GATE CE Formula Sheet for Quick Revision
GATE Civil Engineering important formulas on one page: solid mechanics, structures, RCC, soil mechanics, fluids, hydrology, environmental and transportation.
The formulas that GATE Civil Engineering numerical questions use most, subject by subject. Use this page for revision after you have studied each subject's lesson.
Solid mechanics and structures
| Formula | Meaning |
|---|---|
| σ = P/A, ε = δL/L, E = σ/ε | Stress, strain, Young's modulus |
| δ = PL/(AE) | Elongation of a bar under axial load |
| σ = My/I | Bending stress (flexure formula) |
| τ = Tr/J | Shear stress in a circular shaft |
| I = bd³/12 (rectangle), πd⁴/64 (circle) | Second moment of area about the centroid |
| Simply supported, UDL w: M_max = wL²/8, δ_max = 5wL⁴/(384EI) | Beam formulas |
| Simply supported, central load P: M_max = PL/4, δ_max = PL³/(48EI) | Beam formulas |
| Cantilever, end load P: δ = PL³/(3EI) | Beam formulas |
| P_cr = π²EI/L_e² | Euler buckling load (L_e = effective length) |
| D_s = (m + r) − 2j (plane truss) | Static indeterminacy; m members, r reactions, j joints |
See Mechanics and solid mechanics and Structural analysis.
Concrete and steel (limit state)
| Formula | Meaning |
|---|---|
| x_u,max/d = 0.48 (Fe 415), 0.46 (Fe 500) | Limiting neutral axis depth |
| M_u,lim = 0.138 f_ck b d² (Fe 415) | Limiting moment of resistance |
| 0.36 f_ck b x_u = 0.87 f_y A_st | Neutral axis from force balance |
| Design load = 1.5 × (dead + live) | Load factor for the strength limit state |
See Concrete and steel.
Geotechnical engineering
| Formula | Meaning |
|---|---|
| e = n/(1 − n), Se = wG | Void ratio, porosity, degree of saturation |
| γ_d = Gγ_w/(1 + e) = γ/(1 + w) | Dry unit weight |
| k from Darcy: v = ki, Q = kiA | Flow through soil |
| σ' = σ − u | Effective stress |
| τ = c + σ' tan φ | Mohr–Coulomb strength |
| K_a = (1 − sin φ)/(1 + sin φ), K_p = 1/K_a | Rankine earth pressure coefficients |
| T_v = c_v t / d² | Consolidation time factor |
Fluid mechanics and hydrology
| Formula | Meaning |
|---|---|
| Q = AV, p/γ + V²/2g + z = constant | Continuity and Bernoulli |
| h_f = fLV²/(2gD) | Darcy–Weisbach head loss |
| Re = VD/ν | Laminar pipe flow below about 2000 |
| V = (1/n) R^(2/3) S^(1/2) | Manning's formula |
| Fr = V/√(gy) | Froude number; critical flow at Fr = 1 |
| Q = CIA | Rational method for peak runoff (consistent units) |
| Δ = 8.64 B/D | Duty D (ha/cumec), delta Δ (m), base period B (days) |
See Fluid mechanics and Hydrology.
Environmental and transportation
| Formula | Meaning |
|---|---|
| BOD_t = L₀(1 − e^(−kt)) | BOD exerted in t days (k base e) |
| Detention time = volume / flow | Settling tanks and reactors |
| SSD = vt + v²/(2gf) | Stopping sight distance (v in m/s) |
| e + f = v²/(gR) | Superelevation and side friction on a curve |
| Flow q = k × v | Traffic flow = density × speed |
See Environmental engineering and Transportation.
Check yourself
A simply supported beam of span 6 m carries a UDL of 10 kN/m. What is the maximum bending moment in kN·m?
Show answer and explanation
Answer: 45M_max = wL²/8 = 10 × 36/8 = 45 kN·m.
A soil has porosity 0.375. What is its void ratio?
Show answer and explanation
Answer: 0.6e = n/(1 − n) = 0.375/0.625 = 0.6.
For φ = 30°, what is the Rankine passive earth pressure coefficient K_p?
Show answer and explanation
Answer: 3K_p = (1 + sin 30°)/(1 − sin 30°) = 1.5/0.5 = 3.
A plane truss has 9 members, 3 reactions and 6 joints. What is its degree of static indeterminacy?
Show answer and explanation
Answer: 0D_s = m + r − 2j = 9 + 3 − 12 = 0: statically determinate.
Traffic moves at 40 km/h with a density of 50 vehicles/km. What is the flow in vehicles per hour?
Show answer and explanation
Answer: 2000q = k × v = 50 × 40 = 2000 veh/h.