Civil & Surveying formulas — 68 solvers

Area of a Three-Sided Parcel by Coordinates

A = \tfrac{1}{2}\left[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right]

Civil & SurveyingGeometryShoelace area of a parcel from the coordinates of its three corners, listed counter-clockwise so the result comes out positive.

Asphalt Tonnage from Area and Thickness

M = A\,t\,\rho

Civil & SurveyingConverts a paving area, lift thickness and compacted mix density into the tonnage to order, reported in US short tons.

Back Azimuth

\alpha_b = \alpha \pm 180^{\circ}

Civil & SurveyingTrigonometryReverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside the zero to three sixty range.

Broad-Crested Weir Discharge

Q = C_d \left(\tfrac{2}{3}\right)^{3/2} \sqrt{g} \; b \, H^{3/2}

Fluid MechanicsCivil & SurveyingWater & WastewaterDischarge over a broad-crested weir or a level spillway crest, derived by forcing critical depth over the crest. In SI the constant group is 1.705, so Q = 1.705 C_d b H^1.5 with lengths in metres.

Cantilever Deflection — Uniform Load

\delta = \frac{w L^{4}}{8 E I}

Strength of MaterialsCivil & SurveyingMechanicsTip deflection of a cantilever carrying a uniformly distributed load along its whole length, δ = wL⁴/8EI — three-eighths of the sag the same total load would cause at the tip.

Chezy Equation

V = C \sqrt{R \, S}

Fluid MechanicsCivil & SurveyingWater & WastewaterAntoine Chezy's 1769 uniform-flow equation: mean velocity equals the Chezy coefficient times the square root of the hydraulic radius times the energy slope.

Combined Axial and Bending Stress

\sigma = \frac{P}{A} + \frac{M c}{I}

Strength of MaterialsCivil & SurveyingMechanicsExtreme-fibre stress where an axial force and a bending moment act together, as in an eccentrically loaded column or a beam-column. Enter a negative c for the relieved face.

Concrete Volume with Waste Allowance

V = L\,W\,T\left(1 + \frac{w}{100}\right)

Civil & SurveyingGeometryOrders concrete for a rectangular slab, footing or column by adding a percent waste allowance to the neat volume, in cubic yards.

Crest Vertical Curve Length for Sight Distance

L = \frac{A\,S^{2}}{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}

Civil & SurveyingGeometryCrest curve length needed to see an object over the hill, for the case where sight distance is shorter than the curve.

Critical Depth in a Rectangular Channel

y_c = \left(\frac{Q^2}{g \, b^2}\right)^{1/3}

Fluid MechanicsCivil & SurveyingWater & WastewaterCritical depth of a rectangular channel from the discharge and the bed width — the depth of minimum specific energy, where the Froude number is exactly one.

Departure of a Traverse Leg

\text{Dep} = L\sin\alpha

Civil & SurveyingTrigonometryEast-west component of a traverse course, taken from the measured horizontal distance and the azimuth of the line.

Detention Storage from a Routing Step

\Delta S = (Q_{in} - Q_{out}) \, \Delta t

Water & WastewaterCivil & SurveyingFluid MechanicsOne step of the reservoir continuity equation: the storage a detention basin accumulates is the difference between inflow and outflow multiplied by the length of the step.

Differential Levelling Elevation

E_2 = E_1 + \text{BS} - \text{FS}

Civil & SurveyingCarries an elevation forward one instrument setup, adding the backsight to the known point and subtracting the foresight.

Earthwork Volume by Average End Area

V = \frac{L\,(A_1 + A_2)}{2}

Civil & SurveyingGeometryVolume between two cross sections, averaging their end areas over the distance between them; the answer is reported in cubic yards.

Earthwork Volume by the Prismoidal Formula

V = \frac{L\,(A_1 + 4A_m + A_2)}{6}

Civil & SurveyingGeometrySimpson's rule applied to earthwork, weighting the middle cross section four times the ends; the answer is reported in cubic yards.

Elevation from Grade and Distance

E_2 = E_1 + \frac{G\,L}{100}

Civil & SurveyingGeometryProjects an elevation along a uniform grade, adding the rise over a measured horizontal distance to the known starting elevation.

Elevation on a Parabolic Vertical Curve

E = E_{BVC} + \frac{g_1 x}{100} + \frac{A\,x^{2}}{200\,L}

Civil & SurveyingGeometryElevation at any station on an equal-tangent parabolic vertical curve, measured from the beginning of vertical curve.

