# 19S- Review of shear stresses for steel beams.

## Review Of Shear Stresses for steel beams.

### Review Of Shear Stresses video.

Introduction to shear stress for beams, and proof that both vertical and horizontal shear stresses are equal. This is the link to the video in U tube.

The topics included are a review of the shear stress derivation and how to estimate the shear value for a beam.

### How to derive the expression for shear stress?

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

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

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

Making integration, we get an expression for the stress for shear τ=∫dMydA/(dxt)=V*Q/I*T, where V is the shear.

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

### Horizontal shear stress for beams.

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

It is proven that the value of τh=τv, where τv is the vertical stress due to shear, while τh is the horizontal stress due to shear by multiplying by area and taking moment at the edge.

### The expression for shear stress for a rectangular section.

We are going to substitute in the general equation for stress as τ=VQ/(Ix*t),  then at the neutral axis where the first moment of area V=b*(h/2)*h/4)=b*h^2/8, while Ix=bh^3/12.

At the neutral axis we have  τ= Q*(b*h^2/8)/b(b*h^3/12) =Q/A*(4/3), then τ=1.5* average stress at the neutral axis, the stress distribution is shown in the next slide.

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

This is the pdf file used in the illustration of this post.