Explicit solutions for shearing and radial stresses in curved beams

In this paper the formulae for the shearing and radial stresses in curved beams are derived analytically based on the solution for a Volterra integral equation of the second kind. These formulae satisfy both the equilibrium equations and the static boundary conditions on the surfaces of the beams. As some applications, the resulting solutions are used to calculate the shearing and radial stresses in a cantilevered curved beam subjected to a concentrated force at its free end. The numerical results are compared with other existing approximate solutions as well as the corresponding solutions based on the theory of elasticity. The calculations show a better agreement between the present solution and the one based on the theory of elasticity. The resulting formulae can be applied to more general cases of curved beams with arbitrary shapes of cross-sections.     

Generalized coordinate for warping of naturally curved and twisted beams with general cross-sectional shapes

This paper presents an efficient procedure for analyzing naturally curved and twisted beams with general cross-sectional shapes. The hypothesis concerning the cross-sectional shapes of the beams is abandoned in this analysis, and relatively general equations are derived for the analysis of such a structure. Solving directly such equations for various boundary conditions, which take into account the effects of torsion-related warping as well as transverse shear deformations, can yield the solutions to the problem. The solutions can be used to calculate various internal forces, stresses, strains and displacements of the beams. The present theory will be used to investigate the stresses and displacements of a cantilevered curved beam subjected to action of arbitrary load. The numerical results are very close to the FEM results.     

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研制和开发了曲梁以及复合曲梁测量应力的创新实验装置, 通过该实验的设计、开发和应用,可以验证它的创新性和综合性,找出曲梁、复合曲梁与直 梁的诸多不同之处. 把该实验用于测试由钢制成的、钢与铜两种材料制成的,具有直角梯形 截面的简支梁在拱顶处受垂直集中力作用时的正应力和切应力,计算结果表明,理论解和实 验值吻合得很好.     

On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Due to the conflict between equilibrium and constitutive requirements,Eringen's strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue(i.e., excessive mandatory boundary conditions(BCs) cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.     

Solution of generalized coordinate for warping for naturally curved and twisted beams

A theoretical method for static analysis of naturally curved and twisted beams under complicated loads was presented, with special attention devoted to the solving process of governing equations which take into account the effects of torsion-related warping as well as transverse shear deformations. These governing equations, in special cases, can be readily solved and yield the solutions to the problem. The solutions can be used for the analysis of the beams, including the calculation of various internal forces, stresses, strains and displacements. The present theory will be used to investigate the stresses and displacements of a plane curved beam subjected to the action of horizontal and vertical distributed loads. The numerical results obtained by the present theory are found to be in very good agreement with the results of the FEM results. Besides, the present theory is not limited to the beams with a double symmetric cross-section, it can also be extended to those with arbitrary cross-sectional shape.     

自然弯扭梁的非线性分析

研究了两端边界均为完全约束的自然弯扭梁在小应变，大位移和大转动情况下的非线性性质，并考虑了横向剪切变形和扭转翘曲变形的影响，分析中还包括了拉伸，弯曲和扭转的各种弹性耦合。由最小势能原理可以导出所给问题的平衡方程。这里欧拉角可以用来表示任意大的转动。该方法还可方便地推广到其他各种不完全约束边界的情况。此外，利用上述结果还可以得到该梁在小位移理论中的基本方程和有关公式。     

Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force

In this paper an integral equation solution to the linear and geometrically nonlinear problem of non-uniform in-plane shallow arches under a central concentrated force is presented. Arches exhibit advantageous behavior over straight beams due to their curvature which increases the overall stiffness of the structure. They can span large areas by resolving forces into mainly compressive stresses and, in turn confining tensile stresses to acceptable limits. Most arches are designed to operate linearly under service loads. However, their slenderness nature makes them susceptible to large deformations especially when the external loads increase beyond the service point. Loss of stability may occur, known also as snap-through buckling, with catastrophic consequences for the structure. Linear analysis cannot predict this type of instability and a geometrically nonlinear analysis is needed to describe efficiently the response of the arch. The aim of this work is to cope with the linear and geometrically nonlinear problem of non-uniform shallow arches under a central concentrated force. The governing equations of the problem are comprised of two nonlinear coupled partial differential equations in terms of the axial (tangential) and transverse (normal) displacements. Moreover, as the cross-sectional properties of the arch vary along its axis, the resulting coupled differential equations have variable coefficients and are solved using a robust integral equation numerical method in conjunction with the arc-length method. The latter method allows following the nonlinear equilibrium path and overcoming bifurcation and limit (turning) points, which usually appear in the nonlinear response of curved structures like shallow arches and shells. Several arches are analyzed not only to validate our proposed model, but also to investigate the nonlinear response of in-plane thin shallow arches.     

