FINITE ELEMENT ANALYSIS IN SOIL/ROCK

Finite element analysis (FEA) involves complicated geometric and mathematical models of real-world problems. FEA will provide us with more reliable results if only the modelling and analysis has been carried out with proper care. An attempt has been made to explain how to utilize the FEA to the maximum extent for obtaining reliable results while explaining the importance of conventional methods too. This article is written by Er. Gowtham B, Geotechnical Design Engineer at FGS Engineers & Innovators.

Well, talking mathematically, conventional methods or analytical methods develop the governing?equations for a considered element which are in the form of partial differential equations. These equations can solve real life problems but not some of the complex problems keeping in view the complex geometry and boundary conditions. There comes the need of finite element analysis (FEA) which converts those partial differential equations into simple algebraic equations which are easy to solve to obtain approximate solutions.

?In geotechnical field, soil/rock itself is a complex and heterogenous structure which displays non-linear response under loads. Using FEA, this complexity can be solved using different constitutive models. Notwithstanding the hurdles like complexity in modelling and time consumption, FEA will provide us with the reliable results and simulate the realistic ground conditions compared to the conventional methods which are being mostly used by designers.

?ARE CONVENTIONAL METHODS ALWAYS THE SOLUTION?

• Non-numerical or conventional methods of design are usually quicker to analyse, but they have major assumptions like linear elasticity, isotropy etc.
• They also provide limited results like average settlement of a foundation, limit states etc.
• Despite these assumptions and limited information provided these programs are enough to provide us a satisfactory design in many cases.
• But the need for finite element analysis arises when conventional programs or analytical solutions are unable to solve some of the problems.
• ?Conventional programs may not be helpful while some of the complex problems are to be modelled with these conditions:

>> Complex ground behaviour like non-linear stiffness, anisotropy etc.

>> Ground improvement problems

>> Complex hydraulic conditions

>> Unusual geometry

>> Soil–structure interaction

>> Complex loadings

>> Back-analysis of field trials

>> The effect of construction stages

>> Time effects like creep & consolidation

So, conventional methods are useful only upto a certain level of complexity, but for simulating better ground/field conditions and to get the better conservativeness for more complex problems, finite element analysis has to be applied. The below figure is an example for slope stability analysis using RS2.

HAS FINITE ELEMENT ANALYSIS BEEN UTILISED WELL?

The finite element analysis (FEA) will always provide us with good results if only proper care is taken in modelling with optimum boundary conditions and suitable constitutive models. Below are the few points in which care must be taken in carrying out finite element analysis.

UNDERSTANDING THE PROBLEM AND PROGRAM

• Utilisation of finite element analysis to its throat level depends on understanding the requirements of problem and selecting the right tool for the job at hand.
• Every program has its own strengths and limitations. So, try to choose the program that suits your task by understanding the range of the program.
• Do not try to judge a tool by commencing a complex problem. Always start with a simple model for better understanding the tool.

2D vs 3D

• Even though all the problems are 3D in nature, most of the designers use 2D programs till date which produce satisfactory solutions (but not for all)
• For e.g., analysing tunnels axisymmetrically, or in plain strain condition and using Ground reaction curves for taking 3D effects into the account.
• The 2D analysis will be based on a number of assumptions which may influence the results of the program. If there isn’t any significant effect of the assumptions on the results, one can go for the 2D analysis for saving time and resources.
• So, unless it is symmetrical or close to plain strain condition, it is better to go for 3D analysis for the better results and also for the results to be on conservative side.

?HOW FAR THE BOUNDARIES HAVE TO BE CONSIDERED?

• There isn’t any hard and fast rule for taking the extent of boundaries for the analysis as the real field has indefinite extent of boundaries.
• But considering the boundary extents too close to the analyzing area will significantly affect the results.
• This may be resolved by analyzing the program with different boundary extents and selecting the optimum extents.
• This can also be resolved by varying the fixity conditions (Vertical and Horizontal fixities) and verifying whether the results are varying along with them or not. If the results aren’t changing for range of extent, that can be taken as an optimum extent of boundary.
• Number of thumb rules are also available in research papers for different types of problem (Raft foundation, Pile foundation, excavations, and tunnels) to be considered for the extent of boundary.
• But for the vertical boundary, extent may vary with the above points as the increasing depth can increase the stiffness as well as the strength of ground.
• So, proper care must be taken in selecting the extent of boundary.

?FIXITY CONDITIONS FOR THE BOUNDARIES

• The vertical and horizontal fixities have to be applied to the boundaries so that the respective displacements will be arrested at the boundaries so that the equilibrium will be established for the meshing.
• The most standard fixities are to arrest all the displacements at the bottom boundary and to arrest the horizontal displacements at the vertical boundaries. The top boundary will be kept free to allow displacements in all directions.
• This cannot be ideal for all the cases and user should adapt according to the nature of their respective problem.

?DISCRETIZATION TO BE USED

• Discretization of the model will depend on the number of nodes on elements (mostly 3-noded, 6-noded & 15-noded) and the shape of elements (Mostly triangular and quadrilateral for 2D & tetrahedral for 3D).
• The nodes are where the program will calculate the unknowns (displacements, stresses, and strains).
• So, the smaller element size and greater node number will give us the precision of the results for the program.
• But for the problems require different density of meshes such as for groundwater analysis, the lower order elements are preferable as per the previous studies, and axisymmetric models require higher density meshes.
• So, discretization has to be considered as per the respective problems.

CONSTITUTIVE MODELS TO BE SELECTED

Constitutive models for the respective soil/rock have to be selected so that the soil/rock should behave with sufficient accuracy for all the applied loading conditions. Nowadays there are number of constitutive models which will simulate all the soil/rock behaviour, but one should not complicate the model by giving complex model to a simple problem. The selection of constitutive models mainly depends upon the following criteria:

• Required output from the program
• Expected stress path
• Structure type (i.e., foundations, embankments, tunnels, excavations etc.)
• Soil & Rock type

Some examples for the constitutive models for the conditions of soil/rocks ?

• Generalised Hoek-Brown model - Jointed rocks
• Modified Cam Clay (MCC) model - Compression of Soft Clays etc.

Apart from all these, there are certain other factors which will affect the finite element analysis to a certain extent, such as giving sensible input parameters, assessing the output values etc. So, the user should be aware of the program and problem which has to be solved thoroughly so that the model and the results are sensible.

APPLICATIONS OF FEA IN GEOTECHNICAL FIELD

Tunnels, anchored walls used to stabilize landslides, building foundations, cellular cofferdams, embankment dams, excavation bracing systems, long-span flexible culverts, offshore structures, plastic concrete seepage cut-off walls, reinforced embankments, reinforced slopes, retaining walls, seepage through earth masses, slurry trench seepage barriers, u-frame locks, unbraced excavations etc., are the main areas where FEA will be useful in the geotechnical field.

FEW BOOKS ON FEA TO REFER:

?There are numerous good books available for reference on FEA in Geotechnical analysis. Below are a few good ones:

1. ?Textbook of Finite Element Analysis. P Seshu. PHI Learning Private Ltd.
2. Bathe, K. J. (1982). Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ.
3. Desai, C. S., and Christian, J. T. (1977). Numerical Methods in Geotechnical Engineering. McGraw-Hill, New York.
4. Michael A. Hicks, Ronald B.J. Brinkgreve, Alexander Rohe. Numerical Methods in Geotechnical Engineering. Taylor and Francis Group.
5. Andrew Lees. Geotechnical Finite Element Analysis. ICE publications.