AEE Mechanical Engineering Core · Chapter 1

Strength of Materials

What to remember

Stress and strain: within the elastic limit, stress is proportional to strain (Hooke's law), σ = Eε. The elastic constants E, G, K and Poisson's ratio ν are linked by fixed formulas.
Beams, shafts and columns follow three basic equations: bending M/I = σ/y = E/R, torsion T/J = τ/r = Gθ/L, and Euler's buckling Pcr = π²EI/Le².
Combined stresses are handled by principal stresses and Mohr's circle. Failure of ductile materials is checked by the maximum shear stress or distortion energy theory.

1. Simple stress, strain and elastic constants

Normal stress σ = P/A (N/m² = Pa; 1 MPa = 1 N/mm²). Shear stress τ = P/A over the sheared area.
Normal strain ε = δL/L (no unit). Shear strain γ is the angle of distortion in radians.
Hooke's law: σ = Eε. Young's modulus E = σ/ε. Shear modulus G = τ/γ. Bulk modulus K = p/(ΔV/V).
Poisson's ratio ν = − lateral strain/longitudinal strain. For most metals it is about 0.25 to 0.33. Its upper limit for any material is 0.5 (incompressible).
Relations: E = 2G(1 + ν); E = 3K(1 − 2ν); E = 9KG/(3K + G).
Elongation of a bar: δ = PL/AE. For a bar of varying section, add the elongations of each part. For a uniformly tapering circular bar: δ = 4PL/(πEd₁d₂).
Stress–strain curve of mild steel: proportional limit, elastic limit, upper and lower yield points, ultimate stress, then necking and fracture. Ductile materials show a large plastic region. Brittle materials (cast iron, concrete) fracture with little strain.
Factor of safety = ultimate (or yield) stress/working stress.
Thermal stress in a fully restrained bar: σ = αEΔT. If the bar is free to expand, no stress develops.
Composite bars in parallel share the load with equal strain: P = σ₁A₁ + σ₂A₂ and σ₁/E₁ = σ₂/E₂.
Strain energy U = σ²/(2E) × volume = P²L/(2AE). Resilience is strain energy at the elastic limit. Proof resilience is the maximum energy stored without permanent set.
Loading types: gradual load gives σ = P/A, sudden load gives σ = 2P/A, and falling-weight (impact) load gives σ = (P/A)[1 + √(1 + 2hAE/PL)].

Constant	Symbol	Meaning
Young's modulus	E	Axial stress/axial strain
Shear (rigidity) modulus	G	Shear stress/shear strain
Bulk modulus	K	Pressure/volumetric strain
Poisson's ratio	ν	Lateral strain/axial strain

2. Complex stresses and Mohr's circle

On a plane at angle θ, with stresses σx, σy and shear τxy:
Normal stress: σθ = (σx + σy)/2 + (σx − σy)/2 · cos2θ + τxy sin2θ
Shear stress: τθ = −(σx − σy)/2 · sin2θ + τxy cos2θ
Principal stresses: σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]. The planes on which they act have zero shear. They are 90° apart. Direction: tan2θ = 2τxy/(σx − σy).
Maximum shear stress: τmax = (σ₁ − σ₂)/2 = √[((σx − σy)/2)² + τxy²]. It acts on planes at 45° to the principal planes.
Mohr's circle: centre at ((σx + σy)/2, 0), radius equal to τmax. Points on the circle give normal and shear stress on any plane; angles on the circle are twice the real angles.
Pure shear gives principal stresses +τ and −τ.

3. Thin and thick cylinders

A cylinder is thin if t < d/20.
Thin cylinder: hoop (circumferential) stress σh = pd/2t. Longitudinal stress σl = pd/4t. So hoop stress is twice the longitudinal stress.
Thin sphere: σ = pd/4t in all directions.
Longitudinal joints in boilers are stressed more than circular joints, which is why efficiency of the longitudinal joint governs thickness.
Thick cylinders (Lame's theory): radial stress σr = a − b/r², hoop stress σh = a + b/r². The hoop stress is highest at the inner surface. Compound cylinders and autofrettage reduce the peak stress.
Pressure vessels and boiler drums in steam power stations are designed against these stresses.

