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?
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?
Solution
To determine the tensile and shearing stresses on a plane inclined at an angle θ with the length of a long steel bar under tensile stress, we can use the concepts of normal and shear stress.
(i) The tensile stress on the plane can be calculated using the formula:
σ_tensile = σ * cos^2(θ)
where σ is the tensile stress acting along the length of the bar. The maximum tensile stress occurs when cos^2(θ) is maximum, which happens when θ = 0° or θ = 180°. Therefore, the tensile stress is maximum when the plane is parallel or antiparallel to the length of the bar.
(ii) The shearing stress on the plane can be calculated using the formula:
τ_shear = σ * sin(2θ)
where σ is the tensile stress acting along the length of the bar. The maximum shearing stress occurs when sin(2θ) is maximum, which happens when θ = 45° or θ = 135°. Therefore, the shearing stress is maximum when the plane is inclined at an angle of 45° or 135° with the length of the bar.
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