Nonlinear Structural Problems Research Articles

Energy minimization versus pseudo force technique for nonlinear structural analysis

The effectiveness of using minimization techniques for the solution of nonlinear structural analysis problems is discussed and demonstrated by comparison with the conventional pseudo force technique. The comparison involves nonlinear problems with a relatively few degrees of freedom. A survey of the state-of-the-art of algorithms for unconstrained minimization reveals that extension of the technique to large scale nonlinear systems is possible.

Modal Superposition Method for Computationally Economical Nonlinear Structural Analysis

A modal superposition method for analyzing nonlinear structural dynamic problems involving impact between components is developed and evaluated. The finite-element method is used to express the equations of motion with nonlinearities represented by pseudo force vector. Three test problems are solved to verify this method. This has demonstrated the applicability of this method to seismic analysis of large, complex structural systems. It is concluded that the modal superposition method has a significant cost advantage over the direct integration method for problems with large wave fronts and the source of nonlinearities restricted to a limited portion of the structure.

Extending the convergence domain of the Newton-Raphson method in structural analysis

This paper contains a procedure for continuing the solution of nonlinear structural analysis problems past limit and branch points when using the Newlon-Raphson or the modified Newton-Raphson methods in conjunction with incremental loading. The singular or ill conditioned Jacobian which arises at or in the vicinity of critical load points is eliminated by adding positive values to select diagonal elements of the Jacobian. A technique for determining which elements to modify together with a method for efficiently determining the optimum magnitude of the modification for minimum solution error are developed. The effectiveness of the procedure is illustrated by a number of example problems of hyperelastic bodies at finite strain which experience either snap-buckling, bifurcation, or a soft load-displacement response followed by stiffening. In addition a few cases are presented to illustrate the sensitivity of the particular problems to imperfections.

An analysis of an implicit finite element algorithm for geometrically nonlinear problems of structural dynamics part 1. Stability

A series of implicit finite element algorithms for the geometrically nonlinear structural dynamics problem are proposed. The proposed algorithms require only the solution of a linear system at each time step. Thus, they are computationally efficient. In addition, the algorithms are discrete conservation laws. The conservative nature of the proposed schemes has a positive effect in providing a stable approximation. The stability of the algorithms is analyzed using energy methods. One of the proposed computational methods is shown to be unconditionally stable.

An analysis of an implicit finite element algorithm for geometrically nonlinear problems of structural dynamics part 2. Accuracy

In part 1 ∗ ∗ Comp. Meths. Appl. Mech. Eng. 14 (1978) 377–390. a series of implicit finite element algorithms for the geometrically nonlinear structural problem were proposed. These algorithms were shown to be computationally efficient because they require the solution of a linear system rather than a nonlinear system at each time point. In addition, because of the conservative nature of the algorithms, their stability properties are easily assessed. One algorithm was shown to be unconditionally stable. In part 2 the convergence of the proposed algorithms is demonstrated. It is shown that for the consistent algorithm, additional conditions must be satisfied to insure convergence. In fact, one of the proposed algorithms is shown to be unconditionally stable but conditionally convergent.

An approximate method for structural optimisation

The nonlinear structural optimisation problem is solved by successive approximations in which the objective function and the constraint functions are approximated as single term posynomials about a given design point at each step, and transformed into a linear programming problem in the logarithmic space. Each feasible basic solution is constructed with the aid of pseudo constraint limits, which are obtained either from specified move limits around the design point, or from the activity level of the constraints at the previous approximated solution, making some of the inactive constraints artificially active. The method is demonstrated by some examples.

Buckling of pressure loaded rings and shells by the finite element method

The correct formulations for solving nonlinear structural problems by the finite element method have now been established. Numerous investigators have given the derivation for the solution of problems by the incremental tangent stiffness method and total formulation methods. These derivations have been applied to many problems and the results have been shown to be quite accurate for the problems that have been selected. However there is one area of application that has received practically no attention. This is in the investigation of the buckling strength of pressure loaded rings and shells. The effect of pressure loading where the loading changes direction as the structure deforms has been included in several previous derivations, by what is known as the load stiffness matrix, but to the author's knowledge no one has investigated problems where this effect has been included in the solution procedure. For rings and some buckling modes of shells, the results can be in error by as much as 50%. This paper will describe an iterative process for solving the nonlinear equilibrium equations and correcting the loads to include the effect of changing geometry at each load level. This approach is different from the classical eigenvalue or bifurcation method. Several case studies will be described which were performed on ring and shell problems. The geometry of these example problems were axisymmetric and in order to apply a nonlinear collapse analysis, the structure had to be perturbed out of its axisymmetric pattern into a buckling pattern. Imperfect geometry and very small concentrated loads were used to cause this perturbation and this will be described in the paper. The sensitivity of the computed collapse pressure to the finite element mesh gradation will be discussed. A comparison will be made between results obtained by including the effect of following pressure load and those obtained by not including this effect.

