Interfacial stresses in curved members bonded with a thin plate

The use of steel plates or externally bonded fibre-reinforced polymer laminates for flexural strengthening of concrete, masonry, timber or metallic structures is a technique that has become very popular. The effectiveness of this technique hinges heavily on the performance of the bond between the strengthening plate and the substrate, which has been the subject of many existing studies. In particular, the interfacial stresses between a beam and a soffit plate within the linear elastic range have been addressed by numerous analytical investigations. Surprisingly, none of these investigations has examined interfacial stresses in members with a curved soffit, despite that such members are often found in practice. This paper presents an analytical model for the interfacial stresses between a curved member of uniform section size and a thin plate bonded to its soffit. The governing differential equations for the interfacial shear and normal stresses are formulated and then solved with appropriate simplifying assumptions. Two numerical examples are presented to illustrate the effect of the curvature of the member on the interfacial stress distributions in a simply supported curved beam for the two cases of a point load and a uniformly distributed load. The analytical solution is verified by comparing its predictions with those from a finite element model.     

钢－混凝土组合梁非线性有限元分析

提出了部分剪力连接钢-混凝土组合梁非线性有限元分析的简化模型，该模型同时考虑了钢、混凝土材料的非线性本构关系和剪力连接件的滑移非线性对组合梁刚度和强度的影响，推导了考虑滑移的剪力连接件单元刚度矩阵。利用该模型计算了连续组合梁的极限荷载、挠度、应力以及滑移，与已有的理论计算结果和实验结果吻合，证明本方法分析钢-混凝土组合梁非线性的可靠性和实用性。     

Three-dimensional nonlinear vibrations of composite beams — III. Chordwise excitations

Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Three-dimensional nonlinear vibrations of composite beams — II. flapwise excitations

The nonlinear equations of motion derived in Part I are used to investigate the response of an inextensional, symmetric angle-ply graphite-epoxy beam to a harmonic base-excitation along the flapwise direction. The equations contain bending-twisting couplings and quadratic and cubic nonlinearities due to curvature and inertia. The analysis focuses on the case of primary resonance of the first flexural-torsional (flapwise-torsional) mode when its frequency is approximately one-half the frequency of the first out-of-plane flexural (chordwide) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations to describe the time variation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability and bifurcations of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic solutions of the modulation equations are studied. Chaotic solutions are identified from their frequency spectra, Poincaré sections, and Lyapunov's exponents. The results show that the beam motion may be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Stress distribution in a composite materialwith a row of antiphase periodically curved fibers

The stress distribution in an infinite matrix containing a row of antiphase periodically curved fibers is studied using the piecewise-homogeneous body model and the exact three-dimensional equations of elasticity theory for anisotropic bodies. It is assumed that the midlines of the fibers are located in the same plane and the materials of the fibers are the same. Numerical results are presented for the case where the materials of the fibers and matrix are isotropic and homogeneous Published in Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 136–144, April 2006.     

Non-linear modal analysis for beams subjected to axial loads: Analytical and finite-element solutions

A rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models. Two important loading scenarios were considered, where a structure is subjected to a uniformly distributed axial and a thrust force. These loads are to mimic the main forces acting on an offshore riser, for which an analytical methodology has been developed and applied. In particular, non-linear normal modes (NNMs) and non-linear multi-modes (NMMs) have been constructed by using the method of multiple scales. This is to effectively analyse the transversal vibration responses by monitoring the modal responses and mode interactions. The developed analytical models have been crosschecked against the results from FEM simulation. The FEM model having 26 elements and 77 degrees-of-freedom gave similar results as the low-dimensional (one degree-of-freedom) non-linear oscillator, which was developed by constructing a so-called invariant manifold. The comparisons of the dynamical responses were made in terms of time histories, phase portraits and mode shapes.     

钢-混凝土组合箱梁梁段有限元法研究

组合梁界面滑移将减小组合梁刚度,增大变形,影响构件性能;剪应力沿截面横向分布不均匀,造成其弯曲正应力的横向分布呈曲线形状,即剪力滞效应;同时组合梁往往重载,具有较小的跨高比,剪切变形不可忽略.根据虚功原理,建立了同时考虑滑移效应、剪力滞及剪切变形效应的组合梁单元刚度矩阵及等效节点力向量,并在此基础上编制了组合梁梁段有限元程序.利用本文程序对现有组合梁试件的混凝土顶板应力、钢梁底板应力、跨中挠度和梁端滑移进行了计算,并将本文计算结果与解析法计算结果及试验结果进行了比较.结果表明,本文计算结果与解析法计算结果及试验结果吻合良好;同时,本文计算结果具有较好的稳定性,验证了本文计算方法的正确性.本文所建立的梁单元刚度矩阵同时考虑了剪切变形、剪力滞及滑移效应的影响,符合工程实际,为有限梁段法分析组合箱梁提供了理论基础.     

