Suggestion of a new dimensionless number for dynamic plastic response of beams and plates

Summary A dimensionless number, termed response number in the present paper, is suggested for the dynamic plastic response of beams and plates made of rigid-perfectly plastic materials subjected to dynamic loading. It is obtained at dimensional reduction of the basic governing equations of beams and plates. The number is defined as the product of the Johnson's damage number and the square of the half of the slenderness ratio for a beam; the product of the damage number and the square of the half of the aspect ratio for a plate or membrane loaded dynamically. Response number can also be considered as the ratio of the inertia force at the impulsive loading to the plastic limit load of the structure. Three aspects are reflected in this dimensionless number: the inertia of the applied dynamic loading, the resistance ability of the material to the deformation caused by the loading and the geometrical influence of the structure on the dynamic response. For an impulsively loaded beam or plate, the final dimensionless deflection is solely dependent upon the response number. When the secondary effects of finite deflections, strain-rate sensitivity or transverse shear are taken into account, the response number is as useful as in the case of simple bending theory. Finally, the number is not only suitable to idealized dynamic loads but also applicable to dynamic loads of general shape. Received 17 October 1997; accepted for publication 19 March 1998     

Analysis of the large deflection dynamic plastic response of simply-supported circular plates by the “membrane factor method

Based on energy equilibrium, a new procedure called the Membrane Factor Method is developed to analyze the dynamic plastic response of plates with deflections in the range where both bending moments and membrane forces are important. The final deflection of a simply-supported circular rigid-plastic plate loaded by a uniformly distributed impulse is obtained. In comparison with other approximate solutions, the present results are found to be simpler and in better agreement with the corresponding experimental values reoorded by Florence. The project supported by a fund from the National Educational Committee.     

Large deflection dynamic plastic response of clamped square plates with stiffeners

The large deflection dynamic plastic response of fully clamped square plates with stiffeners under blast load is analyzed in detail in this paper. Various relevant motion patterns and criterions are presented. The formulas of maximum permanent deformation of the plates with stiffeners are derived. The results of. calculation are compared with those of experiment in [3], with good agreement shown in most cases.     

冲击荷载作用下简支圆板的塑性动力响应统一解

采用统一强度理论求解了简支圆板在中等脉冲荷载作用下的动力响应问题,得出了统一的动力塑性极限荷载、内力场和速度场,并给出了上限解和下限解。讨论了静力许可条件和运动许可条件。利用本文的解还得出了简支圆板在静力荷载作用下的极限荷载、内力场和速度场。根据选择不同的拉压比参数,本文所给出的解可以适用于各种拉压异性和拉压同性材料。Tresca解、Mohr Coulomb解和双剪统一屈服准则解是本文的特例,Mises解是本文当=1和b=0.5时的线性逼近。研究结果表明,拉压比和强度理论参数b对动力解的影响要大于对静力解的影响,所以,根据材料的不同选择合适的强度理论,对于更好的发挥材料的强度潜力,减轻结构的重量具有重要的意义。     

The dynamic plastic behavior of a simply supported circular plate in a damping medium with finite-deflections

A theoretical analysis is presented for the dynamic plastic behavior of a simplysupported rigid,perfectly plastic circular plate in damping medium with finite-deflectionssubjected to a rectangular pressure pulse.Analytical solutions of every moving stage underboth medium and high loads are developed.     

用矩量法解阻尼介质中简支方板的塑性动力响应问题

本文用矩量法解薄板塑性动力响应问题，分析阻尼介质对简支方板塑性动力响应的影响，对计算结果进行了讨论．     

Nonlinear dynamic response of beam and its application in nanomechanical resonator

Nonlinear dynamic response of nanomechanical resonator is of very important characteristics in its application. Two categories of the tension-dominant and curvature-dominant nonlinearities are analyzed. The dynamic nonlinearity of four beam structures of nanomechanical resonator is quantitatively studied via a dimensional analysis approach. The dimensional analysis shows that for the nanomechanical resonator of tension-dominant nonlinearity, its dynamic nonlinearity decreases monotonically with increasing axial loading and increases monotonically with the increasing aspect ratio of length to thickness; the dynamic nonlinearity can only result in the hardening effects. However, for the nanomechanical resonator of the curvature-dominant nonlinearity, its dynamic nonlinearity is only dependent on axial loading. Compared with the tension-dominant nonlinearity, the curvature-dominant nonlinearity increases monotonically with increasing axial loading; its dynamic nonlinearity can result in both hardening and softening effects. The analysis on the dynamic nonlinearity can be very helpful to the tuning application of the nanomechanical resonator.     

