CELE Strength of Materials — Stresses in BeamsConcept Map

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

To

Section Modulus and Design

From

Flexural Bending Stress

Strength

strong

Relationship

Section modulus (S = I/c) is the direct design shortcut; σ_max = M/S eliminates need to calculate c and I separately

To

Flexural Bending Stress

From

Moment of Inertia (I) and Centroid

Strength

strong

Relationship

Flexure formula σ = My/I depends directly on I and location of y relative to centroid (neutral axis)

To

Moment of Inertia (I) and Centroid

From

Horizontal Shear Stress

Strength

strong

Relationship

Shear formula τ = VQ/(Ib) requires I and Q (first moment of area) computed about the centroidal axis

To

Horizontal Shear Stress

From

Shear Flow in Built-Up Beams

Strength

strong

Relationship

Shear flow q = VQ/I is directly derived from shear stress τ = VQ/(Ib); connector spacing s = F/q applies this concept

To

Flexural Bending Stress

From

Combined Axial and Bending Stress

Strength

strong

Relationship

Combined stress superimposes axial stress (P/A) and bending stress (Mc/I); uses same bending formula but adds uniform term

To

Moment of Inertia (I) and Centroid

From

Unsymmetric (Non-Rectangular) Sections

Strength

strong

Relationship

Non-rectangular sections require careful centroid location and parallel-axis moment of inertia; NA is not at mid-depth

To

Flexural Bending Stress

From

Beam Design Process

Strength

strong

Relationship

Step 1 of design: calculate S_required = M_max/σ_allow using flexure concept; then select section with sufficient S

To

Horizontal Shear Stress

From

Beam Design Process

Strength

strong

Relationship

Step 2 of design: verify shear capacity using τ_max = VQ/(Ib) ≤ τ_allow; often governs short, heavily loaded beams

To

Flexural Bending Stress

From

Material Assumptions and Linearity

Strength

strong

Relationship

Linear stress-strain relationship (Hooke's Law) and plane-sections assumption are foundations enabling the linear flexure formula

To

Unsymmetric (Non-Rectangular) Sections

From

Section Modulus and Design

Strength

moderate

Relationship

For asymmetric sections, must compute S from I/c using different c_top and c_bottom; yields different S values at top and bottom

To

Moment of Inertia (I) and Centroid

From

Combined Axial and Bending Stress

Strength

moderate

Relationship

Kern of section (middle-third rule for rectangles) depends on centroid location; determines safe eccentricity without tension

To

Material Assumptions and Linearity

From

Flexural Bending Stress

Strength

moderate

Relationship

Validity of σ = My/I is limited to elastic range before yielding; assumes linear elastic behavior and small deflections

To

Shear Flow in Built-Up Beams

From

Horizontal Shear Stress

Strength

strong

Relationship

Shear flow is the extension of horizontal shear stress; determines how much horizontal force per unit length must be transmitted by connectors

To

Combined Axial and Bending Stress

From

Beam Design Process

Strength

moderate

Relationship

For eccentric loads or short columns, design process includes superposing axial and bending effects; must check tension limits

To

Flexural Bending Stress

From

Unsymmetric (Non-Rectangular) Sections

Strength

strong

Relationship

For T-, L-, and I-beams, apply flexure formula σ = My/I at different distances c_top and c_bottom from NA; yields unequal stresses