ALLOWABLE
BEARING CAPACITY
OF
SHALLOW FOUNDATIONS
BASED ON SHEAR
WAVE VELOCITY
Semih S. Tezcan , Ali Keceli , Zuhal Ozdemir
ABSTRACT
Firstly, the
historical background is presented for the determination of ultimate bearing
capacity of shallow foundations. The principles of plastic equilibrium used in
the classical formulation of the ultimate bearing capacity are reviewed,
followed by a discussion about the sources of approximations inherent in the
classical theory.
Secondly, based
on a variety of case histories of site investigations, including extensive bore
hole data, laboratory testing and geophysical prospecting, an empirical
formulation is proposed for the determination of allowable bearing capacity of
shallow foundations. The proposed expression corroborates consistently with the
results of the classical theory and is proven to be reliable and safe, also
from the view point of maximum allowable settlements. It consists of only two
soil parameters, namely, the insitu measured shear wave velocity, and the unit
weight. The unit weight may be also determined with sufficient accuracy, by
means of another empirical expression, using the P-wave velocity. It is
indicated that once the shear and P-wave velocities are measured insitu by an
appropriate geophysical survey, the allowable bearing capacity is determined
reliably through a single step operation. Such an approach, is considerably
cost and time-saving, in practice.
Key words : bearing
capacity, shear wave, foundation design, shallow footings, allowable bearing pressure
1. Introduction
The
ultimate bearing capacity of a particular soil, under a shallow footing, was
investigated theoretically by Prandtl (1921) [5] and Reissner (1924) [6]
using the concept of plastic equilibrium as early as in 1921. The formulation
however is slightly modified, generalised, and updated later by Terzaghi (1925)
[12],
Meyerhof (1956) [4], Hansen (1968) [3], De Beer (1970) [2],
and Sieffert et al. (2000) [7]
.
The
historical bearing capacity formulation, as will be discussed briefly in the
next Section, is still widely used in geotechnical engineering practice.
However, there are various uncertainities in representing the real insitu soil
conditions by means of a few laboratory tested shear strength parameters. The basic soil parameters are cu = cohesion, undrained
shear strength and φ = angle of
internal friction, which can only be determined by laboratory testing of
undisturbed soil samples. It is sometimes impossible to take undisturbed soil
samples especially in sandy and gravelly soils.
The
insitu measured shear wave velocity, vs , however as a single field index
represents the real soil conditions, much more effectively and reliably than
the laboratory tested shear strength parameters. In addition to geophysical
refraction seismic survey, there are several other techniques of measuring the
shear wave velocity at the site as discussed by Stokoe et al. (1972) [11], Tezcan et al. (1975) [14].
Because, the insitu measured shear wave velocity, vs , reflects the true
photograph of the soil, containing
the contributions of the void ratio, effective confining stresses,
stress history, shear and compressive strengths, geologic age etc. As will be seen later in
this study, the shear wave velocity, vs , enables the practicing engineer to
determine the allowable bearing capacity, qa , in a most convenient, reliable and
straight forward manner.
2. Classical
formulation
Using
the principles of plastic equilibrium, the ultimate bearing capacity, qf , of a shallow strip
footing, with a depth of D, from the
surface and with a width of B and length L, ( Figure 1) , is given by Terzaghi (1967) [13] as ,
qf = c Nc sc + g D Nq + 0.5 g B Ng sg (1)
where,
a) Bearing capacity factors;
Nq
= exp (p
tan φ)
tan2 (450 + φ /2)
Nc
= (Nq - 1) cot φ
Ng = 1.8 (Nq -1)
tan φ by
Hansen (1968) [3]
or
Ng = (Nq - 1) tan (1.4 φ) .by
Meyerhof (1956) [4]
b) Shape factors:
sc
= 1 + 0.20 B / L …………………………….……………. (φ ≠ 0
conditions)
sc = [1 + 0.20 B / L] [1 + 0.3 (D / B)0.25 ] ….. (φ = 0 conditions, saturated clays)
sg = 1 - 0.2 (B / L) ………………………. (B / L = footing width to length ratio)
sg = 0.6 ……………………………………………………… (circular
footing)
It is customary to take B / L = 0 for a
strip footing, and B / L = 1 for a square footing. The formulation is
applicable to ‘shallow’ foundations
in which the depth D , is not greater than the
breadth B. The foundation shape factor expression of sc given above for saturated clays under undrained
conditions, where f = 0, is generated using the Nc curves supplied by Skempton (1951) [8].
