Semi-Monocoque Theory: corrective solutions



In [1]:

    
from pint import UnitRegistry
import sympy
import networkx as nx
#import numpy as np
import matplotlib.pyplot as plt
#import sys
%matplotlib inline
from IPython.display import display

Import Section class, which contains all calculations



In [2]:

    
from Section import Section

Initialization of sympy symbolic tool and pint for dimension analysis (not really implemented rn as not directly compatible with sympy)



In [3]:

    
ureg = UnitRegistry()
sympy.init_printing()

Define sympy parameters used for geometric description of sections



In [4]:

    
A, A0, t, t0, a, b, h, L, E, G = sympy.symbols('A A_0 t t_0 a b h L E G', positive=True)

We also define numerical values for each symbol in order to plot scaled section and perform calculations



In [5]:

    
values = [(A, 150 * ureg.millimeter**2),(A0, 250  * ureg.millimeter**2),(a, 80 * ureg.millimeter), \
          (b, 20 * ureg.millimeter),(h, 35 * ureg.millimeter),(L, 2000 * ureg.millimeter), \
          (t, 0.8 *ureg.millimeter),(E, 72e3 * ureg.MPa), (G, 27e3 * ureg.MPa)]
datav = [(v[0],v[1].magnitude) for v in values]

Third example: Simple rectangular section with 7 nodes Non symmetric

Define graph describing the section:

1) stringers are nodes with parameters:

x coordinate
y coordinate
Area

2) panels are oriented edges with parameters:

thickness
lenght which is automatically calculated



In [6]:

    
stringers = {1:[(3*a,h),A],
             2:[(2*a,h),A],
             3:[(a,h),A],
             4:[(sympy.Integer(0),h),A],
             5:[(sympy.Integer(0),sympy.Integer(0)),A],
             6:[(sympy.Rational(7,4)*a,sympy.Integer(0)),A],
             7:[(3*a,sympy.Integer(0)),A]}

panels = {(1,2):t,
          (2,3):t,
          (3,4):t,
          (4,5):t,
          (5,6):t,
          (6,7):t,
          (7,1):t}

Define section and perform first calculations



In [7]:

    
S1 = Section(stringers, panels)



In [8]:

    
S1.cycles









    Out[8]:





$$\left [ \left [ 2, \quad 3, \quad 4, \quad 5, \quad 6, \quad 7, \quad 1, \quad 2\right ]\right ]$$

Plot of S1 section in original reference frame

Define a dictionary of coordinates used by Networkx to plot section as a Directed graph. Note that arrows are actually just thicker stubs



In [9]:

    
start_pos={ii: [float(S1.g.node[ii]['ip'][i].subs(datav)) for i in range(2)] for ii in S1.g.nodes() }



In [10]:

    
plt.figure(figsize=(12,8),dpi=300)
nx.draw(S1.g,with_labels=True, arrows= True, pos=start_pos)
plt.arrow(0,0,20,0)
plt.arrow(0,0,0,20)
#plt.text(0,0, 'CG', fontsize=24)
plt.axis('equal')
plt.title("Section in starting reference Frame",fontsize=16);

Plot of S1 section in inertial reference Frame

Section is plotted wrt center of gravity and rotated (if necessary) so that x and y are principal axes. Center of Gravity and Shear Center are drawn



In [11]:

    
positions={ii: [float(S1.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S1.g.nodes() }



In [12]:

    
x_ct, y_ct = S1.ct.subs(datav)

plt.figure(figsize=(12,8),dpi=300)
nx.draw(S1.g,with_labels=True, pos=positions)
plt.plot([0],[0],'o',ms=12,label='CG')
plt.plot([x_ct],[y_ct],'^',ms=12, label='SC')
#plt.text(0,0, 'CG', fontsize=24)
#plt.text(x_ct,y_ct, 'SC', fontsize=24)
plt.legend(loc='lower right', shadow=True)
plt.axis('equal')
plt.title("Section in pricipal reference Frame",fontsize=16);

Expression of inertial properties in principal reference frame



In [13]:

    
sympy.simplify(S1.Ixx), sympy.simplify(S1.Iyy), sympy.simplify(S1.Ixy), sympy.simplify(S1.θ)









    Out[13]:





$$\left ( - \frac{A}{112 \left(535 a^{2} - 96 h^{2}\right) \sqrt{286225 a^{4} - 102464 a^{2} h^{2} + 9216 h^{4}}} \left(- 286225 a^{4} \sqrt{286225 a^{4} - 102464 a^{2} h^{2} + 9216 h^{4}} + 286225 a^{4} \left\lvert{535 a^{2} - 96 h^{2}}\right\rvert - 102464 a^{2} h^{2} \left\lvert{535 a^{2} - 96 h^{2}}\right\rvert + 9216 h^{4} \sqrt{286225 a^{4} - 102464 a^{2} h^{2} + 9216 h^{4}} + 9216 h^{4} \left\lvert{535 a^{2} - 96 h^{2}}\right\rvert\right), \quad \frac{A}{112 \left(535 a^{2} - 96 h^{2}\right) \sqrt{286225 a^{4} - 102464 a^{2} h^{2} + 9216 h^{4}}} \left(286225 a^{4} \sqrt{286225 a^{4} - 102464 a^{2} h^{2} + 9216 h^{4}} + 286225 a^{4} \left\lvert{535 a^{2} - 96 h^{2}}\right\rvert - 102464 a^{2} h^{2} \left\lvert{535 a^{2} - 96 h^{2}}\right\rvert - 9216 h^{4} \sqrt{286225 a^{4} - 102464 a^{2} h^{2} + 9216 h^{4}} + 9216 h^{4} \left\lvert{535 a^{2} - 96 h^{2}}\right\rvert\right), \quad 0, \quad 0\right )$$



In [14]:

    
S1.symmetry









    Out[14]:





[{'edges': [], 'nodes': []}, {'edges': [], 'nodes': []}]



In [15]:

    
S1.compute_L()



In [16]:

    
S1.L









    Out[16]:





$$\left[\begin{matrix}1 & 2 & - \frac{7}{4} & -3\\-2 & -3 & \frac{7}{4} & 3\\1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & -1 & -1\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$$



In [ ]:

    
S1.compute_H()



In [ ]:

    
S1.H.subs(datav)



In [ ]:

    
S1.compute_KM(A,h,t)



In [ ]:

    
S1.Ktilde



In [ ]:

    
S1.Mtilde.subs(datav)



In [ ]:

    
sol_data = (S1.Ktilde.inv()*(S1.Mtilde.subs(datav))).eigenvects()
sol_data



In [ ]:

    
β2 = [sol[0] for sol in sol_data]
β2



In [ ]:

    
X = []
for sol in sol_data:
    for i in range(len(sol[2])):
        X.append(sympy.N(sol[2][i]/sol[2][i].norm()))
X



In [ ]:

    
λ = [sympy.N(sympy.sqrt(E*A*h/(G*t)*βi).subs(datav)) for βi in β2]
λ



In [ ]: