BEARING CAPACITY FORMURA FOR SHALLOW FOUNDATIONS DURING EARTHQUAKE

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1 3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 004 Paper No. 393 BEARING CAPACITY FORMURA FOR SHALLOW FOUNDATIONS DURING EARTHQUAKE Yoshito MAEDA, Tatsuo IRIE and Yasuyuki YOKOTA 3 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 []. However, this is only valid for a relatively small horizontal seismic coefficient of about 0., as indicated in JRA []. 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 Professor, Kyushu Kyoritsu University, Fukuoka Prefecture, Japan. maeda@kyukyo-u.ac.jp Chief Engineer, CTI Engineering Co. Ltd., Fukuoka Prefecture, Japan. irie@ctie.co.jp 3 Engineer, Oita Prefecture Office, Japan. yokota-yasuyuki@pref.oita.lg.jp

2 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.. 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., 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 Input earthquake and equivalent dynamic model (b) Relationship of inclined load and (a)failure mechanism of whole body active wedge oab Figure Failure mechanism considering inclinations of load and ground

3 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. tanθ = α s / g tan β = α f / g (Equation ) (Equation ) 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., 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.. 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. q = cn c + pn q + γbn N = µ N N c γ = µ cθ γθ c0, Nq = µ qθ Nq0, ( µ N + N ) β γ γ γ (Equation 3) (Equation 3-) Here, µ cθ, µ qθ, µ γθ and µ β are inclination coefficients of load and ground that can be found using equations 4, 5 and 6. N c0 and N q0 are the bearing capacity coefficients when the load and ground are not inclined, i.e. θ=β=0. Since N γ contains both N γ and N γ, N γ becomes N γ0 when θ=β=0 in Equations 3-. Equations 8, 9- and 9- define these coefficients. µ cθ = µ qθ cos δ = cos ( δ θ) (Equation 4) cos δ µ γθ = η cos ( δ θ) (Equation 5) µ β = tanβ (Equation 6)

4 N c0 = tan + sin ( ψ φ) sin ψ sin φ ( ψ φ) {( + sin φ) exp( ω tan φ) } (Equation 7) N q0 = sinψ cos π φ exp 4 cosφ sin ( ψ φ) ( ω tanφ) (Equation 8) N γ sinψ = cosφ sin + ( ψ φ ) ( 9 tan φ + ) + sin + sin sinψ { cosφ [ cos ( ψ φ ) { + ω ) + ω )} exp( 3ω tanφ) + 3tanφ cos ) )} 3tanφ cos sinψ cos π + φ exp 4 ( ) 3ω tanφ (Equation 9-) N γ sinψ = cosφ sin + ( ψ φ) ( 9 tan φ + ) cos + cos [ sin( ψ φ) cos( ψ φ ) + ω ) + ω )} exp( 3ω tanφ) 3tanφ sin ) )} sinψ {{ 3tanφ sin cosφ + sinψ sin π + φ exp 4 ( ) 3ω tanφ (Equation 9-) The bearing capacity component perpendicular to ground surface is expressed in equation 0. q v = q cosθ (Equation 0) In the above equations, η represents the correction factor for ground weight. It corrects the bearing capacity coefficient for ground weight N γ 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 =/, 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

5 study. The objective of this paper is only to check the coefficient N γ defined in the bearing capacity formula proposed herein. However, results revealed that the test values agree with the numerical solutions. A bearing capacity fall rate q/q Test values Kötter Proposed fomula β(deg) 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 [0]. 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 []. 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. Bearing capacity coefficient ratio Nc/Nc0 c Proposed fomula 提案式 Standard for railroad 鉄道 Standard for road 道路 Load inclination angle θ ( θ(deg) ) Load 荷重傾斜角 inclination θ angle ( ) θ(deg) (a) Nc (b) Nγ 支持力係数 Nγの変化 (β=0) Figure 4 Comparison of proposed formula and Japanese design standards Bearing capacity 支持力係数比 coefficient ratio N γ/nγ0 Nγ/Nγ0 Proposed 提案式 fomula Standard 鉄道 for railroad Standard 道路 for road

LABORATORY TEST ON BEARING CAPACITY Experiment Outline The laboratory test for bearing capacity uses a plastic rod, built-up

The loading apparatus and ground can be rotated, making it possible to combine arbitrary inclination angles of ground (β) and

The load position is adjusted so that the resultant of applied load acts at the center of footing's base for all load

