Linear static analysis and finite element modeling for laminated ...
Composite Structures 62 (2003) 27–39 www.elsevier.com/locate/compstruct
Linear static analysis and finite element modeling for laminated composite plates using third order shear deformation theory M. Rastgaar Aagaah, M. Mahinfalah *, G. Nakhaie Jazar Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105, USA
Abstract In this paper, deformations of a laminated composite plate due to mechanical loads are presented. Third order shear deformation theory of plates, which is categorized in equivalent single layer theories, is used to derive linear dynamic equations of a rectangular multi-layered composite plate. Moreover, derivation of equations for FEM and numerical solutions for displacements and stress distributions of different points of the plate with a sinusoidal distributed mechanical load for Navier type boundary conditions are presented. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Third order shear deformation; Laminated composite plate; Finite element
1. Introduction The use of composite materials in structural components are increasing due to their attractive properties such as high strength-to-weight ratio, ability to tailor the structural properties, etc. Plate structures find numerous applications in the aerospace, military and automotive industries. The effects of transverse shear deformation are considerable for composite structures, because of their high ratio of extensional modulus to transverse shear modulus. Most of the structural theories used till now to characterize the behavior of composite laminates fall into the category of equivalent single layer (ESL) theories. In these theories, the material properties of the constituent layers are combined to form a hypothetical single layer whose properties are equivalent to throughthe-thickness integrated sum of its constituents. This category of theories has been found to be adequate in predicting global response characteristics of laminates, like maximum deflections, maximum stresses, fundamental frequencies, or critical buckling loads [1]. Third order shear deformation theory, which is one of the ESL theories, is derived. This theory is based on *
Corresponding author. Tel.: +1-701-231-8839; fax: +1-701-2318913. E-mail address: [email protected] (M. Mahinfalah). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00081-3
the same assumptions as the classical (CLPT) and first order shear deformation plate theories (FSDT), except that the assumption on the straightness and normality of the transverse normal is relaxed [2–4]. Theories higher than third order are not used because the accuracy gained is so little that the effort required to solve the equations is not justified [5]. In single layer displacement-based theories, one single expansion for each displacement component is used through the entire thickness, and therefore, the transverse strains are continuous through the thickness, a strain state appropriate for homogeneous plates [5–7]. In the present work, the equations of motion have been derived for the linear deformation of laminated plates subjected to a mechanical load based on a third order shear deformation plate theory in conjunction with the Von Karman strains. Unlike to the first order shear deformation theory, the higher order theory does not require shear correction factors. Finally the finite element solution for the plate is derived.
2. Elasticity equations The plate considered in this investigation consists of N orthotropic cross-ply and angle-ply layers with a total thickness h. Components of global Cartesian
28
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
coordinates X, that is located at the middle of the plate, are ðx; y; zÞ where x; y are in-plane coordinates, and z is the transverse coordinate. The top layer is at z ¼ h=2 and the bottom layer is located at z ¼ h=2. Layer coordinates of a typical nth layer are Xn and its components are ðxn ; yn ; zn Þ and xn is in the direction of fibers as shown in Fig. 1. The linear constitutive equation of the nth layer when considering thermal expansion effect is given by 8 9 r1 > > > > > > > r2 > > > > > > =
2
6 6 6 6 3 ¼6 6 > r4 > > > 6 > > > > 6 > > > > r 5 > 4 > : ; r6
Q11
Q12 Q22
Sym
Q13 Q23 Q33
0 0 0 Q44
0 0 0 Q45 Q55
9 38 e1 a1 h > Q16 > > > > 7> e a h > > > Q26 7> > 2 2 > > > > 7> = < 7 a h e Q36 7 3 3 > > e4 0 7 > 7> > > > 7> > > > e 0 5> 5 > > > > ; : e Q66 6 ð1Þ
where ai are coefficients of thermal expansion in direction of layer coordinates and h is the change in temperature of each layer. Since the thermal effects cause a volume change, they do not have effect on transverse stresses and strains. Therefore, thermal expansion coefficients for an orthotropic lamina have only three components [8,9]. Let h be the angle between the layer coordinates and the global coordinate, then the following relationships exist between stresses and strains in both coordinates. frg ¼ ½T frg feg ¼ ½T feg
ð2Þ
r and e are prescribed in global coordinate but r and e are components of stress and strain in lamina coordinates. ½T is rotational matrix about the transverse direction z at h and is defined as
2
C2 6 S2 6 6 0 ½T ¼ 6 6 0 6 4 0 CS
S2 C2 0 0 0 CS
0 0 1 0 0 0
0 0 0 C S 0
0 0 0 S C 0
3 2CS 2CS 7 7 7 0 7 7 0 7 5 0 2 2 ðC S Þ
ð3Þ
where C ¼ cosðhÞ and S ¼ sinðhÞ. From Eqs. (1) and (2), the following relation is held between elastic coefficients at two different coordinate systems. ½Q ¼ ½T 1 ½Q ½T
ð4Þ
½Q and ½Q are defined in terms of local coordinate of each layer and global coordinate of plate, respectively. After rotation of coordinates, in-plane thermal expansion coefficient a6 will appear. 9 8 9 2 38 Q11 Q12 Q13 e1 a1 h > r1 > 0 0 Q16 > > > > > > > > 6 > > > > > e2 a2 h > r2 > Q22 Q23 0 0 Q26 7 > > > > > > > 6 7 = < < = 6 7 r3 Q 0 0 Q a h e 33 36 3 3 7 ¼6 6 7 Q44 Q45 0 7> > > 6 > e4 > r4 > > > > > 4 > > > > 0 5> Sym Q55 > > > e5 > r5 > > > > > ; : : ; r6 e6 a6 h Q66 ð5Þ The relationships of material properties at two different coordinate systems are presented in Appendix A. The following displacement field that was introduced by Robbins and Reddy [5], is the displacement field of third order shear deformation plate theory (TRDT). ow0 2 1 o/z 3 u ¼ u0 þ z/x z z C1 þ /x 2 ox ox 1 ouz þ ð6Þ 3 ox 1 o/z ow0 z3 C1 þ /y v ¼ v0 þ z/y z2 2 oy oy 1 ouz þ ð7Þ 3 oy w ¼ w0 þ z/z þ z2 uz
ð8Þ
where 4 ; u0 ¼ uðx; y; 0; tÞ; 3h2 v0 ¼ vðx; y; 0; tÞ and w0 ¼ wðx; y; 0; tÞ
C1 ¼
Fig. 1. Local and global coordinate systems of a laminate.
ð9Þ
ðu0 ; v0 ; w0 Þ are the displacements of transverse normal on plane z ¼ 0 and ð/x ; /y Þ are rotations of transverse normal on plane z ¼ 0. /z is extension of a transverse normal, and uz is interpreted as a higher order rotation of transverse normal. The number of dependent variables in Eqs. (6)–(8) is only 7. The displacement field in Eqs. (6)–(8) accommodates quadratic variation of transverse shear strains (and hence stresses) and vanishing of transverse shear stresses on the top and bottom of a general laminate
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
composed of monoclinic layers [10,5]. Thus there is no need to use shear correction factors in a third order theory. The third order theories provide a slight increase in accuracy relative to the first order shear deformation theory (FSDT) solution, at the expense of a significant increase in computational effort. Moreover, finite element models of third order theories that satisfy the vanishing of transverse shear stresses on the bounding planes have the disadvantage of requiring continuity of C 1 [5]. Using virtual work method, Z t ðdU þ dV dKÞ dt ¼ 0 ð10Þ 0
equilibrium equations of the plate can be derived. U , V , and K are virtual strain energy, virtual work done by applied forces and virtual kinetic energy, respectively, and t is time. Then the linear strains according to displacement field Eqs. (6)–(8) are e1 ¼ e01 þ zðk10 þ zk11 þ z2 k12 Þ e2 ¼ e3 ¼ e4 ¼ e5 ¼ e6 ¼
e02 e03 e04 e05 e06
þ þ þ þ
zðk20 þ zk21 þ z2 k22 Þ zðk30 Þ zðk41 Þ ¼ 2e23 zðk51 Þ ¼ 2e13
ð11Þ
h2
þ r5 de5 þ r6 de6 Þ dz dA þ
Z
Z X0
Z X0
h 2
ðqdwÞ dA !
€ d€ qð€ ud€ u þ €vd€v þ w wÞ dz dA dt ¼ 0
ð12Þ
h2
Plate inertias and stress resultants are defined as follow: N Z znþ1 X I1;...;7 ¼ qn ð1; z; z2 ; z3 ; z4 ; z5 ; z6 Þ dz ð13Þ n¼1
ðNi ; Mi ; Si Þ ¼
zn N X n¼1
Z
ði ¼ 4; 5Þ
ð15Þ
znþ1
ri z2 dz ði ¼ 1; 2; 3; 4; 5; 6Þ
ð16Þ
zn
where I1;...;7 are inertias and Ni , Mi , Si , Qi and Pi are stress resultants. Using fundamental lemma of calculus of variation [10–12], the equation of motion of the plate can be written as du0 : 1 o/€ o€ w0 N1;x þ N6;y ¼ I1 €u0 þ I2 /€x I3 z C1 I4 2 ox ox € 1 o u z C1 I4 /€x I4 3 ox dv0 : 1 o/€ o€ w0 N2;y þ N6;x ¼ I1€v0 þ I2 /€y I3 z C1 I4 2 oy oy € 1 o u C1 I4 /€y I4 z 3 oy
ð17Þ
ð18Þ
C1 S1;xx þ C1 S2;yy þ Q4;y þ Q5;y 3C1 P4;y 3C1 P5;x þ 2C1 S6;xy þ q
þ zðk60 þ k61 þ k62 Þ
X0
N Z X n¼1
ri dz
zn
n¼1
ðPi Þ ¼
znþ1
dw0 :
It is assumed that there is an isothermal condition and temperature change and therefore thermal strains do not exist. In Appendix B, the relationships between strain components and displacement field Eqs. (6)–(8) are presented. By substitution of stresses and strain and distributed force in Eq. (10), the final integral equation for plate elasticity is given by Z T Z Z h2 ðr1 de1 þ r2 de2 þ r3 de3 þ r4 de4 0
N Z X
ðQi Þ ¼
29
ð19Þ
d/x : M1;x C1 S1;x Q5 þ Q5;y þ 3C1 P5 þ M6;y C1 S6;y 1 ¼ ðI2 C1 I4 Þ€u0 þ ðI3 2C1 I5 þ C12 I7 Þ/€x þ ðI2 C1 I4 Þ 2 € o/€z o€ w 1 ou 0 þ ðC1 I5 þ C12 I7 Þ þ ðI5 þ C1 I7 Þ z 3 ox ox ox ð20Þ d/y : M2;x C1 S2;x Q4 þ Q5;y þ 3C1 P4 þ M6;x C1 S6;x ¼ ðI2 C1 I4 Þ€v0 þ ðI3 2C1 I5 þ C12 I7 Þ/€y 1 o/€ o€ w0 þ ðI4 þ C1 I6 Þ z þ ðC1 I5 þ C12 I7 Þ 2 oy oy
znþ1
ri ð1; z; z3 Þ dz
1 o€u0 o€v0 € z þ C1 I4 € 0 þ I2 /€z þ u þ C1 I4 ¼ I1 w 3 ox oy 2 € € € o/ y 1 o/ o/ 1 C1 I6 2z C1 I6 þ C1 I5 x þ C1 I5 2 2 ox oy ox €0 €0 o2 /€z o2 w o2 w o/€ C12 I7 2 C12 I7 2 C12 I7 x 2 oy ox oy ox 2 2 € o/y 1 o u € € 1 ou I7 2z I7 2z C12 I7 3 ox 3 oy oy
ði ¼ 1; 2; 3; 6Þ
zn
ð14Þ
€ 1 ou þ ðI5 þ C1 I7 Þ z 3 oy
ð21Þ
30
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
d/z : 1 1 h P1;xx þ P2;yy þ P6;xy N3 q 2 2 2 o€ u0 o€v0 € z þ C1 I4 € 0 þ I2 /€z þ I3 u þ C1 I4 ¼ I1 w ox oy o/€y o/€ þ ðC1 I5 C12 I7 Þ x þ ðC1 I5 C12 I7 Þ ox oy 2 2 2€ €0 €0 1 ow ow o/ C12 I7 2 C12 I7 2 C1 I6 2z 2 ox oy ox 2€ 2 2 € € 1 o/ 1 ou 1 ou C1 I6 2z I7 2z I7 2z 2 3 ox 3 oy oy
ð22Þ
d/z : 1 1 2 h2 S1;xx þ P2;yy þ P6;xy 2M3 þ q 3 3 3 4 1 o€ u 1 o€v0 0 € z þ I4 € 0 þ I4 /€z þ I5 u þ I4 ¼ I3 w 3 ox 3 oy o/€y o€v0 1 o/€ 1 þ ðI5 C1 I7 Þ x þ ðI5 C1 I7 Þ þ C 1 I4 oy 3 ox 3 oy 2 2€ 2€ 2 € €0 1 o / 1 ow 1 o/ 1 ou C1 I7 2 I6 2z I6 2z I7 2z 3 6 ox 6 oy 9 ox oy € 1 o2 u I7 2z ð23Þ 9 oy where C1 is defined as
88 99 > > > N1 > > > >> => < > > > N > > 2 > > > > > > > > N > > 3 > > > > > : ;> > > > > N 6 > >8 9 > > > > > M > > 1>> > >> > > = < > > > M2 > > > > > 2 > > > > M ½A > > > 3>> > > > ;= 6 4 > > P1 > > > >> => < > > > Sym > > P 2 > > > > > > > > P > > > > > > ;> : 3> > > > > > 8 P6 9 > > > > > > >> > S1 > > >> > > = < > > > S2 > > > > > > > > > S > > > 3> > > > ; :: ;> S6
88 099 > > > e1 > > > >> => < 0> > > e > > 2 > > > > 0 > > > e > > > 3>> > > > : 0;> > > > > > 8 e6 9 > > > > > 0 > > k1 > > >> > > > > = < > > 0 > > k > > 2 > 3> > > 0 > k ½D ½E > > > > 3 >> > > > ; : = < 0 k ½E ½F 7 7 8 69 1 ½F ½H 5> > > k1 > > > >> => < 1> > ½J > > > k > > 2 > > 1 > > > > k > > > > 3 > > > > ; : > > 1 > k > >8 6 9> > > > > 2 > >> > k1 > > >> > > = < > > 2 > k2 > > > > > > > 2 > > k3 > > > > > > > ; :: 2 ;> k6
½B ½D
ð26Þ Eq. (26) provides the relations between stress resultants and strains, which are defined by displacement field parameters. In the Eq. (26), the elements of stiffness matrix can for example be defined as 2 3 A11 A12 A13 A16 6 A22 A23 A26 7 7 ½A ¼ 6 ð27Þ 4 A33 A36 5 Sym A66 Elements of matrices ½A, ½B, ½D, ½E, ½F and ½H are defined as follows: ðAij ; Bij ; Dij ; Eij ; Fij ; Hij ; Jij Þ ¼
Z
h 2
Qij ð1; z; z2 ; z3 ; z4 ; z5 ; z6 ; z7 Þ dz
h2
C1 ¼
4 3h2
ð24Þ
ði; j ¼ 1; 2; 3; 6Þ ð28Þ
and q is distributed transverse load on the top surface.