Energy Loss in a Hydraulic Jump

\Delta E = \frac{(y_2 - y_1)^{3}}{4 \, y_1 \, y_2}

Fluid MechanicsCivil & SurveyingWater & WastewaterSpecific energy destroyed by a hydraulic jump in a rectangular channel, obtained by subtracting the downstream specific energy from the upstream one and simplifying with the momentum equation.

Fixed-End Moment — Fixed-Fixed Beam, Uniform Load

M_F = \frac{w L^{2}}{12}

Strength of MaterialsCivil & SurveyingMechanicsMoment at each built-in end of a fixed-fixed beam under a uniformly distributed load, wL²/12. Midspan carries only wL²/24, half as much again the other way.

Froude Number (Open Channel)

Fr = \frac{V}{\sqrt{g \, D}}

Fluid MechanicsCivil & SurveyingWater & WastewaterFroude number of an open channel: the ratio of flow velocity to the speed of a shallow-water wave, with D the hydraulic depth (flow area divided by top width).

Grade to Slope Angle

\theta = \arctan\!\left(\frac{G}{100}\right)

Civil & SurveyingTrigonometryConverts a percent grade into the slope angle measured from horizontal, and back again with the tangent function.

High or Low Point on a Vertical Curve

x_t = -\frac{g_1 L}{A}

Civil & SurveyingDistance from the BVC to the crest or sag of a parabolic vertical curve, where the profile grade passes through zero.

Horizontal Curve External Distance

E = R\left(\sec\frac{\Delta}{2} - 1\right)

Civil & SurveyingTrigonometryClearance from the point of intersection to the midpoint of the curve, the distance a curve cuts back from the corner.

Horizontal Curve Length from Degree of Curve

L = \frac{100\,\Delta}{D}

Civil & SurveyingGeometryLength of a circular curve in 100 ft stations, from the total deflection angle and the degree of curve, on the arc definition.

Horizontal Curve Long Chord

C = 2R\sin\frac{\Delta}{2}

Civil & SurveyingTrigonometryStraight-line distance from the point of curvature to the point of tangency, the chord that spans the entire circular curve.

Horizontal Curve Middle Ordinate

M = R\left(1 - \cos\frac{\Delta}{2}\right)

Civil & SurveyingTrigonometryOffset from the middle of the long chord to the middle of the arc, the number that governs sight distance around obstructions.

Horizontal Curve Tangent Length

T = R\tan\frac{\Delta}{2}

Civil & SurveyingTrigonometryDistance from the point of intersection back to the point of curvature, from the curve radius and its total deflection angle.

Hydraulic Jump Conjugate Depths

\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^{2}} - 1\right)

Fluid MechanicsCivil & SurveyingWater & WastewaterBelanger's momentum solution for a hydraulic jump in a rectangular channel: the sequent (conjugate) depth downstream of the jump from the depth and Froude number upstream of it.

Hydraulic Radius

R = \frac{A}{P}

Fluid MechanicsCivil & SurveyingWater & WastewaterHydraulic radius of any conduit or channel: the flow area divided by the wetted perimeter, the single geometric number that Manning, Chezy and every friction equation actually consume.

Kirpich Time of Concentration

t_c = 0.0078 \, L^{0.77} \, S^{-0.385}

Water & WastewaterCivil & SurveyingTime of concentration of a small rural catchment by Kirpich's 1940 fit, published for channel length in feet and returned here in minutes whatever units you type.

Latitude of a Traverse Leg

\text{Lat} = L\cos\alpha

Civil & SurveyingTrigonometryNorth-south component of a traverse course, from the measured slope-corrected distance and the azimuth of the line.

LRFD Load Combination (1.2D + 1.6L)

U = 1.2 D + 1.6 L

Civil & SurveyingThe everyday strength-design load combination from ASCE 7 (combination 2) and the National Building Code family: dead load at 1.2 plus live load at 1.6.

Max Moment — Simple Beam, Off-Centre Point Load

M = \frac{P a (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsMaximum bending moment in a simply supported beam with a single point load at distance a from one support, M = Pab/L, occurring directly under the load.

Maximum In-Plane Shear Stress

\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsRadius of Mohr's circle: the largest shear stress on any plane through a plane-stress element, equal to half the difference of the two principal stresses.

Maximum Principal Stress (Mohr's Circle)

\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsLarger of the two principal stresses for a plane-stress element, from the normal stresses σx and σy and the shear τxy — the circle's centre plus its radius.

