Stability and Determinacy of Beams, Trusses, and Frames

To determine whether a structure is stable and statically determinate, we compare the number of unknowns with the number of available equilibrium equations. This helps identify whether:

Both sides equal: the structure is stable and determinate
Left side > right side: the structure is statically indeterminate (degree of indeterminacy = difference)
Right side > left side: the structure is unstable

1. Beams

For beams, we compare:

$$ r = 3n - e_c $$

Or rearranged for checking:

$$ r + e_c \quad \text{vs.} \quad 3n $$

Where:

$r$ = number of external reactions
$n$ = number of connected members (usually 1 for single-span beams)
$e_c$ = equations of condition (1 for internal hinge, 2 for internal roller, etc.)

If $r + e_c = 3n$, the beam is stable and determinate.
If $r + e_c > 3n$, the beam is indeterminate.
If $r + e_c < 3n$, the beam is unstable.

2. Trusses

For planar trusses, use the standard comparison:

$$ m + r \quad \text{vs.} \quad 2j $$

Where:

$m$ = number of members
$r$ = number of external reactions
$j$ = number of joints

If $m + r = 2j$, the truss is stable and determinate.
If $m + r > 2j$, the truss is statically indeterminate.
If $m + r < 2j$, the truss is unstable.

3. Frames

For rigid frames, the check is:

$$ 3b + r \quad \text{vs.} \quad 3j + c $$

Where:

$b$ = number of members (bars)
$r$ = number of external reactions
$j$ = number of joints
$c$ = internal constraints (e.g., internal hinges or known displacements)

If $3b + r = 3j + c$, the frame is stable and determinate.
If $3b + r > 3j + c$, the frame is statically indeterminate.
If $3b + r < 3j + c$, the frame is unstable.

Geometric Instability

Even if a structure satisfies the basic equation for determinacy, it may still be unstable due to its geometry. This is referred to as geometric instability, and it typically arises from poor support layout or improper member arrangement.

A structure that appears statically determinate may still be unstable if it cannot resist motion due to geometric configuration alone.

Common Cases of Geometric Instability:

All Supports are Rollers: A beam supported only by rollers may have 3 reactions, but if all reactions are vertical and parallel, they cannot resist horizontal or rotational movement.
Concurrent Forces: If all reaction forces or applied forces meet at a single point, the structure may become a mechanism and rotate freely about that point.
Colinear Members in Trusses: A triangular truss is stable, but removing a diagonal or aligning all members linearly can cause collapse under load.
Improper Restraint in Frames: A frame with hinges and rollers in poor locations may allow sway or collapse, even if equations are satisfied.

Always check whether the structure can resist: