M = np.zeros((DOF * nx, DOF * nx))
elems = []
# creating beam elements
nids = list(nid_pos.keys())
for n1, n2 in zip(nids[:-1], nids[1:]):
elem = Beam2D()
elem.n1 = n1
elem.n2 = n2
elem.E = E
elem.A1 = elem.A2 = A
elem.Izz1 = elem.Izz2 = Izz
elem.rho = rho
elem.interpolation = 'legendre'
update_K(elem, nid_pos, ncoords, K)
update_M(elem, nid_pos, M, lumped=lumped)
elems.append(elem)
if lumped:
assert np.count_nonzero(M - np.diag(np.diagonal(M))) == 0
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) # defining known DOFs
check = np.isclose(x, 0.)
bk[0::DOF] = check
bk[1::DOF] = check
bk[2::DOF] = check
bu = ~bk # defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
def test_beam2d_displ(plot=False):
# number of nodes
n = 2
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
F = -7000
rho = 7.83e3 # kg/m3
L = 3
x = np.linspace(0, L, n)
y = np.zeros_like(x)
# tapered properties
b_root = 0.05 # m
b_tip = b_root # m
h_root = 0.05 # m
h_tip = h_root # m
A_root = h_root * b_root
A_tip = h_tip * b_tip
Izz_root = b_root * h_root**3 / 12
Izz_tip = b_tip * h_tip**3 / 12
# getting nodes
ncoords = np.vstack((x, y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3 * n, 3 * n))
M = np.zeros((3 * n, 3 * n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A2 = A_root
Izz1 = Izz2 = Izz_root
beam = Beam2D()
beam.interpolation = 'legendre'
beam.n1 = n1
beam.n2 = n2
# Material Lastrobe Lescalloy
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, 0.) # locating node at root
# clamping at root
for i in range(DOF):
bk[i::DOF] = check
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
# test
f = np.zeros(K.shape[0])
f[-2] = F
fu = f[bu]
# solving
uu = solve(Kuu, fu)
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = uu
beam = beams[0]
nplot = 100
xplot = np.linspace(0, L, nplot)
yplot = np.zeros_like(xplot)
uinterp, vinterp = uv(beam, *u[:6], n=nplot)
assert np.allclose(vinterp[0], [
0.00000000e+00, -9.08870216e-05, -3.62319884e-04, -8.12456281e-04,
-1.43945391e-03, -2.24147047e-03, -3.21666364e-03, -4.36319114e-03,
-5.67921065e-03, -7.16287987e-03, -8.81235649e-03, -1.06257982e-02,
-1.26013627e-02, -1.47372077e-02, -1.70314909e-02, -1.94823700e-02,
-2.20880027e-02, -2.48465466e-02, -2.77561595e-02, -3.08149990e-02,
-3.40212230e-02, -3.73729889e-02, -4.08684547e-02, -4.45057778e-02,
-4.82831161e-02, -5.21986273e-02, -5.62504690e-02, -6.04367989e-02,
-6.47557747e-02, -6.92055542e-02, -7.37842949e-02, -7.84901547e-02,
-8.33212912e-02, -8.82758621e-02, -9.33520250e-02, -9.85479378e-02,
-1.03861758e-01, -1.09291644e-01, -1.14835752e-01, -1.20492241e-01,
-1.26259268e-01, -1.32134991e-01, -1.38117568e-01, -1.44205156e-01,
-1.50395913e-01, -1.56687997e-01, -1.63079565e-01, -1.69568776e-01,
-1.76153786e-01, -1.82832754e-01, -1.89603837e-01, -1.96465193e-01,
-2.03414980e-01, -2.10451355e-01, -2.17572476e-01, -2.24776501e-01,
-2.32061587e-01, -2.39425892e-01, -2.46867574e-01, -2.54384790e-01,
-2.61975699e-01, -2.69638457e-01, -2.77371223e-01, -2.85172155e-01,
-2.93039409e-01, -3.00971144e-01, -3.08965517e-01, -3.17020687e-01,
-3.25134810e-01, -3.33306044e-01, -3.41532548e-01, -3.49812478e-01,
-3.58143993e-01, -3.66525251e-01, -3.74954408e-01, -3.83429623e-01,
-3.91949053e-01, -4.00510856e-01, -4.09113189e-01, -4.17754212e-01,
-4.26432080e-01, -4.35144952e-01, -4.43890985e-01, -4.52668338e-01,
