Shear Center Research Articles

Profile beams with adaptive bending–twist coupling by adjustable shear centre location

Semi-active structural elements based on variable stiffness represent a promising approach to the solution of the conflict of requirements between load-carrying capability and shape adaptivity in morphing lightweight structures. In the present work, a structural concept with adaptive bending–twist coupling aiming at a broad adjustment range of coupling stiffness while maintaining high flexural rigidity is investigated by analysis, simulation and experiment.

Structural design optimization of a tiltrotor aircraft composite wing to enhance whirl flutter stability

In order to enhance the aeroelastic stability of a tiltrotor aircraft, a structural optimization framework is developed by applying a multi-level optimization approach. Each optimization level is designed to achieve a different purpose; therefore, relevant optimization schemes are selected for each level. Enhancement of the aeroelastic stability is selected as an objective in the upper-level optimization. This is achieved by seeking the optimal structural properties of a composite wing, including its mass, vertical, chordwise, and torsional stiffness. In the upper-level optimization, the response surface method (RSM), is selected. On the other hand, lower-level optimization seeks to determine the local detailed cross-sectional parameters, such as the ply orientation angles and ply thickness, which are relevant to the wing structural properties obtained at the upper-level. To avoid manufacturing difficulties, only a few discrete ply orientation angles and an integral number of plies are considered as constraints. A genetic algorithm is selected as the optimizer at the lower-level. Use of the upper-level optimization causes a 13–18% increase in the flutter speed when compared to the baseline configuration. In the lower-level optimization, the optimization results were obtained considering the resulting failure margin and the location of the shear center.

Free Vibration Analysis of Cross-Ply Laminated Thin-Walled Beams with Open Cross Sections: Exact Solution

AbstractThe objective of the present paper is to analyze the free vibrations of thin-walled beams with arbitrary open cross section, made of cross-ply laminates with midplane symmetry, by means of an exact solution. The theory of thin-walled composite beams is based on assumptions consistent with Vlasov’s beam theory and classical lamination theory. The governing differential equations for coupled bending-torsional vibrations were obtained using the principle of virtual displacements. To simplify the coupled system of differential equations, an ideal center of gravity and shear center were introduced. In the case of a simply supported thin-walled beam, the closed-form solution for the natural frequencies of free harmonic vibrations was derived. The frequency equation, given in determinantal form, is expanded in an explicit analytical form. To demonstrate the validity of this method, the natural frequencies of nonsymmetric thin-walled beams having coupled deformation modes are evaluated and compared with r...

Notes on the Centre of Shear

Two methods are known for calculating the position of the shear centre of a bar. By using them we obtain different results. Some authors state that the position of the centre of shear depends on the Poisson ratio, whereas others that it does not. The purpose of this paper is to give a new interpretation for this disagreement.

Sleeve connections of cold-formed steel sigma purlins

Cold-formed steel sigma purlins normally possess advantageous structural features such as an enhanced web buckling resistance and a location of shear centre closer to the web. In modern roof construction, typical connections for sigma purlins include butt, continuous and sleeve systems. The sleeve connection is often favoured due to ease of installation and lack of restriction on the length of connected purlins. The inherent characteristic of this type of connection is an important factor affecting the structural behaviour of the entire purlin system. As the rotational stiffness and moment resistance will directly influence the stiffness and ultimate load of the purlin system, the determination of these quantities is therefore imperative for purlin design. This paper reports on an experimental study into the sleeve connection of sigma purlins. A series of laboratory tests were carried out to determine the moment–rotation behaviour as well as the moment resistance of this type of connection. Engineering models were then developed to predict these connection behaviours and results matched the experimental observations well. The experimental results reported in this paper provide a well-documented first-hand data for sleeve connections of cold-formed steel sigma sections in terms of moment resistance, rotational stiffness and typical failure modes. The proposed analytical models provide rational engineering methods that can be used in the design practice. Such design methods are not available in any design codes or specifications.

