Mehanika_1-11_Prostorne_rešetke

[Napomene o programskom kôdu u Sage-u:

Ćelije u kojima je prvi redak %auto ne trebate izvoditi ( evaluate ); one se izvode „same” pri otvaranju radnoga lista.

Funkcije, programska realizacija kojih nije bitna za razumijevanje gradiva Mehanike 1., „skrivene” su.]

%auto
default_viewer = 'threejs'
load (DATA + 'Truss_3D.sage')

Mehanika 1.

Statički određeni prostorni rešetkasti nosači

[Ovaj radni list sadrži program Truss_3D za izračunavanje sila u štapovima statički određenih prostornih rešetkastih nosača i primjere njegove primjene. Teorijske natuknice možete naći u datoteci Jednostavni statički određeni rešetkasti u Bilješkama i skicama s predavanja iz Mehanike 1.]

Primjer 1.

Opis rešetke:

čvorovi, štapovi, ležajevi, opterećenja u čvorovima

Zadavanje čvorova:

rječnik čvorova:

{ oznaka0: (x0, y0, z0), oznaka1: (x1, y1, z1), oznaka2: (x2, y2, z2), ... }

oznakai — oznaka (indeks) čvora i
(xi, yi, zi) — koordinate čvora i (os z je orijentirana „prema gore”)

općenito: rječnik (dictionary) — niz uređenih parova ključ: vrijednost navedenih unutar vitičastih zagrada {}

Zadavanje štapova:

rječnik štapova:

{ oznaka0: (čvor00, čvor10), oznaka1: (čvor01, čvor11), oznaka2: (čvor02, čvor12), ... }

oznakai — oznaka (indeks) štapa i
(čvor0i, čvor1i) — početni i krajnji čvor štapa i

Zadavanje ležajeva:

lista oznaka ležajnih čvorova:

[ oznaka0, oznaka1, ... ]

Zadavanje opterećenja u čvorovima:

rječnik opterećenja:

{ oznaka0: (Fx0, Fy0, Fz0), oznaka1: (Fx1, Fy1, Fz1), oznaka2: (Fx2, Fy2, Fz2), ... }

oznakai — oznaka čvora u kojem djeluje opterećenje
(Fxi, Fyi, Fzi) — komponente čvornog opterećenja

joints = { 0: (0.0, 0.0, 0.0), 1: (3.0, 0.0, 0.0), 2: (0.0, 3.0, 0.0), 3: (0.0, 0.0, 3.0) }
print 'joints:'
print_dict (joints)
print

bars = { 0: (0, 3), 1: (1, 3), 2: (2, 3) }
print 'bars:' 
print_dict (bars)
print

supports = [0, 1, 2]
print 'supports:', supports
print

loads = { 3: (100.0, 100.0, 0.0) }
print 'loads:' 
print_dict (loads)

joints:
   0:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:  (3.00000000000000, 0.000000000000000, 0.000000000000000)
   2:  (0.000000000000000, 3.00000000000000, 0.000000000000000)
   3:  (0.000000000000000, 0.000000000000000, 3.00000000000000)

bars:
   0:  (0, 3)
   1:  (1, 3)
   2:  (2, 3)

supports: [0, 1, 2]

loads:
   3:  (100.000000000000, 100.000000000000, 0.000000000000000)

joints:
   0:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:  (3.00000000000000, 0.000000000000000, 0.000000000000000)
   2:  (0.000000000000000, 3.00000000000000, 0.000000000000000)
   3:  (0.000000000000000, 0.000000000000000, 3.00000000000000)

bars:
   0:  (0, 3)
   1:  (1, 3)
   2:  (2, 3)

supports: [0, 1, 2]

loads:
   3:  (100.000000000000, 100.000000000000, 0.000000000000000)

Grafički prikaz rešetke:

crno — ležajni čvorovi
bijelo — slobodni čvorovi

plot_truss (joints, bars, supports, loads)

Izračunavanje sila u štapovima:

truss3D (joints, bars, supports, loads) — rezultat: rječnik s komponentama oznakai: silai

oznakai — oznaka štapa i (iz rječnika bars)
silai — sila u štapu i

bar_forces = truss3D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  199.99999999999997
   1:  -141.42135623730948
   2:  -141.42135623730948

   0:  199.99999999999997
   1:  -141.42135623730948
   2:  -141.42135623730948

Grafički prikaz rješenja:

plavo — štapovi s vlačnim silama
crveno — štapovi s tlačnim silama
žuto — štapovi bez sila

plot_truss_with_forces (joints, bars, supports, bar_forces)

Rješavanje u koracima:

slobodni čvorovi:

others (a1, a2) — rezultat: komponente liste ili rječnika a2 koje nisu u listi ili rječniku a1

free_joints = others (supports, joints)
free_joints

[3]

[3]

provjera zadovoljenja Maxwellova pravila:

$3\,n - b \,=\, 0$

$n$ — broj slobodnih čvorova

$b$ — broj štapova

maxwells_rule (joints, supports, bars)

3 * 1 - 3  ==  0

3 * 1 - 3  ==  0

„ravnotežna” matrica:

A = equilibrium_matrix (joints, bars, free_joints)
show (A.n (digits = 4))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 0.0000 & 0.7071 & 0.0000 \\ 0.0000 & 0.0000 & 0.7071 \\ -1.000 & -0.7071 & -0.7071 \end{array}\right)$

vektor vanjskih sila:

b = load_vector (loads, free_joints)
b

(-100.0, -100.0, -0.0)

