Answer:
The cross-sectional area of the beam is 50 mm x 150 mm = 7500 mm². The shear stress in the beam is:
τ = V/A = 1200 N / 7500 mm² = 0.16 N/mm²
The total cross-sectional area of the three nails is:
A_nails = 3 x π/4 x d² = 3 x π/4 x (5 mm)² = 58.9 mm²
The shear stress in each nail is:
τ_nail = τ x A / A_nails = 0.16 N/mm² x 7500 mm² / 58.9 mm² = 20.4 N/mm²
The maximum permissible force in each nail is 600 N, so the maximum permissible stress in each nail is:
σ_nail = F_nail / A_nail = 600 N / 58.9 mm² = 10.2 N/mm²
The maximum permissible shear stress occurs when all three nails are carrying the same load, so the total force on the nails is 1200 N. The maximum permissible spacing between the nails is:
s = L / (n-1) ≤ (σ_nail / τ_nail) x d
where L is the length of the beam, n is the number of nails, and d is the diameter of each nail.
For safety, we want to use a spacing that is less than or equal to the maximum permissible spacing:
s ≤ (σ_nail / τ_nail) x d
Substituting the given values, we get:
s ≤ (10.2 N/mm² / 20.4 N/mm²) x 5 mm = 2.5 mm
Therefore, the largest permissible spacing s between the nails is 2.5 mm.