Design Charts for Seismic Bearing Capacity of Shallow Footings
Awad Ali Al-Karni
King Saud University, College of Engineering,
Civil Engineering Department
P.O. Box 800, Riyadh, 11421
Saudi Arabia
Abstract
Seismic loading could reduce the soil bearing capacity and cause large settlement or
failure of the supported foundation. Mathematical analysis was developed in literature
on calculating the seismic bearing capacity. The developed analysis used horizontal
and vertical acceleration coefficients that applied by earthquake shaking in evaluating
the seismic bearing capacity of dry soils. In this paper, design charts based on previous
analysis of seismic bearing capacity by the author were developed. These charts will
be used to evaluate whether the design based on the calculated static bearing capacity
is adequate or not. If not, a new suggested design should be tried and checked again.
Detail description of the procedures of using these design charts was given. Example
problem was provided to show how to implement these procedures. The study shows
that the most effective parameters on seismic bearing capacity are the horizontal
acceleration coefficient (kh), the vertical acceleration coefficient (kv), the static safety
factor (FsS), the factor D(=c/H), and the footing depth of embedment (Df), and the
soil angle of friction ().
Introduction
Several theoretical studies on the seismic bearing capacity of soils supporting shallow
footings have been presented (Sarma and Issofelis, 1990; Richards et. al., 1991,
Richards et. al., 1993; Budhu and Al-Karni, 1993, Al-Karni, 1993). These studies used
a limit equilibrium analysis with various assumptions on the shape of the failure
surface. Richards et. al. (1991) used a simple Coulomb type planar failure surface,
while Sarma and Issofelis (1990), and Budhu and Al-Karni (1993) and Al-Karni
(1993) used logarithmic spiral failure surfaces. Budhu and Al-Karni (1993) presented
the differences between the seismic bearing capacities calculated from these theories.
The intention of this contribution is to present a graphical method for checking the
stability of designed shallow footings when subjected to seismic loads and to suggest
the required minimum safety factor for new design. The developed charts in here are
based on the seismic bearing capacity presented by Budhu and Al-Karni (1993).
Seismic Bearing Capacity Equation
The seismic bearing capacity factors described by Budhu and Al-Karni (1993) can be
used to modify popular static bearing capacity equations (Terzaghi, 1943, Meyerhof,
1963, Hansen, 1970, Vesic, 1973). For example, the Meyerhof bearing capacity
1
equation for vertical load can be modified to become a general equation to include
seismic effects as follows.
quE  CNc S sc dcec  q f NqS sq dqeq  0.5BNS s d e
(1)
where quE is the ultimate seismic bearing capacity, C is soil cohesion, NcS, NqS, NS, are
static bearing capacity factors, s and d are shape and depth factors respectively, qf is
the overburden pressure, B is the footing width,  is the unit weight of the soil, and ec,
eq, and e are the seismic factors calculated from Budhu and Al-karni (1993) as

ec  exp  4.3kh1 D

(2)
  5.3k 1.2 
h 
eq  (1  k v ) exp  
  1  k v 
(3)
  9k 1.1
2
e  (1  k v ) exp   h
3
  1  k v
(4)




where kh is the horizontal acceleration coefficient, kv is the vertical acceleration
coefficient, D = C/(H) and H is the depth of the failure zone from the ground surface
given as
H
0.5B


