Strength of Materials and Stresses

What to remember

• Stress = force/area; strain = change in length/original length; elongation of a bar δ = PL/AE; elastic constants are linked by E = 2G(1+ν) = 3K(1−2ν).
• Bending: M/I = σ/y = E/R. Torsion: T/J = τ/r = Gθ/L. Thin cylinder: hoop stress = pd/2t, longitudinal stress = pd/4t.
• Euler's buckling load P = π²EI/Le², and effective length is L for hinged ends, 2L for fixed-free, L/2 for fixed-fixed and L/√2 for fixed-hinged.

Simple stress, strain and elastic constants

Hooke's law: within the proportional (elastic) limit, stress is proportional to strain. The constant is the modulus of elasticity E (Young's modulus) = σ/ε. Units: stress in N/mm² = MPa; E of steel about 200 GPa; E of concrete from the code formula.

Axial deformation: δ = PL/AE.

• Example: P = 50 kN, L = 2 m, A = 500 mm², E = 200 GPa (2 × 10⁵ N/mm²). δ = 50,000 × 2000/(500 × 200,000) = 1 mm.
• Bar of varying section: add the elongations of each part. Bar under its own weight: δ = WL/2AE (half that of the same load applied at the end).
• Tapering circular bar: δ = 4PL/(π E d₁ d₂).
• Composite bar (two materials, same strain): load shared in proportion to AE. P = P₁ + P₂ and σ₁/E₁ = σ₂/E₂.

Elastic constants:

• Poisson's ratio ν = lateral strain/longitudinal strain (steel about 0.3, theoretical limit 0.5).
• Shear modulus G = τ/γ; bulk modulus K = p/(volumetric strain).
• Volumetric strain = ε₁ + ε₂ + ε₃ (for axial load ε(1−2ν)).
• E = 2G(1+ν) and E = 3K(1−2ν); E = 9KG/(3K+G).
• Example: E = 200 GPa, ν = 0.25: G = 200/(2 × 1.25) = 80 GPa; K = 200/(3 × 0.5) = 133.3 GPa.

Temperature stress: free expansion = αLΔT. If expansion is fully prevented, σ = EαΔT. Example: α = 12 × 10⁻⁶ per °C, ΔT = 50 °C, E = 2 × 10⁵ N/mm²: σ = 120 MPa.

Stress-strain curve of mild steel: proportional limit, elastic limit, upper and lower yield points, strain hardening, ultimate stress, then necking and fracture. High-strength deformed bars have no sharp yield point, so a 0.2% proof stress is used. Factor of safety = ultimate (or yield) stress/working stress.

Strain energy

Strain energy stored U = σ²/(2E) × volume = P²L/(2AE). Resilience is the strain energy stored up to the elastic limit; modulus of resilience is per unit volume; toughness is the area under the full stress-strain curve.

• Gradual load: σ = P/A. Suddenly applied load: σ = 2P/A (twice).
• Falling load (impact) gives still higher stress.

Compound stresses and Mohr's circle

For a plane element with σx, σy and shear τ:

• Principal stresses: σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τ²].
• Maximum shear stress = √[((σx−σy)/2)² + τ²] = (σ₁−σ₂)/2.
• Principal planes: tan 2θ = 2τ/(σx−σy). Shear stress is zero on principal planes. Principal planes are 90° apart; maximum shear planes are at 45° to them.
• Example: σx = 100, σy = 20, τ = 30 MPa: centre = 60, radius = √(40²+30²) = 50. So σ₁ = 110, σ₂ = 10, τmax = 50 MPa.
• Mohr's circle: centre at (σx+σy)/2, radius equal to the maximum shear stress.

Thin shells, torsion and bending

Thin cylinder (t < d/20), internal pressure p:

• Hoop (circumferential) stress = pd/2t. Longitudinal stress = pd/4t (half of hoop).
• Thin sphere: stress = pd/4t in every direction.
• Example: p = 2 MPa, d = 1000 mm, t = 10 mm: hoop = 100 MPa, longitudinal = 50 MPa.

Torsion of circular shafts: T/J = τ/r = Gθ/L. Polar moment J = πd⁴/32 (solid), π(D⁴−d⁴)/32 (hollow). Shear stress τ = 16T/πd³. Power P = 2πNT/60 (N in rpm, T in N·m, P in watts). Example: N = 300 rpm, T = 1000 N·m: P = 31.4 kW. A hollow shaft is more efficient than a solid shaft of the same weight. Shafts in series carry the same torque; shafts in parallel share the same twist.

