Limit Analysis
The concept of a lower bound has been introduced with reference to the work formula approach to analyse deformation. This approach generally results in an underestimate of the required load. Clearly, there also will be an “upper bound”, i.e. an overestimate of the load that needs to be applied to effect a given deformation. The two approaches together are called “limit analysis” since the actual loads required will lie between the lower and upper bounds. In practice, limit analysis is much easier to apply to a problem than the slip-line field approach and can be reasonably accurate. The upper bound is particularly useful for the study of metalworking processes in which it is essential to ensure sufficient forces are applied to cause the required deformation. In contrast, the lower bound is valuable in engineering where failure of a component must be avoided and hence an estimate of the minimum collapse load is needed.
The approach taken for estimating the upper bound is based on suggesting a likely deformation pattern, i.e. lines along which slip would be expected to occur for a given loading situation. Then the rate at which energy is dissipated by shear along these lines can be calculated and equated to the work done by an (unknown) external force. By refining the geometry of the deformation pattern, the minimum upper bound can be determined. Frictional forces can be accommodated in this approach. The approach utilises hodographs, which are self-consistent plots of velocity for different regions within a body being deformed; the different regions are assigned by considering how the overall body will deform for a particular deformation process and their relative velocities are estimated by assuming that the applied external force has unit velocity.
For both upper- and lower-bounds, one of the following two conditions has to be satisfied:
- geometrical compatibility between internal and external displacements or strains. This is usually concerned with kinetic conditions – velocities must be compatible to ensure no gain or loss of material at any point.
- stress equilibrium i.e. the internal stress fields must balance the externally applied stresses (forces).
The basis of limit analysis rests upon two theorems, which can be proved mathematically. In simple terms, these theorems are:
- Lower Bound: any stress system in which the applied forces are just sufficient to cause yielding.
- Upper Bound: Any velocity field that can operate is associated with an upper bound solution.
Example 1: Notched bar in tension.
Example 2: Notched bar in plane bending.