On finite element large displacement and elastic-plastic dynamic analysis of shell structures

Finite element procedures for nonlinear dynamic analysis of shell structures are presented and assessed. Geometric and material nonlinear conditions are considered. Some results are presented that demonstrate current applicabilities of finite element procedures to the nonlinear dynamic analysis of two-dimensional shell problems. The nonlinear response of a shallow cap, an impulsively loaded cylindrical shell and a complete spherical shell is predicted. In the analyses the effects of various finite element modeling characteristics are investigated. Finally, solutions of the static and dynamic large displacement elastic-plastic analysis of a complete spherical shell subjected to external pressure are reported. The effect of initial imperfections on the static and dynamic buckling behavior of this shell is presented and discussed.     

Nonlinear finite element for modeling reinforced concrete columns in three-dimensional dynamic analysis

A new finite element is proposed for slender, flexure-dominated reinforced concrete columns subjected to cyclic biaxial bending with axial load, and its implementation into a program for the nonlinear static or dynamic analysis of structures in three-dimensions, is described. The element belongs to the class of distributed inelasticity discrete models for the nonlinear dynamic response analysis of frame structures to earthquake ground motions. The element tangent flexibility matrix is constructed at each time step by Gauss-Lobatto integration of the section tangent flexibility matrix along the member length. The tangent flexibility matrix of the cross-section relates the increment of the vector of the three normal stress resultants N, M_y, M_z, to the vector increment of the section deformation measures. ε_o, _y, _z, and is constructed on the basis of the bounding surface of the cross-section, which is defined as the locus of points in the space of the normalized N, M_y, M_z, which correspond to ultimate strength. The bounding surface concept enables the model to produce realistic predictions for the nonlinear response of the cross-section to any arbitrary loading path in the space N-M_y-M_z.The bounding surface is introduced and utilized in a very flexible manner, enabling a variety of cross-sectional shapes to be treated in a unified way. As this flexibility is at the expense of computational simplicity and memory size requirements, emphasis is placed on algorithmic techniques to facilitate numerical implementation and to increase computational efficiency.     

Effect of axial force on framework dynamics

The effect of the member axial forces is included in the free and forced vibration of the frameworks. The vibration may be caused by externally applied dynamic forces or support motion. The masses on the framework may be distributed over the elements or lumped at the joints or both. The axial force of the members is considered as static force. The forced vibration of the frameworks are determined by means of modal analysis. Moreover the buckling of the frameworks is also investigated by means of this dynamic analysis. The magnitude of static loading which tends zero the value of the i'th natural circular frequency corresponds to i'th buckling load, and the modal shape for this frequency represents the i'th buckling shape. The dynamic part of the general computer program, STDYNL, is modified to include the effect of member axial force. A set of single beams, a three story and a sixteen story frame are considered as example problems to illustrate the effect of the member axial force on the vibration. The buckling modes of these beams and frames are also investigated.     

Large displacement extension of consistent shell element for static and dynamic analysis

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.     

Static, dynamic and stability analysis of structures composed of tapered beams

A new numerical method is proposed for the static, dynamic and stability analysis of linear elastic plane structures consisting of beams with constant width and variable depth. It is a finite element method based on an exact flexural and axial stiffness matrix and approximate consistent mass and geometric stiffness matrices for a linearly tapered beam element with constant width. Use of this method provides the exact solution of the static problem with just one element per member of a structure with linearly tapered beams and excellent approximate solutions of the dynamic and stability problems with very few elements per member of the structure in a computationally very efficient way. Very detailed comparison studies of the proposed method against a number of other known finite element methods with respect to accuracy and computational efficiency for cantilever tapered beams of rectangular and I cross section clearly favor the proposed method. A continuous beam, a gable frame and a portal frame consisting of tapered members are analyzed by the proposed method as well as by other known methods to illustrate the use of the method to structures composed of tapered beams.     

Nonlinear free vibration of fixed off-shore framed structures

The dynamical behavior of fixed off-shore framed structures is studied using the Wittrick-Williams algorithm to solve the nonlinear eigenvalue problem. The effects of shear deformation and rotary inertia as well as axial static loading are considered in this study of nonlinear free vibration.

The members are assumed to be rigidly connected and the added water mass is assumed equal to the mass of the water displaced. The structural modeling is based on a two-dimensional representation of the three-dimensional tower assuming a constant dimension equal to the base length perpendicular to the plane. The distributed masses of the members in the plane of the frame are computed by summing up the structural mass, the mass of the water contained in the tube, and the mass of the water displaced. The member masses in the plane perpendicular to the frame are assumed to be lumped at the horizontal cross-brace levels.

