Question

a channel and a plate are welded together as shown to form a section that is symmetrical with respect to the y axis. determine the moments of inertia of the combined section with respect to its centroidal

Answer 1

the total moment of inertia of the combined section is:

`Itotal = Ichannelcentroidal + Iplatecentroidal`

To determine the moments of inertia of the combined section with respect to its centroidal y-axis, we can break this problem into two parts: the channel and the plate. Then, we can use the parallel axis theorem to find the moments of inertia about the centroidal y-axis for each part and then combine them to get the total moment of inertia.

Step 1: Calculate the moment of inertia for the channel part about its centroidal y-axis.

Let's assume the channel has the following dimensions:

- Width of the channel (b): Given value in meters.

- Height of the channel (h): Given value in meters.

- Thickness of the channel web (t): Given value in meters.

The moment of inertia of the channel about its centroidal y-axis can be calculated using the formula for a rectangular section:

`Ichannel = (1/12) * b * h^3`

Step 2: Calculate the moment of inertia for the plate part about its centroidal y-axis.

Let's assume the plate has the following dimensions:

- Width of the plate (b_plate): Given value in meters.

- Height of the plate (h_plate): Given value in meters.

The moment of inertia of the plate about its centroidal y-axis can be calculated using the formula for a rectangle:

`Iplate = (1/12) * bplate * hplate^3`

Step 3: Use the parallel axis theorem to find the moments of inertia of each part about the combined section's centroidal y-axis.

The parallel axis theorem states that the moment of inertia about an axis parallel to and a distance "d" away from the centroidal axis can be found by adding the moment of inertia about the centroidal axis and the product of the area and the square of the distance "d."

For the channel:

Let d_channel be the distance between the centroid of the channel and the combined section's centroidal y-axis. It will be half of the channel's width.

d_channel = b/2

`Ichannelcentroidal = Ichannel + (Achannel * dchannel^2)`

For the plate:

Let d_plate be the distance between the centroid of the plate and the combined section's centroidal y-axis. It will be half of the plate's height.

d_plate = h_plate/2

`Iplatecentroidal = Iplate + (Aplate * dplate^2)`

Step 4: Add the moments of inertia of both parts to get the total moment of inertia of the combined section about its centroidal y-axis:

`Itotal = Ichannelcentroidal + Iplatecentroidal`