Semi-Monocoque Theory



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









    



/home/claudio/anaconda2/envs/py35/lib/python3.5/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')
/home/claudio/anaconda2/envs/py35/lib/python3.5/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')

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

First example: Closed 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 [6]:

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

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

Define section and perform first calculations



In [7]:

    
S1 = Section(stringers, panels)

Verify that we find a simply closed section



In [8]:

    
S1.cycles









    Out[8]:





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

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);

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



In [11]:

    
S1.Ixx0, S1.Iyy0, S1.Ixy0, S1.α0









    Out[11]:





$$\left ( \frac{6 A}{5} h^{2}, \quad A \left(a^{2} - 2 a b + 2 b^{2}\right), \quad 0, \quad 0\right )$$

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 [12]:

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



In [13]:

    
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 [14]:

    
S1.Ixx, S1.Iyy, S1.Ixy, S1.θ









    Out[14]:





$$\left ( \frac{6 A}{5} h^{2}, \quad A \left(a^{2} - 2 a b + 2 b^{2}\right), \quad 0, \quad 0\right )$$

Shear center expression



In [15]:

    
S1.ct









    Out[15]:





$$\left[\begin{matrix}0\\- \frac{h \left(10 \left(a - b\right) \left(a^{2} - \left(a - 2 b\right) \left|{a - 2 b}\right|\right) - \left(2 a^{2} - 3 \left(a - 2 b\right)^{2}\right) \left(a + 2 \sqrt{b^{2} + h^{2}} + \left|{a - 2 b}\right|\right)\right)}{10 \left(a + 2 \sqrt{b^{2} + h^{2}} + \left|{a - 2 b}\right|\right) \left(a^{2} - 2 a b + 2 b^{2}\right)}\end{matrix}\right]$$

Analisys of symmetry properties of the section

For x and y axes pair of symmetric nodes and edges are searched for



In [16]:

    
S1.symmetry









    Out[16]:





[{'edges': [((1, 2), (2, 3)), ((3, 4), (5, 1)), (4, 5)],
  'nodes': [(1, 3), (2, 2), (4, 5)]},
 {'edges': [], 'nodes': []}]

Compute axial loads in Stringers in S1

We first define some symbols:



In [17]:

    
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')

Set loads on the section:

Example 1: shear in y direction and bending moment in x direction



In [18]:

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

Compute axial loads in stringers and shear flows in panels



In [19]:

    
S1.compute_stringer_actions()
S1.compute_panel_fluxes();

Axial loads



In [20]:

    
S1.N









    Out[20]:





$$\left \{ 1 : \frac{M_{x}}{3 h}, \quad 2 : \frac{M_{x}}{3 h}, \quad 3 : \frac{M_{x}}{3 h}, \quad 4 : - \frac{M_{x}}{2 h}, \quad 5 : - \frac{M_{x}}{2 h}\right \}$$

Shear flows



In [21]:

    
S1.q









    Out[21]:





$$\left \{ \left ( 1, \quad 2\right ) : \frac{T_{y}}{6 h}, \quad \left ( 2, \quad 3\right ) : - \frac{T_{y}}{6 h}, \quad \left ( 3, \quad 4\right ) : - \frac{T_{y}}{2 h}, \quad \left ( 4, \quad 5\right ) : 0, \quad \left ( 5, \quad 1\right ) : \frac{T_{y}}{2 h}\right \}$$

Example 2: twisting moment in z direction



In [22]:

    
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 [23]:

    
S1.N









    Out[23]:





$$\left \{ 1 : 0, \quad 2 : 0, \quad 3 : 0, \quad 4 : 0, \quad 5 : 0\right \}$$

Panel fluxes



In [24]:

    
S1.q









    Out[24]:





$$\left \{ \left ( 1, \quad 2\right ) : - \frac{M_{z}}{2 h \left(a - b\right)}, \quad \left ( 2, \quad 3\right ) : - \frac{M_{z}}{2 h \left(a - b\right)}, \quad \left ( 3, \quad 4\right ) : - \frac{M_{z}}{2 h \left(a - b\right)}, \quad \left ( 4, \quad 5\right ) : - \frac{M_{z}}{2 h \left(a - b\right)}, \quad \left ( 5, \quad 1\right ) : - \frac{M_{z}}{2 h \left(a - b\right)}\right \}$$

Set loads on the section:

Example 3: shear in x direction and bending moment in y direction



In [25]:

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

Axial loads



In [26]:

    
S1.N









    Out[26]:





$$\left \{ 1 : \frac{M_{y} a}{2 a^{2} - 4 a b + 4 b^{2}}, \quad 2 : 0, \quad 3 : - \frac{M_{y} a}{2 a^{2} - 4 a b + 4 b^{2}}, \quad 4 : - \frac{M_{y} \left(\frac{a}{2} - b\right)}{a^{2} - 2 a b + 2 b^{2}}, \quad 5 : - \frac{M_{y} \left(- \frac{a}{2} + b\right)}{a^{2} - 2 a b + 2 b^{2}}\right \}$$

Panel fluxes Not really an easy expression



In [27]:

    
S1.q









    Out[27]:





$$\left \{ \left ( 1, \quad 2\right ) : \frac{T_{x} \left(a \sqrt{b^{2} + h^{2}} + a \left|{a - 2 b}\right| - b \left|{a - 2 b}\right|\right)}{a^{3} - 2 a^{2} b + 2 a^{2} \sqrt{b^{2} + h^{2}} + a^{2} \left|{a - 2 b}\right| + 2 a b^{2} - 4 a b \sqrt{b^{2} + h^{2}} - 2 a b \left|{a - 2 b}\right| + 4 b^{2} \sqrt{b^{2} + h^{2}} + 2 b^{2} \left|{a - 2 b}\right|}, \quad \left ( 2, \quad 3\right ) : \frac{T_{x} \left(a \sqrt{b^{2} + h^{2}} + a \left|{a - 2 b}\right| - b \left|{a - 2 b}\right|\right)}{a^{3} - 2 a^{2} b + 2 a^{2} \sqrt{b^{2} + h^{2}} + a^{2} \left|{a - 2 b}\right| + 2 a b^{2} - 4 a b \sqrt{b^{2} + h^{2}} - 2 a b \left|{a - 2 b}\right| + 4 b^{2} \sqrt{b^{2} + h^{2}} + 2 b^{2} \left|{a - 2 b}\right|}, \quad \left ( 3, \quad 4\right ) : - \frac{T_{x} \left(a^{2} - a \left|{a - 2 b}\right| + 2 b \left|{a - 2 b}\right|\right)}{2 a^{3} - 4 a^{2} b + 4 a^{2} \sqrt{b^{2} + h^{2}} + 2 a^{2} \left|{a - 2 b}\right| + 4 a b^{2} - 8 a b \sqrt{b^{2} + h^{2}} - 4 a b \left|{a - 2 b}\right| + 8 b^{2} \sqrt{b^{2} + h^{2}} + 4 b^{2} \left|{a - 2 b}\right|}, \quad \left ( 4, \quad 5\right ) : - \frac{T_{x} \left(a^{2} - a b + a \sqrt{b^{2} + h^{2}} - 2 b \sqrt{b^{2} + h^{2}}\right)}{a^{3} - 2 a^{2} b + 2 a^{2} \sqrt{b^{2} + h^{2}} + a^{2} \left|{a - 2 b}\right| + 2 a b^{2} - 4 a b \sqrt{b^{2} + h^{2}} - 2 a b \left|{a - 2 b}\right| + 4 b^{2} \sqrt{b^{2} + h^{2}} + 2 b^{2} \left|{a - 2 b}\right|}, \quad \left ( 5, \quad 1\right ) : - \frac{T_{x} \left(a^{2} - a \left|{a - 2 b}\right| + 2 b \left|{a - 2 b}\right|\right)}{2 a^{3} - 4 a^{2} b + 4 a^{2} \sqrt{b^{2} + h^{2}} + 2 a^{2} \left|{a - 2 b}\right| + 4 a b^{2} - 8 a b \sqrt{b^{2} + h^{2}} - 4 a b \left|{a - 2 b}\right| + 8 b^{2} \sqrt{b^{2} + h^{2}} + 4 b^{2} \left|{a - 2 b}\right|}\right \}$$

Compute Jt

Computation of torsional moment of inertia:



In [28]:

    
S1.compute_Jt()
S1.Jt









    Out[28]:





$$\left[\begin{matrix}\frac{4 h^{2} t \left(a - b\right)^{2}}{a + 2 \sqrt{b^{2} + h^{2}} + \left|{a - 2 b}\right|}\end{matrix}\right]$$

Second example: Open section



In [29]:

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

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

Define section and perform first calculations



In [30]:

    
S2 = Section(stringers, panels)

Verify that the section is open



In [31]:

    
S2.cycles









    Out[31]:





$$\left [ \right ]$$

Plot of S2 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 [32]:

    
start_pos={ii: [float(S2.g.node[ii]['ip'][i].subs(datav)) for i in range(2)] for ii in S2.g.nodes() }



In [33]:

    
plt.figure(figsize=(12,8),dpi=300)
nx.draw(S2.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 [34]:

    
S2.Ixx0, S2.Iyy0, S2.Ixy0, S2.α0









    Out[34]:





$$\left ( A h^{2}, \quad A a^{2}, \quad 0, \quad 0\right )$$

Plot of S2 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 [35]:

    
positions={ii: [float(S2.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S2.g.nodes() }



In [36]:

    
x_ct, y_ct = S2.ct.subs(datav)

plt.figure(figsize=(12,8),dpi=300)
nx.draw(S2.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 [37]:

    
S2.Ixx, S2.Iyy, S2.Ixy, S2.θ









    Out[37]:





$$\left ( A h^{2}, \quad A a^{2}, \quad 0, \quad 0\right )$$

Shear center expression



In [38]:

    
S2.ct









    Out[38]:





$$\left[\begin{matrix}0\\- h\end{matrix}\right]$$

Analisys of symmetry properties of the section

For x and y axes pair of symmetric nodes and edges are searched for



In [39]:

    
S2.symmetry









    Out[39]:





[{'edges': [((1, 2), (3, 4)), (2, 3)], 'nodes': [(1, 4), (2, 3)]},
 {'edges': [(1, 2), (3, 4)], 'nodes': [(1, 2), (3, 4)]}]

Compute axial loads in Stringers in S2

Set loads on the section:

Example 2: shear in y direction and bending moment in x direction



In [40]:

    
S2.set_loads(_Tx=0, _Ty=Ty, _Nz=0, _Mx=Mx, _My=0, _Mz=0)

Compute axial loads in stringers and shear flows in panels



In [41]:

    
S2.compute_stringer_actions()
S2.compute_panel_fluxes();

Axial loads



In [42]:

    
S2.N









    Out[42]:





$$\left \{ 1 : \frac{M_{x}}{2 h}, \quad 2 : - \frac{M_{x}}{2 h}, \quad 3 : - \frac{M_{x}}{2 h}, \quad 4 : \frac{M_{x}}{2 h}\right \}$$

Shear flows



In [43]:

    
S2.q









    Out[43]:





$$\left \{ \left ( 1, \quad 2\right ) : - \frac{T_{y}}{2 h}, \quad \left ( 2, \quad 3\right ) : 0, \quad \left ( 3, \quad 4\right ) : \frac{T_{y}}{2 h}\right \}$$

Set loads on the section:

Example 2: shear in x direction and bending moment in y direction



In [44]:

    
S2.set_loads(_Tx=Tx, _Ty=0, _Nz=0, _Mx=0, _My=My, _Mz=0)
S2.compute_stringer_actions()
S2.compute_panel_fluxes();



In [45]:

    
S2.N









    Out[45]:





$$\left \{ 1 : \frac{M_{y}}{2 a}, \quad 2 : \frac{M_{y}}{2 a}, \quad 3 : - \frac{M_{y}}{2 a}, \quad 4 : - \frac{M_{y}}{2 a}\right \}$$



In [46]:

    
S2.q









    Out[46]:





$$\left \{ \left ( 1, \quad 2\right ) : \frac{T_{x}}{2 a}, \quad \left ( 2, \quad 3\right ) : \frac{T_{x}}{a}, \quad \left ( 3, \quad 4\right ) : \frac{T_{x}}{2 a}\right \}$$

Second example (2): Open section



In [47]:

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

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

Define section and perform first calculations



In [48]:

    
S2_2 = Section(stringers, panels)

Plot of S2 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 [49]:

    
start_pos={ii: [float(S2_2.g.node[ii]['ip'][i].subs(datav)) for i in range(2)] for ii in S2_2.g.nodes() }



In [50]:

    
plt.figure(figsize=(12,8),dpi=300)
nx.draw(S2_2.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 [51]:

    
S2_2.Ixx0, S2_2.Iyy0, S2_2.Ixy0, S2_2.α0









    Out[51]:





$$\left ( A h^{2}, \quad A a^{2}, \quad 0, \quad 0\right )$$

Plot of S2 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 [52]:

    
positions={ii: [float(S2_2.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S2_2.g.nodes() }



In [53]:

    
x_ct, y_ct = S2_2.ct.subs(datav)

plt.figure(figsize=(12,8),dpi=300)
nx.draw(S2_2.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 [54]:

    
S2_2.Ixx, S2_2.Iyy, S2_2.Ixy, S2_2.θ









    Out[54]:





$$\left ( A h^{2}, \quad A a^{2}, \quad 0, \quad 0\right )$$

Shear center expression



In [55]:

    
S2_2.ct









    Out[55]:





$$\left[\begin{matrix}- a\\0\end{matrix}\right]$$