(-100.0, -100.0, -0.0)

rješavanje matričnog sustava i pridruživanje rješenja štapovima:

dict_bar_force (bars, forces) — rezultat: rječnik s komponentama oznakai: silai

oznakai — oznaka štapa i u rječniku bars
silai — uzdužna sila u štapu i u listi forces

x = A \ b
x

(199.99999999999997, -141.42135623730948, -141.42135623730948)

(199.99999999999997, -141.42135623730948, -141.42135623730948)

bfs = dict_bar_force (bars, x)
print_dict (bfs)

   0:  199.99999999999997
   1:  -141.42135623730948
   2:  -141.42135623730948

   0:  199.99999999999997
   1:  -141.42135623730948
   2:  -141.42135623730948

Primjer 2.

joints = { 0: (0.0, 0.0, 0.0), 1: (5.0, 0.0, 0.0), 2: (0.0, 5.0, 0.0), 3: (5.0, 5.0, 0.0), 4: (2.5, 2.5, 10.0), 5: (-1.0, 0.0, 0.0), 6: (0.0, -1.0, 0.0), 7: (0.0, 0.0, -1.0), 8: (-1.0, 5.0, 0.0), 9: (0.0, 5.0, -1.0), 10: (5.0, 5.0, -1.0) }
print 'joints:'
print_dict (joints)
print

bars = { 1: (0, 2), 2: (0, 3), 3: (2, 3), 4: (3, 1), 5: (0, 1), 6: (1, 4), 7: (0, 4), 8: (2, 4), 9: (3, 4), 10: (0, 5), 11: (0, 6), 12: (0, 7), 13: (2, 8), 14: (2, 9), 15: (3, 10) }
print 'bars:' 
print_dict (bars)
print

supports = [5, 6, 7, 8, 9, 10]
print 'supports:', supports
print

loads = { 4: (50.0, -50.0, 0.0), 1: (0.0, 0.0, -100.0) }
print 'loads:' 
print_dict (loads)

joints:
   0:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   2:  (0.000000000000000, 5.00000000000000, 0.000000000000000)
   3:  (5.00000000000000, 5.00000000000000, 0.000000000000000)
   4:  (2.50000000000000, 2.50000000000000, 10.0000000000000)
   5:  (-1.00000000000000, 0.000000000000000, 0.000000000000000)
   6:  (0.000000000000000, -1.00000000000000, 0.000000000000000)
   7:  (0.000000000000000, 0.000000000000000, -1.00000000000000)
   8:  (-1.00000000000000, 5.00000000000000, 0.000000000000000)
   9:  (0.000000000000000, 5.00000000000000, -1.00000000000000)
  10:  (5.00000000000000, 5.00000000000000, -1.00000000000000)

bars:
   1:  (0, 2)
   2:  (0, 3)
   3:  (2, 3)
   4:  (3, 1)
   5:  (0, 1)
   6:  (1, 4)
   7:  (0, 4)
   8:  (2, 4)
   9:  (3, 4)
  10:  (0, 5)
  11:  (0, 6)
  12:  (0, 7)
  13:  (2, 8)
  14:  (2, 9)
  15:  (3, 10)

supports: [5, 6, 7, 8, 9, 10]

loads:
   1:  (0.000000000000000, 0.000000000000000, -100.000000000000)
   4:  (50.0000000000000, -50.0000000000000, 0.000000000000000)

joints:
   0:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   2:  (0.000000000000000, 5.00000000000000, 0.000000000000000)
   3:  (5.00000000000000, 5.00000000000000, 0.000000000000000)
   4:  (2.50000000000000, 2.50000000000000, 10.0000000000000)
   5:  (-1.00000000000000, 0.000000000000000, 0.000000000000000)
   6:  (0.000000000000000, -1.00000000000000, 0.000000000000000)
   7:  (0.000000000000000, 0.000000000000000, -1.00000000000000)
   8:  (-1.00000000000000, 5.00000000000000, 0.000000000000000)
   9:  (0.000000000000000, 5.00000000000000, -1.00000000000000)
  10:  (5.00000000000000, 5.00000000000000, -1.00000000000000)

bars:
   1:  (0, 2)
   2:  (0, 3)
   3:  (2, 3)
   4:  (3, 1)
   5:  (0, 1)
   6:  (1, 4)
   7:  (0, 4)
   8:  (2, 4)
   9:  (3, 4)
  10:  (0, 5)
  11:  (0, 6)
  12:  (0, 7)
  13:  (2, 8)
  14:  (2, 9)
  15:  (3, 10)

supports: [5, 6, 7, 8, 9, 10]

loads:
   1:  (0.000000000000000, 0.000000000000000, -100.000000000000)
   4:  (50.0000000000000, -50.0000000000000, 0.000000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.025) . rotateZ (pi/3) . show (frame = False)

plot_truss (joints, bars, supports, loads, load_scale)

load_scale — vrijednost kojom se množe vrijednosti sila za određivanje duljina strelica; podrazumijeva se vrijednost 0,01

bar_forces = truss3D (joints, bars, supports, loads)
print_dict (bar_forces)

