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1D analysis of laminated composite and sandwich plates using a new fifth-order plate theory

Abstract

In the present study, a new fifth-order shear and normal deformation theory (FOSNDT) is developed for the analysis of laminated composite and sandwich plates under cylindrical bending. The theory considered the effects of transverse shear and normal deformations. To account for the effect of transverse shear deformation, in-plane displacement uses polynomial shape function expanded up to fifth-order in-terms of the thickness coordinate. Transverse displacement uses derivative of shape function to account for the effect of transverse normal deformations. Therefore, the present theory involves six independent unknown variables. The theory satisfies traction free boundary conditions at top and bottom surfaces of the plate and does not require the shear correction factor. The principle of virtual work is used to obtain the variationally consistent governing differential equations and associated boundary conditions. Analytical solutions for simply supported boundary conditions are obtained using Navier’s solution technique. Non-dimensional displacements and stresses obtained using the present theory are compared with existing exact elasticity solutions and lower and higher-order theories to prove the efficacy of the present theory. The comparison shows that the displacements and stresses predicted by the present theory are in good agreement with those obtained by using the exact solution.

Keywords
Fifth-order; shear deformation; normal deformation; laminated; sandwich; bending

1 INTRODUCTION

The demand for high-strength, high-modulus and low density composite materials have generated an increased number of applications in many industries such as in aircraft, spacecraft, civil engineering, mechanical engineering, marine and many more.

The development of plate theory has a long history. Many well-known engineers, scientists, and mathematicians have made their contribution in the development of beam, plate and shell theories such as Jacob (II) Bernoulli, Leonard Euler, Joseph-Louis Lagrange Simeon Denis Poisson, Claude-Louis Navier and Gustav Robert Kirchhoff. The historical review of the development of beam, plate and shell theories is given in Timoshenko and Woinowsky-Krieger (1959) Timoshenko, S., and Woinowsky-Krieger, S. (1959). Theory of plates and shells. McGraw Hill, New York. , Todhunter and Pearson (1960) Todhunter, I., and Pearson, K. (1960). A history of the theory of elasticity and of the strength of materials from Galileo Galilei (1564-1642) to Lord Kelvin (1824-1907). Vols. I, II and III, Dover Publications, Inc., New York. and Carrera et al. (2011) Carrera, E., Giunta, G., and Petrolo, M. (2011). Beam Structures: Classical and Advanced Theories, John Wiley & Sons Ltd U.K. .

Well-known exact elasticity solutions for one dimensional and two dimensional bending of laminated composite and sandwich plates are developed by Pagano (1969 Pagano, N. J. (1969). Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 3: 398-411. , 1970a Pagano, N. J. (1970a). Exact solutions for bidirectional composites and sandwich plates. Journal of Composite Materials 4: 20–34. , 1970b Pagano, N. J. (1970b). Influence of shear coupling in cylindrical bending of anisotropic laminates. Journal of Composite Materials 4: 330-343. ). These solutions serves as benchmark solutions for the comparison of results obtained by using analytical or numerical solutions based on approximate plate theories. Exact elasticity solutions are mathematically difficult and computationally more cumbersome. This led to the development of analytical and numerical solution based on approximate plate theories. The simplest plate theory, based on the displacement field, is the classical plate theory (CPT) developed by Kirchhoff (1850) Kirchhoff, G.R., (1850). Uber das gleichgewicht und die bewegung einer elastischen Scheibe. Journal für die Reine und Angewandte Mathematik (Crelle's Journal) 40: 51-88. in the nineteenth century. But, since shear deformation effect is neglected by the CPT it cannot be applied for the analysis of thick plates where shear deformation effect is more pronounced. Mindlin (1951) Mindlin, R.D. (1951). Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. ASME Journal of Applied Mechanics18: 31-38. has considered the effect of transverse shear deformation for the first time in his first-order shear deformation theory (FSDT). The FSDT suffers from the drawback of constant shear strain condition through the thickness of the plate. Also it requires shear correction factor to properly account the strain energy due to shear deformation. These limitations of CPT and FSDT led to the development of higher-order shear deformation theories. The development of various higher-order plate theories and the solution techniques are recently reviewed by Sayyad and Ghugal (2015a) Sayyad, A. S., and Ghugal, Y. M. (2015a). On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results. Composite Structures 29: 177-201. .

Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. has developed a simple higher-order shear deformation theory (HSDT) for laminated composite beams and plates. This HSDT is further used by many researchers for the solution of various solid mechanics problems. Kant and Kommineni (1994) Kant, T., and Kommineni, J. R. (1994). Nonlinear analysis of angle ply composite and sandwich laminates. ASCEJournal of Aerospace Engineering 7(3): 342-352. have established a refined higher-order shear deformation theory for linear and geometrically non-linear behavior of fiber reinforced angle ply laminated composite and sandwich plates based on finite element formulation using a Lagrangian approach. Soldatos and Watson (1997) Soldatos, K. P., and Watson, P. (1997). A method for improving the stress analysis performance of one-and two-dimensional theories for laminated composites. Acta Mechanica 123(1):163-186. and Shu and Soldatos (2000) Shu, X.P., and Soldatos, K. P. (2000). Cylindrical bending of angle-ply laminates subjected to different sets of edge boundary conditions. International Journal of Solids and Structures 37: 4289-4307. developed the hyperbolic shear deformation theory for the cylindrical bending of cross-ply and angle-ply laminates.

