Determinacy & Indeterminacy of Structures | Structural Analysis - Civil Engineering (CE) PDF Download

Static and Kinematic Indeterminacy

This chapter explains the concepts of static determinacy, static indeterminacy and kinematic indeterminacy for structural systems used in civil engineering. The emphasis is on clear definitions, the relations and formulae used to evaluate degrees of indeterminacy for common structure types (trusses, beams, frames) in two and three dimensions, and worked examples to illustrate calculation procedures. All formulae use conventional symbols; definitions of symbols are provided when first used.

Statically Determinate Structures

• Definition: A structure is statically determinate when its support reactions and internal forces can be found using only the equations of static equilibrium.
• In such structures, the internal bending moment and shear force distributions are independent of member cross-sectional properties and flexural rigidity, provided members remain linearly elastic under loads that do not cause compatibility effects.
• No additional stresses arise from temperature changes, lack of fit, or differential settlement because there are no redundant constraints to produce such restraint stresses.
• Statically determinate structures are typically simpler to analyse and are preferred where predictable deformation behaviour is required.

Statically Indeterminate Structures

• Definition: A structure is statically indeterminate if the equations of static equilibrium alone are insufficient to determine all support reactions and internal forces.
• Such structures require additional compatibility conditions (geometric relations) and constitutive relations (elasticity of members) to determine the redundant reactions or internal forces.
• For statically indeterminate structures, bending moment and shear depend on member flexural rigidity and cross-sectional properties. Additional stresses may result from temperature changes, shrinkage, prestress, lack of fit, and differential settlement.
  

Degree of Static Indeterminacy

Degree of static indeterminacy is the number of redundant static unknowns that cannot be found by equilibrium equations alone. It is denoted by D_S and may be split into external and internal parts:

• D_S = D_Se + D_Si
• Where D_Se is the external static indeterminacy and D_Si is the internal static indeterminacy.

External Static Indeterminacy

External static indeterminacy relates to the support system and is the number of external reaction components in excess of the number of independent static equilibrium equations.

• For planar (2D) structures: D_Se = r_e - 3 where r_e is the total number of external reaction components.
• For spatial (3D) structures: D_Se = r_e - 6.

Internal Static Indeterminacy

Internal static indeterminacy depends on the structure's internal connectivity and degree of restraint (releases). It is related to the number of closed member loops and to the number of internal redundant force components that cannot be obtained from equilibrium alone.

• For plane (2D) structures: D_Si = 3C - r_r where C is the number of independent closed loops and r_r is the number of released internal reaction components.
• For space (3D) structures: D_Si = 6C - r_r.
• The number of released reactions is computed as:
  r_r = Σ(m_j - 1) for 2D
  r_r = 3 Σ(m_j - 1) for 3D
  where m_j is the number of member ends meeting at joint j. Hybrid joints and partial releases must be accounted for appropriately.

Common Formulae for Specific Structure Types

• For a 2D pin-jointed truss: D_S = m + r_e - 2j, where m is number of members and j is number of joints.
• For a 2D truss split into external and internal parts: D_Se = r_e - 3 and D_Si = m - (2j - 3).
• For a 3D truss: D_S = m + r_e - 3j with D_Se = r_e - 6 and D_Si = m - (3j - 6).
• For 2D rigid frames: D_S = 3m + r_e - 3j - r_r.
• For 3D rigid frames: D_S = 6m + r_e - 6j - r_r.
• For 3D rigid frames using loop form: D_S = (r_e - 6) + (6C - r_r).

Kinematic Indeterminacy

Kinematic indeterminacy (also called degree of freedom or degree of kinematic indeterminacy) is the number of independent displacement components that must be specified to define the geometry of the structure. If the number of unknown displacement components exceeds the number of independent compatibility equations, the structure is kinematically indeterminate and additional equilibrium conditions are required to determine displacements and corresponding reactions.

• In general the degree of kinematic indeterminacy is denoted by D_k.
• Typical counts of degrees of freedom per joint:
  Plane pin-jointed frame: each joint has 2 degrees of freedom.
  Space pin-jointed frame: each joint has 3 degrees of freedom.
  Plane rigid-jointed frame: each joint has 3 degrees of freedom (2 translations + 1 rotation).
  Space rigid-jointed frame: each joint has 6 degrees of freedom (3 translations + 3 rotations).