4. Shear force, bending moment and bending stress

Relations: dM/dx = V and dV/dx = −w. Bending moment is maximum where shear force is zero or changes sign.
Bending equation: M/I = σ/y = E/R. Section modulus Z = I/ymax, so σmax = M/Z.
Moment of inertia: rectangle bh³/12 (Z = bh²/6); solid circle πd⁴/64 (Z = πd³/32); hollow circle π(D⁴ − d⁴)/64.
Shear stress in beams: τ = V·A·ȳ/(I·b). For a rectangle, maximum τ = 1.5 × average, at the neutral axis. For a solid circle, maximum τ = 4/3 × average. In an I-section the web carries most of the shear.
Pure bending: the neutral axis passes through the centroid. Stress is zero there and maximum at the outer fibre.

Beam and load	Maximum BM	Maximum deflection
Cantilever, point load W at free end	WL	WL³/(3EI)
Cantilever, UDL w per length	wL²/2	wL⁴/(8EI)
Simply supported, central point load W	WL/4	WL³/(48EI)
Simply supported, UDL w	wL²/8	5wL⁴/(384EI)

Slope at the free end of a cantilever with point load: WL²/(2EI). For a simply supported beam with central load: WL²/(16EI) at the supports.
Methods for deflection: double integration, Macaulay's method, moment-area method and conjugate beam method.
Flexural rigidity is EI. A beam of larger EI deflects less.

5. Torsion and springs

Torsion equation: T/J = τ/r = Gθ/L.
Polar moment of inertia: solid shaft J = πd⁴/32; hollow shaft J = π(D⁴ − d⁴)/32.
Power transmitted: P = 2πNT/60 watts, with N in rpm and T in N·m.
A hollow shaft is more economical than a solid shaft of the same weight and material, because material near the axis is lightly stressed.
Shafts in series: same torque, angles of twist add. Shafts in parallel: same angle of twist, torques add.
Close-coiled helical spring (axial load W, mean radius R, wire diameter d, n coils): τ = 16WR/(πd³), deflection δ = 64WR³n/(Gd⁴). Stiffness k = W/δ = Gd⁴/(64R³n).
Springs in series: 1/k = 1/k₁ + 1/k₂. Springs in parallel: k = k₁ + k₂.

6. Columns and struts

Slenderness ratio λ = Le/r, with radius of gyration r = √(I/A).
Euler's formula: Pcr = π²EI/Le². It applies to long columns. Critical stress σcr = π²E/λ².
Effective length Le depends on end conditions:

End conditions	Effective length
Both ends hinged	L
One end fixed, other free	2L
Both ends fixed	L/2
One end fixed, other hinged	L/√2

Rankine-Gordon formula covers both short and long columns: 1/PR = 1/Pc + 1/PE, where Pc is the crushing load and PE the Euler load.
Euler's formula over-predicts the strength of short columns, so it is valid only above a limiting slenderness ratio.

7. Theories of failure

Maximum principal stress (Rankine): failure when σ₁ reaches the limit. Suits brittle materials.
Maximum shear stress (Tresca): failure when τmax = σy/2. Suits ductile materials. It is conservative.
Maximum distortion energy (von Mises): failure when the equivalent stress equals σy. Best agreement with tests on ductile steel.
Maximum strain energy (Haigh) and maximum principal strain (St. Venant) are older theories.
In pure shear, yield occurs at τ = σy/√3 = 0.577σy by von Mises and at σy/2 by Tresca.

8. Worked examples

Example 1 (bar). P = 50 kN, A = 500 mm², E = 200 GPa, L = 2 m. σ = 50,000/500 = 100 MPa. δ = σL/E = 100 × 2000/200,000 = 1 mm.

Example 2 (beam). Simply supported span 4 m, UDL 10 kN/m, rectangular section 100 mm × 200 mm. Mmax = wL²/8 = 10 × 16/8 = 20 kN·m. Z = 100 × 200²/6 = 666,667 mm³. σ = 20 × 10⁶/666,667 = 30 MPa.

Example 3 (principal stress). σx = 80 MPa, σy = 0, τ = 30 MPa. Centre = 40, radius = √(1600 + 900) = 50. σ₁ = 90 MPa, σ₂ = −10 MPa, τmax = 50 MPa.

Example 4 (thin cylinder). p = 2 MPa, d = 1 m, t = 10 mm. Hoop = 2 × 1000/(2 × 10) = 100 MPa. Longitudinal = 50 MPa.

