Analysis of Collapse Surfaces for Trusses with Hardening and Softening

There are some severe restrictions in the traditional limit analysis. For the load type, the load space is one-dimensional and monotonic by imposing the restriction of uni-directional proportional loading. On the other hand,
only can the constitutive law be perfectly elastic, either
rigid-perfectly plastic or elastic-perfectly plastic. However, the load space is usually high dimensional in true situation, and the hardening/softening behavior prevails in almost all engineering structures. To deal with these problems, it is important to loose the above restrictions.
The greatest advantage of limit analysis is that it can obtain collapse loads directly without giving loading paths.
In the field of limit analysis, the most common approach is applying mathematical programming to calculating the collapse loads of elastoplastic structures, and the problem becomes an optimization problem of maximizing the collapse load. Collapse loads form a collapse surface in high
dimensional load space; we deem a collapse surface to be not merely numerical results of optimization, but an equivalent model of yield surfaces once the structure in equilibrium loses its static indeterminancy and forms mechanisms.
That is, we observe that each piece of a collapse surface represents a collapse mode of the structure and is corresponding to the yield surfaces of those structural members that form a collapse mechanism.Therefore, by using the relationship between the collapse surface and the yield surface in load space, we define the conditions of mechanism vectors which have mathematical and physical meaning. After
searching each mechanism, the model of collapse surface can be constructed, and then we can construct the safety region in load space. Finally, some examples for truss structures with hardening and softening are given to verify this method.