   1:  -75.0
   2:  106.06601717798213
   3:  -25.000000000000014
   4:  -25.0
   5:  -25.0
   6:  106.06601717798213
   7:  -212.1320343559643
   8:  318.19805153394645
   9:  -212.1320343559643
  10:  -0.0
  11:  -50.0
  12:  -200.00000000000003
  13:  50.0
  14:  300.00000000000006
  15:  -200.00000000000003

   1:  -75.0
   2:  106.06601717798213
   3:  -25.000000000000014
   4:  -25.0
   5:  -25.0
   6:  106.06601717798213
   7:  -212.1320343559643
   8:  318.19805153394645
   9:  -212.1320343559643
  10:  -0.0
  11:  -50.0
  12:  -200.00000000000003
  13:  50.0
  14:  300.00000000000006
  15:  -200.00000000000003

plot_truss_with_forces (joints, bars, supports, bar_forces) . rotateZ (pi/3) . show (frame = False)

Rješavanje u koracima:

free_joints = others (supports, joints)
free_joints

[0, 1, 2, 3, 4]

[0, 1, 2, 3, 4]

maxwells_rule (joints, supports, bars)

3 * 5 - 15  ==  0

3 * 5 - 15  ==  0

A = equilibrium_matrix (joints, bars, free_joints)
show (A.n (digits = 3))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrrrrr} 0.000 & 0.707 & 0.000 & 0.000 & 1.00 & 0.000 & 0.236 & 0.000 & 0.000 & -1.00 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 1.00 & 0.707 & 0.000 & 0.000 & 0.000 & 0.000 & 0.236 & 0.000 & 0.000 & 0.000 & -1.00 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.943 & 0.000 & 0.000 & 0.000 & 0.000 & -1.00 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & -1.00 & -0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 1.00 & 0.000 & 0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.943 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 1.00 & 0.000 & 0.000 & 0.000 & 0.000 & 0.236 & 0.000 & 0.000 & 0.000 & 0.000 & -1.00 & 0.000 & 0.000 \\ -1.00 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & -0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.943 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & -1.00 & 0.000 \\ 0.000 & -0.707 & -1.00 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & -0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & -0.707 & 0.000 & -1.00 & 0.000 & 0.000 & 0.000 & 0.000 & -0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.943 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & -1.00 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.236 & -0.236 & -0.236 & 0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & -0.236 & -0.236 & 0.236 & 0.236 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & -0.943 & -0.943 & -0.943 & -0.943 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \end{array}\right)$

b = load_vector (loads, free_joints)
b

(0.0, 0.0, 0.0, -0.0, -0.0, 100.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -50.0, 50.0,
-0.0)

(0.0, 0.0, 0.0, -0.0, -0.0, 100.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -50.0, 50.0, -0.0)

x = A \ b
bfs = dict_bar_force (bars, x)
print_dict (bfs)

   1:  -75.0
   2:  106.06601717798213
   3:  -25.000000000000014
   4:  -25.0
   5:  -25.0
   6:  106.06601717798213
   7:  -212.1320343559643
   8:  318.19805153394645
   9:  -212.1320343559643
  10:  -0.0
  11:  -50.0
  12:  -200.00000000000003
  13:  50.0
  14:  300.00000000000006
  15:  -200.00000000000003

   1:  -75.0
   2:  106.06601717798213
   3:  -25.000000000000014
   4:  -25.0
   5:  -25.0
   6:  106.06601717798213
   7:  -212.1320343559643
   8:  318.19805153394645
   9:  -212.1320343559643
  10:  -0.0
  11:  -50.0
  12:  -200.00000000000003
  13:  50.0
  14:  300.00000000000006
  15:  -200.00000000000003

Primjer 3.

joints = { 1: (0.0, 0.0, 0.0), 2: (5.0, 0.0, 0.0), 3: (0.0, 10.0, 0.0), 4: (5.0, 10.0, 0.0), 5: (0.0, 0.0, 5.0), 6: (5.0, 0.0, 5.0), 7: (0.0, 5.0, 5.0), 8: (5.0, 5.0, 5.0), 9: (0.0, 10.0, 5.0), 10: (5.0, 10.0, 5.0), 11: (0.0, 0.0, 10.0), 12: (5.0, 0.0, 10.0,), 13: (0.0, 5.0, 10.0), 14: (5.0, 5.0, 10.0), 15: (0.0, 10.0, 10.0), 16: (5.0, 10.0, 10.0) }
print 'joints:'
print_dict (joints)
print

bars = { 1: (15, 16), 2: (13, 16), 3: (13, 14), 4: (12, 13), 5: (11, 12), 6: (8, 13), 7: (9, 10), 8: (8, 9), 9: (7, 8), 10: (6, 7), 11: (5, 6), 12: (6, 12), 13: (8, 12), 14: (8, 14), 15: (8, 16), 16: (10, 16), 17: (12, 14), 18: (14, 16), 19: (6, 8), 20: (8, 10), 21: (5, 11), 22: (7, 11), 23: (7, 13), 24: (9, 13), 25: (9, 15), 26: (11, 13), 27: (13, 15), 28: (5, 7), 29: (7, 9), 30: (9, 16), 31: (1, 5), 32: (1, 6), 33: (2, 6), 34: (3, 5), 35: (3, 9), 36: (4, 9) }
print 'bars:' 
print_dict (bars)
print

supports = [1, 2, 3, 4]
print 'supports:', supports
print

loads = { 14: (100.0, 0.0, 0.0) }
print 'loads:' 
print_dict (loads)

