{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "$\\newcommand{\\dS}{\\,\\text{dS}}\n", "\\newcommand{\\dx}{\\,\\text{dx}}\n", "\\newcommand{\\bsig}{\\boldsymbol{\\sigma}}$\n", "\n", "# Computing consistent reaction forces\n", "\n", "One often needs to compute a resulting reaction force on some part of the boundary as post-processing of a mechanical resolution.\n", "\n", "Quite often, such a reaction will be computed using the stress field associated with the computed displacement. We will see that this may lead to slight inaccuracies whereas another more consistent approach using the virtual work principle is more consistent.\n", "\n", "\n", "## A cantilever beam problem\n", "\n", "We reuse here a simple 2D small strain elasticity script of a rectangular domain of dimensions $L\\times H$ representing a cantilever beam clamped on the left hand-side and under uniform body forces $\\boldsymbol{f}=(f_x, f_y)$. $P^2$ Lagrange elements are used for the displacement discretization. \n", "\n", "For the sake of illustration, we are interested in computing the horizontal and vertical reaction forces $R_x$ and $R_y$ on the left boundary as well as the resulting moment $M_z$ around the out-of-plane direction. In the present simple case using global balance equations, they are all given explicitly by:\n", "$$\n", "\\begin{align*}\n", "R_x &= \\int_{x=0} \\boldsymbol{T}\\cdot \\boldsymbol{e}_x \\dS = \\int_{x=0} (-\\sigma_{xx}) dS = -f_x \\cdot L \\cdot H \\\\\n", "R_y &= \\int_{x=0} \\boldsymbol{T}\\cdot \\boldsymbol{e}_y \\dS = \\int_{x=0} (-\\sigma_{xy}) dS = -f_y \\cdot L \\cdot H\\\\\n", "M_z &= \\int_{x=0} (\\vec{\\boldsymbol{OM}}\\times \\boldsymbol{T})\\cdot \\boldsymbol{e}_z \\dS = \\int_{x=0} (y \\sigma_{xx}) dS = f_y \\cdot \\dfrac{L^2}{2} \\cdot H\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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