If the soil
is ‘weak’, or
in other words
is not fairly dense or stiff, i.e. Dr < 0.35 , N60
< 8 , cu < 100 kPa , or
vs < 200 m/sec,
the reduced shear strength parameters cr and fr are
used in Eq.1, instead of the laboratory determined c and
f, as follows [13] :
cr
= 0.67 c (2a)
tan
fr = 0.67 tan f (2b)
3. Sources of
approximations in classical approach
The approximations involved in the derivation and use of the ultimate
bearing capacity, qf ,
given by Eq.1, may be
summarized as follows:
a) The soil mass
is assumed to be purely homogeneous and isotropic, while the soil in nature is extremely heteregenous and
tixotropic, further the classical theory is developed only for a planar case,
while all footings are 3- dimensional in real behaviour.
b) The first term
of Eq.1
represents the shear strength, the
second term is the contribution of the surcharge pressure due to the depth of
foundation, and the third term represents the contribution of the self-weight.
It is only an approximation to superimpose the contributions of various load
cases in an entirely nonlinear plastic stress-strain environment.
c) The
contribution of self-weight can be determined only approximately, by numerical
or graphical means, for which no exact formulation is available.
d) The shear
strength of soil within a depth D , from the
surface is neglected.
e) Depending on
the degree of, compressibility of the soil, there may be three types of failure
modes; (i) general shear, (ii)
local shear, and (iii) punching shear, as shown in Figure
1. The theoretical considerations behind Eq.1, correspond only to the general
shear mode, which is typical for soils of low compressibility, such as
dense sands and stiff clays. In the local
shear failure, only a partial state of plastic equilibrium is developed
with significant compression under the footing. In the punching shear mode, however, direct planar shear failures occur
only along the vertical directions around the edges of the footing. Therefore, Eq.1 is no longer applicable for soils
of high compressibility, such as loose sand and soft clay, which may undergo,
either (ii) the local shear or
(iii)
the punching shear failures.
Consequently, the results of Eq.1
will only be approximate for such soils. In reality, the excessive settlement
and not the shear failure is normally the limiting criterion in high
compressibility soils.
f) The ultimate
bearing capacity calculations are very sensitive to the values of shear
strength parameters c , and f , which are
determined in the laboratory using ‘undisturbed’ soil samples, which may not
necessarily represent the true conditions prevailing at the site.
Unrealistically, high bearing capacity is calculated especially, if the shear
strength parameter, f , is
inappropriately determined to be on the high side in the laboratory. All soil
parameters including the real values of internal angle of friction, water
content, void ratio, confining pressure, presence of boulders or cavities, etc
are not necessarily the same in the soil samples.
g) Customarily,
after a due geotechnical survey, a single value of allowable bearing capacity qa , is assigned in practice, to a
particular construction site. However, minor variations in sizes, shapes and
depths of different foundations at a particular site are overlooked, and the
same qa value is used in
foundation design, through- out the construction site.
h) A factor of
safety of 3 is used normally, in order to obtain the allowable bearing capacity,
qa , which
contains a significant amount of reserve strength in it, accounting for all the
inaccuracies and approximations cited herein.
This significantly large factor of safety represents the degree of
uncertainties and our ‘ignorance’ in determining the real soil conditions.
i)
Last, but not the least, although
some quantitative guidance is available as contained in Section 2, there is quite a bit of intuition in determining whether
the soil is on the ‘strong’ or the ‘weak’ side, for the purpose of using reduced (two
thirds) shear strength parameters, in accordance with Eq 2.
