Understanding Plastic Modulus for Rectangular Beams

The plastic modulus, often denoted as $Z_p$, is a fundamental property of a cross-section used in structural engineering. It quantifies the resistance of a beam to bending when the material yields beyond its elastic limit, reaching a plastic state. For a symmetric cross-section like a rectangle, the plastic neutral axis (PNA) passes through the centroid, dividing the section into two equal areas. The plastic modulus represents the first moment of area of one half of the cross-section about the PNA.

Formula for Rectangular Beam Plastic Modulus

The formula to calculate the plastic modulus ($Z_p$) for a rectangular cross-section with width $b$ and depth $d$ is given by:

$$Z_p = \frac{b d^2}{4}$$

In this formula, '$b$' represents the width of the rectangular beam, and '$d$' represents the depth of the rectangular beam.

Given Beam Dimensions

The problem provides the following dimensions for the rectangular beam:

• Width, $b = 200$ mm
• Depth, $d = 400$ mm

Step-by-Step Calculation of Plastic Modulus

We can calculate the plastic modulus by substituting the given dimensions into the formula. Follow these steps:

1. State the plastic modulus formula:

   $$Z_p = \frac{b d^2}{4}$$

2. Substitute the given values for the width ($b$) and depth ($d$):

   $$Z_p = \frac{(200 \text{ mm}) \times (400 \text{ mm})^2}{4}$$

3. Calculate the square of the depth:

   $$(400 \text{ mm})^2 = 160000 \text{ mm}^2$$

4. Multiply the width by the squared depth:

   $$200 \text{ mm} \times 160000 \text{ mm}^2 = 32000000 \text{ mm}^3$$

5. Divide the result by 4 to find the plastic modulus:

   $$Z_p = \frac{32000000 \text{ mm}^3}{4}$$

   $$Z_p = 8000000 \text{ mm}^3$$

6. Express the final answer in standard scientific notation:

   $$Z_p = 8 \times 10^6 \text{ mm}^3$$

Final Result

The calculated plastic modulus for the specified rectangular beam is $8 \times 10^6$ mm³. This value is essential for determining the ultimate bending strength of the beam in plastic design scenarios.