Update beams on supports

book/design_workflow/general_mechanics/beams_with_multiple_supports.ipynb

@@ -12,7 +12,97 @@
   {
    "cell_type": "markdown",
    "metadata": {},
-   "source": []
+   "source": [
+    "## Beams on multiple supports (concentrated loads)\n",
+    "```{figure} Images/Beams on multiple supports.png\n",
+    "---\n",
+    "scale: 100%\n",
+    "---\n",
+    "\n",
+    "| Type | M(1)  | M2     | M(2)   | M3     | M(3)   | R1     | R2     | R3     | R4     |\n",
+    "|------|--------|--------|--------|--------|--------|--------|--------|--------|--------|\n",
+    "| a    | 0.1562 | -0.1875| 0.1562 | || 0.3125 | 1.3750 | 0.3125 |        |        |        |\n",
+    "| b    | 0.2031 | -0.0938| -0.0469||| 0.4062 | 0.6876 | -0.0938|        |        |        |\n",
+    "| c    | 0.1750 | -0.1500| 0.1000 | -0.1500| 0.1750 | 0.3500 | 1.1500 | 1.1500 | 0.3500 |\n",
+    "| d    | 0.2125 | -0.0750| -0.0750| -0.0750| 0.2125 | 0.4250 | 0.5750 | 0.5750 | 0.4250 |\n",
+    "| e    | -0.0375| -0.0750| 0.1750 | -0.0750| -0.0375| -0.0750| 0.5750 | 0.5750 | -0.0750|\n",
+    "| f    | 0.1625 | -0.0175| 0.1375 | -0.0500| -0.0250| 0.3250 | 1.3000 | 0.4250 | -0.0500|\n",
+    "| g    | 0.2000 | -0.1000| -0.0375| 0.0250 | 0.0125 | 0.4000 | 0.7250 | -0.1500| 0.0250 |\n",
+    "\n",
+    "$M = factor * F * L$  \n",
+    "$R = factor *x* F$\n",
+    "\n",
+    "The numbers in the tables are related to the concentrated loads working on the structure. The bending moments and support reactions in the designated points are found by multiplying the factors in the table with F x L or F.\n",
+    "\n",
+    "For example:\n",
+    "- M(1) is the maximum bending in beam (1).\n",
+    "- M3 is the hogging moment located at support 3.\n",
+    "- R2 is the support reaction in support 2.\n",
+    "\n",
+    "For situation c:\n",
+    "- F = 30 kN\n",
+    "- L = 8 m\n",
+    "\n",
+    "The calculations are as follows:\n",
+    "- M(1) = 0.175 x 30 x 8 kNm\n",
+    "- M3 = -0.15 x 30 x 8 kNm\n",
+    "- R2 = 1.15 x 30 kN\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Beams on multiple supports (distributed load)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "```{figure} Images/Beams on multiple supports 1.png\n",
+    "---\n",
+    "scale: 100%\n",
+    "---\n",
+    "\n",
+    "| Type | M(1) location rel. to 1* | M2     | M(2) location rel. to 2** | M3     | M(3) | R1     | R2     | R3     | R4     |\n",
+    "|------|---------------------------|--------|----------------------------|--------|------|--------|--------|--------|--------|\n",
+    "| h    | 0.0703                    | 0.3750 | -0.1250                    | 0.0703 | 0.6250||| 0.3750 | 1.2500 | 0.3750 |        |\n",
+    "| i    | 0.0957                    | 0.4375 | -0.0625                    | -0.0313| ||      | 0.4375 | 0.6250 | -0.0625|        |\n",
+    "| j    | 0.0800                    | 0.4000 | -0.1000                    | 0.0250 | 0.5000| -0.1000| 0.0800 | 0.4000 | 1.1000 | 1.1000 | 0.4000 |\n",
+    "| k    | 0.1013                    | 0.4500 | -0.0500                    | -0.0500|       | -0.0500| 0.1013 | 0.4500 | 0.5500 | 0.5500 | 0.4500 |\n",
+    "| l    | -0.025                    |        | -0.0500                    | 0.0750 | 0.5000| -0.0500|        | -0.0500| 0.5500 | 0.5500 | -0.0500|\n",
+    "| m    | 0.0735                    | 0.3833 | -0.1167                    | 0.0535 | 0.5830| -0.0333|        | 0.3833 | 1.2001 | 0.4499 | -0.0333|\n",
+    "| n    | 0.0939                    | 0.4333 | -0.0667                    | -0.0250|       | 0.0167 |        | 0.4333 | 0.6501 | -0.1001| 0.0167 |\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$M = factor * q * L^2$  \n",
+    "$R = factor * q * L$\n",
+    "\n",
+    "The numbers in the tables are related to the concentrated loads\n",
+    "working on the structure. The bending moments and support reactions in\n",
+    "the designated points are found by multiplying the factors in the table with\n",
+    "$q * L^2$ or $q * L$.\n",
+    "\n",
+    "For example: M(1) is the maximum bending in beam\n",
+    "(1). M3 is the hogging moment located at\n",
+    "support 3. R2 is the support reaction in support 2.  \n",
+    "  \n",
+    "For situation m:  \n",
+    "$F = 5 kN/m$\n",
+    "$L = 6 m$  \n",
+    "$M(1) = 0,0735 * 5 * 62 kNm$  \n",
+    "$M3 = -0,0333 * 5 * 62 kNm$  \n",
+    "$R2 = 1,2001 * 5 * 6 kN$  \n",
+    "  \n",
+    "*) location of bending moment M(1) in relation to support 1 (distance = factor x L)  \n",
+    "**) location of bending moment M(2) in relation to support 2 (distance = factor\n",
+    "x L)  "
+   ]
   }
  ],
  "metadata": {

