#1. A force F = (3i + 4j – 2k) kN acts at point A(2, -1, 3) m. Determine the moment of this force about a line passing through the origin with direction vector (1, 1, 1).
#2. Three forces act at a point: F₁ = 120 N at α₁ = 30°, F₂ = 80 N at α₂ = 150°, F₃ = 100 N at α₃ = 240°. Determine the direction of the equilibrant force measured counterclockwise from the positive x-axis.
#3. A wrench applies a force F = 50 N at point (0.3, 0.1, 0) m, directed along vector (0, 4, 3). The wrench handle is along the x-axis. Determine the effective moment about the x-axis.
#4. A system of forces is equivalent to a wrench with resultant force R = 500 N and moment M = 200 N·m. If R and M are parallel, determine the pitch of the wrench.
#5. Four forces of magnitudes 10, 20, 30, and 40 N act at the vertices of a square of side 2 m, all directed toward the center. Determine the magnitude of the resultant moment about the center.
#6. A force F = 100 N passes through points A(1, 2, 3) m and B(4, 6, 3) m. Determine the scalar component of the moment about the z-axis through the origin.
#7. A distributed force varies as q(x) = q₀(1 + x/L) over length L = 4 m, where q₀ = 5 kN/m. Determine the location of the resultant from x = 0.
#8. A force system consists of F₁ = 100i N at (0, 2) m and F₂ = 100j N at (3, 0) m. The system can be reduced to a single force at what point along the x-axis?
#9. Two couples M₁ = 50 N·m and M₂ = 80 N·m act in planes making 60° with each other. Determine the magnitude of the resultant couple.
#10. A force F = 200 N acts along a line with direction cosines l = 0.5, m = 0.7, n = 0.5099. Determine the moment about the y-axis if the force passes through (2, 0, 3) m.
#11. A cable supports loads at three points along its span. The horizontal component of tension is constant at 40 kN. If the sag at the lowest point is 2.5 m and the total span is 50 m, determine the maximum cable tension.
#12. The resultant of a force system passes through (2, 3, 1) m with magnitude 500 N in direction (0.6, 0.8, 0). The system also produces a couple moment of 300k N·m. Determine the total moment about the origin.
#13. A parabolic cable carries a uniformly distributed horizontal load of 8 kN/m over a span of 60 m. The supports are at the same level and the sag is 6 m. Determine the total length of the cable (approximate).
#14. A force F acts along the line r = (1 + 2t)i + (2 + t)j + (3 – t)k. If |F| = 120 N, determine the moment about point (1, 2, 3) m.
#15. Three non-coplanar forces of 100, 150, and 200 N act along the edges of a cube of side 1 m from one vertex. Determine the magnitude of the resultant force.
#16. A continuous beam ABC has spans AB = 6 m and BC = 4 m. A uniformly distributed load of 10 kN/m acts on span AB. Using moment distribution or similar methods, determine the support moment at B (approximately).
#17. A propped cantilever beam of length 8 m carries a concentrated load of 40 kN at midspan. Determine the reaction at the prop (simple support).
#18. A fixed-fixed beam of span 10 m carries a uniformly distributed load of 6 kN/m. Determine the maximum bending moment in the beam.
#19. A compound beam consists of beam AB (cantilever, 4 m) connected by a hinge at B to beam BC (simply supported at C, 5 m). A load of 30 kN acts at A. Determine the reaction at C.
#20. A beam on elastic foundation has spring constant k = 500 kN/m² and flexural rigidity EI = 2 × 10⁴ kN·m². Determine the characteristic length parameter λL for a beam of length 10 m.
#21. A simply supported beam of span 12 m has an influence line for bending moment at midspan. When a unit load is placed at 3 m from the left support, determine the influence line ordinate for moment at midspan.
#22. A three-hinged parabolic arch has a span of 24 m and a rise of 6 m. A uniformly distributed load of 15 kN/m covers the entire span. Determine the maximum bending moment in the arch.
#23. A cantilever beam of length L carries a linearly varying load from zero at the free end to w₀ at the fixed end. If w₀ = 12 kN/m and L = 6 m, determine the reaction moment at the fixed support.
#24. A beam fixed at both ends has a span of 8 m and carries a concentrated load of 48 kN at one-third span from the left. Determine the fixed-end moment at the left support.
#25. A simply supported beam of span 10 m carries loads such that the shear force diagram shows +40 kN from left support to 4 m, then -20 kN from 4 m to right support. Determine the maximum bending moment.
#26. A beam ABC has a hinge at B. Span AB = 5 m (simply supported at A), span BC = 3 m (simply supported at C). A moment of 40 kN·m is applied at B. Determine the reaction at A.
#27. A continuous beam with three equal spans of 6 m each carries uniformly distributed load of 8 kN/m on all spans. Using three-moment equation, determine the moment at the first interior support.
#28. A simply supported beam has span 8 m. A moving load system consists of two loads: 20 kN followed by 30 kN at 2 m spacing. Determine the absolute maximum shear force in the beam.
#29. A cantilever beam of length 4 m has a moment of inertia that varies linearly from I₀ at the free end to 2I₀ at the fixed end. A concentrated load P acts at the free end. The deflection at the free end compared to uniform section I₀ is:
#30. A portal frame has columns 4 m high and beam 6 m long. The columns are fixed at the base. A horizontal load of 20 kN acts at the beam level on the left column. Determine the horizontal reaction at the right base.
#31. A statically indeterminate truss has one redundant member. The redundant member has length 5 m, cross-sectional area 1000 mm², and modulus E = 200 GPa. If removing this member causes adjacent joint displacement of 2 mm, determine the force in the redundant member.
#32. Using the method of virtual work, determine the vertical deflection at the center of a simply supported truss of span 12 m and height 3 m under a central load of 60 kN. All members have A = 800 mm² and E = 200 GPa. Top chord length = 6.71 m each, bottom chord = 12 m, diagonals = 4.24 m each.
#33. A space truss is supported at three points forming a triangle. The truss has 9 joints and 18 members. Determine the degree of static indeterminacy.
#34. A K-truss has a span of 20 m with 5 panels of 4 m each and height of 3 m. A concentrated load of 50 kN acts at the center of the bottom chord. Determine the force in the central vertical member.
#35. A three-bar truss has two bars meeting at the load point at 60° to the horizontal and a third bar horizontal. All bars have the same AE. A vertical load P is applied. Determine the force in the horizontal bar.
#36. A tower truss has two diagonal members at a joint, one at +45° and one at -45° from horizontal. If the vertical component of load at the joint is 100 kN downward, determine the force in each diagonal assuming the horizontal member carries no axial load.
#37. A compound truss is formed by connecting two simple trusses with three linking members. The simple trusses are internally stable. Determine the condition for the compound truss to be stable.
#38. A Fink truss has an apex angle of 120° and a span of 18 m. A uniformly distributed load of 6 kN/m acts on the top chord (on both sides). Determine the horizontal thrust at the supports.
#39. A planar frame has 4 members, 4 joints, and is fixed at two support points (6 reactions). Using the formula 3m + r – 3j, determine the degree of static indeterminacy.
#40. A cantilever truss extends 8 m from a wall with a depth of 2 m. It has 3 top chord panels and corresponding bottom chord, diagonals, and verticals. A load of 30 kN acts at the free end on the bottom chord. Determine the force in the top chord panel adjacent to the wall.
#41. In analyzing a statically indeterminate truss by the force method, a member is cut and replaced with redundant forces X. If the flexibility coefficient f₁₁ = 0.002 m/kN and the displacement due to external loads Δ₁ = 0.05 m, determine X.
#42. A symmetric bowstring truss has parabolic top chord with rise 3 m and horizontal bottom chord of span 15 m. Under uniform horizontal loading of 5 kN/m on the bottom chord, determine the maximum compressive force in the top chord.
#43. A frame has three members meeting at a rigid joint. Member 1 is horizontal (4 m), member 2 is at 60° above horizontal (3 m), member 3 is at 60° below horizontal (3 m). A 50 kN horizontal force acts at the joint. Determine the axial force in member 1.
#44. A truss joint connects 4 members: two horizontal (opposing), one vertical downward, and one at 45° upward-right. If the left horizontal member has 80 kN tension and the vertical member has 40 kN compression, determine the force in the 45° member.
#45. A determinate arch is subjected to a moving concentrated load. The influence line for horizontal thrust at the support shows maximum ordinate h/4r where h is load and r is rise. For span 30 m and rise 6 m, determine the maximum thrust due to a 100 kN load.
#46. A Z-section consists of two flanges 60 mm × 10 mm connected by a web 10 mm × 80 mm, with flanges offset in opposite directions. Determine the product of inertia Ixy about centroidal axes.
#47. A rectangular section 50 mm × 100 mm has its moment of inertia determined about an axis at 30° to the horizontal centroidal axis. Determine this moment of inertia using Mohr’s circle.
#48. An unequal angle section 100 mm × 75 mm × 8 mm has its principal axes oriented at an angle θ to the geometric axes. Determine the approximate angle θ.
#49. A composite section consists of two channels placed back-to-back with a gap of 20 mm. Each channel has Ix = 2.5 × 10⁶ mm⁴, A = 2000 mm², and centroid 25 mm from back. Determine the total Ix of the composite section.
#50. A thin-walled tube of mean radius 50 mm and wall thickness 3 mm is subjected to torsion. Determine the polar moment of inertia.
#51. A built-up section consists of a W-section with cover plates welded to both flanges. The W-section has Ix = 300 × 10⁶ mm⁴ and depth 400 mm. Each cover plate is 200 mm × 15 mm. Determine the total Ix.
#52. A semicircular area of radius R has its moment of inertia about the base (diameter). Determine this in terms of R.
#53. The radius of gyration of a hollow circular section with outer radius 60 mm and inner radius 40 mm about a centroidal axis is:
#54. A composite beam has a steel plate (E = 200 GPa) 150 mm × 10 mm attached to a wood section (E = 10 GPa) 150 mm × 200 mm. Using transformed section method, determine the neutral axis location from the bottom of wood.
#55. A quarter-ellipse with semi-axes a = 60 mm and b = 40 mm has its centroid located at:
#56. A T-section has unsymmetric flanges: top flange 120 mm × 15 mm, bottom flange 80 mm × 15 mm, web 15 mm × 150 mm. Determine the horizontal centroidal axis location from the bottom.
#57. Using the perpendicular axis theorem, if Ix = 3 × 10⁶ mm⁴ and Iy = 2 × 10⁶ mm⁴ for a thin plate, determine Iz.
#58. A hexagonal cross-section has a flat-to-flat dimension of 80 mm. Determine the moment of inertia about an axis through the center parallel to two flats.
#59. A plastic section modulus is calculated for a rectangular section 50 mm × 100 mm. Determine the ratio of plastic to elastic section modulus.
#60. An I-section has top flange 200 mm × 20 mm, web 15 mm × 260 mm, bottom flange 150 mm × 25 mm. Determine the distance of centroid from bottom.
#61. A double-threaded square screw has lead 12 mm and mean diameter 40 mm. For coefficient of friction 0.12, determine the torque to raise a 10 kN load.
#62. A vehicle of mass 1500 kg has its center of gravity 0.6 m above ground, 1.2 m behind the front axle, and 1.8 m ahead of the rear axle. The coefficient of friction is 0.7. Determine the maximum deceleration before front wheels lock.
#63. A collar bearing supports an axial load of 8 kN. The inner and outer radii are 30 mm and 60 mm. For uniform wear theory and μ = 0.08, determine the friction torque.
#64. Two wedges, each with angle 10°, are used to position a heavy machine. Coefficient of friction is 0.15 on all surfaces. Determine if the system is self-locking.
#65. A band brake has drum radius 200 mm and angle of contact 270°. The coefficient of friction is 0.25. For tight side tension of 2000 N, determine the braking torque.
#66. A ladder of length L and weight W rests on a floor (μ₁ = 0.3) and against a wall (μ₂ = 0.2). Determine the minimum angle with the floor for equilibrium.
#67. A conical pivot bearing has cone angle 30° and supports a load of 5 kN. The bearing has inner radius 20 mm and outer radius 50 mm. For μ = 0.1 and uniform pressure, determine the friction torque.
#68. A rope passes over a rough cylindrical drum through angles of 90° on each side of the top. If one end carries 500 N and μ = 0.3, determine the range of tension on the other end for equilibrium.
#69. A screw jack with square threads has efficiency 40% when raising a load. Determine the efficiency when lowering the same load.
#70. A block of weight W rests on a plane inclined at 25°. A force P is applied horizontally. If μs = 0.35, determine the ratio P/W for impending motion up the plane.
#71. A V-belt has a groove angle of 38°. If the coefficient of friction is 0.3, determine the effective coefficient of friction.
#72. A differential screw has two threads: outer thread with pitch 5 mm and inner thread with pitch 3 mm, both single-start. Determine the mechanical advantage neglecting friction.
#73. A block resting on a surface starts sliding when the surface is inclined at 35°. When a 50 N horizontal force is applied while the surface is horizontal, the block begins to slide if it weighs at least:
#74. A pivoted brake shoe presses against a rotating drum. The pressure varies as p = p₀ sin θ, where θ is measured from one end. The shoe subtends 90° and has width b and radius r. Determine the normal force in terms of p₀, b, and r.
#75. A thrust bearing consists of 8 pads each subtending 30° at mean radius 100 mm. For axial load 4 kN, μ = 0.04, and uniform pressure, determine the friction power loss at 1500 rpm.
#76. A hemispherical shell of radius R rests on a horizontal surface. A small weight is placed on the rim. Determine the maximum weight (as fraction of shell weight W) that can be placed without tipping.
#77. A uniform rod of length L and weight W is hinged at one end and held at angle θ to the horizontal by a string attached to the other end. If the string is horizontal, determine the string tension.
#78. A floating body has metacentric height 0.5 m. If the body is given a small angular displacement of 5°, determine the righting moment per unit weight of the displaced fluid.
#79. A structure is externally statically indeterminate to degree 2. It has 15 members and 8 joints. Determine the degree of internal indeterminacy.
#80. A uniform plank of length 4 m is placed on two supports. Support A is 0.5 m from one end; support B is 1 m from the other end. If the plank weighs 200 N, determine the load that can be placed at the free end (0.5 m side) before it tips.
#81. A crane has a counterweight of 50 kN at 3 m from the pivot. The lifted load is at a variable distance from the pivot. If the crane boom weighs 10 kN with center of gravity 4 m from the pivot (on the load side), determine the maximum load at 8 m from pivot.
#82. An A-frame consists of two identical legs each 5 m long, joined at the apex, with feet 4 m apart. A horizontal force of 8 kN is applied at the apex. Determine the horizontal component of the reaction at each foot.
#83. A rigid frame has a horizontal beam 6 m long supported by two inclined props at 45° to the horizontal, attached at 2 m from each end. A uniformly distributed load of 10 kN/m acts on the beam. Determine the force in each prop.
#84. A cylindrical vessel has hemispherical ends. The cylinder has length 2 m and radius 0.5 m. It is filled with water (density 1000 kg/m³) and supported horizontally at its centroidal axis. Determine the support reactions when full.
#85. A three-hinged arch has span 24 m and rise 6 m with the crown hinge at mid-span. A horizontal thrust of 40 kN is applied at the left support. Determine the vertical reaction at the right support.
#86. A uniform disk of radius R and weight W rests against a step of height h (h < R). Determine the minimum horizontal force P applied at the center to roll the disk over the step.
#87. A cable supports two concentrated loads of 30 kN and 50 kN at horizontal distances of 8 m and 18 m from the left support, which is 3 m lower than the right support. Total span is 24 m. Determine the horizontal tension component.
#88. A bent rod consists of two perpendicular segments: horizontal (1.5 m) and vertical (1 m). It is pinned at the end of the horizontal segment and carries a 200 N load at the free end of the vertical segment. Determine the pin reaction.
#89. A uniform sphere of weight W rests between two planes: one vertical and one inclined at 60° to the horizontal. If both surfaces are smooth, determine the reaction from the inclined plane.
#90. A composite body consists of a cone (height h₁, base radius R) on top of a cylinder (height h₂, radius R). Both have the same density. If h₁ = h₂ = R, determine the height of the center of gravity from the base of the cylinder.
#91. A simply supported beam of span L carries a uniformly distributed load w. Using virtual work, determine the central deflection if EI is constant. Express in terms of wL⁴/EI.
#92. A pin-jointed frame has 5 members meeting at a joint where a vertical load P is applied. Using the principle of virtual work, if the virtual displacement at the joint is δ, determine the work done by P.
#93. A mechanism has two links of length 0.5 m and 0.3 m. A force of 100 N is applied at the end of the longer link. Using virtual work, determine the force at the end of the shorter link when both links are horizontal.
#94. A cantilever beam of length 3 m carries a point load of 10 kN at the free end. Using Castigliano’s theorem, if EI = 2 × 10⁴ kN·m², determine the deflection at the free end.
#95. A three-bar truss with all members having AE = 10⁵ kN carries a vertical load of 50 kN. Using virtual work, if the member lengths are 3, 4, and 5 m respectively, determine the vertical deflection at the load point.
#96. A toggle mechanism has two arms of equal length L at angle θ to each other. A vertical force P is applied at the apex. Using virtual work, determine the horizontal force Q at the slider when θ = 30°.
#97. For a simply supported beam with concentrated load P at distance ‘a’ from left support and ‘b’ from right (a + b = L), the deflection under the load by Castigliano’s method is:
#98. A portal frame with hinged feet has columns of height h and beam of length L. A horizontal force H is applied at the beam level. Using virtual work, determine the vertical reaction at one support in terms of H, L, and h.
#99. A system of two parallel springs (k₁ = 100 N/m and k₂ = 200 N/m) supports a rigid bar. Using potential energy method, determine the deflection when a 30 N load is applied at the center.
#100. Using the principle of minimum potential energy, a beam on an elastic foundation (k per unit length) with concentrated central load P has deflection at the load point proportional to:
