FE Civil cheat sheet

Earthwork volume by average end area: V = L × (A1 + A2)/2, with A in ft² and L in ft giving V in ft³; divide by 27 for yd³.

Prismoidal correction for three-level sections: V = (L/6)(A1 + 4Am + A2); use when sections differ greatly.

Shrinkage and swell: bank volume × (1 + swell) = loose volume; compacted volume = bank volume / (1 − shrinkage) as a fraction.

Crew production rate = (number of units per cycle × cycles per hour) × efficiency; cost per unit = hourly crew cost / hourly production.

Critical path method: total float = LS − ES = LF − EF; activities with zero total float lie on the critical path.

Concrete mix by absolute volume: cement + water + fine + coarse + air = 27 ft³ per yd³ (or 1 m³).

Water-cement ratio governs strength and durability; lower w/c raises strength but reduces workability.

Slump test (ASTM C143) measures workability; standard slump for most structural concrete is roughly 2–4 in.

Safety: OSHA requires fall protection at 6 ft in construction (29 CFR 1926 Subpart M) and trench protective systems at 5 ft depth.

Effective stress: σ' = σ − u; total stress σ = γz, pore pressure u = γw × h.

Void ratio, porosity, and unit weight: e = n/(1−n); γ = (Gs + e·S)·γw/(1+e); dry unit weight γd = γ/(1+w).

Relative density Dr = (emax − e)/(emax − emin); used to describe sand compactness.

Terzaghi bearing capacity (strip footing): qu = cNc + qNq + 0.5γBNγ; apply shape, depth, and inclination factors for other cases.

Consolidation settlement: S = Cc·H/(1+e0) · log10((σ'0 + Δσ)/σ'0); time factor Tv = cv·t/Hdr².

Atterberg limits: PI = LL − PL; high PI indicates high plasticity and greater compressibility.

Unified Soil Classification: use percent passing No. 200 sieve, LL, and PI to assign group symbols (e.g., CL, SM, GW).

Slope stability factor of safety FS = resisting forces/moment ÷ driving forces/moment; FS > 1.5 is typical for design.

Simply supported beam: max moment wL²/8 (uniform load), PL/4 (center point load); max shear wL/2 or P/2.

Section properties: rectangular I = bh³/12; bending stress σ = Mc/I = M/S; shear stress τ = VQ/(Ib).

ASD vs LRU: LRFD factored load U = 1.2D + 1.6L + 0.5(Lr or S or R); ASD uses service loads against allowable stress.

Euler column buckling: Pcr = π²EI/(KL)²; slenderness ratio KL/r governs buckling mode.

Reinforced concrete: Whitney stress block a = As·fy/(0.85·f'c·b); nominal moment Mn = As·fy·(d − a/2).

Steel tension member: nominal strength Pn = Fy·Ag (yielding) or Fu·Ae (fracture); use the smaller.

Deflection of simply supported beam under uniform load: δmax = 5wL⁴/(384EI) at midspan.

Truss analysis: method of joints for member forces, method of sections for cutting through three members.

Horizontal curve: R = 5729.58/D (D in degrees, R in ft); tangent T = R·tan(Δ/2); length L = R·Δ·(π/180).

Superelevation and side friction: e + f = V²/(15R) with V in mph and R in ft.

Sight distance on a crest vertical curve: L = A·S²/(2158) when S < L, where A = |g2 − g1| in percent.

Stopping sight distance: SSD = 1.47·V·t + V²/(30(f ± G)), with t = 2.5 s reaction time.

Traffic flow relationships: q = k·v; headway h = 3600/q (veh/h); density k = q/v.

Level of service for highways is graded A–F based on density (pc/mi/ln) and flow rate for basic freeway segments.

Pavement design: ESAL (equivalent single axle loads) converts mixed traffic to 18-kip standard axles.

Asphalt mix: optimum asphalt content occurs near maximum stability and unit weight with target air voids (~4%).

Rational method peak flow: Q = CiA; C is runoff coefficient, i is intensity (in/hr), A in acres gives Q in cfs.

Manning's equation: Q = (1.486/n)·A·R^(2/3)·S^(1/2) in US customary units.

Time of concentration by Kirpich: Tc = 0.0078·L^0.77·S^(−0.385) with L in ft and S in ft/ft.

Water treatment train: coagulation → flocculation → sedimentation → filtration → disinfection.

BOD decay: BODt = BODu·(1 − e^(−kt)); ultimate BOD and rate constant k depend on waste characteristics.

Activated sludge: solids retention time SRT = (V·X)/(Qw·Xr + Qe·Xe); F/M ratio controls sludge bulking.

Hardy Cross and continuity: at each junction ΣQin = ΣQout; energy loss hf = f·(L/D)·(V²/2g) by Darcy-Weisbach.

Detention basin storage volume estimated from inflow/outflow hydrographs; required storage = area between curves.

