# 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 of inelastic material that is studied beyond the elastic limit (which can be observed in the stress-strain diagram). The 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 the plastic range. This method of analysis is quite rapid and has a rational approach to the analysis of structure. It controls the economy regarding the 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 a stress-strain curve as perfectly plastic. In this analysis, it is assumed that the 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 the following conditions for plastic analysis

- Mechanism condition
- Equilibrium condition
- Plastic moment condition

### Mechanism condition:

When the ultimate load is reached collapse mechanism is usually formed. The mechanism condition states that a structure must develop a collapse mechanism or a plastic hinge mechanism to achieve its ultimate strength. A collapse mechanism is a specific pattern of plastic deformation that allows the structure to redistribute loads and maintain equilibrium until failure occurs. The formation of plastic hinges at critical locations within the structure is necessary for the development of a collapse mechanism.

### Equilibrium condition:

The equilibrium condition requires that the structure remains in equilibrium at all stages of loading, including the plastic deformation stage. This means that the sum of forces and moments acting on the structure must be balanced at each step of the analysis, even when plastic deformation occurs. The equilibrium condition ensures that the structure can sustain the applied loads without undergoing excessive displacements or failure.

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 (the moment at which plastic hinges form and the structure moves to failure) of the section. The plastic moment condition is based on the concept of plastic hinges. A plastic hinge is a section within a structure where plastic deformation occurs, resulting in a redistribution of moments and forces. The plastic moment condition states that the plastic moments at critical sections, such as beam-to-column connections, must be equal to or greater than the moments that cause yielding in the structural members. In other words, the plastic moment capacity of the critical sections should be greater than or equal to the applied bending moments.

### Plastic moment:

If we consider the case of the simply supported beam when the load is gradually applied to it, the bending moment and stresses increase. As the load is increased, the stresses in the fibers of the 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). A plastic hinge behaves like an ordinary hinge allowing free rotation about itself. The yield moment and plastic moment have a relationship that can be described with the help of the following relation:

My = 2/3 Mp

In the calculation of plastic moments, the term shape factor has its own importance. The shape factor can be defined as the ratio of the plastic moment to yield moment is said to be as the shape factor. Shape factor depend usually on the 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 a body that is elastic (returns to its shape after the load is removed), it will show resistance against deformation, such a body is called to be a structure. On the other hand, if no resistance is shown against the body, then it is known as a mechanism. when plastic hinges equal to n+1 form in the structure, then the structure will collapse(where n is the degree of indeterminacy of structure). It means if the plastic hinges in structures increase in number than 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 a 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 degrees of indeterminacy, its degree of indeterminacy reduces and it becomes a determinate structure if '* n*' number of plastic hinges forms in it.

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