anabel

Force Method - Truss

import ema as em
import matplotlib.pyplot as plt
import numpy as np
import sympy as sp
%config InlineBackend.figure_format = 'svg'
dm = em.Model(2,2) # create instance of model object
n = dm.dnodes 
e = dm.delems

A1 = 10000
Ac = 20000
I = 1
dm.xsection('default', A1, I)
csec = dm.xsection('section-c', Ac, I)

n1 = dm.node('1', 0.0, 0.0)
n2 = dm.node('2', 16., 0.0)
n3 = dm.node('3', 8.0, 6.0)
n4 = dm.node('4', 0.0, 6.0)

a = dm.truss('a', n3, n4) # add truss element to model object
b = dm.truss('b', n1, n3)
c = dm.truss('c', n2, n3, xsec=csec)

dm.pin(n1)
dm.pin(n4)
dm.pin(n2)

dm.numDOF() # Automatically number model dofs

# em.utilities.export.FEDEAS(dm)
[[3, 4], [5, 6], [1, 2], [7, 8]]
prim = em.Model(2,2) # create instance of model object
np = prim.dnodes 
ep = prim.delems

A1 = 10000
Ac = 20000
I = 1
prim.xsection('default', A1, I)
csec = prim.xsection('section-c', Ac, I)

prim.node('1', 0.0, 0.0)
prim.node('2', 16., 0.0)
prim.node('3', 8.0, 6.0)
prim.node('4', 0.0, 6.0)

prim.truss('a', np['3'], np['4']) # add truss element to model object
prim.truss('c', np['2'], np['3'], xsec=csec)

prim.pin(np['1'])
prim.pin(np['4'])
prim.pin(np['2'])

prim.numDOF() # Automatically number model dofs

np['3'].p['x'] = 30
np['3'].p['y'] = 50
# Establish redundant member force
dm.redundant(b, '1')
fig, ax = plt.subplots()
em.plot_structure(dm, ax)
svg

Part 1 : Nodal Loading

Ufp = em.analysis.SolveDispl(prim)
A = em.A_matrix(dm)
A.f@Ufp
$V_{{fffffff}}$
$a_1$ 0.077333
$b_1$ 0.165400
$c_1$ 0.041667 
b.q0['1'] = 1
Uf = em.analysis.SolveDispl(dm)
A = em.A_matrix(dm)
A.f@Uf
$V_{{fffffffff}}$
$a_1$ -0.000361
$b_1$ -0.000718
$c_1$ -0.000141 
# Create model matrices
B = em.B_matrix(dm)
P = em.P_vector(B)
# Define nodal loading
n['3'].p['x'] = 30
n['3'].p['y'] = 50
P.set_item('1', 50)
P.set_item('2', 30)
# Show full B matrix
B
$a_1$ $b_1$ $c_1$
$1$ 1.0 0.8 -0.8
$2$ -0.0 0.6 0.6
$3$ 0.0 -0.8 0.0
$4$ 0.0 -0.6 0.0
$5$ 0.0 0.0 0.8
$6$ 0.0 0.0 -0.6
$7$ -1.0 0.0 0.0
$8$ 0.0 0.0 0.0 
# Show portion of B matrix corresponding to free dofs
B.f
$a_1$ $b_1$ $c_1$
$1$ 1.0 0.8 -0.8
$2$ -0.0 0.6 0.6 
# Show portion of B matrix corresponding to primary system
B.i
$a_1$ $c_1$
$1$ 1.0 -0.8
$2$ -0.0 0.6

Find \(Q_p\)

Qp = B.bari @ P.f
Qp
$Q_{{ffffff}}$
$a_1$ 90.0
$c_1$ 50.0 
# create full 3x1 Qp vector from current 2x1 Qp vector
Qp = em.Q_vector(B, Qp)
Qp
$Q_{{0}}$
$a_1$ 90.0
$b_1$ 0.0
$c_1$ 50.0

find \((V_h)_p\)

# Initialize structure matrices
A = em.A_matrix(dm)
Fs = em.Fs_matrix(dm) 
Fs
$a_1$ $b_1$ $c_1$
$a_1$ 0.0008 0.000 0.0000
$b_1$ 0.0000 0.001 0.0000
$c_1$ 0.0000 0.000 0.0005 
Ve = Fs@Qp
Vhp = B.barx.T @ Ve
Vhp
C:\Users\claud\Anaconda3\lib\site-packages\IPython\core\formatters.py:371: FormatterWarning: text/html formatter returned invalid type <class 'ema.utilities.Structural_Vector'> (expected <class 'str'>) for object: Structural_Vector([-0.1402])
  FormatterWarning





Structural_Vector([-0.1402])

find \((V_{h})_x\)

Ve_x = Fs@B.barx
Ve_x
$b_1$
$a_1$ -0.00128
$b_1$ 0.00100
$c_1$ -0.00050 
Vhx = B.barx.T@Ve_x
Vhx
0
0 0.003548 
Qx = -Vhp/Vhx
Qx
C:\Users\claud\Anaconda3\lib\site-packages\IPython\core\formatters.py:371: FormatterWarning: text/html formatter returned invalid type <class 'ema.utilities.Structural_Vector'> (expected <class 'str'>) for object: Structural_Vector([[39.51521984]])
  FormatterWarning





Structural_Vector([[39.51521984]])

Part 2 : Thermal Loading

Find \((V_h)_p\)

# e['a'].e0['1'] = 100*2e-5
# e['b'].e0['1'] = 100*2e-5
# e['c'].e0['1'] = 100*2e-5
A = em.A_matrix(dm)
A
$1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$a_1$ 1.0 -0.0 0.0 0.0 0.0 0.0 -1.0 0.0
$b_1$ 0.8 0.6 -0.8 -0.6 0.0 0.0 0.0 0.0
$c_1$ -0.8 0.6 0.0 0.0 0.8 -0.6 0.0 0.0 
V0 = em.V0_vector(dm)
V0
$V_{{ffffff}}$
$a_1$ 0.0
$b_1$ 0.0
$c_1$ 0.0 
Vhp = B.barx.T@V0
Vhp
C:\Users\claud\Anaconda3\lib\site-packages\IPython\core\formatters.py:371: FormatterWarning: text/html formatter returned invalid type <class 'ema.utilities.Structural_Vector'> (expected <class 'str'>) for object: Structural_Vector([0.])
  FormatterWarning





Structural_Vector([0.])

Find \(Q_x\)

Qx = -Vhp/Vhx
Qx
C:\Users\claud\Anaconda3\lib\site-packages\IPython\core\formatters.py:371: FormatterWarning: text/html formatter returned invalid type <class 'ema.utilities.Structural_Vector'> (expected <class 'str'>) for object: Structural_Vector([[-0.]])
  FormatterWarning





Structural_Vector([[-0.]])