On the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous Saint–Venant beam theory

The relationship between twist and shear centres in an orthotropic Saint–Venant beam, with fiberwise homogeneous elastic moduli and constant Poisson ratios, is investigated. Arbitrary cross-sections are considered. As a new result the relative position of these points is expressed in terms of the scalar potential whose gradient is the rotated field of twist tangential stresses. Its evaluation requires the solution of n+1 boundary value problems, being n⩾0 the number of holes in the cross-section. In an isotropic and homogeneous beam the potential is Prandtl stress function and known formulae, providing the relative position of twist and shear centres, are recovered. Explicit expressions of sliding-torsional compliance blocks for Timoshenko beams, defined by an energy condition of equivalence with the orthotropic and fiberwise homogeneous Saint–Venant theory, are provided. Coincidence of twist centre and Timoshenko shear centre is proven. Numerical computations on homogeneous and composite orthotropic L-sections are performed.

Multi-material topology optimization of laminated composite beam cross sections

This paper presents a novel framework for simultaneous optimization of topology and laminate properties in structural design of laminated composite beam cross sections. The structural response of the beam is evaluated using a beam finite element model comprising a cross section analysis tool which is suitable for the analysis of anisotropic and inhomogeneous sections of arbitrary geometry. The optimization framework is based on a multi-material topology optimization model in which the design variables represent the amount of the given materials in the cross section. Existing material interpolation, penalization, and filtering schemes have been extended to accommodate any number of anisotropic materials. The methodology is applied to the optimal design of several laminated composite beams with different cross sections. Solutions are presented for a minimum compliance (maximum stiffness) problem with constraints on the weight, and the shear and mass center positions. The practical applicability of the method is illustrated by performing optimal design of an idealized wind turbine blade subjected to static loading of aerodynamic nature. The numerical results suggest that the proposed framework is suitable for simultaneous optimization of cross section topology and identification of optimal laminate properties in structural design of laminated composite beams.

Twist control of aerodynamic profiles by a reactive method (experimental results)

Active twist control of airfoils using embedded actuators has been widely studied during the last decade. To control the twist, an additional torsional moment is applied using these actuators. This active method is efficient but encounters limitations due to the amount of energy necessary to activate the system. An alternative method to this energy costly active control is proposed. Rather than applying an additional torsional moment to control the twist of the structure, the shear center of the profile is shifted using an internal mechanical system. More precisely, the activation of a clutch-like device, able to link or unlink thin internal walls, allows the redistribution of the bending and torsional shear stresses. This process leads to a possible twist control of the airfoil, significantly less energy consuming than an active one. Unlike the active twist control method, this method needs an external force to be effective, so it is called a “reactive’’ control method. Experimental investigations have been performed to verify the efficiency of this method. The results show that it is possible to obtain significant twist angle modifications. Moreover, the measurement of internal longitudinal displacements, correlated with the twist angle, enables the twist control of the structure.

Structural Analysis Programme Based on the Equivalent Column’s Method

The focus of this article is to accomplish a program using MatLab software, which will determine the global behavior of the structure, the stress, stability and frequency of the tall building with n levels. The program starts with the global analyses based on the equivalent column theory. The central core consists of reinforced concrete shear walls of the elevator's tubes and of the staircase, design to resist the lateral loads. Applying the equivalent column's theory, the bracing system will be replaced by an equivalent column, considered to be a vertical cantilever fixed at the base and the loads will be applied in the shear center axis. The program using mathematical and physical relations, has as input dates the geometrical and stiffness characteristics of the plan of the building and as output dates the critical load, the fundamental frequency, the maximum stresses and deformations of the structure.

On torsion and shear of Saint-Venant beams

Torsion and shear stress fields of a Saint-Venant beam and the relative location of shear and twist centres are investigated for sections of any degree of connectedness. The sliding-torsional compliance tensor of a Timoshenko beam is evaluated by an energy equivalence with Saint-Venant theory. Accordingly, the mutual sliding-torsional term is shown to depend linearly on the relative position of shear and twist centres and the standard definition of shear centre in a Timoshenko beam is found to be coincident with Saint-Venant twist centre. Coincidence of shear and twist centres is assessed for sections with vanishing Poisson ratio and for open, closed and multi-cell thin-walled cross sections. The eigenvalues of the shear factors tensor and the torsion factor are shown to be greater than unity, with the principal directions of shearing and bending compliances non necessarily coincident for non-symmetric cross sections. Numerical examples are developed to provide evidences of the location of the centres and of the principal shearing directions, for non-symmetric L-shaped cross sections with various thickness ratios.

