Semi-Monocoque Theory



In [18]:

    
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 [19]:

    
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 [20]:

    
ureg = UnitRegistry()
sympy.init_printing()

Define sympy parameters used for geometric description of sections



In [21]:

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

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



In [22]:

    
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)]
datav = [(v[0],v[1].magnitude) for v in values]

Triangular section

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 [23]:

    
stringers = {1:[(sympy.Integer(0),h),A],
             2:[(sympy.Integer(0),sympy.Integer(0)),A],
             3:[(a,sympy.Integer(0)),A]}

panels = {(1,2):t,
          (2,3):t,
          (3,1):t}

Define section and perform first calculations



In [24]:

    
S1 = Section(stringers, panels)



In [25]:

    
S1.cycles









    Out[25]:





$$\left [ \left [ 2, \quad 3, \quad 1, \quad 2\right ]\right ]$$

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 [26]:

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



In [27]:

    
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);

Expression of Inertial properties wrt Center of Gravity in with original rotation



In [28]:

    
S1.Ixx0, S1.Iyy0, S1.Ixy0, S1.α0









    Out[28]:





$$\left ( \frac{2 A}{3} h^{2}, \quad \frac{2 A}{3} a^{2}, \quad - \frac{A a}{3} h, \quad - \frac{1}{2} \operatorname{atan}{\left (\frac{2 A a h}{2 A a^{2} - 2 A h^{2}} \right )}\right )$$

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 [29]:

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



In [30]:

    
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 [31]:

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









    Out[31]:





$$\left ( - \frac{A}{3 \left(a^{2} - h^{2}\right) \sqrt{a^{4} - a^{2} h^{2} + h^{4}}} \left(- a^{4} \sqrt{a^{4} - a^{2} h^{2} + h^{4}} + a^{4} \left\lvert{a^{2} - h^{2}}\right\rvert - a^{2} h^{2} \left\lvert{a^{2} - h^{2}}\right\rvert + h^{4} \sqrt{a^{4} - a^{2} h^{2} + h^{4}} + h^{4} \left\lvert{a^{2} - h^{2}}\right\rvert\right), \quad \frac{A}{3 \left(a^{2} - h^{2}\right) \sqrt{a^{4} - a^{2} h^{2} + h^{4}}} \left(a^{4} \sqrt{a^{4} - a^{2} h^{2} + h^{4}} + a^{4} \left\lvert{a^{2} - h^{2}}\right\rvert - a^{2} h^{2} \left\lvert{a^{2} - h^{2}}\right\rvert - h^{4} \sqrt{a^{4} - a^{2} h^{2} + h^{4}} + h^{4} \left\lvert{a^{2} - h^{2}}\right\rvert\right), \quad 0, \quad 0\right )$$

Shear center expression

Expressions can be messy, so we evaluate them to numerical values



In [32]:

    
sympy.N(S1.ct.subs(datav))









    Out[32]:





$$\left[\begin{matrix}-12.9681912643707\\-1.04213508398858\end{matrix}\right]$$

Analisys of Loads

We define some symbols



In [33]:

    
Tx, Ty, Nz, Mx, My, Mz, F, ry, ry, mz = sympy.symbols('T_x T_y N_z M_x M_y M_z F r_y r_x m_z')



In [34]:

    
S1.set_loads(_Tx=0, _Ty=Ty, _Nz=0, _Mx=Mx, _My=0, _Mz=0)
#S1.compute_stringer_actions()
#S1.compute_panel_fluxes();

Axial Loads



In [35]:

    
#S1.N

Panel Fluxes



In [36]:

    
#S1.q

Example 2: twisting moment in z direction



In [37]:

    
S1.set_loads(_Tx=0, _Ty=0, _Nz=0, _Mx=0, _My=0, _Mz=Mz)
S1.compute_stringer_actions()
S1.compute_panel_fluxes();

Axial Loads



In [38]:

    
S1.N









    Out[38]:





$$\left \{ 1 : 0, \quad 2 : 0, \quad 3 : 0\right \}$$

Panel Fluxes evaluated to numerical values



In [39]:

    
{k:sympy.N(S1.q[k].subs(datav)) for k in S1.q }









    Out[39]:





$$\left \{ \left ( 1, \quad 2\right ) : 0.000357142857142857 M_{z}, \quad \left ( 2, \quad 3\right ) : 0.000357142857142857 M_{z}, \quad \left ( 3, \quad 1\right ) : 0.000357142857142857 M_{z}\right \}$$

Torsional moment of Inertia



In [40]:

    
S1.compute_Jt()



In [41]:

    
sympy.N(S1.Jt.subs(datav))









    Out[41]:





$$\left[\begin{matrix}38750.2556239891 t\end{matrix}\right]$$



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