Beams Of Arbitrary Cross Section Research Articles

Analysis of composite bridges with intermediate diaphragms & assessment of design guidelines

In this research effort, the generalized warping and distortional problem of straight or horizontally curved composite beams of arbitrary cross section, loading and boundary conditions is presented. An inclined plane of curvature is also considered (with respect to the horizontal plane) in order to account for a slope in the cross-section plane. Additionally, a finite stiffness of diaphragmatic plates has been introduced in the formulation in order to compare with the beam models where rigid diaphragms are considered, as usually is assumed in practice. The numerical method employed for the 1D beam formulation is based on Isogeometric tools (NURBS) while the 3D shell or solid models are developed in Finite Element commercial software for composite cross sections with diaphragms. The influence of friction on the contact interaction between the concrete and steel parts is also considered. The number of intermediate diaphragms is determined according to commonly used bridge design guidelines and the results are compared to the developed arrangements in order to assess the overall structural behavior of bridge decks. Straight or curved beam models with open or closed composite cross sections of full (1D proposed beam) or partial material-bonding (3D solid/shell models) and various arrangements of intermediate diaphragms and materials have been studied.

Beam & Shell Models for Composite Straight or Curved Bridge Decks with Intermediate Diaphragms & Assessment of Design Specifications

In this research effort, the generalized warping and distortional problem of straight or horizontally curved composite beams of arbitrary cross section, loading and boundary conditions is presented. An inclined plane of curvature is considered. Additionally, the stiffness of diaphragmatic plates has been introduced in the formulation in order to compare with the case where rigid diaphragms are assumed. Isogeometric tools (NURBS) are employed in order to obtain the results for the 1D formulation and 3D shell models are developed in FEM commercial software for composite cross sections with diaphragms. The number of intermediate diaphragms according to bridges design specifications is compared to the analyzed diaphragmatic arrangements in order to assess the overall structural behavior of bridges decks. For this purpose, examples of curved beam models with open or closed cross sections and various arrangements of diaphragms have been studied.

Bending by explosion of a multilayered concrete beam on a visco-elastic basis

In this work, the problem of bending of a multilayered concrete beam of arbitrary cross-section by explosive loading on a visco-elastic basis is considered. It is assumed that different grades of concrete can be realized in layers in the cross-section. The property of concrete resistance to tension and compression is considered in work. It is assumed that the dynamic loading is caused by consecutive explosion of two charges over the middle of the span of beam. The distribution of bending moments and deflections of the beam at each time is determined. The time of the end of motion and the residual deflection of beam are found.

Longitudinal cracks in non-linear elastic beams exhibiting material inhomogeneity

Longitudinal fracture behavior of non-linear elastic beam configurations is studied in terms of the strain energy release rate. It is assumed that the beams exhibit continuous material inhomogeneity along the width as well as along the height of the cross-section. The Ramberg-Osgood stress-strain relation is used for describing the non-linear mechanical behavior of the inhomogeneous material. A solution to strain energy release rate is derived that holds for inhomogeneous beams of arbitrary cross-section under combination of axial force and bending moments. Besides, the solution may be applied at any law of continuous distribution of the modulus of elasticity in the beam cross-section. The longitudinal crack may be located arbitrary along the beam height. The solution is used to investigate a longitudinal crack in a beam configuration of rectangular cross-section under four-point bending. The crack is located symmetrically with respect to the beam mid-span. It is assumed that the modulus of elasticity varies continuously according a cosine law in the beam cross-section. The longitudinal fracture behavior of the inhomogeneous beam is studied also by applying the J-integral approach for verification of the non-linear solution to the strain energy release rate derived in the present paper. Effects of material inhomogeneity, crack location along the beam height and non-linear mechanical behavior of the material on the longitudinal fracture behavior are evaluated. Thus, the solution derived in the present paper can be used in engineering design of inhomogeneous non-linear elastic structural members to assess the influence of various material and geometrical parameters on longitudinal fracture.

Distortional Analysis of Beams of Arbitrary Cross Section Using BEM

This paper presents a general formulation for the distortional analysis of beams of arbitrary cross section under arbitrary external loading and general boundary conditions. The nonuniform distortional/warping distributions along the beam length are taken into account by employing independent parameters multiplying suitable deformation modes accounting for in-plane and out-of-plane cross-sectional deformation (distortional/warping functions). The paper proposes a novel procedure for cross-sectional analysis which results in the solution of separate boundary value problems for each resisting mechanism (flexure, torsion) on the cross-sectional domain instead of relying on eigenvalue analysis procedures encountered in the literature. These distortional and warping functions are computed employing a boundary element method (BEM) procedure. Subsequently, sixteen boundary value problems are formulated with respect to displacement and rotation components as well as to independent distortional/warping parameters along the beam length and solved using the analog equation method (AEM), a BEM-based technique. After the establishment of kinematical components, stress components on any arbitrary point of each cross section of the beam can be evaluated, yielding a prediction in good agreement with three-dimensional finite-element method (FEM) solutions, in contrast to conventional beam models.

