Analysis of thick skew laminate with elliptical cutout subjected to non-linear temperature distribution: major axis of ellipse parallel to inclined side.

Kumar, M.S.R. Niranjan ; Sarcar, M.M.M. ; Murthy, V. Bala Krishna 等

Nomenclature

[E.sub.1] = Young's modulus of the lamina in the fibre direction [E.sub.2] = [E.sub.3] = Young's modulus of the lamina in the transverse direction of the fibre [G.sub.12] = [G.sub.13] = Shear modulus in the longitudinal plane of the fibre [G.sub.23] = Shear modulus in the transverse plane of the fibre [v.sub.12] = [v.sub.13] = Poisons ratio in the longitudinal plane of the fibre [v.sub.23] = Poisons ratio in the transverse plane of the fibre [[alpha].sub.1] = Coefficient of thermal expansion in the fiber direction [[alpha].sub.2] = [[alpha].sub.3] = Coefficient of thermal expansion in the transverse direction of the fiber EL = Exact elasticity solution FE = Finite element solution

Normalized [sigma] [sigma]/[p.sub.0][s.sup.s], Normalized [tau]/[p.sub.0]s, Normalized w = 100[E.sub.2]w/[p.sub.0][hs.sup.4]

s = Length of the plate (l) / thickness of the plate (h)

[p.sub.0] = The maximum intensity of sinusoidal load

a = Length and width of the square plate

[p.sub.3] = Major axis of the ellipse parallel to inclined side of the skew plate. 1/2 and 1/3 are the normalized positions along the thickness direction

(Normalized z = 2z / h, z coordinate measured from middle plane of the plate and h=total thickness of the plate)

Introduction

The increasing use of fibre reinforced laminates in space vehicles, aircrafts, automobiles, ships and chemical vessels have necessitated the rational analysis of structures for their mechanical response. In addition, the anisotropy and non-homogeneity and larger ratio of longitudinal to transverse modulii of these new materials demand improvement in the existing analytical tools. As a result, the analysis of laminated composite structures has attracted many research workers, and has been considerably improved to achieve realistic results. In the design of modern high-speed aircraft and missile structures, swept wing and tail surfaces are extensively employed. Moreover some of the structural elements are provided with cutouts of different shapes to meet the functional requirements like (i) for the passage of various cables, (ii) for undertaking maintenance work and (iii) for fitting auxiliary equipment. Depending upon the nature of application, these structural elements are acted upon by mechanical and thermal loads of varied nature. Usually, the anisotropy in laminated composite structures causes complicated responses under different loading conditions by creating complex couplings between extensions, bending, and shear deformation modes. To capture the full mechanical behavior, it must be described by three dimensional elasticity theories.

In solving the three-dimensional elasticity equations of rectangular plates, quite a number of solution approaches have been proposed. Srinivas and Rao [1] and Srinivas et al. [2] presented a set of complete analytical analyses on bending, buckling and free vibration of plates with both isotropic and orthotropic materials. Zhang and Zhang [3] presented a new concise procedure for obtaining the static exact solution of composite laminates with piezo-thermo-elastic layers under cylindrical bending using the basic coupled thermo-electro-elastic differential equations. Setoodeh and Karami [4] employed a three-dimensional elasticity based layer-wise finite element method (FEM) to study the static, free vibration and buckling responses of general laminated thick composite plates. Pagano et al.[5] has given exact solutions for the deflections and stresses of a cross- ply laminated rectangular composites without holes using elasticity theory. Kong and Cheung[6] proposed a displacement-based, three-dimensional finite element scheme for analyzing thick laminated plates by treating the plate as a three-dimensional inhomogeneous anisotropic elastic body. Prasad and Shuart [7] presented a closed form solution for the moment distributions around holes in symmetric laminates subjected to bending moments. Ukadgaonker et al.[8] gave a general solution for bending of symmetric laminates with holes. Morley et al.[9] developed an elementary bending theory for the small displacements of initially flat isotropic skew plates without hole. Karami et al.[10] has applied Differential Quadrature Method (DQM) for static, free vibration, and stability analysis of skewed and trapezoidal composite thin plates without hole. From the review of available literature it is observed that the static analysis of skew plates with cutouts using elasticity theory has not been studied. The behavior of a laminate with skew edges and having various types of cutouts is different from the one without skew edges and/or cut outs. So it is necessary to analyse this kind of problem using elasticity theory based finite element method to evaluate for the most accurate behaviour of thick laminated skew plates with cutouts.

