It may be seen from Fig. 4a that the displacement E along the y direction of the plate increases with the coordinate 7, and is antisymmetric about the center line of the plate for a/b > 1, while in the case of a/b < 1, the
Table 1. Convergence test
Central disnlacement.
8’
x
Element model A
Element model B
Element model C
Ref. [9]
0.190799 x 10-d
0.190807 x 1o-4
0.190810 x 1o-4
0.1914 x 1o-4
Thermal deformation and stress analysis of plates
45
d Ref.4
0
Ref. 4 Modeling A
B
A
Modeling
B
C
n
Modeling
C
s
0
Modeling
n
Modeling
Cl Modeling
GA
A
0.040 0.020 0.000 -0.020 -0.040 -0.080 -0.080 -0.
T;=O.9436
too p
-O-b2po
0.60
0.70
0.80
0.90
1.
0.60
0.70
0.80
0.90
I.
Y Fig.
3a. Stress resultant N,v.
Fig. 3b. Stress resultant N,,.
0.000 I
-0.00
I
o A f7
I
I
Ref.4 Modeling Modeling Modeling
A B C
-0.01 N ‘I -0.01
-0.02
-0.02
-0.03
Fig. 3c. Stress resultant NxY.
displacement ij shows an opposite trend. The variations of the force resultant NYand moment resultant MXYas a function of coordinate J are shown in Fig. 4b and c, respectively. Both NYand h4_, reach an extreme value at the center line of the plate. It may also be observed that NY and MXYinduced in a plate for aspect ratio a Jb = 1 are always higher than for other aspect ratios. In a plate which is heated nonuniformly, the various elements tend to expand with the temperature but are partically prevented from doing so by neighboring elements. Such a phenomenon is particularly significant in a rectangular plate. In Fig. 4a, it is shown that the displacement for the aspect ratio a/f, = 1 is the smallest of all three
aspect ratios, so that the stress and moment resultant
NY and MXYinduced in the square plate become the highest. Figure 5 presents the effects of the boundary conditions on the deformation, force resultant, and moment resultant. Since the clamped plate is completely fixed along the edges, and is more difficult to expand, it is not surprising that the force and moment resultants of the clamped plate are higher than those of the simply supported plate. Plots of shear force and moment resultant vs coordinate J are shown for various lamination angles in Fig. 6. For the lamination angle 8 between 0 and 45”, it is noticed that the larger the value of the lamination angle, the
L.-w.
46
&EN and L.-Y. C&N
Y
(a) Displacement
%
(b)
Force
resultant
N,
7 (c) Moment resultant
M.,
Fig. 4. Distribution of thermal deformation, force and moment resultants (EL/& = 40.0; GUI& = 0.5; ~~~~0.25; a/h = 100.0; G(,= CL~ = 1.0 x 10-6; [-45”/45”],; simply supported).
higher the shear stress and moment resultant become. Figure 7 shows the variation of force and moment resultants against the coordinate y for four different 2 positions. It is clear that the force and moment resultant distributions are symmetric about the center lines of the plate, so that the analysis can be confined to the first quadrant only. However, in further study on thermal buckling, the entire domain of the plate is investigated. Secondly, the laminated plate is analyzed under the temperature distribution T(z). The displacement ii, for various aspect ratios a/b is presented in Fig. 8a. It is observed that the displacement tifdecreases as the
aspect ratio a/b increases and always reaches a minimum value at the center line of the plate. The variation of the moment resultant Mxy as a function of coordinate 7 for various X positions is shown in Fig. 8b. It may be seen that the distribution of Mx, is antisymmetric about the center lines of the plate. Finally, comparisons have been made for the force and moment resultants N, and M, under temperature distributions T(j;_,7) and T(Z) which are symmetric and antisymmetric about the center line and middle surface of the plate, respectively. Figure 9 shows that the symmetric temperature distribution corresponds to the symmetric force and moment resultant distri-
Thermal deformation and stress analysis of plates
simply
0.006
47
bution, while the antisymmetric temperature distribution corresponds to the antisymmetric force and moment resultant distribution.
supported
CONCLUSION
A finite element formulation for laminated composite plates subjected to non-uniform thermal loading is presented. The displacement state for the assembied set of Hermite elements is geometricatly admissible. Monotonic convergence is also ensured. After comparing present solutions with closedform solution for some problems available in the literature. thermal deformations and stresses are in-
~~~~~
;=oT 0.20
.
,
*
*
0.40
0.60
0.80
0.120 1.
I
0.090
-
0.080
-
7 (a)
Displacement
7
0.100 x=0.5564
simply
supported -0.060-
-0.300x=0.25 -0.
00’ b .oo
0.20
0.40
0.80
0.60
1.00
-0.12ol
0.00
u (b)
Force
0.20
0.40
_
0.60
0.80
I
1.00
Y
resultant
N.
(a)
0.009
Force
resultant
N,,
-
simply supported
:~, 1
0.20
0.40
-
0.60
0.80
1,
Y
(c) Moment Fig.
5.
