BEARING CAPACITY FORMURA FOR SHALLOW FOUNDATIONS DURING EARTHQUAKE

Yoshito MAEDA1, Tatsuo IRIE2 and Yasuyuki YOKOTA3

SUMMARY This paper proposes a formula widely applicable for calculating bearing capacity of shallow foundations, which can evaluate both inclined load action of superstructure and inclined bearing stratum during earthquake. The formula is derived using seismic coefficient method and admissible velocity field method from upper bound theorem. Its applicability was verified by series of experiment. In practice, most bearing capacity formulas assume only the influence of load inclination. However, it was found from the newly proposed formula and experimental results that in case of strong earthquake most of the present formulas might have risk of over-evaluating the bearing capacity.

INTRODUCTION Generally, the inertia force during an earthquake varies with time and place. However, it is known that an equivalent dynamic model can be made by considering inclined ground, which corresponds to the degree of inertia (i.e. seismic intensity in seismic coefficient method). On the other hand, it is typical to consider inclined loads only for inertia force of superstructure in seismic load-capacity problem of shallow foundation. This is based on the assumption that the inertia of the superstructure is dominant and the influence of bearing stratum is comparatively small, according to Yamaguchi [1]. However, this is only valid for a relatively small horizontal seismic coefficient of about 0.2, as indicated in JRA [2]. Moreover, the range of horizontal seismic coefficient where the bearing stratum can be ignored has never been studied. The seismic design of foundation is shifting to performance-based-design-method in consequence of recent major earthquakes. Safety is checked for two earthquake levels, i.e., the ordinary earthquake and the rarely occurring earthquake. Therefore, it is very important to study the effect of inertia force of bearing stratum to bearing capacity of foundation. The authors have already proposed a multiple-use bearing capacity formula that considers the inclination of load and ground; refer to Maeda [3]. It uses the dynamic model of seismic coefficient method and applies the admissible velocity field method in the upper bound theory of plasticity. In this paper, the

1 Professor, Kyushu Kyoritsu University, Fukuoka Prefecture, Japan. Email: maeda@kyukyo-u.ac.jp 2 Chief Engineer, CTI Engineering Co. Ltd., Fukuoka Prefecture, Japan. Email: irie@ctie.co.jp 3 Engineer, Oita Prefecture Office, Japan. Email: yokota-yasuyuki@pref.oita.lg.jp

13th World Conference on Earthquake Engineering Vancouver, B.C., Canada

August 1-6, 2004 Paper No. 3293

applicability of the proposed formula, and the influence of load and ground’s inclination to bearing capacity are examined by comparing the formula with bearing capacity test results of a two-dimensional plastic laminated body.

BEARING CAPACITY OF SHALLOW FOUNDATION ACCORDING TO ADMISSIBLE VELOCITY FIELD METHOD

Breaking mechanism and bearing capacity type considering the load and dip of the ground An equivalent dynamic model of the ground when inertia force in the supporting ground is uniform and failure is determined by the horizontal seismic coefficient due to maximum inertia force of superstructure is shown in Fig.1. The ground can be assumed inclined when the inertia force acts opposite to the earthquake direction. Under such a condition, a shallow foundation failure mechanism, as shown in Fig.2, can be obtained. Here, it is assumed that ab and cd are straight lines and bc is logarithmic spiral line. Thus, the bearing capacity, which considers inclinations of load and ground at the same time, can be determined.

Figure 1 Input earthquake and equivalent dynamic model

(a)Failure mechanism of whole body (b) Relationship of inclined load and active wedge oab

Figure 2 Failure mechanism considering inclinations of load and ground

Here, the inclination angles of the superstructure load and ground vary according to the degree of seismic response (i.e., vibration mode, response magnification, etc). The inclinations of load (θ) and ground (β) can be express by the following equations where g is the acceleration due to gravity and, αs and αf are response accelerations of load and ground, respectively.

gs /tan αθ = (Equation 1)

gf /tan αβ = (Equation 2) Moreover, in the case of αs =αf , the responses of the superstructure and bearing stratum are equal, and θ=β . For the failure mechanism illustrated in Fig.2, an upper limit of the bearing capacity is found by equating internal dispersion energy and external work. Also, the admissible velocity field method in this paper assumes an associated flow rule (ν=φ) where the soil’s yield condition is defined by compatibility of the Mohr-Coulomb failure criterion and plastic flow. Internal dispersion energy can be computed as illustrated in Fig.2. The straight-line part is the product of adhesive strength and a discontinuous quantity of admissible velocity. Its sum with dispersion energy of internal area represents the transition zone; refer to Yamaguchi [4]. External work is the sum of the work due to the weight of ground, inclined load Q (q=Q/B) and surcharge load p, where the ground consists of an active wedge zone, transition zone and passive soil pressure zone. In the following, the inclined load q is obtained by equating the total internal dispersion energy and the total external work.

