(a) Show the loading on the beams corresponding to the bending moment diagrams shown below. The beams are simply supported at A…

(a)

The loading follows from the BMD through dM/dx = V and dV/dx = –w (sagging moment positive; V = net upward force to the left of the section). Hence a straight BMD segment means constant shear and no load on it; a parabolic segment means a UDL; a kink means a point load equal to the jump in shear; and a moment at a simple support means an applied couple.

Diagram (i): span 6 m, C at mid-span

1. M jumps from 0 to –1600 N·m at A, so a couple of 1600 N·m (anticlockwise) acts at A.
2. AC is straight with slope [800 – (–1600)]/3 = 800 N, so V = +800 N, there is no load on AC, and R_A = 800 N (upward).
3. CB is parabolic with its peak (+800 N·m) at C, so V(C⁺) = 0. Measured from B, M = R_B·s – w s²/2, with dM/ds = 0 at s = 3 m. This gives R_B = 3w and 3R_B – 4.5w = 800, so w = 177.78 N/m on CB and R_B = 533.33 N (upward).
4. The shear drops from 800 N to 0 at C, so there is a point load of 800 N (downward) at C.

Check: ΣF = 800 + 533.33 – 800 – 177.78 × 3 = 0, and M_B = –1600 + 800 × 6 – 800 × 3 – 177.78 × 4.5 = 0.

Diagram (ii): span 3 m

1. The end moments –1500 and –2300 N·m are applied couples: 1500 N·m anticlockwise at A and 2300 N·m clockwise at B.
2. The BMD is straight with slope (–2300 + 1500)/3 = –266.67 N, so V is constant and there is no load on the span.
3. R_A = –266.67 N, i.e. 266.67 N downward at A, and 266.67 N upward at B. This reaction couple (800 N·m anticlockwise) balances the net clockwise couple of 2300 – 1500 = 800 N·m.

Thus diagram (i) is an end couple, a mid-span point load and a UDL on the right half, while diagram (ii) is a beam bent only by unequal end couples.

(b)

Given: σx = 20 MN/m², σy = 10 MN/m², |τxy| = 7.5 MN/m², E = 210 GN/m² = 210 000 MN/m², ν = 0.3, d = 250 mm.

Assumptions: The plate is thin, so it is in plane stress. The hole is treated in the usual strength-of-materials way: it deforms with the nominal strain of the plate, and the local stress concentration at its edge is neglected. The circle becomes an ellipse whose axes lie along the principal strain directions. For an isotropic material these coincide with the principal stress directions.

Principal stresses. σ₁,₂ = (σx + σy)/2 ± √[((σx – σy)/2)² + τxy²] = 15 ± √(5² + 7.5²) = 15 ± √81.25 = 15 ± 9.014 MN/m²

σ₁ = 24.014 MN/m² and σ₂ = 5.986 MN/m². Both are tensile.

Orientation. In the figure the shear on the +x face acts downward and on the +y face acts to the left, so τxy = –7.5 MN/m². tan 2θp = 2τxy/(σx – σy) = –15/10 = –1.5, so 2θp = –56.31° and θp = –28.15°. The major principal direction therefore lies 28.15° clockwise from the x-axis.

Principal strains (generalised Hooke's law, plane stress). ε₁ = (σ₁ – νσ₂)/E = (24.014 – 0.3 × 5.986)/210 000 = 22.218/210 000 = 1.0580 × 10⁻⁴ ε₂ = (σ₂ – νσ₁)/E = (5.986 – 0.3 × 24.014)/210 000 = –1.218/210 000 = –5.800 × 10⁻⁶

The same values follow from εx, εy and γxy = τxy/G with G = E/2(1 + ν) using strain transformation, which confirms the result.

Axes of the ellipse. Major axis = d(1 + ε₁) = 250 × (1 + 1.0580 × 10⁻⁴) = 250 + 0.02645 = 250.0265 mm, at 28.15° clockwise from x. Minor axis = d(1 + ε₂) = 250 × (1 – 5.800 × 10⁻⁶) = 250 – 0.00145 = 249.9985 mm, perpendicular to the major axis.

Poisson contraction from the larger principal stress outweighs the smaller stress in the ε₂ direction, so the minor axis actually shortens slightly even though both principal stresses are tensile.

(c)

Cast irons are Fe–C–Si alloys with about 2–4% carbon. The form the carbon takes is set by composition (Si promotes graphite), cooling rate (fast cooling favours cementite) and treatment (heat treatment or inoculation). That form fixes the microstructure, and the microstructure fixes the properties.

• Grey: (i) form of carbon: Free graphite as flakes; (ii) microstructure: Graphite flakes in a ferritic, pearlitic or mixed matrix; fracture surface looks grey; (iii) mechanical properties: Low tensile strength and almost no ductility (flake tips act as notches); good compressive strength, excellent damping and machinability.
• White: (i) form of carbon: Combined carbon as cementite (Fe₃C); no free graphite; (ii) microstructure: Massive cementite and pearlite (transformed ledeburite); fracture looks white; (iii) mechanical properties: Very hard and wear-resistant, extremely brittle, practically unmachinable.
• Malleable: (i) form of carbon: Temper-carbon rosettes (irregular nodules), formed by annealing white iron; (ii) microstructure: Graphite aggregates in a ferrite (blackheart) or pearlite matrix; (iii) mechanical properties: Good ductility and toughness; better tensile strength and shock resistance than grey iron.
• Nodular (SG / ductile): (i) form of carbon: Spheroidal graphite, produced by Mg or Ce treatment of the melt; (ii) microstructure: Graphite spheroids in a ferrite and/or pearlite matrix ("bull's-eye" ferrite around nodules); (iii) mechanical properties: Highest strength and ductility of the cast irons, approaching cast steel; good fatigue and impact resistance.

As graphite changes from flakes to nodules to spheres, the notch effect falls and strength and ductility rise. Hence grey iron is used for machine beds, white iron for wear liners, malleable iron for pipe fittings and SG iron for crankshafts.