The six-parameter theory of straight elastic bars with transverse shear–torsion decoupling

Extensional and flexural displacements of straight elastic bars are conventionally treated as independent superposable effects. However, coupling of displacement fields due to boundary effects is an important feature in many realistic configurations. The present paper examines the vibration modes of a beam in which the aforementioned coupling is caused by a roller support that is mounted on an incline. A closed form solution of the coupled dislacement modes is derived. The transcendental characteristic equation is shown to depend solely on the incline angle for the end support, the reduced frequency formed from the bar wave speed, and the ratio of the cross‐section’s radius of gyration to the span length. The characteristic equation is solved numerically. The resulting eigenvalues are used to evaluate the coupled mode shapes, which leads to a quantitative assessment of coupling effects relative to the limiting cases where the incline is horizontal or vertical, for which the displacement fields truly are u...

This analysis reviews the ideas, insights and assumptions that are critical for a comprehensive formulation of a theory for bars. One assumption, the Wagner hypothesis, was thought to be necessary for explaining what appears to be pure torsional buckling when certain bars are compressed. The characterization of the phenomenon as pure torsional buckling is incorrect. The hypothesis is not needed, and without it, the theory returns to a simpler and more rational earlier form. Bar theory had its early beginnings in 1705, when James Bernoulli proposed that the curvature of a bent bar is proportional to the bending moment and that internal resisting couples result from the extensions and contractions of the bar’s longitudinal filaments. Daniel Bernoulli, James’s nephew, communicated the idea to Leonhard Euler in 1742, and Euler, acting on it, solved the elastica problem in 1744 Love 1944 . Later, Euler defined buckling and computed buckling loads. His work laid the foundation of what is known as stability theory. Euler considered a bar as though it were a line of particles that curved in accordance with Bernoulli’s principle. In 1776, Charles Augustin de Coulomb Love 1944 examined the normal planes of a bent bar. By assuming that the profiles did not warp plane sections remain plane and with known or assumed properties for the bar’s material, he was able to determine the neutral axis for bending. From that time, the bar was considered as a threedimensional body for purposes of analysis. It is evident that bar theory rests on “special hypotheses” that make it approximate. Plate and shell theories are also approximate for their dependency on special hypotheses. Such theories are useful and occasionally needed, as when the elasticity solution is not available. They typically supply the stresses, strains, and deformations that most concern the structural engineer and machine designer and are simpler to use than the elasticity theory. The drawback to an approximate theory is that it often produces inconsistent and incomplete results. For example, bar theory is inconsistent when it predicts a transverse shear stress on the normal plane of the bar and neglects the shear strain it should produce. It is incomplete as well because a prismatic bar of irregular cross section will generally not have a true shear center and shear centerline so that it is not possible to know whether transverse loads will twist the bar as well as bend it. S. Timoshenko suggested that for some bars, the error might be small if the centroid line is also taken to be the shear centerline. Elasticity theory is considered to be exact because its results are complete and consistent. However, it includes one glaring inconsistency that is masked by specifying that it is meant for use when strains and displacements are small. An analysis begins with the initial configuration of the body, and that configuration is used to express the equilibrium of the deformed body in its final state. If a truer representation is desired, the loads may be incrementally applied and the configuration determined after each in-

By endowing El Fatmi’s theories of bars with first-order warping functions due to torsion and shear, a family of theories of bars, of various applicability ranges, is effectively constructed. The theories thus formed concern bars of arbitrary cross-sections; they are reformulations of the mentioned theories by El Fatmi and theories by Kim and Kim, Librescu and Song, Vlasov and Timoshenko. The Vlasov-like theory thus developed is capable of describing the torsional buckling and lateral buckling phenomena of bars of both solid and thin-walled cross-sections, which reflects the non-trivial correspondence, noted by Wagner and Gruttmann, between the torsional St.Venant’s warping function and the contour-wise defined warping functions proposed by Vlasov. Moreover, the present paper delivers an explicit construction of the constitutive equations of Timoshenko’s theory; the equations linking transverse forces with the measures of transverse shear turn out to be coupled for all bars of asymmetric cross-sections. The modeling is hierarchical: the warping functions are numerically constructed by solving the three underlying 2D scalar elliptic problems, providing the effective characteristics for the 1D models of bars. The 2D and 1D problems are indissolubly bonded, thus forming a unified scientific tool, deeply rooted in the hitherto existing knowledge on elasticity of elastic straight bars.

