194 lines
6.8 KiB
Markdown
Executable File
194 lines
6.8 KiB
Markdown
Executable File
---
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author: Akbar Rahman
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date: \today
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title:
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tags: []
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uuid: 8f32c8c0-5ed4-49ab-aca8-e5e058262580
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lecture_slides: [ ./lecture_slides/fea ]
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lecture_notes: [ ./lecture_notes/fea ]
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exercise_sheets: []
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---
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# MMME3086 Content
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1. Direct and Energy based formulation of 1d elements (*stiffness matrices*)
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1. Assembly of stiffness matrices to form the *global stiffness matrix*
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1. 2d pin jointed structures
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1. continuum elements
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1. structural elements
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1. practical FEA guidelines
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## Coursework
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- Worth 35% of module
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- Will be set on 2023-11-09
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- Will be due on 2023-11-23
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# Background (Slides 0101)
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- domain is discretised into *finite elements*
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- each element is defined by its corners (*nodes*)
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- typical shapes for elements are triangular/quadrilaterals in 2d problems or
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tetrahedral/hexahedral in 3d problems
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- for each element, the behaviour is described by the displacements of the nodes and material law
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(stress strain relationships)
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- this is usually expressed as the *stiffness* of the element
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- elements are assembled in a mesh and the requirements of continuity and equilibrium between
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neighbouring elements are satisfied
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- this assembly process results in a large system of simultaneous equations
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- boundary conditions are applied to assembly of elements, to yield a unique solution to the
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overall system
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- solution matrices are sparsely populated
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- equations are solved numerically to compute the displacements at each node
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- the displacements of each node can be used to obtain the stresses in each element
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- finite element method is suitable for practical engineering stress analysis of complex geometries
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- to obtain good accuracy in regions of rapidly changing variables, a large number of small (*fine*)
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elements must be used
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## Basic Overview of the Steps Required for FEA
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1. Discretise the domain
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1. Write the element stiffness matrices
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1. Assemble the global stiffness matrix
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1. Apply boundary conditions
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1. Solve matrix
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1. Post-processing---e.g. obtaining additional information like reaction forces and element stresses
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# Stress Analysis Fundamentals (Slides 0102)
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## Uniaxial Loading
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- The engineering definitions for engineering stress and strain assumes that stress is uniform,
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but this is rarely true over large areas.
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- However the definition gets more useful for small elements:
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\begin{equation}
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\sigma = \lim_{\delta A \rightarrow 0} \sigma_\text{engineering}
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\end{equation}
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- For uniaxial loading situations, *engineering (nominal) strain* is:
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\begin{equation}
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\varepsilon_\text{engineering} = \frac{\Delta L}{L_0}
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\end{equation}
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- For uniaxial loading situations, *engineering (nominal) stress* is:
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\begin{equation}
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\sigma_\text{engineering} = \frac{F}{A_0}A
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\end{equation}
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## Multi-Axial (3D) Stress and Strain Definitions
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- In Cartesian axes system, are six components of stresses on the elements:
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- *direct stresses* $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$ (tensile/compressive stresses)
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caused by forces normal to the area
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- *shear stresses* $\sigma_{xy}$, $\sigma_{xz}$, $\sigma_{yz}$ caused by shear forces parallel to
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the area
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- the first subscript refers to the direction of the outward normal to the plane the stress is
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acting on
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- the second subscript refers to the direction of the stress
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- The stress and strain vectors can be expressed as the following:
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\begin{equation}
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\pmb \sigma = \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{xy} \\ \sigma_{xz} \\ \sigma_{yz} \end{bmatrix}
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\pmb \varepsilon = \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\ \varepsilon_{xy} \\ \varepsilon_{xz} \\ \varepsilon_{yz} \end{bmatrix}
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\end{equation}
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where:
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\begin{align*}
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\sigma_{xx} &= \frac{\partial u_x}{\partial x} \\
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\sigma_{xx} &= \frac{\partial u_x}{\partial x} \\
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\sigma_{xx} &= \frac{\partial u_x}{\partial x} \\
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\varepsilon_{xy} &= \frac12 \left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right) \\
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\varepsilon_{xz} &= \frac12 \left(\frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x}\right) \\
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\varepsilon_{yz} &= \frac12 \left(\frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y}\right)
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\end{align*}
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- In computation mechanics, the stress and strain relation is modelled as:
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\begin{equation}
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\pmb \sigma = \pmb d \pmb \varepsilon
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\end{equation}
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### Stress-Strain Relationship (Hooke's Law)
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- Hooke's law can be used to create the following stress-strain relations for isotropic linear
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elastic materials with thermal strain:
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\begin{align}
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\varepsilon_{xx} &= \frac 1 E \left( \sigma_{xx} - \nu\left(\sigma_{yy} + \sigma_{zz}\right)\right) + \alpha\Delta T \\
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\varepsilon_{yy} &= \frac 1 E \left( \sigma_{yy} - \nu\left(\sigma_{xx} + \sigma_{zz}\right)\right) + \alpha\Delta T \\
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\varepsilon_{zz} &= \frac 1 E \left( \sigma_{zz} - \nu\left(\sigma_{yy} + \sigma_{xx}\right)\right) + \alpha\Delta T \\
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\varepsilon_{xy} &= \frac{\sigma_{xy}}{2\mu} \\
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\varepsilon_{xz} &= \frac{\sigma_{xz}}{2\mu} \\
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\varepsilon_{yz} &= \frac{\sigma_{yz}}{2\mu} \\
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\mu &= \frac{E}{2(1+\nu)}
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\end{align}
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where $E$ is Young's modulus, $\nu$ is Poisson's ratio, $\alpha$ is the coefficient of thermal expansion,
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$\Delta T$ is the temperature change, and $\mu$ is the shear modulus
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# Energy Methods (Slides 0103)
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## Stability
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- Strain energy is released upon the removal of applied loads and the body returns to undeformed state:
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\begin{equation}
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U = \frac12 \pmb \sigma \pmb \sigma \times V
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\end{equation}
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- If the material behaviour is non-linear, it can be generalised to:
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\begin{equation}
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U = \int_\nu\int_\varepsilon \sigma \mathrm{d}\varepsilon \mathrm{d}V
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\end{equation}
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- Work done by external forces can be expressed as:
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\begin{equation}
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W = \sum_i F_iu_i
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\end{equation}
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where $i$ is any point where force $F_i$ causes displacement $u_i$
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- The *total potential energy* can be expressed as:
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\begin{equation}
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\text{TPE} = U - W
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\end{equation}
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- The principle of minimum TPE states that when the body is in equilibrium, TPE must be 'stationary'
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with respect to the variables of the problem
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- The equilibrium is **stable if the TPE is minimum**
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- In most FE problems, the displacement $u$ is chosen as the unknown variables of the problem:
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\begin{equation}
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\frac{\partial(\text{TPE})}{\partial u} = 0
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\end{equation}
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# Mathematical Background (Slides 0104)
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See lecture slides,
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[MMME1026 Notes](https://notes.alv.cx/permalink?uuid=16edb140-9946-4759-93df-50cad510fe31),
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and [lecture slides](./lecture_slides/numerical_methods)/[notes](./lectures_notes/numerical_methods) on numerical methods.
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# Simple 1D Finite Elements (Slides 0105)
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## A Simple Uniaxial 1D Pin-Jointed Element
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See lecture slides.
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