#1. A solid circular shaft with a diameter of 40 mm is subjected to an axial tensile load of 75 kN. Determine the tensile stress in the shaft.
#2. A steel rod (E = 200 GPa) has a diameter of 20 mm and a length of 1.5 m. When subjected to an axial tensile load of 50 kN, determine the total elongation.
#3. A plane stress element is subjected to σx = 100 MPa, σy = -20 MPa, and τxy = 40 MPa. Using the von Mises criterion, determine the equivalent stress.
#4. A helical compression spring is made of steel wire (G = 79.3 GPa) with a wire diameter of 5 mm and a mean coil diameter of 40 mm. The spring has 8 active coils. Determine the spring stiffness.
#5. A thin-walled cylindrical pressure vessel has an internal diameter of 600 mm and a wall thickness of 10 mm. The vessel is subjected to an internal pressure of 4 MPa. Determine the hoop stress.
#6. A ball bearing has a basic dynamic load rating of 45 kN. The bearing operates under an equivalent radial load of 6 kN at a speed of 1500 rpm. Determine the L10 bearing life in hours.
#7. A square-threaded power screw has a mean diameter of 30 mm and a lead of 5 mm. The coefficient of friction between the threads is 0.12. Determine the torque required to raise a load of 15 kN.
#8. A flat belt drive transmits 12 kW at 1200 rpm. The driving pulley has a diameter of 250 mm. The coefficient of friction is 0.28 and the angle of contact is 165°. Determine the tight side belt tension.
#9. A fillet weld with a leg size of 6 mm and a length of 120 mm is subjected to a tensile load of 40 kN applied parallel to the weld length. Determine the shear stress in the weld throat.
#10. A shaft and hole have the following dimensions: Shaft: 35.020 mm to 35.045 mm; Hole: 34.995 mm to 35.020 mm. Determine the maximum interference for this fit.
#11. A plane stress element has σx = 90 MPa, σy = -30 MPa, and τxy = 35 MPa. Determine the maximum principal stress σ₁.
#12. A solid circular shaft with a diameter of 55 mm transmits a torque of 1200 N·m. Determine the maximum torsional shear stress in the shaft.
#13. A machined steel shaft has an ultimate tensile strength of 650 MPa. Using Marin factors of ka = 0.80 (surface), kb = 0.88 (size), and kc = 1.0 (load), determine the modified endurance limit for reversed bending.
#14. A thick-walled cylinder has an inner radius of 45 mm and an outer radius of 75 mm. The cylinder is subjected to an internal pressure of 50 MPa with no external pressure. Determine the maximum tangential stress.
#15. An M14 bolt (tensile stress area At = 115 mm²) with Grade 8.8 properties (proof strength Sp = 650 MPa) is used in a bolted joint. Determine the recommended initial preload using a factor of 0.75 of the proof load.
#16. A gear transmits a torque of 150 N·m at a rotational speed of 1200 rpm. Determine the power transmitted by the gear.
#17. A solid circular steel column (E = 200 GPa) has a diameter of 50 mm and a length of 2.2 m. The column has pinned-pinned end conditions (K = 1.0). Determine the critical buckling load.
#18. A rectangular key with a width of 12 mm and length of 50 mm is used to connect a gear to a 45 mm diameter shaft. The shaft transmits a torque of 450 N·m. Determine the shear stress in the key.
#19. A mechanical system consists of four components in series with individual reliabilities of R₁ = 0.97, R₂ = 0.94, R₃ = 0.96, and R₄ = 0.99. Determine the overall system reliability.
#20. A hole is specified with a position tolerance of Ø0.30 mm at MMC. The hole has an MMC size of 18.00 mm and is produced at an actual size of 18.12 mm. Determine the total allowable position tolerance including the MMC bonus.
#21. A simply supported rectangular beam has a width of 60 mm and a height of 120 mm. The beam is subjected to a maximum bending moment of 12 kN·m. Determine the maximum bending stress.
#22. A solid circular steel shaft (G = 79.3 GPa) has a diameter of 40 mm and a length of 700 mm. The shaft transmits a torque of 800 N·m. Determine the angle of twist.
#23. Four helical springs with stiffnesses k₁ = 20 N/mm, k₂ = 30 N/mm, k₃ = 40 N/mm, and k₄ = 60 N/mm are connected in series. Determine the equivalent spring stiffness.
#24. A thin-walled spherical pressure vessel has an inner diameter of 1000 mm and a wall thickness of 12 mm. The vessel is subjected to an internal pressure of 3.2 MPa. Determine the stress in the vessel wall.
#25. A ball bearing is subjected to a radial load of 4500 N and an axial load of 2000 N. Given that the ratio Fa/Fr exceeds the limiting value e, use X = 0.56 and Y = 1.15 to determine the equivalent dynamic bearing load.
