A Refined Formula for the Allowable Soil Pressure Using Shear Wave Velocities
ABSTRACT Based on a variety of case histories of site investigations, including extensive bore hole data, laboratory testing and geophysical prospecting at more than 550 construction sites, an empirical formulation is proposed for the rapid determination of allowable bearing pressure of shallow foundations in soils and rocks. The proposed expression corroborates consistently with the results of the classical theory and is proven to be rapid, and reliable. Plate load tests have been also carried out at three different sites, in order to further confirm the validity of the proposed method. It consists of only two soil parameters, namely, the in situ measured shear wave velocity and the unit weight. The unit weight may be also determined with sufficient accuracy, by means of other empirical expressions proposed, using P or S wave velocities. It is indicated that once the shear and Pwave velocities are measured in situ by an appropriate geophysical survey, the allowable bearing pressure as well as the coefficient of subgrade reaction and many other elasticity parameters may be determined rapidly and reliably.
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 Cited In (0)
 Geophysics 01/1989; 54(1):82. · 1.72 Impact Factor

Article: Evaluation of in situ anisotropy from crosshole and downhole shear wave velocity measurements
Géotechnique 01/1995; 45(2):267238. · 1.48 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The existence of interface waves along fractures depends on the specific stiffness of the fracture. With increasing fracture stiffness, the velocity of the interface wave increases and the spectral content of the signal is altered. We used interface wave measurements and wavelet analysis to examine the effect of normal and shear stresses on fracture shear stiffness. Fracture shear stiffness is more sensitive to changes in shear stress than to normal stress.Compressional waves propagated along a fracture appear to contain a delayed wave that is sensitive to changes in the fracture stiffness. Using acoustic wavefront imaging, we are able to visualize the delayed produced in the compressionalwave by the fracture from the spatial distribution of the arriving energy.Journal of Applied Geophysics 01/1996; · 1.33 Impact Factor
Page 1
The Open Civil Engineering Journal, 2011, 5, 18 1
18741495/11 2011 Bentham Open
Open Access
A Refined Formula for the Allowable Soil Pressure Using Shear Wave Velocities
S. S. Tezcan1,* and Z. Ozdemir2
1Civil Engineering, Bogazici University, Bebek, Istanbul, Turkey
2Bogazici University, Istanbul, Turkey
Abstract: Based on a variety of case histories of site investigations, including extensive bore hole data, laboratory testing
and geophysical prospecting at more than 550 construction sites, an empirical formulation is proposed for the rapid de
termination of allowable bearing pressure of shallow foundations in soils and rocks. The proposed expression corroborates
consistently with the results of the classical theory and is proven to be rapid, and reliable. Plate load tests have been
also carried out at three different sites, in order to further confirm the validity of the proposed method. It consists of only
two soil parameters, namely, the in situ measured shear wave velocity and the unit weight. The unit weight may be also
determined with sufficient accuracy, by means of other empirical expressions proposed, using P or S  wave velocities.
It is indicated that once the shear and Pwave velocities are measured in situ by an appropriate geophysical survey, the
allowable bearing pressure as well as the coefficient of subgrade reaction and many other elasticity parameters may be
determined rapidly and reliably.
Keywords: Shear wave velocity, shallow foundations, allowable bearing pressure, dynamic technique, soils and rocks.
1. INTRODUCTION
soil mechanics and foundation engineering in Germany,
stated in 1943 that “For the determination of allowable bear
ing pressure, the geophysical methods, utilising seismic
wave velocity measuring techniques with absolutely no dis
turbance of natural site conditions, may yield relatively more
realistic results than those of the geotechnical methods,
which are based primarily on bore hole data and laboratory
testing of socalled undisturbed soil samples ”.
Professor Schulze [1], a prominent historical figure in
been made to solving geotechnical problems by means of
geophysical prospecting. The Pwave velocities, for instance,
have been used to determine the unconfined compressive
strengths and modulus of elasticity of soil samples by Coates
[2]. Hardin and Black [3], and also Hardin and Drnevich [4],
based on extensive experimental data, established indispen
sable relations between the shear wave velocity, void ratio,
and shear rigidity of soils. Similarly, Ohkubo and Terasaki
[5] supplied various expressions relating the seismic wave
velocities to weight density, permeability, water content,
unconfined compressive strength and modulus of elasticity.