Chakrabarti and Sheikh (2005) Chakrabarti, A., and Sheikh, A. H. (2005). Analysis of laminated sandwich plates based on interlaminar shear stress continuous plate theory. ASCE Journal ofEngineering Mechanics 131(4): 377-384. have developed a finite element model for the bending analysis of soft core sandwich plates. A study of global-local higher-order theories for laminated composite plates is performed by Zhen and Wanji (2007) Zhen, W., and Wanji, C. (2007). Study of global-local higher order theories for laminated composite plates. Composite Structures 79: 44-54. by presenting the general formulas of nth order global local higher-order theory. Fares and Elmarghany (2008) Fares, M. E., and Elmarghany, M. (2008). A refined zigzag nonlinear first order shear deformation theory of composite laminated plates. Composite Structures 82:71-83. have presented a refined zig-zag nonlinear FSDT of laminated composite plates using the Galerkin method. Ferreira et al. (2011) Ferreira, A. J. M., Roque, C. M. C., Carrera, E. and Cinera, M. (2011). Two higher order zig-zag theories for the accurate analysis of bending, vibration and buckling response of laminated plates by radial basis function collocation and a unified formulation. Journal of Composite Materials 45(24): 2523-2536. applied the Carrera’s unified formulation (CUF) for predicting the free vibration, static deformation and buckling behavior of thin and thick cross-ply laminated plates. Carrera and Zappino (2016) Carrera, E. and Zappino, E. (2016). CUF based variable kinematic models for free-vibration analysis of aircraft structures. AIAA Journal 54(1): 280-292. proposed several models based on 1D, 2D and 3D kinematics for free vibrations of shell structures using Lagrange polynomials. Pagani et al. (2016) Pagani, A., de Miguel, A. G., Petrolo, M. and Carrera, E. (2016). Analysis of laminated beams via unified formulation and Legendre polynomial expansions. Composite Structures 156: 78-92. have developed refined computational model based on layer-wise approach using CUF for the analysis of laminated structures. Sarangan and Singh (2016) Sarangan S., and Singh, B. N. (2016). Higher order closed form solution for the analysis of laminated composite and sandwich plates based on new shear deformation theories. Composite Structures 138: 391-403. have presented higher-order closed form solutions for the static, buckling and free vibration analysis of laminated composite and sandwich plates based on new shear deformation theories using Navier’s closed form solution technique. Kant and Shiyekar (2008) Kant, T., and Shiyekar, S. M. (2008). Cylindrical bending of piezoelectric laminates with a higher order shear and normal deformation theory. Computers and Structures 86: 1594-1603. obtained Navier type closed form solutions for the cylindrical bending of piezoelectric laminates subjected to electro-mechanical loading using higher-order shear and normal deformation theory. Sayyad and Ghugal (2015b) Sayyad, A. S., and Ghugal, Y. M. (2015b). A nth-order shear deformation theory for composite laminates in cylindrical bending. Curved and Layered Structures 2: 290-300. applied a nth order shear deformation theory for the cylindrical bending of composite laminates. Ghugal and Sayyad (2011) Ghugal, Y. M. and Sayyad, A. S. (2011). Free vibration of thick isotropic plates using trigonometric shear deformation theory. Journal of Solid Mechanics 3: 172–182. presented trigonometric shear and normal deformation theory for the free vibration of thick isotropic square and rectangular plate which was further extended by Sayyad and Ghugal (2016) Sayyad, A. S., and Ghugal, Y. M. (2016). Cylindrical bending of multilayered composite laminates and sandwiches. Advances in Aircraft and Spacecraft Sciences 3(2): 113-148. for the cylindrical bending of multilayered composite laminates and sandwiches. A critical review of literature on bending, buckling and free vibration analysis of shear deformable isotropic, laminated composite and sandwich beams based on equivalent single layer theories, layerwise theories, zig-zag theories and exact elasticity solution has recently been presented by Sayyad and Ghugal (2017a) Sayyad, A. S., and Ghugal, Y. M. (2017a). Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature. Composite Structures 171: 486-504. . Sayyad and Ghugal (2017b) Sayyad, A. S., and Ghugal, Y. M. (2017b). A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. International Journal of Applied Mechanics 9(1):1-36. have also developed a displacement based unified shear deformation theory for the analysis of shear deformable advanced composite beams and plates.

1.1 The plate under consideration for the present study

A cross-ply laminated composite plate made of orthotropic fibrous composite material having length ‘a’ and width ‘b’ in the in the x and y directions respectively is considered as shown in Figure 1 . The y direction of the plate is assumed to be infinitely long compared to other two dimensions, therefore, strains in the y direction are assumed to be zero ( εy=γyz=0 ). The thickness of the plate is measured in z-direction and at z=0, the mid plane of the plate is located. The plate under consideration consists of N number of layers bonded together. The plate is carrying an out of plane load q(x), acting on its top surface. i.e. (z=h/2) .

