Bounding theorem

Strictly, shear occurs not just on the shear planes we have drawn, but on a myriad of 45° planes near the indenter. If our assumed geometry for slip is wrong it can be shown rigorously by a theorem called the upper-bound theorem that the value we get for F at yield - the so-called limit load - is always on the high side.)... 

Equation (15.48) governs the performance of the completely segregated reactor, and Equation (15.49) governs the maximum mixedness reactor. These reactors represent extremes in the kind of mixing that can occur between molecules that have different ages. Do they also represent extremes of performance as measured by conversion or selectivity The bounding theorem provides a partial answer ... 

Apply the bounding theorem to the reversible, second-order reaction... 

Another way of stating the upper bound theorem is that during collapse, the work done by the external loads is equal to the work dissipated by the internal loads. The lower bound theorem states that if any internal stress distribution can be found which does not exceed the )deld stress at any point, and which is in equilibrium with the applied sustained loading, then the structure does not collapse. 

Static or lower bound theorem If a load presents a magnitude such that the stress state satisfies the equilibrium and yield conditimis, then the structure will not collapse and the load factor (5i) is less than or equal to failure load factor (5p). In this static approach the failure load factor is determined by searching for the maximum load factor. 

In the static approach the resultant of the compressive stresses is plotted at each cross section of the structure. The line containing aU the positions of the compressive stress resultants is called by thrust line (Fig. 4a). The thrust line can be determined by analytical methods or graphical methods, such as funicular polygon (Kooharian 1952). The stmcture is safe when the thrust line is located totally inside of the geometry of structure and the equitibrium with the external loads can be found. In this conditions and according to the lower bound theorem, the applied load is less... 

Masonry Macro-block Analysis, Fig. 4 Application of the limit analysis to a masonry arch (a) static approach (lower bound theorem) (b) kinematic approach (upper bound thetHem) (c) uniqueness theorem... 

With regard to kinematic approach, when the structure presents an enough number of hinges to transform it into a mechanism (Fig. 4b), the load factor obtained by equating the work of the external loads to zero corresponds to an upper bound limit (upper bound theorem). The load obtained from the kinematic approach is greater than or equal to the failure load. Different positions for the hinges on the structure can be adopted, and several mechanisms, and respective load factors, can be found. The mechanism that presents the lowest load factor corresponds to the collapse mechanism. Thus, the kinematic approach presents also several solutions. 

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