Mehanika_1-11_Ravninske_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
load (DATA + 'Truss_2D.sage')

Mehanika 1.

Statički određeni ravninski rešetkasti nosači

[Ovaj radni list sadrži program Truss_2D za izračunavanje sila u štapovima statički određenih ravninskih rešetkastih nosača i primjere njegove primjene. Teorijske natuknice možete naći u datoteci Jednostavni statički određeni rešetkasti nosači 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), oznaka1: (x1, y1), oznaka2: (x2, y2), ... }

oznakai — oznaka (indeks) čvora i
(xi, yi) — koordinate čvora i (os y 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), oznaka1: (Fx1, Fy1), oznaka2: (Fx2, Fy2), ... }

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

joints = { 0: (0.0, 0.0), 1: (3.0, 0.0), 2: (6.0, 0.0), 3: (1.5, 2.5), 4: (4.5, 2.5), 5: (-0.5, 0.0), 6: (0.0, -0.5), 7: (6.0, -0.5) }
print 'joints:'
print_dict (joints)
print

bars = { 0: (0, 1), 1: (1, 2), 2: (0, 3), 3: (3, 1), 4: (1, 4), 5: (4, 2), 6: (3, 4), 7: (5, 0), 8: (6, 0), 9: (7, 2) }
print 'bars:' 
print_dict (bars)
print

supports = [5, 6, 7]
print 'supports:', supports
print

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

joints:
   0:  (0.000000000000000, 0.000000000000000)
   1:  (3.00000000000000, 0.000000000000000)
   2:  (6.00000000000000, 0.000000000000000)
   3:  (1.50000000000000, 2.50000000000000)
   4:  (4.50000000000000, 2.50000000000000)
   5:  (-0.500000000000000, 0.000000000000000)
   6:  (0.000000000000000, -0.500000000000000)
   7:  (6.00000000000000, -0.500000000000000)

bars:
   0:  (0, 1)
   1:  (1, 2)
   2:  (0, 3)
   3:  (3, 1)
   4:  (1, 4)
   5:  (4, 2)
   6:  (3, 4)
   7:  (5, 0)
   8:  (6, 0)
   9:  (7, 2)

supports: [5, 6, 7]

loads:
   1:  (0.000000000000000, -100.000000000000)

joints:
   0:  (0.000000000000000, 0.000000000000000)
   1:  (3.00000000000000, 0.000000000000000)
   2:  (6.00000000000000, 0.000000000000000)
   3:  (1.50000000000000, 2.50000000000000)
   4:  (4.50000000000000, 2.50000000000000)
   5:  (-0.500000000000000, 0.000000000000000)
   6:  (0.000000000000000, -0.500000000000000)
   7:  (6.00000000000000, -0.500000000000000)

bars:
   0:  (0, 1)
   1:  (1, 2)
   2:  (0, 3)
   3:  (3, 1)
   4:  (1, 4)
   5:  (4, 2)
   6:  (3, 4)
   7:  (5, 0)
   8:  (6, 0)
   9:  (7, 2)

supports: [5, 6, 7]

loads:
   1:  (0.000000000000000, -100.000000000000)

Grafički prikaz rešetke:

crno — ležajni čvorovi
bijelo — slobodni čvorovi

plot_truss (joints, bars, supports, loads)

Izračunavanje sila u štapovima:

truss2D (joints, bars, supports, loads) — rezultat: rječnik s komponentama oznakai: silai

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

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  30.0
   1:  30.0
   2:  -58.3095189485
   3:  58.3095189485
   4:  58.3095189485
   5:  -58.3095189485
   6:  -60.0
   7:  -0.0
   8:  -50.0
   9:  -50.0

   0:  30.0
   1:  30.0
   2:  -58.3095189485
   3:  58.3095189485
   4:  58.3095189485
   5:  -58.3095189485
   6:  -60.0
   7:  -0.0
   8:  -50.0
   9:  -50.0

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

[0, 1, 2, 3, 4]

[0, 1, 2, 3, 4]

provjera zadovoljenja Maxwellova pravila:

$2\,n - b = 0$

$n$ — broj slobodnih čvorova

$b$ — broj štapova

maxwells_rule (joints, supports, bars)

2 * 5 - 10  ==  0

2 * 5 - 10  ==  0

„ravnotežna” matrica:

