Engineering Formulas

Structural Analysis

Moment of Inertia (I_xx, I_yy)

I_xx = bh³/12  |  I = πd⁴/64

Second moment of area for rectangular, circular, triangular, and hollow sections. Includes parallel axis theorem applications.

Pure bending, stress analysis View →

Beam Theory

Simple Bending Equation

σ/M = y/I = E/R

Fundamental flexure formula relating bending stress, moment, neutral axis distance, moment of inertia, modulus of elasticity, and radius of curvature.

Flexural stress, neutral axis View →

Beam Theory

Shear Force & Bending Moment Relations

V = dM/dx  |  w = dV/dx

Differential relationships between load intensity, shear force, and bending moment. Basis for drawing SFD and BMD.

Load-shear-moment relations View →

Deflection

Beam Deflection Methods

EI d²y/dx² = M(x)  |  δ = PL³/3EI

Double integration, Macaulay's method, conjugate beam, and Mohr's moment area methods. Standard deflection formulas for cantilever, simply supported, and fixed beams.

Serviceability limit state View →

Stability

Euler's Column Buckling

P_cr = π²EI/(KL)²

Critical buckling load for columns with different end conditions (pinned-pinned, fixed-fixed, fixed-free, fixed-pinned). Effective length factor K values.

Buckling, slenderness ratio View →

Determinacy

Truss Determinacy & Joint Method

m + r = 2j  |  ΣF_x=0, ΣF_y=0

Static determinacy condition for plane trusses. Method of joints for member force calculation using equilibrium equations at each node.

Statically determinate structures View →

Basic Properties

Soil Phase Relationships

n = e/(1+e)  |  γ_d = γ/(1+w)

Interrelationships among void ratio (e), porosity (n), degree of saturation (S), water content (w), dry unit weight (γ_d), and specific gravity (G).

Mass-volume relationships View →

Seepage

Permeability & Flow Net

k = QL/(Aht)  |  q = kh(N_f/N_d)

Darcy's Law coefficient calculation, constant head and falling head permeability tests. Flow net discharge estimation using flow channels and equipotential drops.

Darcy's law, seepage flow View →

Settlement

Consolidation Settlement

S_c = (C_cH/(1+e₀)) × log(σ'₂/σ'₁)

Primary consolidation settlement calculation using compression index (C_c). Time factor T_v, degree of consolidation U, and time rate of settlement.

Servicability, clay layers View →

Foundations

Terzaghi's Bearing Capacity

q_ult = cN_c + qN_q + 0.5γBN_γ

Ultimate bearing capacity equation for shallow foundations. Terzaghi's bearing capacity factors N_c, N_q, N_γ. Shape and depth corrections.

Shallow foundation design View →

Stability

Factor of Safety — Slopes

FS = τ_f/τ = (cL + Wcosθ tanφ)/(W sinθ)

Factor of safety calculation for infinite slopes and finite slopes using Swedish circle method (method of slices). Bishop's simplified method.

Embankment stability View →

Retaining Walls

Earth Pressure at Rest / Active / Passive

K_a = tan²(45-φ/2)  |  K_p = tan²(45+φ/2)

Rankine's earth pressure coefficients for cohesionless soils. Coulomb's theory with wall friction. At-rest coefficient K₀ ≈ 1 − sinφ.

Lateral earth pressure View →

Linear Measurement

Tape Corrections

C_p = PL/AE  |  C_T = α(T-T₀)L

Correction for pull/tension, temperature, slope, sag, and mean sea level. True distance = Measured distance + Algebraic sum of corrections.

Precise chaining View →

Height Measurement

Height of Instrument & Rise/Fall

HI = RL + BS  |  RL = HI − FS

Two methods for reducing levels: H.I. method (collimation) and rise-and-fall method. Arithmetic checks for closing error and permissible closure.

Elevation differences View →

Traversing

Traverse Balancing & Coordinates

Δx = L cos θ  |  Δy = L sin θ

Latitudes and departures. Bowditch rule and transit rule for closing error adjustment. Coordinate computation from measured distances and bearings.

