Seismic Bearing Capacity of Strip Footings Adjacent to Slopes Using the Upper Bound Limit Analysis   Priyanka Ghosh Research Scholar, Department of Civil Engineering, Indian Institute of Science, Bangalore – 560012, India priyog@civil.iisc.ernet.in and Jyant Kumar Assistant Professor, Department of Civil Engineering, Indian Institute of Science, Bangalore – 560012, India jkumar@civil.iisc.ernet.in

ABSTRACT

By using the upper bound limit analysis, the bearing capacity factor N_g for a rough strip footing placed near the sloping ground surface was computed as a function of horizontal earthquake acceleration coefficient a_h. The soil was assumed to follow the Mohr-Coulomb failure criterion and an associated flow rule. In this present analysis no slip was allowed along the footing-soil interface i.e. perfectly rough condition was considered. The boundary of the radial shear zone was assumed to comprise of an arc of the logarithmic spiral. The results from the present analysis were presented in the form of the bearing capacity factor N_g, as a function of a_h, s/B, b and f; where s is the distance of the footing edge from the slope edge, B is the width of the strip footing and b is the horizontal inclination of the slope. The results from the present analysis were thoroughly compared with the available data in the literature.

Keywords: Bearing capacity, foundations, limit analysis, plasticity, seismic, slopes.

INTRODUCTION

In the last decade a number of investigations have been performed in finding the seismic bearing capacity of foundations by taking into account the pseudo-static earthquake body forces in the horizontal and vertical directions. By adopting the finite element method, it is possible to carry out a dynamic elasto-plastic analysis of the foundations. However, such an analysis is often quite time consuming and, moreover, achieving the convergence of results in such a numerical computation is found to be an extremely difficult task. On the other hand, pseudo-static dynamic analysis provides an easier way to estimate the bearing resistance of foundation for any imposed earthquake acceleration. From the available theories, it is now known that the magnitude of all the bearing capacity factors decreases quite extensively with increase in the values of a_h and a_v, where a_h and a_v are the horizontal and vertical earthquake acceleration coefficients, respectively. As compared to bearing capacity factors N_c and N_q, the reduction is generally much more remarkable for the bearing capacity factor N_g. By using the upper bound limit analysis, the bearing capacity factor N_g for a rough strip footing placed near the sloping ground surface was computed as a function of a_h on the basis of single side failure mechanism. The soil was assumed to follow the Mohr-Coulomb failure criterion and associated flow rule. In this present analysis no slip was allowed along the footing-soil interface i.e. perfectly rough condition was considered. The boundary of the radial shear zone was assumed to comprise of an arc of the logarithmic spiral. However, the expression for the logarithmic spiral was based on an angle f_L which was varied from 0 to f, that is, the shape of the outer boundary arc of the radial shear zone was varied from circular to logarithmic spiral depending on the value of f_L. The whole logarithmic spiral zone was discretized into a number of triangular rigid blocks. Also, the focus position of the logarithmic spiral was varied rather than keeping fixed at the edge of the footing. The optimization performed by the present mechanism was found to be computationally easier than the multi-block mechanism proposed by previous investigators (Michalowski, 1997; Soubra, 1999; Zhu, 2000; and Askari and Farzaneh, 2003). With the application of the upper bound theorem of limit analysis, the magnitude of the failure load was determined by equating the rate of the total work done by the external and body forces to the rate of dissipation of the total internal energy along the block interfaces and the bottom surfaces of all the rigid blocks. The results from the present analysis were presented in the form of the bearing capacity factor N_g, as a function of a_h, s/B, b and f; where s is the distance of the footing edge from the slope edge, B is the width of the strip footing and b is the horizontal inclination of the slope. Sawada et al. (1994) and Askari and Farzaneh (2003) have reported the bearing capacity of c-f soil. However, in presence of a_h and s/B, N_g values were not exclusively reported in their study. The results from the present analysis were thoroughly compared with the data proposed by different researchers.

