Indeterminate Structures: Force Methods - Concept Map
Force (Flexibility) Method for Indeterminate Structures
Related concepts
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
Concept connections
- Degree of IndeterminacyRedundant SelectionDI determines how many redundants must be chosen and removed
- Redundant SelectionPrimary (Released) StructureRedundants are removed to create a stable, determinate primary structure
- Primary (Released) StructureLoad-Induced DeflectionsDeflections δ₀ are calculated on the primary structure under applied loads
- Primary (Released) StructureFlexibility CoefficientsFlexibility δᵢⱼ terms measure how much the primary structure deflects per unit redundant
- Load-Induced DeflectionsCompatibility Equationsδ₀ appears on the left side of compatibility equations (δ₀ + R·δᵢⱼ = Δ)
- Flexibility CoefficientsCompatibility Equationsδᵢⱼ coefficients form the matrix multiplying redundant vector R in compatibility
- Compatibility EquationsMethod of Consistent DeformationSolving compatibility equations is the core step that yields all redundant forces
- Deflection Calculation MethodsLoad-Induced DeflectionsVirtual work, integration, and moment-area methods are used to compute δ₀
- Deflection Calculation MethodsFlexibility CoefficientsSame methods apply to compute δᵢⱼ from unit loads on primary structure
- Three-Moment Equation (Clapeyron)Continuous Beam AnalysisThree-moment equation is the standard force method application for continuous beams
- Propped Cantilever AnalysisMethod of Consistent DeformationPropped cantilever is a classic single-redundant application of consistent deformation
- Fixed-Fixed (Encastré) BeamsMethod of Consistent DeformationFixed-fixed beams have two redundant end moments solved via compatibility equations
- Support Settlements and DisplacementsCompatibility EquationsSettlement Δ modifies the right-hand side of compatibility; δ₀ + R·δᵢⱼ = Δ (not zero)
- Sign Convention and Coordinate SystemCompatibility EquationsCorrect signs for δ₀, δᵢⱼ, R are critical for setting up compatibility equations correctly
- Method of Consistent DeformationPractical Design Applications per NSCP 2015Force method results (M, V, N diagrams) are used to verify code limits (deflection, cracking)
- Three-Moment Equation (Clapeyron)Practical Design Applications per NSCP 2015Three-moment solution yields support moments and reactions needed for NSCP 2015 design checks
- Indeterminate Frame AnalysisMethod of Consistent DeformationFrames extend the force method to multiple (internal and external) redundancies
- Load-Induced DeflectionsSupport Settlements and DisplacementsUnderstanding load deflections is essential for interpreting effects of known settlements Δ
- Flexibility CoefficientsSign Convention and Coordinate Systemδᵢⱼ sign depends on consistent positive direction (down/right); wrong sign = wrong result
- Propped Cantilever AnalysisContinuous Beam AnalysisPropped cantilever is a simplified case of continuous beam (two supports only)
Diagrams
Force Method for Indeterminate Structures – Complete Concept Hierarchy
Comprehensive hierarchical view of the force method, showing how degree of indeterminacy leads to redundant selection, primary structure creation, and the systematic application of flexibility method with deflection calculations and practical solutions like the three-moment equation.
Method of Consistent Deformation – Step-by-Step Procedure
Detailed flowchart showing the logical sequence of the consistent deformation method: identify indeterminacy, select and release redundants, compute deflections under load and unit redundant, set up and solve compatibility equations, then find all reactions and internal forces.
Three-Moment Equation Application for Continuous Beams
Procedure for applying the three-moment (Clapeyron) equation to multi-span continuous beams, including identification of load terms (6Ax̄/L) for UDL and point load cases, assembly of the system, and verification.
Deflection Calculation – Method Selection
Decision tree for selecting the most efficient deflection calculation method based on structure type and complexity. Shows when to use standard formulas, double integration, virtual work, or geometric methods.
Propped Cantilever – Quick Reference Solution
Streamlined solution path for propped cantilevers, showing the key formulas for UDL (Rₚ = 3wL/8) and point load cases, with direct application of compatibility equations.
State of Knowledge: From Indeterminate to Solved Structure
State diagram showing the conceptual journey from an indeterminate structure through redundant selection, deflection calculation, and compatibility equations, culminating in a fully solved, verified structure ready for design.
Force Method Problem Categories in Board Exams
Breakdown of typical force method problems encountered in PRC Civil Engineer exams, with continuous beams being the most frequent, followed by propped cantilevers and fixed-fixed beams. Understanding each category is essential for exam preparation.