轴向受载的Euler-Bernoulli梁的双向弯曲扭转耦合振动研究

本文研究了轴向受载的Euler-Bernoulli梁的双向弯曲扭转耦合自由振动问题.选择梁横截面的剪切中心作为坐标原点,坐标轴平行于梁截面的几何轴;振动微分方程中有关梁截面几何特性的参数均采用相对于几何轴的参数.结合具体的边界条件求解自由振动微分方程组,辅以Mathematica软件计算梁振动的固有频率.针对具体的算例,给出了三种边界条件下梁弯扭耦合振动的固有频率的数值结果,并与Ansys软件的计算结果进行了比较,分析了误差来源以及轴向荷载对弯扭耦合自由振动的影响.数值结果验证了本文方法在其适用范围内的精确性和有效性.本文忽略了翘曲刚度的影响.     

考虑剪切效应的工字梁弯曲应力比拟杆法

随着工字形短深梁和宽翼缘梁结构的发展,截面非线性剪切变形对弯曲应力的影响愈加突出,导致传统设计中所采用的初等梁理论计算结果误差较大,不再适用。本文基于比拟杆法综合考虑剪切效应,推导出工字形梁横力弯曲应力解析计算公式,并与有限元及现有解析计算方法进行对比分析。结果表明:当跨高比较小,翼缘腹板面积比较大时剪切效应对弯曲变形有显著影响。同时相比于现有解析方法,本文计算结果精度较高且适用范围更广,可用于梁结构设计。     

BAR SIMULATION METHOD FOR BENDING NORMAL STRESS OF I-BEAM IN CONSIDERATION OF SHEAR EFFECT 1)

随着工字形短深梁和宽翼缘梁结构的发展,截面非线性剪切变形对弯曲应力的影响愈加突出,导致传统设计中所采用的初等梁理论计算结果误差较大,不再适用。本文基于比拟杆法综合考虑剪切效应,推导出工字形梁横力弯曲应力解析计算公式,并与有限元及现有解析计算方法进行对比分析。结果表明：当跨高比较小,翼缘腹板面积比较大时剪切效应对弯曲变形有显著影响。同时相比于现有解析方法,本文计算结果精度较高且适用范围更广,可用于梁结构设计。     

空间曲梁单元的扭转修正系数研究

三维空间曲梁有限单元模型是模拟曲梁结构的有效数值方法,可以考虑曲梁的弯扭耦合特性,最为符合曲梁的几何和受力特征.由于有限元法采用梁理论的平截面假定,空间曲梁单元上的扭转剪应力分布与实际曲梁截面上的扭转剪应力不同,从而会导致扭转刚度和扭转变形的计算失真.本文基于剪切应变能等效原理,推导了不同长宽比的矩形截面空间曲梁单元的扭转刚度修正系数η和截面边中点处扭转剪应力的修正系数λ,并采用曲线悬臂梁进行了验证.验证结果表明,根据本文提出的η作为校正因子的空间曲梁单元模型,对任意矩形截面曲梁计算的扭转变形均与实体单元模型的结果吻合良好;且只有截面为正方形时,扭转剪应力修正系数η才恰好与弯曲剪应力修正系数(1.2)一致.     

用能量法计算双模量深梁的弯曲挠度

为了得到双模量深梁弯曲变形挠度的实用计算方法,在考虑剪切变形的基础上,采用能量法研究了双模量深梁在外载荷作用下的弯曲变形挠度计算问题,并推导出了双模量深梁弯曲挠度的计算公式.把能量法的挠度计算结果与弹性理论方法的挠度计算结果进行比较,可知用能量法研究双模量深梁的弯曲变形不但计算过程简便,而且计算精度也很高.研究结果表明,双模量深梁的剪切形状因子与双模量材料的拉压弹性模量有关,而各向同性材料深梁的剪切形状因子却与它的弹性模量无关,所以双模量深梁的剪切形状因子与各向同性材料深梁的剪切形状因子有着本质上的区别.     

