Numerical analysis of beam and surface structures under large displacement.

Akmadzic, Vlaho ; Trogrlic, Boris ; Mihanovic, Ante 等

1. INTRODUCTION

The incremental application of the small displacement theory is a typical solution for the large displacement problems (Izzudin, 2001). Updating of the configuration and load is mean differences between small and large displacement approach. In this paper is also presented the null configuration procedure which enables the process of updating. The linear beam finite element modeling consists of modeling of the cross-section and modeling of the comparative body (Trogrlic & Mihanovic, 2008). The cross-section is discretised by quadratic and/or triangular elements, which is represented by filament. Such a filament discretisation enables the monitoring of the normal stresses in the element and hence the state of the cross-section under the action of the longitudinal forces and two bending moments. The equilibrium of the cross-section is obtained by an iterative procedure so that the failure surface is constructed numerically under actions of the longitudinal force and the bending moments. The procedure includes a partial or complete plastification of the filaments or the whole cross-section. A tangential material modulus is applied. The torsional effect and consequently the effect of the shear force of the reinforced concrete element are observed on the so-called comparative body. The analysis of a comparative body gives the main properties of torsional stiffness of a given element including the material non-linearity. Surface structures system is discretised with the isotropic shell finite element (Akmadzic, 2008). The new finite element, four nodes superparametric, is capable to describe double curved surfaces. The curved surface is transformed into the flat one and calculation process goes with the flat shell finite element with the six DOF per node. The shell finite element is made as the connection of the flat membrane and bending finite element. The membrane state possesses the drilling degree of freedom around the surface normal ort, so that the shell finite element, as the beam finite element, has six DOF per node. In order to test how applicable, functional and accurate this algorithm is one example is shown in this paper.

2. THE NULL CONFIGURATION PROCEDURE

At one increment level is used a total Lagrange formulation. After the completed iterative procedure, at the level of the considered i-th increment, after the change of structure position and applying the principle of virtual work the governing equilibrium equation

[K.sub.0] (u) + [K.sub.g] (u) + [K.sub.L] (u) = F (u) (1)

where [K.sub.0] is the basic stiffness matrix, [K.sub.g] is the geometry stiffness matrix and [K.sub.L] is the large displacement stiffness matrix. The total applied load so led to real large displacements [u.sub.i]. However, if we attempt to consider the displacements resulting from a total load over the last tangential stiffness we would obtain fictional displacements

[[bar].u].sub.i] = [K.sub.T] ([u.sub.i]).sup.-1] F ([u.sub.i]) (2)

If we subtract these fictitious displacements from the actual position formed by the initial geometry plus large displacement, a null configuration would be obtained. In the next increment under the incremental load, the tangential stiffness matrix is [K.sub.T] ([u.sub.i+1]). Now, the previous total load acts on a new stiffness without its influence upon that change which actually exists. The new null configuration should be

[[bar].u].sub.i+1] = [K.sub.T] [([u.sub.i+1]).sup.-1] F([u.sub.i]) (3)

where all external and internal changes, caused by the change of stiffness at the level of that increment, have been mapped. This influence can be presented so that, in addition

[sub.1] [DELTA]F([u.sub.i]) = F([u.sub.i]) - [K.sub.T](u.sub.i+1]) [[bar.u].sub.i] (4)

Influence of this member is relatively small due to the first incremental load, but its influence is significant due to the last incremental load. The complete load in the next incremental step becomes

[DELTA]F ([u.sub.i]) = [sub.0] [DELTA]F ([u.sub.i]) + [sub.1][DELTA]F ([u.sub.i]) (5)

where [sub.0][DELTA]F ([u.sub.i]) is value of standard incremental load.

3. EXAMPLE

Program PRONELL, which is used, enables discretisation of the space structures with the beam-column and shell finite elements and uses the large displacement theory.

In this developing level the program can take into account the nonlinear problem analysis. This paper will present one example where we can see the application of the large displacement method. This method results are compared to results obtained with the other methods. A space three-storey one-field frame, presented in Figure 1, has been analyzed. The connections between all elements of model are fixed and first floor elements are fixed at footings.

