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BrianChevalier/Application Problem 4.ipynb

Created March 21, 2017 06:44
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Application Problem 4.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Theory and Background"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem Statement"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The vertical stress under the corner of a rectangular footing subjected to a uniform load of intensity q is given by the solution of Boussinesq’s equation,\n",
"\n",
"\\begin{equation}\n",
"\\sigma = \\frac{q}{4\\pi}\\left[ \n",
"\\frac{2mn\\sqrt{m^2+n^2+1}m^2+n^2+2}{m^2+n^2+1+m^2n^2+1}\n",
"+\\sin^{-1}\\left( \\frac{2mn\\sqrt{m^2+n^2+1}}{m^2+n^2+1+m^2n^2}\\right)\n",
"\\right]\n",
"\\label{Equation}\n",
"\\end{equation}\n",
"\n",
"\n",
"Where m and n are dimensionless ratios with $m=a/z$ and $n=b/z$ , where $a=4.6$ and $b = 14$ are the dimensions of the footing and z is the depth below the footing. If $a = 4.6$ and $b = 14$, use a third-order interpolating polynomial to compute the stress, $\\sigma$, at a depth $z = 10m$ below the corner of a footing that subjected to a total load of 100 tones. Note that you will need to use Boussinesq’s equation to generate enough data points over different depths to create an interpolated polynomial."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Solution Strategy\n",
"To solve this problem, Newton's Divided Difference Interpolation is used. Interpolation creates a continuous function, $f(x)$, of order $n$, that passes through $n+1$ points."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To create a general equation to find the produce an interpolation we will look at the formula for a second order interpolation which is shown in the following equation:\n",
"\n",
"\\begin{align}\n",
"f_2(x)=b_0+b_1(x-x_0)+b_2(x-x_0)(x-x_1)\n",
"\\end{align}\n",
"\n",
"where:\n",
"\n",
"\\begin{equation}\n",
"b_0=f(x_0)\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"b_1= \\frac{f(x_1)-f(x_0)}{x_1-x_0}\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"b_2 = \\frac{\\frac{f(x_2)-f(x_1)}{x_2-x_1}-\\frac{f(x_1)-f(x_0)}{x_1-x_0}}{x_2-x_0}\n",
"\\end{equation}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There is a pattern to this polynomial which is given by:\n",
"\n",
"\\begin{align}\n",
"f_n(x)=b_0+b_1(x-x_0)+...+b_n(x-x_0)(x-x_1)...(x-x_{n-1})\n",
"\\end{align}\n",
"\n",
"where:\n",
"\n",
"\\begin{align}\n",
"b_0&=f[x_0]\\\\\n",
"b_1&=f[x_1, x_0]\\\\\n",
"b_2&=f[x_2, x_1, x_0]\\\\\n",
"\\vdots\\\\\n",
"b_{n-1}&=f[x_{n-1},x_{n-2},...,x_0]\\\\\n",
"b_n&=f[x_n,x_{n-1},...,x_0]\n",
"\\end{align}\n",
"\n",
"and finally the $n+1$ divided difference is given by:\n",
"\n",
"\\begin{align}\n",
"b_{n+1}&=f[x_{n+1},...,x_0]\\\\\n",
"&=\\frac{f[x_{n+1},...,x_1]-f[x_{n},...,x_0]}\n",
"{x_{n+1}-x_0}\n",
"\\end{align}\n",
"\n",
"This same process is followed to solve this problem."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Problem Solving Process"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The first step to solve this problem is to generate data so that a cubic interpolation function can be produced. A MATLAB function is defined to take in some inputs including a, b, z, and q. The code is as follows:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"``` matlab\n",
"function [sigma] = func(a, b, z, q)\n",
" \n",
" %define m and n\n",
" m = a./z;\n",
" n = b./z;\n",
" \n",
" %define some variables to break up the equation\n",
" var1 = q/(4*pi);\n",
" var2 = 2.*m.*n.*sqrt(m.^2+m.^2+1).*m.^2+n.^2+2;\n",
" var3 = m.^2+n.^2+1+m.^2.*n.^2.*m.^2+n.^2+1;\n",
