Theory of Flexural Shear, Bending and Torsion for a Thin-Walled Beam of Open Section

Aspects of the general Vlasov theory are examined separately as applied to a thin-walled channel section cantilever beam under free-end end loading. In particular, the flexural bending and shear that arise under transverse shear and axial torsional loading are each considered theoretically. These analyses involve the location of the shear centre at which transverse shear forces when applied do not produce torsion. This centre, when taken to be coincident with the centre of twist implies an equivalent reciprocal behaviour. That is, an axial torsion applied concentric with the shear centre will twist but not bend the beam. The respective bending and shear stress conversions are derived for each action applied to three aluminium alloy extruded channel sections mounted as cantilevers with a horizontal principal axis of symmetry. Bending and shear are considered more generally for other thin-walled sections when the transverse loading axes at the shear centre are not parallel to the section = s centroidal axes of principal second moments of area. The fixing at one end of the cantilever modifies the St Venant free angular twist and the free warping displacement. It is shown from the Wagner-Kappus torsion theory how the end constrained warping generates an axial stress distribution that varies with the length and across the cross-section for an axial torsion applied to the shear centre. It should be mentioned here for wider applications and validation of the Vlasov theory that attendant papers are to consider in detail bending and torsional loadings applied to other axes through each of the centroid and the web centre. Therein, both bending and twisting arise from transverse shear and axial torsion applied to each position being displaced from the shear centre. Here, the influence of the axis position upon the net axial and shear stress distributions is to be established. That is, the net axial stress from axial torsional loading is identified with the sum of axial stress due to bending and axial stress arising from constrained warping displacements at the fixing. The net shear stress distribution overlays the distributions from axial torsion and that from flexural shear under transverse loading. Both arise when transverse forces are displaced from the shear centre.

Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory

The linear and nonlinear torsional free vibration analyses of functionMly graded micro/nuno-tubes （FGMTs） are analytically investigated based on the couple stress theory. The employed non-classical continuum theory contains one material length scale parameter, which can capture the small scale effect. The FGMT model accounts for the through-radius power-law variation of a two-constituent material. Hamilton＇s principle is used to develop the non-classical nonlinear governing equation. To study the effect of the boundary conditions, two types of end conditions, i.e., fixed-fixed and fixed-free, are considered. The derived boundary value governing equation is of the fourthorder, and is solved by the homotopy analysis method （HAM）. This method is based on the Taylor series with an embedded parameter and is capable of providing very good approximations by means of only a few terms, if the initial guess and the auxiliary linear operator are properly selected. The analytical expressions are developed for the linear and nonlinear natural frequencies, which can be conveniently used to investigate the effects of the dimensionless length scale parameter, the material gradient index, and the vibration amplitude on the natural frequencies of FGMTs.

Saint-Venant torsion analysis of bars with rectangular cross-section and effective coating layers

This paper investigates the torsion analysis of coated bars with a rectangular cross-section. Two opposite faces of a bar are coated by two isotropie layers with different materials of the original substrate that are perfectly bonded to the bar. With the Saint- Venant torsion theory, the governing equation of the problem in terms of the warping function is established and solved using the finite Fourier cosine transform. The state of stress on the cross-section, warping of the cross-section, and torsional rigidity of the bar are evaluated. Effects of thickness of the coating layers and material properties on these quantities are investigated. A set of graphs are provided that can be used to determine the coating thicknesses and material properties so as to keep the maximum von Mises stress on the cross-section below an allowable value for effective use of the coating layer.

环的相对于余遗传Torsion Theory的Jacobson根

本文推广了环论中拟正则左理想的概念，引入τ－拟正则左理想的概念，给出环的相对于余遗传Ｔｏｒｓｉｏｎｔｈｅｏｒｙ的Ｊａｃｏｂｓｏｎ根的元素刻划．

The Cotorsionfree Radical of Relatively Torsion Theory

Let τ be a torsion theory on R-rood and M be a left R-mod. In this paper the τ-cotorsionfree radical Cτ（M） of M is studied. When τ is stable, the construction and the supplementing radical of Cτ （M） are obtained.

Relatively Injective Modules with Respect to Torsion Theory

For a hereditary torsion theory T,this paper mainly discuss properties of A-τ- injective modules,where A is a fixed left R-module.It is proved that if M is an A-τ-injective, B is a submodule of A,then 1)M is A/B-τ-injective;2)M is B-τ-injective when B isτ- dense in A.Furthermore,we show that if A_1,A_2,…,A_n are relatively injective modules, then A_1⊕A_2⊕…⊕A_n is self-τ-injective if and only if A_i is self-τ-injective for each i.

