(a) Determine the coordinates of the centroid (𝑥̅,𝑦ത) of the cross-sectional area of the concrete beam in the figure below. (b) What is the moment of inertia of the beam’s cross sectional area about the x-axis?

To find the centroid and the moment of inertia of a concrete beam's cross-sectional area, we need to follow a systematic approach. Let's break this down into two parts: first, determining the coordinates of the centroid, and then calculating the moment of inertia about the x-axis.

Finding the Centroid Coordinates (𝑥̅, 𝑦̅)

The centroid of a cross-sectional area is the point where the area can be considered to be concentrated. For composite shapes, we can use the formula:

• 𝑥̅ = (Σ(Aᵢ * 𝑥ᵢ)) / ΣAᵢ
• 𝑦̅ = (Σ(Aᵢ * 𝑦ᵢ)) / ΣAᵢ

Here, Aᵢ is the area of each individual section, and (𝑥ᵢ, 𝑦ᵢ) are the coordinates of the centroid of each section. To apply this, we need to know the dimensions of the beam and the locations of the centroids of each section.

Steps to Calculate the Centroid

1. **Divide the cross-section into simpler shapes**: For example, if the beam has a rectangular shape, you can break it down into rectangles or triangles.

2. **Calculate the area of each section**: For a rectangle, the area is width times height.

3. **Determine the centroid of each section**: For a rectangle, the centroid is located at its geometric center.

4. **Apply the centroid formulas**: Substitute the areas and centroids into the formulas for 𝑥̅ and 𝑦̅.

Calculating the Moment of Inertia about the x-axis

The moment of inertia (I) about the x-axis is a measure of an object's resistance to bending or flexural deformation. For composite areas, the moment of inertia can be calculated using the parallel axis theorem:

• I = Σ(Iᵢ + Aᵢ * d²)

Where:

• Iᵢ is the moment of inertia of each section about its own centroidal axis.
• Aᵢ is the area of each section.
• d is the distance from the centroid of the section to the x-axis of the entire shape.

Steps to Calculate the Moment of Inertia

1. **Find the moment of inertia for each section**: Use standard formulas for rectangles, triangles, etc. For a rectangle, I about its centroidal axis is given by:

Iᵢ = (b * h³) / 12

2. **Calculate the distance d**: This is the vertical distance from the centroid of each section to the overall centroid of the beam.

3. **Substitute into the moment of inertia formula**: Add the contributions from each section to get the total moment of inertia about the x-axis.

By following these steps, you can accurately determine both the centroid coordinates and the moment of inertia for the concrete beam's cross-section. If you have specific dimensions or a diagram, we can work through the calculations together for a more concrete example!