Indeterminate Structures: Displacement Methods - Concept Map
Displacement Methods for Indeterminate Structures
Related concepts
Fixed-End Moments (FEM)
Sub concepts
- UDL FEM formula: −wL²/12 and +wL²/12
- Central point load FEM: −PL/8 and +PL/8
- Off-center point load FEM: −Pab²/L² and +Pa²b/L²
- Triangular load FEM variants
- Sign convention (clockwise-positive on member)
Relationship to central
Foundation concept — all displacement methods begin with FEM calculations for given loads on fully fixed members
Slope-Deflection Method
Sub concepts
- Member-end moment equation: M = (2EI/L)(2θ_near + θ_far − 3ψ) + FEM
- End rotations (θ_A, θ_B) as primary unknowns
- Chord rotation (ψ = Δ/L) for support settlement or sidesway
- Joint equilibrium conditions (ΣM = 0)
- Support boundary conditions (pinned end: M = 0; fixed end: free to rotate)
- System solution: write equations for all member ends, impose equilibrium, solve for rotations, back-substitute for moments
Relationship to central
Primary hand method for solving indeterminate frames; uses rotation/displacement unknowns and member stiffness
Moment Distribution Method
Sub concepts
- Distribution factor: DF = k / Σk
- Member stiffness k = 4EI/L (fixed-fixed) or 3EI/L (pinned far end)
- Balance step: distribute −(unbalanced moment) × DF at each joint
- Carry-over: half the distributed moment moves to far end (×1/2)
- Iteration until balance moments are negligible
- Final moments: sum of FEM + all distributed and carried-over increments
Relationship to central
Fast iterative hand method; alternative to slope-deflection with tabular workflow
Stiffness and Flexibility
Sub concepts
- Flexural rigidity EI (elastic modulus × second moment)
- Member stiffness k proportional to EI/L
- Reciprocal relationship: stiffness matrix in computer analysis
- Effect of boundary conditions on effective stiffness (3EI/L vs 4EI/L)
Relationship to central
Core mechanical relationships underlying both displacement methods
Joint and Support Conditions
Sub concepts
- Rigid (moment-resistant) joints: enforce ΣM = 0
- Pinned supports: free to rotate; moment = 0
- Fixed supports: rotation = 0; moment is unknown reaction
- Simple (hinged) supports: reaction exists, moment = 0
- Settlement and sidesway effects on chord rotation ψ
Relationship to central
Boundary constraints that determine the number of unknowns and equations
Continuous Beams and Frames
Sub concepts
- Two-span and multi-span continuous beams
- Symmetric vs asymmetric loading patterns
- Portal frames with lateral loads
- Sidesway-prevented vs sidesway-allowed frames
- Application to load distribution and moment redistribution
Relationship to central
Primary structural systems solved by displacement methods in practice
Comparison: Force Methods vs Displacement Methods
Sub concepts
- Force method: redundants are internal forces; good for structures with few redundancies
- Displacement method: unknowns are rotations/displacements; scalable to large frames
- Computer implementation: displacement methods form the basis of FEM and matrix analysis
- Hand calculation preference: moment distribution fastest for continuous beams; slope-deflection for frames with sidesway
Relationship to central
Contextual understanding of when and why to use displacement methods
Common Pitfalls and Sign Conventions
Sub concepts
- FEM sign errors: must match convention (clockwise-positive) throughout
- Distribution factor at fixed end = 0; at free/pinned end = 1
- Carry-over flows only from balanced joint to far end, never backward
- Chord rotation ψ included only when supports settle or frame sidesways
- Near-end vs far-end moment identification in slope-deflection formula
Relationship to central
Critical accuracy checks for exam and professional practice
Concept connections
- Fixed-End Moments (FEM)Slope-Deflection MethodFEM is added to the stiffness-based moment equation M = (2EI/L)(2θ_near + θ_far − 3ψ) + FEM; every equation includes FEM as the load term
- Fixed-End Moments (FEM)Moment Distribution MethodFEM is the initial entry in the distribution table; all balancing and carry-over operations modify FEM values to reach equilibrium
- Slope-Deflection MethodJoint and Support ConditionsJoint equilibrium (ΣM = 0) and support boundary conditions (pinned M = 0, fixed θ = 0) form the constraint equations that determine the size and solution of the rotation system
- Moment Distribution MethodStiffness and FlexibilityDistribution factors DF = k / Σk directly use member stiffness k = 4EI/L or 3EI/L; stiffness ratios govern how moments are shared at joints
- Slope-Deflection MethodMoment Distribution MethodMoment distribution is a simplified iterative relaxation of the slope-deflection equations; both yield identical final moments but moment distribution avoids explicit matrix solve
