Lecture 11 shear stresses in beamsPresentation Transcript
Unit 2- Stresses in BeamsTopics Covered 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.
Shear Stresses in Beams ofRectangular 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)
Shear Stresses in Beams ofRectangular 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
Vertical & Horizontal Shear Stresses subjected Consider a beam of rectangular cross section to a positive shear force.
Shear Stresses A C Shear forces and bending moments are different across different sections. D Area A B σ σ +dσ dM − ×A×y − I y1 x y €M M+dM dx b − Ay € τ=F× I×b
Shear stress distribution for different section Rectangular SectionA is the area of the x-section cut off by a lineparallel to the neutral axis. is the distance ofthe centroid of A from the neutral axis Parabolic distribution of shear stresses
Shear stress distribution for different sectionRectangular Section The maximum value of shear stress would obviously beat the location y = 0.
Shear stress distribution for different sectionRectangular Section
Shear stress distributionSection Shear stress Max shear Shear Stress stress distribution F ⎛ d 2 2 ⎞ τ = ⎜ − y ⎟ τ max = 1.5τ avg 2I ⎝ 4 ⎠ € € F 2 4 τ = (R − y 2 ) τ max = τ avg 3I 3 € €