CELE Structural Theory & Analysis — Indeterminate Structures: Force MethodsConcept Map

Concept

Degree of Indeterminacy (DI)

Sub Concepts

• Static indeterminacy
• Kinematic indeterminacy
• Redundant selection criteria
• Stability checks

Relationship To Central

Determines the number of redundant unknowns to be released and solved

Concept

Primary (Released) Structure

Sub Concepts

• Determinate structure creation
• Support release methods
• Internal release techniques
• Stability verification

Relationship To Central

Obtained by removing all redundants; serves as the basis for deflection calculations

Concept

Flexibility Coefficients

Sub Concepts

• Deflection per unit load (δ₁₁, δ₂₂, etc.)
• Cross-deflection terms (δ₁₂, δ₂₁, etc.)
• Calculation methods (virtual work, unit load, integration)
• Sign convention (positive downward/rightward)

Relationship To Central

Quantify how much a unit redundant force deflects the primary structure at the release point

Concept

Load-Induced Deflections

Sub Concepts

• Deflection from distributed loads (δ₀)
• Deflection from point loads
• Deflection from temperature changes
• Deflection from support settlement
• Superposition principle

Relationship To Central

Deflections caused by applied loads at the points where redundants are released

Concept

Compatibility Equations

Sub Concepts

• Single redundant: δ₀ + R·δ₁₁ = Δ
• Multiple redundants: matrix form
• Support settlement/displacement
• Geometric constraints
• Solution of simultaneous equations

Relationship To Central

Express that actual deflections must satisfy boundary conditions (e.g., zero at fixed/pinned supports)

Concept

Method of Consistent Deformation

Sub Concepts

• Release redundant step
• Calculate δ₀ (load effect)
• Calculate δᵢⱼ (redundant flexibility)
• Apply compatibility condition
• Solve for redundant magnitude

Relationship To Central

Core procedure that systematically applies force method steps

Concept

Three-Moment Equation (Clapeyron)

Sub Concepts

• Equation form: Mₐ·Lᵢ + 2Mᵦ(Lᵢ + Lᵢ₊₁) + Mᵨ·Lᵢ₊₁ = -6(A₁x̄₁/L₁ + A₂x̄₂/L₂)
• Load term (6Ax̄/L) calculation
• UDL span term: wL³/4
• Point load span term: 3PL²/8
• End conditions (simple = M₀, fixed = unknown)
• Sequential application for multi-span beams

Relationship To Central

Specialized force method for continuous beams; relates moments at three consecutive supports

Concept

Propped Cantilever Analysis

Sub Concepts

• Single redundant (prop reaction)
• UDL case: Rₚᵣₒₚ = 3wL/8, Mfixed = wL²/8
• Point load case
• Distributed load combinations
• Moment and shear diagrams

Relationship To Central

Practical indeterminate structure frequently solved by force method

Concept

Fixed-Fixed (Encastré) Beams

Sub Concepts

• Two degrees of indeterminacy
• UDL loading: end moments = ±wL²/12
• Central point load: end moments = ∓PL/8
• Symmetric vs. asymmetric loading
• Reaction calculations

Relationship To Central

Classic indeterminate structure with two redundant moment unknowns

Concept

Continuous Beam Analysis

Sub Concepts

• Equal-span vs. unequal-span beams
• Support reactions from moment diagrams
• Influence of settlement at interior supports
• Design envelope construction
• Serviceability checks (deflection limits per NSCP 2015)

Relationship To Central

Most practical application of force methods in bridge and building design

Concept

Deflection Calculation Methods

Sub Concepts

• Integration of M/EI (double integration)
• Virtual work (unit load method)
• Moment-area theorems
• Conjugate beam method
• Castigliano's theorem (second derivative)

Relationship To Central

Essential techniques for computing δ₀ and δᵢⱼ terms

Concept

Support Settlements and Displacements

Sub Concepts

• Vertical settlement Δ at fixed/pinned supports
• Horizontal displacement at roller supports
• Rotational settlement/tilt
• Differential settlement effects
• Compatibility: δ₀ + R·δ₁₁ = Δ (known value)

Relationship To Central

Modifies compatibility equation right-hand side; common in indeterminate analysis