General analytical solution to bending of composite laminated beams with delaminations

Based on the first-order shear deformable beam theory, a refined model for composite beams containing a through-the-width delamination is presented, and the deformation at the delamination front is considered. Different from the ordinary delami- nated beam theory, each of the perfectly bonded portions of the new model is constructed as two separated beams along the interface without assuming a plane section at the de- lamination front. The governing equations of the delaminated portions and bonded ones are established, combined with continuity conditions of displacements and internal forces. Solutions of delaminated composite beams with different boundary conditions, delamina- tion locations and sizes axe shown in excellent agreement with the finite element results, showing efficiency and applicability of the present model.     

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.     

Geometric nonlinear formulation for curved beams with varying curvature

Based on exact Green strain of spatial curved beam, the nonlinear strain-displacement relation for plane curved beam with varying curvature is derived. Instead of using the previous straight beam elements, curved beam elements are used to approximate the curved beam with varying curvature. Based on virtual work principle, rigid-flexible coupling dynamic equations are obtained. Physical experiments were carried out to capture the large overall motion and the strain of curved beam to verify the present rigid-flexible coupling formulation for curved beam based on curved beam element. Numerical results obtained from simulations were compared with those results from the physical experiments. In order to illustrate the effectiveness of the curved beam element methodology, the simulation results of present curved beam elements are compared with those obtained by previous straight beam elements. The dynamic behavior of a slider-crank mechanism with an initially curved elastic connecting rod is investigated. The advantage of employing generalized-α method is pointed out and the special nonlinear dynamic characteristics of the curved beam are concluded.     

Analytical solutions to edge effect of composite laminates based on symplectic dual system

In the symplectic space composed of the original variables, displacements, and their dual variables, stresses, the symplectic solution for the composite laminates based on the Pipes-Pagano model is established in this paper. In contrast to the traditional technique using only one kind of variables, the symplectic dual variables include displacement components as well as stress components. Therefore, the compatibility conditions of displacement and stress at interfaces can be formulated simultaneously. After being introduced into the symplectic dual system, the uniform schemes, such as the separation of variables and symplectic eigenfunction expansion method, can be implemented conveniently to analyze composite laminate problems. An analytical solution for the free edge effect of composite laminates is obtained, showing the effectiveness of the symplectic dual method in analyzing composite laminates.     

Stochastic boundary element method for reliability analysis of plate and beams composite structures

I.IntroductionTheengineeringstructuresareoftells,'1>:'rectcdtotheactionofthestochasticloadingthatvarieswiththetime,forexample,theengineeringstructuresactedonbytheearthquake,theoceanstructuresactedonbydynamicpressure,andthevehiclesofthetransportationinflue…     

Shear deformation effect in non-linear analysis of composite beams of variable cross section

In this paper the non-linear analysis of a composite Timoshenko beam with arbitrary variable cross section undergoing moderate large deflections under general boundary conditions is presented employing the analog equation method (AEM), a BEM-based method. 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 beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the AEM. Application of the boundary element technique yields a system of non-linear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples are worked out to illustrate the efficiency, the accuracy, the range of applications of the developed method and the influence of the shear deformation effect.     

Solutions of the integral equations for shearing stresses in two-material curved beams

In the same way as shearing stresses for curved beams made of one material, the problem of evaluating the shearing stresses of composite curved beams is also reduced to one of solving the integral equations. Solving directly two integral equations can derive the formulae of shearing stresses, which satisfy not only the equilibrium equations but also the static boundary conditions on the boundary surfaces of the beams. The present analysis will be used to investigate the shearing stresses of a cantilevered curved beam made of two materials, which is loaded by a concentrated force at its free end. The comparison between the numerical results of shearing stresses obtained using the equations developed in this paper and a three-dimensional finite element analysis shows excellent agreement.     

