FE Civil formula finder

Find how steep a straight line is when you know two points on it.

$m$

slope of the line, rise over runy units per x unit (unitless when both axes share a unit)

$x_1$

x coordinate of the first pointany consistent unit

$y_1$

y coordinate of the first pointany consistent unit

$x_2$

x coordinate of the second pointany consistent unit

$y_2$

y coordinate of the second pointany consistent unit

Watch out: Subtract in the same order on top and bottom (point 2 minus point 1 both times), or the sign of the slope flips.

Find the straight-line distance between two points on a plane.

$d$

straight-line distance between the two pointssame unit as the coordinates

$x_1$

x coordinate of the first pointany consistent unit

$y_1$

y coordinate of the first pointany consistent unit

$x_2$

x coordinate of the second pointany consistent unit

$y_2$

y coordinate of the second pointany consistent unit

Solve any equation of the form ax² + bx + c = 0 for its two roots.

$x$

the roots, the values of x that make the equation equal zeroany consistent unit

$a$

coefficient of the x² term (cannot be zero)any consistent unit

$b$

coefficient of the x termany consistent unit

$c$

constant termany consistent unit

Watch out: Move everything to one side so it equals zero before reading a, b and c, and keep their signs; if b² minus 4ac is negative there are no real roots.

Solve a triangle that is not a right triangle when you know a side and its opposite angle plus one more side or angle.

$a$

length of side a, opposite angle Aany consistent length unit

$b$

length of side b, opposite angle Bany consistent length unit

$c$

length of side c, opposite angle Cany consistent length unit

$A$

interior angle opposite side a° or radians

$B$

interior angle opposite side b° or radians

$C$

interior angle opposite side c° or radians

Watch out: Each side must be paired with the angle across from it, and check that the calculator is in degree mode when the angles are in degrees.

Find the third side of a triangle from two sides and the angle between them, or find an angle when you know all three sides.

$c$

the side opposite angle C, the one you are solving forany consistent length unit

$a$

one of the two sides that form angle Cany consistent length unit

$b$

the other side that forms angle Cany consistent length unit

$C$

the angle between sides a and b° or radians

Watch out: C must be the angle between a and b (the included angle), not one of the other two angles.

Take the derivative of a function inside another function, like sin(3x²).

$f$

the outer functionunitless

$g(x)$

the inner function, which is fed into funitless

$x$

the variable you are differentiating with respect toany consistent unit

$f'(g(x))$

derivative of the outer function, evaluated at the inner functionunitless

$g'(x)$

derivative of the inner functionunitless

Watch out: Do not stop after differentiating the outer function; you still have to multiply by the derivative of the inside.

Integrate a product of two functions, like x times e^x or x times sin x.

$u$

the part you choose to differentiate, ideally one that gets simplerunitless

$dv$

the part you choose to integrateunitless

$du$

the derivative of u times dxunitless

$v$

the integral of dvunitless

Watch out: Pick u as the piece that gets simpler when differentiated (a power of x before e^x or sin x); the wrong pick makes the new integral harder.

Find the angle between two vectors, the projection of one onto the other, or the work done by a force.

$\vec{A}$

first vectorany consistent unit

$\vec{B}$

second vectorany consistent unit

$|A|$

magnitude (length) of vector Asame unit as A

$|B|$

magnitude (length) of vector Bsame unit as B

$\theta$

angle between the two vectors when placed tail to tail° or radians

Watch out: The result is a plain number, not a vector; with components it is simply ax·bx + ay·by + az·bz.

Find a vector perpendicular to two vectors, such as the moment of a force (r × F).

$\vec{A}$

first vectorany consistent unit

$\vec{B}$

second vectorany consistent unit

$|A|$

magnitude (length) of vector Asame unit as A

$|B|$

magnitude (length) of vector Bsame unit as B

$\theta$

angle between the two vectors when placed tail to tail° or radians

$\hat{n}$

unit vector perpendicular to the plane of A and B, direction set by the right-hand ruleunitless

Watch out: Order matters: A × B points the opposite way from B × A.

Predict what a spreadsheet cell formula returns when it mixes operations.

