Mechanics of Materials: cram pack

Formula sheets, the key method steps for every topic and 14 practice problems, with full solutions at the back.

New problems Back to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

1Formulas and methods

Axial stress & strain

Normal stress: σ = F/A
Strain: ε = ΔL/L
Hooke’s law: σ = E·ε
Elongation: ΔL = FL/(EA)
Thermal strain: ε_T = α·ΔT

Torsion of shafts

Shear stress: τ = T·r/J
Solid shaft: J = πd⁴/32
Hollow shaft: J = π(D⁴ − d⁴)/32
Angle of twist: φ = TL/(GJ)
Power: P = T·ω = 2πnT/60

Method

Find the polar moment of area: J = πd⁴/32 for a solid shaft.
Shear stress at the surface: τ = T·r/J, with r the outer radius.
Angle of twist: φ = TL/(GJ) in radians.
If power is given, find the torque first with P = T·ω, ω = 2πn/60.

Shear force & bending moment

Shear and moment: V = ΣF on one side, M = ΣF·d on one side
Bending stress: σ = M·y/I
Rectangle: I = bh³/12
Section modulus: W = I/y_max
Simply supported, uniform load: M_max = wL²/8

Method

Find the support reactions from equilibrium of the whole beam.
Cut the beam at the section you need and keep one side.
Shear force: sum of the vertical forces on that side. Bending moment: sum of their moments about the cut.
For the maximum moment, find where the shear force is zero.

Centroid & second moment of area

Centroid: ȳ = ΣA_iy_i/ΣA_i
Rectangle: I = bh³/12
Parallel-axis theorem: I = I_c + A·d²
Composite section: I = Σ(I_c,i + A_id_i²)

Mohr’s circle & stress transformation

Centre: C = (σ_x + σ_y)/2
Radius: R = √(((σ_x − σ_y)/2)² + τ_xy²)
Principal stresses: σ_1,2 = C ± R
Principal direction: tan 2θ_p = 2τ_xy/(σ_x − σ_y)
Inclined plane: σ_θ = C + (σ_x − σ_y)/2·cos 2θ + τ_xy·sin 2θ

Method

Calculate the centre of Mohr’s circle: C = (σx + σy)/2.
Calculate the radius: R = √(((σx − σy)/2)² + τxy²).
The principal stresses are C ± R and the largest in-plane shear stress is R.
Angles on the circle are double the angles on the element: tan 2θp = 2τxy/(σx − σy).

Beam deflection

Point load at midspan: δ = PL³/(48EI)
Uniform load, simply supported: δ = 5wL⁴/(384EI)
Cantilever, end load: δ = PL³/(3EI)
Cantilever, uniform load: δ = wL⁴/(8EI)
Superposition: δ = δ₁ + δ₂

Column buckling

Euler load: P_cr = π²EI/(KL)²
Effective length factor: K = 1 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2 fixed–free
Critical stress: σ_cr = P_cr/A
Slenderness: λ = KL/r, r = √(I/A)

2Practice problems

1.Axial stress & strainMedium

Find the axial strain ε in the brass bar, in microstrain (µε = 10⁻⁶).

P = 55 kNd = 30 mmL = 2.5 mE = 100 GPa

ε = µε

2.Axial stress & strainHard

The stepped bar is fixed at the wall and loaded at its free end. Find the largest normal stress σ_max in the bar.

P = 35 kNsegment 1: Ø30 mm, 0.5 m, steel (E = 200 GPa)segment 2: Ø20 mm, 2 m, steel (E = 200 GPa)

σ_max = MPa

3.Torsion of shaftsMedium

A solid brass shaft is fixed at the wall and twisted by a torque T at its free end. Find the angle of twist φ of the free end, in degrees.

T = 100 N·md = 40 mmL = 3 mG = 39 GPa

φ = °

4.Torsion of shaftsHard

A solid shaft must transmit 100 kW at 300 rpm without the shear stress exceeding 75 MPa. Find the smallest diameter d that works.

P = 100 kWN = 300 rpmτ_allow = 75 MPa

d = mm

5.Shear force & bending momentMedium

Find the bending moment M_x at the section x = 0.5 m from support A. Use sagging moments positive and upward shear on the left face positive.

