Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.     

Dynamic response of axially loaded monosymmetrical thin-walled Bernoulli–Euler beams

The dynamic bending–torsion coupled vibrations of elastic axially loaded slender thin-walled beams with monosymmetrical cross-sections are investigated by using normal mode method. The Bernoulli–Euler beam theory is employed and the effects of warping stiffness and axial force are included in the present formulations. The theoretical expressions for the displacement response of axially loaded slender thin-walled beams subjected to concentrated or distributed loads are presented. The method is illustrated by its application to two test examples to describe the effects of warping stiffness and axial force on the dynamic behavior of thin-walled beams. The numerical results for the dynamic bending displacements and torsional displacements are given. The proposed theory is fairly general and can be used for thin-walled beam assemblage of arbitrary boundary conditions subjected to various kinds of loads.     

Stress state and elastic buckling of a thin-walled beam with monosymmetrical open cross-section

The subject of this study is a simple thin-walled beam carrying a uniformly distributed transverse load. Strength and stability problems of the beam are resolved on the ground of Vlasov’s theory. Local stability is described according to the theory of thin plates and shells. The results of analytical solution are verified with the use of finite element method. Discrete model of the beam is built on the bases of shell elements SHELL4 of the COSMOS/M system. The results of both methods are compared using diagrams.     

Global static and stability analysis of thin-walled structures with open cross-section using FE shell-beam models

The finite shell-beam models for static and global stability analysis of thin-walled structures with open cross-section are proposed. The discretization using thin-walled beam elements is connected with the space discretization of some parts of the frame. The space joint element, formulated using only translational degrees of freedom on cross-sections connecting the joint with the thin-walled beams and the so-called transition elements, defined between the beam and the shell nodes are used for consistent coupling beams and shell parts.     

Theory of initially twisted, composite, thin-walled beams

An asymptotically correct theory for initially twisted, thin-walled, composite beams has been constructed by the variational asymptotic method. The strain energy of the original, three-dimensional structure is first rigorously reduced to be a two-dimensional energy expressed in terms of shell strains. Then the two-dimensional strain energy is further reduced to be expressed in terms of the classical beam strain measures. The resulting theory is a classical beam model approximating the three-dimensional energy through the first-order of the initial twist. Consistent use of small parameters that are intrinsic to the problem allows a natural derivation for all thin-walled beams within a common framework, regardless of whether the section is open, closed, or strip-like. Several examples are studied using the present theory and the results are compared with a general cross-sectional analysis, VABS, and other published results.     

Nonlinear theory of non-uniform torsion of thin-walled open beams

A nonlinear theory of non-uniform torsion based on finite displacements is developed. Expressions for the finite nonlinear strains in Lagrangian coordinates and the Kirchhoff stresses for thin-walled open beams are presented. Using the principle of stationary total potential, the dual forms of the beam equilibrium equations are derived. For conservatively loaded thin-walled open beams a static stability criterion, based on the positive definiteness of the second variation of the total potential, is presented. The criterion developed takes into account the effects of changes in beam geometry such as initial bending curvature, prior to instability.     

Vibration and elastic instability of thin-walled domes under uniform external pressure

Two thin-walled varying meridional curvature axisymmetric shell elements are presented for the vibration and elastic instability of thin-walled hemi-ellipsoidal domes under uniform external pressure. The theoretical analysis is an extension of previous work carried out by the author, where for the two elements presented in the present report, a cubic and a quadratic variation was assumed for the meridional and the circumferential displacements along the meridian of these elements. In the previous study, only linear variations were assumed for the meridional and circumferential displacements along the meridian of these elements. Comparisons were made between experiment and theory for both buckling and vibration of hemi-ellipsoidal shell domes, which varied from very flat oblate vessels to very long prolate vessels. In general, agreement between experiment and theory was good for the hemi-spherical dome and the prolate vessels, but not very good for the fat oblate vessels. Additionally, the two new elements gave poorer results than the original simpler element for the cantilever mode of vibration, but better results for the lobar modes of vibration.     

Localized support settlements of thin-walled storage tanks

This paper investigates the influence of support settlements on the out-of-plane displacements of thin-walled cylindrical tanks with a fixed top roof. The shell considered is representative of many steel tanks constructed in Puerto Rico and in the United States, and has a ratio between the diameter and the height of the order of 2.5, with slenderness ratio (radius to thickness) of the order of 1,700. The behavior of the tank is investigated using the finite element computer package ABAQUS by means of a geometrically non-linear algorithm for the analysis and linear elastic material behavior. Results are presented for geometrically linear analysis, geometrically nonlinear analysis and bifurcation buckling analysis. It is shown that the equilibrium path is highly non linear and that the shell displays a plateau for a settlement of the order of half the thickness of the shell. Linear results provide a poor indication of the real displacements in the shell, so that geometric nonlinearity should be included in the analysis for working loads.     

