Nonuniform Warping Research Articles

Interaction of combined loads on the lateral stability of thin-walled composite beams

Based on a seven-degree-of-freedom shear deformable beam model, analytical solutions are derived for the lateral stability analysis of cross-ply laminated thin-walled beams subjected to combined axial and bending loads. The model includes shear deformability in a full form, i.e. shear flexibility due to both bending and nonuniform warping is considered. The theory is formulated in the context of large displacements and rotations, considering moderate bending rotations and large twist. Composite is assumed to be made of symmetric balanced laminates and especially orthotropic laminates. The closed-form analytic expressions obtained in this paper are valid for simply supported bisymmetric beams. These fundamental solutions explicitly identify the influence of geometric nonlinear effects due to the prebuckling deformation. The numerical results are compared with the bifurcation loads of the postbuckling response. In addition, the effects of the variation of load height parameter and fiber angle orientation are investigated.

Non uniform warping for beams

A non-uniform warping beam theory including the effects of torsion and shear forces is presented. Based on a displacement model using three warping parameters associated to the three St Venant warping functions corresponding to torsion and shear forces, this theory is free from the classical assumptions on the warpings or on the shears, and valid for any kind of homogeneous elastic and isotropic cross-section. This general theory is applied to analyze, for a representative set of cross-sections, the elastic behavior of cantilever beams subjected to torsion or shear-bending. Numerical results are given for the one-dimensional structural behavior and the three-dimensional stresses distributions; for the stresses in the critical region of the built-in section, comparisons with three-dimensional finite elements computations are presented. The study clearly shows when the effect of the restrained warping is localized or not.

Non uniform warping for beams

A non-uniform warping beam theory including the effects of torsion and shear forces is presented. Based on a displacement model using three warping parameters associated to the three St Venant warping functions corresponding to torsion and shear forces, this theory is free from the classical assumptions on the warpings or on the shears, and valid for any kind of homogeneous elastic and isotropic cross-section. This general theory is applied to analyze, for a representative set of cross-sections, the elastic behavior of cantilever beams subjected to torsion or shear-bending. Numerical results are given for the one-dimensional structural behavior and the three-dimensional stresses distributions; for the stresses in the critical region of the built-in section, comparisons with three-dimensional finite elements computations are presented. The study clearly shows when the effect of the restrained warping is localized or not.

Mechanics of thin-walled curved beams made of composite materials, allowing for shear deformability

In this paper, a new theoretical model is developed for the generalized linear analysis of composite thin-walled curved beams with open and closed cross-sections. In the present model two important concepts concerning to composite thin-walled curved beams are addressed. The first one is the incorporation in the model of what is called full shear deformability, i.e. shear flexibility due to both bending and non-uniform warping is considered. The second feature is connected with the constitutive aspects, and it contemplates the use of different hypotheses that can be adopted in the formulation. These topics are treated in a straightforward way by means of the Linearized Principle of Virtual Work. In order to obtain the motion equations of the model a non-linear displacement field, whose rotations are formulated by means of the rule of semitangential transformation, is employed. This model allows the study of many problems of statics, free and forced vibrations with arbitrary initial stresses and linear stability of composite thin-walled curved beams with general cross-sections. A discussion about the constitutive equations is performed in order to explain characteristic features of the effects included in the theory. This paper presents the theoretical formulation together with finite element procedures that are developed to obtain the numerical approximations to the general equations of thin-walled shear-deformable composite curved beams. For this kind of structural member, iso-parametric finite elements are introduced. Numerical examples are carried out in several topics of statics, dynamics and buckling problems, focusing attention in the validation of the theory with respect to experimental data and with 2D and 3D computational approaches. Also, new parametric studies are performed in order to show the influence of shear deformability on the mechanics of the thin-walled composite curved-beams with open and closed cross-sections as well as to illustrate the utility of the model.

A non-uniform warping theory for beams

This Note proposes a non-uniform warping beam theory including the effects of torsion and shear forces. Based on a displacement model using three warping parameters associated to three St Venant warping functions corresponding to torsion and shear forces, this theory is free from the classical assumptions on the warpings or on the shears, and is valid for any kind of homogeneous elastic and isotropic cross-section. The result on the structural behavior of the beam specifies the effect of the non-symmetry of the cross-section, and the closed form results obtained for the stresses show the contribution of each internal force. Comparison with St Venant beam theory highlights the additional effects due to the non-uniformity of the warping. To cite this article: R. El Fatmi, C. R. Mecanique 335 (2007).

Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications

This two-part contribution presents a beam theory (BT) with a non-uniform warping (NUW) including the effects of torsion, and shear forces and valid for any homogeneous cross-section made of isotropic elastic material. In part I, the governing equations of the NUW-BT has been established and simplified-NUW-BT versions has been deduced, wherein the number of degrees of freedom is reduced. In this part II, these theories are used to analyze, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not), the elastic behavior of cantilever beams subjected to torsion or shear-bending. For bi-symmetrical-CS, torsion and shear-bending are analyzed separately: analytical and numerical results are given for the distributions along the beam axis of the cross-sectional displacements and stresses, for the NUW-BT and its simplified versions. Numerical results are also given for the three-dimensional stress distributions close to the embedded section: the stress predictions of the NUW-BT are compared to those obtained by three-dimensional finite elements computations. It can be drawn from all these results indications that can help to decide when the simplified theories may be applied, and hence when the warping parameters may be reduced. As specified in NUW-BT, torsion and bending are coupled for non-symmetrical-CS, even if the bending moments refer to the centroid while the torsional moment refers to the shear center. To illustrate this coupling effect, the particular example of the channel-CS presented in Kim and Kim [Kim, N.-I., Kim, M.-Y., 2005. Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-Walled Structures 43, 701–734.] is analyzed and the results are compared.

Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory

This two-part contribution presents a beam theory with a non-uniform warping including the effects of torsion and shear forces, and valid for any homogeneous cross-section made of isotropic elastic material. Part I is devoted to the theoretical developments and part II discusses analytical and numerical results obtained for torsion and shear-bending of cantilever beams made of different kinds of cross-section. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters associated to the three warping functions corresponding to torsion and shear forces. Starting from this displacement model and using the principle of virtual work, the corresponding beam theory is derived. For this theory, closed-form results are obtained for the cross-sectional constants and the three-dimensional expressions of the normal and shear stresses. Comparison with classical beam theories is carried out and additional effects due to the non-uniformity of the warping are highlighted. In particular, the contributions of primary and secondary internal forces and the effect of the non-symmetry of the cross-section on the structural behavior of the beam are specified. Simplified versions of this theory, wherein the number of degrees of freedom is reduced, are also presented. The analytical and numerical analyzes presented in part II give responses on the quality of this non-uniform beam theory and indicate also when its simplified versions could be applied.

Mechanics of shear deformable thin-walled beams made of composite materials

In this paper, a new theoretical model is developed for the generalized linear analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form the shear deformability by means of two features. The first one may be addressed as a mechanical aspect where the effect of shear deformability due to both bending and non-uniform warping is considered. The second feature is connected with the constitutive aspects, and it contemplates the use of different hypotheses adopted in the formulation. These topics are treated in a straightforward way by means of the Linearized Principle of Virtual Works. The model is developed by employing a non-linear displacement field, whose rotations are formulated by means of the rule of semitangential transformation. This model allows studying many problems of static's, free vibrations with or without arbitrary initial stresses and linear stability of composite thin-walled beams with general cross-sections. A discussion about the constitutive equations is performed, in order to explain distinctive aspects of the effects included in the theory. This paper presents the theoretical formulation together with finite element procedures that are developed with the aim to obtain solutions to the general equations of thin-walled shear deformable composite beams. A non-locking fourteen-degree-of-freedom finite element is introduced. Numerical examples are carried out in several topics of static's, dynamics and buckling problems, focusing attention in the validation of the theory with respect to experimental data and with 2D and 3D computational approaches. Also, new parametrical studies are performed in order to show the influence of shear flexibility in the mechanics of the thin-walled composite beams as well as to illustrate the usefulness of the model.

Modeling distortional shear in thin-walled elastic beams

In this paper, the theoretical background and the numerical analyses of an advanced beam finite element that relaxes the hypothesis of the cross-section non-deformability are presented. The corresponding new modes, called distortional modes, are added to the modes describing the behavior of a classical thin-walled beam: tension/compression, bending and torsion. For instance, a load acting in a cross-sectional plane of a beam is considered to induce not only bending and torsion but also distortion. The distortion produces non-uniform shear and axial stresses together with a non-uniform warping of the cross-section. These resulting effects, significant for very thin-walled open profiles (and thin-walled closed profiles with high distortional loadings), are shown in this paper to be important when compared to bending and torsion stresses even in simple loading cases.

Stability of composite thin-walled beams with shear deformability

In this paper, a theoretical model is developed for the stability analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form, the shear flexibility (bending and non-uniform warping), featured in a consistent way by means of a linearized formulation based on the Reissner’s Variational Principle. The model is developed using a non-linear displacement field, whose rotations are based on the rule of semi-tangential transformation. This model allows to study the buckling and lateral stability of composite thin-walled beam with general cross-section. A finite element with two-nodes and fourteen-degrees-of-freedom is developed to solve the governing equations. Numerical examples are given to show the importance of the shear flexibility on the stability behavior of this type of structures.

Magnetoresistance and susceptibility study of Sr 2RuO 4 under high magnetic fields

The magnetoresistance and susceptibility of Sr 2RuO 4 were studied in magnetic fields up to 33 T at 50 mK. The magnetoresistance exhibited notable anisotropy within the conducting plane, which we attribute to the non-uniform warping of the Fermi surface. Superconductivity recurrence, predicted for magnetic fields applied parallel to the conducting plane, was not detected up to 33 T.

