# 3b-Elastic and plastic section moduli for any shape.

Last Updated on June 8, 2024 by Maged kamel

## Elastic and plastic section moduli for any shape

This is the continuation of the previous post, Elastic and plastic section moduli for a rectangle. We will proceed to estimate the section modulus Sx and plastic section modulus Zx for any shape.

### The position of the elastic neutral axis for the case of a singly symmetric shape.

In the next slide, there are two examples of situations where the E.N. axis will not divide the shape equally. These shapes are Wt shape and unequal angle.

What will be the steps to find the neutral axis if we have an unsymmetrical shape? We will indicate this in the next slide.

### Section modulus Sx for any shape.

For the shapes where the E.N.axis will not pass by the middle, we will follow the following steps.

Create a datum line for instance passing by the bottom of the shape. Use any arbitrary axis, this axis will divide Area A into A1 and A2, estimate A1 and A2, and find the total area that is equal to A.

Estimate the distances y1, y2, or the Cg distance for A1 and the Cg of the second A2.

Estimate the E.N axis distance from the bottom datum by dividing the sum of A*y over the area.

The elastic neutral axis can be located at a distance from the bottom datum equals (A1*y1+A2*y2)/(A1+A2)

The same procedure which we have used in the case of the rectangular shape will be used.
1-We estimate the value of y- bar for any area by estimating to get the Y bar for the section, by summing the first moment of areas, considering the datum line is at the bottom, the product of A1y1+A2*y2= At*y bar, since the total area is known, we can get Y-bar.

2- We estimate the inertia about the neutral axis, we have just evaluated at a distance y bar from the datum.
3- we estimate y max value,  which will be the maximum value of (y1,y2).
4- Estimate Sx which is the Ix/ymax.

### How to evaluate Zx, Plastic section modulus for any shape?

For the plastic section modulus Zx. Create a datum line for instance passing by the bottom of the shape.

Use any arbitrary axis, this axis will divide Area A into two equal areas each equal A/2. Estimate the distances y1, y2, or the Cg distance for A1 and the Cg of the second A2 about the external datum

Estimate the P.N axis distance from the bottom datum by dividing the moment of area about the area and get the expression of y bar equals 1/2(y1+y2) where y bar is the P.N axis distance from the external datum.

Estimate y1 bar and Y2 bar about the P.N axis.

To estimate the plastic section modulus Zx. Use the expression of Zx equals A/2*(y1 bar +y2 bar).

For the plastic section modulus Zx of a rectangular shape.

In step -1. Let a datum line pass by the lower base, consider another axis located at a distance of kd from the lower datum line. This axis will have a distance from the upper fiber equals (d-kd).

Step 2: Estimate the area A1. Let A1 be the area enclosed by the upper datum axis and the top side of the rectangle. We can estimate the A1 as equal to (b)*(d-kd)=bd(1-k).

The third step is to estimate the second area or A2, it will equal (b)*(kd). Equate A1 and A2.

Bd(1-k)=b*kd, we can get an expression for k as equal to bd/2bd=1/2.

Thus the P.N axis will be located at kd distance which is d/2. Let y1 cg be the distance between the P.N axis and Cg of A1, it will be equal to d/4, similarly, y2cg equals d/4.

Zx can be equal to bd^2/4.

For more details, please refer to the first part via post 3a.
The next post link: 4-Solved problem 4-3 for the elastic and plastic section.

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