Bending: M/I = σ/y = E/R (pure bending, plane sections remain plane). Section modulus Z = I/ymax.

• Parallel axis theorem: I = Ig + Ah². Perpendicular axis theorem: Iz = Ix + Iy (for plane areas).
• Example: rectangle 100 mm × 200 mm, M = 10 kN·m: σ = 6M/bd² = 6 × 10⁷/(100 × 40,000) = 15 MPa.

Shear stress in beams: τ = V A ȳ/(I b). Rectangle: maximum = 1.5 × average, at the neutral axis. Circle: maximum = 4/3 × average. Example: V = 30 kN on 100 × 200: average 1.5 MPa, maximum 2.25 MPa. In an I-section most shear is taken by the web.

Columns

Euler's formula: P = π²EI/Le². The slenderness ratio λ = Le/r where r = √(I/A). Euler's theory holds for long, slender columns; for short columns failure is by crushing (Rankine's formula covers both). Buckling occurs about the axis of least moment of inertia.

A column loaded eccentrically has direct plus bending stress: σ = P/A ± Pe/Z. Core (kern) of a section: the zone within which a compressive load causes no tension. Rectangle: middle third of depth (kern width d/3, e = d/6). Circle: diameter d/4 (radius d/8). Retaining walls and masonry dams use this idea.

Worked examples

• 1. Composite bar: a steel bar (A = 400 mm², E = 200 GPa) and a copper bar (A = 400 mm², E = 100 GPa) are joined side by side and carry 60 kN together with equal strain. Load share is in the ratio of AE, 2:1, so steel takes 40 kN and copper 20 kN. Stresses: steel 100 MPa and copper 50 MPa.
• 2. Shear and bending in a beam: a simply supported beam of span 4 m carries a central load of 20 kN, section 100 × 200 mm. Maximum moment = WL/4 = 20 kN·m, so σmax = 6 × 20 × 10⁶/(100 × 40,000) = 30 MPa.
• 3. Hollow shaft: outer diameter 100 mm, inner 50 mm. J = π(100⁴ − 50⁴)/32 = π × 9.375 × 10⁷/32 ≈ 9.2 × 10⁶ mm⁴. Compare with the solid shaft of 100 mm: J = 9.82 × 10⁶ mm⁴, so the hole removes 25% of the material (area) but only 6% of J.
• 4. Euler's load: a hinged steel column of L = 4 m, I = 8 × 10⁶ mm⁴, E = 2 × 10⁵ N/mm²: P = π² × 2 × 10⁵ × 8 × 10⁶/(4000)² = 987 kN (about). If both ends are fixed, the load becomes four times.
• 5. Strain energy: a bar of 1 m length and 500 mm² area carrying 100 kN stores U = P²L/2AE = (100,000)² × 1000/(2 × 500 × 200,000) = 50,000 N·mm = 50 N·m.

Strength classes and failure theories

• Theories of failure: maximum principal stress theory (Rankine, suits brittle materials), maximum shear stress theory (Tresca) and distortion energy theory (von Mises), the last two suit ductile materials. Von Mises is the least conservative of the shear-based theories and fits steel best.
• Types of loading: axial, shear, bending, torsion; combined loading gives bending plus torsion in shafts. Fatigue is failure under repeated stress below the static strength; the endurance limit is the stress below which fatigue failure does not occur for many steels.
• Hardness and ductility: ductility is measured by percentage elongation and percentage reduction in area; brittle materials such as cast iron break with little elongation.

Exam traps

• E, G and K are linked, but ν for steel is about 0.3 and never above 0.5.
• A suddenly applied load gives twice the stress of the same load applied gradually, not the same stress.
• Longitudinal stress is half the hoop stress in a thin cylinder.
• Maximum shear stress in a rectangle is 1.5 times the average, not 2 times; in a circle it is 4/3.
• Section modulus of a rectangle is bd²/6 and of a circle πd³/32.
• Euler's buckling load is lower for the longer effective length; fixed-fixed carries 4 times the hinged-hinged load.
• Principal planes carry no shear; maximum shear planes carry non-zero normal stress.
• Thermal stress arises only when expansion is restrained.

One-liners

• 1. E = 2G(1+ν).
• 2. E = 3K(1−2ν).
• 3. Strain energy per unit volume = σ²/2E.
• 4. Hoop stress is twice the longitudinal stress.
• 5. J for a solid shaft = πd⁴/32.
• 6. Torsion equation: T/J = τ/r = Gθ/L.
• 7. Bending equation: M/I = σ/y = E/R.
• 8. Maximum shear stress in a circular section is 4/3 times the average.
• 9. Effective length of a fixed-free column is 2L.
• 10. Slenderness ratio = Le/r.
• 11. Kernel of a rectangular section is the middle third.
• 12. Thermal stress = EαΔT.

