Equilibrium condition
The equilibrium condition is a fundamental requirement in both elastic and plastic analysis. It states that the sum of all forces acting on a structure must be zero, and the sum of all moments about any point must also be zero. This condition ensures that the structure is in a state of static equilibrium, meaning that it is not experiencing any external forces or moments that would cause it to move or rotate.

In elastic analysis, the equilibrium condition is used to determine the internal forces and moments within a structure based on the applied loads and the material properties. The analysis assumes that the structure behaves elastically, meaning that it deforms under the applied loads but returns to its original shape once the loads are removed. The equilibrium condition is used to solve for the internal forces and deformations in the structure.

In plastic analysis, the equilibrium condition is used to determine the collapse load of a structure. Plastic analysis assumes that the structure behaves plastically, meaning that it undergoes significant permanent deformation beyond its elastic limit. The equilibrium condition is used to determine the internal forces and moments at the collapse load, which is the maximum load that the structure can support before it becomes unstable.

Yield condition
The yield condition is a condition that must be satisfied in plastic analysis but not in elastic analysis. It states that the internal forces and moments in a structure must not exceed the yield strength of the material. When the yield condition is satisfied, the structure is considered to be in a state of plastic collapse.

The yield condition is used to determine the collapse load of a structure in plastic analysis. It ensures that the structure does not undergo excessive deformation or failure when subjected to the maximum load. If the internal forces and moments exceed the yield strength of the material, the structure will undergo plastic deformation and may fail.

Plastic moment condition
The plastic moment condition is another condition that must be satisfied in plastic analysis but not in elastic analysis. It states that the internal bending moment in a structural member at the collapse load must be equal to or greater than the plastic moment capacity of the section.

The plastic moment condition is used to determine the collapse load of a structure in plastic analysis. It ensures that the bending moment in the structural member does not exceed its plastic moment capacity, which is the maximum bending moment that the section can resist before it undergoes significant plastic deformation. If the bending moment exceeds the plastic moment capacity, the section will undergo plastic deformation and may fail.

Mechanism condition
The mechanism condition is not required to be satisfied in either elastic or plastic analysis. It is a condition that is used in the design of structures to ensure that they have sufficient redundancy and ductility to withstand unexpected loads or failures. The mechanism condition states that the structure should have multiple load paths and alternative load paths that can be activated in case of failure. This condition is not directly related to the equilibrium, yield, or plastic moment conditions, but it is important for the overall safety and performance of the structure.

In summary, the equilibrium condition is the only condition that is required to be satisfied in both elastic and plastic analysis. The yield condition, plastic moment condition, and mechanism condition are specific to plastic analysis and are not required in elastic analysis.