Torsional Warping Research Articles

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.

Cross-section optimal design of composite laminated thin-walled beams

This paper aims to perform optimal design of cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. The finite element method is used in discretizing the theory. For design sensitivity calculations, the cross-section is modelled throughout design elements. Geometrically, these elements may coincide with the laminates that constitute the cross-section. The developed formulation is based on the concept of adjoint structure. After a warping function is calculated for the cross-section, an adjoint problem may be formulated for each of the properties and a corresponding adjoint warping is determined. It can be applied in a unified way to open, closed or hybrid cross-sections. Design optimization is performed by nonlinear programming techniques. Laminate thickness and lamina orientations are considered as design variables.

Theoretical analysis of coupled torsional, warping and distortional waves in a straight thin-walled box beam by higher-order beam theory

When a thin-walled box beam is subjected to a dynamic torsional load, warping and distortional deformations of its cross section are coupled with torsional deformation. Due to the coupling, the wave propagation along the beam axis becomes considerably complicated. However, there appears no in-depth investigation on dispersion phenomena. It is difficult to see the intriguing interconnection of branches of the dispersion curves by means of numerical solutions of three-dimensional elastodynamic equations. Thus, we aim to analytically investigate the dispersion relation and wave reflection phenomena by using a one-dimensional higher-order beam theory that is capable of representing the three aforementioned deformations. Through this study, the interconnections of real, imaginary and complex branches of the dispersion relation for low frequencies are revealed and reflection and transmission phenomena occurring at a joint of two beams of different cross-sectional dimensions are also investigated.

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.

Fully Coupled 10-Degree-of-Freedom Beam Theory for Piecewise Straight Thin-Walled Beams with General Quadrilateral Cross Sections

A fully coupled 10-degree-of-freedom (DOF) beam theory is developed for the analysis of thin-walled closed piecewise straight beams of general quadrilateral cross sections. The developed beam theory can predict the structural response of arbitrarily shaped thin-walled quadrilateral sectioned beams that are curved or connected to each other through a number of angled joints. For accurate prediction, the local effects by four beam sectional deformations such as torsional warping, torsional distortion, bending warping, and bending distortion must be taken into account in addition to the structural responses by standard six translational and rotational DOFs of the Timoshenko beam theory. When two straight beams of general quadrilateral sections meet at an angle, all 10 DOF deformations become fully coupled. Furthermore, it is not easy to derive the exact interface conditions at the angled joint because 10 DOFs have different physical behavior. For the analysis of the beam system in consideration, the section deformation patterns corresponding to all of the 10 DOFs are presented. In particular, the cross-sectional shape functions for bending distortion and bending warping for general quadrilateral sections are derived for the first time. Then, a systematic method to match 10 DOFs of two straight quadrilateral beams meeting at a joint of an arbitrary angle is developed. Several numerical case studies are considered to check the validity of the developed beam analysis.

Analytical Modeling of Cracked Thin-Walled Beams Under Torsion

We present an analytical model, derived using the principle of virtual work, for closed and open thin-walled beams subjected to torsion. Non-classical effects like primary and secondary torsional warping are taken into account. A closed-form expression for Mode II stress intensity factor for thin-walled beams with a longitudinal crack is derived. We start by finding analytical solutions based on the Vlasov’s torsional beam theory for thin-walled beams with any cross-sections. We then use the calc ulated stress of a rectangular thin-walled beam to determine an analytical expression for Mode II stress intensity factor. Finally, we add a correction factor to account for complex beha vior of long cracks. Examples are presented to illustrate the effectiveness of the cu rrent model, and it is validated by comparison with detailed finite element results obt ained using the ABAQUS TM .

Effect of Bracing on Linear Free Vibration Characteristics of Thin-Walled Beams with Open Cross Section

The paper presents an analysis of the coupled vibration of beams with arbitrary thin-walled open cross section, braced with identical transversal header beams uniformly distributed along their length. The explicit form of analytic solution is derived by directly solving the governing differential equations of motion. The development is based on Vlasov theory which includes the effect of flexural-torsion coupling, the constrained torsion warping, and rotary inertia. The governing differential equations for coupled bending-torsional vibrations are performed using the principle of virtual displacements. In the case of simply supported beam, exact explicit expressions are derived to predict the natural frequencies and the corresponding mode shapes. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad. The expressions are concise and very simple and as such convenient to be used by a practicing engineer who does not need to go into detail of thin-walled beam theory. Also, the use of explicit expressions gives significant savings in computing time compared with the alternative numerical methods [finite-element method (FEM), finite strip method, differential transform method, etc.]. To demonstrate the validity of this method the natural frequencies of braced thin-walled beams, having coupled deformation modes, are evaluated and compared with FEM.

