Shishir Kumar Sahu, Ph. D. Professor Civil Engineering Department National Institute of Technology, Rourkela Orissa, India

Shishir Kumar Sahu, Ph. D. Professor Civil Engineering Department National Institute of Technology, Rourkela Orissa, India

Introduction Review Of Literature Aim & Scope Of Present Investigation Finite Element Method Mathematical Formulation Results & Discussion Conclusion References 2

A structural composite is a material system combining two or more phases on a macroscopic scale, whose mechanical performance and properties are designed to be superior of those constituted materials acting independently 3

Ceramic matrix composites(sili con carbide fibers in silicon fiber matrix) Metal matrix composites(bor on fiber in aluminium matrix) Carbon/carbon matrix(carbon fiber in carbon matrix) Fiber reinforced composite polymer matrix composites(e glass fiber with Epoxy matrix) 4

There is a increase in utilization of composite materials in thin walled structural components of aircrafts, submarines, automobiles and other high- performance application areas, when exposed to high temperature and moisture changes in vibration static stability dynamic stability under different loading conditions. 5

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Tooth colored filling materials. Used to repair broken front teeth and replacing defective. 7

Static analysis Author year Method used Paper details Ashton & Whitney Sairam & Sinha 1971 classical plate theory 1991 First Order Shear Deformation Theory (FSDT) Patel et al. 2002 higher-order theory studied the hygrothermal effects on bending, buckling and vibration of composite laminated plates by Ritz method The effect of moisture and temperature on the deflections and stress resultants are presented for simply supported and clamped antisymmetric cross-ply and angle-ply laminates using reduced lamina properties at elevated moisture concentration and temperature. He used an isoparametric quadratic element which takes shear deformation into account. Static and dynamic characteristics of thick composite laminates exposed to hygrothermal environment. He employed a 8 noded quadrilateral isoparametric higher order finite element. He evaluated deflection, buckling & 8

Author year Method used Rao & Sinha 2004 finite element analysis Upadhyay et al. cont Paper details Studied the bending characteristics of thick multidirectional fibrous composite plates 2010 HSDT Studied solution for nonlinear flexural response of elastically supported cross-ply and angle-ply laminated composite plates with different lamination scheme & B.C. under hygrothermal environment. 9

Stability analysis Author Year Theory applied Paper details Sairam & Sinha 1991 FSDT hygrothermal effects on the buckling of composite laminated plates using the finite element method Chao Shyu & 1996 first-order shear deformation plate theory (FSDT). calculate the buckling loads for composite laminated plates under hygrothermal environments, where a micro to- macromechanical analytical model was proposed. Shen 2002 HOSDT hygrothermal conditions on the buckling and post buckling of shear deformable laminated cylindrical panels subjected to axial compression using a micro-to-macromechanical analytical model. He concern the post-buckling behavior of anti-symmetric angle-ply and symmetric cross-ply laminated plates under different sets of environmental conditions. 10

Cont Author Year Method Paper details Barton jr. 2007 Raleigh-Ritz method presented the elastic buckling of rectangular, symmetric angle-ply laminates subjected to a uniform moisture environment. This investigation presents an alternative method of computing the buckling load using eigen values. 11

Author year Method Paper details Bolotin 1964 analytical Studied on dynamic stability of structures Balamurugan et al. 1996 FEM Studied the instability of anisotropic laminated composite plates using 9-noded quadrilateral element & studied the effect of a large amplitude on the dynamic instability for a simply supported laminated composite plate. Results shows increase in aspect ratio reduces the dynamic strength. Ganapathi 2000 FEM Studied the instability of the elastic plates considering geometric non-linearity. Dey & Singha 2006 FEM Studied the dynamic stability characteristics of simply supported laminated composite skew plates subjected to a periodic in-plane load. The formulation includes the effects of transverse shear deformation, in-plane and rotary inertia. The boundaries of the instability regions are 12

Effect of temperature and moisture on buckling of laminated composite plates. Effect of temperature and moisture on dynamic stability analysis of composite plate. Experiment to study the hygrothermal effects on stability of laminated composite plates. The effects of aspect ratio, thickness ratio, ply orientation, load parameters and degree of orthotropy on first few frequencies of the laminated plate subjected to hygrothermal loading. 13