joints:
   1:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   3:  (0.000000000000000, 10.0000000000000, 0.000000000000000)
   4:  (5.00000000000000, 10.0000000000000, 0.000000000000000)
   5:  (0.000000000000000, 0.000000000000000, 5.00000000000000)
   6:  (5.00000000000000, 0.000000000000000, 5.00000000000000)
   7:  (0.000000000000000, 5.00000000000000, 5.00000000000000)
   8:  (5.00000000000000, 5.00000000000000, 5.00000000000000)
   9:  (0.000000000000000, 10.0000000000000, 5.00000000000000)
  10:  (5.00000000000000, 10.0000000000000, 5.00000000000000)
  11:  (0.000000000000000, 0.000000000000000, 10.0000000000000)
  12:  (5.00000000000000, 0.000000000000000, 10.0000000000000)
  13:  (0.000000000000000, 5.00000000000000, 10.0000000000000)
  14:  (5.00000000000000, 5.00000000000000, 10.0000000000000)
  15:  (0.000000000000000, 10.0000000000000, 10.0000000000000)
  16:  (5.00000000000000, 10.0000000000000, 10.0000000000000)

bars:
   1:  (15, 16)
   2:  (13, 16)
   3:  (13, 14)
   4:  (12, 13)
   5:  (11, 12)
   6:  (8, 13)
   7:  (9, 10)
   8:  (8, 9)
   9:  (7, 8)
  10:  (6, 7)
  11:  (5, 6)
  12:  (6, 12)
  13:  (8, 12)
  14:  (8, 14)
  15:  (8, 16)
  16:  (10, 16)
  17:  (12, 14)
  18:  (14, 16)
  19:  (6, 8)
  20:  (8, 10)
  21:  (5, 11)
  22:  (7, 11)
  23:  (7, 13)
  24:  (9, 13)
  25:  (9, 15)
  26:  (11, 13)
  27:  (13, 15)
  28:  (5, 7)
  29:  (7, 9)
  30:  (9, 16)
  31:  (1, 5)
  32:  (1, 6)
  33:  (2, 6)
  34:  (3, 5)
  35:  (3, 9)
  36:  (4, 9)

supports: [1, 2, 3, 4]

loads:
  14:  (100.000000000000, 0.000000000000000, 0.000000000000000)

joints:
   1:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   3:  (0.000000000000000, 10.0000000000000, 0.000000000000000)
   4:  (5.00000000000000, 10.0000000000000, 0.000000000000000)
   5:  (0.000000000000000, 0.000000000000000, 5.00000000000000)
   6:  (5.00000000000000, 0.000000000000000, 5.00000000000000)
   7:  (0.000000000000000, 5.00000000000000, 5.00000000000000)
   8:  (5.00000000000000, 5.00000000000000, 5.00000000000000)
   9:  (0.000000000000000, 10.0000000000000, 5.00000000000000)
  10:  (5.00000000000000, 10.0000000000000, 5.00000000000000)
  11:  (0.000000000000000, 0.000000000000000, 10.0000000000000)
  12:  (5.00000000000000, 0.000000000000000, 10.0000000000000)
  13:  (0.000000000000000, 5.00000000000000, 10.0000000000000)
  14:  (5.00000000000000, 5.00000000000000, 10.0000000000000)
  15:  (0.000000000000000, 10.0000000000000, 10.0000000000000)
  16:  (5.00000000000000, 10.0000000000000, 10.0000000000000)

bars:
   1:  (15, 16)
   2:  (13, 16)
   3:  (13, 14)
   4:  (12, 13)
   5:  (11, 12)
   6:  (8, 13)
   7:  (9, 10)
   8:  (8, 9)
   9:  (7, 8)
  10:  (6, 7)
  11:  (5, 6)
  12:  (6, 12)
  13:  (8, 12)
  14:  (8, 14)
  15:  (8, 16)
  16:  (10, 16)
  17:  (12, 14)
  18:  (14, 16)
  19:  (6, 8)
  20:  (8, 10)
  21:  (5, 11)
  22:  (7, 11)
  23:  (7, 13)
  24:  (9, 13)
  25:  (9, 15)
  26:  (11, 13)
  27:  (13, 15)
  28:  (5, 7)
  29:  (7, 9)
  30:  (9, 16)
  31:  (1, 5)
  32:  (1, 6)
  33:  (2, 6)
  34:  (3, 5)
  35:  (3, 9)
  36:  (4, 9)

supports: [1, 2, 3, 4]

loads:
  14:  (100.000000000000, 0.000000000000000, 0.000000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.025) . show (frame = False)

maxwells_rule (joints, supports, bars)

3 * 12 - 36  ==  0

3 * 12 - 36  ==  0

bar_forces = truss3D (joints, bars, supports, loads)
print_dict (bar_forces)

   1:  0.0
   2:  70.71067811865484
   3:  100.0
   4:  -2.0097183471152322e-14
   5:  0.0
   6:  -212.13203435596432
   7:  -2.842170943040401e-14
   8:  141.42135623730957
   9:  50.0
  10:  -70.71067811865476
  11:  -1.4210854715202004e-14
  12:  -100.00000000000003
  13:  141.42135623730957
  14:  0.0
  15:  70.71067811865476
  16:  0.0
  17:  -100.00000000000001
  18:  -100.00000000000001
  19:  50.0
  20:  -0.0
  21:  100.00000000000006
  22:  -141.4213562373096
  23:  100.00000000000006
  24:  70.71067811865474
  25:  -0.0
  26:  100.00000000000004
  27:  -0.0
  28:  2.6129691639212584e-14
  29:  -150.00000000000003
  30:  -70.71067811865477
  31:  100.00000000000004
  32:  70.71067811865478
  33:  -150.00000000000003
  34:  -2.9213883368193624e-14
  35:  49.99999999999998
  36:  -70.71067811865476

