CELE Structural Theory & Analysis — Analysis of Determinate StructuresConcept Map

Concept

Structural Determinacy

Sub Concepts

• Degree of Static Indeterminacy (DI)
• Stability Assessment
• Equations of Condition
• Internal Releases

Relationship To Central

Foundation – establishes whether a structure can be solved by equilibrium equations alone

Concept

Reactions

Sub Concepts

• Equilibrium Equations (ΣFx, ΣFy, ΣM)
• Support Types (Pin, Roller, Fixed)
• Equations of Condition at Hinges
• Compound Beam Analysis

Relationship To Central

First step after classifying determinacy – solve for support forces and moments

Concept

Internal Forces

Sub Concepts

• Axial Force (N)
• Shear Force (V)
• Bending Moment (M)
• Free Body Diagram Method

Relationship To Central

Primary output – axial, shear, and moment within members from reaction equilibrium

Concept

Three-Hinged Arches

Sub Concepts

• Crown Hinge Equation (ΣMc = 0)
• Horizontal Thrust (H)
• Reaction Determination
• Moment Reduction Effect

Relationship To Central

Special determinate structure type – uses crown hinge condition for horizontal thrust

Concept

Frame Analysis

Sub Concepts

• Portal Frames
• Member Axial-Shear-Moment Distribution
• Sign Conventions
• L-Frames and Complex Geometries

Relationship To Central

Application of determinacy and internal forces to rigid structures

To

Reactions

From

Structural Determinacy

Strength

strong

Relationship

Determinacy confirms that equilibrium equations alone suffice to solve for all reactions—no compatibility equations needed

To

Internal Forces

From

Reactions

Strength

strong

Relationship

Reaction values are the boundary conditions; they anchor the free-body-diagram method used to find internal N, V, M at any section

To

Three-Hinged Arches

From

Equations of Condition

Strength

strong

Relationship

The crown hinge creates one equation of condition; this fourth equation solves for the unique horizontal thrust that makes the arch determinate

To

Frame Analysis

From

Internal Forces

Strength

strong

Relationship

Frame analysis applies the axial-shear-moment method to each member; results are plotted as member-specific diagrams

To

Reactions

From

Equilibrium Equations

Strength

strong

Relationship

Three equations (ΣFx, ΣFy, ΣM) form the core of reaction determination; equations of condition extend this system when hinges exist

To

Degree of Static Indeterminacy

From

Support Types

Strength

strong

Relationship

The number and type of reaction components (r) directly determines DI; fixed supports contribute 3 unknowns each, pins contribute 2, rollers contribute 1

To

Horizontal Thrust

From

Three-Hinged Arches

Strength

strong

Relationship

Horizontal thrust H is the defining output of arch analysis; it emerges from the crown moment condition and makes the arch more efficient than a beam

To

Internal Forces

From

Free Body Diagram Method

Strength

strong

Relationship

FBD of a cut segment is the direct method for finding N, V, M; equilibrium of the cut face yields all three internal forces

To

Equations of Condition

From

Compound Beam Analysis

Strength

moderate

Relationship

Each internal hinge in a compound beam contributes one equation of condition; suspended-span-first analysis exploits these conditions systematically

To

Structural Determinacy

From

Stability Assessment

Strength

strong

Relationship

Stability is checked after determinacy; even a determinate structure (DI=0) is unstable if reactions are concurrent or parallel

To

Frame Analysis

From

Sign Convention

Strength

strong

Relationship

Consistent sign convention across all members is essential for correct internal-force diagrams; confusion here is a common exam error

To

Moment Distribution

From

Portal Frames

Strength

moderate

Relationship

Portal frame behavior depends on whether it is determinate (fixed bases, three members, gives DI=3) or indeterminate; internal moments reflect base conditions