Civil Engineering Formulas and Handbook
Section 1) Surveying
Traverse Surveying
Correction For Sag $$ {C_s} = {{{w^2}{l^3}}\over {24.P^2}} $$ Note:-sag correction is subtractive
Where
w = weight of tape per meter
l= distance between supports
P = Applied Tension
Correction For Tension $$ {C_p} = {{(P-{P_0})l}\over AE} $$
Levelling
Reciprocal levelling
Section 2) Transportation Engineering
Stopping sight distance
Stopping sight distance (SSD in 'metres') of a vehicle is the sum of :
1) The distance travelled by the vehicle during the total reaction time known as lag distance and
$${\text{Lag distance}}=0.278Vt$$
2) The distance travelled by the vehicle after the application of the brakes, to a dead stop position which is known as braking distance.
$${\text{Braking distance}}={{V^2}\over {254(f\pm S \%)}}$$
$$\text{SSD = Lag distance + Braking distance}$$
$$SSD = 0.278Vt + {{V^2}\over {254(f\pm S \%)}}$$
Where S% is gradient '+' sign for ascending gradient & '-' sign for descending gradient
Overtaking Sight Distance (OSD)
Important equation and terminology checks
- Traverse balancing: corrections must oppose the closing errors. The Compass/Bowditch and Transit correction equations above are written with a minus sign for this reason.
- Direction of closing error: use \(\operatorname{atan2}(\sum D,\sum L)\), or determine the quadrant from the signs of the total latitude and departure. A simple \(\tan^{-1}(\sum D/\sum L)\) alone can give the wrong quadrant.
- Area by cross-staff/offsets: for a series of offsets at equal spacing, use the trapezoidal summation \(A=\frac{d}{2}\sum_{i=1}^{n-1}(O_i+O_{i+1})\). A single \(\frac12(L_1+L_2)d\) applies to one interval only.
- Curvature and refraction: the constants shown in the notes assume distance \(d\) in kilometres and correction in metres. Curvature correction is normally subtracted from observed staff readings; combined curvature-refraction correction is approximately \(0.0673d^2\) m under standard assumptions.
- Stopping sight distance: \(SSD=0.278Vt+\frac{V^2}{254(f\pm G)}\) is a commonly used metric form when \(V\) is in km/h, \(t\) in seconds, \(f\) is the longitudinal friction factor, and \(G\) is gradient as a decimal. If the gradient is entered as a percentage number, the denominator convention must be adjusted; do not insert “S%” without clarifying whether it means percent or decimal.
- Overtaking sight distance: the source contains alternative expressions and assumptions for \(d_2\), \(T\), and vehicle speeds. These depend on the design standard and unit convention; verify against the applicable IRC standard before using them for design.
- Scope: these are formatted study notes, not a substitute for checking the applicable surveying/IRC standard or project specifications.
| Terrain Catogory | Over taking Sight Distance | Safe stopping sight for minimum design speed | Safe stopping sight for Rulling design speed |
|---|---|---|---|
| Plain Terrain | 475 | 120 | 170 |
| Rolling Terrain | 310 | 90 | 120 |
| Hilly Terrain | - | 50 | 60 |
| Steep | - | 40 | 50 |
Super Elevation (e)
| emax | Terrain Type |
|---|---|
| 0.07 | Plain and Rolling Terrain |
| 0.10 | Hilly Area |
| 0.04 | Urban roads with frequent intersections |
Ruling minimum radius of the curve
Radii for horizontal curves in meters
| Terrain Catogory | Absolute | Rulling |
|---|---|---|
| Plain Terrain | 250 | 370 |
| Rolling Terrain | 155 | 250 |
| Hilly Terrain | 50 | 80 |
| Steep | 30 | 50 |
Extra Widening (Ew)
Transition Curve
Length of transition curve (L)
(i) According to rate of change of centrifugal acceleration
(ii) According to rate of change of super elevation
(iii) According to empirical formula
To find out offset length at desired chainage 'x' the equation of cubical parabola
Set Back Distance (m)
1. For Single lane road
&
$${\alpha \over 2}={{180.s}\over {2\Pi R}}$$
Where, Lc = Length of Curve & s = SSD
(b) Lc < SSD
$$m=R\left({1-cos{\alpha \over 2}}\right) + {{S - L_c}\over 2} sin{\alpha \over 2}$$
&
$${\alpha \over 2}={{180.L_c}\over {2\Pi R}}$$
2. For two lane road
&
$${\alpha \over 2}={{180.s}\over {2\Pi (R-d)}}$$
Where, Lc = Length of Curve & s = SSD
(b) Lc < SSD
$$m=R-(R-d)\left({1-cos{\alpha \over 2}}\right) + {{S - L_c}\over 2} sin{\alpha \over 2}$$
&
$${\alpha \over 2}={{180.L_c}\over {2\Pi (R-d)}}$$
Grade Compensation
Vertical Curve
Summit curves (Crest curve with convexity upwards):
Valley Curves (Sag curve with Concavity Upward):
Summit curves (Crest curve with convexity upwards):
For CBR ≤ 5%:
$$M_{r,s} = 10 \times CBR$$
For CBR > 5%:
$$M_{r,s} = 17.6 \times CBR^{0.64}$$
Resilient modulus formula from repeated load triaxial test based on recoverable elastic deformation under deviator stress
$$M_{rs} = \frac{2 (1 - \mu^2) \, p_a}{\Delta}$$
Where:
$$ \begin{align*} M_r &= \text{Resilient modulus (MPa)} \\ \mu &= \text{Poisson's ratio of the material (dimensionless)} \\ p_a &= \text{Applied deviator (cyclic) stress (MPa)} \\ \Delta \varepsilon &= \text{Recoverable axial strain (dimensionless)} \end{align*} $$Fatigue Model (IRC:37-2018)
$$N_f = K_1 \left( \frac{1}{\varepsilon_t} \right)^{K_2} \left( \frac{1}{E_{bit}} \right)^{K_3}$$
Where:
$$ \begin{aligned} N_f &= \text{Number of cumulative standard axles to cause fatigue cracking} \\ \varepsilon_t &= \text{Tensile strain at the bottom of bituminous layer (in microstrain)} \\ E_{bit} &= \text{Elastic modulus of bituminous layer (in MPa)} \\ K_1, K_2, K_3 &= \text{Empirical constants} \end{aligned} $$As per IRC:37-2018 (Table 4.3):
$$ \begin{aligned} \text{For 90% reliability upto 30msa } K_1 &= 0.516 \times 10^{-4} \\ \text{For 80% reliability upto 30 msa } K_1 &= 1.4064 \times 10^{-4} \\ K_2 &= 3.89 \\ K_3 &= 0.854 \end{aligned} $$Where (for rutting model):
$$ \begin{aligned} N_d &= \text{Number of cumulative standard axles to cause rutting failure} \\ \varepsilon_v &= \text{Vertical compressive strain on top of subgrade or granular layer} \\ K_4, K_5 &= \text{Empirical constants} \end{aligned} $$As per IRC:37-2018 (Table 4.4):
$$ \begin{aligned} \text{For 90% reliability upto 30 msa:} \\ K_4 &= 1.41 \times 10^{-8} \\ K_5 &= 4.5337 \\ \text{For 80% reliability upto 30 msa:} \\ K_4 &= 1.84 \times 10^{-8} \\ K_5 &= 4.5337 \end{aligned} $$Granular Sub Base
Resilient modulus for granular layer is function of modulus of subgrade and thickness of granular layer
$$M_{rgran} = 0.2 \times (h)^{0.45} \times M_{rSupport}$$
$$M_{rSupport} \text{ is modulus of support, below layer ubder consideration}$$
Section 3) Mechanics of Materials
Stress
Strain
$$\varepsilon = \frac{\Delta L}{L}$$
Hooke's Law
$$\sigma = E \times \varepsilon$$
Shear Stress
Bending Stress (Flexure Formula)
Torsional Shear Stress
Deflection of Simply Supported Beam (Center Load)
Euler's Buckling Load (Pinned-Pinned)
Moment of Inertia (Rectangle)
Torsion Equation
Principal Stresses (2D)
Maximum Shear Stress
Strain Energy (Axial Load)
Castigliano's Theorem (Deflection at point of load P)
Shear Force (Beam Element)
$$\frac{dV}{dx} = -w(x)$$
Bending Moment (Beam Element)
$$\frac{dM}{dx} = V(x)$$
Relation between Moment and Slope
Slenderness Ratio
Radius of Gyration
Hydraulics / Hydrology
This reference collects commonly used equations for partially full circular pipes, weirs, open-channel flow, pipe-friction losses, rainfall-runoff estimation, infiltration, flood frequency, evapotranspiration and reservoir storage. Check the assumptions and units before using any equation for design.
1. Hydraulic Characteristics of a Circular Pipe Flowing Partially Full
Let \(D\) be the internal pipe diameter and \(\theta\) the angle subtended at the pipe centre by the wetted arc. In the equations below, \(\theta\) is expressed in degrees; the sine and cosine functions must use the same angle convention.
where \(y\) is the depth of water measured from the invert.
