Ultimate limit state of pre-stressed reinforced concrete elements

During the last decades the general theory of pre-stressed concrete elements is well developed, but at the same time the ultimate limit state (ULS) of these elements is not enough investigated like their service limit state, which presently forms a basis for design concepts. The problem of pre-stressed concrete elements in ultimate limit state is rather complicated compared to ordinary concrete. This is because the pre-stressed concrete theory provides no prior indication if the section will behave as over- or under-reinforced one. Formation of a plastic hinge in its classic interpretation is practically impossible for pre-stressed concrete elements because development of plastic deformations can begin in pre-stressed steel or in compressed concrete, but not simultaneously. The present study is focused on two cases of stress-strain state of a pre-stressed element in ULS: under-reinforced section (URS) and over-reinforced one (ORS). The first takes place because part of pre-stressed steel carries the pre-stress force as an external load. The failure initiates from the pre-stressed steel yielding, hence the deformations' equation, based on strain compatibility, can be used to solve the ULS problem. In the second case (ORS) the section failure initiates at the compressed concrete, whereas the stresses in the pre-stressed steel are unknown. Therefore it is impossible to use the deformations' equation and the problem has no closed solution. Original Iskhakov's diagrams are proposed for finding ultimate stresses and strains values in concrete and steel for URS and ORS cases. As a result, two bending moment limit values are obtained and their minimum corresponds to the real bearing capacity of the element.