Theoretical analysis on elastic buckling of nanobeams based on stress-driven nonlocal integral model

Several studies indicate that Eringen's nonlocal model may lead to some inconsistencies for both Euler-Bernoulli and Timoshenko beams, such as cantilever beams subjected to an end point force and fixed-fixed beams subjected a uniform distributed load. In this paper, the elastic buckling behavior of nanobeams, including both EulerBernoulli and Timoshenko beams, is investigated on the basis of a stress-driven nonlocal integral model. The constitutive equations are the Fredholm-type integral equations of the first kind, which can be transformed to the Volterra integral equations of the first kind. With the application of the Laplace transformation, the general solutions of the deflections and bending moments for the Euler-Bernoulli and Timoshenko beams as well as the rotation and shear force for the Timoshenko beams are obtained explicitly with several unknown constants. Considering the boundary conditions and extra constitutive constraints, the characteristic equations are obtained explicitly for the Euler-Bernoulli and Timoshenko beams under different boundary conditions, from which one can determine the critical buckling loads of nanobeams. The effects of the nonlocal parameters and buckling order on the buckling loads of nanobeams are studied numerically, and a consistent toughening effect is obtained.     

复合曲梁中剪应力和径向应力的研究

文献［郾］讨论了复合曲梁在弯曲时的正应力问题，在此基础上，本文进一步导出了其剪应力和径向应力的计算公式，从而使复合曲梁的应力问题全部得到解决，作为特例，也可以从上述公式得到单层匀质曲梁的的相应公式，最后给出计算实例。     

Local elastodynamic stresses in the unit cell of a woven fabric composite

Summary A theoretical study of the local elastodynamic stresses of woven fabric composites under dynamic loadings is presented in this article. The analysis focuses on the unit cell of an orthogonal woven fabric composite, which is composed of two sets of mutually orthogonal yarns of either the same fiber (nonhybrid fabric) or different fibers (hybrid fabric) in a matrix material. Using the mosaic model for simplifying woven fabric composites and a shear lag approach to account for the inter-yarn deformation, a one-dimensional analysis has been developed to predict the local elastodynamic and elastostatic behavior. The initial and boundary value problems are formulated and then solved using Laplace transforms. Closed form solutions of the dynamic displacements and stresses in each yarn and the bond shearing stresses at the interfaces between adjacent yarns are obtained in the time domain for any type of in-plane impact loadings. When time tends to infinity, the dynamic solutions approach to their corresponding static solutions, which are also developed in this article. Solutions of certain special cases are identical to those reported in the literature. Lastly, the dynamic stresses and bond shearing stresses of plain weave composites subjected to step uniform impacts are presented and discussed as an example of the general analytical model. Received 3 May 1999; accepted for publication 22 September 1999     

Energy domain integral applied to solve center and double-edge crack problems in three-dimensions

A three-dimensional Boundary Element Method (BEM) implementation of the energy domain integral for the numerical computation of the crack energy release rate is presented in this paper. The domain expression of the energy release rate is naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using the appropriate boundary integral equations. The pointwise crack energy release rate is evaluated along the three-dimensional crack front over a cylindrical domains that surround a segment of the crack front. The accuracy of the implementation is demonstrated by solving several problems, which include geometries containing straight as well as curved crack fronts.     

Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: Equations of motion and experimental validation

The non-linear equations and boundary conditions of non-planar (two bending and one torsional) vibrations of inextensional isotropic geometrically imperfect beams (i.e. slightly curved and twisted beams) are derived using the extended Hamilton's principle. The assumptions of Euler-Bernoulli beam theory are used. The order of magnitude of the natural geometric imperfection is assumed to be the same as the first order of vibrations amplitude. Although the natural imperfection is small, in contrast to the case of straight beams (i.e. geometrically perfect beams), this study shows that the vibration equations are linearly coupled and have linear and quadratic terms in addition to cubic terms. Also, in the case of near-square or near-circular beams, coupling terms between lateral and torsional vibrations exist. Furthermore, a problem of parametric excitation in the case of perfect beams changes to a problem of mixed parametric and external excitation in the case of imperfect beams. The validity of the model is investigated using the existing experimental data.     

