6A-Practice Problem 5-2-2: Find y-bar, Zx, and Zy.

Last Updated on October 19, 2026 by Maged kamel

How can we estimate y-bar, Zx, and Zy for an I-shape symmetric only about the Y-axis? That is the Topic of this Post.

Practice Problem 5-2-2: Find y bar, Zx, and Zy for the unsymmetric section.

There is a video for this Post.

Timestamps for the video.

• 00:00 Introduction to practice Problem 5-2- 2.
• 01:40 Estimation of AT/2. 02:15 Find the vertical distance y1 from P.N.A to the soffit of the upper Flange.
• 02:41 Determine the y-bar distance from the top of the Flange to the P.N.axis.
• 03:40 Start the estimate for the Plastic Moment, Mp.
• 06:04 The Value of Mp.
• 06:54 Start the estimate for the Zy.

Practice Problem-5-2-2 from the Steel Design Handbook.

An unsymmetric flexural member consists of three parts. The upper Flange is 12×0.50 inches, the Web part is 3/8″ x”16”, “nd the lower Flange is 7×1/2”. “here are three requirements: a), b), and c).
For Part a), determine the y-bar, or the distance from the top of the upper Flange to the plastic neutral axis.

First, we estimate the total Area of the three parts: the upper Flange has an Area of 12 x 1/2, the Web has an Area of 16 x 3/8, and the lower Flange has an Area of 7 x 1/2. The total Area is 15.50 in^2.

First, we estimate the total Area of the three parts: the upper Flange has an Area of 12 x 1/2, the Web has an Area of 16 x 3/8, and the lower Flange has an Area of 7 x 1/2. The total Area is 15.50 in^2.

The distance Y-bar from the top of the shape to the P.N. axis

On the next slide, since the Area of the top Flange is less than At/2, the plastic neutral axis cannot cross the upper Flange; it is located below the upper Flange by a distance y1.

After checking, we can find y1 by equating the difference between At/2 and the upper Flange Area to (3/8) y1. y1 will be equal to 4.667 inches.

In the third slide, we will estimate the y-bar distance, which is the distance from the top of the upper Flange to the P.N. axis.

If we add ½ inch to y1, y-bar will be 5.166 inches, or 5.17 inches rounded. We could check the Area of the lower portion to ensure it equals At/2. 

On the next slide, as required by item b), find the Plastic Moment for A572 grade 50 steel. The yield stress Fy equals 50 ksi. We estimate the compression force C as At/2*Fy = 7.75*50 = 387.50 kips.

The distance C1 from the compression force to the P.N. axis.

We will estimate the C1 distance, which is the distance from the Point of application of force c to the P.N.A. We can estimate c1 by taking the first Moment of Area for the upper half of the section about the P.N.A.The Moment of Area equals 33.609 inch3.

When we divide the first Moment of Area by At/2, we find that C1 equals 4.34 inches.

The distance T1 from the Tension force to the P.N. axis.

We move to the next slide to estimate T1, the distance from the Point of application of Force T to the P.N.A. Since there is no symmetry, we expect a Value different from c1.
Again, we will estimate the Moment of Area of the lower half of the section, which is the sum of the Moment of Area of the lower Flange and the Web Area part below the P.N. axis. T1 equals 8.34 inches.

 The next slide shows the values of the C and T forces and the two distances, c1 and T1. Mp equals C*(C1+T1), or C*yct, the vertical distance between forces C and T.

The Plastic Moment about the P-N axis- part b.

The Mp Value equals (387.50 x 12.6868); we divide by 12 to find Mp in Ft-kips. The Mp Value is 409.46 Ft-kips, which meets the requirement for item b).

The Value of the section Modulus about the minor axis y.

Due to symmetry about axis y, we will treat the C-shaped section, find the first Moment of Area about the y-axis, and divide by the sum of the three shapes. We will find that X-bar equals 1.592 inches.

Finally, we will estimate the Yielding Moment My (C*2x bar) and divide by Fy. The final Value of the plastic section Modulus about Y, Zy, equals 24.676 in^3.

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The PDF for the data of this Post can be reviewed or downloaded.

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, “Beaming Members.” A Beginner’s Guide to the Steel Construction Manual, 16^th ed.

The next Post, Post 6b, is Practice Problem 5-2-3: verify Zx for W18x50.

The next Post is “A Guide to Local Buckling Parameters for Steel Beams.