3. Finite element modeling of equations Using approximation equation for displacement field as ui ¼ hui ifN g
/yi ¼ h/yi ifN g
vi ¼ hvi ifN g
/zi ¼ h/zi ifN g
wi ¼ hwi ifN g uzi ¼ huzi ifN g
For in-plane components, the matrix form of stress resultants and strains are 8 9 8 09 Q4 > e4 > > > > > > > = = < < 0 Q ½A ½B ½D ½E 5 ¼ e50 ð29Þ ½D ½E ½F ½H > P4 > > k4 > > > > > ; ; : : P5 k50 where for example A44 A45 ½A ¼ Sym A55
ð25Þ
/xi ¼ h/xi ifN g
ð30Þ
and ðAij ; Bij ; Dij ; Eij ; Fij ; Hij Þ ¼
Z
h 2
Qij ð1; z; z2 ; z3 ; z4 ; z5 ; z6 Þ dz
h2
ði; j ¼ 4; 5Þ and substitution of displacements approximations in Eqs. (13)–(16) and (17)–(23), finite element type of elasticity equations can be derived. For writing the equations in displacement field parameters, following relations have to be used.
ð31Þ Finally, using the finite element analysis equations of motion can be written in compact form as the following equation
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
½MfX€ g þ ½KfX g ¼ fF g
ð32Þ
Using quadratic 6 nodes triangular elements to satisfy C 1 -continuity, and imposing the following boundary conditions for Navier type boundary conditions, governing equations can be solved. Elements of matrices ½K, ½M and fF g are presented in Appendixes C, D and E, respectively. The plate is simply supported at four edges therefore, primary boundary conditions are u0 ðx; 0Þ ¼ u0 ðx; bÞ ¼ 0 /x ðx; 0Þ ¼ /x ðx; bÞ ¼ 0 v0 ð0; yÞ ¼ v0 ða; yÞ ¼ 0 /y ð0; yÞ ¼ /y ða; yÞ ¼ 0
ð33Þ
w0 ðx; 0Þ ¼ w0 ðx; bÞ ¼ w0 ð0; yÞ ¼ w0 ða; yÞ ¼ 0 It is seen that the governing equations are in general dynamic form. To analyze the static behavior of the plate, the stiffness matrix ½K and the force vector fF g are needed.
4. Numerical solutions Tables 1 and 2 contain nondimensionalized mid point deflections and stresses obtained with 3-D elasticity theory (ELS), TSDT, first order shear deformation theory (FSDT), and classical laminate plate theory (CLPT) for the following three problems: Problem 1. A three-ply (0-90-0) square ða=b ¼ 1Þ laminate with equal thickness layers has been subjected to a sinusoidal distributed transverse load on top plane, and the results are presented in Table 1. The material properties of each ply is assumed as E1 ¼ 175 GPa, E2 ¼ 7 GPa, G12 ¼ G13 ¼ 3:5 GPa, G23 ¼ 1:4 GPa and v12 ¼ v13 ¼ 0:25.
31
Problem 2. A four-ply (0-90-90-0) square ða=b ¼ 1Þ laminate with equal thickness layers has been subjected to a sinusoidal distributed transverse load on top plane and the results are presented in Table 2. The material properties are as Problem 1. Problem 3. A comparison of maximum deflection using FSDT with correction factor of k ¼ 5=6 and TRDT for an antisymmetric cross-ply by different number of layers and with the same boundary conditions and load distribution as previous examples. For this case, the material properties are G12 ¼ G13 ¼ 0:5E2 , G23 ¼ 0:2E2 , v12 ¼ 0:25 and a=h ¼ 10. The results are shown in Table 3. FSDT and ELS results are presented in [10,13]. The following nondimensionalized quantities at specific points are presented in Tables and Graphs as a result of TSDT and are compared to CLPT, FSDT and ELS solutions of the problem [10,12]. 2 a b E2 h 3 a b h h ¼ w0 ; ; ; w rxx ¼ rxx 2 2 2 2 2 a4 q0 b2 q0 2 2 a b h h h h ; ; ryy ¼ ryy rxy ¼ rxy 0; 0; 2 2 4 2 b2 q0 b2 q0 a h b h ryz ¼ ryz ; 0; 0 rxz ¼ rxz 0; ; 0 2 bq0 2 bq0 ð34Þ
5. Conclusions Using the Reddy displacement field for third order shear deformation theory, a set of dynamic equations for modeling the behavior of a laminated plate is derived. Third order shears deformation theory (TRDT) of Reddy has 7 parameters in displacement field and
Table 1 Nondimensionalized maximum stresses and deflections at the mid point of a square simply supported (0-90-0) laminate a=h
Method
rxx
ryz
w
4
ELS TSDT FSDT
0.775 0.7392 0.4370
0.217 0.1884 0.1561 0.1968
– 0.0197 0.0177
ELS TSDT FSDT
0.590 0.5713 0.5134
0.123 0.1082 0.0915 0.1108
– 0.0773 0.0669
ELS TSDT FSDT
0.552 0.5426 0.5384
0.094 0.0791 0.0703
– 0.0463 0.0434
CLPT
0.5387
0.0827 –
0.4313
3-D ELS [13] 10
3-D ELS [13] 100
3-D ELS [13] –
32
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
Table 2 Nondimensionalized maximum stresses and deflections at the mid point of a square simply supported (0-90-90-0) laminate a=h
Method
rxx
ryy
ryz
rxz
rxy
w
4
ELS TSDT FSDT
0.720 0.681 0.406
0.663 0.647 0.576
0.292 0.244 0.196 0.280
0.219 0.211 0.140 0.269
0.0467 0.0451 0.0308
0.0195 0.0190 0.0170
ELS TSDT FSDT
0.559 0.551 0.499
0.401 0.394 0.361
0.196 0.163 0.130 0.181
0.301 0.211 0.167 0.318
0.0275 0.0451 0.0241
0.00743 0.00732 0.00663
ELS TSDT FSDT
0.539 0.539 0.538
0.276 0.275 0.270
0.141 0.129 0.101
0.337 0.308 0.178
0.0216 0.0216 0.0213
0.00437 0.00435 0.00435
CLPT
0.539
0.270
0.139 0.139
0.337 0.337
0.0213
0.00432
3-D ELS [13] 10
3-D ELS [13] 100
3-D ELS [13] –
Table 3 Nondimensionalized maximum deflection at a square simply supported antisymmetric cross-ply (0-90-. . .) laminate a=h 4 10 20 50 100 CLPT
N ¼2
N ¼6
FSDT
TSDT
FSDT
TSDT
0.021492 0.012373 0.011070 0.010705 0.010653
0.020102 0.012253 0.010241 0.010701 0.010653
0.015473 0.006354 0.005052 0.004687 0.004635
0.015423 0.006375 0.005055 0.004687 0.004635
0.0106306
satisfies the vanishing of transverse shear stresses on the boundary planes. By deriving the dynamic equation of motion and equations in finite element form, stresses and transverse displacements of different points of plate are defined. From the results, it is clear that the third order theory gives more accurate results for deflections and stresses when compared to the first order shear deformation plate theory (FSDT) with correction factor for shear deformation of k ¼ 5=6. It is known that the shear correction factor k depends on the lamina properties. The fact that no shear correction coefficients are needed in the third order theory makes it more convenient to use. In Table 1, nondimensionalized stresses rxx and ryz are presented. It is seen that for span to thickness ratio a=h ¼ 4, errors for TRDT results are 4.6% and 13.3%, respectively, but for FSDT these errors are 43.6% and 28.1%. For a=h ¼ 10, these errors are reduced to 3.2% and 12.1% for TRDT and 13% and 26% for FSDT. For a=h ¼ 100, these errors are 1.8% and 15.9% for TRDT and 2.5% and 25.5% for FSDT, respectively. It is seen that the TRDT results for ryz are very close to the stresses computed from three-dimensional elasticity of first order shear deformation theory for less amount of
0.004618
a=h. In Table 2, for a symmetric cross-ply, it is seen that the third order theory in comparison with the elasticity solution, predicts deflection by 2% while the first order theory predicts by about 12.8% for a=h ¼ 4. The errors are 1.4% in TSDT and 10.8% in FSDT for a=h ¼ 10. It is seen that the errors are reduced at higher a=h. For a=h ¼ 100, the errors are 0.4% for both theories. Results for stresses are also presented in Table 2. The closer
Fig. 2. Nondimensionalized normal stress rxx through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
33
symmetric cross-ply with 2 and 6 layers are presented. Relative errors for the results of TRDT and FSDT in 2ply and 6-ply plates are 7.5% and 0.4% and for a=h ¼ 4. These errors are near zero for a=h ¼ 100.
