Open Section Beams Research Articles

Coupled deflection and rotation of anisotropic open-section composite stiffeners

Thin-walled open-section beams constructed of panels of laminated composite materials and subjected to transverse loading are considered in this paper. Such beams are used as stiffeners in stiffened-panel construction for aerospace vehicles and in composite structural frames for rotorcraft vehicles. Beams constructed of panels having generally anisotropic mechanical properties are considered. Due to anisotropically induced normal-shear coupling effects these open-section beams will rotate and deflect laterally out-of-plane under the action of in-plane transverse loads in addition to their expected transverse deflection. A one-dimensional modified beam theory is used to describe the coupled deflection and rotation of these beams. The theory, which is intended for preliminary design studies, can be used to obtain inexpensive 'hand-calculation' predictions of the behavior of anisotropic thin-walled beams. .Predictions of the theory are compared with predictions of a 3-D finite element analysis in which the beams are constructed of general laminated shell elements. Good agreement between the theory and the finite element results is shown.

Finite Element Analysis of Thin‐Walled Curved Beams Made of Composites

The equilibrium equation of thin‐walled curved beams of open cross‐sections, based on the principle of virtual work, is presented in variational form. Finite element formulation for static and stability analysis of thin‐walled curved beams of open cross section based on this equilibrium equation is discussed. A straight beam element employing suitable transformations is used to model the curve beams. The geometric stiffness matrix of a thin‐walled curved beam of open cross section has been derived for general loading conditions. Numerical studies have been conducted for in‐ and out‐of‐plane buckling analysis of curved beams made of isotropic and composite sections. The results of the present analysis are compared with the values presented in the literature. Laminated composites of different materials such as glass/epoxy, Kevlar/epoxy, boron/epoxy, and graphite/epoxy with single‐layer 0° composites, two‐layer 45°/−45° composites, and three‐layer 0°/45°/0° composites are considered for static and stability ...

Structural response of composite beams and blades with elastic couplings

The structural behavior of coupled, thin-walled, composite beams of open as well as closed section was analyzed using Vlasov theory and then the results were validated by experiment. The analysis modeled the walls of beams as general composite laminates and accounted for the transverse shear deformation of the cross-section. The out-of-plane warping deformation of the cross-section was included implicitly in this formulation. In order to validate the analysis, graphite-epoxy beams of various cross-sections such as solid rectangular, I-section, single-cell rectangular and two-cell airfoil were fabricated and tested for their structural response under tip bending, torsional and extensional loads. Specialized bending-torsion and extension-torsion couplings were introduced in these beams using proper ply lay-ups. Good correlation between theoretical and experimental results was achieved. Transverse-shear-related couplings were found to influence the structural response of open- as well as closed-section beams. For blades with hygrothermally stable lay-ups, bending-transverse shear coupling increased the bending flexibility by about 50%. The in-plane-bending coupling stiffness [ B] of the walls of the beam generally influenced the structural response of the beams quite significantly; this effect was expecially large for I-beams. The influence of constraining the warping deformation was found to be substantial on the structural response of open-section beams as compared to closed-section beams. A 630% increase in the torsional stiffness due to constrained warping was noticed for graphite-epoxy I-beams of slenderness ratio 30. The feasibility of achieving the desired levels of bending-torsion and extension-torsion couplings in two-cell rotor blades was demonstrated.

Experimental and theoretical analysis of composite I-beams with elastic couplings

This paper presents a theoretical-cum-experimental study on the static structural response of composite I-beams with elastic couplings. A Vlasov type linear theory is developed to analyze composite open section beams made out of general composite laminates, where the transverse shear deformation of the beam cross-section is included. In order to validate this analysis, graphite-epoxy and kevlar-epoxy symmetric I-beams were fabricated using an autoclave molding technique. The beams were tested under tip bending and torsional loads, and their structural response in terms of bending slope and twist measured with a laser optical system. Good correlation between theoretical and experimental results is achieved. A 630 percent increase in the torsional stiffness due to constrained warping is noticed for graphite-epoxy beams with slenderness ratio of 30. Also extension-twist coupling 'B16' of flanges of these I-beams increases the bending-torsion coupling stiffness of

Dynamics of cracked hollow beams

The finite element method with normal quadrilateral plate elements is used for the determination of the bending compliance of orthogonal full and open hollow section beams with transverse cracks. Computed results are compared with those from theory for the full rectangular sections and with experiments performed with rectangular hollow sections. The effect of such cracks on the dynamic response of channel-section beams is investigated. Such analysis is of importance for the identification of cracks in steel structures.