Area of flow
Wetted perimeter
Hydraulic radius
Velocity by Manning's equation
Here \(n\) is Manning's roughness coefficient and \(S\) is the energy slope (dimensionless, m/m). For uniform open-channel flow, the energy slope is commonly approximated by the pipe slope.
Discharge and proportional discharge
For a full circular pipe flowing under the same Manning \(n\) and slope, let \(A_f=\pi D^2/4\), \(R_f=D/4\), and \(Q_f=A_fV_f\). Then:
Equivalently, with \(\theta\) in degrees:
For a detailed derivation, see Hydraulic Characteristics of a Circular Pipe.
2. Broad-Crested Weir
A broad-crested weir has a relatively long crest in the direction of flow. A commonly used general form is:
where \(Q\) is discharge (m³/s), \(L\) is effective crest width (m), \(H\) is upstream total head above the crest (m), and \(C\) is a coefficient whose value and units depend on the adopted equation and flow conditions.
3. Sharp-Crested Weirs
3.1 Rectangular sharp-crested weir
For a fully contracted rectangular sharp-crested weir, the Francis-type metric equation is often written approximately as:
where \(Q\) is in m³/s, \(L\) and \(H\) are in metres, and \(n_c\) is the number of end contractions (0 for a suppressed weir; commonly 2 for a fully contracted rectangular weir). This empirical form is valid only under its associated experimental assumptions; use the governing standard for design or measurement.
3.2 V-notch (triangular) sharp-crested weir
where \(C_d\) is the discharge coefficient, \(g\approx9.81\text{ m/s}^2\), \(\theta\) is the included notch angle, and \(H\) is the head above the notch apex. Use a calibrated coefficient and ensure the approach flow and nappe conditions meet the selected method's requirements.
3.3 Trapezoidal notch
A trapezoidal notch combines rectangular and triangular portions. Its discharge should be calculated using a recognised equation for the particular notch geometry and coefficient convention. The coefficient expression sometimes reproduced as \(\frac{2}{3}(1+\frac{2h}{b})(1-\frac{h}{H})^{3/2}\) is not a generally applicable discharge coefficient and is therefore not presented here as a universal design equation.
4. Chezy Formula
\(V\) = mean velocity (m/s); \(C\) = Chezy coefficient (units depend on the unit system, commonly m\(^{1/2}\)/s in SI); \(R=A/P\) = hydraulic radius (m); and \(S\) = energy slope (m/m).
The Chezy coefficient must be selected or calibrated for the channel and flow conditions. It is not dimensionless in a consistent dimensional analysis.
5. Darcy–Weisbach Friction Head Loss
\(h_f\) = friction head loss (m); \(f_D\) = Darcy friction factor (dimensionless); \(L\) = pipe length (m); \(D\) = internal pipe diameter (m); \(V\) = mean velocity (m/s); and \(g\approx9.81\text{ m/s}^2\).
6. Manning's Formula for Open Channels
\(V\) = mean velocity (m/s); \(n\) = Manning roughness coefficient; \(R\) = hydraulic radius (m); and \(S\) = energy slope (m/m). Discharge is \(Q=AV\), where \(A\) is the flow area (m²).
7. Runoff Coefficient Method (Rational Formula)
For \(Q\) in m³/s, rainfall intensity \(I\) in mm/h and catchment area \(A\) in hectares, the conversion factor 0.00278 is required. \(C\) is the dimensionless runoff coefficient.
8. Time of Concentration — Kirpich Equation
This version estimates \(T_c\) in minutes when \(L\) is the channel length in metres and \(S\) is the dimensionless average slope (m/m). The equation is empirical and should be used only within an appropriate catchment range.
9. Rainfall Intensity–Duration–Frequency (IDF)
\(I\) = rainfall intensity; \(T_r\) = return period (years); \(t\) = storm duration; and \(K,n,b,m\) = locally fitted constants. The units for \(t\), \(I\), and the constants must match the IDF model from which they were derived. This is a general model form, not a universal set of constants.
10. Infiltration Rate — Horton's Equation
\(f(t)\) = infiltration rate at time \(t\); \(f_0\) = initial infiltration rate; \(f_c\) = final constant infiltration rate; and \(k\) = decay constant. If \(t\) is in hours, \(k\) must be in h\(^{-1}\); use consistent units for all infiltration rates.
11. SCS Curve Number Method — Direct Runoff
Potential maximum retention after runoff begins:
Here \(S\) is in millimetres and \(CN\) is the dimensionless curve number (for the metric form).
With the conventional initial-abstraction assumption \(I_a=0.2S\), direct runoff depth is:
\(P\) = storm rainfall depth (mm); \(Q_d\) = direct runoff depth (mm). The original formula is incomplete unless the condition \(P>I_a\) is stated; otherwise runoff is zero.
12. Unit Hydrograph Peak Discharge
This expression appeared in the source material, with \(Q_p\) in m³/s, \(A\) in km² and \(T_r\) in hours. Verify the derivation before use: the constant depends on the assumed hydrograph shape and the definition of \(T_r\) (such as time to peak or effective rainfall duration). It is not a universal unit-hydrograph equation. Use the peak formula corresponding to the adopted unit-hydrograph method and duration.
13. Flood Frequency Analysis — Gumbel Form
\(X_T\) = estimated flood magnitude for return period \(T\); \(\bar X\) = sample mean; \(\sigma\) = sample standard deviation; and \(K_T\) = frequency factor for the selected distribution and return period. For a Gumbel distribution, calculate \(K_T\) using the relevant sample size and frequency-factor procedure; it is not a fixed constant.
14. Probability of Exceedance — Plotting Position
\(m\) = rank of the event in the ordered series and \(N\) = number of observations. State the ranking convention used. For annual maximum floods sorted from largest to smallest, this plotting position estimates exceedance probability; the corresponding return period is often estimated as \(T\approx1/(P_e/100)\) years, subject to the assumptions of the analysis.
15. Evapotranspiration — Blaney–Criddle Method
This is a simplified form of the Blaney–Criddle relationship. \(T\) is mean daily temperature in °C and \(p\) is the percentage fraction of annual daytime hours for the period, expressed according to the convention required by the selected version of the method. The result and time period depend on the method's units and adjustments. For crop-water design, use the applicable calibrated/adjusted method rather than treating this simplified equation as universally sufficient.
16. Reservoir Storage — Prismoidal Formula
\(V\) = volume between sections (m³); \(h\) = vertical interval between the end sections (m); \(A_1\) and \(A_2\) = end areas (m²); and \(A_m\) = area at the midpoint (m²). This formula assumes the midpoint area is available and the area variation is suitably represented by the prismoidal rule.
Important Engineering Checks
- Keep units consistent throughout each calculation.
- Confirm whether angles are in degrees or radians; do not mix conventions.
- Use coefficients from an applicable standard, published method or site calibration.
- Distinguish head above a weir crest from upstream water depth and approach velocity head.
- For design submissions, record assumptions, catchment data, return period, roughness values and the source of coefficients.
This page is a technical reference, not a substitute for project-specific hydraulic design, applicable standards or independent checking.
Section 5) Stresses in Structures
Equation of Pure torsion
$${T\over {I_p}}={\tau \over r}={{G \theta} \over L}$$
Where,
τ --> Shear stress for example shear stress on wall of a tube subjected to torsion
Ө --> Angle of twist
$$G = {E\over {2(1+\mu)}}$$
1) Circular tube subjected to torsion
$$\text{Polar Moment of Inertia }{I_p} = \text{Area of The section }\times{r^2}$$
$$(\pi D t)\times \left({D \over 2}\right)^2 = {{\pi {D^3} t}\over 4}$$
2) Thin Square tube subjected to torsion
Maximum shear stress τ is given by
$$\tau = {T\over {2A.t}}$$
Normal Stress (Axial)
$$ \sigma = \frac{P}{A} $$
Where:
\( \sigma \) = Normal stress (N/mm² or MPa)
\( P \) = Axial load (N)
\( A \) = Cross-sectional area (mm²)
Shear Stress
$$ \tau = \frac{V}{A} $$
Where:
\( \tau \) = Shear stress (N/mm²)
\( V \) = Shear force (N)
\( A \) = Area resisting shear (mm²)
Bending Stress (Flexural Formula)
$$ \sigma = \frac{M y}{I} $$
Where:
\( M \) = Bending moment (Nmm)
\( y \) = Distance from neutral axis (mm)
\( I \) = Moment of inertia (mm⁴)
Torsional Shear Stress
$$ \tau = \frac{T r}{J} $$
Where:
\( T \) = Applied torque (Nmm)
\( r \) = Distance from center to outer fiber (mm)
\( J \) = Polar moment of inertia (mm⁴)
Combined Axial and Bending Stress
$$ \sigma_{max} = \frac{P}{A} \pm \frac{M y}{I} $$
Principal Stresses (2D State)
$$ \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} $$
Where:
\( \sigma_x, \sigma_y \) = Normal stresses
\( \tau_{xy} \) = Shear stress
Maximum Shear Stress
$$ \tau_{max} = \frac{\sigma_1 - \sigma_2}{2} $$
Bearing Stress
$$ \sigma_b = \frac{P}{A_b} $$
Where:
\( A_b \) = Bearing area (mm²)
Hoop (Circumferential) Stress (Thin Cylindrical Pressure Vessel)
$$ \sigma_h = \frac{p d}{2 t} $$
Where:
\( p \) = Internal pressure (N/mm²)
\( d \) = Internal diameter (mm)
\( t \) = Thickness (mm)
Longitudinal Stress (Pressure Vessel)
$$ \sigma_l = \frac{p d}{4 t} $$
Section 6) Steel Design
Design Stress in Steel
Design Tensile Strength (Gross Section)
Design Tensile Strength (Net Section)
Slenderness Ratio
Plastic Section Modulus
Shear Strength (Web Area)
Bolt Shear Strength (Single Plane)
Weld Strength
Factored Load
$$P_u = \gamma_f \times P$$
Bending Moment — Shear Force Relation
$$\frac{dM}{dx} = V(x)$$
Section 7) Concrete Mix Design
Water-Cement Ratio (Abram’s Law)
$$ \frac{f_1}{f_2} = \left(\frac{w_2}{w_1}\right)^n $$
where:
\( f_1, f_2 \) = compressive strengths
\( w_1, w_2 \) = water-cement ratios
\( n \) = constant (usually between 5 to 7)
Volume of Concrete
$$ V = \frac{W + \frac{C}{\rho_c} + \frac{F}{\rho_f} + \frac{C.A}{\rho_{ca}} + \frac{F.A}{\rho_{fa}} + \frac{W}{\rho_w}}{1000} $$
Absolute Volume of Cement
$$ V_c = \frac{W_c}{\rho_c} $$
Absolute Volume of Fine Aggregate
$$ V_{fa} = \frac{W_{fa}}{\rho_{fa}} $$
Absolute Volume of Coarse Aggregate
$$ V_{ca} = \frac{W_{ca}}{\rho_{ca}} $$
Volume of Water
$$ V_w = \frac{W_w}{\rho_w} $$
Air Content (as percentage)
$$ \text{Air \%} = \frac{V_{air}}{V_{total}} \times 100 $$
Aggregate Proportioning by Weight
$$ \frac{W_{ca}}{W_{fa}} = \frac{V_{ca} \times \rho_{ca}}{V_{fa} \times \rho_{fa}} $$
Volume of Cementitious Materials
$$ V_m = \frac{W_c}{\rho_c} + \frac{W_{fa}}{\rho_{fa}} + \frac{W_{ca}}{\rho_{ca}} + \frac{W_w}{\rho_w} $$
Adjustment for Moisture Content
$$ W_{corrected} = W_{SSD} + (MC \times W_{SSD}) $$
Correction for Aggregate Water Absorption
$$ W_{adjusted} = W_{SSD} - (Absorption \times W_{SSD}) $$
Target Mean Strength
$$ f_{ck'} = f_{ck} + t \times s $$
where
\( f_{ck} \) = characteristic strength
\( t \) = tolerance factor (from IS 456 table)
\( s \) = standard deviation
Volume of Entrapped Air
$$ V_{air} = \text{as per IS 10262 recommendations (usually 1-2\%)} $$
Modulus of Elasticity of Concrete
$$ E_c = 5000 \sqrt{f_{ck}} $$
where:
\( E_c \) = modulus of elasticity (MPa)
\( f_{ck} \) = characteristic compressive strength (MPa)
Bulking of Sand (Approximate Correction)
$$ \text{Corrected Sand Volume} = \frac{\text{Specified Volume}}{1 + \frac{\text{Bulking Percentage}}{100}} $$
Mix Proportion by Volume
**For M20 concrete**
1 : 1.5 : 3 (Cement : Sand : Aggregate)
Water-cement ratio as per design requirements.
Fineness Modulus (FM)
$$ FM = \frac{\text{Cumulative \% retained on standard sieves}}{100} $$
Slump Test Value Range
- Very low workability: 0–25 mm
- Low workability: 25–50 mm
- Medium workability: 50–100 mm
- High workability: 100–175 mm
Degree of Workability — Compaction Factor
- Very low: 0.70
- Low: 0.85
- Medium: 0.92
- High: 0.95
Ultimate Creep Strain
$$ \epsilon_{cr} = \phi \times \epsilon_{e} $$
where:
\( \epsilon_{cr} \) = creep strain
\( \phi \) = creep coefficient
\( \epsilon_{e} \) = elastic strain
Maturity of Concrete (Nurse-Saul Equation)
$$ M = \sum (T_a - T_0) \times \Delta t $$
where:
\( T_a \) = average concrete temperature
\( T_0 \) = datum temperature (usually -11°C)
\( \Delta t \) = time interval (hours)
Specific Gravity
$$ G = \frac{\text{Weight of Aggregate in Air}}{\text{Weight of Equal Volume of Water}} $$
Water Absorption of Aggregate
$$ \text{Water Absorption (\%)} = \frac{W_{ssd} - W_{oven}}{W_{oven}} \times 100 $$
where:
\( W_{ssd} \) = saturated surface dry weight
\( W_{oven} \) = oven-dry weight
Segregation Index (SI)
$$ SI = \frac{\text{Difference of densities of top & bottom layers}}{\text{Average density}} $$
Compressive Strength Formula
$$ f_c = \frac{P}{A} $$
where:
\( P \) = failure load (N)
\( A \) = loaded area (mm²)
Cement Content per Bag
**1 bag cement = 50 kg = 0.035 m³**
Section 8) Geotechnical Engineering
Cohesive soils:- Soil consisting of the finer products of rock weathering. Cohesion is derived from large number of water films associated with the fine grained particles in the soil.
Total force of Attraction between two particle (f) =
$$f={{2 {\pi} rT}\over {1+tan \left({\theta}\over 2 \right) }}$$
where r => Radius of Particle
T => Surface Tension
Liquidity index:-The difference between natural moisture content and plastic limit expressed as percentage ratio of the plasticity index.