Elastic lateral buckling of cantilever Litesteel Beams under transverse loading

The LiteSteel Beam (LSB) is a new hollow flange channel section developed by OneSteel Australian Tube Mills using its patented dual electric resistance welding and automated continuous roll-forming technologies. The LSB has a unique geometry consisting of torsionally rigid rectangular hollow flanges and a relatively slender web. Its flexural strength for intermediate spans is governed by lateral distortional buckling characterised by simultaneous lateral deflection, twist and web distortion. Recent research on LSBs has mainly focussed on their lateral distortional buckling behaviour under uniform moment conditions. However, in practice, LSB flexural members are subjected to non-uniform moment distributions and load height effects as they are often under transverse loads applied above or below their shear centre. These loading conditions are known to have significant effects on the lateral buckling strength of beams. Many steel design codes have adopted equivalent uniform moment distribution and load height factors based on data for conventional hot-rolled, doubly symmetric I-beams subject to lateral torsional buckling. The non-uniform moment distribution and load height effects of transverse loading on cantilever LSBs, and the suitability of the current design modification factors to include such effects are not known. This paper presents a numerical study based on finite element analyses of the elastic lateral buckling strength of cantilever LSBs subject to transverse loading, and the results. The applicability of the design modification factors from various steel design codes was reviewed, and suitable recommendations are presented for cantilever LSBs subject to transverse loading.

Opposing Theories and the Buckling Strength of Unequal-Leg Angle Columns

A structural engineer wishing to determine the buckling load of unequal-leg angle columns receives conflicting information from published literature because researchers offer two radically different models for their analyses of thin-walled open-profile bars. One model considers the bar to be a continuum whereas the American Institute of Steel Construction adopts a noncontinuum multifilament model. Furthermore, the continuum model defines the normal planes of the deflected bar with the line of centroids whereas the multifilament model uses the shear center line. A test program based on 5×3×1/4 in. (127×76×6 mm) steel angles is proposed to resolve the issue by determining which model produces computed buckling loads that better correlate with collapse loads obtained from tests. The possibility that neither method produces a buckling load that correlates with collapse loads is not ruled out.

Equivalent Slenderness Ratio Method for Design of T-Struts Subject to Compressive Force

T-struts subject to centroid compression buckle flexural-torsionally about their axis of symmetry. When the force is applied at the shear center of the section, T-struts buckle either flexurally or torsionally without coupling of flexure with twisting. Although the buckling load about the symmetry-axis of shear center loading is greater than that of centroid loading, the design capacity of T-struts with defect such as fabrication error, load eccentricity and residual stress decrease by shifting the working line of a T-section compression chord to the shear center. This feature is not well known to designer of constructional steel. This article presents the equivalent slenderness ratio method, a new method for the design of T-struts subject to compressive force, introduces another three methods including the one presented by Shaofan Chen, the one in code and the one in ANSYS, contrasts the calculation results of those four methods and recommends the implementation of equivalent slenderness ratio method in the design of T-struts subject to compressive force.

Equivalences and Contrasts of Thermo-Elasticity and Thermo-Viscoelasticity: A Comprehensive Critique

Similarities and differences between thermo-elasticity and thermo-viscoelas- ticity are critically examined and evaluated. Topics include, among others, constitutive relations, Poisson's ratio, energy dissipation, temperature effects on material properties, thermal expansions, loading histories, failure criteria, lifetimes, 1–D beams, torsion, columns, plates, motions in time of neutral axes and shear centers, computational issues, wave propagation, torsional divergence, control reversal, aerodynamic derivatives, flutter and experimental determinations of viscoelastic properties. The full, partial or no possible applications of the elastic/viscoelastic correspondence principle, including approximate approaches, are analyzed and discussed.

