钢筋混凝土梁在低速冲击下的变形与破坏研究

通过分析钢筋混凝土梁在低速冲击下的应力状态和变形破坏多阶段的性状,建立了冲击体与梁局部与整体相互作用的刚性离散模型,推导了钢筋混凝土梁各构件在不同阶段的变形刚度,利用拉格朗日方程导出了弹性与塑性阶段的运动方程。分析表明：在低速冲击荷载作用下梁的变形与其动力特性、冲击体与梁的质量比Ms/Mb,冲击速度V0、冲击时间和冲击体形状密切相关。在足够宽的Ms/Mb、V0的变化范围内,钢筋混凝土梁呈现出一致的状态特性。     

边界约束对梁抗爆动力响应的影响(Ⅰ)–理论研究及分析

边界约束的差异会直接影响结构的抗爆动力响应及承载能力,文中建立了复杂约束条件下抗爆梁在弹性阶段和塑性阶段的解析计算方法,并计算分析了竖向弹性与阻尼约束、水平约束刚度、抗弯约束、荷载形式以及屈服弯矩动力强化系数对动力响应的影响。计算表明：竖向弹性与阻尼约束会引起附加惯性力,能够明显降低结构在弹塑性阶段的位移动力系数。水平约束和抗弯约束影响结构的动态响应主要在塑性阶段,水平约束使梁截面在变形过程中产生横向压力,抗弯约束直接限制刚体转动,均有效降低了梁位移动力系数,相对提高结构的承载力。相同约束刚度和荷载峰值条件下,平台荷载下结构的位移动力函数均高于三角形荷载下位移动力函数,说明动荷载的作用时间越长,对结构承载越不利。另外考虑屈服弯矩的动力增强系数时,可提高结构的抗爆潜力。     

考虑有限深度土体运动的Winkler地基梁1/2次谐波共振响应分析

土-结构相互作用系统动力响应的基本特征之一是有限范围内弹性地基与其支承结构共同运动,将土体运动引入系统的动力学方程可体现其对系统动力学特性的影响。基于考虑有限深度土体运动影响的Winkler地基上有限长梁的非线性运动方程,利用Galerkin法和多尺度法,求得弹性地基梁1/2次谐波共振的幅频响应方程和位移的二阶近似解。进而通过数值计算,得到了梁1/2次谐波共振的幅频响应曲线,研究了地基深度、质量、弹性模量、Winkler参数和阻尼等对弹性地基梁1/2次谐波共振响应的影响。研究结果表明:有限深度土体运动对Winkler地基梁1/2次谐波共振响应影响显著。运动方程中引入土体运动的影响后,梁1/2次谐波共振区间明显减小。随地基深度、质量和弹性模量改变,弹性地基梁1/2次谐波共振的幅频响应曲线偏转程度、共振区间和响应幅值等均发生定量改变。当弹性地基刚度增大到一定程度,Winkler地基参数变化对系统1/2次谐波共振响应的影响明显减弱。阻尼对系统动力响应起抑制作用,当参数η增大到一定值后将不会出现1/2次谐波共振响应的非平凡解。     

An intrinsic beam model based on a helicoidal approximation—Part II: Linearization and finite element implementation

The helicoidal beam model developed in the first part of this work is applied here to the development of a mixed finite element for space-curved and twisted beams undergoing large displacements and finite rotations. Starting from the governing weak form expressed by the principle of virtual work, a consistent linearization is obtained in the following and a novel updated Lagrangian finite element implementation is thoroughly discussed. The unique features and the distinguishing properties previously claimed for the helicoidal model are shown here to imply remarkable numerical consequences. For this purpose, meaningful example problems regarding the non-linear static response of beams are addressed in the following and the results are compared with those available from the literature. Furthermore, a finite element in time for the rigid body dynamic problem is developed within the framework of the helicoidal geometry. The underlying philosophy of this novel finite element for dynamics is the realization of the helicoidal decomposition of the rigid body motion within a time step.     