Nonuniform warping torsion of orthotropic composite beams

The proposed work deals with the behaviour of orthotropic composite beams subjected to torsion, taking into consideration the effects due to nonuniform warping. We start from a one-dimensional approach based on the integral parameter method and consider any complex cross-sectional shape. The basic relations obtained lead to numerical computation based on the finite element method. As a special case of the general results obtained, an example of a sandwich beam is worked out in detail.

The Thin‐Walled Rod of an Open Cross‐Section as an Element of a 3D‐Structure

AbstractThe relation for a rectilinear thin‐walled rod of an open cross‐section is established between displacements and corresponding forces applied at the ends of the rod. The problem is solved in a linear setting, taking into account a flexure, a uniform and nonuniform warping, and also an axial deformation. The calculation is conducted in the deformed configuration. Coefficients of the thin‐walled rod relation form a matrix of rigidity of this rod that allows to include a thinwalled rod of an open cross‐section in a library of a finite element method.

せん断ゆがみ関数を用いた薄肉はりの理論に関する研究 : 非一様せん断ゆがみ関数を用いた曲げ解析

In this paper, two analysis models, a uniform shear warping model and a nonuniform shear warping model for the thin-walled beam's flexural behavior are developed, which can show the precise distribution of normal stress and shear stress within the beam section including the shear lag phenomena. The mechanism of boundary condition of the developed shear warping models and their relation for the Timoshenko's beam model are discussed and clearly verified. By using the St. Venant's semi-inverse method, the uniform and nonuniform warping functions are defined. By two examples, the usefulness of the proposed analysis model is shown.

Generalized thin-walled beam models for flexural-torsional analysis

With non-uniform warping being an important mode of deformation, supplementary to the other six modes of stretching, shearing, twisting, and bending, we utilize a fairly comprehensive one-dimensional beam theory for the development of a simple finite element model for the analysis of arbitrary thin-walled beams under general loadings and boundary conditions. The formulation is valid for both open- and closed-type sections, and this is accomplished by using a kinematical description accounting for both flexural and warping torsional effects. To eliminate the shear/warping locking in this C 0-element, a generalized mixed variational principle is utilized, in which both displacement and strain fields are approximated separately. In this, the strain parameters are of the interelement-independent type, and are therefore eliminated on the element level by applying the relevant stationarity conditions of the employed ‘modified’ Hellinger-Reissner functional, thus leading to the standard form of stiffness equations for implementation. A rather extensive set of numerical simulations are given to demonstrate the versatility of the models in practical applications involving usage of such components in their stand-alone forms as well as in plate/shell stiffening.

Limitation of Simplified Hypotheses for the Prediction of Torsional Oscillations for Thin-Walled Beams

The study of uniform torsion of thin walled beams is done fairly easily in the thin wall beam case, the effects of thickness being neglected. Then, the classical Saint-Venant warping function simplifies to the double sectorial area. The dynamical case needs the use of a more refined theory involving nonuniform warping, and characterized by supplementary dynamic torsion constants. The computed values obtained from BIEM for those constants are compared with those deduced from thin walled beam theory, and from the measurements of experimental natural frequencies. It is shown that the relative thickness acts significantly on the results in both static and dynamic torsion cases.

On a Finite Element Formulation of Dynamical Torsion of Beams Including Warping Inertia and Shear Deformation Due to Nonuniform Warping

We start from the energetical expressions of dynamical torsion of beams in terms of angular and warping displacement and velocity. We derive the stiffness and two mass matrices including both secondary effects for torsion: the shear deformation due to nonuniform warping and the warping inertia. The suitability of these matrices for evaluation modified torsional frequencies is investigated in the case of thick, as well as thin-walled, cross section beams by comparison with analytical and experimental results.

Analysis of buildings with interconnected shear walls

A method for analysis of interconnected shear walls is described that accounts for the torsional stiffness caused by non-uniform warping. Floor to floor, plane wall members are idealized as part of the input to a three-dimensional frame analysis program. Where walls in different planes intersect, constraints that ensure common vertical displacements are imposed. These constraints appear in the computer input as special types of 'members'; however, within the solution, the Lagrange multipliers method is used to process the constraint equations. A frontal method that handles the constraints efficiently along with members and loadings is used to organize the analysis.The use of this method of analysis is simple and versatile relative to comparable methods. Most arrangements of shear walls that form open cross sections can be analyzed in conjunction with frames and other walls. Location of the shear centre of groups of walls and their torsional properties are not required.Analysis results are compared to experimental results and the results of other methods of analysis. The accuracy of the proposed method depends upon the number of storeys in the model because force transfer between adjacent wall segments occurs only at floor levels; in reality, a continuous transfer of force occurs through the height of the walls. In the opinion of the authors, using the proposed method for buildings of 20 or more storeys would certainly provide adequate accuracy for design.