Flexural testing of an epoxy oversized strand model under traction

Experimental flexural tests are conducted on an epoxy oversized 1×6 strand; the objectives of these tests are to measure the bending stiffness and the associated strains. Three sets of boundary conditions are applied. The bending stiffness is deduced from the transverse midspan deflection and is found to be dependent on the axial force and the moment distribution. The deduced bending stiffness is compared to values computed from the theoretical bending stiffnesses.     

Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.     

Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson’s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.     

Large deflections of cantilever beams of non-linear elastic material under a combined loading

Large deflection of cantilever beams made of Ludwick type material subjected to a combined loading consisting of a uniformly distributed load and one vertical concentrated load at the free end was investigated. Governing equation was derived by using the shearing force formulation instead of the bending moment formulation because in the case of large deflected member, the shearing force formulation possesses some computational advantages over the bending moment formulation. Since the problem involves both geometrical and material non-linearities, the governing equation is complicated non-linear differential equation, which would in general require numerical solutions to determine the large deflection for a given loading. Numerical solution was obtained by using Butcher's fifth order Runge-Kutta method and are presented in a tabulated form.     

超静定梁?柱的解析解研究

本文采用渐进积分法研究了超静定梁?柱的弯曲问题. 首先建立超静定梁?柱的四阶挠度微分方程, 考虑到边界条件和连续光滑条件, 采用连续分段独立一体化积分法求解得到了挠度的精确解析解. 为了满足工程设计需要, 构造了超静定梁?柱的四阶挠度微分迭代方程, 选取无轴向力作用时超静定梁的挠曲线作为梁的初函数, 将初函数代入梁的四阶挠度微分迭代方程进行积分, 利用边界条件和连续光滑条件确定积分常数, 得到下一次迭代挠度函数, 依次进行迭代积分运算. 计算出了最大挠度、最大转角和最大弯矩等用轴向力放大系数表示的多项式解析函数解. 本文选取了两种边界条件下受分布力作用的超静定梁?柱进行分析, 计算结果表明, 当超静定梁?柱所受的轴向力小于欧拉临界力的1/2时, 迭代六次误差就可以控制在1%以内; 不仅梁?柱最大位移和最大内力的大小随轴向力的增大而增大, 而且其位置也随轴向力的增大而发生迁移. 本文的研究对揭示轴向力对超静定梁?柱变形和内力的影响有重要意义, 为超静定梁?柱的实际设计提供了一定的理论基础.      

Simplified theory and analytical solution for functionally graded thin plates with different moduli in tension and compression

In this study, a simplified theory for functionally graded thin plates with different moduli in tension and compression is proposed. Based on the classical Kirchhoff hypothesis, a mechanical model concerning tension-compression subzone is established, first. Using the geometrical and physical relations and equation of equilibrium, all stress components are expressed in terms of the deflection, in which modulus of elasticity in tensile and compressive zone are regarded as two different functions while Poisson's ratios are taken as two different constants. Via the equilibrium conditions and continuity conditions, the governing equation expressed in terms of the deflection as well as the unknown neutral layer are derived, respectively. Moreover, the application in polar coordinates, the strain energy and the perturbation solution for the unknown neutral layer, are discussed in detail. The results indicate that the bending stiffness derived in this study play an important role while contacting the classical problem and this problem. The analytical solutions from equilibrium conditions and continuity conditions are consistent. Analyses of more general cases for modulus of elasticity and Poisson's ratio also show the applicability of the simplified theory. This study provides a theoretical basis for the subsequent work.     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

中墩斜置对连续箱梁弯扭性能的影响

为了揭示中墩斜支承对连续箱梁力学性能的影响,本文考虑约束扭转和竖向挠曲耦合作用,建立了斜支承连续箱梁的力法方程,并获得了内力和变形的解析式.选取斜支承两跨连续箱梁为数值算例,分别计算了竖向对称和偏心均布荷载作用下的内力和变形,并用ANSYS软件计算了控制截面的弯矩.计算结果表明,本文方法计算的弯矩与ANSYS计算值吻合...     