Minimum Principal Stress (Mohr's Circle)

\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsSmaller of the two principal stresses for a plane-stress element — the circle's centre minus its radius. Often the compressive one, and the value brittle materials care about least.

Moment of Inertia — I-Beam or Built-Up Section

I = \frac{B H^{3} - (B - t_w)(H - 2t_f)^{3}}{12}

Strength of MaterialsCivil & SurveyingGeometryStrong-axis second moment of area of a doubly symmetric I-shape, taken as the full bounding rectangle minus the two rectangular voids beside the web.

Muskingum Channel Storage

S = K\left[X I + (1 - X) O\right]

Water & WastewaterCivil & SurveyingFluid MechanicsStorage held in a river reach by the Muskingum method: a prism term set by the outflow plus a wedge term set by the inflow, weighted by X and scaled by the travel time K.

Parallel Axis Theorem (I = I_c + Ad²)

I = I_c + A d^{2}

Strength of MaterialsCivil & SurveyingGeometrySecond moment of area of a shape about an axis parallel to its own centroidal axis: add A times the offset squared. The transfer term that builds every plate girder and flitch beam.

Percent Grade from Rise and Run

G = \frac{100\,\Delta h}{L}

Civil & SurveyingGeometryPercent grade is one hundred times the vertical rise divided by the horizontal run, measured level, never along the slope.

Plastic Moment Capacity (Mp = Z fy)

M_p = Z f_y

Strength of MaterialsCivil & SurveyingMoment at which a compact steel section is fully plastic, with every fibre at yield. The nominal flexural strength Mn used in LRFD steel design before the φ factor.

Plastic Section Modulus — Rectangle

Z = \frac{b h^{2}}{4}

Strength of MaterialsCivil & SurveyingGeometryPlastic section modulus of a solid rectangle, the first moment of the two half-areas about the equal-area axis. Exactly 1.5 times the elastic section modulus bh²/6.

Radius from Degree of Curve (Arc Definition)

R = \frac{5729.578}{D}

Civil & SurveyingGeometryConverts degree of curve to radius using the arc definition, where D is the central angle subtending one 100 ft station of arc.

Rational Method Peak Runoff

Q = \frac{C \, i \, A}{3.6 \times 10^{6}}

Water & WastewaterCivil & SurveyingFluid MechanicsPeak runoff from a small catchment: the runoff coefficient times the rainfall intensity times the drainage area, in the SI form that takes intensity in mm/h and area in square metres.

Reinforced Concrete Nominal Moment Capacity

M_n = A_s f_y \left(d - \frac{a}{2}\right)

Civil & SurveyingStrength of MaterialsNominal flexural strength of a singly reinforced concrete section: the tension force in the rebar times the internal lever arm to the centroid of the Whitney compression block.

SCS Curve Number Runoff Depth

Q = \frac{(P - 0.2S)^2}{P + 0.8S}, \quad S = \frac{25400}{CN} - 254

Water & WastewaterCivil & SurveyingDirect runoff depth from a storm by the SCS (NRCS) curve-number method, with the potential maximum retention S in millimetres taken from the curve number and the initial abstraction fixed at 0.2S.

SCS Lag Time of Concentration

t_c = \frac{L^{0.8} \, (S + 1)^{0.7}}{1140 \, Y^{0.5}}

Water & WastewaterCivil & SurveyingTime of concentration from the SCS watershed lag equation, in which the curve number carries the retardance: lag equals flow length to the 0.8 power times the retention term over 1900 times the root of the percent slope, and the time of concentration is the lag divided by 0.6.

SCS Triangular Unit Hydrograph Peak

q_p = \frac{2 \, A \, Q}{2.67 \, T_p}

Water & WastewaterCivil & SurveyingFluid MechanicsPeak discharge of the SCS synthetic triangular unit hydrograph, from the runoff volume spread over a base length of 2.67 times the time to peak.

Seismic Base Shear (V = Cs W)

V = C_s W

Civil & SurveyingTotal lateral force an earthquake is designed to put into a building's base, from the ASCE 7 equivalent lateral force procedure: a seismic response coefficient times the effective seismic weight.

Shear Flow (q = VQ/I)

q = \frac{V Q}{I}

Strength of MaterialsCivil & SurveyingMechanicsLongitudinal shear force per unit length that must cross a joint in a built-up beam — the number that sets nail spacing, bolt pitch and weld size.

