Liang-Jenq Leu

Force recovery procedures in nonlinear analysis

Abstract Conventionally, incremental-iterative schemes have been used in solving nonlinear problems. There are two phases involved in such analyses. The first or ‘predictor’ phase relates to solution of the structural displacement increments from the total incremental equations of equilibrium, while the second or ‘corrector’ phase is concerned with recovery of the element forces from the element displacement increments obtained in the first phase. In this paper, it will be demonstrated that accuracy of the numerical solutions depends primarily on the corrector or procedure for recovering the element forces. The expression used in the predictor can only affect the number of iterations required at each incremental step, but not the final shape of the load-deflection curves. A key factor in selecting the procedure for force recovery is that higher order nonlinear effects must be included, based on rigorous continuum mechanics formulations.

Postbuckling analysis of trusses with various Lagrangian formulations

difference between the responses of the two probes proving that the vortices are shed alternately. The reduced frequency of shedding for all the cases vs the angle of attack is plotted in Fig. 4. In this figure we display data obtained from both facilities. There is a very clear dependence of the reduced frequency on the angle of attack. On the other hand, the influence of the Reynolds number is negligible. The evidence presented here indicates that vortices are shed over delta wings at high angles of attack, just like the cases of other flat surfaces or bluff bodies. Once this aerodynamic phenomenon is set in motion, an aircraft will respond, and interaction between the aerodynamics and the wing attitude will lead to wing rock. However, it should be emphasized that this type of wing rock has not been studied so far. The basic difference with the well-known case is that for very large angles of attack, the flow is fully separated, even if the attitude of the aircraft is fixed. Since the submission of this research Note, the present team has continued work on this project. Most recently, it was found that at intermediate angles of attack, simultaneous vortex shedding is also possible. The reader will find more information in a recent conference paper.

Key Considerations in Tracing the Postbuckling Response of Structures with Multi Winding Loops

The postbuckling response of structures with multi winding loops is characterized by the appearance of multi adjacent equilibrium paths, which often makes the iterations difficult to converge to the desired path. In this paper, some key issues for tracing the postbuckling response of a structure using an incremental-iterative approach are discussed. Concerning the finite element equations used, it is essential that the corrector used for recovering the element forces from the element displacements be made as accurate as possible, and that the predictor for computing the structural displacements under given load increments, which are approximate by nature due to linearization involved, be accurate to the level not to misguide the iterations. As for the incremental-iterative scheme, it is required to be: (1) numerically stable when encountering the limit points, (2) adjustable in load increments to reflect the stiffness variation, and (3) self-adaptive in changing the loading direction. To demonstrate the ideas involved, some examples with highly complicated postbuckling responses will be solved in this paper.

Constitutive laws and force recovery procedures in nonlinear analysis of trusses

Abstract Postbuckling analysis of trusses often involves large strain behaviors. For structures with large strains, the use of identical constants in the incremental material laws does not imply identical stress-strain relations, in terms of the 2nd Piola-Kirchhoff stress and Green-Lagrange strain, for formulations with various reference coordinates. Two sets of equations will be derived for recovering the bar forces. One set, referred to as the ‘total form’, is fully nonlinear; and the other, referred to as the ‘incremental form’, is approximate in that the material law is assumed to be piecewise linear. The solutions obtained with the total form are exact and step-size independent, while those obtained with the incremental form are error-prone and step-size dependent for large strain problems.

Non-linear stiffnesses in analysis of planar frames

Abstract This paper presents an approach for analyzing the geometrically non-linear behavior of elastic planar frames based on continuum mechanics principles. With the inclusion of non-linear stiffness matrices, the effects of bowing and rotation are taken into account in the calculation of element forces. Such non-linear stiffness matrices have also enabled the element, which in general is acted upon by a set of self-equilibrating nodal forces, to pass the rigid body test in the non-linear sense. In an incremental-iterative non-linear analysis, distinction must be made between the ‘predictor’ and ‘corrector’ phase. The former deals with the solution of structural displacements from the structure equations, while the latter is concerned with the calculation of element forces from element displacements. It is the latter that governs the accuracy of an incremental-iterative non-linear analysis. Numerical examples demonstrate that equations of higher-order accuracy should be used in the corrector phase to avoid possible errors.

