Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity

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Abstract

This work presents a new multi-layer laminated composite structure model to predict the mechanical behaviour of multi-layered laminated composite structures. As a case study, the mechanical behaviour of laminated composite beam (90°/0°/0°/90°) is examined from both a static and dynamic point of view. The results are compared with the model “Sinus” and finite element method studied by Abou Harb. Results show that this new model is more precise than older ones as compared to the results by the finite element method (Abaqus). To introduce continuity on the interfaces of each layer, the kinematics defined by Ossadzow was used. The equilibrium equations and natural boundary conditions are derived by the principle of virtual power. To validate the new proposed model, different cases in bending, buckling and free vibration have been considered.

Introduction

Now composite materials are used in nearly all phases of structure work, from space craft to marine vessels, from bridges and domes on civic buildings to sporting goods. The significant increase in the use of composite materials structure calls for the development of rigorous mathematical methods capable of modelling, designing and optimising of the composite under any given set of conditions.

One of the major challenges in computational structural mechanics is the development of the advanced models and numerical techniques in order to provide efficient tools exhibiting good interior and edge solutions.

In this paper we are introducing an “exponential function” as a shear stress function; the exponential functions are found to be very much richer than trigonometric Sine and Cosine functions in their Fourier development series. According to the definition of the transverse shear stress function, the existing laminated composite beam is divided into two broad categories; firstly, the global approximation models and secondly the discrete layer approximation models. The equivalent single-layer laminate theories are those in which a heterogeneous laminated plate is treated as a statically equivalent, single layer having a complex constitutive behaviour, reducing the 3D continuum problem to 2D problem.

The equivalent single layer models are:

  • Kirchhoff (1850) and Love (1934) present their theory (or classical theory) in which deformation due to transverse shear is neglected, implies that the normal to the mid-plane remains straight and normal at mid-surface after deformation. This theory can be used for thin beams;

  • Reissner (1945) and Mindlin (1951) present their theory (or first order theory). That the first order deformation theory extends the kinematics of the classical laminated plate theory by including a gross transverse shear deformation in its kinematic assumption, the transverse shear strain remain constant with respect to the thickness coordinate, implies that the normal to the mid-plane remains straight but not normal to mid-surface after deformation due to shear effect. The first order theory requires shear correction factors, which are difficult to determine for arbitrary laminated composite plate;

  • and the higher order models are based on the hypothesis of non-linear stress variation through thickness (Reddy, 1984; Touratier, 1991). These models are able to represent the section warping in the deformed configuration.


However, these theories do not satisfy the continuity conditions of transverse shear stress at layer interfaces. Although the discrete layer approximation theories are accurate, but they are rather complex in solving problems because the order of their governing equations depends on the number of layers.

Di Sciuva, 1987, Di Sciuva, 1993 and then Touratier, 1991, Touratier, 1992 proposed simplified discrete layer model with only five variational unknowns (two membrane displacements, a transverse displacement and two rotations), allowing the section to be represented wrapping in the deformed configuration for the Touratier (1992) model. Nevertheless, in these two cases the compatibility conditions, both layer interfaces and the boundaries, cannot be satisfied. From Touratier’s work, (Beakou, 1991) and (Idlbi, 1995) proposed, respectively, shell and plate models which satisfy both the stress continuity at interfaces and the zero stress conditions at the boundaries.

Finally, He (1994) introduced the Heaviside step function which enables automatic satisfaction of the displacement continuity at interfaces between different layers. The new discrete layer model presented comes from the work of Di Sciuva (1993), He (1994) and Ossadzow et al. (1995), the displacement field is:U1(x1,x3,t)=u10(x1,t)−x3w,1(x1,t)+h1(x31(x1,t)U2=0U3(x1,t)=w(x1,t)with transverse shear function:h1(x3)=g(x3)+∑m=1N−1λ1(m)−x32+f(x3)2+(x3−x3(m))H(x3−x3(m))where, H(x3x3(m)), the Heaviside Step function is defined as:H(x3−x3(m))=1forx3⩾x3(m)0forx3<x3(m)and f(x3) is the shear refinement function, and g(x3) is the membrane refinement function, and the λI(m) are coefficients of the continuity.

