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MMME3086 Content

Direct and Energy based formulation of 1d elements (stiffness matrices)
Assembly of stiffness matrices to form the global stiffness matrix
2d pin jointed structures
continuum elements
structural elements
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

Discretise the domain
Write the element stiffness matrices
Assemble the global stiffness matrix
Apply boundary conditions
Solve matrix
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

I’m not sure entirly what the significance of this slide is.

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.

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