Non-uniform Torsion Research Articles

Third-order geometric stiffness formulation for improved mesh convergence of thin and wide spatial beams in the generalized strain beam formulation

AbstractThis paper presents the stiffness formulation of a beam element with the relevant third-order nonlinear geometric effects for relatively wide and thin rectangular beams, in particular when loaded in the plane and simultaneously deformed out of the plane. The element is initially straight in its undeformed configuration. The formulation is based on Timoshenko beam theory with nonuniform torsion and Wagner effects. The derivation is carried out by means of the Hellinger–Reissner variational principle with custom interpolation functions. The element is incorporated into the generalized strain beam formulation for multibody systems. Numerical simulations of precision flexure mechanisms show that the use of a single third-order element per flexible member can already yield adequate performance, at a significant reduction of the necessary degrees of freedom and the computation time, compared with using multiple second-order elements in the generalized strain beam formulation.

Non-Uniform Torsion Behavior of Thin-Walled Beams According to the Finite Element Method

Non-Uniform Torsion Behavior of Thin-Walled Beams According to the Finite Element Method

Flexural-torsional analysis of steel beam structures using ANSYS

The theories for uniform torsion and non-uniform torsion are studied. The aim of the paper is to study the behaviour of the beams of different sections for combined bending and torsion under various loadings and support conditions. The comparative study of the finite element for beams with different geometry sections of solid and hollow circular sections, rectangular section and symmetric I sections subjected to concentric and eccentric loading conditions with simply supported and cantilever end conditions is presented using ANSYS 18.1 workbench. The length of the beam is kept constant as 4 m. The transverse and axial deformation, the equivalent stresses and strains of beams were studied along with the mode of failure and effect of bending and torsion. It was observed that concentric loading resulted in transverse deflections while the torsional moment loading causes in-plane deformation in form of twisting of the cross-section with maximum stress generation and the eccentric loading results in combined transverse and in-plane axial deformation. From the result obtained it can be said that open sections has lesser efficiency for torsional and eccentric type loading as higher stresses and axial deflections were observed as compared to the hollow circular and rectangular sections.

Experimental research on curved continuous steel-concrete composite twin I-girder bridge

Due to the excellent economy and convenient erection, curved continuous steel–concrete composite twin I-girder (CTIG) bridges have been applied in mountainous bridges and unban overpasses. Rigorous investigation of the bending-torsion interaction is important for the curved continuous CTIG bridge design. A CTIG model bridge with a span arrangement of 17.46 m + 17.46 m and a curved radius of 200 m, was fabricated and tested under a symmetric static load. The mechanical behavior and failure modes of the model bridge were investigated. Results show that the non-uniform torsion and the web distortion effect are the characteristic mechanical behaviours of the curved CTIG bridge. The crossbeams resist the web distortion to ensure the transmission of the torsional moment. The constraint of the web distortion by the crossbeams results in the bottom flanges of the CTIG bridge being subjected to additional lateral bending moments. The ultimate strength of the CTIG bridge studied in this research is reached by a shear failure involving web buckling, formation of diagonal web yielding, and concrete deck crushing. This study contributes to a deeper understanding of the mechanical behavior of the curved continuous CTIG bridge and provides a valuable experimental basis for research regarding the design of bridges of this type.

Interaction of the lateral buckling strength with the axial load for FG micro-sized I-section beam–columns

This article presents the assessment of axial preloading impact on lateral stability capacity of a size-dependent I-section beam–column element with axially varying material properties under transverse load. Material properties of the axially/transversely loaded micro beam obey the power-law function along the longitudinal direction. Nonlocal elasticity theory, which also is identified as Eringen’s theory, is employed in conjunction with Vlasov’s model for non-uniform torsion to extract the governing equations for the lateral displacement and the twisting angle with the aid of energy approach. Then, the endurable buckling load is predicted using the semi-analytical power series technique. To examine the correctness of the present methodology, a comparison between the attained outcomes with those existing in the literature is made. Eventually, in the presence of a prescribed axial force, the interaction between micro beam lateral stability strength and column flexural buckling is examined and discussed. Additionally, the impact of important parameters such as nonlocal and load height ones, power-law index, and axial force is perused on the aforementioned size-dependent structure’s lateral buckling resistance.

