Concrete Engineering
Volume, mix design, and workability formulas.
Concrete Volume
Concrete Volume
V = L × W × H
V = Volume (m³)
L = Length (m)
W = Width (m)
H = Height / Depth (m)
View Calculator → L = Length (m)
W = Width (m)
H = Height / Depth (m)
Concrete Mix Design
Water–Cement Ratio
w/c = Ww / Wc
w/c = Water–cement ratio (unitless)
Ww = Mass of water (kg)
Wc = Mass of cement (kg)
Typical range: 0.35 – 0.60 per ACI 211.1
View Calculator → Ww = Mass of water (kg)
Wc = Mass of cement (kg)
Typical range: 0.35 – 0.60 per ACI 211.1
Concrete Workability
Slump Requirements
S = hmold − hslump
S = Slump (mm or in)
hmold = Height of mold (300 mm)
hslump = Height after slump
Typical: 25–100 mm for structural concrete
Concrete Tools → hmold = Height of mold (300 mm)
hslump = Height after slump
Typical: 25–100 mm for structural concrete
Concrete Mix Design
Target Mean Strength
ftarget = fck + 1.65σ
ftarget = Target mean compressive strength (MPa)
fck = Characteristic compressive strength (MPa)
σ = Standard deviation (MPa)
Typical σ = 4–6 MPa per IS 456 / ACI 214
Concrete Tools → fck = Characteristic compressive strength (MPa)
σ = Standard deviation (MPa)
Typical σ = 4–6 MPa per IS 456 / ACI 214
Concrete Properties
Modulus of Elasticity
Ec = 5000√fck (IS 456) / Ec = 4700√f′c (ACI 318)
Ec = Modulus of elasticity of concrete (MPa)
fck = Characteristic compressive strength (MPa)
f′c = Specified compressive strength (MPa)
IS 456 uses fck; ACI 318 uses f′c
Concrete Tools → fck = Characteristic compressive strength (MPa)
f′c = Specified compressive strength (MPa)
IS 456 uses fck; ACI 318 uses f′c
Concrete Reinforcement
Development Length
Ld = (φ × σs) / (4 × τbd)
Ld = Development length (mm)
φ = Diameter of reinforcement bar (mm)
σs = Stress in bar at design load (MPa)
τbd = Design bond stress (MPa)
Per IS 456:2000 and ACI 318 provisions
Concrete Tools → φ = Diameter of reinforcement bar (mm)
σs = Stress in bar at design load (MPa)
τbd = Design bond stress (MPa)
Per IS 456:2000 and ACI 318 provisions
Concrete Reinforcement
Reinforcement Ratio
ρ = Ast / (b × d)
ρ = Reinforcement ratio (unitless)
Ast = Area of steel reinforcement (mm²)
b = Width of section (mm)
d = Effective depth (mm)
Min ρ ≈ 0.2%, max ρ ≈ 4.0% per ACI 318
Concrete Tools → Ast = Area of steel reinforcement (mm²)
b = Width of section (mm)
d = Effective depth (mm)
Min ρ ≈ 0.2%, max ρ ≈ 4.0% per ACI 318
Structural Analysis
Beam bending, shear, and flexural stress formulas.
Structural Bending
Simple Beam — Max Moment
Mmax = wL² / 8
Mmax = Maximum bending moment (kN⋅m)
w = Uniformly distributed load (kN/m)
L = Span length (m)
For simply supported beam, UDL across full span
View Calculator → w = Uniformly distributed load (kN/m)
L = Span length (m)
For simply supported beam, UDL across full span
Structural Stress
Flexural Stress
σ = M · y / I
σ = Bending stress (MPa or N/mm²)
M = Applied moment (N⋅mm)
y = Distance from neutral axis (mm)
I = Moment of inertia (mm&sup4;)
Structural Tools → M = Applied moment (N⋅mm)
y = Distance from neutral axis (mm)
I = Moment of inertia (mm&sup4;)
Structural Shear
Simple Beam — Max Shear
Vmax = wL / 2
Vmax = Maximum shear force (kN)
w = Uniformly distributed load (kN/m)
L = Span length (m)
Occurs at supports for a simply supported beam
View Calculator → w = Uniformly distributed load (kN/m)
L = Span length (m)
Occurs at supports for a simply supported beam
Structural Buckling
Euler’s Buckling Load
Pcr = π²EI / (KL)²
Pcr = Critical buckling load (N)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
K = Effective length factor
L = Unbraced length (mm)
K = 1.0 pinned–pinned, 0.5 fixed–fixed, 2.0 cantilever
Structural Tools → E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
K = Effective length factor
L = Unbraced length (mm)
K = 1.0 pinned–pinned, 0.5 fixed–fixed, 2.0 cantilever
Structural Properties
Elastic Section Modulus
S = I / y
S = Elastic section modulus (mm³)
I = Moment of inertia (mm&sup4;)
y = Distance from neutral axis to extreme fiber (mm)
Relates bending stress to moment: σ = M / S
Structural Tools → I = Moment of inertia (mm&sup4;)
y = Distance from neutral axis to extreme fiber (mm)
Relates bending stress to moment: σ = M / S
Structural Shear
Shear Stress in Beams
τ = VQ / (Ib)
τ = Shear stress (MPa or N/mm²)
V = Internal shear force (N)
Q = First moment of area about neutral axis (mm³)
I = Moment of inertia (mm&sup4;)
b = Width at section of interest (mm)
Structural Tools → V = Internal shear force (N)
Q = First moment of area about neutral axis (mm³)
I = Moment of inertia (mm&sup4;)
b = Width at section of interest (mm)
Structural Deflection
Beam Deflection — UDL
δmax = 5wL&sup4; / (384EI)
δmax = Maximum deflection (mm)
w = Uniformly distributed load (N/mm)
L = Span length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
For simply supported beam, full-span UDL
Structural Tools → w = Uniformly distributed load (N/mm)
L = Span length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
For simply supported beam, full-span UDL
Structural Deflection
Beam Deflection — Point Load
δmax = PL³ / (48EI)
δmax = Maximum deflection (mm)
P = Concentrated load at midspan (N)
L = Span length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
For simply supported beam, point load at center
Structural Tools → P = Concentrated load at midspan (N)
L = Span length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
For simply supported beam, point load at center
Structural Axial
Axial Deformation
δ = PL / (AE)
δ = Axial deformation (mm)
P = Applied axial load (N)
L = Original length (mm)
A = Cross-sectional area (mm²)
E = Modulus of elasticity (MPa)
Structural Tools → P = Applied axial load (N)
L = Original length (mm)
A = Cross-sectional area (mm²)
E = Modulus of elasticity (MPa)
Structural Properties
Moment of Inertia — Rectangle
I = bh³ / 12
I = Moment of inertia about centroidal axis (mm&sup4;)
b = Base width (mm)
h = Height of section (mm)
For rectangular section about its neutral axis. Use parallel axis theorem for composite shapes.
View Calculator → b = Base width (mm)
h = Height of section (mm)
For rectangular section about its neutral axis. Use parallel axis theorem for composite shapes.
Structural Properties
Radius of Gyration
r = √(I / A)
r = Radius of gyration (mm)
I = Moment of inertia (mm&sup4;)
A = Cross-sectional area (mm²)
Used in slenderness and buckling calculations per AISC 360 and Eurocode 3.
Structural Tools → I = Moment of inertia (mm&sup4;)
A = Cross-sectional area (mm²)
Used in slenderness and buckling calculations per AISC 360 and Eurocode 3.
Structural Buckling
Slenderness Ratio
λ = KL / r
λ = Slenderness ratio (unitless)
K = Effective length factor
L = Unbraced length (mm)
r = Radius of gyration (mm)
K = 1.0 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2.0 cantilever. Per AISC 360 and Eurocode 3.
