“Computational Civil Engineering - "Intersections" International Journal

“Computational Civil Engineering 2005”, International Symposium 71( K + A WA) T ∆U= ∆FT(3)where W is a diagonal matrix of penalty weights, A is the unscaled matrix of thepenalty elements, K T is the global tangent stiffness matrix, ∆U and ∆F are theincremental displacement and force vector, respectively.2.6 Geometry updating and force recoveryUsing an updated Lagrangian formulation (UL) the nonlinear geometrical effectsare considered updating the element forces and geometry configurations at eachload increment. The natural deformation approach (NDA) in conjunction with thegeometrical “rigid body qualified” stiffness matrix [8] is adopted for the elementforce recovery and the web plane vector approach is effectively used to update theframe element coordinates [1].The element incremental displacements can be conceptually decomposed into twoparts: the rigid body displacements and the natural deformations. The rigid bodydisplacements serve to rotate the initial forces acting on the element from theprevious configuration to the current configuration, whereas the naturaldeformations constitute the only source for generating the incremental forces. Theelement forces at the current configuration can be calculated as the summation ofthe incremental forces and the forces at the previous configuration.3. ANALYSIS ALGORITHM AND COMPUTER PROGRAMIn order to trace the equilibrium path, for proportionally and non-proportionallyapplied loads, the proposed model has been implemented in a simple incrementaland incremental- iterative matrix structural-analysis program. In the simpleincremental method, the simple Euler stepping algorithm is used in conjunctionwith constant work-load increments, whereas in the incremental-iterative approach,at each load increment a modified constant arc-length method is applied tocompute the complete nonlinear load-deformation path [1]. Convergence of theiterative process is said to have occurred when within certain tolerances the internalactions are in equilibrium with the applied external loads.Based on the analysis algorithm just described, an object-oriented computerprogram, NEFCAD 3D, has been developed to study the combined effects ofmaterial, geometric and semi-rigid connection nonlinear behaviour on the loadversus-deflectionresponse for spatial framed structures. It combines the structuralanalysis routine with a graphic routine to display the final results [4]. The graphicalinterface allows for easy generation of data files, graphical representation of thedata, and plotting of the deflected shape, bending moments, shear force and axialforce diagrams, load-deflection curves for selected nodes, area of uncracked,cracked and yielded parts of the elements as shown in Fig. 3.