Linear Static Analysis of Laminated Composite Plates with Layerwise Finite Element
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Sigma Journal of Engineering and Natural Sciences 2014, Vol. 32, Issue 3, pp. 297-309; doi.org/10.62051/ytu.sigma-journal-of-engineering-and-natural-sciences-linear-static-analysis-of-laminated-composite-plates-with-layerwise-finite-eleme
Abstract
Keywords: Laminated composite; finite element; zig-zag.
1. Introduction
Composite materials consist of two or more materials which together produce desirable properties such as stiffness, strength, corrosion resistance, thermal properties and fatique life that cannot be achieved with any of the constituents alone. Fiber reinforced composite materials for structural applications are often made in the form of a thin layer, called lamina. Structural elements such as beams or plates are formed by stacking the layers to achieve desired strength and stiffness. *
Corresponding Author/Sorumlu Yazar: e-mail/e-ileti: hasim@itu.edu.tr, tel: (212) 559 64 63
In this section; the plate theories are explained from single layer structures through multilayered ones. First studies in the plate and shell literature are grouped as Love First Approximation Theory (LFAT) by Kirchhoff (1850) and Love (1927) with an assumption that normal to the reference surface Ω remain normal in the deformed states and do not change in length. Likewise, Cauchy (1828) and Poisson (1829) have studied in thin shell assumptions which can be assigned to the first grouping. Reissner (1945) and Mindlin (1951) considered not only the work done by in-plane stresses but also the work done by transverse shear stresses in their studies and they are grouped as Love Second Approximation Theory (LSAT). Extensions of Kirchoff-Love First Approximation Theory to layered structures are known as Classsical Lamination Theory (Jones 1975). Applications of LSAT theories to multilayered structures are referred as the First Order Shear Deformation Theory (FSDT) by Whitney (1969). Thai and Choi (2013) proposed a simple FSDT for bending and free vibration analysis of laminates by making further simplifying assumptions to the existing FSDT, the number of unknowns and governing equations of the present FSDT are reduced by one. However, the drawback of FSDT comes from the representation of the constant transverse shear strains through laminate thickness and this discrepancy between the actual quadratic stress state and the constant stress state predicted by the first order theory is often corrected in computing transverse shear force resultants by multiplying the transverse stress integrals with a shear correction coefficient parameter. Due to the need for shear correction coefficients used in the first order theory, higher order theories such as Reddy’s theory (1997) are developed to have quadratic variation of the transverse shear strains and transverse shear stresses through each layer by expanding the displacement field in terms of the thickness coordinate up to any desired degree. However, its FEM implementation is somewhat hindered by the need to employ a C continuous basis for the transverse displacement. Mendonça et al. (2013) have presented a method to model the bending problem of arbitrary anisotropic laminated composite plates, which allows an arbitrary C continuity under the kinematic hypothesis of the Third Order Plate Theory proposed by Reddy. Thai et al. (2012) have developed a finite element called NS-DSG3 based on a combination of node based smoothing discrete shear gap method with the Higher Order Shear Deformation Plate Theory (HSDT). Bhar et al. (2010) have brought out the significance of using the HSDT over the FSDT for analyzing laminated composite stiffened plates. Following Reddy (1997), these types of theories such as CLT, FSDT or HSDT are grouped as Equivalent Single Layer Theories (ESLM) which have a number of unknown variables that are independent of the number of constitutive layers NL . In addition to their inherent simplicity and low computational cost, the ESL models often provide sufficiently accurate description of global response for thin to moderately thick laminates, e.g., gross deflections, critical buckling loads. However, the ESL models are often incapable of accurately describing the state of stress and strain at the ply level near geometric and material discontinuities. In all equivalent single layer laminate theories based on assumed displacement fields and it is assumed that the displacements are continuous functions of the thickness coordinate. This in turn results in all stresses in ESL models are discontinuous at layer interfaces contrary to the actual transverse stress state.
2. Material Properties Of Laminated Composite Plates
From Figure 1, it can be seen that a laminate is a collection of laminae which could have different material properties and fiber orientation to achieve the desired stiffness and thickness. As a direct consequence of exhibition of different mechanical and physical properties in the thickness direction, layered composite plates show higher transverse shear and transverse normal stress deformability. (GLT /ET GTT /ET 1/10~1/200, where L denotes the fiber directions, while T is direction orthogonal to L) than single layer plates.
From Figure 2, it is noticed that laminated composites are made from two materials: a reinforcement material called fiber and a base material, called matrix material.
