The design of a steel beams can be made considering their ability to yield under the action of bending loads. A fully compact beam cross section develop its plastic bending moment capacity through its entire depth. In this article we will discuss some basics of steel beam design and plastic analysis procedure to compute beam capacity.

## Bending stress in structural steel beams

A random cross-section with a vertical plane of symmetry is used to characterise the moment-rotation properties of a generic cross-section. It is expected that the cross-section in the accompanying figure will be exposed to an increasing bending moment, and changes in stress in this cross-section are shown below.

### 1. Elastic Behavior of steel beams

Stage 1 (Figure b) stresses are all smaller than the material’s yield stress due to the applied moment, which is distributed over the cross-section.

### 2. Yield Moment of steel beams

In step 2 (Figure c), the applied moment is sufficiently high for the material to reach the outermost fibres of the cross-section, which are where it will give. The yield stress is greater than all other stresses in the cross section.

### 3. Elastic-plastic bending of steel beams

In step 3 (Figure d and e), the cross-section is subjected to a moment that is greater than the yield moment. The fibres at the yield stress grew inward toward the beam’s core. Excess rotation of the section occurs as a result of the extra moment being applied, and the moment-rotation curve becomes nonlinear since no stress is greater than the yield stress.

### 4. Bending of steel beams

Since all of the cross-fibers section’s are under yield stress (Figure f), the cross-plastic section’s moment capacity has been attained. It is clear that the neutral axis must experience an infinite strain in order for the whole plastic instant to exist, which is virtually unattainable.

### 5. Strain hardening

The ability to withstand a tiny amount of excess moment is due to the material’s strain hardening. The moment-rotation curve of a typical elastic-plastic material’s cross-section is depicted following figure. According to this hypothetical moment-rotation curve, the cross-section sustains moments linearly up to its plastic moment limit before yielding an unspecified amount in rotation.

## What is plastic hinge

Before we start discussing the plastic analysis process, it is important we also develop an understanding of the plastic. Plastic hinge is that section of beam that may rotate freely once the plastic moment capacity is achieved; in other words, it acts like a hinge. The foundation of plastic analysis is what is known as a “** plastic hinge**.” Stresses at the plastic hinge are constant, although strains and rotations can increase. The stress distribution at the section of plastic hinge is shown in figure of the right.

Read more: How to calculate the design capacity of a steel column? [Step by Step guide]

## Plastic analysis of steel beams

It is carried out using the principle of virtual work which states that:

Total virtual work done by all the forces acting on a system in static equilibrium is zero for a set of infinitesimal virtual displacements from equilibrium

Principal of virtual work by John Bernoulli 1717

The output of the plastic analysis is the plastic moment that will cause the beam to develop a collapse mechanism.

Before we discuss the plastic analysis it is important to understand the collapse mechanism.

### Collapse mechanism of steel beams under bending

Collapse mechanism of a steel beam is a particular arrangement and number of plastic hinges in the beam that once formed will cause the beam to collapse (will make the beam unstable). For example, collapse mechanisms for a steel simply supported steel beam with a point load shown in following figure:

A beam could develop several types of plastic mechanism. The moment that will cause a particular collapse mechanism are calculated using principal of virtual work. That is moment is called plastic moment of Mp. Once Mp is computed for all collapse mechanism, the collapse mechanism with the lowest value of Mp will be the governing one and beam will fail as per that collapse mechanism. Lastly, the Mp value of governing collapse mechanism will be used to design the beam against bending.

### Steps of a plastic analysis

Following are the steps of a plastic analysis.

- Draw all possible collapse mechanism (Hint: Plastic Hinges are usually formed at location of loads and supports). E.g. Following picture shows the collapse mechanism for a beam fixed on one end and simply supported on other)

2.The values of rotations and deflections are calculated using the trigonometry and beam geometry

3. The next step is equate internal and external works. E.g. For first collapse mechanism in this Figure, External work done by loads is loads times the vertical deflections at the location of loads. Internal work done is moment Mn time the total rotations in the plastic hinges (** θ**). Note that ** θ**‘s to be considered for computation of internal work are only for plastic hinges. It should not include the real hinges for example in this case the roller support on the right.

4. Value of Mn is computed for each collapse mechanism

5. Lastly, the minimum value of Mn will be the plastic bending moment, which will be used to design the beam.

Read this to know: How to design a steel beam?