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

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrr} 1.000 & 0.0000 & 0.5145 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -1.000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.8575 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -1.000 & 0.0000 \\ -1.000 & 1.000 & 0.0000 & -0.5145 & 0.5145 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.8575 & 0.8575 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & -1.000 & 0.0000 & 0.0000 & 0.0000 & -0.5145 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.8575 & 0.0000 & 0.0000 & 0.0000 & -1.000 \\ 0.0000 & 0.0000 & -0.5145 & 0.5145 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & -0.8575 & -0.8575 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5145 & 0.5145 & -1.000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.8575 & -0.8575 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

vektor vanjskih sila:

b = load_vector (loads, free_joints)
show (b.n (digits = 4))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.0000,\,0.0000,\,-0.0000,\,100.0,\,0.0000,\,0.0000,\,0.0000,\,0.0000,\,0.0000,\,0.0000\right)$

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

(29.999999999999996, 29.999999999999996, -58.309518948453004, 58.30951894845301,
58.30951894845301, -58.30951894845301, -60.0, -0.0, -49.99999999999999,
-49.99999999999999)

(29.999999999999996, 29.999999999999996, -58.309518948453004, 58.30951894845301, 58.30951894845301, -58.30951894845301, -60.0, -0.0, -49.99999999999999, -49.99999999999999)

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

   0:  30.0
   1:  30.0
   2:  -58.3095189485
   3:  58.3095189485
   4:  58.3095189485
   5:  -58.3095189485
   6:  -60.0
   7:  -0.0
   8:  -50.0
   9:  -50.0

   0:  30.0
   1:  30.0
   2:  -58.3095189485
   3:  58.3095189485
   4:  58.3095189485
   5:  -58.3095189485
   6:  -60.0
   7:  -0.0
   8:  -50.0
   9:  -50.0

Primjer 2.

joints = { 0: (1.6, 3.0), 1: (3*1.6, 3.0), 2: (5*1.6, 3.0), 3: (0.0, 0.0), 4: (3.2, 0.0), 5: (2*3.2, 0.0), 6: (3*3.2, 0.0), 7: (-0.5, 0.0), 8: (0.0, -0.5), 9: (2*3.2, -0.5) }
print 'joints:'
print_dict (joints)
print

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

supports = [7, 8, 9]
print 'supports:', supports
print

loads = { 1: (0.0, -22.0), 6: (22.4, 16.8) }
print 'loads:' 
print_dict (loads)

joints:
   0:  (1.60000000000000, 3.00000000000000)
   1:  (4.80000000000000, 3.00000000000000)
   2:  (8.00000000000000, 3.00000000000000)
   3:  (0.000000000000000, 0.000000000000000)
   4:  (3.20000000000000, 0.000000000000000)
   5:  (6.40000000000000, 0.000000000000000)
   6:  (9.60000000000000, 0.000000000000000)
   7:  (-0.500000000000000, 0.000000000000000)
   8:  (0.000000000000000, -0.500000000000000)
   9:  (6.40000000000000, -0.500000000000000)

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

supports: [7, 8, 9]

loads:
   1:  (0.000000000000000, -22.0000000000000)
   6:  (22.4000000000000, 16.8000000000000)

joints:
   0:  (1.60000000000000, 3.00000000000000)
   1:  (4.80000000000000, 3.00000000000000)
   2:  (8.00000000000000, 3.00000000000000)
   3:  (0.000000000000000, 0.000000000000000)
   4:  (3.20000000000000, 0.000000000000000)
   5:  (6.40000000000000, 0.000000000000000)
   6:  (9.60000000000000, 0.000000000000000)
   7:  (-0.500000000000000, 0.000000000000000)
   8:  (0.000000000000000, -0.500000000000000)
   9:  (6.40000000000000, -0.500000000000000)

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

supports: [7, 8, 9]

loads:
   1:  (0.000000000000000, -22.0000000000000)
   6:  (22.4000000000000, 16.8000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.05)

maxwells_rule (joints, supports, bars)

2 * 7 - 14  ==  0

2 * 7 - 14  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  -14.8266666667
   1:  -17.92
   2:  -15.7533333333
   3:  15.7533333333
   4:  -15.7533333333
   5:  -9.18
   6:  19.04
   7:  -19.04
   8:  29.8133333333
   9:  44.64
  10:  31.36
  11:  22.4
  12:  -13.9
  13:  8.7

   0:  -14.8266666667
   1:  -17.92
   2:  -15.7533333333
   3:  15.7533333333
   4:  -15.7533333333
   5:  -9.18
   6:  19.04
   7:  -19.04
   8:  29.8133333333
   9:  44.64
  10:  31.36
  11:  22.4
  12:  -13.9
  13:  8.7

plot_truss_with_forces (joints, bars, supports, bar_forces)

Primjer 2., još jednom

joints = { 0: (1.6, 3.0), 1: (3*1.6, 3.0), 2: (5*1.6, 3.0), 3: (0.0, 0.0), 4: (3.2, 0.0), 5: (2*3.2, 0.0), 6: (3*3.2, 0.0), 9: (2*3.2, -0.5) }
print 'joints:'
print_dict (joints)
print

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

supports = [3, 9]
print 'supports:', supports
print

loads = { 1: (0.0, -22.0), 6: (22.4, 16.8) }
print 'loads:' 
print_dict (loads)

joints:
   0:  (1.60000000000000, 3.00000000000000)
   1:  (4.80000000000000, 3.00000000000000)
   2:  (8.00000000000000, 3.00000000000000)
   3:  (0.000000000000000, 0.000000000000000)
   4:  (3.20000000000000, 0.000000000000000)
   5:  (6.40000000000000, 0.000000000000000)
   6:  (9.60000000000000, 0.000000000000000)
   9:  (6.40000000000000, -0.500000000000000)

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

supports: [3, 9]

loads:
   1:  (0.000000000000000, -22.0000000000000)
   6:  (22.4000000000000, 16.8000000000000)

joints:
   0:  (1.60000000000000, 3.00000000000000)
   1:  (4.80000000000000, 3.00000000000000)
   2:  (8.00000000000000, 3.00000000000000)
   3:  (0.000000000000000, 0.000000000000000)
   4:  (3.20000000000000, 0.000000000000000)
   5:  (6.40000000000000, 0.000000000000000)
   6:  (9.60000000000000, 0.000000000000000)
   9:  (6.40000000000000, -0.500000000000000)

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

supports: [3, 9]

loads:
   1:  (0.000000000000000, -22.0000000000000)
   6:  (22.4000000000000, 16.8000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.05)

maxwells_rule (joints, supports, bars)

2 * 6 - 12  ==  0

2 * 6 - 12  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  -14.8266666667
   1:  -17.92
   2:  -15.7533333333
   3:  15.7533333333
   4:  -15.7533333333
   5:  -9.18
   6:  19.04
   7:  -19.04
   8:  29.8133333333
   9:  44.64
  10:  31.36
  13:  8.7

   0:  -14.8266666667
   1:  -17.92
   2:  -15.7533333333
   3:  15.7533333333
   4:  -15.7533333333
   5:  -9.18
   6:  19.04
   7:  -19.04
   8:  29.8133333333
   9:  44.64
  10:  31.36
  13:  8.7

plot_truss_with_forces (joints, bars, supports, bar_forces)

Primjer 3.

joints = { 1: (0.0, 0.0), 2: (3.0, 4.0), 3: (0.0, 4.0), 4: (-6.0, 6.5), 5: (-6.0, -0.5), 6: (3.0, 0.0), 7: (10.5, 0.0) }
print 'joints:'
print_dict (joints)
print

bars = { 1: (1, 3), 2: (1, 2), 3: (6, 2), 4: (2, 7), 5: (3, 5), 6: (3, 4) }
print 'bars:' 
print_dict (bars)
print

supports = [4, 5, 6, 7]
print 'supports:', supports
print

loads = { 1: (-30.0, 0.0), 2: (60.0, 0.0), 3: (0.0, -90.0) }
print 'loads:' 
print_dict (loads)

joints:
   1:  (0.000000000000000, 0.000000000000000)
   2:  (3.00000000000000, 4.00000000000000)
   3:  (0.000000000000000, 4.00000000000000)
   4:  (-6.00000000000000, 6.50000000000000)
   5:  (-6.00000000000000, -0.500000000000000)
   6:  (3.00000000000000, 0.000000000000000)
   7:  (10.5000000000000, 0.000000000000000)

bars:
   1:  (1, 3)
   2:  (1, 2)
   3:  (6, 2)
   4:  (2, 7)
   5:  (3, 5)
   6:  (3, 4)

supports: [4, 5, 6, 7]

loads:
   1:  (-30.0000000000000, 0.000000000000000)
   2:  (60.0000000000000, 0.000000000000000)
   3:  (0.000000000000000, -90.0000000000000)

joints:
   1:  (0.000000000000000, 0.000000000000000)
   2:  (3.00000000000000, 4.00000000000000)
   3:  (0.000000000000000, 4.00000000000000)
   4:  (-6.00000000000000, 6.50000000000000)
   5:  (-6.00000000000000, -0.500000000000000)
   6:  (3.00000000000000, 0.000000000000000)
   7:  (10.5000000000000, 0.000000000000000)

bars:
   1:  (1, 3)
   2:  (1, 2)
   3:  (6, 2)
   4:  (2, 7)
   5:  (3, 5)
   6:  (3, 4)

supports: [4, 5, 6, 7]

loads:
   1:  (-30.0000000000000, 0.000000000000000)
   2:  (60.0000000000000, 0.000000000000000)
   3:  (0.000000000000000, -90.0000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.02)

maxwells_rule (joints, supports, bars)

2 * 3 - 6  ==  0

2 * 3 - 6  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   1:  -40.0
   2:  50.0
   3:  -24.0
   4:  -34.0
   5:  -53.5714285714
   6:  46.4285714286

   1:  -40.0
   2:  50.0
   3:  -24.0
   4:  -34.0
   5:  -53.5714285714
   6:  46.4285714286

plot_truss_with_forces (joints, bars, supports, bar_forces)

Primjer 4.

Tvorba rječnika čvorova:

make_joints (j0, generator)

j0 — oznaka prvoga čvora
generator — lista uređenih trojki (nj, (x0, y0), (dx, dy))

nastaje niz čvorova s koordinatama (x0, y0), (x0 + dx, + ), ( + 2 $\cdot$ dx, + 2 $\cdot$ ), ..., ( + (nj-1) $\cdot$ , + (-1) $\cdot$ dy)

joint_gen = [(11, (0.0, 0.0), (1.5, 0.0)), 
             (11, (0.0, 2.0), (1.5, 0.0))]
joints = make_joints (0, joint_gen)
joints[22] = (0.0, -0.5)
joints[23] = (15.0, -0.5)
joints[24] = (-0.5, 2.0)
print_dict (joints)

   0:  [0.000000000000000, 0.000000000000000]
   1:  [1.50000000000000, 0.000000000000000]
   2:  [3.00000000000000, 0.000000000000000]
   3:  [4.50000000000000, 0.000000000000000]
   4:  [6.00000000000000, 0.000000000000000]
   5:  [7.50000000000000, 0.000000000000000]
   6:  [9.00000000000000, 0.000000000000000]
   7:  [10.5000000000000, 0.000000000000000]
   8:  [12.0000000000000, 0.000000000000000]
   9:  [13.5000000000000, 0.000000000000000]
  10:  [15.0000000000000, 0.000000000000000]
  11:  [0.000000000000000, 2.00000000000000]
  12:  [1.50000000000000, 2.00000000000000]
  13:  [3.00000000000000, 2.00000000000000]
  14:  [4.50000000000000, 2.00000000000000]
  15:  [6.00000000000000, 2.00000000000000]
  16:  [7.50000000000000, 2.00000000000000]
  17:  [9.00000000000000, 2.00000000000000]
  18:  [10.5000000000000, 2.00000000000000]
  19:  [12.0000000000000, 2.00000000000000]
  20:  [13.5000000000000, 2.00000000000000]
  21:  [15.0000000000000, 2.00000000000000]
  22:  (0.000000000000000, -0.500000000000000)
  23:  (15.0000000000000, -0.500000000000000)
  24:  (-0.500000000000000, 2.00000000000000)

