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Units,Conversions andSymbols45Contents45.1 Introduction 45/3

45.1.1 Units 45/345.1.2 Conversion factors 45/345.1.3 Properties of materials 45/345.1.4 Mathematical relations and

trigonometrical functions 45/3

45.2 International unit system 45/345.2.1 Base units 45/345.2.2 Supplementary units 45/345.2.3 Notes 45/345.2.4 Derived units 45/445.2.5 Decimal multiples and submultiples 45/445.2.6 Common variations and auxiliary units 45/4

45.3 Conversion factors 45/545.3.1 Systeme International and imperial units 45/545.3.2 Systeme International and US units 45/6

45.4 Symbols 45/645.4.1 Greek alphabet 45/6

45.1 Introduction

45.1.1 UnitsThe Systeme International (SI) system of units used throughoutthis book is the standard system used throughout Europe andmany other countries in the world. It was first accepted at aninternational conference in 1960 and, in 1971, a directive by theEuropean Economic Community required the existing imperialand metric CGS systems to be replaced by SI.

The definitions of, and the symbols for, SI units are given insections 2.1.1 to 2.1.5.

Although, in time, it can be expected that there will be strictadherence to the SI units given in sections 2.1.1 to 2.1.5, thereare at present some cases in which, for convenience or becauseof previously established practice, the units are varied or auxili-ary units are introduced. The most common variations are givenin section 2.1.6 and it should be noted that they do not representa serious departure from the SI system.

45.1.2 Conversion factorsSection 2.2.1 and, in particular, Table 2.4 gives conversionfactors between most common imperial and SI units, togetherwith the reciprocals. Section 2.2.2 draws attention to somedifferences between imperial and US units.

45.1.3 Properties of materialsFor information on the properties of materials, the reader isreferred to the appropriate earlier chapter, as follows:

Aluminium Chapter 14Bituminous materials Chapters 23 and 24Concrete Chapters 4, 12 and 37Masonry Chapter 15Paint Chapter 4Plastics Chapter 4Reinforcement Chapter 12Rock Chapter 10Rubber Chapter 4Soil Chapter 9Timber Chapter 16

45.1.4 Mathematical relations and trigonometricalfunctionsMathematical relations commonly employed in civil engineeringwork, including those for statistical applications, are given inChapter 1.

The reader is referred to any of the standard works whichevaluate mathematical relations and provide tables of trigono-metrical functions for more detailed information.

45.2 International unit system (Sl)The SI is a metric system giving a fully coherent set of units forscience, technology and engineering, involving no conversionfactors. The starting point is the selection and definition of aminimum set of independent 'base' units. From these, 'derived'units are obtained by forming products or quotients in variouscombinations, again without numerical factors. For con-venience, certain combinations are given shortened names. Asingle SI unit of energy (joule = kilogram metre-squared persecond-squared) is, for example, applied to energy of any kind,whether it be kinetic, potential, electrical, thermal, chemical...thus unifying usage throughout science and technology.

The SI system has seven base units, and two supplementaryunits of angle. Combinations of these are derived for all otherunits.

45.2.1 Base unitsDefinitions of the seven base units have been laid down in thefollowing terms. The quantity symbol is given in italic, the unitsymbol (with its standard abbreviation) in roman type. Asmeasurements become more precise, changes are occasionallymade in the definitions.

(1) Length: /, metre (m) The metre was defined in 1983 as thelength of the path travelled by light in a vacuum during atime interval of 1/299 792 458 of a second.

(2) Mass: m, kilogram (kg). The mass of the internationalprototype (a block of platinum preserved at the Internatio-nal Bureau of Weights and Measures, Sevres).

(3) Time: t, second (s) The duration of 9 192 631 770 periodsof the radiation corresponding to the transition between thetwo hyperfine levels of the ground state of the caesium-133atom.

(4) Electric current: i, ampere (A) The current which, main-tained in two straight parallel conductors of infinite length,of negligible circular cross-section and 1 m apart in vacuum,produces a force equal to 2 x 10~7N/m length.

