SF 260 Stabilizer



In [200]:

    
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 [201]:

    
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 [202]:

    
ureg = UnitRegistry()
sympy.init_printing()

Define sympy parameters used for geometric description of sections



In [203]:

    
Ams, Afs, Amid, Atip, tms, tfs, tpt, tpn, hms, hfs, xfs, xtip, L = \
sympy.symbols('A_ms A_fs A_mid A_tip t_ms t_fs t_pn t_pt h_ms h_fs x_sp x_tip L', positive=True)

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



In [204]:

    
L1 = (205.+110.)/2
tg_β = 381.8/1400.

xtip1 = -651.8 + np.arctan(tg_β)*L1/2


# altezza longheroni
hms1 = 45.
hfs1 = 50.
#pos longherone anteriore
xfs1 = -345.
# spessore pannello
tpt1 = 0.81

# punghezza pannello 1
lp1 = np.sqrt((hms1-hfs1)**2+xfs1**2)

# area pannello 1
Ap1 =lp1*tpt1

#spessore longheroni
tms1 = 3.6
tfs1 = 1.471*2

# area correnti
Astr_ms1=2.28760E+02
Astr_fs1=1.471*25.

# AREE
Ams1 = hms1*tms1/6+Ap1/4+Astr_ms1
Afs1 = hfs1*tfs1/6+Ap1/4+Astr_fs1
Amid1 = Ap1/2



In [205]:

    
values = [(Ams, Ams1 * ureg.millimeter**2),(Afs, Afs1  * ureg.millimeter**2), \
          (Amid, Amid1 * ureg.millimeter**2),(Atip, 0 * ureg.millimeter), \
          (tms, tms1 * ureg.millimeter),(tfs, tfs1 * ureg.millimeter), \
          (tpt, tpt1 * ureg.millimeter), (tpn, 0. * ureg.millimeter), \
          (hms, hms1 * ureg.millimeter), (hfs, hfs1 * ureg.millimeter), \
          (xfs, xfs1 * ureg.millimeter), (xtip, xtip1 * ureg.millimeter),
          (L,  L1* ureg.millimeter)]
datav = [(v[0],v[1].magnitude) for v in values]

First 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 [206]:

    
stringers = {1:[(sympy.Integer(0),hms),Ams],
             2:[(xfs/2,(hms+hfs)/2),Amid],
             3:[(xfs,hfs),Afs],
             4:[(xfs,-hfs),Afs],
             5:[(xfs/2,-(hms+hfs)/2),Amid],
             6:[(sympy.Integer(0),-hms),Ams]}

panels = {(1,2):tpt,
          (2,3):tpt,
          (3,4):tfs,
          (4,5):tpt,
          (5,6):tpt,
          (6,1):tms}

Define section and perform first calculations



In [207]:

    
S1 = Section(stringers, panels)

Verify that we find a simply closed section



In [208]:

    
S1.cycles









    Out[208]:





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

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 [209]:

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



In [210]:

    
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 [211]:

    
S1.Ixx0, S1.Iyy0, S1.Ixy0, S1.α0









    Out[211]:





$$\left ( 2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}, \quad \frac{x_{sp}^{2} \left(\frac{A_{fs} A_{mid}}{2} + 2 A_{fs} A_{ms} + \frac{A_{mid} A_{ms}}{2}\right)}{A_{fs} + A_{mid} + A_{ms}}, \quad 0, \quad 0\right )$$

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 [212]:

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



In [213]:

    
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("Sezione 1",fontsize=16);
plt.savefig("sect1.jpg")

Expression of inertial properties in principal reference frame



In [214]:

    
S1.Ixx, S1.Iyy, S1.Ixy, S1.θ









    Out[214]:





$$\left ( 2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}, \quad \frac{x_{sp}^{2} \left(\frac{A_{fs} A_{mid}}{2} + 2 A_{fs} A_{ms} + \frac{A_{mid} A_{ms}}{2}\right)}{A_{fs} + A_{mid} + A_{ms}}, \quad 0, \quad 0\right )$$



In [215]:

    
Ixx1 = S1.Ixx.subs(datav)

Compute axial loads in Stringers in S1

We first define some symbols:



In [216]:

    
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: bending moment in x direction



In [217]:

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

Compute axial loads in stringers and shear flows in panels



In [218]:

    
S1.compute_stringer_actions()
S1.compute_panel_fluxes();

Axial loads



In [219]:

    
S1.N









    Out[219]:





$$\left \{ 1 : \frac{A_{ms} M_{x} h_{ms}}{2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}}, \quad 2 : \frac{A_{mid} M_{x} \left(\frac{h_{fs}}{2} + \frac{h_{ms}}{2}\right)}{2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}}, \quad 3 : \frac{A_{fs} M_{x} h_{fs}}{2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}}, \quad 4 : - \frac{A_{fs} M_{x} h_{fs}}{2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}}, \quad 5 : \frac{A_{mid} M_{x} \left(- \frac{h_{fs}}{2} - \frac{h_{ms}}{2}\right)}{2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}}, \quad 6 : - \frac{A_{ms} M_{x} h_{ms}}{2 A_{fs} h_{fs}^{2} + \frac{A_{mid} h_{fs}^{2}}{2} + A_{mid} h_{fs} h_{ms} + \frac{A_{mid} h_{ms}^{2}}{2} + 2 A_{ms} h_{ms}^{2}}\right \}$$

Shear flows



In [220]:

    
S1.q









    Out[220]:





$$\left \{ \left ( 1, \quad 2\right ) : 0, \quad \left ( 2, \quad 3\right ) : 0, \quad \left ( 3, \quad 4\right ) : 0, \quad \left ( 4, \quad 5\right ) : 0, \quad \left ( 5, \quad 6\right ) : 0, \quad \left ( 6, \quad 1\right ) : 0\right \}$$

Example 2: twisting moment in z direction



In [221]:

    
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 [222]:

    
S1.N









    Out[222]:





$$\left \{ 1 : 0, \quad 2 : 0, \quad 3 : 0, \quad 4 : 0, \quad 5 : 0, \quad 6 : 0\right \}$$

Panel fluxes



In [223]:

    
S1.q









    Out[223]:





$$\left \{ \left ( 1, \quad 2\right ) : - \frac{M_{z}}{2 x_{sp} \left(h_{fs} + h_{ms}\right)}, \quad \left ( 2, \quad 3\right ) : - \frac{M_{z}}{2 x_{sp} \left(h_{fs} + h_{ms}\right)}, \quad \left ( 3, \quad 4\right ) : - \frac{M_{z}}{2 x_{sp} \left(h_{fs} + h_{ms}\right)}, \quad \left ( 4, \quad 5\right ) : - \frac{M_{z}}{2 x_{sp} \left(h_{fs} + h_{ms}\right)}, \quad \left ( 5, \quad 6\right ) : - \frac{M_{z}}{2 x_{sp} \left(h_{fs} + h_{ms}\right)}, \quad \left ( 6, \quad 1\right ) : - \frac{M_{z}}{2 x_{sp} \left(h_{fs} + h_{ms}\right)}\right \}$$

Compute Jt

Computation of torsional moment of inertia:



In [224]:

    
S1.compute_Jt()
S1.Jt









    Out[224]:





$$\left[\begin{matrix}\frac{2 t_{fs} t_{ms} t_{pn} x_{sp}^{2} \left(h_{fs} + h_{ms}\right)^{2}}{h_{fs} t_{ms} t_{pn} + h_{ms} t_{fs} t_{pn} + t_{fs} t_{ms} \sqrt{x_{sp}^{2} + \left(h_{fs} - h_{ms}\right)^{2}}}\end{matrix}\right]$$



In [225]:

    
Jt1 = S1.Jt.subs(datav)[0]
Jt1









    Out[225]:





$$4716931.10419239$$

Section 2



In [226]:

    
L2 = 410. - L1

xtip2 = -651.8 + np.arctan(tg_β)*(L1+L2/2.)