Skew Laminate

The term 'skew' in skew laminate refers to oblique, swept or parallelogram. In case of skew plate the angle between the adjacent sides of the plate is not equal to 900. If opposite sides of the plate are parallel, it becomes a parallelogram and when their lengths are equal, the plate is called a rhombic plate. In the present analysis a rhombic laminated plate is considered by varying the skew angle from 00 to 500 as shown in Fig.1.

[FIGURE 1 OMITTED]

Problem Statement

The research problem deals with the thermoelastic analysis of thick skew laminated plate with elliptical cutout by elasticity theory based on finite element method.

Problem Modeling

Geometric modeling

The Figure.1 shows the in-plane dimensions of the laminate considered for the present analysis. The dimensions for 'l' and 'b' are taken as 20mm. d is the length of the minor axis of the ellipse. Major axis of the ellipse is taken as twice the length of the minor axis.

The value of d is determined from the ratio of d/1 which is varied from 0.1 to 0.4, and the skew angle [alpha] is varied from [0.sup.0] to [50.sup.0], the thickness of the plate is fixed from the length to thickness ratio l/h (s =10). The individual layers are arranged so that the total thickness of the layers oriented in x- direction ([theta] = [0.sup.0]) is equal to the total thickness of the layers oriented in y- direction ([theta] = [90.sup.0]).

Finite Element Modeling

The finite element mesh is generated using a three dimensional brick elements 'SOLID90' and 'SOLID 95' of ANSYS [11]. The 20 node thermal element is applicable to a steady state or transient thermal analysis. The elements consists of has 20 nodes and temperature degree of freedom for 'SOLID90'and X, Y, and Z directional displacement for 'SOLID 95'. 'SOLID90' and 'SOLID 95' have compatibility to transfer the temperatures from thermal analysis to structural analysis This element (Fig.2) is a structural solid element designed based on three dimensional elasticity theory and is used to model thick orthotropic solids.

[FIGURE 2 OMITTED]

Boundary conditions

Thermal

A temperature of [100.sup.0] C on the top face and [25.sup.0] C on the bottom and side faces is applied. The surface of the hole is subjected to convection with film coefficient h = 5 and bulk temperature [25.sup.0] C.

Structural

All the edges of the skew plate are clamped i.e. all the three degrees of freedom (Displacements in global x-, y- and z- directions) of the nodes attached to the side faces of the plate are constrained.

Loading

The output from the thermal analysis is applied as thermal loading.

Material Properties (Graphite-Epoxy)

[K.sup.L] = 36.42 W/m K [E.sup.1] = 172.72 GPa, [G.sup.12] = [G.sup.13] = 3.45 GPa, [[alpha].sub.1] = 0.57 x [10.sup.-6] / [sup.0]C

[K.sub.T] = 0.96 W/m K [E.sub.2] = [E.sub.3] = 6.909 GPa [G.sub.23] = 1.38 GPa, [[alpha].sub.2] = [[alpha].sub.3] = 35.6 x [10.sup.-6] / [sup.0]C

[v.sub.12] = [v.sub.13] = [v.sub.23] = 0.25

Validity of the Present Analysis

To validate the finite element results, a square plate with simply supported edges and subjected to a sinusoidal load of p = p0 sin ([pi]x/a) sin ([pi]y/b), where a and b are the length and width of the plate, is modeled with SOLID95 element. The results obtained from this model are compared with the exact elasticity solution [5] for various lengths to thickness ratios of the plate (Table 1). It is observed that the finite element results are in close agreement with the exact elasticity solution.

Table 1. Presents the results of a square laminate ('a' = 'b'). The location for maximum [[sigma].sub.x] i.e. (a/2, a/2, [+ or -] 1/2) is at the centre and at top and bottom faces of the laminate. Maximum value of [[sigma].sub.y] is observed at the centre and at the interface of outer and its adjacent layers (a/2, a/2, [+ or -]1/3) of the laminate. The value of the shear stress [[tau].sub.yz] is taken at the mid point of one of the vertical sides and at the neutral surface of the laminate (0, a/2, 0). The value of the shear stress [[tau].sub.zx] is taken at the mid point of one of the horizontal sides and at the neutral surface of the laminate (a/2,0,0) and the transverse deflection is obtained at geometric centre of the laminate (a/2,a/2,0).