Figure 1
Geometry and co-ordinate system of the layered plate deform in cylindrical bending

2 FIFTH-ORDER SHEAR AND NORMAL DEFORMATION THEORY

Through-thickness distributions of transverse shear and normal stresses for composite laminates are important for delamination type failure. Therefore, it is essential to understand and calculate transverse shear and normal stress through the thickness of the plate accurately ( Carrera 2005 Carrera, E. (2005). Transverse normal strain effects on thermal stress analysis of homogeneous and layered plates. AIAA Journal 43(10): 2232-2242. ). However, in a whole variety of higher-order plate theories existing in the literature very few researchers have considered the effect of transverse normal stress for developing refined plate theory in view of minimizing the number of unknown variables. In the well-known theory of Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. , thickness coordinate is expanded up to third-order in the in-plane displacement field and the effect of transverse normal deformation is neglected.

The present theory is built upon classical plate theory having following important features.

  1. 1

    The present theory considers the effects of transverse shear and normal deformations (εz0) .

  2. 2

    The axial displacement in the x direction consists of extension, bending and shear components. The extension (u0) and bending (ub) components are analogues to the classical plate theory whereas the shear component (us) contains polynomial shape function expanded up to fifth-order in terms of the thickness coordinate (z/h). Hence the theory is designated as the fifth-order shear and normal deformation theory (FOSNDT).

    u=u0 + ub +us (1)

Where

ub=zdw0dx and us=[z4z33h2]ϕx +[z16z55h4]ψx (2)
  1. 3

    The transverse displacement w in z- direction is assumed to be a function of x and z coordinates to include the effect of transverse normal deformations (εz0) .

    w=w0+(14z2h2)ϕz+(116z4h4)ψz (3)

  2. 4

    The theory enforces the parabolic variation of the transverse shear stress across the thickness of the plate. Thus, the theory obviates the need of the shear correction factor.

  3. 5

    The body forces are not considered in the analysis.

2.1 Kinematics of the present theory

Based on the aforementioned assumptions and features, the displacement field of the present theory (FOSNPT) can be expressed as

u ( x , z ) = u 0 ( x ) z d w 0 d x + [ z 4 z 3 3 h 2 ] ϕ x ( x ) + [ z 16 z 5 5 h 4 ] ψ x ( x ) w ( x , z ) = w 0 ( x ) + ( 1 4 z 2 h 2 ) ϕ z ( x ) + ( 1 16 z 4 h 4 ) ψ z ( x ) (4)

where u and w are the x and z-directional displacements of any point on the plate, u 0 and w0 are the in-plane displacements of mid-plane in x and z-directions respectively; ϕx and ψx are rotations of the normal to the middle plane about y axis which account the effect of transverse shear deformation. ϕz and ψz represent higher-order transverse cross-sectional deformation modes i.e. effect of transverse normal deformations. The non-zero strain components associated with the present displacement field are obtained by using the linear theory of elasticity.

ε x = u x = d u 0 d x z d 2 w 0 d x 2 + ( z 4 z 3 3 h 2 ) d ϕ x d x + ( z 16 z 5 5 h 4 ) d ψ x d x ε z = w z = ( 8 z h 2 ) ϕ z + ( 64 z 3 h 4 ) ψ z γ x z = u z + w x = ( ϕ x + d ϕ z d x ) ( 1 4 z 2 h 2 ) + ( ψ x + d ψ z d x ) ( 1 16 z 4 h 4 ) (5)

2.2 Constitutive Equations

The constitutive equations for the kth lamina are given by

{ σ x σ z τ x z } k = [ Q 11 Q 13 0 Q 13 Q 33 0 0 0 Q 55 ] k { ε x ε z γ x z } k (6)

where Qij are the reduced elastic constants in x-z plane, σx is the normal stress along x-direction, σz is the stress acting along z-direction and τxz is shear stress along z-direction. The following relationship between the reduced elastic constants and the engineering elastic constants are used.

Q 11 = E 1 ( 1 μ 23 μ 32 ) ( 1 μ 12 μ 21 μ 23 μ 32 μ 31 μ 13 2 μ 12 μ 23 μ 31 ) , Q 13 = E 1 ( μ 31 + μ 21 μ 32 ) ( 1 μ 12 μ 21 μ 23 μ 32 μ 31 μ 13 2 μ 12 μ 23 μ 31 ) , Q 33 = E 3 ( 1 μ 12 μ 21 ) ( 1 μ 12 μ 21 μ 23 μ 32 μ 31 μ 13 2 μ 12 μ 23 μ 31 ) , Q 55 = G 13 (7)

where E1, E3 are Young’s moduli, G13 is the shear modulus and μ12 , μ21 , μ13 , μ31 , μ23 , μ3 2 are Poisson’s ratios; the subscripts 1, 2, 3 correspond to x, y , z directions of Cartesian coordinate systems, respectively.