4. Practical
recommendations
Based on the practical experiences
of the writers, the ranges of allowable bearing capacities for different
categories of cohesive and granular soils are summarized in Table 1. For comparisons as well as for
quick reference purposes, the values of SPT counts N60 , shear strength parameters cu , and
f , relative
density Dr , and also the shear wave
velocity vs , for
each soil category are also given in Table
1. The ranges of allowable bearing pressures qa , are
tested to be in conformity with the empirical recommendations of the UBC-97 (1997) [16], the Turkish
Earthquake Code TEC-1998 [15], and the BS 8004 (1986) [1].
5. Use of
shear wave velocity
a ) For control of settlements
Based on numerous case studies, as
discussed in the subsequent Sections, the allowable bearing capacity, qa , under a shallow foundation in units of kPa, may be obtained from the following empirical expressions:
qa = 0.024 g
vs
(3a)
qa = 2.4 ( 10 -4 ) r vs (3b)
where, g = unit weight (kN/m3), r = mass density (kg/m3), and vs = shear wave velocity (m/sec). Since, a proper foundation
design must satisfy not only an assured degree of safety against possible shear
failures of the supporting soil, but also the settlements, and in particular
the differential settlements, should not exceed the tolerable limits as given
by Skempton et al. (1956) [10] . Hence, the coefficient of the
empirical formula in Eq. 3 is so
selected to be on the low side, that no settlement problem will necessarily be
encountered in relatively soft soil conditions. This point has been rigorously
tested and verified for all soft ‘weak’ soil conditions existing in the case
histories given Table 2.
Although, the empirical expressions
of Eq. 3, are proposed by the
writers, on the basis of extensive geotechnical and geophysical soil
investigations at 14 different sites, they should be used with caution. For
relatively important buildings, and especially until a stage when the validity
of these simple empirical expressions are amply tested and calibrated over a
sufficient period of time, the allowable bearing pressure should be determined
also by means of conventional methods using Terzaghi’s soil parameters.
The proposed empirical expressions
are for estimating the allowable bearing pressure only. The settlement
calculations however, should be conducted, especially for soft soil conditions
and for important structures, using either the elastic theory [10] , or the Skempton-Bjerrum method [9] . Because, settlements sometime may
be the dominating factor.
b ) For setting an upper ceiling for
qa
In order to set a practical upper
ceiling for the allowable bearing capacity, qa , especially for the rocky
formations the empirical expression given in Eq. 3, is adjusted to yield gradually reduced values through a
factor sv , for shear wave velocities greater than 500 m/sec, as follows:
qa = 0.024 g
vs sv ≥ 30.6 g
(4)
sv = 1 – 3 x 10 -6 ( vs- 500 ) 1.6 (5)
The
variation of allowable bearing capacity qa , with
shear wave velocity vs , is illustrated in Figure 2, where the reduction factor sv , sets an asymptotic upper limit of qa = 30.6 g
for shear wave velocities vs ≥ 2
000 m/sec.
c ) For calculating unit weights
There
is a direct relationship between the average unit weight g , and the P-wave velocity of a soil
layer. Based on extensive case histories
of laboratory testing, a convenient empirical relationship in this regard, is
proposed by the writers as follows:
gp = go + 0.002
vp
(6)
where, gp = the unit weight in kN/m3 based on P-wave velocity, vp = P-wave velocity in m/sec, and go = the reference unit weight values given as
follows:
go = 16
for loose sandy, silty and clayey soils
go = 17 for dense sand and gravel
go = 18
for mudstone, limestone, claystone, conglomerate, etc.
go = 20 for sandstone, tuff, graywacke, schist, etc.
As seen in Figure 3, the unit weights calculated by the empirical expression
given in Eq.6, are in excellent
agreement with those determined in the laboratory. In the absence of any bore
hole sampling and laboratory testing of soil samples, the above empirical
expression provides a reliable first approximation for the unit weights of
various soils, once the insitu measured P-wave velocities are available. In
fact, the speedy evaluation of unit weights, prior to any soil sampling,
enables the practicing engineer to calculate the allowable bearing capacity qa ,
readily from Eq. 3.
