Keywords: multilevel analysis, reinforced concrete beams, sectional analysis, stiffness degradation, unloading.

This paper presents a nonlinear multilevel analysis method for reinforced concrete beams whereby the members are divided into sub-elements and sectional analysis is utilised to evaluate load-deflection response and stiffness degradation. The concept of the multilevel analysis method based on substructuring technique [1] as a means of solving a discretized continuum problem which involves a large number of unknowns by breaking it down into a series of lower level problems is introduced.

In the present study, the nonlinear multilevel analysis is applied to the analysis of a reinforced concrete member. It comprises two levels of the nonlinear substructuring technique and involves nonlinear member analysis and nonlinear sectional analysis. The member is divided into a number of sub-elements. The sectional analysis is conducted to evaluate the current stiffness of the sub-elements. Having determined the stiffness of all sub-elements, the condensed member is formulated again and assembled into the global structure for the member analysis.

To obtain the actual load-deflection response of reinforced concrete members, the nonlinear stress-strain relations of concrete and reinforcing steel with unloading-reloading characteristics are taken into account at the sectional level. The rationale behind this is that when a reinforced concrete member is subjected to flexure, at cracking load level, the concrete at the tension side of the critical section starts to crack and the neutral axis moves to certain stable position. The adjacent sections which remain uncracked may unload. On further loading, the concrete in the compression zone of the critical section becomes inelastic. After reaching the peak loading capacity, the concrete in the compression zone gradually deteriorates. This causes the neutral axis depth to keep on increasing such that the distance of the tension reinforcement from the neutral axis continuously decreases and eventually the axial strain in the tension reinforcement starts to decrease and thereby causing strain reversal. Such strain reversal could have significant effects on the post-peak behaviour as well as the displacements of the reinforced concrete member [2].

Examples of reinforced concrete beams are presented to demonstrate the applicability of the nonlinear multilevel analysis. The multilevel analysis method can be extended to the analysis of prestressed concrete beams and time-dependent behaviour of beams and frames.

1

Przemieniecki, J.S., "Matrix structural analysis of substructures", AIAA Journal, 1(1), 138-147, 1963. doi:10.2514/3.1483

2

Kwan, A.K.H., Au, F.T.K., Chau, S.L., "Theoretical study on effect of confinement on flexural ductility of normal and high-strength concrete beams", Magazine of Concrete Research, 56(5), 299-309, 2004. doi:10.1680/macr.56.5.299.36313

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