Uniform Torsion Research Articles

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.

Stable Möbius Bands from Isometrically Deformed Circular Helicoids

We consider the problem of producing a ruled Möbius band by subjecting an unstretchable, homogeneous, isotropic, elastic material surface material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have uniform torsion inversely proportional to the pitch of the helicoidal reference configuration. Granted that the energy density of the material surface depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a ruled Möbius band as described reduces, in closed form and without approximation, to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and construct corresponding stable Möbius bands. The only input parameter entering our variational problem is the number ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u $\\end{document} of turns in a helicoidal reference configuration. We only find solutions if ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u $\\end{document} exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraints leads to a multiplicity of coexisting stable solutions with n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n\\ge 3$\\end{document} half twists. For each n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n\\ge 3$\\end{document}, we construct an energetically optimal Möbius band which exhibits n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-fold rotational symmetry. All other energy minima yield Möbius bands which lack symmetry. To our knowledge, this study contains the first examples of stable Möbius bands produced by isometrically deforming reference configurations that are not flat.

A linearization solution for elastic-plastic torsion problems by Edge-based smoothed finite element method

This paper proposes a new scheme for calculating the elastic-plastic uniform torsion of rods considering the hardening behavior of the material. Using PRANDTL-REUSS equation under Hencky proportional loading condition, we develop an elastic-plastic torsion governing equation that can be applied to materials with arbitrary multilinear hardening forms. The edge-based smoothed finite element method is applied to solve the boundary value problem of the above equation. In addition, the incremental scheme is used to calculate the plastic torque. Using the Lagrange interpolation method, the Mises stress is calculated by calculating the gradient in the cell. The nodal stresses are obtained by averaging the stresses at the centroid of the cells. After each load step, the overall domain is divided into elastic and plastic zones according to the yield stresses of the material, and then different governing equations are applied in the different zones. To summarize, this paper provides linearization solutions for nonlinear torsion.

Torsional instability of functionally graded porous toroidal shell segments including elastic edge constraints and elevated temperature

This article aims to analyze the nonlinear instability of toroidal shell segment (TSS) made of functionally graded porous material (FGPM), exposed to elevated temperature and subjected to uniform torsion. The volume fraction of two material constituents is varied through the shell thickness according to a power-law function, while porous distribution into FGPM is modeled in forms of cosine functions. Governing equations in terms of deflection and stress function are established within the framework of classical shell theory incorporating geometrical nonlinearity. Two edges of the TSS are assumed to be simply supported and elastically restrained. Multi-term analytical solutions are assumed to satisfy boundary conditions and Galerkin method is applied to derive closed-form expressions of nonlinear load–deflection relations and buckling loads. Parametric studies are carried out to analyze the separate and combined influences of material and geometry properties, porous coefficient and distributions, degree of edge constraint and elevated temperature on the buckling resistance and postbuckling strength of torsionally loaded FGPM TSSs.

English

Bounds for the torsional rigidity of cylindrically anisotropic bars with one plane of elastic symmetry perpendicular to the axis of bar are derived making of use the two minimum theorems of elasticity. All results of the paper are based on the theory of uniform torsion which was developed by Saint-Venant and Prandtl. Illustrative example shows that the one term approximation leads to relative close bounds.

Thermo-torsional postbuckling of CNT-reinforced composite toroidal shell segments with surrounding elastic media and tangentially restrained edges

An analytical investigation on the nonlinear stability of toroidal shell segment (TSS) made of carbon nanotube (CNT)-reinforced composite, surrounded by an elastic medium, exposed to elevated temperature and subjected to uniform torsion is presented in this paper. The properties of constituents are assumed to be temperature dependent and effective properties of composite are determined using an extended rule of mixture. CNTs are embedded into matrix phase according to functionally graded or uniform distributions. Basic equations in terms of deflection and stress function are established within the framework of classical shell theory including geometrical nonlinearity in von Kármán–Donnell sense and interactive pressure from surrounding elastic medium. Two boundary edges of the shell are assumed to be simply supported and tangentially restrained. Multi-term analytical solutions are assumed and Galerkin method is used to derive expressions of buckling load and nonlinear relation between torsional load and deflection. Parametric studies are carried out to analyze the effects of material and geometry properties, in-plane boundary condition, elevated temperature and surrounding elastic medium on the buckling resistance and postbuckling behavior of torsionally loaded TSS. Novel finding of this study is that tangential edge constraints have no and negative effects on critical torsional loads at room and elevated temperatures, respectively, and profoundly beneficial influences on postbuckling load capacity of TSSs.

