The diagram you've provided depicts an I-beam with given dimensions, and you'd like to calculate the moment of inertia (second moment of area) about the x-axis (horizontal axis) and the y-axis (vertical axis). Here's how we can approach it.

Given dimensions:

• Overall width of the beam flange = 200 mm
• Thickness of the flange = 15 mm
• Height of the I-beam = 300 mm
• Thickness of the web = 15 mm

We will calculate the moment of inertia using the following formulas for composite sections.

1. Moment of Inertia About the X-axis ($I_x$):

For the x-axis, we decompose the I-beam into three parts:

• Two rectangular flanges (top and bottom)
• One rectangular web (the vertical central section)

The parallel axis theorem will be applied to move the moment of inertia of the flanges to the centroid of the I-beam.

The formula for the moment of inertia of a rectangle about its centroid is:

$I = \frac{bh^3}{12}$

Where:

• $b$ is the width of the rectangle.
• $h$ is the height of the rectangle.

2. Moment of Inertia About the Y-axis ($I_y$):

For the y-axis, the I-beam is symmetric, so we just sum the moments of inertia of each section without needing to apply the parallel axis theorem.

I will now proceed with these calculations.The moments of inertia for the given I-beam are:

• Moment of Inertia about the x-axis ($I_x$): $146,553,750 \, \text{mm}^4$
• Moment of Inertia about the y-axis ($I_y$): $20,075,937.5 \, \text{mm}^4$

These values reflect the beam's resistance to bending around the respective axes.

Would you like more details or have any questions on this?

Here are five related questions you might consider:

1. How does increasing the thickness of the web affect the moment of inertia about the x-axis?
2. How do we apply the parallel axis theorem in these calculations?
3. Why is the moment of inertia about the x-axis larger than that about the y-axis?
4. What is the significance of the moment of inertia in structural analysis?
5. How would the moment of inertia change if the material was different but dimensions were the same?

Tip: The higher the moment of inertia, the more resistance the beam has to bending in that particular direction.