Civil engineering cheat sheet

Static equilibrium

A body in static equilibrium has zero resultant force and zero resultant moment. Choose a clear free-body diagram.

ΣF = 0; ΣM = 0

Normal stress

Average normal stress is force divided by cross-sectional area in a simple axial-loading model.

σ = F/A; N/m² = Pa

Strain and stiffness

Axial strain compares length change with original length. In a linear elastic uniaxial model, stress is proportional to strain.

ε = ΔL/L; σ = Eε

Loads and responses

A load is an input; reactions, stresses and displacements are responses. Geometry and boundary conditions affect the model.

A simply supported beam and a cantilever have different constraints.

Free-body diagrams

Isolate one body and draw external forces and moments acting on it. Choose axes and show unknown reactions without also drawing the body’s forces on its surroundings.

A support reaction belongs on the isolated beam; the equal-and-opposite force acts on the support.

Moment and lever arm

A force’s moment about a point is force times perpendicular distance in a planar model. The distance is not always the length of the member.

A 200 N force with a 0.3 m perpendicular lever arm produces a 60 N·m moment.

Distributed loading

For a uniform load w across length L, the equivalent resultant is wL acting at the interval’s midpoint. Other load shapes have different resultants and centroids.

2 kN/m over 4 m gives an 8 kN resultant at the centre of that loaded length.

Water pressure

For a static liquid with approximately constant density, gauge pressure rises with depth: p = ρgh. Pressure and resultant force are different quantities.

At 2 m depth in water with ρ = 1000 kg/m³ and g = 9.8 m/s², p ≈ 19.6 kPa.

Worked example: beam reactions

A simply supported 6 m beam carries one downward 12 kN load at its centre. Symmetry and vertical equilibrium give an upward reaction of 6 kN at each support, neglecting beam weight.

Check 6 + 6 − 12 = 0, then check the moment balance about either support.

Common mistake: stress and force

Stress is force per area; force alone cannot compare members of different cross-sectional area. A simplified stress calculation is not a complete structural-design check.

For 10 kN acting uniformly over 500 mm², average normal stress is 20 N/mm² = 20 MPa.

Check your understanding

What are the SI units of bending moment?

N·m. Moment is force multiplied by perpendicular distance.

Curriculum connections

Sources