Thermo-torsional buckling and postbuckling of thin FGM cylindrical shells with porosities and tangentially restrained edges

This paper aims to analyze the effects of porosities, tangential constraints of boundary edges and elevated temperature on the buckling and postbuckling behaviors of thin functionally graded cylindrical shells subjected to uniform torsion. The porosities are assumed to exist within functionally graded material (FGM) according to even and uneven distributions. The properties of constituents are assumed to be temperature dependent and effective properties of porous FGM are determined using a modified rule of mixture. Mathematical formulations are established within the framework of classical shell theory taking into account von Kármán–Donnell nonlinearity and elasticity of tangential edge constraints. Multi-term analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is used to determine buckling torsional load and load–deflection relation for postbuckling analysis. Numerical results show that porosities have remarkably deteriorative influences on the torsional buckling resistance and postbuckling load capacities of FGM cylindrical shells. This study also reveals that tangential edge constraints have pronouncedly beneficial and detrimental effects on the postbuckling strength of torsion-loaded porous FGM cylindrical shells at room and elevated temperatures, respectively.

Saint-Venant torsion of thin-walled nonhomogeneous elliptical tube

This paper deals with the uniform torsion of thin-walled elliptical tube. The material of the tube is nonhomogeneous and it depends on one of the curvilinear coordinates which defines the cross section of thin-walled bar with closed profile. The approximate solution for the stresses, torsion function and torsional rigidity are obtained by the application of two extreme value theorems of linearized elasticity.

Two Theorems on the Torsional Rigidity of Piezoelectric Beams

In this paper two inequalities are presented for the torsional rigidity of homogeneous monoclinic piezoelectric beams. All results of the paper are based on the Saint-Venant theory of uniform torsion. The cross section of the considered elastic and piezoelectric beams may be simply connected or multiply connected two-dimensional bounded plane domain. Examples illustrate the proven inequality relations.

Assessment of thermal behavior of variable step twist in the elliptical spiral tube heat exchanger

In this paper, the thermal performance of three different models includes novel torsion step-pitch step, three-bladed spiral tube-triple turbinate tube (TTT), and elliptical spiral tube-oval turbinate tube-elliptical helical tube (TOT) are investigated. Through comparing different turbulence models (RNG k-ε, realizable k-ε, RSM, standard k-ω, and SST k-ω) with the experimental value, the SST k-ω model with high precision is selected to estimate Nusselt number and friction coefficient in a triple spiral tube a uniform torsion step. In this model, the maximum error between numerical and experimental results are 2.7% and 4.7%, respectively. This comparison focused on the tube's thermal performance for heat exchangers; therefore, Nusselt number, pressure drop, and PEC number are considered as critical factors for analyses. The results show that the variable twist spiral tube dramatically enhances thermal performance compared to other cases. The study also confirmed that a constant pitch step in the helical tube has better performance than a variable pitch step.

Estimation of the Torsional Rigidity of Orthotropic Solid Cross Section

In this paper, two inequality relations are proven for the torsional rigidity of orthotropic elastic solid cross sections. By using the derived inequality relations, lower and upper bounds can be obtained for the torsional rigidity. All results of the paper follow from the Saint-Venant theory of uniform torsion. The presented bounding formulae are based on the mean value theorem of integral calculus.

A novel analytical solution for warping analysis of arbitrary annular wedge-shaped bars

Structural components with arbitrary cross sections play a key role in several engineering fields. Many engineering structures are subjected to torsional moments, so it is important to design and analyze these type of crucial engineering issues. First, this study offers a novel analytical solution for Prandtl’s stress distribution of arbitrary annular wedge-shaped bars under uniform torsion moment based on eigenfunction expansion. Next, warping function is derived by integration of Cauchy-Riemann type relations in polar coordinate system. The solution encompasses existing solutions for standard wedge-shaped bars as subsets. Finally, accuracy of the proposed analytical method is fully demonstrated through some benchmarks which are available in the literature.

Saint-Venant’s torsion of thin-walled nonhomogeneous open elliptical cross section

This paper deals with the Saint-Venant’s torsion of thin-walled isotropic nonhomogeneous open elliptical cross section whose shear modulus depends on the one of the curvilinear coordinates which define the cross-sectional area of the beam. The approximate solution of torsion problem is obtained by variational method. The usual simplification assumptions are used to solve the uniform torsion problem of bars with thin-walled elliptical cross-sections. An example illustrates the application of the derived formulae of shearing stress and torsional rigidity.

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.

Lateral-distortional buckling of beams with hollow flanges and folded plate webs

This paper is concerned with the effects of distortion on the lateral buckling of steel beams which have rectangular or square hollow flanges and folded plate webs. The hollow flanges potentially make very large contributions to the torsional stiffness, but flange distortion (in a shear mode) causes a significant reduction in the effective torsional stiffness and in the lateral buckling resistance.In this paper, the shear distortion of the hollow flanges is analysed, and the strain energy stored in a distorted flange is compared with the strain energy stored during uniform torsion of the flange. This comparison allows the development of a suitable parameter for use in the evaluation of the effect of flange distortion on the torsional stiffness of a hollow flange beam. Uniform torsion test results are used to confirm the significance of this parameter.Finite strip analyses of simply supported hollow flange beams in uniform bending show that there is a significant interaction between distortion and uniform torsion during lateral buckling, which may be approximated by combining their flexibilities. More accurate finite element analyses show that the finite strip predictions are conservative. An approximate method of predicting the effects of flange distortion on elastic lateral buckling is presented which shows good agreement with the finite element predictions.

