Many researchers analysed the bearing capacity based on the static method in which the bearing capacity coefficients were calculated based on static loads on the footings and the weight of the soil in both active and passive conditions. The classical bearing capacity theories started from Rankine (1857), Prandtl (1921), Terzaghi (1943), Meyerhof (1957), Saran et al. (1989) and many others who extensively studied the bearing capacity of shallow footings for static loading case. Terzaghi's bearing capacity theory (1943) was the first general theory for the bearing capacity of soils. Okabe (1924) and Mononobe and Matsuo (1929) were the pioneers in the inclusion of ‘seismicity’ in the design of structures. IS: 1893–1984(Part-3) has also adopted the Mononobe and Okabe method for the determination of seismic active and passive earth pressure behind the retaining wall. Sarma and lossifelis (1990), Richards et al. (1993), Budhu and Al-Karni (1993) and Kumar and Kumar (2003) considered the seismic forces both on the structures and on the supporting soil mass, which were not considered by Meyerhof (1957). Researchers like Dormieux and Pecker (1995), Paolucci and Pecker (1997), Soubra (1997), Kumar and Rao (2002), Kumar (2003) and Choudhury and Subba Rao (2005) studied the seismic bearing capacity of shallow footings for horizontal ground. Sawada et al. (1994), Sarma (1999) and Askari and Farzaneh (2003) gave the solution for seismic bearing capacity of shallow foundations near the sloping ground. Again, some work for surface footing on the sloping ground was carried out by Zhu (2000), Kumar and Kumar (2003) and Kumar and Rao (2003) using limit equilibrium analysis, method of characteristics, etc. Choudhury and Rao (2006), Castelli and Lentini (2012), Farzaneh and Askari (2013) and Chakraborty and Kumar (2014) determined the seismic bearing capacity of a shallow foundation embedded in sloping ground by using the theorem of limit equilibrium method and limit analysis in conjunction with finite elements and non-linear optimisation technique, respectively. In their analysis, it was found that on increasing slope inclination, the bearing capacity decreased. But the researchers did not analyse the bearing capacity on layered soil. Yamamoto (2010) investigated seismic bearing capacity coefficients of spread and embedded foundations near slope in the analytical method. The pseudo-static approach was used, and the seismic forces consisted of a horizontal load applied to the foundation and inertia of a soil mass. Chakraborty and Kumar (2013) evaluated the bearing capacity factor on the sloping ground by applying lower bound (LB) finite element limit analysis in conjunction with non-linear optimisation. Baazouzi et al. (2016) studied the numerical analysis of the bearing capacity for a strip footing near a cohesionless slope and subjected to a centred load using the finite difference code. Button (1953) was the first to analyse the bearing capacity of strip footing on two layers of clay under static loading conditions. In this analysis, it was postulated that failure surface at the ultimate load is cylindrical, where the centre of the cylindrical curve lies at the edge of the footing. Meyerhof and Hanna (1978) considered the case of footing reposing in various layers overlaying a strong soil deposit. Michalowski and Shi (1995) applied the kinematic approach of limit analysis to account for the limit pressure under footings to ascertain the bearing capacity of footings reposing on two-layered soil. Purushothamaraj et al. (1974) analysed the bearing capacity of shallow substratum utilising the upper bound (UB) limit analysis theorem. From all these literature surveys, it is seen that the bearing capacity of shallow foundation embedded in slope on layered soil is still limited. In the present analysis, the seismic bearing capacity of strip footing embedded in slope on two-layered soil has been analysed by using the limit equilibrium method with the pseudo-static approach.
A strip footing having a width
As shown in Fig. 3, wedge AMKJ is a known active wedge that is posited at the top layer, giving pressure to the passive wedge. The weight of the wedge
Total load acting on the foundation is given by
Total cohesive force (C1) on the slip lines AJ and MK is calculated as
The intensity of load at layer thickness
Conceding to limit equilibrium conditions, the authors can write
After solving Equations 5 and 6 and modifying both, the active pressure can be obtained as given below:
From Fig. 4, active pressure distributing from the top layer to the wedge
The weight of the wedge:
Base shear at the interface between the two layers is given as:
Total cohesive force at the slip lines JE and KE is expressed as:
Conceding to limit equilibrium conditions,
Solving Equations 11 and 12 and simplifying them, we obtain
The details of equations of
Hence, total active pressure from both the layers is given by
Due to active pressure generated in the top layer, the passive zone gives resistance to the active pressure. The weight of the passive wedge, as depicted in Fig. 5, is given as
Extra loadingacting on the foundation is expressed as surcharge load
Total surcharge load in the top layer of the passive zone is given as
Total cohesive force in the slip lines KG and GD of the passive wedge is given as
Using limit equilibrium equations
Solving Equations 19 and 20 and modifying both equations, passive resistance can be expressed as:
Total weight of the passive wedge KGE, as shownin Fig. 6, is calculated as
According to 2 : 1 load distribution method, intensity of surcharge load at depth
Base shear between two passive wedges can be coded as
Cohesive forces in slip lines KE, KG and GE are given as
Case 1: Considering the effective area of active zones
Case 2: Considering the effective area of passive zones
The bearing capacity factor (
The details of the equations a1, b1, e1 and d1 are given in
Where
In seismic condition, (
The bearing capacity factor (
Undrained seismic bearing capacity Ncs for strip footing placed adjacent to two layered slope with i=15°.