# altezza longheroni
hms2 = 42.
hfs2 = 46.
#pos longherone anteriore
xfs2 = -345.
# spessore pannello
tpt2 = 0.81
tpn2 = 0.64

# lunghezza pannelli 2
lpt2 = np.sqrt((hms2-hfs2)**2+xfs1**2)
lpn2 = np.sqrt(hfs2**2+(xfs2-xtip2)**2)

# area pannelli 2
Apt2 =lpt2*tpt2
Apn2 =lpn2*tpn2


#spessore longheroni
tms2 = 1.6
tfs2 = 1.471

# area correnti
Astr_ms2=2.28760E+02
Astr_fs2=1.471*25.

# AREE
Ams2 = hms2*tms2/6+Apt2/4+Astr_ms2
Afs2 = hfs2*tfs2/6+Apt2/4+Astr_fs2+Apn2/2
Amid2 = Apt2/2
Atip2 = Apn2



In [227]:

    
values = [(Ams, Ams2 * ureg.millimeter**2),(Afs, Afs2  * ureg.millimeter**2), \
          (Amid, Amid2 * ureg.millimeter**2),(Atip, Atip2 * ureg.millimeter), \
          (tms, tms2 * ureg.millimeter),(tfs, tfs2 * ureg.millimeter), \
          (tpt, tpt2 * ureg.millimeter), (tpn, tpn2 * ureg.millimeter), \
          (hms, hms2 * ureg.millimeter), (hfs, hfs2 * ureg.millimeter), \
          (xfs, xfs2 * ureg.millimeter), (xtip, xtip2 * ureg.millimeter), \
          (L,  L2* ureg.millimeter) ]
datav = [(v[0],v[1].magnitude) for v in values]



In [228]:

    
stringers = {1:[(sympy.Integer(0),sympy.Float(hms2)),sympy.Float(Ams2)],
             2:[(sympy.Float(xfs2/2),sympy.Float((hms2+hfs2)/2)),sympy.Float(Amid2)],
             3:[(sympy.Float(xfs2),sympy.Float(hfs2)),sympy.Float(Afs2)],
             4:[(sympy.Float(xtip2),sympy.Integer(0)),sympy.Float(Atip2)],
             5:[(sympy.Float(xfs2),sympy.Float(-hfs2)),sympy.Float(Afs2)],
             6:[(sympy.Float(xfs2/2),sympy.Float(-(hms2+hfs2)/2)),sympy.Float(Amid2)],
             7:[(sympy.Integer(0),sympy.Float(-hms2)),sympy.Float(Ams2)]}

panels = {(1,2):sympy.Float(tpt2),
          (2,3):sympy.Float(tpt2),
          (3,4):sympy.Float(tpn2),
          (4,5):sympy.Float(tpn2),
          (5,6):sympy.Float(tpt2),
          (6,7):sympy.Float(tpt2),
          (3,5):sympy.Float(tfs2),
          (7,1):sympy.Float(tms2)}

stringers = {1:[(sympy.Integer(0),hms),Ams], 2:[(xfs/2,(hms+hfs)/2),Amid], 3:[(xfs,hfs),Afs], 4:[(xtip,sympy.Integer(0)),Atip], 5:[(xfs,-hfs2),Afs], 6:[(xfs/2,-(hms+hfs)/2),Amid], 7:[(sympy.Integer(0),-hms),Ams]}

panels = {(1,2):tpt, (2,3):tpt, (3,4):tpn, (4,5):tpn, (5,6):tpt, (6,7):tpt, (3,5):tfs, (7,1):tms}

Define section and perform first calculations



In [229]:

    
S2 = Section(stringers, panels)



In [230]:

    
S2.cycles









    Out[230]:





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

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 [231]:

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



In [232]:

    
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 [233]:

    
S2.Ixx0, S2.Iyy0, S2.Ixy0, S2.α0









    Out[233]:





$$\left ( 2452468.34717176, \quad 54246667.2423998, \quad 0, \quad 0\right )$$

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 [234]:

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



In [235]:

    
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("Sezione 2",fontsize=16);
plt.savefig("sect2.jpg")

Expression of inertial properties in principal reference frame



In [236]:

    
S2.Ixx, S2.Iyy, S2.Ixy, S2.θ









    Out[236]:





$$\left ( 2452468.34717176, \quad 54246667.2423998, \quad 0, \quad 0\right )$$



In [237]:

    
Ixx2 = S2.Ixx.subs(datav)



In [238]:

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



In [239]:

    
S2.compute_Jt()
S2.Jt









    Out[239]:





$$\left[\begin{matrix}4611571.20415473\end{matrix}\right]$$



In [240]:

    
Jt2 = S2.Jt[0]
Jt2









    Out[240]:





$$4611571.20415473$$

Section 3



In [241]:

    
L3 = 330.

xtip3 = -651.8 + np.arctan(tg_β)*(410.+L3/2.)


# altezza longheroni
hms3 = 38.1
hfs3 = 37.4
#pos longherone anteriore
xfs3 = -345.
# spessore pannello
tpt3 = 0.81
tpn3 = 0.64

# lunghezza pannelli 2
lpt3 = np.sqrt((hms3-hfs3)**2+xfs3**2)
lpn3 = np.sqrt(hfs3**2+(xfs3-xtip3)**2)

# area pannelli 2
Apt3 =lpt3*tpt3
Apn3 =lpn3*tpn3


#spessore longheroni
tms3 = 1.6
tfs3 = 1.471

# area correnti
Astr_ms3=1.17760E+02
Astr_fs3=1.471*25.

# AREE
Ams3 = hms3*tms3/6+Apt3/4+Astr_ms3
Afs3 = hfs3*tfs3/6+Apt3/4+Astr_fs3+Apn3/2
Amid3 = Apt3/2
Atip3 = Apn3



In [242]:

    
values = [(Ams, Ams3 * ureg.millimeter**2),(Afs, Afs3  * ureg.millimeter**2), \
          (Amid, Amid3 * ureg.millimeter**2),(Atip, Atip3 * ureg.millimeter), \
          (tms, tms3 * ureg.millimeter),(tfs, tfs3 * ureg.millimeter), \
          (tpt, tpt3 * ureg.millimeter), (tpn, tpn3 * ureg.millimeter), \
          (hms, hms3 * ureg.millimeter), (hfs, hfs3 * ureg.millimeter), \
          (xfs, xfs3 * ureg.millimeter), (xtip, xtip3 * ureg.millimeter), \
          (L,  L3* ureg.millimeter) ]
datav = [(v[0],v[1].magnitude) for v in values]



In [243]:

    
stringers = {1:[(sympy.Integer(0),sympy.Float(hms3)),sympy.Float(Ams3)],
             2:[(sympy.Float(xfs3/2),sympy.Float((hms3+hfs3)/2)),sympy.Float(Amid3)],
             3:[(sympy.Float(xfs3),sympy.Float(hfs3)),sympy.Float(Afs3)],
             4:[(sympy.Float(xtip3),sympy.Integer(0)),sympy.Float(Atip3)],
             5:[(sympy.Float(xfs3),sympy.Float(-hfs3)),sympy.Float(Afs3)],
             6:[(sympy.Float(xfs3/2),sympy.Float(-(hms3+hfs3)/2)),sympy.Float(Amid3)],
             7:[(sympy.Integer(0),sympy.Float(-hms3)),sympy.Float(Ams3)]}

panels = {(1,2):sympy.Float(tpt3),
          (2,3):sympy.Float(tpt3),
          (3,4):sympy.Float(tpn3),
          (4,5):sympy.Float(tpn3),
          (5,6):sympy.Float(tpt3),
          (6,7):sympy.Float(tpt3),
          (3,5):sympy.Float(tfs3),
          (7,1):sympy.Float(tms3)}

Define section and perform first calculations



In [244]:

    
S3 = Section(stringers, panels)



In [245]:

    
S3.cycles









    Out[245]:





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

Plot of S3 section in original reference frame



In [246]:

    
plt.figure(figsize=(12,8),dpi=300)
nx.draw(S3.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 [247]:

    
S3.Ixx0, S3.Iyy0, S3.Ixy0, S3.α0









    Out[247]:





$$\left ( 1438031.67399904, \quad 32004402.8386356, \quad 0, \quad 0\right )$$

Plot of S3 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 [248]:

    
x_ct, y_ct = S3.ct.subs(datav)

plt.figure(figsize=(12,8),dpi=300)
nx.draw(S3.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("Sezione 3",fontsize=16);
plt.savefig("sect3.jpg")

Expression of inertial properties in principal reference frame



In [249]:

    
S3.Ixx, S3.Iyy, S3.Ixy, S3.θ









    Out[249]:





$$\left ( 1438031.67399904, \quad 32004402.8386356, \quad 0, \quad 0\right )$$



In [250]:

    
Ixx3 = S3.Ixx.subs(datav)



In [251]:

    
S3.set_loads(_Tx=0, _Ty=0, _Nz=0, _Mx=0, _My=0, _Mz=0)



In [252]:

    
S3.compute_Jt()
S3.Jt









    Out[252]:





$$\left[\begin{matrix}3231899.87681444\end{matrix}\right]$$



In [253]:

    
Jt3 = S3.Jt[0]

Section 4



In [254]:

    
L4 = 330.

xtip4 = -651.8 + np.arctan(tg_β)*(740.+L4/2.)


# altezza longheroni
hms4 = 32.25
hfs4 = 26.8
#pos longherone anteriore
xfs4 = -345.+ np.arctan(tg_β)*(L4/2.)
# spessore pannello
tpt4 = 0.81
tpn4 = 0.64

# lunghezza pannelli 2
lpt4 = np.sqrt((hms4-hfs4)**2+xfs4**2)
lpn4 = np.sqrt(hfs4**2+(xfs4-xtip4)**2)

# area pannelli 2
Apt4 =lpt4*tpt4
Apn4 =lpn4*tpn4


#spessore longheroni
tms4 = 1.6
tfs4 = 0.

# area correnti
Astr_ms4=2.28760E+02
Astr_fs4=7.74400E+01

# AREE
Ams4 = hms4*tms4/6+Apt4/4+Astr_ms4
Afs4 = hfs4*tfs4/6+Apt4/4+Astr_fs4+Apn4/2
Amid4 = Apt4/2
Atip4 = Apn4



In [255]:

    
values = [(Ams, Ams4 * ureg.millimeter**2),(Afs, Afs4  * ureg.millimeter**2), \
          (Amid, Amid4 * ureg.millimeter**2),(Atip, Atip4 * ureg.millimeter), \
          (tms, tms4 * ureg.millimeter),(tfs, tfs4 * ureg.millimeter), \
          (tpt, tpt4 * ureg.millimeter), (tpn, tpn4 * ureg.millimeter), \
          (hms, hms4 * ureg.millimeter), (hfs, hfs4 * ureg.millimeter), \
          (xfs, xfs4 * ureg.millimeter), (xtip, xtip4 * ureg.millimeter), \
          (L,  L4* ureg.millimeter) ]
datav = [(v[0],v[1].magnitude) for v in values]



In [256]:

    
stringers = {1:[(sympy.Integer(0),sympy.Float(hms4)),sympy.Float(Ams4)],
             2:[(sympy.Float(xfs4/2),sympy.Float((hms4+hfs4)/2)),sympy.Float(Amid4)],
             3:[(sympy.Float(xfs4),sympy.Float(hfs4)),sympy.Float(Afs4)],
             4:[(sympy.Float(xtip4),sympy.Integer(0)),sympy.Float(Atip4)],
             5:[(sympy.Float(xfs4),sympy.Float(-hfs4)),sympy.Float(Afs4)],
             6:[(sympy.Float(xfs4/2),sympy.Float(-(hms4+hfs4)/2)),sympy.Float(Amid4)],
             7:[(sympy.Integer(0),sympy.Float(-hms4)),sympy.Float(Ams4)]}

panels = {(1,2):sympy.Float(tpt4),
          (2,3):sympy.Float(tpt4),
          (3,4):sympy.Float(tpn4),
          (4,5):sympy.Float(tpn4),
          (5,6):sympy.Float(tpt4),
          (6,7):sympy.Float(tpt4),
          (7,1):sympy.Float(tms4)}

