Nonlinear analysis of trusses through energy minimization

https://doi.org/10.1016/j.compstruc.2004.05.008Get rights and content

Abstract

Equilibrium configurations of structures are those that make the potential energy of the system a relative minimum. Based on this principle an adaptive local search procedure is developed where better and better shapes are sought with smaller potential until a relative minimum is reached. The algorithm is applied to trusses and it has been shown that satisfactory results can be obtained for problems with material and geometrical nonlinearities. The method is capable of giving deflected shapes of structures even if they have configurational instability and it can be generalized to be applied to analysis of more general types of structures.

Introduction

Structural problems are generally attacked through formulations that are based on equilibrium considerations. For linear problems this approach results in determination of solutions of a linear matrix equation. For nonlinear problems, the result is a nonlinear matrix equation, where the coefficient matrix is dependent on both the load vector and the displacement vector. These equations are generally solved by successive iterations or by local linearizations (e.g. [1], [2]). There are formulations based on energy theorems. These formulations too result in equations that are similar to those obtained through equilibrium formulation, and thus the solution techniques are quite similar.

A completely different technique would be the application of minimum potential energy principle per se: “Of all the displacements which satisfy the boundary conditions of a structural system, those corresponding to the stable equilibrium configurations make the potential energy a relative minimum” [3], Theorem X. This principle, although well known and applied through illustrative examples in almost every structural analysis book, is not thoroughly exploited except for very few cases [4], [5]. In the present work it has been shown that its application to trusses is very fruitful, enabling analysis of trusses for large deflections, before and after the loss of stability, and even of those ones which are “unstable” by definition. The materials forming the members of the truss may be linear or nonlinear, and all can be treated with equal ease.

The problem, formulated as an application of the minimum potential energy principle, is obviously an optimization problem. For optimization of the total potential, there are various techniques that have been used, like sequential quadratic programming [4], tree search [5], random search method and simulated annealing algorithm [6]. In this analysis, the results obtained by an adaptive local search method are presented. The method has proved to be very successful.

One reason of the neglect of energy minimization in solving structures for such a long time may be its demand for relatively longer time of computation. With the advances in computer capacities this disadvantage is becoming more and more invalid.

Section snippets

Problem formulation

Consider the problem of analyzing a structure. The total potential U for a given state of deformations (characterized by the strains ε within the body, creating the generalized deflections ui coupled with the generalized loads Pi) can be written asU(ε)=∫Ve(ε)dV−∑i=1NPPiuie(ε)=∫0εσ(ε)dεwhere σ and ε are stress and strain which are interrelated through σ=σ(ε); e is the strain energy density; NP is the number of loads; V is the volume of the body.

The integral in (1) gives the strain energy stored

Implementation of geometric and material nonlinearity

The formulation presented above intrinsically takes care of the geometrical nonlinearity since the equations are written already on the deformed shape. This formulation is such that it cannot be put into a simpler form to be valid for small deformations only. This latter case can be dealt with by application of very small loads so that the deformed shape is not remarkably different than the undeformed shape.

Material properties enter into the formulation through stress strain relationship σ=σ(ε

Optimization algorithm

There is a large number of techniques that can be applied to solve the optimization problem formulated above. The one explained below and used in this analysis is a local search method [10]. The technique adopted, adaptive local search method (ALSM), is based on the concept of a variable neighborhood in which search is carried on. If a better solution is found in this neighborhood, it replaces the current solution and a new search neighborhood is defined as centered at that new point and with a

Results and discussions

Several problems are solved to test the procedure, as to its versatility, robustness, convergence speed, correctness and accuracy. The results are more than encouraging. It was possible to solve all the problems analysed, which involved deformations ranging from very small to very large, with material properties involving all kinds of nonlinearities, and yield values. The only weak point of the method seems to be its relatively long run time. This point can be considered as not very important

Conclusions

The method, which rests on very simple but sound principles, is seen to be very powerful in solving any problem on truss statics. Some points to be marked can be listed as follows:

  • Geometric nonlinearity and material nonlinearity can be handled very easily.

  • There is no difference in formulating statically determinate and indeterminate trusses, and even those which have geometric instability.

  • Accuracy can be controlled to the desired degree, and is never lost.

  • The main part of the software is a mere

References (10)

There are more references available in the full text version of this article.

Cited by (56)

  • A robust unsupervised neural network framework for geometrically nonlinear analysis of inelastic truss structures

    2022, Applied Mathematical Modelling
    Citation Excerpt :

    Once the network is trained, the displacement field and corresponding structural responses can be found. As shown in Tables 6 and 7, the obtained solutions reveal that the presented method is highly close to those found in literature [18]. At the same time, it is easily be seen that the results of the linear and geometric nonlinear analysis are practically the same, so the influence of geometry changes is small for this loading.

  • Analyses of plane stress and plane strain through energy minimization

    2021, Structures
    Citation Excerpt :

    The advances of the metaheuristic methods and computer technology have made possible to analysis of relatively more complex structural systems. Geometric and material nonlinearity of plane truss systems were exploited using simulated annealing and local search algorithms by Toklu [9]. Geometrically nonlinear space truss systems were investigated with harmony search, genetic algorithm, and particle swarm optimization algorithms [10,11].

  • Mixing dynamic relaxation method with load factor and displacement increments

    2016, Computers and Structures
    Citation Excerpt :

    This algorithm was not able to pass the load limit points. Some other researchers also studied the tracing of structural equilibrium path [6–14]. Frequently, the iterative methods are divided into two categories: 1 – explicit techniques 2 – implicit techniques.

  • A metaheuristic-based method for analysis of tensegrity structures

    2024, Structural Design of Tall and Special Buildings
View all citing articles on Scopus
View full text