book/design_workflow/general_mechanics/forget_me_nots.ipynb

@@ -10,7 +10,20 @@
   {
    "cell_type": "markdown",
    "metadata": {},
-   "source": []
+   "source": [
+    "```{figure} Images/Forgetmenots 1.jpg\n",
+    "---\n",
+    "scale: 100%\n",
+    "---\n",
+    "```{figure} Images/Forgetmenots 2.jpg\n",
+    "---\n",
+    "scale: 100%\n",
+    "---\n",
+    "```{figure} Images/Forgetmenots 3.jpg\n",
+    "---\n",
+    "scale: 100%\n",
+    "---"
+   ]
   }
  ],
  "metadata": {

book/design_workflow/general_mechanics/mechanic_formulas.ipynb

@@ -12,7 +12,26 @@
   {
    "cell_type": "markdown",
    "metadata": {},
-   "source": []
+   "source": [
+    "| | Kinematic relations | Constitutive relations | Static relations | Differential equation |\n",
+    "|--------|-------------------------------------|--------------------------------------|-----------------------------------|---------------------------------|\n",
+    "| Extension | $\\epsilon = \\frac{du}{dx}$ | $ N = EA \\epsilon $ | $\\frac{dN}{dx} + q_x = 0$ | $EA \\frac{d^2u}{dx^2} + q_x = 0$ |\n",
+    "| Bending | $\\phi = -\\frac{dw}{dx}$;   $\\kappa = \\frac{d\\phi}{dx}$;   $\\kappa=-\\frac{d^2w}{dx^2}$ | $M=EI\\kappa$|$\\frac{dV}{dx}+q_z=0$; $\\frac{dM}{dx}-v=0$; $\\frac{d^2M}{dx^2}+q_z=0$ | $-EI\\frac{d^4w}{dx^4}+q_z=0$|\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\epsilon (z) = \\epsilon + \\kappa_z z$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$\\sigma(z)=\\sigma^N+\\sigma^N=\\frac{N}{A}+\\frac{M_z z}{I_zz}$"
+   ]
   }
  ],
  "metadata": {