DECLARATION 1: PLAGIARISM
2. CHAPTER 2: LITERATURE REVIEW
2.3 Numerical methods
Physically modelling the occurrence of a large scale sinkhole is a time consuming and expensive process. In especially complex applications, it may be difficult, inaccurate and sometimes impossible to physically perform the procedure. In such cases, it may be more suitable to use alternate methods to get the required results, e.g. analytical methods, numerical models. Engineering research and investigations require the use of laboratory tests and analytical models in order to validate theoretical numerical results, to ensure that the assumptions of the numerical model are representative of real life conditions.
Numerical methods provide algebraic expressions of real world situations and scientific concepts (Strand 7 verification manual, 2013). Research in the undergraduate dissertation
“Numerical modelling of the strains in a sand mass when undermined by a developing sub grade void” (Munian, 2010) focused solely on the use of finite element analysis in order to model the formation of a sinkhole. Literature indicated that alternative methods to FEA can be considered, e.g. discrete element methods, finite difference method.
2.3.1 Finite Element Analysis
“Finite element analysis is a numerical method that is used in complex engineering applications where the solution would otherwise have been difficult to obtain” Lee et al. (2006). According to Potts & Zdravkovic (1999) most engineering solutions are based on the solution of the following equations.
1) Equilibrium equations
The equilibrium equations describes the transfer of forces throughout the soil body, this is done by analysing the variation of stress throughout the soil in the x, y and z plane.
+ ! + "# = 0
& !
&' +& !
&( +& #!
&) = 0
& #
&' +& !#
&( +& #
&) + γ = 0
Where , !, #, !, #, #!are the vector components of stress and γ is the unit weight of soil in the z-direction (Potts, 2007).
2) Compatibility equations
This is the strain - displacement relationship, the strain is considered to be the differential of the displacement. The compatibility equations accounted for the relationship between the strains in the soil mass and the displacement that the particles undergo (Potts & Zdravkovic, 1999).
=+ , , ! =+ -!, # = + .#
! = − - − ,! , #= − . − ,# , ! = − .! − -#
Where u, v, w represent the displacement components in the x, y, z direction and , !, #, !,
#, #! are the vector components of strain (Potts, 2007).
3) Constitutive behaviour
The material behaviour of a structural body is mathematically characterised by its constitutive equations or material laws (University of Colorado, 2012).
The relationship between the stress and strain effectively describes the Young’s Modulus of the soil, i.e.
0 =
Young’s Modulus = Stress/ Strain (for 1 D cases) And Poisson’s ratio (ν) = lateral strain/axial strain
For 2D applications the Young’s Modulus takes Poisson’s ratio into account and is modified to 0 1 = 0
1 − 34
For 3D applications, the Young’s modulus is dictated by Poisson’s ratio and is modified to:
0 1 = 0
(1 − 27)(1 + 7)
The 1D expression for Young’s modulus was then modified to take into account material behaviour of the system:
9∆σ< = =>?9∆ <
Where D represents the matrix that describes the material properties of the soil, or in finite element analysis terms, the stiffness matrix.
4) Boundary conditions
The boundary conditions in terms of the force and the displacement must be defined. The Neumann boundary in the model is where normal stresses are given a value, and the Dirichlet boundary is where the displacement are given data (Sayas, 2008).
2.3.1.1 Finite element analysis process
The process of finite element analysis follows 3 general steps:
A) Pre-processing
The input of the models geometry, generation of the mesh and discretization of the boundary and specification of boundary conditions.
• Discretization:
FEA is based on the principle of reducing complex problems by dividing them into smaller problems and solving each one separately (Strand 7 verification manual, 2013). During element discretization, the problem domain is divided into small regions called finite elements.
Elements are defined by nodes. Elements with nodes at the corners are straight, while the presence of mid-side nodes allows the elements to be curved. The various types of elements and their nodal positions are shown in Figure 2.1 below:
Figure 2.1: FEA element types
Looking at the example of a simple beam, a beam can bend in an infinite number of ways, by discretizing the boundary, it is assumed that the beam can only deform into a particular shape, limiting the variables and thus simplifying the problem.
The FEA pre-processing steps are as follows:
• Define the geometry of the model
• Define the type of elements being used
• Define the connectivity of the elements (define the mesh)
• Define the restraints/boundary conditions of the nodes B) Analysis
Specifications of material properties and loading conditions.
• Define the material properties of the elements, e.g. Tensile strength, Young’s modulus
• Define the loading conditions (gravity etc.)
The analysis process involves the selection of the primary variable. Potts (2007) states that most FEA programs select the displacement for this. Secondary quantities, like stress and strain are expressed in terms of the primary displacement quantity (Hutton, 2004). A set of algebraic equations to solve for the displacement are then developed. The deformation behaviour of the elements can be derived from the principle of minimum potential energy where “the static equilibrium position of a loaded linear elastic body is the one that minimises the total potential energy” Potts (2007). These leads to the following expression for the stiffness of the specific element:
=@A?9∆BA}C = {∆DA}
Where [@A] is the element stiffness matrix, {∆BA}Cis the vector of incremental element nodal displacements, and {∆DA} is the vector of incremental element nodal forces (Potts, 2007). The stiffness matrix represents the resistance of the element to change when subjected to external influences (Hutton, 2004).
In linear materials the element equations are then combined to form global equations:
[@E]{∆BE}C = {∆DE}
Where [@E] is the global stiffness matrix, {∆BE}C is the vector of all incremental nodal displacements, and {∆DE} is the vector of all incremental nodal forces (Potts, 2007). The global equations are solved using Gaussian elimination and matrix decomposition (Potts, 2007).
C) Post processing Analysis of results
FEA provides approximate solutions for the system, the accuracy of which depends on the convergence of the equations and the simplification assumptions made. The exact solution to the problem is solved by the program using the partial differential equations stated earlier. FEA will then produce and approximation of the exact solution. When the approximation converges on the exact solution, the computational process is complete. The actual convergence process is brought about by either altering the size of the elements or the order (magnitude) of the matrix. By changing the size of the element, the mesh is being constantly redefined and a more accurate solution can be obtained.
2.3.1.2 Limitations of the finite element analysis method
The process of flow of granular materials is difficult to model using the finite element method, due to the disfiguring of the mesh because of large soil deformations (Wieckowski, 2003). One of the main problems associated with modelling of flow conditions in finite element analysis is modelling the surface from which the soil is flowing. An entirely free surface, with a wide opening will cause free flow conditions in the soil mass, under these conditions the iterations do not converge and the points in the sand mass are in failure. In order to overcome this problem Sanad et al. (2001) initially modelled the discharge boundary as restrained in both the x and y direction. By releasing the restraint on the boundary and providing a downward displacement to the row of nodes in direct contact with the aperture boundary, initial flow conditions could be modelled.
2.3.2 Discrete element analysis
This numerical method is used for dynamic applications and accounts for motion of soil particles. The discrete element method can be used in order to model the motion of granular material as separate particles (Coetzee, et al., 2006). The program functions using Newton’s second law and has the advantage of allowing for simulation of flow in a silo, when the boundary conditions are free. The discrete element code is a medium by which the “inter- particle contact parameters” in the soil structure can be investigated (Gallego, et al., 2010).
When the void is formed, the soil particles are mobilized and hence undergo changes in displacement, shear strength and strength of contact bonds. According to Caudron et al. (2006) the soil properties that govern the particle interactions are:
• The frictional angle of the soil
• The degree of stiffness between contact particles
• The strength of the bond between contact particles
Some of the benefits of using a computer program to predict soil movement as opposed to practical laboratory methods is that in the modelling process all grain displacements and contact forces are known. Also, a specific sample can be tested multiple times and no heavy equipment is required (Van Baars, 1995).
The computational times for this method were however found to be excessive, due to the number of particles. Each particle undergoes a specific displacement and bonds that form between particles can change throughout the interactions. In order to limit computational time, a restriction on the number of soil particles was proposed by Van Baars (1995) where the number of particles would be limited to “no more than a few tens of thousands”. By using discrete element analysis where the actual grain structure is simulated, it was anticipated that results would be more accurate.
In DEM when the particles undergo deformations, grains can break and new contacts will be formed. One of the main considerations in the modelling process is whether to use 2-D or 3-D modelling. 3-D models were found to be time consuming and required very powerful computer software; 2-D methods were more simplistic and yielded similar results. The physical modelling of the granular nature of the particles depends on the creation of particles and the formation of boundary conditions.
After performing the investigations, Van Baars (1995) concluded that models using equilibrium equations (FEM) yielded the same results as models based on equations of motion (DEM) but only when soil is in a quasi-static state and motion is minimal. FEM provides quicker convergence but for completely dynamic applications, DEM must be used. The main limitation in modelling using DEM is the restriction on the number of particles; hence any models performed will only illustrate laboratory models of a small scale.