A new super convergent thin walled composite beam element for analysis of box beam structures

Abstract

In this paper, a new composite thin wall beam element of arbitrary cross-section with open or closed contour is developed. The formulation incorporates the effect of elastic coupling, restrained warping, transverse shear deformation associated with thin walled composite structures. A first order shear deformation theory is considered with the beam deformation expressed in terms of axial, spanwise and chordwise bending, corresponding shears and twist. The formulated locking free element uses higher order interpolating polynomial obtained by solving static part of the coupled governing differential equations. The formulated element has super convergent properties as it gives the exact elemental stiffness matrix. Static and free vibration analyses are performed for various beam configuration and compared with experimental and numerical results available in current literature. Good correlation is observed in all cases with extremely small system size. The formulated element is used to study the wave propagation behavior in box beams subjected to high frequency loading such as impact. Simultaneous existence of various propagating modes are graphically captured. Here the effect of transverse shear on wave propagation characteristics in axial and transverse directions are investigated for different ply layup sequences.

Introduction

Composites are being used widely as construction material in aircraft industries because of their high strength to weight ratio, increased fatigue life and improved damage tolerant nature. Thin walled structures are integral parts of an aircraft. In many structures like rotor blades, wing spars etc they can be modeled as one dimensional beam as the sectional dimensions are much small compared to the length. Several non-classical behavior are exhibited by thin walled composite structures which includes the effect of elastic coupling, transverse shear deformation and restrained torsional warping. These characteristics can be exploited to improve efficiency through proper modeling.

The influence of transverse shear deformation cannot be neglected even in comparatively slender composite beams because of low shear modulus to direct modulus ratio (Davalos et al., 1994; Kant and Gupta, 1988). The effects are more significant for high frequency responses, where Euler Bernoulli beam theory (EBT) gives exorbitantly high wave speeds. In thin walled composite beam, the end restrains causes non uniform out-of-plane torsional warping as opposed to Saint Venant’s assumptions. This effect is predominant in open section beam and in such cases Vlasov theory is normally adopted to incorporate restrained warping effect, which causes considerable change in the effective torsional stiffness.

The box beam is normally analyzed using a 1-D mathematical model, but representing 3-D motion. 1-D approximations are associated with assumption of local displacements in terms of generalized beam displacements namely extension, bending in two directions, shear in two directions and twist. A survey of existing numerical and analytical thin walled composite beam theories was done by Jung et al., 1999a, Jung et al., 1999b and Volovoi et al. (2001). A variational-asymptotic approach has been adopted by several researchers for the above modeling problem. It helps in an efficient reduction of 3-D elasticity problem to 1-D beam problem. Analytical cross-sectional models based on variational- asymptotic formulation were presented by Berdichevsky et al. (1992), Badir et al. (1993) and Volovoi and Hodges (2002). Apart from the analytical modeling of the beam cross-section, asymptotically correct finite element modeling techniques has also been developed. VABS (Variational Asymptotic Beam Section Analysis) was developed by Cesnik and Hodges (1997) which derive the cross-sectional stiffness through finite element discretization. A finite element based cross-sectional analysis using variational asymptotic method and incorporating transverse shear effect is presented by Popescu and Hodges (1999). A first order shear deformable analytical cross-sectional modeling technique was proposed by Jung et al. (2002) without neglecting in-plane bending moments.

In first order shear deformation theory (FSDT) and higher order shear deformation theories (HSDT), finite element formulation requires C^o continuous elements for independent interpolation of transverse displacement and slope. Shear constraints are always associated with these C^o elements. When thin beams are discretized using such elements, they do not yield zero shear strains. This is defined as the shear locking problem. A shear locked element causes considerable under estimation of deformation. With the above inconsistent formulation, the problem of shear locking can be eliminated using selective or reduced integration (Averill and Reddy, 1990).

All constraint media problems, like shear locking problem leads to two sets of stiffness matrix. One from unconstrained strain field and the other from the constrained strain field. For shear deformable elements, the bending stiffness matrix [K_B] comes from the unconstrained strain field, while the shear stiffness matrix [K_S] comes from the constrained strain field. Matrix [K_S] is also called the Penalty matrix. The problem is thus reduced to solving the matrix equation

$$\text{[[K}{}_{\mathrm{<mtext>B</mtext>}}\text{]+α[K}{}_{\mathrm{<mtext>S</mtext>}}\text{]]{u}={f}}$$

where {u} and {f} are the nodal displacements and forces and α is the penalty parameter. In the penalty limit as the beam becomes thin, α value becomes very large and for accurate solution, [K_S] requires to be singular. One way of eliminating the problem is to perform reduced integration on the penalty matrix [K_S] to make it rank deficient. This ensures that [K_S] is singular and proper solutions can be obtained. Hence, numerical integration plays a crucial role in getting proper solutions in the constrained media problems.