#26. A square-threaded power screw has a mean diameter of 36 mm, a lead of 6 mm, and a coefficient of friction of 0.10. Determine the efficiency of the screw when raising a load.
#27. A roller chain drive transmits 8 kW at 500 rpm. The driving sprocket has 19 teeth with a chain pitch of 12.7 mm. Determine the tension in the chain.
#28. A single-lap adhesive joint has an overlap length of 35 mm and a width of 30 mm. The joint is subjected to a tensile load of 10 kN. Determine the average shear stress in the adhesive.
#29. A clearance fit has the following dimensions: Hole: 30.000 mm to 30.025 mm; Shaft: 29.965 mm to 29.980 mm. Determine the maximum clearance for this fit.
#30. A hydraulic cylinder has a bore diameter of 100 mm and a rod diameter of 45 mm. The operating pressure is 18 MPa. Determine the maximum extension force the cylinder can produce.
#31. A notched bar under axial tension has a nominal stress of 75 MPa and a stress concentration factor Kt = 2.4. Determine the maximum stress at the notch.
#32. A machine element is subjected to principal stresses σ₁ = 160 MPa, σ₂ = 50 MPa, and σ₃ = -40 MPa. Using the maximum shear stress theory (Tresca), determine the maximum shear stress.
#33. A cantilever beam with a rectangular cross-section (width = 50 mm, height = 80 mm) and length of 1000 mm is made of steel (E = 200 GPa). A concentrated load of 3000 N is applied at the free end. Determine the maximum deflection.
#34. A helical compression spring has a mean coil diameter of 35 mm and a wire diameter of 5 mm. The spring is made of steel and carries a load of 400 N. Using the Wahl correction factor, determine the shear stress in the spring wire.
#35. A thick-walled cylinder has an inner radius of 35 mm and an outer radius of 60 mm. The cylinder is subjected to an internal pressure of 45 MPa with no external pressure. Determine the radial stress at the inner surface.
#36. An M14 steel bolt (tensile stress area At = 115 mm², E = 200 GPa) has a grip length of 60 mm. Determine the approximate bolt stiffness.
#37. A gear with a pitch diameter of 180 mm rotates at 1000 rpm. Determine the pitch line velocity.
#38. A full-penetration butt weld joins two plates, each 12 mm thick. The weld length is 150 mm, and the joint is subjected to a tensile load of 120 kN applied perpendicular to the weld. Determine the tensile stress in the weld.
#39. An assembly consists of four stacked parts with the following nominal dimensions and tolerances: Part 1: 30.00 ± 0.04 mm, Part 2: 45.00 ± 0.06 mm, Part 3: 20.00 ± 0.05 mm, Part 4: 10.00 ± 0.03 mm. Using the worst-case method, determine the total tolerance of the assembly.
#40. A solid circular shaft with a diameter of 45 mm is subjected to a bending moment of 700 N·m and a torque of 500 N·m simultaneously. Using the von Mises criterion, determine the equivalent stress in the shaft.
#41. A machine component is subjected to a fluctuating load with an alternating stress of 100 MPa and a mean stress of 70 MPa. The material has an ultimate tensile strength of 550 MPa and an endurance limit of 230 MPa. Using the Goodman criterion, determine the factor of safety against fatigue failure.
#42. A rectangular beam with a width of 40 mm and a height of 80 mm is subjected to a shear force of 20 kN. Determine the maximum shear stress in the beam cross-section.
#43. A spring-mass system has a spring stiffness of 12 kN/m and a mass of 3 kg. Determine the natural frequency of the system.
#44. A thin-walled cylindrical pressure vessel has an internal diameter of 450 mm, a wall thickness of 6 mm, and is made of steel with a yield strength of 280 MPa. If the design pressure is 3.5 MPa, determine the factor of safety against yielding based on the hoop stress.
#45. A ball bearing must support an equivalent dynamic load of 5 kN and operate for 6000 hours at 1200 rpm. Determine the required basic dynamic load rating for the bearing.
#46. A power screw with a mean diameter of 28 mm has a coefficient of friction of 0.12. For the screw to be self-locking, what is the maximum allowable lead?
#47. A V-belt drive has a driver pulley diameter of 175 mm and a driven pulley diameter of 450 mm. The center distance between the pulleys is 700 mm. Determine the required belt length.
#48. A lap joint is to be welded using two parallel fillet welds, each 80 mm long. The joint must carry a tensile load of 35 kN with an allowable shear stress of 95 MPa in the weld throat. Determine the minimum required leg size for the fillet welds.
#49. A flatness tolerance of 0.10 mm is specified on a surface. What is the meaning of this tolerance?
#50. A plane stress element is subjected to σx = 110 MPa, σy = -30 MPa, and τxy = 50 MPa. Using Mohr’s circle, determine the maximum in-plane shear stress.