Practice questions

1. A steel bar of area 500 mm2 and length 1.5 m carries an axial pull of 100 kN. With E = 200 GPa the elongation is

   1. 0.15 mm
   2. 15 mm
   3. 1.5 mm
   4. 3 mm
   Answer

   C. 1.5 mm

   delta = PL/AE = 100,000 x 1500/(500 x 200,000) = 1.5 mm.

2. A bar of area 250 mm2 carries an axial load of 50 kN. The stress in it is

   1. 500 N/mm2
   2. 20 N/mm2
   3. 125 N/mm2
   4. 200 N/mm2
   Answer

   D. 200 N/mm2

   Stress = 50,000/250 = 200 N/mm2.

3. The unit of Young's modulus in SI is

   1. N
   2. N/m2 (pascal)
   3. N-m
   4. N/m
   Answer

   B. N/m2 (pascal)

   E = stress/strain; strain is dimensionless, so E has units of stress.

4. The value of Poisson's ratio for most metals lies in the range

   1. 0.25 to 0.33
   2. 1 to 2
   3. 0.5 to 1
   4. 0.01 to 0.05
   Answer

   A. 0.25 to 0.33

   Steel is about 0.3; the theoretical upper limit is 0.5.

5. A bar fixed between rigid walls is heated by 40 C. With alpha = 1.2 x 10^-5 per C and E = 2 x 10^5 N/mm2 the thermal stress is

   1. 192 N/mm2
   2. 96 N/mm2
   3. 9.6 N/mm2
   4. 48 N/mm2
   Answer

   B. 96 N/mm2

   sigma = E alpha dT = 2e5 x 1.2e-5 x 40 = 96 N/mm2.

6. For E = 200 GPa and Poisson's ratio 0.25, the modulus of rigidity G is

   1. 80 GPa
   2. 40 GPa
   3. 100 GPa
   4. 160 GPa
   Answer

   A. 80 GPa

   G = E/[2(1+nu)] = 200/2.5 = 80 GPa.

7. If Poisson's ratio is 1/3, the bulk modulus K equals

   1. E/3
   2. 3E
   3. 2E
   4. E
   Answer

   D. E

   K = E/[3(1-2nu)] = E/(3 x 1/3) = E.

8. A bar elongates with strain 0.001 and Poisson's ratio is 0.3. The lateral strain magnitude is

   1. 0.003
   2. 0.0013
   3. 0.0003
   4. 0.0001
   Answer

   C. 0.0003

   Lateral strain = nu x longitudinal = 0.3 x 0.001.

9. The volumetric strain of a bar under axial strain 0.002 with nu = 0.25 is

   1. 0.002
   2. 0.004
   3. 0.001
   4. 0.0005
   Answer

   C. 0.001

   Volumetric strain = e(1 - 2nu) = 0.002 x 0.5.

10. The elongation of a vertical bar of length L under its own weight W is

    1. WL/4AE
    2. WL/2AE
    3. 2WL/AE
    4. WL/AE
    Answer

    B. WL/2AE

    The load varies from zero to W, so the elongation is half of WL/AE.

11. A load applied suddenly on a bar produces a stress that is how many times the stress under the same load applied gradually?

    1. 2
    2. 1
    3. 1.5
    4. 4
    Answer

    A. 2

    Work done by the load P x delta equals strain energy, giving sigma = 2P/A.

12. Strain energy stored per unit volume in a bar stressed to 100 N/mm2 with E = 2 x 10^5 N/mm2 is

    1. 0.0025 N-mm/mm3
    2. 0.05 N-mm/mm3
    3. 0.25 N-mm/mm3
    4. 0.025 N-mm/mm3
    Answer

    D. 0.025 N-mm/mm3

    u = sigma^2/2E = 10,000/400,000 = 0.025.

13. A composite bar has steel (E = 200 GPa) and copper (E = 100 GPa) of equal area acting together with equal strain. The load taken by steel compared with copper is

    1. equal
    2. four times
    3. twice
    4. half
    Answer

    C. twice

    Load shares are in the ratio of AE, so 2:1.