2.3 Governing Equations and Boundary Conditions

Variationally consistent governing differential equations and associated boundary conditions are derived by using the principle of virtual work. For the plate under consideration, the principle of virtual work takes the following form.

b 0 L h / 2 + h / 2 ( σ x δ ε x + σ z δ ε z + τ x z δ γ x z ) d z d x 0 L q δ w d x = 0 (8)

where is the virtual displacement i.e. infinitesimal change in the position coordinates of the points under consideration. q(x) represents transverse load acting on the top surface of the plate. By substituting virtual strain from Eq. (5) into the Eq. (8) one can obtain

0 L ( N x d δ u 0 d x M x b d 2 δ w 0 d x 2 + M x s 1 d δ ϕ x d x + M x s 2 d δ ψ x d x + Q z S 1 δ ϕ z + Q z s 2 δ ψ z + Q x z 1 δ ϕ x + Q x z 1 d δ ϕ z d x + Q x z 2 δ ψ x + Q x z 2 d δ ψ z d x ) d x = 0 L q δ w d x (9)

where Nx represents the axial force resultant; Mxb , Mxs represent bending moment and higher order moment resultants; Qxz1, Qxz2 represent shear force resultants due to shear deformation; and Qzs1, Qzs2 represent shear force resultants due to normal deformations.

N x = h / 2 + h / 2 σ x d z = A 11 d u 0 d x B 11 d 2 w 0 d x 2 + C 11 d ϕ x d x + D 11 d ψ x d x + I 13 ϕ z + J 13 ψ z M x b = h / 2 + h / 2 σ x z d z = B 11 d u 0 d x A s 11 d 2 w 0 d x 2 + C s 11 d ϕ x d x + D s 11 d ψ x d x + I s 13 ϕ z + J s 13 ψ z M x s 1 = h / 2 + h / 2 σ x f 1 ( z ) d z = C 11 d u 0 d x C s 11 d 2 w 0 d x 2 + C s s 111 d ϕ x d x + C s s 211 d ψ x d x + I s s 113 ϕ z + J s s 113 ψ z M x s 2 = h / 2 + h / 2 σ x f 2 ( z ) d z = D 11 d u 0 d x D s 11 d 2 w 0 d x 2 + C s s 211 d ϕ x d x + D s s 111 d ψ x d x + I s s 213 ϕ z + J s s 213 ψ z Q x z 1 = h / 2 + h / 2 τ x z f 1 ' ( z ) d z = C s s s 155 ϕ x + C s s s 155 d ϕ z d x + C s s s 255 ψ x + C s s s 255 d ψ z d x Q x z 2 = h / 2 + h / 2 τ x z f 2 ' ( z ) d z = C s s s 255 ϕ x + C s s s 255 d ϕ z d x + D s s s 155 ψ x + D s s s 155 d ψ z d x Q z S 1 = h / 2 + h / 2 σ z f 1 " ( z ) d z = I 13 d u 0 d x I s 13 d 2 w 0 d x 2 + I s s 113 d ϕ x d x + I s s 213 d ψ x d x + I s s s 133 ϕ z + I s s s 233 ψ z Q z S 2 = h / 2 + h / 2 σ z f 2 " ( z ) d z = J 13 d u 0 d x J s 13 d 2 w 0 d x 2 + J s s 113 d ϕ x d x + J s s 213 d ψ x d x + I s s s 233 ϕ z + J s s s 133 ψ z (10)

where

f 1 ( z ) = [ z 4 z 3 3 h 2 ] , f 1 ' ( z ) = [ 1 4 z 2 h 2 ] , f 1 " ( z ) = [ 8 z h 2 ] , f 2 ( z ) = [ z 16 z 5 5 h 4 ] , f 2 ' ( z ) = [ 1 16 z 4 h 4 ] , f 2 " ( z ) = [ 64 z 3 h 4 ] (11)

The governing equations can be derived from Eq. (9) by integrating the displacement variables by parts and setting the coefficients of δu0, δw0,δϕx, δψx, δϕz and δψz to zero separately, and the following equations can be obtained:

δ u 0 : d N x d x = A 11 d 2 u 0 d x 2 B 11 d 3 w 0 d x 3 + C 11 d 2 ϕ x d x 2 + D 11 d 2 ψ x d x 2 + I 13 d ϕ z d x + J 13 d ψ z d x = 0 (12)
δ w 0 : d 2 M x b d x 2 + q = B 11 d 3 u 0 d x 3 A s 11 d 4 w 0 d x 4 + C s 11 d 3 ϕ x d x 3 + D s 11 d 3 ψ x d x 3 + I s 13 d 2 ϕ z d x 2 + J s 13 d 2 ψ x d x 2 + q = 0 (13)
δ ϕ x : d M x S 1 d x Q x z 1 = C 11 d 2 u 0 d x 2 C s 11 d 3 w 0 d x 3 + C s s 111 d 2 ϕ x d x 2 + C s s 211 d 2 ψ x d x 2 + I s s 113 d ϕ z d x + J s s 113 d ψ z d x C s s s 155 ϕ x C s s s 155 d ϕ z d x C s s s 255 ψ x C s s s 255 d ψ z d x = 0 (14)
δ ψ x : d M x S 2 d x Q x z 2 = D 11 d 2 u 0 d x 2 D s 11 d 3 w 0 d x 3 + C s s 211 d 2 ϕ x d x 2 + D s s 111 d 2 ψ x d x 2 + I s s 213 d ϕ z d x + J s s 213 d ψ z d x C s s s 255 ϕ x C s s s 255 d ϕ z d x D s s s 155 ψ x D s s s 155 d ψ z d x = 0, (15)
δ ϕ z : d Q x z 1 d x Q z s 1 = C s s s 155 d ϕ x d x + C s s s 155 d 2 ϕ z d x 2 + C s s s 255 d ψ x d x + C s s s 255 d 2 ψ z d x 2 I 13 d u 0 d x + I s 13 d 2 w 0 d x 2 I s s 113 d ϕ x d x I s s 213 d ψ x d x I s s s 133 ϕ z I s s s 233 ψ z = 0 (16)
δ ψ z : d Q x z 2 d x 2 Q z s 2 = C s s s 255 d ϕ x d x + C s s s 255 d 2 ϕ z d x 2 + D s s s 155 d ψ x d x + D s s s 155 d 2 ψ z d x 2 J 13 d u 0 d x + J s 13 d 2 w 0 d x 2 J s s 113 d ϕ x d x J s s 213 d ψ x d x I s s s 233 ϕ z J s s s 133 ψ z = 0 (17)