Bending of beams of arbitrary cross sections - optimal design by analytical formulae

The paper deals with the conceptual design of a beam under bending. The common problem of designing a beam in a state of pure bending is discussed in the framework of Pareto-optimality theory. The analytical formulation of the Pareto-optimal set is derived by using a procedure based on the reformulation of the Fritz John Pareto-optimality conditions. The shape of the cross section of the beam is defined by a number of design variables pertaining to the optimization process by means of efficiency factors. Such efficiency factors are able to describe the bending properties of any beam cross section and can be used to derive analytical formulae. Design performance is determined by the combination of cross section shape, material and process. Simple expressions for the Pareto-optimal set of a beam of arbitrary cross section shape under bending are derived. This expression can be used at the very early stage of the design to choose a possible cross section shape and material for the beam among optimal solutions.

B-splines in the Analog Equation Method for the generalized beam analysis including warping effects

In this paper, the Analog Equation Method (AEM), a boundary element based method, is employed for the analysis of homogenous beams of arbitrary cross section (thin- or thick-walled) taking into account nonuniform warping and shear deformation effects (shear lag due to both flexure and torsion), considering a B-spline approximation for the fictitious loads of a substitute problem. The fictitious loads are established using a BEM-based technique and the solution of the original problem is obtained from the integral representation of the solution of the substitute problem. The beam is subjected to the combined action of arbitrarily distributed or concentrated axial and transverse loading, as well as to bending, twisting and warping moments. Its edges are subjected to the most general boundary conditions, including also elastic support. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, which are obtained by solving corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. Ten one-dimensional boundary value problems are described by second-order differential equations. Integrating B-splines in the AEM technique, the computational cost is efficiently reduced while the results are highly accurate.

Generalized warping effect in the dynamic analysis of beams of arbitrary cross section

In this paper a general formulation for the nonuniform warping dynamic analysis of beams of arbitrary simply or multiply connected cross section, under arbitrary external loading and general boundary conditions is presented taking into account the effects of rotary and warping inertia. The nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, respectively, which are obtained by solving the corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. A shear stress “correction” is also performed in order to improve the stress field arising from the employed kinematical considerations. Ten initial boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameters and solved using the Analog Equation Method, a Boundary Element Method based technique in combination with an appropriate time integration scheme. The warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach.

Improved model for interfacial stresses accounting for the effect of shear deformation in plated beams

A significant increase in strength and performance of reinforced concrete, timber and metal beams may be achieved by adhesively bonding a fibre reinforced polymer composite, or metallic such as steel plate to the tension face of a beam. One of the major failure modes in these plated beams is the debonding of the plate from the original beam in a brittle manner. This is commonly attributed to the interfacial stresses between the adherends whose quantification has led to the development of many analytical solutions over the last two decades. The adherends are subjected to axial, bending and shear deformations. However, most analytical solutions have neglected the effect of shear deformation in adherends. Few solutions consider this effect approximately but are limited to one or two specific loading conditions. This paper presents a more rigorous solution for interfacial stresses in plated beams under an arbitrary loading with the shear deformation of the adherends duly considered in closed form using Timoshenko's beam theory. The solution is general to linear elastic analysis of prismatic beams of arbitrary cross section under arbitrary loading with a plate of any thickness bonded either symmetrically or asymmetrically with respect to the span of the beam.

Generalized warping analysis of curved beams by BEM

In this paper, the analysis of homogenous curved beams of arbitrary cross section (thin- or thick-walled) taking into account the coupling of extension, flexure and torsion, nonuniform warping as well as shear deformation effects (shear lag due to both flexure and torsion) has been studied. The curved beam is subjected to the combined action of arbitrarily distributed or concentrated axial and transverse loading, as well as to bending, twisting and warping moments. Its edges are subjected to the most general boundary conditions. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, which are obtained by solving corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. Ten one-dimensional boundary value problems are described by second-order differential equations and solved employing the Analog Equation Method (AEM), a boundary element based method, using either constant or quadratic elements for the representation of the adopted fictitious loads. Results are compared with FEM solutions employing curved beam, shell or 3-d solid elements.

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

In this paper, a boundary element method is developed for the non-linear flexural–torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderately large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of non-linear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the non-linear effects to the response of the beam.

Nonlinear Effects in Elastic Flexural-Torsional Vibrations of Beams of Arbitrary Cross Section

In this paper a boundary element method is developed for the nonlinear flexural-torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderate large deflections and rotations under general boundary conditions, taking into account the effects of rotary and torsional warping inertia. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the nonlinear effects to the response of the beam.

Postbuckling analysis of beams of arbitrary cross section using BEM

In this paper, the postbuckling analysis of beams of arbitrary cross section is presented taking into account moderate large displacements, large angles of twist and adopting second order approximations for the deflection–curvature relations. The beam is subjected to the combined action of the arbitrarily distributed or concentrated axial, transverse and torsional loading, while it is supported by the most general boundary conditions including elastic support or restraint. Starting from a displacement field without any simplifying assumptions about the angle of twist amplitude and based on the total potential energy principle, four highly coupled nonlinear governing differential equations are derived taking into account the shortening and warping effects and the Wagner’s coefficients due to the asymmetric character of the cross section. The arising four boundary value problems with respect to the transverse displacements, to the axial displacement and the angle of twist are solved using the Analog Equation Method, a BEM based method. The geometric, inertia, torsion and warping constants of the arbitrary beam cross section are evaluated employing only boundary integrals. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint–Venant’s torsional constant. Numerical examples are worked out to illustrate the efficiency, accuracy and range of applications of the developed method. Conclusions of great practical interest are drawn.