Symplectic analysis of thin-walled curved box girders with torsion, distortion and shear lag warping effects

Shear lag effect and torsional and distortional warping can significantly affect the performance of thin-walled curved box girders used in modern bridge engineering. The structural behaviour of these bridges exhibits complexity due to coupled bending and torsion together with warping effects of non-uniform torsion, distortion and shear lag. A practical method of analysis, based on the symplectic approach, with the same perspective as the Higher Order Beam Theories, is presented for overcoming the difficulties of numerical approaches via the Finite Element Method. In this paper, the Hamiltonian Structural Analysis method implements the analysis of the shear lag effect together with non-uniform torsion and distortion, for curved box girder bridges. The examples given can allow engineers to evaluate the effects of the negative shear lag phenomenon and the distribution of axial stresses in box slabs and webs, due to different warping effects, which strongly influence thin-walled box sections subjected to point loads and restrained areas.

Nonuniform torsion analysis in tapered composite bars by including higher-order warping modes

Nonuniform torsion analysis in tapered composite bars by including higher-order warping modes

Approximate analysis of non-uniform torsion

It is difficult to obtain closed form exact solutions for the non-uniform torsion of beams. Advanced finite element methods easily produce very accurate solutions, but simple programs are not widely available. One simple program is validated by comparing its solutions with closed form solutions.A simple approximate method is proposed which combines the uniform and warping torsion stiffness approximations. The determination of the warping stiffness approximation is simplified by using the moment-area method. The approximate twist rotations are compared with the accurate solutions, are shown to be of generally acceptable accuracy, and in most cases to be conservative.Approximate solutions for the central twist rotations of simply supported beams with off-centre torques can be combined using superposition to find solutions for a wide range of torque loadings. A worked example is given.

A harmonic approach to the non-uniform torsion of box girder with edge cantilevers by fully considering the secondary torsional moment deformation effect

The restrained torsion analysis of open and purely closed cross-section thin-walled beams has been well addressed by applying the classical Vlasov and Benscoter non-uniform torsion theory, respectively. However, the warping torsion of the thin-walled box girder with wide overhanging slabs has to date received relatively less attention, although it exhibits notable and complicated shear lag phenomena. In this paper, a semi-analytical approach based on the harmonic analysis is presented to consistently take into account the secondary torsional moment deformation effect (STMDE) in edge cantilevers and closed cell of the box girder. The non-uniform warping and the associated strain field are reasonably determined by fulfilling the following requirements as (i) the longitudinal local equilibrium, (ii) the traction-free boundary conditions, and (iii) the shear flow continuity at joints connecting wall branches of the cross section. The advantage of the proposed method lies in that regardless of the applied loading and boundary conditions each harmonic term of the state variables can be manually determined to save a large amount of computing resources and time. Convergence study shows that the resulted Fourier series of state variables can be made converged if only first few terms are retained. Some benchmark problems, which primarily focus on thin-walled box girders having single- and multi-cellular cross sectional shapes under typical constraints and applied torques, are solved to demonstrate the reliability of the harmonic approach for non-uniform torsion of the box girder with edge cantilevers. It is shown by comparing with the results of shell finite element analysis that the semi-analytical solutions by the harmonic approach generally have better accuracy than the numerical results using 3D beam element with seven nodal degrees of freedom and analytical results by classical thin-tube theory (TTT).

