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 of 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=τ*dx*b=σ’*dA-σ*dA, but (σ’-σ)*dA (dM)*y/Ix, then τ*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 τ=∫dMydA/(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 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 a moment at the edge.
The following 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 links to Boston University’s Mechanical Engineering.
This is a link to the next post, ‘A Solved Problem for Shear Stress.’