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R D H o l t z ~

SI Un i ts in G eo tech nica l ng ineer ing

REFER ENCE : Holtz, R. D. , SI Un its in Geoteehnleal Engineering,

Geotechnical Testing Journal, GTJODJ, Vol . 3 , No. 2 , June 1980, pp.

73-79.

ABSTRACT:

A brief descript ion is presented of the Internat ional

System of U nits (SI) as i t m ight be ap plied o geotechnical engineering.

Base as well as derived SI units that are of interest to geotechnical

engineers are described in detai l , and conversion factors for units in

common usage are given. A few examples of conversions are also

presented.

KEY WO RDS : units of measurement, metr ic system, symbols

W i t h i n t h e s c i en t i fi c a n d e n g i n e e r in g c o m m u n i t y , t h e r e h a s

a l w a y s b e e n s o m e c o n f u s i o n a s t o t h e p r o p e r s y s t e m o f u n i t s f o r

p h y s i c a l m e a s u r e m e n t s a n d q u a n t i t i e s . M a n y s c h e m e s h a v e b e e n

a d v a n c e d t h r o u g h o u t t h e p a s t fe w c e n t u r ie s a n d s o m e , s u c h a s t h e

I m p e r i a l o r B r i t i s h E n g i n e e r i n g s y s t e m , t h e s o - c a l l e d m e t r i c

s y s t e m , a n d a f e w h y b r i d s , h a v e a c h i e v e d m o d e r a t e l y w i d e p o p u l a r

u s a g e . R e c e n t l y , w i t h t h e g r o w t h o f i n t e r n a t i o n a l c o o p e r a t i o n a n d

t r a d e , i t h a s b e c o m e i n c r e a s i n g l y a p p a r e n t t h a t o n e s i n g l e , c o m -

m o n l y a c c e p t e d s y s te m o f u n i t s w o u l d b e n o t o n l y c o nv e n i e n t b u t

a l s o o f t r e m e n d o u s p r a c t i c a l v a l u e .

E v e n t h o u g h g e o t e c h n i c a l e n g i n e e r i n g m a y n o t h a v e t h e g r e a t e s t

c o n f u s i o n o f u n i t s , i t u n d o u b t e d l y r a n k s n e a r t h e t o p o f a l l f i e l d s i n

t h e n u m b e r o f d i f f e r e n t s y s t e m s i n c o m m o n u s a g e . L a b o r a t o r y

e n g i n e e r s , f o l l o w i n g t h e i r c o u n t e r p a r t s i n t h e p h y s i c a l s c ie n c e s ,

h a v e a t t e m p t e d t o u s e s o m e s o r t o f m e t r i c s y s te m , u s u a l l y th e c g s

( c e n t i m e t r e - g r a m - s e c o n d ) s y s te m f o r t h e s i m p l e r l a b o r a t o r y te s t s .

B u t t h e y a l s o a p p l y t h e m k s ( m e t r e - k i lo g r a m - s e c o n d ) s y s te m t o

m e a s u r e m e n t s o f p r e s s u r e a n d s t r e s s i n c o n s o l i d a t i o n a n d t r i a x i a l

t e s t s a n d u s e B r i t i s h E n g i n e e r i n g u n i t s f o r c o m p a c t i o n t e s t s . A s

a n y t e a c h e r o f s o i l m e c h a n i c s c a n t e s t i f y , t h e c o n f u s i o n t o t h e

u n i n i t i a t e d i s t r e m e n d o u s . A t l e a s t p r a c t i c i n g g e o t e c h n i c a l

e n g i n e e r s i n N o r t h A m e r i c a h a v e b e e n s o m e w h a t c o n s i s t e n t i n t h e

u s e o f t h e B r i t i s h E n g i n e e r i n g s y s t e m fo r l a b o r a t o r y a n d f i e l d d e n -

s i t i e s , s t r e s s m e a s u r e m e n t s , a n d t h e l i k e , a l t h o u g h t h e y c o m m o n l y

a l t e r n a t e b e t w e e n p o u n d s p e r s q u a r e f o o t , k i p s p e r s q u a r e f o o t ,

t o n s p e r s q u a r e f o o t , a n d p o u n d s p e r s q u a r e i n c h , d e p e n d i n g o n

h o w t h e y o r t h e i r c l i e n t s f e e l a b o u t t h e s u b j e c t . F o r t u n a t e l y , 1 t o n -

f o r c e p e r s q u a r e f o o t i s w i t h i n 2 % o f 1 k g f / c m 2 , a c o m m o n

l a b o r a t o r y u n i t f o r s tr e s s a n d p r e s s u r e , a n d t h e f o u n d a t i o n

e n g i n e e r u s i n g c o n s o l i d a t i o n t e s t d a t a c a n c o n v e r t d i r e c t l y w i t h l i t -

t l e e r r o r . S t r i c t l y s p e a k i n g , t h e u s e o f f o r c e a s a b a s i c u n i t i s in c o r -

r e c t ; m a s s s h o u l d b e t h e b a s i c u n i t , w i t h f o r c e d e r i v e d a c c o r d i n g t o