View Calculator → K = Effective length factor
L = Unbraced length (mm)
r = Radius of gyration (mm)
K = 1.0 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2.0 cantilever. Per AISC 360 and Eurocode 3.
Structural Stress
Combined Axial & Bending Stress
σ = P/A ± M/S
σ = Combined stress at extreme fiber (MPa)
P = Axial load (N)
A = Cross-sectional area (mm²)
M = Applied bending moment (N⋅mm)
S = Elastic section modulus (mm³)
Superposition of axial and flexural stresses. Used in beam-column design per AISC H1.
Structural Tools → P = Axial load (N)
A = Cross-sectional area (mm²)
M = Applied bending moment (N⋅mm)
S = Elastic section modulus (mm³)
Superposition of axial and flexural stresses. Used in beam-column design per AISC H1.
Structural Torsion
Torsional Shear Stress
τ = Tr / J
τ = Torsional shear stress (MPa)
T = Applied torque (N⋅mm)
r = Radial distance from center (mm)
J = Polar moment of inertia (mm&sup4;)
For circular sections: J = πD&sup4;/32. For rectangular sections, use Saint-Venant torsion constants.
Structural Tools → T = Applied torque (N⋅mm)
r = Radial distance from center (mm)
J = Polar moment of inertia (mm&sup4;)
For circular sections: J = πD&sup4;/32. For rectangular sections, use Saint-Venant torsion constants.
Structural Connections
Bearing Stress
σb = P / Ab
σb = Bearing stress (MPa)
P = Applied bearing force (N)
Ab = Bearing area (mm²)
Critical at bolted connections, column base plates, and concrete bearing surfaces per AISC J8.
Structural Tools → P = Applied bearing force (N)
Ab = Bearing area (mm²)
Critical at bolted connections, column base plates, and concrete bearing surfaces per AISC J8.
Structural Properties
Neutral Axis of Composite Section
¯y = Σ(Ai · yi) / ΣAi
¯y = Distance from reference axis to centroid (mm)
Ai = Area of component i (mm²)
yi = Distance to centroid of component i (mm)
Used to locate the centroidal axis for composite shapes like I-beams, T-beams, and channels.
View Calculator → Ai = Area of component i (mm²)
yi = Distance to centroid of component i (mm)
Used to locate the centroidal axis for composite shapes like I-beams, T-beams, and channels.
Structural Plastic
Plastic Moment Capacity
Mp = Fy · Z
Mp = Plastic moment capacity (N⋅mm)
Fy = Yield stress (MPa)
Z = Plastic section modulus (mm³)
Z ≈ 1.5S for rectangular sections. Compact sections per AISC F2 achieve Mp before LTB.
View Calculator → Fy = Yield stress (MPa)
Z = Plastic section modulus (mm³)
Z ≈ 1.5S for rectangular sections. Compact sections per AISC F2 achieve Mp before LTB.
Structural Deflection
Cantilever Deflection — Point Load
δmax = PL³ / (3EI)
δmax = Maximum deflection at free end (mm)
P = Point load at free end (N)
L = Cantilever length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
Cantilever with point load at free end. Compare to simply supported: 16× more deflection for same P, L.
View Calculator → P = Point load at free end (N)
L = Cantilever length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
Cantilever with point load at free end. Compare to simply supported: 16× more deflection for same P, L.
Structural Deflection
Cantilever Deflection — UDL
δmax = wL&sup4; / (8EI)
δmax = Maximum deflection at free end (mm)
w = Uniformly distributed load (N/mm)
L = Cantilever length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
Full-span UDL on cantilever. Maximum slope θ = wL³/(6EI).
View Calculator → w = Uniformly distributed load (N/mm)
L = Cantilever length (mm)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
Full-span UDL on cantilever. Maximum slope θ = wL³/(6EI).
Structural Bending
Cantilever — Max Moment and Shear
Mmax = wL²/2 Vmax = wL
Mmax = Maximum moment at fixed support (N⋅mm)
Vmax = Maximum shear at fixed support (N)
w = Uniformly distributed load (N/mm)
L = Cantilever length (mm)
For point load at free end: Mmax = PL, Vmax = P.
View Calculator → Vmax = Maximum shear at fixed support (N)
w = Uniformly distributed load (N/mm)
L = Cantilever length (mm)
For point load at free end: Mmax = PL, Vmax = P.
Structural Bending
Fixed-End Beam — UDL Moments
Mend = wL²/12 Mmid = wL²/24
Mend = Moment at fixed supports (N⋅mm)
Mmid = Positive moment at midspan (N⋅mm)
w = Uniformly distributed load (N/mm)
L = Span length (mm)
Fixed-end moments are double the simply supported case. Hogging at supports, sagging at midspan.
View Calculator → Mmid = Positive moment at midspan (N⋅mm)
w = Uniformly distributed load (N/mm)
L = Span length (mm)
Fixed-end moments are double the simply supported case. Hogging at supports, sagging at midspan.
Structural Properties
Bending Stress Formula
σ = M / S
σ = Maximum bending stress (MPa)
M = Applied bending moment (N⋅mm)
S = Elastic section modulus (mm³)
Alternative form of flexure formula: σ = My/I, where S = I/y.
Structural Tools → M = Applied bending moment (N⋅mm)
S = Elastic section modulus (mm³)
Alternative form of flexure formula: σ = My/I, where S = I/y.
Geotechnical Engineering
Bearing capacity and consolidation settlement formulas.
