Finite Element Analysis And Analytical Method

Finite Element Analysis And uninflected MethodStone chromatography pillars be widely aimd as a territory repairment technique especi e very(prenominal)y in construction of school foundations. The main concern in the application of rock poopdy mainstays rely on how well it performs, which involves reducing the everywhereall resolution of the nether region tower.This project generally investigates the resemblance and contrast amidst mortal agent digest and uninflected order in poseurling tilt pit chromatography editorials, whereby resolvings of the sway editorials be analyse whether it is consistent. Finite particle analyses were carried emerge by axisymmetric modelling of the nether region newspaper mainstay utilize 15-noded triangular genes with the comfor control boardw are packet boat PLAXIS. A drained compend was conducted development Mohr-Coulombs measuring for soft system, rock and rolls and sand. Analytical data used to compar e the solution was found according to the formula system published by Heinz J. Priebe (1995). Both regularitys were compared by varying parameters such as modulus of torture of the towboat to sand dimension, field ratio, show, diam, and friction tip of mark newspaper column that signifies dissimilar modify conditions.It is challenging to find a site with acceptable argument conditions for construction of structures such as buildings, bridges, etc. Often the bearing expertness of the lubricating oil would non be sufficient to support the loads of the structures nor would it be in a workable condition for the employees to build the structure. The need for the use of such land with weak cohesive modify strata has been a take turn bulgeion for design engineers. Although the design of piles foundation can meet all the design necessities, extensive lengths of piles needed eventually results in great increase of cost of the overall project. Therefore, it is a ne cessity that the ground conditions mustiness be improved to allow the buildings and heavy construction.A add in concert of ground advance techniques have been developed over the past fifty years. main(prenominal) concern of these techniques includes creating unassailable reinforcing components to the land mass, which results in a dent that has a higher bearing capacity. Out of the various techniques available for ground improvement, the lapidate column has been widely used.Stone columns ( similarly known as granulose-grained columns, granular piles or sand columns) are used to improve soft ground by increase the load bearing pressure sensation of the ground and reducing settlement of the foundation of structures, embankments, etc. Although these structures are permissible for a relatively boast intacty settlement, it is necessary that the settlement be minimized for supreme safety.There have been several ways for installing match columns depending on the design, local practice and availability of equipment. Among which, the intimately general manners are the vibro-replacement method and vibro- shifting or vibro-compaction methods. Vibro-replacement technique of gem column is a process whereby large sized columns of compacted coarse aggregates are installed through the weak soil by means of special in-depth vibrators. This can be carried out either with the run dry or steady process. In the dry process, a hole of desire depth is drilled down in to the ground by feed a vibroflot. Upon extraction of the vibroflot, the borehole must be able to stand open. The crush of the soil will be a result of the vibrator near the after part of the vibroflot. In the wet process, the vibroflot will form a borehole that is of larger diameter than the vibrator and it requires continuous supply of water. As a result the un baptisteryd hole is blush out and filled with granular soil. The main difference amongst wet and dry process is the absence of continu ous jetting water during the initial formation of the borehole in the dry process.The performance of the cavity columns is not mensurable by simple-minded investigations. However, analytically, the efficiency of this composite system that consists of rock music column and soil interactions can be assessed by separate experimental condition of profound parameters as proposed by Priebe (1995) 1.Stone column technique has proven prosperous in improving m any(prenominal) applications. such(prenominal) applications include slope stability of two(prenominal) natural slopes and embankments. Construction of such embankments can commence like a shot after the trigger of stone columns (Vibro Stone Columns, 2009) 2. Other advantages include increasing bearing capacity of ground, reducing total and diametricalial settlements, reducing the liquefaction likely of sands. The main disadvantage of the stone column technique is its ability to cannonball along convex bombure on the upp er part of the stone column. unmoved(p) field tests (cone penetration test and full scale footing test) out front construction and after construction of stone columns have evidencen significant improvements in the soil (J. T. Blackburn, J. K. Cavey, K. C. Wikar, and