exp  tan    D f
  
2

cos  
4
2


(5)
where Df is the depth of embedment of the footing and  is the angle of friction of the
soil.
Critical Acceleration
The critical acceleration is the horizontal acceleration that applied by the seismic
shaking at which the footing will start to move down inside the soil mass. At the
critical acceleration the allowable seismic bearing capacity (qaE) reduces to the
allowable static bearing capacity (qaS), or
qaE  qaS
(6)
By denoting the static safety factor with FsS, and substituting of qaS by quS/ FsS, where
quS is the ultimate bearing capacity, equation (6) becomes,
qaE 
quS
Fs S
(7)
and by replacing qaE by quE/FsE, where quE is the ultimate seismic bearing capacity and
FsE is the seismic safety factor, then
quE
q
 uS
FsE FsS
(8)
where FsE is equal to one at the critical acceleration and by rearranging Eq. (8), then
2
Fs E 
quE  Fs s
quS
(9)
Parametric Study
The parameters that used in calculating the bearing capacity for static and seismic case
include the angle of friction , the factor D (c/H), the ratio Df/B and L/B, kv, kh, and
the static safety factor Fss. The effect of each of these parameters will be presented
below. The object of this parametric study is to show how significant each of these
parameters will affect the seismic safety factor. Some of these parameters may be
excluded due to its insignificant effect.
Effect of Angle of Friction, 
Equation (9) has been solved numerically for different angle of friction and a sample
of the results is shown by Fig. 1. This figure shows a difference of about 0.06 of kh at
FsE=1 for =0 =40. Such a result makes no significant effect of the angle of friction
on the seismic safety factor on most cases. Based on this result, the effect of angle of
friction was noticed to be significant when both c> and Df>0. So, the charts were
developed for angle of friction equal to zero (for c> and Df>0) and angle of friction
grater than zero (for c> and Df>0 soil).
2.4
f=0
f=10
f=20
f=30
f=40
2.0
FsE
1.6
1.2
Df/B=1.0, kv=0.4
D=0.4, FsS=3
0.8
0.4
0.0
0.0
0.2
0.4
0.6
0.8
1.0
kh
Fig. 1. Variation of FsE with kh at different values of angle of friction, .
Effect of D
The value of D is representing mainly the effect of soil cohesion on the seismic
bearing capacity. By assuming that the angle of friction and Df/B are equal to zero to
reduce the bearing capacity equation to one term, the seismic safety factor (FsE ) can
be calculated as
Fs E  e  4.3k h
(1 D )
 Fs s
(10)
3
By plotting this relationship, the variation of FsE with kh can be shown as in Fig. 2a for
different values of D at static safety factor of 3. This figure shows a significant effect
of D on the seismic safety factor, where the critical acceleration changes from 0.4 at
D=0.4 to 0.8 at D=3.2. Therefore, for soil with zero angle of friction, the most
significant parameters that affecting the soil bearing capacity in seismic environment
is the value of D and the static safety factor. The effect of D was found to be
significant also when Df/B>0 as shown in Fig. 2b. Therefore, the design charts will be
developed for different values of D in the range of 0 to 2.0.
3.0
2.0
1.5
Df/B=0, kv=0
FsS=3, =0
2.0
D=0.4
D=0.8
D=1.6
D=3.2
D=6.4
1.5
Df/B=3, kv=0
FsS=3, =30
2.5
FsE
2.5
FsE
3.0
D=0.4
D=0.8
D=1.6
D=3.2
D=6.4
1.0
1.0
0.5
0.5
0.0
0.0
0.0
0.3
0.6
0.9
0.0
1.2
0.2
0.4
0.6
0.8
1.0
kh
kh
(a)
(b)
Fig. 2. Variation of FsE with kh at different values of D (a) at Df/B=0, (b) at Df/B=3.
Effect of Df/B
Fig. 3 shows the effect of Df/B is significant for Df/B between 0 and 1. For example,
at D=0.8, the critical acceleration at FsE equal to one is equal to 0.36 at Df/B=0.0 and
equal to 0.28 at Df/B=1. However, there is no significant changes when Df/B becomes
>1. For example, the difference in the critical acceleration values at Df/B =1 and Df/B
=2 is about 0.03. This resulted in development of design charts considering only Df/B
=0 and Df/B1.
Effect of L/B
An example of the results of the variation of FsE with kh at different values of L/B is
shown in Fig. 4. This Figure shows no significant effect of the L/B on the value of the
seismic safety factor. The parameter L/B will be excluded during the development of