Since that time, various significant contributions have
been extensively studied for the purpose of determining the
properties of soils and rocks by Imai and Yoshimura [6],
Tatham [7], Willkens, et al. [8], Phillips, et al. [9], Keceli
[10, 11], Jongmans [12], Sully and Campanella [13], and
PyrakNolte, et al. [14]. Imai and Yoshimura [6] proposed
The use of geophysical methods in soil mechanics has
*Address correspondence to this author at the Civil Engineering, Bogazici
University, Bebek, Istanbul, Turkey; Tel: +90. 212. 352 65 59;
Fax: +90. 212. 352 65 58; Mobile: +90. 532. 371 03 40;
Email: tezokan @ gmail. com
an empirical expression for the determination of bearing
capacity qf and / or qa as
nqa = qf = Vs
which yields values unacceptably much higher than the clas
sical theory as will be evident in next section. Campanella
and Stewart [15], determined various soil parameters by
digital signal processing, while Butcher and Powell [16],
supplied practical geophysical techniques to assess various
soil parameters related to ground stiffness. An empirical ex
pression is also proposed by Abd ElRahman [17], for the
ultimate bearing capacity of soils, using the logarithm of
shear wave velocity.
A series of guidelines have been also prepared in this
respect by the Technical Committee TC 16 of IRTP, ISS
MGE [18], and also by Sieffert [19]. Keceli [11], Turker
[20], based on extensive case studies, supplied explicit ex
pressions for the allowable bearing pressure, using shear
wave velocity. In this paper, the earlier formula presented by
Tezcan, et al. [21], has been calibrated and improved with
the soil data of 550 construction sites. Massarsch [22] deter
mined deformation properties of finegrained soils from
seismic tests. As to the in situ measurement of P and S –
wave velocities, various alternate techniques are available as
outlined in detail by Stokoe and Woods [23], Tezcan, et al.
[24], Butcher, et al. [25], Richart, et al. [26], Kramer [27],
Santamarina, et al. [28].
2.4 / (1590) (kPa) (1)
2. THEORETICAL BASIS FOR THE EMPIRICAL
EXPRESSION
expression for the allowable soil pressure qa  underneath a
shallow foundation, the systematic boundary value approach
used earlier by Keceli [11] will be followed. The state of
In order to be able to arrive at a particular empirical
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2 The Open Civil Engineering Journal, 2011, Volume 5 Tezcan and Ozdemir
stress and the related elastic parameters of a typical soil col
umn is shown in Fig. (1). Considering a foundation depth of
Df with a unit crosssectional area of A=1, the typical form
of the compressive ultimate bearing capacity at the base of
the foundation nothing but only as a format, may be written
approximately as;
qf = ? Df
qa = qf / n= ? Df / n
where qf = ultimate bearing capacity at failure, ? = unit
weight of soil above the base of the foundation, qa = allow
able bearing pressure, and n= factor of safety. In order to be
able to incorporate the shear wave velocity Vs2 into the above
expressions, the depth parameter Df will be expressed as ve
locity multiplied by time as;
Df = Vs2 t
in which, the Vs2 is purposely selected to be the shear wave
velocity measured under the foundation, t = is an unknown
time parameter. The time parameter t is introduced herein
just as a dummy parameter in order to keep consistency in
appropriate units. Substituting eqn (4) into eqn (3), yields
(2)
(3)
(4)
qa = ? Vs2 t / n
The unknown time parameter t, will be determined on the
basis of a calibration process. For this purpose, a typical
‘hard’ rock formation will be assumed to exist under the
foundation, with the following parameters, as suggested ear
lier by Keceli [11];
qa = 10 000 kN/m2, Vs2 = 4 000 m/sec
? = 35 kN/m3, n = 1.4
Substituting these numerical values into eqn (5), we obtain t
= 0.10 sec, thus;
(5)
(6)
qa = 0.1 ? Vs2 / n
This is the desired empirical expression to determine the
allowable bearing pressure qa , in soils and rocks, once the
average unit weight, ? , for the soil layer above the founda
tion and the in situ measured Vs2  wave velocity for the
soil layer just below the foundation base are available. The
(7)
unit of Vs2 is in m / sec, the unit of ? is in kN / m3, then the
resulting qa – value is in units of kPa. The unit weight values
may be estimated using the empirical expressions;
?p = ?0 + 0.002 Vp1 and ?s = 4.3 Vs1
as proposed earlier by Tezcan et al. [21], and by Keceli [29],
respectively. The second expression is especially recom
mended for granular soils, for which the measured Vs1 values
represent appropriately the degree of water content and / or
porosity. The wave velocities must be in units of m / sec. The
only remaining unknown parameter is the factor of safety, n,
which is assumed to be, after a series of calibration proc
esses, as follows:
0.25 (8a) (8b)
n = 1.4 (for Vs2 ? 4 000 m/sec), (9)
n = 4.0 (for Vs2 ? 750 m/sec)
The calibration process is based primarily on the refer
ence qa – values determined by the conventional Terzaghi
method, for all the data sets corresponding to the 550 – con
struction sites considered. For Vs2 values greater than 750
m/sec and smaller than 4 000 m/sec a linear interpolation is
recommended. The engineering rock formations are assumed
to start for Vs2 > 750 m / sec. The factors of safety, as well as
the empirical allowable bearing pressure expressions, for
various soil (rock) types, are given in Table 1. It is seen that
three distinct ranges of values are assumed for n = factor of
safety. For soil types with Vs2 ? 750 m/sec the factor of
safety is n = 4, for rocks with Vs2 ? 4 000 m/sec it is n= 1.4 .
