Seismic bearing capacity factors for strip footings

Transcription

1 Seismic bearing capacity factors for strip footings Amir H.Shafiee 1, M.Jahanandish 2 1- Former MSc Student of Soil Mech. & Found. Eng., Shiraz University, Shiraz, Iran 2- Associate Professor, Civil Eng. Department, Shiraz University, Shiraz, Iran Shafiee.amirhossein@gmail.com jahanand@shirazu.ac.ir Abstract Bearing capacity failures due to earthquake have detrimental effects. A great majority of available solutions in the literature is analytical. In this paper, the finite element method is used to estimate the seismic bearing capacity of strip footings for a wide range of friction angles and seismic coefficients. Curves relating seismic bearing capacity factors to earthquake acceleration are presented, and compared to available solutions in the literature. Furthermore, the effect of soil inertia on the seismic bearing capacity is investigated in the present paper. The results indicate that soil inertia plays a negligible role compared to the structural seismic load. Keywords: Seismic bearing capacity, Finite element method, Strip footings, Soil inertia 1. INTRODUCTION The earthquakes may cause severe damages to soil and geotechnical structures. The soil may liquefy; the slopes may fail; or the footing may collapse as a result of reduction in bearing capacity of underneath soil. The latter shall be considered in this paper. Some cases of bearing capacity failure due to earthquake were observed in the Miyagihen-Oki (Japan) earthquake of magnitude 7.8 on 1978, where the soil was dry and dense enough to avoid liquefaction [1]. Richards et al. [2] called this phenomenon "seismic shear fluidization". In seismic shear fluidization, the soil flows at finite levels of effective stress and it can take place in dry soil in which no pore water is present [2]. Seismic bearing capacity of strip footings has recently been one of the most interesting issues for researchers and practical engineers. The published studies in this field can be classified into analytical and experimental solutions. The analytical solutions consist of limit equilibrium method as [1, 3-5], upper bound limit analysis as [6-8], and stress characteristics method as [9, 10]. Ghahramani and Berrill [11] used the zero extension line method to determine seismic bearing capacity factors for strip footings. A few experimental studies are also available in the literature [12, 13]. Knappett et al. [13] performed a series of shaking table tests to examine the effect of earthquake magnitude, frequency, and embedment ratio on the slip surfaces behind the footing. Numerical methods have been extensively used recently in various geotechnical problems. The complicated and advanced softwares as well as powerful computers facilitate the solution of rigorous problems. Several researchers employed numerical methods to determine the bearing capacity of footings in static conditions [e.g. 14, 15]. The authors have not found any published paper that study seismic bearing capacity of footings entirely based on numerical solutions. In the present paper, the finite element method is used to estimate the seismic bearing capacity of strip footings on soil which satisfies Mohr-Coulomb strength criterion, for various friction angles and seismic coefficients. The analyses are based on pseudo-static method. The commercially available code, PLAXIS 2D, is used for the finite element analyses. In this paper curves relating seismic bearing capacity factors to seismic coefficient are presented, and compared to available solutions in the literature. The effect of soil inertia on the bearing capacity reduction due to seismic acceleration is also included in this paper. Furthermore, the error induced by using superposition principle for computing seismic bearing capacity is discussed in details.