Appendix A. Relationships of material properties in rotated coordinate systems a1 ¼ a 1 C 2 þ a 2 S 2 a2 ¼ a1 S 2 þ a2 C 2 Fig. 3. Nondimensionalized normal stress ryy through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
a3 ¼ a3 a4 ¼ 0 a5 ¼ 0 a6 ¼ ða1 a2 ÞCS
0.5
Q11 ¼ Q11 C 4 þ 2ðQ12 þ 2Q66 ÞC 2 S 2 þ Q22 S 4 0.25
Q12 ¼ ðQ11 þ Q22 4Q66 ÞC 2 S 2 þ Q12 ðC 4 þ S 4 Þ Q13 ¼ Q13 C 2 þ Q23 S 2
0
Q16 ¼ Q22 CS 3 þ Q11 C 3 S CSðC 2 S 2 ÞðQ12 þ 2Q66 Þ Q22 ¼ Q11 S 4 þ 2ðQ12 þ 2Q66 ÞC 2 S 2 þ Q22 C 4
-0.25
Q23 ¼ Q13 S 2 þ Q23 C 2
-0.5 0
0.4
0.2
0.6
Q26 ¼ Q22 C 3 S þ Q11 CS 3 þ CSðC 2 S 2 ÞðQ12 þ 2Q66 Þ Q33 ¼ Q33
Fig. 4. Nondimensionalized transverse shear stress rxz through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
Q44 ¼ Q44 C 2 þ Q55 S 2 Q45 ¼ ðQ55 Q44 ÞCS Q55 ¼ Q55 C 2 þ Q44 S 2
0.5
Q66 ¼ ðQ11 þ Q22 2Q12 2Q66 ÞC 2 S 2 þ Q66 ðC 4 þ S 4 Þ
0.25
Appendix B. Relationships between strain components and displacement
0 -0.25
ou0 ox o/ k10 ¼ x ox 1 o2 / z k11 ¼ 2 ox2 2 o w0 o/x 1 o2 wz 2 k1 ¼ C1 þ þ 3 ox2 ox2 ox e01 ¼
-0.5 0
0.04
0.08
0.12
0.16
Fig. 5. Nondimensionalized transverse shear stress ryz through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
results can be seen for in-plane stresses of TRDT and the results of elasticity solution. In Figs. 2–5 these through the thickness nondimensionalized stresses of the cross-ply with a=h ¼ 4 are plotted. It is seen that the stresses are discontinuous like other ESL theories due to the continuity of the transverse shear strains through the thickness of lamina. In Table 3, deflection of an anti-
ov0 oy o/ y k20 ¼ oy e02 ¼
k21 ¼
1 o2 / z 2 oy 2
34
k22
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
2 o w0 o/y 1 o2 w z þ þ ¼ C1 3 oy 2 oy 2 oy
e03 ¼ /z k30
K1;1 ¼
Z
K1;2 ¼
Z
K1;3 ¼
k31 ¼ 0
Z
C1 E61 Ny Nxx C1 E62 Ny Nyy 2C1 E66 Ny Nxy Þ dA
ow0 þ /y oy
K1;4 ¼
Z
þ B61 Ny Nx þ B66 Ny Ny C1 E61 Ny Nx
ow0 þ /y oy
k42 ¼ 0 ow0 þ /x ox ow0 1 k5 ¼ 3C1 þ /x ox
C1 E66 Ny Ny Þ dA K1;5 ¼
k52
¼0
e06
ou0 ov0 þ ¼ oy ox
o2 u0 oxoy 2 o w0 o/x o/y 2 o2 uz þ k62 ¼ C1 2 þ þ 3 oxoy oxoy oy ox
Appendix C. Elements of local stiffness matrix
ðB12 Nx Ny þ B16 Nx Nx C1 E12 Nx Ny C1 E16 Nx Nx þ E62 Ny Ny þ B66 Ny Nx C1 E62 Ny Ny
K1;6
o/x o/y þ oy ox
k61 ¼
Z A
e05 ¼
k60 ¼
ðB11 Nx Nx þ B16 Nx Ny C1 E11 Nx Nx C1 E16 Nx Ny A
k40 ¼ 0 ¼ 3C1
ðC1 E11 Nx Nxx C1 E12 Nx Nyy 2C1 E16 Nx Nxy A
k32 ¼ 0
k41
ðA12 Nx Ny þ A16 Nx Nx þ A62 Ny Ny þ A66 Ny Nx Þ dA A
¼ 2uz
e04 ¼
ðA11 Nx Nx þ A16 Nx Ny þ A61 Ny Nx þ A66 Ny Ny Þ dA A
K1;7
C1 E66 Ny Nx Þ dA Z 1 ¼ A13 Nx N D11 Nx Nxx D16 Nx Nxy þ A63 Ny N 2 A 1 1 D61 Ny Nxx D66 Ny Nxy D12 Nx Nyy 2 2 1 D62 Ny Nyy dA 2 Z 1 1 ¼ 2B13 Nx N E11 Nx Nxx E12 Nx Nyy 3 3 A 2 1 E16 Nx Nxy þ 2B63 Ny N E61 Ny Nxx 3 3 1 2 E62 Ny Nyy E66 Ny Nxy dA 3 3
K2;1 ¼
Z
ðA12 Ny Nx þ A26 Ny Ny þ A61 Nx Nx þ A66 Nx Ny Þ dA A
N is the column vector of shape function and as an example for a quadratic triangular element, multiplication of two shape function is written as follow: 8 9 N1 > > > > > > > > N > 2> > = < > N3 NN ¼ h N1 N2 N3 N4 N5 N6 i N4 > > > > > > > > N > > > ; : 5> N6
K2;2 ¼
Z
ðA22 Ny Ny þ A26 Ny Nx þ A62 Nx Ny þ A66 Nx Nx Þ dA A
K2;3 ¼
Z
ðC1 E22 Ny Nyy 2C1 E26 Ny Nxy C1 E61 Nx Nxx A
C1 E12 Ny Nxx C1 E62 Nx Nyy 2C1 E66 Nx Nxy Þ dA K2;4 ¼
Z
ðB12 Ny Nx þ B26 Ny Ny C1 E12 Ny Nx C1 E26 Ny Ny A
þ B61 Nx Nx þ B66 Nx Ny C1 E61 Nx Nx also for simplicity, the following terms are used. oN oN o2 N Ny ¼ Nxx ¼ 2 ox oy ox 2 2 oN oN Nyy ¼ 2 Nxy ¼ oy oxoy
Nx ¼
C1 E66 Nx Ny Þ dA K2;5 ¼
Z
ðB22 Ny Ny þ B26 Ny Nx C1 E22 Ny Ny C1 E26 Ny Nx A
þ B62 Nx Ny þ B66 Nx Nx C1 E62 Nx Ny C1 E66 Nx Nx Þ dA
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
K2;6 ¼
Z A
1 1 A23 Ny N D12 Ny Nxx D22 Ny Nyy 2 2
K3;6 ¼
1 D26 Ny Nxy þ A63 Nx N D61 Nx Nxx 2 1 D62 Nx Nyy D66 Nx Nxy dA 2 K2;7 ¼
K3;1 ¼
Z
1 1 2B23 Ny N E12 Ny Nxx E22 Ny Nyy 3 3 A 2 1 E26 Ny Nxy þ 2B63 Nx N E61 Nx Nxx 3 3 1 2 E62 Nx Nyy E66 Nx Nxy dA 3 3
Z
K3;7 ¼
ðC1 E11 Nxx Nx C1 E12 Nyy Nx C1 E16 Nxx Ny
A
C1 E26 Nyy Ny 2C1 E16 Nxy Nx 2C1 E66 Nxy Ny Þ dA K3;2 ¼
Z
ðC1 E12 Nxx Ny C1 E16 Nxx Nx C1 E22 Nyy Ny
35
Z
1 C1 E13 Nxx N þ C1 H11 Nxx Nxx 2 A 1 þ C1 H12 Nxx Nyy þ C1 H16 Nxx Nxy C1 E23 Nyy N 2 1 1 þ C1 H12 Nyy Nxx þ C1 H22 Nyy Nyy þ C1 H26 Nyy Nxy 2 2 2C1 E36 Nxy N þ C1 H16 Nxy Nxx þ C1 H26 Nxy Nyy þ 2C1 H66 Nxy Nxy dA
Z
1 1 2 C1 J11 Nxx Nxx þ C1 J12 Nxx Nyy þ C1 J16 Nxx Nxy 3 3 3 A 1 1 2C1 F23 Nyy N þ C1 J12 Nyy Nxx þ C1 J22 Nyy Nyy 3 3 2 þ C1 J26 Nyy Nxy 2C1 F13 Nxx N 4C1 F36 Nxy N 3 2 2 þ C1 J16 Nxy Nxx þ C1 J26 Nxy Nyy 3 3 4 2 þ C1 J66 Nxy Nxy dA 3
A
C1 E26 Nyy Nx 2C1 E26 Nxy Ny 2C1 E66 Nxy Nx Þ dA K3;3 ¼
K4;1 ¼
Z
Z
ðC12 J11 Nxx Nxx þ C12 J12 Nxx Nyy þ C12 J12 Nyy Nxx
C1 E11 Nx Nx C1 E16 Ny Nx C1 E66 Ny Ny
þ C12 J22 Nyy Nyy þ 2C12 J16 Nxy Nxx þ 2C12 J26 Nxy Nyy
C1 E16 Nx Ny Þ dA
A
þ 4C12 J66 Nxy Nxy þ 2C12 J16 Nxx Nxy þ A44 Ny Ny þ A45 Ny Nx þ A55 Nx Nx þ A45 Nx Ny þ 2C12 J26 Nyy Nxy
K4;2 ¼
Z
C1 E12 Nx Ny C1 E16 Nx Nx C1 E26 Ny Ny
6C1 D45 Nx Ny þ 9C12 F44 Ny Ny þ 9C12 F45 Ny Nx Z
C1 E66 Ny Nx Þ dA
þ 9C12 F55 Nx Nx þ 9C12 F45 Nx Ny Þ dA ðC1 F11 Nxx Nx C1 F16 Nxx Ny þ C12 J11 Nxx Nx
A
þ
C12 J16 Nxx Ny
þ
C12 J12 Nyy Nx
K4;3 ¼
Z
C1 F12 Nyy Nx C1 F26 Nyy Ny þ
C12 J26 Nyy Ny
2C1 F66 Ny Nxy þ C12 J11 Nx Nxx þ C12 J12 Nx Nyy
2C1 F16 Nxy Nx
þ 2C12 J16 Nx Nyy þ C12 J16 Ny Nxx C1 F12 Nx Nyy C1 F26 Ny Nyy þ C12 J26 Ny Nyy þ 2C12 J66 Ny Nxy
þ A45 Ny N 6C1 D45 Ny N þ A55 Nx N 3C1 D45 Nx N
þ A45 NNy þ A55 NNx 6C1 D45 NNy 3C1 D45 NNx
3C1 D55 Nx N þ 9C12 E45 Ny N þ 9C12 E55 Nx N Þ dA K3;5 ¼
ðC1 F11 Nx Nxx 2C1 F16 Nx Nxy C1 F16 Ny Nxx
A
2C1 F66 Nxy Ny þ 2C12 J16 Nxy Nx þ 2C12 J66 Nxy Ny
Z
ðB12 Nx Ny þ B16 Nx Nx þ B26 Ny Ny þ B66 Ny Nx
A
6C1 D44 Ny Ny 6C1 D45 Ny Nx 6C1 D55 Nx Nx
K3;4 ¼
ðB11 Nx Nx þ B16 Nx Ny þ B66 Ny Ny þ B16 Ny Nx
A
3C1 D55 NNx þ 9C12 F45 NNy þ 9C12 F55 NNx Þ dA
ðC1 F12 Nxx Ny C1 F16 Nxx Nx þ C12 J12 Nxx Ny
A
þ C12 J16 Nxx Nx C1 F22 Nyy Ny C1 F26 Nyy Nx
K4;4 ¼
Z
ðD11 Nx Nx þ D16 Nx Ny
A
þ C12 J22 Nyy Ny þ C12 J26 Nyy Nx 2C1 F26 Nxy Ny
2C1 F11 Nx Nx 2C1 F16 Nx Ny D16 Ny Nx þ D66 Ny Ny
2C1 F66 Nxy Nx þ 2C12 J26 Nxy Ny þ 2C12 J66 Nxy Nx
2C1 F16 Ny Nx 2C1 F66 Ny Ny þ C12 J11 Nx Nx
þ A44 Ny N 6C1 D44 Ny N þ A45 Nx N 3C1 D45 Nx N
þ C12 J16 Nx Ny þ C12 J16 Ny Nx þ C12 J66 Ny Ny þ A55 NN
3C1 D45 Nx N þ 9C12 F44 Ny N þ 9C12 F45 Nx N Þ dA
6C1 D55 NN þ 9C12 F55 NN Þ dA
36
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
K4;5 ¼
Z
ðD12 Nx Ny þ D16 Nx Nx 2C1 F12 Nx Ny
K5;4 ¼
Z
A
K4;6
K4;7
2C1 F16 Nx Nx D26 Ny Ny þ D66 Ny Nx 2C1 F26 Ny Ny
2C1 F26 Ny Ny D16 Nx Nx þ D66 Nx Ny 2C1 F16 Nx Nx
2C1 F66 Ny Nx þ C12 J12 Nx Ny þ C12 J16 Nx Nx
2C1 F66 Nx Ny þ C12 J12 Ny Nx þ C12 J26 Ny Ny
þ C12 J26 Ny Ny þ C12 J66 Ny Nx þ A45 NN 6C1 D45 NN
þ C12 J16 Nx Nx þ C12 J66 Nx Ny þ A45 NN 6C1 D45 NN
þ 9C12 F45 NN Þ dA Z 1 1 ¼ B13 Nx N E11 Nx Nxx E12 Nx Nyy E16 Nx Nxy 2 2 A 1 1 þ B36 Ny N E16 Ny Nxx E26 Ny Nyy E66 Ny Nxy 2 2 1 1 C1 E13 Nx N þ C1 H11 Nx Nxx þ C1 H12 Nx Nyy 2 2 1 þ C1 H16 Nx Nxy C1 E36 Ny N þ C1 H16 Ny Nxx 2 1 þ C1 H26 Ny Nyy þ C1 H66 Ny Nxy dA 2 Z 1 1 ¼ 2D13 Nx N F11 Nx Nxx F12 Nx Nyy 3 3 A
K5;1 ¼
Z
2 1 F16 Nx Nxy þ 2D36 Ny N F16 Ny Nxx 3 3 1 2 F26 Ny Nyy F66 Ny Nxy 2C1 F13 Nx N 3 3 1 1 2 þ C1 J11 Nx Nxx þ C1 J12 Nx Nyy þ C1 J16 Nx Nxy 3 3 3 1 1 2C1 F36 Ny N þ C1 J16 Ny Nxx þ C1 J26 Ny Nyy 3 3 2 þ C1 J66 Ny Nxy dA 3
K5;5 ¼
þ C12 J26 Nx Ny þ C12 J66 Nx Nx þ A44 NN 6C1 D44 NN
K5;6
K5;7
C1 E66 Nx Ny Þ dA ðB22 Ny Ny þ B26 Nx Ny þ B26 Ny Nx þ B66 Nx Nx C1 E22 Ny Ny C1 E26 Nx Ny C1 E26 Ny Nx
K6;1
K5;3 ¼
ðC1 F12 Ny Nxx C1 F22 Ny Nyy 2C1 F26 Ny Nxy
A