   1:  0.0
   2:  70.71067811865484
   3:  100.0
   4:  -2.0097183471152322e-14
   5:  0.0
   6:  -212.13203435596432
   7:  -2.842170943040401e-14
   8:  141.42135623730957
   9:  50.0
  10:  -70.71067811865476
  11:  -1.4210854715202004e-14
  12:  -100.00000000000003
  13:  141.42135623730957
  14:  0.0
  15:  70.71067811865476
  16:  0.0
  17:  -100.00000000000001
  18:  -100.00000000000001
  19:  50.0
  20:  -0.0
  21:  100.00000000000006
  22:  -141.4213562373096
  23:  100.00000000000006
  24:  70.71067811865474
  25:  -0.0
  26:  100.00000000000004
  27:  -0.0
  28:  2.6129691639212584e-14
  29:  -150.00000000000003
  30:  -70.71067811865477
  31:  100.00000000000004
  32:  70.71067811865478
  33:  -150.00000000000003
  34:  -2.9213883368193624e-14
  35:  49.99999999999998
  36:  -70.71067811865476

plot_truss_with_forces (joints, bars, supports, bar_forces) . show (frame = False)

Primjer 4. — Varijacija na temu

joints = { 1: (0.0, 0.0, 0.0), 2: (5.0, 0.0, 0.0), 3: (0.0, 10.0, 0.0), 4: (5.0, 10.0, 0.0), 5: (0.0, 0.0, 5.0), 6: (5.0, 0.0, 5.0), 7: (0.0, 5.0, 5.0), 8: (5.0, 5.0, 5.0), 9: (0.0, 10.0, 5.0), 10: (5.0, 10.0, 5.0), 11: (0.0, 0.0, 10.0), 12: (5.0, 0.0, 10.0,), 13: (0.0, 5.0, 10.0), 14: (5.0, 5.0, 10.0), 15: (0.0, 10.0, 10.0), 16: (5.0, 10.0, 10.0) }
print 'joints:'
print_dict (joints)
print

bars = { 1: (15, 16), 2: (13, 16), 3: (13, 14), 4: (12, 13), 5: (11, 12), 6: (6, 11), 7: (9, 10), 8: (8, 9), 9: (7, 8), 10: (5, 8), 11: (5, 6), 12: (6, 12), 13: (8, 12), 14: (8, 14), 15: (8, 16), 16: (10, 16), 17: (12, 14), 18: (14, 16), 19: (6, 8), 20: (8, 10), 21: (5, 11), 22: (5, 13), 23: (7, 13), 24: (9, 13), 25: (9, 15), 26: (11, 13), 27: (13, 15), 28: (5, 7), 29: (7, 9), 30: (9, 16), 31: (1, 5), 32: (1, 6), 33: (2, 6), 34: (3, 5), 35: (3, 9), 36: (4, 9) }
print 'bars:' 
print_dict (bars)
print

supports = [1, 2, 3, 4]
print 'supports:', supports
print

loads = { 14: (100.0, 0.0, 0.0) }
print 'loads:' 
print_dict (loads)

joints:
   1:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   3:  (0.000000000000000, 10.0000000000000, 0.000000000000000)
   4:  (5.00000000000000, 10.0000000000000, 0.000000000000000)
   5:  (0.000000000000000, 0.000000000000000, 5.00000000000000)
   6:  (5.00000000000000, 0.000000000000000, 5.00000000000000)
   7:  (0.000000000000000, 5.00000000000000, 5.00000000000000)
   8:  (5.00000000000000, 5.00000000000000, 5.00000000000000)
   9:  (0.000000000000000, 10.0000000000000, 5.00000000000000)
  10:  (5.00000000000000, 10.0000000000000, 5.00000000000000)
  11:  (0.000000000000000, 0.000000000000000, 10.0000000000000)
  12:  (5.00000000000000, 0.000000000000000, 10.0000000000000)
  13:  (0.000000000000000, 5.00000000000000, 10.0000000000000)
  14:  (5.00000000000000, 5.00000000000000, 10.0000000000000)
  15:  (0.000000000000000, 10.0000000000000, 10.0000000000000)
  16:  (5.00000000000000, 10.0000000000000, 10.0000000000000)

bars:
   1:  (15, 16)
   2:  (13, 16)
   3:  (13, 14)
   4:  (12, 13)
   5:  (11, 12)
   6:  (6, 11)
   7:  (9, 10)
   8:  (8, 9)
   9:  (7, 8)
  10:  (5, 8)
  11:  (5, 6)
  12:  (6, 12)
  13:  (8, 12)
  14:  (8, 14)
  15:  (8, 16)
  16:  (10, 16)
  17:  (12, 14)
  18:  (14, 16)
  19:  (6, 8)
  20:  (8, 10)
  21:  (5, 11)
  22:  (5, 13)
  23:  (7, 13)
  24:  (9, 13)
  25:  (9, 15)
  26:  (11, 13)
  27:  (13, 15)
  28:  (5, 7)
  29:  (7, 9)
  30:  (9, 16)
  31:  (1, 5)
  32:  (1, 6)
  33:  (2, 6)
  34:  (3, 5)
  35:  (3, 9)
  36:  (4, 9)