一种复合材料旋翼桨叶结构大变形分析方法的改进及验证

提出了一种适用于直升机旋翼复合材料桨叶大变形分析的改进方法。将旋翼桨叶变形分析分解为一维非线性分析和二维剖面特性分析,并考虑横向剪切、翘曲对剖面刚度及弹性耦合的影响;为使方法适用于旋翼气动弹性分析,将应变能中的广义应变用参考轴线处的弹性运动表示,保留所有非线性项,推导出计算复合材料桨叶大变形的公式;采用有限元法处理方程,对梁结构进行了分析,并将大变形状态下的位移计算结果与Princeton梁实验值、Minguet复合材料梁实验值以及中等变形梁理论计算结果进行了比较,验证了大变形状态下本文计算方法的正确性;此外与中等变形梁模型计算结果的对比,验证了本文方法在计算精度上的提高。     

变截面欧拉梁自由振动分析的重采样微分求积法

采用重采样微分求积法求解了变截面欧拉梁的自由振动问题。推导了变截面梁的控制方程离散格式，采用重采样矩阵方法对边界条件进行处理，给出了变截面梁自由振动算法。采用本文方法对不同类型截面形式和不同边界条件的变截面梁进行自由振动分析，并和其他解法进行比较。计算结果表明，本文方法可以适用于不同变截面类型和不同边界条件，计算精度与解析解吻合良好，具有良好的收敛性能。在同等精度条件下网格点数少于现有计算方法。重采样转换矩阵边界处理方法相比于传统边界处理方法具有更快的收敛性能。     

开口薄壁梁的扭转理论与应用

以Vlasov薄壁构件理论为基础, 推导了开口薄壁构件一阶扭转理论. 该理论考虑了翘曲剪应力对截面转角的影响, 截面的转角分为自由翘曲转角和约束剪切转角, 在约束扭转中, St.Venant扭矩仅仅与自由翘曲转角有关, 而翘曲扭矩仅与约束剪切转角有关. 利用半逆解方法求出了约束扭转中薄壁构件的St.Venant扭矩表达公式; 依据能量方法, 建立了约束剪切转角和翘曲扭矩之间的关系, 并提出了翘曲剪切系数概念, 给出了一阶扭转理论的微分方程. 为了有效求解微分方程, 给出了求解微分方程的初参数法方程和相应的影响函数矩阵; 当St.Venant扭矩可以忽略时, 得到与一阶弯曲理论(Timoshenko梁理论)相似的一阶扭转理论简化形式. 最后利用算例证明了一阶扭转理论和简化理论的有效性.      

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

变截面Timoshenko梁动力反应的半解析法

为研究移动荷载下截面剪切变形和转动惯量影响,在推导变截面Timoshenko梁振型正交性的数学表达式的基础上,建立了任意荷载作用下Timoshenko梁动力响应的模态叠加法.然后,将模态摄动法和模态叠加法结合起来,提出了变截面Timoshenko梁动力反应计算的公式.在此基础上,基于矩形截面梁,比较分析了简支Timoshenko梁理论和Euler梁理论动力反应随移动荷载速度、长细比和截面衰减率的变化规律的区别.计算结果表明:由于剪切变形和转动惯量的影响,Timoshenko梁的动力反应将大于Euler梁.当长细比小于10时,Timoshenko梁跨中位移比Euler梁增加25%以上,当长细比大于30后,可采用Euler梁理论进行简化分析.     

A new beam element for analyzing geometrical and physical nonlinearity

Based on Timoshenko＇s beam theory and Vlasov＇s thin-walled member theory, a new model of spatial thin-walled beam element is developed for analyzing geometrical and physical nonlinearity, which incorporates an interior node and independent interpolations of bending angles and warp and takes diversified factors into consideration, such as traverse shear deformation, torsional shear deformation and their coupling, coupling of flexure and torsion, and the second shear stress. The geometrical nonlinear strain is formulated in updated Lagarange （UL） and the corresponding stiffness matrix is derived. The perfectly plastic model is used to account for physical nonlinearity, and the yield rule of von Mises and incremental relationship of Prandtle-Reuss are adopted. Elastoplastic stiffness matrix is obtained by numerical integration based on the finite segment method, and a finite element program is compiled. Numerical examples manifest that the proposed model is accurate and feasible in the analysis of thin-walled structures.     