The results of the study indicate that while the first two frequencies obtained from the nonlinear and linear eigenvalue solutions agree closely, the effect of nonlinear eigenvalue solution is significant for the higher frequencies. The results also highlight the significant effects of the axial static force in the dynamic tangent stiffness matrix in the free vibration study of the off-shore structure. Fields for further research include (i) soil-structure interaction studies for gravity off-shore structures, buried pipelines, and (ii) nuclear power plant structures.     

A corotational procedure that handles large rotations of spatial beam structures

A practical motion process of the three dimensional beam element is presented to remove the restriction of small rotations between two successive increments for large displacement and large rotation analysis of space frames using incremental-iterative methods. In order to improve convergence properties of the equilibrium iteration, an n-cycle iteration scheme is introduced.

The nonlinear formulation is based on the corotational formulation. The transformation of the element coordinate system is assumed to be accomplished by a translation and two successive rigid body rotations: a transverse rotation followed by an axial rotation. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system. The membrane strain along the deformed beam axis obtained from the elongation of the arc length of the beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations. Two methods, referred to as direct method and incremental method, are proposed in this paper to calculate the total deformational rotations.

An incremental-iterative method based on the Newton-Raphson method combined with arc length control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the present method.     

A mixed GA-PSO-based approach for performance-based design optimization of 2D reinforced concrete special moment-resisting frames

A mixed genetic algorithm and particle swarm optimization in conjunction with nonlinear static and dynamic analyses as a smart and simple approach is introduced for performance-based design optimization of two-dimensional (2D) reinforced concrete special moment-resisting frames. The objective function of the problem is considered to be total cost of required steel and concrete in design of the frame. Dimensions and longitudinal reinforcement of the structural elements are considered to be design variables and serviceability, special moment-resisting and performance conditions of the frame are constraints of the problem. First, lower feasible bond of the design variables are obtained via analyzing the frame under service gravity loads. Then, the joint shear constraint has been considered to modify the obtained minimum design variables from the previous step. Based on these constraints, the initial population of the genetic algorithm (GA) is generated and by using the nonlinear static analysis, values of each population are calculated. Then, the particle swarm optimization (PSO) technique is employed to improve keeping percent of the badly fitted populations. This procedure is repeated until the optimum result that satisfies all constraints is obtained. Then, the nonlinear static analysis is replaced with the nonlinear dynamic analysis and optimization problem is solved again between obtained lower and upper bounds, which is considered to be optimum result of optimization solution with nonlinear static analysis. It has been found that by mixing the analyses and considering the hybrid GA-PSO method, the optimum result can be achieved with less computational efforts and lower usage of materials.     

Direct dynamic analysis of shells of revolution using high-precision finite elements

For the transient dynamic analysis of structural systems, the direct numerical integration of the equations of motion may be regarded as an alternative to the mode superposition method for linear problems and a necessity for nonlinear problems. When compared to a modal superposition solution, the direct integration approach is attractive in that the eigenvalue problem is avoided. Depending on the amount of information required from the dynamic analysis, e.g. frequencies, frequencies and mode shapes, and/or a complete time history, a direct integration scheme may prove to be more efficient than a modal superposition solution for some linear problems as well.The purpose of this study is to develop and demonstrate a direct integration algorithm which is compatible with an existing high-precision rotational shell finite element. Excellent comparative efficiency for static problems was achieved with this element by the incorporation of the exact geometry, the utilization of high-order interpolation polynomials and, yet, the retention of only a minimum number of nodal variables in the global formulation. Likewise, accurate and efficient results for the free vibration analysis of rotational shells were facilitated by the inclusion of a consistent mass matrix and the utilization of a rationally justified kinematic condensation procedure. The approach to the direct integration stage is strongly tempered by the established characteristics of this element which enable a given shell to be modeled accurately in the spatial domain with a comparatively coarse discretization.The equations of motion for a shell of revolution under conservative loading are derived from Hamilton's variational principle and specialized for the discretization of a rotational shell into curved shell elements. Degrees of freedom in excess of those required to establish minimum (C°) continuity at the nodal circles are eliminated through kinematic condensation. Some guidance as to the proper order of the polynominal approximations for a dynamic analysis is provided by earlier free vibration studies. Whereas the condensation is exact for static problems, it is only approximate for dynamic response and it was found that the accuracy of the eigenvalues obtained for the reduced problem decreases with increasing order of the condensed functions. This tendency is counted by the desirability of using sufficiently high-order interpolations so as to permit accurate stress computations, both at the nodal circles and between nodes since a coarse discretization is necessary to realize maximum efficiency. It was found that cubic polynomials were generally satisfactory from both standpoints, except in localized regions of high stress gradients where quintic polynomials were employed. The finite element discretizations for the direct integration studies were selected on this basis.For the high-precision finite element at hand, the efficiencies achieved in the space domain are demonstrated by the ability to achieve precise solutions with relatively coarse discretization patterns. The resulting comparatively large elements are not subject to accurate representation by diagonal mass matrices so that an implicit, consistent mass approach is followed. Efficiency in the time domain as well rests on the successful modeling of rotational shells subject to dynamic loading using coarse discretitations in space and large increments in time. Computational efficiency and accuracy are demonstrated for various problems documented in the literature, including a shallow spherical cap subject to a step pulse and a hyperboloidal shell under a simulated dynamic wind pressure.     