N e w t o n ' s S e c o n d L a w o f M o t i o n . U s e o f th e k i l o g r a m a s a u n i t o f

f o r c e i s o n e o f t h e d i f f i c u l t i e s w i t h t h e s o - c a l l e d m e t r i c s y s t e m , a

m o d i f i e d v er s io n o f t h e m k s s y s t e m t h a t w a s c o m m o n a m o n g c o n -

IAssociate professor, School of Civil Engineering, Purdue University,

W . Lafayette, In d. 47907. M ember of ASTM.

t i n e n t a l E u r o p e a n e n g i n e e r s . A t l e a s t t h e y t r i e d t o k e e p t h e d i s t i n c -

t i o n b e tw e e n m a s s a n d f o r c e b y c a l li n g t h e k i l o g r a m - f o r c e a k i l o -

p o n d ( k p ) .

A m o d e r n i z e d v e r s io n o f t h e m e t r i c s y st e m h a s b e e n d e v e l o p in g

o v e r t h e p a s t 3 0 y e a r s . T h e s y s t e m i s k n o w n a s S I , w h i c h s t a n d s f o r

l e S y s t ~ m e I n t e r n a t i o n a l d U n i t d s ( T h e I n t e r n a t i o n a l S y s t e m o f

U n i t s ). I t i s d e s c r ib e d i n d e t a il in A S T M E 3 8 0 , t h e S t a n d a r d f o r

M e t r i c P r a c t i c e , a v a i l a b l e i n t h e b a c k o f e v er y p a r t o f t h e A n n u a l

B o o k o f A S T M S t an d a rd s . T h e s y s t e m m a y s o o n b e c o m e t h e

c o m m o n s y s t e m i n t h e U n i t e d S t a t e s a n d t h e f ew o t h e r c o u n t r i e s

s t i l l u s i n g I m p e r i a l o r B r i t i s h E n g i n e e r i n g u n i t s . I n f a c t , G r e a t

B r i t a i n i t s e l f c o n v e r t e d c o m p l e t e l y t o S I i n 1 9 7 2, a n d A u s t r a l i a ,

C a n a d a , a n d N e w Z e a l a n d a r e p r e s e n t ly w el l a l o n g t h e w a y t o

c o n v e rs i o n. M o s t E u r o p e a n c o u n t r i e s a l r e a d y h a v e d e f a c t o

c o n v e r s i o n t o S I , e s p e c i a l l y i n e n g i n e e r i n g p r a c t i c e .

T h e S I M e t r i c S y s t e m

T h e S I m e t r i c s y s t e m i s a f u l l y c o h e r e n t a n d r a t i o n a l i z e d s y s t e m .

I t i s f o u n d e d o n s e v e n b a s i c u n i t s : f o r l e n g t h ( m e t r e , m ) , m a s s

( k i l o g r a m , k g ) , t i m e ( s e c o n d , s ) , e lec t r ic curren t ( a m p e r e , A ) ,

t h e r m o d y n a m i c t e m p e r a t u r e ( k e l v i n , K ) , l u m i n o u s i n t e n s i t y

( c a n d e l a , c d ) , a n d a m o u n t o f s u b s t a n c e ( m o l e ) . A l l o f t h e s e b a s i c

u n i t s h a v e p r e c i s e d e f i n i t i o n s , n a m e s , a n d s y m b o l s . U n i t s f o r a l l

o t h e r p h y s i c a l q u a n t i t i e s c a n b e d e r i v e d in t e r m s o f t h e s e b a s i c

u n i t s . S o m e t i m e s t h e d e r i v e d q u a n t i t i e s a r e g i v e n s p e c if i c n a m e s ,

s u c h a s t h e n e w t o n ( N ) f o r f o r c e a n d t h e w a t t ( W ) f o r p o w e r . T h e

d e r i v e d u n i t o f f o r c e r e p l a c e s t h e k i l o g r a m - f o r c e (k g f ) o f t h e m k s

s y s t e m s o t h a t t h e n a m e o f t h e u n i t i n d i c a t e s t h a t i t i s a u n i t o f

f o r c e , n o t m a s s . A g r e a t a d v a n t a g e i s t h a t o n e a n d o n l y o n e u n i t

e x i s ts f o r e a c h p h y s i c a l q u a n t i t y , a n d a l l o t h e r m e c h a n i c a l q u a n -

t i t ie s s u c h a s v e l o c i ty , f o r c e , w o r k , a n d s o o n c a n b e d e r i v e d f r o m

t h e b a s i c u n i t s . I n a d d i t i o n , t h e S I u n i t s fo r fo r c e , e n e r g y , a n d

p o w e r a r e i n d e p e n d e n t o f t h e n a t u r e o f t h e p h y s i c a l p r o c e s s ,

w h e t h e r m e c h a n i c a l , e l e c t r i c a l , o r c h e m i c a l .

