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 site distance
Stopping site 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 Site Distance (OSD)
| 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):
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
Section 4) Hydraulics
The depth at partial flow can be evaluated from equation -
$$ d = ({D\over 2}-{D\over 2}.cos{\alpha\over 2}) $$Discharge when pipe is running partially full,
q =av
Proportionate discharge = $$ {q\over Q} = {{a.v}\over {A.v'}}$$
=$$ {q\over Q} =({\alpha\over 360^\circ}-{sin\alpha\over 2\pi}).{(1-{{{360^\circ}sin\alpha}\over {2 \pi \alpha}})}^{2\over 3}$$
for detailed derivation go to link
The broad-crested weir is a type of weir used in open channel flow measurements, which has a rectangular notch on top. The discharge over a broad-crested weir can be calculated using the following equation:
$$Q = CLH^{3/2}$$
where:
- Q is the discharge over the weir [m^3/s],
- C is the discharge coefficient, which depends on the shape and size of the weir,
- L is the width of the weir [m],
- H is the height of the water above the weir crest [m].
To calculate the value of C, you can use one of the following equations:
- For a rectangular weir:
$$C = 1.84 - 0.5\left(\frac{H}{L}\right)^{1.5} + 0.25\left(\frac{H}{L}\right)^3$$
For Trapezoidal weir:-
$$C = \frac{2}{3}\left(1 + \frac{2h}{b}\right)\left(1 - \frac{h}{H}\right)^{3/2}$$
where:
- V is the average velocity of flow [m/s],
- C is the Chezy coefficient, which is a dimensionless constant that depends on the roughness of the channel,
- R is the hydraulic radius, which is the cross-sectional area of flow divided by the wetted perimeter [m],
- S is the slope of the energy line, which is the change in energy per unit length of the channel [m/m].
The loss of head due to friction is an important concept in fluid mechanics and is often used in the design of pipelines and other fluid transport systems. The loss of head due to friction is given by the Darcy-Weisbach equation:
$$h_f = \frac{fLV^2}{2gd}$$
where:
- h_f is the loss of head due to friction [m],
- f is the Darcy-Weisbach friction factor, which depends on the Reynolds number and the roughness of the pipe,
- L is the length of the pipe [m],
- V is the average velocity of flow in the pipe [m/s],
- g is the acceleration due to gravity (approximately 9.81 m/s^2), and
- d is the diameter of the pipe [m].
where:
- V is the average velocity of flow [m/s],
- n is the Manning's roughness coefficient, which is a dimensionless constant that depends on the roughness of the channel,
- R is the hydraulic radius, which is the cross-sectional area of flow divided by the wetted perimeter [m],
- S is the slope of the energy line, which is the change in energy per unit length of the channel [m/m].
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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