Lecture 12 Notes - Chapter 4: Pure Bending

Chapter 4 - Pure Bending

4.1 Symmetric Members in Pure Bending

• Bending moments and stress relations crucial for understanding beam behavior under loads.
• Important definitions: bending moment, bending stress, neutral axis, bending stress distribution.

4.1A Internal Moment and Stress Relations

• The bending moment (M) in a beam is affected by applied loads and reactions.
• Internal stress is determined using the flexural formula: $\sigma = \frac{My}{I}$
  • where M = moment, y = distance from the neutral axis to the point of interest, I = moment of inertia.

4.1B Deformations

• In bending, deformations occur, causing the beam to curve and experience stresses.
• Important concepts include:
  • Radius of curvature (R)
  • Curvature ((\kappa = \frac{1}{R}))

4.2 Stress and Deformations in the Elastic Range

• The material must remain in the elastic range for Hooke’s Law to apply; therefore:
  • $\sigma = E\epsilon$

- where E is the modulus of elasticity, and (\epsilon) is the strain.

4.4 Members Made of Composite Materials

• Composite materials may have different properties; hence stresses for compounds are calculated based on modular ratios and transformed sections.

4.7 Eccentric Axial Loading

• Loading that does not act through the centroid can introduce additional moments.
• The effects of eccentricity must be considered:
  • General equations relate axial load, bending moment, and stress distributions:
  • $\sigma = P/A + \frac{My}{I}$

4.9 Analysis of General Cases of Eccentric Axial Loading

• Analyze structures with varying directional loading and calculate stress distributions accordingly.

Key Definitions and Terminologies

• Bending Moment: The moment that causes a beam to bend.
• Bending Stress: Stress experienced in the beam due to bending moments.
• Neutral Axis: Line (or plane) within the beam where no longitudinal stress occurs.
• Section Modulus (S): Geometric property that quantifies the strength of a beam's cross-section, defined as $S = \frac{I}{c}$
  • where c is the distance from the neutral axis to the outermost fiber.

Important Equations

• Flexural formula: $\sigma = \frac{My}{I}$ (for bending stress)
• Curvature: $\kappa = \frac{1}{R}$
• Stress due to axial and bending loads:
  $\sigma = \frac{P}{A} + \frac{My}{I}$

Examples in Practice

• Example 1 & 2: Demonstrates drawing shear and moment diagrams and applying the corresponding equations to find stresses beneath different conditions.
• Calculations of moments and reactions to determine maximum tensile and compressive stresses are critical steps.

Conclusions

• In designing and analyzing beams, consider both axial and bending effects to ensure safety and stability.
• Equations linking moment, stress, shear, and area are vital in structural assessments and should be mastered for exams.