#1. A tapered circular steel rod (E = 200 GPa) has diameter varying linearly from 30 mm at one end to 15 mm at the other over a length of 600 mm. If an axial tensile load of 25 kN is applied, determine the total elongation.
#2. A solid circular shaft is subjected to combined bending moment of 4 kN·m and torque of 6 kN·m. Using the maximum shear stress theory (Tresca), determine the equivalent torque.
#3. A compound cylinder is formed by shrinking an outer cylinder (inner radius 60 mm, outer radius 80 mm) onto an inner cylinder (inner radius 40 mm, outer radius 60 mm). If the interface pressure is 30 MPa, determine the maximum hoop stress in the inner cylinder.
#4. A cantilever beam of length L carries a linearly varying load from zero at the free end to w₀ at the fixed end. Express the maximum deflection in terms of w₀L⁴/EI.
#5. A steel beam (E = 200 GPa) with I-section (I = 85 × 10⁶ mm⁴) has a span of 8 m and is simply supported. If the beam carries a moving concentrated load of 50 kN, determine the absolute maximum bending moment.
#6. At a point in a stressed body, the principal stresses are σ₁ = 180 MPa, σ₂ = 60 MPa, and σ₃ = -30 MPa. If the yield strength is 280 MPa, determine the factor of safety using the von Mises criterion.
#7. A hollow circular shaft (outer diameter 100 mm, inner diameter 70 mm) transmits 250 kW at 400 RPM. Determine the maximum shear stress in the shaft.
#8. A continuous beam ABC has spans AB = 6 m and BC = 4 m. A UDL of 12 kN/m acts over span AB only. Using the three-moment equation, determine the bending moment at support B.
#9. A steel bar (E = 200 GPa, A = 400 mm²) of length 1 m is dropped from a height of 50 mm onto a rigid stop. Determine the maximum instantaneous stress in the bar. (Neglect the weight of the bar during impact)
#10. A beam-column with length 3 m is subjected to an axial compressive load of 200 kN and a transverse UDL of 8 kN/m. If E = 200 GPa and I = 30 × 10⁶ mm⁴, the critical load is 658 kN. Using the amplification factor method, determine the maximum bending moment.
#11. A thin-walled closed box section has outer dimensions 100 mm × 60 mm with uniform wall thickness 4 mm. If subjected to a torque of 3 kN·m, determine the shear stress in the walls.
#12. A simply supported beam (E = 200 GPa, I = 60 × 10⁶ mm⁴) of span 6 m carries two concentrated loads of 30 kN each at the third points. Determine the maximum deflection.
#13. A steel specimen shows the following Rockwell hardness: HRC 45. Using the approximation S_ut ≈ 3.41 × HB (where HB ≈ 420 for HRC 45), estimate the ultimate tensile strength.
#14. A shaft of length 2 m is subjected to distributed torque varying linearly from 0 at one end to 2 kN·m/m at the other. If G = 80 GPa and d = 60 mm (solid), determine the total angle of twist.
#15. At a point, the stress state is σₓ = 100 MPa, σᵧ = -50 MPa, τₓᵧ = 60 MPa. Determine the octahedral shear stress.
#16. A propped cantilever beam of length 5 m carries a concentrated load of 40 kN at midspan. Determine the reaction at the prop.
#17. A thick-walled cylinder with inner radius 30 mm and outer radius 50 mm is subjected to internal pressure of 80 MPa and external pressure of 20 MPa. Determine the maximum hoop stress.
#18. A rotating steel shaft (ρ = 7850 kg/m³) of diameter 80 mm and length 1.5 m spins at 3000 RPM. Determine the centrifugal stress due to self-weight (solid shaft, uniform rotation).
#19. A built-up section consists of two channels placed back-to-back with a gap. If each channel has A = 2500 mm², Iₓ = 25 × 10⁶ mm⁴ about its own centroid, and the centroids are 80 mm apart, determine the total moment of inertia about the combined centroidal x-axis.
#20. A steel wire (E = 200 GPa, A = 50 mm²) of length 3 m supports a weight of 5 kN at its end and is then subjected to a sudden additional load of 2 kN. Determine the maximum stress in the wire.
#21. A circular shaft has a keyway that reduces its torsional strength. If the shaft diameter is 60 mm and the stress concentration factor for the keyway is 1.6, determine the effective polar moment of inertia for stress calculations.
#22. A welded connection has two 8 mm fillet welds, each 150 mm long, oriented parallel to the applied load of 180 kN. Determine the shear stress in the welds.