Geotechnical Foundations
Ultimate Bearing Capacity
(Terzaghi — Strip Footing)
qult = cNc + γDNq + 0.5γBNγ
qult = Ultimate bearing capacity (kN/m²)
c = Cohesion (kN/m²)
γ = Unit weight of soil (kN/m³)
D = Depth of foundation (m)
B = Width of footing (m)
Nc, Nq, Nγ = Bearing capacity factors
Foundation Tools → c = Cohesion (kN/m²)
γ = Unit weight of soil (kN/m³)
D = Depth of foundation (m)
B = Width of footing (m)
Nc, Nq, Nγ = Bearing capacity factors
Geotechnical Settlement
Consolidation Settlement
(Normally Consolidated Clay)
Sc = H · Cc / (1 + e0) · log((σ′0 + Δσ) / σ′0)
Sc = Consolidation settlement (m)
H = Thickness of clay layer (m)
Cc = Compression index
e0 = Initial void ratio
σ′0 = Initial effective stress (kN/m²)
Δσ = Stress increment (kN/m²)
Foundation Tools → H = Thickness of clay layer (m)
Cc = Compression index
e0 = Initial void ratio
σ′0 = Initial effective stress (kN/m²)
Δσ = Stress increment (kN/m²)
Geotechnical Retaining Walls
Active Earth Pressure
(Rankine Theory)
Pa = 0.5 × Ka × γ × H²
Pa = Active earth pressure resultant (kN/m)
Ka = Active earth pressure coefficient, tan²(45 − φ/2)
γ = Unit weight of soil (kN/m³)
H = Height of wall (m)
φ = Internal friction angle (degrees)
Foundation Tools → Ka = Active earth pressure coefficient, tan²(45 − φ/2)
γ = Unit weight of soil (kN/m³)
H = Height of wall (m)
φ = Internal friction angle (degrees)
Geotechnical Retaining Walls
Passive Earth Pressure
(Rankine Theory)
Pp = 0.5 × Kp × γ × H²
Pp = Passive earth pressure resultant (kN/m)
Kp = Passive earth pressure coefficient, tan²(45 + φ/2)
γ = Unit weight of soil (kN/m³)
H = Height of wall (m)
φ = Internal friction angle (degrees)
Foundation Tools → Kp = Passive earth pressure coefficient, tan²(45 + φ/2)
γ = Unit weight of soil (kN/m³)
H = Height of wall (m)
φ = Internal friction angle (degrees)
Geotechnical Compaction
Proctor Compaction — Dry Density
ρdry = ρwet / (1 + w/100)
ρdry = Dry density (kg/m³ or g/cm³)
ρwet = Wet (bulk) density (kg/m³ or g/cm³)
w = Moisture content (%)
Used in standard & modified Proctor compaction tests
Foundation Tools → ρwet = Wet (bulk) density (kg/m³ or g/cm³)
w = Moisture content (%)
Used in standard & modified Proctor compaction tests
Geotechnical Permeability
Permeability — Constant Head
k = (Q × L) / (A × h × t)
k = Hydraulic conductivity (cm/s)
Q = Volume of water collected (cm³)
L = Length of soil sample (cm)
A = Cross-sectional area of sample (cm²)
h = Constant head difference (cm)
t = Time of collection (s)
Foundation Tools → Q = Volume of water collected (cm³)
L = Length of soil sample (cm)
A = Cross-sectional area of sample (cm²)
h = Constant head difference (cm)
t = Time of collection (s)
Geotechnical Foundations
Meyerhof Bearing Capacity
(General Shear)
qult = cNcscdc + qNqsqdq + 0.5γBNγsγdγ
qult = Ultimate bearing capacity (kN/m²)
c = Cohesion (kN/m²)
q = Overburden pressure at base = γD (kN/m²)
γ = Unit weight of soil (kN/m³)
B = Width of footing (m)
Nc, Nq, Nγ = Bearing capacity factors
sc, sq, sγ = Shape factors
dc, dq, dγ = Depth factors
View Calculator → c = Cohesion (kN/m²)
q = Overburden pressure at base = γD (kN/m²)
γ = Unit weight of soil (kN/m³)
B = Width of footing (m)
Nc, Nq, Nγ = Bearing capacity factors
sc, sq, sγ = Shape factors
dc, dq, dγ = Depth factors
Geotechnical Settlement
Immediate (Elastic) Settlement
(Schmertmann Method)
Si = C1C2qnetΣ(IzΔz / Es)
Si = Immediate settlement (mm)
C1 = Depth correction factor = 1 − 0.5(σ′vo/qnet)
C2 = Creep correction factor = 1 + 0.2log(t/0.1)
qnet = Net foundation pressure (kN/m²)
Iz = Strain influence factor
Es = Modulus of elasticity of soil (kN/m²)
View Calculator → C1 = Depth correction factor = 1 − 0.5(σ′vo/qnet)
C2 = Creep correction factor = 1 + 0.2log(t/0.1)
qnet = Net foundation pressure (kN/m²)
Iz = Strain influence factor
Es = Modulus of elasticity of soil (kN/m²)
Geotechnical Phase Relations
Void Ratio & Porosity
e = Vv / Vs n = e / (1 + e)
e = Void ratio (unitless)
n = Porosity (decimal)
Vv = Volume of voids (m³)
Vs = Volume of solids (m³)
Typical e: 0.3–0.9 for sands, 0.5–1.5 for clays. Porosity n = e/(1+e) × 100%.
Soil Tools → n = Porosity (decimal)
Vv = Volume of voids (m³)
Vs = Volume of solids (m³)
Typical e: 0.3–0.9 for sands, 0.5–1.5 for clays. Porosity n = e/(1+e) × 100%.
Geotechnical Phase Relations
Degree of Saturation
S = Vw / Vv × 100%
S = Degree of saturation (%)
Vw = Volume of water (m³)
Vv = Volume of voids (m³)
S = 0% for dry soil, 100% for fully saturated. Related: w = Se/Gs where Gs = specific gravity.
Soil Tools → Vw = Volume of water (m³)
Vv = Volume of voids (m³)
S = 0% for dry soil, 100% for fully saturated. Related: w = Se/Gs where Gs = specific gravity.
Geotechnical Permeability
Hydraulic Gradient & Darcy’s Law
i = Δh / L v = k · i Q = A · k · i
i = Hydraulic gradient (m/m)
Δh = Head difference (m)
L = Flow path length (m)
v = Discharge velocity (m/s)
k = Hydraulic conductivity (m/s)
Q = Flow rate (m³/s)
A = Cross-sectional area (m²)
View Calculator → Δh = Head difference (m)
L = Flow path length (m)
v = Discharge velocity (m/s)
k = Hydraulic conductivity (m/s)
Q = Flow rate (m³/s)
A = Cross-sectional area (m²)
Geotechnical Settlement
Total Foundation Settlement
Stotal = Si + Sc + Ss
Stotal = Total settlement (mm)
Si = Immediate (elastic) settlement (mm)
Sc = Primary consolidation settlement (mm)
Ss = Secondary compression settlement (mm)
Allowable total settlement typically 25 mm for isolated footings, 50 mm for rafts per IS 1904.
View Calculator → Si = Immediate (elastic) settlement (mm)
Sc = Primary consolidation settlement (mm)
Ss = Secondary compression settlement (mm)
Allowable total settlement typically 25 mm for isolated footings, 50 mm for rafts per IS 1904.
Hydraulics & Water Resources
Open channel, pipe flow, and continuity equations.
Hydraulics Open Channel
Manning’s Equation
V = (1 / n) · R2/3 · S1/2
V = Flow velocity (m/s)
n = Manning’s roughness coefficient
R = Hydraulic radius = A / P (m)
S = Channel slope (m/m)
For Q = A · V, multiply by cross-sectional area
View Calculator → n = Manning’s roughness coefficient
R = Hydraulic radius = A / P (m)
S = Channel slope (m/m)
For Q = A · V, multiply by cross-sectional area
Hydraulics Pipe Flow
Hazen–Williams Equation
V = 0.849 · C · R0.63 · S0.54
V = Flow velocity (m/s)
C = Hazen–Williams roughness coefficient
R = Hydraulic radius (m)
S = Hydraulic slope (m/m)
C ≈ 140 for PVC, 100 for steel, 120 for concrete
Hydraulics Tools → C = Hazen–Williams roughness coefficient
R = Hydraulic radius (m)
S = Hydraulic slope (m/m)
C ≈ 140 for PVC, 100 for steel, 120 for concrete
Hydraulics Flow
Flow Continuity Equation
Q = A · V
Q = Volumetric flow rate (m³/s)
A = Cross-sectional flow area (m²)
V = Mean flow velocity (m/s)
Applies to any pipe or channel flow
Hydraulics Tools → A = Cross-sectional flow area (m²)
V = Mean flow velocity (m/s)
Applies to any pipe or channel flow
Hydraulics Pipe Flow
Darcy–Weisbach Equation
hf = f × (L/D) × V² / (2g)
hf = Head loss due to friction (m)
f = Darcy friction factor (unitless)
L = Pipe length (m)
D = Pipe internal diameter (m)
V = Flow velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
Hydraulics Tools → f = Darcy friction factor (unitless)
L = Pipe length (m)
D = Pipe internal diameter (m)
V = Flow velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
Hydraulics Energy
Bernoulli Equation
P1/γ + V1²/2g + z1 = P2/γ + V2²/2g + z2 + hL
P = Pressure at section (kN/m²)
γ = Specific weight of fluid (kN/m³)
V = Flow velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
z = Elevation head (m)
hL = Head loss between sections (m)
Hydraulics Tools → γ = Specific weight of fluid (kN/m³)
V = Flow velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
z = Elevation head (m)
hL = Head loss between sections (m)
Hydraulics Open Channel
Weir Flow — Rectangular Sharp-Crested
Q = Cd × (2/3) × √(2g) × B × H3/2
Q = Flow rate (m³/s)
Cd = Discharge coefficient (≈ 0.62)
g = Acceleration due to gravity (9.81 m/s²)
B = Weir crest width (m)
H = Head over the weir crest (m)
Hydraulics Tools → Cd = Discharge coefficient (≈ 0.62)
g = Acceleration due to gravity (9.81 m/s²)
B = Weir crest width (m)
H = Head over the weir crest (m)
Hydraulics Pumps
Pump Power
P = γ × Q × H / η
P = Brake power (W or kW)
γ = Specific weight of fluid (N/m³)
Q = Flow rate (m³/s)
H = Total dynamic head (m)
η = Pump efficiency (decimal)
η typically 0.60–0.90 for centrifugal pumps
Hydraulics Tools → γ = Specific weight of fluid (N/m³)
Q = Flow rate (m³/s)
H = Total dynamic head (m)
η = Pump efficiency (decimal)
η typically 0.60–0.90 for centrifugal pumps
Hydraulics Flow Regime
Reynolds Number
Re = ρVD / μ = VD / ν
Re = Reynolds number (unitless)
ρ = Fluid density (kg/m³)
V = Flow velocity (m/s)
D = Pipe diameter or hydraulic depth (m)
μ = Dynamic viscosity (Pa⋅s)
ν = Kinematic viscosity (m²/s)
Re < 2000 laminar, 2000–4000 transitional, > 4000 turbulent. Pipe flow criterion.