A n o t h e r m a j o r a d v a n t a g e o f S I i s t h a t i t is a fu l l y c o h e r e n t s y s t e m .

T h i s m e a n s t h a t a p r o d u c t o r q u o t i e n t o f a n y tw o u n i t q u a n t i t i e s i s

a u n i t o f t h e r e s u l t i n g q u a n t i t y . F o r e x a m p l e , u n i t l e n g t h s q u a r e d

s h o u l d b e u n i t a r e a , a n d u n i t f o r c e s h ou l d b e u n i t m a s s t i m e s u n i t

a c c e l e r a t i o n . O b v i o u s l y , m a n y o f t h e e n g i n e e r i n g u n i t s i n c o m m o n

u s e (f o r e x a m p l e , a c r e , p o u n d - f o r c e , o r k i l o g r a m - f o r c e ) , a r e n o t

c o h e r e n t u n i t s . A l s o , u n i t s t h a t m i g h t b e r e l a t e d t o b a s i c u n i ts b y

p o w e r s o f t e n a r e n o t c o n s i s t e n t w i t h i n t h e S I s y s t e m . A g o o d e x -

a m p l e i s t h e l i t r e ( L ), w h i c h i s a c u b i c d e c i m e t r c . T h e e q u i v a l e n t

v o l u m e o f t h e l i t r e h a s b e e n d e f i n e d a s e x a c t l y 10 - 3 m 3 ( 1 0 0 0 cm 3 ) .

A d d i t i o n a l a d v a n t a g e s o f S I i n c l u d e th e u s e o f u n i q u e a n d w e l l-

d e f i n e d s y m b o l s a n d a b b r e v i a t i o n s a n d t h e c o n v e n i e n t d e c i m a l

r e l a ti o n b e t w e e n m u l t i p l e s a n d s u b m u l t i p l e s o f t h e b a s i c u n i t s .

© 1980 by the Am er ican Socie ty for Test ing and Ma ter ia ls 0149 6115180/0006 0073500.40

73

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7 GEOTECHNIC L TESTING JOURN L

Basle and Derived SI Metric Units

B a s e Un i t s

The three base units of interest to geotechnical engineers are

length, mass, and time. The SI units for these quantities are the

me t r e , the k i l o g r a m, and the second . Temperature, which might

also be of interest, is expressed in kelvins, although the system does

allow for use of the degree Celsius (°C), which has the same inter-

val. Electric current is expressed in a m p e r e s . Supplementary units

include the rad ian (rad) and s t e rad ian (sr), the units of plane and

solid angle, respectively.

As mention ed, these basic SI units have precise physical defini-

tions. For example, contrary to a popular misconception, the

metre is not the distance between two bars in Paris, but r ather has

been defined as bein g exactly equal to a certa in nu mbe r of wave-

lengths of radiation correspondi ng to a specific transition level in

krypton 86. The s tanda rd kilogram is equal to the mass of the in-

ternational prototype kilogram, a cylinder of platinum-iridium

alloy preserved in a vault at Le Burea u Internat ional des Poids et

Mesures at S~vres, France. Similar standar d kilograms can also be

found at the U.S. National Bureau of Standards near Washington,

D.C. The second has been defined as the duration of a certain

num ber of periods of the radiat ion corresponding to a specific

transiti on state in cesium 133.

D e r i v e d U n i t s

De r i v e d u n i t s geoteehnical engineers might use are listed in

Table 1. Prefixes are used to indicate multiples and submultiples

of the basic and derived units. SI prefixes are listed in Table 2. T he

prefixes should be applie d to indicate orders of magnitud e of the

basic or derived units and to reduce redundant zeros so that

numerica l values lie between 0.1 a nd 1000. T hey should not be ap-

plied to the denomi nator of compound units (kilogram is an excep-

tion since it is a basic unit in the SI system). Note that spaces, not

commas, should be used to separate groups of zeros. (Thi s latter

item was a concession to the Europeans, so that they would stop

using a comm a where Americans would use a decimal poin t.)

To ma inta in he coherence of the system, it is recommended that

only basic units be used to form derived units. For example, the

unit of force, the newton, is derived according to Newton's Second

TABLE 2--Pref ixes fo r I uni ts .

