Keywords: non-linear beam theory, non-linear material, reinforced concrete, stress field integration, Newton's iteration, rate of convergence.

In the finite-element analysis of geometrically non-linear spatial beams and frame structures made of non-linear material, the non-linearity of the constitutive law plays an important role in the rate of convergence of the global iteration procedure. In this paper we study the influence of solving the non-linear constitutive equations in the global or local manner.

The discretized governing equations of the non-linear spatial beams must be solved iteratively. In the finite-element formulation, the choice of the primary unknowns seems to be crucial for the efficiency of the overall numerical algorithm. The constitutive equations can be eliminated from the set of the governing equations of the beam. This reduces the size of the global system of linearized equations that needs to be solved in each iteration. When the non-linear material is taken into account, such a formulation must be capable of solving locally the constitutive equations in each step of the global iteration. Non-linear constitutive equations must be solved iteratively. This implies that, in each step of the global iteration, several local iterations in integration points need to be executed. One would expect that such an approach would not dramatically increase the number of global iterations compared to linear material; this turns out to be true, yet the computational times are often substantially bigger. If the constitutive equations are taken to be the part of the global governing equations, the size of the global linearized system of equations is much larger, but the local iterations are not needed. Such an approach may somewhat increase the number of global iterations, but the overall computational cost might be lower.

When solving the governing equations of the beam, the constitutive equations can be treated in several ways. Three different approaches will be analyzed in the paper:

i)

direct global approach where the constitutive stress-resultant force and moment vectors are evaluated directly from the known strains in each step of the global iteration;

ii)

indirect global approach where the strain vectors are obtained iteratively from the equilibrium stress-resultant force and moment vectors in each step of the global iteration;

iii)

partly reduced approach where the constitutive equation for forces is eliminated from the system of governing equations.

A short outline of each approach will be presented. We will also briefly describe three different numerical methods which will be used for comparisons of the convergence rate and the overall computational cost of the above approaches. In order to make the comparisons unbiased, the computer code for all of the formulations shares the same pre- and postprocessing algorithms and all of the common parts are programed identically. Computer programs were developed and tested using Matlab [1], based on the algorithms, presented in [2,3]. We used the same type of the discretization in all formulations.

The influence of different approaches on the rate of convergence and the overall computational costs will be examined through several numerical examples. For each particular case we will compare not only the number of iterations but also the total number of floating point operations. The total number of floating point operations reveals the actual computational cost of the algorithm independently on the computer equipment used for the studies. We have chosen to compare the three formulations on realistic problems using realistic material model. The data were obtained from well documented experimental tests of reinforced concrete beams and frames. As the differences in the computational costs between the approaches grow if the applied loads approach the critical ones, all the tests are performed in the neighbourhood of the critical loading.

1

Matlab, The language of technical computing, The Mathworks, Inc., 1999.

2

D. Zupan, M. Saje, "The three-dimensional beam theory: finite element formulation based on curvature", Comput. Struct. 81, 1875-1888, 2003. doi:10.1016/S0045-7949(03)00208-6

3

D. Zupan, M. Saje, "Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures", Comput. Methods Appl. Mech. Engrg. 192, 5209-5248, 2003. doi:10.1016/j.cma.2003.07.008

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