Hydraulics Tools → ρ = Fluid density (kg/m³)
V = Flow velocity (m/s)
D = Pipe diameter or hydraulic depth (m)
μ = Dynamic viscosity (Pa⋅s)
ν = Kinematic viscosity (m²/s)
Re < 2000 laminar, 2000–4000 transitional, > 4000 turbulent. Pipe flow criterion.
Hydraulics Flow Regime
Froude Number
Fr = V / √(gD)
Fr = Froude number (unitless)
V = Mean flow velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
D = Hydraulic depth = A/T (m)
Fr < 1 subcritical, Fr = 1 critical, Fr > 1 supercritical flow. Used in open channel design.
View Calculator → V = Mean flow velocity (m/s)
g = Acceleration due to gravity (9.81 m/s²)
D = Hydraulic depth = A/T (m)
Fr < 1 subcritical, Fr = 1 critical, Fr > 1 supercritical flow. Used in open channel design.
Hydraulics Open Channel
Hydraulic Radius
R = A / P
R = Hydraulic radius (m)
A = Cross-sectional flow area (m²)
P = Wetted perimeter (m)
Key parameter in Manning's and Hazen-Williams equations. For full pipe: R = D/4.
Hydraulics Tools → A = Cross-sectional flow area (m²)
P = Wetted perimeter (m)
Key parameter in Manning's and Hazen-Williams equations. For full pipe: R = D/4.
Hydraulics Pipe Flow
Pipe Head Loss — General
hL = hf + Σhm
hL = Total head loss (m)
hf = Major (friction) head loss = fLV²/(2gD) (m)
Σhm = Sum of minor losses = ΣK(V²/2g) (m)
K = Minor loss coefficient (per fitting)
Minor losses include bends, valves, tees, entrances, and exits. K values from engineering handbooks.
Hydraulics Tools → hf = Major (friction) head loss = fLV²/(2gD) (m)
Σhm = Sum of minor losses = ΣK(V²/2g) (m)
K = Minor loss coefficient (per fitting)
Minor losses include bends, valves, tees, entrances, and exits. K values from engineering handbooks.
Hydraulics Open Channel
Open Channel Uniform Flow
Q = (1/n) · A · R2/3 · S1/2
Q = Flow rate (m³/s)
n = Manning’s roughness coefficient
A = Cross-sectional flow area (m²)
R = Hydraulic radius = A/P (m)
S = Channel slope (m/m)
Uniform flow occurs when flow depth and velocity are constant. Normal depth is found by solving for y.
View Calculator → n = Manning’s roughness coefficient
A = Cross-sectional flow area (m²)
R = Hydraulic radius = A/P (m)
S = Channel slope (m/m)
Uniform flow occurs when flow depth and velocity are constant. Normal depth is found by solving for y.
Highway & Transportation
Vertical and horizontal curve geometry formulas.
Highway Vertical Curve
Vertical Curve Length
L = K · A
L = Length of vertical curve (m)
K = Rate of curvature (m / %)
A = Algebraic difference in grades (%), A = |G1 − G2|
K values from AASHTO tables per design speed
Highway Tools → K = Rate of curvature (m / %)
A = Algebraic difference in grades (%), A = |G1 − G2|
K values from AASHTO tables per design speed
Highway Horizontal Curve
Horizontal Curve — Tangent Length
T = R · tan(Δ / 2)
T = Tangent length (m)
R = Curve radius (m)
Δ = Intersection (deflection) angle
Also: L = πRΔ/180 (arc length)
Highway Tools → R = Curve radius (m)
Δ = Intersection (deflection) angle
Also: L = πRΔ/180 (arc length)
Highway Horizontal Curve
Superelevation
e + f = V² / (127R)
e = Rate of superelevation (m/m)
f = Side friction factor (unitless)
V = Vehicle speed (km/h)
R = Curve radius (m)
AASHTO max e = 0.04–0.12 depending on conditions
Highway Tools → f = Side friction factor (unitless)
V = Vehicle speed (km/h)
R = Curve radius (m)
AASHTO max e = 0.04–0.12 depending on conditions
Highway Safety
Stopping Sight Distance
SSD = 0.278Vt + V² / (254f)
SSD = Stopping sight distance (m)
V = Initial speed (km/h)
t = Perception–reaction time (s), typically 2.5 s
f = Coefficient of friction between tires and road
AASHTO uses t = 2.5 s, f varies with speed (0.30–0.41)
Highway Tools → V = Initial speed (km/h)
t = Perception–reaction time (s), typically 2.5 s
f = Coefficient of friction between tires and road
AASHTO uses t = 2.5 s, f varies with speed (0.30–0.41)
Highway Vertical Curve
Crest Vertical Curve Length
L = A × S² / (200(√h1 + √h2)²)
L = Curve length (m), for S < L
A = Algebraic difference in grades (%), |G1 − G2|
S = Stopping sight distance (m)
h1 = Eye height of driver (m), typically 1.07 m
h2 = Object height (m), typically 0.15 m
Highway Tools → A = Algebraic difference in grades (%), |G1 − G2|
S = Stopping sight distance (m)
h1 = Eye height of driver (m), typically 1.07 m
h2 = Object height (m), typically 0.15 m
Highway Safety
Passing Sight Distance
PSD = d1 + d2 + d3 + d4
PSD = Passing sight distance (m)
d1 = Distance during perception/reaction and acceleration (m)
d2 = Distance while occupying left lane (m)
d3 = Safety clearance distance (m)
d4 = Distance traveled by opposing vehicle (m)
AASHTO PSD values range from 200–800 m for speeds 50–120 km/h. Critical for two-lane highway design.
Highway Tools → d1 = Distance during perception/reaction and acceleration (m)
d2 = Distance while occupying left lane (m)
d3 = Safety clearance distance (m)
d4 = Distance traveled by opposing vehicle (m)
AASHTO PSD values range from 200–800 m for speeds 50–120 km/h. Critical for two-lane highway design.
Transportation Traffic
Traffic Flow Relationship
q = k · v
q = Traffic flow rate (vehicles/h)
k = Traffic density (vehicles/km)
v = Space-mean speed (km/h)
Fundamental diagram of traffic flow. Capacity occurs at critical density. Per HCM methodology.