#23. A fixed-fixed beam of span 8 m carries a UDL of 15 kN/m over its entire length. Determine the maximum positive bending moment (at midspan).
#24. A steel column (E = 200 GPa) has a W-section with A = 5000 mm², Iₓ = 120 × 10⁶ mm⁴, Iᵧ = 40 × 10⁶ mm⁴. The column is 6 m long, pinned at both ends, with lateral bracing about the weak axis at midheight. Determine the critical buckling load.
#25. A simply supported beam carries a concentrated moment M₀ at its center. If the span is L, E is the modulus, and I is the moment of inertia, determine the maximum deflection in terms of M₀L²/EI.
#26. A steel plate (40 mm wide, 10 mm thick) has a 10 mm diameter central hole. If the plate is subjected to an axial tensile load of 30 kN and the stress concentration factor is 2.3, determine the maximum stress at the hole edge.
#27. A composite beam consists of wood (80 mm × 200 mm, E = 12 GPa) with a steel plate (80 mm × 10 mm, E = 200 GPa) attached at the bottom. Determine the transformed width of the steel in terms of wood.
#28. A cantilever beam of length 2 m with rectangular cross-section (40 mm × 80 mm) has a vertical load of 6 kN at the free end applied at the top edge (eccentrically). Determine the maximum tensile stress considering combined bending and torsion.
#29. A thin ring (mean radius 200 mm, cross-sectional area 400 mm²) is subjected to a diametral compressive load of 5 kN. Determine the maximum bending stress in the ring.
#30. A shaft transmits power through a flange coupling with 6 bolts on a pitch circle diameter of 150 mm. If the shaft transmits 100 kW at 500 RPM and each bolt has a diameter of 16 mm, determine the shear stress in each bolt.
#31. A bi-metallic strip consists of steel (E = 200 GPa, α = 12 × 10⁻⁶/°C, t = 2 mm) bonded to aluminum (E = 70 GPa, α = 23 × 10⁻⁶/°C, t = 3 mm). If the temperature increases by 50°C, which way does the strip bend?
#32. A column with one end fixed and the other end free has an effective length factor K = 2. If the same column has both ends fixed (K = 0.5), what is the ratio of critical buckling loads (fixed-fixed to fixed-free)?
#33. A pressure vessel has hemispherical ends and a cylindrical middle section. If the internal pressure is p, diameter is d, and wall thickness is t, what is the ratio of hoop stress in the cylinder to the stress in the hemisphere?
#34. At a point on the surface of a shaft under combined bending and torsion, σₓ = 150 MPa (bending) and τₓᵧ = 80 MPa (torsion). Using the maximum principal stress theory, determine the equivalent bending moment factor if Mₑ = M/2 + ½√(M² + T²) and the actual M/T ratio is 15/8.
#35. A helical compression spring has 8 active coils, mean coil diameter 50 mm, wire diameter 8 mm, and G = 80 GPa. Determine the spring stiffness.
#36. A continuous beam with three equal spans of 4 m each carries a UDL of 10 kN/m over all spans. Determine the bending moment over the first interior support.
#37. A steel strut (E = 200 GPa) has a hollow circular cross-section with outer diameter 80 mm and inner diameter 60 mm. If the strut is 3 m long with pinned ends, determine the maximum eccentricity of load application such that the stress does not exceed 150 MPa when carrying 200 kN.
#38. A steel wire rope has an effective diameter of 20 mm and is wound around a drum of diameter 800 mm. If E = 100 GPa (effective for wire rope), determine the bending stress induced in the rope.
#39. A T-beam with flange 200 mm × 25 mm and web 25 mm × 175 mm carries a shear force of 100 kN. If the centroid is 145 mm from the bottom, determine the shear stress at the junction of flange and web.
#40. A cantilever beam (EI = 4 × 10⁶ N·m²) of length 3 m has a rotational spring (k_θ = 2 × 10⁶ N·m/rad) at the fixed end. If a load of 10 kN is applied at the free end, determine the deflection at the free end.
#41. A steel bar (cross-section 30 mm × 30 mm) is subjected to an axial tensile load P and a torque T simultaneously. If P = 50 kN and T = 200 N·m, determine the maximum principal stress at the surface.
#42. A riveted lap joint has 3 rivets in a row with pitch 60 mm. The plate is 12 mm thick and 180 mm wide. If the rivet diameter is 18 mm, the plate material has allowable tensile stress of 140 MPa, and the rivet material has allowable shear stress of 100 MPa, determine the efficiency of the joint.
