notes/uni/mmme/3086_computer_modelling_techniques/fea.md

author	date	title	tags	uuid	lecture_slides	lecture_notes	exercise_sheets
Akbar Rahman	\today			8f32c8c0-5ed4-49ab-aca8-e5e058262580	./lecture_slides/fea	./lecture_notes/fea

MMME3086 Content

1. Direct and Energy based formulation of 1d elements (stiffness matrices)
2. Assembly of stiffness matrices to form the global stiffness matrix
3. 2d pin jointed structures
4. continuum elements
5. structural elements
6. practical FEA guidelines

Coursework

• Worth 35% of module
• Will be set on 2023-11-09
• Will be due on 2023-11-23

Background (Slides 0101)

• domain is discretised into finite elements

• each element is defined by its corners (nodes)

• typical shapes for elements are triangular/quadrilaterals in 2d problems or tetrahedral/hexahedral in 3d problems

• for each element, the behaviour is described by the displacements of the nodes and material law (stress strain relationships)

  • this is usually expressed as the stiffness of the element

• elements are assembled in a mesh and the requirements of continuity and equilibrium between neighbouring elements are satisfied

  • this assembly process results in a large system of simultaneous equations

• boundary conditions are applied to assembly of elements, to yield a unique solution to the overall system

• solution matrices are sparsely populated

• equations are solved numerically to compute the displacements at each node

  • the displacements of each node can be used to obtain the stresses in each element

• finite element method is suitable for practical engineering stress analysis of complex geometries

• to obtain good accuracy in regions of rapidly changing variables, a large number of small (fine) elements must be used

Basic Overview of the Steps Required for FEA

1. Discretise the domain
2. Write the element stiffness matrices
3. Assemble the global stiffness matrix
4. Apply boundary conditions
5. Solve matrix
6. Post-processing---e.g. obtaining additional information like reaction forces and element stresses

Stress Analysis Fundamentals (Slides 0102)

Uniaxial Loading

• The engineering definitions for engineering stress and strain assumes that stress is uniform, but this is rarely true over large areas.

• However the definition gets more useful for small elements:

  \begin{equation} \sigma = \lim_{\delta A \rightarrow 0} \sigma_\text{engineering} \end{equation}

• For uniaxial loading situations, engineering (nominal) strain is:

  \begin{equation} \varepsilon_\text{engineering} = \frac{\Delta L}{L_0} \end{equation}

• For uniaxial loading situations, engineering (nominal) stress is:

  \begin{equation} \sigma_\text{engineering} = \frac{F}{A_0}A \end{equation}

Multi-Axial (3D) Stress and Strain Definitions

• In Cartesian axes system, are six components of stresses on the elements:

  • direct stresses \sigma_{xx}, \sigma_{yy}, \sigma_{zz} (tensile/compressive stresses) caused by forces normal to the area

  • shear stresses \sigma_{xy}, \sigma_{xz}, \sigma_{yz} caused by shear forces parallel to the area

    • the first subscript refers to the direction of the outward normal to the plane the stress is acting on
    • the second subscript refers to the direction of the stress

• The stress and strain vectors can be expressed as the following:

  \begin{equation} \pmb \sigma = \begin{bmatrix} \sigma_{xx} \ \sigma_{yy} \ \sigma_{zz} \ \sigma_{xy} \ \sigma_{xz} \ \sigma_{yz} \end{bmatrix} \pmb \varepsilon = \begin{bmatrix} \varepsilon_{xx} \ \varepsilon_{yy} \ \varepsilon_{zz} \ \varepsilon_{xy} \ \varepsilon_{xz} \ \varepsilon_{yz} \end{bmatrix} \end{equation}

  where:

  \begin{align*} \sigma_{xx} &= \frac{\partial u_x}{\partial x} \ \sigma_{xx} &= \frac{\partial u_x}{\partial x} \ \sigma_{xx} &= \frac{\partial u_x}{\partial x} \ \varepsilon_{xy} &= \frac12 \left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right) \ \varepsilon_{xz} &= \frac12 \left(\frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x}\right) \ \varepsilon_{yz} &= \frac12 \left(\frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y}\right) \end{align*}

• In computation mechanics, the stress and strain relation is modelled as:

  \begin{equation} \pmb \sigma = \pmb d \pmb \varepsilon \end{equation}

Stress-Strain Relationship (Hooke's Law)

• Hooke's law can be used to create the following stress-strain relations for isotropic linear elastic materials with thermal strain:

  \begin{align} \varepsilon_{xx} &= \frac 1 E \left( \sigma_{xx} - \nu\left(\sigma_{yy} + \sigma_{zz}\right)\right) + \alpha\Delta T \ \varepsilon_{yy} &= \frac 1 E \left( \sigma_{yy} - \nu\left(\sigma_{xx} + \sigma_{zz}\right)\right) + \alpha\Delta T \ \varepsilon_{zz} &= \frac 1 E \left( \sigma_{zz} - \nu\left(\sigma_{yy} + \sigma_{xx}\right)\right) + \alpha\Delta T \ \varepsilon_{xy} &= \frac{\sigma_{xy}}{2\mu} \ \varepsilon_{xz} &= \frac{\sigma_{xz}}{2\mu} \ \varepsilon_{yz} &= \frac{\sigma_{yz}}{2\mu} \ \mu &= \frac{E}{2(1+\nu)} \end{align}

  where E is Young's modulus, \nu is Poisson's ratio, \alpha is the coefficient of thermal expansion, \Delta T is the temperature change, and \mu is the shear modulus

Energy Methods (Slides 0103)

Stability

• Strain energy is released upon the removal of applied loads and the body returns to undeformed state:

  \begin{equation} U = \frac12 \pmb \sigma \pmb \sigma \times V \end{equation}

  • If the material behaviour is non-linear, it can be generalised to:

    \begin{equation} U = \int_\nu\int_\varepsilon \sigma \mathrm{d}\varepsilon \mathrm{d}V \end{equation}

• Work done by external forces can be expressed as:

  \begin{equation} W = \sum_i F_iu_i \end{equation}

  where i is any point where force F_i causes displacement u_i

• The total potential energy can be expressed as:

  \begin{equation} \text{TPE} = U - W \end{equation}

  • The principle of minimum TPE states that when the body is in equilibrium, TPE must be 'stationary' with respect to the variables of the problem

  • The equilibrium is stable if the TPE is minimum

  • In most FE problems, the displacement u is chosen as the unknown variables of the problem:

    \begin{equation} \frac{\partial(\text{TPE})}{\partial u} = 0 \end{equation}

Mathematical Background (Slides 0104)

See lecture slides, MMME1026 Notes, and lecture slides/notes on numerical methods.

Simple 1D Finite Elements (Slides 0105)

A Simple Uniaxial 1D Pin-Jointed Element

See lecture slides.