Seismic Bearing Capacity of Shallow Foundations

The seismic risk mitigation is one of the greatest challenges of the Civil Engineering and an important contribution toward this challenge can be given by the Geotechnical Earthquake Engineering. Lesson learned by recent destructive earthquakes (January 2010 Port-au-Prince region of Haiti and March 2011 Tohoku Japan), confirms that local soil conditions can play a significant role on earthquake ground motions. Earthquake-induced damage in Port-au-Prince was devastating and widespread. Yet, there were clearly areas of the city where little to no damage occurred, and areas of the city where an overwhelming majority of the buildings were severely damaged or destroyed. These types of damage patterns are common in earthquakes, and a wide number of factors need to be considered in order to conclusively piece together the causes. For a given earthquake, these factors include, but are not limited to: (a) relative distance from the fault rupture plane, (b) construction type and quality, (c) local soil conditions (i.e. strength/stiffness of the soil foundation, depth to bedrock, impedance contrasts, geology), (d) topography (topographic and basin effects), and (e) near fault effects (rupture directivity, fling step, hanging wall effects, polarity effects, etc.). Often several of these factors work together and it can be difficult to identify the primary cause of damage. Design of foundations in seismic areas needs special considerations compared to the static case. The inadequate performance of structures during recent earthquakes has motivated researchers to revise existing methods and to develop new methods for seismic-resistant design. This includes new design concepts, such as, performance-based design (PBD) (Priestley et al., 2005) and new measures of the structure performance based on energy concepts and damage indexes (Park et al., 1987; Moustafa, 2011). Similarly, the widespread damage and inadequate performance of code-designed structures during the 1994 Northridge (California) and the 1995 Kobe (Japan) earthquakes have prompted seismologists and engineers of the essential importance of characterizing and modelling near-field ground motions with impulsive nature (Moustafa & Takewaki, 2010). For foundations of structures built in seismic areas, the demands to sustain load and deformation during an earthquake will probably be the most severe in their design life. As stressed by Hudson (1981) the soil-structure interaction is a crucial point for the evaluation of the seismic response of structures.


Introduction
The seismic risk mitigation is one of the greatest challenges of the Civil Engineering and an important contribution toward this challenge can be given by the Geotechnical Earthquake Engineering. Lesson learned by recent destructive earthquakes (January 2010 Port-au-Prince region of Haiti and March 2011 Tohoku Japan), confirms that local soil conditions can play a significant role on earthquake ground motions. Earthquake-induced damage in Port-au-Prince was devastating and widespread. Yet, there were clearly areas of the city where little to no damage occurred, and areas of the city where an overwhelming majority of the buildings were severely damaged or destroyed. These types of damage patterns are common in earthquakes, and a wide number of factors need to be considered in order to conclusively piece together the causes. For a given earthquake, these factors include, but are not limited to: (a) relative distance from the fault rupture plane, (b) construction type and quality, (c) local soil conditions (i.e. strength/stiffness of the soil foundation, depth to bedrock, impedance contrasts, geology), (d) topography (topographic and basin effects), and (e) near fault effects (rupture directivity, fling step, hanging wall effects, polarity effects, etc.). Often several of these factors work together and it can be difficult to identify the primary cause of damage. Design of foundations in seismic areas needs special considerations compared to the static case. The inadequate performance of structures during recent earthquakes has motivated researchers to revise existing methods and to develop new methods for seismic-resistant design. This includes new design concepts, such as, performance-based design (PBD) (Priestley et al., 2005) and new measures of the structure performance based on energy concepts and damage indexes (Park et al., 1987;Moustafa, 2011). Similarly, the widespread damage and inadequate performance of code-designed structures during the 1994 Northridge (California) and the 1995 Kobe (Japan) earthquakes have prompted seismologists and engineers of the essential importance of characterizing and modelling near-field ground motions with impulsive nature (Moustafa & Takewaki, 2010). For foundations of structures built in seismic areas, the demands to sustain load and deformation during an earthquake will probably be the most severe in their design life. As stressed by Hudson (1981) the soil-structure interaction is a crucial point for the evaluation of the seismic response of structures.
Due to seismic loading, foundations may experience a reduction in bearing capacity and increase in settlement. Two sources of loading must be taken into consideration: "inertial" loading caused by the lateral forces imposed on the superstructure, and "kinematic" loading caused by the ground movements developed during the earthquake. Part 5 of Eurocode 8 (2003) states that foundations shall be designed for the following two loading conditions : a. inertia forces on the superstructure transmitted on the foundations in the form of axial, horizontal forces and moment ; b. soil deformations arising from the passage of seismic waves. In the last years the seismic action has increased in many National Codes according to recent records which show values up to 0.8 g for very destructive earthquakes. The upgrading of the seismic action requires accurate analyses taking into account all the boundary conditions including the presence of surcharges, sloping ground, depth factors and so on. With the aim to investigate the influence of these factors on the seismic stability of a shallow foundation, a model based on the limit equilibrium method has been developed. Many analytical and numerical solutions are today available to evaluate seismic bearing capacity of shallow foundations, and cover area such as the limit equilibrium method, limit analysis, methods of characteristics, finite element analysis and other areas for the computation of the seismic bearing capacity factors required for the design of a foundation. Nevertheless, pseudo-static approaches are more attractive because they are simple, when compared to difficult and more complex dynamic analyses. Thus, a pseudo-static model to account for reduction in bearing capacity due to earthquake loading is presented. In this model the loading condition consists in normal and tangential forces on the foundation and inertial forces into the soil. An upper bound solution of the limit load of the shallow foundation is found. Results of the proposed analysis are given in terms of the ratios between seismic and static bearing capacity factors N c * /N c , N q * /N q and N  * /N  . Results are also compared with those deduced by other authors using different methods of analysis.

