BEM PCE Civil & Structural cheat sheet

Earthwork volume by average end area: V = L × (A1 + A2)/2; for three or more sections use the prismoidal formula V = L/6 × (A1 + 4Am + A2), where Am is the mid-section area.

Horizontal curve geometry: T = R tan(Δ/2), L = RΔ (Δ in radians) = πRΔ°/180, M = R[sec(Δ/2) − 1], E = R[sec(Δ/2) − 1] measured to the external point; degree of curve (arc definition) D = 5729.58/R.

Stopping sight distance: SSD = 0.278 V t + V²/(254(f ± G)), with V in km/h, t in seconds, f the longitudinal friction factor, and G the grade in decimal (plus for upgrade).

Vertical curve: elevation at any point y = y_PVC + g1x + (g2 − g1)x²/(2L); the high/low point occurs at x = −g1L/(g2 − g1), valid only when 0 ≤ x ≤ L.

Superelevation and side friction: e + f = V²/(127R) with V in km/h and R in metres; AASHTO-style design distributes e between 0.02 and 0.08 (0.04–0.06 typical where ice is possible).

Storm sewer design uses the rational method Q = CiA (Q in m³/s when i is m/s and A is m²; with i in mm/h, Q = CiA/3.6×10⁶); Manning's equation for pipe capacity Q = (1/n) A R^(2/3) S^(1/2).

Traffic flow relationships: q = k × v (flow = density × speed), and the fundamental diagram gives capacity where the tangent from the origin touches the speed–density curve, typically near 0.5 free-flow speed.

Pavement and compaction control: relative compaction = field dry density / laboratory maximum dry density (standard Proctor or modified Proctor), with acceptance typically ≥ 95% standard or ≥ 92% modified in the upper layers.

Soil classification and drainage: Atterberg limits define the plasticity index PI = LL − PL; the group index GI = (F − 35)[0.2 + 0.005(LL − 40)] + 0.01(F − 15)(PI − 10), with F the percent passing the 0.075 mm sieve.

Ultimate strength of a singly reinforced rectangular section: Mn = As fy (d − a/2) with a = As fy/(0.85 f'c b); the tension-controlled limit corresponds to a tensile strain of at least 0.005 at nominal strength.

Balanced and minimum reinforcement: ρb = 0.85 β1 (f'c/fy) × [0.003/(0.003 + εy)]; a practical minimum for beams is As,min ≈ max(0.25√f'c/fy, 1.4/fy) × bw × d (SI, MPa).

Shear design: nominal concrete shear Vc ≈ 0.17 λ √f'c bw d (SI, MPa, N); the required stirrup spacing s = Av fy d / (Vs) with Vs = Vu/φ − Vc and φ = 0.75 for shear.

Serviceability: deflection limits are span/360 for floors under live load and span/240 for roofs; long-term deflection includes a multiplier λΔ = ξ/(1 + 50ρ') applied to the sustained-load immediate deflection.

Column behaviour: short columns fail by material crushing; slender columns are governed by the slenderness ratio kL/r and magnified moments; the Euler buckling load is Pcr = π²EI/(kL)².

Steel tension members: nominal strength is the lesser of yielding on the gross section (Fy Ag) and fracture on the net section (Fu Ae), with Ae = U An and shear lag factor U from the connection geometry.

Steel beam design: plastic moment Mp = Fy Zx; lateral-torsional buckling reduces capacity between Lp and Lr, where Lp ≈ 1.76 ry √(E/Fy) for compact sections.

Bolted and welded connections: bolt shear strength Rn = Fnv Ab per shear plane, bearing strength depends on the thinner plate and edge distance, and weld strength is Rn = 0.6 FEXX × throat area, with the throat = 0.707 × leg size for a 45° fillet.

Load combinations and factors: strength design uses 1.2D + 1.6L + 0.5(Lr or S or R) and 1.2D + 1.0W + L + 0.5(Lr or S or R); service combinations (D + L) govern deflection and settlement checks.