$\texttt{=}$

starts a formula, so the cell shows the result instead of the textnone

$\texttt{A1}$

the value in column A, row 1whatever the cell holds

$\texttt{A2}$

the value in column A, row 2whatever the cell holds

$\texttt{A3}$

the value in column A, row 3whatever the cell holds

$\texttt{+}$

addnone

$\texttt{*}$

multiplynone

Watch out: =A1+A2*A3 multiplies A2 by A3 first and then adds A1; you need =(A1+A2)*A3 to add first.

Make a spreadsheet cell return one value when a condition is true and another when it is false.

$\texttt{test}$

a comparison that is either true or false, like A1>100true or false

$\texttt{value if true}$

what the cell returns when the test is truewhatever the cell holds

$\texttt{value if false}$

what the cell returns when the test is falsewhatever the cell holds

Watch out: In nested IFs the first test that is true wins, so check the order of the conditions.

Find a root of f(x) = 0 numerically by improving a guess one step at a time.

$x_{j+1}$

the next, improved estimate of the rootsame unit as x

$x_j$

the current estimate of the rootsame unit as x

$f(x_j)$

the function evaluated at the current estimatesame unit as f

$f'(x_j)$

the derivative of the function evaluated at the current estimatef units per x unit

Watch out: The starting guess must be near the root, and the method fails where the derivative is zero.

Find the average of a set of measured values.

$\bar{x}$

sample mean, the average of the valuessame unit as the data

$x_i$

each individual value in the samplesame unit as the data

$n$

number of values in the sampleunitless (a count)

Measure how spread out a sample of values is around its mean.

$s$

sample standard deviationsame unit as the data

$x_i$

each individual value in the samplesame unit as the data

$\bar{x}$

sample meansame unit as the data

$n$

number of values in the sampleunitless (a count)

Watch out: A sample divides by n minus 1; a whole population divides by N, so pick sx, not σx, on the calculator for a sample.

Find the chance of exactly k successes in n independent tries, each with the same chance of success.

$P(X=k)$

probability of exactly k successesunitless, 0 to 1

$n$

number of independent trialsunitless (a count)

$k$

number of successes you are asking aboutunitless (a count)

$p$

probability of success on one trial (1 minus p is the probability of failure)unitless, 0 to 1

$\binom{n}{k}$

number of ways to choose k successes out of n trials, n! divided by k!(n minus k)!unitless

Watch out: The formula gives exactly k; an "at least" or "at most" question needs a sum of terms, or 1 minus the complement.

Convert a value from any normal distribution to the unit normal so you can read probabilities from the table.

$z$

number of standard deviations the value sits above (or below, if negative) the meanunitless

$x$

the value you are checkingsame unit as the data

$\mu$

population meansame unit as the data

$\sigma$

population standard deviationsame unit as the data

Watch out: The handbook table has four columns: F(x) is the area to the left, R(x) the area to the right, W(x) the area between minus x and x; read the one the question asks for.

Measure how closely two sets of paired data follow a straight line.

$r$

correlation coefficient, from minus 1 (perfect falling line) through 0 (no linear trend) to plus 1 (perfect rising line)unitless

$x_i$

each x value in the paired datasame unit as x

$y_i$

each y value paired with x_isame unit as y

$\bar{x}$

mean of the x valuessame unit as x

$\bar{y}$

mean of the y valuessame unit as y

Watch out: r is not the slope of the line, and the coefficient of determination R² is r squared.

Find what one amount of money today grows to after n periods at interest rate i.

$F$

future worth, the amount at the end of period ndollars

$P$

present worth, the amount today (time zero)dollars

$i$

interest rate per period, as a decimalunitless (per period)

$n$

number of compounding periodsperiods (years, months)

Watch out: i and n must use the same period, so a monthly rate goes with a number of months.

Find what one amount received in the future is worth today.

$P$

present worth, the value today (time zero)dollars

$F$

future amount, received at the end of period ndollars

$i$

interest rate per period, as a decimalunitless (per period)

$n$

number of compounding periodsperiods (years, months)

Watch out: i and n must use the same period, so a monthly rate goes with a number of months.

Turn an amount today into equal payments at the end of each period, like a loan payment.

$A$

uniform amount paid at the end of each perioddollars per period

$P$

present worth, the amount at time zerodollars

$i$

interest rate per period, as a decimalunitless (per period)

$n$

number of periods (payments)periods (years, months)

Watch out: The first payment A comes one period after P, not at time zero.

Decide whether a project is worth doing: it is justified when benefits are at least equal to costs.