the design charts.
4
3.0
3.0
1.5
Fss=3, kv=0.0
D=0.4, =30
FsE
2.5
L/B=0
L/B=1
L/B=2
L/B=3
L/B=4
2.5
2.0
FsE
2.0
Df/B=0
Df/B=1
Df/B=2
Df/B=3
Df/B=4
Fss=3, kv=0.0
D=0.4, =
1.5
1.0
1.0
0.5
0.5
0.0
0.0
0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
kh
0.6
0.8
1.0
kh
Fig. 4. Variation of FsE with kh at
different values of L/B.
Fig. 3. Variation of FsE with kh at
different values of Df/B.
Effect of kv
The effect of kv on seismic bearing capacity was found to be insignificant when Df is
equal zero as shown in Fig. 5a. However, the difference could be significant when
significant when Df is greater than zero as shown in Fig. 5b. For example, the
difference between the values of the critical acceleration at kv =0.4 and kv =0 is about
0.065 at Df=1 and reduces to 0.0 at Df =0. The difference could reach about 0.12 in
some cases. It was found that the effect of kv will not change much when Df becomes
grater than 1. Therefore, the vertical component of acceleration can not be excluded in
developing the design charts for Df>0. Thus the design charts were developed at values
of kv in the range of 0.0 to 0.4 in which the common value of kv falls.
3.0
3.0
kv=0.0
kv=0.1
kv=0.2
kv=0.3
kv=0.4
FsE
2.0
2.5
2.0
FsE
2.5
D=0.4, =30
FsS=3, Df/B=0
1.5
1.0
0.5
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
D=0.4, =30
FsS=3, Df/B=1
1.5
1.0
0.0
kv=0.0
kv=0.1
kv=0.2
kv=0.3
kv=0.4
0.0
kh
0.2
0.4
0.6
0.8
1.0
kh
(a)
(b)
Fig. 5. Variation of FsE with kh at different values of kv (a) at Df/B=0, (b) at Df/B=1.
5
Effect of Static Safety Factor
The value of static safety factor was found to be a major control variable in developing
the design charts. The significant effect can be noticed as shown in Fig. 6. From this
figure, the critical acceleration increased from 0.28 at FsS= 3 to 0.46 at FsS=7. The
design charts were developed at values of static safety factor in the range of 3 to 7.
6.0
4.0
Fss=3
Fss=4
Fss=5
Fss=6
Fss=7
3.0
Df/B=1, kv=0.2
D=0.8, =30
FsE
5.0
2.0
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
kh
Fig. 6. Example of variation of FsE with kh at different values of static safety factor.
Design Charts
The proposed design charts represent the variation of FsE with kh for different variables
including the angle of friction , the factor D, the ratio of Df/B, kv, and the static safety
factor Fss. At certain values of the parameters mentioned above, the seismic safety
factor was calculated at different values of kh and a certain value of Fss. This process
was repeated for different values of Fss to be able to draw the curves of the variation of
FsE with kh for different values of Fss. These procedures will be repeated using
different values of the parameters above. Some variables will be eliminated for their
little effect on the value of the critical acceleration as discussed above.
According to the parametric study above, the design charts were found to be
classified into
1. Charts for D>0, Df/B =0, and 0, for any kv.
2. Charts for D0, Df/B  1, and =0, for different kv.
3. Charts for D0, Df/B  1, and >0, for different kv.
4. Charts for D=0, Df/B =0, and >0, for any kv.
The design charts were given by Appendix A. Thus, for a given problem, the
corresponding charts should be used. If the problem is not fallen in any of the category
above (i.e., Df/B=0.25), the closest one should be used since the margin of error is not
that significant.
Design Procedure
6
The design charts can be used for new design of foundation or for an existing one. It is
suggested here to design the foundation first for static case and then check the design
using the charts here. If the seismic safety factor is greater than or equal to the required
one by the designer, then the static design is safe. If not, a new design should be tried.
The following procedure could be followed for using the charts:
1. Design the foundation for static case, or get the design parameters for the
existing foundation.
2. Calculate the value of D, and Df/B.
3. Inter to the proper design chart using the value of kh and the design static safety
factor.
4. Find the value of FsE and check with the required minimum one.