A co-rotational weak-form quadrature planar beam element for geometric nonlinear static and dynamic analysis

The co-rotational formulation of quadrature planar beam element undergoing large displacement and large rotation is presented. A local frame co-rotates with the differential element and decomposes the motion into a rigid body movement and a strain-producing deformation. General explicit formulations of elemental vectors and matrices, including internal force vector, external force vector, tangent stiffness matrix, and mass matrix, are derived via the numerical integration together with the differential quadrature law. Thus, the element nodes and numerical integration method can be chosen arbitrarily based on the accuracy requirement and problem type. A number of case studies on the static, postbuckling, and dynamic response of beams and frame structures are conducted. The convergence study shows that the co-rotational quadrature element has an exponential rate of convergence and the reduced Gauss integration yield the highest accuracy. It is seen that the proposed co-rotational quadrature beam element is simple in formulations, computationally efficient, and capable of capturing the complex nonlinear behavior of beam and frame structures with high precision.     

Vibrations of non-uniform functionally graded MWCNTs-polystyrene nanocomposite beams under action of moving load

Free and forced vibrations of non-uniform functionally graded multi-walled carbon nanotubes (MWCNTs)-polystyrene nanocomposite beams are investigated via Timoshenko beam theory. Different MWCNTs distributions in the thickness direction are introduced to improve fundamental natural frequency and dynamic behavior of non-uniform polymer composite beam under action of moving load. So, linear distribution patterns of carbon nanotubes (CNTs) in the thickness direction which can readily be achieved in practice are studied. The effects of shear deformation, rotary inertia, non-uniformity of the cross-section are also considered in the formulation. The finite element method is employed to obtain a numerical approximation of the motion equation. The non-uniform beam is approximated by another beam consisting of n elements with piecewise constant thickness so that the volume remains constant for each element. The effects of non-uniformity parameters, material distributions, velocity of the moving load and boundary conditions on the dynamic behavior are investigated. It is found that the symmetrical linear distribution of MWCNTs results in an increase in the fundamental natural frequency of nanocomposite beams which are higher than those of beams with uniform and unsymmetrical MWCNTs distributions.     

Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through non-Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and non-Kolmogorov spectrum, the analytical expression for the effective radius of curvature of partially coherent Hermite–Gaussian (PCHG) beams propagating through non-Kolmogorov turbulence is derived, and the relative effective radius of curvature is used to describe the effect of turbulence on the effective radius of curvature. It is shown that the effective radius of curvature of PCHG beams depends on the beam and non-Kolmogorov turbulence parameters and on the propagation distance. The variation of relative effective radius of curvature with increasing generalized exponent parameter α of non-Kolmogorov turbulence is non-monotonic. The longer the propagation distance is, the larger the effect of turbulence on the effective radius of curvature of PCHG beams is. The effective radius of curvature of PCHG beams with shorter wavelength, smaller beam order, larger beam waist width or better spatial coherence is more affected by the non-Kolmogorov turbulence. The results are interpreted physically.     

A Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams

The objective of this paper is to develop constitutive equations of a Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic–plastic and elastic–viscoplastic beams with rigid cross-sections. Specifically, attention is limited to response of a material with constant yield strength. A yield function is proposed which couples the inelastic responses of tension and shear. Another yield function is proposed for bending which depends on a hardening variable that models motion of the elastic–plastic boundary in the beam’s cross-section. Evolution equations are proposed for elastic strains and the hardening variable and an overstress-type formulation is used for elastic–viscoplastic response. In contrast, with standard finite element approaches the CPE model needs no integration through the element region. Also, an implicit scheme is developed to integrate the evolution equations without iteration. Examples of transient large motions of beams, which are impulsively loaded, indicate that the CPE produces reasonably accurate response relative results in the literature and full three-dimensional calculations using ABAQUS.     

Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load

This paper studies the dynamic response of functionally graded beams with an open edge crack resting on an elastic foundation subjected to a transverse load moving at a constant speed. It is assumed that the material properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory to account for the transverse shear deformation. The cracked beam is modeled as an assembly of two sub-beams connected through a linear rotational spring. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of dynamic equations through Galerkin’s procedure. The natural frequencies and dynamic response with different end supports are obtained. Numerical results are presented to investigate the influences of crack location, crack depth, material property gradient, slenderness ratio, foundation stiffness parameters, velocity of the moving load and boundary conditions on both free vibration and dynamic response of cracked functionally graded beams.     