A simple triangular facet shell element with applications to linear and non-linear equilibrium and elastic stability problems

This paper has two main objectives. First to describe a very simple facet triangular plate and shell finite element called TRUMP which includes, if required, transverse shear deformation and is based on physical lumping ideas with a simple mechanical interpretation [ 1,2,4,5 ]. Second to give an account of some non-trivial numerical examples of large deflection and post-buckling of shells. There are two types of non-linear structural problems which give rise to particularly delicate numerical experimentation. They are those involving deflections of the order of the structural dimensions, such as three-dimensional elastica, and the instability phenomena of the type leading to dynamic snapthrough, e.g. in cylindrical panels. To tackle such problems using a highly sophisticated shell element such as SHEBA is neither easy nor inexpensive. It is shown that the TRUMP element with only 18 displacement and rotation degrees of freedom is relatively economical to use and yet capable of engineering accuracy. The paper makes use of the theory of simplified geometrical stiffness based on the natural mode method which has been described fully in previous publications [1,2].

An alternative procedure in dynamic relaxation

In the usual analytical methods for structural analysis, the structure is assumed to be loaded and the internal forces are present in it from the beginning of the calculations. The unknown forces are then found by assembhng and solving a set of simultaneous equations. The conventional methods in structural analysis have been radically altered by the advent of high speed digital computers. A new approach to the calculation of internal forces, made practically possible by this situation, is to follow their development in the structure from its initially unloaded state. This idea has been developed into a method of structural analysis which depends on following the dynamic behaviour of the structure until it it is damped down to its static con&ration. This method, which is presently known as Dynamic Relaxation, was originally conceived by Day [l] during the course of his investigations of complex prestressed concrete pressure vessels for nuclear reactors. The method of dynamic Relaxation (DR) is essentially a step-by-step integration of critically damped vibration using viscous damping to ensure the attainment of a steady-state solution. Dynamic terms involving inertia and viscous damping are added to the static equations of equilibrium. These dynamic equations of equilibrium, after being written in an explicit finite difference form, are solved using a standard substitution technique with the help of a high speed digital computer. When the damping coefficients are chosen to be critical the oscillations die out in the quickest possible time and the function converges to the static value. In practice, a value slightly lower than the critical value is chosen so as to get an oscillatory convergence which approaches the static value in the final limit and also gives bounds between which the true solution is found. The entire formulation is linear in time and consequently the precedure only requires a direct substitution of the value of the function at a previous iteration to calculate the new value of the function. During this direct substitution, it is relatively simple and straightforward to include any nonlinear term appearing in the formulation. For the sake of clarity the application of DR technique to the simple problem of bending of a beam is discussed in Appendix. Dynamic relaxation technique is presently a well established technique to solve linear and nonlinear structural problems. There is considerable amount of literature on its application to the analysis of plates, shells and three-dimensional solids [2-Q. Dynamic relaxation scheme is an explicit formulation and as such the time-wise integration procedure which

Incremental Solution Procedure for Nonlinear Structural Dynamics Problems

A general incremental formulation is presented for solving nonlinear structural dynamic problems. The formulation introduces a new term which is not obtained in the usual finite element tangent stiffness approach. The generality of the derivation is used to show how the computation of the geometric stiffness matrix can be eliminated and its contribution included through a simple addition of the estimated incremental displacements to the equilibrium correction term. This approximation thus has the potential for saving significant computing time in the assemblage of higher order finite elements. Two simple numerical experiments are conducted and it is shown that accuracy is still retained when this approximation is used.

A unified approach to three finite element theories for geometric nonlinearity

The paper considers theoretical and numerical questions associated with finite element analysis of geometrically nonlinear structural problems. A variety of finite element theories have recently arisen leading to geometric stiffness matrices, initial displacement matrices, and other similar forms. The present analysis determines an integrated unified approach to these geometrically nonlinear problems. The development leads very readily to the resolution of questions pertaining to highly nonlinear problems, and the interrelations of recent emergent theories become apparent. The investigation includes evaluation of alternative approaches for numerical computations of incremental and single-step analyses and advocates specific basic approximations for particular applications.

A comparative study of different numerical solution techniques as applied to a nonlinear structural problem

This paper describes and compares a number of different numerical methods frequently used in solution of nonlinear problems in structural mechanics. A simple mechanical system is chosen. By varying the geometry and the stiffness of the members, different types of nonlinear structural behavior may be simulated. A comparative study of various solution techniques applied to the sample problem is given. The solution paths followed by these methods are illustrated by use of map plots of the strain energy and the total energy of the system. Also, a new method for automatic computation of steps for incremental techniques is suggested.

Monte Carlo Solution of Nonlinear Vibrations

A Monte Carlo technique is presented which can effectively be used for nonlinear response analysis of a structure subjected to a random pressure field undergoing large deflections. The pressure field is idealized as a multidimensional Gaussian process with mean zero and homogeneous both in time and space. The response analysis is performed in the time domain by numerically simulating generalized forces rather than in the frequency domain. The solution satisfies the boundary conditions and the differential equation in a Galerkin sense. Two numerical examples involving large deflections of a string and a plate are worked out. The result indicates that the present method indeed provides a powerful tool in solving nonlinear structural response problems under random excitations.