Nonlinear reinforced concrete frame element with torsion

Abstract In this paper an element for nonlinear analysis of reinforced concrete framed structures subjected to torsion is developed using a force-based formulation. The interaction between the axial force and bending moment is taken into account by adopting the Navier–Bernoulli assumption and using the fibre element approach. The torsional DOFs are formulated independently and the effect of normal and tangential forces on the torsional stiffness of section is accounted for by modifying the torque–twist curve of a section under pure torsion. Furthermore, the torsional warping effects on the section stiffness are considered by adding an extra strain field to the section axial strain caused by normal generalised forces. The formulation accuracy and efficiency are verified by some numerical examples.

GBT-based local, distortional and global buckling analysis of thin-walled steel frames

This paper reports research work concerning the use of Generalised Beam Theory (GBT) to analyse the local, distortional and global buckling behaviour of thin-walled steel frames. After a brief review of the main concepts and procedures involved in performing a GBT buckling analysis, one addresses the formulation and implementation of a GBT-based beam finite element that incorporates local, distortional and global deformation modes – in particular, one describes (i) the definition of joint elements, which involves providing a relation between the connected member GBT degrees of freedom and the joint generalised displacements, and (ii) the kinematical models adopted to simulate the torsion warping transmission. Next, one uses evidence gathered from shell finite element analyses to establish kinematical constraint conditions that ensure cross-section warping and in-plane displacement compatibility at the frame joints connecting two non-aligned plain or lipped channel members. Finally, one presents and discusses numerical results that make it possible to illustrate the application and show the capabilities of the above GBT-based finite element formulation and implementation. For validation purposes, some GBT-based results (critical buckling loads and mode shapes) are compared with values yielded by shell finite element analyses carried out in the code ANSYS.

Finite element analysis of thin-walled composite laminated beams with geometrically nonlinear behavior including warping deformation

A finite element model for structural analysis of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including torsion warping deformation, is presented. A general continuum formulation considering the updated Lagrangean procedure and a generalized displacement control method, are used to describe the deformation of the structure. The beam cross-section geometry is discretized by quadratic isoparametric finite elements to determine its bending-torsion properties. The structural discretization is performed throughout three-dimensional two-node Hermitean finite beam elements, with seven degrees-of-freedom per node. Several applications are presented, addressing the influence of lamina orientation on the structural behavior.

Buckling and free vibration analysis of stiffened panels

The paper presents the application of the new stiffener element with seven degrees of freedom per node and the subsequent application in determining frequencies, mode shapes and buckling loads of different stiffened panels. In structural modelling, the stiffener and the plate/shell are treated as separate elements where the displacement compatibility transformation between the seven and six degrees of freedom nodes of these two types of elements takes into account the torsion–flexural coupling in the stiffener and the eccentricity of internal (contact) forces between the beam–plate/shell parts. The model becomes considerably more flexible due to this coupling technique. The development of the stiffener is based on a general beam theory, which includes the constraint torsional warping effect and the second-order terms of finite rotations. Numerical tests are presented to demonstrate the importance of torsion warping. As part of the validation of the results, complete shell and the usual six degrees of freedom per node shell–beam finite element analyses were made for stiffened panels.

The axial–torsional vibration of pretwisted beams

We investigate the coupled axial–torsional vibration of pretwisted beams. The equations of motion governing the extension, torsion, and cross-sectional warping of pretwisted beams are derived from Hamiltons principle, and the common assumptions used to simplify the equations are carefully examined through scaling analysis. Inconsistencies in previous works—such as the neglect of spatial and time derivatives of torsion in the Lagrangian functional—are remedied, giving rise to fourth-order terms in the equations of motion that are significant for higher harmonics. Furthermore, scaling analysis of the governing equations yields a set of objective criteria for checking the validity of the frequently misused assumption that the warping function under pretwist is locally identical to that of a prismatic beam; we show that this simplification is responsible for the common poor prediction of axial resonance frequencies for pretwisted beams.

Dynamic stiffness matrix of thin-walled composite I-beam with symmetric and arbitrary laminations

For the spatially coupled free vibration analysis of thin-walled composite I-beam with symmetric and arbitrary laminations, the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented. For this, a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force–displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters. Then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force–displacement relationships. In addition, the finite-element (FE) procedure based on Hermitian interpolation polynomials is developed. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with analytical solutions, the results from available references and the FE analysis using the thin-walled Hermitian beam elements. Particular emphasis is given in showing the phenomenon of vibrational mode change, the effects of increase of the modulus and the bending–twisting coupling stiffness for beams with various boundary conditions.