   0:  [0.000000000000000, 0.000000000000000]
   1:  [1.50000000000000, 0.000000000000000]
   2:  [3.00000000000000, 0.000000000000000]
   3:  [4.50000000000000, 0.000000000000000]
   4:  [6.00000000000000, 0.000000000000000]
   5:  [7.50000000000000, 0.000000000000000]
   6:  [9.00000000000000, 0.000000000000000]
   7:  [10.5000000000000, 0.000000000000000]
   8:  [12.0000000000000, 0.000000000000000]
   9:  [13.5000000000000, 0.000000000000000]
  10:  [15.0000000000000, 0.000000000000000]
  11:  [0.000000000000000, 2.00000000000000]
  12:  [1.50000000000000, 2.00000000000000]
  13:  [3.00000000000000, 2.00000000000000]
  14:  [4.50000000000000, 2.00000000000000]
  15:  [6.00000000000000, 2.00000000000000]
  16:  [7.50000000000000, 2.00000000000000]
  17:  [9.00000000000000, 2.00000000000000]
  18:  [10.5000000000000, 2.00000000000000]
  19:  [12.0000000000000, 2.00000000000000]
  20:  [13.5000000000000, 2.00000000000000]
  21:  [15.0000000000000, 2.00000000000000]
  22:  (0.000000000000000, -0.500000000000000)
  23:  (15.0000000000000, -0.500000000000000)
  24:  (-0.500000000000000, 2.00000000000000)

Tvorba rječnika štapova:

make_bars (b0, generator)

b0 — oznaka prvog štapa
generator — lista uređenih trojki (nb, (a0, b0), (as, bs))

nastaje niz štapova sa čvorovima oznake kojih su (a0, b0), (a0 + as, b0 + bs), (a + 2 $\cdot$ as, b0 + 2 $\cdot$ bs), ..., (a0 + (nb-1) $\cdot$ as, b0 + (nb-1) $\cdot$ bs)

bar_gen = [(10, (0, 1), (1, 1)),
           (10, (11, 12), (1, 1)),
           (11, (0, 11), (1, 1)),
           (10, (0, 12), (1, 1))]
bars = make_bars (0, bar_gen)
bars[41] = (22, 0)
bars[42] = (23, 10)
bars[43] = (24, 11)
print_dict (bars)

   0:  [0, 1]
   1:  [1, 2]
   2:  [2, 3]
   3:  [3, 4]
   4:  [4, 5]
   5:  [5, 6]
   6:  [6, 7]
   7:  [7, 8]
   8:  [8, 9]
   9:  [9, 10]
  10:  [11, 12]
  11:  [12, 13]
  12:  [13, 14]
  13:  [14, 15]
  14:  [15, 16]
  15:  [16, 17]
  16:  [17, 18]
  17:  [18, 19]
  18:  [19, 20]
  19:  [20, 21]
  20:  [0, 11]
  21:  [1, 12]
  22:  [2, 13]
  23:  [3, 14]
  24:  [4, 15]
  25:  [5, 16]
  26:  [6, 17]
  27:  [7, 18]
  28:  [8, 19]
  29:  [9, 20]
  30:  [10, 21]
  31:  [0, 12]
  32:  [1, 13]
  33:  [2, 14]
  34:  [3, 15]
  35:  [4, 16]
  36:  [5, 17]
  37:  [6, 18]
  38:  [7, 19]
  39:  [8, 20]
  40:  [9, 21]
  41:  (22, 0)
  42:  (23, 10)
  43:  (24, 11)

   0:  [0, 1]
   1:  [1, 2]
   2:  [2, 3]
   3:  [3, 4]
   4:  [4, 5]
   5:  [5, 6]
   6:  [6, 7]
   7:  [7, 8]
   8:  [8, 9]
   9:  [9, 10]
  10:  [11, 12]
  11:  [12, 13]
  12:  [13, 14]
  13:  [14, 15]
  14:  [15, 16]
  15:  [16, 17]
  16:  [17, 18]
  17:  [18, 19]
  18:  [19, 20]
  19:  [20, 21]
  20:  [0, 11]
  21:  [1, 12]
  22:  [2, 13]
  23:  [3, 14]
  24:  [4, 15]
  25:  [5, 16]
  26:  [6, 17]
  27:  [7, 18]
  28:  [8, 19]
  29:  [9, 20]
  30:  [10, 21]
  31:  [0, 12]
  32:  [1, 13]
  33:  [2, 14]
  34:  [3, 15]
  35:  [4, 16]
  36:  [5, 17]
  37:  [6, 18]
  38:  [7, 19]
  39:  [8, 20]
  40:  [9, 21]
  41:  (22, 0)
  42:  (23, 10)
  43:  (24, 11)

supports = [22, 23, 24]
loads = { 0: (-12.0, 0.0), 13: (0.0, -16.0), 17: (0.0, 18.0), 19: (12.0, -16.0) }

plot_truss (joints, bars, supports, loads, load_scale = 0.075)

maxwells_rule (joints, supports, bars)

