样条配点法计算动响应

本文以3次B样条函数作为时域的试函数,用双配点法建立起一个分析结构动响应的单步计算格式.通过对算法精度和稳定性的详尽分析,找到了两个值得推荐的方案:条件稳定的四阶算法(C1)及无条件稳定的二阶算法(C2).这两个方案的算法特性与一些熟知算法作了比较.     

计算动响应的若干无条件稳定的二阶算法的比较

本文分析比较若干二阶的直接积分算法,主要是Zienkiewicz,Wood等的SS32算法(1984)和邵慧萍,蔡承文的三参数算法(1987)。文中指出,当参数符合一定对应关系时两族算法具有几乎相同的精度。但在参数空间中的无条件稳定区,后者的大于前者,而且三参数法避免了SS32所固有的严重超越现象。     

Thermal post buckling behavior of rectangular antisymmetric cross-ply composite plates

Summary The thermal post buckling behavior of simply-supported, antisymmetric cross-ply plates with immovable edges is investigated in this paper. For this purpose, a one term approximation for the inplane and transverse displacements is assumed and Rayleigh-Ritz method is used to obtain the equations of equilibrium. The question, whether the true thermal buckling (bifurcation buckling under thermal loads) exists for such plates is also addressed to. The conditions under which the thermal buckling in the classical sense occurs are derived. It is shown that nonlinear bending stiffness for such plates is direction dependent, because of an extra quadratic nonlinear term in addition to the usual cubic nonlinear term in the governing equilibrium equation. Results are presented for various plate configurations.     

State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

A fourth‐order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase‐space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth‐order. An optional high‐frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth‐order in the frequency. Thus, the low‐frequency response is virtually undamped, and the algorithm remains third‐order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front. Copyright © 2005 John Wiley & Sons, Ltd.     

Accuracy of a composite implicit time integration scheme for structural dynamics

A comprehensive study of the two sub‐steps composite implicit time integration scheme for the structural dynamics is presented in this paper. A framework is proposed for the convergence accuracy analysis of the generalized composite scheme. The local truncation errors of the acceleration, velocity, and displacement are evaluated in a rigorous procedure. The presented and proved accuracy condition enables the displacement, velocity, and acceleration achieving second‐order accuracy simultaneously, which avoids the drawback that the acceleration accuracy may not reach second order. The different influences of numerical frequencies and time step on the accuracy of displacement, velocity, and acceleration are clarified. The numerical dissipation and dispersion and the initial magnitude errors are investigated physically, which measure the errors from the algorithmic amplification matrix's eigenvalues and eigenvectors, respectively. The load and physically undamped/damped cases are naturally accounted. An optimal algorithm‐Bathe composite method (Bathe and Baig, 2005; Bathe, 2007; Bathe and Noh, 2012) is revealed with unconditional stability, no overshooting in displacement, velocity, and acceleration, and excellent performance compared with many other algorithms. The proposed framework also can be used for accuracy analysis and design of other multi‐sub‐steps composite schemes and single‐step methods under physical damping and/or loading. Copyright © 2016 John Wiley & Sons, Ltd.     

Extended state‐space time integration with high‐frequency energy dissipation

The introduction of algorithmic dissipation into conservative time integration algorithms is addressed in two steps. The problem is formulated in state‐space, and it is demonstrated that the use of two balanced dissipation terms—one in the equation of motion and the other in the relation between the state‐space variables—can produce dissipation that increases monotonically with the modal frequency. This formulation is only partially satisfactory, because it leads to dissipation also in the low‐frequency range. A substantially improved form of algorithmic dissipation is then developed by representing the two balanced dissipation terms via auxiliary state‐space variables related to the displacement and velocity increments by a first‐order filter. The filter technique retains the high‐frequency behavior of the original balanced algorithm, but reduces the low‐frequency dissipation from first order in the frequency to third order. Energy balance equations are derived, and the behavior of the two algorithms is described by detailed spectral analysis with explicit asymptotic results directly in terms of the damping parameter of the algorithms. Both algorithms satisfy full symmetry of displacement and normalized velocity variables for the free algorithmically damped vibration of any single mode and, therefore, exhibit monotonic transients without ‘overshoot’ effects. It is also demonstrated how the algorithm can be arranged in ‘single solve’ format, where only the displacement update requires a matrix equation solution, while the velocity and the auxiliary state‐space variables are updated subsequently in each time step by simple vector relations with scalar coefficients. Copyright © 2007 John Wiley & Sons, Ltd.     