The Role of Beam Flexibility and Ground Contact Model in the Clattering of Deformable Beams

It has been shown in previous research that, relative to (the usually considered case of) a single impact, multiple impacts (clattering) of rigid casings can greatly enhance the probability of failure of fragile components mounted on or inside them. This paper addresses the important issues of the roles of casing flexibility and contact model in the above situation. A finite element analysis of clattering of a Timoshenko beam is carried out here. Dependence of the maximum change in average velocity due to impact, on the beam stiffness and coefficient of restitution, are studied here.

A reduction method for boundary element reanalysis

Abstract This paper is concerned with reanalysis for boundary element systems. An efficient and accurate reduction method is proposed for reanalyzing such systems. The new method has several merits. In particular, the reduced system has been uncoupled by a Gram–Schmidt orthonormalization procedure. Also, a computation-inexpensive convergence criterion is proposed for the automatic determination of the number of basis vectors required to approximate the solution within a specified level of accuracy. Several example problems are illustrated to verify the accuracy and efficiency of the proposed reduction method.

Stiffness Matrices for Linear and Buckling Analyses of Composite Beams with Partial Shear Connection

This paper is concerned with linear and buckling analyses of composite beams with partial shear connection (partial composite beams) using the finite element method. Two elements derived from different types of shape functions are proposed in this study. The first element, referred to as exact, is based on the exact shape functions obtained by solving the differential equations governing the transverse displacement and the slip of the shear connector layer of a partial composite beam. The second element, referred to as approximate, is based on the conventional linear and cubic shape functions used in conventional axial and beam elements. By making use of these two types of shape functions, the elastic and geometric stiffness matrices can be derived explicitly from the strain energy and the load potential, respectively. Both types of elements can be used to carry out linear static and buckling analyses. As expected, the exact element is more accurate than the approximate element if the same discretization is adopted. However, the approximate element has the advantage of easy implementation since the expressions of its elastic and geometric stiffness matrices are very simple. Also, the solutions obtained from the approximate element converge very fast; with reasonable discretization, say 8 elements per member, very accurate solutions can be obtained.

Flexibility based formulation for nonlinear analysis of reinforced concrete frames considering the effects of finite length inelastic zones

Abstract This paper is concerned with geometrically and materially nonlinear analysis of planar reinforced concrete (RC) frames with a particular emphasis on how to model efficiently and accurately the effects of material nonlinearity. To this end, an inelastic stiffness matrix is developed using a flexibility formulation on the basis of cross sectional generalized stress‐strain relationships. The derived stiffness matrix takes into account the effects of cracking of concrete and yielding of reinforcement that may occur within a finite‐length zone located at the ends of a member. Its implementation is easy and efficient since the number of degrees of freedom is the same as that of the elastic stiffness matrix. Geometric nonlinearity is incorporated into the analysis through the use of the conventional geometric stiffness matrix. The accuracy and efficiency of the derived inelastic stiffness matrix are verified by several example problems.

Topology Optimization of Elastic-Plastic Structures

The evolutionary structural optimization method is improved and extended to elastic-plastic topology optimization for the first time. An adaptive rejection ratio is proposed to control the number of removal elements without destroying the symmetric pattern in each evolution. Two performance indices suitable for elastic-plastic topology optimization are also proposed and examined. The performance indices can be used to investigate the material efficiency of structures in different evolutionary stages, and to serve as stop criteria in the evolutionary process. Moreover, an interactive special purpose computer analysis and graphics system is developed to visualize the topology in the displacement in an elastic-plastic analysis on the obtained topology are discussed.