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Damp heat effect in stress analysis of composite laminated beams

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https://www.eduzhai.net International Journal of Composite M aterials 2013, 3(3): 46-55 DOI: 10.5923/j.cmaterials.20130303.02 Hygro-Thermal Effects on Stress Analysis of Tapered Laminated Composite Beam Debabrata Gayen, Tarapada Roy* Department of M echanical Engineering, National Institute of Technology Rourkela, Rourkela, 769008, India Abstract The present work deals with an analytical method in order to determine the stress distributions (such as axial in-plane stresses and inter-laminar shear stresses) in mu ltilayered sy mmetric and anti-symmetric circular tapered laminated composite beams under hygro and thermal loadings. In the present analysis, all derivations for calculat ion of stresses have been developed based on the modification of conventional la mination and paralle l a xis theories. The hygro-thermal loads are considered as a linear function of coordinates in the planes of each layer. Hygro-thermally induced stresses are obtained for various types of laminates with cantilevered boundary conditions at different moisture concentrations and temperatures. A complete code has been developed using MATLAB program and validation of the present formulat ionshas been done by comparison with available solutions in the literature. Various results have also been found for tapered laminated cantilever beams of carbon/epoxy and graphite/epoxy materials. It has been observed that effects of stacking sequence, fiber orientation, coefficient of thermal expansion (CTE) and coefficient of mo isture expansion (CM E) have significance roles in the change of inter-laminar shear and axial in-plane stresses distribution through the laminate thickness. Keywords Tapered laminated composite, Conventional lamination theory, laminated plate approach, Inter-laminar shear stresses, axial in-plane stresses, Hygro-thermal load 1. Introduction Laminated co mposite structures/systems are increasingly used in aerospace and other engineering applications due to their high specific strength, stiffnessand good corrosion resistance, low coefficients of thermal expansion and of hygro-thermal expansion. They are subjected to different environmental condit ions during service life. It has been seen that moisture and temperature have an adverse effect on the performance of co mposites (viz. stiffness and strength are reduced with the increase in moisture concentration and temperature). Although due to thermal expansion mismatch between layers of different fiber orientations, thermal stresses induced within and between laminas and thermal behaviors of the laminated composite material sare mo re pronounced than that of isotropic materials. Accurate evaluation of bending stiffness is an important for better prediction of axial in-plane stresses, inter-laminar shear stresses and deflection, bulking and vibration responses of structures. Some of the important works in the directions are presented in the follo wing paragraph. Whitney and Ashton[1] considered the effect of environment on the bending of symmetric and un-symmetric * Corresponding author: tarapada@nitrkl.ac.in (Tarapada Roy) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved laminated composite plates. Pipes et al.