$$LI={{m-PL}\over {LL-PL}}\times 100$$
m => moisture content
$$\text{Plasticity Index}={\text{Liquid Limit }-\text{ Plastic Limit}} $$
Water Content
$$ w = \frac{W_w}{W_s} \times 100 $$
Specific Gravity of Soil Solids
$$ G = \frac{W_s}{W_s - W_w} $$
Void Ratio
$$ e = \frac{V_v}{V_s} $$
Porosity
$$ n = \frac{V_v}{V} \times 100 $$
Degree of Saturation
$$ S_r = \frac{V_w}{V_v} \times 100 $$
Dry Density
$$ \rho_d = \frac{W_s}{V} $$
Bulk Density
$$ \rho = \frac{W}{V} $$
Saturated Density
$$ \rho_{sat} = \frac{W_{sat}}{V} $$
Submerged Density
$$ \rho' = \rho_{sat} - \rho_w $$
Unit Weight Relationships
$$ \gamma_d = \frac{G \times \gamma_w}{1+e} $$
Liquidity Index
$$ LI = \frac{w - PL}{LL - PL} \times 100 $$
Plasticity Index
$$ PI = LL - PL $$
Flow Index
$$ FI = \frac{w_1 - w_2}{\log(N_2/N_1)} $$
Consistency Index
$$ CI = \frac{LL - w}{PI} $$
Shear Strength (Coulomb’s Equation)
$$ \tau = c + \sigma \tan \phi $$
Terzaghi’s Bearing Capacity Equation
$$ q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma $$
Settlement of Soil
$$ S = \frac{C_c}{1+e_0} \log \left( \frac{\sigma'_0 + \Delta \sigma}{\sigma'_0} \right) \times H $$
Factor of Safety (Slope Stability)
$$ FOS = \frac{\text{Resisting forces}}{\text{Driving forces}} $$
Section 9) Sewerage and Sewage disposal
Chezy’s Formula
$$ V = C \sqrt{R S} $$Where:
= Velocity of flow (m/s)
= Chezy's coefficient
= Hydraulic radius (m)
= Slope of the energy line (m/m)
Manning’s Formula
$$ V = \frac{1}{n} R^{2/3} S^{1/2} $$
Where:
= Manning’s roughness coefficient
= Hydraulic radius (m)
= Slope of the channel (m/m)
Hydraulic Radius
$$ R = \frac{A}{P} $$
Where:
= Area of flow (m²)
= Wetted perimeter (m)
Discharge in Open Channel
$$ Q = A \times V $$
Where:
= Discharge (m³/s)
= Area of flow (m²)
= Velocity (m/s)
Population Forecast (Arithmetic Increase Method)
$$ P_n = P_0 + n \times I $$
Where:
= Population after decades
= Present population
= Average increase per decade
= Number of decades
Population Forecast (Geometric Increase Method)
$$ P_n = P_0 (1 + r)^n $$
Where:
= Population after decades
= Present population
= Number of decades
Sewage Flow Estimation
$$ Q = q \times P $$
Where:
= Sewage discharge (litres/day)
= Per capita sewage generation (litres/person/day)
$$ P_n = P_0 + n \times I $$
= Population
BOD First Order Reaction
$$ BOD_t = BOD_0 \left(1 - e^{-k t}\right) $$
Where:
= Ultimate BOD
= Time in days
Oxygen Demand for Nitrification
$$ O_2 = 4.57 \times NH_3 $$
Detention Time in Sedimentation Tank
$$ t = \frac{V}{Q} $$
Where:
Detention Time in Sedimentation Tank
$$ t = \frac{V}{Q} $$
Where:
\( t \) = Detention time (sec or hours)
\( V \) = Volume of tank (m³)
\( Q \) = Flow rate (m³/sec)
Surface Loading Rate (Overflow Rate)
$$ \text{SLR} = \frac{Q}{A} $$
Where:
\( Q \) = Flow rate (m³/day)
\( A \) = Surface area of tank (m²)
BOD (Biochemical Oxygen Demand) Removal
$$ BOD_t = BOD_0 \left(1 - e^{-k t}\right) $$
Where:
\( BOD_0 \) = Initial BOD (mg/L)
\( k \) = Reaction rate constant (day⁻¹)
\( t \) = Time (days)
BOD Loading on Aeration Tank
$$ \text{BOD Loading} = \frac{Q \times BOD}{V} $$
Where:
\( Q \) = Flow rate (m³/day)
\( BOD \) = Concentration (mg/L)
\( V \) = Volume of aeration tank (m³)
Food to Microorganism Ratio (F/M Ratio)
$$ F/M = \frac{Q \times S_0}{V \times X} $$
Where:
\( Q \) = Flow rate (m³/day)
\( S_0 \) = Influent BOD (mg/L)
\( V \) = Volume of aeration tank (m³)
\( X \) = MLSS (Mixed Liquor Suspended Solids) (mg/L)
Mean Cell Residence Time (MCRT)
$$ \theta_c = \frac{V \times X}{Q_w \times X_w + Q_e \times X_e} $$
Where:
\( Q_w \) = Waste sludge flow rate (m³/day)
\( X_w \) = Concentration of solids in waste sludge (mg/L)
\( Q_e \) = Effluent flow rate (m³/day)
\( X_e \) = Effluent solids concentration (mg/L)
Sludge Volume Index (SVI)
$$ SVI = \frac{\text{Settled Sludge Volume (mL/L)}}{\text{MLSS (mg/L)}} \times 1000 $$
Where:
SVI in mL/g
Air Requirement for Aeration
$$ Q_{air} = \frac{Q \times BOD \times Y}{C} $$
Where:
\( Q_{air} \) = Air required (m³/hr)
\( Y \) = Oxygen requirement per kg of BOD (kg O₂/kg BOD)
\( C \) = Oxygen transfer efficiency
Chlorine Dose
$$ \text{Dose (mg/L)} = \frac{\text{Chlorine applied (mg)}}{\text{Volume (L)}} $$
pH Definition
$$ pH = -\log[H^+] $$
Where:
\( [H^+] \) = Hydrogen ion concentration (mol/L)
Section 10) Water Supply and Purification
Population Forecasting (Arithmetic Increase Method)
$$ P_n = P_0 + n \times I $$
Per Capita Demand
$$ q = \frac{Q}{P} $$
Detention Time in Sedimentation Tank
$$ t = \frac{V}{Q} $$
Surface Loading Rate (Overflow Rate)
$$ SLR = \frac{Q}{A} $$
Filter Rate (for Rapid Sand Filter)
$$ FR = \frac{Q}{A} $$
Chlorine Dose
$$ C_d = C_r + D_c $$
Disinfection Contact Time (CT Concept)
$$ C \times T = \text{constant for effective disinfection} $$
Water Pipe Flow (Hazen-Williams Equation)
$$ V = 0.85 \times C \times R^{0.63} \times S^{0.54} $$
Loss of Head in Filter
$$ h_f = \frac{Q^2}{K} $$
Water Demand Formula (General)
$$ Q = q \times P $$
Section 11) Building Materials
| SR. No. | Items | Unit | Materials | ||
|---|---|---|---|---|---|
| Particulars | Quantity | Unit | |||
| 1 | Brick Work | 100 cu.m | Bricks | 50000 | Nos |
| Mortar | 30 | cu.m | |||
| 2 | Rubble stone Masonry | 100 cu.m | Stone | 125 | cu.m |
| Mortar | 42 | cu.m | |||
| 3 | 25 mm Cement Concrete 1:2:4 | 100 sq.m | Stone grit | 2.4 | cu.m |
| sand | 1.2 | cu.m | |||
| cement | 24 | Bags | |||
| 4 | 12 mm thick plastering | 100 sq.m | Mortar | 2 | cu.m |
| 5 | Pointing in brickwork | 100 sq.m | Mortar | 0.6 | cu.m |
| 6 | White washing one coat | 100 sq.m | Lime | 10 | Kg |
| 7 | Distempring | ||||
| i) First coat | 100 sq.m | Dry distemper | 6.5 | Kg | |
| ii) Second coat | 100 sq.m | Dry distemper | 5 | Kg | |
| 8 | Oil Painting one coat | 100 sq.m | Ready-made paint | 10 | Litres |
| 9 | Oil Painting one coat | 100 sq.m | Paint stiff | 10 | Kg |
| 10 | 20 mm thick Damp-proof course of 1:2:4 cement Mortar | 100 sq.m | Cement | 27 | Bags |
| Sand | 1.8 | Cu.m | |||
| Imperno | 27 | Kg | |||
| 11 | Painting with bitumen | ||||
| i) First coat | 100 sq.m | Bitumen | 150 | Kg | |
| ii) Second coat | 100 sq.m | Bitumen | 100 | Kg | |
| 12 | C.G.I. Sheet Roofing | 100 sq.m | C.G.I. sheets | 128 | Sq.m |
| 13 | A.C. Sheet Roofing | 100 sq.m | A.C. sheets | 115 | Sq.m |
| 14 | Panneled door shutter 40 mm thick | 100 sq.m | Timber | 4.5 | Cu.m |
| 15 | Battened door shutter 40 mm thick | 100 sq.m | Timber | 5 | Cu.m |
| 16 | Partialy Panneled and Glazed shutter 40 mm thick | 100 sq.m | Timber | 3 | Cu.m |
| 17 | Fully glazed shutter 40 mm thick | 100 sq.m | Timber | 2 | Cu.m |
Fineness Modulus (for Aggregates)
$$ \text{FM} = \frac{\text{Cumulative retained on standard sieves}}{100} $$
(Fineness modulus indicates average particle size.)
Compressive Strength (for Bricks, Cement Cubes etc.)
Modulus of Rupture (for Concrete)
Cement Soundness Test
Standard Consistency of Cement
Initial and Final Setting Time (Cement)
Section 12) Irrigation Engineering
Section 13) Construction Management
Time-Cost Trade Off (Crash Cost / Slope of Cost Curve)
Total Float (TF)
Free Float (FF)
Independent Float (IF)
Critical Path Method (CPM)
Cost-Time Optimization Slope
Resource Allocation Formula
Labour Productivity
Earned Value (EV)
Schedule Variance (SV)
Cost Variance (CV)
Performance Indices
Bar Bending Schedule (Unit Weight of Steel)
Number of Bricks Required
Material Requirement in Concrete
Section 14) Fluid Mechanics
Pressure Intensity
Hydrostatic Pressure at Depth
Pascal’s Law
Continuity Equation
Bernoulli’s Equation
Discharge (Flow Rate)
Reynolds Number
Manning’s Formula (Open Channel Flow)
Chezy’s Formula
Drag Force
Section 15) Structural Analysis
Bending Moment — Shear Force Relation
Shear Force — Load Relation
Moment of Inertia for Rectangular Section
Bending Stress (Flexure Formula)
Deflection for Cantilever Beam (Point Load at Free End)
Deflection for Simply Supported Beam (Central Point Load)
Slope at Free End of Cantilever (Point Load)
Moment Area Theorem (First Theorem)
Moment Area Theorem (Second Theorem)
Principle of Superposition
Section 16) Reinforced Cement Concrete
Modular Ratio
Moment of Resistance (Limit State of Collapse — Flexure)
Limiting Depth of Neutral Axis
Lever Arm
In Reinforced Cement Concrete (RCC) design, the j value represents the lever arm factor, which is the distance between the centroid of the tensile reinforcement and the line of action of the compressive force in the concrete. It is used in the calculation of bending moments and stresses.
