An elasticity solution of functionally graded beams with different moduli in tension and compression

In this study, we analytically solved the problem of a functionally graded beam with different moduli in tension and compression under the action of uniformly distributed loads. By determining the location of the unknown neutral layer of the beam, we first established a simplified mechanical model based on complete partition of tension and compression. Using boundary conditions and continuity conditions of the neutral layer, we obtained an elasticity solution of the problem, in which grade functions of tensile and compressive moduli of elasticity are assumed to be two different exponential expressions while Poisson's ratio is unchanged. The numerical results and comparison also verified the validity of the analytical solution. By changing the grade parameters of the material, the stress and displacement of the beam in three cases, i.e., the tensile modulus is greater than, equal to, or less than the compressive modulus, are discussed, respectively. The result shows that due to the introduction of bimodular effect and functional grade of materials, the maximum tensile and compressive bending stresses may not take place at the bottom and top of the beam, which should be given more attention in the analysis and design of structures made of functionally graded materials with bimodular effect.     

Deflection of thick beams of multimodular materials

A transfer-matrix analysis is presented for determining the static behaviour of thick beams of ‘multimodular materials’ (i.e. materials which have different elastic behaviour in tension and compression, with nonlinear stress–strain curves approximated as piecewise linear, with four or more segments). To validate the transfer-matrix method results, a closed-form solution is also presented for cases in which the neutral-surface location is constant along the beam axis. Numerical results for axial displacement, transverse deflection, bending slope, bending moment, transverse shear, axial force and location of neutral surface are presented for multimodular and bimodular models of unidirectional aramid cordrubber. The transfer-matrix method results agree very well with the closed-form solutions.     

Kerbspannungen sind Biegespannungen – was sind gute und böse Kerben?

Notch stresses are bending stresses – what are good and bad notches? The origin of a stress concentration in an engineering component can be explained by the imagination of superimposed bending stress. The curved surface of a notch can also be seen as the concave side of a heavily curved beam under bending. The nonlinear stress distribution of the beam under bending superimposes with the nominal stress and causes the stress concentration. In order to avoid such stress concentrations the increase of stress along the notch contour has to compensate with the decrease of the nominal stress. A multi‐linearized model demonstrates the interaction between the flow of stresses and the increase of the cross‐section of the component.     

双向梯形夹芯板柱面弯曲成形回弹分析

整体弯曲成形是制造曲面夹芯板高效且经济的方法,其成形特点与回弹预测是重点研究方向。采用结合有限元的半解析法对双向梯形夹芯的力学参数进行推导,获得夹芯等效弹性常数,分析上、下面板不等厚夹芯板柱面弯曲成形时面板与夹芯的变形特点及应力中性层的变化,在此基础上建立夹芯板平面应变弯曲回弹理论计算模型,预测夹芯板弯曲成形的应力分布与回弹,并与数值模拟及多点弯曲成形实验结果进行对比。结果表明:夹芯板回弹量与中厚板十分接近,回弹量较小,易于控制成形精度;理论预测的横截面切向应力与回弹都偏大,其中上面板应力相对误差在2.9%以内,下面板应力相对误差在6.5%以内,下面板纵向中心截面线误差在1.0mm范围内,各项误差均在很小范围内,验证了本工作回弹计算模型的准确性。     

基于修正单层梁理论的夹层梁最大弯曲正应力计算

目的计算夹层梁横截面的最大弯曲正应力。方法将夹层梁等效成等截面均质单层梁,进而推导出了理论计算公式,并在此基础上进行了三点弯曲试验的算例研究。结果当破坏载荷与夹层梁横截面的尺寸一定时,随着芯层与总厚度比的增加,修正单层梁理论计算的最大正应力值逐渐增加,而单层梁理论计算的结果为恒定值。对于同样结构的夹层梁,随着芯层弹性模量与表层模量比的增加,修正单层梁理论计算的最大正应力与单层梁理论的差异值越来越小。结论修正单层梁理论与层合梁理论计算的结果是一致的,该方法可有效进行最大弯曲正应力的预测与计算。     

