# Basics and Principles of Plastic Analysis

## Definition:

Plastic analysis is defined as the analysis in which the criterion for the design of structures is the ultimate load. We can define it as the analysis inelastic material is studied beyond the elastic limit (which can be observed in stress strain diagram). Plastic analysis derives from a simple mode failure in which plastic hinges form. Actually the ultimate load is found from the strength of steel in plastic range. This method of analysis is quite rapid and has rational approach for analysis of structure. It controls the economy regarding to weight of steel since the sections required by this method are smaller than those required by the method of elastic analysis. Plastic analysis has its application in the analysis and design of indeterminate structures.

## Basics of Plastic analysis:

Plastic analysis is usually based on the idealization of stress strain curve as perfectly plastic. In this analysis it is assumed that width thickness ratio of plate elements is small so the local buckling does not occur. Broadly speaking the section will be declared as perfectly plastic. Keeping in mind these assumptions, it can be said that section will reach its plastic moment capacity and after that will be subjected to considerable moment at applied moments.

## Principles of Plastic analysis:

There are following conditions for plastic analysis

- Mechanism condition
- Equilibrium condition
- Plastic moment condition

### Mechanism condition:

When the ultimate load is reached collapse mechanism usually formed.

### Equilibrium condition:

Σ FX=0, Σ FY=0, Σ Mxy=0

### Plastic moment condition:

The bending moment at any section in the structure should not be more than the full plastic moment (moment at which plastic hinges form and structure moves to failure) of the section.

### Plastic moment:

If we consider the case of simply supported beam, when the load is gradually applied on it, bending moment and stresses increases. As the load is increased, the stresses in fibers of beam reach to yield stress. At this stage the moment which has converted the stresses into the yield stress is said to be as Plastic moment. it is usually denoted by Mp.at this stage the beam member cannot take up any additional moment but may maintain this moment for some amount of rotation and acts like a plastic hinge(hinge means having no capacity to resist moment). Plastic hinge behaves like an ordinary hinge allowing free rotation about itself. The yield moment and plastic moment has relationship which can be described by help of following relation:

My = 2/3 Mp

In calculation of plastic moments the term shape factor has its own importance. Shape factor can be defined as the ratio of plastic moment to yield moment is said to be as the shape factor. Shape factor depend usually on shape of the cross section.

For rectangular cross section the plastic moment can be calculated as:

Yield stress x (bh2/4)

When the load is applied on the body which is elastic (return to its shape after the load is removed), it will show resistance against deformation, such a body is called to be as structure. On the other hand if no resistance is shown against the body, then it is known as mechanism. when plastic hinges equal to n+1 form in the structure, then the structure will collapse(where n is degree of indeterminacy of structure). It means if the plastic hinges in structures increases in number than the their degree of indeterminacy, structures move towards collapse.

## Plastic hinge and degree of indeterminacy:

Whenever plastic hinge forms in the structure, equilibrium is obtained. As the result the degree of static indeterminacy reduces by one with the formation of one plastic hinge. We can say that if the structure has 'n' number of degree of indeterminacy, its degree of indeterminacy reduces and it becomes determinate structure if '* n*' number of plastic hinges forms in it.

Keywords: Elastic Plastic Analysis, Elasto-Plastic Analysis of structures, Design of Steel Structures