where the extension, bending, bending-extension, bending-twisting stiffnesses used in the equations (12) - (17) can be obtained as

( A i j , B i j , A s i j ) = Q i j h / 2 + h / 2 ( 1, z , z 2 ) d z , ( C i j , C s i j , C s s 1 i j , C s s 2 i j , I s s 1 i j , J s s 1 i j ) = Q i j h / 2 + h / 2 f 1 ( z ) [ 1, z , f 1 ( z ) , f 2 ( z ) , f 1 " ( z ) , f 2 " ( z ) ] d z , ( D i j , D s i j , D s s 1 i j , I s s 2 i j , J s s 2 i j ) = Q i j h / 2 + h / 2 f 2 ( z ) [ 1, z , f 2 ( z ) , f 1 " ( z ) , f 2 " ( z ) ] d z , ( C s s s 1 i j , C s s s 2 i j ) = Q i j h / 2 + h / 2 f 1 ' ( z ) [ f 1 ' ( z ) , f 2 ' ( z ) ] d z , ( D s s s 1 i j ) = Q i j h / 2 + h / 2 f 2 ' ( z ) f 2 ' ( z ) d z , ( I i j , I s i j , I s s s 1 i j , I s s s 2 i j ) = Q i j h / 2 + h / 2 f 1 " ( z ) [ 1, z , f 1 " ( z ) , f 2 " ( z ) ] d z , ( J i j , J s i j , J s s s 1 i j ) = Q i j h / 2 + h / 2 f 2 " ( z ) [ 1, z , f 2 " ( z ) ] d z (18)

The boundary conditions along edges (x = 0, x = a) are of the following form:

N x = 0 o r u 0 = 0 ; M x b = 0 o r d w 0 / d x = 0 ; d M x b / d x = 0 o r w 0 = 0 ; M x s 1 = 0 o r ϕ x = 0 M x s 2 = 0 o r ψ x = 0 ; Q x z 1 = 0 o r ϕ z = 0 ; Q x z 2 = 0 o r ψ z = 0 (19)

2.4 Closed form solutions

For a simply supported laminated composite plate, the kinematic boundary conditions are given below:

w 0 = 0, N x = 0, M x b = 0, M s 1 = 0, M s 2 = 0 (20)

To determine the unknown displacement variables, the Navier’s solution technique is implemented. To satisfy the aforementioned boundary conditions the displacements and rotations are assumed in Fourier trigonometric form

( u 0 , ϕ x , ψ x ) = m = 1 ( u m , ϕ x m , ψ x m ) cos ( m π x a ) ( w 0 , ϕ z , ψ z ) = m = 1 ( w m , ϕ z m , ψ z m ) sin ( m π x a ) (21)

where um, wm, ϕxm, ψxm, ϕzm and ψzm are the unknowns to be determined. According to Navier’s solution scheme, transverse load is also expanded in Fourier trigonometric form

q ( x ) = m = 1 q m sin ( m π x a ) (22)

where qm is the coefficient of Fourier series expansion and m is the positive integer. For sinusoidal load, qm=q0 and m=1. Substitution of Eqs. (21) and (22) into governing equations (12) through (17) leads to the following form

[ K 11 K 12 K 13 K 14 K 15 K 16 K 22 K 23 K 24 K 25 K 26 K 33 K 34 K 35 K 36 K 44 K 45 K 46 s y m m e t r i c K 55 K 56 K 66 ] { u m w m ϕ x m ψ x m ϕ z m ψ z m } = { 0 q m 0 0 0 0 } (23)

where [Kij] are the elements of stiffness matrix

K 11 = A 11 ( m 2 π 2 a 2 ) , K 12 = B 11 ( m 3 π 3 a 3 ) , K 13 = C 11 ( m 2 π 2 a 2 ) , K 14 = D 11 ( m 2 π 2 a 2 ) ; K 15 = I 13 ( m π a ) , K 16 = J 13 ( m π a ) , K 22 = A s 11 ( m 4 π 4 a 4 ) , K 23 = C s 11 ( m 3 π 3 a 3 ) , K 24 = D s 11 ( m 3 π 3 a 3 ) , K 25 = I s 13 ( m 2 π 2 a 2 ) , K 26 = J s 13 ( m 2 π 2 a 2 ) , K 33 = C s s 111 ( m 2 π 2 a 2 ) C s s s 155 , K 34 = [ C s s 211 ( m 2 π 2 a 2 ) + C s s s 255 ] , K 35 = I s s 113 ( m π a ) C s s s 155 ( m π a ) , K 36 = J s s 113 ( m π a ) C s s s 255 ( m π a ) , K 45 = I s s 213 ( m π a ) C s s s 255 ( m π a ) , K 46 = J s s 213 ( m π a ) D s s s 155 ( m π a ) , K 56 = C s s s 255 ( m 2 π 2 a 2 ) I s s s 233 , K 66 = D s s s 155 ( m 2 π 2 a 2 ) J s s s 133 (24)

