Indeterminacy (Determinate and indeterminate structures)

Hi,  There are the structures which can be easily analysed using the conditions of the equilibrium of the structure. These structures are easy to find their support reactions and the internal forces generated in the different members of these structures.

  • Beams: 

The simply supported beam, beam with one end hinged one with roller support, perfect frames etc are the structures which are determinate. There are structures which have other support conditions, like if the beam has both ends fixed, or both ends hinged then the structure is statically indeterminate.
There are four reaction components if the beam is hinged at both the ends, and the conditions of the equilibrium are only three. So with these three conditions of equilibrium I can only find the 3 support reactions. So the structure is indeterminate.
if, R= total nos. of support reactions
    r = nos. of conditions of equilibrium.
then, the indeterminacy = R-r.
so in case of the fixed ends beam the indeterminacy is 1.

  • Pinned frames:

In case of  frames, there are two kinds of indeterminacy. One is external and another is internal.
In case of the pinned frames, the external indeterminacy is R-r,
 but the internal indeterminacy is found by the formula:
I = m- (2j-r)
I = Internal indeterminacy
j= nos. of joints in the frame,
r= nos. of conditions of equilibrium.

  • Rigid Frames:

In case rigid frames, the external indeterminacy is again R-r but the internal indeterminacy is equal to
I = 3a
a= nos. of the closed areas in the frame structure,
I = Internal indeterminacy.
Note: In any structure if a pin is introduced then it increases the numbers of equilibrium equations by 1. The equation is governed by the fact that the sum of all the moments is zero at the pin.

Thanks for your visit.