Logarithmic stress singularities at clamped-free corners of a cantilever orthotropic beam under flexure

The rectangular orthotropic beam under flexure is studied by decomposing the problem into an interior problem and a boundary problem. The interior problem is required to satisfy field equations and boundary conditions at the lateral faces, whereas the boundary conditions at the beam ends are imposed in the average sense. The boundary problem, which reestablishes the pointwise boundary conditions at the beam ends, is solved via eigenfunction expansion under the assumption of transverse inextensibility. It is shown that logarithmic stress singularities are present at the corners of the clamped end section and the corresponding stress-intensity factor is computed.     

A sixth-order multiderivative method for two beam problems

A sixth-order method is developed and analysed for the numerical solution of the linear fourth-order boundary value problem for which the boundary conditions are given in terms of functional values and (i) first-order derivatives (the clamped-clamped beam problem) or (ii) second-order derivatives (the simple-simple beam problem). The method is tested on a number of problems from the literature.     

Numerical transfer-method with boundary conditions for arbitrary curved beam elements

A novel numerical transfer-method is presented to solve a system of linear ordinary differential equations with boundary conditions. It is applied to determine the structural behaviour of the classical problem of an arbitrary curved beam element. The approach of this boundary value problem yields a unique system of differential equations. A Runge–Kutta scheme is chosen to obtain the incremental transfer expression. The use of a recurrence strategy in this equation permits to relate both ends in the domain where boundary conditions are defined. Semicircular arch, semicircular balcony and elliptic–helical beam examples are provided for validation.     

Hydroelastic behaviour of compound floating plate in waves

The paper deals with the plane problem of the hydroelastic behaviour of floating plates under the influence of periodic surface water waves. Analysis of this problem is based on hydroelasticity, in which the coupled hydrodynamics and structural dynamics problems are solved simultaneously. The plate is modeled by an Euler beam. The method of numerical solution of the floating-beam problem is based on expansions of the hydrodynamic pressure and the beam deflection with respect to different basic functions. This makes it possible to simplify the treatment of the hydrodynamic part of the problem and at the same time to satisfy accurately the beam boundary conditions. Two approaches aimed to reduce the beam vibrations are described. In the first approach, an auxiliary floating plate is added to the main structure. The size of the auxiliary plate and its elastic characteristics can be chosen in such a way that deflections of the main structure for a given frequency of incident wave are reduced. Within the second approach the floating beam is connected to the sea bottom with a spring, the rigidity of which can be selected in such a way that deflections in the main part of the floating beam are very small. The effect of the vibration reduction is quite pronounced and can be utilized at the design stage.     

An inverse mode problem for the continuous beam

Given three eigenvalues and their corresponding eigenfunctions of the beam. It is assumed that the total mass of a beam is known. The problem of constructing the physical elements of the system from the known data is considered. The sufficient conditions for the construction of a physical realizable system with positive continuous variable: bending rigidity function and density function are established. If these conditions are satisfied, the system may be constructed uniquely.     

Rotation effects in the global/local analysis of cantilever beam contact problems

Summary A modification of a previously developed method for solving the static contact problem of a rigid, cylindrical punch indenting an isotropic cantilever beam is presented. The boundary conditions at the contact site are modified by including beam upper-surface rotation effects. Local contact stress distributions, as well as global beam displacements, are computed for various ratios of contact width to beam thickness and for various positions of the indenter. The results are compared to those of a previous study in which the rotation effects were not considered, to Hertz theory of contact stress, and to beam theory solutions. It is noted that the rotation effects became prominent as the contact width approached the thickness of the beam.     

