BENDING IN BEAM

    3. BEAMS: STRAIN, STRESS, DEFLECTIONS
        The beam, or flexural member, is frequently encountered in structures and
machines, and its elementary stress analysis constitutes one of the more interesting facets
of mechanics of materials. A beam is a member subjected to loads applied transverse to
the long dimension, causing the member to bend. For example, a simply-supported beam
loaded at its third-points will deform into the exaggerated bent shape shown in Fig. 3.1
       
        Cantilever beams and simple beams have two reactions (two forces or one force
and a couple) and these reactions can be obtained from a free-body diagram of the beam
by applying the equations of equilibrium.          Such beams are said to be statically
determinate since the reactions can be obtained from the equations of equilibrium.
Continuous and other beams with only transverse loads, with more than two reaction
components are called statically indeterminate since there are not enough equations of
 equilibrium to determine the reactions.

               Figure 3.1 Example of a bent beam (loaded at its third points)
   

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