Last Updated on June 7, 2024 by Maged kamel

## Review Of Shear Stresses for steel beams.

Introduction to shear stress for beams and proof that both vertical and horizontal shear stresses are equal.

If we take two sections at a beam apart by a distance, at elevation =y, there will be two compression forces due to M and M+dM due to the moment difference.

The stress developed from the known formula f=M*y/I will be multiplied by dA, creating two compressive forces. These forces are not equal due to the current difference, which is=dM. To create a balance, a shear force acting horizontally will be created.

As the sum of Fx =0, the shear force is developed to create a balance; this force=shear stress* area=Ï„*dx*b=Ïƒ’*dA-Ïƒ*dA, but (Ïƒ’-Ïƒ)*dA=(dM)*y/Ix, then Ï„*dx*t=dM*y/I.

**How do we derive the expression for shear stress?**

Making integration, we get an expression for the stress for shear Ï„=âˆ«dM*y*dA/(dx**t)=V**Q/I*T, where V is the shear.

Q is the first moment of the area at the point of interest, I is the moment of inertia for the section, and finally,t is the breadth of the section.

**Horizontal shear stress for beams. **

*For the balance of a beam element, the horizontal stress of shear should be accompanied by vertical stress due to shear*, as shown in the next slide.

It is proven that the value of Ï„h=Ï„v, where Ï„v is the vertical stress due to shear, while Ï„h is the horizontal stress due to shear by multiplying by area and taking a moment at the edge.

The next post will contain a solved problem for stress estimation for different shapes.

For more detailed shear in bending, please read link-Shear in Bending

This is a link for the next post, A solved problem for shear stress.