2 2. PROBLEM DEFINITION Terzaghi [16] showed that the ultimate bearing capacity (q u ) of a shallow strip foundation of width B and depth D can be represented by the expression: q u = cn c + qn q + 0.5γBN γ (1) where c = cohesion; q = equivalent surcharge; γ = soil unit weight. In Eq.1 N c, N q and N γ are static bearing capacity factors and assumed to be functions of the soil friction angle only. Eq.1 is valid for a centrally and vertically loaded shallow, strip foundation. For seismic bearing capacity and pseudo-static method, the seismic bearing capacity factors N ce, N qe, and N γe replace N c, N q, and N γ in Eq. (1). From the available studies, it is known that N ce, N qe, and N γe factors not only depend on the soil friction angle, but also depend on the seismic coefficient. The pseudo-static approach is adopted in the present analyses. In this method the seismic forces are considered as the horizontal loads applying to the foundation, surcharge and the underlying soil. The horizontal load is obtained by multiplying the horizontal seismic coefficient (k h ) by foundation load (P), surcharge (q), and soil weight (W). The schematic diagram of the problem is shown in Fig. 1.Only the horizontal seismic coefficient (k h ) is accounted in the present study. The earthquake acceleration for soil, foundation, and surcharge is assumed to be the same. The term (k h.w) which is in fact the pseudo-static force applying to the soil mass, is called "soil inertia". k h.p Soil inertia Fig. 1. Schematic diagram of the problem 3. NUMERICAL MODELING AND ANALYSIS PLAXIS 2D [17] is a commercially available two-dimensional program used to perform deformation and stability analyses for various types of geotechnical applications. The 15-node triangular elements were selected to generate the meshes. These meshes may look disorderly, but the numerical performance of such meshes is usually better than for regular meshes [17]. An updated mesh analysis was used in calculation process. In this type of calculation the stiffness matrix is based on the deformed geometry. PLAXIS uses a series of calculation steps to solve the problems. Within each step, the calculation program continues to carry out iterations until the calculated errors are smaller than tolerated error, which is user-defined. In numerical simulation, the plane strain elastic-plastic, Mohr-Coulomb model encoded in PLAXIS was used. The plane strain model indicates the strip foundation. The elastic parameters of E = kpa and ν = 0.3 were assumed. The values of c = 10 kpa, q = 10 kpa, and γ = 16 kn/m 3 were considered in order to compute N ce, N qe, and N γe, respectively. The friction angle φ is varied from 15 to 40 in 5 increments. It is assumed that the soil is completely dry and obeys associated flow rule. In the numerical model, the width B of the footing is 1.5 m. The typical model is shown in Fig. 2. Several trial analyses were performed to choose an appropriate size for the soil domain. Finally, a soil domain with the depth of 12B and length of 24B was selected. It was seen that the selected domain is large enough to vanish boundary effects. The domain is divided into 2869 triangular elements. As seen in Fig. 2, finer meshes are used in the footing neighborhood. The boundary conditions consist of vertical rollers in vertical sides and fixed boundary at the bottom. 2

3 Fig. 2. The typical model in PLAXIS The loading of rigid strip footing is modeled by imposing uniform inclined stresses at the surface nodes below the footing base. The inclination of imposed stresses is equal to the horizontal seismic coefficient (k h ). The loading process is continued incrementally until (a) soil relative stiffness reaches zero and (b) the nodal out of balance forces get solved. The term "nodal out of balance forces" refers to the difference between the external loads and the forces that are in equilibrium with the current stresses. The stress vertical component that satisfies the above two conditions is the ultimate bearing capacity of the footing. The finite element analyses are performed for 0 k h 0.6. The bearing capacity factors were computed based on the superposition principle. In order to calculate N ce, the surcharge (q) and unit weight (γ) were set to zero. It means that the N ce factor is only dependent on the structural inclined load. Having calculated the ultimate bearing pressure (q u ), the N ce factor was obtained for a particular friction angle (φ) and seismic coefficient (k h ) by using: N ce = q u / c (2) Factor N qe was obtained by setting cohesion (c) and unit weight (γ) to zero. But, in order to avoid numerical difficulties, a small amount of cohesion (c = 0.1 kpa) was used in the analyses. The effect of cohesion was then eliminated by subtracting the term cn ce from ultimate bearing pressure. The term N qe depends on both structural inclined load and inclined surcharge. Following the computation of the ultimate bearing capacity, the N qe value was calculated using: N qe = q u / q (3) PLAXIS is able to carry out pseudo-static analysis by applying horizontal acceleration to the soil weight. In order to compute N γe, the surcharge (q) was kept equal to zero. Again a small value of c was chosen to prevent numerical instabilities. The calculations were performed in two separate phases. In the first phase the pseudo-static analysis was performed and in the second phase, the external inclined stresses were applied to the foundation and the ultimate bearing pressure (q u ) was computed. The N γe value was computed by the expression: N γe = 2q u / γb (4) 3