矩形截面梁受多项式分布载荷的应力解

本文根据弹性力学半逆解法，对矩形截面梁的主要边界上受高次多项式分布力，提出一个统一的应力函数模式，运用此应力函数可以求解矩形截面梁受５次以内多项式分布载荷时的应力分量．文章以矩形截面梁受５次分布截荷为例，给出了材料力学弯曲正应力公式的修正项．     

Elastic bending analysis of bilayered beams containing a gradient layer by an alternative two-variable method

Aifantis’s strain gradient elasticity theories and Zhang’s two-variable method are used to study elastic bending problems of bilayered micro-cantilever beams, containing a gradient layer, subjected to a transverse concentrated load. The differential element method is used to obtain differential governing equations. The variational method is employed to overcome the difficulty in deriving nonlocal natural boundary conditions, which could not be automatically fulfilled in gradient theories, not like that in classical theories. Then the differential governing equations subjected to the related boundary conditions are solved analytically to obtain the deformation field, which could be degenerated to that in classical elasticity theories. The gradient parameters of epoxy polymeric resin and copper single crystals in the present model are provided by fitting Lam’s and Demir’s experiments. The influences of length and layer thickness on normalized deflection and effective rigidity are discussed in a representative case of a Cu/epoxy polymeric resin beam. Results show that size effect makes the effective rigidity vary more prominently with shorter beam length or larger layer thickness. For given materials, although size effect exists, classical elasticity theories are still valid in some particular combination of three geometric parameters: beam length, upper and lower layer thicknesses.     

A Displacement Solution to Transverse Shear Loading of Composite Beams by BEM

In this paper the boundary element method is employed to develop a displacement solution for the general transverse shear loading problem of composite beams of arbitrary constant cross section. The composite beam (thin or thick walled) consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal bending ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of a warping function. The shear deformation coefficients are obtained from the solution of two boundary value problems with respect to warping functions appropriately arising from the aforementioned one using only boundary integration, while the coordinates of the shear center are obtained from these functions using again only boundary integration. Three boundary value problems are formulated with respect to corresponding warping functions and solved employing a pure BEM approach. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the obtained values of the resultant transverse shear stresses compared with those obtained from an exact solution is remarkable.     

不同模量简支梁均布荷载下的弹性力学解

利用半逆解法,寻求了不同模量简支梁在均布荷载下的弹性力学解,并比较了现有的近似解。分析表明:由于材料不同模量的引入,应力较大程度地重新分布,使得材料力学解答中的最大正应力被低估;解答误差随着材料的不同模量上下波动,深梁情况下还会加剧。     

Equilibrium of an elastic finite cylinder: Filon's problem revisited

This paper deals with an analytical solution of the axisymmetric boundary-value problem of the theory of elasticity for a finite circular cylinder with free ends and arbitrary loaded curved surface. The object of this paper is to employ the method of superposition to obtain accurate values of the stress field near the boundaries. The classical Filon (1902) problem of uniformly distributed tangential load applied along two rings at the curved surface is addressed in full detail. The distribution of stresses along some typical sections of the cylinder are shown graphically.     

基于物理中面FGM梁的非线性力学行为

当坐标面置于功能梯度材料(FGM)梁的物理中面上时,其本构方程中,面内力与弯矩并不耦合,这使得问题的控制方程以及边界条件得以简化.该文利用物理中面概念,基于一阶非线性梁理论,导出了FGM梁的基本方程,分析研究了热载荷作用下FGM梁的过屈曲、弯曲以及在这些构形上的振动等问题.假设功能梯度材料性质只沿梁厚度方向,并按成分含...     