" var4 = asin((2.*m.*n.*sqrt(m.^2+n.^2+1))./...\n",
" (m.^2+n.^2+1+m.^2.*n.^2));\n",
" \n",
" %compute sigma based on Boussinesq’s equation\n",
" sigma = var1.*(var2./var3+var4);\n",
" \n",
"end %function\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, the constants must be defined. The variables a, and b are given in the problem statement. The variable q is given in metric tons which is a mass and not a force so it must be converted:\n",
"\n",
"\\begin{align}\n",
"q&=\\frac{F}{A}\\\\\n",
"&=\\frac{(100Mg)(9.81m/s^2)}{(4.61m)(14m)}\\\\\n",
"&=\\frac{981kN}{64.4 m^2}\\\\\n",
"&=15.2329kPa\n",
"\\end{align}\n",
"\n",
"The code to initialize these variables is as follows:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"a = 4.6;\n",
"b = 14;\n",
"q = 981/(a*b);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next the function is plotted to visualize the the function. Note that that the variable z in the scatter command has a mius sign in front. This is to show that z is a depth into the ground."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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ZeeU788qdTxhKOVYqhCCToDouarij8ePHf/XVV6dPnxZCVFVVHTlyJCIiQgixZ8+egoIC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\nkAKBBACQAoEEAJDC/wNjBNWM1EnbSAAAAABJRU5ErkJggg==\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"z = linspace(8, 12, 8); %Set input array for function file\n",
"\n",
"fig1=figure(1); clf; grid on; axis square; hold on;\n",
"p = scatter(func(a, b, z, q), -z); hold on;\n",
"xlabel('\\sigma(z)'); ylabel('z'); title('z vs. \\sigma(z)');\n",
"legend('\\sigma(z)');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This plots shows the section of the function where the interpolation will occur. Since a third order interpolation function must be used four points must be selected. The points *closest* to the point $z=10$ must be selected."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"z = z(1,3:6); %redefine z to be vals. closest to z=10\n",
"T = func(a, b, z, q); %find the corresponding outputs"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Therefore the following data will be used:\n",
"\n",
"z = [9.1429, 9.7143, 10.2857, 10.8571]\n",
"\n",
"T = [1.7597, 1.7023, 1.6522, 1.6084]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Calculate $b_0$:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Therefore b0 is 1.75969\n"
]
}
],
"source": [
"b0 = T(1);\n",
"fprintf('Therefore b0 is %.5f\\n', b0);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Calculate $b_1$:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Therefore b1 is -0.10041\n"
]
}
],
"source": [
"b1 = (T(2)-T(1))/(z(2)-z(1));\n",
"fprintf('Therefore b1 is %.5f\\n', b1);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Calculate $b_2$:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false,
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Therefore b2 is 0.01116\n"
]
}
],
"source": [
"%T[z2, z1]\n",
"var1 = (T(3)-T(2))/(z(3)-z(2));\n",
"\n",
"%T[z1, z0]\n",
"var2 = (T(2)-T(1))/(z(2)-z(1));\n",
"\n",
"b2 = (var1-var2)/(z(3)-z(1));\n",
"fprintf('Therefore b2 is %.5f\\n', b2);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Calculate $b_3$:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Therefore b3 is -0.00094\n"
]
}
],
"source": [
"%T[z3 ,z2, z1]\n",
"var1 = (T(4)-T(3))/(z(4)-z(3));\n",
"var2 = (T(3)-T(2))/(z(3)-z(2));\n",
"\n",
"temp1 = (var1-var2)/(z(4)-z(2));\n",
"\n",
"%T[z2 ,z1, z0]\n",
"var1 = (T(3)-T(2))/(z(3)-z(2));\n",
"var2 = (T(2)-T(1))/(z(2)-z(1));\n",
"\n",
"temp2 = (var1-var2)/(z(3)-z(1));\n",
"\n",
"b3 = (temp1-temp2)/(z(4)-z(1));\n",
