Mechanical properties of structural materials
Pull a bar slowly and plot stress (σ = F/A) against strain (ε = ΔL/L). At first the line is straight: Hooke's law, σ = E·ε. The slope E is Young's modulus, the stiffness of the material. Typical values: steel 200 GPa, concrete about 30 GPa, timber along the grain about 10 GPa.
- Elastic: the bar returns to its length when the load is removed.
- Yield strength: stress where steel starts to stretch permanently.
- Ultimate strength: the highest stress before it breaks.
- Ductile materials (steel) stretch a lot before breaking and warn us. Brittle ones (concrete in tension, glass) snap suddenly.
Extension of a bar: ΔL = F·L / (A·E).
Section properties
The shape of the cross-section decides how well a member resists bending. Key numbers:
- Area A: for pulling and pushing. Rectangle A = b·h.
- Centroid: the balance point of the shape; the neutral axis passes through it.
- Second moment of area I: rectangle I = b·h³/12; circle I = π·d⁴/64.
- Section modulus Z = I / y, where y is the distance to the outer edge. Rectangle: Z = b·h²/6.
Bending stress: σ = M / Z. Depth enters as h³ in I and as h² in Z, so the same amount of material is far stronger when placed deep (I-section, box section).
Deformation of beams and members
Under load, members deform. A bar changes length by ΔL = FL/(AE). A beam bends and its centre moves down by the deflection δ:
- Simply supported, central load P: δ = P·L³ / (48·E·I)
- Simply supported, even load w: δ = 5·w·L⁴ / (384·E·I)
- Cantilever, tip load P: δ = P·L³ / (3·E·I)
E·I is the flexural rigidity. Doubling the span makes the sag 8 times bigger (point load); doubling the depth makes it 8 times smaller. Codes limit sag to about span/250 so floors do not feel bouncy and plaster does not crack.
Key formulas and definitions
- σ = F / A, ε = ΔL / L
- E = σ / ε (Hooke: σ = E·ε)
- ΔL = F·L / (A·E)
- Rectangle: I = b·h³ / 12, Z = b·h² / 6
- Circle: I = π·d⁴ / 64
- Bending stress: σ = M / Z
- Central load: δ = P·L³ / (48·E·I); even load: δ = 5·w·L⁴ / (384·E·I); cantilever tip: δ = P·L³ / (3·E·I)
Worked examples
1. A steel bar 2 m long and 20 mm diameter (A = 314 mm²) carries a pull of 62.8 kN. E = 200 000 N/mm². Find σ, ε and the extension.
σ = 62 800 / 314 = 200 MPa. ε = σ/E = 200 / 200 000 = 0.001. ΔL = ε × L = 0.001 × 2000 = 2 mm.
2. A rectangular section is 100 mm wide and 200 mm deep. Find I and Z.
I = bh³/12 = 100 × 200³ / 12 = 66.7 × 10⁶ mm⁴. Z = I / (h/2) = 66.7 × 10⁶ / 100 = 0.667 × 10⁶ mm³.
3. A 50 × 150 mm plank is used flat (h = 50) and on edge (h = 150). Compare I.
On edge: I = 50 × 150³ / 12 = 14.06 × 10⁶ mm⁴. Flat: I = 150 × 50³ / 12 = 1.56 × 10⁶ mm⁴. Ratio = 9, so on edge it is 9 times stiffer.
4. A 4 m simply supported beam carries 20 kN at mid-span; Z = 0.667 × 10⁶ mm³. Find the bending stress.
M = PL/4 = 20 × 4 / 4 = 20 kN·m = 20 × 10⁶ N·mm. σ = M/Z = 20 × 10⁶ / 0.667 × 10⁶ = 30 MPa.
5. The same beam is steel (E = 200 000 N/mm²) with I = 66.7 × 10⁶ mm⁴. Find the mid-span deflection.
δ = PL³ / (48EI) = 20 000 × 4000³ / (48 × 200 000 × 66.7 × 10⁶) = 1.28 × 10¹⁵ / 6.4 × 10¹⁴ = 2.0 mm. Span/δ = 2000, well inside span/250.
6. A steel cantilever of 1.5 m has I = 10 × 10⁶ mm⁴ and a 2 kN tip load. Find the tip deflection (E = 200 000 N/mm²).
δ = PL³/(3EI) = 2000 × 1500³ / (3 × 200 000 × 10 × 10⁶) = 6.75 × 10¹² / 6 × 10¹² = 1.125 mm.
Common mistakes
- Using h and b the wrong way round in I = bh³/12. The cubed side is the one in the direction of bending (depth).
- Mixing mm and m. I in mm⁴, E in N/mm² and forces in N give deflection in mm.
- Thinking a stiffer material is the same as a stronger one. E is stiffness; yield strength is strength.
- Forgetting that the neutral axis carries zero bending stress, so the middle of a solid beam is wasted material.