Shrinkage: Compacted Volume from Bank Volume

V_C = V_B\left(1 - \frac{S_h}{100}\right)

Civil & SurveyingSoil MechanicsConverts bank volume into the compacted volume it fills in an engineered embankment, using the soil's percent shrinkage.

Slope Ratio (H:V) to Percent Grade

G = \frac{100}{n}

Civil & SurveyingSoil MechanicsConverts an embankment slope quoted as n horizontal to one vertical into the equivalent percent grade, and back.

Specific Capacity of a Well

S_c = \frac{Q}{s}

Water & WastewaterCivil & SurveyingSpecific capacity of a water well: the pumping rate divided by the drawdown it produces, reported in cubic metres per day per metre of drawdown.

Specific Energy in an Open Channel

E = y + \frac{V^2}{2g}

Fluid MechanicsCivil & SurveyingWater & WastewaterSpecific energy: the total head measured from the channel bed, being the flow depth plus the velocity head. It is the quantity that stays constant across a smooth transition.

Stadia Distance from Rod Intercept

D = K\,s + C

Civil & SurveyingHorizontal distance from a stadia rod intercept, with the instrument's stadia interval factor and additive constant.

Stockpile Volume (Truncated Pyramid)

V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)

Civil & SurveyingGeometryVolume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer is reported in cubic yards.

Stopping Sight Distance

d = v\,t_r + \frac{v^{2}}{2a}

Civil & SurveyingDistance to stop from a design speed, adding the reaction distance travelled before braking to the braking distance itself.

Submerged Culvert Inlet (Orifice) Headwater

Q = C_d \, A \sqrt{2g\left(HW - \tfrac{D}{2}\right)}

Civil & SurveyingFluid MechanicsWater & WastewaterCapacity of a culvert in inlet control with the inlet submerged, treated as an orifice discharging under the head between the headwater surface and the centroid of the barrel opening.

Superelevation Rate for a Horizontal Curve

\frac{e}{100} + f = \frac{v^{2}}{g\,R}

Civil & SurveyingBalances banking and side friction against centripetal demand on a horizontal curve, using standard gravity of 9.80665 m/s².

Support Reaction — Simple Beam, Off-Centre Point Load

R_A = \frac{P (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsReaction at the near support of a simply supported beam with one point load at distance a from it. The far reaction is the remainder, P − R_A = Pa/L.

Swell: Loose Volume from Bank Volume

V_L = V_B\left(1 + \frac{S}{100}\right)

Civil & SurveyingSoil MechanicsConverts undisturbed bank volume into the loose volume the same soil occupies once excavated, using its percent swell.

Thiem Steady-State Well Drawdown

s = \frac{Q}{2 \pi T} \ln\frac{R}{r}

Water & WastewaterCivil & SurveyingSteady-state drawdown around a well fully penetrating a confined aquifer, by the Dupuit-Thiem equilibrium solution. Transmissivity is taken in square metres per day and the pumping rate is converted to a daily volume inside the brain.

Transmissivity from Conductivity and Thickness

T = K \, b

Water & WastewaterCivil & SurveyingTransmissivity of a confined aquifer as the hydraulic conductivity multiplied by the saturated thickness, returned in square metres per day for a conductivity entered in any speed unit.

Transverse Shear Stress (τ = VQ/Ib)

\tau = \frac{V Q}{I b}

Strength of MaterialsCivil & SurveyingMechanicsShear stress at any height in a beam cross-section, peaking at the neutral axis. For a rectangle it works out to exactly 1.5 times the average V/A.

Traverse Closure Error

E_c = \sqrt{\left(\Sigma\text{Lat}\right)^{2} + \left(\Sigma\text{Dep}\right)^{2}}

Civil & SurveyingGeometryLinear misclosure of a closed traverse, combining the residual sums of the latitudes and departures as a right triangle.

Traverse Precision Ratio

N = \frac{P}{E_c}

Civil & SurveyingExpresses traverse accuracy as one part in N, dividing the total perimeter walked by the linear closure error.

Vertical Curve Length from K Value

L = K\,A

Civil & SurveyingLength of a parabolic vertical curve as the design K value times the algebraic grade change, K being length per percent of grade.

Wind Velocity Pressure (qz = 0.613 Kz Kzt Kd V²)

q_z = 0.613 \, K_z K_{zt} K_d V^{2}

Civil & SurveyingVelocity pressure at height z from the ASCE 7 wind provisions, SI form: 0.613 times the exposure, topographic and directionality factors times the basic wind speed squared, in pascals.