THE SAINT-VENANT PROBLEM OF PLANE BAR UNDER AN AXIAL FORCE

In the paper, the solution of Saint-Venant problem is obtained through assumption of principal stress curves by means of the equilibrium equations which were deduced in paper [1]. The results show that the speed of shear approaching to zero is a(3)/y(3) and axial stress approaching to constant is a(2)/y(2).

A New Solution with Torsion in Model of dS Gauge Theory of Gravity

有在严肃的 de 保姆计量器理论的模型的扭转的一个新静态的 de 保姆答案被获得。扭转仅仅包含 O (3 ) 对称的张肌部分根据无法缩减的分解。新答案的一些性质被讨论。

基于板-梁理论的FGM双轴对称箱形截面组合扭转理论与有限元验证

基于板梁理论研究FGM双轴对称箱形截面组合扭转理论与有限元验证.首先,选取弹性模量变化函数,建立能量变分模型,引入无量纲参数,获得双轴对称箱形截面组合扭转的近似解析解;然后,利用微分方程模型计算双轴对称箱形截面组合扭转的精确解析解,给出微分方程解答;最后,利用ANSYS有限元分析软件进行验证,将本文研究所得解析解和ANSYS有限元分析软件得出的解析解进行对比,对比结果验证了本文理论推导及所给出的自由扭转刚度、约束扭转刚度表达式的正确性.

基于板-梁理论的FGM矩形截面杆扭转振动的理论研究

基于板-梁理论和Saint-Venant扭转理论的研究成果,将纵向位移假设为含有待定系数的表达式,通过几何方程推导了功能梯度材料矩形截面杆的总应变能方程,并利用最小应变能原理确定待定系数,得到自由扭转刚度的理论公式。在此基础上对矩形截面杆的自由振动出现纯扭转模态时的动能进行研究,求出在夹支条件下的扭转振动时的动能表达式、总势能的表达式及频率表达式。利用有限元软件ANSYS建立具有功能梯度材料特性的矩形截面杆的扭转振动的有限元模型,通过后处理命令查看对应边界条件的振动频率。通过有限元软件验证表明:基于板-梁理论推导得出的理论解与有限元解吻合较好,最大误差在4%以内。

圆孔工字型蜂窝梁扭转模态自由振动研究

从能量等效角度,提出了一种圆孔工字型截面蜂窝梁连续型模型,应用新的薄壁构件扭转理论:板-梁理论,建立了构件扭转模态自由振动能量泛函模型,明确了圆孔工字型截面蜂窝梁名义约束扭转惯性矩(翘曲常数)、自由扭转常数、转动惯量。通过变分运算,获得了平衡微分方程模型。基于扭转模态试函数,推导了圆孔工字型截面蜂窝梁自由振动扭转模态振动圆频率,并与有限元结果对比校验理论模型预测精度。开展了悬臂圆孔工字型蜂窝梁受集中扭矩作用的端部截面转角理论分析与有限元计算,进一步检验了基于板-梁理论建立的蜂窝梁连续模型的适用性。应用板-梁理论对圆孔工字型截面蜂窝梁自由振动扭转的分析过程及推导步骤,为其他开孔类型工字型蜂窝梁扭转模态自由振动研究提供了直接参考。

基于板-梁理论的双轴对称三角形管翼缘工字形梁组合扭转的理论研究及有限元验证

基于板-梁理论推导双轴对称三角形管翼缘工字形悬臂梁的总应变能方程,得到梁的翘曲刚度和自由扭转刚度,然后通过外力势能和总应变能得到总势能方程。再依据能量变分模型、微分方程模型及边界条件得出悬臂梁端部最大扭转角公式,将求出的最大扭转角与用ANSYS建立的有限元模型计算结果进行对比。结果表明,悬臂梁在端扭矩作用下的扭转角理论值与有限元值较为接近,误差最小的绝对值为2.40%,最大的绝对值为4.44%,可为工程设计提供参考。

Compensation of stress intensity factors in hollow cylinders containing several cracks under torsion by electro-elastic coating

In this article, a formulation for a hollow cylinder reinforced with an electroelastic layer is investigated. The hollow cylinder and its electro-elastic coating are under the Saint-Venant torsional loading. First, the solution to the problem containing a Volterra-type screw dislocation is obtained by using the Fourier transform. The problem is then reduced to a set of Cauchy singular integral equations by the distributed dislocation method. Finally, several examples are presented to show the effect of the electro-elastic coating on the reduction of the stress intensity factors at the crack tips.

NEW BOUNDARY ELEMENT METHOD FOR TORSION PROBLEMS OF CYLINDER WITH CURVILINEAR CRACKS

The Saint-Venant torsion problems of a cylinder with curvilinear cracks were considered and reduced to solving the boundary integral equations only on cracks. Using the interpolation models for both singular crack tip elements and other crack linear elements, the boundary element formulas of the torsion rigidity and stress intensity factors were given. Some typical torsion problems of a cylinder involving a straight, kinked or curvilinear crack were calculated. The obtained results for the case of straight crack agree well with those given by using the Gauss-Chebyshev integration formulas, which demonstrates the validity and applicability of the present boundary element method.