Factor Prefix Symbol

1018 exa E

1015 peta P

1012 tern T

109 giga G

106 mega M

103 kilo k

102 hecto h

101 deka da

10 - I deci d

10-2 centi c

10 -3 milli m

10 -6 micro /~

10 -9 nano n

10-12 pico p

10-15 femto f

10 - Is atto a

Law, F = M a , where the mass M is in kilograms, an d the accelera-

tion a is in m/ s 2, all basic units. For derived combinatio nal units

such as pressure or stress (pascals or newtons per square metre),

multiples and submultiples of the basic metric uni ts (in this case

metres) should be avoided. Fo r example, N/c m 2 and N /m m 2 are

wrong; the appropriate prefix should be used with the numerator

to indicate larger or smaller quantities, for example, kPa (kN/ m 2)

or MPa (MN/m2).

SI Units of Interest to Geoteehnleal ngineers

L e n g t h

The SI unit for length, the metre, should already be familiar.

(By the way, metre, not meter, is the recommended ASTM spell-

ing.) Useful SI length multiples and subm ultiples are the kilometre

(kin), millimetre (ram), micrometre (#m), and nanom etre (nm).

Conversion factors for common British Engin eering and mks units

are given in Table 3.

Good SI practice suggests that multiple and submul tiple metric

units be used in increm ents of 1000, for example, millimetre,

metre, and kilometre. Use of the centimetre, especially for lengths

unde r 300 mm, should be avoided.

TABLE 1--Derived SI uni ts .

Quantity Unit Symbol Formula

acceleration metre per second square m/s2 ...

area square metre m2

area hectare ha hm2 '104 m2

density kilogram per cubic metre kg/m3 ...

force newton N kg- m/s2

frequency hertz Hz 1/s

moment or torque newton metre N m k g m 2 / s 2

power watt W J/s

pressure pascal Pa N/m 2

stress pascal Pa N/m 2

unit weight newton per cubic metre N/m3 kg/s2- m2

velocity metre per second m/s . ..

voltage volt V W/A

volume cubic metre m3

volume litre L dm3 = 10-3 m3

work (energy) joule J N m

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TABLE 3--Conversion actors for units of length.

Unit Sl Equivalent

1 inch

1 foot

1 yard

1 mile (U.S. statute)

1 mile (nautical)

1 angstrom

mil

25.4 mm = 0.0254 m

0.3048 m

0.9144 m

1.609 × 103 m : 1.609 km

1.852 × 103 m = 1.852 km

1 X 10 -I °m =0 .1 nm

2.54 × 10 - s m = 0.0254 m m = 25.4/~m

M a s s

It will be recalled from physics that the inertia or mass of a

physical object, for which the SI uni t is the kilogram, is a measure

of the property that controls the response of that object to an ap-

plied force. It is convenient o measure the mass in terms of the ac-

celeration of an object produced by a unit force, as related by

Newton's Second Law of Motion. Thus, a unit force causes a 1-kg

mass to accelerate at 1 m/s 2. The mass then is an appropriate

measure of the amou nt of matter an object contains. The mass re-

mains the same even if the object's temperature, shape, or other

physical attribut es change. Unlike weight, which will be discussed

later, t he mass of an object does not depend on the local gravita-

tional attraction, and thus it is also independent of the object's

location in the universe.

Among all the SI units, the kilogram is the only one whose

name, for historical reasons, contains a prefix. The names of

multiples and submu ltiples of the kilogram are formed by at-

taching prefixes to the word gram rather than to kilogra m. In

other words, 10 -6 kg is not a microkilogram, but a milligram,

10 -3 g. Similarly, 1000 kg is not 1 kilokilogram but is equivalen t

to 1 megagram (Mg); 1000 kg is also the metric ton, sometimes

spelled tonn e to avoid confusion with the British ton, which is

equal to 2000 lb. ASTM recommends that metric ton be restricted

to commercial usage, and that the term tonne be avoided

altogether. Practical uni ts of mass in engineering practice are the

megagram, and in laboratory work, the kilogram and gram.

Some useful relationships and conversion factors for units of

mass are given in Table 4.

T i m e

Although the second is the basic SI time unit, minutes (min),

hours (h), days, a nd the like may be used where convenient, even

though they are not decimally related. (Maybe some day we will

even have a decimal time system; see Carrigan [1].)

F o r c e

As mentioned, the SI unit of force is derived from F = M a and

it is called the newton, equal to 1 kg- m/ s 2. Conversion factors for

common enginee ring force uni ts are given in Table 5.

TABLE 4--Conversion

actors fo r units o f mass.

Un~ SI Equivalent

1 pound mass (avoirdupois)

1 British (short) ton (2000 Ibm)

1 gram

1 metric ton

1 slug (1 lb-force per ft /s 2)

0.4536 kg

907.2 kg

10 -3 kg

103kg = 106g = 1Mg

14.59 kg

HOLTZ ON 81 UNITS 75

TABLE5--Conversion factors fo r units of orce.

Unit SI Equivalent

1 lb-force

1 British ton-force

1 kg-force (kp)

1 kip (1000 lb-force)

1 metric ton-force (1000 kg-force)

1 dyne (g'cm/s2)

4.448 N

8.896 × 103 N = 8.896 kN

9.807 N

4.448 × 103 N = 4.448 kN

9.807 × 103 N = 9.807 kN

10 -s N = 10/~N

It is obvious that the numbe rs in newtons for such items as col-

umn loads would be very large indeed a nd consequently somewhat

awkward. Therefore, consistent with the rules for application of

prefixes, it is simple to adjust these rather large numbers to more

manageable quantiti es for engineeringwork. The common prefixes

would be kilo-, mega-, and giga-, so that engineering forces would

be expressed in kilonewtons, kN, meganewtons, MN, and

giganewtons, GN. (The symbol for mega is M to avoid confusion

with the symbol for milli, m.) T hus, since 1 ton-force is 8.9 kN,

1000 ton-force would be 8.9 MN. Some useful relationsh ips using

these prefixes are: 2

1 kilonewton (kN) = 103 newton

1 meganewton (MN) = 106 newton : 103 kN

1 figanewton (FN) = 108 newton = 105 kN = 102 MN

1 giganewton (GN) = 109 newton = 106 kN = 103 MN

3 giganewtons = 30 figanewtons = 1 boxafiganewtons3

14.4 giganewtons = 1 grossafiganewtons

The correct unit to express the weight of an object is the newton,

since weight is the gravitational force that causes a downward ac-

celeration of that object. This can be expressed by saying that

weight W equals M g where M is the mas s of the object and g is the

acceleration of gravity. It will be recalled that the acceleration of

gravity varies with latit ude an d elevation; thus SI re commends that

weight be avoided and that mass be used instead. If weight must be

used, it is suggested that the location and gravitational accelera-

tion also be stated. However, for most ordinary engineering pur-

poses, the difference in accelera tion (about 0.5%) can be

neglected, and as long as we express the weight in newtons, the

units will be consistent.

Another p roblem with weight is that it is commonly used when

we really mean the mass of an object. For example, in the

laboratory when we weigh an object on a laboratory balance, we

really are com paring two masses, t hat of the un known object with

that of an object of known mass. Even scales or balances that

displace linear springs are calibrated against objects of known

mass.

Further ambiguity occurs of course because common units of

mass, such as the pound or kilogram, are often used in engineering

practice as a uni t of force. If pound is used as a unit of force, then

depending on the resulting accelerations, different mass units are

defined. For example, if t lbf causes an acceleration of 1 ft/s 2,

then the mass is 1 lbf's2/ft, which is called a slug. In other words,

1 lbf = 1 slug X 1 ft/s 2. Using slugs as unit s of mass avoids the

confusion with pounds as mass, and this un it has been commonly

used in aerodynamics and fluid mechanics.

2Kovacs, W. D., Conversion factors for kilonewtons per square meter

and common engineering stress units, Purdue University, 1974 (un-

published).

3This unit is only a constant before the box is opened.

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7 GEOTECHNIC L TESTING JOURN L

If we wanted instead to use a pound-mass system, we would

define a unit of force called the p o u n d a l , where 1 poundal = 1 lb-

mass × 1 f t / s 2 . Poundals are apparently only found in physics

books.

Some examples i llustrating conversion between different mass

units are given in the Appendix to this paper.

D e n s i t y a n d U n i t W e i g h t

Density is defined in physics as mass per uni t volume. Its unit s

in the SI system are kilograms per cubic metre (kg/m3). In m any

cases in geotechnical engineering, it may be more convenient o ex-

press density in mega grams per cubic metre. Conversions from the

common laboratory an d field densities are:

S t r e s s a n d P r e s s u r e

The SI unit for stress and pressure is the p a s c a l (Pa), which is

exactly equal to 1 N/m 2. There has been some objection, especially

in Europe, to the use of the pascal as the basic unit of stress and

pressure, because it is so small. The Ger mans and French, for ex-

ample, often use the bar, which is exactly 105 Pa. However, the

pascal is more logical since it is a coherent unit; that is to say,

equations involving the pascal with other SI un its can be written

without coefficients of proportionality being required. Conversion

factors for some common engineeri ng units for stress and pressure

are given in Table 6.