Nowadays, the mechanical characterization of 3-D spacer fabrics has attracted the interest of many textile researchers. These Spacer fabrics present special mechanical and physical characteristics compared to conventional textiles due to their wonderful porous 3-D structures. These fabrics, produced by warp knitting method, have extensive application in automobile, locomotive, aerospace, building and other industries. In these applications, the compressibility behaviour plays a significant role in the fabric structural stability. This compressibility behaviour could be affected by different knitting parameters such as density of pile yarn, fabric thickness, texture design etc. The aim of this paper is to introduce and develop an appropriate elastic theoretical model to predict the compressibility behaviour of warp knitted spacer fabric (WKSF). Three theoretical models are proposed, based on modelling pile yarns as the curved bars and are improved in three steps: a) with same curvatures in weft and warp directions (model A), b) curved bar for warp direction and cantilever bar for weft direction (model B), and c) curved bars with two different curvatures in weft and warp directions considering the curvature variations under loading (model C: improved model). The results obtained by the proposed models have been compared with previous model based on simply cantilever bars theory in literature. The results show that the simulation data obtained by the model C are closer to the experimental results comparing to the models A and B. Model C based on different weave parameters could better predict the elastic compressibility behaviour of this kind of WKSF in order to compare with previous models.

Influence of adjacent segments on the torsional load-bearing behaviour of assembled half-shell towers

The solution of the impact - problems of a rigid body against an elastic bar arc before based on the classical longitudinal vibration theory of elastic bar. In this paper, we are using a more precise theory to deal with the impact problem of a rigid body against a half -infinite elastic bar with an elastic gasket. This problem is solved completely.

The solutions of the impact problems of a rigid body against an elastic bar formerly are based on the classical longitudinal vibration theory of an elastic bar. In this paper, we based on the more precise theory to deal with the impact problem of a rigid body against a finite elastic bar set on a foundation.

This study investigates the buckling of an elastic bar with a linearly varying rectangular cross-section under thermomechanical loading. The research focuses on such straight bars' stress-strain state and stability, addressing large displacement scenarios using a nonlinear system of ordinary differential equations. The mathematical model considers axial and transverse displacements, rotation angles, and internal force factors, creating a variable temperature field along the length of the bar. A numerical solution is obtained using MathCad’s built-in computational tools, enabling precise determination of displacement components, internal forces, and bending moments. The numerical calculation determines the value of the first critical force, which is essential for calculating the strength of the structure. The critical load is determined through numerical iteration from the compressive load.

The impacting and rebounding behaviors of straight elastic components are investigated and a unified approach is proposed to analytically predict the whole process of the collision and rebounding of straight elastic bars and beams after each of them impinges on ideal (massless) elastic spring(s). The mathematical problems with definitive solution are formulated, respectively, for both the constrained-motion and free-motion stages, and the method of mode superposition, which is concise and straightforward especially for long-time interaction and multiple collision cases, is successfully utilized by repeatedly altering boundary and initial conditions for these successive stages. These two stages happen alternatively and the collision process terminates when the constrained motion no longer occurs. In particular, three examples are investigated in detail; they are: a straight bar impinges on an ideal elastic spring along its axis, a straight beam vertically impinges on an ideal elastic spring at the beam's midpoint, and a straight beam vertically impinges on two ideal springs with the same stiffness at the beam's two ends. Numerical results show that the coefficient of restitution (COR) and the nondimensional rebounding time (NRT) only depend on the stiffness ratio between the ideal spring(s) and the elastic bar/beam. Collision happens only once for the straight bar impinging on spring, while multiple collisions occur for the straight beam impinging on springs in the cases with large stiffness ratio. Once multiple collisions occur, COR undergoes complicated fluctuation with the increase of stiffness ratio. Approximate analytical solutions (AASs) for COR and NRT under the cases of small stiffness ratio are all derived. Finally, to validate the proposed approach in practical collision problems, the influence of the springs' mass on the collision behavior is demonstrated through numerical simulation.

The paper deals with stability problems of straight elastic bars made of a homogenous isotropic material. The approach concerns both the bars of compact cross-sections and of thin-walled cross-sections, the transverse distortions being neglected. The stability analysis method developed for thin-walled bars in the paper: L. Zhang and G.S. Tong, “Flexural-torsional buckling of thin-walled beam members based on shell buckling theory”, Thin-Walled Structures, vol. 42 (2004), pp. 1665–1687 is here extended to the bars whose deformations obey the assumptions of the Vlasov-like theory. The approach proposed makes it possible to assess the values of critical loads causing: axial forces, bending moments, transverse forces and torques, possibly simultaneously. The main task is to perform maximization of the relevant Rayleigh quotient; its form is given for all rational shapes of the bar’s cross section. The paper shows how to assess critical axial buckling loads and lateral buckling loads of straight elastic bars made of a homogenous isotropic material.

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.

A general analytical approach of the non-linear problem of a linear elastic and isotropic straight bar of variable stiffness is indicated. The linear equivalence method, introduced by one of the authors, is applied to two fundamental cases for the isostatic straight bar, i.e. the cantilever bar (a Cauchy type problem) and the simply supported bar (a bilocal problem). Some numerical examples concerning moderate deformations and rotations are presented.

On the nonlinear bending of a hyperstatic bar

The non-linear theory of elastic thin-walled bars of open cross-sections proposed by the author~[1] is applied to the study of large torsion of such bars. The fundamental equations are simplified for the case of bisymmetrical and central symmetrical cross-sections. For non-symmetrical cross-sections, it is generally impossible to obtain pure torsion without bending in the non-linear theory. The problem is solved by a perturbation method. Two specific examples are considered.

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.