6. Case histories
a ) Field investigations
In order to establish a sound and
reliable relationship between the allowable bearing capacity qa , and
the shear wave velocity vs , a
series of case histories have been studied as summarized in Table 2. For each case, in-depth
geotechnical and geophysical site investigations have been conducted and a
comprehensive set of insitu and laboratory tested soil parameters have been
determined. Most of the basic soil
parameters, for each typical soil layer, are shown in Figures 4 through Figure 6. If
however, for any particular soil parameter in any typical soil layer, multiple
values were available, from various bore hole and seismic survey measurements,
only the average of these multiple values have been indicated.
b ) Allowable bearing capacities by
the classical theory
The first column in these Figures
contain the insitu measured SPT data, N30,
the laboratory tested values of cu = undrained shear strength (kPa) , f’ =
effective internal angle of friction, gn = unit weight (kN/m3), and also the
qf = ultimate bearing capacities (kPa), and qa = allowable bearing capacities (kPa) calculated using the classical
approach of Eq. 1.
If, a particular soil layer is considered to be ‘weak’ in accordance with Terzaghi’s (1967) [13] recommendations, two thirds of shear strength parameters have
been utilized in the bearing pressure capacity calculations, as given in Eq. 2.
c ) Allowable bearing capacity by vs
The second column contains, the
insitu measured vs and vp-
velocities (m/sec), n = Poisson ratio, gp = unit weights (kN/m3) determined on the basis of P-wave velocities
given in Eq. 6, qa = allowable bearing capacities (kPa) based on shear wave velocities, in
accordance with Eq. 3. In all case
histories, the shear wave velocity, vs ,
and the P-wave velocity, vp , have been measured insitu by means
of seismic refraction method, using low level explosives. The propagating waves
have been recorded by means of a 12-channel Smart
Seis Geometrics instrument, which is capable of producing very high
resolution of signal/noise ratio, due
to its instant analogue and/or digital signal analyses and automatic filtering
process.
In practice, the geophysical
explorations are not daily business in foundation engineering, therefore, there
is a necessity for experienced technical staff for such a purpose. The shear
wave velocities may be measured, through impact energy methods, during the bore
hole drilling, or using the cross-hole technique [11] , [14].
Realizing that, the bearing capacity
is correlated with large strains at failure, while the shear wave velocity is
associated with ‘zero strain’ levels, the proposed empirical expressions are
adjusted effectively in order to accommodate the differences in strain levels.
For each case history, the allowable
bearing capacities obtained by the classical theory have been compared in Figure 7, with those determined by Eq. 3, using the shear wave velocities.
It is seen that there is a very good agreement between these two different sets
of values. The allowable bearing capacities qa ,
based on the shear wave velocities are more uniform in distribution,
exhibiting no erratic variation and further, they provide an inherent factor of
safety against shear failure and intolerable settlements. The empirical
allowable bearing pressure expression given in Eq. 3, ensures for all foreseeable soft soil conditions, including
those of the case studies that, the maximum allowable settlement is not
exceeded.
7. Conclusions
a)
The determination of adequately
safe allowable bearing capacity of a soil layer under a shallow foundation is a
problem of vital importance in geotechnical engineering. The classical approach
is not only costly and time consuming due to extensive insitu and laboratory
testing required, but also involves significant approximations and intuitive
judgements. Despite the ‘exact’ nature of the classical theory, a huge factor
safety, on the order of 300 percent, is recommended in order to account for the
unexpected inaccuracies and our ‘ignorance’
of the real soil conditions.
b)
The proposed empirical shear
wave velocity approach however, is surprisingly cost effective, and time
saving. The insitu measured shear wave velocity, vs, as an
indispensable single field index, is capable of representing the real soil
conditions at the site, including the
true influence of a family of soil parameters like water content,
confining pressure, relative density, void ratio, nonuniformity, discontinuity,
nonhomogeneity, shear and compressive strength, etc. The complications and
misrepresentations associated with soil sampling, sample disturbance, accurate
simulations in the laboratory testing, etc. are all avoided. Shear wave
velocity measurement at a site however, calls for additional cost and expert
geophysical personnel.
c)
The depth, width and length
of a foundation plays a significant role especially in granular soils, in the
derivation of mathematical formulation when following the classical approach.