Closed loop adjustment View →

Horizontal Curves

Simple Circular Curve Elements

R = 1718.9/D (degree)  |  T = R tan(Δ/2)

Tangent length, curve length, long chord, apex distance, versed sine. Rankine's tangential angles method and two-theodolite method.

Road/railway curves View →

Computation

Area & Volume Calculations

A = (O₁+Oₙ)/2 + ΣOᵢ × d  |  V = A × L

Trapezoidal rule and prismoidal rule for area from offsets and volume from cross-sections. Simpson's rule for irregular boundaries.

Earthwork computation View →

Pipe Systems

Darcy-Weisbach Head Loss

h_f = fLV²/(2gD)  |  Re = ρVD/μ

Major energy loss in pipes due to friction. Friction factor f from Moody diagram based on Reynolds number and relative roughness. Minor losses at fittings.

Pipeline design View →

Open Channel

Manning's Formula

V = (1/n)R^(2/3)S^(1/2)  |  Q = AV

Mean velocity and discharge in open channels (canals, rivers, sewers). Manning's roughness coefficient n values for various materials. Hydraulic radius R.

Critical flow, normal depth View →

Flow Measurement

Discharge over Weirs & Notches

Q = (2/3)C_db√(2g)H^(3/2)  |  V = √(2gh)

Rectangular, trapezoidal (Cipolletti), and triangular (90°) notches. Coefficient of discharge C_d, velocity of approach correction.

Flow measurement View →

Critical Flow

Specific Energy & Critical Depth

E = y + V²/(2g)  |  y_c = (q²/g)^(1/3)

Specific energy diagram for open channel flow. Critical depth y_c where Froude number F_r = 1. Subcritical vs supercritical flow classification.

Hydraulic jump View →

Machines

Pump & Turbine Efficiency

η = Output/Input  |  P = ρgQH

Power and efficiency calculations for centrifugal pumps and turbines. Manometric head, specific speed, cavitation parameters.

Water supply systems View →

Safety

Stopping & Overtaking Sight Distance

SSD = 0.278Vt + V²/(254f)  |  OSD = 0.278V(t₁+t₂+t₃)

Stopping sight distance based on driver reaction time t and friction factor f. Overtaking sight distance components: perception, maneuver, and clear zone.

Horizontal alignment design View →

Alignment

Super-elevation Design

e + f = V²/(127R)  |  e ≤ 0.07 (IRC)

Required super-elevation for horizontal curves considering vehicle speed V and curve radius R. IRC limits: max e = 7%, f = 0.15 for mixed traffic.

Curve safety, banking View →

Pavements

CBR & Flexible Pavement Thickness

T = √(P/p)(a/V)² − d²  |  CBR = (Pt/Pstd) × 100

California Bearing Ratio test interpretation. Burmister rigid/flexible pavement thickness design charts. Group Index method for subgrade evaluation.

Subgrade support View →

Traffic Engineering

Traffic Flow Parameters

q = kV  |  k_j = density at jam

Fundamental diagram of traffic flow: volume (q), density (k), and speed (V) relationships. Space-mean vs time-mean speed. Jam density concept.

Capacity analysis View →

CPM/PERT

Critical Path Method & PERT

EFT = EST + t  |  LF = LFT - t

Forward pass (EST, EFT) and backward pass (LST, LFT). Slack = LF − EF. PERT expected time te = (a + 4m + b)/6 and variance σ² = ((b-a)/6)².

Activity-on-node networks View →

Optimization

Project Crashing & Cost-Time Trade-off

Cost Slope = (CC - NC)/(NT - CT)

Crash cost per day (slope) for critical activities. Determine optimal crash duration considering direct cost increase vs indirect cost reduction.

Time-cost optimization View →

Equipment

Equipment Production Rate

Q = C × S × F × E  |  Cycle Time = Load + Travel + Dump

Excavator, bulldozer, and dump truck productivity. Bucket factor F, job efficiency E, swing angle correction, haul road grade resistance.

Site planning View →

Quantities

Centre Line & Long-Wall Method

Total Length = 2(L+B) − No. of junctions × thickness

Quantity estimation for buildings using centre-line, long-wall-short-wall, and part-by-part methods. Deduction rules for openings and junctions.

Bill of quantities View →