FAILURE MECHANISM

A non-plastic triangular trapped wedge (ABC) was considered below the footing base as shown in Figure 1. The collapse mechanism consists of a logarithmic spiral radial shear zone BCD sandwiched in between two rigid zones, namely triangular block ABC and quadrilateral block BDEG. At collapse, the footing and the underlying triangular rigid block ABC were assumed to move as a single rigid body with velocity V0, as shown in Figure 1. The focus (F) of the logarithmic spiral arc (CD) was moved along the straight line DBF passing through the right footing edge (B). The equation of the logarithmic spiral was defined by means of an expression: where r₁ = distance FD, r₀ = distance FC, and q = ÐCFD. The value of f_L was varied from 0 to f, that is, the shape of the outer boundary arc (CD) of the radial shear zone (BCD) was varied from circular to logarithmic spiral depending on the value of of f_L. The whole logarithmic spiral zone was discretized into a number (n) of triangular rigid blocks as shown in Figure 1. The collapse mechanism is very similar to that adopted by Kumar (2004). For doing the computations, the number (n) of rigid blocks was varied from 500 for lower values of of f to 700 for higher values of f. For the sake of illustration, however, the radial shear zone BCD in Figure 1 has been discretized into just three triangular blocks. In order that the logarithmic spiral arc (CD) of the failure surface meets tangential to the straight lines AC and DE of the rupture lines, the values of angles (ACF and (BDE were kept equal to (p/2 - f_L) and (p/2 + f_L), respectively. The geometry of the chosen failure mechanism can be completely defined by means of four independent variables, namely, a₁, h, f_L, and l; where l is the distance of the logarithmic spiral focus F from the right footing edge B, and a₁ and h are the values of ÐBAC and ÐDBG (Figure 1).

Figure 1. Collapse mechanism and velocity hodograph

VELOCITY HODOGRAPH

The velocity hodograph triangles have also been drawn in Figure 1 for three blocks (n = 3) of the radial shear zone. In these figures V₁, V₂, V₃ and V₄ are the respective absolute velocities of the blocks BCC₁, BC₁C₂, BC₂D and BDEG. V₁₀ is the relative velocity of the block BCC₁ with respect to the block ABC; V₂₁ is the relative velocity of the block BC₁C₂ with respect to the block BCC₁; V₃₂ is the relative velocity of the block BC₂D with respect to the block BC₁C₂; V₄₃ is the relative velocity of the block BDEG with respect to the block BC₂D. The interfaces of all the triangular sectors were treated as velocity discontinuity lines. Since the soil mass was assumed to obey Mohr-Coulomb failure criterion and an associated flow rule, the directions of the velocities V₀, V₁, V₂, V₃ and V₄ make an angles f with the corresponding rupture lines. Likewise, the directions of the relative velocities V₁₀, V₂₁, V₃₂ and V₄₃ are also inclined at an angle f with the corresponding velocity discontinuity lines BC, BC₁, BC₂ and BD, respectively. The directions of the velocities V₀, V₁, V₂, V₃, V₄, V₁₀, V₂₁, V₃₂ and V₄₃ are fixed, and therefore, by using the velocity hodograph triangles, the magnitudes of the velocities V₁, V₂, V₃, V₄, V₁₀, V₂₁, V₃₂ and V₄₃ can be computed in terms of the velocity (V₀) of the footing and the underlying block (ABC). It should be mentioned that the velocity hodograph triangles were based on the consideration of velocity discontinuities along the interfaces of various rigid blocks not along the radial lines.