$B$

benefits of the projectdollars (present or annual worth)

$C$

costs of the projectdollars (present or annual worth)

Watch out: B and C must be on the same basis (both present worth or both annual worth), and to choose between projects compare them incrementally, not by the biggest ratio.

Spread the loss in value of an asset evenly over its life.

$D_j$

depreciation in year j, the same every yeardollars per year

$\text{Cost}$

initial cost of the assetdollars

$\text{Salvage}$

expected salvage value at the end of its lifedollars

$n$

depreciable life of the assetyears

Watch out: Subtract the salvage value before dividing; book value in a later year is cost minus all depreciation taken so far.

Combine the interest rate and the inflation rate into one rate for finding present worth of inflated cash flows.

$d$

inflation-adjusted interest rate per periodunitless (per period)

$i$

interest rate per period, without inflationunitless (per period)

$f$

general inflation rate per periodunitless (per period)

Watch out: Multiply the decimals, not the percents: i = 5% and f = 3% give d = 0.05 + 0.03 + 0.0015 = 8.15%, not just 8%.

Find the interest rate at which a project just breaks even, then compare it with the minimum acceptable rate (MARR).

$PW_{benefits}$

present worth of all benefits (income) at the trial interest ratedollars

$PW_{costs}$

present worth of all costs at the trial interest ratedollars

Watch out: The unknown is the interest rate, so you solve by trying rates (or reading the factor tables) until the two sides match; accept the project when that rate is at least the MARR.

Find unknown forces and support reactions on a body that is not moving.

$F_x$

each force component in the x (horizontal) directionlbf, kips, N or kN

$F_y$

each force component in the y (vertical) directionlbf, kips, N or kN

$M$

each moment about a point you choose, including couplesft·lbf, ft·kips, N·m or kN·m

Watch out: Take moments about a point where unknown forces act so they drop out, and keep one sign convention for every term.

Find the centroid of a shape built from simple pieces like rectangles, triangles and circles.

$\bar{x}$

x location of the centroid of the whole shape, from your reference axisin, ft, mm or m

$A_i$

area of each simple piecein², ft², mm² or m²

$x_i$

x distance from the reference axis to the centroid of each piecein, ft, mm or m

Watch out: Treat holes and cut-outs as negative areas, and measure every x_i from the same reference axis.

Move a moment of inertia from a centroidal axis to any parallel axis, usually to build up a composite section.

$I_x$

moment of inertia about the new (parallel) axisin⁴, ft⁴, mm⁴ or m⁴

$\bar{I}_x$

moment of inertia about the axis through the shape's own centroidin⁴, ft⁴, mm⁴ or m⁴

$A$

area of the shapein², ft², mm² or m²

$d$

perpendicular distance between the centroidal axis and the new axisin, ft, mm or m

Watch out: You can only transfer from the centroidal axis; to go between two non-centroidal axes, first come back to the centroid.

Check whether a body will slide, or find the force that starts it sliding.

$F$

friction force along the contact surfacelbf or N

$\mu_s$

coefficient of static frictionunitless

$N$

normal force, pressing the two surfaces together, perpendicular to the contactlbf or N

Watch out: F equals μs·N only when slip is about to happen; on a slope N is W cos θ, not the full weight W.

Find the turning effect of a force about a point.

$M$

moment of the force about the pointft·lbf, ft·kips, N·m or kN·m

$r$

position vector from the point to any point on the force's line of actionft, in, m or mm

$F$

the forcelbf, kips, N or kN

Watch out: In the scalar form M = F·d, d is the perpendicular distance to the line of action, not the straight distance to where the force is applied.

Find the force a lever or simple machine delivers, from moment balance about the pivot.

$F_{out}$

force the lever delivers (output, or load)lbf or N

$F_{in}$

force applied to the lever (input, or effort)lbf or N

$a_{in}$

moment arm of the input force, perpendicular distance from the pivotin, ft, mm or m

$a_{out}$

moment arm of the output force, perpendicular distance from the pivotin, ft, mm or m

Watch out: A longer input arm gives a bigger output force; flipping the ratio is the usual slip.

Find speed after a given time when acceleration does not change.

$v$

velocity at time tft/s or m/s

$v_0$

initial velocity, at time zeroft/s or m/s

$a$

constant acceleration (g for free fall)ft/s² or m/s²

$t$

time elapseds

Watch out: Only valid when acceleration is constant; for free fall pick a direction as positive and give g its sign.