Modifications to beam theory for bending and twisting of open-section composite beams

Modifications to beam theory for bending and twisting of open-section composite beams

The influence of shear deformations on the stability of thin-walled beams under non-conservative loading

This paper analyses the influence of shear deformations on the stability of thin-walled beams of open section subjected to follower-type loads. The warping of the cross-sections is assumed to depend on the shear forces as well as on both primary and secondary torsion. The internal stresses are consistently deduced from the kinematic field. An algorithm is presented which works in the framework of the least squares method making use of a vector iteration technique. An extensive comparison is performed with the results offered by the classical models.

Numerical method for analysing open thin-walled structures under interaction of bending and torsion

A computer program is developed to analyse concrete beams of open thin-walled sections, at different stages of loading from zero load to failure. The program is divided into two parts; the first part deals with the beam from zero load to cracking. Of course, loading is combined bending, shear and torsion (warping torsion and St Venant's torsion). In this part of the program, Vlassov's theory has been used. The cracking load is defined as that load which causes principal tensile stresses equal to the tensile strength of concrete. The second part of the program deals with all post-cracking stages of loading from cracking point to failure. An iteration procedure is used until full convergence occurs at a particular cross-section. The geometrical properties are calculated; these include the contribution of steel in the cross-section and that of concrete in the compressive zones. The mathematical model is given. The computer results are compared with earlier experimental results, and the two sets of results show reasonable agreement. The program is written in FORTRAN.

Modifications to beam theory for bending and twisting of open-section composite beams

Modifications to beam theory are presented to describe the combined bending and twisting of anisotropic composite material open-section beams subjected to pure bending and transverse loading. The beams are constructed of thin panels of laminated fiber reinforced composite materials which have either anisotropic or orthotropic in-plane mechanical properties. Due to the anisotropic coupling between normal and shear effects, three modes of beam deformation under transverse loading are obtained; bending (deflection), twisting (rotation), and axial extension or contraction (displacement). The theory is formulated in terms of effective laminate properties and describes overall beam deformation. Numerical examples are presented in which the theory is applied to describe the behavior of typical open cross-sections under a variety of loadings.

On coupled bending and torsional vibration of uniform beams

It is known that the Bernoulli-Euler theory of beam flexure may be combined with the Saint-Venant theory of torsion to describe the coupled antisymmetric free vibration of uniform beams having a plane of symmetry [1]. The principal modes may be expressed in terms of tabulated functions. This paper extends the theory to allow for warping of the beam cross-section, for this can make a large difference to results for thin-walled beams of open section.

Torsional vibrations and buckling of thin-walled beams on elastic foundation

The problem of free torsional vibration and buckling of doubly symmetric thin-walled beams of open section, subjected to an axial compressive static load and resting on continuous elastic foundation, is discussed in this paper. An analytical method based on the dynamic stiffness matrix approach is developed, including the effects of warping. The resulting transcendental equation is solved for thin-walled beams clamped at one end and simply supported at the other. A computer program is developed, based on the dynamic stiffness matrix approach. The software consists of a master program to set up the dynamics stiffness matrix and to call specific subroutines to perform various system calculations. Numerical results for natural frequencies and buckling load for various values of warping and elastic foundation parameter are obtained and presented.