Finite element analysis of the dynamic response of clamped sandwich beams subject to shock loading

The finite element (FE) method is employed to analyse the response of clamped sandwich beams subject to shock loadings. Pressure versus time histories representative of shock loadings are applied uniformly to the outer face of the sandwich beam; an impulse applied uniformly to the outer face of the sandwich beam is shown to model adequately shock loadings. Material elasticity and strain hardening representative of structural steels have only a minor effect upon the beam response. Further, the magnitude of the compressive strength of the core has only a limited influence upon the dynamic response of the sandwich beam for the representative range of core strengths considered. The FE results for the deflections and structural response time agree well with the rigid ideally-plastic analytical predictions of Fleck and Deshpande (J. Appl. Mech. (2003), in press).     

Characteristic-value analysis for plastic dynamic buckling of columns under elastoplastic compression waves

The characteristic-value analysis of plastic dynamic buckling is presented for columns under the action of elastoplastic compression wave caused by an axial-step load. Two critical conditions constituting a dynamic instability criterion are derived on the basis of transformation and conservation of energy. The governing equations, the boundary conditions and the continuity conditions derived by the use of the first critical condition are the same as those given by the adjacent-equilibrium criterion and are insufficient for determining two characteristic parameters involved in the governing equations. A supplementary restraint equation for buckling deformations at the plastic-wave front and the elastic-wave front is derived by the use of the second critical condition. Then, a couple of characteristic equations for two characteristic parameters are derived on the condition that the governing equations have non-trivial solutions satisfying the boundary conditions, the continuity conditions and the supplementary restraint equation. The critical-load parameters, dynamic characteristic parameter (exponent parameter of inertia term) and dynamic buckling modes are calculated from the solutions of the characteristic equations.     

Analysis of interlaminar stress and nonlinear dynamic response for composite laminated plates with interfacial damage

By considering the effect of interfacial damage and using the variation principle, three-dimensional nonlinear dynamic governing equations of the laminated plates with interfacial damage are derived based on the general sixdegrees-of-freedom plate theory towards the accurate stress analysis. The solutions of interlaminar stress and nonlinear dynamic response for a simply supported laminated plate with interfacial damage are obtained by using the finite difference method, and the results are validated by comparison with the solution of nonlinear finite element method. In numerical calculations, the effects of interfacial damage on the stress in the interface and the nonlinear dynamic response of laminated plates are discussed.     

Optimal control of the dynamic response of an anisotropic plate with various boundary conditions

The problem of minimising the dynamic response of an anisotropic rectangular plate with minimum possible expenditure of force is presented for various cases of boundary conditions. The plate has a principal direction of anisotropy rotated at an arbitrary angle relative to the coordinate axes. This orientation angle has been taken as an optimisation design parameter. The control problem is formulated as an optimisation problem by using a performance index, which comprises a weight sum of the control objective and penalty function of the control force. The explicit solutions for the closed-loop distributed control function is obtained by means of Liapunov-Bellman theory. To assess the present solution, numerical results are presented to illustrate the effect of anisotropy ratio, orientation angle, aspect ratio and boundary conditions on the control process.     

Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates

The governing equation of motion of gradient elastic flexural Kirchhoff plates, including the effect of in-plane constant forces on bending, is explicitly derived. This is accomplished by appropriately combining the equations of flexural motion in terms of moments, shear and in-plane forces, the moment–stress relations and the stress–strain equations of a simple strain gradient elastic theory with just one constant (the internal length squared), in addition to the two classical elastic moduli. The resulting partial differential equation in terms of the lateral deflection of the plate is of the sixth order instead of the fourth, which is the case for the classical elastic case. Three boundary value problems dealing with static, stability and dynamic analysis of a rectangular simply supported all-around gradient elastic flexural plate are solved analytically. Non-classical boundary conditions, in additional to the classical ones, have to be utilized. An assessment of the effect of the gradient coefficient on the static or dynamic response of the plate, its buckling load and natural frequencies is also made by comparing the gradient type of solutions against the classical ones.     

球壳塑性变形下的应变增长现象

应变增长现象会对容器安全形成威胁。以往研究涉及的应变增长现象大多在壳体弹性变形范围内,本文中实验观察到球壳塑性变形时的应变增长现象,应变增长系数(最大应变值与第一个应变峰的比值)最大值达到1.16。实验还获得了容器内壁压力-时间曲线,并利用球壳响应理论分析出应变增长现象是由容器内壁的周期性多脉冲载荷引起的,该载荷存在3个较明显的脉冲,前两个脉冲对应变增长现象起主要作用。     