A Minimal GBT Model for Distortional-Twist Elastic Analysis of Box-Girder Bridges

A simple and efficient method is proposed for the analysis of twist of rectangular box-girder bridges, which undergo distortion of the cross section. The model is developed in the framework of the Generalized Beam Theory and oriented towards semi-analytical solutions. Accordingly, only two modes are accounted for: (i) the torsional mode, in which the box-girder behaves as a Vlasov beam under nonuniform torsion, and, (ii) a distortional mode, in which the cross section behaves as a planar frame experiencing skew-symmetric displacements. By following a variational approach, two coupled, fourth-order differential equations in the modulating amplitudes are obtained. The order of magnitude of the different terms is analyzed, and further reduced models are proposed. A sample system, taken from the literature, is considered, for which generalized displacement and stress fields are evaluated. Both a Fourier solution for the coupled problem and a closed-form solution for the uncoupled problem are carried out, and the results are compared. Finally, the model is validated against finite element analyses.

Three-dimensional beam element for pre- and post-buckling analysis of thin-walled beams in multibody systems

This paper presents a three-dimensional beam element for stability analysis of elastic thin-walled open-section beams in multibody systems. The beam model is based on the generalized strain beam formulation. In this formulation, a set of independent deformation modes is defined which are related to dual stress resultants in a co-rotational frame. The deformation modes are characterized by generalized strains or deformations, expressed as analytical functions of the nodal coordinates referred to the global coordinate system. A nonlinear theory of non-uniform torsion of open-section beams is adopted for the derivation of the elastic and geometric stiffness matrices. Both torsional-related warping and Wagner’s stiffening torques are taken into account. Second order approximations for the axial elongation and bending curvatures are included by additional second order terms in the expressions for the deformations. The model allows to study the buckling and post-buckling behaviour of asymmetric thin-walled beams with open cross-section that can undergo moderately large twist rotations. The inertia properties of the beam are described using both consistent and lumped mass formulations. The latter is used to model rotary and warping inertias of the beam cross-section. Some validation examples illustrate the accuracy and computational efficiency of the new beam element in the analysis of the buckling and post-buckling behaviour of thin-walled beams under various loads and (quasi)static boundary conditions. Finally, applications to multibody problems are presented, including the stability analysis of an elementary two-flexure cross-hinge mechanism.

A numerical method to compute global resonant vibrations of ships at forward speed in oblique waves

Resonant wave-induced vibrations of a ship hull girder, known as “springing”, is an important issue when addressing fatigue aging of the steel structure. To compute resonant wave-induced vibrations, fluid-structure interactions and associated structural and hydrodynamic properties need to be addressed. This paper introduces a time-domain numerical method that predicts higher-order springing taking into account forward speed. Structural dynamics were computed based on a beam element approach that considers vertical and horizontal bending as well as nonuniform torsion. Furthermore, mass and stiffness matrices accounted for strong coupling effects between hull girder bending and torsion. The hydrodynamic solver coupled the fully nonlinear stationary free surface flow with the oscillatory flow and considered geometrical nonlinearities caused by the changing wetted surface due to the incident waves, nonlinear rigid body motions and linear elastic vibrations. Numerical predictions were validated against model tests of a post-Panamax containerships at forward speed. Dry and wet natural frequencies and midship vertical bending, horizontal bending and torsional moments induced by higher order springing vibrations compared favourably to experimental measurements.

Non-uniform torsion analysis of functionally graded beams with solid or thin-walled section using hierarchical Legendre expansion functions

In this paper, an accurate yet computationally efficient beam model based on hierarchical Legendre expansion functions is developed for the analysis of non-uniform or restrained torsion problems of Functionally Graded (FG) beams with solid or thin-walled section. The mechanical properties of the FG beams studied in this paper, such as Young’s modulus and shear modulus, are assumed to continuously vary along either the length or thickness direction following a power-law distribution. The proposed beam model is based on the assumption that the beam’s cross-section is infinitely rigid in its own plane. However, the longitudinal displacement field over the beam’s cross-section is enriched in an element-wise manner by the unknown longitudinal displacement parameters multiplying with hierarchical Legendre expansion functions. The proposed modeling methodology has two novel aspects: First, it allows the torsional or twisting angle, which is a priori defined as an unknown kinematic variable, to be directly captured without a post-processing recovery step, even if there exists a strong flexual-torsional coupling within the beam; second, the longitudinal warping response of the beam, triggered by stretching, bending, twisting, or the coupling between them, can be captured without the pre-determination of warping modes and at a lower level of DOFs. The strong-form governing equations of non-uniform or restrained torsion problems of the FG beam are derived based on the principle of minimum potential energy and is directly solved by the high-quality general-purpose ordinary differential equation (ODE) solver, i.e. COLSYS ODE solver. Also, a more efficient Rayleigh-Ritz energy method is applied to provide the weak-form solutions. The resulting beam model is suitable to a more general cross-section, such as solid section, branched open section, or half open-half closed section. The accuracy and efficiency of the proposed beam model are validated extensively by comparing with the previously published results in literature. Effects of the power-law index, taper ratio, and section-type on the torsional response of FG beams with material gradation along length or thickness direction are studied with various numerical examples.

