{
"cells": [
{
"cell_type": "markdown",
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"source": [
"# **Simple beam - Stress and Strain**\n",
"In this fourth notebook, the stresses and strains inside the element are studied. For any civil engineer, the deep understanding of the behaviour of stress and strain inside an element is essential. For this reason, after completing this DEMO, you'll be equipped with the preliminary and fundamental knowledge to be able to tackle the next important concepts and theories. \n",
"The system, geometry and actions are the same presented in the previous DEMO \"03-sb_diagrams\"."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## **Goals**\n",
"\n",
"* Describe the differences between the various strains and stresses\n",
"* Explain the relations between them\n",
"* Identify which parameters influence their behaviour"
]
},
{
"cell_type": "markdown",
"metadata": {
"tags": []
},
"source": [
"## **Tasks and exercises**\n",
"\n",
"Answer to the following questions by using and playing with the interactive visualisation tool (bottom of the notebook).\n",
"\n",
"1. What are the differences between the centroid and the neutral axis? Is there a way to superimpose them in this studied case?\n",
"\n",
" Solution:
\n",
" The centroid is dependent only on the geometry of the section; on the other hand, the neutral axis is function of the centroid and the internal forces.\n",
" \n",
" Yes, if the external horizontal force P is removed and the axial force N becomes zero, the neutral axis is in the same position of the centroid.\n",
"
\n",
"\n",
"2. Does the cross-section influence the maximal stresses and strains? By changing the length of the beam to 8 m, do you ascertain any changes?\n",
"\n",
" Solution:
\n",
" Yes; the internal forces do not change, but the resistance of the section (area and inertia) is directly influenced by h and b. Note that increasing h has a bigger impact only on the bending strain and bending stress (power of 2); on the other hand b and h have the same influence, because the axial stress and strain is dependent only on the area: $\\sigma_N = N/A$ and $\\epsilon_N = \\sigma_N/E$ and the shear stress is also only dependent on the area: $\\tau_{max} = \\frac{V S}{I_y b} = \\frac{V\\frac{bh^2}{8}}{\\frac{bh^3}{12}b} = \\frac{3V}{2bh} = \\frac{3V}{2A}$.\n",
" \n",
" Yes; the parameters responsible for the resistance of the section (area and inertia) do not change, but the amplitude of the shear and the bending moment increase, thus the stress and strain inside the section increase, except for the axial strain and stress, that are independent of the length of the beam and they remain the same.\n",
"
\n",
"\n",
"3. Consider the starting configuration; assume that the beam is composed of a material with a yield strength of 27.2 MPa.\n",
" 1. Will the beam remain elastic everywhere?\n",
" Solution:
\n",
" No, at the position where the bending moment is maximum, the stress in the top compressed fiber is equal to 27.5 MPa, that is bigger than 27.2 MPa.\n",
"
\n",
" \n",
" 2. Where will it yield?\n",
" Solution:
\n",
" At the center of the beam, where the bending moment is the highest.\n",
"
\n",
" \n",
" 3. If you can change only one parameter to assure that the beam remains always elastic, what will you do?\n",
"\n",
" Solution:
\n",
" The possible solution are:\n",
" \n",
"
\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## **Build the interactive visualisation tool**"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"
\\n\"+\n", " \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n", " \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n", " \"
\\n\"+\n", " \"\\n\"+\n",
" \"from bokeh.resources import INLINE\\n\"+\n",
" \"output_notebook(resources=INLINE)\\n\"+\n",
" \"
\\n\"+\n",
" \"\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"
\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"
\\n\"+\n \"Geometrical and mechanical parameters:
\\n h = 200 mmForces and Moments:
\\n P = 10 kNGeometrical and mechanical parameters:
\\n h = `+h+` mmForces and Moments:
\\n P = `+P+` kNGeometrical and mechanical parameters:
\\n h = `+h+` mmForces and Moments:
\\n P = `+P+` kNForces and Moments:
\\n P = `+P+` kN