The governing equations have been developed using first order deformation theory (FSDT). The constitutive relation for the plate, when subjected to moisture and temperature, are given by {F}= [D]{ε}-{F N }.. (1) Where {F}= {N x, N y, N xy, M x, M y, M xy, Q x, Q y } T {F N } = {N x N, N yn, N xyn, M xn, M yn, M xyn, 0, 0} T {ε} ={ε x, ε y, γ xy, κ x, κ y, κ xy, φ x, φ y } T 14

Diagram 15

Cont Where, N x,n y,n xy = in-plane internal force resultants per unit length. M x,m y,m xy = internal moment resultants per unit length. Q x,q y = transverse shear resultants. N xn,n yn,n N xy = in-plane non mechanical force resultants per unit length due to moisture and temperature. M xn,m yn,m N xy = non-mechanical moment resultants per unit length due to moisture and temperature. ε x, ε y, γ xy = in-plane strains of the mid-plane κ x,κ y, κ xy = curvature of the plate Φ x, φ y = shear rotations in X-Z and Y-Z planes respectively. 16

Cont A ij, B ij, D ij and S ij are the extensional, bendingstretching coupling, bending and transverse shear stiffnesses. α = shear correction factor 17

n _ A ij = (Z k -Z k-1 ) B ij = (Z 2 k-z 2 k-1) D ij = (Q) (Z 3 k-z 3 k k-1) For i,j = 1,2,6 k S ij = α k n k n 1 1 k 1 n (Q) _ (Q) 1 k k (Q) k Cont (Z k -Z k-1 ) For i,j = 4,5 (2) Z k,z k-1 = bottom and top distance of lamina from mid-plane 18

The non-mechanical force and moment n _ resultants are expressed as (Q) { N xn, N N y, N N xy } T k k 1 = {e} k (Z k -Z k-1 ) {M xn,m yn,m xyn } T = k n k(z 2 k-z 2 k-1) Where {e} k = {e x, e y, e xy } T = [T]{β 1, β 2 } kt (C-C 0 )+ [T]{α 1 α 2 } kt (T-T 0 ) 1 _ (Q) k 19

T cos sin 2 2 sin 2 sin cos 2 2 cos2 e x, e y, e xy = non-mechanical strains due to moisture and temperature β 1, β 2 = moisture coefficients along 1 and 2 axes of a lamina, respectively α 1,α 2 = thermal coefficients along 1 and 2 axes of a lamina, respectively T,T 0 = elevated and reference temperatures C,C 0 = elevated and reference moisture concentrations 20

k n 1 _ (Q) k in equations (2) and (3) is defined as (Q) =[T 1 ] T k [Q ij ] k [T 1 ] (Q) = [T 1 ] T k [Q ij ][T 1 ] For i,j =1,2,6 k n 1 _ k n 1 _ n k 1 _ (Q) k = [T 2 ] T [Q ij ][T 1 ] For i,j = 4,5 (4) 21

Eight nodded isoparametric element is used for static stability analysis of woven fiber composite plates subjected to hygrothermal loading. Five degrees of freedom u, v, w, θ x and θ y are considered at each node. The stiffness matrix, the initial stress stiffness matrix, the mass matrix & the nodal load vectors of the element are derived by using the principle of minimum potential energy. Arbitrarily oriented laminated plate Geometry of an N-layered laminate 22

The stiffness matrix, the initial stress stiffness matrix, the mass matrix and the load vectors of the element, given by equations (9) and (12)-(15), are evaluated by first expressing the integrals in local natural co-ordinates, of the element and then performing numerical integration by using Gaussian quadrature. Then the element matrices are assembled to obtain the respective global matrices [K], [K σ ], [M],{P}, {P N }. The first part of the solution is to obtain the initial stress resultants induced by the external transverse static load and by moisture and temperature in static conditions. [K]{δ i } = {P}+{p N }.(16) 23

Then the initial stress resultants N x i, N y i,n xyi, M xi, M yi, M xyi, Q x i and Q y i are obtained from equations (1) and (8). The second part of the solution involves determination of natural frequencies from the condition. [[K] +[K σ ]]-ω n2 [M] = 0 ω n = natural frequency 24

A computer program is developed by using MATLAB environment to perform all the necessary computations. The element stiffness and mass matrices are derived using a standard procedure. Numerical integration technique by Gaussian quadrature is adopted for the element matrix. Subspace iteration method is adopted throughout to solve the eigenvalue problems. 25