[2] analyzed the stresses in laminated composite plates subjected to the combined effect of elevated temperature and absorbed mo isture. Sims and Wilson[3] derived an approximate elasticity solution for the transverse shearing stresses in a mu ltilayered anisotropic composite beam. Sai Ram and Sinha[4] presented the effects of moisture and temperature on the bending characteristics of laminated composite plates using finite element method (FEM ). Sai Ram and Sinha[5] investigated the effects of moisture and temperature on the free v ibration of laminated co mposite plates.Sai Ram and Sinha [6]also investigated the effect of mo isture and temperature on the buckling analysis of laminated composite plates. Chanand Demirhan[7] developed two analytical closed-form solutions, such as laminated plate approach and laminated shell approach, to evaluate the stiffness matrices and bending stiffness of the laminated composite circular tube. Tarn and Wang[8] presented a state space approach for the analysis of extension, torsion, bending, shearing and pressuring of laminated composite tube. Patel et al.[9] presented the static and dynamic characteristics of thick composite laminates exposed to hygro-thermal environ ments using eight-nodded C0 isoparamet ric h igher o rder quadrilat eral plate elements. Zenkour[10] investigated the static thermo-elastic response of symmetric and anti-symmetric cross-ply laminated plates considering unified shear deformation plate theory. Naidu and Sinha[11] analy zed the large deflection bending behaviour of laminated co mposite International Journal of Composite M aterials 2013, 3(3): 46-55 47 cylindrical shell panels subjected to hygro-thermal environ ments using FEM. Wang et al.[12] studiedhygro-the rmal e ffect on the response of dynamic inter-la minar stresses distribution in laminated plates with piezoelectric actuator of the beam and tanα = ∆R and difference between the large L and small rad ii of the beam is ∆R = RL − RS . layers. Naidu and Sinha[13] analyzed free v ibration 2.2. Laminated Constituti ve Equation behaviour of laminated composite shell subjected to hygro-thermal environ ments using FEM. Bahrami and Co mposite laminate consists of a set of mult iple layers Nosier[14] emp loyed the displacement fields of general stacking together with d ifferent fiber orientations. It is very cross-ply, symmetric, and anti-symmetric angle-p ly importance to analyze this problem based on layer by layer laminates under thermal and hygroscopic loadings and analysis. To circu mvent or encompass this tedious approach, developed elasticity based solution for predicting the the laminate is often analyzed based upon its mid -plane axes inter-laminar normal and shear stress distributions. Su[15] as a reference axis. The properties of each layer are developed closed-form analytical solutions for evaluating transformed to the reference axis. Now the general the thermal induced stresses in laminated composite tubular load-deformation relation of laminate co mposite is given as and rectangular beams. Rao[16] developed closed-form expressions based on modifying the laminated plate approach to determine the displacement and twisting angle N  M    =MN    + N M HT HT    = BA B  ε0  D   κ   (1) of tapered composite tube under axial tension and torsion. Where, [ ] ∑ ∫ [ ] [ ] ∑ ∫ [ ] Shao and Ma[17] carried out the thermo-mechanical analysis of functionally graded hollow circu lar cy linders s= ubjected to N k zk+1 = σ k .dz, M k zk+1 σ k .z.dz (2) mechanical loads and linearly increasing bo= undary i 1 = zk i 1 zk temperature. Syed et al.