Method of analysis
The prediction of the bearing capacity of a shallow foundation is a very important problem in Geotechnical Engineering, and in the last decades solutions using limit analysis, slip-line, limit equilibrium and, recently, numerical methods (i.e. finite element and difference finite methods) have been developed. The problem of static bearing capacity of shallow foundations has been extensively studied in the past by Terzaghi (1943), Meyerhoff (1963), Vesic (1973 and many others. The ultimate load that the foundation soil can sustain is expressed by the linear combination of the three bearing capacity factors N c , N q and N  which depend uniquely on the friction angle of the soil. Further solutions for the bearing capacity were given successively in a more general form, taking into account, by means of corrective factors, of the shape of the foundation, of the load and ground inclination and of the depth and inclination of the bearing surface. In all these studies, the bearing capacity evaluation is based on the assumption that a failure surface can develop beneath the foundation, according to the well known failure surfaces given by the limit equilibrium method or by the limit analysis. Most foundation failures during earthquakes occur due to liquefaction phenomena, even if failures due to reduction in bearing capacity have been observed during Naigata earthquake (1964) Japan and Izmit earthquake (1999) in Turkey (Day, 2002). Liquefiable soils are categorized by all seismic codes as extreme ground conditions, where, following a positive identification of this hazard, the construction of shallow footings is essentially allowed only after proper soil treatment. More specifically, liquefaction-induced shear strength degradation of the foundation subsoil may result in post-shaking static bearing capacity failure, while excessive seismic settlements may also accumulate. However, the accurate estimation of the degraded bearing capacity and the associated dynamic settlements could potentially ensure a viable performance-based design of shallow footings. Richards et al. (1993) observed seismic settlements of foundations on partially saturated dense or compacted soils. These settlements were not associated with liquefaction or densification and could be easily explained in terms of seismic bearing capacity reduction. In fact, the inertial forces applied on the foundation and in the soil mass reduce the static bearing capacity. Thus, many authors have investigated the seismic bearing capacity giving results in terms of the ratio of the seismic to the static bearing capacity factors N c * /N c , N q * /N q and N  * /N  . The pseudo-static approach is being used to determine bearing capacity of the foundations subjected to seismic loads in non-liquefying soils, considering also the depth effects for an embedded footing and the effect of a sloping ground located at some distance from the footing. Dynamic nature of the load and other factors which affect the dynamic response are not being accounted for. Ground factors and bearing capacity ratios N c * /N c , N q * /N q and N  * /N  are presented as a function of the friction angle of soil ', of the ratio H/B between the embedment depth H and the width of the footing B, of a slope angle  and of the ratio d/B being d the distance from the edge of the slope. The inertial and kinematic effects due to seismic loading have been analyzed in the evaluation of the seismic bearing capacity.

