CTC / MTC 222 Strength of Materials Chapter 8 Stress Due to Bending

Chapter Objectives Compute the maximum bending stress in a beam using the flexure formula. Compute the bending stress at any point in a beam cross section. Determine the appropriate design stress and design beams to carry a given load safely

The Flexure Formula Positive moment – compression on top, bent concave upward Negative moment – compression on bottom, bent concave downward Maximum Stress due to bending (Flexure Formula) σmax = M c / I Where M = bending moment, I = moment of inertia, and c = distance from centroidal axis of beam to outermost fiber For a non-symmetric section distance to the top fiber, ct , is different than distance to bottom fiber cb σtop = M ct / I σbot = M cb / I Conditions for application of the flexure formula Listed in Section 8-3, p. 308

The Flexure Formula Conditions for application of flexure formula Beam is straight, or nearly straight Cross-section is uniform Beam is not subject to torsion Beam is relatively long and narrow in proportion to its depth Beam is made from a homogeneous material with equal moduli of elasticity in tension and compression Stress is below the proportional limit No part of beam fails from instability Location where stress is computed is not close to point of application of a concentrated load

Stress Distribution For a positive moment, top of beam is in compression and bottom is in tension Stress at centroidal axis (neutral axis) is zero Stress distribution within cross section is linear Magnitude of stress is directly proportional to distance from neutral axis Stress at any location in cross section due to bending σ = M y / I , where y = distance from neutral axis of beam to location in cross section

Section Modulus, S Maximum Stress due to bending σmax = M c / I Both I and c are geometric properties of the section Define section modulus, S = I / c Then σmax = M c / I = M / S Units for S – in3 , mm3 Use consistent units Example: if stress, σ, is to be in ksi (kips / in2 ), moment, M, must be in units of kip – inches For a non-symmetric section S is different for the top and the bottom of the section Stop = I / ctop Sbot = I / cbot

Stress Concentration Factors Changes in cross-section of a member can cause stress concentrations Stress concentration factor KT Depends on geometry of the member Can be measured experimentally, or by computerized analyses KT = σmax / σnom Solving for and σmax using σnom = M / S σmax = M KT / S See Sections 3-9, 8-9 and Appendix A-21

Flexure Center For the flexure formula to apply, the applied load must not cause the beam to twist The loads must pass through the flexural center or shear center, of the beam If section has an axis of symmetry, the shear center is on that axis If section doesn’t have an axis of symmetry, the shear center may be outside the limits of the section itself Example - Channel

Calculation of Design Stress Design stress (or allowable stress), σd Sometimes based on yield strength: σd = Sy / N Sometimes based on ultimate strength σd = Su / N Ductile Materials - >5% elongation before failure Static loads - σd = Sy / N , N = 2 Repeated loads - σd = Su / N , N = 8 Impact or shock load - σd = Su / N , N = 12 Brittle Materials - <5% elongation before failure Static loads - σd = Su / N , N = 6 Repeated loads - σd = Su / N , N = 10 Impact or shock load - σd = Su / N , N = 15 Design stresses for specific materials are specified in design codes Examples: AISC or Aluminum Association