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The Open Civil Engineering Journal, 2011, 5, 1-8 1

1874-1495/11 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 P-wave 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 so-called undisturbed soil samples ”.

Professor Schulze [1], a prominent historical figure in

been made to solving geotechnical problems by means of

geophysical prospecting. The P-wave 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

Pyrak-Nolte, 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;

E-mail: 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 El-Rahman [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 fine-grained 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 cross-sectional 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

2005-10. At each construction site, by virtue of City by-law,

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 =4.6–8.10-4 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

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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