In cohesive soils, the geometry of foundation does not play a significant role
anyhow. Nevertheless in classical theory, the soil is idealized into an
isotropic, homogeneous and uniform elasto-plastic planar geometrical medium. In
the shear wave velocity approach however, there is absolutely no need to
consider the foundation size and depth, even in granular soils, since the
influence of all these parameters are inherently incorporated in the insitu
measured vs – values. The
classical approach is further handicapped by the layered conditions. In shear wave
velocity approach however, the bearing capacity of a single layer, immediately
under the foundation, is directly determined, as a one step operation.
d)
The empirical formulations
proposed for calculating both the allowable bearing capacity qa , and the unit weight g, are proven to be safe and reliable as verified consistently by 14
different laboratory tested case
histories. The validity and reliability of the proposed scheme will be better
established however, as the proposed empirical method is constantly calibrated
by conventional method at more and more sites.
8. Acknowledgments
The authors gratefully acknowledge
the assistance and cooperation extended by Mr. Tufan Durgunoglu, and Mr.
Abdullah Calisir of the Geotechnics Co., Istanbul Suleyman Demirel
University
9. References
[1]. British Standard 8004 (1986). Code of Practice for Foundations,
British Standards Institution, London
[2]. DeBeer, E.E. (1970). “Experimental determination of the shape
factors and the bearing capacity factors of sand”, Geotechnique, Vol. 20,
pp. 387-411.
[3]. Hansen, J.B. (1968). “A revised extended formula for bearing
capacity”, Danish Geotechnical Institute Bulletin, No. 28.
[4]. Meyerhof, G.G. (1956). “ Penetration tests and bearing capacity of
cohesionless soils”, Proceedings ASCE, Vol. 82, No. SM1, Paper 866, pp. 1-19.
[5]. Prandtl, L. (1921). “Über die Eindringungsfestigkeit (Härte) plastischer Baustoffe und die Festigkeit von Schneiden” (On the penetrating
strengths (hardness) of plastic construction materials and the strength of
cutting edges), Zeit. Angew. Math. Mech., 1, No.1,
pp.15-20.
[6]. Reissner, H. (1924). “Zum Erddruckproblem” (Concerning the
earth-pressure problem), Proc. 1st Int. Congress of Applied Mechanics, Delft
[7]. Sieffert, J.G., and Ch. Bay-Gress
(2000). “Comparison of the European
bearing capacity calculation methods for shallow foundations”, Geotechnical
Engineering, Institution of Civil Engineers, Vol. 143, pp. 65-74.
[8]. Skempton, A. W. (1951). “The bearing capacity of clays’’ ,
Proceedings, Building Research Congress,
1, 180-9.
[9]. Skempton, A. W. and Bjerrum, L. (1957). “A contribution to the settlement analysis of foundation on clay”,
Geotechnique, Vol. 7, pp. 168-178.
[10]. Skempton, A. W. and MacDonald, D. H. (1956). Allowable settlement of buildings, Proceedings ICE, 5, Part 3, pp. 727-68.
[11]. Stokoe, K. H., and Woods, R. D.
(1972). “Insitu Shear Wave Velocity by
Cross-Hole Method”, Journal of the Soil Mechanics and Foundation Divison,
ASCE, Vol. 98, No. SM5, pp. 443-460.
[12]. Terzaghi, K. (1925). “Structure and volume of voids of soils”,
Pages 10, 11, 12, and part of 13 of Erdbaumechanik
auf Bodenphysikalisher Grundlage, translated by A. Casagrande in From theory to practice in soil mechanics,
New York, John Wiley and Sons, 1960, pp. 146-148.
[13]. Terzaghi, K., and Peck, R. B. (1967). “Soil Mechanics in Engineering Practice”,
2nd edn, John Wiley and Sons, New York
[14]. Tezcan, S. S., Erden, S. M., and Durgunoğlu, H. T.
