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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:

Or rearranged for checking:

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.)

2. Trusses

For planar trusses, use the standard comparison:

Where:

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

3. Frames

For rigid frames, the check is:

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)

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.

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:

• Translation in both X and Y directions
• Rotation about any point

Problem: L-frame

For the frame shown, determine the stability and determinacy.

The reactions of the given frame all meet at the hinge. Therefore, the beam is unstable due to concurrent reactions.
$$3b+r=3j+c$$ b=2; r=3; j=3; c=0
$$3(2)+3=3(3)+0$$ Note: While using the formula for the determinacy of the frame returns $0=0$ making the structure determinate, it is still classified as unstable.

Problem 2: Compound Structure (Beam & Frame)

The bars are denoted by the purple lines; joints by the blue dots; and support reactions by the green symbols. We analyze only the encircled region since the bottom part of the structure becomes determinate if the internal reactions of the roller support are obtained. $$3b+r=3j+c$$ b=6; r=6; j=7; c=1 (one internal hinge)
$$3(6)+(6)=3(7)+1$$ $$I=\text{(left-hand side) - (right-hand side)}$$ $$I=3(6)+(6)-(3(7)+1)$$ $$I=24-22=\boxed{\text{indeterminate to the 2º & stable}}$$

Problem: Truss with Cross-Bracing

Evaluate the Stability and Determinacy of the truss shown.

The members are indicated by the purple lines; joints by the blue dots; and reactions by the green arrows. $$m+r=2j$$ m=6; r=5; j=4
$$6+5=2(4)$$ $$11>8$$ $$I=11-8=\boxed{\text{indeterminate to the 3º & stable}}$$

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