Find position after a given time when acceleration does not change.

$s$

position at time t, along the line of travelft or m

$s_0$

initial position, at time zeroft or m

$v_0$

initial velocity, at time zeroft/s or m/s

$a$

constant acceleration (g for free fall)ft/s² or m/s²

$t$

time elapseds

Watch out: Only valid when acceleration is constant; use g = 32.2 ft/s² or 9.81 m/s² with the sign of your chosen direction.

Relate speed and distance when time is not given, like a braking or stopping distance.

$v$

velocity at position sft/s or m/s

$v_0$

initial velocity, at position s_0ft/s or m/s

$a$

constant acceleration (negative when slowing down)ft/s² or m/s²

$s$

final positionft or m

$s_0$

initial positionft or m

Watch out: Convert mph or km/h to ft/s or m/s first (1 mph = 1.467 ft/s), and give deceleration a negative sign.

Find the acceleration a set of forces gives a body, or the force needed for a given acceleration.

$\sum F$

sum of all forces acting on the body, in the direction of motionlbf or N

$m$

mass of the bodyslug (lbf·s²/ft) or kg

$a$

acceleration of the bodyft/s² or m/s²

Watch out: In US units a weight in lbf is not a mass; divide by g = 32.2 ft/s² to get slugs.

Relate speed and height (or spring stretch) at two points when no energy is lost to friction.

$T_1$

kinetic energy at state 1, one half m v²ft·lbf or J

$V_1$

potential energy at state 1, from height (mgh) and springs (one half k s²)ft·lbf or J

$T_2$

kinetic energy at state 2ft·lbf or J

$V_2$

potential energy at state 2ft·lbf or J

Watch out: With friction or other non-conservative forces, add the work term U1→2 (negative for friction) to the right side.

Find velocities after two bodies collide, when no outside force acts during the impact.

$m_1$

mass of body 1slug or kg

$m_2$

mass of body 2slug or kg

$v_1$

velocity of body 1 just before impactft/s or m/s

$v_2$

velocity of body 2 just before impactft/s or m/s

$v_1'$

velocity of body 1 just after impactft/s or m/s

$v_2'$

velocity of body 2 just after impactft/s or m/s

Watch out: Velocities carry signs, so a body moving the other way is negative; momentum is conserved but kinetic energy is not, unless the impact is perfectly elastic (e = 1).

Find how fast an undamped spring and mass system vibrates on its own.

$\omega_n$

undamped natural circular frequencyrad/s

$k$

spring stiffness (spring constant)lbf/ft or N/m

$m$

vibrating massslug or kg

Watch out: The answer is in rad/s; divide by 2π for cycles per second (Hz), and use mass, not weight, for m.

Stress in a bar, rod or column loaded straight along its axis, in tension or compression.

$\sigma$

normal stress on the cross section (tension positive)psi or ksi, Pa or MPa

$P$

axial load carried by the memberlb or kips, N or kN

$A$

cross-sectional area perpendicular to the loadin² or mm², m²

How much a bar stretches or shortens under an axial load while the material stays elastic.

$\delta$

change in length of the member (elongation or shortening)in. or mm

$P$

axial load in the memberlb or kips, N or kN

$L$

original length of the memberin. or mm

$A$

cross-sectional areain² or mm²

$E$

modulus of elasticity (Young's modulus) of the materialpsi or ksi, MPa or GPa

Watch out: Keep every length in the same unit as E: with E in ksi, use kips and inches, not feet.

Maximum shear stress at the outer surface of a solid or thick-walled circular shaft carrying a torque.

$\tau$

shear stress at distance c from the center (the maximum when c is the outer radius)psi or ksi, MPa

$T$

torque (twisting moment) on the shaftin·lb or kip·in, N·m

$c$

radius from the shaft center to the point, usually the outer radiusin. or mm, m

$J$

polar moment of inertia of the cross section (πd⁴/32 for a solid shaft)in⁴ or mm⁴, m⁴

Watch out: c is the radius, not the diameter, and J is the polar moment (twice I for a circle), not I.

Maximum normal stress at the top or bottom fiber of a beam under a bending moment.