- Chord Rotation (ψ)Slope-Deflection MethodChord rotation term −3ψ appears in the slope-deflection equation; non-zero when supports settle (ψ = Δ/L) or frame sidesways
- Continuous Beams and FramesMoment Distribution MethodContinuous beams are the primary hand-calculation targets for moment distribution; the tabular method is fastest for symmetric multi-span layouts without sidesway
- Continuous Beams and FramesSlope-Deflection MethodFrames with sidesway (lateral loads, asymmetric geometry) require explicit sidesway analysis; slope-deflection equations include chord rotation term necessary for frames
- Common Pitfalls and Sign ConventionsFixed-End Moments (FEM)The most common error is flipping the FEM sign; inconsistent sign convention propagates through entire solution and yields wrong final moments
- Common Pitfalls and Sign ConventionsMoment Distribution MethodDistribution factor at a fixed end is 0 (no distribution to a fixed support); at a pinned far end use k = 3EI/L instead of 4EI/L to account for zero carry-over
- Stiffness and FlexibilitySlope-Deflection MethodThe factor 2EI/L in the slope-deflection equation is twice the stiffness k/2; higher EI or shorter L increases moment resistance to rotation
- Member Stiffness kDistribution Factor DFDF = k / Σk; members with higher stiffness (larger EI or shorter L) receive larger fraction of unbalanced moment at a joint
- Support Boundary ConditionsMoment Distribution MethodFixed end: DF = 0 (moment carried over only from that end, not to it); pinned end: use 3EI/L stiffness; fixed-fixed: use 4EI/L
- Carry-Over PrincipleMoment Distribution MethodCarry-over = 1/2 × distributed moment; flows one direction only (from balanced joint to far end); fundamental to the iteration convergence
- Comparison: Force Methods vs Displacement MethodsSlope-Deflection MethodDisplacement method solves for rotations/displacements directly; scales well to multi-story frames; unlike force method, no need to identify or release redundancies
- Comparison: Force Methods vs Displacement MethodsMoment Distribution MethodDisplacement method avoids the explicit flexibility or stiffness matrix inversion; moment distribution uses purely mechanical iteration accessible to hand calculation
Diagrams
Displacement Methods: Complete Concept Hierarchy
Comprehensive mind map showing the hierarchical structure of displacement methods, from foundational FEM concepts through slope-deflection and moment distribution workflows, structural applications, and critical error types. This visual organizes all major topics and their relationships for quick reference during exam preparation.
Slope-Deflection Method: Step-by-Step Solution Process
Flowchart showing the complete workflow for the slope-deflection method, from problem identification through FEM computation, equation setup, system solution, back-substitution, and verification. Each step includes key actions and decision points typical on the PRC exam.
Moment Distribution Method: Iterative Balance and Carry-Over
Flowchart detailing the moment distribution iterative process, including stiffness and DF calculation, tabular setup, and the balance-distribute-carry-over cycle. Highlights convergence checking and final summation—the exact workflow used in hand calculations and on the licensure exam.
Fixed-End Moment Selection Logic: Which FEM Formula to Use?
Decision tree for selecting the correct fixed-end moment formula based on loading type. Essential reference for avoiding sign errors and formula misapplication—a frequent source of exam mistakes. Includes superposition principle for complex loads.
Stiffness and Equilibrium Flow in Slope-Deflection
Sequence diagram showing the cause-and-effect chain in slope-deflection analysis: loads create FEM, FEM unbalances joints, stiffness generates rotations to balance, rotations create member moments. Illustrates the feedback loop between rotations and moments that reaches equilibrium.
Moment Distribution Convergence States
State diagram showing the progression through moment distribution cycles, from initial FEM setup through repeated balance-distribute-carry-over steps to convergence. Each state represents a calculation checkpoint and decision stage in the iterative process.
Typical Role of Displacement Methods in Structural Analysis Practice
Pie chart showing the distribution of practical applications where displacement methods are most commonly chosen—either by hand (continuous beams, moment distribution) or by computer (large frames, FEM). Reflects the workflow priorities in professional and exam contexts.