[18]presented the analytical expressions for co mputing thermal stresses in fiber ∑ ∫ { } reinforced co mposite beam with rectangular and hat sections under uniform temperature environ ment. Zenkour[19] studied the hygro-thermal bending analysis of functionally ∑ ∫ { } graded plate resting on elastic foundations. Singh and =  N HT  = M HT  [ ] [ ] k zk+1 Qk α k x, y ∆T + β k x, y ∆C dz i=1 zk [ ] [ ] k zk+1 Qk i=1 zk α k x, y ∆T + β k x, y ∆C zdz (3) ∑∑ ∫∫ ∑ ∫ cwwClsaohhoimtnAearhsakiklnirtsrhaaahtdtoebeaecunadsfotgrothsmcirb[tmoper2liamaeut0ietnnt]piraooaftndtiisceouoialtnrtndreehare.eilpreloeydloavrnytieee.ofswtfHuhicuteersiereieivhnnneygttpaegllCarrae-no0tdlae-etmfthfihmnfeiainorcittamdieeraeanllelstolheithmaemoinagpefrahlnelreysetmrstsriueeoesanssrrdistnceoeehgds= fr = In the ab[[DoAv]]e=eiqi= k=ku11a= zzztkzkk+ik+1o1 nQQ, kk dzz2,d[zB] i k1 zk +1 zk Qk zdz (4) distribution and axial in-plane stress distribution analysis of laminated composite structures and also it is still remain as [ε0 ]and [κ ], N  and M  ,[ N ]and [M ], N HT  and M HT  important area on research. In the present work a tapered matrices are mid-p lane strain and curvature, net resultant laminated composite beam model has been considered and forces and mo ments, applied load and mo ment, the details analysis has been carried out under bothhygro and hygro-thermal induced force and mo ment respectively. thermal loadings. [ A],[ B], [ D],[ β ]k x, y and [α ]k x, y matrices are in-p lane or 2. Hygro-Thermo-Mechanical Analysis of a Tapered Laminated Composite Beam extensional stiffness, extensional-bending coupling stiffness, bending stiffness, coefficient of thermal expansion (CTE) and coefficient of moisture expansion (CM E)of kth ply in the laminate x-y coordinate system. Now ∆T and ∆C are In the present model, the cross-sectional plane of the beam remains plane after deformation. 2.1. Geometry of Tapered Laminated Composite Beam A tapered laminated co mposite beam is considered with larger end radius RL , smaller end radius RS , Length L and a beam thickness t as shown in Fig. 1.The radius of the tapered beam at a distance x fro m the larger end, Rx will be expressed as R=x RL − x tanα .Where α is the taper angle the change of temperature and mo isture concentration respectively. Here, zk and zk+1 are the inner and outer coordinate of the kth ply respectively. Now, constitutive (i.e. stress-strain) relation for a lamina in the principal material d irections are presented as follows, σ1  Q11 Q12 0   ε1  σ 2   = Q21 Q22 0     ε2   (5) τ12   0 0 Q66  γ12  48 Debabrata Gayen et al.: Hygro-Thermal Effects on Stress Analysis of Tapered Laminated Composite Beam Now reduced stiffness matrix [Q]k for kth ply defined as, = Q11 1= −νE112ν 21 , Q22 E2 1 −ν12ν 21 Q=12 Q=21 ν 21E1= 1 −ν12ν 21 ν12 E2 1 −ν12ν 21 (6) Q66 = G12 Where E1, E2 ,G12 and ν12 are the elastic moduli along longitudinal, transverse direction, shear modulus in 1-2 planes and Poisson’s ratio along the principal material d irectio n res p ectiv ely . 