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Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams

Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams. 6.2 Bending Deformation and Strain. Key Points: Bending moment causes beam to deform. X = longitudinal axis Y = axis of symmetry Neutral surface – does not undergo a change in length. Key Points:

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Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams

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  1. Chapter 6Section 3,4Bending Deformation, Strain and Stress in Beams

  2. 6.2 Bending Deformation and Strain • Key Points: • Bending moment causes beam to deform. • X = longitudinal axis • Y = axis of symmetry • Neutral surface – does not undergo a change in length

  3. Key Points: • Internal bending moment causes beam to deform. • For this case, top fibers in compression, bottom in tension.

  4. Key Points: • Neutral surface – no change in length. • All cross-sections remain plane and perpendicular to longitudinal axis. • Neglect deformation of cros section within its own plane.

  5. Radius of curvature Length decreases Length = constant

  6. Says normal strain is linear Maximum at outer surface (where y = c)

  7. 6.2 Bending Stress – The Flexure Formula What about Stress???? Recall from section 6.1: Therefore, it follows that

  8. Sum moments about cut: This is the moment of inertia, I

  9. The Flexure Formula: Internal bending moment, lb-in Max bending stress, psi Distance from NA to outer fiber, in Moment of inertia, in4 Or in general:

  10. Examples: • Find maximum moment • Find area properties, I and c • Calculate stress

  11. WHERE IS BENDING STRESS MAXIMUM??? • Answer: • Outer surface (furthest away from Neutral Axis) • Value of x along length where moment is maximum!!

  12. Example: The T-shape beam is subjected to the loading below. • Draw shear and moment diagram. Identify location and magnitude of Mmax. • Determine location and magnitude of maximum bending stress and draw stress profile. Is the beam safe if the material is aluminum w/ sy = 15 ksi? • What is the largest internal moment the beam can resist if sallow = 2 ksi?

  13. 45K

  14. Statics: Example 1 - Pliers Given: Typical household pliers as shown. Find: Force applied to wire and force in pin that connects the two parts of the pliers. Do this for homework. See solution Link Side: what is the shear stress in pin and bending stress in handle? SofM

  15. Statics: Example 2 – Crane Structure Given: Crane structure as shown. Find: Forces and FBD’s for cables A-B and A-E, boom DEF and post AFC. Do this for homework. See solution Link Side: what is the normal stress in cables (average normal only) and normal stress in boom and post (combined loading)? SofM

  16. Example 4: Determine the resultant internal loadings acting on the cross sections located through points D and E of the frame. (1-114)

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