Convergence Study Comparison with Previous Studies Numerical Results 26

, Convergence of non-dimensional free vibration frequencies for SSSS 4 layer plates for different ply orientations at 325K temperature a/b=1, a/t=100, na 2 6 2 28.1 10 / / E t 2 2 12 0 K 0.3 6 0 1 0.3 10 / K E1 130 10 9 E2 9.5 10 9 9 G23 0. 5G12 G 12 G12 6 10 13 G Mesh Division Non-dimensional frequencies at Temperature (0/90/90/0) (45/-45/-45/45) 4 4 6 6 8 8 10 10 Sairam & Sinha(1992) 8.079 8.039 8.036 8.036 ( 8.088) 11.380 10.785 10.680 10.680 _ 27

Table 1: Comparison of non-dimensional free vibration frequencies for simply supported four layer (0/90/90/0) cross-ply plates a/b=1,a/t=100, At T=300K, E 1 =130x10 9 N/m 2, E 2 =9.5x10 9 N/m 2,G 13 =G 12 G 12 =6x10 9 N/m 2, G 23 = 0.5G 12, υ 12 =0.3, α 1 =-0.3x10-6 / 0 K, α 2 =28.1x10-6 / 0 K Non dimensional frequency, λ=ω n a 2 t 2 Methods Non- dimensional frequencies at 325K Temperature 1 2 3 4 Sairam & Sinha (1991) 8.088 19.196 39.324 45.431 Patel & Ganapathi(2002) 8.0531 - - - Present FEM 8.0791 19.1002 39.3358 45.3505 28

Table 2: Comparison of non-dimensional free vibration frequencies for simply supported four layer (0/90/90/0) cross-ply plates a/b=1,a/t=100 At C=0.00, E 1 =130x10 9 N/m 2, E 2 =9.5x1 0 9 N/m 2,G 13 =G 12 G 12 =6x10 9 N/m 2, G 23 = 0.5G 12, υ 12 =0.3, β 1 = 0, β 2 = 0.44 Non dimensional frequency, λ=ω n a 2 t 2 Methods Non- dimensional frequencies at C = 0.1% 1 2 3 4 Sairam Sinha(1992) & 9.429 20.679 40.068 46.752 Patel Ganapathi(2002) & 9.3993 _ Present FEM (9.4223) (20.5974) (40.0842) (46.7083) 29

E 130Gpa 1 Methods E2 9. 5Gpa G 6Gpa 12 6 0 0.3 10 K 28.1 10 6 0 K 1 / N xcr N xcr 2 / C 0%, T 300K,,,,,,,,, G13 G 12 23 0. 5G12 T 300K G 0. 3 1 0 2 0. 44 Non-dimensional critical load 12 Sairam & Sinha Present FEM Patel,Ganapathi& Makhecha Present FEM At T=325K 0.4488 (0.4481) 0.4466 (0.4457) At C=0.1% 0.6099 (0.6095) 0.6084 (0.6078) 30

Frequency Effect of temperature on non-dimensional frequency of (45/-45/45/- 45) laminate 16 14 vibration 12 10 8 6 4 2 0 0 10 20 30 40 50 60 Temperature 31

Frequency Effect of moisture on non-dimensional frequency of (45/-45/45/- 45) laminate 16 14 12 vibration 10 8 6 4 2 0 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 Moisture 32

Frequency Frequency 1.6 buckling 1.6 1.4 buckling 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 20 40 60 Temperature 1.2 1 0.8 0.6 0.4 0.2 0 0 0.001 0.002 0.003 Moisture 33

Dynamic load factor 1.2 1 0.8 temp=0k temp=325k temp=350k temp=375k temp=400k 0.6 0.4 0.2 0 0 20 40 60 80 100 Non-dimensional excitation frequency 34

Dynamic load factor 1.2 1 0.8 0.6 mois=0% mois=0.1% mois=0.25% mois=0.5% 0.4 0.2 0 0 20 40 60 80 100 Non-dimensional excitation frequency 35

Dynamic load factor 1.2 1 0.8 0.6 0.4 0.2 0 plate cylindrical spherical 0 20 40 60 80 100 120 Non-dimensional excitation frequency 36

Dynamic load factor 1.2 1 a/b=1,b/t=100 a/b=2,b/t=100 a/b=3,b/t=100 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 Non-dimensional excitation frequency 37

Dynamic load factor 1.2 1 a/b=1,b/t=100 a/b=2,b/t=100 a/b=3,b/t=100 0.8 0.6 0.4 0.2 0 0 50 100 150 200 Non-dimensional excitation frequency 38