Common Formulae for Degree of Kinematic Indeterminacy

• For 2D rigid frame when all members are axially extensible: D_k = 3j - r_e.
• For 2D rigid frame if m members are axially rigid (inextensible): D_k = 3j - r_e - m.
• For 2D rigid frame when J′ hybrid joints exist: D_k = 3(j + j′) - r_e - m + r_r.
• For 3D rigid frame: D_k = 6(j + j′) - r_e - m + r_r.
• For 2D pin-jointed truss: D_k = 2(j + j′) - r_e - m + r_r.
• For 3D pin-jointed truss: D_k = 3(j + j′) - r_e - m + r_r.

Interpretation and Practical Notes

• When D_S = 0 the structure is statically determinate. When D_S > 0 it is statically indeterminate by D_S degrees. When D_S < 0 the structure is a mechanism (unstable) and cannot resist some forms of loading.
• When D_k = 0 the structure is kinematically determinate. When D_k > 0 the structure has redundant displacements (kinematically indeterminate) and when D_k < 0 the structure is a mechanism (insufficient constraints to prevent rigid body motion).
• Static and kinematic determinacies are related but distinct. A structure may be statically determinate yet kinematically indeterminate and vice versa; however, for a stable, well-posed structural system the counts must be consistent (no incompatibilities between equilibrium and compatibility).
• Released connections (hinges, pinned supports, internal releases) reduce internal restraint and change both static and kinematic counts. Hybrid joints (partially restrained joints) must be modelled carefully to account for the correct number of restraints and freed degrees.
• For complex continuous or indeterminate structures, methods such as force method (consistent deformations), displacement method (stiffness/matrix methods), slope-deflection, moment distribution, and finite element analysis are used to obtain reactions and internal actions.

Worked Examples

Example 1 - Degree of Static Indeterminacy of a 2D Truss

Problem: Determine D_S for a pin-jointed plane truss with m = 15 members, j = 8 joints and external reactions r_e = 3.

Sol.
Use the formula for a 2D pin-jointed truss: D_S = m + r_e - 2j.
Substitute the given values.
Compute m + r_e - 2j.
m + r_e - 2j = 15 + 3 - 2(8).
Evaluate the arithmetic.
15 + 3 - 16 = 2.
Therefore the truss is statically indeterminate to degree 2.

Example 2 - Degree of Static Indeterminacy of a 2D Rigid Frame

Problem: For a planar rigid frame with m = 4 members, j = 4 joints, external reaction components r_e = 6, and no releases (r_r = 0), find D_S.

Sol.
Use the formula for a 2D rigid frame: D_S = 3m + r_e - 3j - r_r.
Substitute the given values.
Compute 3m + r_e - 3j - r_r.
3m + r_e - 3j - r_r = 3(4) + 6 - 3(4) - 0.
Evaluate the arithmetic.
12 + 6 - 12 = 6.
Therefore the frame is statically indeterminate to degree 6.

Example 3 - Degree of Kinematic Indeterminacy for a Small Structure

Problem: Consider a small plane rigid-jointed frame with j = 3 joints, external reaction components r_e = 3, and m = 0 axially rigid members to be subtracted. Find D_k assuming members are axially extensible.

Sol.
Use the 2D rigid frame formula for axial extensibility: D_k = 3j - r_e.
Substitute values.
3j - r_e = 3(3) - 3.
Evaluate the arithmetic.
9 - 3 = 6.
Therefore the structure has six kinematic degrees of freedom.

Applications and Design Considerations

• Statically indeterminate structures are common in building and bridge design because they provide redundancy and allow load sharing; however, they require more complex analysis and a clear account of material and geometric properties.
• Redundancy improves robustness - if one member fails, loads can redistribute - but it also makes the structure sensitive to differential settlements, temperature variations, and construction tolerances.
• Choice of modelling as pin-connected or rigid-connected should reflect actual connection behaviour; modelling errors in connectivity or releases leads to incorrect counts of D_S and D_k.
• Matrix stiffness methods and finite element analysis inherently handle indeterminacy by assembling global equilibrium with compatibility and constitutive relations; for manual analysis the force and displacement methods remain essential tools.

Summary. Determinacy and indeterminacy are fundamental to structural analysis. Use the provided formulae to compute D_S and D_k for typical 2D and 3D structures. Identify external reactions, member counts, joints, releases and closed loops carefully when applying these formulae. For statically indeterminate systems, incorporate compatibility and material stiffness when finding internal forces and reactions.