# Why is geotechnical engineering among the last engineering fields to use FEM?

Posted on 9th August 2022

A short history of geotechnical engineering

Geotechnical engineering is the specialised branch of civil engineering that uses the principles of soil and rock mechanics to study the behaviour of earth materials. The very first activities in this field were the irrigation and flood control measures that the Mesopotamian civilisation was using around 2000 B.C.E. Later on, foundations started to be created by Ancient Greeks in the form of pad footings, and strip-and-raft foundations. However, until the eighteenth century, geotechnical engineering was seen as an art rather than a science.

The scientific-based approach to analysing the subsurface has started with foundation issues in structures such as the Tower of Pisa. In 1717, Gautier recognized the “natural slope” of different soils, later known as the angle of repose, and in 1773 Coulomb initialized the classical geotechnical mechanics. The concepts that developed in this period are still used today.

However, modern geotechnical engineering started later in 1925 with Terzaghi’s contribution to the study of fundamental properties of clay and the failure due to piping beneath dams. Since then, numerous developments took place, and include subfields such as earthquake engineering, soil dynamics, and numerical modelling.

The evolution of computer technology in numerical analysis

Over the past few decades, applied soil mechanics has seen numerous developments, with the introduction of computer technology in the construction sector playing a major role. Almost all processes nowadays can be mathematically modelled and simulated using a set of partial differential equations (PDEs). The PDE can be used to analyse a continuous element (i.e., a finite element of soil). Therefore, computers are used as they can numerically solve differential equations using the finite element method.

Although the finite element method has been used in many fields of engineering practice for over thirty years, its use in analysing geotechnical problems has only recently begun. Because many complex issues have only recently been solved, there are only a few books that cover the application of the finite element method to geotechnical engineering. This method, however, is only a tool that must be used correctly. This necessitates a thorough knowledge of both the computer software and the geotechnical problem.

The main design goals of the software calculations are to ensure the structure's safety and stability. Based on the form of stability, the analysis method must solve for unknown forces expressed in terms of stresses, as well as movements in the soil domain expressed as displacements and strains.

For general three-dimensional (3D) conditions of the soil domain, the stress vector has 6 components: 3 direct stresses (σx, σy, σz) in the three coordinate directions (x, y, z) and three shears stresses (𝜏xy, 𝜏xz, 𝜏yz). Similarly, the corresponding strain vector consists of 3 direct strains (εx, εy, εz) and 3 shear strains (γxy, γxz, γyz). These are calculated from the displacement components u, v and w acting in the three coordinate directions. Consequently, the total number of unknowns that the method of analysis has to solve for is 15: 6 stresses, 6 strains and 3 displacements.

For an analysis method to accurately calculate the above 15 quantities for a given geotechnical problem, it must satisfy the following theoretical requirements:

equilibrium

compatibility

constitutive behaviour

boundary conditions

Equilibrium represents the balance between any external forces acting on the soil domain (e.g. self-weight, distributed load, concentrated forces and moments) and internal forces (in terms of stresses). This condition provides in general 3 equations for the solution process, which represent equilibrium between internal and external forces in three coordinate directions:

∑𝐹𝑥 = 𝜕𝜎𝑥/𝜕𝑥 + 𝜕𝜏𝑥𝑦/𝜕𝑦 + 𝜕𝜏𝑧𝑥/𝜕𝑧

∑𝐹𝑦 = 𝜕𝜏𝑥𝑦/𝜕𝑥 + 𝜕𝜎𝑦/𝜕𝑦 + 𝜕𝜏𝑧𝑦/𝜕𝑧

∑𝐹𝑧 = 𝜕𝜏𝑥𝑧/𝜕𝑥 + 𝜕𝜏𝑦𝑧/𝜕𝑦 + 𝜕𝜎𝑧/𝜕𝑧

In the equations above, the left-hand side represents the sum of all external forces in the x, y and z directions respectively, whereas the right-hand side represents the sum of stresses (internal forces) in the respective coordinate directions. If deformations in the soil domain are defined by continuous displacements functions u, v and w in the x, y and z directions respectively, such deformations are considered to be compatible, and the resulting strains are defined as:

𝜀𝑥 = 𝜕𝑢/𝜕𝑥

𝜀𝑦 = 𝜕𝑣/𝜕𝑦

𝜀𝑧 = 𝜕𝑤/𝜕𝑧

𝛾𝑥𝑦 = 𝜕𝑣/𝜕𝑥 + 𝜕𝑢/𝜕𝑦

𝛾𝑥𝑧 = 𝜕𝑤/𝜕𝑥 + 𝜕𝑢/𝜕𝑧

𝛾𝑦𝑧 = 𝜕𝑤/𝜕𝑦 + 𝜕𝑣/𝜕𝑧

The compatibility condition, therefore, produces 6 additional equations for the solution process. Together with the 3 equilibrium equations, this makes 9 equations and 15 unknowns and therefore further 6 equations are needed for an accurate solution. These are provided through the requirement that the analysis method must satisfy the constitutive behaviour of the material, which, in general, describes the relationship between the stresses and strains in that material.

The finite element method meets all theoretical and design requirements and is becoming an increasingly preferred analysis tool. However, since there is no agreed methodology within the international numerical community on how to properly implement constitutive models in software, which numerical solver is the most accurate, or how to implement certain boundary conditions. Therefore, different software will solve the problem differently and the user must understand how the computer black box works, in order to produce a meaningful solution.

Current limitations

Since it first emerged in the 1930s, soil mechanics is still a relatively new scientific field. Its application has seen significant changes over the last few decades, as has computer technology. Initially, graphical and closed-form solutions were used between the 1930s and 1960s. Later years brought digital software to assist engineers in solving problems with the last 30 years being focused on PDE solvers. The ability of these PDE solvers to use techniques such as automatic mesh generation and adaptive mesh refining to ensure the convergence of very nonlinear equations was the most notable feature. Recent advancements have focused on the fast and efficient construction of 3D numerical models.

Because of theoretical and software limitations, geotechnical engineering has only recently begun to use finite element analysis or BIM methods. Part of the reason for computer software limitations is our lack of understanding of how to apply soil mechanics principles to the unsaturated soil zone near the ground surface. Determining the moisture flux boundary conditions that must be applied at the ground surface is also a challenge. Future research will aid in our understanding of theory and the development of more precise analysis techniques.

Another limitation was the intersection of the 3D topology and the geo-strata surfaces. This problem is currently in development, and engineers can now create conceptual models in minutes by importing DXF designs or borehole data directly into the software.

One geotechnical software that tries to boost the geotechnical field is GEO5. GEO5 is a geotechnical software that is making a step forward in the industry by offering a user-friendly platform where engineers can efficiently perform analysis. The focus is put on efficiency and productivity, by integrating different features such as the top-down workflow interface in all individual programs. GEO5 also supports BIM principles and applies them successfully, creating 3D subsoil models easily and rapidly. Creating models quickly and accurately helps engineers reduce the cost of the projects and allows for a better assessment of the design approaches.

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