The strain energy release rates in adhesively bonded balanced and unbalanced specimens and lap joints

An analytical model is developed to determine the strain energy release rate in adhesive joints of various configurations such as the double-cantilever beam and single-lap joints. The model is based on asymptotic analysis of adhesive layer stresses and Irwin’s crack closure integral. Closed-form solutions are presented for balanced and unbalanced joints under mode I, II and mixed-mode I/II that take into account the influence of the shear force on the adhesive stresses, and its influence on the strain energy release rate. The accuracy of the model is tested against the classical beam theory expressions for double-cantilever beam and end-notch flexure specimens. In fact, classical beam theory’s expressions are found to be the lower bound of the proposed model solutions, and the two methods converge as the adhesive layer thickness decreases. Analysis of single-lap joints reveals the influence of edge shear forces on the total strain energy release rate, and more importantly on the ratio between modes I and II. Results from the proposed analytical model are in good agreement with finite element results and with analytical models found in the literature.     

采用有限单元法计算梁任意形状截面特性

采用将梁截面离散化的方式,用数值积分计算截面的几何特性,并根据梁剪切变形和扭转理论,利用变分原理建立截面的有限元法方程,求解任意形状截面的扭转常数、剪切中心以及剪切面积修正系数等特性.本方法适用于各种形式的截面,具有计算精度高及适应性强的特点.根据上述理论编制了相应程序,按照不同的单元划分方式,分别计算出矩形截面截面特性,与理论解进行比较;又对舟山市定海长峙至岙山预应力混凝土连续箱梁截面进行了计算,并与Ansys结果进行比较,均证明采用本文的计算方法能得到满意的结果,且该方法适用于各种形状的截面形式.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

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.     

均布力偶作用下细长梁力学分析

通过铁木辛柯梁理论分析了反向均布表面剪应力——等效均匀分布力偶作用下的等截面均质细长梁挠度和应力分布规律,并与有限元法的计算结果对比发现：当边界条件中剪力不为零时,弯曲挠度和正应力分析必须考虑剪力的影响,即Euler梁理论不能满足分析的要求;若存在剪力为零边界时,可使用Euler梁分析弯曲挠度和正应力;剪应力分布向通常规律一样,仍沿高度方向呈抛物线分布,即使对于剪力为零的横截面也可能存在剪应力,这是由于表面剪应力的影响使得梁的上下表面存在剪应力,并且剪应力在横截面内正负可以发生变化。     

多亚层柔性节点模型及其在双材料4ENF试件界面分析中的应用

提出了多亚层柔性节点模型用于分析双材料裂纹尖端的应力和变形。该模型考虑了胶层的变形,各亚层视为独立的剪切变形梁,采用两个界面柔度系数考虑界面应力对各亚层界面变形的影响,界面变形包括双材料界面和胶层的变形。通过对FRP-混凝土末端切口四点弯试件(Four-point bending end-notched flexure specimen,简称4ENF)进行界面分析,并与其他模型和有限元分析对比表明:刚性节点模型忽略了裂纹尖端的应力和变形集中,只能粗略地估计构件的整体变形和界面应力;半刚性节点容许裂纹尖端的转动,对裂纹尖端的变形估计优于刚性节点模型,但精度依然不高;多亚层柔性节点模型反映了裂纹尖端的应力和变形集中,与数值分析结果吻合很好,该研究对进行双材料结构的工程设计具有理论指导和参考价值。     

Finite element analysis of interfacial stresses in FRP–RC hybrid beams

The interfacial stresses in fiber reinforced plastic (FRP)–reinforced concrete (RC) hybrid beams were studied by the finite element method. The mesh sensitivity test shows that the finite element results for interfacial stresses are not sensitive to the finite element mesh. The finite element analysis then is used to calculate the interfacial stress distribution and evaluate the effect of the structural parameters on the interfacial behavior. It is shown that both the normal and shear stresses at the interface are influenced by the material and geometry parameters of the composite beam. This research is helpful for the understanding on mechanical behavior of the interface and design of the FRP–RC hybrid structures.     

膜-基复合材料界面剪应力的三维半解析计算

 采用影响系数法对膜-基复合材料的界面剪应力三维半解析进行 了分析研究.利用三维有限元方法对薄膜的影响系数进行了计算. 将 基体作为半无限大体，利用其平面边界作用单位力时的位移场解析 解，得到基体的影响系数. 结果表明，对膜-基复合材料界面剪应力 进行三维半解析计算，克服了完全用三维有限元对其进行计算的限 制，为该类问题的分析提供了新的途径.     

Phenomenological invariants and their application to geometrically nonlinear formulation of triangular finite elements of shear deformable shells

A phenomenological definition of classical invariants of strain and stress tensors is considered. Based on this definition, the strain and stress invariants of a shell obeying the assumptions of the Reissner–Mindlin plate theory are determined using only three normal components of the corresponding tensors associated with three independent directions at the shell middle surface. The relations obtained for the invariants are employed to formulate a 15-dof curved triangular finite element for geometrically nonlinear analysis of thin and moderately thick elastic transversely isotropic shells undergoing arbitrarily large displacements and rotations. The question of improving nonlinear capabilities of the finite element without increasing the number of degrees of freedom is solved by assuming that the element sides are extensible planar nearly circular arcs. The shear locking is eliminated by approximating the curvature changes and transverse shear strains based on the solution of the Timoshenko beam equations. The performance of the finite element is studied using geometrically linear and nonlinear benchmark problems of plates and shells.     