Shear deformation effect in flexural-torsional buckling analysis of beams of arbitrary cross section by BEM

In this paper a boundary element method is developed for the general flexural-torsional buckling analysis of Timoshenko beams of arbitrarily shaped cross section. The beam is subjected to a compressive centrally applied concentrated axial load together with arbitrarily axial, transverse and torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-walled theory and the significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.

Lateral buckling analysis of beams of arbitrary cross section by BEM

In this paper, the lateral buckling analysis of beams of arbitrary cross section is presented taking into account moderate large displacements and employing nonlinear relationships between bending moments and curvatures. The beam is supported by the most general boundary conditions including elastic support or restraint. Starting from a displacement field without any simplifying assumptions about the angle of twist amplitude and based on the total potential energy principle, the stability criterion is formulated taking into account the warping effects, the prebuckling displacements and the Wagner’s coefficients due to the asymmetric character of the cross section. The stability criterion is based on the positive definiteness of the second variation of the total potential energy and is established using the Analog Equation Method (AEM), a BEM based method. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint–Venant’s torsional constant. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The effects of both the load height and prebuckling deflections as well as the discrepancy of the Eurocode 3 standard solutions are discussed.

Flexural–torsional postbuckling analysis of beams of arbitrary cross section

In this article, the postbuckling analysis of axially compressed elements of arbitrary cross section is presented taking into account moderately large displacements, moderately large angles of twist and employing nonlinear relationships between bending moments and curvatures. The elements are supported by the most general boundary conditions including elastic support or restraint. Based on Galerkin’s method and approximating the displacement field of the element by polynomial expressions the governing differential equations lead to a nonlinear algebraic system. The geometric, inertia, torsion and warping constants of the arbitrary beam cross section are evaluated employing the boundary element method. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint–Venant’s torsional constant. Both the Wagner’s coefficients and the shortening effect are taken into account, while their influence is examined and discussed. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method.

Image effects on the transport of intense nonaxisymmetric charged beams

The effect of conducting pipes on the equilibrium of intense nonaxisymmetric continuous beams of charged particles is investigated. For a cylindrical pipe and an elliptical beam, we obtain an exact closed form analytical expression for the electrostatic potential. Using a variational principle, we then explore the distortions that equilibrium beams suffer due to the conducting channel. Finally, we present an exact proof that despite the nonlinear forces acting on beams of arbitrary cross section inside conducting pipes of arbitrary shape, the density of these beams remains homogeneous and their cross sectional area remains the same as the one in free space.

Transverse Vibrations of Laminated Beams in Three-Dimensional Formulation

A new asymptotic method is proposed to describe the free and forced transverse vibrations of elastic laminated beams of arbitrary cross section using the three-dimensional elastic equations without additional hypotheses and constraints. For beams with layers of equal Poisson's ratio, the zero-order natural frequencies are equal to the natural frequencies predicted by classical beam theory based on the Bernoulli hypothesis. The method makes it possible to calculate the frequencies of free vibrations and amplitudes of forced vibrations with prescribed accuracy for the first natural modes

Vibration and dynamic response control of cantilevers carrying externally mounted stores

The vibrational and dynamic response control of cantilevers carrying externally mounted stores is investigated. The cantilevered structure is modeled as a thin-walled beam of arbitrary cross section and incorporates a number of nonclassical effects such as transverse shear, secondary warping, anisotropy of constituent materials, and heterogeneity of the construction. The control is carried out via a dynamic bending moment applied at the tip of the structure. A feedback control law relating the boundary moment with one of the kinematical variables characterizing the response of the beam is implemented, and its results upon the closed-loop eigenfrequencies and dynamic response are highlighted. The obtained numerical results emphasize the efficiency of this control methodology to enhance, without weight penalties, vibrational and dynamic response behavior and inhibit and even suppress the occurrence of the resonance phenomenon.

Thin‐Walled Space Frames. I: Large‐Deformation Analysis Theory

A finite element formulation for the large‐deformation analysis of space‐frame structures is presented. The formulation is based on second‐order geometric nonlinear theory and Vlasov's theory for thin‐walled beams (i.e., large displacement of members with small strains, which includes the warping deformation influence). The influence of member‐distributed loading on the geometric nonlinear response of space‐frame structures also is included. An updated Lagrangian formulation is used to model large joint translations and rotations. Prismatic beams of arbitrary cross section are considered. Rodriguez's modified rotation vector is used to represent angular deformations, which avoids rotational discontinuities at the joints of deformed space‐frame structures. Numerical results and algorithmic details are presented in a companion paper.