Lesson Wednesday September 18th

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# Matplotlib compatibility patch for Pyodide
import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
    matplotlib.RcParams._get = dict.get

Lesson Wednesday September 18th#

During today’s lesson you’ll work on a complex exercise on the topic of SymPy. Please ask your questions regarding the homework as well!

Exercise SymPy#

Given the following structure.

../../_images/11.svg

  1. Find the normal force distribution

Given the following structure.

../../_images/21.svg

  1. Find the bending moment distribution

  2. Find the shear force distribution

  3. Find the displacement distribution

Solution assignment 1

Given the following coordinate system:

../../_images/a_assen.svg
../../_images/ans1.svg
\[N_\text{AC} = 8 x - 16 \text{ } \left(\text{ kN}\right)\]
\[N_\text{CD} = 32 \text{ } \left(\text{ kN}\right)\]
\[N_\text{DB} = -28 \text{ } \left(\text{ kN}\right)\]

Solution assignment 2

Given the following coordinate system:

../../_images/2_assen.svg
../../_images/ans2.svg
\[M_\text{AB} = \cfrac{x^{3}}{2} - \cfrac{21 x^{2}}{2} + \cfrac{147 x}{4} - \cfrac{95}{4} \text{ } \left(\text{kNm}\right)\]
\[M_\text{BS} = \cfrac{x^{3}}{2} - \cfrac{21 x^{2}}{2} + \cfrac{183 x}{2} - \cfrac{595}{2} \text{ } \left(\text{kNm}\right)\]
\[M_\text{SD} = 18 x - 126 \text{ } \left(\text{kNm}\right)\]
\[M_\text{DC} = 144 - 12 x \text{ } \left(\text{kNm}\right)\]

Solution assignment 3

Given the following coordinate system:

../../_images/2_assen.svg
../../_images/ans3.svg
\[V_\text{AB} = \cfrac{3 x^{2}}{2} - 21 x + \cfrac{147}{4} \text{ } \left(\text{kN}\right) \]
\[V_\text{BS} = \cfrac{3 x^{2}}{2} - 21 x + \cfrac{183}{2} \text{ } \left(\text{kN}\right)\]
\[V_\text{SD} = 18 \text{ } \left(\text{kN}\right)\]
\[V_\text{DC} = -12 \text{ } \left(\text{kN}\right)\]

Solution assignment 4

Given the following coordinate system:

../../_images/2_assen.svg
../../_images/ans4.svg
\[w_\text{AB} = - \cfrac{x^{5}}{120000} + \cfrac{7 x^{4}}{24000} - \cfrac{49 x^{3}}{24000} + \cfrac{19 x^{2}}{4800} \text{ } \left(\text{m}\right)\]
\[w_\text{BS} = - \cfrac{x^{5}}{120000} + \cfrac{7 x^{4}}{24000} - \cfrac{61 x^{3}}{12000} + \cfrac{119 x^{2}}{2400} - \cfrac{73 x}{320} + \cfrac{73}{192} \text{ } \left(\text{m}\right)\]
\[w_\text{SD} = - \cfrac{x^{3}}{1000} + \cfrac{21 x^{2}}{1000} - \cfrac{20537 x}{150000} + \cfrac{7549}{25000} \text{ } \left(\text{m}\right)\]
\[w_\text{DC} = \cfrac{x^{3}}{1500} - \cfrac{3 x^{2}}{125} + \cfrac{40213 x}{150000} - \cfrac{11413}{12500} \text{ } \left(\text{m}\right)\]

If you don’t have Python and SymPy installed, click –> Live Code to activate live coding and use the cells below:

import sympy as sym
sym.init_printing()

x = sym.symbols('x')
C1, C2, C3, C4 = sym.symbols('C1 C2 C3 C4')

q = 

V = sym.integrate(-q,x)+C1
M = sym.integrate(V,x)+C2
kappa = M / EI
phi = sym.integrate(kappa,x)+C3
w = sym.integrate(-phi,x)+C4

Eq1 = sym.Eq(

sol = sym.solve((Eq1,Eq2,Eq3,Eq4,Eq5,Eq6,Eq7,Eq8,Eq9,Eq10,Eq11,Eq12), (C1,C2,C3,C4,C5,C6,C7,C8,C9,C10,C11,C12))
display(sol)

M.subs(sol)

import numpy as np
import matplotlib.pyplot as plt

%config InlineBackend.figure_formats = ['svg']

def plot(w_list, x_range_list,ylabel):
    plt.figure()
    for i, w in enumerate(w_list):
        # check if x is the only symbol in the expression
        if len(w.free_symbols) > 1:
            raise ValueError('The expression must be a function of x only.')
        
        w_numpy = sym.lambdify(x, w)
        x_vals = np.linspace(x_range_list[i][0], x_range_list[i][1], 100)
        
        # if the expression is a constant, we need to make sure that it is broadcasted correctly
        if isinstance(w_numpy(x_vals),float) or isinstance(w_numpy(x_vals),int):
            w_numpy = np.vectorize(w_numpy)
            plt.plot([x_range_list[i][0], x_range_list[i][1]],[w_numpy(x_vals),w_numpy(x_vals)])
        else:
            plt.plot(x_vals,w_numpy(x_vals))

        plt.plot(x_vals,w_numpy(x_vals))
        plt.xlabel('$x$')
        plt.ylabel(ylabel)
    ax = plt.gca()
    ax.spines['right'].set_color('none')
    ax.spines['top'].set_color('none')
    ax.spines['bottom'].set_position('zero')
    ax.spines['left'].set_position('zero')
    ax.invert_yaxis()

plot(
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