Shear Center of thing circular section

[Mish4444]

Hi there, i have a problem trying to derive the equation for the shear centre of a thin walled slit circular section of uniform section.

I have attached a photo of the question bellow. it is taken from mechanics of materials. I hope someone can help.

 

[nvn]

Mish4444: You must list relevant equations yourself, and show your work. And then someone might check your math. We are not allowed to give you the relevant equations for your homework. Hint 1: Do you think you should first compute second moment of area, Iz, of the cross section? Do you think Iz will be needed to solve this problem?

 

[Mish4444]

Hi NVN thanks for the reply.

This is where i have got to so far:

 

[nvn]

Mish4444: Some of your current work is rather sloppy, and I cannot read some of it. How do you expect people to read characters if you do not make them legible? Also, the current work is poorly labeled, and is not making much sense yet. What is Z*D in the denominator of tau_D, and why is it there? What is Zo = t? Why did you integrate statical moment of area in the tau_D equation from zero to pi, instead of from zero to alpha? And what does subscript D mean? Hint 2: Do you think you also need to integrate moment from zero to pi? Hint 3: Iz is currently wrong; try again. Can you make your work clearer? And label your parameters more carefully, so we know what you mean? Try it again.

 

[DeeNos]

The shear center of a thin circular section is an important concept in mechanics of materials, as it determines the location where shear forces can be applied without causing any twisting or bending of the section. This is particularly important in structures that experience shear forces, such as beams and columns.

To derive the equation for the shear center of a thin walled slit circular section, we first need to understand the geometry of the section. The section consists of a circular ring with a slit cut along its diameter. This creates two separate sections, each with a rectangular cross section.

To determine the shear center, we can use the principle of superposition. This means that we can consider the two rectangular sections separately and then combine their effects to determine the overall shear center.

First, we consider one of the rectangular sections. We can apply the formula for the shear center of a rectangular section, which is given by:

yc = (Iy1 * y1 + Iy2 * y2) / (Iy1 + Iy2)

Where:
yc = distance from the neutral axis to the shear center
Iy1 and Iy2 = moments of inertia of the two rectangular sections about their respective neutral axes
y1 and y2 = distances from the neutral axis of the section to the centroid of each rectangular section

Next, we consider the second rectangular section and apply the same formula. Once we have the values for yc for each section, we can use the principle of superposition to determine the overall shear center, which is given by:

y = (A1 * yc1 + A2 * yc2) / (A1 + A2)

Where:
y = distance from the neutral axis to the overall shear center
A1 and A2 = areas of the two rectangular sections
yc1 and yc2 = distances from the neutral axis to the shear centers of each rectangular section

By applying this formula, we can determine the location of the shear center for a thin walled slit circular section of uniform section. I hope this helps in solving your problem. If you have any further questions, feel free to reach out.