Frames, Machines and Cables - Concept Map
Structural Analysis of Determinate Systems: Frames, Machines, and Cables
Related concepts
Frames
Sub concepts
- Multi-force members (loaded at ≥3 points or carrying moments)
- Two-force members (identify to reduce unknowns)
- Pin connections (Newton's third law — equal and opposite)
- Dismembering method (FBD of each member separately)
- External reactions (from whole frame FBD first)
- Internal pin forces (member-to-member interactions)
- Equilibrium equations (ΣFx = 0, ΣFy = 0, ΣM = 0 per member)
Relationship to central
Stationary structures with multi-force members; analyzed by dismembering and applying equilibrium
Machines
Sub concepts
- Input force (effort)
- Output force (load/resistance)
- Mechanical advantage (M.A. = load/effort)
- Effort arm vs load arm
- Lever principle (M.A. = effort arm / load arm)
- Pin-jointed linkages
- Dismembering and equilibrium (same as frames)
- Force multiplication or displacement advantage
Relationship to central
Moving/articulated structures that transmit or modify forces; analyzed like frames but with input–output force relationship focus
Cables
Sub concepts
- Tension only (no compression or bending)
- Concentrated loads → funicular polygon (straight segments)
- Uniform horizontal load → parabolic shape
- Self-weight along cable length → catenary shape
- Horizontal component H (constant throughout cable)
- Vertical component (varies with slope angle)
- Maximum tension at steepest point (supports)
- Sag-to-span ratio (determines shape and tension distribution)
- Cable geometry relationships
Relationship to central
Perfectly flexible members carrying only tension; shape determined by load type
Equilibrium Principles
Sub concepts
- Force equilibrium (ΣFx = 0, ΣFy = 0)
- Moment equilibrium (ΣM = 0 about any point)
- Free body diagrams (FBD)
- Newton's third law (action–reaction at pins)
- Two-force member condition (force along member)
- Three-force member condition (forces concurrent or parallel)
Relationship to central
Foundation for analyzing all three types; applies to rigid bodies in static equilibrium
Load Types and Shapes
Sub concepts
- Concentrated point loads
- Distributed loads (uniform, triangular, etc.)
- Horizontal load distribution (suspension bridges)
- Load along cable length (self-weight)
- Funicular polygon (concentrated load cable)
- Parabolic cable equation
- Catenary equation (advanced)
Relationship to central
Determines how cables deform and how forces distribute in structures
Key Formulas and Relationships
Sub concepts
- Mechanical advantage: M.A. = F_load / F_effort
- Lever M.A.: M.A. = L_effort / L_load
- Parabolic cable: H = wL² / 8d
- Max tension in parabolic cable: T_max = √[H² + (wL/2)²]
- Cable tension at angle: T = H / cos(θ)
- Funicular cable: H = constant, T = H / cos(θ) per segment
- Moment equilibrium: ΣM = 0
Relationship to central
Quantitative tools for solving frames, machines, and cables
Analysis Methods
Sub concepts
- Dismembering technique (frames/machines)
- Section method (for internal forces)
- Method of joints vs method of members
- Graphical funicular polygon
- Analytical resolution (angles and components)
- Step-by-step equilibrium application
Relationship to central
Systematic procedures for solving structure problems
Common Exam Mistakes
Sub concepts
- Treating multi-force members as two-force members
- Forgetting Newton's third law at pins
- Confusing cable shapes (parabola vs catenary)
- Placing max tension at midspan instead of supports
- Incorrect sign convention in equilibrium equations
- Not identifying two-force members first
- Errors in FBD orientation and direction
Relationship to central
Pitfalls to avoid in PRC Civil Engineer Licensure Exam
Concept connections
- FramesEquilibrium PrinciplesFrames are solved by applying equilibrium equations (ΣF = 0, ΣM = 0) to each dismembered member.
- FramesTwo-Force MembersIdentifying two-force members in a frame immediately specifies force direction, simplifying equilibrium analysis.
- MachinesFramesMachines are analyzed using the same dismembering and equilibrium methods as frames, but focus on input–output force relationship.
- MachinesMechanical AdvantageMachines are defined by their ability to change force magnitude and direction; M.A. quantifies this benefit.
- CablesTension OnlyCables, being perfectly flexible, can only carry tension; this fundamental property determines analysis methods.
- CablesLoad Types and ShapesThe shape of a cable (parabola, catenary, or funicular polygon) is determined by the type of load applied.
- Key Formulas and RelationshipsCablesParabolic cable formula H = wL²/8d and T_max = √[H² + (wL/2)²] are essential tools for cable problems.
- Key Formulas and RelationshipsMachinesMechanical advantage formulas (M.A. = F_load/F_effort, M.A. = L_effort/L_load) directly solve machine problems.
- Equilibrium PrinciplesAnalysis MethodsAll analysis methods—dismembering, section method, funicular polygon—are grounded in equilibrium principles.
- FramesAnalysis MethodsFrames use the dismembering technique: draw FBD of whole structure, then each individual member, and apply equilibrium systematically.
- CablesEquilibrium PrinciplesCable tension is found by applying force equilibrium (ΣF = 0) at load points and using the constant horizontal component.
- Common Exam MistakesFramesCommon pitfall: treating multi-force members as two-force members, leading to incorrect results.
- Common Exam MistakesCablesCommon pitfall: confusing parabolic cable (uniform horizontal load) with catenary (self-weight), or placing max tension at midspan instead of supports.
- Two-Force MembersEquilibrium PrinciplesTwo-force member condition (force along member) is a direct consequence of moment equilibrium about each end.
- Mechanical AdvantageEquilibrium PrinciplesM.A. relationships are derived from moment equilibrium at the machine's pivot(s): Fe × Le = FL × la.
Diagrams
Frames, Machines, and Cables — Complete Concept Hierarchy
Comprehensive mind map showing all major topics, subtopics, and key concepts within the Frames, Machines, and Cables chapter. Branches represent the three main structure types plus supporting equilibrium principles.
Frame and Machine Analysis — Step-by-Step Procedure
Systematic flowchart for analyzing frames and machines. Shows the sequence: global equilibrium first, then dismembering, individual FBD analysis, and verification using Newton's third law. Critical decision points highlight when to identify two-force members.
Cable Analysis — Load Type to Solution Path
Decision flowchart for cable problems. Routes analysis based on load type: concentrated loads (funicular), uniform horizontal load (parabola), or self-weight (catenary). Each path leads to appropriate formulas and solution methods.
Mechanical Advantage and Machine Force Analysis
Flowchart for analyzing mechanical advantage in machines. Shows how to identify effort and load arms, calculate M.A., and verify using moment equilibrium. Handles both simple levers and multi-stage machines.
Cable Transition Between Load Types and Shapes
State diagram showing how cable problems transition from load type identification through shape determination to final tension solutions. Illustrates the three primary paths: concentrated loads (funicular), distributed horizontal (parabolic), and distributed along cable (catenary).
Two-Force Member Identification and Application
Flowchart for identifying two-force members in frames and machines. Critical first step that simplifies analysis. Shows decision criteria: exactly two load points (pins), no applied moment, no other loads. Once identified, the force direction is along the member axis.
Typical PRC Licensure Exam Problem Distribution — Frames, Machines, Cables
Approximate distribution of problem types in the PRC Civil Engineer Licensure Examination for this chapter. Frame dismembering and cable analysis are the most frequent; understanding these two areas is essential for exam success.