2.3. In-Plane Stress Calcul ation For hygro-thermal stress analysis of tapered laminated composite beam, laminated plate approach is chosen due to its simp licity. Laminated plate approach is derived based on the conventional lamination theory and translation of laminate axis. Here infin itesimal element, which is inclined an angle θ with respect to the axis of the beam, z is rotated about the x axis as shown in Fig. 1. The total extensional, coupling and bending stiffness matrices of an n ply laminate with a ply thickness tply and mid-thickness radius of the tapered tube at a distance x from the larger end, Rx are obtained as follo w 2π k ∫ ∑ [ ] ( ) =  Ax,y R= x A ' x, y dθ 0 Rx n=1 Q x, y z 'k +1 − z 'k ∫ ∑ ( ) [ ] = Bx,y 2π R= x B ' x, y dθ 0 1 2 Rx k n=1 Q  x, y z 'k2+1 − z 'k2 (7) ∫ ∑ ( ) [ ] = Dx,y 2π R= x D ' x, y dθ 0 1 3 Rx k n=1 Q  x , y z '3k +1 − z '3k Where transformed reduced stiffness matrix o f kth ply Q  k x, y is given in appendix. [ A ]' x,y ,[ A ']x,y and [D ']x,y can be found out by paralle l a xis theorem and these can be written  A'x, y =  Ax, y B= 'x, y Bx, y + Rx cosθ  Ax, y (8) D 'x,y = Dx,y + 2Rxcosθ Bx,y + ( Rx cosθ )2  Ax,y 2.4. Determination of Hygro-thermal Force and Moment For finding the hygro-thermal force and mo ment expression the following steps are followed. In x-y coordinate system, CTE i.e. [α ]k andCME i.e. [β ]k of x, y x, y kth ply of the composite beam are obtained fro m the trans formation of CTE and CM E of the lamina in 1-2 coordinate. Now fro m Fig. 1, these are first rotated about the x axis with an angle θ , rotated about z axis with the angle of fiber orientation −φ . So the transformat ion of CTE and CM E in x-y coordinatesystem can be written as, [α ]k x, y = T (−φ )εk [α '] 1,2 = T (−φ )εk T (θ )εk [α ] 1,2 [β ]k x, y = T (−φ )εk [β ]' 1,2 = T (−φ )εk T (θ )εk [β ] 1,2 (9) [T ]ε is given in appendix. The hygro-thermal induced force and mo ment are obtained by integrating thermal strain through the thickness of lamina, and can be expressed as, ∑ [ ] = N Hx',Ty' k n =1 Q1k,2 ∆T  [α ]k x, y T  + ∆C [ β ]k x, y T  {  z 'k +1 − z 'k } (10) ∑ { } [ ] [ ] [ ] M HT= x ', y ' k n =1 Q1k,2  ∆T   2   α k T x,y  + ∆C 2   β k x, y T     z '2k+1 − z '2k Where ∆T and ∆C are the difference of temperature and mo isture content respectively. The forces and mo ments of the mid plane of the lamina obtained using equation (10) can be transformed as, Figure 1. Geometry of the tapered laminated composite beam International Journal of Composite M aterials 2013, 3(3): 46-55 49 ∑ [ ] [ ] [ ] = N Hx,Ty k n =1 Q 1k, 2 ∆T    α k T x,y  + ∆C   β k x, y T     ∂σ k x + ∂τ k xy + ∂τ k xz = 0 ∂x ∂y ∂z { } ( z 'k+1+ Rx cosθ ) − ( z 'k + Rx cosθ ) [ ] = N Hx',Ty' ∑ [ ] = M Hx,Ty k n =1 Q 1k,2  ∆T   2 [α ]k x, y T  + ∆T 2 [ β ]k x, y T     (11) { } ( z 'k+1+ Rx cosθ )2 − ( z 'k + Rx cosθ )2 ∂τ k xy + ∂σ k y + ∂τ k yz = 0 (16) ∂x ∂y ∂z ∂τ k xz + ∂τ k yz + ∂σ k z = 0 ∂x ∂y ∂z The lamina is assumed to be under plane stress condition [ ] [ ] = M HT x ', y ' + Rx cosθ N HT x ', y ' The overall hygro-thermal induced force and mo ment can be expressed as, ∫ [ ] ∫ [ ] = N Hx,Ty 2= π N Hx,Ty Rx dθ , M  Hx,Ty 2π M HT x, y Rx dθ (12) 0 0 The total ply strain at any point in the kth ply of the laminate can be obtained by using the following relations ε ε Total x Total y k   = γ Total xy     ε ε 0 x 0 y    + ( Rx + z ') cosθ    κ κ x y    (13) γ 0 xy  κ xy  (i.e. stresses σ z ,τ yz ,τ xz are assumed to be zero ). Therefore, for finding the inter-laminar stress components beam is consider as narrow, so the stresses are assumed to be independent of y. For mult i-layer la minate with τ 0 xz =0 , above equation (16) can be reduced as, ∑ ∫ τ k xz = k − zi+1 ∂σ i x dz i=1 zi ∂x (17) The laminated constitutive equation including hygro-thermal effect can be exp ress as, ε 0   a   κ   x, y = bT b   N + N HT  d   M + M HT   x, y (18) [ ] Where, ε Total  k x, y , ε 0  x, y and κ x, y rep res en t the total ply strain, mid-plane strain and mid -plane curvature respectively.  