For other intermediate values of shear wave velocity, linear
interpolation is recommended. The validity of these values
has been extensively checked and calibrated by the soil data
at 550 construction sites. The relatively higher value of fac
tor of safety assumed for soils is deemed to be appropriate to
compensate the inaccuracies and gaps existing in the meas
ured values of shear wave velocity. In fact, Terzaghi and
Peck [30] states that “The factor of safety of the foundation
with respect to breaking into the ground should not be less
than about 3”.
It is determined by Terzaghi and Peck [30] also that the
width of footing, B, has a reducing influence on the value of
allowable bearing pressure for granular soils. Therefore, a
correction factor ?? is introduced into the formula, for sandy
Fig. (1). Soil column and related parameters.
qa = Allowable stress (qa = qf / n)
qf = Ultimate bearing pressure
qf = γ γ Df ; ; qa = γ γ Df / n
d =Settlement of layer H
d = Pa H / AE = qa H/ E
d = qa / ks ks = qa / d
Assuming d = 0.025 m
ks = qa / 0.025 = 40qa (kN / m3)
qa = ks d
Pa = Aqa
Pf = A qf
d
Df
(Vp2 / Vs2)
Layer 1
Foundation
depth
H
Layer 2
(Vp1 / Vs1)
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A Refined Formula for the Allowable Soil Pressure Using Shear Wave Velocities The Open Civil Engineering Journal, 2011, Volume 5 3
soils only, as shown in the third line of Table 1. The pro
posed values of this correction factor, for different founda
tion width B, are as follows:
? = 1.00
? = 1.13 – 0.11 B
? = 0.83 – 0.01 B
for (0 ? B ? 1.20 m)
for (1.2 ? B ? 3.00 m)
for (3.0 ? B ? 12.0 m)
(10)
3. COEFFICIENT OF SUBGRADE REACTION
determine ks=coefficient of subgrade reaction of the soil
layer just beneath the foundation base by making use of
the expressions given in Fig. (1). The coefficient of subgrade
reaction ks , is defined, similar to the definition of spring
constant in engineering mechanics, to be the necessary verti
cal pressure to produce a unit vertical displacement and ex
pressed as;
The shear wave velocity may be used successfully to
ks = qa / d
For shallow foundations, the total vertical displacement
is restricted to 1 inch =0.025 m, as prescribed by Terzaghi
and Peck [30]. When, d=0.025 m is substituted in eqn (11),
the coefficient of subgrade reaction becomes in units of
kN/m3;
(11)
ks = 40 qa or,
4. ELASTICITY PARAMETERS
ks = 4? Vs2 / n (12, 13)
by geophysical means, for the soil layer No.2 just under the
foundation, several parameters of elasticity, such as G =
Shear modulus, Ec = Constraint modulus of elasticity, E =
Modulus of elasticity (Young’s modulus), Ek = Bulk
modulus, and μ = Poisson’s ratio may be obtained easily.
