Mobility of a statically indeterminate structure is

Mobility of Statically Indeterminate Structures

The term mobility, often interchanged with the degree of freedom (DOF), is a fundamental concept in structural analysis. It defines the number of independent parameters required to fully specify the position or configuration of a system or structure. Understanding mobility helps in classifying structures as mechanisms, statically determinate, or statically indeterminate.

Understanding Structural Mobility Classifications

The value of mobility (M) directly indicates the type and behavior of a structure:

- If mobility (M) > 0: The structure is considered a mechanism. This means it possesses more degrees of freedom than constraints, allowing it to move without being subjected to external forces. Such structures are inherently unstable.
- If mobility (M) = 0: The structure is statically determinate and stable. In this case, the structure has precisely the right number of constraints to prevent any movement, and its internal forces and reactions can be determined solely using the equations of static equilibrium.
- If mobility (M) < 0: The structure is classified as statically indeterminate. This condition indicates that the structure has an excess of constraints beyond what is necessary for static equilibrium. It is often referred to as being "over-constrained" or having "redundant constraints."

Statically Indeterminate Structures and Their Mobility

A statically indeterminate structure is a structure where the number of unknown reactions or internal forces exceeds the number of available static equilibrium equations. This excess of constraints leads directly to a negative value for its mobility.

- The presence of redundant constraints means that some of the supports or members provide more restraint than strictly needed to maintain stability. While these extra constraints increase the structure's stiffness and load-carrying capacity, they make the structure indeterminate.
- This redundancy is precisely why the mobility value becomes negative. The absolute value of this negative mobility, $|M|$, directly represents the degree of indeterminacy of the structure. For example, a mobility of $-1$ means the structure is indeterminate to the first degree, $-2$ means it's indeterminate to the second degree, and so forth.
- Therefore, for a structure to be categorized as statically indeterminate, its mobility (M) must be less than or equal to $-1$. This can be expressed as $\text{M}\le -1$.

Conclusion on Mobility for Statically Indeterminate Structures

In summary, the relationship between mobility and structural classification is clear:

- A mechanism exhibits positive mobility ($\text{M}>0$).
- A statically determinate structure has zero mobility ($\text{M}=0$).
- A statically indeterminate structure possesses negative mobility ($\text{M}<0$), specifically $\text{M}\le -1$.

Therefore, the mobility of a statically indeterminate structure is indeed less than or equal to $-1$.