Stresses in Beams - Concept Map
Internal Stresses Produced by Bending Moment and Shear Force in Elastic Beams
Related concepts
Flexural (Bending) Stress
Sub concepts
- Neutral Axis (NA) location at centroid
- Zero stress at NA; maximum at extreme fibers
- Flexure formula: σ = My/I
- Maximum bending stress: σ_max = Mc/I = M/S
- Section Modulus S = I/c
- Sagging moment: bottom fiber in tension, top in compression
Relationship to central
Primary stress type; linear variation through beam depth governs member sizing
Horizontal Shear Stress
Sub concepts
- Shear stress formula: τ = VQ/(Ib)
- First moment of area Q about neutral axis
- Zero at top and bottom fibers; maximum at NA
- Rectangular section: τ_max = 3V/(2A)
- Circular section: τ_max = 4V/(3A)
- Wide-flange approximation: τ ≈ V/A_web
Relationship to central
Secondary stress type; parabolic distribution; complements bending stress analysis
Section Modulus and Design
Sub concepts
- Definition: S = I/c
- Rectangular section: S = bh²/6
- Circular section: S = πd³/32
- Design formula: S_required = M_max/σ_allow
- Efficiency principle: deep sections are superior in bending
- I-beam principle: flanges maximize S while minimizing weight
Relationship to central
Key design parameter linking moment to allowable stress; governs section selection
Moment of Inertia (I) and Centroid
Sub concepts
- Rectangular: I = bh³/12
- Circular: I = πd⁴/64
- Parallel-axis theorem for composite sections
- Centroid location for unsymmetric (T-, L-) sections
- Different I values for major and minor axes
Relationship to central
Geometric properties defining stress distribution; essential for flexure and shear formulas
Shear Flow in Built-Up Beams
Sub concepts
- Shear flow definition: q = VQ/I
- Connector capacity and spacing: s = F/q
- Nailed wood built-up sections
- Bolted plate girders
- Welded I-beam and box-beam fabrication
- NSCP 2015 spacing limits
Relationship to central
Application of shear stress formula; determines connector spacing in fabricated sections
Combined Axial and Bending Stress
Sub concepts
- Combined stress formula: σ = P/A ± Mc/I
- Eccentric load: M = Pe (eccentricity e)
- Kern (core) of a section
- Middle-third rule for rectangles (e ≤ h/6)
- Tension limit for masonry and footings
- Short-column behavior under off-center loads
Relationship to central
Superposition principle; eccentric loading produces both normal and bending stress
Beam Design Process
Sub concepts
- Step 1: Calculate M_max and V_max from SFD/BMD
- Step 2: Flexure design using S_required = M/σ_allow
- Step 3: Shear verification using τ_max ≤ τ_allow
- Step 4: Serviceability check (deflection — Chapter 5)
- Flexure-governed vs. shear-governed beams
- Allowable stress selection per NSCP 2015, ACI 318, AISC 360
Relationship to central
Systematic approach integrating both flexure and shear checks; ensures safe member sizing
Unsymmetric (Non-Rectangular) Sections
Sub concepts
- Neutral axis not at mid-depth
- Different c_top and c_bottom distances
- Composite parallel-axis moment of inertia
- Stress at top vs. stress at bottom
- Identification of governess (critical) fiber
- Application to RC T-beams and steel plate girders
Relationship to central
Extended analysis for T-beams, L-beams, and composite sections; different top and bottom stresses
Material Assumptions and Linearity
Sub concepts
- Linear elastic material behavior
- Plane sections remain plane
- Stress proportional to strain (Hooke's Law)
- Small deflection assumption
- Homogeneous, isotropic material
- Limit before yielding (plastic behavior)
Relationship to central
Foundation assumptions enabling the flexure formula; limit range of validity
Concept connections
- Flexural Bending StressSection Modulus and DesignSection modulus (S = I/c) is the direct design shortcut; σ_max = M/S eliminates need to calculate c and I separately
- Moment of Inertia (I) and CentroidFlexural Bending StressFlexure formula σ = My/I depends directly on I and location of y relative to centroid (neutral axis)
- Horizontal Shear StressMoment of Inertia (I) and CentroidShear formula τ = VQ/(Ib) requires I and Q (first moment of area) computed about the centroidal axis
- Shear Flow in Built-Up BeamsHorizontal Shear StressShear flow q = VQ/I is directly derived from shear stress τ = VQ/(Ib); connector spacing s = F/q applies this concept
- Combined Axial and Bending StressFlexural Bending StressCombined stress superimposes axial stress (P/A) and bending stress (Mc/I); uses same bending formula but adds uniform term
