STATICALLY INDETERMINATE PROBLEMS

 Statically indeterminate problems.
Members for which reaction forces and internal forces can be found out from static
equilibrium equations alone are called statically determinate members or structures.
Problems requiring deformation equations in addition to static equilibrium equations to
solve for unknown forces are called statically indeterminate problems.


















The reaction force at the support for the bar ABC in figure  can be determined
considering equilibrium equation in the vertical direction.




Now, consider the right side bar MNO in figure 1.22 which is rigidly fixed at both the ends.
From static equilibrium, we get only one equation with two unknown reaction forces R1 and
R2.

Hence, this equilibrium equation should be supplemented with a deflection equation which
was discussed in the preceding section to solve for unknowns.
If the bar MNO is separated from its supports and applied the forces , then
these forces cause the bar to undergo a deflection
R1,R2 and P
δMO that must be equal to zero.

δMN and δNO are the deflections of parts MN and NO respectively in the bar MNO.
Individually these deflections are not zero, but their sum must make it to be zero.
Equation 1.19 is called compatibility equation, which insists that the change in length of the
bar must be compatible with the boundary conditions.
Deflection of parts MN and NO due to load P can be obtained by assuming that the
material is within the elastic limit,

 

Substituting these deflections in equation

Combining both the  equations one can get,

 

From these reaction forces, the stresses acting on any section in the bar can be easily
determined.