2205/X65双金属管道JCO成形有限元研究

目的研究2205/X65双金属管道JCO成形过程应力应变的演变及其影响因素。方法通过对2205/X65双金属管道JCO成形过程进行有限元模拟,分析了双金属管道成形后应力应变分布与演变情况,得到双金属板材自由弯曲变形特点及不同材料之间的界面应力分布特点,在此基础上研究了最大剪切应力随下压量、下模跨距以及模具半径的变化规律。结果 2205/X65双金属管成形后应力呈分段分布,除了最后一次下压位置,其他各段的应力大小和分布都是一样的,成形后的管坯由圆弧段和直管段相隔循环构成。2205/X65双金属管道成形后中性层在靠近屈服强度更大的2205一侧;在模具下压到最低点时板材上下表面屈服,中性层应力最小;卸载完成后管道厚度方向中间部位有最大残余应力分布,上下表面的环向应力都呈压应力分布。结论不同材料之间界面应力存在大梯度过渡,双金属板材弯曲成形中性层向一侧偏移。2205/X65双金属管道成形过程中最大剪切应力随着下压量的增大呈线性增大,当下压量为30mm时,最大剪切应力达到实验测试的双金属板材的剪切强度;随着下模跨距的增加,最大剪切应力不断减小;随着上模半径的增加,最大剪切应力不断增加,增加幅度较大;随着下模半径的增加,最大剪切应力不断增加,但增加幅度较小。成形后回弹角随着下压量的增大先减小再增大,最后趋于稳定。     

轴向往复运动抽油杆柱在弯曲井眼内横向振动的仿真模型

抽油杆柱横向振动的研究是分析油井杆管偏磨现象的基础。该文将定向井抽油杆柱在弯曲井眼内的横向振动问题简化为具有初弯曲的纵横弯曲梁在油管约束下的横向振动问题。除考虑交变轴向载荷对抽油杆柱横向振动的激励外,首次提出弯曲井眼也是轴向往复运动抽油杆柱横向振动的主要激励。综合考虑弯曲井眼对轴向往复运动抽油杆柱横向振动的激励,以及交变轴向载荷对抽油杆柱横向振动的激励,应用弹性碰撞理论描述油管对抽油杆柱横向振动的约束,基于弹性体振动理论建立了具有初弯曲的抽油杆柱在弯曲井眼内的横向振动仿真模型。采用有限差分法与Newmark法实现了对定向井抽油杆柱横向振动的仿真计算。仿真结果表明:仿真结果与现场实际偏磨情况相符,说明在分析抽油杆柱横向振动时,轴向运动导致的弯曲井眼激励是一项不可忽略的因素;算例油井中,杆管偏磨的危险点主要出现在油井的造斜段与抽油杆柱的受压段。该文所建立定向井抽油杆柱横向振动仿真模型可用于指导抽油杆扶正器的优化配置。     

ㄇ形梁剪力滞效应的解析解

精确分析ㄇ形梁在纵横向荷载共同作用下,其横断面上正应力分布规律对于计算其有效宽度有重要意义,应用力法原理,先将ㄇ形梁和翼板截开成矩形截面梁和平面应力板,在截面上代之以赘余的分布剪力,对于平面应力板,通过利用板变形的对称性来简化其边界条件,然后假设一个满足板的控制方程的Airy应力函数求得板的应力和位移,再利用Timoshenko梁理论求得梁的挠度和转角,根据截面上梁与板的纵向位移相等的变形协调条件便可最终确定截面上的分布剪力,给出的数值算例验证了方法的有效性,并与铜陵长江公路大桥主梁的模型有限元结果和试验结果作了对比,解析解法还可用来检验其他各种数值计算方法的精度,并可推广到其他多跨薄壁结构梁桥的膜应力分析中。     

A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration

A locking-free meshfree curved beam formulation based on the stabilized conforming nodal integration is presented. Motivated by the pure bending solutions of thin curved beam, a meshfree approximation is constructed to represent pure bending mode without producing parasitic shear and membrane deformations. Furthermore, to obtain the exact pure bending solution (bending exactness condition), the integration constraints corresponding to the Galerkin weak form are derived. A nodal integration with curvature smoothing stabilization that satisfies the integration constraints is proposed under the Galerkin weak form for shear deformable curved beam. Numerical examples demonstrate that the resulting meshfree formulation can exactly reproduce pure bending mode with arbitrary dicretizations, and the method is stable and free of shear and membrane locking. Computational efficiency and accuracy are achieved simultaneously in the proposed formulation     