Torsional post-buckling of thin-walled open section beams resting on a continuous elastic foundation

The post-buckling behaviour of thin-walled beams of open section subjected to an axial compressive load and resting on a Winkler-type continuous elastic foundation is discussed. We assume the strains to be small and elastic, shear deformations and the in-plane cross-sectional deformations to be negligible. The post-buckling paths are determined for simply supproted and clamped beams with constant cross-section. The points of bifurcation in both cases are found to be symmetric and stable for various values of warping and foundation parameters.

A Vlasov beam element

A finite element for the analysis of thin-walled open section beam structures is presented. The element is based on Vlasov's beam theory. In linear analyses the element is very effective since it does not require the use of numerical integration and only one element can exactly model simple cases of nonuniform torsion.

Geometric stiffness of thin‐walled open section beams using a semiloof beam formulation

AbstractA geometric stiffness formulation is presented for thin‐walled open section beams using semiloof elements. The validity of the formulation is confirmed by performing torsional and lateral initial stability (buckling) analyses of beams. Results are compared with classical solutions.

Finite element analysis of stability of thin-walled beams of open section

A finite element formulation is presented for the calculation of the buckling loads for elastic straight thin-walled beams of (curved) open section subjected to any distribution of conservative static loads. The stability criterion, based on the positive definiteness of the second variation of the potential energy, is derived from a previous formulation by the authors. Flexural-extensional and torsional-extensional coupling, as well as the geometric effects of an applied torque, are accounted for. Cubic Hermite shape functions are adopted, resulting in a fully compatible element. A variety of example problems illustrate the accuracy and the convergence properties of the proposed formulation.

Warping torsion in thin‐walled open section beams using the semiloof beam element

AbstractMany structural members including metallic, reinforced polymer and concrete members must be modelled, in the simplest approximation, as thin‐walled beams whose cross‐sections exhibit significant, out‐of‐plane warping owing to torsion. The elastic capabilities of the semiloof beam element are extended to include warping torsion of thin‐walled open section beams. The performance of the extended element is compared with that of the original formulation.

A consistent approach to linear stability of thin-walled beams of open section

This paper presents a stability criterion for thin-walled beams of curvilinear section, based on the positive definiteness of the second variation of the potential energy. The internal stresses are derived according to a shell model which incorporates the kinematic restraints of thin-walled beams. In addition to the classical terms of the potential energy, the geometric effects of an applied torque, as well as flexural-extensional and torsional-extensional coupling terms, are accounted for.

Finite element analysis of laminated anisotropic thin-walled open-section beams by R. K. Gupta, A. Venkatesh and K. P. Rao

Finite element analysis of laminated anisotropic thin-walled open-section beams by R. K. Gupta, A. Venkatesh and K. P. Rao

Postbuckling Analysis for the Bending of a Long Beam With a Thin, Open, Circular Cross Section

A longitudinal bending moment applied to a thin, open section beam induces limit point buckling, due to a second-order effect that flattens the leaves of the cross section. It is shown here, for the case of a circular cross section, that the leaves themselves may exhibit multiple equilibrium configurations. Thus, the postbuckling behavior of the beam is not uniquely defined, and perturbations may shift the system to an unexpected loading path. In this paper, the various equilibrium configurations are examined and judged for stability. A “perfect” bifurcating geometry is identified, and a form of imperfection sensitivity is observed.

Finite element analysis of laminated anisotropic thin-walled open-section beams

A finite element analysis of thin-walled open-section laminated anisotropic beams is presented herein. A two-noded, 8 degrees of freedom per node thin-walled open-section laminated anisotropic beam finite element has been developed and used. The displacements of the element reference axes are expressed in terms of one-dimensional first order Hermite interpolation polynomials and line member assumptions are invoked in the formulation of the stiffness matrix. The problems of: 1. (a) an isotropic material Z section straight cantilever beam, and 2. (b) a single-layer (0°) composite Z section straight cantilever beam, for which continuum solutions (exact/approximate) are possible, have been solved in order to evaluate the performance of the finite element. Its applicability has been shown by solving the following problems: 3. (c) a two-layer (45°/−45°) composite Z section straight cantilever beam, 4. (d) a three-layer (0°/45°/0°) composite Z section straight cantilever beam.