Dynamic load factor 1.2 1 a/b=1,ry=2.5 a/b=2,ry=2.5 a/b=3,ry=2.5 0.8 0.6 0.4 0.2 0 0 50 100 150 200 250 Non-dimensional excitation frequency 39

Dynamic load factor 1.2 1 a/b=1,ry=2.5 a/b=2,ry=2.5 a/b=3,ry=2.5 0.8 0.6 0.4 0.2 0 0 100 200 300 400 Non-dimensional excitation frequency 40

Dynamic load factor 1.2 1 0.8 E1/E2=10 E1/E2=20 E1/E2=40 0.6 0.4 0.2 0 0 20 40 60 80 100 120 140 Non-dimensional excitation frequency 41

Dynamic load factor 1.2 1 E1/E2=10 E1/E2=20 E1/E2=40 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 120 140 Non-dimensional excitation frequency 42

The excitation frequencies of laminated composite panels decrease with increase of temperature due to reduction of stiffness for all laminates. The excitation frequencies of laminated composite panels also decrease substantially with increase of moisture concentration for all laminates. Due to static component of load, the onset of instability shifts to lower frequencies with wide instability regions of the laminated composite panels. 43

Increasing the thickness of the panels results in better dynamic stability strength. Increasing the aspect ratio, shifts the frequencies of instability region to higher values and reduces the dynamic stability strength. The width of dynamic instability region is smaller for square panels than rectangular panels. 44

[1] Adams DF, Miller AK. Hygrothermal micro stress in unidirectional composite exhibiting inelastic materials behavior. Journal of Composite Material 1977;11:285 99. [2] Balamurugan,V., Ganapathi,M.,and Varadan,T.K.Nonlinear dynamic instability of laminated composite plates using finite element method.computers and Structures,60(1),125-130,1996. [3] Bolotin, V.V.The Dynamic stability of elastic systems.holdenday,sanfrancisco,1964 [4]Ganapati, M., Patel,B.P., Boisse,P., and Touratier,M.Nonlinear dynamic stability characteristics of elastic plates subjected to periodic in-plane load. International journal Non-linear Mechanics, 35,467-480,2000. [5] Ishikawa T, Koyama K, Kobayayaski S. Thermal expansion coefficients of unidirectional composites. Composite Material 1978;12:53 168. 45

[6] Lee SY, Yen WJ. Hygrothermal effects on the stability of a cylindrical composite shell panel. Journal of Computers and Structures 1989;33:551}9. [7] Lee SY, Chou CJ, Jang JL, Lim JS. Hygrothermal effects on the linear and nonlinear analysis of symmetric angle-ply laminated plates. Journal of Composite Structure 1992;21-1:41 8. [8] Patel BP, Ganapathi M, Makhecha DP. Hygrothermal effects on the structural behavior of thick composite laminates using higher order theory. Journal of Composite Structure 2002;56:25 34. [9] Qatu MS, Leissa AW. Buckling or transverse deformations of unsymmetrically laminated plates subjected to in-plane loads. AIAA Journal 1993;31:189}94. [10] Ram KSS, Sinha PK. Hygrothermal effects on the buckling of laminated composite plates. Journal of Composite Structures 1992;21:233}47. [11] Rao VVS, Sinha PK. Bending characteristic of thick multidirectional composite plates under hygrothermal environment. Reinforced Plastics Composite 2004;23:1481 95. 46

[12] Reddy JN. Mechanics of laminated composite plates: theory and analysis. Boca Raton, FL: CRC Press, 1997. [13] Sai Ram KS, Sinha PK. Hygrothermal effects on the bending characteristics on laminated composite plates. Journal of Composite Structures 1991;40:1009 15. [14] Shen HS. Hygrothermal effects on the nonlinear bending of shear deformable laminated plates. Journal of Engineering Mechanics 2002;128:493 6. [15] Snead JM, Palazotto AN. Moisture and temperature effects on the instability of cylindrical composite panels. Journal of Aircraft 1983;20:777}83. [16] Strife JR, Prewo KM. The thermal expansion behavior of unidirectional and bidirectional Kevlar/epoxy composites. Journal of Composite Material 1979;13:264 7. [17] Tsai SW, Hahn HT. Introduction to composite materials. Westport, CT: Technomic Publishing Co, 1980. [18] Whitney JM, Ashton JE. Effect of environment on the elastic response of layered composite plates. AIAA 1971;9:1708 13. 47

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