An elastic–plastic thermal stress analysis of aluminum metal–matrix composite beams

In this study, an elastic–plastic thermal stress analysis is carried out on steel fiber-reinforced aluminum metal–matrix composite beams. Temperature is chosen to vary linearly. It is zero and T₀ at the upper and lower surfaces, respectively. The beam is fixed by two rigid planes at the ends. The solution is performed at 0°, 30°, 45°, 60° and 90° orientation angles. The plastic region is expanded at the lower side of the beam. It is found that the intensity of the residual stress component of σ_x and the equivalent plastic strain are maximum at lower surface of the beam. The residual stress is found to be greatest for the 0° orientation angle. In addition, the intensity of the equivalent plastic strain is the greatest for the same angle.     

定轴转动与基础激励下梁的非线性动力学

采用Kane方程，建立了含耦合的几何及惯性非线性项的定轴转动与轴向基础激励联合作用下柔性梁的非线性动力学控制方程组，该方程组不仅包含二次及三次非线性项，而且体现了参数激励与外激励的联合作用。运用多尺度法，研究了匀转速，顺臂安装下悬臂梁的一阶模态主参激共振与外激励1/2次亚谐共振同时作用时梁的一阶近似稳态响应。结果表明，梁的一阶模态幅频特性将受到转速，旋转半径和激励幅值等参数变化的显著影响。     

Vertical dynamic responses of a simply supported bridge subjected to a moving train with two‐wheelset vehicles using modal analysis method

The vertical dynamic responses of a simply supported bridge subjected to a moving train are investigated by means of the modal analysis method. Each vehicle of train is modelled as a four‐degree‐of‐freedom mass–spring–damper multi‐rigid body system with a car body and two wheelsets. The bridge, together with track, is modelled as a simply supported Bernoulli–Euler beam. The deflection of the beam is described by superimposing modes. The train and the beam are regarded as an entire dynamic system, in which the contact forces between wheelset and beam are considered as internal forces. The equations of vertical motion in matrix form with time‐dependent coefficients for this system are directly derived from the Hamilton's principle. The equations of motion are solved by Wilson‐θ method to obtain the dynamic responses for both the support beam and the moving train. Compared with the results previous reported, good agreement between the proposed method and the finite element method is obtained. Finally, the effects of beam mode number, vehicle number, beam top surface, and train velocity on the dynamic responses of the entire train and bridge coupling system are studied, and the dynamic responses of beam are given under the train moving with resonant velocity. Copyright © 2005 John Wiley & Sons, Ltd.     

功能梯度梁静力与动力分析的格林函数法

功能梯度梁静动态响应的数值分析方法一般局限于有限元法,存在有限元法的固有缺点,有必要发展新的数值求解方法。将功能梯度梁静力分析的控制微分方程转化为与匀质材料梁静力分析控制微分方程相一致的形式,并利用匀质材料梁静力问题的格林函数,开展功能梯度梁的静力分析。在此基础上,进一步推导获得功能梯度梁的柔度矩阵,据此建立功能梯度梁的运动方程,开展功能梯度梁的动力特性分析和动力响应分析。数值算例表明,采用格林函数法可以高效准确地分析功能梯度梁的静力响应与动力行为,验证了方法的计算精度与效率。     

Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load

In the present study, linear dynamic analysis of an axially functionally graded (AFG) beam with simply-supported edges due to a moving harmonic load has been analyzed by using Euler–Bernoulli beam theory. Elasticity modulus and mass density of the beam vary continuously in the axial direction of the beam according to a power–law form. The equation of motion is derived by using Lagrange’s equations. The unknown functions denoting the transverse deflections of the AFG beam is expressed in modal form, and Newmark method is employed to find the dynamic responses of AFG beam. In this study, the influences of material distribution, velocity of the moving load and excitation frequency on the dynamic response of the beam are investigated. In order to establish the accuracy of the present formulation and results, the first three free vibration frequencies are obtained, and compared with the published results available in the literature. Good agreement is observed. Results indicate that the above-mentioned effects play a very important role on the dynamic responses of the beam, and it is believed that new results are presented for non-linear dynamics of FG beams under moving loads which are of interest to the scientific and engineering community in the area of FGM structures.     

Dynamic buckling of elastic–plastic beams including effects of axial stress waves

Buckling of axially compressed elastic–plastic beams is discussed. The load is applied instantaneously and remains unaltered during the motion. The effect of stress waves travelling along the beam is taken into account. It is assumed that the material of the beam has linear-strain hardening. A method of solution, based on the Galerkin technique, is proposed; this method is applicable to an arbitrary number of degrees of freedom. Numerical examples are presented.     