(5) Thermodynamic temperature: T, kelvin (K) The fraction1/273.16 of the thermodynamic (absolute) temperature ofthe triple point of water.

(6) Luminous intensity: /, candela (cd) The luminous intensityin the perpendicular direction of a surface of 1/600 000 m2 ofa black body at the temperature of freezing platinum undera pressure of 101 325 N/m2.

(7) Amount of substance: Q, mole (mol) The amount ofsubstance of a system which contains as many elementaryentities as there are atoms in 0.012kg of carbon-12. Theelementary entity must be specified and may be an atom, amolecule, an ion, an electron . . . or a specified group of suchentities.

45.2.2 Supplementary unitsPlane angle: : a, ft, . . . radian (rad) The plane angle betweentwo radii of a circle which cuts on the circumference of the circlean arc of length equal to the radius.

Solid angle: Q, steradian (sr) The solid angle which, having itsvertex at the centre of a sphere, cuts off an area of the surface ofthe sphere equal to a square having sides equal to the radius.

45.2.3 NotesTemperature At O K, bodies possess no thermal energy. Speci-fied points (273.16 and 373.16 K) define the Celsius (centigrade)scale (O and 10O0C). In terms of intervals, I0C= 1 K. In terms oflevels, a scale Celsius temperature 9 corresponds to(9 + 273.16) K.

Force The SI unit is the newton (N). A force of 1 N endows amass of 1 kg with an acceleration of 1 m/s2.

Weight The weight of a mass depends on gravitational effect.The standard weight of a mass of 1 kg at the surface of the Earthis 9.807 N.

45.2.4 Derived unitsAll physical quantities have units derived from the base andsupplementary SI units, and some of them have been givennames for convenience in use. Base, supplementary and some ofthe derived units are listed in Table 45.1.

Table 45.1 Systeme International base, supplementary and derivedunits

Quantity Unit name Derivation Unitsymbol

Baselength Metre mmass Kilogram kgtime Second selectric current Ampere Athermodynamic Kelvin K

temperatureluminous intensity Candela cdamount of Mole mol

substanceSupplementary

plane angle Radian radsolid angle Steradian sr

Derivedforce Newton kgm/s2 Npressure, stress Pascal N/m2 Paenergy Joule N m, W s Jpower Watt J/s Welectric charge, Coulomb As C

fluxmagnetic flux Weber Vs Wbelectric potential Volt J/C Vmagnetic flux Tesla Wb/m2 T

densityresistance Ohm V/A Qinductance Henry Wb/A, Vs/A Hcapacitance Farad C/V, A s/V Fconductance Siemens A/V Sfrequency Hertz s ' Hzluminous flux Lumen cd sr Imilluminance Lux lm/m2 Ixradiation activity Becquerel s ' Bqabsorbed dose Gray J/kg Gymass density Kilogram per

cubic metre kg/m3dynamic viscosity Pascal-second Pa sconcentration Mole per cubic

metre mol/m3linear velocity Metre per second m/slinear acceleration Metre per second-

squared m/s2angular velocity Radian per second rad/sangular Radian per second-

acceleration squared rad/s2torque Newton metre N mcurrent density Ampere per square

metre A/m2resistivity Ohm metre Q mconductivity Siemens per metre S/mthermal capacity Joule per kelvin J/Kspecific heat Joule per kilogram

capacity kelvin J/(kg K)thermal Watt per metre

conductivity kelvin W/(m K)luminance Candela per square

metre cd/m2

42.2.5 Decimal multiples and submultiplesDecimal multiples and submultiples of SI units are indicated bythe prefix letters given in Table 45.2. Thus, MN is meganewtonand us is microsecond. Prefixes for the kilogram are expressed interms of the gram, i.e. 1000kg= 1 Mg, not 1 kkg. There is apreference to express stress as 1 N/mm2 instead of 1 MN/mm2.Table 45.2 Decimal prefixes

