Torsional Warping Research Articles

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).

Torsional load-bearing capacity of half-shell segments for prestressed concrete towers

This article deals with the torsional load-bearing behaviour of concrete half-shell structures. The analogy between the bending stressed tension members according to second order theory and torsional members with warping hindered cross-section leads to a significantly simplified calculation of the additionally occurring warping stresses. This eliminates the need for the laborious solution of complex differential equations of the classical torsional warping theory to determine the load-bearing behaviour of warped cross-sections. The load redistribution approach projects the global torsion load of the segment tower onto the individual half-shell. The combination of the two approaches enables the analytical calculation of the torsional load-bearing behaviour of the concrete half-shell construction as a function of the global torsional moment and the system variables of the half-shell.The objective is to investigate and describe the torsional load-bearing behaviour of segmented half-shell structures in an engineering model. Compared to closed circular ring segments, the vertical joints have a significant influence, especially on the torsional rigidity. In order to determine the horizontal joint bearing capacity of these innovative constructions, it is essential to understand the load transfer of the torsional load. The presented approach allows to separate the individual effects of the load transfer in half-shell constructions from each other and thus to identify the factors influencing the torsional load-bearing behaviour.

Free vibration behaviour of thin-walled concrete box-girder bridge using Perspex sheet experimental model

Purpose: Curved box-girder bridges offers an excellent solution to the problems associated with traffic congestion. However, owing to their complex geometry, they are subjected to shear lag, torsional warping and cross-sectional distortion, which must be assessed in their study and design. Furthermore, the dynamic behaviour of curved bridges adds to the complexity of the issue, emphasizing the importance of studying free vibration. The purpose of this study is to numerically model the concrete curved box-girder bridge considering torsional warping, distortion and distortional warping effects and to identify key parameters that influence the free vibration response of the box-girder bridge by validating it with experimental and analytical studies. Design/methodology/approach: The concrete bridge is numerically modelled by means of computationally effective thin-walled box-beam finite elements that consider torsional warping, distortion and distortional warping, which are prominent features of thinwalled box-girders. The free vibration analysis of the concrete curved box-girder bridge is performed by developing a finite element based MATLAB program. Findings: The identification of critical parameters that influence the free vibration behaviour of curved thin-walled concrete box-girder bridges is one of the main findings of the study. Each parameter and its effect has been extensively discussed. Research limitations/implications: The study limits for the preliminary design phase of thin-walled box-girder bridge decks, where a complete three-dimensional finite element analysis is unnecessary. The current approach can be extended to future research using a different method, such as finite element grilling technique on multi-span curved bridges having unequal span.

Three-dimensional nonlinear mixed 6-DOF beam element for thin-walled members

This paper presents a three-dimensional mixed beam element formulation for fully nonlinear distributed plasticity analysis of members composed of sections with no significant torsional warping such as steel angles and tees. This formulation is presented using a corotational total Lagrangian approach and implemented in the OpenSees corotational framework. In this context, a basic coordinate system is lined up with the element chord and translates and rotates as the element deforms. The element tangent stiffness matrix and resisting forces in the basic system are derived through linearization of the two-field Hellinger-Reissner variational principle. The displacement shape functions are cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation. The generalized stress resultant shape functions are linear for moments and constant for axial force and torque with the P - δ effect considered, which are developed from equilibrium equations. The fiber section method with uniaxial constitutive laws is adopted to account for material nonlinearity. Since the degrees-of-freedom in the basic system are defined with respect to different reference points, all element responses are transformed to acting about the shear center before conducting the corotational transformation. The mixed element is validated through a number of experimental and numerical examples.