-4.61475168e-01, -4.70309632e-01, -4.79169888e-01, -4.88054095e-01,
-4.96960409e-01, -5.05886988e-01, -5.14831990e-01, -5.23793574e-01,
-5.32769895e-01, -5.41759113e-01, -5.50759384e-01, -5.59768868e-01,
-5.68785720e-01, -5.77808099e-01, -5.86834163e-01, -5.95862069e-01
])
if plot:
# plotting
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.gca().set_aspect('equal')
plt.plot(x, y, '-k')
plt.plot(xplot + uinterp[0], yplot + vinterp[0], '--r')
plt.show()
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A[nid_pos[n1]]
A2 = A[nid_pos[n2]]
Izz1 = Izz[nid_pos[n1]]
Izz2 = Izz[nid_pos[n2]]
beam = Beam2D()
beam.n1 = n1
beam.n2 = n2
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M, lumped=False)
elements.append(beam)
I = np.ones_like(M)
# applying boundary conditions
# uroot = 0
# vroot = unknown
# betaroot = 0
# utip = unknown
# vtip = prescribed displacement
# betatip = unknown
known_ind = [0, 2, (K.shape[0] - 1) - 1]
bu = np.logical_not(np.in1d(np.arange(M.shape[0]), known_ind))
bk = np.in1d(np.arange(M.shape[0]), known_ind)
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A[nid_pos[n1]]
A2 = A[nid_pos[n2]]
Izz1 = Izz[nid_pos[n1]]
Izz2 = Izz[nid_pos[n2]]
beam = Beam2D()
beam.n1 = n1
beam.n2 = n2
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M, lumped=True)
elements.append(beam)
I = np.ones_like(M)
# applying boundary conditions
# uroot = 0
# vroot = unknown
# betaroot = 0
# utip = unknown
# vtip = prescribed displacement
# betatip = unknown
known_ind = [0, 2, (K.shape[0] - 1) - 1]
bu = np.logical_not(np.in1d(np.arange(M.shape[0]), known_ind))
bk = np.in1d(np.arange(M.shape[0]), known_ind)
def test_nat_freq_cantilever_beam(plot=False, mode=0):
n = 100
L = 3 # total size of the beam along x
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
rho = 7.83e3 # kg/m3
x = np.linspace(0, L, n)
# path
y = np.ones_like(x)
# tapered properties
b = 0.05 # m
h = 0.05 # m
A = h * b
Izz = b * h**3 / 12
# getting nodes
ncoords = np.vstack((x, y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3 * n, 3 * n))
M = np.zeros((3 * n, 3 * n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A2 = A
Izz1 = Izz2 = Izz
beam = Beam2D()
beam.n1 = n1
beam.n2 = n2
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, 0.) # locating node at root
# clamping at root
for i in range(DOF):
bk[i::DOF] = check
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
eigvals, U = eigh(a=Kuu, b=Muu)
omegan = eigvals**0.5
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = U[:, mode]
# reference answer
alpha123 = np.array([1.875, 4.694, 7.885])
omega123 = alpha123**2 * np.sqrt(E * Izz / (rho * A * L**4))
print('Theoretical omega123', omega123)
print('Numerical omega123', omegan[0:3])
assert np.allclose(omega123, omegan[0:3], rtol=0.01)
if plot:
# plotting
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.plot(x, y, '-k')
plt.plot(x + u[0::DOF], y + u[1::DOF], '--r')
plt.show()
M = np.zeros((DOF * nx, DOF * nx))
elems = []
# creating beam elements
nids = list(nid_pos.keys())
for n1, n2 in zip(nids[:-1], nids[1:]):
elem = Beam2D()
elem.n1 = n1
elem.n2 = n2
elem.E = E
elem.A1 = elem.A2 = A
elem.Izz1 = elem.Izz2 = Izz
elem.rho = rho
elem.interpolation = 'legendre'
update_K(elem, nid_pos, ncoords, K)
update_M(elem, nid_pos, M)
elems.append(elem)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) # defining known DOFs
check = np.isclose(x, 0.)
