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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.

  1. Find the normal force distribution

Given the following structure.

  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:

NAC=8x−16 ( kN)N_\text{AC} = 8 x - 16 \text{ } \left(\text{ kN}\right)
(1)
NCD=32 ( kN)N_\text{CD} = 32 \text{ } \left(\text{ kN}\right)
(2)
NDB=−28 ( kN)N_\text{DB} = -28 \text{ } \left(\text{ kN}\right)
(3)
Solution assignment 2

Given the following coordinate system:

MAB=x32−21x22+147x4−954 (kNm)M_\text{AB} = \cfrac{x^{3}}{2} - \cfrac{21 x^{2}}{2} + \cfrac{147 x}{4} - \cfrac{95}{4} \text{ } \left(\text{kNm}\right)
(4)
MBS=x32−21x22+183x2−5952 (kNm)M_\text{BS} = \cfrac{x^{3}}{2} - \cfrac{21 x^{2}}{2} + \cfrac{183 x}{2} - \cfrac{595}{2} \text{ } \left(\text{kNm}\right)
(5)
MSD=18x−126 (kNm)M_\text{SD} = 18 x - 126 \text{ } \left(\text{kNm}\right)
(6)
MDC=144−12x (kNm)M_\text{DC} = 144 - 12 x \text{ } \left(\text{kNm}\right)
(7)
Solution assignment 3

Given the following coordinate system:

VAB=3x22−21x+1474 (kN)V_\text{AB} = \cfrac{3 x^{2}}{2} - 21 x + \cfrac{147}{4} \text{ } \left(\text{kN}\right)
(8)
VBS=3x22−21x+1832 (kN)V_\text{BS} = \cfrac{3 x^{2}}{2} - 21 x + \cfrac{183}{2} \text{ } \left(\text{kN}\right)
(9)
VSD=18 (kN)V_\text{SD} = 18 \text{ } \left(\text{kN}\right)
(10)
VDC=−12 (kN)V_\text{DC} = -12 \text{ } \left(\text{kN}\right)
(11)
Solution assignment 4

Given the following coordinate system:

wAB=−x5120000+7x424000−49x324000+19x24800 (m)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)
(12)
wBS=−x5120000+7x424000−61x312000+119x22400−73x320+73192 (m)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)
(13)
wSD=−x31000+21x21000−20537x150000+754925000 (m)w_\text{SD} = - \cfrac{x^{3}}{1000} + \cfrac{21 x^{2}}{1000} - \cfrac{20537 x}{150000} + \cfrac{7549}{25000} \text{ } \left(\text{m}\right)
(14)
wDC=x31500−3x2125+40213x150000−1141312500 (m)w_\text{DC} = \cfrac{x^{3}}{1500} - \cfrac{3 x^{2}}{125} + \cfrac{40213 x}{150000} - \cfrac{11413}{12500} \text{ } \left(\text{m}\right)
(15)

If you don’t have Python and SymPy installed, click rocket --> 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(
Activities
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