Statics: practice problems and exam questions

Engineering mechanics: statics. Equilibrium of rigid bodies, support reactions, trusses, internal forces in beams and the properties of cross-sections. Almost every exam question starts with a correct free-body diagram.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Vectors: dot & cross productmagnitude and components · dot product · cross product and momentscalculations
  2. Beam support reactions1 point load · 2 point loads · point load + UDLcalculations
  3. Truss: method of jointssymmetric · asymmetric · vertical + horizontal loadcalculations
  4. Frames & machinestwo-force members and simple machines · pin forces and linkages · multi-member framescalculations
  5. Shear force & bending momentpoint load · full-span UDL · point load + UDLcalculations
  6. Centroid & second moment of areacentroid · second moment of area · bending stresscalculations

Exam tips for Statics

  • Draw the free-body diagram first and replace every support with its reaction forces.
  • Take moments about a point where unknown forces act, so they drop out of the equation.
  • Check your result with the equation you did not use: ΣF = 0 in the other direction is a quick test.

Worked examples

Vectors: dot & cross productEasy
xy135°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 110 N at 0° and F₂ = 190 N at 135°, both measured anticlockwise from the positive x-axis. Find the direction φ of their resultant, measured anticlockwise from the positive x-axis (between 0° and 360°).

F1 = 110 N, θ1 = 0°F2 = 190 N, θ2 = 135°

Find φ (°)

Show hints
  1. Split each force into x- and y-components.
  2. Add the x-components and the y-components separately.
  3. Rₓ = −24.35 N, Ry = 134.4 N.
Show the answer and the step-by-step solution

Vectors: a vector a = (ax, ay, az) has magnitude |a| = √(ax² + ay² + az²), and the unit vector in its direction is û = a/|a|. A vector of magnitude F at an angle θ measured anticlockwise from the positive x-axis has components Fx = F·cos θ and Fy = F·sin θ. Vectors add component by component.

Components of the resultant

Rₓ = 110·cos 0° + 190·cos 135° = −24.35 N
Ry = 110·sin 0° + 190·sin 135° = 134.4 N

Direction

Rₓ is negative, so the resultant points into the second or third quadrant: add 180° to the calculator angle.

arctan(Ry/Rₓ) = arctan(134.4/−24.35) = −79.73°
φ = 100.3°

Answer

φ = 100.3°
Beam support reactionsMedium
P1​ = 15 kNP2​ = 30 kNABRA​RB​ = ?2 m1 m3 m

The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RB.

L = 6 mP1 = 15 kN at 2 mP2 = 30 kN at 3 m

Find RB (kN)

Show hints
  1. Use static equilibrium: ΣFy = 0 and ΣM = 0. Two unknowns, two equations.
  2. Take moments about A. RA passes through A, so it drops out and you get RB directly.
  3. Total downward load is 45 kN, so RA + RB = 45 kN.
Show the answer and the step-by-step solution

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Taking moments about one support removes that support’s reaction from the equation.

P1​ = 15 kNP2​ = 30 kNABRA​RB​2 m1 m3 m

Free-body diagram

Replace the supports with reaction forces. A roller gives one vertical reaction; with only vertical loads the pin gives one vertical reaction too.

Equilibrium

Moments about A, counter-clockwise positive. Then vertical force balance.

ΣMA = 0: RB·6 = 15·2 + 30·3
RB = 120/6 = 20 kN
ΣFy = 0: RA = 45 − 20 = 25 kN

Answer

RB = 20 kN

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Beam reaction calculatorShear force and bending moment diagram calculatorMoment of inertia calculator (second moment of area)Bending stress calculator (σ = My/I)Centroid calculator (composite shapes and polygons)Truss calculator (method of joints)

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