a b   A B −1 Where, bT d   =   B D The mechanica l strain is obtained by subtracting the thermal Hence, in the present analysis tapered cantilever beam is strain fro m the total strain. Finally, the p ly stress can be considered as a narrow beam, q is the transverse shear load obtained by multiply ing the matrix of the ply with the per unit width and the in -plane forces are not considered on mechanica l strain. It can be e xpressed as, ( ) [ ] ε Total  k = x, y ε 0 x, y + Rx + z ' cosθ κ x, y [ ] [ ] ε= M kx,y ε Total  k x, y − α k ∆T − x, y β (14) k ∆C x, y So, a xia l in-plane stress equation will be [ ]σ k x, y = Q  k x, y ε M  k x, y = Q  k x , y ε 0 x,y  − [α ]k x, y + ( Rx ∆T − + z ')cosθ [ ]β k x, y ∆C [κ ] x, y     (15) 2.5. Inter-l aminar Shear Stress Formul ati on under Trans verse Load beam. So, N=x N=y N=xy M=y M=xy 0 and ∂M x = −q . ∂x After putting the equations (15) and (18) into equation (17), obtained the follo wing exp ression  Q1i1    ∂ε 0 x ∂x + (Rx + zi ) cosθ ∂κ x ∂x         ∑ ∫ τ k xz k = − i =1 zk +1 zk +Q1i2    ∂ε 0 y ∂x + (Rx + zi ) cosθ ∂κ y ∂x   dz  (19)  +Q1i6    ∂γ 0 xy ∂x + (Rx + zi ) cosθ ∂κ xy ∂x  Since for the present case N HT  and M HT  not Sims and Wilson[3] developed an analytical solution of the inter-laminar shear stresses of the laminated co mposite beam subjected to a transverse load q. For analytical solution, two assumptions are made in derivation and these are, stresses not depend on y direction and σ z ,τ yz are small and depend on x, hence, equation (19) becomes, ∑ ( ) τ k ∫ xz = k  zk+1 Q1i1b11 + Q1i2b12 + Q1i6b16   qdz ( ) i=1 zk +(Rx + zi ) cosθ Q1i1d11 + Q1i2d12 + Q1i6d16   ∑ ∫ { [ ] [ ] } can be neglected. So, the layer of the laminate can equations be written at equilib as, riu m fo r th= e k th τ xkz k zk+1 i=1 zk Qi 1,2 B 1,2 + (Rx + zi ) cosθ Qi 1,2 D 1,2 qdz 50 Debabrata Gayen et al.: Hygro-Thermal Effects on Stress Analysis of Tapered Laminated Composite Beam ∑ [ ] ( )    i k =1 Qi 1,2 B 1,2 zk +1 − zk    ∑ [ ] τ k xz = q +Rx cosθ i=k1 Qi 1,2 D 1,2 ( zk +1 − zk )  (20) ∑ ( ) [ ]  +  1 2 cos θ k i =1 Qi 1,2 D 1,2 zk2+1 − zk2    3. Results and Discussion The present analysis has been discussed based on the axial in-plane and inter-laminar shear stresses distribution of tapered laminated cantilever beam subjected to different combination of mo istures and temperatures. Based on the above formulat ions a complete MATLA B code has been developed and validated. Various results for different fiber orientations and stacking sequences (viz. symmetric and anti-symmetric laminates) havealso been obtained and presented in the follo wing subsections. 3.1. Vali dati on of Developed MATLAB Code In order to verify the developed code the dimensions and mechanical propert ies considered for carbon/epoxy (AS4/ 3501-6) rectangular laminated cantilever beamare presented in[15].Fig. 2 shows the inter-laminar shear stresses distribution of this beam under thermal environment. It has been observed that result from the present code is excellent agreement with the already published results of Su[15] in case of thermal loading. observed fro m Fig. 3 that the inter-laminar stresses have a significant effect on the cross sections of tapered co mposite beam. Stress (Pa) 2500 2000 R=0.013 at x=L/2 R=0.0165 at x=L/4 1500 R=0.0095 at x=3L/4 1000 500 0 -500 -1000 -1500 -2000 -2500 2 Figure 3. 4 6 8 10 12 14 16 Number of ply Interlaminar shear stress distribution of tapered beam The Fig. 4 shows that the normalized axial stress distributionsof tapered laminated cantilever beam considering different co mposite materials with tply=0.001321m, R=0.0130m, delT= 325K and delC=1. It is also observed from Fig. 4 that thenormalized stress is more pronounced forgraphite/epoxy laminated beam. x 109 4 3 2 Graphite/epoxy Carbon/epoxy Kevlar/epoxy 1 Stress (Pa) 25 0 -1 20 -2 15 -3 Stress (Pa) 10 5 Analytical model with temperature effect of Carbon/epoxy (AS4/3501-6) 0 0 2 4 6 8 10 12 14 16 Number of ply Figure 2. Inter-laminar shear stress distribution for beam Three different types of laminates with sixteen layered (such as carbon/epoxy (AS4/3501-6), Kev lar/epo xy (Aramid 149/epo xy) and graphite/epo xy (GY-70/ 934)) have been considered for the analysis of tapered circular laminated cantilever beam having radius of 0.013m and thickness (tply) of each ply is 0.001321m.The material properties used are available in[4]. In the present analysis, the stacking sequence of laminate considered[? 5/90/0/90/0/? 5]S . The variations of inter-laminar shear stresses across the thickness of thetapered circulargraphite/epo xy beam at different cross sections are depicted in Fig. 3.It is clearly -4 2 Figure 4. 4 6 8 10 12 14 16 Number of ply Normalized σ x stress distribution of tapered beam The normalized axial stress distribution for tapered graphite/epoxy laminated cantilever beam with different combinations of moisture concentrations and constant temperature (0 K)is shown in Fig. 5. It has been observed fro m Fig. 5 that the mo isture concentrations have a significant effect on the a xia l stress distribution. The Fig. 6 shows the normalized stress distribution (σ x ) of tapered beam having different types of stacking sequences with tply=0.001321m, R=0.0130m,delT=325K and delC=1. It is clearly observed that the induced stress is more in case of anti-symmetric laminate. The comparison of increased stress in 0o ply due to hygro-thermal loading is presented in Table 1. Fro m the Table 1, it is observed that the fiber orientation of the tapered laminated beam p lays an important role in the variations of hygro-thermal induced stresses and it is also International Journal of Composite M aterials 2013, 3(3): 46-55 51 cleared that the constituent of0° and90° p lies demonstrates the greatest significant due to hygo-thermal effect. x 109 5 4 3 delC=0.25 delC=0.75 delC=1.25 2 Stress (Pa) 1 0 -1 -2 -3 -4 -5 2 4 6 8 10 12 14 16 Number of ply Figure 5. Normalized σ x stress distribution with different moisture concentrations and Constant temperature x 109 8 6 4 Stress (Pa) 2 0 -2 -4 [±45/90/0/90/0/±45]s [±45/90/90/±45/0/0]s -6 [±45/90/90/±45/0/0]2T -8 2 4 6 8 10 12 14 16 Number of ply Figure 6. Normalized σ x distribution for different stacking sequences for graphite/epoxy tapered beam Table 1. Comparisons of increased st ress in 0o ply Stacking sequence Max.σ x (MPa) due tohyro and thermal loadings Max.