A quasi-3D finite element model for the analysis of thin-walled beams under axial–flexural–torsional loads

A quasi-3D finite element model (FEM) with a low computational cost is presented for coupled bending and torsional–warping analysis of thin-walled beams. The presented FEM converts the problem of 3D analysis of thin-walled beams into separated 2D cross-sectional and 1D modeling. The warping function over the cross-section of the thin-walled beams is firstly approximated, via a 2D FEM. Then, a 1D FEM is developed for computing the axial variations of the displacement field variables. For describing the kinematic of the thin-walled beam due to axial extension and bending, the concept of hypothetical equivalent layered composite cross-section (HELCS) is employed. Using the HELCS, the non-classical effects like transverse shear flexibility can be easily incorporated into the formulation. A refined sinus theory is employed to describe the displacement fields of the equivalent layered cross-section. In contrast to other refined models of literature, the present model does not require the calculation of any geometric, torsional and warping constants. For validation of the proposed model, comparisons are made with the results of 2D/3D finite element analysis as well as the numerical and analytical results reported by other researchers. The proposed model not only gives accurate results but also reduces the number of involved degrees-of-freedom (DOFs) significantly. Considering the effects of the restrained warping due to torsion, no need to use the shear correction factor, satisfaction of zero-conditions of shear stresses on the exterior surfaces of beams are other advantages of the proposed FEM.

DYNAMIC PROBLEM FOR A THIN-WALLED BAR WITH A MONOSYMMETRIC PROFILE

The paper presents an analytical solution to the dynamic problem for a thin-walled elastic rod, thecross-section of which has one axis of symmetry. The solution is constructed for an arbitrary dynamic load and two types of boundary conditions: hinged support in constrained torsion and free warping of the end sections of the rod; rigid fastening with constrained torsion and absence of warping. The peculiarity of the mathematical model lies in the fact that the differential equations of motion contain a complete system ofinertial terms. Spectral expansions obtained as a result of using the method of integral transformations are represented as an effective method for solving linear non-stationary problems in mechanics. The structuralalgorithm of the method of finite multicomponent integral transformations proposed by Yu.E. Senitsky is used.

The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion

The paper discusses the influence of load and support conditions on the behaviour of sandwich panels subjected to torsion. 3-D numerical models are presented, in which various boundary conditions have been defined. The case of the load causing the concentrated torque in the span is analyzed, and the load definition affects the structural response. The numerical results were compared with the results obtained for the analytical beam model, which included both free torsion and secondary warping torsion. The conditions under which the models achieve a high agreement between the results were determined, but the significant sensitivity of the solution was also indicated. In each case of the considered load and boundary conditions, the structural response shows appropriate symmetry.

Enriched beam finite element models with torsion and shear warping for the analysis of thin-walled structures

This paper presents three beam Finite Element (FE) formulations developed for the analysis of thin-walled structures. These account for out-of-plane cross-section warping by removing the classical rigid body cross-section hypothesis and capture the interaction of axial/bending stress components with shear and torsion.The beam FE models rely on different kinematic assumptions to describe out-of-plane cross-section deformations. Indeed, warping displacement field is interpolated in the element volume according to different approaches, with increasing level of accuracy and detail. First two models adopt a coarse warping description, where warping displacement field is defined as the linear combination of assumed warping profiles and unknown kinematic parameters. In the first model, these are considered as equal to the generalized cross-section torsional curvature and shear strains and a classical displacement-based formulation is adopted to derive the element governing equations. In the second model, warping parameters are assumed as independent kinematic quantities and a mixed approach is considered to derive the FE formulation. Third model, also relying on a mixed formulation, independently interpolates warping by introducing additional degrees of freedom on the cross-section plane, thus, resulting in a richer description of the out-of-plane deformations. This latter is also adopted to propose a numerical procedure for the warping profile evaluation of thin-walled beams subjected to torsional and shear forces, for general cross-section geometry.The efficiency and accuracy of the proposed FE formulations are validated by simulating the response of thin-walled structures under torsion and coupled torsion/shear actions and the influence of the kinematic assumptions characterizing each formulation is discussed.