6 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=50cm wide, H=50cm high and L=3cm 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=0cm wide and L=0cm 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 δ V0, δ H0, and at base of footing δ V, δ H are expressed in the following equations with reference to Fig.6. δv = δv0 cosθ-δh0 sinθ (Equation ) δh = δv0 sinθ+δh0 cosθ (Equation ) Figure 5 Bearing capacity test apparatus

7 Figure 6 Illustration of loading apparatus The plastic rod has a diameter of.6mm, length of L=0cm and unit weight of γ d =.5kN/m 3. Its internal friction angle and adhesive strength determined from laboratory shear testing are φ= and c=0kn/m, respectively. A displacement control system using screw jack is applied where the speed of inclined load application is set to.5mm/min. Twenty five tests were performed, using combinations of β=0, 5, 0, 0 and θ=0, 0, 0, 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 load Pv(kPa) 垂直荷重 P V (kpa) 垂直荷重 P V (kpa) 水平 ( 平行 ) 荷重 P (kpa) H Horizontal(parallel) load PH(kPa) 垂直変位 δ 垂直変位 V (cm) δ V (cm) 水平 ( 平行 ) 変位水平 ( δ 平行 H (cm) ) 変位 δ H (cm) Vertical displacementδv(cm) Horizontal(parallel) displacement δh(cm) Figure 7 Evaluation method of bearing capacity 水平 ( 平行 ) 荷重 P (kpa) H

8 Comparison of Experimental Results and Proposed Formula In this experiment, the bearing capacity coefficient is related only to the weight of ground N γ, since the dry plastic rod is set without embedment. Therefore, comparison according to weight of ground N γ is carried out as follows. The figures below show the influence of load and ground inclination to bearing coefficient N γ. The vertical axis represents the ratio of bearing capacity coefficient N γt corresponding to load inclination angle θ and bearing capacity coefficient N γ0 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=0 and less. For θ=0, 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=, the foundation will starts to slide at β=0. 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 N γ (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=0.

9 Figure 8(b) Influence of ground inclination to N γ (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 N γ (β=θ)

Figure 9 shows the failure condition of the ground when the load inclination angle, θ, is0

This explains that the load due to active wedge zone acts in the direction of ground

capacity envelope curve for θ=β, where load P is reduced to its vertical and horizontal

The figure reveals that the bearing capacity envelope curve becomes small when inclination

10 Figure 9 shows the failure condition of the ground when the load inclination angle, θ, is0 and the ground inclination angle, β, is 0, 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=0 ) Figure 0 shows the bearing capacity envelope curve for θ=β, where load P is reduced to its vertical and horizontal components (i.e., normal load P V and horizontal load P H ). 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 0 Bearing capacity envelope curve

11 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.. 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.. 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 N γ 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 N γ, 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. Yamaguchi H. Soil mechanics (revised edition). Gihodo Publication, 976: Japan Road Association. Specifications for highway bridges (e.g. Part IV Substructure, Part V Seismic Design), Maeda Y., Ochiai H., Yokota Y. Bearing capacity equation for shallow foundation considering inclinations of load and ground, JSCE Journal No.75/III-60, Yamaguchi H. Mechanics of soil, Kyoritsu Publication, 976: Ministry of Construction Public Works Res. Inst. Study on ultimate bearing capacity of shallow rigid foundation, Public Works Res. Inst. Material no.6, Maeda Y. Study on application of velocity field method in evaluation of bearing capacity of foundation, Kyushu University doctoral thesis, Shioiri M., Yamaguchi H., Kimura M. Bearing capacity on inclined ground based on centrifugal loading, Proceedings of 3st JSCE Annual Conference III-05, 975: Japan Road Association. Specifications for highway bridges Part IV Substructure, Railway Technical Res. Inst. Railroad structural design standards, Komada K. Calculation diagram of soil bearing capacity under two-dimensional inclined load, Public Works Res. Inst. Report no.35, Meyerhof G: The bearing capacity of foundation under eccentric and inclined loads, Proc.3rd Int. Conf. Soil Mech. and Found. Eng., 953: 4-4.

STUDY ON SEIMIC BEARING CAPACITY OF GROUPED PILES WITH BATTERED PILES

3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 394 STUDY ON SEIMIC BEARING CAPACITY OF GROUPED PILES WITH BATTERED PILES Akira KOMATSU, Yoshito MAEDA, Takahiro