View Calculator → k = Traffic density (vehicles/km)
v = Space-mean speed (km/h)
Fundamental diagram of traffic flow. Capacity occurs at critical density. Per HCM methodology.
Transportation Traffic
Traffic Density & Spacing
k = 1000 / s s = s0 + tv + cv²
k = Traffic density (vehicles/km)
s = Average center-to-center spacing (m)
s0 = Minimum bumper-to-bumper spacing (m)
t = Perception-reaction time (s)
v = Speed (m/s)
c = Braking coefficient
Jam density kj occurs when vehicles are bumper-to-bumper. Typical kj ≈ 160 veh/km.
View Calculator → s = Average center-to-center spacing (m)
s0 = Minimum bumper-to-bumper spacing (m)
t = Perception-reaction time (s)
v = Speed (m/s)
c = Braking coefficient
Jam density kj occurs when vehicles are bumper-to-bumper. Typical kj ≈ 160 veh/km.
Transportation Traffic
Vehicle Delay (Webster’s Formula)
d = (C(1 − λ)²) / (2(1 − λx)) + (x²) / (2q(1 − x)) − 0.65(C/q²)1/3x(2+5λ)
d = Average delay per vehicle (s)
C = Cycle length (s)
λ = Proportion of effective green time = g/C
x = Degree of saturation = q/(λs)
q = Arrival flow rate (veh/s)
s = Saturation flow rate (veh/s)
Webster’s model for signalized intersection delay. HCM 2016 uses a similar formulation.
Transportation Tools → C = Cycle length (s)
λ = Proportion of effective green time = g/C
x = Degree of saturation = q/(λs)
q = Arrival flow rate (veh/s)
s = Saturation flow rate (veh/s)
Webster’s model for signalized intersection delay. HCM 2016 uses a similar formulation.
Transportation Pavement
ESAL — Equivalent Single Axle Load
ESAL = Σ(Fi × ni × LDF × TDF)
ESAL = Equivalent 80-kN single axle loads
Fi = Load equivalency factor for axle type i
ni = Number of axles of type i
LDF = Lane distribution factor
TDF = Truck directional distribution factor
AASHTO design method. Fi = (Wi/80)4 for flexible pavements (4th power rule).
Transportation Tools → Fi = Load equivalency factor for axle type i
ni = Number of axles of type i
LDF = Lane distribution factor
TDF = Truck directional distribution factor
AASHTO design method. Fi = (Wi/80)4 for flexible pavements (4th power rule).
Transportation Pavement
CBR Pavement Thickness Design
T = √(W × 106 / (CBR × π × p))
T = Total pavement thickness above subgrade (mm)
W = Wheel load (N)
CBR = California bearing ratio of subgrade (%)
p = Tyre contact pressure (MPa)
CBR method per IRC 37 and AASHTO. Typical subgrade CBR: 2–8% for clay, 10–30% for sand/gravel.
Highway Tools → W = Wheel load (N)
CBR = California bearing ratio of subgrade (%)
p = Tyre contact pressure (MPa)
CBR method per IRC 37 and AASHTO. Typical subgrade CBR: 2–8% for clay, 10–30% for sand/gravel.
Transportation Geometric Design
Road Gradient & Asphalt Quantity
G = (Δh / L) × 100% Wa = A × t × ρ
G = Road gradient (%)
Δh = Elevation difference (m)
L = Horizontal distance (m)
Wa = Asphalt weight required (tonnes)
A = Pavement area (m²)
t = Asphalt layer thickness (m)
ρ = Asphalt density (typically 2.35 t/m³)
Max gradient: 3–6% for highways, 6–10% for urban roads per AASHTO. Asphalt quantity = V × ρ.
Highway Tools → Δh = Elevation difference (m)
L = Horizontal distance (m)
Wa = Asphalt weight required (tonnes)
A = Pavement area (m²)
t = Asphalt layer thickness (m)
ρ = Asphalt density (typically 2.35 t/m³)
Max gradient: 3–6% for highways, 6–10% for urban roads per AASHTO. Asphalt quantity = V × ρ.
Steel Structures
Axial stress and section property formulas.
Steel Axial
Axial Stress
σ = P / A
σ = Axial stress (MPa or N/mm²)
P = Applied axial load (N)
A = Cross-sectional area (mm²)
Tension if P pulls, compression if P pushes
Steel Tools → P = Applied axial load (N)
A = Cross-sectional area (mm²)
Tension if P pulls, compression if P pushes
Steel Sections
I-Beam Cross-Section Area
A = bf · tf · 2 + d · tw
A = Total cross-sectional area (mm²)
bf = Flange width (mm)
tf = Flange thickness (mm)
d = Web depth (mm)
tw = Web thickness (mm)
Steel Tools → bf = Flange width (mm)
tf = Flange thickness (mm)
d = Web depth (mm)
tw = Web thickness (mm)
Steel Buckling
Euler’s Buckling Load
Pcr = π²EI / (KL)²
Pcr = Critical buckling load (N)
E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
K = Effective length factor
L = Unbraced length (mm)
Steel Tools → E = Modulus of elasticity (MPa)
I = Moment of inertia (mm&sup4;)
K = Effective length factor
L = Unbraced length (mm)
Steel Shear
Shear Capacity
(AISC φVn)
φVn = φ × 0.6 × Fy × Aw × Cv
φVn = Factored shear strength (kN)
φ = Resistance factor (1.00 for shear per AISC)
Fy = Specified yield stress (MPa)
Aw = Web area = d × tw (mm²)
Cv = Web shear coefficient
Steel Tools → φ = Resistance factor (1.00 for shear per AISC)
Fy = Specified yield stress (MPa)
Aw = Web area = d × tw (mm²)
Cv = Web shear coefficient
Steel Flexure
Moment Capacity
(AISC φMn, Compact Section)
φMn = φ × Mp = φ × Fy × Zx
φMn = Factored flexural strength (kN⋅m)
φ = Resistance factor (0.90 for flexure per AISC)
Mp = Plastic moment capacity (kN⋅m)
Fy = Specified yield stress (MPa)
Zx = Plastic section modulus (mm³)
Steel Tools → φ = Resistance factor (0.90 for flexure per AISC)
Mp = Plastic moment capacity (kN⋅m)
Fy = Specified yield stress (MPa)
Zx = Plastic section modulus (mm³)
Steel Tension
Tensile Capacity — Yield
(AISC Gross Section)
φPn = φ × Fy × Ag
φPn = Factored tensile strength — yield (kN)
φ = Resistance factor (0.90 for yielding per AISC)
Fy = Specified yield stress (MPa)
Ag = Gross cross-sectional area (mm²)
Steel Tools → φ = Resistance factor (0.90 for yielding per AISC)
Fy = Specified yield stress (MPa)
Ag = Gross cross-sectional area (mm²)
Steel Tension
Tensile Capacity — Rupture
(AISC Net Section)
φPn = φ × Fu × Ae
φPn = Factored tensile strength — rupture (kN)
φ = Resistance factor (0.75 for rupture per AISC)
Fu = Specified tensile strength (MPa)
Ae = Effective net area (mm²)
Steel Tools → φ = Resistance factor (0.75 for rupture per AISC)
Fu = Specified tensile strength (MPa)
Ae = Effective net area (mm²)
Soils & Foundations
Stress distribution and foundation analysis formulas.