#43. A simply supported beam of span 6 m has a rectangular cross-section. The beam must carry a concentrated load at midspan. If the allowable bending stress is 12 MPa and the maximum allowable shear stress is 1.5 MPa, and depth must be twice the width, determine the minimum width for a 20 kN load.
#44. A cylindrical bar of aluminum (E = 70 GPa, ν = 0.33) has diameter 40 mm and length 500 mm. When subjected to an axial compressive load of 100 kN, determine the change in diameter.
#45. A closed coiled helical spring with mean radius 40 mm and wire diameter 10 mm has n turns. If G = 80 GPa and the spring must have a stiffness of 50 N/mm, determine the required number of active turns.
#46. A steel plate (E = 200 GPa, ν = 0.3) is subjected to plane stress: σₓ = 120 MPa, σᵧ = 80 MPa, τₓᵧ = 40 MPa. Determine the principal strain ε₁.
#47. A shaft with diameter ratio (inner/outer) of 0.6 transmits the same torque as a solid shaft. Determine the ratio of weights of the hollow shaft to the solid shaft.
#48. A propped cantilever beam of length L carries a concentrated load P at distance ‘a’ from the fixed end. If the prop reaction is R_B, express R_B in terms of P when a = L/3.
#49. A steel beam (E = 200 GPa) with moment of inertia I = 50 × 10⁶ mm⁴ is subjected to a temperature gradient through its depth of 100 mm such that the top is 40°C hotter than the bottom (α = 12 × 10⁻⁶/°C). If the beam is simply supported over a span of 4 m, determine the maximum deflection due to the temperature gradient.
#50. A rectangular beam 60 mm wide and 120 mm deep is bent about its weak axis (60 mm dimension vertical). If the bending moment is 5 kN·m, determine the maximum bending stress.
#51. A thin-walled tube with mean radius 50 mm and thickness 3 mm is subjected to internal pressure p and axial tensile force F. If p = 2 MPa and the maximum allowable stress is 100 MPa, determine the maximum allowable axial force.
#52. A simply supported beam with overhang has supports at A and B (span AB = 5 m) with overhang BC = 2 m. A UDL of 12 kN/m acts over the overhang BC only. Determine the reaction at support A.
#53. A solid circular shaft of diameter d is replaced by a hollow shaft with the same outer diameter. If the inner diameter is 0.5d, by what percentage does the torsional strength decrease?
#54. A steel structure member is subjected to repeated loading between 0 and 180 MPa (tension only). If the ultimate strength is 500 MPa and the endurance limit is 240 MPa, determine the factor of safety using the Goodman criterion.
#55. A column fixed at the base and free at the top carries an eccentric load P at eccentricity e. The secant formula gives the maximum stress. If the slenderness ratio is 100 and E = 200 GPa, what is the Euler stress?
#56. A crane hook carries a load of 50 kN. The hook has a trapezoidal cross-section with inner width 40 mm, outer width 20 mm, and height 80 mm. If the center of curvature is at the inner edge, approximate the maximum stress.
#57. A beam of length 6 m is fixed at one end and simply supported at the other. It carries a UDL of 20 kN/m. Determine the maximum positive bending moment.
#58. A steel specimen is tested in fatigue with a stress amplitude of 200 MPa and a mean stress of 100 MPa. If S_ut = 600 MPa and S_e = 280 MPa, determine the safety factor using the Soderberg criterion (S_y = 400 MPa).
#59. A circular plate of radius R and thickness t is simply supported at its edge and carries a uniform pressure p. The maximum deflection at the center is given by (5+ν)pR⁴/64D(1+ν) where D = Et³/12(1-ν²). If ν = 0.3, p = 0.5 MPa, R = 100 mm, t = 5 mm, and E = 200 GPa, estimate the maximum deflection.
#60. A compression member has a yield strength of 350 MPa and slenderness ratio of 80. Using the Euler-Johnson parabolic formula with transition at slenderness ratio of 120, determine the critical stress. (E = 200 GPa)
#61. A steel bar (E = 200 GPa, ρ = 7850 kg/m³) of length 15 m and cross-sectional area 600 mm² hangs vertically. Determine the maximum stress at the top due to self-weight.
#62. A circular shaft has two pulleys mounted on it. Pulley A is at the left end and receives 15 kW at 500 RPM. Pulley B is 0.8 m from A and delivers 9 kW. Pulley C is 0.5 m from B and delivers the remaining power. Determine the maximum torque in the shaft.
#63. A steel plate (E = 200 GPa) of thickness 20 mm is bent into a cylinder of inner radius 500 mm. Determine the maximum bending stress induced in the plate.