After knowing the values of unknown displacement variables u0, w0,ϕx,ψx,ϕz and ψz from Eq. (23) , one can obtain all the displacements and stress components within the laminated composite plate using equations (4) through (6).

2.5 Estimation of transverse shear stress and normal stress

Through-thickness distributions of transverse shear and normal stresses for composite laminates are important for delamination type failure. The evaluation of transverse shear stresses from the constitutive relations leads to discontinuity at the inter face of two adjacent layers of a laminate and thus violates the equilibrium conditions. Thus, elasticity equilibrium equation neglecting the body force is used to derive expression for the transverse stress in the kth lamina of composite laminate.

τ x z k = h k h k + 1 σ x k x d z + C (25)

From equation (25) the transverse stress ( τxz ) can be evaluated through integration with respect to the laminate thickness coordinate (z). The in-plane stress ( σx ) obtained by using equation (4) is substituted in equation (25) . The constants of integrations (C) can be determined by substituting the boundary conditions. It is expected that this procedure will produce an accurate transverse shear stresses.

3.0 NUMERICAL RESULTS AND DISCUSSION

Aluminum alloy and fibrous composite materials are being used increasingly for numerous space applications.

3.1 Aluminum alloy: Aluminum is one of the most widely used metals in modern aircraft construction. It is vital to the aviation industry because of its high strength to weight ratio and its comparative ease of fabrication. The outstanding characteristic of aluminum is its light weight. Aluminum melts at the comparatively low temperature of 12500F. It is nonmagnetic and is an excellent conductor. Following material properties (Aluminum 3003-H14) are used for numerical study.

Material 1 ( Krishna Murty, 1984 Krishna Murty, A. V. (1984). Toward a consistent beam theory. AIAA Journal 22 (6): 811-816. ): E1=E2=E3=E=69 GPa and G12=G13=G23=G=26 GPa

3.2 Fibrous composite materials: Engineers are interested in these materials because of their favorable mechanical characteristic of high strength/high stiffness to weight ratio and potential for zero or near-zero coefficient of thermal expansion. The use of high modulus Graphite-Epoxy composite parts for space applications is already well established. Using Graphite-Epoxy parts for space vehicles and structures has many advantages including: 1) Critical weight savings 2) Improved control of thermal distortions 3) Increased structural stiffness. Following properties of Graphite-Epoxy composite material are used for the numerical study.

Material 2 ( Pagano, 1969 Pagano, N. J. (1969). Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 3: 398-411. ): E1=172.5 GPa, E2=E3=6.9 GPa, G12=G13=3.45 GPa , G23=1.38 GPa, μ12=μ13=μ23=0.25

Material 3 ( Kant and Swaminathan, 2000 Kant, T., and Swaminathan, K. (2000). Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory. Composite Structures 56:329-344. ): E1=131.1 GPa, E2=E3=6.9 GPa , G12=G13=3.588 GPa, G23=3.088 GPa, μ12=μ13=0.32, μ23=0.49

Material 4 ( Kapuria et al., 2004 Kapuria, S., Dumir, P.C., and Jain, N. K. (2004). Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Composite Structures 64: 317-327. ): E1=0.2208 MPa, E2=0.2001 MPa, E3=2760 MPa , G12=16.56 MPa, G23=455.4 MPa, G31=545.1 MPa, μ12=0.99, μ13=μ23=0.00003

For the validity of the present theory, following examples are solved for the numerical study.

  1. a

    Cylindrical bending of two-layer (00/900) antisymmetric cross-ply laminated composite plates. ( Figure 2a )

    Figure 2
    Simply supported laminated plates subjected to sinusoidal load

  2. b

    Cylindrical bending of three-layer (00/900/00 ) symmetric cross-ply laminated composite plates. ( Figure 2b )

  3. c

    Cylindrical bending of three-layer (00/core/0 0) symmetric sandwich plates. ( Figure2C )

Displacements and stresses for laminated composite and sandwich plates under cylindrical bending obtained by using the present theory (FOSNDT) are presented in Tables 1 - 4 and compared with the those obtained by using the classical plate theory (CPT), FSDT of Mindlin (1951) Mindlin, R.D. (1951). Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. ASME Journal of Applied Mechanics18: 31-38. , HSDT of Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. , sinusoidal shear and normal deformation theory of Sayyad and Ghugal (2016) Sayyad, A. S., and Ghugal, Y. M. (2016). Cylindrical bending of multilayered composite laminates and sandwiches. Advances in Aircraft and Spacecraft Sciences 3(2): 113-148. . Exact elasticity solution developed by Pagano (1969) Pagano, N. J. (1969). Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 3: 398-411. is considered as a benchmark solution for comparison. The displacements and stresses are calculated at typical important locations and presented in the following non-dimensional form.

