In this tutorial we investigate shear stresses induced in a beam by bending.
Analysis of shear bending stress is more complex than normal bending stress and this tutorial provides a basic introduction valid for beams of rectangular cross-section with height significantly greater than breadth.
The diagram below illustrates the origin of shear stress in bending.
![source of shear stress in bending](images/images%20tut%2008/diagram01.jpg)
The diagram shows a beam of rectangular cross-section with height h and width b.
A small element of the beam height y1 above the neutral axis x-x with vertical faces A and B is subject to shearing forces V. The bending moment M varies along the beam and from our previous study we know:
![](images/images%20tut%2008/math01.gif)
Also from our study of normal bending stress we know σ is directly proportional to M.
![](images/images%20tut%2008/math02.gif)
It follows that the mean normal stress on face B must be greater than on face A and thus the net the force FB on face B must be greater than the net force FA on face A.
For the element to be in equilibrium the sum of horizontal forces must equal zero. Thus a horizontal force FC is exerted on the bottom face of the element resulting in a shear stress τ on this face.
![](images/images%20tut%2008/math03.gif)
Note that Iz is the second moment of area of the entire cross-section about the neutral z axis.
![](images/images%20tut%2008/math04.gif)
![](images/images%20tut%2008/math05.gif)
We know that the shear stress on a plane in a biaxial element is accompanied by shear stresses of equal magnitude on the other three orthogonal planes with directions such that the element is in equilibrium. Hence a shear stress of the same magnitude exists on the other faces of the element as shown below.
(In this model it was assumed that V has the same value on both vertical faces of the element, which is true for beams with point loading. For distributed loads, V is not constant but when the length of the element → 0 the expression above for τ is valid.)
![element with shear stresses](images/images%20tut%2008/diagram02.gif)
We now examine the distribution of shear stress on the cross-sectional y-z plane in the case of the rectangular section below.
![Beam cross-section](images/images%20tut%2008/diagram03.jpg)
![](images/images%20tut%2008/math06.gif)
![](images/images%20tut%2008/math07.gif)
Distribution of τ across the section for values of y1 from -c to +c is a quadratic function as shown below. τmax occurs at the neutral axis where y1 = 0 and τ = 0 where y1 = ±c. Note that locations of minimum and maximum shear bending stresses are the reverse of normal bending stresses.
![distribution of shear stress across beam section](images/images%20tut%2008/diagram04.gif)
![](images/images%20tut%2008/math08.gif)
Compare this value with the assumption of uniformly distributed shear stress over the cross-section for shearing force V giving:
![](images/images%20tut%2008/math09.gif)
i.e. τmax is greater by a factor of 3/2. Similar factors for solid circular and hollow circular sections are 4/3 and 2 respectively.
Next: Deflection of beams in bending
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