appendix a

There has always been some confusion with regards to the system of units to be used in engineering practices and ...... Christian, J.T., and Carrier, W.D. (1978). "Janbu .... "Discussion of Dams and Sand Foundations," by A.C. Koenig, Trans ASCE,. Vol. 73. ..... (1971). "Design Manual - Soil Machanics, Foundations, and Earth.
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APPENDIX A SI UNITS IN GEOTECHNICAL ENGINEERING Introduction There has always been some confusion with regards to the system of units to be used in engineering practices and other commercial transactions. FPS (Foot-pound-second) and MKS (MeterKilogram-second) systems are still in use in many parts of the world. Sometimes a mixture of two or more systems are in vogue making the confusion all the greater. Though the SI (Le System International d'Unites or the International System of Units) units was first conceived and adopted in the year 1960 at the Eleventh General Conference of Weights and Measures held in Paris, the adoption of this coherent and systematically constituted system is still slow because of the past association with the FPS system. The conditions are now gradually changing and possibly in the near future the SI system will be the only system of use in all academic institutions in the world over. It is therefore essential to understand the basic philosophy of the SI units. The Basics of the SI System The SI system is a fully coherent and rationalized system. It consists of six basic units and two supplementary units, and several derived units. (Table A.I) Table A.1

Basic units of interest in geotechnical engineering

Quantity

Unit

SI symbol

1.

Length

Meter

m

2.

Mass Time Electric current Thermodynamic temperature

Kilogram Second Ampere Kelvin

kg S A

3. 4. 5.

K

987

Appendix A

988

Supplementary Units The supplementary units include the radian and steradian, the units of plane and solid angles, respectively.

Derived Units The derived units used by geotechnical engineers are tabulated in Table A.2. Prefixes are used to indicate multiples and submultiples of the basic and derived units as given below. Factor Prefix Symbol mega M 106 3 10 kilo k io-36 milli m micro (i io~

Table A. 2 Derived units Quantity

Unit

SI symbol

Formula

acceleration area density force pressure stress moment or torque unit weight frequency volume volume work (energy)

meter per second squared square meter

-

m/sec2 m2 kg/m3

N Pa Pa N-m

kg-m/s2 N/m 2 N/m 2 kg-m 2 /s 2 kg/s2m2 cycle/sec 10-3m3 N-m

kilogram per cubic meter newton pascal pascal newton-meter newton per cubic meter hertz cubic meter liter joule

N/m 3

Hz m3 L J

Mass Mass is a measure of the amount of matter an object contains. The mass remains the same even if the object's temperature and its location change. Kilogram, kg, is the unit used to measure the quantity of mass contained in an object. Sometimes Mg (megagram) and gram (g) are also used as a measure of mass in an object.

Time Although the second (s) is the basic SI time unit, minutes (min), hours (h), days (d) etc. may be used as and where required.

Force As per Newton's second law of motion, force, F, is expressed as F = Ma, where, M = mass expressed in kg, and a is acceleration in units of m/sec2. If the acceleration is g, the standard value of which is 9.80665 m/sec2 ~ 9.81 m/s2, the force F will be replaced by W, the weight of the body. Now the above equation may be written as W = Mg.

SI Units in Geotechnical Engineering

989

The correct unit to express the weight W, of an object is the newton since the weight is the gravitational force that causes a downward acceleration of the object. Newton, N, is defined as the force that causes a 1 kg mass to accelerate 1 m/s2 .kg-m or 1N = 1—E-r— s2

Since, a newton, is too small a unit for engineering usage, multiples of newtons expressed as kilonewton, kN, and meganewton, MN, are used. Some of the useful relationships are 1 kilonewton, kN = 103 newton = 1000 N meganewton, MN = 106 newton = 103 kN = 1000 kN Stress and Pressure The unit of stress and pressure in SI units is the pascal (Pa) which is equal to 1 newton per square meter (N/m2). Since a pascal is too small a unit, multiples of pascals are used as prefixes to express the unit of stress and pressure. In engineering practice kilopascals or megapascals are normally used. For example, 1 kilopascal = 1 kPa = 1 kN/m2 = 1000 N/m2 1 megapascal = 1 MPa = 1 MN/m2 = 1000 kN/m2 Density Density is defined as mass per unit volume. In the SI system of units, mass is expressed in kg/m3. In many cases, it may be more convenient to express density in megagrams per cubic meter or in gm per cubic centimeter. The relationships may be expressed as 1 g/cm3 = 1000 kg/m3 = 106 g/m3 = 1 Mg/m3 It may be noted here that the density of water, p^ is exactly 1.00 g/cm3 at 4 °C, and the variation is relatively small over the range of temperatures in ordinary engineering practice. It is sufficiently accurate to write pw = 1.00 g/cm3 = 103 kg/m3 = 1 Mg/m3 Unit weight Unit weight is still the common measurement in geotechnical engineering practice. The relationship between unit weight, 7 and density p, may be expressed as 7= pg. For example, if the density of water, pw = 1000 kg/m3, then ,..,„, = ,000 4 x 9*1 i = 9810 4 -!| mj s2 m3 s 2 Iro rn

N

Since, 1N = 1 -£—, y =9810—-j = 9.81 kN/m3 s2 m

990

Appendix A Table A.3

SI to FPS

To convert

Length

Conversion factors

FPS to SI

From

To

Multiply by

From

To

Multiply by

m m cm mm

ft in in in

3.281 39.37 0.3937 0.03937

ft in in in

m m cm mm

0.3048 0.0254 2.54 25.4

m2 m2 cm2 mm'

ft2 in2 in2 in2

10.764 1550 0.155 0.155x 10~2

ft2 in2 in2 in2

m2 m2 cm2 mm2

929.03 xlO^ 6.452x10^ 6.452 645.16

m3 m3 cm3

ft3 in3 in3

35.32 61,023.4 0.06102

ft3 in3

in3

m3 m3 cm3

28.317xlO- 3 1 6.387 x 10~6 16.387

Force

N kN kN kN

Ib Ib kip US ton

0.2248 224.8 0.2248 0.1124

Ib Ib kip US ton

N kN kN kN

4.448 4.448 x 10'3 4.448 8.896

Stress

N/m2 kN/m2 kN/m2 kN/m2 kN/m2

lb/ft2 lb/ft2 US ton/ft2 kip/ft2 lb/in2

20.885 xlO-3 20.885 0.01044 20.885 x 10-3 0.145

lb/ft2 lb/ft2 US ton/ft2 kip/ft2 lb/in2

N/m2 kN/m2 kN/m2 kN/m2 kN/m2

47.88 0.04788 95.76 47.88 6.895

Unit weight

kN/m3 kN/m3

lb/ft3 lb/in3

6.361 0.003682

lb/ft3 lb/in3

kN/m3 kN/m3

0.1572 271.43

Moment

N-m N-m

Ib-ft Ib-in

0.7375 8.851

Ib-ft Ib-in

N-m N-m

1.3558 0.11298

Moment of inertia

mm4 m4

in4 in4

2.402 x KT6 2.402 x 106

in4 in4

mm4 m4

0.4162 x 106 0.4162 x KT6

Section modulus

mm3 m3

in3 in3

6.102 x 1Q-5 6.102 x 104

in3 in3

mm3 m3

0.16387 x 105 0.16387 x 10-4

Hydraulic conductivity

m/min cm/min m/sec cm/sec

ft/min ft/min ft/sec in/sec

3.281 0.03281 3.281 0.3937

ft/min ft/min ft/sec in/sec

m/min cm/min m/sec cm/sec

0.3048 30.48 0.3048 2.54

Coefficient of consolidation

cm2/sec m2/year cm2/sec

in2/sec inVsec ft2/sec

0.155 4.915 x 10-5 1.0764 x lO'3

in2/sec in2/sec ft2/sec

cm2/sec m2/year cm2/sec

6.452 20.346 x 103 929.03

Volume

SI Units in Geotechnical Engineering Table A.4

991

Conversion factors —general

To convert from

To

Multiply by

Angstrom units

inches feet millimeters centimeters meters

3.9370079 10~9 3.28084 x 10-'° 1 xio- 7 1 x io~8 1 x io-10

Microns

inches

3.9370079 x 1(T5

US gallon (gal)

cm3 m3 ft3 liters

3785 3.785 x io-3 0.133680 3.785

Pounds

dynes grams kilograms

4.44822 x io5 453.59243 0.45359243

Tons (short or US tons)

kilograms pounds kips

907.1874 2000 2

Tons (metric)

grams kilograms pounds kips tons (short or US tons)

1 x IO 6 1000 2204.6223 2.2046223 1.1023112

kips/ft2

lbs/in2 lbs/ft2 US tons/ft2 kg/cm2 metric ton/ft2

6.94445 1000 0.5000 0.488244 4.88244

gms/cm3 kg/m3 lbs/ft3

27.6799 27679.905 1728 4

Pounds/in

3

Poise

kN-sec/m2

millipoise

poise kN-sec/m2 gm-sec/cm2

ioio-73 ioicr6

ft/mm

ft/day ft/year

1440 5256 x IO 2

ft/year

ft/min

1.9025 x 1Q-6

cm/sec

m/min ft/min ft/year

0.600 1.9685 1034643.6

APPENDIX B SLOPE STABILITY CHARTS AND TABLES As per Eq.( 10.43), the factor of safety Fs is defined as Fs = m-nru where, m, n = stability coefficients, and ru = pore pressure ratio. The values of m and n may be obtained from Figs. B.I to B.I4

3:1 4:1 Slope cot/?