The formula for the j value is:
$$ j = 1 - \frac{k}{3} $$
Where:
-
k is the neutral axis depth factor, calculated as:
$$ k = \frac{M_u}{b \cdot d^2 \cdot f_{ck}} $$
Here:
- M_u = Ultimate moment of resistance
- b = Breadth of the beam
- d = Effective depth of the beam
- f_{ck} = Characteristic compressive strength of concrete
The j value typically ranges between 0.85 to 0.95 for practical RCC design, depending on the section and material properties.
Let me know if you'd like further clarification or assistance! 😊
Ultimate Load on Axially Loaded Short Column
Development Length
Design Bond Stress
Minimum Reinforcement in Slab
Nominal Cover (IS 456:2000 Table 16)
Shear Strength of Concrete (without shear reinforcement)
Axial Load Capacity
Pure Moment Capacity (No Axial Load)
Interaction Equation for Combined Axial Load and Bending (Limit State Method — Clause 39.6, IS 456:2000)
Moment of Resistance under Combined Load
Minimum Eccentricity
Slenderness Ratio
Radius of Gyration
Load on Slab
Short Span and Long Span Ratio
Moment Coefficients (from IS 456:2000 Table 26)
Effective Depth (d)
Area of Steel (A_s)
Minimum Reinforcement as per IS 456:2000
Check for Deflection Control
Distribution Steel
Spacing of Bars
Factored Load on Footing
Required Area of Footing
Size of Square Footing
Bending Moment at Face of Column
Effective Depth (d) for Bending
One-Way Shear Check (at distance d from face)
Nominal shear:
Check:
Two-Way (Punching) Shear Check
Critical perimeter:
Nominal shear stress:
Check:
Area of Steel (A_s)
Minimum Reinforcement
Development Length
Rise and Tread (As per IS 456:2000)
Number of Steps
Horizontal Span of Staircase
Length of Waist Slab
Self-Weight of Waist Slab
Load of Steps
Live Load (as per IS 875 Part 2)
Total Load per Meter on Waist Slab
Bending Moment at Mid-Span (Simply Supported)
Effective Depth (d) for Bending
Area of Steel (A_s)
Minimum Reinforcement
Development Length
Section 17) Prestressed Concrete (PSC)
Loss of Prestress due to Elastic Shortening
Loss of Prestress due to Creep of Concrete
Loss of Prestress due to Shrinkage of Concrete
Loss of Prestress due to Relaxation of Steel
Total Loss of Prestress
Final Prestress in Tendon
Ultimate Moment Capacity (Limit State)
Eccentricity of Prestressing
Section 18) Bridge Engineering
Discharge by Manning’s Formula
$$ Q = \frac{1}{n} A R^{2/3} S^{1/2} $$
Where: \( Q \) = Discharge (m³/s) \( n \) = Manning’s roughness coefficient \( A \) = Flow area (m²) \( R \) = Hydraulic radius (m) \( S \) = Bed slope
Hydraulic Radius (for any section)
$$ R = \frac{A}{P} $$
Where: \( P \) = Wetted perimeter (m)
Area of Rectangular Waterway
$$ A = b \times d $$
Where: \( b \) = Bottom width (m) \( d \) = Depth of flow (m)
Wetted Perimeter for Rectangular Section
$$ P = b + 2d $$
Velocity by Chezy’s Formula
$$ V = C \sqrt{R S} $$
Where: \( C \) = Chezy’s constant
Discharge through a Culvert (Rectangular)
$$ Q = A \times V $$
Head Loss through Culvert
$$ h_L = K \times \frac{V^2}{2g} $$
Where: \( K \) = Loss coefficient \( g \) = 9.81 m/s²
Maximum Permissible Velocity (Non-Erosive)
$$ V_{max} = \text{as per IS 4745:1968 depending on soil type} $$
Flood Discharge Estimation (Rational Formula)
$$ Q = C \times I \times A $$
Where: \( C \) = Runoff coefficient \( I \) = Rainfall intensity (mm/hr) \( A \) = Catchment area (hectares)
Freeboard Provision
$$ F = H_{max} \times f $$
Where: \( H_{max} \) = Maximum afflux/head (m) \( f \) = Freeboard factor (typically 0.25 to 0.5)
Bearing Pressure
$$ p = \frac{P}{A} $$Where:
\( p \) = Bearing pressure (MPa)
\( P \) = Vertical load (kN)
\( A \) = Loaded area (mm^2)
Shape Factor (S)
$$ S = \frac{b \times d}{2t (b + d)} $$Where:
\( b, d \) = Plan dimensions of bearing (mm)
\( t \) = Thickness of elastomeric layer (mm)
Shear Strain
$$ \gamma = \frac{\Delta}{t} $$Where:
\( \gamma \) = Shear strain
\( \Delta \) = Horizontal displacement (mm)
\( t \) = Total elastomer thickness (mm)
Shear Stress
$$ \tau = \frac{H}{A} $$Where:
\( \tau \) = Shear stress (MPa)
\( H \) = Horizontal load (kN)
\( A \) = Loaded area (mm^2)
Rotation Capacity
$$ \theta = \frac{6 \Delta}{t^2} $$Where:
\( \theta \) = Rotation (radians)
\( \Delta \) = Horizontal displacement (mm)
\( t \) = Elastomer thickness (mm)
Compressive Stress Check
$$ p_{actual} \leq p_{allowable} $$Where:
\( p_{actual} \) = Actual bearing pressure (MPa)
\( p_{allowable} \) = Permissible pressure (MPa)
Minimum Plan Area Required
$$ A = \frac{P}{p_{allowable}} $$Where:
\( A \) = Required plan area (mm^2)
Total Horizontal Movement
$$ \Delta_{total} = \Delta_{creep} + \Delta_{shrinkage} + \Delta_{thermal} $$All values in mm.
Permissible Shear Strain Limit
$$ \gamma \leq 0.7 $$As per IRC:83 Part II.
Shape Factor Requirement
$$ S \geq 2.5 $$As per IRC:83 Cl. 5.5.
Total Horizontal Movement
$$ \Delta_{total} = \Delta_{creep} + \Delta_{shrinkage} + \Delta_{thermal} $$All values in mm.
Thermal Movement
$$ \Delta_{thermal} = L \times \alpha \times \Delta T $$Where:
\( L \) = Length of bridge deck (mm)
\( \alpha \) = Coefficient of thermal expansion (per °C)
\( \Delta T \) = Temperature variation (°C)
Creep Movement
$$ \Delta_{creep} = L \times C_{cr} $$Where:
\( C_{cr} \) = Creep strain (typically 0.0003–0.0005)
Shrinkage Movement
$$ \Delta_{shrinkage} = L \times C_{sh} $$Where:
\( C_{sh} \) = Shrinkage strain (typically 0.0002–0.0003)
Minimum Gap Width at Installation
$$ g = \Delta_{total} + 2 \times \Delta_{construction} $$Where:
\( \Delta_{construction} \) = Allowance for construction tolerance (mm)
Expansion Joint Gap for Temperature Effect
$$ g_{temp} = L \times \alpha \times \Delta T $$At one end of the deck, values in mm.
Allowable Joint Gap as per IRC:83
$$ g_{max} \leq g_{allowable} $$As per IRC:83 guidelines depending on movement capacity.
Sealant Movement Capacity Check
$$ M_{capacity} \geq \frac{\Delta_{total}}{2} $$Where:
\( M_{capacity} \) = Movement capacity of sealant (mm)
Bearing Movement & Joint Synchronization
$$ \Delta_{joint} \geq \Delta_{bearing} $$Expansion joint movement must match or exceed bearing movement.