Some practical procedures for the solution of nonlinear finite element equations

Procedures for the solution of incremental finite element equations in practical nonlinear analysis are described and evaluated. The methods discussed are employed in static analysis and in dynamic analysis using implicit time integration. The solution procedures are implemented, and practical guidelines for their use are given.     

Finite element analysis of static and dynamic stresses around suddenly punched holes in plates and shells

The magnitude and distribution of stresses around suddenly punched holes in initially stressed plates and shells is of interest to insure that cracks will not precipitate from stress concentration. This problem is of practical interest to pressure vessel designers to preclude catastrophic failure when holes are punched in vessels to release gas. This paper presents a finite element analysis of several problems investigating static and dynamic stress fields around suddenly punched circular holes.

The first problem deals with the investigation of the radial and tangential stress fields in the vicinity of a suddenly punched hole in a stretched, elastic, isotropic plate subjected to an initial hydrostatic stress field. The wave propagation from a punched hole in the plate under a hydrostatic state of stress was solved analytically, using transform techniques, by Miklowitz; the finite element analysis of this problem presented in this paper confirms the analytical solution. Two grid meshes were investigated and results are presented to show the effect of grid mesh on solution accuracy and the power of finite element techniques for solving stress unloading problems. A formula for determining integration step size is found to be a function of the minimum element length and the wave propagation velocity. A similar investigation into the stress effects around a suddenly punched hole in the plate subjected to an initial uniaxial state of stress was also carried out as a prerequisite for the final problem studied.

The last problem is an anisotropic composite shell of varying thickness under an initial stress field due to internal pressure. The static and dynamic stress fields are computed from an unloading wave that radiates outward from a reinforced circular hole that is cut in the shell in 20 μs. A finite-element model of the shell is developed using quadrilateral and triangular plate elements and both in-plane and bending stiffness is included in the analysis as is nonlinear differential stiffening incorporated into the analysis as a single step approximation. Both bending and in-plane waves radiate outward from the cut hole and the dynamic stresses around the hole edge are computed for both unloading waves. The effects of the unloading waves are temporally spaced due to different wave velocities.

The paper demonstrates that fast response stress problems are readily amenable to finite-element analysis. For holes other than circular, the power of finite-element methods is apparent since these shapes lead to mathematically intractable problems if closed form solutions are attempted.     

悬索桥主缆初应力对桥梁动力特性的影响

为掌握主缆初应力对桥梁动力特性的影响,以悬索桥为例,采用静力非线性分析方法计算应力变化过程中悬索桥的跨中挠度、缆索轴力及加劲梁的轴力变化;将得到的内力作为结构的预加应力进行有预应力的模态分析. 应用ANSYS软件进行分析,其中有限元建模时采用4种单元类型:空间梁单元BEAM4用于模拟加劲梁和塔;空间杆单元LINK10用于模拟主缆及吊杆;通过设定BEAM4和LINK10单元初应变进行有预应力的模态分析;采用MASS21单元模拟横隔板、吊杆锚固装置和桥面上的栏杆,并分别考虑质量和质量惯性矩的作用. 分析表明,悬索桥主梁竖弯振动频率受主缆初应力的影响较大,而侧弯振动频率和扭转振动频率受此影响较小. 该结果为同类桥梁的动力特性分析提供参考.     

Nonlinear dynamic response structural optimization using equivalent static loads

It is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non-gradient based optimization methods because many nonlinear dynamic analyses are required. Therefore, it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization. The equivalent static loads (ESLs) method is newly proposed and applied to nonlinear dynamic response optimization. The equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis. The ESLs are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization. Then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization. The updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied. Several examples are solved to validate the method. The results are compared to those of the conventional method with sensitivity analysis using the finite difference method.     