5. Repeat the design if FsE is below the minimum one.
Example
Fig. 6 shows a typical shallow foundation carrying a load of 3500 kN. It is required to
check the stability of this foundation against earthquake loading with kh=0.25 and
kv=0.1.
Fig. 6. Typical shallow foundation subject to kh=0.25 and kv=0.1.
Solution
According to the given soil parameters and foundation dimensions, the ultimate
bearing capacity (qu) according to Vesic (1972) procedures is equal to 2195 kN. Since
qapp is equal to 700 kN/m2, then the static safety factor is equal to 3.13 (i.e. =2195/700)
which is considered as enough for static design. For seismic design, the value of D
should be determined first. From Eq. 6, the value of H is equal to 6.95, and then D
(=c/H) is equal to 0.2. According to the design chart given by Fig. A.5.2 at (the effect
of D in this case is not significant) FsS=3.13, Df/B=1, kh=0.25 and kv=0.1, the seismic
7
factor of safety below one and equal about 0.95. Then the design is not enough to
resist the applied seismic load and redesign is recommended.
Conclusion
Seismic bearing capacity design charts were developed here. These charts were
divided into four groups due to the soil properties and the design dimensions of the
foundations. These groups include:
1. Charts for D>0, Df/B =0, and 0, for any kv.
2. Charts for D0, Df/B  1, and =0, for different kv.
3. Charts for D0, Df/B  1, and >0, for different kv.
4. Charts for D=0, Df/B =0, and >0, for any kv.
Using these chart lead to a quick answer whether the designed foundation is
safe or not. In case, of new design, the charts can be used for finding a suggested
design value of the static safety factor that satisfy the required minimum value of
seismic safety factor.
For a given problem, the corresponding charts for the design parameters should
be used. If the problem is not fallen in any of the category above (i.e., Df/B=0.25), the
closest one should be used since the margin of error is not that significant. In case of
more precise results, the calculation should be done according to the given equations
above. However, seeking very precise results is not recommended due to the
approximation of the method.
References
Al-Karni, A.A., and Budhu, M. (1994). “ Seismic Settlement of Shallow Footings on
Sand”, Vertical and Horizontal Deformations of Foundations and Embankments,
Geotechnical Special Publication No. 40, Settlement '94, College Station, USA, pp.
748 - 759.
Budhu, M., and Al-Karni, A. A. (1993). "Seismic bearing capacity of soils."
Geotechnique, 43 (1) , 181-187.
D'Appolonia, D.J., D'Appolonia, E.D. and Brissette, R.F. (1968). "Settlement of
spread footings on sand." J. SM &. Fnds. Div., ASCE, 94, 1011-1053.
Hansen, J. B. (1970). "A revised and extended formula for bearing capacity." Danish
Geotechnical Institute, Bulletin 28, Copenhagen.
Meyerhof, G. C., (1963). "Some recent research on the bearing capacity of
foundations." Canadian Geotechnical Journal, 1 (1), 16-26.
Richards, R., Elms, D.G., and Budhu, M. (1991). "Soil fluidization and foundation
behavior." Proc. Second International Conference on Recent Advances in
8
Geotechnical Earthquake Engineering and Soil Dynamics, Rolla,Missouri, I, 719723.
Richards, R., Elms, D.G., and Budhu, M. (1993). "Seismic bearing capacity and
settlement of foundations." J. Geotech. Engrg., 119 (4), ASCE, 662-674.
Sarma, S. K., and Iossifelis, I. S. (1990). "Seismic bearing capacity factors of shallow
strip footings." Geotechnique, 40 (2), 265-273.
Terzaghi, K., (1943). Theoretical Soil Mechanics, Wiley, New York.
Vesic, A.S. (1973). "Analysis of ultimate loads of shallow foundation." J. Geotech.
Engrg., ASCE, 99 (1), 43-73.
9
Appendix A
The Design Charts for Seismic Bearing Capacity
10
Fig. A.1. Design charts for D>0, Df/B=0, and 0, for any kv.
11
Fig. A. 2. 1. Design charts for D0, Df/B  1, and =0, for kv=0.
12
Fig. A. 2. 2. Design charts for D0, Df/B  1, and =0, for kv=0.2.
13
Fig. A. 2. 3. Design charts for D0, Df/B  1, and =0, for kv=0.4.
14
Fig. A. 3. Design charts for D0, Df/B  1, and >0, for different kv.
15
Fig. A. 4. Design charts for D=0, Df/B =0, and >0, for any kv.
16