C1 F16 Nx Nxx C1 F26 Nx Nyy 2C1 F66 Nx Nxy þ C12 J12 Ny Nxx þ C12 J22 Ny Nyy þ 2C12 J26 Ny Nxy þ
C12 J16 Nx Nxx
þ
C12 J26 Nx Nyy
þ
2C12 J66 Nx Nxy
þ A45 NNx þ A44 NNy 6C1 D44 NNy 6C1 D45 NNx þ 9C12 F44 NNy þ 9C12 F45 NNx Þ dA
þ 9C12 F44 NN Þ dA Z 1 1 ¼ B23 Ny N E12 Ny Nxx E22 Ny Nyy E26 Ny Nxy 2 2 A 1 1 þ B36 Nx N E16 Nx Nxx E26 Nx Nyy E66 Nx Nxy 2 2 1 1 C1 E23 Ny N þ C1 H12 Ny Nxx þ C1 H22 Ny Nyy 2 2 1 þ C1 H26 Ny Nxy C1 E36 Nx N þ C1 H16 Nx Nxx 2 1 þ C1 H26 Nx Nyy þ C1 H66 Nx Nxy dA 2 Z 1 1 ¼ 2D23 Ny N F12 Ny Nxx F22 Ny Nyy 3 3 A 2 1 F26 Ny Nxy þ 2D36 Nx N F16 Nx Nxx 3 3 1 2 F26 Nx Nyy F66 Nx Nxy 2C1 F23 Ny N 3 3 1 1 2 þ C1 J12 Ny Nxx þ C1 J22 Ny Nyy þ C1 J26 Ny Nxy 3 3 3 1 1 2C1 F36 Nx N þ C1 J16 Nx Nxx þ C1 J26 Nx Nyy 3 3 2 þ C1 J66 Nx Nxy dA 3 Z ¼ A13 NNx þ A36 NNy D16 Nxy Nx D66 Nxy Ny A
C1 E66 Nx Nx Þ dA Z
ðD22 Ny Ny þ D26 Ny Nx 2C1 F22 Ny Ny
2C1 F66 Nx Nx þ C12 J22 Ny Ny þ C12 J26 Ny Nx
ðB12 Ny Nx þ B16 Nx Nx þ B26 Ny Ny þ B66 Nx Ny
A
þ 9C12 F45 NN Þ dA
2C1 F26 Ny Nx D26 Nx Ny þ D66 Nx Nx 2C1 F26 Nx Ny
C1 E12 Ny Nx C1 E16 Nx Nx C1 E26 Ny Ny
K5;2 ¼
Z
A
A
Z
ðD12 Ny Nx þ D26 Ny Ny 2C1 F12 Ny Nx A
K6;2
1 1 1 D11 Nxx Nx D16 Nxx Ny D12 Nyy Nx 2 2 2 1 D26 Nyy Ny dA 2 Z ¼ A23 NNy þ A36 NNx D26 Nxy Ny D66 Nxy Nx A
1 1 1 D12 Nxx Ny D16 Nxx Nx D22 Nyy Ny 2 2 2 1 D26 Nyy Nx dA 2
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
K6;3 ¼
Z A
1 1 C1 H11 Nxx Nxx þ C1 H12 Nxx Nyy þ C1 H16 Nxx Nxy 2 2
K7;1 ¼
1 1 þ C1 H12 Nyy Nxx þ C1 H22 Nyy Nyy þ C1 H26 Nyy Nxy 2 2 þ C1 H16 Nxy Nxx þ C1 H26 Nxy Nyy þ 2C1 H66 Nxy Nxy C1 E13 NNxx C1 E23 NNyy 2C1 E36 NNxy dA K6;4 ¼
K6;5 ¼
K6;6 ¼
Z
1 1 1 E11 Nxx Nx E16 Nxx Ny þ C1 H11 Nxx Nx 2 2 2 A 1 1 1 þ C1 H16 Nxx Ny E12 Nyy Nx E26 Nyy Ny 2 2 2 1 1 þ C1 H12 Nyy Nx þ C1 H26 Nyy Ny E16 Nxy Nx 2 2 E66 Nxy Ny þ C1 H16 Nxy Nx þ C1 H66 Nxy Ny þ B13 NNx þ B36 NNy C1 E13 NNx C1 E36 NNy dA
K7;3 ¼
1 1 1 D13 Nxx N þ F12 Nxx Nyy þ F11 Nxx Nxx 2 4 4
1 1 1 þ F16 Nxx Nxy D23 Nyy N þ F12 Nyy Nxx 2 2 4 1 1 þ F22 Nyy Nyy þ F26 Nyy Nxy D36 Nxy N 4 2 1 1 þ F16 Nxy Nxx þ F26 Nxy Nyy þ F66 Nxy Nxy þ A33 NN 2 2 1 1 D13 NNxx D23 NNyy D36 NNxy dA 2 2 Z 1 1 ¼ E13 Nxx N þ H12 Nxx Nyy þ H11 Nxx Nxx 6 6 A 1 1 þ H16 Nxx Nxy E23 Nyy N þ H12 Nyy Nxx 3 6 1 1 þ H22 Nyy Nyy þ H26 Nyy Nxy 2E36 Nxy N 6 3 1 1 2 þ H16 Nxy Nxx þ H26 Nxy Nyy þ H66 Nxy Nxy 3 3 3 1 1 þ 2B33 NN E13 NNxx E23 NNyy 3 3 2 E36 NNxy dA 3
1 1 1 E11 Nxx Nx E16 Nxx Ny E12 Nyy Nx 3 3 3 A 1 2 2 E26 Nyy Ny E16 Nxy Nx E66 Nxy Ny 3 3 3 þ 2B13 NNx þ 2B36 NNy dA
Z
1 1 1 E12 Nxx Ny E16 Nxx Nx E22 Nyy Ny 3 3 3 A 1 2 2 E26 Nyy Nx E26 Nxy Ny E66 Nxy Nx 3 3 3 þ 2B13 NNy þ 2B36 NNx dA
Z
1 1 C1 J11 Nxx Nxx þ C1 J12 Nxx Nyy 3 3
2 1 1 þ C1 J16 Nxx Nxy C1 J12 Nyy Nxx þ C1 J22 Nyy Nyy 3 3 3 2 2 2 þ C1 J26 Nyy Nxy þ C1 J16 Nxy Nxx þ C1 J26 Nxy Nyy 3 3 3 4 þ C1 J66 Nxy Nxy 2C1 F13 NNxx 2C1 F23 NNyy 3 4C1 F36 NNxy dA
1 1 1 E12 Nxx Ny E16 Nxx Nx þ C1 H12 Nxx Ny 2 2 2 A 1 1 1 þ C1 H16 Nxx Nx E22 Nyy Ny E26 Nyy Nx 2 2 2 1 1 þ C1 H22 Nyy Ny þ C1 H26 Nyy Nx E26 Nxy Ny 2 2 E66 Nxy Nx þ C1 H26 Nxy Ny þ C1 H66 Nxy Nx þ B23 NNy þ B36 NNx C1 E23 NNy C1 E36 NNx dA
Z
Z
A
Z
A
K6;7
K7;2 ¼
37
K7;4 ¼
Z A
1 1 1 F11 Nxx Nx F16 Nxx Ny þ C1 J11 Nxx Nx 3 3 3
1 1 1 þ C1 J16 Nxx Ny F12 Nyy Nx F26 Nyy Ny 3 3 3 1 1 2 þ C1 J12 Nyy Nx þ C1 J26 Nyy Ny F16 Nxy Nx 3 3 3 2 2 2 F66 Nxy Ny þ C1 J16 Nxy Nx þ C1 J66 Nxy Ny 3 3 3 þ 2D13 NNx þ 2D36 NNy 2C1 F13 NNx 2C1 F36 NNy dA K7;5 ¼
Z
1 1 1 F12 Nxx Ny F16 Nxx Nx þ C1 J12 Nxx Ny 3 3 3 A 1 1 1 þ C1 J16 Nxx Nx F22 Nyy Ny F26 Nyy Nx 3 3 3 1 1 2 þ C1 J22 Nyy Ny þ C1 J26 Nyy Nx F26 Nxy Ny 3 3 3 2 2 2 F66 Nxy Nx þ C1 J26 Nxy Ny þ C1 J66 Nxy Nx 3 3 3 þ 2D23 NNy þ 2D36 NNx 2C1 F23 NNy 2C1 F36 NNx dA
38
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
Z
K7;6
1 1 1 ¼ E13 Nxx N þ H11 Nxx Nxx þ H12 Nxx Nyy 3 6 6 A 1 1 1 þ H16 Nxx Nxy E23 Nyy N þ H12 Nyy Nxx 3 3 6 1 1 2 þ H22 Nyy Nyy þ H26 Nyy Nxy E36 Nxy N 6 3 3 1 1 þ H16 Nxy Nxx þ H26 Nxy Nyy 3 3 2 þ H66 Nxy Nxy þ 2B33 NN E13 NNxx E23 NNyy 3 2E36 NNxy
K7;7 ¼
m2;7 ¼
A
m3;1 ¼
2 1 1 F13 Nxx N þ J11 Nxx Nxx þ J12 Nxx Nyy 3 9 9 A 2 2 1 þ J16 Nxx Nxy F23 Nyy N þ J12 Nyy Nxx 9 3 9 1 2 4 þ J22 Nyy Nyy þ J26 Nyy Nxy F36 Nxy N 9 9 3 2 2 4 þ J16 Nxy Nxx þ J26 Nxy Nyy þ J66 Nxy Nxy 9 9 9 2 2 þ 4D33 NN F13 NNxx F23 NNyy 3 3 4 F36 NNxy dA 3
Z
1 I4 NNy 3
dA
ðC1 I4 Nx N Þ dA
A
m3;2 ¼
Z
ðC1 I4 Ny N Þ dA
A
m3;3 ¼
Z
ðC12 I7 Nx Nx þ C12 I7 Ny Ny þ I1 NN Þ dA
A
m3;4 ¼
Z
ðC1 ðI5 C1 I7 ÞNx N Þ dA
A
dA
Z
Z
m3;5 ¼ m3;6 m3;7 m4;1
Z
ðC1 ðI5 C1 I7 ÞNy N Þ dA A Z 1 1 C1 I6 Nx Nx þ C1 I6 Ny Ny þ I2 NN dA ¼ 2 2 A Z 1 1 C1 I7 Nx Nx þ C1 I7 Ny Ny þ I3 NN dA ¼ 3 3 A Z ¼ ððI2 C1 I4 ÞNN Þ dA A
m4;2 ¼ 0 Z m4;3 ¼ ðC1 ðI5 C1 I7 ÞNNx Þ dA A Z m4;4 ¼ ððI3 2C1 I5 þ C12 I7 ÞNN Þ dA A
Appendix D. Elements of local mass matrix Z m1;1 ¼ ðI1 NN Þ dA A
m1;2 ¼ 0 Z m1;3 ¼ ðC1 I4 NNx Þ dA A Z m1;4 ¼ ððI2 C1 I4 ÞNN Þ dA A
m1;5 ¼ 0 Z 1 m1;6 ¼ I3 NNx dA 2 A Z 1 m1;7 ¼ I4 NNx dA 3 A m2;1 ¼ 0 Z m2;2 ¼ ðI1 NN Þ dA A Z m2;3 ¼ ðC1 I4 NNy Þ dA A
m2;4 ¼ 0 Z m2;5 ¼ ððI2 C1 I4 ÞNN Þ dA A
m2;6 ¼
Z A
1 I3 NNy 2
dA
m4;5 ¼ 0 Z 1 ð I4 þ C1 I6 ÞNNx dA m4;6 ¼ 2 A Z 1 ð I5 þ C1 I7 ÞNNx dA m4;7 ¼ 3 A m5;1 ¼ 0 Z m5;2 ¼ ððI2 C1 I4 ÞNN Þ dA A Z m5;3 ¼ ðC1 ðI5 C1 I7 ÞNNy Þ dA A
m5;4 ¼ 0 Z m5;5 ¼ ððI3 2C1 I5 þ C12 I7 ÞNN Þ dA A Z 1 ð I4 þ C1 I6 ÞNNy dA m5;6 ¼ 2 A Z 1 ð I5 þ C1 I7 ÞNNy dA m5;7 ¼ 3 A Z 1 m6;1 ¼ I3 Nx N dA 2 A Z 1 m6;2 ¼ I3 Ny N dA 2 A Z 1 1 C1 I6 Nx Nx þ C1 I6 Ny Ny þ I2 NN dA m6;3 ¼ 2 2 A
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
1 1 ¼ I4 Nx N þ C1 I6 Nx N dA 2 2 A Z 1 1 ¼ I4 Ny N þ C1 I6 Ny N dA 2 2 A Z 1 1 I5 Nx Nx þ I5 Ny Ny þ I3 NN dA ¼ 4 4 A Z 1 1 I6 Nx Nx þ I6 Ny Ny þ I4 NN dA ¼ 6 6 A Z 1 ¼ I4 Nx N dA 3 A Z 1 ¼ I4 Ny N dA 3 A Z 1 1 C1 I7 Nx Nx þ C1 I7 Ny Ny þ I3 NN dA ¼ 3 3 A Z 1 1 ¼ I5 Nx N þ C1 I7 Nx N dA 3 3 A Z 1 1 ¼ I5 Ny N þ C1 I7 Ny N dA 3 3 A Z 1 1 I6 Nx Nx þ I6 Ny Ny þ I4 NN dA ¼ 6 6 A Z 1 1 I7 Nx Nx þ I7 Ny Ny þ I5 NN dA ¼ 9 9 A Z
m6;4 m6;5 m6;6 m6;7 m7;1 m7;2 m7;3 m7;4 m7;5 m7;6 m7;7
Appendix E. Elements of local force vector f1 ¼ f2 ¼ 0 Z f3 ¼ ðhqN Þ dA A
39
f4 ¼ f5 ¼ f6 ¼ 0 f7 ¼
Z A
h3 qN dA 4
References [1] Green AE, Naghdi PM. A theory of laminated composite plates. J Appl Math 1982;29(1):35–46. [2] Bose P, Reddy JN. Analysis of composite plates using various plate theories. Part 1: Formulation and analytical solutions. Struct Eng Mech 1998;6(6):583–612. [3] Liberescu L, Hause T. Recent developments in the modeling and behavior of advanced sandwich constructions: a survey. Compos Struct 2000;48:1–17. [4] Lo KH, Christensen RM, Wu EM. A higher-order theory of isotropic elastic plates. J Appl Mech 1977;18:31–8. [5] Robbins DH, Reddy JN. Structural theories and computational models for composite laminates. Appl Mech Rev 1994;47(61):147–70. [6] Noor AK, Scott Burton W. Three-dimensional solutions for antisymmetrically laminated anisotropic plates. J Appl Mech, Trans ASME 1989;1:7. [7] Noor AK, Scott W. Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 1989;42(1):1– 12. [8] Whitney JM, Leissa AW. Analysis of heterogeneous anisotropic plates. J Appl Mech 1969;6:261–266. [9] Whitney JM, Pagano NJ. Shear deformation in heterogeneous anisotropic elastic plates. J Appl Mech 1970;37:1031–6. [10] Reddy JN. Mechanics of laminated composite plates. New York: CRC Press; 1997. [11] Ochao OO, Reddy JN. Finite element analysis of composite laminate. Netherlands: Kluwer Academic Publishers; 1997. [12] Reddy JN. Energy and Variational methods in applied mechanics. New York: John Wiley & Sons; 1984. [13] Pagano NJ. Exact solutions for rectangular bi-directional composites and sandwich plates. AIAA J 1972;10:931–3.