Table 1
Comparison of In-Plane Displacement, Transverse Displacement, In-Plane Normal Stress and Transverse Shear Stress for Aluminum Alloy Plate Subjected to Sinusoidal Load under Cylindrical Bending
Table 4
Comparison of In-Plane Displacement, Transverse Displacement, In-Plane Normal Stress and Transverse Shear Stress for Three-Layer (00/Core/900) Symmetric Laminated Composite Plate Subjected to Sinusoidal Load under Cylindrical Bending
u ¯ ( 0, h 2 ) = b E 3 u q 0 h , w ¯ ( a 2 ,0 ) = 100 E 3 w h 3 b q 0 a 4 , σ ¯ x ( a 2 , h 2 ) = b σ x q 0 , τ ¯ x z ( 0,0 ) = b τ x z q 0 (26)

The through-the-thickness profiles for in-plane displacement ( u¯ ), in-plane normal stress ( σ¯x ) and transverse shear stress ( τ¯xz ) for laminated and sandwich plates subjected to a sinusoidal load are plotted in Figures 3 through 14.

Figure 3
Through thickness variation of in-plane displacement ( u¯ ) at (x=0, z) for two-layer (00/900) antisymmetric laminated composite plate subjected to sinusoidal load. (a/h=4)

High-strength aluminum alloy is an important airframe material since 1920s. Therefore, the present theory is tested for the plate made of aluminum alloy (material 1). Comparison of non-dimensional displacements and stresses of aluminum alloy plate subjected to sinusoidal load are tabulated in Table 1 . For the comparison purpose, numerical results by using HSDT of Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. , FSDT of Mindlin (1951) Mindlin, R.D. (1951). Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. ASME Journal of Applied Mechanics18: 31-38. and CPT are obtained. The numerical results are presented for thick (a/h = 4), moderately thick (a/h = 10) and thin plates (a/h = 100). From Table 1 , it is pointed out that numerical results obtained by using the present theory and HSDT of Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. are in excellent agreement with each other whereas FSDT and CPT underestimate the displacements and stresses due to neglect of shear and normal deformations.

The comparison of non-dimensional displacements and stresses for the two-layer (00 /900) laminated composite plates is shown in Table 2 . The plate is subjected to a sinusoidal load ( Figure 2a ) and made up of orthotropic material 2. Both the layers are of equal thickness i.e. h /2. Through-the-thickness distributions of in-plane displacement and stresses are plotted in Figures 3 - 5 and variation of transverse displacement with respect to a/h ratio is plotted in Figure 6 . Exact elasticity solutions presented by Pagano (1969) Pagano, N. J. (1969). Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 3: 398-411. are taken as basis for the comparison of numerical results obtained by using the present theory (FOSNDT), HSDT of Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. , SSNPT of Sayyad and Ghugal (2016) Sayyad, A. S., and Ghugal, Y. M. (2016). Cylindrical bending of multilayered composite laminates and sandwiches. Advances in Aircraft and Spacecraft Sciences 3(2): 113-148. , FSDT of Mindlin (1951) Mindlin, R.D. (1951). Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. ASME Journal of Applied Mechanics18: 31-38. and CPT. HSDT, FSDT and CPT do not consider the effect of transverse normal deformation (ε z=0) whereas the present theory and SSNPT considers the effect of transverse normal deformation (εz≠0). It can be observed from Table 2 that the present theory shows considerable improvement in the in-plane displacement and stresses compared to those obtained by using HSDT and SSNPT. The percentage error predicted using the present theory is less in many cases as compared to HSDT, SSNPT, FSDT and CPT. This is in fact due to inclusion of fifth order term in-terms of the thickness coordinate in the displacement field. Figures 4 and 5 shows stresses are always maximum in 00 layer and minimum in 900 layers. The transverse shear stress ( τ¯xz ) which is an important indicator to the onset of delamination are obtained using equations of equilibrium to ascertain the continuity at the layer interface. Through-the-thickness distribution of transverse displacement is not uniform when itis obtained using the present theory and SSNPT whereas it is uniform when obtained by using HSDT, FSDT and CPT.

Table 2
Comparison of In-Plane Displacement, Transverse Displacement, In-Plane Normal Stress and Transverse Shear Stress for Two-Layer (00/900) Antisymmetric Laminated Composite Plate Subjected to Sinusoidal Load under Cylindrical Bending.
Figure 5
Through thickness variation of transverse shear stress ( τ¯xz ) at (x=0, z) for two-layer (00/900) antisymmetric laminated composite plate subjected to sinusoidal load (a/h=4)
Figure 6
Variation of transverse displacement ( w¯ ) with respect to aspect ratio for two-layer (00/900) antisymmetric laminated composite plate subjected to sinusoidal load.
Figure 4
Through thickness variation of in-plane normal stress ( σ¯x ) at (x=a/2, z) for two-layer (00/900) antisymmetric laminated composite plate subjected to sinusoidal load (a/h=4)

Table 3 compares numerical values of non-dimensional displacements and stresses obtained by using the present theory and other higher-order theories for three-layer (00/90 0/00) symmetric laminated composite plate subjected to a sinusoidal load (see Figure 2b ). The plate is made of material 2 and overall thickness is equally distributed among all the layers i.e. h/3. The examination of Table 3 reveals that present results are in excellent agreement with those obtained by using the exact elasticity solution of Pagano (1969) Pagano, N. J. (1969). Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 3: 398-411. . In this problem also considerable improvement in the results is observed due to refinement of the polynomial shape function. Large percentage error is observed when these quantities are obtained by using FSDT and CPT due to neglect of shear and normal deformations. Through-the-thickness distributions of in-plane displacement and stresses are plotted in Figures 7 - 9 . Variation of transverse displacement with respect to a/h ratios is plotted in Figure 10 .