M. R. Demcsak., 2010) 3. In a mull of the behaviour of stone columns, (Mitchell J.K., and Huber T.K., 1985) 4, by utilize exhaustible atom summary, had proved that the installation of stone columns petabits to a 30-40% reduction in settlement of the look ons expected that of an untreated ground.1.2 ObjectivesThe main objective of this project is to show that the analytical method used to design stone columns and the delimited element method used to model the stone column numerically, has comparable total and differential settlement. The analysis in like manner post the understanding of the influence on settlement by varying parameters such as modulus of contortion of the column to sand ratio (Ec/Es), scen e of action ratio (Ac/A), stress 0, diameter D, and friction angle of stone column c, and terminally comparing them against the Priebe analytical approach.The objectives of the project are tostudy the existing analytical and numerical theories related to stone column modellingdevelop an axisymmetric pretext of the stone columns by utilize impermanent element method, andcompare the settlement difference with the analytical results by altering various parameters related to settlement change.This project uses the limited element bundle package PLAXIS to simulate the stone column numerically and the design method proposed by Heinz J. Priebe (1995) 1 for the analytical results.1.3 transcription of the research paperIn addition to the abstract, list of figures and notation, acknowledgement, and table of contents, this sermon is divided to six chaptersThe first chapter consists of introduction and background of stone columns where it briefly summarizes the installation methods, s ome of the advantages and disadvantages of the stone columns.The second chapter describes the study of existing analytical and numerical theories regarding modelling stone columns. In this chapter, other than the main findings from the theories, the full procedure of Priebe (1995) method of modelling stone column has been reviewed.Third chapter describes how the stone column was modelled using the PLAXIS software, including the assumptions made and technical data used in different models.The fourth chapter shows the results obtained from the analysis compared to the analytical method proposed by Priebe (1995). The results are presented using necessary graphs and charts.The fifth chapter includes the conclusion of the project and provides recommendations for notwithstanding studying.The last chapter lists out the references used in this project.The Appendix contains documents such as the risk Assessment, Diary of the work progress, and the any additional tables and figures of the analysis.CHAPTER TWO2. LITERATURE surveilMany researchers in this field have made their effortless percentage studying the behaviour of stone columns numerically and analytically. Most of the numerical analyses were conducted using finite element analysis, whereas analytical method is derived from a serial publication of have-to doe withitys. some(a) of the main findings from researchers related to this study are reviewed on a lower floor.2.1 Analytical types2.1.1 Alamgir, Miura, Poorooshasb, and Madhav, (1996)Alamgir et al. (1995) proposed a simple theoretical approach to evaluate the deformation behaviour of uniformly stung ground reinforced by columnar inclusions. The displacements of the soil and stone columns are obtained by considering the elastic deformation of both soil and column. A typical column-reinforced ground and column soil building block (Fig. 2.1) where the column is considered to be cylinder, of top side H and diameter of dc (=2a where a is the radius)Th e deformation at a deal section at heart the column, wcz, is assumed to be constant end-to-end whereas the deformation of the meet soil, wrz, increases from the soil column surface towards the outer marge of the unit electric cell (Fig. 2.2). This denotes that since the column soil interface is elastic and no slip occurs, the displacements of the soil and the column at interface can be assumed to be equal. The deformation of the surrounding ground, wrz, is assumed to followwhere wrz is the displacement of the soil element at a depth z and at a radial distance r, wcz is the displacement of the column element at a depth z, cz and c are the displacement parameters, a and b are the radii of column and unit cell, respectively, r is the radial distance mensural from the center of the column.The column and the surrounding soil were discretized in to a number of elements as shown in Fig. 2.3. The interaction cut back stresses and stresses on the column and the soil were obtained by u sing equilibrium of vertical forces within the medium (Fig. 2.4). in turn the displacement of the column and soil were obtained by solving equations by applying the running(a) deformation characteristics of the soil. Therefore, the deformation of the jth element of the column, Wcj was obtained aswhere H is the height of a single element, Es and Ec are the modulus of deformations of soil and column fabric respectively, vs is the Poissons ratio of the soil, and cj is the normal