#64. A riveted joint has two rows of rivets with 4 rivets in each row. The pitch in each row is 80 mm and the rows are staggered. The plate is 10 mm thick with allowable tensile stress 150 MPa. If rivet diameter is 20 mm, determine the allowable load for the joint based on plate tearing along a zigzag path.
#65. A prestressed concrete beam has a rectangular section 300 mm × 500 mm. A prestressing force of 800 kN is applied at an eccentricity of 100 mm below the centroid. Determine the stress at the top fiber.
#66. A shaft coupling uses 8 bolts on a pitch circle diameter of 200 mm. If the shaft transmits 150 kW at 300 RPM and each bolt has an allowable shear stress of 60 MPa, determine the minimum bolt diameter required.
#67. A thin-walled pressure vessel has an elliptical cross-section with semi-major axis a = 150 mm and semi-minor axis b = 100 mm, with uniform wall thickness t = 5 mm. If the internal pressure is 1.5 MPa, estimate the maximum hoop stress.
#68. A beam with varying moment of inertia I(x) = I₀(1 + x/L) is simply supported over length L and carries a UDL w. Using energy methods, the deflection at midspan is approximately 5wL⁴/384EI_avg. If L = 4 m, w = 10 kN/m, E = 200 GPa, and I₀ = 40 × 10⁶ mm⁴, estimate the midspan deflection.
#69. A steel plate with a double-edge notch is subjected to tension. The plate width is 60 mm, notch depth is 10 mm on each side, and notch radius is 2 mm. If the stress concentration factor is 2.4 and the applied load is 80 kN per unit thickness (mm), determine the maximum stress per mm thickness at the notch root.
#70. A stepped shaft with diameters 80 mm and 50 mm is subjected to bending. The fillet radius at the step is 5 mm. Using the stress concentration factor chart approximation (Kt ≈ 1 + 2√(D/r-1) for D/d=1.6), estimate Kt.
#71. A composite shaft consists of an inner brass core (d = 40 mm, G = 35 GPa) and an outer steel tube (OD = 60 mm, ID = 40 mm, G = 80 GPa). The composite transmits a torque of 4 kN·m. Determine the maximum shear stress in the steel portion.
#72. A cantilever beam with a rectangular cross-section carries a concentrated load at the free end. The load is applied at the top edge of the cross-section (not through the shear center). If the section is 40 mm wide × 100 mm deep, determine the torsional shear stress if P = 5 kN and the distance from shear center to load point is 50 mm.
#73. A reinforced concrete beam has effective depth d = 400 mm and width b = 250 mm. The steel reinforcement area is 1500 mm² at the bottom. If the modular ratio n = 10 and the beam is in pure bending, determine the depth of the neutral axis from the top.
#74. A curved beam with a rectangular cross-section (30 mm × 60 mm, with 60 mm in radial direction) has an inner radius of 40 mm. If the beam is subjected to a bending moment that tends to reduce the curvature (straighten the beam), determine the ratio of maximum stress at inner fiber to the maximum stress at outer fiber.
#75. A steel column (E = 200 GPa, σ_y = 300 MPa) has a rectangular section 100 mm × 50 mm and length 4 m with both ends pinned. Using the tangent modulus theory at the point where E_t = E/2, determine the critical stress.
#76. A beam is subjected to combined bending and axial compression. The cross-section is rectangular (100 mm × 150 mm), bending moment is 20 kN·m about the strong axis, and axial load is 200 kN. Determine the maximum compressive stress.
#77. A steel cable (E = 150 GPa, A = 500 mm²) of length 20 m supports a load at its end. If the cable sags by 25 mm under a load of 10 kN, determine the actual length of the cable along its catenary (approximate using parabolic assumption).
#78. A circular disk of outer radius 150 mm has a central hole of radius 30 mm. If the disk rotates at 300 rad/s (ρ = 7800 kg/m³, ν = 0.3), determine the maximum tangential stress (occurs at inner radius).
#79. A bolted connection uses high-strength friction-grip bolts. Each bolt is pretensioned to 170 kN. If the coefficient of friction between the plates is 0.35 and the joint is in double shear (two friction surfaces per bolt), determine the slip resistance per bolt.
#80. A simply supported beam of span L carries a moving load system consisting of two equal loads P spaced at distance ‘a’. For maximum bending moment under one of the loads, the loads should be positioned such that:
#81. A thin cylindrical shell of diameter 1000 mm and thickness 10 mm is subjected to internal pressure of 2 MPa and an axial tensile force of 500 kN. Determine the von Mises equivalent stress.