The impact of section properties on thin walled beam sections with restrained torsion

In this paper the influence of section properties on thin walled beam sections with restrained torsion is studied by considering the angle of twisting for non-uniform torsion. The study is based on the governing equation for non-uniform torsion of thin-walled beam of open and closed cross-sections, including the restrained behavior. The hyperbolic shape functions is used to generate the general solution for homogenous equation as it gives the exact results in accordance with the analytical solution. This study is focused on a bar subjected to torsion with warping of wide-ranging open and closed cross-sections with different value of characteristics number for torsion. The impact of the characteristics number for torsion on different section type is studied and evaluated. The obtained results are compared based on the influence of section properties on the components of the total applied torsion and presented graphical for different value of characteristics number for torsion.

Influence of spatially varying material properties on the bimoment normal and shear stresses by warping torsion of FGM beams

In this contribution, which is an extension of author’s research [1,2], an influence of the in two and three directions varying material properties on non-uniform torsion of the Functionally Graded Material (FGM) thin-walled beams is originally investigated. Based on the semi-analytical solution of the fourth order differential equation for non-uniform torsion, the local finite element equations of the twisted FGM beam are presented, considering the non-uniform torsion with effect of warping and secondary deformations due to the angle of twist. The warping part of the first derivative of the twist angle caused by the bimoment is considered as an additional degree of freedom at the beam nodes. The Multi-Layers Method (MLM) [3], and the Reference Beam Method (RBM) [4] are extended for homogenization of the spatially varying material properties and relevant stiffnesses in the real beam onto effective longitudinally varying ones in the homogenized beam for the load case of non-uniform torsion.The focus of the numerical investigation, with consideration of the warping and Deformation Effect due to the Secondary Torsional Moment (STMDE), is on elastostatic analysis of straight FGM beams with rectangular hollow cross-sections. The influence of the variation of the material properties with respect to two or three directions on the angle of twist and the bimoment normal and torsional shear stresses is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid finite elements. This FGM warping non-uniform torsion beam finite element can be used not only in torsional analysis of general thin-walled beams but also in the design of FGM sensors and actuators in mechatronics.

The mechanics of built-up cold-formed steel members

Abstract The paper presents methods of analysis of built-up sections in which the discrete locations of fasteners is accounted for explicitly, rather than by smearing their effect using continuous shear flexibility as in current approaches. By considering fasteners at discrete points, it is possible to analyse the effects of placing additional fasteners at the ends (end fastener groups), to account directly for actual support conditions and to determine the optimum locations of fasteners. The paper first outlines the linear analysis of beams in flexure and introduces the notion of the effective flexural rigidity to account for partial composite actions. Closed form solutions are provided for five load and end support cases to demonstrate the application of the analysis. Next, the paper describes the linear analysis of built-up sections in torsion, considering first uniform torsion followed by non-uniform torsion. Closed form solutions are obtained for the effective torsion rigidity (GJeff) of built-up sections featuring closed loops. A framework is also presented for determining the effective torsion rigidity (EIw,eff) of open built-up sections in non-uniform torsion. The paper concludes with the analysis of built-up sections subject to flexural buckling. A general variational buckling equation is derived followed by an energy-type method for calculating buckling loads for common end support conditions, including columns supported on flexible end tracks. Closed form solutions are presented for up to seven rows of fasteners longitudinally. While by nature approximate, the solutions are shown to be highly accurate. Comparisons are made between the presented closed form solutions and buckling load predictions obtained using current design provisions.