Round-robin analysis of the RILEM TC 162-TDF beam-bending test: Part 2—Approximation of δ from the CMOD response

A round robin test programme was carried out using the beam-bending test recommended by RILEM TC 162-TDF [1]. The test programme included both plain and steel fibre reinforced concrete (SFRC) beams. A detailed analysis was carried out to investigate the influence of different test configurations on measurements of the crack mouth opening displacement (CMOD). Linear elastic fracture mechanics (LEFM) and non-linear fracture mechanics (NLFM) methods were utilised to investigate the problem analytically. From the analytical studies carried out, it is proposed that the CMOD should not be measured at a distance more than 5mm from the bottom fibre of the beam. A larger distance than 5mm will cause the deviation between the measured CMOD and the true CMOD to reach an unacceptable level. A simple rigid body model has been proposed to relate the CMOD to the mid-span deflection. The NLFM analysis and experimental results were compared for both the plain and SFRC beam results and it was found that results based on the basis of CMOD can be compared to those based on deflections, for practical purposes, using the simple rigid body model. The experimental results strongly suggest that the rigid body model could be effectively applied for all the types of materials tested in the round robin test programme. In addition it was found that the conversion from CMOD to the equivalent mid-span deflection, δ_c, revealed good agreement between the load-average mid-span deflection (P-δ) curve and the load-equivalent mid-span deflection (P-δ) curve especially for the SFRC specimens. It is proposed that the load-CMOD (P-CMOD) curve be used to calculate the proposed RILEM design parameters (as opposed to the P-δ curve) via the use of a correction factor determined using the simple rigid body model.     

火灾场冲击波荷载作用下简支钢梁动力响应

根据火灾场不同时刻温度的变化,考虑火灾高温对结构钢弹性模量、屈服强度和钢梁塑性弯矩的影响,分析了火灾场中冲击波荷载作用下简支钢梁各个阶段不同时刻的动力响应,通过具体算例计算了火灾场工字形简支钢梁在冲击波荷载作用下弹性阶段和塑性阶段的变形,分析表明在计算火灾场冲击波荷载作用下简支钢梁的动力响应时,火灾高温对结构钢的弹性模量、屈服强度以及梁的塑性弯矩和动力响应有显著的影响。     

Development of a Novel Beam Model and Its Dynamic Characteristics

Rotatory inertia due to the shear deformation is usually neglected for all engineering beam models. This article, however, introduces a novel beam model that further considers the rotatory inertia caused by the shear deformation, based on the Timoshenko beam model. The equation of motion for the proposed beam model is derived. Its dynamic characteristics, including wave and vibration characteristics, are studied in detail. Numerical simulations are conducted and related issues are discussed. It is found that the improvement is negligible when the wave number or vibration order is small for thin beams. However, the improvement is evident when the wave number or vibration intensity is fairly large for thick beams. This research finding reveals that only one frequency spectrum, one set of the wave phase velocities, and one set of group velocities should be considered in Timoshenko-type beams. This finding can help settle the debate in the literature on how many frequency spectra should be considered for these analyses. This research study shows that the proposed new beam model is more reasonable, accurate, and it is a more realistic model.     

运动刚体激励下弹性梁的振动响应分析

本文研究了与弹性梁有两点接触的运动刚体对梁激励时的响应，以悬臂梁为例进行了数值计算及实验研究．发现运动质量在梁上运动时，梁呈非周期振动；而且，最先产生响应的位置大约在梁的中部，而不是在质量作用点或梁端部．实验验证了计算结果．     

Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration

This paper initiates the theoretical analysis of nonlinear microbeams and investigates the static bending, postbuckling and free vibration. The nonlinear model is conducted within the context of non-classical continuum mechanics, by introducing a material length scale parameter. The nonlinear equation of motion, in which the nonlinear term is associated with the mean axial extension of the beam, is derived by using a combination of the modified couple stress theory and Hamilton’s principle. Based on this newly developed model, calculations have been performed for microbeams simply supported between two immobile supports. The static deflections of a bending beam subjected to transverse force, the critical buckling loads and buckled configurations of an axially loaded beam, and the nonlinear frequencies of a beam with initial lateral displacement are discussed. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio. Our results also indicate that the nonlinearity has a great effect on the static and dynamic behaviors of microscale beams. To attain accurate and reliable characterization of the static and dynamic properties of microscale beams, therefore, both the microstructure-dependent parameters and the nonlinearities have to be incorporated in the design of microscale beam devices and systems.