Investigation of the Warping Torsion of a Press Machine

In this article, the investigation of a press machine with 30 tons of pressing weight is presented. The beam of this machine is an I-beam, which has an open cross-section. It is known that this version of cross-section is sensitive to torsional stress. The stress from warping torsion is normal stress, so the opened cross-section is more sensitive to this type of stress. The bimoment that causes normal stress can also be very high, so dealing with this stress is very important.

Major axis steel joint with additional plates subjected to torsion: Stiffness characterization

This investigation deals with the characterization of the torsional behaviour of major axis steel joints with doubled extended end-plate and additional plates welded to the column flanges at the height of the beam flanges. One aim of this research is to provide the real joint torsional boundary conditions under which the beam elements behave for frame analysis and stability checks, thus avoiding the need for simplifications such as pinned or rigid joint conditions. Another objective is to characterize the stiffness of 3D joints, which requires knowledge of each of the individual components, such as the ones studied in this investigation.With this purpose two experimental tests were done, followed by a parametric study performed with calibrated finite element models that led to the analytical expressions of the components that characterize the torsional stiffness of this kind of joints. After a detailed analysis, these joints were found to behave as semi-rigid and full-strength. The semi-rigid behaviour is due to the torsional warping effects transmitted by the beam. The results are also compared with those of the same joints but without additional plates (described in a recent publication) and it is shown that the additional plates increase the joint stiffness significantly.

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.

One-dimensional finite element analysis of thin-walled box-girder bridge

Presented herein is a finite element approach for the static analysis of thin-walled box-girder bridge subjected to Indian railway loading using a three-noded one-dimensional beam element. The one-dimensional model is favoured over the three-dimensional model due to its simplicity and cost-effectiveness during the initial design and analysis phase of the box-girder bridge. The complex structural actions of thin-walled box-girders such as torsional warping and cross-sectional distortion are taken into consideration in addition to extension, bending and torsion. Keeping in mind these complex actions, the beam element incorporates three extra degrees of freedom in addition to the familiar six degrees of freedom per node. MATLAB coding is done to obtain various response parameters for single as well as double cell box girders. Furthermore, the variation of different response parameters due to changes in radius, span length, load position, varying cross-section and various combinations of Indian rail loads has also been studied. The finite element approach used in this study is validated by solving two numerical examples which are in excellent agreement with the results of previous studies.

Procedures of Fatigue Analysis by Supporting Direct Load Application on Midship Sections

Fatigue evaluation of ship structures using direct calculation methods to calculate fatigue loads are standard practice today. There are several numerical codes available for use in analyses of these fatigue loads. In addition to the varying degrees of computational complexity associated with fatigue prediction methods, the inherent uncertainties of these procedures are also large. This paper introduces the procedure for stochastic fatigue analysis of typical midship models with direct load transfer applied, where an oil tanker is considered. It also covers a comparison of the results with the component-based approach included in the DNVGL Class Note 30.7: “Fatigue Assessment of Ship Structures”. The "real" case analysis includes both internal pressure loads from tank fluids as well as external pressure adjusted for wet and dry surfaces in the waterline area, according to DNVGL Class Note 30.7. Local fine mesh models of fatigue details have been analysed using the sub-modelling technique. The procedure performs well on a typical midship model apart from the file sizes of the generated load transfer files. With 25 wave periods and 12 different headings, the analysed 3-cargo-hold model (1/2 + 1 + 1/2) in the midship area had to be split into four super elements in order to get the analysis through finite element analyses. The procedure is suitable for vessels where warping (torsion) is of less importance. The described procedures are supported by a developed tool to be used in the analysis procedures. Three details of local fine mesh models such as deck erection butt weld, longitudinal stiffener through web-frame, and bottom erection butt weld have been analysed. The results have been compared with the component-based approach. For some of the details there are comparable results, but for others the results vary significantly. The typical trend is that the details heavily influenced by the external pressure (side longitudinal) give less comparable results than e.g. a detail in the main deck mainly influenced by global loads. A comparison of the effect of reducing versus not reducing the pressure amplitude in the waterline on the fatigue life has also been performed and discussed.