Linear static analysis and finite element modeling for laminated composite plates using third order shear deformation theory M. Rastgaar Aagaah, M. Mahinfalah *, G. Nakhaie Jazar Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105, USA
Abstract In this paper, deformations of a laminated composite plate due to mechanical loads are presented. Third order shear deformation theory of plates, which is categorized in equivalent single layer theories, is used to derive linear dynamic equations of a rectangular multi-layered composite plate. Moreover, derivation of equations for FEM and numerical solutions for displacements and stress distributions of different points of the plate with a sinusoidal distributed mechanical load for Navier type boundary conditions are presented. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Third order shear deformation; Laminated composite plate; Finite element
1. Introduction The use of composite materials in structural components are increasing due to their attractive properties such as high strength-to-weight ratio, ability to tailor the structural properties, etc. Plate structures find numerous applications in the aerospace, military and automotive industries. The effects of transverse shear deformation are considerable for composite structures, because of their high ratio of extensional modulus to transverse shear modulus. Most of the structural theories used till now to characterize the behavior of composite laminates fall into the category of equivalent single layer (ESL) theories. In these theories, the material properties of the constituent layers are combined to form a hypothetical single layer whose properties are equivalent to throughthe-thickness integrated sum of its constituents. This category of theories has been found to be adequate in predicting global response characteristics of laminates, like maximum deflections, maximum stresses, fundamental frequencies, or critical buckling loads [1]. Third order shear deformation theory, which is one of the ESL theories, is derived. This theory is based on *
Corresponding author. Tel.: +1-701-231-8839; fax: +1-701-2318913. E-mail address: [email protected] (M. Mahinfalah). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00081-3
the same assumptions as the classical (CLPT) and first order shear deformation plate theories (FSDT), except that the assumption on the straightness and normality of the transverse normal is relaxed [2–4]. Theories higher than third order are not used because the accuracy gained is so little that the effort required to solve the equations is not justified [5]. In single layer displacement-based theories, one single expansion for each displacement component is used through the entire thickness, and therefore, the transverse strains are continuous through the thickness, a strain state appropriate for homogeneous plates [5–7]. In the present work, the equations of motion have been derived for the linear deformation of laminated plates subjected to a mechanical load based on a third order shear deformation plate theory in conjunction with the Von Karman strains. Unlike to the first order shear deformation theory, the higher order theory does not require shear correction factors. Finally the finite element solution for the plate is derived.
2. Elasticity equations The plate considered in this investigation consists of N orthotropic cross-ply and angle-ply layers with a total thickness h. Components of global Cartesian
28
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
coordinates X, that is located at the middle of the plate, are ðx; y; zÞ where x; y are in-plane coordinates, and z is the transverse coordinate. The top layer is at z ¼ h=2 and the bottom layer is located at z ¼ h=2. Layer coordinates of a typical nth layer are Xn and its components are ðxn ; yn ; zn Þ and xn is in the direction of fibers as shown in Fig. 1. The linear constitutive equation of the nth layer when considering thermal expansion effect is given by 8 9 r1 > > > > > > > r2 > > > > > > =
2
6 6 6 6 3 ¼6 6 > r4 > > > 6 > > > > 6 > > > > r 5 > 4 > : ; r6
Q11
Q12 Q22
Sym
Q13 Q23 Q33
0 0 0 Q44
0 0 0 Q45 Q55
9 38 e1 a1 h > Q16 > > > > 7> e a h > > > Q26 7> > 2 2 > > > > 7> = < 7 a h e Q36 7 3 3 > > e4 0 7 > 7> > > > 7> > > > e 0 5> 5 > > > > ; : e Q66 6 ð1Þ
where ai are coefficients of thermal expansion in direction of layer coordinates and h is the change in temperature of each layer. Since the thermal effects cause a volume change, they do not have effect on transverse stresses and strains. Therefore, thermal expansion coefficients for an orthotropic lamina have only three components [8,9]. Let h be the angle between the layer coordinates and the global coordinate, then the following relationships exist between stresses and strains in both coordinates. frg ¼ ½T frg feg ¼ ½T feg
ð2Þ
r and e are prescribed in global coordinate but r and e are components of stress and strain in lamina coordinates. ½T is rotational matrix about the transverse direction z at h and is defined as
2
C2 6 S2 6 6 0 ½T ¼ 6 6 0 6 4 0 CS
S2 C2 0 0 0 CS
0 0 1 0 0 0
0 0 0 C S 0
0 0 0 S C 0
3 2CS 2CS 7 7 7 0 7 7 0 7 5 0 2 2 ðC S Þ
ð3Þ
where C ¼ cosðhÞ and S ¼ sinðhÞ. From Eqs. (1) and (2), the following relation is held between elastic coefficients at two different coordinate systems. ½Q ¼ ½T 1 ½Q ½T
ð4Þ
½Q and ½Q are defined in terms of local coordinate of each layer and global coordinate of plate, respectively. After rotation of coordinates, in-plane thermal expansion coefficient a6 will appear. 9 8 9 2 38 Q11 Q12 Q13 e1 a1 h > r1 > 0 0 Q16 > > > > > > > > 6 > > > > > e2 a2 h > r2 > Q22 Q23 0 0 Q26 7 > > > > > > > 6 7 = < < = 6 7 r3 Q 0 0 Q a h e 33 36 3 3 7 ¼6 6 7 Q44 Q45 0 7> > > 6 > e4 > r4 > > > > > 4 > > > > 0 5> Sym Q55 > > > e5 > r5 > > > > > ; : : ; r6 e6 a6 h Q66 ð5Þ The relationships of material properties at two different coordinate systems are presented in Appendix A. The following displacement field that was introduced by Robbins and Reddy [5], is the displacement field of third order shear deformation plate theory (TRDT). ow0 2 1 o/z 3 u ¼ u0 þ z/x z z C1 þ /x 2 ox ox 1 ouz þ ð6Þ 3 ox 1 o/z ow0 z3 C1 þ /y v ¼ v0 þ z/y z2 2 oy oy 1 ouz þ ð7Þ 3 oy w ¼ w0 þ z/z þ z2 uz
ð8Þ
where 4 ; u0 ¼ uðx; y; 0; tÞ; 3h2 v0 ¼ vðx; y; 0; tÞ and w0 ¼ wðx; y; 0; tÞ
C1 ¼
Fig. 1. Local and global coordinate systems of a laminate.
ð9Þ
ðu0 ; v0 ; w0 Þ are the displacements of transverse normal on plane z ¼ 0 and ð/x ; /y Þ are rotations of transverse normal on plane z ¼ 0. /z is extension of a transverse normal, and uz is interpreted as a higher order rotation of transverse normal. The number of dependent variables in Eqs. (6)–(8) is only 7. The displacement field in Eqs. (6)–(8) accommodates quadratic variation of transverse shear strains (and hence stresses) and vanishing of transverse shear stresses on the top and bottom of a general laminate
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
composed of monoclinic layers [10,5]. Thus there is no need to use shear correction factors in a third order theory. The third order theories provide a slight increase in accuracy relative to the first order shear deformation theory (FSDT) solution, at the expense of a significant increase in computational effort. Moreover, finite element models of third order theories that satisfy the vanishing of transverse shear stresses on the bounding planes have the disadvantage of requiring continuity of C 1 [5]. Using virtual work method, Z t ðdU þ dV dKÞ dt ¼ 0 ð10Þ 0
equilibrium equations of the plate can be derived. U , V , and K are virtual strain energy, virtual work done by applied forces and virtual kinetic energy, respectively, and t is time. Then the linear strains according to displacement field Eqs. (6)–(8) are e1 ¼ e01 þ zðk10 þ zk11 þ z2 k12 Þ e2 ¼ e3 ¼ e4 ¼ e5 ¼ e6 ¼
e02 e03 e04 e05 e06
þ þ þ þ
zðk20 þ zk21 þ z2 k22 Þ zðk30 Þ zðk41 Þ ¼ 2e23 zðk51 Þ ¼ 2e13
ð11Þ
h2
þ r5 de5 þ r6 de6 Þ dz dA þ
Z
Z X0
Z X0
h 2
ðqdwÞ dA !
€ d€ qð€ ud€ u þ €vd€v þ w wÞ dz dA dt ¼ 0
ð12Þ
h2
Plate inertias and stress resultants are defined as follow: N Z znþ1 X I1;...;7 ¼ qn ð1; z; z2 ; z3 ; z4 ; z5 ; z6 Þ dz ð13Þ n¼1
ðNi ; Mi ; Si Þ ¼
zn N X n¼1
Z
ði ¼ 4; 5Þ
ð15Þ
znþ1
ri z2 dz ði ¼ 1; 2; 3; 4; 5; 6Þ
ð16Þ
zn
where I1;...;7 are inertias and Ni , Mi , Si , Qi and Pi are stress resultants. Using fundamental lemma of calculus of variation [10–12], the equation of motion of the plate can be written as du0 : 1 o/€ o€ w0 N1;x þ N6;y ¼ I1 €u0 þ I2 /€x I3 z C1 I4 2 ox ox € 1 o u z C1 I4 /€x I4 3 ox dv0 : 1 o/€ o€ w0 N2;y þ N6;x ¼ I1€v0 þ I2 /€y I3 z C1 I4 2 oy oy € 1 o u C1 I4 /€y I4 z 3 oy
ð17Þ
ð18Þ
C1 S1;xx þ C1 S2;yy þ Q4;y þ Q5;y 3C1 P4;y 3C1 P5;x þ 2C1 S6;xy þ q
þ zðk60 þ k61 þ k62 Þ
X0
N Z X n¼1
ri dz
zn
n¼1
ðPi Þ ¼
znþ1
dw0 :
It is assumed that there is an isothermal condition and temperature change and therefore thermal strains do not exist. In Appendix B, the relationships between strain components and displacement field Eqs. (6)–(8) are presented. By substitution of stresses and strain and distributed force in Eq. (10), the final integral equation for plate elasticity is given by Z T Z Z h2 ðr1 de1 þ r2 de2 þ r3 de3 þ r4 de4 0
N Z X
ðQi Þ ¼
29
ð19Þ
d/x : M1;x C1 S1;x Q5 þ Q5;y þ 3C1 P5 þ M6;y C1 S6;y 1 ¼ ðI2 C1 I4 Þ€u0 þ ðI3 2C1 I5 þ C12 I7 Þ/€x þ ðI2 C1 I4 Þ 2 € o/€z o€ w 1 ou 0 þ ðC1 I5 þ C12 I7 Þ þ ðI5 þ C1 I7 Þ z 3 ox ox ox ð20Þ d/y : M2;x C1 S2;x Q4 þ Q5;y þ 3C1 P4 þ M6;x C1 S6;x ¼ ðI2 C1 I4 Þ€v0 þ ðI3 2C1 I5 þ C12 I7 Þ/€y 1 o/€ o€ w0 þ ðI4 þ C1 I6 Þ z þ ðC1 I5 þ C12 I7 Þ 2 oy oy
znþ1
ri ð1; z; z3 Þ dz
1 o€u0 o€v0 € z þ C1 I4 € 0 þ I2 /€z þ u þ C1 I4 ¼ I1 w 3 ox oy 2 € € € o/ y 1 o/ o/ 1 C1 I6 2z C1 I6 þ C1 I5 x þ C1 I5 2 2 ox oy ox €0 €0 o2 /€z o2 w o2 w o/€ C12 I7 2 C12 I7 2 C12 I7 x 2 oy ox oy ox 2 2 € o/y 1 o u € € 1 ou I7 2z I7 2z C12 I7 3 ox 3 oy oy
ði ¼ 1; 2; 3; 6Þ
zn
ð14Þ
€ 1 ou þ ðI5 þ C1 I7 Þ z 3 oy
ð21Þ
30
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
d/z : 1 1 h P1;xx þ P2;yy þ P6;xy N3 q 2 2 2 o€ u0 o€v0 € z þ C1 I4 € 0 þ I2 /€z þ I3 u þ C1 I4 ¼ I1 w ox oy o/€y o/€ þ ðC1 I5 C12 I7 Þ x þ ðC1 I5 C12 I7 Þ ox oy 2 2 2€ €0 €0 1 ow ow o/ C12 I7 2 C12 I7 2 C1 I6 2z 2 ox oy ox 2€ 2 2 € € 1 o/ 1 ou 1 ou C1 I6 2z I7 2z I7 2z 2 3 ox 3 oy oy
ð22Þ
d/z : 1 1 2 h2 S1;xx þ P2;yy þ P6;xy 2M3 þ q 3 3 3 4 1 o€ u 1 o€v0 0 € z þ I4 € 0 þ I4 /€z þ I5 u þ I4 ¼ I3 w 3 ox 3 oy o/€y o€v0 1 o/€ 1 þ ðI5 C1 I7 Þ x þ ðI5 C1 I7 Þ þ C 1 I4 oy 3 ox 3 oy 2 2€ 2€ 2 € €0 1 o / 1 ow 1 o/ 1 ou C1 I7 2 I6 2z I6 2z I7 2z 3 6 ox 6 oy 9 ox oy € 1 o2 u I7 2z ð23Þ 9 oy where C1 is defined as
88 99 > > > N1 > > > >> => < > > > N > > 2 > > > > > > > > N > > 3 > > > > > : ;> > > > > N 6 > >8 9 > > > > > M > > 1>> > >> > > = < > > > M2 > > > > > 2 > > > > M ½A > > > 3>> > > > ;= 6 4 > > P1 > > > >> => < > > > Sym > > P 2 > > > > > > > > P > > > > > > ;> : 3> > > > > > 8 P6 9 > > > > > > >> > S1 > > >> > > = < > > > S2 > > > > > > > > > S > > > 3> > > > ; :: ;> S6
88 099 > > > e1 > > > >> => < 0> > > e > > 2 > > > > 0 > > > e > > > 3>> > > > : 0;> > > > > > 8 e6 9 > > > > > 0 > > k1 > > >> > > > > = < > > 0 > > k > > 2 > 3> > > 0 > k ½D ½E > > > > 3 >> > > > ; : = < 0 k ½E ½F 7 7 8 69 1 ½F ½H 5> > > k1 > > > >> => < 1> > ½J > > > k > > 2 > > 1 > > > > k > > > > 3 > > > > ; : > > 1 > k > >8 6 9> > > > > 2 > >> > k1 > > >> > > = < > > 2 > k2 > > > > > > > 2 > > k3 > > > > > > > ; :: 2 ;> k6
½B ½D
ð26Þ Eq. (26) provides the relations between stress resultants and strains, which are defined by displacement field parameters. In the Eq. (26), the elements of stiffness matrix can for example be defined as 2 3 A11 A12 A13 A16 6 A22 A23 A26 7 7 ½A ¼ 6 ð27Þ 4 A33 A36 5 Sym A66 Elements of matrices ½A, ½B, ½D, ½E, ½F and ½H are defined as follows: ðAij ; Bij ; Dij ; Eij ; Fij ; Hij ; Jij Þ ¼
Z
h 2
Qij ð1; z; z2 ; z3 ; z4 ; z5 ; z6 ; z7 Þ dz
h2
C1 ¼
4 3h2
ð24Þ
ði; j ¼ 1; 2; 3; 6Þ ð28Þ
and q is distributed transverse load on the top surface.