ANALYSIS

With the application of the upper bound theorem of limit analysis, the magnitude of the failure load was determined by equating the rate of the total work done by the external and body forces to the rate of dissipation of the total internal energy along velocity discontinuity lines. It should be mentioned that the rigid block ABC was considered as a part of the footing, and therefore, no relative movement was permitted between the footing and the underlying block ABC. Since the soil mass was assumed to obey an associated flow rule, the rate of dissipation of internal energy per unit area along all the velocity discontinuities becomes simply equal to cVcosf where V is the magnitude of the velocity jump along any velocity discontinuity and c is cohesion value of the soil. The magnitude of the failure load, P_u per unit length of strip footing, was computed by the following expression:

(1)

The velocity hodographs and the collapse mechanism are shown in Figure 1 for three rigid blocks (n = 3) of the radial shear zone. In the above expression, a₀ = length AC, a_i = base length of any i-th triangular block within the radial shear zone BCD (i varies from 1 to n), a_n+1 = length DE, b_i is the length of i-th interface of the two adjacent blocks (i varies from 0 to n). V_i is the absolute velocity of the i-th block in the radial shear zone BCD (i varies from 1 to n); V_n+1 is the absolute velocity of the block BDEG. The superscripts "v" and "h" attached to any velocity term define the components of the corresponding velocity in the vertical and horizontal directions, respectively. The vertical component of the velocity was taken as positive in the downward direction. On the other hand, the horizontal component of the velocity towards the direction of a_h (left to right of Figure 1) was considered positive. W₀, W_n+1 and W_i are the respective weight of rigid block ABC, BDEG and i-the block in the radial shear zone BCD. Therefore, by using the above expression the magnitude of the failure load P_u can be determined for a chosen collapse mechanism. By varying independently the four variables, namely, f_L and f_l, the magnitude of the force P_u was then minimized. It was ensured that the chosen collapse mechanism remains always kinematically admissible. For a chosen collapse mechanism to be kinematically admissible, the magnitudes of all the velocity terms, in terms of V₀, must remain always positive. While doing the computations, the values of angles f_L, h and a₁ were varied with minimum interval of 1^o. The minimum interval for l/B was chosen equal to 0.005.

RESULTS

Bearing Capacity Factor N_g

The magnitude of the collapse load P_ug per unit length of the footing, for the unit weight of soil mass, was determined in terms of bearing capacity factor N_g. The bearing capacity equation for cohesionless soil (f = 0) is given below:

pug=0.5gBNg

(2)

where, p_ug is the average magnitude of the ultimate pressure for the unit weight of soil mass; p_ug = P_ug/B. The variation of the bearing capacity factor N_g with the changes in a_h for different values of f, b and s/B is presented in Figures. 2, 3 and 4. It can be observed that the magnitudes of the bearing capacity factor N_g decrease continuously with increase in the magnitude of a_h for a particular value of s/B, f and b. The magnitudes of the bearing capacity factor N_g become smaller for greater inclinations of slope inclination angle (b). There was always a particular value of a_h beyond which the slope, with cohesioless soil, becomes always unstable. The maximum value of a_h up to which the cohesionless sloping ground will remain stable is tan(f - b). As a result, the values of the bearing capacity factor N_g were obtained only up to a_h = tan(f - b). The values of N_g increase quite extensively with increase in the s/B ratio and become constant at s/B = s_max/B, where s_max is the value of s beyond which there will be no effect of slope angle on the bearing capacity factor N_g. The variation of s_max/B with a_h and f is given in Table 1. It can be observed from Table 1 that for a particular value of f the values of s_max/B decrease continuously with the increase in a_h to reach the minimum value and then increase again. The values of N_g become greater for higher values of f.

Table 1. The variation of s_max/B with f and a_h

Figure 2. The variation of N_g with a_h and s/B for f = 10^o and 20^o

Figure 3. The variation of N_g with a_h and s/B for f = 30^o and 40^o

Figure 4. The variation of N_g with a_h and s/B for f = 50^o

Failure Patterns

The failure patterns have also been drawn by keeping the scale same both for the horizontal and vertical axes. These failure patterns were sketched for f = 30^o and 40^o, s/B = 0.5, 1.0, 2.0 and 3.0, f = 5^o, 10^o and 20^o, and for different values of a_h. The failure patterns are illustrated in Figure 5. It was noted that with increase in a_h, the geometry of the collapse mechanism shifts towards the direction of a_h and the overall size of the collapse zone becomes smaller. An increase in the s/B ratio also leads to an increase in the overall size of the collapse zone.