"fprintf('Therefore b3 is %.5f\\n', b3);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, plug in the solved values into the interpolation function:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{align*}\n",
"f_n(x)&=b_0+b_1(x-x_0)+...+b_n(x-x_0)(x-x_1)...(x-x_{n-1})\\\\\n",
"f_3(x)&=b_0 + b_1(x-x_0) + b_2(x-x_0)(x-x_1) + b_3(x-x_0)(x-x_1)(x-x_2)\\\\\n",
"f_3(x)&=1.75969 - 0.10041(x-9.1429) \\\\\n",
" & +0.01116(x-9.1429)(x-9.7143)\\\\\n",
" & - 0.00094(x-9.1429)(x-9.7143)(x-10.2857)\n",
"\\end{align*}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This can be implemented in a MATLAB anonymous function to be plotted and to solve the problem."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"f_3 = @(x) 1.75969 - 0.10041.*(x-9.1429) + ...\n",
"0.01116.*(x-9.1429).*(x-9.7143)- ...\n",
"0.00094.*(x-9.1429).*(x-9.7143).*(x-10.2857);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Results"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem Solution\n",
"To find the stress at depth $z = 10$ the interpolation function is evaluated at this point with the following:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Therefore the stress at z = 10 is 1.67643\n"
]
}
],
"source": [
"stress = f_3(10);\n",
"fprintf('Therefore the stress at z = 10 is %.5f\\n', stress);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, the interpolation function can be plotted on top of the data to visualize the interpolation and to verify that it is accurate. "
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false,
"scrolled": false
},
"outputs": [
{
"data": {
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4cXExtZGQkJCQkMBQaQBAH+0SDNZj+7u+KgwCAqlvumkEABZF1Sip37fB2mkiLhrRAIEE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\ncTG1MWPGjDNnzshkMnt7e90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HMACZTDZixAhra+sux4uLi99+++0LFy7Y2Nj0+Ky5c+f+\n+9//njRpEi1lArAaTtkBGMCoUaO6pxEhxN/ff+LEiVlZWT0+6+jRo8HBwUgjAAoCCcC44uPjv/zy\ny+7HOzo6cnJytPO/AQCn7AAAgBUwQgIAAFZAIAEAACsgkAAAgBUQSAAAwAr/H4uFbSxO0TybAAAA\nAElFTkSuQmCC\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig1 = figure(1); clf; grid on; axis square; hold on;\n",
"p = scatter(func(a, b, z, q), -z);\n",
"xlabel('\\sigma(z)'); ylabel('z'); title('z vs. \\sigma(z)');\n",
"legend('\\sigma(z)');\n",
"\n",
"x = linspace(z(1), z(4), 10); %Set input array for function file\n",
"p1 = plot(f_3(x), -x);"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## Discussion\n",
"Most of the MATLAB work in this problem is hard coded. This script would be more useful if the code is taken and generalized. Newton's divided difference method is recursive which mean that the computations from the previous step are used in the next step. This fact could be taken advantage of to create a recursive script that could generate an interpolation function given the order of the function. This nice thing about this script, however, is that any function can be inputed for the `func`. The code would interpolate a third order function as long as the appropriate data is user selected."
]
}
],
"metadata": {
"celltoolbar": "Hide code",
"kernelspec": {
"display_name": "Matlab",
"language": "matlab",
"name": "matlab"
},
"language_info": {
"codemirror_mode": "octave",
"file_extension": ".m",
"help_links": [
{
"text": "MetaKernel Magics",
"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
}
],
"mimetype": "text/x-matlab",
"name": "matlab",
"version": "0.14.2"
},
"latex_metadata": {
"author": "Julius C. F. Schulz",
"title": "Amazing Article"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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