3. Finite element modeling of equations Using approximation equation for displacement field as ui ¼ hui ifN g
/yi ¼ h/yi ifN g
vi ¼ hvi ifN g
/zi ¼ h/zi ifN g
wi ¼ hwi ifN g uzi ¼ huzi ifN g
For in-plane components, the matrix form of stress resultants and strains are 8 9 8 09 Q4 > e4 > > > > > > > = = < < 0 Q ½A ½B ½D ½E 5 ¼ e50 ð29Þ ½D ½E ½F ½H > P4 > > k4 > > > > > ; ; : : P5 k50 where for example A44 A45 ½A ¼ Sym A55
ð25Þ
/xi ¼ h/xi ifN g
ð30Þ
and ðAij ; Bij ; Dij ; Eij ; Fij ; Hij Þ ¼
Z
h 2
Qij ð1; z; z2 ; z3 ; z4 ; z5 ; z6 Þ dz
h2
ði; j ¼ 4; 5Þ and substitution of displacements approximations in Eqs. (13)–(16) and (17)–(23), finite element type of elasticity equations can be derived. For writing the equations in displacement field parameters, following relations have to be used.
ð31Þ Finally, using the finite element analysis equations of motion can be written in compact form as the following equation
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
½MfX€ g þ ½KfX g ¼ fF g
ð32Þ
Using quadratic 6 nodes triangular elements to satisfy C 1 -continuity, and imposing the following boundary conditions for Navier type boundary conditions, governing equations can be solved. Elements of matrices ½K, ½M and fF g are presented in Appendixes C, D and E, respectively. The plate is simply supported at four edges therefore, primary boundary conditions are u0 ðx; 0Þ ¼ u0 ðx; bÞ ¼ 0 /x ðx; 0Þ ¼ /x ðx; bÞ ¼ 0 v0 ð0; yÞ ¼ v0 ða; yÞ ¼ 0 /y ð0; yÞ ¼ /y ða; yÞ ¼ 0
ð33Þ
w0 ðx; 0Þ ¼ w0 ðx; bÞ ¼ w0 ð0; yÞ ¼ w0 ða; yÞ ¼ 0 It is seen that the governing equations are in general dynamic form. To analyze the static behavior of the plate, the stiffness matrix ½K and the force vector fF g are needed.
4. Numerical solutions Tables 1 and 2 contain nondimensionalized mid point deflections and stresses obtained with 3-D elasticity theory (ELS), TSDT, first order shear deformation theory (FSDT), and classical laminate plate theory (CLPT) for the following three problems: Problem 1. A three-ply (0-90-0) square ða=b ¼ 1Þ laminate with equal thickness layers has been subjected to a sinusoidal distributed transverse load on top plane, and the results are presented in Table 1. The material properties of each ply is assumed as E1 ¼ 175 GPa, E2 ¼ 7 GPa, G12 ¼ G13 ¼ 3:5 GPa, G23 ¼ 1:4 GPa and v12 ¼ v13 ¼ 0:25.
31
Problem 2. A four-ply (0-90-90-0) square ða=b ¼ 1Þ laminate with equal thickness layers has been subjected to a sinusoidal distributed transverse load on top plane and the results are presented in Table 2. The material properties are as Problem 1. Problem 3. A comparison of maximum deflection using FSDT with correction factor of k ¼ 5=6 and TRDT for an antisymmetric cross-ply by different number of layers and with the same boundary conditions and load distribution as previous examples. For this case, the material properties are G12 ¼ G13 ¼ 0:5E2 , G23 ¼ 0:2E2 , v12 ¼ 0:25 and a=h ¼ 10. The results are shown in Table 3. FSDT and ELS results are presented in [10,13]. The following nondimensionalized quantities at specific points are presented in Tables and Graphs as a result of TSDT and are compared to CLPT, FSDT and ELS solutions of the problem [10,12]. 2 a b E2 h 3 a b h h ¼ w0 ; ; ; w rxx ¼ rxx 2 2 2 2 2 a4 q0 b2 q0 2 2 a b h h h h ; ; ryy ¼ ryy rxy ¼ rxy 0; 0; 2 2 4 2 b2 q0 b2 q0 a h b h ryz ¼ ryz ; 0; 0 rxz ¼ rxz 0; ; 0 2 bq0 2 bq0 ð34Þ
5. Conclusions Using the Reddy displacement field for third order shear deformation theory, a set of dynamic equations for modeling the behavior of a laminated plate is derived. Third order shears deformation theory (TRDT) of Reddy has 7 parameters in displacement field and
Table 1 Nondimensionalized maximum stresses and deflections at the mid point of a square simply supported (0-90-0) laminate a=h
Method
rxx
ryz
w
4
ELS TSDT FSDT
0.775 0.7392 0.4370
0.217 0.1884 0.1561 0.1968
– 0.0197 0.0177
ELS TSDT FSDT
0.590 0.5713 0.5134
0.123 0.1082 0.0915 0.1108
– 0.0773 0.0669
ELS TSDT FSDT
0.552 0.5426 0.5384
0.094 0.0791 0.0703
– 0.0463 0.0434
CLPT
0.5387
0.0827 –
0.4313
3-D ELS [13] 10
3-D ELS [13] 100
3-D ELS [13] –
32
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
Table 2 Nondimensionalized maximum stresses and deflections at the mid point of a square simply supported (0-90-90-0) laminate a=h
Method
rxx
ryy
ryz
rxz
rxy
w
4
ELS TSDT FSDT
0.720 0.681 0.406
0.663 0.647 0.576
0.292 0.244 0.196 0.280
0.219 0.211 0.140 0.269
0.0467 0.0451 0.0308
0.0195 0.0190 0.0170
ELS TSDT FSDT
0.559 0.551 0.499
0.401 0.394 0.361
0.196 0.163 0.130 0.181
0.301 0.211 0.167 0.318
0.0275 0.0451 0.0241
0.00743 0.00732 0.00663
ELS TSDT FSDT
0.539 0.539 0.538
0.276 0.275 0.270
0.141 0.129 0.101
0.337 0.308 0.178
0.0216 0.0216 0.0213
0.00437 0.00435 0.00435
CLPT
0.539
0.270
0.139 0.139
0.337 0.337
0.0213
0.00432
3-D ELS [13] 10
3-D ELS [13] 100
3-D ELS [13] –
Table 3 Nondimensionalized maximum deflection at a square simply supported antisymmetric cross-ply (0-90-. . .) laminate a=h 4 10 20 50 100 CLPT
N ¼2
N ¼6
FSDT
TSDT
FSDT
TSDT
0.021492 0.012373 0.011070 0.010705 0.010653
0.020102 0.012253 0.010241 0.010701 0.010653
0.015473 0.006354 0.005052 0.004687 0.004635
0.015423 0.006375 0.005055 0.004687 0.004635
0.0106306
satisfies the vanishing of transverse shear stresses on the boundary planes. By deriving the dynamic equation of motion and equations in finite element form, stresses and transverse displacements of different points of plate are defined. From the results, it is clear that the third order theory gives more accurate results for deflections and stresses when compared to the first order shear deformation plate theory (FSDT) with correction factor for shear deformation of k ¼ 5=6. It is known that the shear correction factor k depends on the lamina properties. The fact that no shear correction coefficients are needed in the third order theory makes it more convenient to use. In Table 1, nondimensionalized stresses rxx and ryz are presented. It is seen that for span to thickness ratio a=h ¼ 4, errors for TRDT results are 4.6% and 13.3%, respectively, but for FSDT these errors are 43.6% and 28.1%. For a=h ¼ 10, these errors are reduced to 3.2% and 12.1% for TRDT and 13% and 26% for FSDT. For a=h ¼ 100, these errors are 1.8% and 15.9% for TRDT and 2.5% and 25.5% for FSDT, respectively. It is seen that the TRDT results for ryz are very close to the stresses computed from three-dimensional elasticity of first order shear deformation theory for less amount of
0.004618
a=h. In Table 2, for a symmetric cross-ply, it is seen that the third order theory in comparison with the elasticity solution, predicts deflection by 2% while the first order theory predicts by about 12.8% for a=h ¼ 4. The errors are 1.4% in TSDT and 10.8% in FSDT for a=h ¼ 10. It is seen that the errors are reduced at higher a=h. For a=h ¼ 100, the errors are 0.4% for both theories. Results for stresses are also presented in Table 2. The closer
Fig. 2. Nondimensionalized normal stress rxx through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
33
symmetric cross-ply with 2 and 6 layers are presented. Relative errors for the results of TRDT and FSDT in 2ply and 6-ply plates are 7.5% and 0.4% and for a=h ¼ 4. These errors are near zero for a=h ¼ 100.
Appendix A. Relationships of material properties in rotated coordinate systems a1 ¼ a 1 C 2 þ a 2 S 2 a2 ¼ a1 S 2 þ a2 C 2 Fig. 3. Nondimensionalized normal stress ryy through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
a3 ¼ a3 a4 ¼ 0 a5 ¼ 0 a6 ¼ ða1 a2 ÞCS
0.5
Q11 ¼ Q11 C 4 þ 2ðQ12 þ 2Q66 ÞC 2 S 2 þ Q22 S 4 0.25
Q12 ¼ ðQ11 þ Q22 4Q66 ÞC 2 S 2 þ Q12 ðC 4 þ S 4 Þ Q13 ¼ Q13 C 2 þ Q23 S 2
0
Q16 ¼ Q22 CS 3 þ Q11 C 3 S CSðC 2 S 2 ÞðQ12 þ 2Q66 Þ Q22 ¼ Q11 S 4 þ 2ðQ12 þ 2Q66 ÞC 2 S 2 þ Q22 C 4
-0.25
Q23 ¼ Q13 S 2 þ Q23 C 2
-0.5 0
0.4
0.2
0.6
Q26 ¼ Q22 C 3 S þ Q11 CS 3 þ CSðC 2 S 2 ÞðQ12 þ 2Q66 Þ Q33 ¼ Q33
Fig. 4. Nondimensionalized transverse shear stress rxz through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
Q44 ¼ Q44 C 2 þ Q55 S 2 Q45 ¼ ðQ55 Q44 ÞCS Q55 ¼ Q55 C 2 þ Q44 S 2
0.5
Q66 ¼ ðQ11 þ Q22 2Q12 2Q66 ÞC 2 S 2 þ Q66 ðC 4 þ S 4 Þ
0.25
Appendix B. Relationships between strain components and displacement
0 -0.25
ou0 ox o/ k10 ¼ x ox 1 o2 / z k11 ¼ 2 ox2 2 o w0 o/x 1 o2 wz 2 k1 ¼ C1 þ þ 3 ox2 ox2 ox e01 ¼
-0.5 0
0.04
0.08
0.12
0.16
Fig. 5. Nondimensionalized transverse shear stress ryz through the thickness of a 0-90-90-0 cross-ply with a=h ¼ 4.