2 * 22 - 44  ==  0

2 * 22 - 44  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  17.4
   1:  22.8
   2:  16.2
   3:  9.6
   4:  3.0
   5:  -3.6
   6:  3.3
   7:  10.2
   8:  5.1
   9:  -2.6645352591e-15
  10:  0.0
  11:  -5.4
  12:  -10.8
  13:  -4.2
  14:  2.4
  15:  9.0
  16:  15.6
  17:  8.7
  18:  -10.2
  19:  -5.1
  20:  -0.0
  21:  7.2
  22:  -8.8
  23:  -8.8
  24:  -8.8
  25:  -8.8
  26:  9.2
  27:  9.2
  28:  -6.8
  29:  -6.8
  30:  -6.8
  31:  -9.0
  32:  -9.0
  33:  11.0
  34:  11.0
  35:  11.0
  36:  11.0
  37:  -11.5
  38:  -11.5
  39:  8.5
  40:  8.5
  41:  -7.2
  42:  -6.8
  43:  -0.0

   0:  17.4
   1:  22.8
   2:  16.2
   3:  9.6
   4:  3.0
   5:  -3.6
   6:  3.3
   7:  10.2
   8:  5.1
   9:  -2.6645352591e-15
  10:  0.0
  11:  -5.4
  12:  -10.8
  13:  -4.2
  14:  2.4
  15:  9.0
  16:  15.6
  17:  8.7
  18:  -10.2
  19:  -5.1
  20:  -0.0
  21:  7.2
  22:  -8.8
  23:  -8.8
  24:  -8.8
  25:  -8.8
  26:  9.2
  27:  9.2
  28:  -6.8
  29:  -6.8
  30:  -6.8
  31:  -9.0
  32:  -9.0
  33:  11.0
  34:  11.0
  35:  11.0
  36:  11.0
  37:  -11.5
  38:  -11.5
  39:  8.5
  40:  8.5
  41:  -7.2
  42:  -6.8
  43:  -0.0

plot_truss_with_forces (joints, bars, supports, bar_forces)

Primjer 5.

joints = { 0: (0, 0), 1: (5, 1.0), 2: (10.0, 0.0), 3: (1.0, 3.0), 4: (9.0, 3.0), 5: (5.0, 5.0), 6: (-1.0, 0.0), 7: (0.0, -1.0), 8: (10.0, -1.0) }
print 'joints:'
print_dict (joints)
print

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

supports = [6, 7, 8]
print 'supports:', supports
print

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

joints:
   0:  (0, 0)
   1:  (5, 1.00000000000000)
   2:  (10.0000000000000, 0.000000000000000)
   3:  (1.00000000000000, 3.00000000000000)
   4:  (9.00000000000000, 3.00000000000000)
   5:  (5.00000000000000, 5.00000000000000)
   6:  (-1.00000000000000, 0.000000000000000)
   7:  (0.000000000000000, -1.00000000000000)
   8:  (10.0000000000000, -1.00000000000000)

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

supports: [6, 7, 8]

loads:
   1:  (0.000000000000000, -50.0000000000000)

joints:
   0:  (0, 0)
   1:  (5, 1.00000000000000)
   2:  (10.0000000000000, 0.000000000000000)
   3:  (1.00000000000000, 3.00000000000000)
   4:  (9.00000000000000, 3.00000000000000)
   5:  (5.00000000000000, 5.00000000000000)
   6:  (-1.00000000000000, 0.000000000000000)
   7:  (0.000000000000000, -1.00000000000000)
   8:  (10.0000000000000, -1.00000000000000)

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

supports: [6, 7, 8]

loads:
   1:  (0.000000000000000, -50.0000000000000)

plot_truss (joints, bars, supports, loads, load_scale = 0.025)

maxwells_rule (joints, supports, bars)

2 * 6 - 12  ==  0

2 * 6 - 12  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  -21.24591464
   1:  -21.24591464
   2:  -32.9403922934
   3:  -46.5847495312
   4:  -46.5847495312
   5:  -32.9403922934
   6:  41.6666666667
   7:  32.9403922934
   8:  32.9403922934
   9:  -2.26933941565e-15
  10:  -25.0
  11:  -25.0