Consistent finite element can be alternatively formulated using interpolating polynomials that are exact solutions to the governing equations. This approach was implemented to obtain shape functions for an isotropic three-dimensional Timoshenko beam (Bazoune et al., 2003) and in deriving exact stiffness matrix in higher order isotropic beam (Eisenberger, 2003), in FSDT asymmetric composite beams (Chakraborty et al., 2001), for higher order isotropic rod (Gopalakrishnan, 2000), for first order shear deformable isotropic beams (Friedman and Kosmatka, 1993), for first and higher order shear deformable isotropic beams (Khedir and Reddy, 1997; Reddy, 1997). In these elements, some constants of the interpolating polynomials are dependent on material and cross-sectional properties. Here, the degrees of interpolation functions depends on the orders of governing equations and as the beam becomes thin, all these material dependent constants transform themselves in such a manner that elementary solutions are recovered. The advantage is that the user need not know whether the shear deformation is significant. With similar physical implication, interpolation functions in terms of series were used in (Eisenberger, 1994) that reduced to continuum solution when higher number terms were considered in the solution. In this paper, this approach is adopted to derive the exact stiffness matrix of a thin walled composite beam.

In the present paper a generic composite thin walled beam element having arbitrary cross-section with open and closed contour is developed. The element uses higher order interpolating polynomials that are derived by solving the static homogeneous coupled governing differential equation and hence predicts the exact elemental stiffness matrix. Each node has 6 degrees of freedom including extension, two in bending in spanwise and chordwise directions, corresponding shears and twist. First order theory is used for transverse shear deformation and out-of-plane torsional warping is modeled using Vlasov theory. Higher order interpolating polynomial for twist eliminates the need of separate degree of freedom for restrained torsional warping resulting in 6 × 6 elemental stiffness matrix compared to 7 × 7 stiffness matrix required by conventional finite element.

The dynamic analysis of thin walled composite beams are generally performed by extending the cross-sectional models, particularly to study the effect of various cross-sectional parameters, ply layup sequences on the free vibration responses. Free vibration analyses of composite beams was done by Hodges et al. (1991) and Song and Librescu (1997) for non rotating and closed cross-section rotating beams respectively. They studied the influence of ply orientation and elastic couplings on natural frequencies. Jung et al. (2001) performed dynamic analyses of rotating and non rotating beam from an analytical cross-sectional model proposed by Jung et al., 1999a, Jung et al., 1999b. Effect of wall thickness and transverse shear on natural frequencies was discussed.

The modeling approach can be briefly outlined as (1) local displacements are obtained from generalized beam displacements, which are functions of spanwise coordinate x and time t; (2) plane stress assumption is used to derive the constitutive relation; (3) strain and kinetic energies are evaluated in terms of beam displacements; (4) Hamilton’s principle is used to derive the governing differential equations; (5) The static homogeneous differential equations are solved to obtain higher order shape functions; (6) the derived shape functions results in exact elemental stiffness matrix and an approximate consistent mass matrix. The formulated element is used to study the static and free vibration behavior of various flat and box beam configurations. The beams used for numerical experiments has varied material and geometric properties exhibiting different elastic coupling.

Composite beam structures subjected to high velocity impact load vibrates at higher modes that includes various local 3-D modes, apart from the beam modes. Very few literature are available in this area though such structures may be very often subjected to highly transient loading such as tool drop and other kind of impact. These relates to wave propagation problems and can be differentiated from conventional dynamic response problem by the (1) high frequency content of loading history and (2) phase transformation during propagation. The dynamics of higher order beam structures subjected to high frequency loading or impact introduces certain effect which are absent in their elementary counterparts. These effects may produce new propagating modes. Finite element formulation for wave propagation problems require large system size to capture all the higher modes as the load has a high frequency content. Hence the element size has to be comparable to wavelengths, which are very small at high frequencies. These problems are usually solved in the frequency domain and one such method is the spectral element method (SEM). SEM has been used for wave propagation problem in isotropic Timoshenko beam (Gopalakrishnan et al., 1992) and in composite Euler–Bernoulli beam (Mahapatra et al., 2000). Essential features of beam transient dynamics can be more easily captured using SEM. However, spectral element formulation, which is based on exact solutions of the governing wave equations in the transformed frequency domain, is available only for few structural elements like rods, beams, cylinders etc. As stated earlier, impact loads with high frequency content excites many local 3-D modes, even in beam like structures. Thus the dynamic model of the beam should be capable of capturing these 3-D modes. Use of 3-D finite element for such problem is not computationally viable. Efficient cross-sectional models e.g. VABS can produce accuracy comparable to 3-D finite element (Yu et al., 2002). Such cross-sectional stiffness model can be used in beam analysis that can decrease the system size considerably. Though the present element formulation is not capable of capturing the 3-D effects, wave propagation analysis are performed to get an insight into the high frequency response of composite thin walled structures and effects of various design parameters on it. In this paper, wave propagation characteristics in longitudinal and transverse directions are studied for box beams with different ply orientation and is compared with EBT neglecting the effect of transverse shear. One of the fundamental feature associated with the mechanics of the symmetric/antisymmetric box beam is the existence of axial-shear/axial-torsion coupling in axial response and bending-torsion/ bending-shear coupling in flexural response. These are graphically captured using a high frequency modulated pulse.

This paper is organized as follows. First the governing equations for a thin walled beam of arbitrary cross-section is derived using Hamilton’s principle. The finite element formulation is given next followed by numerical experiments involving static, free vibration and wave propagation analysis. The numerical results are compared with the results available in the existing literature. The paper ends with some important conclusions and future scope of further studies in making faster and cost effective finite element analysis.