#51. A component experiences an alternating stress of 80 MPa and a mean stress of 50 MPa. The material has a yield strength of 380 MPa and an endurance limit of 190 MPa. Using the Soderberg criterion, determine the factor of safety.
#52. A solid circular steel column has a diameter of 55 mm and a length of 2.5 m. The column has fixed-pinned end conditions (K = 0.7). Determine the slenderness ratio.
#53. A compression spring with a stiffness of 600 N/m is compressed by 120 mm. Determine the potential energy stored in the spring.
#54. A pressure pipe with an inner diameter of 180 mm must handle an internal pressure of 6.5 MPa. The allowable stress is 125 MPa and the weld joint efficiency is 0.80. Using the thin-wall approximation t = pD/(2SE), determine the minimum required wall thickness.
#55. A spur gear has a module of 5 mm and 28 teeth. Determine the pitch diameter of the gear.
#56. A single-shear riveted joint has 5 rivets, each with a diameter of 12 mm. The allowable shear stress for the rivet material is 75 MPa. Determine the maximum load the joint can carry based on rivet shear strength.
#57. A system consists of four identical components connected in parallel, each with a reliability of 0.82. Determine the overall system reliability.
#58. A flange coupling connects two shafts using 8 bolts of 10 mm diameter arranged on a bolt circle of 120 mm diameter. The allowable shear stress in the bolts is 55 MPa. Determine the maximum torque the coupling can transmit.
#59. A machined surface has a roughness specification of Ra = 1.6 µm. What is the equivalent surface roughness in microinches (µin)?
#60. A short rectangular column (50 mm × 70 mm) carries a compressive load of 40 kN with an eccentricity of 20 mm along the 70 mm dimension. Determine the maximum compressive stress in the column.
#61. A worm gear set has a single-threaded worm driving a gear with 36 teeth. If the worm rotates at 1500 rpm, determine the output speed of the gear.
#62. An M18 bolt (tensile stress area At = 192 mm²) is made of Grade 10.9 steel with a proof strength of 580 MPa. Determine the proof load of the bolt.
#63. A component follows a Weibull failure distribution with shape parameter β = 2.0 and characteristic life η = 4000 hours. Determine the B10 life (time at which 90% of components survive).
#64. A lead screw actuator is driven by a motor that produces 3.0 N·m of torque. The lead screw has a lead of 6 mm and an efficiency of 80%. Determine the linear thrust force produced by the actuator.
#65. A shaft has a circular runout tolerance of 0.04 mm specified relative to a datum axis. If the shaft has an eccentricity of 0.025 mm relative to the datum, what is the full indicator movement (FIM) that would be measured?
#66. A steel rod (E = 200 GPa) has a cross-sectional area of 300 mm² and a length of 1.8 m. When subjected to a tensile load of 45 kN, determine the strain energy stored in the rod.
#67. A stress element has principal stresses σ₁ = 130 MPa, σ₂ = 40 MPa, and σ₃ = -20 MPa. Determine the octahedral shear stress.
#68. A torsion spring is made from 3.0 mm diameter steel wire (E = 200 GPa) wound to a mean coil diameter of 25 mm with 8 active coils. Determine the angular deflection when a torque of 1.0 N·m is applied.
#69. A press fit connection has an interface pressure of 50 MPa between a 45 mm diameter shaft and a hub. The engagement length is 35 mm and the coefficient of friction is 0.12. Determine the torque capacity of the connection.
#70. A planetary gear set has a sun gear with 24 teeth and a ring gear with 72 teeth. The ring gear is fixed. If the sun gear rotates at 1800 rpm, determine the speed of the planet carrier.
#71. An M10 bolt (At = 58 mm²) in a bolted joint has an initial preload of 18 kN. The joint stiffness constant is C = 0.22. The external load varies from 4 kN to 12 kN. Determine the alternating stress in the bolt.
#72. A hole is located at a position that deviates from its nominal true position by 0.12 mm in the X-direction and 0.16 mm in the Y-direction. Determine the actual true position deviation (diametral value).
#73. A steel plate (ν = 0.3) is subjected to plane strain conditions with σx = 180 MPa and σy = 60 MPa. Determine the out-of-plane stress σz.
#74. A simply supported beam with a moment of inertia of 6.0 × 10⁶ mm⁴ and a span of 2.5 m is made of steel (E = 200 GPa). A concentrated load of 8 kN is applied at the center. Determine the maximum deflection.
#75. A helical compression spring has a wire diameter of 4 mm, mean coil diameter of 32 mm, and 10 active coils. The spring is made of steel (G = 79.3 GPa). A mass of 3 kg is attached to the spring. Determine the natural frequency of the spring-mass system.