Factor by which Prefixunit is multiplied Name Symbol

1018 exa E1015 peta P1012 tera T109 giga G106 mega M103 kilo k102 hecto h10' deca da10 ' deci d10 2 centi c10 3 milli m10 6 micro u,10 9 nano n10 l2 pico p10 15 femto f10 18 atto a

45.2.6 Common variations and auxiliary unitsThe main variations that are commonly applied to civil engi-neering are:

Stress expressed as N/mm instead of pascals (Pa) (1 N/mm= 1 MN/m= IMPa)Pressure, e.g. underwater, expressed as bar instead of pascals(1 bar= 10OkPa, 1 mbar = 0.1 kPa)Temperature expressed as 0C (Celsius or centigrade) insteadof K (kelvin) (O0C = 273.16K, 10O0C = 373.16 K)Mass expressed as tonne instead of kilograms (kg)( I t =1000 kg)

Some of the common variations from the strict SI system arelisted in Table 45.3 and are here termed 'auxiliary' units.

Table 45.3 Auxiliary unitsQuantity Symbol SI

Angledegree () TC/180 radminute (') second (") - -

Areaacre a 100 m2hectare ha 0.01 km2barn barn 10 28 m2

Energyerg erg 0.1 uJcalorie cal 4.186 Jelectron-volt eV 0.160 aJgauss-oersted GaOe 7.96 uJ/m3

Forcedyne dyn 10 uN

LengthAngstrom A 0.1 um

Table 45.3 (continued)Quantity Symbol SI

Masstonne t 1000 kg

Nucleonics, Radiationbecquerel Bq 1.0 s"1gray Gy 1.0 J/kgcurie Ci 3.7 x 10' Bqrad rd 0.01 Gyroentgen R 2 . 6 x 1 0 C/kg

Pressurebar b 100 kPatorr Torr 133.3 Pa

Timeminute min 60 shour h 3600 sday d 86400 s

Volumelitre lo r L 1.0 dm3

45.3 Conversion factors

45.3.1 Systeme International and imperial unitsAlthough SI is now the standard system in use throughoutEurope and much of the rest of the world, imperial units areused occasionally in some specialized areas and many publica-tions prior to about 1980 were in imperial units.

Conversion factors between SI and imperial units are given inTable 45.4. Column 1 gives the imperial units, column 2 the SIequivalent and column 3 the reciprocal.

Table 45.4 Conversion factors: imperial to SlImperial SI Reciprocal

Length (m)in 25.40 mm 0.0394ft 0.3048m 3.2800yd 0.9144m 1.0940fathom 1.829m 0.5470mile 1.6093km 0.6210nautical mile 1.852km 0.5400

Area (m2)l in 2 645.2mm2 1.550OxIO-3l f t 2 0.0929m2 10.7600lyd2 0.8361m2 1.2000!acre 4047m2 0.247OxIQ-3lmile2 2.590km2 0.3860

Volume (m3)1 in3 16.39 x 103 mm3 0.0610 x 10~31 ft3 0.0283 m3 35.300lyd3 0.7646m3 1.310IUK gal 4.546dm3 0.220

Second moment of area (m4)lin4 416 x l O 3 mm4 2.40 x 10~6

Velocity (m/s, rad/s)Acceleration (m/s2, rad/s2)

If t /s 0.3048 m/s 3.2800

Table 45.4 (continued)Imperial SI Reciprocal

Imile/h 0.4470 m/s 2.2370!knot 0.5144 m/s 1.9440Ideg/s 17.45mrad/s 0.0573Irev/s 6.283 rad/s 0.1590lft/s2 0.3048 m/s2 3.2810

Mass (kg)1 oz 28.35 g 0.03531 Ib 0.454 kg 2.2000Icwt 50.80kg 0.01971 UK ton 1016 kg 0.9840 x 10~3

Energy (J), Power (W)l f t lb f 1.356 J 0.737IBtu 1055 J 0.948 x l O 31 therm 105.5 kJ 9.478 x 10"3IkWh 3.60 MJ 0.278IBtu/h 0.293 W 3.413If t lbf / s 1.356 W 0.737lhp 745.9 W 1 . 3 4 I x I O 3

