Structural Analysis
What is Structural Analysis?
A structure whose support reactions and internal forces can be determined solely by the equations of static equilibrium.
Key formula / rule: Degree of Indeterminacy (Truss, 2D)
Key points
- Differentiate between statically determinate and indeterminate structures.
- Apply equilibrium equations to determine reactions and internal forces in determinate structures.
- Calculate the ° of indeterminacy for various structural systems.
- Understand the necessity of compatibility equations for indeterminate structures.
Common exam trap
Confusing the number of members/supports with determinacy without considering constraints.
Definitions
- Term
Statically Determinate Structure
- Meaning
A structure whose support reactions and internal forces can be determined solely by the equations of static equilibrium.
- Term
Statically Indeterminate Structure
- Meaning
A structure for which the support reactions and internal forces cannot be uniquely determined by the equations of static equilibrium alone, requiring additional compatibility conditions.
- Term
Degree of Indeterminacy (DOI)
- Meaning
The number of unknown forces or reactions in a structure that exceed the number of available static equilibrium equations.
- Term
Equilibrium Equations
- Meaning
Mathematical statements that express the conditions of static equilibrium for a body (e.g., ΣFx = 0, ΣFy = 0, ΣM = 0).
- Term
Compatibility Equations
- Meaning
Equations that express the continuity of displacements and rotations at joints and along members, necessary for analyzing indeterminate structures.
Learning objectives
Differentiate between statically determinate and indeterminate structures.
Apply equilibrium equations to determine reactions and internal forces in determinate structures.
Calculate the ° of indeterminacy for various structural systems.
Understand the necessity of compatibility equations for indeterminate structures.
Formulae
- Name
Degree of Indeterminacy (Truss, 2D)
- Note
m = number of members, r = number of reactions, j = number of joints
- Expression
DOI = m + r - 2j
- Name
Degree of Indeterminacy (Frame, 2D)
- Note
ri = number of restraints at internal joint i, r = number of external reactions, N = number of members. Alternatively, DOI = (Σ(3-ri)) + (No. of external reactions) - 3 * (No. of members).
- Expression
DOI = Σ(3-ri) + r - 3N
Prerequisites
Statics (equilibrium of rigid bodies)
Basic mechanics of materials
Understanding of structural elements (beams, columns, trusses)
Common mistakes
Confusing the number of members/supports with determinacy without considering constraints.
Assuming all structures with supports are indeterminate.
Incorrectly applying equilibrium equations to indeterminate structures without considering compatibility.
Overlooking the degrees of freedom at joints in complex structures.
Miscalculating the ° of indeterminacy for trusses and rigid frames.
Keywords
Statically Determinate
Statically Indeterminate
Equilibrium Equations
Compatibility Equations
Degree of Indeterminacy
Redundancy
Structural Analysis
Truss Analysis
Frame Analysis
Practice preview
For a plane frame with 6 members, 5 joints (including supports), and 2 fixed supports, what is the degree of kinematic indeterminacy if axial deformations are neglected?…
hard
What does static indeterminacy of a structure represent?…
easy
Which of the following is a statically determinate beam?…
easy