Bridge Approach Slab Design
Length of Approach Slab (IRC: SP:58 & IRC:87)
$$ L = (5 \text{ to } 7) \times h $$Where:
\( h \) = Thickness of road crust at abutment face (m)
Dead Load of Approach Slab
$$ W_{dead} = \gamma_c \times b \times t \times 1 $$Where:
\( \gamma_c \) = Density of concrete (kN/m³)
\( b \) = Width of slab (m)
\( t \) = Thickness of slab (m)
Live Load Bending Moment (as simply supported slab)
$$ M_{LL} = \frac{W \times L}{4} $$Where:
\( W \) = Live load (kN)
\( L \) = Span of slab (m)
Dead Load Bending Moment
$$ M_{DL} = \frac{W_{dead} \times L^2}{8} $$Total Bending Moment
$$ M_{total} = M_{DL} + M_{LL} $$Required Effective Depth (d)
$$ d = \sqrt{\frac{M_{total} \times 10^6}{R \times b}} $$Where:
\( R \) = Moment of resistance factor (N/mm²)
\( b \) = Width of slab (mm)
Area of Steel (Main Reinforcement)
$$ A_{st} = \frac{M_{total} \times 10^6}{0.87 \times f_y \times j \times d} $$Where:
\( f_y \) = Yield strength of steel (N/mm²)
\( j \) = Lever arm factor (≈0.9)
Distribution Steel (IRC:21)
$$ A_{sd} = 0.12 \times t \times 1000 $$Where:
\( t \) = Slab thickness (mm)
Factor of Safety against Settlement Differential
$$ FS = \frac{q_{allow}}{q_{actual}} $$Where:
\( q_{allow} \) = Allowable bearing capacity (kN/m²)
\( q_{actual} \) = Actual bearing pressure under slab (kN/m²)
Reinforcement Spacing
$$ s = \frac{1000 \times A_{bar}}{A_{st}} $$Where:
\( A_{bar} \) = Area of one bar (mm²)
Vertical Load on Well
$$ P = DL + LL + WL $$Where:
\( DL \) = Dead Load (kN)
\( LL \) = Live Load (kN)
\( WL \) = Wind Load (kN)
Lateral Earth Pressure (Rankine's Theory)
$$ P_a = \frac{1}{2} \times K_a \times \gamma \times H^2 $$Where:
\( K_a \) = Coefficient of active earth pressure
\( \gamma \) = Unit weight of soil (kN/m³)
\( H \) = Depth of embedment (m)
Passive Earth Pressure (for Stability)
$$ P_p = \frac{1}{2} \times K_p \times \gamma \times D^2 $$Where:
\( K_p \) = Coefficient of passive earth pressure
\( D \) = Depth of foundation (m)
Factor of Safety against Overturning
$$ FS_{OT} = \frac{M_{resisting}}{M_{overturning}} $$Factor of Safety against Sliding
$$ FS_{SL} = \frac{\mu \times (W + P_p)}{P_a + H_{water}} $$Where:
\( \mu \) = Coefficient of friction between base and soil
\( H_{water} \) = Water current force (if any) (kN)
Vertical Bearing Capacity Check
$$ q_{max} = \frac{P}{A} \left(1 + \frac{6e}{B} \right) $$Where:
\( A \) = Base area (m²)
\( e \) = Eccentricity (m)
\( B \) = Base width (m)
Eccentricity Check
$$ e = \frac{M_{resisting} - M_{overturning}}{P} $$Should satisfy:
\( e \leq \frac{B}{6} \)
Scour Depth Calculation (Lacey’s Formula)
$$ d_s = 0.473 \left( \frac{Q}{f} \right)^{1/3} $$Where:
\( Q \) = Design discharge (m³/s)
\( f \) = Lacey’s silt factor
Grip Length (Minimum Embedment Depth)
$$ D \geq 1.33 \times d_s $$Where:
\( d_s \) = Maximum scour depth (m)
Peak Discharge from Rational Method
$$ Q = C \times I \times A $$Where:
\( Q \) = Peak discharge (m³/s)
\( C \) = Runoff coefficient
\( I \) = Rainfall intensity (mm/hr)
\( A \) = Catchment area (hectares)
Unit Hydrograph Principle
$$ Q = \frac{R}{T} $$Where:
\( Q \) = Direct runoff (m³/s)
\( R \) = Rainfall excess (mm)
\( T \) = Time base (hr)
Runoff Volume (Direct Runoff Hydrograph)
$$ V = \frac{\sum (Q \times \Delta t)}{1000} $$Where:
\( V \) = Volume (ha-m)
\( Q \) = Discharge (m³/s)
\( \Delta t \) = Time interval (s)
Time of Concentration (Kirpich’s Formula)
$$ T_c = 0.0195 \times L^{0.77} \times S^{-0.385} $$Where:
\( T_c \) = Time of concentration (min)
\( L \) = Length of channel (m)
\( S \) = Slope of channel bed (m/m)
Baseflow Separation (Straight-Line Method)
$$ Q_b = \text{Intercept at recession limb start and end connected linearly} $$Synthetic Unit Hydrograph (Snyder’s Method)
$$ Q_p = C_p \times A^{0.75} / T_p $$Where:
\( Q_p \) = Peak discharge (m³/s)
\( C_p \) = Peak coefficient
\( A \) = Catchment area (km²)
\( T_p \) = Time to peak (hr)
Lag Time (Snyder’s Method)
$$ T_l = C_t \times (L \times L_c)^{0.3} $$Where:
\( T_l \) = Lag time (hr)
\( C_t \) = Time constant
\( L \) = Stream length (km)
\( L_c \) = Distance from outlet to centroid (km)
Peak Discharge of Flood Hydrograph
$$ Q_p = \frac{V}{T_r} $$Where:
\( Q_p \) = Peak flood discharge (m³/s)
\( V \) = Total runoff volume (m³)
\( T_r \) = Time to recession (hr)
Continuity Equation
$$ \frac{dS}{dt} = I - O $$
Where: \( S \) = Storage (m³) \( I \) = Inflow (m³/s) \( O \) = Outflow (m³/s)
Muskingum Routing Equation
$$ O_2 = C_0 I_2 + C_1 I_1 + C_2 O_1 $$
Where: \( O_2 \) = Outflow at current time step \( I_1, I_2 \) = Inflow at previous and current time steps \( O_1 \) = Outflow at previous time step \( C_0, C_1, C_2 \) = Muskingum routing coefficients
Muskingum Coefficients
$$ C_0 = \frac{-K X + 0.5 \Delta t}{K (1-X) + 0.5 \Delta t} $$
$$ C_1 = \frac{K X + 0.5 \Delta t}{K (1-X) + 0.5 \Delta t} $$
$$ C_2 = \frac{K (1-X) - 0.5 \Delta t}{K (1-X) + 0.5 \Delta t} $$
Where: \( K \) = Storage constant (time) \( X \) = Weighting factor (0 ≤ X ≤ 0.5) \( \Delta t \) = Time step (s)
Storage-Discharge Relationship (Reservoir Routing)
$$ S = f(O) $$
Where: \( S \) = Storage (m³) \( O \) = Outflow (m³/s)
Modified Puls Method
$$ \frac{(2S_2 + O_2 \Delta t)}{2} = \frac{(2S_1 + O_1 \Delta t)}{2} + \frac{\Delta t}{2}(I_1 + I_2) $$
Where: \( S_1, S_2 \) = Storage at previous and current steps \( O_1, O_2 \) = Outflow at previous and current steps \( I_1, I_2 \) = Inflow at previous and current steps \( \Delta t \) = Time interval
Runoff Volume (Direct Runoff Hydrograph - DRH)
$$ V = \frac{A \times R}{1000} $$
Where: \( V \) = Runoff volume (m³) \( A \) = Catchment area (km²) \( R \) = Rainfall excess (mm)
Peak Discharge by Rational Formula
$$ Q = 0.278 \times C \times I \times A $$
Where: \( Q \) = Peak discharge (m³/s) \( C \) = Runoff coefficient \( I \) = Rainfall intensity (mm/hr) \( A \) = Area (km²)
Time of Concentration (Kirpich’s Formula)
$$ T_c = 0.0195 \times L^{0.77} \times S^{-0.385} $$
Where: \( T_c \) = Time of concentration (minutes) \( L \) = Maximum flow length (m) \( S \) = Slope of catchment (m/m)
Unit Hydrograph Ordinates (UH Method)
$$ Q_u = \frac{V}{T_r} $$
Where: \( Q_u \) = Discharge per unit rainfall depth (m³/s) \( V \) = Runoff volume (m³) \( T_r \) = Duration of effective rainfall (hr)
Base Flow Separation (Straight Line Method)
Draw a straight line joining the start and end of baseflow on the hydrograph to separate surface runoff.