Frame mode method for thin-walled structures

The Rayleigh-Ritz method is extended to thin-walled structures on continuous supports by means of the combinations of frame modes transversely and continuous beam modes longitudinally. Effects of warping, distortion and shear-lag are considered simultaneously. The thin-walled structure problem is reduced to simpler plane frame and continuous beam problems. Numerical results are compared to the finite element and finite strip methods. The present method is advantageous over both the finite element and finite strip methods in the reduced number of generalised coordinates and its ability to use the existing frame programmes to analyse thin-walled structures. No new elements are required to be generated.     

Coupled axial–torsional vibration of thin-walled Z-section beam induced by boundary conditions

The coupled axial–torsional vibration of thin-walled Z-section beam induced by the boundary conditions is investigated. The value of the warping function is not zero at centroid for Z-section beam. If the axial displacement of the pin end is restrained at the centroid of the Z-section for thin-walled Z-section beam, the axial vibration and torsional vibration may be coupled. The governing equations for linear axial and torsional vibration of a thin-walled Z-section beam are derived by the d’Alembert principle and the virtual work principle. The bending vibration is uncoupled from axial and torsional vibrations and is not dealt with in this paper. For harmonic vibration, the general solution of these equations with undetermined constant coefficients may be obtained. Substituting the general solution into the displacement and force boundary conditions, a set of homogeneous equations can be obtained. The natural frequencies and the coefficients of the general solution may be obtained by solving the homogeneous equations using the bisection method.Numerical examples are studied to verify the accuracy of the proposed method and to investigate the effect of boundary conditions and the value of warping function at centroid on the coupled axial and torsional natural frequency of Z-section beam.     

An efficient formulation for thin-walled beams curved in plan

An efficient formulation is developed for the elastic analysis of thin-walled beams curved in plan. Using a second-order rotation tensor, the strain values of the deformed configuration are calculated in terms of the displacement values and the initial curvature by adopting the right extensional strain measure. The principle of virtual work is then used to obtain the nonlinear equilibrium equations, based on which a finite element beam formulation is developed. The accuracy of the method is confirmed through comparison with test results, shell finite element formulations and other curved beam formulations from the literature. It is also shown that the results of the developed formulation are very accurate for the cases where initial curvature is large.     

Non-linear model for stability of thin-walled composite beams with shear deformation

A geometrically non-linear theory for thin-walled composite beams is developed for both open and closed cross-sections and taking into account shear flexibility (bending and warping shear). This non-linear formulation is used for analyzing the static stability of beams made of composite materials subjected to concentrated end moments, concentrated forces, or uniformly distributed loads. Composite is assumed to be made of symmetric balanced laminates or especially orthotropic laminates. In order to solve the non-linear differential system, Ritz's method is first applied. Then, the resulting algebraic equilibrium equations are solved by means of an incremental Newton–Rapshon method. This paper investigates numerically the flexural–torsional and lateral buckling and post-buckling behavior of simply supported beams, pointing out the influence of shear–deformation for different laminate stacking sequence and the pre-buckling deflections effect on buckling loads. The numerical results show that the classical predictions of lateral buckling are conservative when the pre-buckling displacements are not negligible, and a non-linear buckling analysis may be required for reliable solutions.     

GBT-based buckling analysis of thin-walled members with non-standard support conditions

This paper reports on the use of a recently developed Generalised Beam Theory (GBT) formulation, and corresponding finite element implementation, to analyse the local and global buckling behaviour of thin-walled members with arbitrary loading and support conditions — this formulation takes into account longitudinal normal stress gradients and the ensuing pre-buckling shear stresses. After presenting an overview of the main concepts and procedures involved in the performance of a GBT-based (beam finite element) member buckling analysis, one addresses in detail the incorporation of non-standard support conditions, such as (i) full or partial localised displacement or rotation restraints, (ii) rigid or elastic intermediate supports or (iii) end supports corresponding to angle connections. In order to illustrate the application and capabilities of the proposed GBT-based approach, one presents and discusses numerical results concerning cold-formed steel (i) lipped channel beams and (ii) lipped I-section beams and columns with various “non-standard” support conditions — while the beams are acted by uniformly distributed or mid-span point loads, applied at the shear centre axis, the columns are subjected to uniform compression. In particular, it is possible to assess the influence of the different support conditions on the beam and column buckling behaviour (critical buckling loads and mode shapes). For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the code Ansys.     

