We have in detail dealt about Static Indeterminacy and its related terminologies. Now we will discuss how to find out the static indeterminacy of the structures that are a plane, or in space or pin jointed.In the case of a structure that is having static indeterminacies, it is essential to bring extra compatibility conditions so that additional equations can be derived in order to solve for unknowns that are greater than 3 ( Greater than a number of equilibrium equations).
Degree of Static Indeterminacies
Now the extra number of equations that is necessary other than the equations that can be obtained from the equilibrium equations ( ƩFx = 0; ƩFy = 0; ƩFz = 0 ), are called as the degree of indeterminacies. The initial step in an analysis of an indeterminate structure is by determining the static indeterminacy of the structure. There are two determinacies,
- External Indeterminacy
- Internal Indeterminacy
The external indeterminacy shows the number of extra supports that are provided on the structure than needed. The internal indeterminacy shows the number of additional members that added to the structure, then the required amount.
The total static determinacy can be given as Ds, which is given as:
Ds = Dse + Dsi eq.(1)
Here the external indeterminacy is given by Dse and the internal indeterminacy is given by Dsi. The Dse is related to the support system of the structure and the Dsi is related to the internal redundancy of the structure.
Calculation of Static Indeterminacies
When we go through the analysis of structures, we encounter with plane structures and space structures.
External Indeterminacy calculation Dse
The plane frames have 3 sets of equilibrium equations, they are
ƩFx = 0 ; ƩFy = 0 ; ƩFz = 0
when it comes to space structures, there are 6 set of equilibrium equations, they are
ƩFx = 0 ; ƩFy = 0 ; ƩFz = 0; ƩMx = 0; ƩMy = 0; ƩMz = 0 ;
Now when the number of external reaction in a structure is greater than the above-mentioned number of equilibrium equations, then the structure is externally indeterminacy. The degree of external indeterminacy Dse is
- For plane structure with 3 set of equilibrium equations
Dse = r - 3 ;
- For space structures with 6 set of equilibrium equations,
Dse = r - 6 ;
Internal Indeterminacy Calculation Dsi
If the number of members in the structure is represented by 'm' and 'j' represents the number of joints in the structure, the condition for internal indeterminacy
- For plane- frame is given by
Dsi = m - (2j - 3)
- For space frames, is given by
Dsi = m - (3j - 6)
Final Static Indeterminacy Ds
The static indeterminacy is the sum of external and the internal indetermiancy, i.e. from eq(1).
- For plane frame structure,
Ds = (m + r ) - 2j
- For Space frames,
Ds = (m + r ) -3j