一种预测层合板层间剪应力的新后处理方法

层合板是航空航天领域典型的承力构件,过大的层间应力是导致其分层失效的主要原因.准确的层间应力预测往往依赖于三维平衡方程后处理方法(TPM).然而,该方法需要计算面内应力的一阶导,使得基于C0型板理论构造的线性单元无法使用TPM计算横向剪应力.本文在三维平衡方程后处理方法的基础上,提出了一种新后处理方法(NPM).新后处理方法通过虚功等效法消除了三维平衡方程后处理方法中产生的位移参数的高阶导.基于提出的新后处理方法和C0型板理论,仅需使用线性单元就可以预测层合板的横向剪应力.为了验证所提方法的有效性,本文基于修正锯齿理论(RZT)和所提方法构造了一种C0连续的三节点三角形线性板单元.数值算例表明,所提方法和三维平衡方程后处理方法具有相同的计算精度,提出的板单元能够准确高效地预测层合板的横向剪应力.此外,所提方法便于结合现有的有限元商用软件使用,基于商用软件中板壳单元获得的节点位移,使用新后处理方法极易获得准确的层间剪应力.     

钢筋混凝土双肢剪力墙静力弹塑性分析

建立了由墙肢单元模型、连粱单元模型和连接单元模型组成的钢筋混凝土双肢剪力墙的静力弹塑性分析计算模型。墙肢单元采用以有限元为基础的宏模型;按是否出现对角线剪切破坏,分别建立短连粱计算模型和长连粱计算模型;为计及连粱与墙肢连接界面的相对位移,建立用复合弹簧模拟的连接单元计算模型;给出了确定模型参数的方法。对有关文献的短连粱和长连粱双肢剪力墙试件进行了静力弹塑性分析,分析结果与试验结果符合较好。     

复合材料厚梁精化锯齿理论及有限元分析

由于具有预先满足层间应力连续的优点,锯齿理论被广泛研究和应用。然而,至今锯齿理论仍然存在如下难题：基于锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。如果这些问题不被重视和解决,应用此类理论分析复合材料力学问题可能得出不恰当的结论。通过发展高精度的考虑横法向应变的C0型锯齿理论,本文将克服已有锯齿理论遇到的上述难题。基于发展的锯齿理论,构造三节点梁单元验证发展理论模型的性能。     

A NEW SPATIAL THIN-WALLED BEAM ELEMENT INCLUDING TRANSVERSE AND TORSIONAL SHEAR DEFORMATION

基于Timoshenko梁及Benscoter薄壁杆件理论,建立了考虑剪切变形、弯扭耦合以及翘曲剪应力影响的空间任意开闭口薄壁截面梁单元. 通过引入单元内部结点,对弯曲转角和翘曲角采用三节点Lagrange独立插值的方法,考虑了剪切变形和翘曲剪应力的影响并避免了横向剪切锁死问题;借助载荷作用下薄壁梁的截面运动分析,在位移和应变方程中考虑了弯扭耦合的影响. 通过数值算例将该单元的计算结果与理论解以及商用有限元软件和其他文献中的数值解进行对比和验证,结果对比表明该薄壁梁单元具有良好的精度和收敛性.     

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 new zig-zag theory and C<Superscript>0</Superscript> plate bending element for composite and sandwich plates

A higher-order zig-zag theory for laminated composite and sandwich structures is proposed. The proposed theory satisfies the interlaminar continuity conditions and free surface conditions of transverse shear stresses. Moreover, the number of unknown variables involved in present model is independent of the number of layers. Compared to the zig-zag theory available in literature, the merit of present theory is that the first derivatives of transverse displacement have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To obtain accurately transverse shear stresses by integrating three-dimensional equilibrium equations within one element, a six-node triangular element is employed to model the present zig-zag theory. Numerical results show that the present zig-zag theory can predict more accurate in-plane displacements and stresses in comparison with other zig-zag theories. Moreover, it is convenient to obtain transverse shear stresses by integration of equilibrium equations, as the C⁰ shape functions is only used when implemented in a finite element.     

A three-parameter elastic foundation model for interface stresses in curved beams externally strengthened by a thin FRP plate

Externally bonding of fiber reinforced polymer (FRP) plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP–concrete interface are of great importance to the effectiveness of this type of strengthening because high stress concentration along the FRP–concrete interface can lead to the FRP debonding from the concrete beam. In this study, we develop an analytical solution of interface stresses in a curved structural beam bonded with a thin plate. A novel three-parameter elastic foundation model is used to describe the behavior of the adhesive layer. This adhesive layer model is an extension of the two-parameter elastic foundation commonly used in existing studies. It assumes that the shear stress in the adhesive layer is constant through the thickness, and the interface normal stresses along two concrete/adhesive and adhesive/FRP interfaces are different. Closed-form solutions are obtained for these two interfacial normal stresses, shear stress within the adhesive layer, and beam forces. The validation of these solutions is confirmed by finite element analysis.     

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.     

Inelastic axial-flexure–shear coupling in a mixed formulation beam finite element

In this paper a beam element that accounts for inelastic axial-flexure–shear coupling is presented. The mathematical model is derived from a three-field variational form. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the beam. No approximation for the beam displacement field is necessary in the formulation. The coupling of the section forces is achieved through the numerical integration of an inelastic multi-axial material model over the cross-section. The proposed element is free from shear-locking. Examples confirm the accuracy and numerical robustness of the proposed element and showcase the interaction between axial force, shear, and bending moment.     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.