When a structure experiences *static* loads, it can either be statically determinate or indeterminate. The former can be analyzed using static equilibrium equations ONLY, while the latter requires more than those equations.

How do we know if a structure is such? We first introduce two concepts: the Number of Unknowns \(N_u\) and the Number of Known Conditions \(N_k\) of a structure.

### Number of Unknowns

The number of unknowns \(N_u\) refers to the number of reactions and internal forces we need to solve. It is the sum of (1) external reaction components \(r\) and (2) internal force components.

Below is a table of unknowns depending on the type of structure. The variable \(m\) refers to the number of members in the model.

### Number of Known Conditions

The number of known conditions \(N_k\) refers to the number of equations we can use to solve the structure. It is the sum of (1) equilibrium equations and (2) conditional equations due to internal connections \(e_c\).

Below is a table of these conditions depending on the type of structure. The variable \(j\) refers to the number of joints in the model.

## How to Classify Determinacy?

To classify if a structure is determinate or not, we need to compare the number of unknowns \(N_u\) and the number of known conditions \(N_v\):

- If \(N_u = N_k\), the number of unknowns is the same as the number of known equations. The equilibrium principle is sufficient to solve the structure; hence, it is determinate.
- If \(N_u > N_k\), there are more unknowns than known equations. We will need more equations to solve the structure; hence, it is indeterminate.
- If \(N_u < N_k\), there are fewer unknowns than known equations. It violates external stability; hence, the structure is externally unstable.

### Degree of Indeterminacy

If a structure is indeterminate, how many more equations do we need to study its behavior? The degree of indeterminacy \(D_i\) tells us exactly that. It's simply the difference between the unknowns and known equations:

\(D_i = N_u -N_k\)

Again, there are three possible scenarios of \(D_i\). For example,

- If we have a \(D_i\) of two, the number of unknowns is greater than the number of known conditions. We would need two more equations to solve the structure.
- If \(D_i\) is equal to zero, then we won't need additional equations to solve it; hence, we can say that it is determinate.
- Lastly, if \(D_i\) is a negative number, the number of unknowns is less than the number of known conditions. Meaning it would violate the stability of the structure.

We can summarise this example as follows:

- \(D_i > 0\) means the structure is indeterminate; the number tells us how many additional equations we need more
- \(D_i = 0\) means the structure is determinate
- \(D_i < 0\) means the structure is externally unstable

This equation can also be an excellent strategy to determine whether a structure is determinate or indeterminate rather than comparing \(N_u\) and \(N_k\).

## Summary

Statically determinate structures can be analyzed using static equilibrium equations only. Conversely, statically indeterminate structures cannot be analyzed using such equations only.

The number of unknowns \(N_u\) refers to the number of reactions and internal forces of the structure we need to solve.

The number of known conditions \(N_k\) refers to the number of equations we know to solve the structure.

Comparing \(N_u\) and \(N_k\) will give the determinacy \(D\) of the structure.

The degree of indeterminacy \(D_i\) can answer the number of additional equations needed for indeterminate structures and is equal to \(D_i = N_u - N_k\).

The equation \(D_i = N_u - N_k\) can help us tell whether the structure is determinate, indeterminate, or unstable.