It is obvious that the pascal is a small unit, b ut, as with SI units

of force, it is easy to add prefixes to make the large numbers more

manageable. Thus, 1 psi in the above table is more conveniently

expressed as 6.9 kPa th an as 6.9 × 103 Pa. For ordinary triaxial

testing of soils, for example, hydrostatic cell pressures rarely ex-

ceed 200 or 300 psi (1379 kPa or 2068 kPa). Or, if all the pressures

in a test series are in this range, it might be convenien t to use 1.4

MPa or 2.1 MPa. And, as with other systems of units , a rounded or

even interval may be more convenient; in this case, 1.5 MPa a nd

2.0 MPa.

Similar examples could be given for engineer ing stresses. Either

kilopaseals or megapascals, kPa or MPa, will become commonly

used for foundation stresses, lateral earth pressures, allowable

bearing values, and the like. In the laboratory, force is measured

by a proving ring or load cell and then converted to stress (for ex-

ample, in the unc onfine d compression or direct shear tests), so the

computati onal process will be no more complicated than it is now.

Similarly, with electrical pressure transducers, a ca libration factor

must be used to convert millivolts (mV) of output to pressure in

whatever units are used.

A convenient approximation, part of which is already in use in

geotechnical engineering practice, is the following:

1 British ton-force/ ft2 = 1 kgf/c m 2 = 1 atm

= 10 metric ton-f orce/ m2 = 100 kPa

The error involved is between 2 and 4% which is certainly less tha n

ordinary engineering accuracy requirements.

TABLE 6--Conversion factor s fo r units o f pressure and stress.

Unit SI Equivalent

1 psi (lb-force/in. 2)

1 arm at STPa

1 kg-foree/cm2

1 metric ton-force/m 2

1 bar

1 ksi (kip/in.2)

1 British ton-foree/ft 2

1 lb-force/ft 2

6.895 × 103 Pa or 6.895 kPa

1.013 × 105 Pa or 101.3 kPa

9.807 × 104 Pa or 98.07 kPa

9.807 × 103 Pa or 9.807 kPa

1 × 10s Paor 100kPa

6.895 × 106 Pa or 6.895 MPa

95.76 × 103 Pa or 95.76 kPa

47.88 Pa

aStandard temperature and pressure, not a motor oil additive or Soil

Test Probe.

lb-mass/ft3 = 16.018 kg/ m 3

1 g/c m 3 = 103 kg /m 3 ---- 1 Mg /m 3

It will be recalled that the density of water Pw is exactly 1.000

g/c m 3 at 4°C, and the vari ation is relatively small over the rang e of

temperature s encounte red in ordinary engineering practice.

Therefore, it is usually sufficiently accurate to take Pw = 103

kg/ m 3 = 1 Mg/m 3, which considerably simplifies phase computa-

tions, for example. It is also useful to know that 1000 kg/ m 3 is

equal to 62.4 lb-mass/ft3.

Typical densities that might be encountered in geotechnical

practice are 1.2 M g/m 3 (74.8 lb/ft3), 1.6 Mg/ m 3 (100 lb/ft3), and

2.0 M g/m 3 (125 lb/ft3). The commonly used density for concrete,

150 lb/f t 3, is almost exactly 2.4 Mg/m 3.

Note that all mass and volume ratios common in geotechnical

engineering practice are not affected by the use of SI units. For ex-

ample, void ratio or water content of any given soil still has the

same numeri cal value.

So far, unit weight or weight per uni t volume has been the com-

mon measurement in geotechnical engineering. Since weight

should be avoided in technical work for all the reasons discussed

earlier, then unit weight also should be avoided. For this reason,

ASTM Committee D-18 on Soil and Rock recently voted to replace

the st andard definitions for unit weight with the app ropriate

definitions of density in ASTM D 653, Defi nitions of Terms and

Symbols Relating to Soil and Rock Mechanics. If density must be

converted to uni t weight, then simply use 3' = Pg. Thus the ap-

propriate value for the acceleration of gravity will have to be con-

sidered. The stan dard value of g is 9.807 m/s 2 (32.17 ft/s2),

which can be used with sufficient accuracy for ordinary engineer-

ing work on the earth. If work is to be carried out on the moon or

some other planet, then the local value for g must be used.

For com puta tions of geostatic stresses, the uni t weights of the

various soil layers can be easily replaced by the pg of the layers.

The usual formula

ffv = i~=l Yigi

then becomes

av -- ~ ==1p i g z i

where

ov = total vertical stress at some depth

P l = density of each layer

z i = thickness of each layer

If pg is a constant throughout the depth h, then

o v = p g h

By analogy, computation of the static pore water pressure u o at

some depth h w below the gro und water table is

U o = p w g h w

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Similarly, to ob tain the effective ve rtical overbu rden stress, the ef-

fective orbu oya nt density p' for each layer below the gr ound water

table can be used, or perhaps more simply, O'vo = Ovo -- uo .

Dimensiona l analysis of these equat ions for stress shows th at if

the densities are expressed in Mg/ m 3, then stresses automatically

come out in kPa. Or

(Mg/m3)(m/s2)m = 1000 (kg.m)/ (s2-m2) = 1000 N/ m 2 = 1 kPa

An example of geostatic stress c omputations using SI units can be

found in the Appendix.