Earthwork volume by average end area: V = L × (A1 + A2)/2, with A in ft² and L in ft giving V in ft³; divide by 27 for yd³.

Prismoidal correction for three-level sections: V = (L/6)(A1 + 4Am + A2); use when sections differ greatly.

Shrinkage and swell: bank volume × (1 + swell) = loose volume; compacted volume = bank volume / (1 − shrinkage) as a fraction.

Crew production rate = (number of units per cycle × cycles per hour) × efficiency; cost per unit = hourly crew cost / hourly production.

Critical path method: total float = LS − ES = LF − EF; activities with zero total float lie on the critical path.

Concrete mix by absolute volume: cement + water + fine + coarse + air = 27 ft³ per yd³ (or 1 m³).

Water-cement ratio governs strength and durability; lower w/c raises strength but reduces workability.

Slump test (ASTM C143) measures workability; standard slump for most structural concrete is roughly 2–4 in.

Safety: OSHA requires fall protection at 6 ft in construction (29 CFR 1926 Subpart M) and trench protective systems at 5 ft depth.

Effective stress: σ' = σ − u; total stress σ = γz, pore pressure u = γw × h.

Void ratio, porosity, and unit weight: e = n/(1−n); γ = (Gs + e·S)·γw/(1+e); dry unit weight γd = γ/(1+w).

Relative density Dr = (emax − e)/(emax − emin); used to describe sand compactness.

Terzaghi bearing capacity (strip footing): qu = cNc + qNq + 0.5γBNγ; apply shape, depth, and inclination factors for other cases.

Consolidation settlement: S = Cc·H/(1+e0) · log10((σ'0 + Δσ)/σ'0); time factor Tv = cv·t/Hdr².

Atterberg limits: PI = LL − PL; high PI indicates high plasticity and greater compressibility.

Unified Soil Classification: use percent passing No. 200 sieve, LL, and PI to assign group symbols (e.g., CL, SM, GW).

Slope stability factor of safety FS = resisting forces/moment ÷ driving forces/moment; FS > 1.5 is typical for design.

Simply supported beam: max moment wL²/8 (uniform load), PL/4 (center point load); max shear wL/2 or P/2.

Section properties: rectangular I = bh³/12; bending stress σ = Mc/I = M/S; shear stress τ = VQ/(Ib).

ASD vs LRFD: LRFD factored load U = 1.2D + 1.6L + 0.5(Lr or S or R); ASD uses service loads against allowable stress.

Euler column buckling: Pcr = π²EI/(KL)²; slenderness ratio KL/r governs buckling mode.

Reinforced concrete: Whitney stress block a = As·fy/(0.85·f'c·b); nominal moment Mn = As·fy·(d − a/2).

Steel tension member: nominal strength Pn = Fy·Ag (yielding) or Fu·Ae (fracture); use the smaller.

Deflection of simply supported beam under uniform load: δmax = 5wL⁴/(384EI) at midspan.

Truss analysis: method of joints for member forces, method of sections for cutting through three members.

Horizontal curve: R = 5729.58/D (D in degrees, R in ft); tangent T = R·tan(Δ/2); length L = R·Δ·(π/180).

Superelevation and side friction: e + f = V²/(15R) with V in mph and R in ft.

Sight distance on a crest vertical curve: L = A·S²/(2158) when S < L, where A = |g2 − g1| in percent.

Stopping sight distance: SSD = 1.47·V·t + V²/(30(f ± G)), with t = 2.5 s reaction time.

Traffic flow relationships: q = k·v; headway h = 3600/q (veh/h); density k = q/v.

Level of service for highways is graded A–F based on density (pc/mi/ln) and flow rate for basic freeway segments.

Pavement design: ESAL (equivalent single axle loads) converts mixed traffic to 18-kip standard axles.

Asphalt mix: optimum asphalt content occurs near maximum stability and unit weight with target air voids (~4%).

Rational method peak flow: Q = CiA; C is runoff coefficient, i is intensity (in/hr), A in acres gives Q in cfs.

Manning's equation: Q = (1.486/n)·A·R^(2/3)·S^(1/2) in US customary units.

Time of concentration by Kirpich: Tc = 0.0078·L^0.77·S^(−0.385) with L in ft and S in ft/ft.

Water treatment train: coagulation → flocculation → sedimentation → filtration → disinfection.

BOD decay: BODt = BODu·(1 − e^(−kt)); ultimate BOD and rate constant k depend on waste characteristics.

Activated sludge: solids retention time SRT = (V·X)/(Qw·Xr + Qe·Xe); F/M ratio controls sludge bulking.

Hardy Cross and continuity: at each junction ΣQin = ΣQout; energy loss hf = f·(L/D)·(V²/2g) by Darcy-Weisbach.

Detention basin storage volume estimated from inflow/outflow hydrographs; required storage = area between curves.