results can be seen for in-plane stresses of TRDT and the results of elasticity solution. In Figs. 2–5 these through the thickness nondimensionalized stresses of the cross-ply with a=h ¼ 4 are plotted. It is seen that the stresses are discontinuous like other ESL theories due to the continuity of the transverse shear strains through the thickness of lamina. In Table 3, deflection of an anti-
ov0 oy o/ y k20 ¼ oy e02 ¼
k21 ¼
1 o2 / z 2 oy 2
34
k22
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
2 o w0 o/y 1 o2 w z þ þ ¼ C1 3 oy 2 oy 2 oy
e03 ¼ /z k30
K1;1 ¼
Z
K1;2 ¼
Z
K1;3 ¼
k31 ¼ 0
Z
C1 E61 Ny Nxx C1 E62 Ny Nyy 2C1 E66 Ny Nxy Þ dA
ow0 þ /y oy
K1;4 ¼
Z
þ B61 Ny Nx þ B66 Ny Ny C1 E61 Ny Nx
ow0 þ /y oy
k42 ¼ 0 ow0 þ /x ox ow0 1 k5 ¼ 3C1 þ /x ox
C1 E66 Ny Ny Þ dA K1;5 ¼
k52
¼0
e06
ou0 ov0 þ ¼ oy ox
o2 u0 oxoy 2 o w0 o/x o/y 2 o2 uz þ k62 ¼ C1 2 þ þ 3 oxoy oxoy oy ox
Appendix C. Elements of local stiffness matrix
ðB12 Nx Ny þ B16 Nx Nx C1 E12 Nx Ny C1 E16 Nx Nx þ E62 Ny Ny þ B66 Ny Nx C1 E62 Ny Ny
K1;6
o/x o/y þ oy ox
k61 ¼
Z A
e05 ¼
k60 ¼
ðB11 Nx Nx þ B16 Nx Ny C1 E11 Nx Nx C1 E16 Nx Ny A
k40 ¼ 0 ¼ 3C1
ðC1 E11 Nx Nxx C1 E12 Nx Nyy 2C1 E16 Nx Nxy A
k32 ¼ 0
k41
ðA12 Nx Ny þ A16 Nx Nx þ A62 Ny Ny þ A66 Ny Nx Þ dA A
¼ 2uz
e04 ¼
ðA11 Nx Nx þ A16 Nx Ny þ A61 Ny Nx þ A66 Ny Ny Þ dA A
K1;7
C1 E66 Ny Nx Þ dA Z 1 ¼ A13 Nx N D11 Nx Nxx D16 Nx Nxy þ A63 Ny N 2 A 1 1 D61 Ny Nxx D66 Ny Nxy D12 Nx Nyy 2 2 1 D62 Ny Nyy dA 2 Z 1 1 ¼ 2B13 Nx N E11 Nx Nxx E12 Nx Nyy 3 3 A 2 1 E16 Nx Nxy þ 2B63 Ny N E61 Ny Nxx 3 3 1 2 E62 Ny Nyy E66 Ny Nxy dA 3 3
K2;1 ¼
Z
ðA12 Ny Nx þ A26 Ny Ny þ A61 Nx Nx þ A66 Nx Ny Þ dA A
N is the column vector of shape function and as an example for a quadratic triangular element, multiplication of two shape function is written as follow: 8 9 N1 > > > > > > > > N > 2> > = < > N3 NN ¼ h N1 N2 N3 N4 N5 N6 i N4 > > > > > > > > N > > > ; : 5> N6
K2;2 ¼
Z
ðA22 Ny Ny þ A26 Ny Nx þ A62 Nx Ny þ A66 Nx Nx Þ dA A
K2;3 ¼
Z
ðC1 E22 Ny Nyy 2C1 E26 Ny Nxy C1 E61 Nx Nxx A
C1 E12 Ny Nxx C1 E62 Nx Nyy 2C1 E66 Nx Nxy Þ dA K2;4 ¼
Z
ðB12 Ny Nx þ B26 Ny Ny C1 E12 Ny Nx C1 E26 Ny Ny A
þ B61 Nx Nx þ B66 Nx Ny C1 E61 Nx Nx also for simplicity, the following terms are used. oN oN o2 N Ny ¼ Nxx ¼ 2 ox oy ox 2 2 oN oN Nyy ¼ 2 Nxy ¼ oy oxoy
Nx ¼
C1 E66 Nx Ny Þ dA K2;5 ¼
Z
ðB22 Ny Ny þ B26 Ny Nx C1 E22 Ny Ny C1 E26 Ny Nx A
þ B62 Nx Ny þ B66 Nx Nx C1 E62 Nx Ny C1 E66 Nx Nx Þ dA
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
K2;6 ¼
Z A
1 1 A23 Ny N D12 Ny Nxx D22 Ny Nyy 2 2
K3;6 ¼
1 D26 Ny Nxy þ A63 Nx N D61 Nx Nxx 2 1 D62 Nx Nyy D66 Nx Nxy dA 2 K2;7 ¼
K3;1 ¼
Z
1 1 2B23 Ny N E12 Ny Nxx E22 Ny Nyy 3 3 A 2 1 E26 Ny Nxy þ 2B63 Nx N E61 Nx Nxx 3 3 1 2 E62 Nx Nyy E66 Nx Nxy dA 3 3
Z
K3;7 ¼
ðC1 E11 Nxx Nx C1 E12 Nyy Nx C1 E16 Nxx Ny
A
C1 E26 Nyy Ny 2C1 E16 Nxy Nx 2C1 E66 Nxy Ny Þ dA K3;2 ¼
Z
ðC1 E12 Nxx Ny C1 E16 Nxx Nx C1 E22 Nyy Ny
35
Z
1 C1 E13 Nxx N þ C1 H11 Nxx Nxx 2 A 1 þ C1 H12 Nxx Nyy þ C1 H16 Nxx Nxy C1 E23 Nyy N 2 1 1 þ C1 H12 Nyy Nxx þ C1 H22 Nyy Nyy þ C1 H26 Nyy Nxy 2 2 2C1 E36 Nxy N þ C1 H16 Nxy Nxx þ C1 H26 Nxy Nyy þ 2C1 H66 Nxy Nxy dA
Z
1 1 2 C1 J11 Nxx Nxx þ C1 J12 Nxx Nyy þ C1 J16 Nxx Nxy 3 3 3 A 1 1 2C1 F23 Nyy N þ C1 J12 Nyy Nxx þ C1 J22 Nyy Nyy 3 3 2 þ C1 J26 Nyy Nxy 2C1 F13 Nxx N 4C1 F36 Nxy N 3 2 2 þ C1 J16 Nxy Nxx þ C1 J26 Nxy Nyy 3 3 4 2 þ C1 J66 Nxy Nxy dA 3
A
C1 E26 Nyy Nx 2C1 E26 Nxy Ny 2C1 E66 Nxy Nx Þ dA K3;3 ¼
K4;1 ¼
Z
Z
ðC12 J11 Nxx Nxx þ C12 J12 Nxx Nyy þ C12 J12 Nyy Nxx
C1 E11 Nx Nx C1 E16 Ny Nx C1 E66 Ny Ny
þ C12 J22 Nyy Nyy þ 2C12 J16 Nxy Nxx þ 2C12 J26 Nxy Nyy
C1 E16 Nx Ny Þ dA
A
þ 4C12 J66 Nxy Nxy þ 2C12 J16 Nxx Nxy þ A44 Ny Ny þ A45 Ny Nx þ A55 Nx Nx þ A45 Nx Ny þ 2C12 J26 Nyy Nxy
K4;2 ¼
Z
C1 E12 Nx Ny C1 E16 Nx Nx C1 E26 Ny Ny
6C1 D45 Nx Ny þ 9C12 F44 Ny Ny þ 9C12 F45 Ny Nx Z
C1 E66 Ny Nx Þ dA
þ 9C12 F55 Nx Nx þ 9C12 F45 Nx Ny Þ dA ðC1 F11 Nxx Nx C1 F16 Nxx Ny þ C12 J11 Nxx Nx
A
þ
C12 J16 Nxx Ny
þ
C12 J12 Nyy Nx
K4;3 ¼
Z
C1 F12 Nyy Nx C1 F26 Nyy Ny þ
C12 J26 Nyy Ny
2C1 F66 Ny Nxy þ C12 J11 Nx Nxx þ C12 J12 Nx Nyy
2C1 F16 Nxy Nx
þ 2C12 J16 Nx Nyy þ C12 J16 Ny Nxx C1 F12 Nx Nyy C1 F26 Ny Nyy þ C12 J26 Ny Nyy þ 2C12 J66 Ny Nxy
þ A45 Ny N 6C1 D45 Ny N þ A55 Nx N 3C1 D45 Nx N
þ A45 NNy þ A55 NNx 6C1 D45 NNy 3C1 D45 NNx
3C1 D55 Nx N þ 9C12 E45 Ny N þ 9C12 E55 Nx N Þ dA K3;5 ¼
ðC1 F11 Nx Nxx 2C1 F16 Nx Nxy C1 F16 Ny Nxx
A
2C1 F66 Nxy Ny þ 2C12 J16 Nxy Nx þ 2C12 J66 Nxy Ny
Z
ðB12 Nx Ny þ B16 Nx Nx þ B26 Ny Ny þ B66 Ny Nx
A
6C1 D44 Ny Ny 6C1 D45 Ny Nx 6C1 D55 Nx Nx
K3;4 ¼
ðB11 Nx Nx þ B16 Nx Ny þ B66 Ny Ny þ B16 Ny Nx
A
3C1 D55 NNx þ 9C12 F45 NNy þ 9C12 F55 NNx Þ dA
ðC1 F12 Nxx Ny C1 F16 Nxx Nx þ C12 J12 Nxx Ny
A
þ C12 J16 Nxx Nx C1 F22 Nyy Ny C1 F26 Nyy Nx
K4;4 ¼
Z
ðD11 Nx Nx þ D16 Nx Ny
A
þ C12 J22 Nyy Ny þ C12 J26 Nyy Nx 2C1 F26 Nxy Ny
2C1 F11 Nx Nx 2C1 F16 Nx Ny D16 Ny Nx þ D66 Ny Ny
2C1 F66 Nxy Nx þ 2C12 J26 Nxy Ny þ 2C12 J66 Nxy Nx
2C1 F16 Ny Nx 2C1 F66 Ny Ny þ C12 J11 Nx Nx
þ A44 Ny N 6C1 D44 Ny N þ A45 Nx N 3C1 D45 Nx N
þ C12 J16 Nx Ny þ C12 J16 Ny Nx þ C12 J66 Ny Ny þ A55 NN
3C1 D45 Nx N þ 9C12 F44 Ny N þ 9C12 F45 Nx N Þ dA
6C1 D55 NN þ 9C12 F55 NN Þ dA
36
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
K4;5 ¼
Z
ðD12 Nx Ny þ D16 Nx Nx 2C1 F12 Nx Ny
K5;4 ¼
Z
A
K4;6
K4;7
2C1 F16 Nx Nx D26 Ny Ny þ D66 Ny Nx 2C1 F26 Ny Ny
2C1 F26 Ny Ny D16 Nx Nx þ D66 Nx Ny 2C1 F16 Nx Nx
2C1 F66 Ny Nx þ C12 J12 Nx Ny þ C12 J16 Nx Nx
2C1 F66 Nx Ny þ C12 J12 Ny Nx þ C12 J26 Ny Ny
þ C12 J26 Ny Ny þ C12 J66 Ny Nx þ A45 NN 6C1 D45 NN
þ C12 J16 Nx Nx þ C12 J66 Nx Ny þ A45 NN 6C1 D45 NN
þ 9C12 F45 NN Þ dA Z 1 1 ¼ B13 Nx N E11 Nx Nxx E12 Nx Nyy E16 Nx Nxy 2 2 A 1 1 þ B36 Ny N E16 Ny Nxx E26 Ny Nyy E66 Ny Nxy 2 2 1 1 C1 E13 Nx N þ C1 H11 Nx Nxx þ C1 H12 Nx Nyy 2 2 1 þ C1 H16 Nx Nxy C1 E36 Ny N þ C1 H16 Ny Nxx 2 1 þ C1 H26 Ny Nyy þ C1 H66 Ny Nxy dA 2 Z 1 1 ¼ 2D13 Nx N F11 Nx Nxx F12 Nx Nyy 3 3 A
K5;1 ¼
Z
2 1 F16 Nx Nxy þ 2D36 Ny N F16 Ny Nxx 3 3 1 2 F26 Ny Nyy F66 Ny Nxy 2C1 F13 Nx N 3 3 1 1 2 þ C1 J11 Nx Nxx þ C1 J12 Nx Nyy þ C1 J16 Nx Nxy 3 3 3 1 1 2C1 F36 Ny N þ C1 J16 Ny Nxx þ C1 J26 Ny Nyy 3 3 2 þ C1 J66 Ny Nxy dA 3
K5;5 ¼
þ C12 J26 Nx Ny þ C12 J66 Nx Nx þ A44 NN 6C1 D44 NN
K5;6
K5;7
C1 E66 Nx Ny Þ dA ðB22 Ny Ny þ B26 Nx Ny þ B26 Ny Nx þ B66 Nx Nx C1 E22 Ny Ny C1 E26 Nx Ny C1 E26 Ny Nx