半群的Goldie Torsion理论和Lambek Torsion理论

在文［１］，［３］的基础上，讨论了ＧｏｌｄｉｅＴｏｒｓｉｏｎ理论和ＬａｍｂｅｋＴｏｒｓｉｏｎ理论的一些性质以及它们间的关系，刻画了相应的右同余拟滤子；并对这两种遗传Ｔｏｒｓｉｏｎ理论，研究了环和其乘法半群之间等Ｔｏｒｓｉｏｎ性问题，获得了一些较有意义的结论．

Analysis of complete plasticity assumption for solid circular shaft under pure torsion and calculation of shear stress

The distribution of shear stress on the cross-section of plastic metal solid circular shaft under pure torsion yielding,the applicability of complete plastic model assumption and the shear stress formula were researched.Based on the shear stress formula of circular shaft under pure torsion in elastic stage,the formula of torque in elastic stage and the definition of yield,it is obtained that the yielding stage of plastic metal shaft under pure torsion is only a surface phenomenon of torque-torsion angle relationship,and the distribution of shear stress is essentially different from that of tensile stress when yielding under uniaxial tension.The pure torsion platform-torsion angle and the shape of torque-torsion angle curve cannot change the distribution of shear stress on the shaft cross-section.The distribution of shear stress is still linear with the maximum shear stress τs.The complete plasticity model assumption is not in accordance with the actual situation of shaft under torsion.The experimental strength data of nine plastic metals are consistent with the calculated results of the new limiting strain energy strength theory(LSEST).The traditional yield stress formula for plastic shaft under torsion is reasonable.The shear stress formula based on the plane assumption in material mechanics is applicable for all loaded stages of torsion shaft.

Backgrounds Constrain on the Torsion Detection

The sources of torsion,macroscopic rotation and aligned spin are discussed.In the case of Dirac spin particle as a direct sensor for torsion f the detection of torsion-spin effect is cons trained by the larger backgrounds stronglyt which are magnetic-spin and rotation-spin.Increasing the source of the torsion is equivalent to increasing the background itself.So,the conclusion is that Dirac particle spectra experiments have no hope to give the evidence for torsion directly in some special torsion gravity theory.

Torsion in Groups of Integral Triangles

Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0<x<1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.

The Efficiency of CP-Violating <i>α</i>2-Dynamos from Primordial Cosmic Axion Oscillation with Torsion

Recently torsion fields were introduced in CP-violating cosmic axion a2-dynamos [Garcia de Andrade, Mod Phys Lett A, (2011)] in order to obtain Lorentz violating bounds for torsion. Here instead, oscillating axion solutions of the dynamo equation with torsion modes [Garcia de Andrade, Phys Lett B (2012)] are obtained taking into account dissipative torsion fields. Magnetic helicity torsion oscillatory contribution is also obtained. Note that the torsion presence guarantees dynamo efficiency when axion dynamo length is much stronger than the torsion length. Primordial axion oscillations due to torsion yield a magnetic field of 109 G at Nucleosynthesis epoch. This is obtained due to a decay of BBN magnetic field of 1015 G induced by torsion. Since torsion is taken as 10–20 s–1, the dynamo efficiency is granted over torsion damping. Of course dynamo efficiency is better in the absence of torsion. In the particular case when the torsion is obtained from anomalies it is given by the gradient of axion scalar [Duncan et al., Nuclear Phys B 87, 215] that a simpler dynamo equation is obtained and dynamo mechanism seems to be efficient when the torsion helicity, is negative while magnetic field decays when the torsion is positive. In this case an extremely huge value for the magnetic field of 1015 Gauss is obtained. This is one order of magnitude greater than the primordial magnetic fields of the domain wall. Actually if one uses tDW ~ 10-4 s one obtains BDW ~ 1022 G which is a more stringent limit to the DW magnetic primordial field.

The Mathematical Foundations of Gauge Theory Revisited

We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies (despite using the same group theoretical methods) and the well known couplings existing between elasticity and electromagnetism (piezzo electricity, photo elasticity, streaming birefringence). The purpose of this paper is to avoid such contradictions by using new mathematical methods coming from the formal theory of systems of partial differential equations and Lie pseudo groups. These results finally allow unifying the previous independent tentatives done by the brothers E. and F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by using respectively the group of rigid motions of space or the conformal group of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra and algebraic analysis. We insist on the fact that these results could not have been obtained before 1975 as the corresponding tools were not known before.