kh | Df/B0 | c1/c2 | c1/ |
kh | Df/B0 | c1/c2 | c1/ |
||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 4 | 6 | 1 | 2 | 4 | 6 | ||||||
0 | 0.25 | 0.25 | 5.52 | 5.54 | 5.56 | 5.56 | 0 | 0.50 | 0.25 | 4.54 | 4.60 | 4.62 | 4.64 |
0.5 | 5.52 | 5.54 | 5.56 | 5.56 | 0.5 | 4.52 | 4.60 | 4.61 | 4.63 | ||||
0.75 | 5.42 | 5.37 | 5.43 | 5.46 | 0.75 | 4.52 | 4.60 | 4.60 | 4.62 | ||||
1 | 4.74 | 4.40 | 4.41 | 4.42 | 1 | 4.36 | 4.42 | 4.44 | 4.47 | ||||
1.5 | 3.05 | 3.17 | 3.22 | 3.22 | 1.5 | 3.32 | 3.45 | 3.52 | 3.54 | ||||
2 | 2.4 | 2.52 | 2.58 | 2.6 | 2 | 1.55 | 2.82 | 2.90 | 2.92 | ||||
3 | - | 1.80 | 1.85 | 1.88 | 3 | – | 2.12 | 2.22 | 2.24 | ||||
4 | - | 0.6 | 1.51 | 1.53 | 4 | – | 0.91 | 1.84 | 1.90 | ||||
5 | - | – | 1.29 | 1.31 | 5 | – | – | 1.61 | 1.64 | ||||
0.75 | 0.25 | 4.50 | 4.46 | 4.25 | 4.49 | 1 | 0.25 | 4.40 | 4.49 | 4.52 | 4.53 | ||
0.5 | 4.5 | 4.48 | 4.53 | 4.59 | 0.5 | 4.39 | 4.49 | 4.52 | 4.53 | ||||
0.75 | 4.4 | 4.4 | 4.54 | 4.58 | 0.75 | 4.38 | 4.49 | 4.52 | 4.53 | ||||
1 | 4.35 | 4.2 | 4.45 | 4.49 | 1 | 4.38 | 4.44 | 4.47 | 4.48 | ||||
1.5 | 4.05 | 3.84 | 3.93 | 3.96 | 1.5 | 3.88 | 4.2 | 4.3 | 4.33 | ||||
2 | 3.45 | 3.24 | 3.36 | 3.4 | 2 | 1.87 | 3.62 | 3.76 | 3.82 | ||||
3 | - | 2.44 | 2.62 | 2.68 | 3 | – | 2.7 | 3.07 | 3.13 | ||||
4 | - | 1.12 | 2.20 | 2.28 | 4 | – | 1.38 | 2.64 | 2.71 | ||||
5 | - | – | 1.92 | 2.0 | 5 | – | – | 2.27 | 2.42 | ||||
1.25 | 0.25 | 4.42 | 4.46 | 4.50 | 4.52 | 1.5 | 0.25 | 4.44 | 4.50 | 4.52 | 4.53 | ||
0.5 | 4.41 | 4.43 | 4.48 | 4.51 | 0.5 | 4.41 | 4.49 | 4.52 | 4.53 | ||||
0.75 | 4.40 | 4.42 | 4.47 | 4.50 | 0.75 | 4.41 | 4.49 | 4.52 | 4.54 | ||||
1 | 4.39 | 4.41 | 4.45 | 4.48 | 1 | 4.38 | 4.44 | 4.47 | 4.48 | ||||
1.5 | 4.01 | 3.59 | 3.74 | 3.85 | 1.5 | 4.17 | 4.49 | 4.52 | 4.44 | ||||
2 | 3.67 | 3.2 | 3.42 | 3.45 | 2 | 2.34 | 4.18 | 4.31 | 4.37 | ||||
3 | - | 2.64 | 2.92 | 2.97 | 3 | – | 3.1 | 3.54 | 3.74 | ||||
4 | - | 1.67 | 2.59 | 2.67 | 4 | – | 1.92 | 3.3 | 3.43 | ||||
5 | - | – | 2.36 | 2.48 | 5 | – | – | 2.87 | 3.12 | ||||
0.1 | 0.25 | 0.25 | 4.84 | 4.79 | 4.8 | 4.81 | 0.1 | 0.5 | 0.25 | 4.12 | 4.2 | 4.23 | 4.24 |
0.5 | 4.82 | 4.78 | 4.8 | 4.81 | 0.5 | 4.13 | 4.2 | 4.24 | 4.24 | ||||
0.75 | 4.64 | 4.79 | 4.8 | 4.81 | 0.75 | 4.13 | 4.2 | 4.24 | 4.24 | ||||
1 | 4.12 | 4.15 | 4.2 | 4.21 | 1 | 4.06 | 4.15 | 4.19 | 4.21 | ||||
1.5 | - | 3.03 | 3.09 | 3.11 | 1.5 | – | 3.37 | 3.46 | 3.5 | ||||
2 | - | 2.42 | 2.49 | 2.51 | 2 | – | 2.76 | 2.87 | 2.91 | ||||
3 | - | – | 1.82 | 1.87 | 3 | – | – | 2.21 | 2.25 | ||||
4 | - | – | 1.47 | 1.51 | 4 | – | – | 1.85 | 1.89 | ||||
5 | - | – | – | 1.28 | 5 | – | – | – | 1.65 | ||||
0.75 | 0.25 | 4.11 | 4.14 | 4.18 | 4.2 | 1 | 0.25 | 4.06 | 4.15 | 4.18 | 4.19 | ||
0.5 | 4.08 | 4.13 | 4.18 | 4.19 | 0.5 | 4.05 | 4.14 | 4.19 | 4.19 | ||||
0.75 | 4.06 | 4.14 | 4.18 | 4.19 | 0.75 | 4.04 | 4.13 | 4.18 | 4.19 | ||||
1 | 4.04 | 4.15 | 4.2 | 4.21 | 1 | 4.02 | 4.11 | 4.14 | 4.17 | ||||
1.5 | - | 3.74 | 3.85 | 3.89 | 1.5 | – | 4.07 | 4.18 | 4.2 | ||||
2 | - | 3.13 | 3.31 | 3.36 | 2 | – | 3.48 | 3.71 | 3.79 | ||||
3 | - | – | 2.61 | 2.68 | 3 | – | – | 3.01 | 3.1 | ||||
4 | - | – | 2.14 | 2.28 | 4 | – | – | 2.34 | 2.67 | ||||
5 | - | – | – | 2.02 | 5 | – | – | – | 2.38 | ||||
1.25 | 0.25 | 4.08 | 4.14 | 4.18 | 4.19 | 1.5 | 0.25 | 4.05 | 4.14 | 4.18 | 4.19 | ||
0.5 | 4.06 | 4.14 | 4.18 | 4.2 | 0.5 | 4.04 | 4.14 | 4.18 | 4.19 | ||||
0.75 | 4.05 | 4.13 | 4.18 | 4.19 | 0.75 | 4.05 | 4.14 | 4.18 | 4.19 | ||||
1 | 4.04 | 4.15 | 4.2 | 4.21 | 1 | 4.06 | 4.14 | 4.19 | 4.2 | ||||
1.5 | - | 4.14 | 4.19 | 4.2 | 1.5 | – | 4.14 | 4.19 | 4.2 | ||||
2 | - | 3.84 | 4.07 | 4.16 | 2 | – | 3.95 | 4.18 | 4.2 | ||||
3 | - | – | 3.37 | 3.48 | 3 | – | – | 3.67 | 3.82 | ||||
4 | - | – | 2.54 | 3.05 | 4 | – | – | 2.72 | 3.38 | ||||
5 | - | – | – | 2.68 | 5 | – | – | – | 2.98 |
Undrained seismic bearing capacity Ncs for strip footing placed adjacent to two layered slope with i=30°.
kh | Df/B0 | c1/c2 | c1/ |
kh | Df/B0 | c1/c2 | c1/ |
||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 4 | 6 | 1 | 2 | 4 | 6 | ||||||
0 | 0.25 | 0.25 | 5.12 | 5.16 | 5.18 | 5.19 | 0 | 0.5 | 0.25 | 4.06 | 4.14 | 4.21 | 4.23 |
0.5 | 5.12 | 5.16 | 5.18 | 5.19 | 0.5 | 4.05 | 4.13 | 4.19 | 4.22 | ||||
0.75 | 4.82 | 4.91 | 4.96 | 4.97 | 0.75 | 4.03 | 4.11 | 4.17 | 4.20 | ||||
1 | 3.8 | 3.96 | 4.03 | 4.05 | 1 | 3.8 | 3.96 | 4.06 | 4.08 | ||||
1.5 | 2.23 | 2.81 | 2.9 | 2.93 | 1.5 | 2.32 | 3.01 | 3.10 | 3.12 | ||||
2 | – | 2.18 | 2.3 | 2.33 | 2 | – | 2.45 | 2.61 | 2.65 | ||||
3 | – | 1.25 | 1.68 | 1.72 | 3 | – | 1.44 | 2 | 2.06 | ||||
4 | – | – | 1.33 | 1.38 | 4 | – | – | 1.65 | 1.72 | ||||
5 | – | – | 1 | 1.17 | 5 | – | – | 1.24 | 1.5 | ||||
0.75 | 0.25 | 3.82 | 3.97 | 4.06 | 4.09 | 1 | 0.25 | 3.80 | 3.92 | 4.04 | 4.06 | ||
0.5 | 3.83 | 3.97 | 4.06 | 4.09 | 0.5 | 3.80 | 3.92 | 4.04 | 4.06 | ||||
0.75 | 3.82 | 3.97 | 4.07 | 4.09 | 0.75 | 3.80 | 3.91 | 4.04 | 4.06 | ||||
1 | 3.8 | 3.96 | 4.06 | 4.09 | 1 | 3.82 | 3.96 | 4.03 | 4.05 | ||||
1.5 | 2.51 | 3.25 | 3.37 | 3.41 | 1.5 | 2.64 | 3.44 | 3.59 | 3.64 | ||||
2 | – | 2.73 | 2.9 | 2.95 | 2 | – | 2.96 | 3.17 | 3.23 | ||||
3 | – | 1.67 | 2.31 | 2.38 | 3 | – | 1.86 | 2.56 | 2.67 | ||||
4 | – | – | 1.92 | 2.05 | 4 | – | – | 2.17 | 2.34 | ||||
5 | – | – | 1.5 | 1.8 | 5 | – | – | 1.74 | 2.07 | ||||
1.25 | 0.25 | 3.83 | 3.96 | 4.03 | 4.06 | 1.5 | 0.25 | 3.8 | 3.96 | 4.03 | 4.05 | ||
0.5 | 3.83 | 3.96 | 4.03 | 4.05 | 0.5 | 3.79 | 3.96 | 4.02 | 4.04 | ||||
0.75 | 3.83 | 3.96 | 4.03 | 4.05 | 0.75 | 3.8 | 3.96 | 4.03 | 4.05 | ||||
1 | 3.83 | 3.96 | 4.03 | 4.05 | 1 | 3.8 | 3.96 | 4.03 | 4.05 | ||||
1.5 | 2.74 | 3.65 | 3.84 | 3.9 | 1.5 | 2.8 | 3.84 | 4.01 | 4.04 | ||||
2 | – | 3.2 | 3.45 | 3.53 | 2 | – | 3.41 | 3.71 | 3.8 | ||||
3 | – | 2.1 | 2.9 | 3.02 | 3 | – | 2.33 | 3.18 | 3.32 | ||||
4 | – | – | 2.43 | 2.64 | 4 | – | – | 2.69 | 2.94 | ||||
5 | – | – | 2.02 | 2.35 | 5 | – | – | 2.28 | 2.64 | ||||
0.1 | 0.25 | 0.25 | 4.25 | 4.31 | 4.34 | 4.34 | 0.1 | 0.5 | 0.25 | 3.57 | 3.68 | 3.74 | 3.76 |
0.5 | 4.25 | 4.31 | 4.34 | 4.35 | 0.5 | 3.57 | 3.69 | 3.74 | 3.76 | ||||
0.75 | 4.25 | 4.31 | 4.33 | 4.33 | 0.75 | 3.57 | 3.69 | 3.74 | 3.76 | ||||
1 | 3.44 | 3.6 | 3.67 | 3.7 | 1 | 3.44 | 3.61 | 3.68 | 3.71 | ||||
1.5 | – | 2.6 | 2.7 | 2.73 | 1.5 | – | 2.82 | 2.94 | 2.97 | ||||
2 | – | 2.03 | 2.15 | 2.19 | 2 | – | 2.29 | 2.46 | 2.5 | ||||
3 | – | – | 1.57 | 1.61 | 3 | – | – | 1.89 | 1.95 | ||||
4 | – | – | 1.23 | 1.3 | 4 | – | – | 1.48 | 1.64 | ||||
5 | – | – | – | 1.1 | 5 | – | – | – | 1.4 | ||||
0.75 | 0.25 | 3.45 | 3.61 | 3.68 | 3.7 | 1 | 0.25 | 3.44 | 3.61 | 3.68 | 3.7 | ||
0.5 | 3.45 | 3.61 | 3.68 | 3.7 | 0.5 | 3.44 | 3.61 | 3.68 | 3.7 | ||||
0.75 | 3.45 | 3.61 | 3.68 | 3.7 | 0.75 | 3.45 | 3.61 | 3.68 | 3.71 | ||||
1 | 3.45 | 3.61 | 3.68 | 3.7 | 1 | 3.45 | 3.61 | 3.68 | 3.7 | ||||
1.5 | – | 3.04 | 3.18 | 3.22 | 1.5 | – | 3.23 | 3.44 | 3.5 | ||||
2 | – | 2.55 | 2.75 | 2.81 | 2 | – | 2.78 | 3.04 | 3.12 | ||||
3 | – | – | 2.2 | 2.28 | 3 | – | – | 2.5 | 2.62 | ||||
4 | – | – | 1.74 | 1.96 | 4 | – | – | 2.03 | 2.27 | ||||
5 | – | – | – | 1.74 | 5 | – | – | – | 1.97 | ||||
1.25 | 0.25 | 3.45 | 3.61 | 3.68 | 3.67 | 1.5 | 0.25 | 3.44 | 3.61 | 3.68 | 3.7 | ||
0.5 | 3.44 | 3.61 | 3.68 | 3.67 | 0.5 | 3.44 | 3.61 | 3.68 | 3.7 | ||||
0.75 | 3.44 | 3.61 | 3.68 | 3.67 | 0.75 | 3.44 | 3.61 | 3.68 | 3.7 | ||||
1 | 3.45 | 3.61 | 3.68 | 3.7 | 1 | 3.45 | 3.62 | 3.68 | 3.7 | ||||
1.5 | – | 3.47 | 3.68 | 3.71 | 1.5 | – | 3.61 | 3.68 | 3.7 | ||||
2 | – | 2.95 | 3.3 | 3.38 | 2 | – | 3.13 | 3.59 | 3.68 | ||||
3 | – | – | 2.76 | 2.91 | 3 | – | – | 3.06 | 3.24 | ||||
4 | – | – | 2.18 | 2.54 | 4 | – | – | 2.33 | 2.86 | ||||
5 | – | – | – | 2.22 | 5 | – | – | – | 2.53 |
Merifield et al. (1999) investigated the bearing capacity of a strip footing resting on a two-layer clay deposit with a horizontal ground surface and proposed a modified bearing capacity factor
Similarly, a dimensionless undrained bearing capacity factor
Ranges of various parameters are given as follows:
Since the heuristic algorithms give us low ramification and high execution and these methods are relatively new, they can be applied in the geotechnical problem. Out of these methods, a brief discussion on PSO is given here as it is used in the analysis.
Kennedy and Eberhart (1995) developed PSO as a simulation of birds swarm. A swarm is a group of individuals with defined rules for individual behaviours and communication. The ability of each individual to deal with the previous experiences of the swarm is called swarm intelligence. This capability guides the swarm towards its optimum goal. PSO is a population-based search technique where a population of particles start their journey in a space concerning the current best position (Hossain and EI-Shafie 2014; Hajihassani et al. 2017). Reynolds (1987) described three simple rules for the behaviours of individuals inside a swarm, which were used as one of the basic concepts of PSO by Kennedy and Eberhart (1995). Although these simple rules model the behaviour of individuals, their combination produces a complicated behaviour for the swarm.
Individuals avoid collision with others
Individuals go towards the goal of the swarm
Individuals go to the centre of the swarm
The process of decision-making related to individuals is another basic concept of PSO. Each individual of the swarm makes decision based on the following two factors:
Own experiences of the individual that is its bestresults so far The experiences of other individuals in the swarm that is the best results in the whole swarm
Figure 10 illustrates the standard flowchart of PSO. At the starting step of the original PSO, a certain number of individuals, called particles, are distributed in the search space by using a random pattern (Kennedy and Eberhart 1995; Cheng et al. 2007; Aote et al. 2013). Each particle is representative of a feasible solution. Figure 11 shows the schematic structure of a particle in PSO involving three divided parts as its current position, best position and velocity. The current position, best position and velocity of particles record the current coordinates, best coordinates and velocity vectors of a particle in D-dimensional space, respectively, where D starts from 1 (Kalatehjari 2013). Consequently, for a particle in D-dimensional space, a 3D-dimensional particle is desirable. PSO aims to meet the termination criteria which are defined as the criteria for terminating the iterative search process. To select an appropriate termination criterion, it should be noted that the termination condition does not cause a premature converge and it should protect against oversampling of the fitness (Engelbrecht 2007). The following termination criteria are frequently used in PSO:
Termination when the maximum number ofiterations is exceeded Termination when a satisfactory solution isfound based on the condition of each problem Termination when no improvement is achievedover a certain number of iterations
These criteria are applied to ensure that PSO can converge on a feasible solution. Although PSO has some limitation which is explained by Gbenga et al. (2016) and Aote et al. (2013), PSO tries to make the objective function as a minimum or maximum dependingon the problem to be solved. To lead the swarm towards this aim, the fitness value of each particle is determined by evaluating its current position by the objective function. After evaluation offitness of all particles, Equation 39 (velocity equation) is used to calculate the velocity of particles based on their best position and the position of the best particle in the swarm. Using Equation 40, particle positions can be updated according to their current positions and velocities. This iterative process continues until reaching the termination criteria. Equations 39 and 40 are as follows (Kennedy and Eberhart 1995):
Aparametric study was done for the variation of pseudo-static seismic bearing capacity coefficients with different soil parameters as shown in Figs 12–18.
Figure 12 shows the variation of seismic bearing capacity coefficient (
Figure 13 depicts the variations of seismic bearing capacity coefficient (
Figure 14 shows the variations of seismic bearing capacity coefficient (
Figure 15 shows the variations of seismic bearing capacity coefficient (
Figure 16 shows the variation of seismic bearing capacity coefficient (
Figure 17 shows the variation of
Figure 18 shows the variation of
Figure 19 depicts the variations of seismic bearing capacity coefficient (
Figure 20 shows the variations of seismic bearing capacity coefficient (
Figure 21 shows the variations of (
The bearing capacity coefficient (
Comparison of variation of seismic bearing capacity (
Sawada et al. (1994) | Askari and Farzaneh (2003) | Yamamoto (2010) | Present Analysis | |||||
---|---|---|---|---|---|---|---|---|
kh | ||||||||
0.1 | 1798 | 3321 | 1066 | 1856 | 1013 | 1795 | 1614.14 | 3139.92 |
0.2 | 1770 | 3269 | 829 | 1307 | 755 | 1283 | 654 | 1167 |
Comparison of the variation in Ncs with different cu/
cu/ |
Ncs (Present study) | Ncs (Wu et al. 2020) | ||||||
---|---|---|---|---|---|---|---|---|
Df/B0=0.5, c1/c2=0.5 | Df/B0=1.5, c1/c2=0.5 | Df/B0=1.5, c1/c2=1.5 | Df/B0=0.5, c1/c2=1.5 | Df/B0=0.5, c1/c2=0.5 | Df/B0=1.5, c1/c2=0.5 | Df/B0=1.5, c1/c2=1.5 | Df/B0=0.5, c1/c2=1.5 | |
1 | 4.06 | 3.78 | 2.78 | 2.34 | 4.05 | 3.79 | 2.8 | 2.34 |
2 | 4.15 | 3.92 | 3.05 | 2.95 | 4.16 | 3.96 | 3.01 | 3.01 |
4 | 4.20 | 4.02 | 3.11 | 3.04 | 4.19 | 4.02 | 3.12 | 3.12 |
6 | 4.22 | 4.04 | 3.22 | 3.10 | 4.2 | 4.04 | 3.15 | 3.15 |
8 | 4.24 | 4.06 | 3.23 | 3.14 | 4.21 | 4.05 | 3.17 | 3.17 |
10 | 4.25 | 4.08 | 3.24 | 3.16 | 4.22 | 4.6 | 3.17 | 3.17 |
Comparison of the variation in Ncs with different kh and c1/c2 for i=0° and Df/B0.
kh | Ncs (Present study) | Ncs (Jahani et al. 2019) | ||||
---|---|---|---|---|---|---|
c1/c2=0.25 | c1/c2=0.50 | c1/c2=0.75 | c1/c2=0.25 | c1/c2=0.50 | c1/c2=0.75 | |
0 | 5.62 | 3.94 | 3.06 | 5.67 | 4 | 3.1 |
0.1 | 5.20 | 3.48 | 2.75 | 5.25 | 3.58 | 2.73 |
0.2 | 4.04 | 3.04 | 2.37 | 4.44 | 3.12 | 2.4 |
0.3 | 3.38 | 2.65 | 2.12 | 3.47 | 2.7 | 2.15 |
Comparison of the variation in Ncs for i=15°.
h1/B0 | Df/B0 | c1/c2 | Xiao et al. (2019) | Present study | ||||||
---|---|---|---|---|---|---|---|---|---|---|
c1/ |
c1/ |
|||||||||
1 | 2 | 4 | 6 | 1 | 2 | 4 | 6 | |||
0.5 | 0.25 | 0.25 | 5.71 | 5.73 | 5.74 | 5.74 | 5.68 | 5.70 | 5.71 | 5.72 |
0.5 | 5.71 | 5.73 | 5.74 | 5.74 | 5.68 | 5.70 | 5.71 | 5.72 | ||
0.75 | 5.53 | 5.57 | 5.58 | 5.59 | 5.49 | 5.55 | 5.56 | 5.57 | ||
1 | 4.51 | 4.58 | 4.61 | 4.63 | 4.49 | 4.53 | 4.58 | 4.60 | ||
0.50 | 0.25 | 4.70 | 4.75 | 4.77 | 4.77 | 4.68 | 4.71 | 4.74 | 4.74 | |
0.5 | 4.70 | 4.75 | 4.77 | 4.77 | 4.68 | 4.72 | 4.75 | 4.75 | ||
0.75 | 4.70 | 4.74 | 4.77 | 4.77 | 4.68 | 4.71 | 4.75 | 4.75 | ||
1 | 4.51 | 4.57 | 4.61 | 4.62 | 4.45 | 4.51 | 4.59 | 4.59 | ||
1 | 0.25 | 0.25 | 5.71 | 5.73 | 5.73 | 5.75 | 5.68 | 5.70 | 5.71 | 5.72 |
0.5 | 5.71 | 5.73 | 5.73 | 5.74 | 5.68 | 5.70 | 5.70 | 5.71 | ||
0.75 | 5.52 | 5.57 | 5.58 | 5.59 | 5.50 | 5.55 | 5.56 | 5.57 | ||
1 | 4.51 | 4.58 | 4.61 | 4.63 | 4.48 | 4.55 | 4.59 | 4.61 | ||
0.50 | 0.25 | 4.70 | 4.75 | 4.77 | 4.78 | 4.68 | 4.72 | 4.74 | 4.75 | |
0.5 | 4.71 | 4.75 | 4.77 | 4.78 | 4.67 | 4.74 | 4.75 | 4.76 | ||
0.75 | 4.70 | 4.75 | 4.77 | 4.78 | 4.66 | 4.74 | 4.74 | 4.75 | ||
1 | 4.51 | 4.58 | 4.62 | 4.62 | 4.49 | 4.55 | 4.60 | 4.61 | ||
2 | 0.25 | 0.25 | 5.71 | 5.73 | 5.74 | 5.73 | 5.69 | 5.72 | 5.72 | 5.73 |
0.5 | 5.71 | 5.73 | 5.74 | 5.74 | 5.69 | 5.71 | 5.72 | 5.73 | ||
0.75 | 5.52 | 5.57 | 5.58 | 5.59 | 5.50 | 5.55 | 5.56 | 5.57 | ||
1 | 4.51 | 4.58 | 4.62 | 4.63 | 5.49 | 5.53 | 5.54 | 5.55 | ||
0.50 | 0.25 | 4.70 | 4.74 | 4.77 | 4.79 | 4.69 | 4.72 | 4.75 | 4.77 | |
0.5 | 4.70 | 4.76 | 4.77 | 4.77 | 4.68 | 4.70 | 4.73 | 4.75 | ||
0.75 | 4.70 | 4.74 | 4.76 | 4.78 | 4.67 | 4.68 | 4.70 | 4.72 | ||
1 | 4.51 | 4.58 | 4.61 | 4.62 | 4.47 | 4.54 | 4.59 | 4.60 | ||
3 | 0.25 | 0.25 | 5.70 | 5.73 | 5.74 | 5.74 | 5.69 | 5.71 | 5.72 | 5.72 |
0.5 | 5.70 | 5.73 | 5.74 | 5.74 | 5.69 | 5.71 | 5.72 | 5.72 | ||
0.75 | 5.52 | 5.56 | 5.58 | 5.58 | 5.50 | 5.55 | 5.56 | 5.57 | ||
1 | 4.51 | 4.58 | 4.61 | 4.62 | 4.48 | 4.55 | 4.58 | 4.60 | ||
0.50 | 0.25 | 4.71 | 4.75 | 4.77 | 4.78 | 4.69 | 4.72 | 4.74 | 4.77 | |
0.5 | 4.70 | 4.75 | 4.77 | 4.78 | 4.68 | 4.72 | 4.74 | 4.77 | ||
0.75 | 4.70 | 4.75 | 4.77 | 4.77 | 4.67 | 4.71 | 4.73 | 4.73 | ||
1 | 4.51 | 4.58 | 4.61 | 4.63 | 4.49 | 4.55 | 4.59 | 4.60 |
Comparison of the variation in Ncs for i=30°.
h1/B0 | Df/B0 | c1/c2 | Xiao et al. (2019) | Present study | ||||||
---|---|---|---|---|---|---|---|---|---|---|
c1/ |
c1/ |
|||||||||
1 | 2 | 4 | 6 | 1 | 2 | 4 | 6 | |||
0.5 | 0.25 | 0.25 | 5.15 | 5.20 | 5.23 | 5.23 | 5.11 | 5.16 | 5.21 | 5.21 |
0.5 | 5.15 | 5.20 | 5.22 | 5.23 | 5.11 | 5.16 | 5.21 | 5.20 | ||
0.75 | 4.86 | 4.95 | 5.00 | 5.01 | 4.81 | 4.88 | 4.91 | 4.96 | ||
1 | 3.88 | 4.01 | 4.07 | 4.09 | 3.82 | 4.01 | 4.03 | 4.05 | ||
0.50 | 0.25 | 4.09 | 4.18 | 4.23 | 4.24 | 4.05 | 4.16 | 4.20 | 4.21 | |
0.5 | 4.09 | 4.18 | 4.23 | 4.24 | 4.02 | 4.14 | 4.18 | 4.20 | ||
0.75 | 4.09 | 4.18 | 4.23 | 4.24 | 4.01 | 4.12 | 4.16 | 4.18 | ||
1 | 3.88 | 4.00 | 4.07 | 4.09 | 3.84 | 3.95 | 4.02 | 4.05 | ||
1 | 0.25 | 0.25 | 5.15 | 5.20 | 5.22 | 5.23 | 5.10 | 5.15 | 5.20 | 5.21 |
0.5 | 5.15 | 5.20 | 5.22 | 5.23 | 5.09 | 5.13 | 5.19 | 5.20 | ||
0.75 | 4.85 | 4.95 | 5.00 | 5.01 | 4.80 | 4.85 | 4.90 | 4.93 | ||
1 | 3.83 | 3.99 | 4.06 | 4.08 | 3.80 | 3.92 | 3.99 | 4.01 | ||
0.50 | 0.25 | 4.10 | 4.19 | 4.24 | 4.25 | 4.07 | 4.11 | 4.20 | 4.22 | |
0.5 | 4.10 | 4.19 | 4.23 | 4.25 | 4.05 | 4.10 | 4.19 | 4.20 | ||
0.75 | 4.10 | 4.19 | 4.23 | 4.25 | 4.04 | 4.10 | 4.18 | 4.19 | ||
1 | 3.83 | 3.99 | 4.06 | 4.08 | 3.80 | 3.95 | 4.04 | 4.06 | ||
2 | 0.25 | 0.25 | 5.15 | 5.20 | 5.22 | 5.23 | 5.11 | 5.17 | 5.20 | 5.21 |
0.5 | 5.15 | 5.20 | 5.22 | 5.23 | 5.10 | 5.17 | 5.20 | 5.21 | ||
0.75 | 4.85 | 4.95 | 5.00 | 5.01 | 4.82 | 4.90 | 4.96 | 4.99 | ||
1 | 3.83 | 3.99 | 4.06 | 4.08 | 3.81 | 3.95 | 4.01 | 4.04 | ||
0.50 | 0.25 | 4.09 | 4.19 | 4.23 | 4.25 | 4.07 | 4.15 | 4.20 | 4.22 | |
0.5 | 4.09 | 4.19 | 4.23 | 4.25 | 4.05 | 4.14 | 4.20 | 4.21 | ||
0.75 | 4.09 | 4.19 | 4.23 | 4.25 | 4.05 | 4.14 | 4.20 | 4.21 | ||
1 | 3.82 | 3.98 | 4.05 | 4.07 | 3.80 | 3.94 | 4.01 | 4.04 | ||
3 | 0.25 | 0.25 | 5.16 | 5.20 | 5.22 | 5.23 | 5.11 | 5.18 | 5.20 | 5.21 |
0.5 | 5.16 | 5.20 | 5.22 | 5.23 | 5.11 | 5.17 | 5.20 | 5.21 | ||
0.75 | 4.85 | 4.95 | 5.00 | 5.01 | 4.81 | 4.92 | 4.99 | 5.00 | ||
1 | 3.83 | 3.99 | 4.06 | 4.08 | 3.80 | 3.96 | 4.03 | 4.06 | ||
0.50 | 0.25 | 4.10 | 4.19 | 4.23 | 4.25 | 4.07 | 4.15 | 4.20 | 4.23 | |
0.5 | 4.09 | 4.19 | 4.23 | 4.24 | 4.06 | 4.14 | 4.20 | 4.22 | ||
0.75 | 4.10 | 4.19 | 4.23 | 4.24 | 4.05 | 4.12 | 4.18 | 4.20 | ||
1 | 3.83 | 3.99 | 4.06 | 4.08 | 3.80 | 3.93 | 4.03 | 4.06 |
Comparison of the variation in Ncs for i=45°.
h1/B0 | Df/B0 | c1/c2 | Xiao et al. (2019) | Present study | ||||||
---|---|---|---|---|---|---|---|---|---|---|
c1/ |
c1/ |
|||||||||
1 | 2 | 4 | 6 | 1 | 2 | 4 | 6 | |||
0.5 | 0.25 | 0.25 | 4.57 | 4.63 | 4.66 | 4.67 | 4.54 | 4.60 | 4.62 | 4.64 |
0.5 | 4.57 | 4.64 | 4.66 | 4.68 | 4.53 | 4.60 | 4.61 | 4.63 | ||
0.75 | 4.23 | 4.34 | 4.40 | 4.41 | 4.21 | 4.31 | 4.37 | 4.40 | ||
1 | 3.44 | 3.57 | 3.63 | 3.65 | 3.41 | 3.53 | 3.55 | 3.59 | ||
0.50 | 0.25 | 3.49 | 3.62 | 3.67 | 3.70 | 3.45 | 3.58 | 3.62 | 3.67 | |
0.5 | 3.49 | 3.61 | 3.67 | 3.70 | 3.44 | 3.59 | 3.60 | 3.67 | ||
0.75 | 3.49 | 3.61 | 3.68 | 3.70 | 3.43 | 3.56 | 3.59 | 3.66 | ||
1 | 3.44 | 3.57 | 3.63 | 3.65 | 3.42 | 3.53 | 3.56 | 3.64 | ||
1 | 0.25 | 0.25 | 4.57 | 4.63 | 4.66 | 4.67 | 4.55 | 4.60 | 4.63 | 4.65 |
0.5 | 4.57 | 4.63 | 4.66 | 4.68 | 4.54 | 4.58 | 4.62 | 4.64 | ||
0.75 | 4.13 | 4.28 | 4.35 | 4.37 | 4.11 | 4.25 | 4.33 | 4.35 | ||
1 | 3.16 | 3.41 | 3.51 | 3.54 | 3.11 | 3.38 | 3.50 | 3.52 | ||
0.50 | 0.25 | 3.49 | 3.61 | 3.68 | 3.69 | 3.45 | 3.59 | 3.64 | 3.67 | |
0.5 | 3.49 | 3.61 | 3.68 | 3.69 | 3.44 | 3.58 | 3.63 | 3.67 | ||
0.75 | 3.49 | 3.61 | 3.68 | 3.69 | 3.43 | 3.57 | 3.63 | 3.67 | ||
1 | 3.16 | 3.41 | 3.51 | 3.54 | 3.12 | 3.38 | 3.50 | 3.52 | ||
2 | 0.25 | 0.25 | 4.58 | 4.64 | 4.66 | 4.68 | 4.54 | 4.60 | 4.62 | 4.66 |
0.5 | 4.58 | 4.63 | 4.67 | 4.68 | 4.53 | 4.60 | 4.61 | 4.64 | ||
0.75 | 4.11 | 4.28 | 4.35 | 4.38 | 4.10 | 4.25 | 4.32 | 4.32 | ||
1 | 3.15 | 3.41 | 3.50 | 3.54 | 3.11 | 3.40 | 3.48 | 3.50 | ||
0.50 | 0.25 | 3.49 | 3.61 | 3.67 | 3.69 | 3.44 | 3.58 | 3.62 | 3.64 | |
0.5 | 3.49 | 3.61 | 3.68 | 3.69 | 3.42 | 3.56 | 3.61 | 3.63 | ||
0.75 | 3.49 | 3.61 | 3.67 | 3.69 | 3.41 | 3.54 | 3.58 | 3.61 | ||
1 | 3.15 | 3.41 | 3.51 | 3.54 | 3.10 | 3.36 | 3.52 | 3.52 | ||
3 | 0.25 | 0.25 | 4.58 | 4.64 | 4.67 | 4.67 | 4.55 | 4.62 | 4.64 | 4.65 |
0.5 | 4.58 | 4.63 | 4.67 | 4.68 | 4.54 | 4.60 | 4.62 | 4.64 | ||
0.75 | 4.11 | 4.28 | 4.35 | 4.37 | 4.08 | 4.24 | 4.31 | 4.35 | ||
1 | 3.15 | 3.40 | 3.51 | 3.54 | 3.12 | 3.36 | 3.41 | 3.48 | ||
0.50 | 0.25 | 3.49 | 3.61 | 3.67 | 3.69 | 3.44 | 3.60 | 3.62 | 3.65 | |
0.5 | 3.48 | 3.61 | 3.67 | 3.69 | 3.43 | 3.60 | 3.61 | 3.63 | ||
0.75 | 3.49 | 3.61 | 3.67 | 3.67 | 3.42 | 3.58 | 3.60 | 3.63 | ||
1 | 3.15 | 3.40 | 3.51 | 3.54 | 3.13 | 3.38 | 3.49 | 3.51 |
This paper has investigated the seismic bearing capacity of shallow foundations near slope using pseudo-static limit equilibrium analysis. Linear failure mechanism has been proposed to obtain pseudo-static bearing capacity coefficients of embedded strip footing near slope using limit equilibrium analysis. The PSO technique has been applied to obtain minimum bearing capacity coefficients.
Based on the present investigation, the following conclusions can be drawn:
The seismic bearing capacity coefficient The minimum pseudo-static bearing capacity coefficients are presented in the form of design table for practical use in geotechnical engineering. It has been observed that the magnitude of bearing capacity coefficients decreases with an increase in slope inclination. The seismic bearing capacity decreases with greater undrained shear strength ratio ( The value of