Nonlinear viscoelastic analysis of thin-walled beams in composite material

Laminated composites of polymeric matrix show anisotropic viscoelastic behaviour, enhanced by temperature and humidity effects. The consideration of anisotropy and viscoelasticity are important for the determination of deformations and, as a consequence, of deformation-related phenomena, as elastic and creep buckling. This paper studies the behaviour of thin-walled beams of composite material under flexure and buckling, taking account of creep effects. The analysis uses a nonlinear viscoelastic finite element code with shell elements, whose basic formulation is given. The use of shell elements allows a better representation of constitutive properties and boundary conditions. Comparison with available analytical results is made for several cases like flexure of an I beam, buckling of beam columns and lateral buckling of this beams. The results show good correlation.     

Flexural–torsional buckling of thin-walled composite box beams

Buckling of an axially loaded thin-walled laminated composite is studied. A general analytical model applicable to the flexural, torsional and flexural–torsional buckling of a thin-walled composite box beam subjected to axial load is developed. This model is based on the classical lamination theory, and accounts for the coupling of flexural and torsional modes for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite bar. Governing buckling equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled composites addressing the effects of fiber angle, anisotropy and boundary conditions on the critical buckling loads and mode shapes of the composites.     

Torsion warping transmission at thin-walled frame joints: Kinematics, modelling and structural response

This paper reports on the use of simple kinematic models to simulate the torsion warping restraint and transmission at thin-walled frame joints in the context of beam finite element structural analysis. After reviewing the main concepts involved in the torsional behaviour of thin-walled members, the paper addresses the development of kinematic models aimed at simulating the torsion warping transmission at frame joints connecting two or more non-aligned plain channel (U-section) or I-section members. Finally, numerical results are presented and discussed, in order to illustrate the application and show the capabilities of the above kinematic models, which make it possible to use beam finite element models accounting for the joint torsion warping behaviour. For validation purposes, the beam finite element results obtained are compared with values yielded by rigorous shell finite element analyses.     

Shape optimization of thin-walled beam-like structures

This article presents a methodology for optimizing the shape of thin-walled structures having a beam-like dynamic behavior. The equivalent nominal beam characteristics (quadratic moments of inertia, torsional rigidity, etc.) of a refined finite element model are determined from a direct calculation based on explicit equations which are functions of the nodal coordinates defining the cross-sectional geometry. A subset of these coordinates is then taken as the design variables for a nonlinear optimization problem where new target physical properties are sought. A complementary model reduction procedure is introduced to improve the precision of the proposed method for beams with variable cross-section or topological accidents.     

Modeling distortional shear in thin-walled elastic beams

In this paper, the theoretical background and the numerical analyses of an advanced beam finite element that relaxes the hypothesis of the cross-section non-deformability are presented. The corresponding new modes, called distortional modes, are added to the modes describing the behavior of a classical thin-walled beam: tension/compression, bending and torsion. For instance, a load acting in a cross-sectional plane of a beam is considered to induce not only bending and torsion but also distortion. The distortion produces non-uniform shear and axial stresses together with a non-uniform warping of the cross-section. These resulting effects, significant for very thin-walled open profiles (and thin-walled closed profiles with high distortional loadings), are shown in this paper to be important when compared to bending and torsion stresses even in simple loading cases.     

On using Legendre polynomials and amended spline transformations in the SFSM for buckling and free vibrations of plates and thin-walled beams

This paper presents two modifications to the spline finite strip method (SFSM) for plate and thin-walled beam buckling and free vibrations; these being augmenting of the flexural buckling displacements with Legendre polynomials which represent additional bubble modes, and a modification of the dedicated amended splines that are conventionally needed to handle boundary conditions. A simple matrix technique to remove the need for amended splines is described, and the efficacy of the method is demonstrated in which it is shown that inclusion of the Legendre polynomials enhances the performance of the SFSM significantly when benchmarked against other techniques.     

Variational design of open cross-section thin-walled beam under stability constraints

A thin-walled beam is in pure bending subjected to couples M₀. The open cross-section profile has two ribs, with cross-section A₀ and it is shaped symmetrically towards the plane perpendicular to the bending plane. The ribs are located at the profile ends. The shape of the profile line is searched for. Criterion is the minimal value of the cross-section area A₁ of the beam. The problem is described by means of variational calculus. Within the numerical calculations a Runge–Kutta method is used. The optimal shapes of beam profiles are shown graphically.