supports: [1, 2, 3, 4]

loads:
  14:  (100.000000000000, 0.000000000000000, 0.000000000000000)

joints:
   1:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   3:  (0.000000000000000, 10.0000000000000, 0.000000000000000)
   4:  (5.00000000000000, 10.0000000000000, 0.000000000000000)
   5:  (0.000000000000000, 0.000000000000000, 5.00000000000000)
   6:  (5.00000000000000, 0.000000000000000, 5.00000000000000)
   7:  (0.000000000000000, 5.00000000000000, 5.00000000000000)
   8:  (5.00000000000000, 5.00000000000000, 5.00000000000000)
   9:  (0.000000000000000, 10.0000000000000, 5.00000000000000)
  10:  (5.00000000000000, 10.0000000000000, 5.00000000000000)
  11:  (0.000000000000000, 0.000000000000000, 10.0000000000000)
  12:  (5.00000000000000, 0.000000000000000, 10.0000000000000)
  13:  (0.000000000000000, 5.00000000000000, 10.0000000000000)
  14:  (5.00000000000000, 5.00000000000000, 10.0000000000000)
  15:  (0.000000000000000, 10.0000000000000, 10.0000000000000)
  16:  (5.00000000000000, 10.0000000000000, 10.0000000000000)

bars:
   1:  (15, 16)
   2:  (13, 16)
   3:  (13, 14)
   4:  (12, 13)
   5:  (11, 12)
   6:  (6, 11)
   7:  (9, 10)
   8:  (8, 9)
   9:  (7, 8)
  10:  (5, 8)
  11:  (5, 6)
  12:  (6, 12)
  13:  (8, 12)
  14:  (8, 14)
  15:  (8, 16)
  16:  (10, 16)
  17:  (12, 14)
  18:  (14, 16)
  19:  (6, 8)
  20:  (8, 10)
  21:  (5, 11)
  22:  (5, 13)
  23:  (7, 13)
  24:  (9, 13)
  25:  (9, 15)
  26:  (11, 13)
  27:  (13, 15)
  28:  (5, 7)
  29:  (7, 9)
  30:  (9, 16)
  31:  (1, 5)
  32:  (1, 6)
  33:  (2, 6)
  34:  (3, 5)
  35:  (3, 9)
  36:  (4, 9)

supports: [1, 2, 3, 4]

loads:
  14:  (100.000000000000, 0.000000000000000, 0.000000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.025) . show (frame = False)

maxwells_rule (joints, supports, bars)

3 * 12 - 36  ==  0

3 * 12 - 36  ==  0

bar_forces = truss3D (joints, bars, supports, loads)
print_dict (bar_forces)

   1:  0.0
   2:  -35.35533905932738
   3:  100.0
   4:  -106.06601717798213
   5:  75.0
   6:  -106.06601717798213
   7:  3.552713678800501e-15
   8:  35.35533905932738
   9:  -0.0
  10:  -35.35533905932738
  11:  25.0
  12:  -25.0
  13:  35.35533905932738
  14:  -0.0
  15:  -35.35533905932738
  16:  0.0
  17:  50.0
  18:  50.0
  19:  0.0
  20:  -0.0
  21:  75.0
  22:  35.35533905932738
  23:  -0.0
  24:  -35.35533905932738
  25:  1.7763568394002505e-14
  26:  0.0
  27:  -0.0
  28:  3.552713678800501e-15
  29:  3.552713678800501e-15
  30:  35.355339059327356
  31:  100.0
  32:  70.71067811865476
  33:  -150.0
  34:  0.0
  35:  49.999999999999986
  36:  -70.71067811865474

   1:  0.0
   2:  -35.35533905932738
   3:  100.0
   4:  -106.06601717798213
   5:  75.0
   6:  -106.06601717798213
   7:  3.552713678800501e-15
   8:  35.35533905932738
   9:  -0.0
  10:  -35.35533905932738
  11:  25.0
  12:  -25.0
  13:  35.35533905932738
  14:  -0.0
  15:  -35.35533905932738
  16:  0.0
  17:  50.0
  18:  50.0
  19:  0.0
  20:  -0.0
  21:  75.0
  22:  35.35533905932738
  23:  -0.0
  24:  -35.35533905932738
  25:  1.7763568394002505e-14
  26:  0.0
  27:  -0.0
  28:  3.552713678800501e-15
  29:  3.552713678800501e-15
  30:  35.355339059327356
  31:  100.0
  32:  70.71067811865476
  33:  -150.0
  34:  0.0
  35:  49.999999999999986
  36:  -70.71067811865474

plot_truss_with_forces (joints, bars, supports, bar_forces) . show (frame = False)