#76. A thick-walled cylinder with an inner radius of 40 mm and outer radius of 80 mm is subjected to an internal pressure of 70 MPa. Determine the maximum shear stress in the cylinder wall.
#77. A power screw supports an axial load of 12 kN. The thrust collar has a mean diameter of 50 mm and a coefficient of friction of 0.08. Determine the torque required to overcome collar friction.
#78. A spur gear tooth is subjected to a tangential load of 2500 N. The gear has a module of 5 mm, face width of 25 mm, and Lewis form factor Y = 0.34. Using the Lewis equation, determine the bending stress at the root of the tooth.
#79. A single-lap adhesive joint connects two steel plates. The overlap length is 45 mm, the joint width is 20 mm, and the joint is subjected to a tensile load of 7 kN. If the allowable shear stress of the adhesive is 10 MPa, determine the factor of safety against adhesive shear failure.
#80. An assembly consists of four parts with bilateral tolerances of ±0.06, ±0.05, ±0.07, and ±0.04 mm. Using the statistical (RSS) method, determine the assembly tolerance.
#81. A component is subjected to cyclic loading at three different stress levels: 8,000 cycles at level 1 (N₁ = 40,000), 20,000 cycles at level 2 (N₂ = 80,000), and 40,000 cycles at level 3 (N₃ = 400,000). Using Miner’s rule, determine the cumulative damage fraction.
#82. A thin-walled circular tube has a mean radius of 35 mm and a wall thickness of 2.5 mm. The tube is subjected to a torque of 1800 N·m. Determine the shear stress in the tube wall.
#83. A rectangular key has a height of 10 mm and a length of 60 mm. It is used to connect a gear to a 50 mm diameter shaft transmitting a torque of 600 N·m. Determine the crushing stress in the key.
#84. A compound gear train has two stages. Stage 1 has a 24-tooth driver meshing with a 72-tooth driven gear. Stage 2 has a 20-tooth driver meshing with an 80-tooth output gear. Determine the overall gear ratio.
#85. A transverse fillet weld with a leg size of 10 mm and a length of 120 mm is subjected to a tensile load of 80 kN perpendicular to the weld axis. Determine the stress on the weld throat.
#86. A hub with inner radius 25 mm and outer radius 50 mm is shrink-fitted onto a solid shaft with an interface pressure of 45 MPa. Determine the tangential stress at the inner surface of the hub.
#87. A system consists of three components in series with individual MTBFs of 4000, 6000, and 10000 hours. Determine the system MTBF.
#88. A perpendicularity tolerance of 0.08 mm is specified for a 40 mm long cylindrical feature. What is the approximate maximum angular deviation from perpendicular that this tolerance allows?
#89. A solid steel shaft with a diameter of 40 mm is press-fitted into a steel hub with an outer diameter of 80 mm. The interface pressure is 35 MPa. Determine the maximum tangential stress in the hub at the inner surface.
#90. A helical compression spring with a stiffness of 20 N/mm is compressed from its free length by 35 mm. Determine the potential energy stored in the spring.
#91. A steel pipe (E = 200 GPa, α = 12 × 10⁻⁶ /°C) is fully constrained at both ends. If the temperature increases by 60°C, determine the thermal stress developed in the pipe.
#92. A tapered roller bearing is subjected to a radial load of 10 kN and an axial load of 5 kN. Using X = 0.4, Y = 1.5, and e = 0.38, determine the equivalent dynamic load.
#93. A clevis joint has a pin diameter of 16 mm passing through two clevis arms, each 10 mm thick. The joint carries a load of 18 kN. Determine the bearing stress on the clevis.
#94. A system has a mean time between failures (MTBF) of 3000 hours and a mean time to repair (MTTR) of 12 hours. Determine the system availability.
#95. A hole is specified as Ø10.0 +0.20/-0 mm with a position tolerance of Ø0.25 mm at MMC. Determine the virtual condition (maximum material boundary) of the hole.
#96. A simply supported beam with a span of 3 m carries a uniformly distributed load of 8 kN/m. The beam has a rectangular cross-section with width 70 mm and height 150 mm. Determine the maximum bending stress.
#97. A double-shear lap joint has 3 bolts of 14 mm diameter. The allowable shear stress for the bolt material is 85 MPa. Determine the maximum load the joint can carry.
#98. A solid steel shaft (G = 79.3 GPa) has a diameter of 35 mm and is 500 mm long. Determine the angle of twist when a torque of 450 N·m is applied.
#99. A cantilever beam made of steel (E = 200 GPa) has a length of 1.5 m and a circular cross-section with diameter 60 mm. A point load of 4 kN is applied at the free end. Determine the maximum deflection.
#100. A thin-walled cylindrical vessel with an inner diameter of 700 mm and wall thickness of 14 mm is subjected to an internal pressure of 5.5 MPa. Calculate the longitudinal stress in the vessel wall.