bk[0::DOF] = check
bk[1::DOF] = check
bk[2::DOF] = check
bu = ~bk # defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
# solving
def test_beam2d_displ(plot=False):
# number of nodes
n = 2
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
F = -7000
rho = 7.83e3 # kg/m3
L = 3
x = np.linspace(0, L, n)
y = np.zeros_like(x)
# tapered properties
b_root = 0.05 # m
b_tip = b_root # m
h_root = 0.05 # m
h_tip = h_root # m
A_root = h_root*b_root
A_tip = h_tip*b_tip
Izz_root = b_root*h_root**3/12
Izz_tip = b_tip*h_tip**3/12
# getting nodes
ncoords = np.vstack((x ,y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3*n, 3*n))
M = np.zeros((3*n, 3*n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A2 = A_root
Izz1 = Izz2 = Izz_root
beam = Beam2D()
beam.interpolation = 'legendre'
beam.n1 = n1
beam.n2 = n2
# Material Lastrobe Lescalloy
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, 0.) # locating node at root
# clamping at root
for i in range(DOF):
bk[i::DOF] = check
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
# test
f = np.zeros(K.shape[0])
f[-2] = F
fu = f[bu]
# solving
uu = solve(Kuu, fu)
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = uu
beam = beams[0]
nplot = 10
xplot = np.linspace(0, L, nplot)
yplot = np.zeros_like(xplot)
h = np.linspace(h_root, h_tip, nplot)
exxbot = exx(-h/2, beam, *u[:6], n=nplot)
exxtop = exx(+h/2, beam, *u[:6], n=nplot)
ref = np.array([-0.00496552, -0.00441379, -0.00386207, -0.00331034, -0.00275862, -0.0022069,
-0.00165517, -0.00110345, -0.00055172, 0.])
assert np.allclose(exxbot[0], ref)
assert np.allclose(exxtop[0], -ref)
if plot:
# plotting
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.plot(xplot, exxbot[0])
plt.plot(xplot, exxtop[0])
plt.show()
def test_static_curved_beam(plot=False):
# number of nodes
n = 24
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
F = 7000
rho = 7.83e3 # kg/m3
thetas = np.linspace(np.pi / 2, 0, n)
radius = 1.2
x = radius * np.cos(thetas)
y = -1.2 + radius * np.sin(thetas)
# path
print('y_tip', y[-1])
# tapered properties
b_root = 0.05 # m
b_tip = b_root # m
h_root = 0.05 # m
h_tip = h_root # m
A_root = h_root * b_root
A_tip = h_tip * b_tip
Izz_root = b_root * h_root**3 / 12
Izz_tip = b_tip * h_tip**3 / 12
# getting nodes
ncoords = np.vstack((x, y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3 * n, 3 * n))
M = np.zeros((3 * n, 3 * n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A2 = A_root
Izz1 = Izz2 = Izz_root
beam = Beam2D()
beam.interpolation = 'legendre'
beam.n1 = n1
beam.n2 = n2
# Material Lastrobe Lescalloy
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, x.min())
# clamping curved beam at the left extremity
for i in range(DOF):
bk[i::DOF] = check
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
# test
f = np.zeros(K.shape[0])
f[-2] = F
fu = f[bu]
# solving
uu = solve(Kuu, fu)
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = uu
u = u.reshape(n, -1)
print('u_tip', u[-1])
ref_from_Abaqus = 5.738e-2, 4.064e-2
print('Reference from Abaqus: u_tip', ref_from_Abaqus)
assert np.allclose(u[-1, 0:2], ref_from_Abaqus, rtol=0.01)
if plot:
# plotting
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.gca().set_aspect('equal')
plt.plot(x, y, '-k')
plt.plot(x + u[:, 0], y + u[:, 1], '--r')
plt.show()
def test_cantilever_beam(plot=False):
n = 100
L = 3 # total size of the beam along x
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
rho = 7.83e3 # kg/m3
x = np.linspace(0, L, n)
# path
y = np.ones_like(x)
# tapered properties
b_root = 0.05 # m
b_tip = b_root # m
h_root = 0.05 # m
h_tip = h_root # m
A_root = h_root * b_root
A_tip = h_tip * b_tip
Izz_root = b_root * h_root**3 / 12
Izz_tip = b_tip * h_tip**3 / 12
# getting nodes
ncoords = np.vstack((x, y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3 * n, 3 * n))
M = np.zeros((3 * n, 3 * n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A_root + (A_tip - A_root) * x1 * A_tip / (L - 0)
A2 = A_root + (A_tip - A_root) * x2 * A_tip / (L - 0)
Izz1 = Izz_root + (Izz_tip - Izz_root) * x1 * Izz_tip / (L - 0)
Izz2 = Izz_root + (Izz_tip - Izz_root) * x2 * Izz_tip / (L - 0)
beam = Beam2D()
beam.n1 = n1
beam.n2 = n2
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, 0.) # locating node at root
# clamping at root
for i in range(DOF):
bk[i::DOF] = check
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
# test
Fy = 700
f = np.zeros(K.shape[0])
f[-2] = Fy
fu = f[bu]
# solving
uu = solve(Kuu, fu)
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = uu
u = u.reshape(n, -1)
a = L
deflection = Fy * a**3 / (3 * E * Izz_root) * (1 + 3 * (L - a) / 2 * a)
print('Theoretical deflection', deflection)
print('Numerical deflection', u[-1, 1])
assert np.isclose(deflection, u[-1, 1])
if plot:
# plotting
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.plot(x, y, '-k')
plt.plot(x + u[:, 0], y + u[:, 1], '--r')
plt.show()
def test_nat_freq_curved_beam():
n = 100
# comparing with:
# https://www.sciencedirect.com/science/article/pii/S0168874X06000916
# see section 5.4
E = 206.8e9 # Pa
rho = 7855 # kg/m3
A = 4.071e-3
Izz = 6.456e-6
thetabeam = np.deg2rad(97)
r = 2.438
thetas = np.linspace(0, thetabeam, n)
x = r * np.cos(thetas)
y = r * np.sin(thetas)
# getting nodes
ncoords = np.vstack((x, y)).T
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
n1s = nids[0:-1]
n2s = nids[1:]
K = np.zeros((3 * n, 3 * n))
M = np.zeros((3 * n, 3 * n))
beams = []
for n1, n2 in zip(n1s, n2s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
x1, y1 = ncoords[pos1]
x2, y2 = ncoords[pos2]
A1 = A2 = A
Izz1 = Izz2 = Izz
beam = Beam2D()
beam.n1 = n1
beam.n2 = n2
beam.E = E
beam.rho = rho
beam.A1, beam.A2 = A1, A2
beam.Izz1, beam.Izz2 = Izz1, Izz2
update_K(beam, nid_pos, ncoords, K)
update_M(beam, nid_pos, M)
beams.append(beam)
# applying boundary conditions
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, x.min()) | np.isclose(
x, x.max()) # locating nodes at both ends
# simply supporting at both ends
bk[0::DOF] = check # u
bk[1::DOF] = check # v
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
eigvals, U = eigh(a=Kuu, b=Muu)
omegan = eigvals**0.5
omega123 = [396.98, 931.22, 1797.31]
print('Reference omega123', omega123)
print('Numerical omega123', omegan[0:3])
assert np.allclose(omega123, omegan[0:3], rtol=0.01)