轴向扩散下简支饱和多孔弹性梁的大挠度分析

基于多孔介质理论和弹性梁的大挠度理论,并考虑轴向变形,在孔隙流体仅沿轴向扩散的假设下,建立了微观不可压饱和多孔弹性梁大挠度弯曲变形的一维非线性数学模型.在此基础上,忽略饱和多孔弹性梁的轴向应变,并利用Galerkin截断法,研究了两端可渗透的简支饱和多孔弹性梁在突加横向均布载荷作用下的拟静态弯曲,给出了饱和多孔梁弯曲时挠度、弯矩和轴力以及孔隙流体压力等效力偶等沿轴线的分布曲线.揭示了大挠度非线性和小挠度线性模型的结果差异,指出大挠度非线性模型的结果小于相应小挠度线性模型的结果,并且这种差异随着载荷的增大而增大.计算表明:当无量纲载荷参数q>5时,应该采用大挠度非线性数学模型进行研究.     

Debonding analysis of FRP–concrete interface between two balanced adjacent flexural cracks in plated beams

A cohesive interface modeling approach to debonding analysis of adhesively bonded interface between two balanced adjacent flexural cracks in conventional material (e.g., concrete or wood) beams strengthened with externally bonded FRP plates is presented. Both the strengthened beam and strengthening FRP are modeled as two linearly elastic Euler–Bernoulli beams bonded together through a thin adhesive layer. A bi-linear cohesive model, which is commonly used in the literature, is adopted to characterize the stress-deformation relationship of the FRP–concrete interface. Completely different from the single-lap or double-shear pull models in which only the axial pull force is considered, the present model takes the couple moment and transverse shear forces in both the substrates into account to study the second type of intermediate crack-induced debonding (IC debonding) along the interface. The whole debonding process of the FRP–concrete interface is discussed in detail, and closed-form solutions of bond slip, interface shear stress, and axial force of FRP in different stages are obtained. A rotational spring model is introduced at locations of the two adjacent flexural cracks to model the local flexibility of the cracked concrete beam, with which the relationship between the local bond slip and externally applied load is established and the real bond failure process of the FRP-plated concrete beam with the increasing of the externally applied load is revealed. Parametric studies are further conducted to investigate the effect of the thickness of adhesive layer on the bond behavior of FRP–concrete interface. The present closed-form solution and analysis on the local bond slip versus applied load relationship for the second type of IC debonding along the interface shed light on the bond failure process of structures externally strengthened with FRP composite plates and can be used effectively and efficiently to predict ductility and ultimate load of FRP-strengthened structures.     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

端部受斜冲击的刚塑性悬臂梁的双铰模型

斜冲击载荷作用在刚塑性悬臂梁的端部,引起作用在梁横截面处的弯矩以及轴力;在发生塑性变形截面处,弯矩及轴力满足交互作用屈服条件。广义应力在移行铰的邻域不违背屈服条件,屈服函数可在移行铰的背面取极大值,移行铰处的剪切力不必为零。如果悬臂梁足够长,在响应的初始阶段移行铰处非零的剪力会在梁上引起多铰变形。通过对双铰模型与单铰模型的比较发现,双铰模型计算的结果与单铰模型计算的结果很接近,单铰模型作为一个近似模型具有一定的合理性。     

纤维增强聚合物布加固裂纹木梁的稳定性

梁中横向裂纹等效为无质量内部转动弹簧，假定纤维增强聚合物(FRP)布与梁表面紧密粘贴，建立了考虑轴向压力二阶效应FRP 布加固裂纹梁线性弯曲的控制方程，并得到其显式解析通解．在此基础上，研究了FRP加固简支裂纹木梁的稳定性，通过数值求解方程，分析了纤维增强聚合物(CFRP)布含量、裂纹深度和位置以及数量等因素对CFRP 布加固简支裂纹杉木梁临界载荷的影响，结果表明：CFRP 加固可明显减小裂纹深度和数量等对裂纹杉木梁临界载荷的影响，且裂纹处弯矩较大或裂纹较深时加固效应愈加显著；CFRP 加固裂纹木梁临界载荷随CFRP 布加固层含量的增加而增加，但当CFRP 布含量达到一定值后，进一步增加CFRP 含量对CFRP加固裂纹梁临界载荷提高并不明显．     

