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Concept# Finite element method in structural mechanics

Résumé

The finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at discrete points called nodes. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young's modulus, shear modulus and Poisson's ratio.
History
The origin of finite method can be traced to the matrix analysis of structures where the concept of a displacement or stiffness matrix approach was introduced. Finite element concepts were developed based on engineering methods in 1950s. The finite element method obtained its real impetus in the 1960s and 1970s by John Argyris, and co-workers; at the University of Stuttgart, by Ray W. Clough; at the University of California, Berkeley, by Olgierd Zienkiewicz, and co-workers Ernest Hinto

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Concepts associés (6)

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Cours associés (10)

ME-484: Numerical methods in biomechanics

Students understand and apply numerical methods (FEM) to answer a research question in biomechanics. They know how to develop, verify and validate multi-physics and multi-scale numerical models. They can analyse and comment results in an oral presentation and a written report.

MATH-451: Numerical approximation of PDEs

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions.

ME-373: Finite element modelling and simulation

L'objectif de ce cours est d'apprendre à réaliser de manière rigoureuse et critique des analyses par éléments finis de problèmes concrets en mécanique des solides à l'aide d'un logiciel CAE moderne.

The recent development of high-strength (HSS) weldable steels has enlarged the range of design alternatives for the optimization of high-head steel-lined pressure tunnels and shafts (SLPT&S) in the hydropower industry. With the liberalization of the European energy market and increasing contribution of new renewable volatile energies in the electricity grid due to high subsidies, storage hydropower and pumped-storage plants are subject to more and more severe operation conditions resulting in more frequent transients. The use of HSS allows the design of thinner and thus more economic steel liners. However, welded HSS do not provide higher fatigue resistance than lower steel grades, and may be particularly subject to the risk of cold cracking in the weld material as dramatically illustrated by the failure of the Cleuson-Dixence pressure shaft in 2000. Fatigue behavior may become the leading limit state criterion. This research project aims at improving the comprehension of the mechanical behavior of SLPT&S and at developing a framework for probabilistic fatigue crack growth and fracture assessment of crack-like flaws in the weld material of longitudinal butt welded joints, considering all possible steel grades for high-head hydropower schemes. The influence of anisotropic rock behavior and geometrical imperfections at the longitudinal joints on the structural stresses have been studied by means of the finite element method accounting for the interaction with the backfill concrete-rock multilayer system. Parametric correction factors have been derived to estimate stress concentrations and structural stresses in steel liners with ease in practice, allowing the use of $S$-$N$ based engineering fatigue assessment approaches. Stress intensity factors (SIF) for axial cracks in the weld material of the longitudinal joints have also been obtained by means of computational linear elastic fracture mechanics (LEFM). The use of the previously developed parametric equations in the classical formulas for SIF in cracked plated structures has been validated, and new parametric equations for the weld shape correction have been proposed. A probabilistic model for fatigue crack growth assessment has been developed in the framework of LEFM in combination with the Paris-Erdogan law. The probability of failure is estimated by means of the Monte Carlo simulation procedure, in which the crack growth rate parameters and the crack shape ratio are defined as stochastic variables. A week-long normalized loading spectrum derived from prototype measurements on an alpine pumped-storage hydropower plant in Switzerland is used. This approach provides relative and quantitative results through parametric studies, giving new insights on the fatigue behavior of steel liners containing cracks in the weld material of the longitudinal joints. Finally, a fatigue assessment case study is presented, detailing the entire calculation procedures developed in this research. It aims at ensuring the transfer of knowledge toward practitioners.

Thomas Keller, Anastasios Vassilopoulos, Congzhe Wang

The two-dimensional (2D) delamination growth in fiber-reinforced polymer (FRP) laminates with in-plane isotropy under Mode I loading condition was numerically investigated using finite element analyses. Two sizes of plate models were developed, focusing on different fracture stages. Cohesive elements were employed to simulate the fracture behavior in the presence of large-scale bridging (LSB). The influences of the pre-crack shape/area, loading zone shape/area and fracture resistance were parametrically studied. It was found that either a flatter pre-crack shape or a flatter loading zone shape could result in higher initial structural stiffness and less uniform distribution of the strain energy release rate (SERR) along the pre-crack perimeter during crack initiation and early propagation. However, they had only a minor effect on the stiffness after full fiber bridging development in all directions. The plates finally achieved constant stiffness, which increased linearly with the fracture resistance. The final crack shape was dependent on the loading zone shape and area, but the effects were relatively weak.

2021Séances de cours associées (23)