4-A Solved problem 4-3 for elastic and plastic sections.

Last Updated on February 18, 2026 by Maged kamel

Solved problem 4-3 for elastic and plastic sections.

This is solved example 4.3 from Prof. Alan Williams’s reference manual. We need to estimate three items: Sx, Zx, and the shape factor for Sx. First, we need to evaluate Ybar, Ymax, and Ix. The next photos show the two methods for estimating the Ix value. These methods will lead to the same result.

The first method for Ix estimation.

The first method is to divide the T-section into a flange and a stem. We will estimate the area and the Cg value for each area, and an outer datum is set at the bottom of the stem.

The first moment of area estimates the distance of the elastic neutral axis from the datum line by determining the difference between the CGs of the flanges and stems and the datum line.

The y-bar distance is 14.883″, which is the distance from the lower datum. Now, we will estimate Y1 top and Y2 bottom from the elastic neutral axis to the upper and lower fibers. The total moment of inertia will be estimated for the two areas about the Elastic neutral axis. The steps can be reviewed in the next slide.

The total moment of inertia is 2275.54 in^4. To get the elastic section modulus, we will use the following form: Sx = 153 in^3, which is the first requirement for solving problem 4-3.

The second method for Ix estimation.

This is the second method for estimating the Ix value for the T section. After finding the Ybar distance from the lower datum, we estimate ix for the T section.

The T section can be represented by a big rectangle of width=15 inches and overall depth =20”, from which we can deduct two rectangles of breadth=7.50” and height=17”. As shown on the next slide, we will estimate the moment of inertia of the T-section about the neutral axis. Ix final value=2275.53 inch4, which is the same value as estimated earlier.

The third method for Ix estimation.

There is another way to determine Ix: consider the datum at the lower base of the stem and estimate the moment of inertia for A1 (the flange area) and A2 (the stem area).Y1 is the distance from the CG of the upper flange to the datum line, while Y2 is the the distance from the CG of the stem part to the datum line. There are y1′,Y2′ distances from the Cg of A1 to E.N.A and Cg of A2 to E.N.A

The following slide shows the equations for Ix1 and Ix2.

We will adjust the Ix equation to include Ixcg for area 1, as well as A1*y1 and A2*y2 terms.

Determine the values for the first and second terms of the Ix equation.

Estimate the third term of the Ix equation.

The final Ix value based on the third method.

How to estimate Zx?

Now, for estimating Zx in problem 4-3. The following pictures will show the procedure for Zx, the plastic neutral axis or P.N.A, which divides the area into two equal areas.

Assuming we have y distance from the top fiber to the plastic section axis, we equate the two areas to find y, since the plastic section modulus divides one area into two halves.

The assumption should be that A1=A2=At/2, where At is the total area. The total area is .15*3+17*1.5=70.50 inch2. Half of this area is 0.50*70.50 = 35.25 in^2. The plastic neutral axis cuts the T flange at a distance of 2.35″ from the top. Please refer to the next slide image.

Check the remaining area, which will give At/2 value for the solved problem 4-3

1-Once the location of the plastic neutral axis is finalized, we will determine y1 and y2.

2-Estimate  the  plastic section modulus from the relation Zx=At/2*(Y1+Y2), where (At) is the total area, y₁ is the distance from the plastic neutral axis to the upper fiber, while y₂ is the distance from the plastic neutral axis to the Lower fiber

Estimation of the Shape factor.

3- Estimate the shape factor by dividing Zx/Sx, Zx value=278 inch4, Sx=153 inch4, and the shape factor=278/153=1.817.

The PDF containing this post’s content can be reviewed and downloaded from the following link.

Here is the link to Chapter 8, “Bending Members.” A Beginner’s Guide to the Steel Construction Manual, 14^th ed.

Here is the link to Chapter 8, “Bending Members.” A Beginner’s Guide to the Steel Construction Manual, 15^th ed.

Here is the link to Chapter 8, “Bending Members.” A Beginner’s Guide to the Steel Construction Manual, 16^th ed.

This is a link for the next post,5-Solved problem 5-1 for Sx&Zx.