Nonlinear Analysis of Reinforced Concrete Frames by a Combined Method

A more realistic and hence nonlinear analysis of reinforced concrete structures is becoming increasingly important. A combination of the displacement method, the transfer matrix method, and a crosssection module is suggested which leads to an effective analysis method for reinforced and prestressed concrete frames. The combined method considers both material and geometrical nonlinearities including large displacements and rotations. The computation of the system is incrementally and iteratively carried out by the displacement method. At element level, an extended transfer matrix method is used. Thus neither displacement nor force shape functions are required. Instead, the axial strain and curvature distributions along the element are segmentally approximated by polynomials. The transfer matrix method provides both the element forces and the element stiffness matrix. It is recursively applied to the deformed element, which is discretised into individual segments whose number and lengths depend on the stiffness gradient. The cross section module is based on cross-sectional integration. It takes into account nonlinear material behaviour including cracking, softening and yielding of reinforcement. The combined method is presented for plane frames but can be extended to spatial systems. Electronic Journal of Structural Engineering, 9 (2009) 30 gram. A complex procedure is required for deriving the element stiffness matrix and element resisting forces. Spacone et al. (1996) propose an iterative element state determination by adjusting the element forces until the predetermined displacements are achieved. The stiffness matrix follows from the inverted flexibility matrix. Neuenhofer & Filippou (1997) present a method for directly determining the element state thus avoiding an iterative procedure at element level. By means of interpolation functions for the curvature, that approach is adapted to geometrical nonlinearity considering small deformations but is restricted to material linearity (Neuenhofer & Filippou 1998). The approach presented here allows for geometrical nonlinearity including large displacements and rotations and material nonlinearity. Neither displacement nor force shape functions are required. Instead, the axial strain and curvature distributions along the element are segmentally approximated by polynomials. The respective advantages of three methods are combined. Olsen (1986) and Pfeiffer (2004) explore similar ideas and can be considered precursors to the approach presented here. It is suitable for the computation of general reinforced or prestressed concrete frames, but also for steelconcrete composite frames and others. In contrast to lumped models, which concentrate the nonlinearity at element ends, the combined method represents a distributed approach. At system level, the computation is incrementally and iteratively carried out using the displacement method. The advantage of the displacement method lies in the ease of representation of a frame's topology by elements. System nodes are mainly defined at the beamand column end and connection points, and at cross-sectional changes. At element level, an extended transfer matrix method is used (Wallmichrath & Starossek 2004). In the transfer matrix scheme, the state variables are recursively transferred over a chosen number of discrete segments from one end of the element to the other. The state variables comprise the internal forces and the displacement quantities. The partitioning of an element into segments depends on the local stiffness gradient. Based on given element end node displacements, which follow from the first level computation at system level, the internal forces at the element end nodes and at the segments are determined. Having obtained the internal forces, the tangent stiffness matrix is calculated using difference quotients. The remaining unbalance forces at the element end nodes enter into the next iteration step at system level and decrease with each further iteration step until achieving convergence. Material nonlinearity along the element is considered within the cross section module via uniaxial fiber stress-strain relations. Curvature and axial strain for a given set of internal forces are iteratively determined by cross-sectional integration. In this way, the strain state corresponds exactly to the internal forces. The axial strain and curvature are the basis for the computation of displacements over the segment length by the transfer matrix method. This procedure is contrary to the classic displacementbased method, where the strain state is obtained via the derivations of the displacement shape functions. The reader is referred to Figure 1 for a schematic description of the hierarchical structure of the method. For a detailed discussion see Wallmichrath (2007).