SHEAR STRESS PROBLEM:

Problem 1: Derivation of Shear stress in rectangular crosssection

problem 1.

         Derive an expression for the shear stress distribution in a beam of solid rectangular cross-section transmitting a vertical shear V.
The cross sectional area of the beam is shown in the figure. A longitudinal cut through the beam at a distance y1, from the neutral axis, isolates area klmn. (A1).
Shear stress,
τ=  VQ
        It

 =   V / It ∫     y.dA
               A1
     
  
                    d/2
 =     V / Ib ∫ by dy
                   y1

=  V / 2I{( d / 2 )*( d / 2 )− ( y1 ) *− ( y1 ) }        ---------------------- (1)
 
The shear stress is shown as below

Max Shear Stress occurs at the neutral axis and this can be found by putting y = 0 in the
equation 1.
        
τmax =  Vd *d
                8I
      
     =  3 V
         2 bh
    
     =  3V
          2A