Design Capacity of T-Strut Subject to Compressive Force

T-struts subject to centroid compression buckle flexural-torsionally about their axis of symmetry. When the force is applied at the shear center of the section, T-struts buckle either flexurally or torsionally without coupling of flexure with twisting. Although the buckling load of shear center loading is greater than that of centroid loading, i.e. T-struts in elastic stage provide larger resistance to buckling about their axis of symmetry when the compression is applied at the shear center in lieu of the centroid of the section, T-struts with defect such as fabrication error, load eccentricity and residual stress always buckles in the elastic-plastic range actually, and the design capacity decrease by shifting the working line of a T-section compression chord to the shear center. That peculiarity is verified by the nonlinear buckling analysis of T-struts models in ANSYS.

Buckling of shear deformable thin-walled members—I. Variational principle and analyticalsolutions

A variational formulation for the buckling analysis of thin-walled members is developed based on the principle of stationary potential energy. The formulation is based on non-orthogonal coordinates and captures shear deformation effects due to bending and warping. It is applicable to members of doubly symmetric cross-sections subject to general axial and transverse forces and naturally incorporates the effect of load position relative to the shear centre. Applying the conditions of neutral stability to the variational expression, the governing differential equations of neutral stability and associated boundary conditions are formulated. The resulting field equations are exactly solved for benchmark cases involving column flexural buckling, column torsional buckling, and lateral-torsional buckling for beams, and the results are compared to closed form solutions based on classical and other modern theories.

Nonlinear analysis of shear deformable beam-columns partially supported on tensionless Winkler foundation

In this paper, a boundary element method is developed for the nonlinear analysis of shear deformable beam-columns of arbitrary doubly symmetric simply or multiply connected constant cross section, partially supported on tensionless Winkler foundation, undergoing moderate large deflections under general boundary conditions. The beam-column is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member as well as the shear forces along the span induced by the applied axial loading. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy and the range of applications of the developed method. Keywords: Nonlinear Analysis, Large Deflections, Timoshenko Beam, Shear center, Shear deformation coefficients; Boundary element method, Winkler foundation, Tensionless foundation

An Exact Three-Dimensional Beam Element With Nonuniform Cross Section

Abstract In this paper, the exact stiffness matrix of curved beams with nonuniform cross section is derived using direct method. The considered element has two nodes and 12 degrees of freedom, with three forces and three moments applied at each node. The noncoincidence effect of shear center and center of area is also considered in this element. The deformations of the beam are due to bending, torsion, tensile, and shear loads. The line passing through center of area is a general three-dimensional curve and the cross section properties may change arbitrarily along it. The method is extended to deal with distributed loads on the curved beams. The stiffness matrix of some selected types of beams is determined by this method. The results are compared (where possible) with previously published results, simple beam finite element analysis and analytic solution. It is shown that the determined stiffness matrix is exact and that any type of beam can be analyzed by this method.

Graphical Determination of the Shear Center in Thin-Walled Asymmetrical U Profiles

This note recalls the circumstances leading to the late introduction of the idea of a shear center (or flexural center). It presents several graphs allowing a direct determination of the position of the shear center and of the corresponding warping constant for thin-walled asymmetrical U sections found as compression elements of channels and of similar profiles.

Elastic lateral-torsional buckling strength of I-girder with trapezoidal web corrugations using a new warping constant under uniform moment

While I-girders with trapezoidal web corrugations have been used in various structural applications and bridges, lateral torsion–flexure of I-girders with trapezoidal web corrugations still needs to be investigated, especially the general formulation for cross-section properties. This paper presents the results of theoretical and finite element analyses of the lateral-torsional buckling of an I-girder with trapezoidal web corrugations under uniform moment. By using the numerical equations and method, the location of shear center and warping constant are proposed. The elastic lateral-torsional buckling strength is then calculated via proposed cross-section properties. The results are compared with previous studies in the literature. A series of finite element analyses with different corrugation profiles and lengths are investigated and the results are compared with proposed formulas. Through comparative numerical studies, the proposed formulas are successfully verified. The effects of corrugation profiles on elastic lateral-torsional buckling strength are also investigated and discussed.