Module 2: Stability and Determinacy

Module 2: Stability and Determinacy

Before analyzing the internal forces of any structure, we must first establish if the structure is stable under load and if its forces can be determined purely by the laws of statics. This module covers the mathematical criteria for stability and external/internal determinacy.

Structural Stability

A structure is considered stable if it remains rigidly fixed in space under the action of any system of loads without moving as a rigid body. An unstable structure will undergo large, uncontrollable displacements (translation or rotation) under minimal load.

Conditions for Stability of Structures

A stable structure must be adequately supported. The support reactions must prevent both translation and rotation. Even if a structure is geometrically stable internally, improper support conditions can lead to external instability.

Checklist

The number of reaction components ( $r$ ) must be equal to or greater than the number of equations of equilibrium available (typically 3 for planar structures: $\Sigma F_x = 0$ , $\Sigma F_y = 0$ , $\Sigma M = 0$ ).
Geometric Instability (Parallel Reactions): The reactions must not be parallel (all pointing in the same direction). If all reaction forces are parallel, there is no resistance to a force applied perpendicular to them, leading to instability in that direction. For example, a beam supported only by three vertical rollers is geometrically unstable against any horizontal load.
Geometric Instability (Concurrent Reactions): The reactions must not be concurrent (all passing through a single point). If all reaction lines of action intersect at a single point, the structure has no resistance to rotation around that specific point. For example, if a structure is supported by a pin and a roller, and the line of action of the roller reaction passes directly through the pin, it is geometrically unstable and will rotate.
Internal Stability: The structure must be composed of stable sub-assemblies (e.g., triangles in a truss) to prevent internal collapse.
Mechanisms: A structure is considered a mechanism and is internally unstable if there are too many internal hinges. For example, a three-bar frame connected entirely by pins without diagonal bracing is a mechanism and will collapse sideways under minimal lateral load.

Kinematic Determinacy

While static determinacy relates to unknown forces, kinematic determinacy relates to unknown displacements (degrees of freedom).

Degrees of Freedom (DOF)

The number of independent displacements and rotations that are required to fully define the deformed shape of the structure. A structure is kinematically determinate if its displacements are known (usually zero at fixed supports). It is kinematically indeterminate if joint displacements are unknown before analysis. This concept is fundamental to Displacement Methods like Slope-Deflection and Matrix Stiffness.

Key Takeaways

Stability requires adequate support to prevent rigid body translation or rotation.
A structure is geometrically unstable if its reaction forces are all parallel or all concurrent, regardless of how many supports it has.

Improper Constraints

Even if a structure has enough support reactions (

r \geq 3

), it can still be unstable if those supports are improperly arranged. This occurs when all reactions are either parallel or concurrent, causing geometric instability.

Determinacy

Determinacy refers to whether the internal forces and reactions of a structure can be uniquely determined using only the equations of static equilibrium.

External Determinacy

A structure is statically determinate externally if all its support reactions can be calculated solely by applying the three equations of equilibrium (

\Sigma F_x = 0

\Sigma F_y = 0

\Sigma M = 0

) along with any condition equations from internal hinges.

Internal Determinacy

A structure is statically determinate internally if, after determining the external reactions, the internal forces in all its members (axial force, shear, and bending moment) can be calculated using the equations of equilibrium applied to isolated parts of the structure.

Key Takeaways

A structure is determinate if all external reactions and internal forces can be found using only the equations of static equilibrium.

Degree of Indeterminacy

If a structure has more unknown forces (reactions and internal member forces) than the available equations of equilibrium, it is statically indeterminate. The degree of indeterminacy represents the number of redundant forces that must be removed to make the structure determinate.

Degree of Indeterminacy for Planar Trusses (2D)

For a planar truss with

j

joints,

m

members, and

r

reaction components:

If $m + r = 2j$ , the truss is statically determinate.
If $m + r > 2j$ , the truss is statically indeterminate to the $(m + r - 2j)$ th degree.
If $m + r < 2j$ , the truss is unstable.

Degree of Indeterminacy for Space Trusses (3D)

For a 3D space truss, each joint provides 3 equilibrium equations (

\Sigma F_x = 0

\Sigma F_y = 0

\Sigma F_z = 0

If $m + r = 3j$ , the space truss is statically determinate.
If $m + r > 3j$ , the space truss is statically indeterminate to the $(m + r - 3j)$ th degree.
If $m + r < 3j$ , the space truss is unstable.

Degree of Indeterminacy for Planar Frames (2D)

For a planar frame or continuous beam with

j

rigid joints,

m

members, and

r

reaction components:

If $3m + r = 3j + e$ (where $e$ is the number of condition equations, e.g., an internal hinge provides $e=1$ equation $\Sigma M = 0$ ), the structure is statically determinate.
If $3m + r > 3j + e$ , the structure is statically indeterminate to the $(3m + r - 3j - e)$ th degree.
If $3m + r < 3j + e$ , the structure is unstable.

Note on Condition Equations ( $e$ ): An internal hinge connecting 2 members provides $e = 1$ . If an internal hinge connects $n$ members, it provides $e = n - 1$ condition equations.

Degree of Indeterminacy for Space Frames (3D)

For a 3D space frame, each rigid joint provides 6 equilibrium equations (3 force, 3 moment), and each member has 6 unknown internal forces:

If $6m + r = 6j + e$ , the space frame is statically determinate.
If $6m + r > 6j + e$ , the space frame is statically indeterminate to the $(6m + r - 6j - e)$ th degree.
If $6m + r < 6j + e$ , the space frame is unstable.

Determinacy Calculator

Use this interactive calculator to quickly determine the stability and degree of indeterminacy of planar trusses, beams, and frames. Enter the parameters for members, joints, reactions, and condition equations.

Truss Determinacy Calculator

Members (m)5

Joints (j)4

Reactions (r)3

Equation: m + r = 2j

m + r = 5 + 3 = 8

2j = 2(4) = 8

Statically Determinate

Assumes internal arrangement is stable and reactions are non-concurrent/non-parallel.

Beam Stability & Support Reactions

Key Takeaways

The degree of indeterminacy quantifies the number of redundant unknown forces in a statically indeterminate structure. These require compatibility equations to solve.
Mathematical formulas involving members ( $m$ ), joints ( $j$ ), and reactions ( $r$ ) provide a quick check for determinacy of trusses (2D: $m + r = 2j$ , 3D: $m + r = 3j$ ) and frames (2D: $3m + r = 3j + e$ , 3D: $6m + r = 6j + e$ ).

PreviousModule 1: Introduction to Structural Analysis and Loads - Examples & Applications

NextModule 2: Stability and Determinacy - Examples & Applications

Structural Stability

Conditions for Stability of Structures

Checklist

Kinematic Determinacy

Degrees of Freedom (DOF)

Improper Constraints

Determinacy

External Determinacy

Internal Determinacy

Degree of Indeterminacy

Degree of Indeterminacy for Planar Trusses (2D)

Degree of Indeterminacy for Space Trusses (3D)

Degree of Indeterminacy for Planar Frames (2D)

Degree of Indeterminacy for Space Frames (3D)

Determinacy Calculator

Truss Determinacy Calculator

Equation: m + r = 2j

Beam Stability & Support Reactions

Left Support

Right Support