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^
=