Thermal quantities (W, J, kg, K)lBtu/(ft2h) 3.155 W/m2 0.31701 Btu/(ft3 h) 10.35 W/m3 0.96601 Btu/(ft h 0F) 1.731 W/(m K) 0.5780If t lbf / lb 2.989 J/kg 0.33401 Btu/lb 2326 J/kg 0 . 4 3 0 O x I O 3IBtu/ft3 37.26 KJ/m3 0.02681 ft lbf/(lb 0F) 5.380 J/(kg K) 0.18601 Btu/(lb 0F) 4.187 kJ/(kg K) 0.23901 Btu/(ft3 0F) 67.07 kJ/m3 K 0.0149

Density (kg/m3)lib/in3 27.68 Mg/m3 0.0361lib/ft3 16.02 kg/m3 0.06241 ton/yd3 1329 kg/m3 0.7520 x IQ-3

Flow rate (kg/s, m3/s)llb/h 0.1260g/s 7.93601 ton/h 0.2822 kg/s 3.54401 Ib/s 0.4536 kg/s 2.2046Ift3 /h 7.866 cm3/s 0.1270Ift3 /s 0.0283 m3/s 35.3360Igal/h 1.263cm3/s 0.7920Igal/min 75.77 cm3/s 0.0312Igal/s 4.546 dm3/s 0.2200

Force (N), pressure (Pa)ldyn 10.OuN 0.10001 lbf 4.445 N 0.2250ltonf 9.964 kN 0.10041 lbf/ft2 47.88 Pa 0.0209llbf/in2 6.895 kPa 0.14501 tonf/ft2 107.2 kPa 9.3280 x 10~31 tonf/in2 15.44 MPa 0.0648linHg 3.386 kPa 0.29501 inH2O 149.1 Pa 6.707Ox 10~3

Torque (N m)1 lbf in 0.113Nm 8.84901 lbf ft 1.356Nm 0.7370ltonf ft 3.307 kNm 0.3020

Inertia (kg m2)Momentum (kg m/s, kg m2/s)

l ib in2 0.293 gm2 3.4130

Table 45.4 (continued)Imperial SI Reciprocal

l ib ft2 0.0421 kg m2 23.75301 ton ft2 94.30 kg m2 0.0106l lbf t / s 0.138kgm/s 7.24601 Ib ft2/s 0.042 kg m2/s 23.8100

Viscosity (Pa s, m2/s)1 poise 9.807 Pa s 0.1020llbfs/ft2 47.88 Pa s 0.02091 lbf h/ft2 172.4 kPa s 5.8000 x 10~31 stokes I.0cm2/s 1.0000Iin2/s 6.452 cm2/s 0.1550Ift2 /s 929.0 cm2/s 1 . 0 7 6 O x I O 3

Illumination (cd, Im)llm/ft2 10.76 lm/m2 0.0929led/ft2 10.76 cd/m2 0.09291 cd/in2 1550 cd/m2 0.645 x IQ-3

45.3.2 Systeme International and US unitsUnited States units differ from imperial units in respect of liquidmeasurement and mass.

1 US gal. = 0.8332 imperial gal. (reciprocal 1.200)= 3.788 dm3 (reciprocal 0.264)= 3.7881

1 US long ton =1.02Ot (SI)I U S short ton = 0.9091 (SI)(1 imperial ton = 1.0161 (SI)

45.4 Symbols45.4.1 The Greek alphabetAlthough very little use is made of Greek letters for symbols inSI, the Greek alphabet is, of course, widely used in mathemati-

cal and other applications in civil engineering. Table 45.5 givesthe Greek alphabet in the form used throughout the text of thisbook.

Table 45.5 The Greek alphabetCapital Lower case Name English

transliteration

A a alpha aB P beta bF y gamma gA 6 delta dE E epsilon eZ C zeta zH r\ eta e9 6 theta th/ / iota iK K kappa kA A lambda 1M n mu mTV v nu nE xi xO o omicron on n pi pP p rho rE a sigma sT i tau tY D upsilon u