The lower-bound and upper-bound collapse load theorems are fundamental concepts in structural engineering, particularly in the field of plastic analysis. These theorems help determine the collapse load of a structure – the load at which a structure will fail due to plastic hinge formation. Here's a simple explanation of each:

** Lower bound collapse load** is the minimum load that causes equilibrium to the structure and just causes any section to yielding (just before yielding). This is the load we can guarantee that the structure will not collapse. The lower bound theorem is a conservative method for the safety design of structure.

**is the maximum load that causes the structure to form any collapse mechanism by plastic (ductile) deformation. This is the load the structure can't reach in real practice. The upper bound theorem is the most efficient method for structural design using plastic analysis.**

__Upper bound collapse load__Both theorems work together to bracket the actual collapse load. The lower bound gives a safe estimate (the load at which the structure is guaranteed not to fail), and the upper bound gives the maximum possible load (beyond which the structure will definitely fail). By applying both theorems, engineers can find a range within which the true collapse load lies, ensuring both safety and efficiency in structural design. Here is an example of a simply supported beam subjected to a concentrated load at midspan.

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