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Question

Consider a long steel bar under a tensile-stress due to forces F acting at the edges along the length of the bar (fig.). Consider a plane making an angle θ with the length. What are the tensile and shearing stresses on this plane?
(i) At what angle is the tensile stress is maximum?
(ii) At what angle is the shearing stress is maximum?
773559_2fbbaaa3029d4b49b125b511083b491d.PNG

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Solution

According Problem F is applied along horizontal so resolving into two component one is parallel to inclined one and other is perpendicular to inclined me
F parallel =F11=Fcosθ
F perpendicular =F=Fsinθ
F Producer tensile stren , F11 producer shear stren
Area of focus aa1 then sinθ=AA1A1=A/sinθ
Tensile stren on the plane aa1
aa1=FA1FsinθA/sinθ=FAsin2θ
Shearing stren on the plane aa1
Shearing stren =Parallel forcearea
=f11A1=FcosθA/sinθ=FsinθcosθA=F(2sinθcosθ)2A
=Fsin2θ2A
(a) For tensile stren to be maximum
sin2θ=1 sinθ=1 θ=π2or 90o
(b) For shearing stren to be maximum
sin2θ=1 2θ=π2 θ=π4 or 45o

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