ABOUT THE COURSE:The course aims to bridge the gap between theoretical geomechanics and its applications in real-life geotechnical engineering problems. You will learn to develop mathematical models based on constitutive relations and numerical techniques, e.g., numerical differentiation and integration, which are crucial for the formulation of mathematical problems and data analysis. You will learn to solve ODEs, Eigen Value Problems, and PDEs. You will also learn how to use the finite difference and finite element methods to solve problems related to slope stability, retaining structures, shallow and deep foundation bearing capacity, consolidation settlement, and seepage analysis. Students will develop their own code and use open-source computational tools like LimitStateGEO and OpenSees. We will share assignment problems each week and conduct a pen and pencil exam at the end of the course. This course equips students and practitioners with a fundamental understanding of geomechanics and the necessary numerical skills to tackle complex geomechanics problems. The course will be helpful for students preparing for competitive exams, as well as design practitioners enhancing their professional careers.INTENDED AUDIENCE: UG and PG students. Research Scholars and Practicing Engineers can also take this course.PREREQUISITES: Exposure to Soil Mechanics, Numerical Methods and basic programmingINDUSTRY SUPPORT: Most design companies working in Geotechnical Engineering like L&T, TCE, Reliance Infra, AFCONS, HCC, Keller, Golder Associates, etc

Syllabus

Week 1: Introduction to course, Mathematical modeling and Problem Solving, Programming and software, Floating point arithmetic and Numerical error: Round-Off and Truncation errors, Convergence and stability
Week 2:Physical laws and governing equations, elasticity and plasticity, constitutive law, plane stress, plane strain, axi-symmetry, yield criteria, flow rule, post yield deformation
Week 3:Numerical Differentiation in Geomechanics: Stress-strain relations, hydraulic gradient, rate of settlement; Methods of Numerical differentiation: Taylor Series Expansion, Forward, Backward, Central Difference scheme, Higher order differential, order of error, Interpolation techniques: Newton’s Divided difference, Lagrange Interpolating Polynomials and its application: Shape functions, soil profiles from borehole data:
Week 4:Numerical Integration in Geomechanics, estimation of total settlement, seepage loss, stiffness matrix, Trapezoidal rule, Simpsons rule, Newton-Cotes quadrature rules, Gaussian Quadrature
Week 5:Ordinary differential equation: Euler’s Method, Runge Kutta Method, Systems of Equations, Stiffness and Multi-step Method, Boundary and Eigen Value Problem, Case Studies, Computer Implementation
Week 6:Finite difference methods and Solutions of simultaneous linear algebraic equations: Gauss Elimination, LU Decomposition, Matrix inversion, Gauss-Seidel, Computer Implementation
Week 7:Partial Differential Equations: Parabolic and Elliptic Equations, Explicit, Simple Implicit and Crank Nicolson Method, Consolidation and Seepage analysis
Week 8:Finite difference method and its application: Beam on elastic foundation, Axially and laterally loaded pile, Sheet pile wall, Soil-reinforcement interaction, Computer Implementation
Week 9:Introduction to finite element method, Steps in FEM, Method of weighted residual: Point Collocation, Least Square and Galerkin Method
Week 10:Energy approach, Rayleigh Ritz method, Principle of minimum potential energy, Variational Approach, interpolation functions, Local and Global coordinate system
Week 11:Direct Approach, Principle of Virtual Work, Spring, Bar and Beam element, Element equations, assembly of element equations, transformation matrix, Truss Problem, Computer Implementation
Week 12:Finite element method of 2D element, CST element, Isoparameteric element, Applications: Flow through porous media, Computer Implementation