Limit equilibrium analysis
The method of analysis is based on the limit equilibrium technique. The failure mechanism, as shown in Figure 1, is a circular surface which from the foundation propagates until the ground surface is reached (Castelli & Motta, 2010;2011). A similar model was proposed by Castelli & Motta (2003) for a bearing capacity analysis of a strip footing resting on a soil of limited depth. The seismic forces are considered as pseudo-static forces acting both on the footing and on the soil below the footing. The ultimate load can be found by a moment equilibrium respect to the centre of the circular surface. Referring to Figure 1 a moment equilibrium can be written and the mobilizing moment is : The resisting moment given by the shear strength S i acting along the base of the slices is : The force N i resultant of the normal stress distribution acting at the base of the slice can be derived by the Bishop's method of slices (1955) with an equilibrium equation in the vertical direction, so one obtains (see Figure 1), for the slices under the footing where i = 1 to n 1 : and for the remaining slices (n 1 +1  i  n tot ) : (1 ) tan cos sin tan where :  q lim = vertical limit load acting on the footing;  c = soil cohesion;  x i = width of the i th slice;  n 1 = number of slices under the footing;  n tot = total number of slices;  k h1 = horizontal seismic coefficient for the limit load;  k h2 = horizontal seismic coefficient for the soil mass;  k v = vertical seismic coefficient for the soil mass;  b wi = distance of the weight W i of the i th slice to the centre of the circular failure surface;  b whi = distance of the inertia force k h2 W i of the i th slice to the centre of the circular failure surface;  b qli = distance of the limit load q lim acting on the i th slice to the centre of the circular failure surface;  b qhli = distance of the shear limit force k h1 q lim acting on the i th slice to the centre of the circular failure surface. Substituting the following terms : Even if the failure mechanism adopted is quite simple, it allows to investigate a variety of loading and geometric conditions that could have been troublesome using other failure mechanisms and results are in a very good agreement with those obtained by other authors. In fact, referring to the kinematic effect due to the inertia of the soil mass on the seismic bearing capacity, Figure 2 shows a comparison between the results of the present study (for k v = 0), those produced by the method proposed by Paolucci & Pecker (1997) and those found by Cascone et al. (2004) with the method of characteristics.

Fig. 2. Seismic ratios as a function of the soil mass inertia
The reduction of the bearing capacity is presented in terms of the ratio N  * /N  as a function of the seismic coefficient k h2 in the soil mass. Despite of the different methods, the results obtained are in good agreement. However, for low values of k h2 , the limit equilibrium approach seems to give the greatest reduction thus it is on the safe side.

Parametric analysis
To investigate the influence of the depth factor on the seismic stability of a shallow foundation, the model proposed has been applied and an upper bound solution of the limit load is found. Results of the analysis are given in terms of the ratios between seismic and static bearing capacity factors N c * /N c , N q * /N q and N  * /N  . Ground factors and bearing capacity ratios are presented as a function of the friction angle of soil ' and of the ratio H/B between the embedment depth H and the width of the footing B. The inertial and kinematic effects due to seismic loading have been analyzed in the evaluation of the seismic bearing capacity. For a shallow foundation resting on a cohesionless soil, with horizontal ground surface and in absence of surcharge, the limit load can be expressed by :  N  = bearing capacity factor;  i i = load inclination factor due to the inertia of the structure;  i k = reduction factor due to the inertia of the soil mass (kinematic interaction factor);  d  = depth factor. The load inclination factor related to the inertia of the structure (i i ) has been discussed by some authors (Pecker & Salencon, 1991;Budhu & Al-Karni, 1993;Dormieux & Pecker, 1995;Paolucci & Pecker, 1997;Fishmann et al., 2003), while less information are available on the depth factor (d  ) and on the reduction factor due to the inertia of the soil mass (i k = kinematic interaction factor). Conventionally, the depth factor (d  ) is assumed equal to unit (Brinch Hansen, 1970). Nevertheless, in an analysis in which the effects due to the inertia of the soil mass are taken into consideration, it is also necessary to take into account the inertia of the soil mass corresponding to the embedment depth H of the footing.

Depth factor evaluation
In static conditions the depth factor d  has been evaluated by a parametric analysis, for both drained (' = 20°, 30°, 40°) and undrained conditions (= 0), varying the ratio H/B between the embedment depth H and the width of the footing B.
In the present analysis, the depth factor d  for drained conditions is defined as the ratio between the bearing capacity factors N  ' of a shallow foundation with embedment H and the conventional bearing capacity N  of a shallow foundation with an embedment equal to 0 : Similarly, for undrained conditions the depth factor d   is defined as the difference between the bearing capacity factors N   ' of a shallow foundation with an embedment H and the conventional bearing capacity N   of a shallow foundation with an embedment equal to 0 : being N ° in undrained conditions, as known, equal to 0. With reference to equation (10) in Table 1  In Figure 3 are reported the values of the depth factor d  for drained conditions versus H/B.
For the values of the friction angle of soil ' taken into consideration, curves shown approximately a linear trend, thus it is possible to express the depth factor d  as a linear function of the ratio H/B according to the equation : For undrained conditions the results obtained ( Figure 4) can be conveniently expressed by the following linear equation : that, obviously, is valid for H/B > 0.125.