K6;1
K5;3 ¼
ðC1 F12 Ny Nxx C1 F22 Ny Nyy 2C1 F26 Ny Nxy
A
C1 F16 Nx Nxx C1 F26 Nx Nyy 2C1 F66 Nx Nxy þ C12 J12 Ny Nxx þ C12 J22 Ny Nyy þ 2C12 J26 Ny Nxy þ
C12 J16 Nx Nxx
þ
C12 J26 Nx Nyy
þ
2C12 J66 Nx Nxy
þ A45 NNx þ A44 NNy 6C1 D44 NNy 6C1 D45 NNx þ 9C12 F44 NNy þ 9C12 F45 NNx Þ dA
þ 9C12 F44 NN Þ dA Z 1 1 ¼ B23 Ny N E12 Ny Nxx E22 Ny Nyy E26 Ny Nxy 2 2 A 1 1 þ B36 Nx N E16 Nx Nxx E26 Nx Nyy E66 Nx Nxy 2 2 1 1 C1 E23 Ny N þ C1 H12 Ny Nxx þ C1 H22 Ny Nyy 2 2 1 þ C1 H26 Ny Nxy C1 E36 Nx N þ C1 H16 Nx Nxx 2 1 þ C1 H26 Nx Nyy þ C1 H66 Nx Nxy dA 2 Z 1 1 ¼ 2D23 Ny N F12 Ny Nxx F22 Ny Nyy 3 3 A 2 1 F26 Ny Nxy þ 2D36 Nx N F16 Nx Nxx 3 3 1 2 F26 Nx Nyy F66 Nx Nxy 2C1 F23 Ny N 3 3 1 1 2 þ C1 J12 Ny Nxx þ C1 J22 Ny Nyy þ C1 J26 Ny Nxy 3 3 3 1 1 2C1 F36 Nx N þ C1 J16 Nx Nxx þ C1 J26 Nx Nyy 3 3 2 þ C1 J66 Nx Nxy dA 3 Z ¼ A13 NNx þ A36 NNy D16 Nxy Nx D66 Nxy Ny A
C1 E66 Nx Nx Þ dA Z
ðD22 Ny Ny þ D26 Ny Nx 2C1 F22 Ny Ny
2C1 F66 Nx Nx þ C12 J22 Ny Ny þ C12 J26 Ny Nx
ðB12 Ny Nx þ B16 Nx Nx þ B26 Ny Ny þ B66 Nx Ny
A
þ 9C12 F45 NN Þ dA
2C1 F26 Ny Nx D26 Nx Ny þ D66 Nx Nx 2C1 F26 Nx Ny
C1 E12 Ny Nx C1 E16 Nx Nx C1 E26 Ny Ny
K5;2 ¼
Z
A
A
Z
ðD12 Ny Nx þ D26 Ny Ny 2C1 F12 Ny Nx A
K6;2
1 1 1 D11 Nxx Nx D16 Nxx Ny D12 Nyy Nx 2 2 2 1 D26 Nyy Ny dA 2 Z ¼ A23 NNy þ A36 NNx D26 Nxy Ny D66 Nxy Nx A
1 1 1 D12 Nxx Ny D16 Nxx Nx D22 Nyy Ny 2 2 2 1 D26 Nyy Nx dA 2
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
K6;3 ¼
Z A
1 1 C1 H11 Nxx Nxx þ C1 H12 Nxx Nyy þ C1 H16 Nxx Nxy 2 2
K7;1 ¼
1 1 þ C1 H12 Nyy Nxx þ C1 H22 Nyy Nyy þ C1 H26 Nyy Nxy 2 2 þ C1 H16 Nxy Nxx þ C1 H26 Nxy Nyy þ 2C1 H66 Nxy Nxy C1 E13 NNxx C1 E23 NNyy 2C1 E36 NNxy dA K6;4 ¼
K6;5 ¼
K6;6 ¼
Z
1 1 1 E11 Nxx Nx E16 Nxx Ny þ C1 H11 Nxx Nx 2 2 2 A 1 1 1 þ C1 H16 Nxx Ny E12 Nyy Nx E26 Nyy Ny 2 2 2 1 1 þ C1 H12 Nyy Nx þ C1 H26 Nyy Ny E16 Nxy Nx 2 2 E66 Nxy Ny þ C1 H16 Nxy Nx þ C1 H66 Nxy Ny þ B13 NNx þ B36 NNy C1 E13 NNx C1 E36 NNy dA
K7;3 ¼
1 1 1 D13 Nxx N þ F12 Nxx Nyy þ F11 Nxx Nxx 2 4 4
1 1 1 þ F16 Nxx Nxy D23 Nyy N þ F12 Nyy Nxx 2 2 4 1 1 þ F22 Nyy Nyy þ F26 Nyy Nxy D36 Nxy N 4 2 1 1 þ F16 Nxy Nxx þ F26 Nxy Nyy þ F66 Nxy Nxy þ A33 NN 2 2 1 1 D13 NNxx D23 NNyy D36 NNxy dA 2 2 Z 1 1 ¼ E13 Nxx N þ H12 Nxx Nyy þ H11 Nxx Nxx 6 6 A 1 1 þ H16 Nxx Nxy E23 Nyy N þ H12 Nyy Nxx 3 6 1 1 þ H22 Nyy Nyy þ H26 Nyy Nxy 2E36 Nxy N 6 3 1 1 2 þ H16 Nxy Nxx þ H26 Nxy Nyy þ H66 Nxy Nxy 3 3 3 1 1 þ 2B33 NN E13 NNxx E23 NNyy 3 3 2 E36 NNxy dA 3
1 1 1 E11 Nxx Nx E16 Nxx Ny E12 Nyy Nx 3 3 3 A 1 2 2 E26 Nyy Ny E16 Nxy Nx E66 Nxy Ny 3 3 3 þ 2B13 NNx þ 2B36 NNy dA
Z
1 1 1 E12 Nxx Ny E16 Nxx Nx E22 Nyy Ny 3 3 3 A 1 2 2 E26 Nyy Nx E26 Nxy Ny E66 Nxy Nx 3 3 3 þ 2B13 NNy þ 2B36 NNx dA
Z
1 1 C1 J11 Nxx Nxx þ C1 J12 Nxx Nyy 3 3
2 1 1 þ C1 J16 Nxx Nxy C1 J12 Nyy Nxx þ C1 J22 Nyy Nyy 3 3 3 2 2 2 þ C1 J26 Nyy Nxy þ C1 J16 Nxy Nxx þ C1 J26 Nxy Nyy 3 3 3 4 þ C1 J66 Nxy Nxy 2C1 F13 NNxx 2C1 F23 NNyy 3 4C1 F36 NNxy dA
1 1 1 E12 Nxx Ny E16 Nxx Nx þ C1 H12 Nxx Ny 2 2 2 A 1 1 1 þ C1 H16 Nxx Nx E22 Nyy Ny E26 Nyy Nx 2 2 2 1 1 þ C1 H22 Nyy Ny þ C1 H26 Nyy Nx E26 Nxy Ny 2 2 E66 Nxy Nx þ C1 H26 Nxy Ny þ C1 H66 Nxy Nx þ B23 NNy þ B36 NNx C1 E23 NNy C1 E36 NNx dA
Z
Z
A
Z
A
K6;7
K7;2 ¼
37
K7;4 ¼
Z A
1 1 1 F11 Nxx Nx F16 Nxx Ny þ C1 J11 Nxx Nx 3 3 3
1 1 1 þ C1 J16 Nxx Ny F12 Nyy Nx F26 Nyy Ny 3 3 3 1 1 2 þ C1 J12 Nyy Nx þ C1 J26 Nyy Ny F16 Nxy Nx 3 3 3 2 2 2 F66 Nxy Ny þ C1 J16 Nxy Nx þ C1 J66 Nxy Ny 3 3 3 þ 2D13 NNx þ 2D36 NNy 2C1 F13 NNx 2C1 F36 NNy dA K7;5 ¼
Z
1 1 1 F12 Nxx Ny F16 Nxx Nx þ C1 J12 Nxx Ny 3 3 3 A 1 1 1 þ C1 J16 Nxx Nx F22 Nyy Ny F26 Nyy Nx 3 3 3 1 1 2 þ C1 J22 Nyy Ny þ C1 J26 Nyy Nx F26 Nxy Ny 3 3 3 2 2 2 F66 Nxy Nx þ C1 J26 Nxy Ny þ C1 J66 Nxy Nx 3 3 3 þ 2D23 NNy þ 2D36 NNx 2C1 F23 NNy 2C1 F36 NNx dA
38
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
Z
K7;6
1 1 1 ¼ E13 Nxx N þ H11 Nxx Nxx þ H12 Nxx Nyy 3 6 6 A 1 1 1 þ H16 Nxx Nxy E23 Nyy N þ H12 Nyy Nxx 3 3 6 1 1 2 þ H22 Nyy Nyy þ H26 Nyy Nxy E36 Nxy N 6 3 3 1 1 þ H16 Nxy Nxx þ H26 Nxy Nyy 3 3 2 þ H66 Nxy Nxy þ 2B33 NN E13 NNxx E23 NNyy 3 2E36 NNxy
K7;7 ¼
m2;7 ¼
A
m3;1 ¼
2 1 1 F13 Nxx N þ J11 Nxx Nxx þ J12 Nxx Nyy 3 9 9 A 2 2 1 þ J16 Nxx Nxy F23 Nyy N þ J12 Nyy Nxx 9 3 9 1 2 4 þ J22 Nyy Nyy þ J26 Nyy Nxy F36 Nxy N 9 9 3 2 2 4 þ J16 Nxy Nxx þ J26 Nxy Nyy þ J66 Nxy Nxy 9 9 9 2 2 þ 4D33 NN F13 NNxx F23 NNyy 3 3 4 F36 NNxy dA 3
Z
1 I4 NNy 3
dA
ðC1 I4 Nx N Þ dA
A
m3;2 ¼
Z
ðC1 I4 Ny N Þ dA
A
m3;3 ¼
Z
ðC12 I7 Nx Nx þ C12 I7 Ny Ny þ I1 NN Þ dA
A
m3;4 ¼
Z
ðC1 ðI5 C1 I7 ÞNx N Þ dA
A
dA
Z
Z
m3;5 ¼ m3;6 m3;7 m4;1
Z
ðC1 ðI5 C1 I7 ÞNy N Þ dA A Z 1 1 C1 I6 Nx Nx þ C1 I6 Ny Ny þ I2 NN dA ¼ 2 2 A Z 1 1 C1 I7 Nx Nx þ C1 I7 Ny Ny þ I3 NN dA ¼ 3 3 A Z ¼ ððI2 C1 I4 ÞNN Þ dA A
m4;2 ¼ 0 Z m4;3 ¼ ðC1 ðI5 C1 I7 ÞNNx Þ dA A Z m4;4 ¼ ððI3 2C1 I5 þ C12 I7 ÞNN Þ dA A
Appendix D. Elements of local mass matrix Z m1;1 ¼ ðI1 NN Þ dA A
m1;2 ¼ 0 Z m1;3 ¼ ðC1 I4 NNx Þ dA A Z m1;4 ¼ ððI2 C1 I4 ÞNN Þ dA A
m1;5 ¼ 0 Z 1 m1;6 ¼ I3 NNx dA 2 A Z 1 m1;7 ¼ I4 NNx dA 3 A m2;1 ¼ 0 Z m2;2 ¼ ðI1 NN Þ dA A Z m2;3 ¼ ðC1 I4 NNy Þ dA A
m2;4 ¼ 0 Z m2;5 ¼ ððI2 C1 I4 ÞNN Þ dA A
m2;6 ¼
Z A
1 I3 NNy 2
dA
m4;5 ¼ 0 Z 1 ð I4 þ C1 I6 ÞNNx dA m4;6 ¼ 2 A Z 1 ð I5 þ C1 I7 ÞNNx dA m4;7 ¼ 3 A m5;1 ¼ 0 Z m5;2 ¼ ððI2 C1 I4 ÞNN Þ dA A Z m5;3 ¼ ðC1 ðI5 C1 I7 ÞNNy Þ dA A
m5;4 ¼ 0 Z m5;5 ¼ ððI3 2C1 I5 þ C12 I7 ÞNN Þ dA A Z 1 ð I4 þ C1 I6 ÞNNy dA m5;6 ¼ 2 A Z 1 ð I5 þ C1 I7 ÞNNy dA m5;7 ¼ 3 A Z 1 m6;1 ¼ I3 Nx N dA 2 A Z 1 m6;2 ¼ I3 Ny N dA 2 A Z 1 1 C1 I6 Nx Nx þ C1 I6 Ny Ny þ I2 NN dA m6;3 ¼ 2 2 A
M. Rastgaar Aagaah et al. / Composite Structures 62 (2003) 27–39
1 1 ¼ I4 Nx N þ C1 I6 Nx N dA 2 2 A Z 1 1 ¼ I4 Ny N þ C1 I6 Ny N dA 2 2 A Z 1 1 I5 Nx Nx þ I5 Ny Ny þ I3 NN dA ¼ 4 4 A Z 1 1 I6 Nx Nx þ I6 Ny Ny þ I4 NN dA ¼ 6 6 A Z 1 ¼ I4 Nx N dA 3 A Z 1 ¼ I4 Ny N dA 3 A Z 1 1 C1 I7 Nx Nx þ C1 I7 Ny Ny þ I3 NN dA ¼ 3 3 A Z 1 1 ¼ I5 Nx N þ C1 I7 Nx N dA 3 3 A Z 1 1 ¼ I5 Ny N þ C1 I7 Ny N dA 3 3 A Z 1 1 I6 Nx Nx þ I6 Ny Ny þ I4 NN dA ¼ 6 6 A Z 1 1 I7 Nx Nx þ I7 Ny Ny þ I5 NN dA ¼ 9 9 A Z
m6;4 m6;5 m6;6 m6;7 m7;1 m7;2 m7;3 m7;4 m7;5 m7;6 m7;7
Appendix E. Elements of local force vector f1 ¼ f2 ¼ 0 Z f3 ¼ ðhqN Þ dA A
39
f4 ¼ f5 ¼ f6 ¼ 0 f7 ¼
Z A
h3 qN dA 4
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