Primjer 5. — Još jedna varijacija

joints = { 1: (0.0, 0.0, 0.0), 2: (5.0, 0.0, 0.0), 3: (0.0, 10.0, 0.0), 4: (5.0, 10.0, 0.0), 5: (0.0, 0.0, 5.0), 6: (5.0, 0.0, 5.0), 7: (0.0, 5.0, 5.0), 8: (5.0, 5.0, 5.0), 9: (0.0, 10.0, 5.0), 10: (5.0, 10.0, 5.0), 11: (0.0, 0.0, 10.0), 12: (5.0, 0.0, 10.0,), 13: (0.0, 5.0, 10.0), 14: (5.0, 5.0, 10.0), 15: (0.0, 10.0, 10.0), 16: (5.0, 10.0, 10.0) }
print 'joints:'
print_dict (joints)
print

bars = { 1: (15, 16), 2: (13, 16), 3: (13, 14), 4: (12, 13), 5: (11, 12), 6: (6, 11), 7: (9, 10), 8: (8, 9), 9: (7, 8), 10: (6, 7), 11: (5, 6), 12: (6, 12), 13: (8, 12), 14: (8, 14), 15: (8, 16), 16: (10, 16), 17: (12, 14), 18: (14, 16), 19: (6, 8), 20: (8, 10), 21: (5, 11), 22: (7, 11), 23: (7, 13), 24: (9, 13), 25: (9, 15), 26: (11, 13), 27: (13, 15), 28: (5, 7), 29: (7, 9), 30: (9, 16), 31: (1, 5), 32: (1, 6), 33: (2, 6), 34: (3, 5), 35: (3, 9), 36: (4, 10) }
print 'bars:' 
print_dict (bars)
print

supports = [1, 2, 3, 4]
print 'supports:', supports
print

loads = { 14: (100.0, 0.0, 0.0) }
print 'loads:' 
print_dict (loads)

joints:
   1:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   3:  (0.000000000000000, 10.0000000000000, 0.000000000000000)
   4:  (5.00000000000000, 10.0000000000000, 0.000000000000000)
   5:  (0.000000000000000, 0.000000000000000, 5.00000000000000)
   6:  (5.00000000000000, 0.000000000000000, 5.00000000000000)
   7:  (0.000000000000000, 5.00000000000000, 5.00000000000000)
   8:  (5.00000000000000, 5.00000000000000, 5.00000000000000)
   9:  (0.000000000000000, 10.0000000000000, 5.00000000000000)
  10:  (5.00000000000000, 10.0000000000000, 5.00000000000000)
  11:  (0.000000000000000, 0.000000000000000, 10.0000000000000)
  12:  (5.00000000000000, 0.000000000000000, 10.0000000000000)
  13:  (0.000000000000000, 5.00000000000000, 10.0000000000000)
  14:  (5.00000000000000, 5.00000000000000, 10.0000000000000)
  15:  (0.000000000000000, 10.0000000000000, 10.0000000000000)
  16:  (5.00000000000000, 10.0000000000000, 10.0000000000000)

bars:
   1:  (15, 16)
   2:  (13, 16)
   3:  (13, 14)
   4:  (12, 13)
   5:  (11, 12)
   6:  (6, 11)
   7:  (9, 10)
   8:  (8, 9)
   9:  (7, 8)
  10:  (6, 7)
  11:  (5, 6)
  12:  (6, 12)
  13:  (8, 12)
  14:  (8, 14)
  15:  (8, 16)
  16:  (10, 16)
  17:  (12, 14)
  18:  (14, 16)
  19:  (6, 8)
  20:  (8, 10)
  21:  (5, 11)
  22:  (7, 11)
  23:  (7, 13)
  24:  (9, 13)
  25:  (9, 15)
  26:  (11, 13)
  27:  (13, 15)
  28:  (5, 7)
  29:  (7, 9)
  30:  (9, 16)
  31:  (1, 5)
  32:  (1, 6)
  33:  (2, 6)
  34:  (3, 5)
  35:  (3, 9)
  36:  (4, 10)

supports: [1, 2, 3, 4]

loads:
  14:  (100.000000000000, 0.000000000000000, 0.000000000000000)

joints:
   1:  (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:  (5.00000000000000, 0.000000000000000, 0.000000000000000)
   3:  (0.000000000000000, 10.0000000000000, 0.000000000000000)
   4:  (5.00000000000000, 10.0000000000000, 0.000000000000000)
   5:  (0.000000000000000, 0.000000000000000, 5.00000000000000)
   6:  (5.00000000000000, 0.000000000000000, 5.00000000000000)
   7:  (0.000000000000000, 5.00000000000000, 5.00000000000000)
   8:  (5.00000000000000, 5.00000000000000, 5.00000000000000)
   9:  (0.000000000000000, 10.0000000000000, 5.00000000000000)
  10:  (5.00000000000000, 10.0000000000000, 5.00000000000000)
  11:  (0.000000000000000, 0.000000000000000, 10.0000000000000)
  12:  (5.00000000000000, 0.000000000000000, 10.0000000000000)
  13:  (0.000000000000000, 5.00000000000000, 10.0000000000000)
  14:  (5.00000000000000, 5.00000000000000, 10.0000000000000)
  15:  (0.000000000000000, 10.0000000000000, 10.0000000000000)
  16:  (5.00000000000000, 10.0000000000000, 10.0000000000000)

bars:
   1:  (15, 16)
   2:  (13, 16)
   3:  (13, 14)
   4:  (12, 13)
   5:  (11, 12)
   6:  (6, 11)
   7:  (9, 10)
   8:  (8, 9)
   9:  (7, 8)
  10:  (6, 7)
  11:  (5, 6)
  12:  (6, 12)
  13:  (8, 12)
  14:  (8, 14)
  15:  (8, 16)
  16:  (10, 16)
  17:  (12, 14)
  18:  (14, 16)
  19:  (6, 8)
  20:  (8, 10)
  21:  (5, 11)
  22:  (7, 11)
  23:  (7, 13)
  24:  (9, 13)
  25:  (9, 15)
  26:  (11, 13)
  27:  (13, 15)
  28:  (5, 7)
  29:  (7, 9)
  30:  (9, 16)
  31:  (1, 5)
  32:  (1, 6)
  33:  (2, 6)
  34:  (3, 5)
  35:  (3, 9)
  36:  (4, 10)

supports: [1, 2, 3, 4]

loads:
  14:  (100.000000000000, 0.000000000000000, 0.000000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.025) . show (frame = False)

bar_forces = truss3D (joints, bars, supports, loads)
print_dict (bar_forces)