A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics

A comprehensive study of A-stable linear two-step time integration methods for structural dynamics analysis is presented in this paper. An optimal A-stable linear two-step (OALTS) time integration method is revealed with desirable performance on low-frequency accuracy and high-frequency numerical dissipation properties. The OALTS time integration method is implemented in a direct integration manner for the second-order equations of structural dynamics; is implicit, A-stable, and second-order accurate in displacement, velocity, and acceleration, simultaneously; is easily started; and is numerical dissipation controllable. The OALTS time integration method shows desirable performance on spectral radius distribution, dissipation and dispersion errors, and overshooting behavior, where the results of some typical algorithms in the literature are also compared. Benchmark examples with/without physical damping are performed to validate the accuracy, stability, and efficiency of the OALTS time integration method.     

Dynamic buckling of an elastoplastic column

Dynamic buckling of columns under axial step loading which produces plastic behaviour is investigated. An experimental study is described and an elastoplastic model is developed in order to analyse the buckling process. Good agreement is obtained between the theoretical predictions and experimental results which justifies the main assumptions underlying the model.     

Stability and post buckling of articulated columns

Summary. This paper considers articulated columns which are composed of connected rigid segments with rotational springs at the joints. The problem is formulated in terms of nonlinear difference equations. The buckling forces and post-buckling properties are found for five different combinations of end conditions. The fixed-hinged column shows limit load and snap through characteristics.     

精细直接积分法的积分方法选择

讨论了精细直接积分法中积分方法选择问题。通过理论推导和数值试验,指出为保持精细算法的高精度,应根据荷载的性质选择合适的积分方法,并得出激励为多项式形式时应选择代数精度高的积分方法的结论,指出科茨积分、高斯积分是保持精细算法高精度的较好积分方法。     

Dynamic bifurcation buckling of an impacted column

In this paper, the dynamic counterpart of the celebrated Euler buckling problem is formulated and solved by considering the case of a slender column that is impacted by a falling mass. We introduce a new notion, that of the time to buckle, “t^∗”, which is the corresponding critical quantity analogous to the critical load in static Euler buckling. A set of experimental results previously presented by Gladden et al. [J.R. Gladden, N.Z. Handzy, A. Belmonte, E. Villermaux, Dynamic buckling and fragmentation in brittle rods, Physical Review Letters 94 (3) (2005) 035503], are used to illustrate the notions we introduce and as a means for comparison against the predictions of our work.     

Properties of time integration with first order filter damping

Introduction of algorithmic damping by increasing the parameter values in the Newmark algorithm leads to undesirable low‐frequency damping and reduced order of accuracy. It is demonstrated, how these effects can be removed by introducing an extra damping term in the form of a first order linear filter. When the linear filter is discretized in time, the state variable associated with the filter can be eliminated, leading to a weighted average of the equations of motion at two consecutive times. The filter procedure contains the known versions of alpha weighted Newmark methods as special cases, but gives a different and improved weighting of the excitation terms. A complete analysis of the properties of the algorithm when used on linear systems is given, including the frequency response properties. It is demonstrated that the effect of ‘overshoot’ is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system, and analytic results are presented for the magnitude of the effect. Copyright © 2005 John Wiley & Sons, Ltd.     

A theory of development and design of generalized integration operators for computational structural dynamics

A standardized formal theory of development/evolution, characterization and design of a wide variety of computational algorithms emanating from a generalized time weighted residual philosophy for dynamic analysis is first presented with subsequent emphasis on detailed formulations of a particular class relevant to the so‐called time integration approaches which belong to a much broader classification relevant to time discretized operators. Of fundamental importance in the present exposition is the evolution of the theoretical design and the subsequent characterization encompassing a wide variety of time discretized operators, and the proposed developments are new and significantly different from the way traditional modal type and a wide variety of step‐by‐step time integration approaches with which we are mostly familiar have been developed and described in the research literature and in standard text books over the years. The theoretical ideas and basis towards the evolution of a generalized methodology and formulations emanate under the umbrella and framework and are explained via a generalized time weighted philosophy encompassing single‐field and two‐field forms of representations of the semi‐discretized dynamic equations of motion. Therein, the developments first leading to integral operators in time, and the resulting consequences then systematically leading to and explaining a wide variety of generalized time integration operators of which the family of single‐step time integration operators and various widely recognized and new algorithms are subsets, the associated multi‐step time integration operators and a class of finite element in time integration operators, and their relationships are particularly addressed. The generalized formulations not only encompass and explain a wide variety of time discretized operators and the recovery of various original methods of algorithmic development, but furthermore, naturally inherit features for providing new avenues which have not been explored an/or exploited to‐date and permit time discretized operators to be uniquely characterized by algorithmic markers. The resulting and so‐called discrete numerically assigned [DNA] markers not only serve as a prelude towards providing a standardized formal theory of development of time discretized operators and forum for selecting and identifying time discretized operators, but also permit lucid communication when referring to various time discretized operators. That which constitutes characterization of time discretized operators are the so‐called DNA algorithmic markers which essentially comprise of both (i) the weighted time fields introduced for enacting the time discretization process, and (ii) the corresponding conditions these weighted time fields impose (dictate) upon the approximations (if any) for the dependent field variables in the theoretical development of time integrators and the associated updates of the time discretized operators. Furthermore, a single analysis code which permits a variety of choices to the analyst is now feasible for performing structural dynamics computations on modern computing platforms. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis and implementation of a new constant acceleration subcycling algorithm

An algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps. With integer time step ratios, the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle. The algorithm is shown to have good accuracy, and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms. The implementation of a generalised form of the algorithm with non-integer time step ratios is presented. © 1997 John Wiley & Sons, Ltd.     