Damping of coupled bending-torsion beam vibrations by spatially filtered warping position feedback

Damping of coupled bending-torsion beam vibrations is realized by local actuators, operating axially on an end cross-section of a fully three-dimensional structure by a collective positive position feedback control with a spatial filter that specifically targets the out-of-plane warping component of the coupled response. To reduce the computational effort, a beam element model is introduced, in which a single control bimoment is supplemented by a fictitious stiffness that accounts for the inability of the local actuators to fully restrain warping. This fictitious stiffness is conveniently calibrated by frequency matching in the limit of infinite control gain and it enables accurate estimation of both complex roots and the stability limit from beam element analysis. It is demonstrated by a numerical example that substantial modal damping is attained by positive position feedback of the torsional warping component from the fully coupled response.

Torsional warping elastostatic analysis of FGM beams with longitudinally varying material properties

In this paper, the influence of torsional warping of the cross-sections of thin-walled twisted Functionally Graded Material (FGM) beams on their elastostatic behavior is investigated. The material properties are assumed to vary longitudinally. The variation is described by a polynomial. Secondary deformations, resulting from the angle of twist, are considered. The transfer matrix method serves as the analysis tool. The transfer relations are derived and used for establishing finite element equations for twisted FGM beams in local coordinate systems. 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 nodes of the beam elements. The numerical investigation is performed with or without consideration of the Secondary Torsional Moment Deformation Effect (STMDE). It is focused on elastostatic analysis of straight cantilever FGM beams with doubly symmetric open as well as closed cross-sections. The influence of the longitudinal variation of the material properties and of the secondary torsion moment on the deformation and stress state is investigated. The results are compared with the ones obtained by a very fine mesh of standard solid and shell as well as warping beam finite elements.

Modelling of steel frames using advanced beam and joint elements with interfaces governed by beam modes

For decades, engineers have assessed and analysed steel frames using simple joints between beams and columns. These joints are often based on oversimplified assumptions using hinges or a direct transfer of beam displacements without any relative displacements. More seldom is the use of spring models that allow relative beam and column displacements at the joints. This despite the standardised component method approach, which can be used to determine the rotational spring stiffness of the relative rotation in a joint. This paper gives a background overview of essential developments in joint modelling and generalised thin-walled beam modelling, including torsional, distortional and related warping effects. For particular situations, some recent proposals for joint models can be applied to joints between thin-walled beams. On this basis, this paper presents a novel idea and a generic methodology that allows the interface between an extended number of generalised beam displacement modes and joints that are modelled using shell elements. The main novelty is the idea to transform from standard degrees of freedom of the interface into a reduced number of beam displacement mode related degrees of freedom. Thus, the number of degrees of freedom of the joint can be reduced to the corresponding total sum of beam modes that have been chosen for the modelling of each of the connected beam elements. The total number of degrees of freedom used for modelling the complete framework will depend on the selected number of modes in each beam element and on the number of extra internal modes chosen in the joint models. For enhanced structural analysis with advanced beam elements and joints that allow relevant distortions and built-in refined connection components, it is believed that this methodology will enable the full detailed analysis of large steel frameworks with a reasonable number of degrees of freedom.

On the Access to Transverse Shear Stiffnesses and to Stiffness Quantities for Non-Uniform Warping Torsion in FGM Beam Structures

Abstract The application of first order shear beam theory in the analysis of beam structures made of functionally graded materials requires the access to homogenized stiffness quantities. These quantities depend on the cross-sectional shape and on the spatial variation of constitutive parameters. Some of these stiffness quantities can be evaluated easily by simple integration, however, the access to transverse shear stiffnesses and to stiffness quantities regarding warping torsion is typically cumbersome. In this contribution a novel approach for their evaluation is proposed, which is based on a reference beam problem. Here, we restrict ourselves to double symmetric cross-sections, however, a generalization of the proposed method to the arbitrary case is obvious. Besides that, a novel approach to cover non-uniform warping torsion is included. The proposed method is efficient, since the discretization of the cross-section suffices, and accurate as can be shown in challenging bench mark problems.

