Evaluating the Subgrade Reaction Modulus Variations with Soil Grains Shape in Coarse-Grained Soils Using Genetic Algorithm

Subgrade reaction modulus (Ks) is one of the main factors in evaluating engineering properties of soils for structural calculations and operations. So, many studies have been performed on the effect of other soil geotechnical parameters on it. One is the effect of soil grains shape on engineering properties of soils, especially Ks. The aim of the present research is to evaluate the effect of soil grains shape on Ks for coarse-grained soils of the west of Mashhad, Iran. For this purpose, 20 PLTs were performed on coarse-grained soils of the west of Mashhad and Ks amounts were determined. Then, flakiness and elongation of the samples measured and changes of Ks by soil grain shape were evaluated. The results showed the strength dependency of Ks to grain forms which an increase in flakiness and elongation indices leads to a decrease in Ks. Therefore, it is necessary to reduce Ks estimated form empirical relationships for flaky and elongated soils. So, by writing a genetic algorithm-based program to find the optimal relationship between the grain shape and the subgrade reaction coefficient, a valid equation for correcting the results from previous empirical equations was presented.


Introduction
In granular soils, the overall geotechnical behavior is strongly affected by grain properties such as size, shape and crushability. Among the properties, grain shape has a considerable influence on the bulk of geotechnical properties [1] [2]. fied as flat, elongated, flaky, spherical, platy, elliptic and… In coarse grain soil, size distribution and particle shape including sphericity, angularity and roughness have pivotal role in determining soil behavior [3]. Flakiness and elongation of particles directly affect permeability, strength, shear strength, residual resistance, elastic modulus and Ks of the coarse-grained soils.
Ellipticity and plainness (as opposed to sphericity) promote inherent anisotropy and affect the evolution of stress-induced anisotropy. Plainness decreases stiffness and residual friction angle. Angularity and roughness promote a decrease in small-strain stiffness, an increase in high strain strength and affect the evolution of stress-induced anisotropy [4] [5] [6].
Ks is one of the most important soil engineering parameters which its accurate determination leads to optimization of construction operations. Ks is a conceptual relation between soil pressure and displacement, which has an important role in the calculation of the elements of strip and raft foundations and piles [7].
Equation (1) is used to calculate Ks: where q is applied load and δ is the settlement rate caused by the applied load.
Recognizing the effect of grain shape on Ks not only can relate the soil geometrical parameters, which can be evaluated in the first step, with engineering properties but also it can present an accurate and initial recognition about the process of performing construction activities and calculations to optimize the construction operations.
Genetic algorithms are search and optimization algorithms based on the principles of natural evolution, which were first introduced by john Holland in 1970 [8]. Genetic algorithms also implement the optimization strategies by simulating evolution of species through natural selections.
Genetic algorithm is generally composed of two processes. First process is selection of individual for the production of next generation and second process is manipulation of the selected individual to form the next generation by crossover and mutation techniques [9]. The selection mechanism determines which individual are chosen for reproduction and how many offspring each selected individual produce. The main principle of selection strategy is the better is an individual; the higher is its chance of being parent.
Considering the influence of particle shape on soil engineering parameters such as K s and lack of practical relations for mentioned corrections, there is a need for accurate equation to determine real K s in flaky and elongated soils.
The present study focused on the evaluation of dependency of Ks values to grain shapes for coarse-grained sediments in the west of Mashhad. The coarsegrained sediments in the west of Mashhad originated from slate and phyllite metamorphic outcrops and contain a large percent of flaky grains that affect the soil properties as well as Ks [10]. In this paper, after some experiments, the optimal relation between the shape of the grains and Ks using genetic algorithm determined and an equation to correct the subgrade reaction coefficient in elongated and flaky soils proposed.

Methodology
Plate load tests performed in this article were based on regulations of the Ministry of Energy of Iran due to compliance of devices and availability of equipment.
Also, flakiness and elongation tests, relative density (D r ), and in situ density (sand bottle test) were conducted according to American Society for Testing and Materials (ASTM) because of its validity and generality.
At first, by 20 plate load tests Ks amounts were calculated. Figure 1 shows the location of the test sites. Table 1 shows several methods to carry out the Plate load tests [11]:  To have more valid results, the PLTs were performed after removing surficial soil and placing and leveling the plates on the subsoil [12]. In addition to perform PLT with the power of 500 KN jack and the circular rigid plate with diameter of 30 cm, a loaded truck was used to apply the surcharge ( Figure 2).

Plate Load Tests Methods
The device was calibrated by national standard organization with coverage factor k = 2 correspond to confidence level 95%.
The required data were collected after installing equipment, applying load on a steel plate and recording load and settlement rate simultaneously by load and settlement gauges [13].
The loading process was incremental and its incremental steps were about one-fifth to one-fourth of the final estimated load.
The load was controlled in the desired increment and the next load increment was applied under the previous load after the settlement reached a stable state [14].

Elongation and Flakiness Tests
Then, the grains shape factors including elongation and flakiness coefficients of the grains of soil samples taken from each studied area were determined in the laboratory. According to the definitions, the flakiness coefficient is the weight percentage of grains that their thickness is less than 0.6 of mean grains size and elongation coefficient is the weight percentage of grains that their length are more than 1.8 of mean grains size [15]. It is of notes that mean grains size is the average mesh size of two continuous sieves that are used for aggregates gradation.
For this, first, two sieves size of 1 inch and 3/4 inch were used for sieving the studied samples. Then residual grains on 3/4 inch sieve were weighted. The same process was done for sieves with sizes of 3/4 inch and 1/2 inch as well as 1/2 inch and 3/8 inch.
Then, in order to determine flakiness and elongation indices, the remained aggregates on 3/4 inch, 1/2 inch and 3/8 inch sieves were passed through a wooden device with nails on it. The distance between two nails called specific distance, which means 1.8 times of the mean size of that soil group in gradation. The elongation index is calculated by Equation (2) as follow: ( ) the weight of long grains elongation index EI 100 overall weight of grains = × (2) In the next step, the aggregate were re-mixed and passed through the grooves in the floor of a special metal frame and the weight of passed grains was calculated. The flakiness index is obtained by Equation (3) as follow: ( ) the weight of passed grains through grooves flakiness index FI 100 overall weight of grains = × (3) Figure 3 shows the tests.
Moreover, for each of the studied areas, in-situ density test was carried out by using sand bottle method and the dominant soil texture in the studied depth was determined [16]. Considering the effect of soil hardness on Ks and in order to determine only the effect of grain shape on this parameter, based on Equation 4 the amounts of d γ , min d γ and max d γ were specified in the laboratory and relative density quantities were calculated [17].
Then, a genetic algorithm based program in Matlab was written to calculate the actual subgrade reaction modulus (ks) values according to the grain shape.

Results of Research
The samples were taken from about 0.2 m -1 m depth that can be presumed as load effect limit and bears only 5% of surficial stress based on boussinesq studies by circularly load on the range of stress effect [7].
In spite of changing the soil properties in different directions due to the anisotropy of the soil, in order to the loading of the structure in the vertical direction, the subgrade reaction modulus is also calculated in this direction, Sampling and speculation were done in the same direction. Table 2 presents the soil properties of 20 stations. As can be seen, the dominant  texture of the soil is coarse-grained gravel and silty sand.
After performing PLTs and collecting the required data, by considering the following items, the Ks was calculated for each station using the pressure-settlement curve: • Loading did not continue until the soil rupture.
• The amount of applied stress on plates and settlement were measured.
• Loading was performed on coarse-grained soils with low cohesion. For unsaturated soils such as studied samples, Equation (5) is used to calculate effective stress [19]: The difference a u δ − is referred to the net normal stress, the difference w a u u − is matric suction and the effective stress parameter x is a material variable that is generally considered to vary between zero and unity.
Previous studies show low importance of suction in coarse-grained soils [20].
Due to the dominant texture of the soil (coarse-grained gravel and silty sand) in approximately dried studied soils, suction was not considered and δ δ ′ = .   After determining Ks, the flakiness and elongation coefficients of studied grains were calculated to find the relation between the Ks and soil grains shape (Table 4)

Discussion
Due to the effect of soil hardness on the subgrade reaction modulus and focus on the effect of soil elongation and flakiness on Ks, by calculating relative density values, statistical fittings were taken only on approximately similar relative density specimens, That is, 15 samples with a relative density between 60% and 75% and almost in the range of stiff soils [7] [21]. Table 5 show the relationship of elongation and flakiness indexes with Ks. Generally, the results demonstrate the effectiveness of soil grain shape on Ks and an increase in elongation and flakiness indices leads to a reduction in Ks.

Figures 5-7 and
In Table 5 indices are well justified [22]. The negative line slopes also show a reverse relation of elongation and flakiness indices with Ks [23].
As seen in Table 5      Finally, a genetic algorithm based program in Matlab was written to calculate real Ks values according to grain shape. Figure 8 shows the most optimal correlation between EI (average of three sieves), FI (average of three sieves) and Ks based on the fitted genetic algorithm.
In Figure 8, it can be seen that by 40% increase in elongation coefficients, about 20% reduction is observed in subgrade reaction modulus and the mentioned coefficients reach to almost 80% of their previous values. Although, doubling flakiness coefficients leads to 15% reduction in Ks. Figure 9 shows the results of fitted genetic algorithm on in-situ and laboratory   The following equation based on the genetic algorithm is used to correct the subgrade reaction modulus obtained from the previous equations for flaky and elongated samples in coarse-grained soils (Equation (6)).

Conclusion
The