the external and internal forces (support reactions and member forces, respectively) in a system, then the system
is said to be statically indeterminate . A statically determinate system, as against an indeterminate one, is that for
which one can obtain all the support reactions and internal member forces using only the static equilibrium
equations.
The equilibrium equations are described as the necessary and sufficient conditions to maintain the
equilibrium of a body. However, these equations are not always able to provide all the information needed to obtain
the unknown support reactions and internal forces. The number of external supports and internal members in a
system may be more than the number that is required to maintain its equilibrium configuration. Such systems are
known as indeterminate systems and one has to use compatibility conditions and constitutive relations in addition to
equations of equilibrium to solve for the unknown forces in that system.
For an indeterminate system, some support(s) or internal member(s) can be removed without disturbing its
equilibrium. These additional supports and members are known as redundants . A determinate system has the exact
number of supports and internal members that it needs to maintain the equilibrium and no redundants. If a system
has less than required number of supports and internal members to maintain equilibrium, then it is considered
unstable.
An indeterminate system is often described with the number of redundants it posses and this number is known as its
degree of static indeterminacy . Thus, mathematically:
Degree of static indeterminacy = Total number of unknown (external and internal)
forces
- Number of independent equations of equilibrium (1.21)
It is very important to know exactly the number of unknown forces and the number of independent equilibrium
equations. Let us investigate the determinacy/indeterminacy of a few two-dimensional pin-jointed truss systems.
Let m be the number of members in the truss system and n be the number of pin (hinge) joints connecting these
members. Therefore, there will be m number of unknown internal forces (each is a two-force member) and 2 n
numbers of independent joint equilibrium equations ( and for each joint, based on its free
body diagram). If the support reactions involve r unknowns, then:
Total number of unknown forces = m + r
Total number of independent equilibrium equations = 2 n
So, degree of static indeterminacy = ( m + r ) - 2 n.
Sometimes, these two different types of redundancy are treated differently; as internal indeterminacy and
external indeterminacy . Note that a structure can be indeterminate either externally or internally or both externally
and internally.
We can group external and internal forces (and equations) separately, which will help us understand easily the cases
of external and internal indeterminacy. There are r numbers of external unknown forces, which are the support
reactions components. We can treat 3 system equilibrium equations as external equations. This will lead us to:
Degree of external static indeterminacy = r - 3.
The number of internal unknown forces is m and we are left with (2 n -3) equilibrium equations. The 3 system
equilibrium equations used earlier were not independent of joint equilibrium equations, so we are left with (2 n - 3)
equations instead of 2 n numbers of equations. So:
Degree of internal static indeterminacy = m - (2 n - 3).
Please note that the above equations are valid only for two-dimensional pin-jointed truss systems. For example, for
three-dimensional ( “space” ) pin-jointed truss systems, the degree of static indeterminacy is given by ( m + r - 3 n
). Similarly, the expression will be different for systems with rigid (fixed) joints, frame members, etc.
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