elastc model modified

bmcs_course/1_1_elastic_stiffness_of_the_composite.ipynb

@@ -12,8 +12,10 @@
     "# Example: Calculate elastic stiffness of a given composite\n",
     "\n",
     "## Task\n",
-    "What is the initial stiffness of a reinforced concrete cross section shown in the Figure with the thickness of 10~mm and width of 100 mm.\n",
-    "The cross-section is reinforced with 6 layers of textile fabrics made of CAR-EP3300 specified in the table below ![image.png](../fig/mixture_rule_elastic.png)"
+    "Predict the tensile stiffness of a reinforced concrete cross section shown in the Figure with the thickness of 10~mm and width of 100 mm.\n",
+    "The cross-section is reinforced with 6 layers of textile fabrics made of CAR-EP3300 specified in the table below\n",
+    "\n",
+    "![image.png](../fig/mixture_rule_elastic.png)"
    ]
   },
   {
@@ -81,9 +83,23 @@
     "\\end{array}"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# How to evaluate an expression?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Calculator:** Let us use Python language as a calculator and evalute the mixture rule for the exemplified cross-section.m"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -98,19 +114,134 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 16,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "37082.18859138533"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "A_composite = width * thickness\n",
     "n_rovings = width / spacing\n",
     "A_layer = n_rovings * A_roving\n",
     "A_carbon = n_layers * A_layer \n",
     "A_concrete = A_composite - A_carbon \n",
-    "E_composite = (E_carbon * A_carbon + E_concrete +A_concrete) / A_composite\n",
+    "E_composite = (E_carbon * A_carbon + E_concrete * A_concrete) / A_composite\n",
     "E_composite"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Thus, the composite has an effective stiffness of 37 GPa."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# How to construct a model?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "To explore the multitude of compbinations let us provide a model which can be interactively used to study the available design options"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In contrast to the previously performed numerical evaluation, we now express the derived equations as mathematical symbols. To do this, let us use a Python package `sympy` to do symbolic algebra."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sympy as sp"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The parameters of the model are now introduced as `sympy.symbols`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A_roving = sp.Symbol('A_r')\n",
+    "n_layers = sp.Symbol('n_l')\n",
+    "spacing = sp.Symbol('d')\n",
+    "thickness = sp.Symbol('h')\n",
+    "width = sp.Symbol('b')\n",
+    "E_carbon = sp.Symbol('E_car')\n",
+    "E_concrete = sp.Symbol('E_c')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The above derived equations can be rephrased in the form"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\frac{A_{r} E_{car} n_{l} - E_{c} \\left(A_{r} n_{l} - d h\\right)}{d h}$"
+      ],
+      "text/plain": [
+       "(A_r*E_car*n_l - E_c*(A_r*n_l - d*h))/(d*h)"
+      ]
+     },
+     "execution_count": 53,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A_composite = width * thickness\n",
+    "n_rovings = width / spacing\n",
+    "A_layer = n_rovings * A_roving\n",
+    "A_carbon = n_layers * A_layer \n",
+    "A_concrete = A_composite - A_carbon \n",
+    "E_composite = (E_carbon * A_carbon + E_concrete * A_concrete) / A_composite\n",
+    "sp.simplify(E_composite)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**The first model:**\n",
+    "Instead of a number, we have now a symbolic expression showing the influence of the individual parameters on a design characteristic, i.e. on the material stiffness. This expression is a model that was constructuted using the conditions of compatibility, equilibrium and constitutive laws.\n",
+    "Using this model, we can explore the behavior of the composite."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -122,26 +253,23 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    " - Install anaconda environment\n",
-    " - Download this notebook from the moodle folder Jupyter notebooks\n",
-    " - Start jupyter\n",
-    " - Open the notebook\n",
-    " - Learn the basics of how to interact with Jupyter notebook"
+    " - Login to jupyter.rwth-aachen.de\n",
+    " - Navigate to this mixture rule example\n",
+    " - Evaluate the cells by issueing the [Shift+Return] key combination"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Further readings\n",
-    " - To get more support in installing the environment you can check this page\n",
-    "  [Quick Start Guide](https://jupyter-notebook-beginner-guide.readthedocs.io/en/latest/index.html)\n",
-    " - To solve our problems we will use packages that are included in the Python bundle of containing \n",
-    "   packages that efficiently support scientific-computing tasks. To motivate you to dig in deeper in\n",
-    "   Python check this race of programming languages over the last 50 years and wait till the end of it ;-)  [Most Popular Programming Languages 1965 - 2019](https://www.youtube.com/watch?v=Og847HVwRSI)\n",
-    " - Basic features of Jupyter, Python, and of the packages that we will use \n",
-    "   for plotting, linear algebra, algebraic manipulations, and data array manipulations will be shortly\n",
-    "   explained.  the that we will use to support our model development will be addressed in a separate notebook."
+    "# Why Jupyter Lab? Why Python?\n",
+    " - [Jupyterlab introduction](https://youtu.be/A5YyoCKxEOU) [7 mins] video explaining the basic features of jupyter notebook within jupyter lab \n",
+    " - Check this race of programming languages over the last 50 years and wait till the end of it ;-)  [Most Popular Programming Languages 1965 - 2019](https://www.youtube.com/watch?v=Og847HVwRSI)\n",
+    " - Useful packages\n",
+    "   - `matplotlib` - plotting  \n",
+    "   - `sympy` - algebraic manipulations\n",
+    "   - `numpy` - data array manipulations\n",
+    "   will be shortly explained."
    ]
   },
   {
@@ -168,7 +296,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.2"
+   "version": "3.9.1"
   },
   "toc": {
    "base_numbering": 1,