Plastic Analysis of Structures: Theories and Mechanisms
Introduction
Unlike elastic analysis, which assumes materials behave linearly according to Hooke’s Law, plastic analysis evaluates the behavior of steel structures beyond their yield point up to collapse. This method offers a realistic estimation of the ultimate load capacity of redundant frames.
Plastic Hinges and Plastic Moment
When a structural steel member is subjected to bending, the outer fibers yield first. As the bending moment increases, yielding spreads inward until the entire cross-section is plastically deformed. At this point, the section cannot resist any additional moment; it acts as a **plastic hinge**, rotating freely under a constant plastic moment ($M_p$).
Collapse Mechanisms and Theorems
A structure collapse occurs when enough plastic hinges form to turn the structure into a kinematic mechanism. Plastic analysis uses two core theorems to determine the collapse load factor ($lambda_c$):
- Lower Bound (Static) Theorem: A calculated collapse load is safe if a bending moment distribution can be found that is in equilibrium with the loads and does not exceed $M_p$ anywhere.
- Upper Bound (Kinematic) Theorem: A calculated collapse load is unsafe or equal to the true collapse load if it is determined by equating external work done by loads to internal work dissipated at plastic hinges during a virtual displacement.