New modifying truss model and numerical simulation of steel fiber reinforced concrete under pure torsion

This paper presents a new strut-and-tie model on steel fiber reinforced concrete (SFRC) members under uniform (Saint Venant) torsion. This truss model is intended to reproduce the structural behavior in ultimate limit state, once concrete is cracked. The proposed model is based on previous formulations developed by other researchers; the main novelty is the use of average value linear interpolation methods. The developed model can be considered for constant section members; can be applied to rectangular, T, L, circular, thin-walled, and single-cell and multi-cell hollow sections, among other shapes. Comparison with experimental and numerical results (developed through accurate finite element analysis) shows that the proposed model is more accurate than the proposed formulations.

05.29: Interaction curves for local and distortional buckling of polygonal tubes in combined torsion and axial loading

ABSTRACTThis paper investigates the interaction of uniform axial loading (compressive or tensile) and uniform torsion on the linear elastic buckling behaviour of long tubes with regular convex polygonal cross‐sections. The analyses pertain to tubes of sufficient length such that the results are independent of end boundary conditions. Using the generalised constrained finite strip method (cFSM), pure local buckling is examined for triangular through octagonal tubes. The interaction curves for pure local buckling are akin to those previously generated from the exact finite strip method and a square and octagonal tube are analysed in‐depth to assess the changes in interaction behaviour as the number of sides in the section is increased. The pure distortional buckling behaviour is examined for pentagonal through octagonal tubes using recent developments in Generalised Beam Theory (GBT) and an almost linear interaction between torsion and compression is found, as has been noted previously for circular tubes. The distortional interaction curves are essentially identical for all analysed tubes.

Investigations on Torsion of the Two-Chords Single Laced Members

The paper presents experimental and numerical studies to determine the equivalent second moment of area of the uniform torsion of the two-chord steel single laced members. The members are used as poles of railway traction network gates, and steel columns of framed buildings as well. The stiffness of uniform torsion of this kind of columns allows to the determine the critical loads of the spatial stability. The experimental studies have been realized on a single - span members with rotation arrested at their ends, loaded by a torque applied at the mid-span. The relationship between angle of rotation of the considered cross-section and the torque has been determined. Appropriate numerical model was created in the ABAQUS program, based on the finite element method. A very good compatibility has been observed between experimental and numerical studies. The equivalent second moment of area of the uniform torsion for analysed members has been determined by comparing the experimental and analytical results to those obtained from differential equation of non-uniform torsion, based on Vlasov’s theory. Additionally, the parametric analyses of similar members subjected to the uniform torsion, for the richer range of cross-sections have been carried out by the means of SOFiSTiK program. The purpose of the latter was determining parametrical formulas for calculation of the second moment of area of uniform torsion.

Lateral-torsional dynamic instability of uniform and double-tapered rectangular beams under harmonic shear deformation

Abstract This paper analyzes lateral-torsional dynamic instability of elastic beams with rectangular cross section having constant thickness, with the depth symmetric with respect to the midpoint and either uniform or tapered linearly in each half. Free vibrations are also investigated. The ends of the beam are prevented from twisting and are pinned with respect to bending in the weak direction. In the strong direction, one end of the beam is fixed and the other is subjected to transverse harmonic motion. This problem was motivated by “butterfly-shaped links” proposed for use in seismic mitigation. Uniform torsion is assumed. Frequencies of free vibration are computed, critical excitation frequencies for lateral-torsional instability are determined, and critical combinations of excitation amplitude and frequency are obtained. The effects of geometric parameters of double-tapered beams on instability are presented. Lateral-torsional instability may occur for very small amplitudes of the moving end of the beam, as is typical for such problems involving parametric resonance.

Modal buckling behaviour of long polygonal tubes in uniform torsion using the generalised cFSM

The buckling behaviour of thin-walled tubes with regular convex polygonal sections (hereafter the “convex” will be implied) has been analysed a number of times over the last half a century, yet literature on such members in torsion has traditionally been lacking. Despite recent advances, there is still significant scope to further expand understanding of the behaviour of these members in torsion. Using the generalised constrained finite strip method (cFSM) and a semi-analytical finite strip method (FSM) that utilises an augmented longitudinal displacement field of sines and cosines, the elastic buckling behaviour of long regular polygonal tubes in uniform torsion is assessed. The analysed polygons all have the same centreline perimeter and uniform thickness of constituent plate elements. It is found that the signature curves of critical stress vs. buckling half-wavelength converge as the number of sides in the tube is increased and that notable differences between the signature curves of tubes with consecutive numbers of sides occur at shorter and shorter half-wavelengths as the number of sides is increased, due to the decreasing width of the individual flats and also the occurrence of an increasing number of distortional modes. At very long half-wavelengths flexural buckling occurs in a helical-like manner, similar to the twisting of a hosepipe. The mechanics of this buckling mode are briefly investigated and it is noted that it is necessary to include all of the non-linear components of the in-plane shear strain, as opposed to only the flexural components, in order to obtain accurate solutions for this mode. Results for pure local and pure distortional buckling are also compared to those obtained via Generalised Beam Theory (GBT).