14. A plane element has sigma x = 100, sigma y = 20 and tau = 30 N/mm2. The major principal stress is

    1. 110 N/mm2
    2. 130 N/mm2
    3. 120 N/mm2
    4. 100 N/mm2
    Answer

    A. 110 N/mm2

    Centre 60; radius = sqrt(40^2 + 30^2) = 50; sigma1 = 110.

15. For the same element (100, 20, 30 N/mm2) the maximum shear stress is

    1. 30 N/mm2
    2. 60 N/mm2
    3. 40 N/mm2
    4. 50 N/mm2
    Answer

    D. 50 N/mm2

    tau max = radius of Mohr's circle = 50.

16. The shear stress on a principal plane is

    1. maximum
    2. zero
    3. equal to half of the principal stress
    4. equal to the principal stress
    Answer

    B. zero

    Principal planes are planes of zero shear.

17. A thin cylinder has p = 2 N/mm2, d = 1000 mm and t = 10 mm. The hoop stress is

    1. 100 N/mm2
    2. 200 N/mm2
    3. 25 N/mm2
    4. 50 N/mm2
    Answer

    A. 100 N/mm2

    Hoop stress = pd/2t = 2 x 1000/20 = 100 N/mm2.

18. The ratio of hoop stress to longitudinal stress in a thin cylindrical shell is

    1. 1
    2. 0.5
    3. 4
    4. 2
    Answer

    D. 2

    pd/2t divided by pd/4t = 2.

19. The stress in a thin spherical shell of diameter d, thickness t and pressure p is

    1. pd/2t
    2. pd/4t
    3. pd/t
    4. pd/8t
    Answer

    B. pd/4t

    A sphere has equal stress in all directions equal to pd/4t.

20. A shaft transmits torque 1000 N-m at 600 rpm. The power transmitted is nearly

    1. 100 kW
    2. 31.4 kW
    3. 62.8 kW
    4. 6.28 kW
    Answer

    C. 62.8 kW

    P = 2 pi N T/60 = 2 x 3.1416 x 600 x 1000/60 = 62.8 kW.

21. The polar moment of inertia of a solid circular shaft of diameter d is

    1. pi d^4/32
    2. pi d^3/32
    3. pi d^4/64
    4. pi d^3/16
    Answer

    A. pi d^4/32

    J = pi d^4/32; the bending I is pi d^4/64.

22. The section modulus of a rectangular section b x d about its neutral axis is

    1. bd^3/12
    2. bd^2/12
    3. bd^3/6
    4. bd^2/6
    Answer

    D. bd^2/6

    Z = I/(d/2) = (bd^3/12)/(d/2) = bd^2/6.

23. A rectangular beam 100 mm wide and 200 mm deep carries a bending moment of 10 kN-m. The maximum bending stress is

    1. 7.5 N/mm2
    2. 30 N/mm2
    3. 15 N/mm2
    4. 60 N/mm2
    Answer

    C. 15 N/mm2

    sigma = 6M/bd^2 = 6 x 10^7/(100 x 40,000) = 15 N/mm2.

24. The maximum shear stress in a rectangular beam section compared with the average shear stress is

    1. 2 times
    2. 1.5 times
    3. same
    4. 4/3 times
    Answer

    B. 1.5 times

    tau max = 1.5 V/(bd), at the neutral axis.

25. The maximum shear stress in a solid circular section compared with the average is

    1. 3/2 times
    2. 2 times
    3. 1.5 times
    4. 4/3 times
    Answer

    D. 4/3 times

    tau max = 4V/3A for a circle.

26. The effective length of a column fixed at one end and free at the other end of length L is

    1. L
    2. 2L
    3. L/2
    4. L/root 2
    Answer

    B. 2L

    Cantilever column buckles with an effective length of 2L.

27. A column fixed at both ends compared with the same column hinged at both ends carries an Euler load that is

    1. same
    2. twice
    3. four times
    4. half
    Answer

    C. four times

    Le = L/2, so P is proportional to 1/Le^2, giving 4 times.

28. A column of effective length 3 m has least radius of gyration 30 mm. The slenderness ratio is

    1. 100
    2. 10
    3. 90
    4. 300
    Answer

    A. 100

    Le/r = 3000/30 = 100.

29. The core (kern) of a solid circular section of diameter d is a concentric circle of diameter

    1. d/2
    2. d/4
    3. d/3
    4. d/8
    Answer

    B. d/4

    The kern has radius d/8, i.e. diameter d/4.

