WIXLIB

M 22-24Highway EngineeringField Formulas

Metric (SI) or US UnitsUnless otherwise stated the formulas shownin this manual can be used with any units.The user is cautioned not to mix units withina formula. Convert all variables to one unitsystem prior to using these formulas.Significant DigitsFinal answers from computations should berounded off to the number of decimal placesjustified by the data. The answer can be nomore accurate than the least accuratenumber in the data. Of course, roundingshould be done on final calculations only. Itshould not be done on interim results.Persons with disabilities may request thisinformation be prepared in alternate formsby calling collect (360) 664-9009. Deaf andhearing impaired people call 1-800-833-6388(TTY Relay Service).1998Engineering PublicationsTransportation BuildingOlympia, WA 98504360-705-7430

CONTENTSNomenclature for Circular Curves .Circular Curve Equations .Simple Circular Curve .Degrees of Curvature to Various Radii .Nomenclature for Vertical Curves .Vertical Curve Equations .Nomenclature for Nonsymmetrical Curves .Nonsymmetrical Vertical Curve Equations .Determining Radii of Sharp Curves .Dist. from Fin. Shld. to Subgrade Shld. .Areas of Plane Figures .Surfaces and Volumes of Solids .Trigonometric Functions for all Quadrants .Trigonometric Functions .Right Triangle .Oblique Triangle .Conversion Factors .Metric Conversion Factors .Land Surveying Conversion Table .Steel Tape Temperature Corrections .Temperature Conversion .Less Common Conversion Factors .Water Constants .Cement Constants .Multiplication Factor Table .Recommended Pronunciations .Reinforcing Steel .245678101112131418232425262830313131323232333334

Nomenclature ForCircular CurvesPOTPoint On Tangent outside theeffect of any curvePOCPoint On a circular CurvePOSTPoint On a Semi-Tangent (withinthe limits of a curve)PIPoint of Intersection of a backtangent and forward tangentPCPoint of Curvature - Point ofchange from back tangent tocircular curvePTPoint of Tangency - Point ofchange from circular curve toforward tangentPCCPoint of Compound Curvature Point common to two curves in thesame direction with different radiiPRCPoint of Reverse Curve - Pointcommon to two curves in oppositedirections and with the same ordifferent radiiLTotal Length of any circular curvemeasured along its arcLcLength between any two points ona circular curveRRadius of a circular curve Total intersection (or central) anglebetweenbackandforwardtangents2

Nomenclature ForCircular Curves (Cont.)DCDeflection angle for full circularcurve measured from tangent atPC or PTdcDeflection angle required fromtangent to a circular curve to anyother point on a circular curveCTotal Chord length, or long chord,for a circular curveC Chord length between any twopoints on a circular curveTDistance along semi-Tangent fromthe point of intersection of theback and forward tangents to theorigin of curvature (From the PI tothe PC or PT)txDistance along semi-tangent fromthe PC (or PT) to the perpendicularoffset to any point on a circularcurve. (Abscissa of any point on acircular curve referred to thebeginning of curvature as originand semi-tangent as axis)tyThe perpendicular offset, orordinate, from the semi-tangent toa point on a circular curveEExternal distance (radial distance)from PI to midpoint on a simplecircular curve3

Circular Curve EquationsEquationsUnitsR 180 L π m or ft. 180 L π RdegreeL π R 180m or ft.T R tanRE cos 2 2C 2R sinm or ft. Rm or ft. , or 2R sin DC2m or ft. MO R 1 cos 2 DC dc m or ft. 2degreeLc L 2 degreeC' 2R sin( dc)m or ft.C 2 R sin( DC )m or ft.tx R sin( 2 dc)m or ft.ty R[1 cos( 2dc) ]m or ft.4

Simple Circular CurveConstant for π 3.141592655

Degree of Curvature forVarious Lengths of RadiiExact for Arc Definition180 100 π 18000D RπRWhere D is Degree of CurvatureLength of Radii for VariousDegrees of Curvature180 100 π 18000R DπDWhere R is Radius Length6

Nomenclature ForVertical CurvesG1 & G2 Tangent Grade in percentAThe absolute of the Algebraicdifference in grades in percentBVCBeginning of Vertical CurveEVCEnd of Vertical CurveVPIVertical Point of IntersectionLLength of vertical curveDHorizontal distance to any point onthe curve from BVC or EVCEVertical distance from VPI to curveeVertical distance from any point onthe curve to the tangent gradeKDistance required to achieve a 1percent change in gradeL1Length of a vertical curve which willpass through a given pointD0Distance from the BVC to thelowest or highest point on curveXHorizontal distance from P' to VPIHA point on tangent grade G1 tovertical position of point P'P and P'Points on tangent grades7

Symmetrical VerticalCurve EquationsA (G 2 ) (G1 )E AL800E 1 Elev.BVC Elev.EVC Elev. VPI 22e 4ED2L2Notes: All equations use units of length (notstations or increments)The variable A is expressed as an absolutein percent (%)Example:If G 1 4% and G 2 -2%Then A 68

Symmetrical Vertical CurveEquations (cont.)e AD 2200LL1 2( AX 200 e 20 AXe 100e2 )AD 0 G1LAX 100( ElevH ElevP' )AK LA9

Nomenclature ForNonsymmetrical VerticalCurvesG1 & G2 Tangent Grades in percentAThe absolute of the Algebraicdifference in grades in percentBVCBeginning of Vertical CurveEVCEnd of Vertical CurveVPIVertical Point of Intersectionl1Length of first section of verticalcurvel2Length of secondvertical curveLLength of vertical curveD1Horizontal distance to any point onthe curve from BVC towards theVPID2Horizontal distance to any point onthe curve from EVC towards theVPIe1Vertical distance from any point onthe curve to the tangent gradebetween BVC and VPIe2Vertical distance from any point onthe curve to the tangent gradebetween EVC and VPIEVertical distance from VPI to curve10sectionof

Nonsymmetrical VerticalCurve EquationsA (G 2 ) (G 1 )L l1 l2E l 1l 2A200( l 1 l 2 ) D e1 m 1 l1 2 D e 2 m 2 l2 112

Determining Radii ofSharp Curves by FieldMeasurementsBC2 BD 2BD2ACBC 2R Note:Points A and C may be any twopoints on the curveExample:Measure the chord length from A to CAC 18.4 then BC 9.2Measure the middle ordinate length B to DBD 3.5R 9.2 2 3.5 13.87. 0212

Distance From FinishedShld. to Subgrade Shld.and Slope EquivalentsEquation: x 100BAA Algebraic difference in % between shld. slopeand subgrade slopeB Depth of surfacing at finished shoulderx Distance from finished shld. to subgrade :81:10EquivalentRate of 67%12.50%10.00%EquivalentVertical Angle33 41'24"29 44'42"26 33'54"21 48'05"18 26'06"14 02'10"11 18'36"9 27'44"7 07'30"5 42'38"SubgradeSlope.020 / 1.025 / 1.030 / 1.035 / 1.040 / 1.050 / 1EquivalentRate of cal Angle1 08'45"1 25'56"1 43'06"2 00'16"2 17'26"2 51'45"13

Areas of Plane FiguresNomenclatureA Areah HeightR RadiusP PerimeterTrianglebh2P a b cA CircleA πR 2P 2 πREllipseA π ab14

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Areas of Plane FiguresSegmentA πR 2 R2 Sin 2360 0Sector 3600 P 2R ( 2πR )360 0A πR 2Fillet A RT πR 2 3600 When: 90 0 , A 0.2146R 216

Areas of Plane FiguresParallelogramA bhA ah'P 2( a b)TrapezoidA ( a b) h2PolygonDivide into trianglesA Sum of all triangles17

Areas of Plane FiguresAnnulus(Circular Ring)A π 2D d24()Irregular Figure a j A L b c d e f g h i 2 18

Surfaces\Volumes of SolidsNomenclatureS Lateral surface areaV VolumeA Area of section perpendicular to sidesB Area of baseP Perimeter of basePA Perimeter of section perpendicular to itssidesR Radius of sphere or circleL Slant height or lateral lengthH Perpendicular HeightC Circumference of circle or sphereParallelepipedS PHS PALV BH ALPyramid or ConeRight or RegularS 1PL2V 1BH319

Surfaces\Volumes of SolidsPyramid or Cone, Right orOblique, Regular or IrregularV 1BH3Prism: Right or Oblique,Regular or IrregularS PH PA LV BH ALCylinder: Right or Oblique,Circular or EllipticS PH PA LV BH AL20

Surfaces\Volumes of SolidsFrustum of any Prism or CylinderV V BH1A ( L2 L1 )2Frustum of Pyramid or ConeRight and Regular, Parallel EndsS 1L( P p )2V p perimeter of top(1H B b Bb3)b area of topFrustum of any Pyramid or Cone,with Parallel EndsV (1H B b Bb3)b area of top21

Surfaces\Volumes of SolidsSphereV S 4πR 24πR 33Spherical SectorS 1πR( 4H C)2V 2πR 2H3Spherical SegmentS 2π RH (1π 4H 2 C 24)1π H 2 ( 3R H )3V 22

Surfaces\Volumes of SolidsSpherical ZoneS 2π RH1V πH 3C12 3C 2 4H 224()Circular RingS 4π 2RrV 2π 2Rr2Prismoidal FormulaV H( B b 4M)6M Area of section parallel to bases,Midway between themb area of top23