Soils Stress Distribution
Boussinesq Vertical Stress
(Point Load)
Δσz = (3P) / (2πz²) × 1 / (1 + (r/z)²)5/2
Δσz = Vertical stress increase at point (kN/m²)
P = Point load (kN)
z = Depth below ground surface (m)
r = Horizontal distance from load to point (m)
Foundation Tools → P = Point load (kN)
z = Depth below ground surface (m)
r = Horizontal distance from load to point (m)
Structural Loads
Wind, live load, and load distribution formulas.
Structural Loads Wind
Wind Velocity Pressure
(ASCE 7)
qz = 0.613 × Kz × Kzt × Kd × V²
qz = Velocity pressure at height z (N/m²)
Kz = Velocity pressure exposure coefficient
Kzt = Topographic factor
Kd = Wind directionality factor
V = Basic wind speed (m/s)
Structural Tools → Kz = Velocity pressure exposure coefficient
Kzt = Topographic factor
Kd = Wind directionality factor
V = Basic wind speed (m/s)
Structural Loads Live Load
Live Load Reduction
(ASCE 7)
L = L0 × (0.25 + 15 / √(KLL × AT))
L = Reduced live load (kN/m²)
L0 = Unreduced design live load (kN/m²)
KLL = Live load element factor
AT = Tributary area (m²)
L ≥ 0.50L0 for members, 0.40L0 for columns
Structural Tools → L0 = Unreduced design live load (kN/m²)
KLL = Live load element factor
AT = Tributary area (m²)
L ≥ 0.50L0 for members, 0.40L0 for columns
Reinforced Concrete Design
Flexural, shear, and serviceability design formulas per ACI 318, IS 456, and Eurocode 2.
RC Design Materials
Concrete Compressive Strength
f′c = Pmax / A
f′c = Specified compressive strength of concrete (MPa)
Pmax = Maximum load at failure (N)
A = Cross-sectional area of test specimen (mm²)
28-day cylinder strength per ASTM C39. fck per IS 456 uses 150 mm cubes.
Concrete Tools → Pmax = Maximum load at failure (N)
A = Cross-sectional area of test specimen (mm²)
28-day cylinder strength per ASTM C39. fck per IS 456 uses 150 mm cubes.
RC Design Flexure
Balanced Reinforcement Ratio
ρb = 0.85β1f′c/fy × 600/(600 + fy)
ρb = Balanced reinforcement ratio (unitless)
β1 = Stress block depth factor (0.85 for f′c ≤ 28 MPa)
f′c = Concrete compressive strength (MPa)
fy = Steel yield strength (MPa)
ρb is the ratio where concrete crushes and steel yields simultaneously per ACI 318. Typically 2.5–4.0%.
RC Tools → β1 = Stress block depth factor (0.85 for f′c ≤ 28 MPa)
f′c = Concrete compressive strength (MPa)
fy = Steel yield strength (MPa)
ρb is the ratio where concrete crushes and steel yields simultaneously per ACI 318. Typically 2.5–4.0%.
RC Design Flexure
Ultimate Moment Capacity
Mu = φAsfy(d − a/2)
Mu = Factored moment capacity (kN⋅m)
φ = Strength reduction factor (0.90 for flexure)
As = Area of tension reinforcement (mm²)
fy = Yield strength of steel (MPa)
d = Effective depth (mm)
a = Depth of equivalent stress block = Asfy/(0.85f′cb) (mm)
Per ACI 318 §22.3. φ = 0.90 for tension-controlled sections.
View Calculator → φ = Strength reduction factor (0.90 for flexure)
As = Area of tension reinforcement (mm²)
fy = Yield strength of steel (MPa)
d = Effective depth (mm)
a = Depth of equivalent stress block = Asfy/(0.85f′cb) (mm)
Per ACI 318 §22.3. φ = 0.90 for tension-controlled sections.
RC Design Flexure
Nominal Moment Capacity
Mn = Asfy(d − a/2)
Mn = Nominal moment capacity (kN⋅m)
As = Area of tension reinforcement (mm²)
fy = Yield strength of steel (MPa)
d = Effective depth (mm)
a = Depth of stress block = Asfy/(0.85f′cb) (mm)
Design strength Mu = φMn. φ varies from 0.65 (compression-controlled) to 0.90 (tension-controlled).
View Calculator → As = Area of tension reinforcement (mm²)
fy = Yield strength of steel (MPa)
d = Effective depth (mm)
a = Depth of stress block = Asfy/(0.85f′cb) (mm)
Design strength Mu = φMn. φ varies from 0.65 (compression-controlled) to 0.90 (tension-controlled).
RC Design Reinforcement
Lap Splice Length
ld = (fyψtψeψs) / (2.1√f′c) × db
ld = Tension development length (mm)
fy = Specified yield strength (MPa)
ψt = Casting position factor (1.3 for top bars, 1.0 otherwise)
ψe = Epoxy coating factor (1.2–1.5)
ψs = Bar size factor (0.8 for ≤20 mm, 1.0 for ≥22 mm)
db = Bar diameter (mm)
Class A splice = 1.0ld, Class B splice = 1.3ld per ACI 318 §25.5.
RC Tools → fy = Specified yield strength (MPa)
ψt = Casting position factor (1.3 for top bars, 1.0 otherwise)
ψe = Epoxy coating factor (1.2–1.5)
ψs = Bar size factor (0.8 for ≤20 mm, 1.0 for ≥22 mm)
db = Bar diameter (mm)
Class A splice = 1.0ld, Class B splice = 1.3ld per ACI 318 §25.5.
RC Design Serviceability
Crack Width Calculation
w = 2.2βεsdcA¯ / h
w = Maximum crack width (mm)
β = Ratio of distances from neutral axis to tension face
εs = Steel strain at service load
dc = Cover from extreme tension fiber to bar center (mm)
A¯ = Average effective tension area per bar (mm²)
h = Overall section depth (mm)
Gergely-Lutz equation per ACI 318. Max crack width: 0.41 mm interior, 0.33 mm exterior exposure.
View Calculator → β = Ratio of distances from neutral axis to tension face
εs = Steel strain at service load
dc = Cover from extreme tension fiber to bar center (mm)
A¯ = Average effective tension area per bar (mm²)
h = Overall section depth (mm)
Gergely-Lutz equation per ACI 318. Max crack width: 0.41 mm interior, 0.33 mm exterior exposure.
RC Design Reinforcement
Minimum Reinforcement Ratio
ρmin = max(0.25√f′c/fy , 1.4/fy)
ρmin = Minimum reinforcement ratio (unitless)
f′c = Concrete compressive strength (MPa)
fy = Steel yield strength (MPa)
ACI 318 §9.6.1. For fy = 420 MPa: ρmin ≈ 0.33%. IS 456: As,min = 0.85bd/fy.
RC Tools → f′c = Concrete compressive strength (MPa)
fy = Steel yield strength (MPa)
ACI 318 §9.6.1. For fy = 420 MPa: ρmin ≈ 0.33%. IS 456: As,min = 0.85bd/fy.
RC Design Reinforcement
Maximum Reinforcement Ratio
ρmax = 0.75ρb (ACI) ρmax = 0.04 (IS 456) ρmax = 0.04Ac (EC2)
ρmax = Maximum reinforcement ratio (unitless)
ρb = Balanced reinforcement ratio
ACI 318 limits to 0.75ρb for tension-controlled sections. IS 456: 4% of gross area. EC2: 4% of concrete area.
RC Tools → ρb = Balanced reinforcement ratio
ACI 318 limits to 0.75ρb for tension-controlled sections. IS 456: 4% of gross area. EC2: 4% of concrete area.
RC Design Shear
RC Shear Capacity
φVn = φ(Vc + Vs)
φVn = Factored shear strength (kN)
Vc = Nominal shear strength provided by concrete (kN)
Vs = Nominal shear strength provided by stirrups (kN)
φ = Strength reduction factor (0.75 for shear per ACI 318)
If Vu > φVc/2, minimum stirrups required. Vs ≤ 0.66√f′cbwd.
