Deformation of a Circular Bar (Statically Indeterminate)

 

Example:  Bar  ABCD  shown below is clamped at both ends and subjected to applied loads
P and 2P.    The bar has cross-sectional area, A and modulus of elasticity is E.  Find the
restraint loads at each end and the maximum normal stress in the bar.

                                                             
                             

 

Strategy:  Use a free body diagram to identify the forces of constraint at A and at D.  Write
the equation of equilibrium.   So more information is needed since there are two unknown

support reactions.  Next apply the load-deflection relation given by  ∆i = FiLi/AiEi .  Apply
the geometrical relation that the net deflection is zero.  Solve for the unknown reactions.

Finally determine the axial force in each section of the bar.  Use it to calculate the normal

Stress in each section as force per unit area.

 

The FBD is as follows:

                                  

 

          →ΣFx = 0    – FA – P  + 2P  +  FD  = 0   and for the data:         FA  –  FD  =  P       (1)

 

Let   ∆  denote axial deflection.  Now the sum of the relative deflections must be zero as

follows:                        ∆B/A  +   ∆C/B  +   ∆  D/C  =  0

 

The axial forces in each section are  F1,  F2,  and  F3  as depicted in the figure below.

 

                                       

 

For A to B:  F1  =  FA,        For B to C:  F2  =  FA + P,       For C to D:  F3  =  FA - P

 

 

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