L = 10 mw = 10 kN/m from 0 to 10 mx = 0.5 m

M_x = kN·m

6.Shear force & bending momentHard

The marks A to G split the beam into six equal parts. At which mark is the bending moment largest in size, sagging or hogging? Click it in the diagram.

55 kN @ 0 m20 kN @ 12 m55 kN @ 8 m6 kN/m, 2–10 m

1A (x = 0 m)
2B (x = 2 m)
3C (x = 4 m)
4D (x = 6 m)
5E (x = 8 m)
6F (x = 10 m)
7G (x = 12 m)

7.Centroid & second moment of areaMedium

A symmetric I-section has flanges 160 mm wide and 12 mm thick, an overall depth of 260 mm and a web 8 mm thick. Find its elastic section modulus S_x for bending about the horizontal centroidal axis x.

B = 160 mmH = 260 mmt_f = 12 mmt_w = 8 mm

S_x = × 10³ mm³

8.Centroid & second moment of areaHard

A 150 mm × 300 mm plate has a rectangular cut-out 60 mm wide and 40 mm high, centred on the vertical centre line with its centre 105 mm above the bottom. Find the second moment of area I_x about the horizontal axis through the centroid.

b = 150 mmh = 300 mmcut-out: 60 × 40 mmhole centre: y = 105 mm

I_x = × 10⁶ mm⁴

9.Mohr’s circle & stress transformationMedium

A point has σₓ = −40 MPa, σᵧ = 70 MPa and τₓᵧ = −30 MPa (tension positive). Find the angle θ_s, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

σ_x = −40 MPaσ_y = 70 MPaτ_xy = −30 MPa

θ_s = °

10.Mohr’s circle & stress transformationHard

A point in a loaded part has σₓ = −30 MPa, σᵧ = −50 MPa and τₓᵧ = 50 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 60° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σ_x = −30 MPaσ_y = −50 MPaτ_xy = 50 MPaθ = 60°

τ_θ = MPa

11.Beam deflectionMedium

A cantilever of length L is fixed at A and carries a point load P at a distance a = 2 m from A. Find the deflection δ_P under the load.

P = 18 kNa = 2 mL = 3 mE = 200 GPa (steel)I = 23 × 10⁶ mm⁴

δ_P = mm

12.Beam deflectionHard

A simply supported beam carries a uniformly distributed load w. Its deflection may not exceed span/360. Find the smallest second moment of area I_req that meets this limit.

w = 4 kN/mL = 3 mE = 200 GPa (steel)

I_req = × 10⁶ mm⁴

13.Column bucklingMedium

A column of length L made of steel is fixed at the base and pinned at the top. Using Euler’s formula, find its critical buckling load P_cr.

L = 2 mE = 200 GPa (steel)d = 60 mm

P_cr = kN

14.Column bucklingHard

A solid round column of length L made of aluminium is fixed at the base and free at the top. It must carry 150 kN with a factor of safety of 3 against buckling. Find the smallest diameter d.

P = 150 kNn = 3L = 1.5 mE = 70 GPa (aluminium)

d = mm

3Solutions

1.Axial stress & strain

Answer: ε = 778.1 µε

Hooke’s law: within the elastic range σ = E·ε. Combining with σ = P/A and ε = δ/L gives δ = PL/(AE).

Stress

A = π·30²/4 = 706.9 mm²

σ = P/A = 55·1000/706.9 = 77.81 MPa

Strain (Hooke’s law)

E for brass = 100 GPa = 100000 MPa.

ε = σ/E = 77.81/100000 = 0.0007781 = 778.1 µε

Answer

ε = 778.1 µε

2.Axial stress & strain

Answer: σ_max = 111.4 MPa

Axially loaded members: each prismatic segment obeys δ = PL/(AE). For members in series the elongations add.

Same force in both segments

The free end carries P, so every section of the bar carries the same axial force P. Only the areas and materials change.

A₁ = π·30²/4 = 706.9 mm², A₂ = π·20²/4 = 314.2 mm²

σ₁ = 49.51 MPa

σ₂ = 111.4 MPa

Largest stress

The thinner segment has the smaller area, so it carries the larger stress.