View Calculator → Vc = Nominal shear strength provided by concrete (kN)
Vs = Nominal shear strength provided by stirrups (kN)
φ = Strength reduction factor (0.75 for shear per ACI 318)
If Vu > φVc/2, minimum stirrups required. Vs ≤ 0.66√f′cbwd.
RC Design Shear
Concrete Shear Strength
Vc = 0.17√f′c × bw × d
Vc = Concrete shear strength (N)
f′c = Concrete compressive strength (MPa)
bw = Web width (mm)
d = Effective depth (mm)
Simplified method per ACI 318 Eq. 22.5.5.1. For members with axial compression, Vc increases.
View Calculator → f′c = Concrete compressive strength (MPa)
bw = Web width (mm)
d = Effective depth (mm)
Simplified method per ACI 318 Eq. 22.5.5.1. For members with axial compression, Vc increases.
RC Design Shear
Punching Shear — Two-Way Slabs
Vn = 0.33√f′c × bo × d
Vn = Nominal punching shear strength (N)
f′c = Concrete compressive strength (MPa)
bo = Perimeter of critical section at d/2 from column face (mm)
d = Effective depth (mm)
ACI 318 §22.6. Check at d/2 from column face. φ = 0.75. If Vu > φVn, add drop panels or shear reinforcement.
View Calculator → f′c = Concrete compressive strength (MPa)
bo = Perimeter of critical section at d/2 from column face (mm)
d = Effective depth (mm)
ACI 318 §22.6. Check at d/2 from column face. φ = 0.75. If Vu > φVn, add drop panels or shear reinforcement.
RC Design Reinforcement
Effective Depth
d = h − cover − φstirrup − φbar/2
d = Effective depth to centroid of tension reinforcement (mm)
h = Overall section height (mm)
cover = Clear cover to stirrups (mm)
φstirrup = Stirrup bar diameter (mm)
φbar = Main tension bar diameter (mm)
For beams: cover = 40 mm (IS 456), 38 mm (ACI 318 interior). For slabs: d ≈ h − 25 mm.
RC Tools → h = Overall section height (mm)
cover = Clear cover to stirrups (mm)
φstirrup = Stirrup bar diameter (mm)
φbar = Main tension bar diameter (mm)
For beams: cover = 40 mm (IS 456), 38 mm (ACI 318 interior). For slabs: d ≈ h − 25 mm.
RC Design Flexure
Flexural Reinforcement Area
As,req = Mu / (φfy(d − a/2))
As,req = Required tension reinforcement area (mm²)
Mu = Factored bending moment (N⋅mm)
φ = Strength reduction factor (0.90 for flexure)
fy = Steel yield strength (MPa)
d = Effective depth (mm)
a = Depth of stress block (mm), solved iteratively
Iterative solution: assume a, compute As, check a = Asfy/(0.85f′cb), repeat.
View Calculator → Mu = Factored bending moment (N⋅mm)
φ = Strength reduction factor (0.90 for flexure)
fy = Steel yield strength (MPa)
d = Effective depth (mm)
a = Depth of stress block (mm), solved iteratively
Iterative solution: assume a, compute As, check a = Asfy/(0.85f′cb), repeat.
Surveying Engineering
Leveling, traversing, area, and earthwork volume formulas.
Surveying Coordinates
Distance Between Coordinates
D = √((x2 − x1)² + (y2 − y1)²)
D = Horizontal distance between points (m)
x1, y1 = Coordinates of point 1 (m)
x2, y2 = Coordinates of point 2 (m)
Pythagorean distance formula used in all coordinate surveying and total station computations.
Surveying Tools → x1, y1 = Coordinates of point 1 (m)
x2, y2 = Coordinates of point 2 (m)
Pythagorean distance formula used in all coordinate surveying and total station computations.
Surveying Leveling
Height of Instrument Method
HI = RL + BS RLnew = HI − FS
HI = Height of instrument (m)
RL = Reduced level of benchmark (m)
BS = Backsight reading (m)
RLnew = Reduced level of new point (m)
FS = Foresight reading (m)
Level difference: Δh = BS − FS. Rise if Δh > 0, fall if Δh < 0.
Surveying Tools → RL = Reduced level of benchmark (m)
BS = Backsight reading (m)
RLnew = Reduced level of new point (m)
FS = Foresight reading (m)
Level difference: Δh = BS − FS. Rise if Δh > 0, fall if Δh < 0.
Surveying Leveling
Rise and Fall Method
RLn = RLn−1 ± (BSn − FSn)
RLn = Reduced level of point n (m)
RLn−1 = Reduced level of previous point (m)
BSn = Backsight at point n (m)
FSn = Foresight at point n (m)
Check: ΣBS − ΣFS = last RL − first RL. Used for precise leveling with many intermediate points.
Surveying Tools → RLn−1 = Reduced level of previous point (m)
BSn = Backsight at point n (m)
FSn = Foresight at point n (m)
Check: ΣBS − ΣFS = last RL − first RL. Used for precise leveling with many intermediate points.
Surveying Traverse
Traverse Adjustment — Bowditch
CΔx,i = −(ΣΔx) × Li / ΣL CΔy,i = −(ΣΔy) × Li / ΣL
CΔx,i = Correction to Δx for leg i (m)
CΔy,i = Correction to Δy for leg i (m)
ΣΔx, ΣΔy = Closure errors in x and y (m)
Li = Length of leg i (m)
ΣL = Total perimeter length (m)
Bowditch rule: correction proportional to leg length. Check: precision = 1:ΣL/√(ΣΔx²+ΣΔy²).
Surveying Tools → CΔy,i = Correction to Δy for leg i (m)
ΣΔx, ΣΔy = Closure errors in x and y (m)
Li = Length of leg i (m)
ΣL = Total perimeter length (m)
Bowditch rule: correction proportional to leg length. Check: precision = 1:ΣL/√(ΣΔx²+ΣΔy²).
Surveying Traverse
Bearing & Coordinate Calculation
Δx = L · sinθ Δy = L · cosθ θ = tan&supmin;¹(Δx/Δy)
Δx, Δy = Coordinate differences (m)
L = Horizontal distance (m)
θ = Bearing angle from north (degrees)
Quadrant rules: NE (θ=0–90), SE (90–180), SW (180–270), NW (270–360). Forward bearing ± 180° = back bearing.
Surveying Tools → L = Horizontal distance (m)
θ = Bearing angle from north (degrees)
Quadrant rules: NE (θ=0–90), SE (90–180), SW (180–270), NW (270–360). Forward bearing ± 180° = back bearing.
Surveying Area
Area by Coordinates (Shoelace)
A = ½|Σ(xiyi+1 − xi+1yi)|
A = Enclosed area (m²)
xi, yi = Coordinates of point i (m)
Also called surveyor’s formula. For n points, repeat first point at end. Sign convention: clockwise gives negative area.
View Calculator → xi, yi = Coordinates of point i (m)
Also called surveyor’s formula. For n points, repeat first point at end. Sign convention: clockwise gives negative area.
Surveying Earthwork
Earthwork Volume — Average End Area
V = (A1 + A2) / 2 × L
V = Volume between sections (m³)
A1, A2 = Cross-sectional areas at ends (m²)
L = Distance between sections (m)
Prismoidal formula: V = L(A1 + 4Am + A2)/6 for greater accuracy.
Earthwork Tools → A1, A2 = Cross-sectional areas at ends (m²)
L = Distance between sections (m)
Prismoidal formula: V = L(A1 + 4Am + A2)/6 for greater accuracy.