30. A short column with eccentric load P and eccentricity e: the stress at the extreme fibre is

    1. P/A only
    2. P/A plus or minus Pe/Z
    3. Pe/A
    4. P/Z
    Answer

    B. P/A plus or minus Pe/Z

    Direct stress plus bending stress from moment P x e.

31. Consider the statements. 1. E = 2G(1 + nu). 2. E = 3K(1 - 2nu). 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 relations between the elastic constants.

32. Consider the statements. 1. In a thin cylinder the longitudinal stress is twice the hoop stress. 2. Hoop 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

    B. 2 only

    Hoop stress is twice the longitudinal stress.

33. Consider the statements on Mohr's circle. 1. The centre is at the mean of the two normal stresses. 2. The radius equals the maximum shear stress. 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

    Centre = (sigma x + sigma y)/2; radius = tau max.

34. Consider the statements on torsion. 1. A hollow shaft is more efficient than a solid shaft of the same weight. 2. Shear stress is maximum at the centre of a solid shaft. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Shear stress in a solid shaft is zero at the centre and maximum at the surface.

35. Consider the statements on columns. 1. Euler's formula is valid for short stocky columns. 2. A column buckles about the axis of greatest moment of inertia. 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

    Buckling is about the axis of least moment of inertia.

36. Consider the statements on strain energy. 1. Resilience is strain energy stored up to the elastic limit. 2. Toughness is the area under the entire stress-strain curve. 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 definitions are standard.

37. Consider the statements on thermal stress. 1. It develops when expansion is restrained. 2. It is independent of the modulus of elasticity. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    sigma = E alpha dT depends on E.

38. Consider the statements on Poisson's ratio. 1. It can exceed 0.5 for ordinary metals. 2. It is the ratio of lateral to longitudinal strain. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    The theoretical upper limit is 0.5.

39. Consider the statements on bending theory. 1. Bending stress is maximum at the neutral axis. 2. Shear stress in a rectangular beam is maximum at the extreme fibres. 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

    Bending stress is zero at the neutral axis and maximum at the extreme fibres.

40. Consider the statements. 1. The kern of a rectangular section is the middle third of the depth. 2. A load inside the kern causes tension anywhere in the section. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    A load inside the core produces compression only.

41. A circular bar of diameter d is subjected to a torque T. The maximum shear stress is

    1. 8T/(pi d^3)
    2. 32T/(pi d^3)
    3. 16T/(pi d^4)
    4. 16T/(pi d^3)
    Answer

    D. 16T/(pi d^3)

    tau = T r/J = T(d/2)/(pi d^4/32) = 16T/(pi d^3).

42. Consider the statements. 1. Ductility is measured by percentage elongation. 2. A brittle material shows large plastic deformation before fracture. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Brittle materials fracture with little plastic deformation.

43. Consider the statements on the principal stresses. 1. The sum of the normal stresses sigma x + sigma y is unchanged by rotation of axes. 2. Principal planes are at 45 degrees to each other. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Principal planes are 90 degrees apart; the sum of normal stresses is invariant.

44. Match the quantity with the formula. P) Torsion equation Q) Bending equation R) Euler load S) Hoop stress

    1. P-M/I = sigma/y, Q-T/J = tau/r, R-pd/2t, S-pi^2 EI/Le^2
    2. P-pd/2t, Q-pi^2 EI/Le^2, R-T/J = tau/r, S-M/I = sigma/y
    3. P-T/J = tau/r, Q-M/I = sigma/y, R-pi^2 EI/Le^2, S-pd/2t
    4. P-pi^2 EI/Le^2, Q-pd/2t, R-M/I = sigma/y, S-T/J = tau/r
    Answer

    C. P-T/J = tau/r, Q-M/I = sigma/y, R-pi^2 EI/Le^2, S-pd/2t

    Each is the standard formula.

45. Match the section with its section modulus. P) Rectangle b x d Q) Solid circle d R) Square side a S) Hollow circle (D, d)

    1. P-pi(D^4 - d^4)/(32D), Q-a^3/6, R-pi d^3/32, S-bd^2/6
    2. P-pi d^3/32, Q-bd^2/6, R-pi(D^4 - d^4)/(32D), S-a^3/6
    3. P-a^3/6, Q-pi(D^4 - d^4)/(32D), R-bd^2/6, S-pi d^3/32
    4. P-bd^2/6, Q-pi d^3/32, R-a^3/6, S-pi(D^4 - d^4)/(32D)
    Answer

    D. P-bd^2/6, Q-pi d^3/32, R-a^3/6, S-pi(D^4 - d^4)/(32D)

    Z = I/ymax in each case.