Concept

Sign Convention and Coordinate System

Sub Concepts

• Positive direction (down/right)
• Redundant force positive sense
• Bending moment convention (sagging positive in FBD)
• Deflection sign (positive with load direction)
• Consistency checks in solver

Relationship To Central

Critical for correct setup of force method equations

Concept

Indeterminate Frame Analysis

Sub Concepts

• Degree of indeterminacy for frames
• Internal releases (hinges, pins)
• Lateral load effects
• Sidesway possibilities
• Multiple compatibility equations

Relationship To Central

Extension of force method to frames with multiple redundancies

Concept

Practical Design Applications per NSCP 2015

Sub Concepts

• Load factors and combinations
• Deflection limits: L/240 to L/180 (per NSCP 2015)
• Combined moment and shear design
• Control of cracking and serviceability
• Verification against code-specified limits

Relationship To Central

Integration of force method results into Philippine building code requirements

To

Redundant Selection

From

Degree of Indeterminacy

Strength

strong

Relationship

DI determines how many redundants must be chosen and removed

To

Primary (Released) Structure

From

Redundant Selection

Strength

strong

Relationship

Redundants are removed to create a stable, determinate primary structure

To

Load-Induced Deflections

From

Primary (Released) Structure

Strength

strong

Relationship

Deflections δ₀ are calculated on the primary structure under applied loads

To

Flexibility Coefficients

From

Primary (Released) Structure

Strength

strong

Relationship

Flexibility δᵢⱼ terms measure how much the primary structure deflects per unit redundant

To

Compatibility Equations

From

Load-Induced Deflections

Strength

strong

Relationship

δ₀ appears on the left side of compatibility equations (δ₀ + R·δᵢⱼ = Δ)

To

Compatibility Equations

From

Flexibility Coefficients

Strength

strong

Relationship

δᵢⱼ coefficients form the matrix multiplying redundant vector R in compatibility

To

Method of Consistent Deformation

From

Compatibility Equations

Strength

strong

Relationship

Solving compatibility equations is the core step that yields all redundant forces

To

Load-Induced Deflections

From

Deflection Calculation Methods

Strength

strong

Relationship

Virtual work, integration, and moment-area methods are used to compute δ₀

To

Flexibility Coefficients

From

Deflection Calculation Methods

Strength

strong

Relationship

Same methods apply to compute δᵢⱼ from unit loads on primary structure

To

Continuous Beam Analysis

From

Three-Moment Equation (Clapeyron)

Strength

strong

Relationship

Three-moment equation is the standard force method application for continuous beams

To

Method of Consistent Deformation

From

Propped Cantilever Analysis

Strength

strong

Relationship

Propped cantilever is a classic single-redundant application of consistent deformation

To

Method of Consistent Deformation

From

Fixed-Fixed (Encastré) Beams

Strength

strong

Relationship

Fixed-fixed beams have two redundant end moments solved via compatibility equations

To

Compatibility Equations

From

Support Settlements and Displacements

Strength

strong

Relationship

Settlement Δ modifies the right-hand side of compatibility; δ₀ + R·δᵢⱼ = Δ (not zero)

To

Compatibility Equations

From

Sign Convention and Coordinate System

Strength

strong

Relationship

Correct signs for δ₀, δᵢⱼ, R are critical for setting up compatibility equations correctly

To

Practical Design Applications per NSCP 2015

From

Method of Consistent Deformation

Strength

moderate

Relationship

Force method results (M, V, N diagrams) are used to verify code limits (deflection, cracking)

To

Practical Design Applications per NSCP 2015

From

Three-Moment Equation (Clapeyron)

Strength

moderate

Relationship

Three-moment solution yields support moments and reactions needed for NSCP 2015 design checks

To

Method of Consistent Deformation

From

Indeterminate Frame Analysis

Strength

moderate

Relationship

Frames extend the force method to multiple (internal and external) redundancies

To

Support Settlements and Displacements

From

Load-Induced Deflections

Strength

moderate

Relationship

Understanding load deflections is essential for interpreting effects of known settlements Δ

To

Sign Convention and Coordinate System

From

Flexibility Coefficients

Strength

moderate

Relationship

δᵢⱼ sign depends on consistent positive direction (down/right); wrong sign = wrong result

To

Continuous Beam Analysis

From

Propped Cantilever Analysis

Strength

weak

Relationship

Propped cantilever is a simplified case of continuous beam (two supports only)