Summary

The SI system is a fully coherent and rational system of units,

well suited to the measurements ordinarily made in geotechnical

engineering practice. The basic units in the system have precise

names, definitions, and symbols, and the units for all other physi-

cal quantitie s can be derived in terms of these basic units. Products

or quotients of any two unit quantities are also units of the

resulting quantity. Use of prefixes to indicate multiples and sub-

multiples of units helps to make the numbers more manageable.

One fact of particular interest to geotechnical engineers is that t he

SI units of force, stress, and pressure have ind epend ent and

precisely defined names and symbols. Use of density instead of

unit weight is not only more correct physically, but also has the ad-

vantage that the density of water is unity (in Mg/m3). The only

minor disadvantage to the use of SI units in geotechnical engineer-

ing practice is that a const ant value for the acceleration of gravity

must be included in the computations of geostatic stresses.

Acknowledgments

This article has been adapted by permission from an appendix

to a n i ntroductory textbook on geotechnical engineering by R. D.

Holtz and W. D. Kovacs, which will be publi shed in 1981 by

Prentice-Hall, Inc. The original version was written in 1969 while

the author was a gradua te st udent at Northwestern University. The

support an d encour agement of Prof. R. J. Krizek is gratefully

acknowledged. The text was typed by Catherine Minth and the

drawings were made by Margaret McFarren.

Reference

[1] Carrigan, R. A., Decimal Time, American Scientist, Vol. 66, No. 3,

May-June 1978, pp. 305-313.

OLTZ ON SI UNITS

Since 1 slug = 14.59 kg, his mass is 68.03 kg. Another way to

calculate his mass is to convert his weight to newtons; then divide

by g:

W = 150 lbf(4.448 N/1 lbf) = 667.20 N or 667.2(kg-m )/s2

M = W/g= (667.2 kg. m/s2)/(9 .807 m/ s 2) = 68.03 kg

Next, we have to either ask an astronomer or look up in the

Handbook of Chemistry and Physics the gravitational acceleration

on the surface of the moon. We find that gmoon ~ 1.67 m/s 2.

Thus,

Wmoon : Mgmoon : 68.03 kg (1.67 m/ s 2) = 1t3.62 N

Or, since 4.448 N = 1 lbf,

Wraoon = 113.62 N (1 1bf/4.448 N) = 25.54 lbf

Check: On earth,

667 N(1.67/9.81) = 113.6 N on the moon

Example 2

Given: The density of water Pw = 1 Mg/m 3.

Required: Calculate t he density of water in (a) g/c m 3 and (b)

lb/ ft 3.

Solution: Set up an equation as follows for Part a.

1 Mg/m 3 ----- 1 Mg/m 3 (106 g / 1 Mg)(1 m/ 100 era) 3 = 1 g/c m 3

For Part b:

1 Mg/m 3 = 1 Mg/ m 3- (103 kg/1 Mg)

(1 1bm/0.4536 kg)(0.3048 m/ 1 ft) 3 = 62.43 lbm /f t3

where Ibm = pound-mass.

Anoth er way to solve Part b is to recall tha t 1 ibm/f t 3 = 16.018

kg/m3; so

1 Mg/ m 3 = 1 Mg/m 3. (103 kg/1 Mg)

× (1 lbm/ft 3/16.0 18 kg /m 3) = 62.43 lbm /ft 3

PPENDIX

omputat ions and onversions That Use SI

U n s

Example 1

Given: Neff Armst rong weighs 150 lb on ea rth.

Required: How much does he weigh on the surface of the

moon?

Solution: First, we have to calculate Mr. Armstrong's mass

on earth. Unless he had healt h problems duri ng the voyage, his

mass will be the same on t he mo on.

M = W/g = (1501bf)/(32.17 ft /s 2)

= 4.66(lbf-s2)/ft = 4.66 slugs

Example 3

Given: The density of water Pw = 1 Mg/m 3.

Required: Convert this density to unit weight in (a) SI a nd (b)

British Engineering units.

Solution: (a) SI units: We know that 3' = Pg; so

3' = 1 Mg/ m 3 (103 kg/1 Mg). 9.807 m/s 2

= 9807(kg. m)/(m 3- s2)

Since 1 N = 1 kg. m/ s 2, then

= 9807 N/ m 3 = 9.807 kN/m 3

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= 2 0 0 8 p o u n d a l s / f t 3

F o r t h e s a n d b e l o w t h e w a t e r t a b l e :

Psat = Ps + p w e ) / 1 + e )

= 2 . 6 5 + 1 . 0 - 0 . 5 ) / 1 + 0 . 5 ) = 2 . 1 0 M g / m 3

w h e r e

Psat

s a t u r a t e d d e n s i t y . F o r t h e c l a y :

I f l b f a r e u s e d , f r o m P a r t a ,

7 = 9 . 8 07 k N / m 3 - 1 0 0 0 N / l k N ) 1 1 b f /4 . 4 48 N )

0

s a t = [Ps 1 + w) ] / 1 + e ) =

[ 2 . 7 1 + 0 . 3 7 ) ] / 1 + 1 . 0 ) = 1 . 8 5 M g / m 3

x 0 . 3 0 4 8 m / 1 f t ) 3 = 6 2 . 4 3 l b f / f t 3

T h i s i s o f c o u r s e t h e f a m i l i a r v a l u e f o r t h e u n i t w e i g h t o f w a t e r .

l ) e p ~ , m

E x a m p l e 4

G i v e n : T h e s o il p r o f i l e s h o w n i n F i g . 1 .