结构动力响应精细时程法的一种并行算法

结构动力响应精细时程法的并行算法分为两类：基于特解的并行算法和基于直接积分法的并行算法;后者因为不需知道荷载的具体形式而更具应用价值。精细时程法的时程积分由齐次方程的通解和非齐次项的积分构成,基于直接积分法的并行算法很好地并行了非齐次项的积分,而对通解项采用串行计算。设计了一种不均衡步数的负载分配策略,能够减少处理器等待自身初值的时间,相对均衡步数的分配策略,能够获得更高的加速比,给出了相应的证明和算例验证。     

The impulsive response of sandwich beams: Analytical and numerical investigation of regimes of behaviour

An analytical model is developed to classify the impulsive response of sandwich beams based on the relative time-scales of core compression and the bending/stretching response of the sandwich beam. It is shown that an overlap in time scales leads to a coupled response and to the possibility of an enhanced shock resistance. Four regimes of behaviour are defined: decoupled responses with the sandwich core densifying partially or completely, and coupled responses with partial or full core densification. These regimes are marked on maps with axes chosen from the sandwich beam transverse core strength, the sandwich beam aspect ratio and the level of blast impulse. In addition to predicting the time-scales involved in the response of the sandwich beam, the analytical model is used to estimate the back face deflection, the degree of core compression and the magnitude of the support reactions. The predictions of the analytical model are compared with finite element (FE) simulations of impulsively loaded sandwich beams comprising an anisotropic foam core and elastic, ideally plastic face-sheets. The analytical and numerical predictions are in good agreement up to the end of core compression. However, the analytical model under-predicts the peak back face deflection and over-predicts the support reactions, especially for sandwich beams with high strength cores. The FE calculations are employed to construct design charts to select the optimum transverse core strength that either minimises the back face deflections or support reactions for a given sandwich beam aspect ratio or blast impulse. Typically, the value of the transverse core strength that minimises the back face deflection also minimises the support reactions. However, the optimal core strength depends on the level of blast impulse, with higher strength cores required for greater blasts.     

Wave scattering method for dynamic response analysis of structures with stochastic parameters

The wave scattering method is presented to analyze dynamic response of frameworks with stochastic parameters. First, with the uncertain physical, geometric, and loading properties in consideration, the stochastic waveguide equations containing the axial, torsional and flexural wave modes are established. Second, the stochastic wave scattering equation and wave translation matrix are derived to obtain the wave modes. Third, the methodology to extract the generalized displacements and forces from stochastic wave modes is proposed. Finally, a cantilever beam, a planar framework, and a space framework have been presented as numerical examples to illustrate the e?ciency of the proposed method. It is found that the results obtained by the proposed method with higher computational e?ciency show an excellent agreement with those by Monte Carlo simulation method. Furthermore, the influences of stochastic parameters on dynamic response are revealed.     

Vertex solution theorem for the upper and lower bounds on the dynamic response of structures with uncertain-but-bounded parameters

The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the dynamic response of structures. By combining the interval mathematics and the finite element analysis, the mass matrix, damping matrix, stiffness matrix and the external loads are represented as interval matrices and vector. With the help of the optimization theory, we present the vertex solution theorem for determining both the exact upper bounds or maximum values and the exact lower bounds or minimum values of the dynamic response of structures, in which these parameters reach their extreme values on the boundary of the interval mass, damping, stiffness matrices and the interval extemal loads vector. Three examples are used to illustrate the computational aspects of the presented vertex solution theorem.     

一种Birkhoff形式下结构动响应问题的保辛中点格式

结构动响应预测是结构设计的基础,是结构振动控制、载荷识别的前提。本文在辛体系下针对结构动响应问题,提出了一种Birkhoff形式下的保辛中点格式。首先引入状态变量,并基于摄动方法将结构动响应方程转化为线性自治Birkhoff方程的形式,进一步利用中心差分推导出线性自治Birkhoff方程的中点格式,其证明是保辛的。该格式不要求Birkhoff方程系数矩阵非奇异,因此适用于奇数维系统。两个不同数值算例的结果充分验证了本文方法的卓越性,也凸显了相对于传统算法在计算精确度和稳定性方面的明显优势。     

On the convergence and causality of a frequency domain method for dynamic structural analysis

Venanico-Filho et al. developed an elegant matrix formulation for dynamic analysis by frequency domain (FD), but the convergence, causality and extended period need further refining. In the present paper, it was argued that: (1) under reasonable assumptions (approximating the frequency response function by the discrete Fourier transform of the discretized unitary impulse response function), the matrix formulation by FD is equivalent to a circular convolution; (2) to avoid the wraparound interference, the excitation vector and impulse response must be padded with enough zeros; (3) provided that the zero padding requirement satisfied, the convergence and accuracy of direct time domain analysis, which is equivalent to that by FD, are guaranteed by the numerical integration scheme; (4) the imaginary part of the computational response approaching zero is due to the continuity of the impulse response functions. The English text was polished by Yunming Chen     

The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.