σ_max = 111.4 MPa

Answer

σ_max = 111.4 MPa

3.Torsion of shafts

Answer: φ = 1.754 °

Torsion of circular shafts: shear stress grows linearly from zero at the centre to τ_max = T·r/J at the surface, with J = πd⁴/32 for a solid shaft and J = π(d_o⁴ − d_i⁴)/32 for a hollow one. The angle of twist is φ = TL/(GJ) in radians. Power and torque are linked by P = T·ω.

Polar second moment of area

J = π·40⁴/32 = 251327 mm⁴

Angle of twist

φ = TL/(GJ) = 100·1000·3000/(39000·251327) = 0.03061 rad

φ = 0.03061·180/π = 1.754°

Answer

φ = 1.754°

4.Torsion of shafts

Answer: d = 60.01 mm

Torque

ω = 2πN/60 = 2π·300/60 = 31.42 rad/s

T = P/ω = 100000/31.42 = 3183 N·m

Diameter from the allowable stress

d = ∛(16T/(π·τ_allow)) = ∛(16·3.183 × 10⁶/(π·75)) = 60.01 mm

Answer

d = 60.01 mm

In practice you would round up to the next standard size.

5.Shear force & bending moment

Answer: M_x = 23.75 kN·m

Method of sections: cut the beam where you want the internal forces and keep one side. The shear force V balances the vertical forces on that side, and the bending moment M balances their moments about the cut.

Support reactions

Equilibrium of the whole beam.

UDL resultant: W = w·ℓ = 10·10 = 100 kN, acting at x̄ = 5 m

ΣM_A = 0: R_B·10 = 10·10·5

R_B = 500/10 = 50 kN

ΣF_y = 0: R_A = 100 − 50 = 50 kN

Cut at x = 0.5 m and keep the left part

Only forces left of the cut appear. A UDL portion acts as one force at the middle of that portion.

M = R_A·x − (each load left of the cut × its distance to the cut)

M_x = 50·0.5 − 10·0.5·0.25 = 23.75 kN·m

Answer

M_x = 23.75 kN·m

6.Shear force & bending moment

Answer: B (x = 2 m)

Largest bending moment: M changes slope wherever the shear force changes, so its peaks sit where the shear force crosses zero: under a point load, inside a distributed load, or over a support of an overhang. At a fixed support of a cantilever the moment is largest at the wall. Compare the size of M at those places, sagging or hogging.

Support reactions

Take moments about one support, then use the vertical force balance.

R₁ = 101.5 kN, R₂ = 76.5 kN

Bending moment at every mark

Sum the moments of all forces on one side of the cut. Sagging is positive, hogging negative.

M_A = 0, M_B = −110 kN·m, M_C = −29, M_D = 28, M_E = 61, M_F = −40, M_G = 0

Answer

Mark B: M_B = −110 kN·m

7.Centroid & second moment of area

Answer: S_x = 521.9 × 10³ mm³

Second moment of area: for a rectangle about its own centroidal axis I = bh³/12, and about its base bh³/3 (parallel-axis theorem I = I_c + A·d²). A hole is handled by subtracting its I, as long as both parts share the same axis. The elastic section modulus is S = I/c, where c is the distance to the extreme fibre.

Outer rectangle minus the two side cut-outs

Both cut-outs together are (B − t_w) wide and h_w = H − 2t_f high, and share the axis x.

h_w = 260 − 2·12 = 236 mm

I_x = (B·H³ − (B − t_w)·h_w³)/12 = (160·260³ − 152·236³)/12

I_x = 6.785 × 10⁷ mm⁴

Section modulus

c = H/2 = 130 mm

S_x = I/c = 6.785 × 10⁷/130

S_x = 521944 mm³ = 521.9 × 10³ mm³

Answer

S_x = 521.9 × 10³ mm³

8.Centroid & second moment of area

Answer: I_x = 332 × 10⁶ mm⁴

Sections with holes: treat a hole as a negative area. The centroid is ȳ = (A₁y₁ − A₂y₂)/(A₁ − A₂), with A₁, y₁ for the full plate and A₂, y₂ for the hole. For the second moment of area about the centroidal axis, use the parallel-axis theorem for each part and subtract the hole: I = (I_c1 + A₁d₁²) − (I_c2 + A₂d₂²), where d is the distance from each part's own centroid to the axis through ȳ. A rectangle has I_c = bh³/12 and a circle I_c = πd⁴/64.