(1975). “Insitu Measurement of Shear Wave Velocity at
Boğaziçi University Campus”, Proceedings of International Conference on
Soil Mechanics and Foundation Engineering, Vol. 2, April 1975, pp. 157-164, Istanbul
Technical University, Ayazağa, Istanbul, Turkey.
[15]. Turkish Earthquake Code (TEC), (1998). < www. koeri.boun.edu.tr >.
[16]. Uniform Building 5360 Workman Mill Road Whittier , California , USA .
Figure 1. – Failure mechanisms under a strip footing
Figure 2. – Allowable bearing capacity of soils based on vs
 Figure 3. – Unit weights based on vp - velocities  |
Figure 4.- Allowable bearing capacities for case histories No.1 through No.4
(Units are; g = kN/m3, c = kN/m2, qa = kN/m2, vs , vp = m/sec)
Figure 5.- Allowable bearing capacities for case histories No.5, 6, 7, and 9
(Units are; g = kN/m3, c = kN/m2, qa = kN/m2, vs , vp = m/sec)
Figure 6.- Allowable bearing capacities for case histories No.10 through No.14
(Units are; g = kN/m3, c = kN/m2, qa = kN/m2, vs , vp = m/sec)
Figure 7. – Comparisons of allowable bearing capacities
(Numerals beside the data points are the case study numbers)
Table 1.- Recommended ranges of
allowable bearing capacities (kPa)
.jpg)
Allowable bearing capacity, qa
in kPa
|
Table 2.- Locations and the scope of investigations for each case study
No
|
Building Identity
|
Number
Of
surveys
|
||||||
Number
|
m
|
m
|
a
|
b
|
(c)
|
(d)
|
||
1
|
Atatürk Primary School Building
Babaeski, Kırklareli , Western Turkey
|
2
|
15.30
|
4.00
|
2
|
2
|
281
|
287
|
2
|
Residential Apartments
Yeşilçay Cooperative, Çay, Afyon
|
4
|
9.50
|
2.50
|
3
|
3
|
110
|
147
|
3
|
Zeki Örnek, Housing complex, Göktürk Village, Eyüp, Istanbul
|
2
|
20.00
|
3.00
|
1
|
3
|
150
|
203
|
4
|
Oztas Apartments, Florya
Şenlik, Bakırkoy, Istanbul
|
2
|
20.00
|
3.00
|
1
|
3
|
146
|
164
|
5
|
Oil tankı(†) , Haramidere,
Istanbul
|
3
|
8.00
|
2.50
|
3
|
3
|
165
|
113
|
6
|
Oil tanks, Samsun, Black Sea
|
6
|
25.00
|
2.50
|
3
|
2
|
215
|
224
|
7
|
Oil tanks, Mudanya, Bursa
|
4
|
20.7
|
2.50
|
3
|
3
|
100
|
133
|
8
|
Oil tanks, Çubuklu, Istanbul
|
3
|
12.00
|
1.00
|
3
|
4
|
115
|
100
|
9
|
Oil tanks, Iskenderun
|
5
|
5.50
|
1.50
|
3
|
3
|
520
|
374
|
10
|
Oil tanks, Mersin
|
8
|
26.10
|
2.50
|
3
|
3
|
187
|
218
|
11
|
Oil tanks, Derince , Kocaeli
|
7
|
21.00
|
1.50
|
3
|
3
|
110
|
86
|
12
|
Oil tanks, Derince, Kocaeli
|
7
|
21.00
|
7.00
|
3
|
3
|
222
|
205
|
13
|
Oil tanks, Aliağa, Izmir
|
6
|
19.20
|
2.50
|
3
|
4
|
231
|
234
|
14
|
Suleyman Demirel University,
Isparta, Southern Turkey
|
2
|
12.00
|
4.00
|
2
|
2
|
120
|
124
|
(a) seismic
refraction surveys,
(b) geophysical soil layers,
(c) the classical Terzaghi approach
(Eq. 1), (d) the shear wave velocity approach
(Eq. 3).
(†) Oil tanks belong to the Turkish Petroleum
Office Co., Ankara , Turkey
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