Traceback (click to the left of this block for traceback)
...
numpy.linalg.LinAlgError: Matrix is singular.

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_38.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("YmFyX2ZvcmNlcyA9IHRydXNzM0QgKGpvaW50cywgYmFycywgc3VwcG9ydHMsIGxvYWRzKQpwcmludF9kaWN0IChiYXJfZm9yY2VzKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp3qM9L8/___code___.py", line 2, in <module>
    bar_forces = truss3D (joints, bars, supports, loads)
  File "<string>", line 85, in truss3D
  File "sage/matrix/matrix_double_dense.pyx", line 1760, in sage.matrix.matrix_double_dense.Matrix_double_dense.solve_right (build/cythonized/sage/matrix/matrix_double_dense.c:12195)
  File "/opt/SageMath/local/lib/python2.7/site-packages/scipy/linalg/basic.py", line 216, in solve
    _solve_check(n, info)
  File "/opt/SageMath/local/lib/python2.7/site-packages/scipy/linalg/basic.py", line 31, in _solve_check
    raise LinAlgError('Matrix is singular.')
numpy.linalg.LinAlgError: Matrix is singular.

Primjer 6. — Schwedlerova „kupola” (prve vrste)

Tvorba rječnika čvorova, rječnika štapova i liste ležajnih čvorova:

schwedler_1 (r0, [h1, h2, ...], n)

- r0 — radijus ležajnog prstena
- h1, h2, ... — visine na kojima su prsteni (0 $<$ h1 $<$ h2 $<$ ...); uzima se da je za ležajni prsten h0 = 0; posljednji h je visina sferina odsječka na kojemu su čvorovi; posljednji h mora biti $\le$ r0
- n — broj polja; n mora biti paran broj
- rezultat: redom rječnik čvorova, rječnik štapova, lista oznaka ležajnih čvorova

joints, bars, supports = schwedler_1 (10., [4.25, 7.75, 9.25, 10.], 8)

plot_truss (joints, bars, supports) . rotateZ (pi/36) . show (frame = False)

maxwells_rule (joints, supports, bars)

3 * 24 - 72  ==  0

3 * 24 - 72  ==  0

Tvorba rječnika opterećenja:

make_loads (generator)

generator — lista uređenih trojki (nl, j0, step, (Fx, Fy, Fz))

nl — broj čvorova opterećenih silom (Fx, Fy, Fz)
j0 — oznaka prvoga u nizu opterećenih čvorova
step — opterećuju se čvorovi s oznakama j0, j0 + step, j0 + 2$\cdot$step, ... , j0 + (nl-1)$\cdot$step; podrazumijeva se vrijednost 1

load_gen = [(8, 8, (0.0, 0.0, -90.0)), 
            (8, 16, (0.0, 0.0, -75.0)),
            (8, 24, (0.0, 0.0, -50.0))]
loads = make_loads (load_gen)

plot_truss (joints, bars, supports, loads, pull = False, load_scale = 0.05) . rotateZ (pi/36) . show (frame = False)

bar_forces = truss3D (joints, bars, supports, loads)

plot_truss_with_forces (joints, bars, supports, bar_forces) . rotateZ (pi/36) . show (frame = False)

Primjer 7. — Schwedlerova „kupola” (prve vrste)

joints, bars, supports = schwedler_1 (12., [2.725, 4.75, 5.75, 6.25], 8)

plot_truss (joints, bars, supports) . rotateZ (pi/36) . show (frame = False)

maxwells_rule (joints, supports, bars)

3 * 24 - 72  ==  0

3 * 24 - 72  ==  0

load_gen = [(8, 8, (0.0, 0.0, -90.0)), 
            (8, 16, (0.0, 0.0, -75.0)),
            (8, 24, (0.0, 0.0, -50.0))]
loads = make_loads (load_gen)

plot_truss (joints, bars, supports, loads, pull = False, load_scale = 0.05) . rotateZ (pi/36) . show (frame = False)

bar_forces = truss3D (joints, bars, supports, loads)

plot_truss_with_forces (joints, bars, supports, bar_forces) . rotateZ (pi/36) . show (frame = False)

loads = { 23: (0.0, 0.0, -100.0) }

plot_truss (joints, bars, supports, loads, pull = False, load_scale = 0.0325) . rotateZ (pi/36) . show (frame = False)

bar_forces = truss3D (joints, bars, supports, loads)

plot_truss_with_forces (joints, bars, supports, bar_forces) . rotateZ (pi/36) . show (frame = False)

Mehanika_1-11_Prostorne_rešetke_(v0.5)

1457 days ago by fresl