γγ++= BN2

1pNcNq qc

(Equation 3)

( )21

00 ,,

γγβγθγ

θθ

µµ

µµ

NNN

NNNN qqqccc

+=

==

(Equation 3-1) Here, µcθ, µqθ, µγθ and µβ are inclination coefficients of load and ground that can be found using equations 4, 5 and 6. Nc0 and Nq0 are the bearing capacity coefficients when the load and ground are not inclined, i.e. θ=β=0. Since Nγ contains both Nγ1 and Nγ2, Nγ becomes Nγ0 when θ=β=0 in Equations 3-1. Equations 8, 9-1 and 9-2 define these coefficients.

( )θ−δδ=µ=µ θθ cos

cosqc

(Equation 4)

( )θ−δδη=µ γθ cos

cos

(Equation 5)

β=µβ tan (Equation 6)

( )

( ) ( ) ( ){ }1tan2expsin1sinsin

sin

tan

1N

1

0c

−φωφ+φφ−ψ

ψ+

φ−ψ=

(Equation 7)

( )( )φψφ

φωφπψ

−

⎟⎠⎞⎜

⎝⎛ −

=sincos

tan2exp24cossin2 12

0qN (Equation 8)

( ) [ ( )

( ) ( ){{

( )} ( ) ( )( )}

( )⎥⎦⎤

⎟⎠⎞⎜

⎝⎛ +−

−+

−++−+

+−−+

+

−−×−

=

φωφπψ

ψφ

ψφφφωωψφ

ωψφφφφ

ψ

φψφψφ

ψγ

tan3exp24cossin

sin

costan3tan3expsin

costan3cos1tan9

sin

cossincos

sin

1

11

12

21N

(Equation 9-1)

( ) ( )[ ( )

( ) ( ){{

( )} ( ) ( )( )}

( )⎥⎦⎤

⎟⎠⎞⎜

⎝⎛ ++

−+

−−+−−

+−+

+

−−−×−

=

φωφπψ

ψφ

ψφφφωωψφ

ωψφφφφ

ψ

φψφψφψφ

ψγ

tan3exp24sinsin

cos

sintan3tan3expcos

sintan3cos1tan9

sin

cossinsincos

sin

1

11

12

2N

(Equation 9-2) The bearing capacity component perpendicular to ground surface is expressed in equation 10.

θcos⋅= qqv (Equation 10) In the above equations, η represents the correction factor for ground weight. It corrects the bearing capacity coefficient for ground weight Nγ that is overestimated when using the general bearing capacity equation based on Prandtl’s failure mechanism, as compared to precise values determined by stress characteristic curve methods; refer to PWRI [5]. This paper assumes η =1/2, which is suggested in Maeda [6]. Comparison to past study results Figure 3 shows the calculation results of this study in comparison to bearing capacity test results of centrifugal loading of sand, according to Shioiri et al. [7] and numerical solutions of Kötter equation used in the stress characteristic curve method. Here, the inclination angles of load and ground are the same (θ=β) and the angle of internal friction of the ground, φ, is 46° to enable direct comparison with the past

study. The objective of this paper is only to check the coefficient Nγ defined in the bearing capacity formula proposed herein. However, results revealed that the test values agree with the numerical solutions.

1.0

0.5

0 5 10 15 20

Test values Kötter Proposed fomula

β(deg)

Abe

arin

gca

paci

tyfa

llra

teq/

q0

Figure 3 Relation of bearing capacity reduction rate and inclination angle; ref. to Shioiri et al. [7]

Comparison to Japanese design standard The computed results, according to bearing capacity equation proposed herein, are compared to Japanese design standards for highway bridges (ref. to JRA [8]) and railroads (ref. to MTRB [9]), for the case of level ground. For highway bridges, the bearing capacity coefficient is used. It considers the effect of load inclination based on results of Komada [10]. However, in the case of railroad, the general bearing capacity coefficient is multiplied by a correction factor to account for load inclination, applying the results of Meyerhof [11]. Figure 4 shows the bearing capacity factor ratio when the load inclination angle θ is in the range of 0° to 30° and the internal friction angle φ is 30°.