Centroid

ȳ = (45000·150 − 2400·105)/(45000 − 2400) = 152.5 mm

Full plate about ȳ

150·300³/12 + 45000·−2.535² = 337.8 × 10⁶ mm⁴

Hole about ȳ

60·40³/12 + 2400·−47.54² = 5.743 × 10⁶ mm⁴

Subtract

I_x = 332 × 10⁶ mm⁴

Answer

I_x = 332 × 10⁶ mm⁴

9.Mohr’s circle & stress transformation

Answer: θ_s = 59.31 °

Plane stress transformation: on a plane rotated by θ, σ = (σₓ + σᵧ)/2 + (σₓ − σᵧ)/2·cos 2θ + τₓᵧ·sin 2θ. Mohr’s circle shows all planes at once: its centre is at C = (σₓ + σᵧ)/2 and its radius is R = √(((σₓ − σᵧ)/2)² + τₓᵧ²). The principal stresses are C ± R, and the largest in-plane shear stress is R.

Angle on the circle

On Mohr’s circle the top point (+R) is 90° from the principal point σ₁; on the element that is 45°.

cos 2θ_s = −30/62.65 = −0.4789, sin 2θ_s = −(−55)/62.65 = 0.8779

2θ_s = atan2(55, −30) = 118.6°

Angle on the element

θ_s = 59.31°

check: θ_s − θ_p = 59.31 − (−75.69) = 135°

Answer

θ_s = 59.31°

Positive angles are counter-clockwise.

10.Mohr’s circle & stress transformation

Answer: τ_θ = −33.66 MPa

Transformation equation

τ_θ = −10·0.866 + 50·−0.5 = −33.66 MPa

Answer

τ_θ = −33.66 MPa

11.Beam deflection

Answer: δ_P = 10.43 mm

Beam deflection: deflections come from integrating the curvature M/(EI) twice. Standard cases are tabulated, for example PL³/(3EI) for a cantilever with an end load and 5wL⁴/(384EI) for a simply supported beam under a full uniform load. Deflections from separate loads add (superposition).

Consistent units

Work in N and mm: P in N, w in N/mm (kN/m = N/mm), L in mm, E in N/mm² (GPa × 1000), I in mm⁴.

P = 18000 N, a = 2000 mm, L = 3000 mm

EI = 4.6 × 10¹² N·mm²

Deflection under the load

δ_P = Pa³/(3EI) = 18000·2000³/(3·4.6 × 10¹²) = 10.43 mm

Answer

δ_P = 10.43 mm

12.Beam deflection

Answer: I_req = 2.531 × 10⁶ mm⁴

Deflection limit

δ_allow = L/360 = 3000/360 = 8.333 mm

Required second moment of area

I = 5wL⁴/(384·E·δ_allow) = 5·4·3000⁴/(384·200000·8.333) = 2.531 × 10⁶ mm⁴

I_req = 2.531 × 10⁶ mm⁴

Answer

I_req = 2.531 × 10⁶ mm⁴

13.Column buckling

Answer: P_cr = 640.7 kN

Euler buckling: a slender column becomes unstable at P_cr = π²EI/(KL)², where KL is the effective length: K = 1 for pinned–pinned, 2 for fixed–free, 0.7 for fixed–pinned and 0.5 for fixed–fixed. It buckles about the axis with the smallest I. The formula holds for slender columns, whose critical stress stays below the yield stress.

Smallest second moment of area

A round section has the same I about every axis.

I = πd⁴/64 = π·60⁴/64 = 636173 mm⁴

Effective length

KL = 0.7·2000 = 1400 mm

Critical load

P_cr = π²EI/(KL)² = π²·200000·636173/1400² = 640691 N = 640.7 kN

Answer

P_cr = 640.7 kN

14.Column buckling

Answer: d = 104.5 mm

Required critical load

P_cr = n·P = 3·150000 = 450000 N

KL = 2·1500 = 3000 mm

Solve for the diameter

d = (64·n·P·(KL)²/(π³E))^(1/4) = (64·450000·3000²/(π³·70000))^(1/4) = 104.5 mm

Answer

d = 104.5 mm

Check: the slenderness KL/r is 114.8, so the column is slender and Euler applies.

ExamProblems · examproblems.com/courses/mechanics-of-materials