Deriving the equation for Shear Stress for Different cross-sections

[Nexus305]

Hi I know the general equation for shear stress in beams is this:
Shear Stress τ = F/Ib ∫y dA

Depending on the shape of the cross section (ie. If its an I beam, T-beam, hollow square/circle) , after integration we will end up with a different equation. I want to know the exact procedure to do the integration for specific cross sections.
I want to know the derivations for the following cross sections: I-beam, T-beam, rectangular, hollow rectangular,circular and hollow circular).

I really need some help with this because I don't understand this derivation at all
Thank you so much!

 

[Navin A S]

The derivation of the shear stress in beams is based on the concept of shear flow. We can use this concept to derive the equations for shear stress in different cross sections. For an I-beam, we can use the equation: τ = F/2Ib ∫y2 dA where I is the second moment of area of the cross section, b is the width of the beam, and F is the applied force. The integral is then taken over the area of the beam. For a T-beam, we can use the equation: τ = F/2Ib ∫(y2 + x2) dA where I is the second moment of area of the cross section, b is the width of the beam, and F is the applied force. The integral is then taken over the area of the beam. For a rectangular beam, we can use the equation: τ = F/4Ib ∫(y2 + x2) dA where I is the second moment of area of the cross section, b is the width of the beam, and F is the applied force. The integral is then taken over the area of the beam. For a hollow rectangular beam, we can use the equation: τ = F/8Ib ∫(y2 + x2) dA where I is the second moment of area of the cross section, b is the width of the beam, and F is the applied force. The integral is then taken over the area of the beam. For a circular beam, we can use the equation: τ = F/2πIb ∫(y2 + x2) dA where I is the second moment of area of the cross section, b is the radius of the beam, and F is the applied force. The integral is then taken over the area of the beam. For a hollow circular beam, we can use the equation: τ = F/4πIb ∫(y2 + x2) dA where I is the second moment of area of the cross section, b is the radius of the beam, and F is the applied force. The integral is then taken over the area of the beam.