Example 5 (Euler). Hinged both ends, L = 2 m, E = 200 GPa, I = 10⁻⁶ m⁴. Pcr = π² × 2 × 10⁵/4 ≈ 493.5 kN.

Exam traps

E = 2G(1 + ν) links E and G. E = 3K(1 − 2ν) links E and K. Do not swap them.
Thin cylinder: hoop stress pd/2t is larger than longitudinal stress pd/4t.
Thin sphere stress is pd/4t, not pd/2t.
Maximum shear stress in a rectangular beam is 1.5 times average. For a circle it is 4/3.
Column fixed at both ends has Le = L/2. A cantilever column has Le = 2L.
Euler's load is proportional to EI and inversely proportional to the square of effective length.
Tresca uses σy/2. Von Mises uses σy/√3 in shear.
Mohr's circle angles are double the real plane angles.
Thermal stress arises only when expansion is prevented.

One-liners

1. Hooke's law: stress is proportional to strain within the elastic limit.
2. Poisson's ratio for most metals is about 0.3, and cannot exceed 0.5.
3. Sudden loading produces twice the stress of gradual loading.
4. Bending moment is maximum where shear force is zero.
5. Section modulus Z = I/ymax.
6. A hollow shaft is more efficient than a solid one of equal weight.
7. Power in a shaft is 2πNT/60.
8. Spring stiffness in series: 1/k = Σ(1/k).
9. Slenderness ratio is Le/r.
10. Euler's formula suits long columns only.
11. Von Mises theory best fits ductile yielding.
12. Principal planes carry zero shear stress.