Define section and perform first calculations



In [257]:

    
S4 = Section(stringers, panels)



In [258]:

    
S4.cycles









    Out[258]:





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

Plot of S4 section in original reference frame



In [259]:

    
positions={ii: [float(S4.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S4.g.nodes() }



In [260]:

    
plt.figure(figsize=(12,8),dpi=300)
nx.draw(S4.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 [261]:

    
S4.Ixx0, S4.Iyy0, S4.Ixy0, S4.α0









    Out[261]:





$$\left ( 1083976.5834516, \quad 26065177.0036062, \quad 0, \quad 0\right )$$

Plot of S4 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 [262]:

    
positions={ii: [float(S4.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S4.g.nodes() }



In [263]:

    
x_ct, y_ct = S4.ct.subs(datav)

plt.figure(figsize=(12,8),dpi=300)
nx.draw(S4.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("Sezione 4",fontsize=16);
plt.savefig("sect4.jpg")

Expression of inertial properties in principal reference frame



In [264]:

    
S4.Ixx, S4.Iyy, S4.Ixy, S4.θ









    Out[264]:





$$\left ( 1083976.5834516, \quad 26065177.0036062, \quad 0, \quad 0\right )$$



In [265]:

    
Ixx4 = S4.Ixx.subs(datav)



In [266]:

    
S4.set_loads(_Tx=0, _Ty=0, _Nz=0, _Mx=0, _My=0, _Mz=0)



In [267]:

    
S4.compute_Jt()
S4.Jt









    Out[267]:





$$\left[\begin{matrix}1510459.46974866\end{matrix}\right]$$



In [268]:

    
Jt4 = S4.Jt[0]
Jt4









    Out[268]:





$$1510459.46974866$$

Section 5



In [269]:

    
L5 = 330.

xtip5 = -651.8 + np.arctan(tg_β)*(1070.+L5/2.)


# altezza longheroni
hms5 = 25.5
hfs5 = 24.
#pos longherone anteriore
xfs5 = -345.+ np.arctan(tg_β)*(L4+L5/2.)
# spessore pannello
tpt5 = 0.81
tpn5 = 0.64

# lunghezza pannelli 2
lpt5 = np.sqrt((hms5-hfs5)**2+xfs5**2)
lpn5 = np.sqrt(hfs5**2+(xfs5-xtip5)**2)

# area pannelli 2
Apt5 =lpt5*tpt5
Apn5 =lpn5*tpn5


#spessore longheroni
tms5 = 1.6+2.9
tfs5 = 0.

# area correnti
Astr_ms5=1.90260E+02
Astr_fs5=7.74400E+01

# AREE
Ams5 = hms5*tms5/6+Apt5/4+Astr_ms5
Afs5 = hfs5*tfs4/6+Apt5/4+Astr_fs5+Apn5/2
Amid5 = Apt5/2
Atip5 = Apn5



In [270]:

    
values = [(Ams, Ams5 * ureg.millimeter**2),(Afs, Afs5  * ureg.millimeter**2), \
          (Amid, Amid5 * ureg.millimeter**2),(Atip, Atip5 * ureg.millimeter), \
          (tms, tms5 * ureg.millimeter),(tfs, tfs5 * ureg.millimeter), \
          (tpt, tpt5 * ureg.millimeter), (tpn, tpn5 * ureg.millimeter), \
          (hms, hms5 * ureg.millimeter), (hfs, hfs5 * ureg.millimeter), \
          (xfs, xfs5 * ureg.millimeter), (xtip, xtip5 * ureg.millimeter), \
          (L,  L5* ureg.millimeter) ]
datav = [(v[0],v[1].magnitude) for v in values]



