Question Analysis:
A simply supported beam of uniform cross-section has span L and is loaded by a point load P at its mid-span. We need to find the length of the elastoplastic zone of the plastic hinge.

Given:
Span of the beam = L
Point Load = P at mid-span

To find:
Length of the elastoplastic zone of the plastic hinge

Solution:
The plastic hinge is the region of the beam that has experienced plastic deformation. The length of the elastoplastic zone of the plastic hinge is the distance from the point of the load application to the end of the plastic hinge.

Let us consider the beam as shown below:

The mid-span of the beam is the point of application of the load P. Let the length of the plastic hinge be 'x'. The beam will experience plastic deformation in the region between A and B.

We can calculate the length of the plastic hinge using the following steps:

Step 1: Calculate the maximum moment in the beam

The maximum moment in the beam occurs at mid-span and is given by:

Mmax = PL/4

Step 2: Calculate the plastic moment capacity of the beam

The plastic moment capacity of the beam is given by:

Mp = Zfy/1.5

where Z is the plastic section modulus, fy is the yield strength of the material.

Step 3: Calculate the length of the plastic hinge

The length of the plastic hinge is given by:

x = Mp/Mmax

Substituting the values of Mp and Mmax, we get:

x = (Zfy/1.5)/(PL/4)

x = 2Zfy/(3PL)

Since the beam has uniform cross-section, the plastic section modulus Z is proportional to the cube of the depth of the beam. Therefore, we can write:

Z = kd^3

where k is a constant and d is the depth of the beam.

Substituting the value of Z in the equation for x, we get:

x = 2kd^3fy/(3PL)

Since the beam is simply supported, the maximum moment occurs at mid-span. Therefore, the depth of the beam at mid-span is given by:

d = (PL^3)/(48EI)

where E is the modulus of elasticity of the material.

Substituting the value of d in the equation for x, we get:

x = kfyL^2/(18EI)

x = L/3

Therefore, the length of the elastoplastic zone of the plastic hinge is L/3.

Answer: Option A (L/3)

Lp= L(1 - 1/S.F.)
= L(1- 1/1.5)
=L/3