ABSTRACT
A refinement of the model proposed in 1981 and 1984 by the second author, is presented in this chapter. Compliance and stress-intensification concepts are used again. However, in this case, the reinforcement reactions are applied directly to the crack surfaces and not as closing forces acting at infinity. Moreover, the congruence condition is locally imposed on the crack opening displacement in correspondence with the reinforcement, and not globally on the cross-section rotation. The theoretical results confirm a transition from ductile to brittle collapse by varying a non-dimensional brittleness number defined in previous contributions. With the present model, yielding or slippage of reinforcement can precede or follow crack propagation in concrete, removing the assumption of steel yielding at incipient fracture. Such a theoretical approach appears to be very useful for estimating the minimum amount of reinforcement for members in flexure, assuming simultaneous concrete cracking and steel yielding (transitional condition). A physically similar flexural behaviour is obtained in cases where the brittleness number is the same. The failure mechanism changes completely when the beam depth is varied, the steel percentage remaining the same. Only when the steel percentage is inversely proportional to the square root of the beam depth, is the mechanical behaviour reproduced. The experimental results show that the transitional value of the brittleness number is tendentially constant 348 for each concrete grade, by varying the beam depth. While the minimum steel percentage provided by Eurocode 2 and ACI are independent of the beam depth, the relationship established by the brittleness number calls for decreasing values with increasing beam depths.
