From bec5809694bae9e16929104160d3e32908a5ef01 Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Tue, 31 Oct 2023 16:02:13 +0000 Subject: [PATCH] add notes, slides, lecture notes on fea --- .../3086_computer_modelling_techniques/fea.md | 193 ++++++++++++++++++ ...ot-2023-10-31T15:36:02,741218558+00:00.png | Bin 0 -> 43222 bytes .../lecture_slides/fea/FE_0101_Intro.pdf | Bin 0 -> 2015822 bytes .../fea/FE_0102_StressAnalysis.pdf | Bin 0 -> 530864 bytes .../fea/FE_0103_EnergyApproaches.pdf | Bin 0 -> 194771 bytes .../fea/FE_0104_Mathematical.pdf | Bin 0 -> 483432 bytes .../lecture_slides/fea/FE_0105_1DFE.pdf | Bin 0 -> 1007765 bytes .../fea/Notes_FE-01-Stress Analysis.pdf | Bin 0 -> 350310 bytes .../fea/Notes_FE-02-Simple 1D Elements.pdf | Bin 0 -> 233047 bytes 9 files changed, 193 insertions(+) create mode 100755 uni/mmme/3086_computer_modelling_techniques/fea.md create mode 100644 uni/mmme/3086_computer_modelling_techniques/images/vimscrot-2023-10-31T15:36:02,741218558+00:00.png create mode 100644 uni/mmme/3086_computer_modelling_techniques/lecture_slides/fea/FE_0101_Intro.pdf create mode 100644 uni/mmme/3086_computer_modelling_techniques/lecture_slides/fea/FE_0102_StressAnalysis.pdf create mode 100644 uni/mmme/3086_computer_modelling_techniques/lecture_slides/fea/FE_0103_EnergyApproaches.pdf create mode 100644 uni/mmme/3086_computer_modelling_techniques/lecture_slides/fea/FE_0104_Mathematical.pdf create mode 100644 uni/mmme/3086_computer_modelling_techniques/lecture_slides/fea/FE_0105_1DFE.pdf create mode 100644 uni/mmme/3086_computer_modelling_techniques/lectures_notes/fea/Notes_FE-01-Stress Analysis.pdf create mode 100644 uni/mmme/3086_computer_modelling_techniques/lectures_notes/fea/Notes_FE-02-Simple 1D Elements.pdf diff --git a/uni/mmme/3086_computer_modelling_techniques/fea.md b/uni/mmme/3086_computer_modelling_techniques/fea.md new file mode 100755 index 0000000..65bd3a8 --- /dev/null +++ b/uni/mmme/3086_computer_modelling_techniques/fea.md @@ -0,0 +1,193 @@ +--- +author: Akbar Rahman +date: \today +title: +tags: [] +uuid: 8f32c8c0-5ed4-49ab-aca8-e5e058262580 +lecture_slides: [ ./lecture_slides/fea ] +lecture_notes: [ ./lecture_notes/fea ] +exercise_sheets: [] +--- + +# MMME3086 Content + +1. Direct and Energy based formulation of 1d elements (*stiffness matrices*) +1. Assembly of stiffness matrices to form the *global stiffness matrix* +1. 2d pin jointed structures +1. continuum elements +1. structural elements +1. 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 +1. Write the element stiffness matrices +1. Assemble the global stiffness matrix +1. Apply boundary conditions +1. Solve matrix +1. 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.](./images/vimscrot-2023-10-31T15:36:02,741218558+00:00.png) + +- 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](https://notes.alv.cx/permalink?uuid=16edb140-9946-4759-93df-50cad510fe31), +and [lecture slides](./lecture_slides/numerical_methods)/[notes](./lectures_notes/numerical_methods) on numerical methods. + +# Simple 1D Finite Elements (Slides 0105) + +## A Simple Uniaxial 1D Pin-Jointed Element + +See lecture slides. diff --git a/uni/mmme/3086_computer_modelling_techniques/images/vimscrot-2023-10-31T15:36:02,741218558+00:00.png b/uni/mmme/3086_computer_modelling_techniques/images/vimscrot-2023-10-31T15:36:02,741218558+00:00.png new file mode 100644 index 0000000000000000000000000000000000000000..6c67ba85ed04107eff4e1b46c1da864297656e40 GIT binary patch literal 43222 zcmeFZ^Lt%U+dr5(Zqyi!+t@~9G-zzwwwgA!jmEa!*hypCw)I`-dER%fnQQ)pnf%b^ zoGUqd?}hvRXeIE6ln4SWHY^kr6oQzjpez*B8xJU`SGLga!8?2cRYKsuw|4wu^3dSR z9a=vCyvDK@Qnr_~GPHNnu{D4)vb3@=ptjSqH88NWGq$ondDF%N1@#$9Ops6BIpuK4 z$yr|E{_ojI%;mD1^iQaan`OS7FRwELS}KD|mzJic92^|H4(1(7wBI4;h+1WQfF>a% z@rm#OKb?N}gv@{v_3dXOiOPd9Wrg}J-NI4 zP(HhKV)u~QCG@H1k3-KzLyLX&Lse7A(i6tP0YL*d7Re$yTlzoK96)J5F`0qOgWh}I z=04Im>~CXJO?obm@^!C411|}wq%TD7KNqvn7&B%f$=O75Ba5vbazBR^FDN!u_4+$B ziTT0Jta%Y{Z($*M**guhK*2Sfuy5rzvsN4Y&z2I>yLUNk?DM-V_=J~yQ9Q4H%FZ#- zXk<;<+mQTcommVHeAON9dE>fL=W3*Zw{SxD=c7C6m^3 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