Interfacial shear stress in FRP-plated RC beams under symmetric loads

A recently popular method for retrofitting reinforced concrete (RC) beams is to bond fibre-reinforced polymer (FRP) plates to their soffits. An important failure mode of such plated beams is debonding of the FRP plates from the concrete due to high level interfacial stresses near the plate ends. A closed-form rigorous solution for the interfacial stresses in simply supported beams bonded with thin plates and subjected to arbitrary loads has been found, in which a non-uniform stress distribution in the adhesive layer was taken into account. This paper uses the rigorous solution to investigate the impact of symmetric loading configurations on the interfacial shear stress distributions, and concludes that the bending moments on the cross sections at the plate ends play a significant role in terms of stress concentration, while the shear forces on the same cross-section contribute little to the concentration. On the basis of this observation, this paper proposes a simplified approximate solution to the shear stress along the interface between concrete and adhesive layer. Compared with the rigorous and other approximate solutions, the simplified solution exhibits sufficient accuracy in terms of stress distribution and stress concentration localized near the plate ends. Due to its compact feature, the simplified solution is more suitable for engineering applications using a portable calculator and to be adopted in the codes of practices.     

2-D elasticity solution of layered composite beams with viscoelastic interlayers

This paper focuses on the mechanical properties of layered composite beams with viscoelastic interlayers. The exact two-dimensional elasticity theory is used to represent the deformation of each beam layer. The viscoelastic interlayer is described by the Maxwell–Wiechert model through the quasi-elastic approximation, which greatly simplifies the analytical process. The stress function with a series of undetermined coefficients depending on the time variable is derived for each beam layer. No matter how many layers the beam includes, the total solution can be obtained rapidly and efficiently by using the recursive matrix technique. The present method can give the exact stress and deformation distributions in the beam, which cannot be predicted by the approximate theories such as the one-dimensional Euler–Bernoulli theory. The convergence of the solution is numerically verified. A comparison study indicates that the present results are in agreement with those obtained from the finite element method; however, they have obvious differences from the results based on the Euler–Bernoulli theory for thick beams. Finally, the variations of stresses and displacements with respect to time in a five-layer beam are discussed in detail.     

A refined beam theory based on the refined plate theory

Summary Based on the refined plate theory, a refined theory of rectangular beams is derived by using the Papkovich-Neuber solution and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. The solutions based on the new theory are the same as the exact solutions of elasticity theory. In three examples it is shown that the new theory provides as good or better results than Levinson’s beam theory when compared to those obtained from the linear theory of elasticity.     

Stresses at joints and cracks in highway and airport pavements

The weight function method is used to determine the stress intensity factors for cracks and joints in highway and airport pavements. The relationship between the stress intensity factors and the deflection caused by a wheel load are then used to obtain the stresses in the pavement. The new method that combines the theory of elasticity and fracture mechanics is called EFM. The results from EFM for the deflection and bending stress at the midslab position along the longitudinal edge are compared with the theoretical solution of Westergaard, solutions using the finite element method, and with the experimental data from the AASHO road test. The deflection caused by an axle load at the corner of a transverse joint for a rigid pavement determined by EFM is also compared with the test data from the AASHO road test. There is good agreement in all cases.     

Antiplane deformation of a cracked composite strip

We obtain the analytic solution of an antiplane problem of the theory of elasticity for a cracked layer made of a composite material whose cross section is formed by a periodic array of repeated rectangular elements. Each element, in turn, contains four rectangular cells of different types. A crack is located on one of the interfaces of materials of these cells. The distributions of displacements on the outer surfaces of the layer are regarded as given. The numerical analyses are performed for the stress intensity factors depending on the mechanical properties of materials and the sizes of the cells.