Lecture 11 shear stresses in beams

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Lecture 11 shear stresses in beams

  1. 1. Unit 2- Stresses in Beams   Lecture -1 – Review of shear force and bending moment diagram   Lecture -2 – Bending stresses in beams   Lecture -3 – Shear stresses in beams   Lecture -4- Deflection in beams   Lecture -5 – Torsion in solid and hollow shafts. Topics Covered
  2. 2. Shear Stresses in Beams of Rectangular Cross Section   In the previous chapter we examined the case of a beam subjected to pure bending i.e. a constant moment along axis .   When a beam is in pure bending, the only stress resultants are the bending moments and the only stresses are the normal stresses acting on the cross sections.   Most beams are subjected to loads that produce both bending moments and shear forces (non-uniform bending)
  3. 3. Shear Stresses in Beams of Rectangular Cross Section   In these cases, both normal and shear forces are developed in the beam.   Normal stresses are calculated with the Flexure Formula.   We will now look at the Shear Stresses
  4. 4. Vertical & Horizontal Shear Stresses  Consider a beam of rectangular cross section subjected to a positive shear force.
  5. 5. Shear Stresses τ = F × A y − I × b x σ +dσ M M+dM σ b y1 Area A € dM I × A × y − dx € y − B A C D Shear forces and bending moments are different across different sections.
  6. 6. Shear stress distribution for different section A is the area of the x-section cut off by a line parallel to the neutral axis. is the distance of the centroid of A from the neutral axis Rectangular Section Parabolic distribution of shear stresses
  7. 7. Shear stress distribution for different section The maximum value of shear stress would obviously beat the location y = 0. Rectangular Section
  8. 8. Shear stress distribution for different section Rectangular Section
  9. 9. Section Shear stress Max shear stress Shear Stress distribution Shear stress distribution € τ = F 2I d2 4 − y2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ € τmax =1.5τavg € τmax = 4 3 τavg € τ = F 3I R2 − y2 ( )

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