Analysis of linear elastic wave propagation in piping systems by a combination of the boundary integral equations method and the finite element method

The combined methodology of boundary integral equations and finite elements is formulated and applied to study the wave propagation phenomena in compound piping systems consisting of straight and curved pipe segments with compact elastic supports. This methodology replicates the concept of hierarchical boundary integral equations method proposed by L. I. Slepyan to model the time-harmonic wave propagation in wave guides, which have components of different dimensions. However, the formulation presented in this article is tuned to match the finite element format, and therefore, it employs the dynamical stiffness matrix to describe wave guide properties of all components of the assembled structure. This matrix may readily be derived from the boundary integral equations, and such a derivation is superior over the conventional derivation from the transfer matrix. The proposed methodology is verified in several examples and applied for analysis of periodicity effects in compound piping systems of several alternative layouts.     

Stress rate integral equations of elastoplasticity

The stress rate integral equations of elastoplaticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent. The project supported by the National Natural Science Foundation of China     

基于无网格法的Timoshenko曲梁动力分析

曲梁具有外形美观、受力性能良好的优点,故在工程中得到广泛应用。本文基于移动最小二乘近似和一阶剪切变形理论,提出一种对Timoshenko曲梁自由振动和受迫振动进行分析的无网格方法。通过一系列离散点离散曲梁,建立曲梁无网格模型,然后推导曲梁势能和动能方程,通过哈密顿原理给出曲梁自由振动和受迫振动的控制方程,因为本文方法不能直接施加边界条件,所以使用完全转换法处理本质边界条件,最后求解方程得到频率和振动模态。文末通过算例验证了本文方法的有效性,且通过收敛性分析表明本文方法具有较好的收敛性,并进一步分析了不同边界条件、跨高比和变截面变曲率对曲梁自由振动和受迫振动的影响,将计算结果与文献解或ABAQUS解进行对比分析,表明本文方法具有较高的精度,且适用于实际工程情况。     

3D dynamic stress intensity factors around two parallel square cracks in an infinite elastic medium under impact load

Summary  Transient stresses around two parallel cracks in an infinite elastic medium are investigated in the present paper. The shape of the cracks is assumed to be square. Incoming shock stress waves impinge upon the two cracks normal to tzheir surfaces. The mixed boundary value equations with respect to stresses and displacements are reduced to two sets of dual integral equations in the Laplace transform domain using the Fourier transform technique. These equations are solved by expanding the differences in the crack surface displacements in a double series of a function that is equal to zero outside the cracks. Unknown coefficients in the series are calculated using the Schmidt method. Stress intensity factors defined in the Laplace transform domain are inverted numerically to the physical space. Numerical calculations are carried out for transient dynamic stress intensity factors under the assumption that the shape of the upper crack is identical to that of the lower crack. Received 2 February 2000; accepted for publication 10 May 2000     

功能梯度变曲率曲梁的几何非线性模型及其数值解

基于弹性曲梁平面问题的精确几何非线性理论,建立了功能梯度变曲率曲梁在机械和热载荷共同作用下的无量纲控制方程和边界条件,其中基本未知量均被表示为变形前的轴线坐标的函数。以椭圆弧曲梁为例,采用打靶法求解非线性常微分方程的两点边值问题,获得了两端固定功能梯度椭圆弧曲梁在横向非均匀升温下的热弯曲变形数值解,分析了材料梯度指数、温度参数、结构几何参数等对曲梁受力及变形的影响。     

穿过反平面圆形夹杂界面的曲线型刚性线

利用复变函数方法和叠加原理建立了求解刚性线夹杂问题的弱奇积分方程,利用Ｃａｕｃｈｙ型奇异积分方程主部分方法,研究了穿过反平面圆夹杂界面的曲线型刚性线夹杂在界面交点处点处的奇性应力指数以及交点处角形域内的奇性应力,并定义了交点处的应力奇性因子。利用所得的奇性应力指数,通过对弱奇异积分方程的数值求解,得出了刚性线端点和交点处的应力奇性因子。     

Hybrid stress analysis of perforated composites using strain gages

A strain gage hybrid method is described for determining individual stresses on the boundary and in the neighborhood of cutouts in orthotropic composites. Results agree with independent measurements and finite element analysis. Few measured strain data are needed, and the measured strains originate away from the hole. Ability to determine the stresses on the edge of a cutout from nonboundary measurements recognizes the difficulties in obtaining reliable measurements very near an edge while circumventing the challenge of attempting to bond gages to the transverse curved surface of a small hole or notch. The method also alleviates the problem of not knowing a priori where the most serious stress will occur on the geometric boundary and, hence, where to locate strain gages.     

Generalized variational principle on nonlinear theory of naturally curved and twisted closed thin-walled composite beams

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