R e q u i r e d : C o m p u t e a n d p l o t th e t o t al , n e u t r a l , a n d e f f e c t i v e

v e r t i c a l s t r e s s e s w i t h d e p t h .

S o l u t i o n : F i r s t , c a l c u l a t e t h e a p p r o x i m a t e s o il d e n s i t ie s . F o r t h e

s a n d a b o v e t h e w a t e r t a b l e :

P d = P s / 1 + e ) = 2 . 6 5 / 1 + 0 . 5 ) = 1 . 7 7 M g / m 3

p = Pd 1 4- w ) = 1 . 7 7 1 . 0 6 ) = 1 .8 7 M g / m 3

w h e r e

P d

: d r y d e n s i t y ,

Ps =

d e n s i t y o f s o i l s o l i d s ,

e = v o i d r a t i o , a n d

w = w a t e r c o n t e n t .

- 2

- 2

- 6

- 4

~ J n : 6

V

.--4-

P s = 2 . 6 5 M g / m 3

S A N D • n = 1 9

e : O . 5

C L A Y

= 2 . 7 M g / m 3

¢ ~n : 3 7

e : l . O

D e p t h , m

0

- I 4 / / ~ y / X ~ X / /~ F /

F I G . 1 - - S o i l p r o f i le f o r E x a m p l e 4 .

Tota l Ver t ica l S t ress, O 'vo , kPa

I 0 0 2 0 0

3 7

Neutral Stress, ¢./-o,RPa

0 I 0 0

i

- 6

- 8

- I 0

- 1 2

- 1 4

2 6 4

b ) B r i ti s h E n g i n e e r i n g u n i t s: F r o m E x a m p l e 2 w e k n o w t h a t 1

M g / m 3 = 6 2 .4 3 l b m / f t 3 . T h e r e f o r e ,

118

Ef fe ct ive Ver t ica l S t ress, cr~ to , k Pa

0 l eO 2

i

7 = 6 2 . 4 3 l b m / f t 3 . 3 2 . 1 7 f t / s 2 = 2 0 0 8 l b m - f t ) / s 2 - f t 3 )

78 G E O TE C H N IC A L TE S T IN G JO U R N A L

F I G .

2 - -T o t a l n e u t ra l, a n d e f fe c t i v e v e r t ic a l s t r e s s p ro f i l e s o r Ex a m p l e 4 .

1 4 6

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  OTTZ ON SI UNITS 9

S e c o n d , c a l c u l a t e t h e t o t a l v e r t i c a l s t r e ss o vo a t a f e w c o n v e n i e n t

p o i n t s i n t h e p r o f i le . A t - - 2 m :

o~ = pgz = ( 1 .8 7 M g / m 3 ) ( 9 . 8 1 m / s 2 ) 2 m =

3 6 . 6 9 ( M g • m ) / ( s 2 . m 2 )

R e c a ll th a t 1 k g . m / s 2 = 1 N a n d 1 N / m 2 = 1 P a . S o 1

( M g - m ) / ( s 2 m 2 ) --- 1 k P a . T h e r e f or e , 3 6 . 6 9 ( M g ' m ) / ( s 2 . m 2 ) =

3 6 . 6 9 k P a o r 3 7 k P a . N o t e t h a t i f d e n s i t ie s a re e x p r e s s e d i n

M g / m 3, v e r ti c a l s t re s s c o m e s o u t a u t o m a t i c a l l y i n k P a . A t - - 6 m :

Ovo = 37 kPa + 2 . 1 ) 9 . 8 1 ) 4 ) = 1 1 9 k P a , a n d s o o n . P o r e w a t e r

p r e s s u r e s a r e c a l c u l a t e d u s i n g u o = pwg hw F o r e x a m p l e , a t

- - 6 m :

u o =

( 1 M g / m 3 ) ( 9 . 8 1 m / s 2 ) ( 4 m ) = 3 9 k P a

T h e c o m p l e t e t o t a l , n e u t r a l , a n d e f f e c t i v e v e r t i c a l s t r e s s p r o f i l e s

a r e s h o w n i n F i g . 2 .

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