Peak Discharge by Snyder’s Synthetic Unit Hydrograph
$$ Q_p = C_p \times A^{0.75} / T_p $$
Where: \( Q_p \) = Peak discharge (m³/s) \( C_p \) = Regional constant \( A \) = Area (km²) \( T_p \) = Time to peak (hr)
Lag Time (Snyder’s Method)
$$ T_l = C_t \times (L L_c)^{0.3} $$
Where: \( T_l \) = Lag time (hr) \( C_t \) = Coefficient based on region \( L \) = Main stream length (km) \( L_c \) = Distance from catchment outlet to centroid (km)
Section 19) Reservoirs
Volume Between Two Elevations
$$ V = \frac{(A_1 + A_2)}{2} \times (H_2 - H_1) $$
Where: \( V \) = Volume between two contours (m³) \( A_1, A_2 \) = Water spread areas at respective elevations (m²) \( H_2 - H_1 \) = Elevation difference (m)
Total Storage Capacity (Summation Method)
$$ V_{total} = \sum \frac{(A_n + A_{n+1})}{2} \times (H_{n+1} - H_n) $$
Where: \( V_{total} \) = Total capacity up to FRL (m³) \( A_n \) = Area at elevation \( H_n \)
Area-Elevation Curve Equation (Assuming Parabolic Relation)
$$ A = k \times H^2 $$
Where: \( A \) = Area at elevation H (m²) \( k \) = Constant based on reservoir geometry \( H \) = Elevation (m)
Reservoir Capacity-Elevation Curve Equation (Using Area-Elevation Curve)
$$ V = \frac{k}{3} \times (H^3 - H_0^3) $$
Where: \( V \) = Volume between elevations \( H_0 \) and \( H \) (m³)
Reservoir Capacity Percentage (for a given Elevation)
$$ \% \, \text{Full} = \frac{V_H}{V_{FRL}} \times 100 $$
Where: \( V_H \) = Volume up to elevation H (m³) \( V_{FRL} \) = Total capacity up to Full Reservoir Level (FRL) (m³)
Storage Capacity by Trapezoidal Rule
$$ V = \sum \frac{(A_i + A_{i+1})}{2} \times \Delta h $$
Where: \( \Delta h \) = Elevation difference between successive contours (m)
Storage Capacity by Simpson’s Rule (if even number of intervals)
$$ V = \frac{\Delta h}{3} \left[ A_0 + 4(A_1 + A_3 + A_5 + \ldots) + 2(A_2 + A_4 + A_6 + \ldots) + A_n \right] $$
Where: \( n \) = Number of intervals (even) \( \Delta h \) = Elevation difference (m)
Discharge over an Ogee Spillway
$$ Q = C L H_e^{1.5} $$
Where: \( Q \) = Discharge (m³/s) \( C \) = Discharge coefficient \( L \) = Effective crest length (m) \( H_e \) = Total head over the spillway crest (m)
Total Head over Crest
$$ H_e = H + \frac{V^2}{2g} $$
Where: \( H \) = Static head (m) \( V \) = Approach velocity (m/s) \( g \) = Acceleration due to gravity (9.81 m/s²)
Effective Length of Spillway (including pier and abutment contraction corrections)
$$ L = L' - 2 N K_p H_e - K_a H_e $$
Where: \( L' \) = Gross length of spillway crest (m) \( N \) = Number of piers \( K_p \) = Pier contraction coefficient \( K_a \) = Abutment contraction coefficient \( H_e \) = Total head over crest (m)
Energy Dissipation – Tail Water Depth (For Hydraulic Jump)
$$ y_2 = \frac{y_1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right) $$
Where: \( y_1 \) = Initial depth before jump (m) \( Fr_1 \) = Froude number before jump \( y_2 \) = Depth after jump (m)
Froude Number at Spillway Toe
$$ Fr_1 = \frac{V_1}{\sqrt{g y_1}} $$
Where: \( V_1 \) = Velocity at toe (m/s) \( y_1 \) = Depth before jump (m)
Spillway Capacity for Design Flood (Simplified)
$$ Q = C_d L H^{1.5} $$
Where: \( C_d \) = Discharge coefficient (may vary with head & shape) \( L \) = Spillway length (m) \( H \) = Head over crest (m)
Discharge Capacity of Canal Escape
$$ Q = A \times V $$
Where: \( Q \) = Discharge capacity (m³/s) \( A \) = Flow area (m²) \( V \) = Velocity of flow (m/s)
Flow Area of Rectangular Escape Channel
$$ A = b \times d $$
Where: \( b \) = Bottom width (m) \( d \) = Depth of flow (m)
Velocity by Chezy’s Formula
$$ V = C \sqrt{R S} $$
Where: \( C \) = Chezy’s constant \( R \) = Hydraulic radius (m) \( S \) = Bed slope
Hydraulic Radius for Rectangular Section
$$ R = \frac{A}{P} $$
Where: \( P \) = Wetted perimeter (m)
Wetted Perimeter for Rectangular Section
$$ P = b + 2d $$
Froude Number Check (For Flow Type)
$$ Fr = \frac{V}{\sqrt{g d}} $$
Where: \( g \) = 9.81 m/s² \( d \) = Depth of flow (m)
Energy Loss through Escape Fall
$$ h_L = K \times \frac{V^2}{2g} $$
Where: \( h_L \) = Head loss (m) \( K \) = Loss coefficient (typically 0.5 to 1.0)
Escape Structure Capacity Check
Ensure: $$ Q_{escape} \geq Q_{maximum\ expected\ discharge} $$
📖 Water Pipeline
1️⃣ Discharge by Hazen-Williams Formula
$$ Q = 0.278 \times C \times D^{2.63} \times S^{0.54} $$
Where: \( Q \) = Discharge (m³/s) \( C \) = Hazen-Williams Coefficient \( D \) = Pipe diameter (m) \( S \) = Hydraulic gradient (head loss per unit length)
2️⃣ Discharge by Darcy-Weisbach Formula
$$ h_f = f \times \frac{L}{D} \times \frac{V^2}{2g} $$
Where: \( h_f \) = Head loss (m) \( f \) = Friction factor \( L \) = Length of pipe (m) \( D \) = Pipe diameter (m) \( V \) = Flow velocity (m/s) \( g \) = 9.81 m/s²
3️⃣ Flow Velocity
$$ V = \frac{Q}{A} $$
Where: \( A = \frac{\pi D^2}{4} \)
4️⃣ Continuity Equation (Multiple Pipes)
$$ Q_1 = Q_2 = Q_3 = \cdots $$
5️⃣ Head Loss in Pipe Junction
$$ h = K \times \frac{V^2}{2g} $$
Where: \( K \) = Loss coefficient
6️⃣ Pressure Head at Any Point
$$ H = \frac{p}{\gamma} $$
Where: \( p \) = Pressure (N/m²) \( \gamma \) = Unit weight of water (N/m³)
7️⃣ Pipe Flow Reynolds Number
$$ Re = \frac{V D}{\nu} $$
Where: \( \nu \) = Kinematic viscosity (m²/s)
8️⃣ Pump Power Required
$$ P = \gamma \times Q \times H_p $$
Where: \( H_p \) = Pump head (m)
9️⃣ Economical Diameter of Pipe
$$ D_e = \left(\frac{Q^2}{C}\right)^{1/5} $$
Where: \( C \) = Coefficient dependent on material and cost factors
📖 Sewage Pumping Station Layout Design
1️⃣ Design Discharge
$$ Q = \frac{P \times q}{1000 \times 24 \times 60 \times 60} $$
Where: \( Q \) = Sewage flow (m³/s) \( P \) = Design population \( q \) = Per capita sewage flow (Lpcd)
2️⃣ Wet Well Volume
$$ V = Q \times t_s $$
Where: \( V \) = Volume of wet well (m³) \( t_s \) = Retention time (sec) (usually 20 to 30 minutes)
3️⃣ Area of Wet Well
$$ A = \frac{V}{H} $$
Where: \( A \) = Plan area (m²) \( H \) = Effective depth of wet well (m)
4️⃣ Number of Pumps
$$ N = n + 1 $$
Where: \( n \) = Number of working pumps \( 1 \) = Standby pump (as per CPHEEO norms)
5️⃣ Total Pumping Head
$$ H_t = h_s + h_f + h_m $$
Where: \( h_s \) = Static head (m) \( h_f \) = Friction losses (m) \( h_m \) = Minor losses (m)
6️⃣ Pump Power Required
$$ P = \frac{\gamma \times Q \times H_t}{\eta} $$
Where: \( \gamma \) = Unit weight of sewage (N/m³) \( \eta \) = Pump efficiency (decimal)
7️⃣ Minimum Rising Main Velocity
Recommended: $$ 0.75\, \text{m/s} \leq V \leq 2.5\, \text{m/s} $$
8️⃣ Diameter of Rising Main
$$ D = \sqrt{\frac{4Q}{\pi V}} $$
Where: \( D \) = Diameter (m) \( Q \) = Discharge (m³/s) \( V \) = Velocity (m/s)
9️⃣ Detention Time Check
$$ t_d = \frac{V}{Q} $$
Where: \( t_d \) = Detention time (sec)
🔟 Sump Bottom Level
$$ \text{Sump Bottom RL} = \text{Inlet RL} - \text{Maximum Depth of Wet Well} $$
📦 Construction Work Machinery
1️⃣ Output of Power Shovel
$$ Q = \frac{q \times N \times E}{T_c} $$
Where:
\( Q \) = Output per hour (m³/hr)
\( q \) = Capacity per cycle (m³)
\( N \) = Number of cycles per hour
\( E \) = Job efficiency (decimal)
\( T_c \) = Cycle time (minutes)
2️⃣ Output of Excavator
$$ Q = q \times n \times E $$
\( q \) = Capacity per cycle (m³)