Static and dynamic analysis of transversely isotropic,moderately thick sandwich beams by analogy

Summary A flexural theory of elastic sandwich beams is derived which renders quite precise results within a wide range of ratios of dimensions, mass densities, and elastic constants of the core and faces. The assumptions of the Timoshenko theory of shear-deformable beams are applied to each of the homogeneous, linear elastic, transversely isotropic layers individually. Core and faces are perfectly bonded. The principle of virtual work is applied to derive the equations of motion of a symmetrically designed three-layer beam and its boundary conditions. By definition of an effective cross-sectional rotation the complex problem is reduced to a problem of a homogeneous beam with effective stiffnesses and with corresponding boundary conditions. Thus, methods of classical mechanics become directly applicable to the higher-order problem. Excellent agreement of the results of illustrative examples is observed when compared to solutions of other higher-order laminate theories as well as to exact solutions of the theory of elasticity.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

A class of higher-order continuum theories, such as modified couple stress, nonlocal elasticity, micropolar elasticity (Cosserat theory) and strain gradient elasticity has been recently employed to the mechanical modeling of micro- and nano-sized structures. In this article, however, we address stability problem of micro-sized beam based on the strain gradient elasticity and couple stress theories, firstly. Analytical solution of stability problem for axially loaded nano-sized beams based on strain gradient elasticity and modified couple stress theories are presented. Bernoulli–Euler beam theory is used for modeling. By using the variational principle, the governing equations for buckling and related boundary conditions are obtained in conjunctions with the strain gradient elasticity. Both end simply supported and cantilever boundary conditions are considered. The size effect on the critical buckling load is investigated.     

Finite differences/elements in classical beam problems: derivation of feasibility conditions under parametric inequality constraints with the help of Reduce and REDLOG

The solution of the classical fourth-order ordinary differential equation for static beam problems by using the finite difference method is reconsidered, but this time for the derivation of feasibility conditions in cases of validity of parametric linear inequality constraints with respect to the loading/geometry of the beam. To this end, the computer algebra system Reduce has been used, but supplemented by its recent REDLOG (REDuce LOGic) package incorporating the efficient Weispfenning computational quantifier elimination algorithms. A particular problem for a finite beam loaded by a triangular loading has been employed as the vehicle for the illustration of the present approach and the derived feasibility conditions are displayed. The finite element method has also been used (instead of the finite difference method) in the same problem. The present results can also be generalized to problems of beams on an elastic foundation, to two-dimensional problems, to optimization problems, etc.     

Analytical solution of a cantilever functionally graded beam

The problem of a cantilever functionally graded beam subjected to different loads is studied. In terms of Airy stress function a general two-dimensional solution is presented for a cantilever functionally graded beam, assuming that all the elastic moduli of the material have the same variations along the beam-thickness direction. Explicit expressions of analytical solutions to some specific examples under different boundary conditions are obtained to demonstrate the usefulness of the proposed general solution technique. This solution will be useful in analyzing functionally graded beam with arbitrary variations of material properties and it can serve as a basis for establishing simplified functionally graded beam theories.     

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.     

Free vibration analysis of beams by using a third-order shear deformation theory

In this study, free vibration of beams with different boundary conditions is analysed within the framework of the third-order shear deformation theory. The boundary conditions of beams are satisfied using Lagrange multipliers. To apply the Lagrange’s equations, trial functions denoting the deflections and the rotations of the cross-section of the beam are expressed in polynomial form. Using Lagrange’s equations, the problem is reduced to the solution of a system of algebraic equations. The first six eigenvalues of the considered beams are calculated for different thickness-to-length ratios. The results are compared with the previous results based on Timoshenko and Euler-Bernoulli beam theories.     

基于传递函数方法的约束层阻尼梁动力学分析

主要采用传递函数方法对约束层阻尼梁进行动力学分析。首先应用Hamilton原理推导出梁的六阶微分方程和边界条件，然后引入状态向量，使用分布参数传递函数方法建立系统的状态空间方程，得到了系统的固有频率、损耗因子和频响曲线。算例韵计算结果和相关文献比较，吻合良好，表明该方法的正确性。且传递函数方法计算简捷，无需求解复杂变系数的微分方程组，更适宜分析粘弹性材料力学性能随频率变化的结构动力学问题。     

Regular wave impact onto an elastic plate

A computational analysis of an elastic plate dropped against regular long waves is presented. The problem is considered within the linear potential-flow theory. The liquid flow is two-dimensional and the plate is modelled by an Euler beam. The analysis is based on the normal-mode method with hydroelastic behavior of the plate being of main interest. Different impact conditions are considered to study the dependence of the total energy of the plate–liquid system on impact geometry and plate properties. The contributions of kinetic and potential energies to the total energy are analyzed. It is shown that the kinetic part of the system energy is small at the instant of time when bending stresses in the beam approach their maximum values. Estimations of both the total energy and the maximum of bending stresses are presented. Most of the calculations are performed for the conditions of experiments carried out in MARINTEK. A range of the problem parameters is also considered, to reveal peculiarities of the unsteady interaction between a falling elastic plate and surface waves.     

