Engineering Formula Sheet

Engineering Formula SheetStatisticsModeMeanPlace data in ascending order.Mode most frequently occurring value xµ mean valueΣxi sum of all data values (x1, x2, x3, n number of data values (xMedianPlace data in ascending order.If n is odd, median central valueIf n is even, median mean of two central valuesStandard Deviation If two values occur at the maximum frequency thedata set is bimodal.If three or more values occur at the maximumfrequency the data set is multi-modal.)n number of data valuesσ standard deviationxi individual data value ( x1, x2, x3, Rangen number of data valuesxmax maximum data valuexmin minimum data valueRange xmax - xminProbabilityIndependent EventsP (A and B and C) PAPBPCFrequencyP (A and B and C) probability of independentevents A and B and C occurring in sequencePA probability of event AxxxxMutually Exclusive Eventsfx relative frequency of outcome xnx number of events with outcome xn total number of eventsPx probability of outcome xfa frequency of all eventsBinomial Probability (order doesn’t matter)P (A or B) PA PBP (A or B) probability of either mutually exclusiveevent A or B occurring in a trialPA probability of event AΣxi sum of all data values (x1, x2, x3, n number of data valuesConditional ProbabilityPk binomial probability of k successes in n trialsp probability of a successq 1 – p probability of failurek number of successesn number of trialsPLTW, Inc.( )( )( )( )( )( )( )P (A D) probability of event A given event DP(A) probability of event A occurringP( A) probability of event A not occurringP(D A) probability of event D given event A did not occurEngineering FormulasIED POEDECEAAEBECIM EDD1

Plane GeometryEllipseRectangle2bCirclePerimeter 2a 2bArea ab2aBTriangleParallelogramhArea bha b c – 2bc·cos A222b a c – 2ac·cos B222c a b – 2ab·cos CC2ch2AbsRegular PolygonsRight Triangle2a2b2Area ½ bhf2c a bcan number of sidesθbahTrapezoidArea ½(a b)hhhbhSolid GeometryCubeSpheres3Volume s2Surface Area 6sr3sVolumerSurface Area 4sr2Rectangular PrismCylinderrhVolume wdhSurface Area 2(wd wh dh)dwh2Volume r hSurface Area 2r h 2r2Right Circular ConehIrregular Prismr hVolume AhA area of basePyramidhA area of basePLTW, Inc.Constants2g 9.8 m/s 32.27 ft/s-1132G 6.67 x 10 m /kg·sπ 3.14159Engineering FormulasIED POEDE2CEAAEBECIM EDD2

ConversionsMassForceArea21 acre 4047 m2 43,560 ft2 0.00156 mi1 kg 2.205 lbm1 slug 32.2 lbm1 ton 2000 lbm1N1 kipEnergy 0.225 lbf 1,000 lbf1J 0.239 cal-4 9.48 x 10 Btu 0.7376 ft·lbf1kW h 3,6000,000 JPressureLength1m1 km1 in.1 mi1 yd1 atmVolume 3.28 ft 0.621 mi 2.54 cm 5280 ft 3 ft1L1mL 0.264 gal3 0.0353 ft 33.8 fl oz3 1 cm 1 cc1psi 1.01325 bar 33.9 ft H2O 29.92 in. Hg 760 mm Hg 101,325 Pa 14.7 psi 2.31 ft of H2ODefined Units1J1N1 Pa1V1W1W1 Hz1F1HTimeTemperature Change1K 1 ºC 1.8 ºF 1.8 ºR1d1h1 min1 yr 24 h 60 min 60 s 365 dPower1W 3.412 Btu/h 0.00134 hp 14.34 cal/min 0.7376 ft·lbf/s 1 N·m 1 kg·m / s2 1 N / m2 1W/A 1J/s 1V/A 1 s-1 1 A·s / V 1 V·s / VSI PrefixesNumbers Less Than OnePower of ozeptoyocto-EquationsdcmµnpfazyNumbers Greater Than OnePower of 11024TemperatureTK TC 273Mass and WeightM VDmTR TF MGTPEZYForceF maF forcem massa accelerationW mgW VDwV volumeDm mass densitym massDw weight densityg acceleration due to gravityPLTW, Inc.Equations of Static EquilibriumTK temperature in KelvinTC temperature in CelsiusTR temperature in RankinTF temperature in FahrenheitEngineering FormulasΣFx 0ΣFy 0ΣMP 0Fx force in the x-directionFy force in the y-directionMP moment about point PIED POEDECEAAEBECIM EDD3

Equations (Continued)Energy: WorkElectricityOhm’s LawFluid MechanicsV IRP IVW workF forced distanceRT (series) R1 R2 ··· Rn’L’ L(Guy-LPowerP1V1 P2V2P powerE energyW workt timeτ torquerpm revolutions per minuteEfficiencyyPout useful power outputPin total power inputB y ’ LKirchhoff’s Current LawQ AvIT I1 I2 ··· In orA1v1 A2v2Kirchhoff’s Voltage LawVT V1 V2 ··· Vn orabsolute pressure gauge pressure atmospheric pressureP absolute pressureF ForceA AreaV volumeT absolute temperatureQ flow ratev flow velocityV voltageVT total voltageI currentIT total currentR resistanceRT total resistanceP powerThermodynamics′Mechanics(where acceleration 0) T Energy: Potential(where acceleration 0)U potential energym massg acceleration due to gravityh heightLLA1v1 A2v2v v0 atEnergy: Kinetic2d d0 v0t ½at22v v0 2a(d – d0)K kinetic energym massv velocityEnergy: ThermalQ thermal energym massc specific heat T change in temperaturePLTW, Inc.τ dFsinθs speedv velocitya accelerationX ranget timed distanceg acceleration due to gravityd distanceθ angleτ torqueF forceEngineering FormulasP rate of heat transferQ thermal energyA Area of thermal conductivityU coefficient of heat conductivity(U-factor) T change in temperatureR resistance to heat flow ( R-value)k thermal conductivityv velocityPnet net power radiated 5.6696 x 10-8e emissivity constantT1, T2 temperature at time 1, time 2v flow velocityPOE 4 DE 4

Section PropertiesMoment of InertiaRectangle CentroidhxxxxbIxx moment of inertia of a rectangular sectionabout x-x axis x and y̅and y̅Right Triangle Centroidx̅and y̅Semi-circle CentroidComplex Shapes Centroidx̅x̅x̅ yy̅ x̅ xy̅ yxi x distance to centroid of shape iyi y distance to centroid of shape iAi Area of shape ix̅ xy̅ yStructural AnalysisMaterial PropertiesBeam FormulasReactionStress (axial)BLMomentDeflection stressF axial forceA cross-sectional areaLxBLMomentLxReaction strainL0 original lengthδ change in lengthMomentx(between loads)Deflectionx( L LMomentxE modulus of elasticity stress strainA cross-sectional areaF axial forceδ deformationPLTW, Inc.andDeformation: AxialδLδ deformationF axial forceL0 original lengthA cross-sectional areaE modulus of elasticityEngineering Formulas ) (at center)BL(at Point of Load)LDeflection(at((at center)BReactionModulus of Elasticity(at center)xDeflectionL(at point of load)LReactionStrain (axial)(at point of load)x() ()())Truss Analysis2J M RJ number of jointsM number of membersR number of reaction forcesPOE 5 AE 4 CEA 4

Simple MachinesInclined PlaneMechanical Advantage (MA)y (L)WedgeIMA Ideal Mechanical AdvantageAMA Actual Mechanical AdvantageDE Effort DistanceDR Resistance DistanceFE Effort ForceFR Resistance ForceLLeverScrew1stClassIMA Pitch 2ndClassC Circumferencer radiusPitch distance betweenthreadsTPI Threads Per Inch3rdClassCompound MachinesMATOTAL (MA1) (MA2) (MA3) . . .Wheel and AxleGears; Sprockets with Chains; and Pulleys withBelts RatiosEffort at Axle()Compound GearsBGRTOTAL ( ) (Effort at WheelPulley SystemsIMA Total number of strands of a single stringsupporting the resistanceIMA PLTW, Inc.)GR Gear Ratioin Angular Velocity - driverout Angular Velocity - drivenNin Number of Teeth - driverNout Number of Teeth - drivendin Diameter - driverdout Diameter - drivenin Torque - driverout Torque - drivenEngineering FormulasPOE 6