[FIGURE 1 OMITTED]

The horizontal forces H=10.0 kN and vertical load q=10 kN/m' are applied, as shown on the Figure 1. Few characteristic cases are analyzed. The original numerical example, with the geometrical characteristic of the beams and columns are taken from the literature (Trogrlic, 2003). It analyzed the space three-storey one-field frame according to the theory of small displacement. So, with the program PRONELL (Mihanovic et al., 2008) we analyzed the same frame with the theory of large displacement. After that we placed into the model on each flat the story plates without the weight. The thickness of the plate in the first case was 22 cm, and after that was 30 cm. The geometrical nonlinearity of the beam and plate is taken into account, while the material nonlinearity is only used in the beam elements. The reinforced concrete plates has Young's modulus E=30 [10.sup.6] kN/[m.sup.2] and Poisson coefficient [upsilon]=0.2. The columns were discretised with 6 rectilinear two node finite elements, while the beams were discretised with the 14 rectilinear two node finite elements. Cross section of the column was discretised with the orthogonal mesh 8x8, the beam with 14x14 finite elements. The plate discretisation was defined with the beam discretisation. Comparison of solutions for different cases has been done, as well as comparison of the results derived from the usage of the PRONELL program and results obtained by another authors (Fig. 2.).

[FIGURE 2 OMITTED]

The influence of the large displacement theory on the space three-storey one-field frame is obvious (Fig. 2.). The system collapses much before according to the theory of large displacement. The frame without story plates is losing the bearing capacity according to the theory of small displacement at load factor 1.35. If we use the large displacement theory it happens at load factor 0.85.

The most interesting thing is happening when the reinforced concrete plates without the self weight are placed into the frame. The behavior of the frame is changed. The system is stronger, less flexible. For the same displacement the bearing capacity is much higher and it is closer to the linear-elastic model. This phenomenon for the frame with the plate thickness h=22cm is happening till the load factor 0.60. After that the system is quickly getting extremely large displacements. For the plate thickness of h=30cm the bearing capacity of the space three-storey one-field frame with plates is increased and the system collapse will be a lot after the collapse of the system without the plates.

4. CONCLUSION

The development of the global numerical model is based on the theory of large displacements. It is applicable in engineering practice for stability and bearing capacity analyzes of the surface and beam structures.

The model is based upon the authentic universal approach to the updating of the system including the loads, gravitational and/or follower, employing the method of the null configuration. It also includes the authentic model of the comparative body for simulating the torsion and shear. The authentic shell finite element, four nodes superparametric, is capable to describe double curved surfaces. The model ensures a high degree of accuracy, as could be seen in the reference. Indirectly, the model also ensures precise monitoring of the influence of large displacements, including at the same simultaneously both the basic, geometrical stiffness and the stiffness of large displacements with a realistic description of material nonlinearity in the beam cross-section. In this step of development, the model is not capable to describe the material nonlinearity of the surface element.

5. REFERENCES

Akmadzic, V. (2008). Model velikih pomaka u analizi plosnih i linijskih konstrukcija (Model of large displacement in analysis of surface and beam structures), Dissertation, Faculty of Civil Engineering University of Mostar, Mostar, B&H

Figueiras, J. A. & Owen, D. R. J. (1984). Analysis of elastoplastic and geometrically nonlinear anisotropic plates and shells, Finite element software for plates and shells, eds. E. Hinton, D. R. J. Owen, Swansea, U. K., pp 235-322

Izzuddin, B. A. (2001). Conceptual issues in geometrically nonlinear analysis of 3D framed structures, Comp. Meth. in Appl. Mech. and Eng., 191: 1029-53

Mihanovic, A., Trogrlic, B. & Akmadzic, V. (2008). Program PRONELL for nonlinear analyses analyses of large and small displacement of the surface and beam structures, Faculty of Civil Engineering University of Mostar, Mostar, B&H

Trogrlic, B. (2003). Nelinearni numericki model stabilnosti i nosivosti prostornih armirano betonskih linijskih konstrukcija (Nonlinear numerical model of stability and bearing capacity of spatial reinforced-concrete linear structures), Dissertation, Faculty of Civil Eng. and Arch. University of Split, Split, Croatia

Trogrlic, B. & Mihanovic, A. (2008). The comparative body model in material and geometric nonlinear analysis of space R/C frames, Eng. Comp., pp 155-171