4 The N γe factor is the only one that depends on the soil inertia as well as structural inclined load. Computation of N γe was cumbersome and in some cases the solution did not converge. In such cases, we increased the accuracy of the finite element analyses to avoid numerical divergence. 4. RESULTS AND DISCUSSION Results of present numerical analyses are presented in a combination of curves and tables. At the beginning, the results of seismic bearing capacity factors are presented in graphs. Then, the effect of soil inertia on the seismic bearing capacity is discussed in details. At the last, the error induced by using superposition principle in seismic bearing capacity computations is described SEISMIC BEARING CAPACITY FACTORS A large amount of numerical analyses were performed to predict seismic bearing capacity factors N ce, N qe, and N γe values for a wide range of seismic coefficient k h. The results are presented in Figs 3(a-c). Richards et al. [2] pointed out that for a dry cohesionless soil, the seismic shear fluidization occurs when k h is greater than tan (φ). In this stage, soil acts like a viscous flow. Thus, the upper limit of k h is set to tan (φ) for computations of N qe, and N γe for friction angles 15 to 30. For φ = 35, 40, the maximum value of k h is 0.6. (a) (b) 4

5 φ = 40 (c) Fig. 3. Variations of (a)n ce (b)n qe (c)n γe with k h for a wide range of friction angles As seen in Figs. 3 (a-c), the values of seismic bearing capacity factors decrease drastically with increase in k h. The factor N γe has the most significant decrease. For example, for φ = 30 and a h = 0.3, the values of N ce, N qe, and N γe are 59%, 45%, and 15% of their static values, respectively EFFECT OF SOIL INERTIA The effect of soil inertia on the seismic bearing capacity of strip footings has been a challenge among researchers. Sarma and Iossifelis [3] concluded that for earthquake accelerations smaller than 0.1g, this effect is not large and can be accommodated within the factor of safety. Dormieux and Pecker [6] observed that the reduction in bearing capacity is mainly caused by load inclination, and hence the soil inertia forces can be neglected. In the present paper, all the numerical analyses were performed for two general cases: including soil inertia; and neglecting soil inertia. It is worthy to remind that N γe factor is the only one that depends on the soil inertia. The results of present analyses are shown in Fig. 4 for friction angles of 30, 35, and 40. Fig. 4. Effect of soil inertia on the seismic bearing capacity 5

6 As seen in Fig.4, the effect of soil inertia on the seismic bearing capacity is small. Generally, in all analyses performed in the present study for all friction angles and seismic coefficients, the effect of soil inertia was found less than 12.5%. It means that the main reason of reduction in bearing capacity belongs to foundation inclined load. The present conclusion agrees with that of [3, 6] 4.3. EVALUATION OF SUPERPOSITION PRINCIPLE ERROR Using superposition principle to compute bearing capacity according to Eq. (1) is not sufficiently accurate. The reason is that the soil behaves nonlinearly specially under high pressures. Lundgren and Mortensen [18], by using theory of plasticity, found that the range of error induced by applying superposition principle for rough bases is 17-20%. Davis and Booker [19] concluded that the error is always on the safe side. In the present paper, a number of analyses were carried out to evaluate the superposition error in both static and seismic conditions. The results are shown in Table 1. Table 1. Effect of superposition on static/seismic bearing capacity φ k h c (kpa) q (kpa) γ (kn/m 3 ) q u (kpa) q u assuming superposition (kpa) Superposition error % (+ safe) Table 1 depicts that the superposition error is always on the safe side and the error percentage corresponding to seismic case is greater than static case. In other words, using superposition principle to determine bearing capacity in seismic case is more conservative than static case. 5. COMPARISON OF RESULTS Comparison of seismic bearing capacity factors obtained by the present numerical study with those obtained by other studies for φ = 30 is shown in Figs. 5(a-c). (a) 6