Scattering of a spheroidal particle illuminated by a gaussian beam

An approach to expanding a Gaussian beam in terms of the spheroidal wave functions in spheroidal coordinates is presented. The beam-shape coefficients of the Gaussian beam in spheroidal coordinates can be computed conveniently by use of the known expression for beam-shape coefficients, g(n), in spherical coordinates. The unknown expansion coefficients of scattered and internal electromagnetic fields are determined by a system of equations derived from the boundary conditions for continuity of the tangential components of the electric and magnetic vectors across the surface of the spheroid. A solution to the problem of scattering of a Gaussian beam by a homogeneous prolate (or oblate) spheroidal particle is obtained. The numerical values of the expansion coefficients and the scattered intensity distribution for incidence of an on-axis Gaussian beam are given.     

Vibrations of Layered Beams under Conditions of Damaging and Slipping Interfaces

A nonlinear laminate layerwise beam theory is developed to simulate the effect of inelastic interlayer slip on the stiffness degradation of layered beam structures. Layerwise continuous and linear in-plane displacement fields are implemented. It is shown that by definition of an effective cross-sectional rotation the complex problem reduces to the simpler case of a homogenized shear-deformable beam with effective stiffness and a corresponding set of boundary conditions. Inelastic defects of the interlayer material are equivalent to eigenstrains in an identical but unlimited elastic background structure of the homogenized beam with proper effective virgin stiffness. Cross-sectional resultants of these eigenstrains are defined. Since the incremental response of the background to the given load and to the properly imposed eigenstrain increments is considered to be linear within a given time step, effective solution methods of the linear theory of flexural vibrations become applicable. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two-to-one internal resonances in a shallow curved beam resting on an elastic foundation

Summary Vibrations of shallow curved beams are investigated. The rise function of the beam is assumed to be small. Sinusoidal and parabolic curvature functions are examined. The immovable end conditions result in mid-plane stretching of the beam which leads to nonlinearities. The beam is resting on an elastic foundation. The method of multiple scales, a perturbation technique, is used in search of approximate solutions of the problem. Two-to-one internal resonances between any two modes of vibration are studied. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed, and a bifurcation analysis of the amplitude and phase modulation equations are given. Conditions for internal resonance to occur are discussed, and it is found that internal resonance is possible for the case of parabolic curvature but not for that of sinusoidal curvature.     

利用不完备实测模态修正杆系结构约束边界条件

该文针对杆系结构约束边界条件模型修正问题进行了研究。通过边界单元刚度矩阵的建立,将现有的子结构校正因子修正方法发展到边界条件的修正中。通过一种迭代的修正过程,仅用有限的低阶实测频率就可以实现对边界条件的修正。另外,该文提出了一种分步式整体修正方法,利用低阶不完备实测模态,实现对边界和上部结构的整体修正,有效地避免了修正方程组的病态问题。论文首先通过对悬臂梁有限元模型进行数值研究,验证了边界条件修正方法和分步式整体修正方法的正确性。最后通过悬臂梁物理模型试验,验证了方法的可行性。     

Separation of delamination modes in composite beams with symmetric delaminations

Delaminated composite beam under general edge loading conditions is studied. Based on a technical engineering theory an analytical procedure for calculation of strain energy release rate and its separation into modes I and II of delamination is presented. By choosing a suitable displacement field based on second-order shear-thickness deformation theory and using the principle of minimum total potential energy, the equations of equilibrium are obtained along with the appropriate boundary conditions. The J integral and its definition for different modes of fracture is used for calculation of strain energy release rate and its separation into different modes. Double cantilever beam (DCB) problem is a special case of this general problem in which loading is in mode I of fracture. The results of this method shows good agreement with FEM (i.e., finite element method) results and experimental data.