On Problem of Torsional Characteristics of Thin-walled Steel Beams with Web Openings

The current trend in the design of steel structures leads, due to the saving of the material, to the frequent use of thin-walled cold formed steel sections. The thin-walled cold-formed steel profiles are often manufactured with web holes. In the design of such a thin-walled steel members with web openings arises a question of correct determination of the real crosssectional properties. This paper focuses on the problem of the real torsional characteristics determination, which are parameters needed for the design of members subjected to the bending with respect to lateral torsional buckling or for the design of the compressed members prone to torsional buckling or flexural torsional buckling. These torsional characteristics include St. Venant torsion constant and warping constant. In the forthcoming European Standard specifying the rules for the design of beams with holes, it is recommended to use the cross-sectional characteristics of the most weakened section of the member with web openings. This paper deals with the possibility of introduction of “substitute cross-section” whose cross-sectional characteristics are determined as the weighted average of the properties of full section and the most weakened section. The solution with substitute cross-section is validated by the series of tests focusing on the experimental verification of the both real torsional stiffness –- St. Venant torsion stiffness and warping torsion stiffness. Based on the results of test series with twisted beams freely supported in torsion, the St. Venant torsion constant is being derived, respectively from the results of the test series with twisted beams fixed in torsion (the warping is restrained at both beam ends), the warping constant is being derived. Both torsional characteristics are verified by tests executed on three different lengths (L=2.0m, L=3.0m and L=4.0m) of beams with sigma cross-section with large circular web openings.

Monitoring of Real-Time Complex Deformed Shapes of Thin-Walled Channel Beam Structures Subject to the Coupling Between Bi-Axial Bending and Warping Torsion

Structural health monitoring (SHM) is a research focus involving a large category of techniques performing in-situ identification of structural damage, stress, external loads, vibration signatures, etc. Among various SHM techniques, those able to monitoring structural deformed shapes are considered as an important category. A novel method of deformed shape reconstruction for thin-walled beam structures was recently proposed by Xu et al. [1], which is capable of decoupling complex beam deformations subject to the combination of different loading cases, including tension/compression, bending and warping torsion, and also able to reconstruct the full-field displacement distributions. However, this method was demonstrated only under a relatively simple loading coupling cases, involving uni-axial bending and warping torsion. The effectiveness of the method under more complex loading cases needs to be thoroughly investigated. In this study, more complex deformations under the coupling between bi-axial bending and warping torsion was decoupled using the method. The set of equations for deformation decoupling was established, and the reconstruction algorithm for bending and torsion deformation were utilized. The effectiveness and accuracy of the method was examined using a thin-walled channel beam, relying on analysis results of finite element analysis (FEA). In the analysis, the influence of the positions of the measurement of surface strain distributions on the reconstruction accuracy was discussed. Moreover, different levels of measurement noise were added to the axial strain values based on numerical method, and the noise resistance ability of the deformation reconstruction method was investigated systematically. According to the FEA results, the effectiveness and precision of the method in complex deformation decoupling and reconstruction were demonstrated. Moreover, the immunity of the method to measurement noise was proven to be considerably strong.

Torsional warping eigenmodes of FGM beams with longitudinally varying material properties

In this paper, the influence of torsional warping of thin-walled cross-sections of twisted Functionally Graded Material (FGM) beams with a longitudinal polynomial variation of the material properties on their eigenvibrations is investigated, considering the secondary deformations due to the angle of twist. The transfer relations needed for the transfer matrix method are derived. Based on them, the local finite element equations of the twisted FGM beam are established. 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 focus of the numerical investigation, with and without consideration of the Deformation due to the Secondary Torsional Moment (STMDE), is on modal analysis of straight cantilever FGM beams with doubly symmetric open and closed cross-sections. The influence of the longitudinal variation of the material properties and the secondary torsion moment on the eigenfrequencies is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid and warping beam finite elements.