σ x (MPa) due to thermal loading Increased σ x (MPa) due to hygro loading 0o 0o 0o [±45/90/0/90/0/±45]S 933.6 204.8 728.8 [±45/902/±45/02]S 1657.0 113.76 1543.4 4. Conclusions An analytical method has been modified in order to predict the stress distribution within the layers and interfaces ofcircular tapered laminated beam under d istributed, hygro and thermal loadings. Fro m the present study, the following important conclusions may be made a) The axial in-plane stresses increases with the increases of moisture concentrations and temperatures. b) The co mbinations of d ifferent mo isture concentrations and constant temperature have a significant effect on the axial in -plane stresses compared to the combination of different temperatures and constant moisture concentration. c) The fiber orientation of the plies plays an important role on the hygro-thermally induced axial in-p lane stresses. This is mainly caused by the hygro-thermal induced mo ments which occur due to the mis match of coefficients of axial therma l e xpansion and moisture e xpansion in each ply of the la minate APPENDIX Transformati ons of Stiffness Matri x Stress and Strain Transformation Matrices:For plane stress condition, 2D stress and Strain transformat ion matrix rotated a positive angle θ about x-axis can be written as, 1 0 0  1 0 0  Tσ (θ )x = 0 mx2 0   and Tε (θ ) x = 0 mx2 0   0 0 mx  0 0 mx  For plane stress condition, the 2D stress transformat ion matrix rotated a positive angle φ about z-axis can be obtained as, = Tσorε (φ )z    mz2 nz2 −mz nz nz2 mz2 mz nz 2mz nz −2mz nz    mz2 − nz2  Where, mx = cosθ , mz = cosφ and nz = sinφ .Now, general transformation equation of stiffness matrix fro m material to laminate coordinate system can be written as, Q = [Tσ ]−1 [Q][Tε ] Where,[Q] and Q are the reduced stiffness matrices 52 Debabrata Gayen et al.: Hygro-Thermal Effects on Stress Analysis of Tapered Laminated Composite Beam of lamina which represent the stress/strain relationship with respect to material (1-2) coordinate system and laminate (x-y) coordinate system, respectively. Transformati on of CTE and CME: [α ] x, y [ ] [ ] [ ] α = Tε −1 z Tε −1 x 1, 2 , [ β ] x, y [ ] [ ] [ ]β = Tε −1 z Tε −1 x 1, 2 = α x mz2α1 + mx2nz2α2 = βx mz2 β1 + mx2nz2 β2 = α x nz2α1 + mx2mz2α2 and= βx nz2 β1 + mx2mz2 β2 = α xy 2mz nz (α1 − mx2α2= ) βxy 2mz nz (β1 − mx2 β2 ) Transformat ion of stiffness matrix for the lamina that first rotated θ about x-axis, then about φ about z-axis can be written as, ( ) Q11 =mz4Q11 + 2mz2nz2 mx2Q12 + 2mx2Q66 + mx4nz4Q22 = Q12 mz2nz2 (Q11 + mx4Q22 − 4mx2Q66 ) + (mz4 + nz4 )mx2Q12 Q22 =nz4Q11 + 2mz2nz2 (mx2Q12 + 2mx2Q66 ) + mz4mx2Q22 Q1=6 mz3nz (Q11 − mx2Q12 − 2mx2Q66 ) + mz nz3 (mx2Q12 − mx4Q22 + 2mx2Q66 ) Q26 = mz nz3 (Q11 − mx2Q12 − 2mx2Q66 ) + mz3nz (mx2Q12 − mx4Q22 + 2mx2Q66 ) = Q66 mz2nz2 (Q11 + mx4Q22 − 2mx2Q12 − 2mx2Q66 ) + (mz4 + nz4 )mx2Q66 Total Extensional, Coupling and Bending Stiffness Matrices are, ∑ =  A11 2π Rx k n=1 mz4Q11 + mz2nz2 (Q12 + 2Q66 ) + 3 8 nz4Q22   ( z 'k +1− z 'k ) ∑ =  A12  2π Rx k n=1 mz2nz2 (Q11 + 3 8 Q22 − 2Q66 ) + 1 2 (mz4 + nz4 )Q12   ( z 'k +1 − z 'k ) ∑ =  A22  2π Rx k n=1 nz4Q11 + mz2nz2 (Q12 + 2Q66 ) + 3 8 mz4Q22   ( z 'k +1 − z 'k ) ∑ =  A16  2π Rx k n=1 m3z nz (Q11 − 1 2 Q12 − Q66 ) + mz n3z ( 1 2 Q12 − 3 8 Q22 + Q66 ) ( z 'k +1 − z 'k ) ∑ ∑ =  A26  2π Rx k n= 1 k n= 1 mz n3z (Q11 − 1 2 Q12 − Q66 ) + m3z nz (1 2 Q12 − 3 8 Q22 + Q66 ) ( z 'k +1− z 'k ) ∑ =  A66  2π Rx k n=1 mz2nz2 (Q11 + 3 8 Q22 − Q12 − Q66 ) + 1 2 (mz4 + nz4 )Q66   ( z 'k +1− z 'k ) ∑ ( ) = B11 π Rx k n=1 mz4Q11 + mz2nz2 (Q12 + 2Q66 ) + 3 8 nz4Q22   z 'k2+1− z 'k2 ∑ ( ) = B12  π Rx k n=1 mz2nz2 (Q11 + 3 8 Q22 − 2Q66 ) + 1 2 (mz4 + nz4 )Q12   z 'k2+1− z 'k2 ∑ ( ) = B22  π Rx k n=1 nz4Q11 + mz2nz2 (Q12 + 2Q66 ) + 3 8 mz4Q22   z 'k2+1− z 'k2 International Journal of Composite M aterials 2013, 3(3): 46-55 53 ∑ ( ) = B16  π Rx k n=1 m3z nz (Q11 − 1 2 Q12 − Q66 ) + mz n3z (1 2 Q12 − 3 8 Q22 + Q66 ) z 'k2+1− z 'k2 ∑ ( ) = B26  π Rx k n=1 mz n3z (Q11 − 1 2 Q12 − Q66 ) + m3z nz ( 1 2 Q12 − 3 8 Q22 + Q66 ) z '2k +1− z '2k ∑ ( ) = B66  π Rx k n=1 mz2nz2 (Q11 + 3 8 Q22 − Q12 − Q66 ) + 1 2 (mz4 + nz4 )Q66   z 'k2+1− z 'k2 ∑ ( ) = D11 2π Rx 3 k n=1 mz4Q11 + mz2nz2 (Q12 + 2Q66 ) + 3 8 nz4Q22   z '3k +1− z '3k ∑ + 2π Rx3 k n=1   1 2 mz4Q11 + 3 4 mz2nz2 (Q12 + 2Q66 ) + 5 16 nz4Q22   ( z 'k +1− z 'k ) ∑ ( ) = D12  2π Rx 3 k n=1 mz2nz2 (Q11 + 3 8 Q22 − 2Q66 ) + 1 2 (mz4 + nz4 )Q12   z '3k +1− z '3k ∑ + 2π Rx3 k n=1 mz2nz2 (1 2 Q11 + 5 16 Q22 − 3 2 Q66 ) + 3 8 (mz4 + nz4 )Q12   ( z 'k +1 − z 'k ) ∑ ( ) = D22  2π Rx 3 k n=1 nz4Q11 + mz2nz2 (Q12 + 2Q66 ) + 3 8 mz4Q22   z '3k +1− z '3k ∑ + 2π Rx3 k 1 n=1 2 nz4Q11 + 3 4 mz2nz2 (Q12 + 2Q66 ) + 5 16 mz4Q22   ( z 'k +1 − z 'k ) D16  ∑ ( ) 2π Rx k   m3z nz  (Q11 − 1 2 Q12 − Q66 )    3 n=1   +mz n3z (1 2 Q12 − 3 8 Q22 + Q66 ) z '3k +1− z '3k ∑ + 2π Rx3 k n=1 m3z nz  (1 2 Q11 − 3 8 Q12 − 3 4 Q66 ) + mz n3z ( 3 8 Q12 − 5 16 Q22 + 3 4 Q66    ) ( z 'k +1 − z 'k ) D26  ∑ ( ) 2π Rx k mz n3z (Q11  − 1 2 Q12 − Q66 )    3 n=1+m3z nz (1 2 Q12 − 3 8 Q22 + Q66 ) z '3k +1− z '3k ∑ + 2π Rx3 k n=1 mz  n3z ( 1 2 +m3z nz ( Q11 − 3 8 Q12 − 3 4 Q66 ) 3 8 Q12 − 5 16 Q22 + 3 4 Q66    ) ( z 'k +1 − z 'k ) ∑ ( ) = D66  2π Rx 3 k n=1   mz2 nz2 (Q11 + 3 8 Q22 − Q12 − Q66 ) + 1 2 (mz4 + nz4 )Q66   z '3k +1− z '3k ∑ + 2π Rx3 k n=1 mz2nz2 (1 2 Q11 + 5 16 Q22 − 3 4 Q12 − 3 4 Q66 ) + 3 8 (mz4 + nz4 )Q66   ( z 'k +1− z 'k ) Overall Hygro-Thermal Induced Force and Moment of the tapered Beam: 54 Debabrata Gayen et al.: Hygro-Thermal Effects on Stress Analysis of Tapered Laminated Composite Beam ∫  N  HT x 2π = [N ]Hx T Rxdθ 0 ( ( )) = π Rx (∆T + ∆C)    mz6 + mz2nz4 + 2mz4nz2    +  nz6 + mz4nz2 + 2mz2nz4  2Q11 (α1 + β1 )    + 3 4 Q12 (α2 + β2 )      Q12 (α1  + 5 8 Q22 + β1 ) (α2 + β 2 ) ( z 'k +1− z 'k ) ∫  N  HT y 2π = [ N ]Hy T Rxdθ 0 ( ( ) ) = π Rx (∆T + ∆C)   nz6 + mz4nz2 + 2mz2nz4  2Q11 (α1 + β1 ) + 3 4 Q12 (α2 + β2 )    (  +  mz6 + mz2nz4 + 2mz4nz2 Q12 (α1 + β1 ) + 5 8 Q22 (α2 + β2 )  z 'k +1− z 'k ) ∫  N  HT xy 2π = [N ]HxyT Rxdθ 0 = π Rx (∆T + ∆C)  m5z nz + mz +2m3z n3z n5z    (2Q11 − Q12 ) (α1 + β1  +( 3 4 Q12 − 5 8 Q22 ) (α2 ) + β2  ) ( z 'k +1 − z 'k ) ∫  M  HT x 2π = [M ]Hx T Rxdθ 0 ( ) π Rx (∆T  + ∆C )   mz6 + mz2nz4 + 2mz4nz2 2Q11 (α1 + β1 ) + 3 4 Q12 (α2 + β2 )     ( ) ( ) 2  +  nz6 + mz4nz2 + 2mz2nz4 Q12 (α1 + β1 ) + 5 8 Q22 (α2 + β2 )  z 'k2+1− z 'k2 ∫  M  HT y 2π = [M ]Hy T Rxdθ 0 ( ) π Rx (∆T  + ∆C)   nz6 + mz4nz2 + 2mz2nz4 2Q11 (α1 + β1 ) + 3 4 Q12 (α2 + β2 )     ( ) ( ) 2  +  mz6 + mz2nz4 + 2mz4nz2 Q12 (α1 + β1 ) + 5 8 Q22 (α2 + β2 )  z '2k +1− z '2k ∫ M  HT xy = 2π [M ]HxyT Rxdθ 0 ( ) π Rx (∆T 2 + ∆C)  m5z nz + mz n5z +2m3z n3z    (2Q11 − Q12 ) (α1 + β1  +( 3 4 Q12 − 5 8 Q22 ) (α 2 ) + β2  ) z 'k2+1− z 'k2 International Journal of Composite M aterials 2013, 3(3): 46-55 55 hygrothermalenvironment, Journal of composites structures, vol. 69, pp. 387-395, 2005. 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