Limit Analysis
Introduction
When designing an engineering structure or a member of that structure, structural stiffness, and structural strength need to be considered. Structural stiffness is required in order to satisfy a serviceability limit state (SLS) requirement, e.g., the maximum in service deflection needs to be acceptable. Since most structures remain elastic during normal operation, structural stiffness is assessed through a linear elastic analysis. Structural strength is required in order to satisfy an ultimate limit state (ULS) requirement, e.g., that the structure can remain functional during and following an extreme event such as an earthquake.
In assessing the strength of a structure, the engineer generally needs to identify collapse mechanisms where structural stability is lost and then to rank these appropriately. One method of ranking is to use the idea of a load factor which expresses the load at collapse divided by the design or operating load. The collapse mechanism with the lowest load factor is then the critical mechanism.
In structural engineering, ductile materials, that exhibit the ability to yield and therefore fail in a gradual and observable fashion, are generally favoured over brittle materials which may fail with little or no warning. Note, however, even when a ductile material is used, the failure mode might appear with little warning and might be deemed as a brittle failure mode, e.g., shear failure of a reinforced concrete (RC) beam or slab.
For ductile structures or members, one of the most important modes of collapse is that of plastic collapse. This mode of failure involves the gradual degradation of structural integrity, with increasing load, through the progressive development of a mechanism whereby the structure ceases to retain any structural integrity and therefore becomes a mechanism in the sense that no further load can be taken. Other failure modes that need to be considered, particularly is a structural member is slender and takes loads that lead to compressive stresses, is that of buckling.
Most materials used in structural engineering offer some degree of ductility. Structural steels, for example, have stress/strain curves that are, post yield, relatively flat and extend some considerable distance along the strain axis prior to failure. The ductile nature of materials used in structural engineering thus allows structural engineers to make use of ideas laid down in the plasticity theorems to place a rational and sensible load factor on plastic collapse. In accepting this load factor as a minimum it is clear that other modes of failure, e.g., buckling need also to be considered.
Thus, the determination of the plastic limit load becomes an important factor in structural design. This load factor will be greater than the elastic limit load which expresses the load at which the highest stressed point in the structure reaches the yield stress. It should also be noted that the plastic collapse load will, because of a structural material's ability to strain harden, be greater than the plastic limit load, i.e., the plastic limit load is generally a conservative prediction of the plastic collapse load.
There are a number of ways in which the plastic limit load may be determined. The basic assumption is that the material exhibits small elastic strains and that behaviour beyond yield is perfectly plastic. As such, in limit analysis, a rigid-perfectly plastic material model is assumed. Stresses are assumed to be limited by the yield stress with corresponding strains allowed to be unlimited. Clearly, for a real material, this is not appropriate as there will be some limiting strain beyond which the material looses any capacity to take stress.
Project Details
This is a longstanding project for RMA and involves the development of EFE for the limit analysis of reinforced concrete slabs and metallic plates.