stress acting at the top of the jth element of the column.Due to the symmetry of load and geometry, the shear stress at the orthogonal boundary of the unit cell is zero, which subsequently hightail its to an equation for cFurther more, the crunch of the soil element adjacent to the boundary of unit cell (N,jth element of the soil), wsNj was derived aswhere sNj is the normal stress acting at the top of the element, n is the set ratio b/a, R is r/a and r is (b-a)/n.By using the displacement compatibility a nd substituting r/a=n-R/2, Eq. 2.1 can be written asFinally, solving the equations 2.2, 2.3, 2.4, and 2.5 can lead to the displacement parameterThe settlement profiles, the shear stress distribution, and the load share from the higher up mention method was compared against a simple finite element analysis as shown in Fig. 2.5, Fig. 2.6, and Fig. 2.7. It is seen that the results obtained shows a reasonable agreement amidst the two methods and can be used as a usable method to determine the settlement of the stone columns.2.1.2 Priebe (1995)Priebe (1995) proposed a design method to assess the behaviour of stone columns that uses an improvement compute which stone columns improve the performance of the subsoil in comparisons to the state without columns. The above statement was outgo described using the adjacent familyAccording to this improvement operator, the deformation modulus of the composite system is increased respectively settlements are reduced.A unit cell of field o f force A is considered which consists of a single column with the subdue section orbit Ac. Calculation of the improvement divisor was done by take for granted thatThe stone column to be of incompressible tangibleThe stone column is installed within a rigid stageThe intensity densities of the stone column and soil are also neglected.Hence, according to Priebes approach, column cannot fail in end bearing and any settlement of the load battlefield results in a bulging of the column, which remains constant all over its length.The improvement of a soil achieved by the presence of stone columns is evaluated found on the assumption that the column stuff shears from the beginning whilst the surrounding soil reacts elastically. Additionally, the coefficient of earth pressure amounts to K=1 by assuming that the soil to be displaced already during the column installation to such a degree that its preliminary resistance corresponds to the liquid state. Using the above criterion the canonical improvement calculate n0 is expressed aswhere= Improvement factorAc = Area of the stone columnA = Grid plain of the single unit= Poissons ratio= Coefficient of active earth pressure for the stone column natural= Friction angle of the stone column materialSince a Poissons ratio of 1/3 is adequate for the state of final settlement in nigh cases, the results of the evaluation is expressed as basic improvement factor n0 and substituting 1/3 as Poissons ratio, which leads to the pastime equation.The relation between the improvement factor n0, the firmament ratio A/Ac and the friction angle of the backfill material is illustrated in figure 2.8 below.The compacted backfill material of the stone column is still compressible. Due to this reason, apply load of any amount will lead to settlements that are unconnected with bulging of the columns. Subsequently, compressibility of the column is integrated by adding up an additional area ratio (A/Ac) as a function of the constrai ned moduli of the columns and soil Dc/Ds and is provided in the Fig. 2.9.The improvement factor as a result of the consideration of the column compressibility is represented by n1, as shown in the equationwhereandFurthermore, for =1/3 can be found using the equation belowThe additional loads due to the bulk densities of the soil and columns falling off the pressure difference asymptotically and reduce the bulging correspondingly. Subsequently, multiplying the basic improvement factor by a depth factor could incorporate the military force of the bulk slow-wittedness, which is given bywhere,fd = Depth factorK0C = Coefficient of earth pressure at rest for stone column material= Bulk density of the soil= Layer thicknessPc = Pressure within the column along the depth encrypt 2.10 shows the influence factor y as a function of the Area ratio A/Ac and can be used to approximate the depth factor. The figure considers the same bulk density for the columns and soil, which may not be true in most cases. Therefore as a safety measure, the lower honour of the soil should be always considered.Using the above depth factor fd, a more enhanced improvement factor can be delineate that considers the tacks of the overburden pressure, and therefore is represented by n2 where it can be related by the following equationThe depth factor is especial(a) so that the settlement of the columns resulting from their inherent compressibility does not exceed the settlement of the composite system. This is because as the depth increases, the support by the soil reaches such an extent that the column do not bulge anymore. The first compatibility authority where the depth factor is limited is applied when the existing soil is stiff or dense and is given byThe second compatibility defy is unavoidable since should not be considered even if it may result from the calculation. This second control relates to the maximum value of the improvement factor nmax and is applied when the existing so il is loose or soft.Both compatibility controls can be determined using figure 2.11 below.Finally, the total settlement of a single or a strip footing can be assessed using the above series of equations. The design results from the performance of an immeasurable column grid below an unlimited load area. For the unimproved ground, the settlement can be found using the equationwhere,s = Total settlementp = Pressure exerted by the above structured = Depth of the stone columnDs = Constrained modulus of the soil besides, the total settlement of the improved ground, where the improvement factor is incorporated, can be found by dividing the settlement by n2, which is shown belowThis method is one of the most common and well-known method of blueprint stone columns and has been widely used all over the world because of its simplicity. Moreover, in comparison with the other methods, it shows a much wider behaviour of the stone column by assuming the stone column and surrounding soil as a co mposite system.2.2 Numerical Models2.2.1 A.P. Ambily and Shailesh R. Gandhi (2007)Ambily and Shailesh (2007) studied the behaviour of stone columns by comparing experimental and Finite Element analysis on a single stone column and a group of 7 columns. research lab experiments were carried out on a stone column of 100mm diameter adjoin by soft clay in cylindrical tanks of 500mm high with diameter varying from 210 to 420 mm for a single column test and from 210 835 mm for a group of 7 columns. This represents the required unit cell area of soft clay slightly each stone column. Pressure cells given over to the loading plate were used to measure the stress intensity of the column and the soil as shown in figures 2.12 and 2.13. Furthermore, it is also assumed the stone columns are installed in a triangular pattern.The load deformation behaviour of the column/treated soil was studied by applying vertical load for both cases column only loading and entire area loading, and observed fo r equal intervals of settlements until failure occurs. After a series of procedure, the shapes of the tested columns are obtained. It is distinctly seen in Fig. 2.14 that bulging mode of failure only occurs in the case of column alone fill, and not in the case of entire area loaded.Finite Element analysis was conducted using 15-noded triangular elements with the software package PLAXIS, to compare the load-settlement behaviour with the model test and the laboratory experiment. The analysis was carried out using a stone column of diameter 25 mm and 225 mm high, which was made at the center of the clay bed and loaded with a plate of diameter two clocks the diameter of the stone column. The axisymmetric finite element mesh to represent the single stone column and the group of stone columns are shown in Fig. 2.15 and Fig. 2.16 respectively.Likewise the laboratory experiment, finite element analyses were done for column alone loaded and entire area loaded case for s/d=3. The results o f these simulations (Fig. 2.17) shows that failure by bulging occurs in column alone loaded case, which also agrees with the results from laboratory experiment.The comparison of the experimental results and finite element analysis data shows significant eubstance in both methods. The comparisons made by A.P. Ambily and Shailesh R. Gandhi include the effect of shear strength, Cu (Fig. 2.18) and the effect of s/d (Fig. 2.19) on the behaviour of stone columns. Additionally, the effect of surcharge on stress settlement behaviour (Fig. 2.20) and effect of s/d and on the stiffness improvement factor (Fig. 2.21) was compared between both methods. These tests have also shown similar behaviour. The stiffness improvement factor () was calculated as the ratio of the stiffness of treated and untreated ground, and beyond s/d = 3, it shows no significant improvement.The analysis was extended to study the effect of the angle of internal friction of stones by varying the as 35, 40, 43, and 45o f or varying values of s/d ranging from 1.5 4. From the results shown in Fig. 2.22, it is confirmed that this relationship is valid for any shear strength values of surrounding soil.Furthermore, the comparisons between a single column and group of 7 columns were found as in Fig. 2.23.Both experimental and finite element method results erupt comparable behaviour regarding the ultimate load and load deformation relationship. To regard that this proposed design method agrees with the existing theories, this study was compared with the existing theories as shown in Fig. 2.24 and Fig. 2.25. The result shows a slightly higher stiffness improvement factor () for an area ratio more than 4 and a lower value for an area ratio less than 4 compared to Priebe (1995).2.3 SummaryThe studies mentioned above show comparable results and have been adopted by many an(prenominal) engineers and contractors. However, not many researchers had compared Priebes analytical model with finite element method. Therefore, the finite element analysis carried out in this project will be compared to the design method proposed by Priebe (1995), since it gives a much broader overview of the composite system consisting of the stone column and soil interactions and moreover it is the most common and improved analytical method used by the design engineers around the globe.CHAPTER THREE3. METHODOLOGY3.1 introductionDifferent methods of modelling stone columns numerically have been implemented in the past. Among those, the most simplest and common type of numerical modelling is using finite element method. In fact, studies have shown that the settlements predicted from the finite element analysis shows comparable results that of the values gained from actual field tests (Kirsch, F. 2009). Numerical calculations are usually complicated and most of the time is impossible to conduct without means of dedicated software. Likewise, in this research project, PLAXIS software is used to carry out the finit e element analyses.3.2 PLAXIS softwareThe main computer software used in this fact-finding project is PLAXIS Professional Version 8.2. PLAXIS is a comprehensive package for finite element analyses for geotechnical applications. It allows simulating the soil behaviour by using soil models. The software employs a graphical user interface that makes it simple to use and also provide the ability to input the necessary parameters such as different soil layers, structural elements, variety of loadings, and boundary conditions through CAD drafting procedures. It allows discretizing the soil component into either 6-noded or 15-noded triangular elements whereby 15-noded triangles provides high stress results for complex problems. The software also allows automatic generation of 2D finite element meshes that can be further re bewitchingd according to the pick of analysis. In addition to that, the software comes with a very useful mark named Staged Construction. This feature allows the mod els to be simulated at different stages by activating and deactivating clusters of elements, application of loads, etc. One of the advantages of this software is the ability to set out the results quickly with minimum errors. The output results include values for stresses, strains, settlements, and structural forces together with the plots of different curves such as, load-displacement curve, stress-strain diagrams, and time-settlement curve.3.3 Finite Element ModellingFinite element analysis was conducted to compare the load-settlement behaviour of the stone column. A two dimensional axisymmetric analysis was carried out since the investigation concerns a single unit of stone column using Mohr-Coulombs criterion for clay and stone column. 15-noded discretization was used for more precise results. The initial vertical stress due to gravity has been considered in this analysis. Similarly, the stress due to column installation, which often depends on the method of construction, is al so considered in this analysis.Assumptions made in the finite element modellingThe soil is assumed to be homogenous, infinite and behaves as Mohr-Coulomb model.The ground water table is at the same level as the stone column and clay layer, meaning the stone column and clay layer is submerged in the water. Hence, effect of ground water condition should be taken into account.The fore of the clay layer is rigid, i.e., full fixity at the base of the geometry (ux=0, uy=0) and tumbler pigeon conditions at the vertical sides (ux=0, uy=free) boundary conditions are shown in Figure 3.1(a). fancied that deformation of the column is mainly by radial bulging and no significant shear is possible. Therefore, interface element between stone column and clay has not been used.Mitchell, J. K., and Huber, T. R. (1985) also carried out similar type of finite element analysis without the inclusion of the interface element.3.4 Geometrical ParametersThe dimensions of the PLAXIS model are shown in Figur e 3.1(b). H is the height of the column, which varies between 10m, 20m, and 30m. D is the diameter of the stone column, which has a typical value of 1m, in all the models except for the model to check the influence of diameter and spacing. Equivalent diameter De depends on the spacing between stone columns as well as the system of rules pattern of the columns. The value of De was calculated by considering the following Influence Area methods.3.4.1 Influence Area MethodsThere are several methods for calculating the similar diameter around the stone column, which depends greatly on the spacing, diameter, and pattern of installation of the stone column. Two methods were considered in this investigation.3.4.1.1 Equivalent Area methodThe uniform area method simply equates the area of the grid spacing with that of the cross sectional area of column to find the influence area around the stone column. The following example gives a better understanding of the above statement.ExampleGrid s pacing of the column = 1.5 X 1.5 meters (square grid)Therefore, Diameter of stone column =Finally,Where, De is the uniform diameter around the stone column.3.4.1.2 Unit cell method (Balaam Booker, 1981)Unit cell consists of the column and the surrounding soil within the zone of influence of the column. The unit cell has the same area as the actual domain and its perimeter is shear free and undergoes no squint displacement. Balaam Booker (1981) relates the diameter of the unit cell to the spacing of the columns aswhere, De is the equivalent diameter(for square grid) S is the spacing of the stone columnSimilarly the different geometrical patterns due to column arrangements are shown in the Figure 3.2.Both methods reviewed above gives relatively similar magnitudes. However, Priebes analytical method concerns more on unit cell area. Hence, for this investigation Equivalent Area method is used to model the influence are in PLAXIS.3.5 Mesh justness runnelMesh generation has a great influence in the accuracy of the model. Generally, the all rightr the mesh the more accurate the result would be. However, this is not true for every case. Therefore a simple test using PLAXIS was conducted to check the effect of mesh refinement.Initially, mesh generation was set to coarse (around 100 elements), utilized as global coarseness of model. The test was carried out by comparing it with the refined mesh (around 500 elements). Moreover, the mesh is further refined which in PLAXIS is set to very fined (around 1000 elements). The generated meshes are shown in Figure 3.3. followed by the time-displacement graph showing the comparison between coarse, medium, fine and very fine mesh refinements. (Figure 3.4)From the above graph it can be seen that the four curves gives comparable results. However, the coarse, medium, and fine meshes give very similar results compared to the very fine mesh refinement. The objective here was to get the lowest value for the displacement since the improved ground due to the installation of stone column would eventually lead to a reduced settlement. Therefore, the finest mesh refinement gives the most precise result.Even though it takes a substantial amount of time to simulate using the most finest meshing, for this investigation, models had been simulated using the very fine mesh option.3.6 Input ParametersVarying the soil parameters can alter soil characteristics. Most important outcome by altering these parameters is deformation that leads to settlement. Such parameters that have major impact on settlement includes, material type, spacing of stone columns, diameter of influence area, diameter of stone column, elastic modulus of both column and soil, depth of the soil layer, Poissons ratio for both column material and soil, Unit weights of the materials, cohesion, friction angle, etc. Soil and material properties are shown in Table 3.1. Note that the effective stress cohesion, c of the stone column is given a small nonzero v alue to avoid numerical complications.The absolute majority of the above parameters are considered for only one type of test model and are varied for different model tests. The varied parameters such as elastic modulus of soil and column, friction angle, spacing between columns and influence area around the stone column are reviewed in the following section.3.7 Test ModelsThe main objective of this project is comparing both analytical and numerical method using Priebes analytical approach and finite element analysis as numerical solution. This can only be achieved by evolution multiple models and simulations to obtain a range of values to compare with, which would lead to a more solid conclusion.Three constitutive models were considered for the representation of the following three cases.A clay layer of 30 m, which has a stone column of height 10 m installed.A clay layer of 30 m, which has a stone column of height 20 m installed.A clay layer of 30 m, which has a stone column of he ight 30 m installed.Note that 1 and 2 are floating columns that are not extended to bedrock or sternly layer, which in stone column installation is a rare case, yet installed occasionally.Each of the above tests was carried out by varying the spacing between columns, which would alter the s/d relationship together with the Ac/A ratio. Further tests were carried out to check the influence of stress 0, diameter D, modulus of deformation of the column to sand ratio Ec/Es and friction angle of stone column c using the third case and compared them against the Priebe analytical approach.The summary of test models is given in the Tables 3.2. All the tests were carried out in 3 stages.Install the stone column Just after the stone column is installedApply Load Just after the load is applied to the columnConsolidation After the consolidation process completed to a minimum pore pressure of 1kPaIn the all cases the materials were idealized as the Mohr-Coulomb model with the characteristic line ar-elastic-perfectly plastic behaviour and the failure criteria defined by the strength parameters given in tables below.Table 3.2 Summary of Model testsModel TestDescriptionConstantsVariables1Influence of column height on settlement(case 1, 2, and 3)0 = 100 kPaAc/A = 0.2c = 40oEc/Es = 20Heigh