$\sigma$

bending (normal) stress at the extreme fiberpsi or ksi, MPa

$M$

bending moment at the sectionin·lb or kip·in, N·m or kN·m

$c$

distance from the neutral axis to the outermost fiberin. or mm

$I$

moment of inertia of the cross section about the neutral axisin⁴ or mm⁴

Watch out: Moments are often given in kip·ft; multiply by 12 to get kip·in before dividing by I in in⁴.

Shear stress at a given height inside a beam cross section caused by the internal shear force.

$\tau$

transverse shear stress at the level you are checkingpsi or ksi, MPa

$V$

internal shear force at the sectionlb or kips, N or kN

$Q$

first moment of the area above (or below) that level about the neutral axis, area times distance to its centroidin³ or mm³

$I$

moment of inertia of the whole cross section about the neutral axisin⁴ or mm⁴

$b$

width of the cross section at the level you are checkingin. or mm

Watch out: b is the width at the cut, so in an I-beam web use the web thickness, not the flange width.

Maximum deflection, at midspan, of a simply supported beam with one point load at the middle.

$\delta_{max}$

maximum deflection, at midspanin. or mm

$P$

concentrated load at midspanlb or kips, N or kN

$L$

span length between supportsin. or mm

$E$

modulus of elasticity of the beam materialpsi or ksi, MPa

$I$

moment of inertia of the beam cross sectionin⁴ or mm⁴

Watch out: L is cubed, so leaving the span in feet while E and I are in inches makes the answer 1,728 times too small.

Largest and smallest normal stresses at a point under 2D (plane) stress, on planes where the shear is zero.

$\sigma_{1,2}$

the two in-plane principal stresses (plus sign gives the larger)psi or ksi, MPa

$\sigma_x$

normal stress on the face perpendicular to x (tension positive)psi or ksi, MPa

$\sigma_y$

normal stress on the face perpendicular to y (tension positive)psi or ksi, MPa

$\tau_{xy}$

shear stress on the x and y facespsi or ksi, MPa

Watch out: Enter compression as a negative number; the square root term alone is the maximum in-plane shear stress.

Axial load at which a long, slender column buckles elastically.

$P_{cr}$

critical (buckling) axial loadlb or kips, N or kN

$E$

modulus of elasticity of the column materialpsi or ksi, MPa

$I$

moment of inertia about the axis the column buckles around, normally the weak axisin⁴ or mm⁴

$K$

effective-length factor for the end supports (pinned-pinned 1.0, fixed-pinned 0.7, fixed-fixed 0.5, fixed-free 2.0)unitless

$L$

unbraced length of the columnin. or mm

Watch out: Use the smaller moment of inertia unless the weak axis is braced, and K = 2.0 for a flagpole (fixed-free) column.

Bending stress in a beam made of two materials, such as steel plates on wood, after converting it to one material.

$\sigma_1$

bending stress in the stiffer Material 1psi or ksi, MPa

$n$

modular ratio, E₁ divided by E₂unitless

$M$

bending moment at the sectionin·lb or kip·in, N·m

$y$

distance from the neutral axis of the transformed section to the fiberin. or mm

$I_T$

moment of inertia of the transformed sectionin⁴ or mm⁴

$E_1$

modulus of elasticity of Material 1 (the stiffer one)psi or ksi, MPa

$E_2$

modulus of elasticity of Material 2psi or ksi, MPa

Watch out: Find the neutral axis on the transformed section first, and multiply by n only for stress in the stiffer material.

Bending strength of a compact steel beam when the whole section has yielded (the yielding limit state, Mn = Mp).

$M_p$

plastic moment, the moment when the entire cross section has yieldedkip·in or kip·ft, kN·m

$F_y$

specified minimum yield stress of the steel (50 ksi for A992)ksi or MPa

$Z$

plastic section modulus about the bending axis (Zx in the AISC tables)in³ or mm³

Watch out: Use Z, not the elastic section modulus S; S times Fy is only the first-yield moment, about 10 to 15% lower for a W shape.

Stress from a tension test, using the specimen area before it was loaded.

$\sigma$

engineering stresspsi or ksi, MPa

$F$

applied forcelb or kips, N or kN

$A_0$

original (initial) cross-sectional area of the specimenin² or mm²

Watch out: Use the original area; dividing by the shrunken (necked) area gives true stress, which is larger.

Link stress and strain while the material is still elastic (below the proportional limit).