Practice questions

Young's modulus E, shear modulus G and Poisson's ratio ν are related by
1. E = G(1 + 2ν)
2. E = G/(2(1 + ν))
3. E = 2G(1 + ν)
4. E = 2G(1 − ν)
Answer
C. E = 2G(1 + ν)
Standard relation between the elastic constants.
The upper limit of Poisson's ratio for any elastic material is
1. 0.33
2. 0.25
3. 1.0
4. 0.5
Answer
D. 0.5
At ν = 0.5 the material is incompressible (K becomes infinite).
In a thin cylindrical shell under internal pressure, the ratio of hoop stress to longitudinal stress is
1. 0.5
2. 1
3. 2
4. 4
Answer
C. 2
σh = pd/2t and σl = pd/4t.
For a beam of rectangular section, the maximum shear stress is how many times the average shear stress?
1. 1.33
2. 1.5
3. 2
4. 1.0
Answer
B. 1.5
τmax = 1.5 V/A at the neutral axis.
The effective length of a column with one end fixed and the other end free is
1. 2L
2. L/2
3. L
4. L/√2
Answer
A. 2L
A cantilever column buckles with an effective length of twice its actual length.
The section modulus of a rectangular section of width b and depth h is
1. bh³/12
2. bh²/12
3. bh³/6
4. bh²/6
Answer
D. bh²/6
Z = I/ymax = (bh³/12)/(h/2).
The polar moment of inertia of a solid circular shaft of diameter d is
1. πd³/16
2. πd⁴/32
3. πd³/32
4. πd⁴/64
Answer
B. πd⁴/32
J = 2I for a circle = 2 × πd⁴/64.
On principal planes the shear stress is
1. zero
2. equal to half the normal stress
3. maximum
4. equal to the normal stress
Answer
A. zero
By definition, principal planes have no shear stress.
A load applied suddenly on a bar produces a stress that is how many times the stress from the same load applied gradually?
1. 1
2. 1.5
3. 2
4. 4
Answer
C. 2
Work done by the load equals strain energy: P·δ = σ²AL/2E gives σ = 2P/A.
The maximum bending moment in a simply supported beam carrying a uniformly distributed load w over span L is
1. wL²/4
2. wL²/2
3. wL/4
4. wL²/8
Answer
D. wL²/8
The maximum occurs at midspan.
Which failure theory is based on the maximum distortion (shear) strain energy?
1. von Mises theory
2. Rankine theory
3. St. Venant theory
4. Tresca theory
Answer
A. von Mises theory
von Mises is the distortion energy theory for ductile materials.
The maximum deflection of a cantilever of length L carrying a UDL w per unit length is
1. wL⁴/(3EI)
2. wL⁴/(8EI)
3. 5wL⁴/(384EI)
4. wL³/(3EI)
Answer
B. wL⁴/(8EI)
Standard result by double integration.
The effective length of a column fixed at both ends is
1. 2L
2. L/2
3. L/√2
4. L
Answer
B. L/2
Fixed ends reduce the buckling length to half.
For a solid circular section, the maximum shear stress in a beam is how many times the average shear stress?
1. 3/2
2. 2
3. 4/3
4. 1
Answer
C. 4/3
τmax = (4/3)(V/A) at the neutral axis.
For the same weight and material, a hollow shaft compared with a solid shaft
1. has the same strength
2. has lower polar modulus
3. can transmit less torque
4. can transmit more torque
Answer
D. can transmit more torque
Material away from the axis is more effective in carrying shear.
Which relation connects Young's modulus E, bulk modulus K and Poisson's ratio ν?
1. E = 3K(1 − 2ν)
2. E = 2K(1 − 3ν)
3. E = K(1 − 2ν)
4. E = 3K(1 + 2ν)
Answer
A. E = 3K(1 − 2ν)
Standard elastic constant relation.
The Rankine-Gordon formula for columns is used because it
1. covers both short and long columns
2. applies only to very long columns
3. applies only to brittle materials
4. ignores end conditions
Answer
A. covers both short and long columns
1/PR = 1/Pc + 1/PE combines crushing and Euler loads.
A load of 80 kN acts on a bar of cross-section 400 mm². The axial stress is
1. 20 MPa
2. 320 MPa
3. 200 MPa
4. 2 MPa
Answer
C. 200 MPa
σ = 80,000/400 = 200 N/mm².
A bar of area 200 mm² and length 1.5 m carries 40 kN axial load. With E = 200 GPa, the elongation is
1. 15 mm
2. 3 mm
3. 0.15 mm
4. 1.5 mm
Answer
D. 1.5 mm
δ = PL/AE = 40,000 × 1500/(200 × 200,000) = 1.5 mm.
A steel bar (α = 12 × 10⁻⁶ per °C, E = 200 GPa) is fully restrained and heated by 50 °C. The thermal stress is
1. 60 MPa
2. 120 MPa
3. 12 MPa
4. 240 MPa
Answer
B. 120 MPa
σ = αEΔT = 12 × 10⁻⁶ × 200,000 × 50 = 120 MPa.
A material has E = 200 GPa and ν = 0.25. Its shear modulus is
1. 80 GPa
2. 160 GPa
3. 40 GPa
4. 100 GPa
Answer
A. 80 GPa
G = E/(2(1 + ν)) = 200/2.5 = 80 GPa.
For E = 200 GPa and ν = 0.25, the bulk modulus is about
1. 80 GPa
2. 200 GPa
3. 267 GPa
4. 133 GPa
Answer
D. 133 GPa
K = E/(3(1 − 2ν)) = 200/1.5 = 133.3 GPa.
At a point, σx = 60 MPa, σy = −20 MPa and τxy = 30 MPa. The maximum principal stress is
1. 50 MPa
2. 60 MPa
3. 70 MPa
4. 90 MPa
Answer
C. 70 MPa
Centre = 20, radius = √(40² + 30²) = 50, so σ₁ = 70 MPa.
A thin cylindrical shell of diameter 800 mm and thickness 8 mm carries internal pressure 1.5 MPa. The hoop stress is
1. 7.5 MPa
2. 75 MPa
3. 37.5 MPa
4. 150 MPa
Answer
B. 75 MPa
σh = pd/2t = 1.5 × 800/16 = 75 MPa.
A simply supported beam of rectangular section 100 mm × 200 mm (depth 200 mm) has a maximum bending moment of 12 kN·m. The maximum bending stress is
1. 9 MPa
2. 36 MPa
3. 24 MPa
4. 18 MPa
Answer
D. 18 MPa