SHEAR STRESS ANALYSIS OF TIMOSHENKO''S BEAM WITH MULTIPLY CONNECTED CROSS SECTION

In this paper, a finite element method is developed to numerically evaluate the shear coefficient of Timoshenko's beam with multiply connectd cross section. With focus on analyzing shear stresses distributed at the neutral axis of the beam, an improved definition of the shear coefficient is presented. Based on this definition, a Galerkin-type finite element formulation is proposed to analyze the shear stresses and shear deflections. Numerical solutions of the examples for some typical cross-sections are compared with the theoretical results. The shear coefficient of tower sections of the Tsing Ma Bridge is calculated by use of the proposed approach, so that the finite element modeling of the bridge can be developed with the accurate values of the sectional properties.     

基于新型广义位移的箱形梁扭转单元及应用

为了改进变截面连续箱梁桥的扭转分析理论，将截面总扭转角分解为自由翘曲扭转角和约束剪切扭转角，选取自由翘曲转角扭率作为广义位移，提出一个2节点8自由度的扭转梁段单元。从约束扭转控制微分方程出发，推导单元刚度矩阵及等效节点荷载列阵。引入应力增大系数，以反映约束扭转对初等梁应力的增大效应。数值算例验证了本文梁段单元的可靠性。最后对一个三跨变截面连续箱梁桥进行分析，结果表明，双力矩影响线与弯矩影响线较为类似，按双力矩影响线进行最不利荷载加载时最大应力值偏小；应力增大系数在集中荷载作用截面出现极值，均发生在腹板与顶板交点处；利用偏载放大系数来考虑扭转附加效应时，不宜考虑弯曲正应力较小及翘曲正应力出现极值的梁段区域。     

A BEM solution to transverse shear loading of composite beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of composite beams of arbitrary constant cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio 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 ones. The transverse shear loading is applied at the shear centre of the cross-section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.     

Effective properties of a composite beam subjected to torsion

The problem of finding the effective characteristics of a rectilinear beam under pure torsion is considered. The problem can be reduced to determining the torsional stress function from the solution of a boundary-value problem in a cross section of the beam for a partial differential equation with variable coefficients. Two special boundary-value problems are formulated to find the effective characteristics. It is shown that the effective coefficients are reciprocal in the case of torsion of a layer with nonuniform thickness. In the two-dimensional case, the problem is solved by a finite element method. The cases of a square beam with single and multiple inclusions are discussed. The dependence of the effective characteristics on the inclusion volume fraction is analyzed.     

HOMOGENIZATION—BASED TOPOLOGY DESIGN FOR PURE TORSION OF COMPOSITE SHAFTS

In conjunction with the homogenization theory and the finite element method, the mathematical models for designing the corss-section of composite shafts by maximizing the torsion rigidity are developed in this paper. To obtain the extremal torsion rigidity, both the cross-section of the macro scale shaft and the representative microstructure of the composite material are optimized using the new models. The micro scale computational model addresses the problem of finding the periodic microstructures with extreme shear moduli. The optimal microstructure obtained with the new model and the homogenization method can be used to improve and optimize natural or artificial materials. In order to be more practical for engineering applications, cellular materials rather than ranked materials are used in the optimal process in the existence of optimal bounds for the elastic properties. Moreover, the macro scale model is proposed to optimize the cross-section of the torsional shaft based on the tailared composites. The validating optimal results show that the models are very effective in obtaining composites with extreme elastic properties, and the cross-section of the composite shaft with the extremal torsion rigidity. The project supported by the National Natural Science Foundation of China (10172078 and 10102018)     

GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.     

A new finite element of spatial thin-walled beams

Based on the theories of Timoshenko＇s beams and Vlasov＇s thin-walled members, a new spatial thin-walled beam element with an interior node is developed. By independently interpolating bending angles and warp, factors such as transverse shear deformation, torsional shear deformation and their Coupling, coupling of flexure and torsion, and second shear stress are considered. According to the generalized variational theory of Hellinger-Reissner, the element stiffness matrix is derived. Examples show that the developed model is accurate and can be applied in the finite element analysis of thinwalled structures.     

直接基于单元平衡的梁非线性分析方法

以单元力平衡条件为基础建立的梁单元在近年来受到广泛关注,与传统的以单元位移相容条件为基础建立的梁单元相比,基于平衡的梁单元可获得较高的精度。针对基于平衡的梁单元,给出了一种几何非线性分析方法,使用该方法进行几何非线性分析时不需进行单元内部位移场的构造,避免了使用现有位移场构造方法时可能引起的振荡问题,同时降低了有限元程...