In the present work the transverse deflection and stresses (including the inter-laminar stresses at the free edge of the elliptical cutout) of a clamped skew laminated plate with elliptical cutout at the centre of the plate and subjected to a non-linearly varying temperature loading is evaluated by varying the size of the elliptical cutout and skew angle.

Results and Discussion

Numerical results are obtained for temperature loading as mentioned above. Variation of the stresses and deflection with respect to the skew angle ([alpha]) for different d/l ratio's is shown in Figs. 3-9. The following observations are made.

Effect of Skew Angle

The in-plane normal stress, [[sigma].sub.x] (Fig.) decreases with increase in skew angle. The increase in skew angle increases the length of the longer diagonal and decreases the length of the shorter diagonal of the skew plate. The first factor (increase in the length of the longer diagonal) increases the flexibility of the plate where as the second factor (decrease in the length of the shorter diagonal) increases the stiffness of the plate. The reduction in the stress [[sigma].sub.x] is due to the domination of stiffness effect. (Fig). The in-plane normal stress, [[sigma].sub.y] increases with increase in skew angle [alpha] for d/l = 0.1 and 0.2. The increase in the stress [[sigma].sub.y] is due to the domination of flexibility effect. For d/l = 0.3 and 0.4 this stress increases up to [alpha] = [40.sup.0] and then decreases. Here the flexibility effect is dominating up to [alpha] = [40.sup.0], from [alpha] = [40.sup.0] onwards the stiffness factor is dominating (Fig).The in-plane shear stress, [[tau].sub.xy] increases with increase in skew angle for d/l = 0.1. For d/l= 0.2 this stress increases up to [alpha] = [40.sup.0] and then decreases. For d/l= 0.3 and 0.4 this stress increases up to [alpha] = 300 and then decreases (Fig).

The Inter laminar normal stress, [[sigma].sub.z] at the free edge of the cutout decreases with the increase in the skew angle for all d/l ratios except for d/l = 0.4. For d/l = 0.4 this stress decreases up to [alpha] = [40.sup.0] and then increases (Fig).The Inter laminar shear stress, [[tau].sub.yz] increases with the increase in skew angle for all d/l ratios (Fig).The Inter laminar shear stress, [[tau].sub.zx] decreases with the increase in skew angle for d/l = 0.1. For the remaining d/l ratios there is no significant variation in this stress with respect to [alpha] (Fig).

There is no significant variation in transverse deflection 'w' with increase in skew angle [alpha] (Fig.).

Effect of d/l Ratio

When the size of the ellipse increases, the area of the hole boundary increases providing more scope for free expansion of the plate. Due to this factor, the stresses will decrease. At the same time, the resisting volume of the material decreases and as a result the induced stresses will increase. The resultant effect of these factors is discussed below.

The in-plane normal stress [[sigma].sub.x] decreases with increase in d/l ratio up to d/l = 0.2 and then increases for skew angle [alpha] = [0.sup.0]. For all other values of skew angle this stress increases up to d/l = 0.3 and then decreases. The second factor (decrease in resisting volume) is dominating up to d/l = 0.3 and later the first factor (increase in area of the free edges) is dominating leading to the decrease in the stresses (Fig.3). The in-plane normal stress [[sigma].sup.y] increases with increase in d/l ratio up to d/l = 0.2 and then decreases for all values of skew angle [alpha] except for [alpha] = [50.sup.0]. For [alpha] = [50.sup.0] this stress decreases with increase in d/l ratio (Fig. 4).The in-plane shear stress [[tau].sup.xy] increases up to d/l = 0.2 and then decreases for all values of skew angle [alpha] (Fig. 5).

The inter laminar normal stress [[sigma].sub.z] and shear stress [[tau].sub.yz] at the free edge of the cutout increase with increase in d/l ratio for all values of skew angle [alpha]. The forces causing the interlaminar stresses form in couples to balance the forces for equilibrium. When the size of the cutout increases, the moment arm of these forces decreases and this may be the reason for increase in interlaminar stresses (Figs. 6and 7). There is no significant variation of inter laminar shear stress [[tau].sub.zx] with respect to d/l ratio for all skew angles.

The transverse deflection 'w' decreases with increase in d/l ratio for all values of [alpha] (Fig. 9).