- Unsymmetric (Non-Rectangular) SectionsMoment of Inertia (I) and CentroidNon-rectangular sections require careful centroid location and parallel-axis moment of inertia; NA is not at mid-depth
- Beam Design ProcessFlexural Bending StressStep 1 of design: calculate S_required = M_max/σ_allow using flexure concept; then select section with sufficient S
- Beam Design ProcessHorizontal Shear StressStep 2 of design: verify shear capacity using τ_max = VQ/(Ib) ≤ τ_allow; often governs short, heavily loaded beams
- Material Assumptions and LinearityFlexural Bending StressLinear stress-strain relationship (Hooke's Law) and plane-sections assumption are foundations enabling the linear flexure formula
- Section Modulus and DesignUnsymmetric (Non-Rectangular) SectionsFor asymmetric sections, must compute S from I/c using different c_top and c_bottom; yields different S values at top and bottom
- Combined Axial and Bending StressMoment of Inertia (I) and CentroidKern of section (middle-third rule for rectangles) depends on centroid location; determines safe eccentricity without tension
- Flexural Bending StressMaterial Assumptions and LinearityValidity of σ = My/I is limited to elastic range before yielding; assumes linear elastic behavior and small deflections
- Horizontal Shear StressShear Flow in Built-Up BeamsShear flow is the extension of horizontal shear stress; determines how much horizontal force per unit length must be transmitted by connectors
- Beam Design ProcessCombined Axial and Bending StressFor eccentric loads or short columns, design process includes superposing axial and bending effects; must check tension limits
- Unsymmetric (Non-Rectangular) SectionsFlexural Bending StressFor T-, L-, and I-beams, apply flexure formula σ = My/I at different distances c_top and c_bottom from NA; yields unequal stresses
Diagrams
Stresses in Beams — Complete Concept Hierarchy
Comprehensive mind map showing the complete hierarchy of concepts in the 'Stresses in Beams' chapter, organized by major topics (flexural stress, shear stress, design process) and their sub-concepts. This map displays the interconnected nature of geometric properties, formulas, and design applications.
Beam Stress Analysis and Design Workflow
Step-by-step design workflow for analyzing and sizing beams under bending and shear. Shows the iterative process of calculating geometric properties, designing for flexure, verifying shear capacity, and confirming serviceability — a typical board-exam problem flow.
Stress Distribution and Formula Selection
Decision tree showing how to select the correct formulas for bending and shear stress based on beam section type (rectangular, circular, I-beam, T-beam). Emphasizes that flexural stress is linear while shear stress is parabolic — a critical distinction for board exams.
Combined Axial and Bending Stress Analysis
Analysis of combined axial compression and bending moment, showing how to apply superposition to find extreme fiber stresses and determine whether tension exceeds safe limits — essential for short columns and footings per NSCP 2015.
Shear Flow and Connector Spacing in Built-Up Beams
Workflow for sizing connectors (nails, bolts, welds) in built-up timber and steel beams. Shows how shear flow (q = VQ/I) determines spacing (s = F/q) to safely transmit horizontal shear between components — a common board-exam and design office task.
Stress States in Beam Cross-Section During Bending
State-transition diagram illustrating how bending stress and shear stress vary through the beam depth. Shows that flexural stress is linear (zero at NA, maximum at fibers) while shear stress is parabolic (maximum at NA, zero at fibers) — opposite distributions that must be considered together in design.
Locating Neutral Axis in Unsymmetric Sections
Detailed procedure for analyzing non-rectangular (T-, L-, and composite) beam sections. Emphasizes the centroid calculation, parallel-axis theorem application, and identification of the critical (governing) stress that dictates design — a frequently tested topic in PRC exams.
Common Pitfalls and Verification Checklist
Critical error-checking flowchart highlighting five common board-exam pitfalls: incorrect Q in shear formula, forgetting the 1.5 factor for rectangular max shear, unit inconsistency, mixing I and S, and sign errors in combined stress. Essential review before licensure exams.