Calculation of higher natural frequencies of simply supported elastic composite beams using Riccati matrix method

A composite beam is composed of an upper slab and a lower beam connected at the interface by shear transmitting studs. In this paper, an improved and efficient numerical model for the calculation of higher natural frequencies of an elastic composite beam is presented. The numerical model uses the Riccati transfer matrix method. First, the exact field transfer matrix for an element of the beam is represented using the combination of the Analog Beam Method and the transfer matrix method. Second, applying Riccati method to the beam system, the natural frequencies can be easily calculated. The advantage of the present model is to overcome the numerical instabilities of the ordinary transfer matrix method, especially when calculating the higher natural frequencies of structures. A numerical example is given to illustrate and compare the results with those available from other methods. Finally, a parametric study is given to examine the effect of various parameters of the elastic composite beam on its free vibration behavior.     

钢板-混凝土组合加固钢筋混凝土梁的非线性有限元分析

以10根采用钢板-混凝土组合加固技术加固的钢筋混凝土梁的试验为基础,建立了钢板-混凝土组合加固钢筋混凝土梁的非线性有限元模型.沿用了钢板-混凝土组合梁有限元分析中栓钉的荷载-滑移曲线及断裂模型,提出了新老混凝土界面的剪切-滑移模型,可以较好地模拟界面受力性能.采用有限元软件ABAQUS,模拟了一次加固、不同损伤加固及持载加固等加载情况下加固构件受力全过程和受弯破坏以及新老混凝土剥离破坏等破坏形态下加固试件的受力性能.在验证了有限元模型的准确性后,分析了钢板厚度、加固部分高度及长度等参数对加固构件承载能力及刚度的影响.最后提出了几点对钢板-混凝土组合加固RC梁的设计建议.     

Local buckling of wide-flange thin-walled anisotropic composite beams

The present contribution deals with the onset of local buckling of compressively loaded thin-walled beams with open I, C, Z, T and L-cross-sections made of laminated composite materials. The method employs a discrete plate analysis approach in the course of which each structural subelement of interest—which presently is the flange—of the thin-walled cross-section is considered as a separate composite plate with elastic rotational restraints at those edges where an adjacent substructural element is located. While in many investigations the lamination schemes of webs and flanges are considered to be purely orthotropic, in the present paper the laminate layups are allowed to be of an arbitrary non-orthotropic nature, which also allows for the analysis of laminates with inherent bending–torsion coupling. The analysis of the buckling loads of the flanges of thin-walled composite beams is performed using the Ritz-method for which some especially adjusted displacement shape functions are employed. For the case of pure orthotropy, a novel closed-form solution is described. The accuracy of the employed approaches is established by comparison with accompanying finite element simulations of thin-walled composite beams. It is revealed that the presented methodology is highly efficient in terms of computational effort and yet performs with satisfying accuracy, which makes it very attractive for actual practical applications whenever the local stability behaviour of wide-flange thin-walled composite beams is to be considered.     

钢-混凝土组合梁负弯矩区畸变屈曲弯矩计算公式

侧向弯曲屈曲及侧向弯扭屈曲均为钢-混凝土组合梁负弯矩区钢梁的重要屈曲模式,而现有计算方法通常只考虑到侧向弯曲屈曲,未考虑到侧向弯扭屈曲,因此有一定局限性。本文在钢梁腹板对钢梁下翼缘的转动约束刚度及侧向约束刚度的计算公式上,结合弹性介质中薄壁杆件的屈曲理论推导了工形钢-混凝土组合梁负弯矩区的侧向弯曲屈曲弯矩及侧向弯扭屈曲弯矩计算公式。实例分析表明,现有计算方法均存在一定理论缺陷,本文计算方法更为合理;同时,本文计算式比现有文献中同类型计算式更为简洁,便于手算,实用性强并适于工程应用。     

Vibration of composite beams with two overlapping delaminations

Delaminations in composite laminates may develop from small cracks due to fabrication and impact loading, or from places of high stress concentration. The locations of the delaminations are not determinate. In this research, an analytical solution for the free vibration of a composite beam with two overlapping delaminations is presented. The delaminated beam is analyzed as seven interconnected beams using the delaminations as their boundaries. The continuity and equilibrium conditions are satisfied between the adjoining regions of the beams. Classical beam theory is applied to each of the beams. Complex vibration behaviors emerge for different sizes and locations of the delaminations. Comparison with analytical results reported in the literature verifies the validity of the present solution.Christian N Della wishes to thank Nangyang Technological University for the scholarship.