Affinity properties of truss model nodes

Abstract

In the application of lower bound theory to reinforced concrete beams, strut and tie models are trusses that consist of axial force members, which react on one another and on external supports through nodes. In a line diagram of struts and ties, they intersect in points, which ensure that moment equilibrium is satisfied at the nodes. The paper examines the implications of replacing the line diagram with members that are of full width – that is compression members at compression yield and anchorages for tension members which act on concrete at compression yield. The nodes will be in hydrostatic compression; for nodes where more than three forces meet, however, the resulting detail becomes unnecessarily complicated. The paper demonstrates that if such a node is chosen as a shape which is affine to the vector diagram of the forces that act on it, the forces will not be concurrent but moment equilibrium will nonetheless be satisfied. Although the general multimember node would be unstable, in the practical case of the support point of a simply supported beam, the constraints of the location of external reactions and of the reinforcement detailing are shown to reduce the analysis of the node to a determinate problem.