$\sigma$

normal stresspsi or ksi, MPa

$E$

modulus of elasticity (Young's modulus), the slope of the straight part of the stress-strain curvepsi or ksi, MPa or GPa

$\varepsilon$

normal strain, change in length divided by original lengthunitless (in./in. or mm/mm)

Watch out: Strain is often given in percent or microstrain; convert to a plain decimal before multiplying by E.

Quick estimate of the tensile strength of a plain carbon steel from its Brinell hardness number.

$TS$

ultimate tensile strength of the steelMPa (the US form is TS in psi ≈ 500 × BHN)

$BHN$

Brinell hardness number from the indentation testunitless (a hardness index)

Watch out: The factor near 3.5 gives MPa; for psi the handbook uses 500 × BHN, and both apply to plain carbon steels only.

Check whether a crack of a given size will run (fast fracture) at a given applied stress in a brittle material.

$K_{IC}$

fracture toughness, the critical stress intensity at which the crack propagatesMPa·√m or ksi·√in.

$Y$

geometry factor (1.1 for an exterior edge crack, 1.0 for an interior crack)unitless

$\sigma$

applied engineering stressMPa or ksi

$a$

crack length for an edge crack, or half the total length for an interior crackm or in.

Watch out: An interior crack of total length 2a uses only a, and a must be in meters (or inches) to match the units of K.

Change in length of a member that is free to move when its temperature changes.

$\Delta L$

change in lengthin. or mm

$\alpha$

coefficient of thermal expansion1/°F or 1/°C

$L$

original lengthin. or mm

$\Delta T$

change in temperature (final minus initial)°F or °C

Watch out: Match α to the temperature unit: a value per °C used with a change in °F is off by a factor of 1.8.

The main factor that controls concrete strength: lower W/C gives stronger, more durable concrete.

$W/C$

weight of mixing water divided by weight of cement in the same batchunitless

Watch out: It is a ratio by weight, not volume; adding water without adding cement raises W/C and lowers strength.

Stiffness of a fiber composite loaded parallel to long, aligned fibers.

$E_c$

Young's modulus of the compositepsi or ksi, GPa

$f_1$

volume fraction of material 1unitless (0 to 1)

$E_1$

Young's modulus of material 1psi or ksi, GPa

$f_2$

volume fraction of material 2 (f₁ + f₂ = 1)unitless (0 to 1)

$E_2$

Young's modulus of material 2psi or ksi, GPa

Watch out: Use volume fractions, not weight fractions; this is the upper bound, and loading across the fibers uses the inverse rule (1/E_c = Σ f/E).

Bulk (oven-dry) specific gravity of coarse aggregate from the soaking test (ASTM C127), used in concrete and asphalt mix design.

$G_{sb}$

bulk specific gravity of the aggregate on an oven-dry basisunitless

$A$

mass of the oven-dry sample in airg or lb

$B$

mass of the saturated surface-dry (SSD) sample in airg or lb

$C$

mass of the saturated sample weighed submerged in waterg or lb

Watch out: Putting B on top gives the SSD specific gravity and A/(A minus C) gives the apparent one, so check which numerator the question wants.

How much water an aggregate soaks into its pores, used to correct mix water in concrete.

$\text{Absorption}$

water held in the pores at SSD, as a percent of the dry masspercent

$W_{SSD}$

weight of the sample in the saturated surface-dry conditiong or lb

$W_{OD}$

weight of the oven-dry sampleg or lb

Watch out: Divide by the oven-dry weight, not the SSD weight.

Percent of a compacted asphalt mix volume that is air, the main Superpave design check (target about 4%).

$V_a$

air voids in the compacted mix (voids in total mix)percent

$G_{mm}$

theoretical maximum specific gravity of the loose mix with no air (Rice value)unitless

$G_{mb}$

bulk specific gravity of the compacted mix specimenunitless

Watch out: G_mm is always the larger number; swapping the two gives a negative air void value.

Percent of the space between aggregate particles (VMA) that is filled with asphalt binder rather than air.

$VFA$

voids filled with asphaltpercent

$VMA$

voids in the mineral aggregate, the space between aggregate particles (air plus effective binder)percent of total mix volume

$V_a$

air voids in the compacted mixpercent of total mix volume

Watch out: Enter VMA and V_a as percents (16, not 0.16) so the factor of 100 gives a percent.

Water in a piece of wood as a percent of its oven-dry weight, which controls its strength and shrinkage.