In [271]:

    
stringers = {1:[(sympy.Integer(0),sympy.Float(hms5)),sympy.Float(Ams5)],
             2:[(sympy.Float(xfs5/2),sympy.Float((hms5+hfs5)/2)),sympy.Float(Amid5)],
             3:[(sympy.Float(xfs5),sympy.Float(hfs5)),sympy.Float(Afs5)],
             4:[(sympy.Float(xtip5),sympy.Integer(0)),sympy.Float(Atip5)],
             5:[(sympy.Float(xfs5),sympy.Float(-hfs5)),sympy.Float(Afs5)],
             6:[(sympy.Float(xfs5/2),sympy.Float(-(hms5+hfs5)/2)),sympy.Float(Amid5)],
             7:[(sympy.Integer(0),sympy.Float(-hms5)),sympy.Float(Ams5)]}

panels = {(1,2):sympy.Float(tpt5),
          (2,3):sympy.Float(tpt5),
          (3,4):sympy.Float(tpn5),
          (4,5):sympy.Float(tpn5),
          (5,6):sympy.Float(tpt5),
          (6,7):sympy.Float(tpt5),
          (7,1):sympy.Float(tms5)}

Define section and perform first calculations



In [272]:

    
S5 = Section(stringers, panels)



In [273]:

    
S5.cycles









    Out[273]:





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

Plot of S5 section in original reference frame



In [274]:

    
positions={ii: [float(S5.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S5.g.nodes() }



In [275]:

    
plt.figure(figsize=(12,8),dpi=300)
nx.draw(S5.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 [276]:

    
S5.Ixx0, S5.Iyy0, S5.Ixy0, S4.α0









    Out[276]:





$$\left ( 614624.862490852, \quad 12644461.7235401, \quad 0, \quad 0\right )$$

Plot of S5 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 [277]:

    
positions={ii: [float(S5.g.node[ii]['pos'][i].subs(datav)) for i in range(2)] for ii in S5.g.nodes() }



In [278]:

    
x_ct, y_ct = S5.ct.subs(datav)

plt.figure(figsize=(12,8),dpi=300)
nx.draw(S4.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("Sezione 5",fontsize=16);
plt.savefig("sect5.jpg")

Expression of inertial properties in principal reference frame



In [279]:

    
S5.Ixx, S5.Iyy, S5.Ixy, S5.θ









    Out[279]:





$$\left ( 614624.862490852, \quad 12644461.7235401, \quad 0, \quad 0\right )$$



In [280]:

    
Ixx5 = S5.Ixx.subs(datav)



In [281]:

    
S5.set_loads(_Tx=0, _Ty=0, _Nz=0, _Mx=0, _My=0, _Mz=0)



In [282]:

    
S5.compute_Jt()
S5.Jt









    Out[282]:





$$\left[\begin{matrix}782674.000135508\end{matrix}\right]$$



In [283]:

    
Jt5 = S5.Jt[0]
Jt5









    Out[283]:





$$782674.000135508$$



In [284]:

    
E = 73100.
ν = .33
G = E/(2*(1+ν))

Compute torsion angle



In [285]:

    
Mz = 1000.



In [286]:

    
ϕ = Mz/G*(L1/Jt1+L2/Jt2+L3/Jt3+L4/Jt4+L5/Jt5)

overall torsion angle



In [287]:

    
ϕ









    Out[287]:





$$3.02155295537087 \cdot 10^{-5}$$

Compute tip displacement and rotation



In [288]:

    
L = np.array([L1, L2, L3, L4, L5])
Mx = 1000.
Jxx = np.array([Ixx1, Ixx2, Ixx3, Ixx4, Ixx5])



In [289]:

    
θ = Mx*L/(E*Jxx)
sum(θ)









    Out[289]:





$$1.68842916711248 \cdot 10^{-5}$$



In [290]:

    
Lc = np.zeros(6)
Lc[1:] = L



In [291]:

    
Lt = np.cumsum(Lc)



In [292]:

    
δ = [Mx*(Lt[i]-Lt[i-1])/(E*Jxx[i-1])*((Lt[i]+Lt[i-1])/2-Lt[-1]) for i in range(1,len(Lt))]



In [293]:

    
sum(δ)









    Out[293]:





$$-0.0085281991034386$$