Chain Vibration and Dynamic Stress in Three-Dimensional Multibody Tracked Vehicles

A three-dimensional computational finite element procedure for the vibration and dynamic stress analysis of the track link chains of off-road vehicles is presented in this paper. The numerical procedure developed in this investigation integrates classical constrained multibody dynamics methods with finite element capabilities. The nonlinear equations of motion of the three-dimensional tracked vehicle model in which the track link s are considered flexible bodies, are obtained using the floating frame of reference formulation. Three-dimensional contact force models are used to describe the interaction of the track chain links with the vehicle components and the ground. The dynamic equations of motion are first presented in terms of a coupled set of reference and elastic coordinates of the track links. Assuming that the structural flexibility of the track links does not have a significant effect on their overall rigid body motion as well as the vehicle dynamics, a partially linearized set of differential equations of motion of the track links is obtained. The equations associated with the rigid body motion are used to predict the generalized contact, inertia, and constraint forces associated with the deformation degrees of freedom of the track links. These forces are introduced to the track link flexibility equations which are used to calculate the deformations of the links resulting from the vehicle motion. A detailed three-dimensional finite element model of the track link is developed and utilized to predict the natural frequencies and mode shapes. The terms that represent the rigid body inertia, centrifugal and Coriolis forces in the equations of motion associated with the elastic coordinates of the track link are described in detail. A computational procedure for determining the generalized constraint forces associated with the elastic coordinates of the deformable chain links is presented. The finite element model is then used to determine the deformations of the track links resulting from the contact, inertia, and constraint forces. The results of the dynamic stress analysis of the track links are presented and the differences between these results and the results obtained by using the static stress analysis are demonstrated.     

Technical overview of the equivalent static loads method for non-linear static response structural optimization

Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.     

Large deformation static and dynamic finite element analysis of extensible cables

Approximate numerical integration of the element total potential energy with polynomial interpolation of the displacements creates high order nonlinear, extensible, cable finite elements. Successful computations of static and dynamic large displacement cable problems are carried out with the element.     

Geometric nonlinear effects on the planar dynamics of a pivoted flexible beam encountering a point-surface impact

Flexible-body modeling with geometric nonlinearities remains a hot topic of research by applications in multibody system dynamics undergoing large overall motions. However, the geometric nonlinear effects on the impact dynamics of flexible multibody systems have attracted significantly less attention. In this paper, a point-surface impact problem between a rigid ball and a pivoted flexible beam is investigated. The Hertzian contact law is used to describe the impact process, and the dynamic equations are formulated in the floating frame of reference using the assumed mode method. The two important geometric nonlinear effects of the flexible beam are taken into account, i.e., the longitudinal foreshortening effect due to the transverse deformation, and the stress stiffness effect due to the axial force. The simulation results show that good consistency can be obtained with the nonlinear finite element program ABAQUS/Explicit if proper geometric nonlinearities are included in the floating frame formulation. Specifically, only the foreshortening effect should be considered in a pure transverse impact for efficiency, while the stress stiffness effect should be further considered in an oblique case with much more computational effort. It also implies that the geometric nonlinear effects should be considered properly in the impact dynamic analysis of more general flexible multibody systems.     

Comparison of numerical integration methods in the analysis of impulsively loaded elasto-plastic and viscoplastic structures

Direct integration methods for ordinary differential equations encountered in the finite element analysis of dynamic problems of structural mechanics are studied and compared. Examples considered are a single-degree-of-freedom oscillator and a transversal impact on a beam. The material behavior is assumed to be elastic, elasto-plastic or viscoplastic. Both geometrically linear and nonlinear cases are treated. The time-integrators used include both explicit and implicit, single- and multistep methods, e.g. methods of Runge-Kutta, Euler, Newmark, Houbolt, Wilson, the central difference method and stiffly stable methods by Gear. Methods are compared in terms of computation time and accuracy and ease of formulation.     

A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods

A comprehensive survey of direct time-integration methods and computational solution procedures for easier computer implementation is given in four parts for dynamic analysis of linear and nonlinear structures.

Part I is exclusively devoted to explicit methods. Popular second order central difference methods (formulation, step-by-step solution procedures, recent developments, computational and stability aspects) are described in detail. Other explicit methods, viz. Runge-Kutta methods, stiffly stable methods, Predictor-Corrector methods and Taylor series schemes are also presented. Techniques for stabilizing numerical computations are given.

In Part II, conventional implicit methods, viz. the Newmark, Wilson-θ and Houbolt methods and their step-by-step solution procedures are given with reference to solution of linear and nonlinear structural dynamics problems. Also presented are Trujillo's modified Newmark-beta method and implicit formulae via weighted residual approach. Computational and stability aspects, desirable characteristics of an ideal solution procedure and salient features of conventional implicit algorithms are discussed.

Part III reviews further developments in implicit methods. In Part IV, mixed implicit-explicit finite element methods and operator-splitting methods are described.

Numerical solution methods surveyed here will be of much use to practicing computational/finite element/structural engineers working in the area of dynamics of structures.