Static Indeterminancy of Structures

To understand the concept behind static indeterminacy, we need to first understand the concept of static equilibrium of the structural system. 

Fig.1.A Concrete Framework having fixed joints- An example of static Indeterminacy in Structures

Static Equilibrium

A body in the state of rest at initial stage under the action of a system of forces is said to be in static equilibrium.  In order to have such a state, it is essential that the resultant of all these forces won't bring a resultant force or torque, that can cause the body to move i.e. the condition of static equilibrium is changed. Hence the only possible condition to have is that the net sum of forces acting on the body at respective directions( i.e. X, Y and Z directions ) has to be equal to zero. If the sum of forces along X direction is ƩFx, the sum of forces along Y direction is ƩFy and the sum of forces along Z directions is ƩFz, Then for the body to stay in static equilibrium the condition to be satisfied is:

ƩFx = 0   ;  ƩFy = 0   ; ƩFz = 0

The above equations are called as equations of static equilibrium of a planar structure that is subjected to a system of forces.

Static Determinacy and Static Indeterminacy

For a structure to be in the state of static equilibrium, it is necessary to satisfy the above-mentioned equilibrium equations. The value of unknown reaction components in the structure can be obtained by solving the equilibrium equations of that structure.

If a stable planar structure possesses three unknown force or reaction components, the solution can be obtained from the set of three equilibrium equations. This state of the structure is static determinacy or the structure is said to be statically determinate.

If there are more than three unknown reactions, the structure cannot be solved by the set of equilibrium equations. This state of the structure is called static indeterminacy or the structure is said to be statically indeterminate. These unknown reactions are called as redundants.

The structure that is statically indeterminate, is indeterminate to an extent equal to the unknown number of reactions that exceed the number of equilibrium equations ie.3.

The redundants can be determined by using additional equations that is obtained from geometrical compatibility.

While analyzing indeterminate structure:

  1. Initially, the redundants are removed from the structure.
  2. This will make the structure determinate, which is called as the cut-back structure
  3. External redundants exist among the external reactions
  4. Internal Redundants remain among the member forces

Note: In addition to these three equilibrium equations, additional equations can also be written where situations like the presence of internal hinge or pins are present in the structure. The condition is that the net bending moment at the hinges is equal to zero.