Surveying Earthwork
Cut and Fill Volumes
Vcut = Vtotal × Scut/(Scut+Sfill) Vfill = Vtotal × Sfill/(Scut+Sfill)
Vcut = Cut volume (m³)
Vfill = Fill volume (m³)
Scut, Sfill = Cut and fill areas at section (m²)
Borrow volume = (Fill − Cut) with shrinkage/swell factors. Typical swell for common earth: 20–30%.
View Calculator → Vfill = Fill volume (m³)
Scut, Sfill = Cut and fill areas at section (m²)
Borrow volume = (Fill − Cut) with shrinkage/swell factors. Typical swell for common earth: 20–30%.
Surveying Mapping
Contour Interval & Interpolation
CI = (max − min) / N x = (H − ha) / (hb − ha) × L
CI = Contour interval (m)
max, min = Maximum and minimum elevations (m)
N = Number of contour intervals desired
x = Distance from point a to contour elevation (m)
H = Contour elevation (m)
ha, hb = Elevations at points a and b (m)
L = Distance between points a and b (m)
Standard CI: 0.5–2 m for flat terrain, 5–20 m for steep terrain. Use linear interpolation for contour location.
Surveying Tools → max, min = Maximum and minimum elevations (m)
N = Number of contour intervals desired
x = Distance from point a to contour elevation (m)
H = Contour elevation (m)
ha, hb = Elevations at points a and b (m)
L = Distance between points a and b (m)
Standard CI: 0.5–2 m for flat terrain, 5–20 m for steep terrain. Use linear interpolation for contour location.
Construction Engineering
Material quantity, cost estimation, and productivity formulas.
Construction Materials
Concrete Material Quantity
C = V × ρ / Σratio × ratioc S = V × ρ / Σratio × ratios
C = Cement mass (kg)
S = Fine aggregate mass (kg)
V = Concrete volume required (m³)
ρ = Concrete density (typically 2400 kg/m³)
ratio = Mix proportion parts (e.g., 1:1.5:3 for M20)
Dry volume factor: 1.54–1.57 (IS 456). Wet concrete = 1.54× dry materials total. Add 5–10% for waste.
Concrete Tools → S = Fine aggregate mass (kg)
V = Concrete volume required (m³)
ρ = Concrete density (typically 2400 kg/m³)
ratio = Mix proportion parts (e.g., 1:1.5:3 for M20)
Dry volume factor: 1.54–1.57 (IS 456). Wet concrete = 1.54× dry materials total. Add 5–10% for waste.
Construction Materials
Cement & Sand Quantity per Bag
No. of bags = C / 50 Vsand = C/ρcem × ratios/ratioc
No. of bags = Number of 50 kg cement bags
C = Total cement mass required (kg)
Vsand = Sand volume required (m³)
ρcem = Cement density (1440 kg/m³)
1 bag cement = 0.035 m³. Typical mix for 1 m³ M20 concrete: 8 bags cement, 0.42 m³ sand, 0.84 m³ aggregate.
View Calculator → C = Total cement mass required (kg)
Vsand = Sand volume required (m³)
ρcem = Cement density (1440 kg/m³)
1 bag cement = 0.035 m³. Typical mix for 1 m³ M20 concrete: 8 bags cement, 0.42 m³ sand, 0.84 m³ aggregate.
Construction Materials
Steel & Rebar Weight
W = D²/162 × L Wplate = ρ × A × t
W = Weight of rebar (kg)
D = Rebar diameter (mm)
L = Total rebar length (m)
Wplate = Steel plate weight (kg)
ρ = Density of steel (7850 kg/m³)
A = Plate area (m²)
t = Plate thickness (m)
D²/162 gives kg/m. Example: 12 mm bar = 0.888 kg/m, 16 mm = 1.579 kg/m, 20 mm = 2.469 kg/m.
View Calculator → D = Rebar diameter (mm)
L = Total rebar length (m)
Wplate = Steel plate weight (kg)
ρ = Density of steel (7850 kg/m³)
A = Plate area (m²)
t = Plate thickness (m)
D²/162 gives kg/m. Example: 12 mm bar = 0.888 kg/m, 16 mm = 1.579 kg/m, 20 mm = 2.469 kg/m.
Construction Productivity
Productivity Rate
Productivity = Total Quantity / (No. of Workers × Time)
Productivity = Output per worker-hour (units/hr)
Total Quantity = Work completed (m³, m², kg, etc.)
No. of Workers = Number of laborers
Time = Duration (hours)
Typical: concreting 0.8–1.2 m³/hr/mason, formwork 5–8 m²/hr/carpenter, rebar 50–80 kg/hr/laborer.
All Calculators → Total Quantity = Work completed (m³, m², kg, etc.)
No. of Workers = Number of laborers
Time = Duration (hours)
Typical: concreting 0.8–1.2 m³/hr/mason, formwork 5–8 m²/hr/carpenter, rebar 50–80 kg/hr/laborer.
Construction Cost
Labor Cost Estimation
Labor Cost = Σ(Ni × Ri × D)
Labor Cost = Total labor cost ($)
Ni = Number of workers of type i
Ri = Daily wage rate of worker type i ($/day)
D = Duration (days)
Include 20–30% overhead for benefits, insurance, and tools. Productivity factor: effective hours = 7–8 per day.
All Calculators → Ni = Number of workers of type i
Ri = Daily wage rate of worker type i ($/day)
D = Duration (days)
Include 20–30% overhead for benefits, insurance, and tools. Productivity factor: effective hours = 7–8 per day.
Construction Equipment
Equipment Productivity
Pe = C × V × Ef × 60 / Ct
Pe = Equipment productivity (m³/hr)
C = Bucket/heap capacity (m³)
V = Volumetric efficiency (0.8–0.95)
Ef = Job efficiency factor (0.5–0.85)
Ct = Cycle time (minutes)
Excavator: Ct ≈ 0.3–0.6 min. Bulldozer productivity = 0.8L²/Ct. Haul truck cycle = load + haul + dump + return.
Earthwork Tools → C = Bucket/heap capacity (m³)
V = Volumetric efficiency (0.8–0.95)
Ef = Job efficiency factor (0.5–0.85)
Ct = Cycle time (minutes)
Excavator: Ct ≈ 0.3–0.6 min. Bulldozer productivity = 0.8L²/Ct. Haul truck cycle = load + haul + dump + return.
Construction Cost
Construction Cost Estimate
Total Cost = Q × Rate + Overhead + Profit
Total Cost = Project total cost ($)
Q = Total material quantity
Rate = Unit rate ($/unit)
Overhead = Site overhead + HO overhead (15–25%)
Profit = Expected profit margin (8–15%)
Rate = material cost + labor + equipment + tools + transport. Bill of quantities (BOQ) + rate analysis = project estimate.
All Calculators → Q = Total material quantity
Rate = Unit rate ($/unit)
Overhead = Site overhead + HO overhead (15–25%)
Profit = Expected profit margin (8–15%)
Rate = material cost + labor + equipment + tools + transport. Bill of quantities (BOQ) + rate analysis = project estimate.
Construction Materials
Material Waste Allowance
Qordered = Qnet × (1 + w/100)
Qordered = Total material to order
Qnet = Net material quantity from design
w = Waste allowance percentage (%)
Typical waste: concrete 5–10%, rebar 5–8%, steel sections 3–5%, formwork 10–15%, tiles 10–15%, paint 10–20%.
All Calculators → Qnet = Net material quantity from design
w = Waste allowance percentage (%)
Typical waste: concrete 5–10%, rebar 5–8%, steel sections 3–5%, formwork 10–15%, tiles 10–15%, paint 10–20%.
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