Figure 1. A laminate made up of plies with different fiber orientations
Figure 2. Fiber and matrix materials in laminated composite plates The moduli and Poisson’s ratio of a laminated composite plate can be expressed in terms of the moduli, Poisson’s ratios and volume fractions of the matrix and fiber materials. Then it can be shown that the lamina engineering constants are given by E E
where E is the longitudinal modulus, E is transverse modulus, 12 is the major Poisson’s ratio, G is the shear modulus, E is modulus of the fiber, E is modulus of the matrix, V is the fiber volume fraction, V is the matrix volume fraction, f and m are the Poisson’s ratios of the fiber and matrix respectively. The stress and strain relationship of a typical orthotropic kth (k = 1,…,n) layer in the local co-ordinate system is σ σ σ
Von Karman (1910) linear strain displacement relation is in the following form. ε ε γ
Figure 3. A lamina with global and material coordinate systems From Figure 3, it can be seen that the laminate is made of several orthotropic layers, with their material axes oriented arbitrarily with respect to the laminate coordinates, the constitutive equations of each layer given in Eq. (2) must be transformed to the laminate coordinates (global-x,y,z). For this aim, transformed stiffness matrix Q is formulated as Q
0 0 sin 2θ sin θ cos θ 0 0 sin 2θ sin θ cos θ 0 sin θ cos θ sin θ cos θ cos θ sin θ 0 cos θ sin θ 0 0 0 sin θ cos θ 0 0 0 By introducing Eq. (5) into Eq. (2), the stress-strain relationship becomes, ε Q Q Q γ τ Q Q ε Q Q Q γ τ Q Q γ Q Q Q
2.1. Zigzag Effect
Transverse discontinuous mechanical properties cause displacement fields u = ( , , in the thickness direction which can exhibit a rapid change of their slopes in correspondence to each layer interface. Figure 4 shows how the scenarios of displacement u distributions in a laminated plate could appear in the exact solution or experiments. This displacement distribution is known as zig-zag effect in the literature defined by Carrera (1997).
Figure 4. Zig-zag effect in the laminated composite plates In all equivalent single layer laminate theories assume that the displacements are continuous functions of the thickness coordinate contrary to the actual zig-zag form of the laminated plates. Therefore; a possible, natural manner of including the zig-zag effect could be implemented by applying CLT, FSDT of TSDT at a layer level, that is, each layer is seen as an independent plate which is known as layerwise theory in the literature.
3. Layerwise Finite Element Formulation
When the main emphasis of the analysis is to determine the overall global response of the laminated plates, for example, gross deflections, critical buckling loads, such global behavior can often be accurately determined using equivalent single layer laminate theories (ESL) especially for very thin laminates. As laminated composite materials undergo the transition from secondary structural components to primary critical structural components which are thicker, then the simple ESL theories are incapable of accurately determining the 3-D stress field at the ply level as a result of zig-zag effect explained in the previous section. As can be seen in Fig. 5, a 3-D layerwise serendipity finite element which has four nodes (cubic) per side in plan and three nodes(quadratic) in the thickness direction is developed in this study. The layerwise finite element has two in-plane degrees of freedom (u , v (I=1,..,12) (j=u,o,a) per plan and thickness nodes and a one (w) out of plane degrees of freedom which = 0. means Figure 6 shows that a three layered layerwise finite element has a total of 180 nodal displacements per element.
Figure 6. Nodal degrees of freedom in 3 layered layerwise finite element Lagrange cubic interpolation functions which are given in Reddy (2005) are used in plan as follows: N N N
s 1 r r 1 s s 9 r 10 ; a 1,4,9,12 9 1 9r r 1 s s 1 r ; a 2,3,10,11 32 1 9s s 1 r r 1 s ; a 5,6,7,8
where r and s are local coordinates in plan. By separating r and s functions from each other, Eq. (8) becomes: N r, s
where T , L , S (I=1,2) functions are defined as 1 1 1 9r 1 r ; T r 1 9r 1 r 16 16 1 1 1 9s 1 s ; T s 1 9s 1 s 16 16 9 9 1 r 1 3r ; S r 1 r 1 3r 16 16 9 9 1 s 1 3s ; S s 1 s 1 3s 16 16 1
Quadratic interpolation functions in the thickness direction are obtained as follows:
where ζ is the local coordinate in the thickness direction. Displacement fields (u, v, w) could be obtained by using Eqs. (9-11). For a displacement state (i = 1, … ,7) where only node 1 displacements exist in a one layered layerwise finite element, the displacement fields could be obtained by using interpolation functions as follows : uü vü u v u v w
i=1 state i=2 state i=3 state i=4 state i=5 state i=6 state i=7 state
N r, s ψü ζ N r, s ψü ζ N r, s ψ ζ N r, s ψ ζ N r, s ψ ζ N r, s ψ ζ w N r, s
3.1. First Order Stiffness Matrix of the Layerwise Element
According to the principle of virtual displacement (PVD), stiffness matrix terms k could be obtained by the work done by the stresses of the ith displacement state with the strains of the jth state such as: / /
By substituting Eqs. (4 – 7) in Eq. (12) and for i = j = 1 state, Eq. (12) yields, k
Displacement fields in Eq. (13) could be rearranged in form of interpolation functions by considering Eq. (11) for i = 1 state yields, k
As can be seen from Eq. (14), integrand contains derivatives with respect to the global coordinates (x, y, z) however; interpolation (shape) functions are written in local coordinate system (r, s, ζ). The relation N
where J is called the Jacobian matrix of the transformation : x y N N N x y … N N N x
The element area dA = dx dy in Eq. (14) is also transformed to dx dy
∂N r, s ∂r ∂N r, s ∂N r, s ∂N r, s J J J ψ ζ ψü ζ ∂s ∂r ∂s ü ∂N r, s ∂N r, s ∂N r, s ∂N r, s J J ψü ζ J J ψü ζ ∂r ∂s ∂r ∂s ∂N r, s ∂N r, s ∂N r, s ∂N r, s J J ψü ζ J J ψü ζ ∂r ∂s ∂r ∂s ∂N r, s ∂N r, s ∂N r, s ∂N r, s J J ψü ζ J J ψü ζ ∂r ∂s ∂r ∂s Q
Integration through thickness (z) direction is calculated using Table 1. Table 1. X-Y ψü
The area integration of the stiffness matrix in Eq. (18) is evaluated by numerical integration using the Gauss quadrature formula with 4x4 sampling points.
