Rigid body moment–rotation mechanism for reinforced concrete beam hinges

Abstract

Structural engineers have long recognised the importance of the ductility of reinforced concrete members in design, that is the ability of the reinforced concrete member to rotate and consequently: redistribute moments; give prior warning of failure; absorb seismic, blast and impact loads; and control column drift. However, quantifying the rotational behaviour through structural mechanics has been found over a lengthy period of time to be a very complex problem so that empirical solutions have been developed which for a safe design are limited by the bounds of the test parameters from which they were derived. In this paper, a rigid body moment–rotation mechanism is postulated that is based on established shear-friction and partial-interaction research; it is shown to give reasonable correlation with test results as well as incorporating and quantifying the three major limits to rotation of concrete crushing and reinforcing bar fracture and debonding.

Introduction

A reinforced concrete slab that has been subjected to a blast load [8] is shown in Fig. 1. Typically, the slab can be considered to consist of two distinct regions: the small hinge region where concrete crushing is visible, where wide flexural cracks occur, and where most of the permanent rotation is concentrated around the wide flexural cracks so that the trend of the moment distribution has little effect; and the non-hinge region which applies to most of the length so that it is affected by the trend of the moment distribution, where there are much narrower cracks, where, in particular, concrete crushing does not occur and where standard procedures of equilibrium and compatibility can be applied [9], [10], [11], [12].

The importance of the ductility of reinforced concrete members, such as that shown in Fig. 1, has been recognised for over fifty years [1], [7] and there is still much ongoing research on the topic [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. However, there has been a major difficulty in developing a mechanism that can quantify the rotation of the hinge because of the concrete softening shown in the stress profile ${\sigma }_{\mathrm{soft}}$ in Fig. 1[23], [24], [25]. It can be shown [11], [10] that this reduction in the material concrete stress in the softening branch requires a hinge of zero length; this problem of using a concrete compressive stress that reduces with increasing strain was recognised in the sixties by Barnard [26], Barnard and Johnson [27], and also by Wood [28] who referred to the problem as “Some controversial and curious developments in the plastic theory of structures”.

To find a solution to the zero hinge length problem, research has mainly concentrated on quantifying the hinge length empirically with examples shown in chronological order in Table 1. The aim was to find a length of hinge over which the curvature from a full-interaction analysis could be integrated to give the correct rotation. It can be seen that the major hinge length variables in Table 1 are the depth of slab $d$, diameter of the reinforcing bar $d_b$ and yield strength of the reinforcing bar which probably control the rotation of the hinge region in Fig. 1, and the span $z$ or $L$ which denotes the contribution to the rotation in the non-hinge region. No doubt these empirical hinge lengths give good agreement within the bounds of the test results from which they were derived but Panagiotakos and Fardis [6] showed that they have to be used with caution outside their experimental bounds. Recently, a more advanced moment–curvature approach has been developed by Fantilli et al. [29] as shear-friction has been used in simulating the moment–curvature post peak behaviour and, furthermore, as shear-friction has also been used to quantify the hinge length in terms of the dimensions of the softening wedge.

There is a limit to the accuracy in using a combination of moment–curvature with hinge length to determine the rotation. This is because moment–curvature is a measure of the sectional ductility of a beam cross section and not of the member ductility that is the rotation which is required. As an illustration, the moment curvature relationship is generally derived assuming full-interaction between all components of the beam so that there is a uni-linear strain profile. This is fine up to concrete cracking but once a crack forms the crack can only widen, as in Fig. 1, if there is slip between the reinforcing bar and the concrete. This opening shown as $\theta$ concentrates the rotation around the flexural crack [30], [12] which now depends on the bond characteristics between the reinforcing bar and the concrete which moment–curvature relationships cannot cope with. Hence, there is a need for a model that can directly simulate the moment–rotation about this primary crack which is the subject of this paper.

A rigid body rotation mechanism which uses the established shear-friction and partial-interaction techniques is first described and this is then compared with test results to show that it can simulate not only the moment–rotation of a hinge but also the limits imposed on this rotation due to concrete wedge failure, and reinforcing bar fracture or debonding. It needs to be emphasised that the aim of this paper is to illustrate a novel moment–rotation model and that it is recognised that further research is required in quantifying the material properties used in the model.