Table 3
Comparison of In-Plane Displacement, Transverse Displacement, In-Plane Normal Stress and Transverse Shear Stress for Three-Layer (00/900/0 0) Symmetric Laminated Composite Plate Subjected to Sinusoidal Load under Cylindrical Bending
Figure 7
Through thickness variation of in-plane displacement ( u¯ ) at (x=0, z) for three-layer (00/900/00) symmetric laminated composite plate subjected to sinusoidal load. (a/h=4)
Figure 9
Through thickness variation of transverse shear stress ( τ¯xz ) at (x=0, z) for three-layer (00/900/00) symmetric laminated composite plate subjected to sinusoidal load. (a/h=4)
Figure 10
Variation of transverse displacement ( w¯ ) with respect to aspect ratio for three-layer (00/900/0 0) symmetric laminated composite plate subjected to sinusoidal load

Table 4 compares the numerical values of non-dimensional displacement and stresses of three-layer (00/core/900) symmetric sandwich plate subjected to a sinusoidal load (see Figure 2c ). Thickness of top and bottom face sheets is 0.1h each whereas thickness of middle soft core is 0.8h. Face sheets of the plate are made of a fibrous composite material 3 whereas the core is made of material 4. For the sandwich plates in cylindrical bending, the exact elasticity solution is not available in the literature; hence present results are compared with published results. Present results are in good agreement with the HSDT of Reddy (1984) Reddy, J. N. (1984). A simple higher order theory for laminated composite plates. ASMEJournal of Applied Mechanics 51:745-752. and SSNPT of Sayyad and Ghugal (2016) Sayyad, A. S., and Ghugal, Y. M. (2016). Cylindrical bending of multilayered composite laminates and sandwiches. Advances in Aircraft and Spacecraft Sciences 3(2): 113-148. . Figure 10 shows variation of transverse displacement with respect to aspect ratio for the three-layer (00/900/00) symmetric laminated composite plate subjected to sinusoidal load. Figures 11 - 13 plots the through-the-thickness distributions of in-plane displacement, in-plane normal stress and transverse shear stress. The examination of Figure 12 reveals that the in-plane normal stress developed in the middle core is very small compared to that in top and bottom face sheets. This is in fact due to core material is soft compared to material of face sheets. The transverse shear stress is obtained using equations of equilibrium of the theory of elasticity to ascertain the stress continuity at the layer interface. Variation of transverse displacement with respect to a/h ratios is plotted in Figure 14 .

Figure 11
Through thickness variation of in-plane displacement ( u¯ ) at (x=0, z) for three-layer (00/core/00) symmetric sandwich plate subjected to sinusoidal load. (a/h=4)
Figure 13
Through thickness variation of transverse shear stress ( τ¯xz ) at (x=0, z) for three-layer (00/core/00) symmetric sandwich plate subjected to sinusoidal load. (a/h=4)
Figure 12
Through thickness variation of in-plane normal stress ( σ¯x ) at (x=a/2, z) for three- layer (00/core/00) symmetric sandwich plate subjected to sinusoidal load. (a/h=4)
Figure 14
Variation of transverse displacement ( w¯ ) with respect to aspect ratio for three-layer (00/core/900) symmetric sandwich plate subjected to sinusoidal load.
Figure 8
Through thickness variation of in-plane normal stress ( σ¯x ) at (x=a/2, z) for three-layer (00/900/00) symmetric laminated composite plate subjected to sinusoidal load (a/h=4)

4.0 CONCLUSIONS

A new fifth-order shear and normal deformation theory for the cylindrical bending of laminated composite and sandwich plates have been developed in this paper. To account for the effect of transverse shear deformation, in-plane displacement uses polynomial shape function expanded up to fifth-order in-terms of the thickness coordinate. The present theory involves six-degrees-of-freedom. The theory satisfies traction free boundary conditions at top and bottom surfaces of the plate and does not required the shear correction factor. For simplicity, this theory is applied for the analysis of laminated composite and sandwich plates deformed in cylindrical bending. Non-dimensional displacements and stresses obtained using the present theory are compared with existing exact elasticity solutions and lower and higher-order theories. From the comparison of numerical results, it is concluded that the present theory is in good agreement with exact elasticity solution of Pagano and shows considerable improvement in the numerical results obtained by using higher-order shear deformation theory of Reddy. This validate that the effect of transverse shear and normal deformations both plays important role in the analysis of laminated composite structures.

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Publication Dates

  • Publication in this collection
    2018

History

  • Received
    26 Apr 2017
  • Reviewed
    23 Sept 2017
  • Accepted
    10 Oct 2017
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