4.1. Square Plates
As can be seen in Fig. 7, simply supported [0/90/0] and [0/90/90/0] layered cross ply square π π plates with length L and thickness h subjected to doubly sinusoidal loading q q Sin L Sin L are studied. The material properties are: E ⁄E 25.0, G G 0.5E , G 0.2E , υ υ
0.25. The whole plate is modeled with 6 x 6 meshes.
Linear Static Analysis of Laminated Composite Plates … Calculations w
at point (0,L/2,0) for the 3-layer plate and the 4-layer plate.
The results are compared with the exact solutions from Pagano and Hatfield (1972) and Pagano (1969) and those obtained from layerwise finite element (Genson) with 6x6 mesh in Table 2 for the 3 layered square plate and in Table 3 for the 4 layered square plate.
Figure 7.Simply supported cross ply 3 and 4 -layer square plates subjected to doubly sinusoidal loading Table 2. Normalized central deflection and stresses of a simply supported 3 -layer square plate under doubly sinusoidal loading FSDT L/h
3-D elasticity solution of Pagano (1969). The second line corresponds to stresses computed from 3-D elasticity equilibrium equations.
Table 3. Normalized central deflection and stresses of a simply supported 4 layer square plate under sinusoidal loading L/h
4.2. Circular Plates
As can be seen in Figure 8, isotropic and orthotropic single layer circular plates with radius r = 6 [m] subjected to uniform pressure are studied. Plates are analyzed using both simply supported E 30x10 psi , ν and clamped boundary conditions. Isotropic material properties: E E 0.25, G G G 12x10 psi Orthotropic material properties: E ν
30x10 psi , E 0.75x10 psi , ν 0.25, ν 6.25x10 , G G 0.45x10 psi , G 0.375x10 psi . The whole plate is modeled with 20 elements. The material fibers angle θ 0 with respect to the global x axis is assumed. The results of isotrop one layered circular plates for clamped and simply supported boundary conditions under uniform pressure are compared with the exact solutions from Ugural(1981) and those obtained from layerwise finite element (Genson) in Figs. 9-10.
3-D elasticity solution of Pagano and Hatfield (1972). Equilibrium -derived stresses.
Figure 9. Transverse displacements of single layer isotropic circular plate under uniform pressure (r/h = 10)
Figure 10. Transverse displacements of single layer isotropic circular plate under uniform pressure (r/h = 50)
Table 4. Normalized central deflection of one layered clamped circular plates
Normalized center deflection, w Number of elements 12 0.1159 0.1242 0.1373
The results of orthotropic single layer clamped circular plates under uniform pressure for various plate aspect ratios (r/h) are compared with the finite element solutions from Wilt et al. (1990) in Table 4. Note that the quantities in the following tables are normalized center deflections, w , i.e. w wD/q a where D=3(D D 2 D 2D and D , D , D and D are bending stiffnesses. The results of orthotropic single layer clamped circular plates for various plate aspect ratios (r/h) under uniform pressure are also compared with closed form solution by Leiknithski (1968) and Murthy and Lakshminarayana (1984) in Figs. 11-12.
Figure 11. Transverse displacements of single layer orthotropic circular plate under uniform pressure (r/h = 25)
Figure 12. Transverse displacements of single layer orthotropic circular plate under uniform pressure (r/h = 50)
5. Conclusions
A 3-D layerwise serendipity finite element which has four nodes (cubic) per side in plan and three nodes(quadratic) in the thickness direction is developed in this study to analyze the linear static response of the laminated plates. The elements are simple, shear locking free and fast convergent. Numerical results show that the present elements are accurate and efficient compared
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HAŞİM, K.A.; SAYGUN, A.I. Linear Static Analysis of Laminated Composite Plates with Layerwise Finite Element. Sigma Journal of Engineering and Natural Sciences 2014, Vol. 32, pp. 297-309. https://doi.org/10.62051/ytu.sigma-journal-of-engineering-and-natural-sciences-linear-static-analysis-of-laminated-composite-plates-with-layerwise-finite-eleme