Figure 5. The geometry of the failure patterns for f = 30^o and 40^o for different values of a_h, b and s/B

COMPARISONS

For a_h = 0.0, b = 0^o and s/B = 0.0, the values of N_g obtained from the present analysis are compared thoroughly in Table 2. It can be observed from Table 2 that the present values of N_g match well with the existing data in the literature. Kumar and Rao (2002) and Kumar (2003) used the method of stress characteristics to determine the values of N_g, which generally gives the lower bound solution. Kumar (2003) considered a non-plastic curved wedge below the footing, which starts tangentially at the footing edge.

Table 2. A comparison of variation of N_h with f for a_h = 0.0, b = 0^o and s/B = 0

For a_h = 0.0, the present N_g values are compared with the results given by various researchers in presence of s/B and b. The comparison is shown in Table 3. It can be seen from Table 3 that the present N_g values compare reasonably well with the different available theories for a_h = 0.0. The N_g values given by Saran et al. (1989) and Saran and Reddy (1990) are always found to be higher than the present values, whereas the N_g values reported by Meyerhof (1957) are found to match reasonably well with the present values.

Table 3. A comparison of variation of N_g with f, b and s/B for a_h = 0.0

As mentioned, Sawada et al. (1994) used the upper bound limit analysis and determined the seismic bearing capacity of a rough strip footing near a downhill slope by considering a simple logarithmic spiral radial shear zone, whereas on the basis of upper bound limit analysis, Askari and Farzaneh (2003) obtained the seismic bearing capacity of rough strip footing adjacent to a slope by considering multi-block mechanism. The magnitudes of bearing capacity (p_u) with a_h and g obtained from the present analysis are compared with those given by Sawada et al. (1994) and Askari and Farzaneh (2003) in Table 4 where p_u is given by the following expression:

pu = cNc + 0.5 g B Ng

(3)

where N_c is the bearing capacity factor for the cohesion of soil mass. The value of N_c was determined separately by considering the method of superposition by simply substituting c = 0. However, in this note the magnitudes of N_c were not reported. In Table 4, the comparison has been drawn for the case of f = 30^o, b = 20^o, c = 9.8 kPa, B = 10 m and s = 20 m. It can be seen from Table 4 that the results of present analysis compare reasonably well with the values given by Askari and Farzaneh (2003), whereas the values reported by Sawada et al. (1994) are found to be the highest.

Table 4. A comparison of variation of bearing capacity (kPa) with a_h and g (kN/m³)
for the case of f = 30^o, b = 20^o, c = 9.8 kPa, B = 10 m and s = 20 m

CONCLUSIONS

The effect of horizontal earthquake acceleration on the bearing capacity factor N_g was examined for a footing placed on a sloping ground surface where slope was considered to start at some distance away from the footing edge. The analysis was carried out by using the upper bound limit analysis. While choosing the collapse mechanism, the periphery of the radial shear zone was assumed to be bounded by logarithmic spiral arc; however, the equation of the logarithmic spiral arc was based on an angle f_L which was varied from 0 to f. Also, the position of the logarithmic spiral focus was varied rather than fixing at the footing edge. The values of N_g were found to increase extensively with increase in the value of s/B ratio for different values of ground inclination b, a_h and soil friction angle f. The effect of slope angle on the bearing capacity factor N_g does not exist beyond s_max/B. It was also noticed that with the increase in a_h, the geometry of the failure surface shifts gradually towards the direction of a_h.

REFERENCES

1. Askari, F., and O. Farzaneh (2003) “Upper-bound Solution for Seismic Bearing Capacity of Shallow Foundations Near Slopes,” Géotechnique, Vol. 53, No. 8, pp. 697-702.
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