Signs of TrigonometricFunctions for AllQuadrantsNote:When using a calculator to computetrigonometricfunctionsfromNorthAzimuths, the correct sign will be displayed24

Trigonometric FunctionsyP (X,Y)(hypotenuse)ry (opposite )θOx (adjacent)xSinθ yopposite r hypotenusecosθ xadjacent r hypotenuseSineCosinetanθ y opposite x adjacentcot θ x adjacent y oppositeTangentCotangentsec θ r hypotenuse xadjacentcscθ r hypotenuse yoppositeSecantCosecantsin θ ReciprocalRelations1csctan θ cos θ 1secX r cosθRectangulary r sinθr (x2 y2θ arctanPolar25yx)1cot θ

Right TrianglesBcAabSCA B C 180 0K AreaPythagoreana2 b 2 c 2TheoremA and B are complementary anglessin A cos B tan A cot B sec A csc Bcos A sin Bcot A tan Bcsc A sec BGivenToFinda, cA, B,b, Ka, bA, B,c, KA, aB, b,c, KA, bB, a,c, KA, cB, a,b, KEquationsinA b acc2 a 2tanA abc a 2 b2B 90 0 Aac sin AB 90 0 Abc cos AB 90 0 Ab c cos A26cos B aca 2c a22btanB aabK 2b a cot Aa 2 cot Ak 2a b tan Ab 2 tan AK 2a c sin Ac 2 sin 2 AK 4K

Oblique TrianglesBacCAbLaw of SinesLaw of Cosinesabc sin A sin B sin Ca 2 b 2 c 2 2bc cos Ab 2 a 2 c 2 2ac cos Bc 2 a 2 b 2 2ab cos CSum of AnglesK AreaGivenA B C 180 0a b cs 2ToFindEquationAsina, b, cA2cos(s b)(s c) A2bc s(s a )bc( s b)(s c)tan 2s(s a )A27

Oblique TrianglesGivenToFindEquationsina, b, cBcosB(s a )(s c) 2acB2 s(s b)ac(s a )(s c)tan 2s(s b)Bsina, b, cCcosC(s a)(s b) 2abC2 s (s c )ab(s a)(s b)tan 2s(s c)Ca, b, cKa, A, Bb, ca, A, BKa, b, ABa, b, AcK a sin Ba sin( A B)c sin Asin Aab sin C a 2 sin B sin CK 22 sin Ab sin Asin B aa sin C b sin Cc sin Asin Bb c a, b, AKa, b, CAa, b, Cc(a2) b 2 2 ab cos Cab sin C2a sin Ctan A b a cos Ca sin( A B)c sin AK c a, b, Cs(s a)(s b)(s c )(a2 b2 2ab cos CK K28ab sin C2)

Conversion FactorsClassmultiply:by:to 2222222

Conversion FactorsClassVolumemultiply:by:to 0

Metric Conversion FactorsClassmultiply:by:to myd0.7646mgal3.785Lgal0.0038mfl oz29.574mLacre 233333lb0.4536kgkip(1000 lb)0.4536metric ton(1000 kg)short ton2000 lb907.2kgshort ton0.9072metric ton31

Land SurveyingConversion FactorsClassmultiply:by:to get:Areaacre4046.8726macre0.404692ha210000 mLengthft12/39.37*m* Exact, by definition of the U.S. Survey footSteel TapeTemperature CorrectionsC 1166. 10 6 ( TC 20) L morC 6.45 10 6 ( TF 68)L fWhere:C TC LM TF Lf CorrectionTemperature in degrees CelsiusLength in metersTemperature in degrees FahrenheitLength in feetTemperature ConversionFahrenheit to CelsiusCelsius to Fahrenheit5( F 32)9 9 C 32 5 32

Less CommonConversion FactorsClassmultiply:Densityby:to 33Pressureksilb/ftVelocity233Water ConstantsFreezing point of water 0 C (32 F)Boiling point of water under pressure of oneatmosphere 100 C (212 F)The mass of one cu. meter of water is 1000 kgThe mass of one liter of water is 1 kg (2.20lbs)1 cu. ft. of water @60 F 62.37 lbs (28.29 kg)1 gal of water @60 F 8.3377 lbs (3.78 kg)Cement Constants1 sack of cement (appx.) 1 ft3 0.028 m 31 sack of cement 94 lbs. 42.64 kg1 gallon water 8.3453 lbs. @39.2 F1 gallon water 3.7854 kg @4 C33

Multiplication Factor TableMultiple91 000 000 000 1061 000 000 1031 000 102100 10110 10-10.1 10-20.01 10-30.001 10-60.000 001 10-90.000 000 001 10* Avoid when oPronunciationjig’a (i as in jig, a as in a-boutas in mega-phonekill’ ohheck’ toedeck’ a (a as in a-boutas in centi-pedeas in mili-taryas in micro-phonenan’ oh34