Solved Problems

Problem: A solid steel shaft with a diameter of $50 \text{ mm}$ is subjected to a torque of $2 \text{ kN}\cdot\text{m}$. Determine the maximum shear stress in the shaft.

Problem: A hollow aluminum shaft ($G = 26$ GPa) has an outer diameter of $80 \text{ mm}$ and an inner diameter of $60 \text{ mm}$. If the shaft is $2 \text{ m}$ long and subjected to a torque of 1.5 kN$\cdot$m, determine the angle of twist in degrees.

Problem: A solid steel shaft transmits $50 \text{ kW}$ of power at $4 \text{ Hz}$ ($240 \text{ rpm}$). If the allowable shear stress is $60 \text{ MPa}$, determine the minimum required diameter of the shaft.

Problem: A composite shaft consists of a steel segment (length $1 \text{ m}$, diameter $50 \text{ mm}$, $G_{\text{st}} = 80$ GPa) and an aluminum segment (length $1.5 \text{ m}$, diameter $40 \text{ mm}$, $G_{\text{al}} = 26$ GPa) connected end-to-end. The shaft is fixed at one end and subjected to a torque of 800 N$\cdot$m at the free end. Determine the total angle of twist.

Problem: A thin-walled rectangular steel tube has outer dimensions of $100 \text{ mm} \times 150 \text{ mm}$ and a uniform wall thickness of $5 \text{ mm}$. It is subjected to a torque of $5 \text{ kN}\cdot\text{m}$. Determine the average shear stress in the wall.

Problem: A solid steel shaft ($G = 80 \text{ GPa}$) is $1 \text{ m}$ long and has a diameter of $40 \text{ mm}$. It is fixed at both ends (A and B). A torque of $500 \text{ N}\cdot\text{m}$ is applied at point C, which is $0.4 \text{ m}$ from A. Determine the reaction torques at A and B.

Problem: A flanged bolt coupling consists of six $10\text{-mm}$ diameter steel bolts on a bolt circle $300 \text{ mm}$ in diameter. The bolts connect two solid steel shafts that transmit 60 kW of power at $150 \text{ rev/min}$. Calculate the maximum shear stress developed in the bolts.

Problem: A closely coiled helical spring is subjected to an axial load of $300 \text{ N}$. The spring wire has a diameter of $10 \text{ mm}$ and the mean coil diameter is $100 \text{ mm}$. Assuming the spring has 15 active coils and a shear modulus $G = 83 \text{ GPa}$, determine the maximum shear stress and the elongation of the spring. Use the Wahl correction factor.

Problem: A solid cylindrical shaft made of cast iron (a brittle material) and another made of mild steel (a ductile material) are both subjected to torsion until failure. Describe and compare the characteristic fracture surfaces of each.

Problem: Compare the torsional resistance (stiffness and stress) of a closed circular thin-walled tube and an open circular tube (a tube with a longitudinal slit) of the same radius ($r$) and thickness ($t$).

Problem: A rotating solid steel shaft transmits power to a gear via a square key. To accommodate the key, a rectangular keyway is milled into the shaft. Discuss the effect of the keyway on the shaft's stress distribution and overall strength.

Problem: A torque $T$ is applied to the end of a solid circular shaft using a rigid wrench. The wrench applies the torque via localized contact forces on the surface of the shaft. Explain how the shear stress distribution changes along the length of the shaft based on Saint-Venant's Principle.