Distribution
resultant
M.,
of thermal deformations, force and
ii=O.,%
*
,
*
0.20
0.40
0.80
0.80
F
(b) Moment resultant
1
hL,
Fig. 6. Distribution of thermal force and moment resultants
moment resultants (EL/E, = 40.0; G,,/E,=OS;v,,=O.25; (EJET=40.0; G,,/E,=OS;v,,= 0.25;a/h = lOO.O;a/b = a/h= 100.0; a/b= 1.0; at =a*= 1.0x 10e6; [-30"/30"],).1.0; a,=a*= 1.01IO-&; [-ff"/P],; simply supported). CM 35/1-D
L.-W. CHENand L.-Y. CHEN
48
r
0.008 0.006
0,004
0.002 M" 0.000
f
-o*oo2/ -0.004
-0.006
-0.t:6
00
Force
0.40
0.80
0.80
1
0
Y
Y
(a)
0.20
(b) Moment
resultant N,
vestigated. It is shown that thermal stresses induced in the clamped plate are higher than those in the simply supported plate. The temperature distribution has a significant infiuence on the dist~bution of thermal stress. Thermal defo~ations induced in a
resultant
I&
square rectangular plate are always smaller than those in a rectangular plate which is not square. Thermal buckling of laminated plates subjected to God-unifo~ temperature ~st~butio~ is a further topic to be investigated.
0.0018
0.0015
0.0012 K 0.0009
9.0008
a.0003
0.0000 0.00
0.20
0.40 ;; 0.80
0.80
0.80
1.00 (b) Moment
Y
result ant a,
Fig. 8. Distribution of thermal deformation and moment resultant (EL&= 40.0; CAT/&= 0.3 tt&+=0.25; n/h = 100.0; a/b = 1.0; DL,= a2 = I.0 x IO-*; [-30”/3#“],; simply supported).
1. 0
Thermal deformation and stress analysis of plates
49
REIXRENCRS 1. J.
0.020
-
N.
-0.040-
-0.090x=0.75 -0.
oeol 0.00
0.20
0.40
0.80
0.90
1. 00
Y
(a) Force resultant
N.
0.009
0.000
0.003 M., 0.000
Lo.o03-
T(q -o.oos-
%=0.75 ,3
-0.009-
T
‘“‘8’.%0
0.20)
0.40
0.60
0.80
I. 00
F
Moment resultant h& Fig. 9. Distribution of thermal force and moment resuitant (E‘ jar = 40.0; G,,/Er = 0.5; vLT= 0.25; a/h = 100.0; a/b = 1.0; LX,= o+= 1.0 x 10m6; [-30”/30”],; simply supported). (b)
L. Maulbetsch, Thermal stresses in plates. J. appl. Mech., ASME 2, Al41-Al46 (1935). 2. I. S. Sokolnikoff and E. S. Sokolnikoff, Thermal stresses in elastic plates. Am. Math. Sot. Trans. 45, 235-255 (1939). 3. B. J. Aleck, Thermal stresses in a rectangular plate clamped along an edge. J. appl. Mech., ASME 16, 11%i22 (1949). 4. R. R. Heldenfels and W. M. Roberts, Experimental and theo~ti~ly dete~ination of thermal stresses in a flat plate. NACA TN 2769 (1952). 5. 8. A. Boley, The determination of temperature, stresses, and deflection in two-dimensional thermoelastic problems. J. Aeronaut. Sci. 23, 67-75 (1962). 6. Y. C. Das and D. R. Navaratna, Thermal bending of rectangular plate. J. Aerospace Sci. 29, 1397-1399 (1962). 7. K. T. Sundara Raja Iyengar and K. Chandrashe~ra, Thermal stresses in restrained rectangular plates. J. Engng Mech. Div., ASCE 94, I-21 (1968). 8. C. W. Lee, Thermal stresses in a thick plate. Innt. J. Solids Struct. 6, 6055615 (1970). 9. Y. C. Das and B. K. Rath, Thermal bending of moderately thick rectangular plate. AIAA Jnl 10, 1349-1352 (1972). 10. I. S. Alvarez and A. R. Cooper Jr, Thermal stress distribution for a thin plate with temperature variation through the thickness. J. appi. mech., ASME 42, 513-514 (1975). 11. M. N. Bapu Rao, Thermal bending of thick rectangular plates. Nucl. Engng Des. 54, 115-l 18 (1979). 12, Y. Stavsky, Thermoelasticity of heterogeneous aeolotropic plates. J. Engng Mech. Div., ASCE SP, 89-105 (1963). 13. C. H. Wu and T. R. Tauchert, Thermoelastic analysis of laminated plates. 1. Symmetric specially orthotropic laminates. J. therm. Stresses 3, 247-259 (1980). 14. C. H. Wu and T. R. Tauchert, Thermoelastic analysis of laminated plates. 2. Antisymmetric cross-ply and angle-ply laminates. J. therm. Srresses 3, 365-378 (1980). 15. F. Weinstein, S. Putter and Y. Stavslcy, Thermoelastic stress analysis of anisotropic composite sandwich plates by finite element method. Compur. Srruct. 17, 31-36 (1983). 16. N. Ganesan and M. S. Dhotarad, Thermal stress analysis of skew plates by finite element method. Compur. Strucr. 21, 1013-1023 (1985). 17. F. K. Bogner, R. L. Fox and L. A. Schmit, A cylindri~i shell discete element. AIAA J&5, 745-750 (1967). 18. E. Hinton and D. R. J. Owen, Finire EZement Programming. Academic Press, London (1979)