New multi-layered laminated composite structures model (“KAM”):

In this work a new multi-layered laminated composite structure model is presented by using exponential function as:f(z)=ze−2(z/h)2g(z)=−ze−2(z/h)2for a multi-layered beam Ω, of uniform thickness ‘h’ and Ω is referred to the co-ordinate system R=(0/x1,x2,x3=z) with z being normal to the plate’s mid-surface Σ, Γ is the frontier of Ω. Then, the domain Ω is such that:Ω⊂R3,Ω=Σ×h2,h2;−h2⩽z⩽h2/M(x1,x2,z)∈Ω,M0(x1,x2,0)∈Σ,φ≫Max(z)where φ is the diameter of the Ω. and the closed domain Ω̈ is set such that:Ω̈=Ω∪Γ/Γ=Γedge∪Γz=±h/2From the beginning our objective was so clear, to find out a transverse shear stress function f(z), which gives the mechanical behaviour of the composite laminated structures as much close as possible to the exact 3D solution by Pagano (1970) or finite element analysis in 2D (stress, strain plane), and better representation of the transverse shear stress in the thickness of the laminated structure. Since different higher order polynomial and trigonometric function already has been tried which are as follow;

Ambartsumian (1958) where;f(z)=z2h24z23

Kaczkowski (1968), Panc (1975) and Reissner (1975) where;f(z)=54z1−4z23h2

Levinson (1980), Murthy (1981) and Reddy (1984) where;f(z)=z1−4z23h2and finally Touratier (1991), where;f(z)=hπsinπzhSo, we took a start with an exponential function, because an exponential function has all even and odd power in its expansion unlike Sine function, which have only odd power. So an exponential function is much richer than a Sine function. If we take a look on the expansions of different transverse shear stress functions as;

Reddy (1984):f(z)=z1−4z23h2=z−1.33z3h2

Touratier (1991):f(z)=hπsinπzh=z−1.645z3h2+0.812z5h4−0.191z7h6+0.0261z9h8

Present model:f(z)=ze−2(z/h)2=z−2z3h2+2z5h4−1.333z7h6+0.666z9h8As it is clear from expansions of the transverse shear stress functions, that the coefficient of successive terms in ‘Sine’ functions are decreasing more rapidly than present exponential function which are the main responsible to gives different mechanical behaviour of laminated structures.

For the transverse shear stress behaviour, it is very important that the first derivative of the transverse shear stress function must give a parabolic response in the thickness direction of the laminate and satisfy the boundary conditions.

Section snippets

Governing equations

From the virtual power principle, the equations of motion and the natural boundary conditions can be obtained. The calculations are made in small perturbations. According to the principle of virtual power:P(a)*=P(i)*+P(e)*But the virtual power of the acceleration quantities are:P(a)*=∫ΩρU*TÜdΩwe suppose:Iw=∫−h/2h/2ρdx3,Iuw=−∫−h/2h/2ρx3dx3Iw=∫−h/2h/2ρx32dx3,I=∫−h/2h/2ρh1(x3)dx3Iω=∫−h/2h/2ρh12(x3)dx3,Iωw=−∫−h/2h/2ρx3h1(x3)dx3so, Eq. (6) becomes (see Appendix A for the mathematical detail):P(

Bending analysis

The static bending analysis is studied, so the virtual power of acceleration quantities are cancelled. Three different bending analyses have been developed for three different specific boundary conditions. For the simply supported conditions, the unknown variables are deduced directly by the equation of motions. For clamped conditions, kinematic boundary conditions are used and, finally, in a free edge case, natural boundary conditions are employed.

The beam studied has a length of L=6.35 m, a

Conclusion

The new multi-layered structure exponential model satisfies exactly and automatically the continuity condition of displacements and transverse shear stresses at interfaces, as well as the boundary conditions for a laminated composite with the help of the Heaviside step function (Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7). For the new proposed model the results are compared with the existing model (Karama et al., 1993) like Sine model proposed by Touratier (1991) and by the finite

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