#82. A steel rod (E = 200 GPa, α = 12 × 10⁻⁶/°C) of diameter 20 mm and length 300 mm is placed between two rigid walls with an initial compressive stress of 50 MPa. If the temperature drops by 30°C, determine the final stress in the rod.
#83. A beam of unsymmetric I-section is subjected to a bending moment M in the vertical plane. If the principal axes are inclined at angle α to the geometric axes, the beam will bend:
#84. A cantilever beam of length 2 m carries a UDL over its entire span. At 1 m from the fixed end, a prop is introduced that deflects 2 mm when it comes in contact with the beam. If EI = 5 × 10⁶ N·m² and w = 10 kN/m, determine the reaction at the prop.
#85. A shaft of 60 mm diameter rotates at 600 RPM. A fatigue failure analysis indicates that the endurance limit in reversed bending is 280 MPa and in reversed torsion is 180 MPa. Using the ellipse criterion for combined fatigue loading, if the alternating bending stress is 140 MPa, what is the maximum alternating torsional stress?
#86. A steel beam (E = 200 GPa, I = 40 × 10⁶ mm⁴) of span 4 m is fixed at both ends. A settlement of 5 mm occurs at the right support. Determine the moment induced at the left support due to this settlement.
#87. A solid steel shaft and a hollow aluminum shaft have the same length, outer diameter, and transmit the same torque. If the hollow shaft has inner diameter = 0.6 × outer diameter and G_steel = 80 GPa, G_aluminum = 27 GPa, determine the ratio of angles of twist (aluminum/steel).
#88. A rectangular plate (a × b, thickness t) is simply supported on all edges and carries a uniform pressure p. The maximum bending stress at the center is approximately σ_max = βpb²/t² where β depends on a/b ratio. For a square plate (a = b), β ≈ 0.287. If a = b = 200 mm, t = 8 mm, and p = 0.1 MPa, determine the maximum stress.
#89. A continuous beam ABC has spans AB = 5 m and BC = 4 m. Support B settles by 10 mm. If EI = 20 × 10⁶ N·m², determine the bending moment at B due to settlement alone.
#90. A wood beam (E = 10 GPa) reinforced with a steel plate at the bottom (E = 200 GPa) is subjected to bending. The wood section is 100 mm × 200 mm and the steel plate is 100 mm × 10 mm. Using transformed section method, determine the maximum bending stress in steel if M = 15 kN·m.
#91. A thin-walled open section (channel) has flanges 60 mm × 5 mm and web 90 mm × 5 mm. Determine the torsional constant J using J = Σ(bt³/3).
#92. A beam is subjected to unsymmetric bending with M_y = 10 kN·m and M_z = 5 kN·m applied simultaneously. If I_y = 40 × 10⁶ mm⁴ and I_z = 15 × 10⁶ mm⁴, and the corner of the rectangular section is at y = 60 mm, z = 40 mm, determine the stress at that corner.
#93. A solid circular shaft is subjected to pure torsion. At what fraction of the outer radius does the shear stress equal 75% of the maximum shear stress?
#94. A simply supported beam carries a sinusoidally distributed load q(x) = q₀ sin(πx/L) over span L. The maximum bending moment at midspan is:
#95. A frame consists of two vertical columns connected by a horizontal beam at the top. If a horizontal load P is applied at the beam level, and each column has EI and height h, determine the lateral stiffness of the frame (assume columns are fixed at base and rigidly connected to beam).
#96. A steel bar (E = 200 GPa, α = 12 × 10⁻⁶/°C) is initially stress-free at 20°C. If one end is fixed and the other end is restrained by a spring of stiffness k = 500 kN/mm, determine the stress when temperature rises to 80°C. The bar has A = 500 mm² and L = 1 m.
#97. A thin-walled tube with mean radius R and thickness t is subjected to torque T. If a longitudinal slit is cut along the tube, by what factor does the torsional stiffness reduce (approximately)? Assume t << R.
#98. A horizontal beam of span 6 m is supported by a cable at its center. The cable has A = 200 mm², E = 200 GPa, and length 4 m. If the beam is rigid and carries a UDL of 20 kN/m, determine the tension in the cable.
#99. A shaft is designed to transmit a maximum torque with a factor of safety of 2. If the actual load varies from 0.5T to 1.5T (where T is the design torque), and using the Soderberg criterion with S_y in shear = 150 MPa and S_e in shear = 100 MPa, determine the allowable mean shear stress.
#100. A steel plate with a central crack of length 2a is subjected to tension. If the fracture toughness K_IC = 50 MPa√m, the crack length 2a = 20 mm, and the geometric factor Y = 1.12, determine the critical stress for fracture.