An improved analysis of a STB specimen for fracture characterization of laminates and foam-cored sandwich composites under mode III loads

This work presents the development of an improved data reduction method for the shear-torsion-bending (STB) test designed for monolithic laminates. The analytical derivation extends the applicability of the STB rig to composite sandwich specimens subjected to an out-of-plane shear loading. The data reduction method consists of an analytical model that expresses the global energy release rate in terms of applied loads, specimen geometry and material properties. The mathematical derivation of the energy release rate relies on first order shear deformation theory, Vlasov theory for non-uniform torsion of beams and near tip effects are also taken into consideration by the analytical model. Face sheets and core are modelled as homogeneous, linear elastic and orthotropic materials. The analytical expression is verified using the energy release rate extracted from a high-fidelity 3D FE based fracture mechanics model of the specimen. A compliance based method is used to generate global predictions, while local predictions are extracted using a displacement-based mode separation method. Local predictions are used to discuss accuracy and limitations of the approximate analytical model.

Finite beam element with exact shape functions for torsional analysis in thin-walled single- or multi-cell box girders

In this paper, a new formulation of beam-type finite element for the non-uniform torsion of simple or continuous thin-walled single- or multi-cell box girder with open and closed cross-sections is developed, considering both torsional warping and shear deformation effects (primary shear deformation due to Saint-Venant torsion and secondary shear deformation due to restrained torsion). The warping displacement of the cross-section is defined as the product of warping function representing the warping moment along the girder axis and a series of segmental shape functions describing the warping shape of each wall of the cross-section. Two different warping functions haven been defined and incorporated in Model A and Model B. The girder is subjected to arbitrary distributed or concentrated twisting moment and restrained by the general torsional-bending boundary conditions. The governing differential equations pertaining to torsion and torsional warping are obtained through the principle of minimum potential energy and the proposed finite beam element is refined by selecting closed-form homogeneous solutions of the differential equations as interpolation functions. Numerical examples are presented regarding rectangular hollow section (RHS) girders and single- or multi-cell box girders with open and closed cross-sections and the results obtained are compared with those provided by pioneering work or examined by using commercial finite-element models to validate the proposed beam-type element and to demonstrate the wide range of applicability and the convenience of using it.

Non-uniform torsion of functionally graded anisotropic bar of an elliptical cross section

An analytical solution is given to the problem of non-uniform torsion of an elliptical cylinder made of functionally graded anisotropic linear elastic material. The material moduli of the considered anisotropic non-homogeneous elastic bar are smooth functions of the axial coordinate. The contour of the elliptical cross section depends on the elastic constants. This dependence provides the zero warping property of the considered elliptical cross section. The obtained stress field is independent of the axial coordinate as in the case of Saint-Venant’s torsion problem, but the rate of twist depends on the axial coordinate.

Torsional vibration technique for the acoustoelastic characterization of concrete

There is no nondestructive evaluation method capable of determining stresses in concrete structural members in situ. Here, we propose and evaluate a torsional-vibration testing technique that offers an approach to estimate the stress state in concrete specimens by means of characterizing the material nonlinearity. We axially load three prismatic specimens comprising different concrete mixtures and measure their torsional vibration frequencies during four loading cycles. The fundamental torsional frequency shows a positive correlation with applied compressive stress for both loading and unloading stages after correcting for the effects of non-uniform torsion, geometric nonlinearity and changing boundary conditions. To quantify this behavior, we define the nonlinear parameter $$\beta_{G}$$ to characterize the material nonlinearity (acoustoelasticity). The values of $$\beta_{G}$$ of the initial loading cycle are lower than those of the subsequent loading cycles. However, the latter values of our concrete mixtures are consistent and similar to the values computed from previously published results. An estimate of the $$\beta_{G}$$ parameter of a concrete structural member provides a pathway for nondestructive assessment of in situ compressive stress in the member.