5:1

3:1 4:1 Slope cot /3

Figure B.1 Stability coefficients m and n for c'lyH = 0 (Bishop and Morgenstern, 1960) 993

994

Appendix B = >'40° = J>40°

2:1

3:1 4:1 Slope cot

5:1

2:1

ft

3:1 4:1 Slope cot /?

5:1

Figure B.2 Stability coefficients for c'ljH = 0.025 and nd = 1.00 (Bishop and Morgenstern, 1960)

40°

2:1

Figure B.3

3:1

4:1

5:1

5

2:1

3:1

4:1

5:1

Stability coefficients m and /? for c'lyH = 0.025 and nd = 1.25 (Bishop and Morgenstern, 1960)

Slope Stability Charts

995

3:1

4:1

5:1

2:1

3:1

cot/J

Figure B.4

2:1

cot/3

Stability coefficients m and n for c'/yH = 0.05 and nd = 1.00 (Bishop and Morgenstern, 1960)

3:1

4:1 cot/?

Figure B.5

4:1

5:1

2:1

3:1

4:1

5:1

cot/3

Stability coefficients m and n for c'lyH = 0.05 and nd = 1.25 (Bishop and Morgenstern, 1960)

996

Appendix B 40°

4

30

Figure B.6

5:1

2:1

5:1

Stability coefficients m and n for c'lyH = 0.05 and nd = 1.50 (Bishop and Morgenstern, 1960)

40° 35° 30" 25° 20°

5

40°

4

35°

n

30° 3

25° 20°

3

4

cot/?

Figure B.7

3

4

cot/?

Stability coefficients m and /? for c'lyH = 0.075 toe circles (O'Connor and Mitchell, 1977)

997

Slope Stability Charts

40°

35° 30°

25° 20°

0

3

4

5

cot/3 Figure B.8

0

"2

3

4

5

cot/3

Stability coefficients m and n for c'lyH = 0.075 and nd = 1.00 (O'Connor et al., 1977)

3

4

5

cot/? Figure B.9

Stability coefficients m and n for c'lyH = 0.075 and nd = 1.25 (O'Connor and Mitchell, 1977)

998

Appendix B

35°

30°

25°

20°

40°

5

35°

4

n

30° 3

25° 20°

2 =~

3

4

3

Figure B.10

4

cot/3

cot 3

Stability coefficients m and A? for c'lyH = 0.075 and nd = 1.50 (O'Connor and Mitchell, 1977) 40°

35 C

40°

30°

35°

25° 30° 20°

25° 20°

3

4

cot/?

Figure B.11

3

4

cot/3

Stability coefficients m and n for c'ljH = 0.100 toe circles (O'Connor and Mitchell, 1977)

Slope Stability Charts

999 40° 35° 30° 25° /u 90°

n

7

2

3

4

5

cot/3

Figure B.12 Stability coefficients m and n for c'/yH = 0.100 and nd = 1.00 (O'Connor and Mitchell, 1977)

40°

35°

30° 25° 20°

2

3

4

5

cot/3

Figure B.13

Stability coefficients m and n for c'lyH = 0.100 and nd = 1.25 (O'Connor and Mitchell, 1977)

1000

Appendix B

40°

35°

30° 25° 20°

5

2

3

4

5

cot/3

Figure B.14

Stability coefficients m and n for c'lyH = 0.100 and nd = 1.50 (O'Connor and Mitchell, 1977)

Slope Stability Charts

1001

Bishop and Morgenstern (1960) Stability Coefficients are Presented in Tabular Form F - m -n.r c' _ Table B1 Stability coefficients m and n for ~77 yh ~ ° Stability coefficients for earth slopes Slope 2:1 0'

Slope 4:1

Slope 3:1

Slope 5:1

m

n

m

n

m

n

m

n

10.0 12.5 15.0 17.5

0.353 0.443 0.536 0.631

0.441 0.554 0.670 0.789

0.529 0.665 0.804 0.946

0.588 0.739 0.893 1.051

0.705 0.887 1.072 1.261

0.749 0.943 1.139 1.340

0.882 1.109 1.340 1.577

0.917 1.153 1.393 1.639

20.0 22.5 25.0 27.5

0.728 0.828 0.933 1.041

0.910 1.035 .166 .301

1.092 1.243 1.399 1.562

1.213 1.381 1.554 1.736

1.456 1.657 1.865 2.082

1.547 1.761 1.982 2.213

1.820 2.071 2.332 2.603

1.892 2.153 2.424 2.706

30.0 32.5 35.0 37.5

1.155 1.274 1.400 1.535

.444 .593 .750 1.919

1.732 1.911 2.101 2.302

1.924 2.123 2.334 2.588

2.309 2.548 2.801 3.069

2.454 2.708 2.977 3.261

2.887 3.185 3.501 3.837

3.001 3.311 3.639 3.989

40.0

1.678

2.098

2.517

2.797

3.356

3.566

4.196

4.362

Table B2 Stability coefficients m and n for ~TJ - 0-025 Slope 2:1

Slope 3:1

anc| n^

=