The Shear modulus, G, and the Constraint modulus, Ec , are
related to the shear and P wave velocities by the following
expressions, respectively;
Once, Vp2 and Vs2 seismic wave velocities are measured,
G = ? Vs
where, ? = mass density given by ? =? / g . From the Theory
of Elasticity, it is known that, E = the Young’s modulus of
elasticity is related to Ec = the Constraint modulus and also
to G = the Shear modulus by the following expressions:
E = Ec (1 + μ ) (1 – 2μ ) / (1  μ)
E =2 (1 + μ) G
Utilising eqn (14) and (15) and also substituting ? , as
? = Ec / G = (Vp / Vs)2
into eqn (16) and (17), we obtain
2????????????????and Ec = ??? Vp
2 (14, 15)
(16)
(17)
(18)
μ = (? – 2) / 2 (? – 1) or, ? = (2μ – 2) / (2μ – 1) (19, 20)
The modulus of elasticity is directly obtained from eqn (17)
as;
E = (3? – 4) G / (? – 1)
The Constraint modulus Ec , may be also obtained in terms of
? as ;
Ec =? (? 1) E / (3? – 4) or, Ec = ? Vp
The Bulk modulus Ek, of the soil layer, may be expressed,
from the theory of elasticity, as
(21)
2 / g (22, 23)
Ek = E / 3 (1 – 2μ) … (24)
Ek = (?  1) E / 3 = ? ( Vp
5. CASE STUDIES
2 ? 4 Vs
2 / 3) / g (25)
mined at more than 550 construction sites in and around the
Kocaeli and Istanbul Provinces in Turkey, between the years
200510. At each construction site, by virtue of City bylaw,
appropriate number of bore holes were drilled, SPT counts
conducted, undisturbed soil samples were taken for labora
tory testing purposes, where shear strength c, the internal
angle of friction ?, unconfined compression strength qu and
unit weight ? were determined. Subsequently, following the
classical procedure of Terzaghi and Peck [30], the ultimate
capacity and also the allowable bearing pressures were de
termined, by assuming the factor of safety as n=3. For
granular soils, immediate settlement calculations were also
conducted, in order to determine whether the shear failure
mechanism or the maximum settlement criterion would con
trol the design.
The numerical values of the allowable bearing pressures,
qa , determined in accordance with the conventional Terzaghi
theory, are shown by a triangular (?) symbol, in Fig. (2),
where the three digit numbers refer to the data base file
numbers of specific construction sites. Parallel to these clas
sical soil investigations, the P and S wave velocities have
been measured in situ, right at the foundation level for the
purpose of determining the allowable bearing pressures, qa ,
which are shown by means of a circle (o), in Fig. (2). Two
separate linear regression lines were also shown in Fig. (2),
for the purpose of indicating the average values of allowable
bearing pressures determined by ‘dynamic’ and ‘conven
tional’ methods. In order to obtain an idea about the relative
conservatism of the two methods, the ratios of allowable
bearing pressures (r = qad / qac), as determined by the ‘dy
namic’ and ‘conventional’ methods, have been plotted
against the Vs – values in Fig. (3).
The allowable bearing pressures have been also deter
Table 1. Factors of Safety, n, for Soils and Rocks(1)
Soil Type Vs – Range (m/sec) n qa (kN/m2)
‘Hard’ rocks
‘Soft’ rocks
Soils
75 Vs ? 4 000
750? Vs ? 4 000
750 ?Vs
n = 1.4
n =4.6–8.104 Vs
n = 4.0
qa = 0.071 ? Vs
qa = 0.1 ? Vs / n
qa = 0.025 ? Vs ?
(1)Linear interpolation is applied for 750 ? Vs ? 4 000 m/sec.
? , correction factor is used for sands only (eqn 10).
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4 The Open Civil Engineering Journal, 2011, Volume 5 Tezcan and Ozdemir
Vs – values smaller than 400 m/sec a narrow band of r = 1.03
to r = 1.12, which should be regarded as quite acceptable.
The ‘dynamic’ method proposed herein yields allowable
bearing pressures slightly (on the order of 3 to 10 percent)
greater than those of the ‘conventional’ method for Vs –
values smaller than 400 m / sec. In fact, the ‘conventional’
method fails to produce reliable and consistent results for
relatively strong soils and soft rocks, because it is difficult to
determine the appropriate soil parameters c, and ? for use in
the ‘conventional’ method. At construction site Nos: 133,
134, 138, 139, 206, 207, 214, 215, 219, 502, 507 and 544,
where the soil conditions have been mostly weathered ande
site, granodiorite arena, greywacke, limestone, etc did not
allow for the measurement of c and ?  values. Therefore, the
use of ‘dynamic’ method becomes inevitable for such strong
soils with Vs2 > 400 m / sec.
It is seen that the linear regression line indicates for
by laboratory testing through geotechnical prospecting, as
well as the in situ measured Vp and Vs – velocities at each of
the 550 construction sites, are too voluminous to be included
herein. Those researchers interested to have access to
these particular data base, may inquire from internet
<tezokan@superonline.com>, <www.tezokan.com>.
The list of soil parameters determined by in situ and also
6. SEISMIC WAVE VELOCITIES
– and S – geophones by means of a 24 – Channel Geometrics
Abem – Pasi seismic instrument, capable of noise filtering.
The P – waves have been generated by hitting 6 – blows
vertically, with a 0.15 kN hammer, onto a 250 x 250 x 16 mm
size steel plate placed horizontally on ground. For the pur
pose of generating S – waves however, an open ditch of size
1.4 x 1.4 x 1.4 m was excavated and then two steel plates
The seismic wave velocities have been measured using P
Fig. (2). Comparative results of ‘Conventional’ and ‘Dynamic’ methods.
Fig. (3). Ratios of allowable bearing pressures ( qa,d / qa,c ) as determined by the ‘dynamic’ and the ‘conventional’ methods.
350
300
250
200
150
100
50
0
250 300 350 400 450 500 550 600 650 700
Vs2 = shear wave velocity, m/sec
qa = allowable bearing pressure, kPa
133
133
128
128
502
493 497
205
498
218
502
210
421
544
355
314
178
178 145
192
137
385
188
Dynamic data points
Conventional data points
by Conventional Method (Terzaghi and Peck, [30])
by Dynamic Method (Tezcan, et.al )
Linear regression lines :
r = qa,d / qa,c
Vs2 = shear wave velocity, m/sec
2,00
1,75
1,50
1,25
1,00
0,75
0,50
200 300 400 500 700 600
133
St 499
St 502
St 128
St 178
St 188
Linear regression line
Page 5
A Refined Formula for the Allowable Soil Pressure Using Shear Wave Velocities The Open Civil Engineering Journal, 2011, Volume 5 5
were placed on opposite vertical faces of this ditch parallel to
the conterline of the geophones. Using the same 0.15 kN
hammer, 6 heavy horizontal blows were applied onto each of
these vertical steel plates. The necessary polarity of the S –
wawes was achieved by hitting these vertical steel plates
horizontally in opposite directions, nonconcurrently.
7. PLATE LOAD TESTING
sures determined by various methods, plate loading tests
have been carried out at three particular construction Sites
For purposes of correlating the allowable bearing pres
Nos: 335, 502 and 544. The soil parameters c, qu , and ? as
determined by laboratory testing, as well as the P and S –
wave velocities measured at site by geophysical prospecting
are all shown in Table 3. A thick steel bearing plate of 316.2
mm x 316.2 mm = 0.10 square meter in size is used under the
test platform of size 1.50 m by 1.50 m. The tests are carried
out right at the bottom elevations of foundations. One half of
the bearing pressure ?0 , which produced a settlement of s =
12.7 mm was selected as the allowable pressure qa as shown
in Fig. (4). It is seen clearly in Table 2 that the results of the
proposed ‘dynamic’ method using P and S – wave velocities
are in very close agreement with those of the plate load test
Fig. (4). Load test results at Sites No: 335, 502, and 544.
Table 2. Comparative Evaluation of Allowable Pressures
Various Soil Parameters (? ? = 0)
qa = Allowable Pressure
qu
(1) Df c
? ?lab Vp2 Vs2
Terzaghi(2)
Eq. 26
Tezcan, et al.(3)
Eq. 7
Load test
Fig. 4
Site No
Owner
Lot Nos
(soil type)
kPa m kPa kN/m3 m/sec m/sec
kPa
kPa
kPa
335
Suleyman Turan
8 Paft./A/930 Pars.
(silty clay)
172 1.50 86
18.9
?0 = 16
896 390 157
173
180
544
Ayhan Dede
G22B / 574 / 11
(weathered diorite)
190 1.50 95
18.0
?0 = 16
1 020 453 172
204
208
502
Ebru Çınar
30 L1C / 440 / 8
(clay stone)
147 1.00 140
22.7
?0 = 20
1 210 489 248
274
280
(1) qu = unconfined compressive strength.
(2) Terzaghi and Peck (1976).
(3) qa = 0.025 ?p Vs (Eq.7), n = 4.
σ = pressure under the test plate, kPa
(σ0 = 2 qa = pressure, which produces s = 12.7 mm )
s = settlement , mm
0 100 200 300 400 500 600 700 800 900 1000
24
20
6
2
8
4
0
qa = 0.5 σ σ0
x
σ σ0 =560
(s = 0.5 inch = 12.7 mm)
σ σ0 =416
σ σ0 =360
x
x
x
x