   0:  -21.24591464
   1:  -21.24591464
   2:  -32.9403922934
   3:  -46.5847495312
   4:  -46.5847495312
   5:  -32.9403922934
   6:  41.6666666667
   7:  32.9403922934
   8:  32.9403922934
   9:  -2.26933941565e-15
  10:  -25.0
  11:  -25.0

plot_truss_with_forces (joints, bars, supports, bar_forces)

loads = { 2: (50, 0) }
print 'loads:' 
print_dict (loads)
print

loads:
   2:  (50, 0)

loads:
   2:  (50, 0)

plot_truss (joints, bars, supports, loads, load_scale = 0.025)

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  42.4918292799
   1:  42.4918292799
   2:  -13.1761569174
   3:  -18.6338998125
   4:  -18.6338998125
   5:  -13.1761569174
   6:  16.6666666667
   7:  13.1761569174
   8:  13.1761569174
   9:  50.0
  10:  1.2490009027e-15
  11:  -1.2490009027e-15

   0:  42.4918292799
   1:  42.4918292799
   2:  -13.1761569174
   3:  -18.6338998125
   4:  -18.6338998125
   5:  -13.1761569174
   6:  16.6666666667
   7:  13.1761569174
   8:  13.1761569174
   9:  50.0
  10:  1.2490009027e-15
  11:  -1.2490009027e-15

plot_truss_with_forces (joints, bars, supports, bar_forces)

Primjer 6.

Točna aritmetika:

a = 5 / sqrt (3)
b = 5
c = 10 / sqrt (3)

joints = { 0: (0, 0), 1: (b, -a), 2: (2*b, 0), 3: (0, c), 4: (2*b, c), 5: (b, a+c), 6: (-1, 0), 7: (0, -1), 8: (2*b, -1) }
print 'joints:'
print_dict (joints)
print

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

supports = [6, 7, 8]
print 'supports:', supports
print

loads = { 1: (0, -50) }
print 'loads:' 
print_dict (loads)

joints:
   0:  (0, 0)
   1:  (5, -5/3*sqrt(3))
   2:  (10, 0)
   3:  (0, 10/3*sqrt(3))
   4:  (10, 10/3*sqrt(3))
   5:  (5, 5*sqrt(3))
   6:  (-1, 0)
   7:  (0, -1)
   8:  (10, -1)

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

supports: [6, 7, 8]

loads:
   1:  (0, -50)

joints:
   0:  (0, 0)
   1:  (5, -5/3*sqrt(3))
   2:  (10, 0)
   3:  (0, 10/3*sqrt(3))
   4:  (10, 10/3*sqrt(3))
   5:  (5, 5*sqrt(3))
   6:  (-1, 0)
   7:  (0, -1)
   8:  (10, -1)

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

supports: [6, 7, 8]

loads:
   1:  (0, -50)

plot_truss (joints, bars, supports, loads, load_scale = 0.025)

maxwells_rule (joints, supports, bars)

2 * 6 - 12  ==  0

2 * 6 - 12  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

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

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_49.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("YmFyX2ZvcmNlcyA9IHRydXNzMkQgKGpvaW50cywgYmFycywgc3VwcG9ydHMsIGxvYWRzKQpwcmludF9kaWN0IChiYXJfZm9yY2VzKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp4PU8IR/___code___.py", line 2, in <module>
    bar_forces = truss2D (joints, bars, supports, loads)
  File "<string>", line 82, in truss2D
  File "sage/matrix/matrix_double_dense.pyx", line 1758, in sage.matrix.matrix_double_dense.Matrix_double_dense.solve_right (/home/sage/SageMath/src/build/cythonized/sage/matrix/matrix_double_dense.c:13465)
  File "/home/sage/SageMath/local/lib/python2.7/site-packages/scipy/linalg/basic.py", line 103, in solve
    raise LinAlgError("singular matrix")
numpy.linalg.linalg.LinAlgError: singular matrix

Pseudorealna aritmetika:

a = 2.88675    # '5 / sqrt (3)'   s točnošću od 6 značajnih znamenaka 
b = 5.
c = 5.77350    # '10 / sqrt (3)'  s točnošću od 6 značajnih znamenaka 

joints = { 0: (0., 0.), 1: (b, -a), 2: (2*b, 0.), 3: (0., c), 4: (2*b, c), 5: (b, a+c), 6: (-1., 0.), 7: (0., -1.), 8: (2*b, -1.) }
print 'joints:'
print_dict (joints)
print

joints:
   0:  (0.000000000000000, 0.000000000000000)
   1:  (5.00000000000000, -2.88675000000000)
   2:  (10.0000000000000, 0.000000000000000)
   3:  (0.000000000000000, 5.77350000000000)
   4:  (10.0000000000000, 5.77350000000000)
   5:  (5.00000000000000, 8.66025000000000)
   6:  (-1.00000000000000, 0.000000000000000)
   7:  (0.000000000000000, -1.00000000000000)
   8:  (10.0000000000000, -1.00000000000000)

joints:
   0:  (0.000000000000000, 0.000000000000000)
   1:  (5.00000000000000, -2.88675000000000)
   2:  (10.0000000000000, 0.000000000000000)
   3:  (0.000000000000000, 5.77350000000000)
   4:  (10.0000000000000, 5.77350000000000)
   5:  (5.00000000000000, 8.66025000000000)
   6:  (-1.00000000000000, 0.000000000000000)
   7:  (0.000000000000000, -1.00000000000000)
   8:  (10.0000000000000, -1.00000000000000)

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

   0:  -1.08397709278e+17
   1:  -1.08397709278e+17
   2:  -1.08397671373e+17
   3:  -1.08397709278e+17
   4:  -1.08397709278e+17
   5:  -1.08397671373e+17
   6:  1.08397671373e+17
   7:  1.08397709278e+17
   8:  1.08397709278e+17
   9:  1.64882056843e-15
  10:  -27.7926204058
  11:  -22.2073795942

   0:  -1.08397709278e+17
   1:  -1.08397709278e+17
   2:  -1.08397671373e+17
   3:  -1.08397709278e+17
   4:  -1.08397709278e+17
   5:  -1.08397671373e+17
   6:  1.08397671373e+17
   7:  1.08397709278e+17
   8:  1.08397709278e+17
   9:  1.64882056843e-15
  10:  -27.7926204058
  11:  -22.2073795942

# obratite pozornost na veličine intenziteta sila !

Primjer 7:

b = 5
c = 4

joints = { 0: (0, 0), 1: (b, 0), 2: (2*b, 0), 3: (0, c), 4: (2*b, c), 5: (b, c), 6: (-1, 0), 7: (0, -1), 8: (2*b, -1) }
print 'joints:'
print_dict (joints)
print

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

supports = [6, 7, 8]
print 'supports:', supports
print

loads = { 1: (0, -50) }
print 'loads:' 
print_dict (loads)

joints:
   0:  (0, 0)
   1:  (5, 0)
   2:  (10, 0)
   3:  (0, 4)
   4:  (10, 4)
   5:  (5, 4)
   6:  (-1, 0)
   7:  (0, -1)
   8:  (10, -1)

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

supports: [6, 7, 8]

loads:
   1:  (0, -50)

joints:
   0:  (0, 0)
   1:  (5, 0)
   2:  (10, 0)
   3:  (0, 4)
   4:  (10, 4)
   5:  (5, 4)
   6:  (-1, 0)
   7:  (0, -1)
   8:  (10, -1)

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

supports: [6, 7, 8]

loads:
   1:  (0, -50)

plot_truss (joints, bars, supports, loads, load_scale = 0.025)

maxwells_rule (joints, supports, bars)

2 * 6 - 12  ==  0

2 * 6 - 12  ==  0

bar_forces = truss2D (joints, bars, supports, loads)
print_dict (bar_forces)

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

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_56.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("YmFyX2ZvcmNlcyA9IHRydXNzMkQgKGpvaW50cywgYmFycywgc3VwcG9ydHMsIGxvYWRzKQpwcmludF9kaWN0IChiYXJfZm9yY2VzKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpAwWN1c/___code___.py", line 2, in <module>
    bar_forces = truss2D (joints, bars, supports, loads)
  File "<string>", line 82, in truss2D
  File "sage/matrix/matrix_double_dense.pyx", line 1758, in sage.matrix.matrix_double_dense.Matrix_double_dense.solve_right (/home/sage/SageMath/src/build/cythonized/sage/matrix/matrix_double_dense.c:13465)
  File "/home/sage/SageMath/local/lib/python2.7/site-packages/scipy/linalg/basic.py", line 103, in solve
    raise LinAlgError("singular matrix")
numpy.linalg.linalg.LinAlgError: singular matrix

Mehanika_1-11_Ravninske_rešetke_(v0.3)

2419 days ago by fresl