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book/week_14/session_3/intro.ipynb

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+{
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+    "```{index} Failure models; Class exercise\n",
+    "```\n",
+    "\n",
+    "(lesson14.3)=\n",
+    "# Lesson December 6th\n",
+    "\n",
+    "During today's lesson you'll work on a complex exercise on the topic of the Failure models. Please ask your questions regarding the [homework](homework14.3) as well!\n",
+    "\n",
+    "## Exercise Failure models\n",
+    "\n",
+    "Given is the following structure. It represents the Prinses Amaliabrug, a bicycle bridge near Dordrecht.\n",
+    "\n",
+    "```{figure} intro_data/structure.svg\n",
+    ":align: center\n",
+    "```\n",
+    "\n",
+    "1. Find the relevant cross-sectional properties.\n",
+    "2. Find the bending- en torsional moment distribution.\n",
+    "3. Find the shear- and normal forces in $\\text{E}$\n",
+    "4. Draw the stress state in point $\\text{E}$ in the principal stress plane including Tresca's failure criterium for unknown yield stress.\n",
+    "5. What should be the minimum yield stress according to Tresca?\n",
+    "\n",
+    "````{admonition} Solution assignment 1\n",
+    ":class: tip, dropdown\n",
+    "\n",
+    "$A_m = 3.23 \\text{ m}^2$\n",
+    "\n",
+    "````\n",
+    "\n",
+    "````{admonition} Solution assignment 3\n",
+    ":class: tip, dropdown\n",
+    "\n",
+    "- $\\sigma_\\text{E} = -0.0174 \\text{ MPa}$\n",
+    "- $\\tau_\\text{E} = 12.53 \\text{ MPa}\n",
+    "\n",
+    "````\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "1.190478974327377 1.1862348425115932 1.7825372573309306\n",
+      "0.14135562546019287 0.07870482085247055\n",
+      "0.22006044631266342\n",
+      "c_2= 0.2791116352472578\n",
+      "d= 0.8201010126776671\n",
+      "c_4= 0.7909893774304093\n",
+      "c_3= 0.5279906282646423\n",
+      "Iyy_totaal= 0.2381341578344103\n",
+      "Bv= 391.408\n",
+      "Mb= 1406.25\n",
+      "Mc= -7.829999999999927\n",
+      "782.816 811.456\n",
+      "Am= 3.237521601035994 , tau= 12.532055380577805\n",
+      "sigma= -0.017426731375872792\n",
+      "12.523345044031144 -12.540771775407018\n"
+     ]
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "t = 10e-3\n",
+    "r_outer = 2.8+t/2\n",
+    "r_inner =  2.80 - t/2\n",
+    "A_outer = np.pi*r_outer**2/2\n",
+    "A_inner = np.pi*r_inner**2/2\n",
+    "A_circle = A_outer-A_inner\n",
+    "c_outer = (4*r_outer/(3*np.pi))\n",
+    "c_inner = (4*r_inner/(3*np.pi))\n",
+    "c_1 = (A_outer * c_outer - A_inner * c_inner) / (A_outer - A_inner)\n",
+    "print(c_outer,c_inner,c_1)\n",
+    "\n",
+    "Ix_1 = np.pi*r_outer**4/16+A_outer*(c_outer-c_1)**2-(np.pi*r_inner**4/16+A_inner*(c_inner-c_1)**2)\n",
+    "Iy_1 = Ix_1\n",
+    "Ixy_1 = 1/8*r_outer**4+A_outer*(c_outer-c_1)**2-(1/8*r_inner**4+A_inner*(c_inner-c_1)**2)\n",
+    "print(Ix_1,Ixy_1)\n",
+    "\n",
+    "Iyy = (Ix_1+Iy_1)/2+(Ix_1-Iy_1)/2*np.cos(2*np.pi/4)+Ixy_1*np.sin(2*np.pi/4)\n",
+    "print(Iyy)\n",
+    "c_2 = 2.8 - c_1*np.sqrt(2)\n",
+    "print('c_2=',c_2)\n",
+    "d = 2.8-2.8/np.sqrt(2)\n",
+    "print('d=',d)\n",
+    "c_4 = 0.25+d-c_2\n",
+    "print('c_4=',c_4)\n",
+    "\n",
+    "A_rechthoek = 4*t\n",
+    "A_klein = 0.25*t*2\n",
+    "c_3 = (A_klein*0.125+A_circle*c_4)/(A_rechthoek+A_klein+A_circle)\n",
+    "print('c_3=',c_3)\n",
+    "\n",
+    "Iyy_totaal = 1/12*(4*t**3)+A_rechthoek*c_3**2+1/12*(2*t*0.25**3)+A_klein*(c_3-0.125)**2+Iyy+A_circle*(c_4-c_3)**2\n",
+    "print('Iyy_totaal=',Iyy_totaal)\n",
+    "\n",
+    "Bv = (12.5*40*20-14.32*15)/25\n",
+    "print('Bv=',Bv)\n",
+    "Mb = 12.5*15*7.5\n",
+    "print('Mb=',Mb)\n",
+    "Mc = 12.5*25*12.5-Bv*10\n",
+    "print('Mc=',Mc)\n",
+    "\n",
+    "Mtb = Bv*2\n",
+    "Mtc = Mtb+14.32*2\n",
+    "print(Mtb,Mtc)\n",
+    "\n",
+    "Am = np.pi*2.8**2 / 4 - 2.8*2.8*0.5+0.25*4\n",
+    "tau = Mtc / (2*Am*t*1000)\n",
+    "print('Am=',Am,', tau=',tau)\n",
+    "\n",
+    "sigma = Mc * (250+280)/Iyy_totaal / 1e6\n",
+    "print('sigma=',sigma)\n",
+    "\n",
+    "sigma1 = sigma/2+np.sqrt((sigma/2)**2+tau**2)\n",
+    "sigma2 = sigma/2-np.sqrt((sigma/2)**2+tau**2)\n",
+    "print(sigma1,sigma2)"
+   ]
+  }
+ ],
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