Structural DesignSteel Beam Design: ShearSteel Beam Design: MomentSpread Footing Designqnet qallowable - pfootingVn 0.6FyAwMn FyZxVa allowable shear strengthVn nominal shear strengthΩv 1.5 factor of safety for shearFy yield stressAw area of webMa allowable bending momentMn nominal moment strengthΩb 1.67 factor of safety forbending momentFy yield stressZx plastic section modulus aboutneutral axisStorm Water RunoffStorm Water DrainageQ CfCiA3Q peak storm water runoff rate (ft /s)Cf runoff coefficient adjustmentfactorC runoff coefficienti rainfall intensity (in./h)A drainage area (acres)Runoff CoefficientAdjustment FactorReturnPeriodCf1, 2, 5, 10 1.0251.1501.21001.25Water SupplyHazen-Williams FormulaLhf head loss due to friction (ft of H2O)L length of pipe (ft)Q water flow rate (gpm)C Hazen-Williams constantd diameter of pipe (in.)Dynamic HeadRational Method Runoff CoefficientsCategorized by 7—0.85Concrete0.8—0.95Shingle roof0.75—0.95Lawns, well drained (sandy soil)Up to 2% slope0.05—0.12% to 7% slope0.10—0.15Over 7% slope0.15—0.2Lawns, poor drainage (clay soil)Up to 2% slope0.13—0.172% to 7% slope0.18—0.22Over 7% orized by 1—0.3Parks0.1—0.25Cemeteries0.1—0.25Railroad yard0.2—0.40Playgrounds0.2—0.35(except asphaltor y .9qnet net allowable soilbearing pressureqallowable total allowable soilbearing pressurepfooting soil bearing pressuredue to footing weighttfooting thickness of footingq soil bearing pressureP column load appliedA area of footingdynamic head static head – head lossPLTW, Inc.Engineering FormulasCEA 5

PLTW, Inc.Engineering FormulasCEA 6Equivalent Length of (Generic) FittingsHazen-Williams Constants

555 Timer Design EquationsT 0.693 (RA 2RB)CBy yBT periodf frequencyRA resistance ARB resistance BC capacitanceBoolean AlgebraBoolean TheoremsCommutative LawConsensus TheoremsX 0 0X Y Y X̅X 1 XX Y Y X̅̅̅X X XAssociative Law̅X(YZ) (XY)ZX 0 XX (Y Z) (X Y) ZX 1 1̅̅̅̅̅̅̅̅DeMorgan’s TheoremsX X XDistributive Law̅̅̅̅̅ ̅ ̅̅X(Y Z) XY XZ̅̅̅̅̅̅̅ ̅ ̅̿̅(X Y)(W Z) XW XZ YW YZSpeeds and Feeds()fm ft·nt·NPlunge Rate ½·fmN spindle speed (rpm)CS cutting speed (in./min)d diameter (in.)fm feed rate (in./min)ft feed (in./tooth)nt number of teethPLTW, Inc.Engineering FormulasDE 5CIM 4

Aerospace EquationsPropulsion(Orbital Mechanics)Forces of Flight LLCL coefficient of liftCD coefficient of dragL liftD dragA wing areadensityRe Reynolds numberv velocityl length of fluid travel fluid viscosityF forcem massg acceleration due to gravityM momentd moment arm (distance fromdatum perpendicular to F)FN net thrustW air mass flowvo flight velocityvj jet velocityI total impulseFave average thrust forcet change in time (thrustduration)Fnet net forceFavg average forceFg force of gravityvf final velocitya accelerationt change in time (thrustduration)NOTE: Fave and Favg areeasily confused.Energy eccentricityb semi-minor axisa semi-major axisT orbital perioda semi-major axisgravitational parameterF force of gravity between twobodiesG universal gravitation constantM mass of central bodym mass of orbiting objectr distance between center of twoobjectsBer oulli’s L w()()PS static pressurev velocityyAtmosphere ParametersK kinetic energym massv velocityU gravitational potential energyG universal gravitation constantM mass of central bodym mass of orbiting objectR Distance center main body tocenter of orbiting objectE Total Energy of an orbitPLTW, Inc.Engineering Formulas[(()])T temperatureh heightp pressuredensityAE 5

PLTW, Inc. Engineering Formulas y Mechanical Advantage (MA) IMA Ideal Mechanical Advantage AMA Actual Mechanical Advantage D E Effort Distance D R Resistance Distance F E Effort Force F R Resistance Force Wedge L Pulley Systems N IMA Total number of strands of a single string supporting the resistance d d IMA Wheel and Axle Effort at Axle Effort at Wheel Lever 1st Class 2nd .