Keywords: non-linear geometric structural analysis, stability of asymmetric three dimensional frames, bracing of structures.

Non-linear -delta behaviour of three-dimensional frames with irregular plan geometry is studied, using a parametric variation of the geometry and stiffness formerly selected. The results obtained using the author's software and using established commercial software are compared. Using the exact total stiffness formulation of the non-linear geometric analyses in the developed software, enables the degree of precision in selected calibration examples to be studied, compared with the exact analytical results as well as the commercial software results. A parametric study of the critical load factor of asymmetric three-dimensional frames, un-braced and braced, permits the characterisation of their carrying capacity with respect to their overall structural stability.

Understanding the behaviour of each structural configuration of the braced structures, allows the selection of the best solution. To proceed with the parametric comparisons and characterization of the structural performance, use is made of a software of automatic calculation of the 2D or 3D frames carrying capacity (INST3D) already developed and presented by Cesar and Barros [1], based on the exact formulation of the structural stiffness matrix of bi-dimensional frame members in the displacement formulation of the incremental balance. The algorithm included in the software INST3D follows the methodology based on the formulation of the exact stiffness matrix with the stability functions proposed by Livesley and Chandler [2], and it enables the critical parameters for the diverse structural configurations as well as the respective instability modes to be obtained.

The use of such bi-dimensional exact stiffness matrix for every prismatic bar of a plane beam-column model is dependent on the stability functions , and originally developed by Livesley and Chandler [2], but can also be related to the four stability functions used by Reis and Camotim [3], Barros [4] and Barros et al. [5].

Appropriate assemblage of such member stiffness allows obtaining the global stiffness matrix to be obtained in the developed software INST3D, from which the critical load is determined through an iterative process in the resolution of the incremental equilibrium equation. Such an iterative determination of the critical parameter also involves the knowledge of the lower (for sidesway or displaceable joints portal frames) and upper (for un-sway, sidesway prevented or un-displaceable joints portal frames) load parameter bounds, between which the critical value is looked upon. In the analysis associated with INST3D, the laminar elements - floor slabs - of the portal frames are modelled as rigid diaphragms, a procedure which can also be modelled in the commercial software SAP 2000 by the use of diaphragm constraints.

A parametric study of the load carrying capacity of a certain asymmetric 3D frames is successfully conducted, which also permits the two methodologies used to be reviewed. The critical load factors of these 3D frames were calculated, for both un-braced and braced configurations (Cesar and Barros [6]). It was concluded that the positioning of the braces could considerably change the structural behaviour and the carrying capacity of the 3D asymmetric frames with respect to the buckling load. The rational use of the bracing elements controls the structural performance of distinct structural designs.

1

Cesar, M.T., Barros, R.C., "Desempenho Não Linear Geométrico de Pórticos Metálicos Contraventados sob Acções Verticais", in "Proceedings of the 4 Congresso de Construção Metálica e Mista", Tema 4 - Análise e Verificação da Segurança, pp. 367-378, Lisboa, Portugal, 2003.

2

Livesley, R.K., Chandler, D.B., "Stability Functions for Structural Frameworks", Manchester University Press, Manchester, UK, 1956.

3

Reis, A.J., Camotim, D., "Estabilidade Estrutural", McGraw-Hill, Portugal, 2001.

4

Barros, R.C., "Apontamentos sobre Instabilidade de Estruturas", Conjunto de trabalhos impressos e manuscritos de apoio ao Mestrado em Estruturas de Eng. Civil, Secção de Estruturas, Departamento de Eng Civil, F.E.U.P., Porto, 1999-2002.

5

Barros, R.C., Oliveira, S., "Análise Elástica de 2 Ordem de Pórticos Planos em Calculadora Programável", in "Proceedings of the 3 Congresso Luso-Moçambicano de Engenharia: Engenharia e Inovação para o Desenvolvimento", Ed: J. Silva Gomes et al., Volume I, Tema C - Materiais e Estruturas, pp. 435-446, Maputo, Mozambique, 2003.

6

Cesar, M.B., Barros, R.C., "Modelos de Comportamento Não Linear Geométrico de Pórticos Assimétricos Tridimensionais", in "Proceedings of the Conference Métodos Computacionais em Engenharia", APMTAC, Lisboa, Portugal, 2004.

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