Conclusions

Thermoelastic analysis of a thick laminated composite skew plate with an elliptical cutout at the centre of the plate has been carried out in the present work. The transverse deflection, maximum in plane stresses and maximum interlaminar stresses at the free edge of the cutout have been evaluated using three-dimensional theory of elasticity based finite element analysis. The results obtained for non-linearly varying temperature loading are analyzed for the variation of skew angle of the plate, size of the ellipse. The magnitude of the in-plane major normal stress [[sigma].sub.x] due to temperature loading is greatly affected by the skew angle variation and their magnitude is observed to be minimum at higher value of the skew angle. The in-plane stresses [[sigma].sub.y] and [[tau].sub.xy] and the transverse deflection 'w' are observed to be minimum at higher d/l ratios. The magnitudes of inter laminar stresses are observed to be minimum at lower d/l ratio.

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[FIGURE 5 OMITTED]

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[FIGURE 9 OMITTED]

References

[1] Srinivas, S. and Rao, A.K. "Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates". Int. J. Solids Structures, vol.6, 1970, pp.1463-81.

[2] Srinivas, S., Rao, C.V. and Rao, A.K. "An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates". J. Sound Vibrations, vol.12, 1970, pp.187-99.

[3] Zhang, C., Di, S. and Zhang, N. "A new procedure for static analysis of thermo-electric laminated composite plates under cylindrical bending". Composite Structures, vol.56, 2002, pp.131-40.

[4] Setoodeh, A. R. and Karami, G. "Static, free vibration and buckling analysis of anisotropic thick laminated composite plates on distributed and point elastic supports using a 3-D layer-wise FEM". Engineering Structures, vol. 26, 2004, pp. 211-20.

[5] Pagano, N.J. and Hatfield, S.J. "Elastic behavior of multilayered bidirectional composites". AIAA Journal, vol.10, 1972, pp.931 - 33.

[6] Kong J. and Cheung, Y. K. "Three-dimensional finite element analysis of thick laminated plates". Computers and Structures, vol. 57, 1995, pp.1051-62.

[7] Prasad, C.B. and Shuart, M.J. "Moment distributions around holes in symmetric composite laminates subjected to bending moments". AIAA Journal, vol.28, 1990, pp.877-82.

[8] Ukadgaonker, V.G. and Rao, D.K.N. "A general solution for moments around holes in symmetric laminates". Comp. Structures, vol.49, 2000, pp.41-54.

[9] Morley, L. S. D. "Skew Plates and Structures. Division I: Solid and Structural Mechanics. Int. Series of Monographs on Aeronautics and Astronautics", Pergamon Press, NewYork, 1963.

[10] Karami, G, Shahpari, S.A and Malekzadeh, P. "DQM analysis of skewed and trapezoidal laminated plates". Comp. Structures., vol.59, 2003, pp.393-402.

[11] ANSYS Reference Manual 2006.

MSR Niranjan Kumar (a1), MMM Sarcar (b), V. Bala Krishna Murthy (c) and K. Mohana Rao (d)

(a) Production Engineering Department, V.R. Siddhartha Engineering College, Vijayawada-520 007, India.

(b) Mech. Engg. Dept., College of Engineering, Andhra University, Visakhapatnam, India.

(c) Mech. Engg. Dept., P.V.P. Siddhartha Institute of Technology, Vijayawada-520 007, India.

(d) Principal, C. R. Reddy College of Engineering, Eluru, A.P., India.

(1) Corresponding Author: Email: m_niranjankumar@rediffmail.com

Table 1: Comparison of present work with exact elasticity theory [5]

                Normalized                Normalized
              [[sigma].sub.x]           [[sigma].sub.y]
S = l/h   (a/2,a/2, [+ or -] 1/2)   (a/2,a/2, [+ or -] 1/3)

10               EL 0.545                  EL 0.430
                   -0.545                    -0.432
                 FE 0.537                  FE 0.431
                   -0.536                    -0.431
20               EL 0.539                  EL 0.380
                   -0.539                    -0.380
                 FE 0.534                  FE 0.377
                   -0.535                    -0.378

          Normalized       Normalized       Normalized
          [[tau].sub.yz]   [[tau].sub.zx]       w
S = l/h   (0,a/2,0)        (a/2,0,0)        (a/2,a/2,0)

10        EL   0.223       EL   0.258       EL   0.677
          FE   0.209       FE   0.212       FE   0.692

20        EL   0.212       EL   0.268       EL   0.4938
          FE   0.218       FE   0.271       FE   0.4838