有阻尼动力方程显式积分方法的精度研究

在动力问题分析中,一种好的显式积分方法不仅要具有良好的稳定性,而且还要具有良好的计算精度。对几种有阻尼动力方程显式积分方法的精度问题进行了分析,研究了各自的局部截断误差、数值阻尼比和相对周期误差。这一研究将有助于更加全面地了解这些积分格式的性能。     

Subcycling first- and second-order generalizations of the trapezoidal rule

Subcycling algorithms which employ multiple timesteps have been previously proposed for explicit direct integration of first- and second-order systems of equations arising in finite element analysis, as well as for integration using explicit/implicit partitions of a model. The author has recently extended this work to implicit/implicit multi-timestep partitions of both first- and second-order systems. In this paper, improved algorithms for multi-timestep implicit integration are introduced, that overcome some weaknesses of those proposed previously. In particular, in the second-order case, improved stability is obtained. Some of the energy conservation properties of the Newmark family of algorithms are shown to be preserved in the new multi-timestep extensions of the Newmark method. In the first-order case, the generalized trapezoidal rule is extended to multiple timesteps, in a simple way that permits an implicit/implicit partition. Explicit special cases of the present algorithms exist. These are compared to algorithms proposed previously. © 1998 John Wiley & Sons, Ltd.     

Energy‐conserving and decaying Algorithms in non‐linear structural dynamics

A generalized formulation of the Energy‐Momentum Methodwill be developed within the framework of the Generalized‐α Methodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response. Furthermore, the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the Constraint Energy‐Momentum Algorithm. The goal of this general approach of implicit energy‐conserving and decaying time integration schemes is, to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties. Numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non‐linear structural dynamics. Among the methods considered will be the Newmark Method, the classical α‐methods, the Energy‐Momentum Methodwith and without numerical dissipation, the Constraint Energy‐Momentum Algorithm and the Constraint Energy Method. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of the elastic buckling of a plywood column

Buckling tests were conducted on specimens of 5-ply lauan plywood for a range of slenderness ratios to measure its buckling stress. Three-dimensional finite element calculations of buckling stress were performed and their validity examined by comparison with experimental results. Both experimental and calculated results revealed that buckling stress is influenced by Young’s modulus values (a measure of stiffness) obtained not only under flexural loading but also under axial loading. When the axial Young’s modulus is larger than the flexural Young’s modulus, the buckling stress is measured as larger than that obtained using the flexural Young’s modulus alone. Inversely, when the axial Young’s modulus is smaller than the flexural Young’s modulus, the buckling stress is measured as smaller than that obtained using the flexural Young’s modulus alone. Therefore, both the Young’s modulus values should be taken into account for determining the buckling stress of a plywood column.     

A methodology for the formulation of error estimators for time integration in linear solid and structural dynamics

In this article, we present a novel methodology for the formulation of a posteriori error estimators applicable to time‐stepping algorithms of the type commonly employed in solid and structural mechanics. The estimators constructed with the presented methodology are accurate and can be implemented very efficiently. More importantly, they provide reliable error estimations even in non‐smooth problems where many standard estimators fail to capture the order of magnitude of the error. The proposed methodology is applied, as an illustrative example, to construct an error estimator for the Newmark method. Numerical examples of its performance and comparison with existing error estimators are presented. These examples verify the good accuracy and robustness predicted by the analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

Uncertainty modelling of critical column buckling for reinforced concrete buildings

Buckling is a critical issue for structural stability in structural design. In most of the buckling analyses, applied loads, structural and material properties are considered certain. However, in reality, these parameters are uncertain. Therefore, a prognostic solution is necessary and uncertainties have to be considered. Fuzzy logic algorithms can be a solution to generate more dependable results. This study investigates the material uncertainties on column design and proposes an uncertainty model for critical column buckling reinforced concrete buildings. Fuzzy logic algorithm was employed in the study. Lower and upper bounds of elastic modulus representing material properties were defined to take uncertainties into account. The results show that uncertainties play an important role in stability analyses and should be considered in the design. The proposed approach is applicable to both future numerical and experimental researches. According to the study results, it is seen that, calculated buckling load values are stayed in lower and upper bounds while the load values are different for same concrete strength values by using different code formula.