Z = 100 × 200²/6 = 666,667 mm³; σ = 12 × 10⁶/666,667 = 18 MPa.
A simply supported beam carries a central point load. If the span is doubled with the same load and section, the central deflection becomes
1. 2 times
2. 4 times
3. 8 times
4. 16 times
Answer
C. 8 times
δ = WL³/(48EI) varies with L³, so 2³ = 8.
A shaft transmits a torque of 500 N·m at 600 rpm. The power is about
1. 31.4 kW
2. 3.14 kW
3. 18.8 kW
4. 314 kW
Answer
A. 31.4 kW
P = 2πNT/60 = 2π × 600 × 500/60 = 31,416 W.
If the diameter of a solid shaft is doubled, the torque it can carry at the same maximum shear stress becomes
1. 16 times
2. 8 times
3. 4 times
4. 2 times
Answer
B. 8 times
T = πτd³/16, which varies with d³.
Two springs of stiffness 200 N/mm and 300 N/mm are connected in series. The combined stiffness is
1. 500 N/mm
2. 120 N/mm
3. 250 N/mm
4. 60 N/mm
Answer
B. 120 N/mm
1/k = 1/200 + 1/300 = 5/600, so k = 120 N/mm.
A column of circular section, diameter 50 mm and effective length 2 m, has slenderness ratio
1. 40
2. 320
3. 80
4. 160
Answer
D. 160
r = d/4 = 12.5 mm; λ = 2000/12.5 = 160.
A beam of rectangular section 100 mm × 150 mm carries a shear force of 30 kN. The maximum shear stress is
1. 3 MPa
2. 2 MPa
3. 1.5 MPa
4. 4.5 MPa
Answer
A. 3 MPa
Average = 30,000/15,000 = 2 MPa; maximum = 1.5 × 2 = 3 MPa.
A cantilever of length 2 m carries a point load of 5 kN at its free end. The maximum bending moment is
1. 5 kN·m
2. 2.5 kN·m
3. 10 kN·m
4. 20 kN·m
Answer
C. 10 kN·m
M = W × L = 5 × 2 = 10 kN·m at the fixed end.
Consider the statements about Poisson's ratio. 1. It cannot exceed 0.5 for an elastic material. 2. For most metals it is about 0.3. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard facts.
Consider the statements about a thin cylinder under internal pressure. 1. The hoop stress is pd/2t. 2. The longitudinal stress is pd/2t. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Longitudinal stress is pd/4t.
Consider the statements about a rectangular beam in bending and shear. 1. The maximum shear stress occurs at the neutral axis. 2. The maximum bending stress occurs at the neutral axis. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Bending stress is zero at the neutral axis and maximum at the outer fibre.
Consider the statements about column formulae. 1. Euler's formula is valid for short columns. 2. Rankine's formula can be used for both short and long columns. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Euler's formula applies to long columns only.
Consider the statements. 1. Principal planes carry zero shear stress. 2. Planes of maximum shear stress are inclined at 45° to the principal planes. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both follow from Mohr's circle.
Consider the statements about thermal stress. 1. Stress arises in a bar that is free to expand. 2. Stress in a fully restrained bar is αEΔT. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
If a bar can expand freely there is no stress.
Consider the statements about shafts. 1. The polar moment of inertia of a hollow shaft is π(D⁴ − d⁴)/32. 2. Shafts connected in series carry the same angle of twist. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Series shafts carry the same torque; parallel shafts have the same angle of twist.
Consider the statements about failure theories. 1. The Tresca theory is more conservative than the von Mises theory. 2. The Rankine theory is best suited to ductile materials. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Rankine suits brittle materials.
Consider the statements. 1. Angles on Mohr's circle are equal to the real angles of the planes. 2. A column with one end fixed and one end free has an effective length of L/2. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
Mohr's circle angles are double the real angles; fixed-free effective length is 2L.
Consider the statements about a simply supported beam with a UDL. 1. The maximum bending moment is at midspan. 2. The maximum shear force is at the supports. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both statements are correct.
Consider the statements about springs. 1. Springs in series have a combined stiffness equal to the sum of the stiffnesses. 2. Springs in parallel have a combined stiffness equal to the sum of the stiffnesses. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
In series, 1/k = Σ1/k.
Which pair of failure theory and criterion is correctly matched?
1. von Mises – maximum shear stress
2. Tresca – maximum distortion energy
3. St. Venant – maximum shear strain energy
4. Rankine – maximum principal stress
Answer
D. Rankine – maximum principal stress
Tresca uses maximum shear stress; von Mises uses distortion energy; St. Venant uses maximum principal strain.
Which pair of column end condition and effective length is correct?
1. Both ends hinged – 2L
2. Fixed and free – L
3. Both ends fixed – L/2
4. Fixed and hinged – L/4
Answer
C. Both ends fixed – L/2
Fixed-fixed has Le = L/2.

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