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40

θ ( °

)

c

提案式

鉄道

道路

Proposed fomula Standard for railroad Standard for road

Load inclination angle θ(deg) Bea

ring

capa

city

coe

ffici

ent r

atio

N

c/N

c0

(a) Nc 支持力係数Nγの変化(β＝０)

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40

荷重傾斜角θ (° )

支持力

係数比

　N

γ/N

γ0

提案式

鉄道

道路

(b) Nγ

Proposed fomula Standard for railroad Standard for road

Bea

ring

capa

city

coe

ffici

ent r

atio

Nγ

/Nγ

0

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Load inclination angle θ(deg)

Figure 4 Comparison of proposed formula and Japanese design standards

LABORATORY TEST ON BEARING CAPACITY Experiment Outline The laboratory test for bearing capacity uses a plastic rod, built-up in a two-dimensional soil layer with dimensions of W=150cm wide, H=50cm high and L=23cm deep as shown in Fig.5. The loading apparatus and ground can be rotated, making it possible to combine arbitrary inclination angles of ground (β) and load (θ). The footing is B=10cm wide and L=20cm deep. The load position is adjusted so that the resultant of applied load acts at the center of footing's base for all load inclination angles θ, as illustrated in Fig.6. In order not to restrain displacement at right angles to the load axis, a load is set beforehand to balance the weight of the apparatus for load inclination adjustment. The displacements taken at point of measurement δV0, δH0, and at base of footing δV, δH are expressed in the following equations with reference to Fig.6. δV = δV0･cosθ－δH0･sinθ (Equation 11) δH = δV0･sinθ＋δH0･cosθ (Equation 12)

Figure 5 Bearing capacity test apparatus

Figure 6 Illustration of loading apparatus

The plastic rod has a diameter of 1.6mm, length of L=20cm and unit weight of γd=11.5kN/m3. Its internal friction angle and adhesive strength determined from laboratory shear testing are φ=21° and c=0kN/m2, respectively. A displacement control system using screw jack is applied where the speed of inclined load application is set to 2.5mm/min. Twenty five tests were performed, using combinations of β=0, 5, 10, 20° and θ=0, 10, 20, 30, 40°. The method used to evaluate bearing capacity in this paper is shown in Fig.7. In this figure, bearing capacity is defined as the intersection of fit curve and the line that bisects the angle of intersection of two lines tangent to the curve.

Vertical displacementδv(cm)

Ver

tical

load

P

v(kP

a)

0 1 2 3

0

1

2

3

4

5

1 2 3

垂直荷

重　P V

(kPa)

垂直変位　δV(cm)

0

0.2

0.4

0.6

0.8

1 2

水平(平行)荷

重　

P H(kPa)

水平(平行)変位　δH(cm)

0

1

2

3

4

5

1 2 3

垂直荷

重　P V

(kPa)

垂直変位　δV(cm)

0

0.2

0.4

0.6

0.8

1 2

水平(平行)荷

重　

P H(kPa)

水平(平行)変位　δH(cm)

1

5

4

3

2

Horizontal(parallel) displacement δH(cm)

0

1

2

3

4

5

1 2 3

垂直荷

重　P V

(kPa)

垂直変位　δV(cm)

0

0.2

0.4

0.6

0.8

1 2

水平(平行)荷

重　

P H(kPa)

水平(平行)変位　δH(cm)

0

1

2

3

4

5

1 2 3

垂直荷

重　P V

(kPa)

垂直変位　δV(cm)

0

0.2

0.4

0.6

0.8

1 2

水平(平行)荷

重　

P H(kPa)

水平(平行)変位　δH(cm)

0 1 2

0.2

0.4

0.6

0.8

Hor

izon

tal(p

ara

llel)

loa

d P

H(k

Pa)

Figure 7 Evaluation method of bearing capacity

Comparison of Experimental Results and Proposed Formula In this experiment, the bearing capacity coefficient is related only to the weight of ground Nγ, since the dry plastic rod is set without embedment. Therefore, comparison according to weight of ground Nγ is carried out as follows. The figures below show the influence of load and ground inclination to bearing coefficient Nγ. The vertical axis represents the ratio of bearing capacity coefficient Nγt corresponding to load inclination angle θ and bearing capacity coefficient Nγ0 when θ=0. Figure 8(a) illustrates the variation of bearing capacity coefficient with respect to load inclination angle θ where ground inclination angle is β=0°. It is clear from the figure that bearing capacity decreases as load inclination angle increases. In addition, the figure shows the test values, computed values according to the proposed equation and prescribed values in Specifications for Highway Bridges. Comparison reveals that these three values coincide when θ=10° and less. For θ=20°, test values exceed computed values and prescribed values. It is conceivable that the reason for this is the unstableness of the foundation. Since the internal friction angle of plastic is φ=21°, the foundation will starts to slide at β=20°. The prescribed value is slightly greater than the computed value, which implies that Specifications for Highway Bridges gives safer values of bearing capacity coefficient. This may be explained by the difference in slip planes.

Figure 8(a) Influence of load inclination to Nγ (β=0°)

Figure 8(b) illustrates the variation of bearing capacity coefficient with respect to ground inclination angle β where load inclination angle is θ=0°. It shows that the bearing capacity coefficient decreases as ground inclination angle β increases. This suggests that the influence of the ground's inertial force is well evaluated. The reduction rate of bearing capacity coefficient due to ground inclination angle β is small compared to that of load inclination angle θ. Moreover, the test and computed values coincide well, except when β=20°.

Figure 8(b) Influence of ground inclination to Nγ (θ=0°)

Figure 8(c) illustrates the variation of bearing capacity coefficient when θ=β. It is clear from this that bearing capacity decreases in the case where inclination angles of load and ground are allowed to increase independently. Moreover, the reduction ratio of the bearing capacity coefficient ratio increases compared to the case where load and ground are allowed to vary independently. This is because the bearing capacity coefficient, which is reduced according to inclinations of load and ground, is a summation of results. Moreover, test values agree well with computed values.

Figure 8(c) Influence of ground inclination to Nγ (β=θ)

Figure 9 shows the failure condition of the ground when the load inclination angle, θ, is10° and the ground inclination angle, β, is 10°, as an example of test results. This explains that the load due to active wedge zone acts in the direction of ground inclination

Figure 9 Condition of ground failure (θ=0°, β=10°)

Figure 10 shows the bearing capacity envelope curve for θ=β, where load P is reduced to its vertical and horizontal components (i.e., normal load PV and horizontal load PH). The figure reveals that the bearing capacity envelope curve becomes small when inclination angles of load (θ) and ground (β) become large, and that the bearing capacity is dependent on these two factors.

Figure 10 Bearing capacity envelope curve

CONCLUSIONS In this paper, the properties and applicability of the bearing capacity equation are investigated in order to evaluate the bearing capacity characteristics of shallow foundation during earthquake loading. The method uses a dynamic model for seismic coefficient method and failure mechanism according to admissible velocity field method. Results are summarized as follows. 1. A multi-application bearing capacity equation is proposed. It assumes a failure mechanism that

considers inclinations of load and ground, and applies admissible velocity field method. Using this equation, evaluation of bearing capacity becomes possible, by considering the degree of inertia force acting on the superstructure and bearing stratum.

2. The applicability of the proposed equation is confirmed, since the decrease of bearing capacity

coefficient according to the equation agrees with the test results from a two-dimensional model. 3. Based on two-dimensional bearing capacity model test results, the bearing capacity coefficient Nγ

decreases according to inclinations of load and ground, similar to results from past studies. The rate of decrease becomes large as the inclination angle increases.

4. In regards to reduction rate of bearing capacity Nγ, the effect of load inclination is greater than that of

ground inclination. Furthermore, the reduction rate becomes slightly bigger in the case when inclinations of load and ground are both considered, compared to the case when only inclination of load is considered.

ACKNOWLEDGEMENT

The authors’ gratitude is indebted to Prof. Dawn A. Shuttle of The University of British Columbia for her great efforts and cooperation in reading and checking the contents of this paper.

REFERENCES 1. Yamaguchi H. “Soil mechanics (revised edition).” Gihodo Publication, 1976: 390. 2. Japan Road Association. “Specifications for highway bridges” (e.g. Part IV Substructure, Part V

Seismic Design), 1996. 3. Maeda Y., Ochiai H., Yokota Y. “Bearing capacity equation for shallow foundation considering

inclinations of load and ground,” JSCE Journal No.715/III-60, 2002. 4. Yamaguchi H. “Mechanics of soil,” Kyoritsu Publication, 1976: 96-97. 5. Ministry of Construction Public Works Res. Inst. “Study on ultimate bearing capacity of shallow

rigid foundation,” Public Works Res. Inst. Material no.1611, 1981. 6. Maeda Y. “Study on application of velocity field method in evaluation of bearing capacity of

foundation,” Kyushu University doctoral thesis, 2003. 7. Shioiri M., Yamaguchi H., Kimura M. “Bearing capacity on inclined ground based on centrifugal

loading,” Proceedings of 31st JSCE Annual Conference III-205, 1975: 365-366. 8. Japan Road Association. “Specifications for highway bridges Part IV Substructure,” 1996. 9. Railway Technical Res. Inst. “Railroad structural design standards,” 2000. 10. Komada K. “Calculation diagram of soil bearing capacity under two-dimensional inclined load,”

Public Works Res. Inst. Report no.135, 1969. 11. Meyerhof G: “The bearing capacity of foundation under eccentric and inclined loads,” Proc.3rd Int.

Conf. Soil Mech. and Found. Eng., 1953: 4-24.