$MC\%$

moisture content on an oven-dry basispercent

$W_{wet}$

weight of the wood as tested, with its moistureg or lb

$W_{OD}$

weight of the same wood after oven dryingg or lb

Watch out: Divide by the oven-dry weight, so green wood can show a moisture content above 100%.

Convert a fluid density into weight per unit volume, the quantity used in pressure and force problems.

$\gamma$

specific weight, weight per unit volume (water is 62.4 lbf/ft³ or 9.81 kN/m³)lbf/ft³ or N/m³

$\rho$

density, mass per unit volume (water is 1.94 slug/ft³ or 1,000 kg/m³)slug/ft³ or kg/m³

$g$

acceleration of gravity32.2 ft/s² or 9.81 m/s²

Watch out: In US units use density in slug/ft³; 62.4 is the specific weight of water, not its density in slugs.

Shear stress in a Newtonian fluid layer from its velocity gradient, for example oil between a moving plate and a fixed one.

$\tau$

shear stress in the fluidlbf/ft² or Pa (N/m²)

$\mu$

absolute (dynamic) viscositylbf·s/ft² or Pa·s (N·s/m²)

$\frac{dv}{dy}$

velocity gradient, change in velocity per distance normal to the wall1/s

Watch out: For a thin film with a straight-line profile, dv/dy is the plate speed divided by the film thickness, and use μ, not the kinematic viscosity ν.

Pressure change between two points in a fluid at rest that differ in depth.

$P_2$

pressure at the lower pointpsf or psi, Pa or kPa

$P_1$

pressure at the upper pointpsf or psi, Pa or kPa

$\gamma$

specific weight of the fluidlbf/ft³ or N/m³

$h$

vertical depth of point 2 below point 1ft or m

Watch out: γh in lbf/ft³ times ft gives psf; divide by 144 for psi.

Total water force on a flat submerged surface such as a gate or wall (atmospheric pressure on both sides).

$F_R$

resultant hydrostatic force on the surfacelbf or kips, N or kN

$\gamma$

specific weight of the fluidlbf/ft³ or N/m³

$h_C$

vertical depth from the free surface to the centroid of the submerged areaft or m

$A$

area of the submerged surfaceft² or m²

Watch out: h_C is the vertical depth, not the slant distance along an inclined gate, and the force acts at the center of pressure, below the centroid.

Velocity at one section of a pipe or channel from the velocity at another, for steady incompressible flow.

$A_1$

flow area at section 1ft² or m²

$v_1$

average velocity at section 1ft/s or m/s

$A_2$

flow area at section 2ft² or m²

$v_2$

average velocity at section 2ft/s or m/s

Watch out: Area goes with diameter squared, so halving a pipe diameter makes the velocity four times larger, not two.

Trade pressure, velocity and elevation head between two points along a flow with no friction loss and no pump or turbine.

$P_1$

pressure at point 1lbf/ft² or Pa

$P_2$

pressure at point 2lbf/ft² or Pa

$\gamma$

specific weight of the fluidlbf/ft³ or N/m³

$v_1$

average velocity at point 1ft/s or m/s

$v_2$

average velocity at point 2ft/s or m/s

$g$

acceleration of gravity32.2 ft/s² or 9.81 m/s²

$z_1$

elevation of point 1 above a chosen datumft or m

$z_2$

elevation of point 2 above the same datumft or m

Watch out: Use the same pressure basis at both points (a free surface or open jet is 0 gauge), and add head loss or pump head when the problem has them.

Tell whether pipe flow is laminar or turbulent, and read the Moody chart.

$Re$

Reynolds number, inertial forces over viscous forces (laminar below 2,100, turbulent above 10,000 in a pipe)unitless

$\rho$

fluid densityslug/ft³ or kg/m³

$v$

average flow velocityft/s or m/s

$D$

pipe diameter, or characteristic lengthft or m

$\mu$

dynamic viscositylbf·s/ft² or Pa·s

$\nu$

kinematic viscosity, μ divided by ρft²/s or m²/s

Watch out: Use ρvD/μ with dynamic viscosity or vD/ν with kinematic viscosity, never mix the two, and convert D from inches to feet.

Friction head loss along a straight pipe for any fluid and any flow regime.

$h_f$

head loss due to frictionft or m

$f$

Darcy (Moody) friction factor, from the Moody chart using Re and ε/Dunitless

$L$

length of pipeft or m

$D$

inside diameter of the pipeft or m

$v$

average velocity in the pipeft/s or m/s

$g$

acceleration of gravity32.2 ft/s² or 9.81 m/s²

Watch out: f here is the Darcy factor, four times the Fanning factor; for laminar flow f = 64/Re.

Force needed to change the speed or direction of a flowing stream, such as at a pipe bend, nozzle or jet hitting a plate.

$\Sigma F$

sum of external forces on the fluid in the control volume, in one directionlbf or N

$\rho$

fluid densityslug/ft³ or kg/m³

$Q$

volumetric flow rateft³/s or m³/s

$v_2$

velocity component leaving the control volume, in the direction of the forceft/s or m/s

$v_1$

velocity component entering the control volume, in the same directionft/s or m/s

Watch out: Write it separately for x and y with signed components; it gives the force on the fluid, and the force on the bend is equal and opposite.

Break a traverse course into its north-south (latitude) and east-west (departure) components.

$\text{Lat}$

latitude, the north-south component of the course (north positive)ft or m

$\text{Dep}$

departure, the east-west component of the course (east positive)ft or m

$L$

horizontal length of the courseft or m

$\theta$

direction of the course measured from north (bearing angle or azimuth)degrees

Watch out: The angle is measured from north, so latitude uses cosine; with bearings, set the signs from the quadrant (S is negative latitude, W is negative departure).

Area enclosed by a closed traverse when you know the coordinates of every corner.

$A$

area enclosed by the polygonft² or m²

$x_i$

x coordinate (easting) of corner ift or m

$y_i$

y coordinate (northing) of corner ift or m

$x_{i+1}$

x coordinate of the next corner around the polygonft or m

$y_{i+1}$

y coordinate of the next corner around the polygonft or m

Watch out: List the corners in order around the shape and close back to the first one; divide square feet by 43,560 for acres.

Earthwork volume between two cross sections a known distance apart.

$V$

volume of earth between the two sectionsft³ or m³

$L$

distance between the two cross sectionsft or m

$A_1$

cross-sectional area at the first stationft² or m²

$A_2$

cross-sectional area at the second stationft² or m²

Watch out: Earthwork answers are usually in cubic yards, so divide cubic feet by 27.

More accurate earthwork volume between two cross sections when the area at the midpoint is also known.

$V$

volume of earth between the two end sectionsft³ or m³

$L$

distance between the two end sectionsft or m

$A_1$

cross-sectional area at the first endft² or m²

$A_m$

cross-sectional area at the mid-section, halfway between the endsft² or m²

$A_2$

cross-sectional area at the second endft² or m²

Watch out: A_m is the area of the actual middle section (from averaged dimensions), not the average of A_1 and A_2.

Distance from the PC (or PT) to the PI of a simple circular horizontal curve.

$T$

tangent distance, from the PC or PT to the PIft or m

$R$

radius of the curveft or m

$I$

intersection (deflection) angle between the two tangents, also written Δdegrees

Watch out: Use half the intersection angle; stationing to the PT runs along the curve (PC plus L), not along the tangents.

Convert between curve radius and degree of curve (arc definition) in US highway work.

$D$

degree of curve, the central angle that subtends a 100 ft arcdegrees

$R$

radius of the curveft

Watch out: The constant 5,729.58 only works with R in feet and a 100 ft arc; it does not apply in meters.

Elevation of any point on a parabolic vertical curve at a horizontal distance x from the PVC.

$Y$

elevation of the curve at distance xft or m

$Y_{PVC}$

elevation of the PVC, the start of the curveft or m

$g_1$

grade of the back (entering) tangent, percent divided by 100ft/ft (decimal)

$g_2$

grade of the forward (leaving) tangent, percent divided by 100ft/ft (decimal)

$L$

horizontal length of the curve, PVC to PVTft or m

$x$

horizontal distance from the PVC to the pointft or m

Watch out: Grades as decimals need L and x in feet; if you keep grades in percent, L and x must be in stations (100 ft).

Distance and direction between two points when you know their coordinates.

$L$

horizontal distance between the two pointsft or m

$\Delta E$

change in easting, second point minus firstft or m

$\Delta N$

change in northing, second point minus firstft or m

$Az$

azimuth from the first point to the second, clockwise from northdegrees (0 to 360)