考虑畸变影响的箱梁横向弯矩计算方法

为研究偏心垂向荷载作用下梯形截面单室箱梁的横向弯矩,对框架分析法计算箱梁横向弯矩的方程进行优化,并在刚性支承法的基础上提出一种更加简单的横向弯矩计算方法;与框架分析法不同,横向弯矩可采用能量变分法求解,建立以箱梁顶板剪力差为未知量的四阶控制微分方程,采用比拟的弹性地基梁解法解出剪力差,得出梯形截面单室箱梁横向弯矩的能量变分法解。对几种箱梁横向弯矩计算方法用两个算例进行验证,结果表明,能量变分法解将箱梁底板上的弯矩误差绝对值由15.41%降到了9.68%;本文方法得出的横向弯矩结果和有限元结果吻合较好,弯矩误差绝对值最大不超过6.01%;本文方法和能量变分法可有效降低箱梁底板上的弯矩误差,计算精度得到提高。     

功能梯度中厚圆/环板轴对称弯曲问题的解析解

基于一阶剪切变形板理论，导出了热/机载荷作用下，位移形式的功能梯度 中厚圆/环板轴对称弯曲问题的控制方程，获得了问题的位移和内力的一般解析解. 作为特 例，分别研究了边缘径向固定和可动的夹紧和简支的4种实心功能梯度圆板，给出了它们的 解，并分析了热/机载荷作用下解的形态，讨论了横向剪切变形、材料梯度常数和边界条件， 对板的轴对称弯曲行为的影响.     

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

具有内核伸出段的套管构件在线接触阶段的受力性能研究

针对具有内核伸出段的轴压套管构件,建立了线接触阶段的力学模型;利用小挠度理论推导出了内核和套筒的剪力、弯矩、挠度、分布接触力、集中接触力等物理量的计算公式。通过与无内核伸出段套管构件的对比算例验证了本文相关物理量推导的正确性,分析了内核与套管在不同刚度比下相关物理量的变化情况。结果表明:由于内核伸出段的影响,点接触会提前向线接触变化,使套管构件的轴向承载力显著下降,端部的接触反力及内核的剪力、弯矩有显著的增加。对具有内核伸出段的套管构件进行了力学性能分析,结果表明:内核与套筒的间隙对线接触长度没有影响,内核的弯矩、剪力随着间隙的增加而增大;随着内核伸出段长度的增加,线接触长度及内核的弯矩、剪力增大。     

Analysis of bending and buckling of pre-twisted beams: A bioinspired study

Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.     

Bending of simply-supported circular timber beam strengthened with fiber reinforced polymer简

The bending behavior of the circular cross-section timber beam strengthened with a fiber reinforced polymer (FRP) sheet is investigated. The tight bonding is on the interface between the surface of a timber beam and the reinforcement layer of the FRP sheet. An analytical expression for the bending moment-curvature relation is presented, and its failure modes are analyzed. The governing equation for nonlinear small deflection of the FRP-strengthened circular timber beam is established, and the corresponding numerical method is given. The bending deformation of the simply-supported circular timber beam strengthened with the carbon fiber reinforced polymer (CFRP) sheet subject to a uniform load is studied numerically. The influence of the angle and thickness of the CFRP layer as well as the timber strength on the bending deflection of the FRP-strengthened circular timber beam is examined. It is revealed that, with the increases of the thickness and angle, the deflection of the CFRP-strengthened circular timber beam is decreased, and its carrying capacity and ductility are increased. However, when the angle of the layer reaches a certain value, the deflection will no longer decrease with the increase of the angle. At the same time, the nonlinear bending moment-curvature relation of the CFRP-strengthened circular timber beam is simplified as an approximate bilinear constitutive model. The approximate deflections of the simply-supported circular timber beam strengthened with the CFRP sheet are obtained. The results are compared with the linearly elastic bending and nonlinear bending models, showing that the mid-span deflections of a CFRP-strengthened circular timber beam with the approximate bilinear constitutive model are greater than those with the nonlinear constitutive model. The results of its stiffness analysis are on the safe side.     

球壳轴对称弯曲问题共轭二阶挠度微分方程

提出球壳轴对称弯曲问题共轭二阶挠度微分方程并给出了初等函数解. 球壳微分方程是薄壳理论三大壳之一旋转壳的典型方程. 共轭二阶挠度微分方程是球 壳中微分方程形式最简单的, 是人们最喜爱的挠度微分方程. 挠度微分方程满足边 界条件非常简单, 使球壳的计算得到很大的简化.