Kinematic interaction factor evaluation
The kinematic interaction factor i k has been evaluated by a parametric analysis only for drained conditions, varying the friction angle of soil in the range ' = 20°, 30° and 40° and the horizontal seismic coefficient for the soil mass k h2 between 0.1 up to 0.3. The kinematic interaction factor i k is defined in this study as the ratio between the bearing capacity factor N  * derived for a given value of the horizontal seismic coefficient k h2 , and the conventional bearing capacity factor N  : The numerical analyses have been carried out assuming a vertical seismic coefficient for the soil mass k v equal to ½ k h2 . In Figure 5 are reported the values of the kinematic interaction factor i k obtained for the soil friction angles taken into consideration.
It is simple to verify that for k h2 = tan ' the kinematic interaction factor i k is equal to 0.

Bearing capacity of strip footings near slopes
When a shallow foundation is placed near the edge of a sloping ground the bearing capacity may be reduced both in static (De Buhan & Gaernier, 1988;Saran et al., 1989;Narita & Yamaguchi, 1990;Shields et al., 1990;Jao et al., 2001) and seismic conditions (Sawada et al., 1994;Pecker, 1996;Sarma & Iossifelis, 1990;Sarma & Chen, 1996;Kumar & Rao, 2003). In this case, the failure mechanism is influenced by the distance d of the foundation from the edge of the sloping ground ( Figure 6). If the shallow foundation is far enough from the edge, the failure mechanism will be not affected by the slope. In the last decades extensive studies have been made for two dimensional problems of a strip footing resting on inclined slope surface so that different methods of analysis are available.
In these studies, the bearing capacity evaluation is based on the assumption that a failure surface can develop beneath the foundation, according to a general shear failure surfaces given by the limit equilibrium method or by kinematic mechanisms. General shear failure is characterized by the existence of a well-defined failure pattern (Terzaghi, 1943), which consists of a continuous slip surface from one edge of the footing to the horizontal or inclined ground surface. Referring to Figure 6, the mobilizing moment is :  The resisting moment M res and the shear strength S i acting along the base of the slices are expressed by equation (2) and (3) respectively. For the slices i = 1 to n 1 the force N i resultant of the normal stress distribution acting at the base of the slice, derived by the Bishop's method of slices (1955) with an equilibrium equation in the vertical direction, is given by equation (4), while for the remaining slices (n 1 +1  i  n tot ) is : where :  q lim = vertical limit load acting on the footing;  n 1 = number of slices under the footing;  n tot = total number of slices;  k h1 = horizontal seismic coefficient for the limit load;  k h2 = horizontal seismic coefficient for the soil mass;  k h3 = horizontal seismic coefficient for the surcharge;  k v = vertical seismic coefficient for the soil mass and the surcharge;  b wi = distance of the weight W i of the i th slice to the centre of the circular failure surface;  b whi = distance of the inertia force k h2 W i of the i th slice to the centre of the circular failure surface;  b qvi = distance of the surcharge force q v x i of the i th slice to the centre of the circular failure surface;  b qhi = distance of the horizontal surcharge force k h3 q v x i of the i th slice to the centre of the circular failure surface;  b qli = distance of the limit load q lim acting on the i th slice to the centre of the circular failure surface;  b qhli = distance of the shear limit force k h1 q lim acting on the i th slice to the centre of the circular failure surface. Substituting the following terms :

Seismic analysis
The recommendations of Eurocode 8 -Part 5 (2003) state that in the calculation of the bearing capacity of a shallow foundation one should include the load inclination and eccentricity arising from the inertia forces of the structure, as well as the possible effects of the inertia in the soil. Thus, the seismic analysis was carried out considering the following seismic coefficients: k h1 = 0.1 and 0.2 for the inertia of the structure; k h2 = 0.2 and 0.4 for the inertia of the soil mass. The value of k h1 was chosen lower than k h2 because the Eurocode 8 (2003) allows to reduce the seismic action by a behaviour factor associated with the ductility classification of the structures. This consideration takes to the conclusion that the kinematic effect, and the consequent reduction in bearing capacity due to the soil inertia, cannot be neglected and in some circumstances it's reduction could be more significant than the reduction due to the inertia of the structure (Cascone et al., 2006). In this study the seismic coefficient k h3 of the surcharge was assumed equal to k h1 . The friction angle of soil was chosen in the range 0° up to 40°, while the angle of the slope near the footing was varied in the range 5° to 35°. In the seismic analysis the angle of the sloping ground is affected by further limitations, because simple equilibrium considerations, for a cohesionless soil (c ' = 0), take to the following : where : , 1 tan 1 In the following,   No significant difference in the threshold distance was found when the inertia of the structure or the inertia of the soil mass is considered. Furthermore the combined effects of soil and structure inertia can be taken into account by using the superposition of the effects principle. In this case, at the same way as found by Paolucci & Pecker (1997) and Cascone et al. (2004), the bearing capacity of the soil self weight under both the seismic loading due to the coefficients k h1 and k h2 , can be evaluated through the following equation : where : N e bearing capacity factor reduced by both acting the coefficients k h1 and k h2 ; N  = static bearing capacity factor; e i = N 1 * /N  bearing capacity ratio for structure inertia only (k h1 > 0, k h2 = k h3 = 0); e k = N   /N  bearing capacity ratio for soil mass inertia only (k h2 > 0, k h1 = k h3 = 0). Figure 17 shows a comparison between the N e /N   ratio and the product e i . e k for, as an example, k h1 = 0.1, 0.2 and 0.3 and k h2 = 0.1, 0.2 and 0.3.
In particular, Figure 17 shows that when the seismic coefficients k h1 and k h2 are small, there is not a significant difference between the N e /N   ratio and the product e i . e k . On the contrary, when the seismic coefficients are high enough to produce a great reduction of the limit load, one can find a great difference in using the N e /N   ratio instead of the product e i . e k .
As example, for ' = 20°, k h1 = 0.1 and k h2 = 0.1, we have : This happens because, when the limit load approaches to zero, the critical surface associated to the simultaneous presence of both the inertial and the kinematic effects is significantly different from that deduced when one considers separately the inertial and the kinematic effects. In this case the superposition of the effects principle may lead to an unconservative design, being the product e i . e k significantly greater than the N e /N   ratio.

Conclusions
The design of shallow foundations subject to different static loadings has been an important area of research for geotechnical engineers. The devastating effects of recent earthquakes on shallow foundations has increased the complexity of the problem. Consequently, it is useful to obtain closed-form solutions for the earthquake resistant design of foundations. Many analytical and numerical solutions are available for the computation of the seismic bearing capacity factors required for the design of shallow foundations. In the present study the seismic bearing capacity of shallows foundation has been evaluated with the limit equilibrium method. Numerical analysis shows that, by considering pseudo-static seismic forces, design solutions can be found for the computing of seismic bearing capacity factors for shallow foundations embedded in both horizontal and sloping ground. Seismic bearing capacity factors with respect to cohesion, surcharge and unit weight components have been computed for a wide range of variation in parameters such as soil friction angle ('), horizontal and vertical seismic coefficients (k h and k v ). An "upper bound" approach of the limit load was adopted to evaluate the seismic reduction factors to take into account the embedment depth of the footing (i i ) and the inertia of the soil mass (i k ), as well as, the bearing capacity ratio for structure inertia only (e i ) and the bearing capacity ratio for soil mass inertia only (e k ). Some considerations can be formulated:  In the evaluation of the bearing capacity due to the soil weight it has been observed that the depth of the embedment depth may play a significant role especially for low values of the friction angle.


In some cases the seismic reduction in the bearing capacity for the soil inertia (kinematic effect) cannot be ignored, being of about the same amount of that produced by the inertia of the structure.  The bearing capacity factors decrease appreciably with increases in both k h and k v .
Bearing capacity decreases as ground inclination  increases and as the embedment depth H increases.  The superposition of the effects principle can be applied to determine the reduced bearing capacity caused by both the seismic actions. However, when the seismic reduction is great, due to high seismic coefficients k h1 and k h2 , the superposition of the effects principle may lead to an unconservative design.  By the simple limit equilibrium method modified bearing capacity factors and simple relations have been proposed which can be used for the practical design of shallow foundations embedded in both horizontal and sloping soil. In many cases the solutions obtained compare well with the previous static results and available results for the seismic conditions.