question_answer 45)

                  Consider a long steel bar under a tensile stress due to forces F acting at the edges along the length of the bar. Consider a plane making an angle  \[\theta \]  with the length. What are the tensile and shearing stresses oil this plane? (a) For what angle is the tensile strees a maximum? (b) For what angle is the shearing strees a maximum?                

Answer:

                  Normal component of force, \[{{F}_{y}}=F\sin \theta \] and component of force along the place, \[{{F}_{x}}=F\cos \theta \]                 Let A = area of the bar, then Area of face a a?.                 \[A'\,=\,\frac{A}{\sin \theta }\]                 \[\therefore \] Tensile strees \[=\,\frac{{{F}_{y}}}{A'}=\,\left( \frac{F}{A} \right)\,{{\sin }^{2}}\theta \]  ?. (i)                 Shearing strees \[=\,\frac{{{F}_{x}}}{A'}=\,\frac{F\,\cos \theta \,\sin \,\theta }{A}\]                 \[=\,\frac{F(2\,\,\sin \theta \,\cos \theta )}{2A}\,\,=\frac{F\,\sin \,2\theta }{2A}\]                     ?. (ii) (a) Tensile strees will be maximum, if sin \[\theta =1\]in eqn. (i) of \[\theta ={{90}^{o}}\]. (b) Shearing strees will be maximum, if sin \[2\theta \,=1\,\] in eqn. (ii) i.e., \[2\theta \,=\,{{90}^{o}}\] or \[\theta =\,{{45}^{o}}\].