\( n \) = Number of cycles per hour
\( E \) = Efficiency (usually 0.8–0.9)
3️⃣ Output of Dumper
$$ Q = \frac{C \times N \times E}{T} $$
\( C \) = Capacity per trip (m³)
\( N \) = Number of trips per hour
\( E \) = Efficiency
\( T \) = Total cycle time per trip (hr)
4️⃣ Output of Bulldozer
$$ Q = \frac{S \times W \times D \times E}{T_c} $$
\( S \) = Speed (m/min)
\( W \) = Blade width (m)
\( D \) = Depth of cut (m)
\( E \) = Efficiency
\( T_c \) = Cycle time (min)
5️⃣ Roller Compaction Output
$$ Q = \frac{W \times S \times C \times E}{T} $$
\( W \) = Width of roller (m)
\( S \) = Speed (m/min)
\( C \) = Number of passes
\( E \) = Efficiency
\( T \) = Time (min)
6️⃣ Tower Crane Lifting Capacity
$$ C = \frac{M}{R} $$
\( C \) = Lifting capacity (kNm)
\( M \) = Moment (kNm)
\( R \) = Radius (m)
7️⃣ Concrete Mixer Output
$$ Q = C_m \times N \times E $$
\( C_m \) = Mixer drum capacity (m³)
\( N \) = Number of batches per hour
\( E \) = Efficiency (decimal)
8️⃣ Truck Haul Capacity
$$ Q = \frac{C_t \times N \times E}{T_c} $$
\( C_t \) = Capacity per trip (m³)
\( N \) = Trips per hour
\( E \) = Efficiency
\( T_c \) = Total cycle time per trip (min)
📦 Quality Control Tests — Civil Engineering
Compressive Strength of Concrete
$$ f_c = \frac{P}{A} $$
\( f_c \) = Compressive Strength (MPa)
\( P \) = Ultimate Load (N)
\( A \) = Area of specimen (mm²)
Water-Cement Ratio
$$ W/C = \frac{W}{C} $$
\( W \) = Weight of water (kg)
\( C \) = Weight of cement (kg)
Slump Test
$$ S = H_1 - H_2 $$
\( S \) = Slump (mm)
\( H_1 \) = Height of mould (mm)
\( H_2 \) = Height of concrete after slump (mm)
Aggregate Impact Value (AIV)
$$ AIV = \frac{W_2}{W_1} \times 100 $$
\( W_1 \) = Weight of oven-dry sample (g)
\( W_2 \) = Weight of sample passing 2.36 mm sieve after impact (g)
Fineness Modulus of Aggregates
$$ FM = \frac{\text{Cumulative retained % on standard sieves}}{100} $$
Bulking of Sand
$$ \text{Bulking \%} = \frac{V_2 - V_1}{V_1} \times 100 $$
\( V_1 \) = Volume of sand without water (ml)
\( V_2 \) = Volume after water added (ml)
Modulus of Elasticity of Concrete
$$ E_c = 5000 \sqrt{f_{ck}} $$
\( f_{ck} \) = Characteristic compressive strength (MPa)
Specific Gravity of Cement
$$ G = \frac{\frac{W_1}{W_1-W_2}}{\text{Volume of liquid displaced}} $$
\( W_1 \) = Weight of cement + flask (g)
\( W_2 \) = Weight of flask with kerosene and cement (g)
Soundness of Cement (Le Chatelier Method)
$$ \text{Expansion} = L_2 - L_1 $$
\( L_1 \) = Distance between indicator points before boiling (mm)
\( L_2 \) = Distance after boiling (mm)
Flakiness Index
$$ FI = \frac{W_f}{W_t} \times 100 $$
\( W_f \) = Weight of flaky particles (g)
\( W_t \) = Total weight of sample (g)
Bitumen Testing
Penetration Value
Measured in tenths of mm under standard load, time & temperature.
Softening Point (Ring and Ball Method)
No formula — temperature at which bitumen softens under load.
Ductility Test
$$ D = \frac{L}{T} $$
\( D \) = Ductility (cm)
\( L \) = Distance at break (cm)
\( T \) = Test length (normally 5 cm)
Viscosity Test
$$ \eta = \frac{W}{t} $$
\( \eta \) = Viscosity
\( W \) = Weight of flow (g)
\( t \) = Time (sec)
Soil Testing
Water Content (Oven Drying Method)
$$ w = \frac{W_w}{W_s} \times 100 $$
\( W_w \) = Weight of water
\( W_s \) = Weight of dry soil
Dry Density
$$ \rho_d = \frac{W_s}{V} $$
\( W_s \) = Weight of dry soil (g)
\( V \) = Volume of soil (cm³)
Atterberg Limits
- $$ LI = \frac{w - PL}{LL - PL} $$
- $$ PI = LL - PL $$
\( w \) = Natural moisture content
\( LL \) = Liquid Limit
\( PL \) = Plastic Limit
Compaction (Proctor Test)
$$ \gamma_d = \frac{W_s}{V} $$
\( W_s \) = Dry weight of soil (g)
\( V \) = Volume of mould (cm³)
Steel testing
Tensile Strength
$$ f_t = \frac{P}{A} $$
\( P \) = Load at fracture (N)
\( A \) = Cross-sectional area (mm²)
Elongation %
$$ e = \frac{L_f - L_i}{L_i} \times 100 $$
\( L_f \) = Final gauge length (mm)
\( L_i \) = Initial gauge length (mm)
Modulus of Elasticity
$$ E = \frac{\sigma}{\epsilon} $$
\( \sigma \) = Stress (MPa)
\( \epsilon \) = Strain
Concrete Workability
Slump Test
$$ S = H_1 - H_2 $$
\( H_1 \) = Height of mould (mm)
\( H_2 \) = Concrete height after slump (mm)
Compaction Factor Test
$$ CF = \frac{W_1}{W_2} $$
\( W_1 \) = Weight of partially compacted concrete
\( W_2 \) = Weight of fully compacted concrete
Tests on Cement
Specific Gravity (Le Chatelier)
$$ G = \frac{\frac{W_1}{W_1-W_2}}{\text{Volume displaced}} $$
Fineness (Blaine Method)
$$ S = \frac{k}{t} $$
\( k \) = Constant
\( t \) = Time (s)
Setting Time
Initial and Final setting times recorded using Vicat apparatus (No formula)
📦 Asphalt Mix
Bulk Density of Mix
$$ \rho_b = \frac{W_d}{V} $$
Where:
\( \rho_b \) = Bulk density (g/cm³ or kg/m³)
\( W_d \) = Weight of dry specimen (g)
\( V \) = Volume of specimen (cm³)
Maximum Theoretical Density (MTD)
Measured via pycnometer method — no formula, but recorded directly as per ASTM D2041.
Air Voids (Va)
$$ V_a = \frac{\rho_{MTD} - \rho_b}{\rho_{MTD}} \times 100 $$
Where:
\( \rho_{MTD} \) = Maximum Theoretical Density
\( \rho_b \) = Bulk Density
Voids in Mineral Aggregate (VMA)
$$ VMA = \frac{V_v + V_s}{V_t} \times 100 $$
Or simplified:
$$ VMA = 100 - \frac{G_{mb} \times P_s}{G_{sb}} $$
Where:
\( G_{mb} \) = Bulk Specific Gravity of Mix
\( P_s \) = % Aggregate by total weight
\( G_{sb} \) = Bulk Specific Gravity of Aggregate
Voids Filled with Bitumen (VFB)
$$ VFB = \frac{VMA - V_a}{VMA} \times 100 $$
Stability (Marshall Method)
Maximum load carried by the mix specimen at 60°C until failure (in kN) — no formula.
Flow Value
Deformation (in mm) corresponding to maximum stability load under Marshall test.
Optimum Bitumen Content (OBC)
Selected corresponding to 4% Air Voids or best balanced values of Stability, VMA, and VFB based on plotted graphs — no direct formula, but determined graphically.
✅ Note: These QC formulas are as per MORTH & ASTM standards, suitable for PWD/IRC roadworks and NHAI projects.
🧱 Cube Compressive Strength Test (As per IS 456:2000)
Compressive Strength of Cube (fck)
$$ f_{ck} = \frac{P}{A} $$
Where:
\( f_{ck} \) = Compressive strength (MPa)
\( P \) = Load at failure (N)
\( A \) = Loaded area of cube face (mm²)
For standard 150 mm cube:
\( A = 150 \times 150 = 22500 \ mm^2 \)
📊 Average Compressive Strength of 3 Cubes
$$ f_{avg} = \frac{f_1 + f_2 + f_3}{3} $$
✅ Acceptance Criteria (IS 456:2000, Clause 16 & Table 11)
A concrete batch shall be accepted if:
- The average strength of any 3 consecutive specimens ≥ specified characteristic strength.
- No individual cube strength is less than:
- \( f_{ck} - 3 \ MPa \) for M15, M20 grades.
- 0.8 × \( f_{ck} \) for M25 and higher grades.
🎯 Target Mean Strength (fcm)
$$ f_{cm} = f_{ck} + k \times s $$
Where:
\( f_{ck} \) = Characteristic compressive strength (MPa)
\( k \) = Statistical constant (1.65 for 5% probability of failure)
\( s \) = Standard deviation (as per IS 10262:2019)
📈 Coefficient of Variation (COV)
$$ COV = \frac{s}{f_{avg}} \times 100 $$
📝 Note: Minimum 3 cubes must be tested at each desired curing age (7, 14, and 28 days) for reliable quality assessment. Acceptance shall strictly conform to IS 456:2000, Table 11 requirements.
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