Nonlinear free vibration analysis of thin-walled curved beam with non-symmetric open cross section

A finite element formulation was presented for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies were derived by the virtual principle. The energy function includes the effect of flexural-torsional coupling, the torsion warping and the shear centre location. For finite element analysis, cubic polynomials were utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem was solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.

A Series Solution for the Free Vibrations of Thin-Walled Composite Beams

Based on series solutions, the free vibration analysis of thin-walled composite beam with channel sections is performed. For this, the dynamic stiffness matrix is presented using the homogeneous form of the simultaneous ordinary differential equations. The general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force-displacement relationships are derived from the energy principle and explicit expressions for displacement parameters are presented based on power series expansions of displacement components. Then the exact dynamic stiffness matrix is determined using force-displacement relationships. In addition, the finite element model based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the analytical solutions from other researchers and the finite element solutions using the Hermitian beam elements. Particularly, the effects of modulus orthotropy and boundary conditions on the vibrational behavior of thin-walled composite beam with various fiber orientations are investigated.

A beam finite element for non-linear analyses of thin-walled elements

The aim of the present paper is to investigate a theoretical and numerical model which is able to study the behaviour of thin-walled beams with open cross section in presence of large torsion. The presented model takes into account for large torsion, linear and non-linear warping currently named shortening effects, pre-buckling deformation and flexural–torsional coupling. In numerical analysis, a 3D beam with two nodes and seven degrees of freedom per node is adopted. The equilibrium equations and the material behaviour are derived in discrete form without assumption on torsion angle amplitude. Due to large torsion context, all the equilibrium equations are non-linear and highly coupled. The linear behaviour is made possible by disregarding non-linear terms. For non-linear behaviour and stability, the tangent stiffness matrix is carried out. Due to large torsion context, new matrices are present. The element is incorporated in a homemade finite element code. Newton–Raphson iterative methods are used with different control parameters. In order to prove the efficiency of the model many examples are presented in linear and non-linear behaviour with presence of bifurcations.

Fatigue life prediction of horizontally curved thin walled box girder steel bridges

The fatigue damage accumulation rates of horizontally curved thin walled box-girder bridge have been estimated from vehicle-induced dynamic stress history using rain flow cycle counting method in the time domain approach. The curved box-girder bridge has been numerically modeled using computationally efficient thin walled box-beam finite elements, which take into account the important structural actions like torsional warping, distortion and distortional warping in addition to the conventional displacement and rotational degrees of freedom. Vehicle model includes heave-pitch-roll degrees of freedom with longitudinal and transverse input to the wheels. The bridge deck unevenness, which is taken as inputs to the vehicle wheels, has been assumed to be a realization of homogeneous random process specified by a power spectral density (PSD) function. The linear damage accumulation theory has been applied to calculate fatigue life. The fatigue life estimated by cycle counting method in time domain has been compared with those found by estimating the PSD of response in frequency domain. The frequency domain method uses an analytical expression involving spectral moment characteristics of stress process. The effects of some of the important parameters on fatigue life of the curved box bridge have been studied.

Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open cross section

A finite element formulation is present for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies are derived by the virtual principle. The energy functional includes the effect of flexural–torsional coupling, the torsion warping and the shear center location. For finite element analysis, cubic polynomials are utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem has been solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.

Dynamic behavior of braced thin-walled beams

The present paper considers the dynamic behavior of beams with open thin-walled cross-sections along their length with bracings (connecting beams and truss). The effect of the constrained torsion warping, rotary inertia, and flexural-torsion coupling due to nonsymmetric cross-sections is included. In the case of simply supported beams, closed-form solutions for determining coupled natural frequencies and corresponding mode shapes are newly derived. The frequency equation, given in a determinantal form, is expanded in an explicit analytical form and then solved using the Mathcad 2001 Professional symbolic computing package. Some illustrative examples on the application of the present theory are given for coupled bending-torsion vibrations of braced thin-walled beams. As compared with FEM, numerical results demonstrate the accuracy and effectiveness of the proposed method

Vibration analysis of 3-D composite beam elements including warping and shear deformation effects

In this paper, the dynamic analysis of 3-D composite beam elements restrained at their edges by the most general boundary conditions and subjected in arbitrarily distributed dynamic loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14×14 stiffness matrix and the nodal load vector of a member of an arbitrarily shaped composite cross section, taking into account both torsional warping and shear deformation effects, which together with the corresponding mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach. Both free and forced vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy.