Journal of Engineering Geology

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Introduction
Bearing capacity is very important in geotechnical engineering, which depends on factors such as footing shape, stress distribution under footing and failure mechanism of soil. Construction of the footing near a slope affects the behavior of footing and reduces the bearing capacity. Also, construction of structures on soft soil usually involves problems such as excessive settlement, deformation and stability problems. In order to increase the bearing capacity, especially in soft soils, one method is adding stone columns to soils. In this method 15 to 35 percent of unsuitable soil volume is replaced with appropriate material. In this research, the bearing capacity and settlement of a strip footing on a clayey slope reinforced with stone columns is investigated. For this purpose, a series of small-scale model tests was performed on the slope reinforced with both types of ordinary and vertical encased stone columns. The effects of length of stone column and location of stone column on the behavior of footing was studied and the optimum length of column and best location for column were determined. Also, some tests were performed on the effect of group stone columns on the footing and the efficiency of columns was investigated.
Material and methods
In order to determine properties of clay soil, stone column and encasement material, some preliminary standard tests were performed. The stone column material was selected with aggregate size ranging from 2-10 mm considering the scale effect. The performance of stone column depends on the lateral confinement provided from the surrounding soil and this lateral confinement represents undrained shear strength of the soil. In very soft soils (cu<15 kPa), the lateral confinement is not adequate and the stone column cannot perform well in carrying the required bearing capacity. For this reason, a series of undrained shear strength standard tests were carried out on clay samples with different water contents. According to these tests, the amount of water content of clay related to cu-15kPa was equal to 25%; while the natural water content of the clay was 4%. Therefore, the additional amount of water was weighted and added to clay. The apparatus of this research was consisted of two main parts including a test box and a hydraulic loading system. The test box dimensions should be such that for all states of the tests, the stress in the soil applied from the loading would be almost zero at all boundaries of the box. Thus, a box was built to accommodate the clay slope with 150 cm×120 cm×30 cm dimensions. The test box was built using steel material and steel belts were welded around it to prevent the deformation at high loads. The front side of the box was made from two pieces of tempered glass and a 10 cm×10 cm grid was drawn on them, for making the slope during construction and observation of deformations during the loading easier. The model strip footing dimensions were 29 cm length, 10cm width and 4cm height and it was made from steel to have no deformation during the loading. The displacement of the footing was measured using two dial gauges with accuracy of 0.01 mm.
The clay was filled in the test box in 5 cm thick layers and compacted with a special 6.8 kg weight tamper. All model stone columns were constructed using the replacement method. In this method, a 10 cm diameter open ended steel pipe was inserted into the soil and the clay within the pipe was excavated. Then the stone column material charged into the hole in 5 cm layers and each layer was compacted using a 2.7 kg special circular steel tamper with 10 blows. The 5cm compactions were repeated until the construction of ordinary stone column was completed. For construction of vertical encased stone columns, the cylindrical encasement mesh should be constructed first. Then, after excavating the hole, the prepared encasement mesh was placed inside the hole and the aggregates were charged into the hole in 5 cm layers and compacted.
Results and discussion
The loading method used in all tests was a stress control method. Bearing capacity values were determined from pressure-displacement diagrams using tangent method. All test results show that when any type of stone columns was added to slope, the bearing capacity of adjacent footing was increased. Vertical encasing of stone columns leads to a further improvement in the behavior of the footing. Influence of length of ordinary stone columns on the behavior of strip footing near clayey slope, was studied for four different lengths. Results show that, the optimum length of stone columns giving the maximum performance is about 4 times their diameter. Also, the location of column for both ordinary and vertical encased stone columns was studied using a series of laboratory tests and results show that the best location for the stone column is right beneath the footing. Also, group stone column tests resulted that for both ordinary and vertical encased types of stone columns, the group of two columns had a better efficiency than the group of three columns.
Conclusion
In this investigation, some model tests with 1/10 model scale on a strip footing near a clayey slope reinforced with stone columns were performed and the effects of different parameters such as stone column length and location were studied. Based on results from experiments on different states of stone columns, the following concluding remarks may be mentioned:
- The maximum encasement influence was observed when the encased stone column is placed under the footing.
- The optimum length of ordinary stone columns which are placed beneath the strip footing gives the maximum performance more than 4 times to their diameter.
-Bulging failure mode governs when the stone column is placed under the footing. When stone column is not beneath the footing, the failure mode was lateral deformation.
- Comparing the different locations of stone columns in the slope shows that for both ordinary and vertical encased stone columns, the best location having the most influence on the strip footing is under the footing and with increasing the spacing between column and footing, the bearing capacity is reduced.
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Introduction
Studying the effect of slope angle on bearing capacity of foundations on the slope in urban areas is a challenging problem that has been investigated by researchers for years. In general, the analytical approaches for solving this problem can be categorized into limit equilibrium, characteristics and limit analysis methods. In recent years, there have been studies for using the limit analysis within the framework of finite element method for geomaterials. In these studies, the soil mass is not considered as rigid and there is no need to predefine a failure surface for the slope. In the performed research, using the upper bound finite element limit analysis, bearing capacity of strip foundation on slope have been studied. This analytical method enables the use of the advantages of both methods of limit analysis and finite element analysis. In this method, the slip between the two elements is considered. In order to find the critical state of the failure, the rate of power internally dissipated is linearly optimized, by using the interior points method. The advantages of this method are the high convergence rate in comparison with other analytical optimization methods. The effect of different upstream and downstream slopes and foundation depths and also the influence of various mesh discretizations have been evaluated. Finally, the results are compared with those obtained from previous methods available in the literature.
Methods
The finite element limit analysis method is based on nodal velocities. Considering the principals of the finite element method and having the nodal velocities, the velocity at each node of the element can be obtained from corresponding shape functions. The rate of power internally dissipated in each element is defined by multiplying the strain rate on stress in each element. In this method, the slip between the two elements and the rate of internal power dissipated at each discontinuity of two adjacent elements is considered. For this purpose, in each node, four new unknowns’ velocities are defined. To remove the stress from the equations, and provide a linear relationship for linear optimization, a linear approximation to the yield function has been used. For this purpose, the Mohr-Coulomb yield criterion is estimated with a polygon in the stress space. Also, using the reduced strength parameter, the effect of the dilation angle is considered. According to the principles of upper bound limit analysis, the value of plastic strain rate is calculated from the flow rule. The velocity field in elements and discontinuities must satisfy the set of constraints imposed by an associated flow rule. In order to have an acceptable kinematics field, the velocity vectors have to satisfy the boundary conditions. These conditions include zero kinematics velocities along the vertical and horizontal boundaries of the geometry as well as negative vertical unit velocities and zero horizontal velocities at points underneath the rigid foundation.
Results and discussion
In order to calculate the bearing capacity of foundation, a set of different uniform and non-uniform mesh has been examined. The results obtained from different uniform mesh sizes indicate a certain divergence in the course of analysis. However, the results between the fine and very fine non-uniform mesh are closely related to each other and are converged. The obtained results show that, by increasing the internal friction angle, the bearing capacity has been increased. At high angles of modified friction, the effect of increasing the internal friction angle on the increase in bearing capacity is more in slopes with lower angles. By increasing the downstream foundation depth, the bearing capacity has been increased. This increase is more important in the case of slopes with lower angles. However, the upstream depth variations didn't present a significant effete on bearing capacity. In order to investigate the effect of upstream angle on the bearing capacity, the upstream mesh is also refined similar to the downstream. The obtained results indicate that variations of the upstream angle have a minor effect on the bearing capacity. This is of course true if the upstream slope is fully stable. The results of the proposed method in this study are an upper bound for the results reported by the limit equilibrium and displacement finite element methods. As seen in Figure 1, the suggested method predicts lower bearing capacities compared to rigid block limit analysis method and is indeed a lower bound for the classical limit analysis method. The finite element limit analysis with linear optimization has resulted in more bearing capacity than cone optimization. The bearing capacities, obtained from characteristic lines method depending to the slope angles, in some cases is more and in some cases less than those explored by the proposed method.
In this paper, the bearing capacity of foundation located on slope was evaluated by finite element limit analysis method. In this regard, the effects of different downstream and upstream angles of slope and foundation depths and also, the effect of various mesh discretizations on the bearing capacity were studied. It is shown that an increase in the downstream angle causes a decrease in the bearing capacity and an increase in the downstream foundation depth leads to an increase in the bearing capacity.  However, the upstream angle and upstream foundation depth were not much effective on the bearing capacity.