7 (b) (c) Fig. 5. Comparison of (a)n ce (b)n qe (c)n γe of the present study with some of the available solutions in the literature As seen in Figs. 5a and 5b, the values of N ce and N qe established from the present analyses are slightly higher than other solutions. According to Fig. 5c, it is clear that the values of N γe from the present study are in a good agreement with other solutions. For higher values of seismic coefficient (say k h > 0.3), all of the solutions for N γe are almost coincident. 6. CONCLUSIONS The finite element method has been used to investigate seismic bearing capacity of strip footings for a wide range of friction angles and seismic coefficients. The pseudo-static approach has been adopted in the analyses. The seismic bearing capacity factors were presented in graphs and compared to a number of available solutions in the literature. The results showed that seismic bearing capacity factors decrease considerably with increase in seismic coefficient. Factor N γe had the most significant decrease. Moreover, it was observed that the soil inertia does not have any significant impact on the seismic bearing capacity. It was also shown that the error of superposition principle in conventional bearing capacity equation is on the safe side. Furthermore, using superposition principle to compute bearing capacity for the seismic case is more conservative than static case. 7

8 7. REFERENCES 1. Richards, R. Elms, D.G. Budhu, M., (1993), Seismic bearing capacity and settlement of foundations, J. Geotech. Eng. ASCE, 119(4), pp Richards, R. Elms, D.G. Budhu, M., (1990), Dynamic fluidization of soils, J. Geotech. Eng. ASCE, 116(5), pp Sarma, S.K. and Iossifelis, I.S., (1990), Seismic bearing capacity factors of shallow strip footings, Geotechnique, 40(2), pp Choudhury, D. and Subba Rao, K.S., (2005), Seismic bearing capacity of shallow strip footings, Geotech. Geol. Eng., 23(4), pp Budhu, M. and Al-Karni, A., (1993), Seismic bearing capacity of soils, Geotechnique, 43(1), pp Dormieux, L. and Pecker, A., (1995), Seismic bearing capacity of foundations on cohesionless soil. J. Geotech. Eng. ASCE, 121(3), pp Soubra, A.H., (1999), Upper bound solutions for bearing capacity of foundations, J. Geotech. Geoenviron. Eng. ASCE, 125(1), pp Ghosh, P., (2008), Upper bound solutions of bearing capacity of strip footing by pseudo-dynamic approach, Acta Geotechnica, 3, pp Kumar, J. and Rao, V.B.K.M., (2002), Seismic bearing capacity factors for spread foundations, Geotechnique, 52(2), pp Kumar, J. and Rao, V.B.K.M., (2003), Seismic bearing capacity of foundations on slopes, Geotechnique, 53(3), pp Ghahramani, A. and Berrill, J.B., (1995), Seismic bearing capacity factors by zero extension line method, Pacific Conf. on EQ. Eng., pp Zeng, X. and Steedman, R.S., (1998), Bearing capacity failure of shallow foundations in earthquakes, Geotechnique, 48(2), pp Knappett, J.A. Haigh, S.k. Madabushi, S.P.G., (2006), Mechanisms of failure for shallow foundations under earthquake loading, Soil Dyn. Earthquake Eng., 26, pp Frydman, S. and Burd, H.J., (1997), Numerical studies of bearing capacity factor N γ, J. Geotech. Geoenviron. Eng. ASCE, 123(1), pp Shafiee, A.H. and Shafiee A., (2009), Computation of bearing capacity factors using finite element method, 8 th Int. Congr. on Civil Eng., Shiraz, Iran. 16. Terzaghi, K., (1943), Theoretical soil mechanics, Wiley, New York. 17. PLAXIS BV., (2002), User s manual of PLAXIS, A.A. Balkema Publishers. 18. Lundgren, H. and Mortensen, K., (1953), Determination by the theory of plasticity of the bearing capacity of continuous footings on sand, Proc. 3rd Int. Conf. on Soil Mechanics and Foundation Eng., Zurich, 1, pp Davis, E.H. and Booker, J.R., (1971), The bearing capacity of strip footings from the standpoint of plasticity theory, 1st Australian-New Zealand conf. in geomechanics, pp