def test_dofset_reset(self):
dof = DofSet()
self.assertTrue(dof.count() == 0)
dof.reset(True)
self.assertTrue(dof.count() == 6)
dof.reset()
self.assertTrue(dof.count() == 0)
def test_dofset_reset(self):
dof = DofSet()
self.assertTrue(dof.count() == 0)
dof.reset(True)
self.assertTrue(dof.count() == 6)
dof.reset()
self.assertTrue(dof.count() == 0)
class Beam(BaseBeam):
'''
2-Node Structural 3D Beam
'''
def __init__(self, n1, n2, material = None, properties = None):
BaseBeam.__init__(self)
self.nodeCount = 2
self.name = 'Structural 3D Beam'
self.dofSet = DofSet()
self.dofSet.reset(True)
self.nodes = (n1,n2)
self.material = material
self.loads = List()
if properties is None:
properties = Properties()
self.properties = properties
def validate(self):
'''
Validate beam
'''
if self.length() < TINY:
raise FEError('Beam length is to small')
# check material
for name in ('E', 'G', 'nu'):
if name not in self.material:
raise FEError('Material definition not complete')
if self.material[name] < TINY:
raise FEError("Material parameter '%s' not correct" % name)
# check profile data
for name in ('Area', 'Iyy', 'Izz'):
if name not in self.properties:
raise FEError('Properties definition not complete')
if self.properties[name] < TINY:
raise FEError("Properties parameter '%s' not correct" % name)
def calcTe(self):
'''
Eval element tranformation matrix
'''
# Eval coordinate transformation matrix
Tl = self.calcT()
# Build the final transformation matrix - a 12x12 matrix
T = np.zeros((12,12), dtype=float)
for i in range(4):
for j in range(3):
for k in range(3):
T[j+3*i,k+3*i] = Tl[j,k]
return T
def calcM(self, lumped = False):
'''
Eval elemenent consistent mass matrix in global coordinates
'''
mat = self.material
prop = self.properties
L = self.length()
LL = L*L
rho = mat["Density"]
Ax, Iy, Iz = prop["Area"], prop["Iyy"], prop["Izz"]
J = prop.get("Ixx", 0.)
if J == 0.:
# Simplification only valid for circular sections
J = Iy + Iz
M = np.zeros((12,12), dtype=float)
t = rho*Ax*L
ry = rho*Iy
rz = rho*Iz
po = rho*J*L
if lumped:
t *= .5
ry *= .5
rz *= .5
po *= .5
M[0][0] = M[1][1] = M[2][2] = M[6][6] = M[7][7] = M[8][8] = t
M[3][3] = M[9][9] = po + ry + rz
M[4][4] = M[10][10] = po + ry + rz
M[5][5] = M[11][11] = po + ry + rz
M[3][4] = M[4][3] = M[9][10] = M[10][9] = po + ry + rz
M[3][5] = M[5][3] = M[9][11] = M[11][9] = po + ry + rz
M[4][5] = M[5][4] = M[10][11] = M[11][10] = po + ry + rz
else:
M[0][0] = M[6][6] = t/3.
M[1][1] = M[7][7] = 13.*t/35. + 6.*rz/(5.*L)
M[2][2] = M[8][8] = 13.*t/35. + 6.*ry/(5.*L)
M[3][3] = M[9][9] = po/3.
M[4][4] = M[10][10] = t*LL/105. + 2.*L*ry/15.
M[5][5] = M[11][11] = t*LL/105. + 2.*L*rz/15.
M[4][2] = M[2][4] = -11.*t*L/210. - ry/10.
M[5][1] = M[1][5] = 11.*t*L/210. + rz/10.
M[6][0] = M[0][6] = t/6.
M[7][5] = M[5][7] = 13.*t*L/420. - rz/10.
M[8][4] = M[4][8] = -13.*t*L/420. + ry/10.
M[9][3] = M[3][9] = po/6.
M[10][2] = M[2][10] = 13.*t*L/420. - ry/10.
M[11][1] = M[1][11] = -13.*t*L/420. + rz/10.
M[10][8] = M[8][10] = 11.*t*L/210. + ry/10.
M[11][7] = M[7][11] = -11.*t*L/210. - rz/10.
M[7][1] = M[1][7] = 9.*t/70. - 6.*rz/(5.*L)
M[8][2] = M[2][8] = 9.*t/70. - 6.*ry/(5.*L)
M[10][4] = M[4][10] = -LL*t/140. - ry*L/30.
M[11][5] = M[5][11] = -LL*t/140. - rz*L/30.
T = self.calcTe()
# Evaluate Transformation M = T^T * M * T
return np.dot(T.T,np.dot(M,T))
def calcKe(self):
'''
Eval element stiffness matrix in local coordinates.
Includes transverse shear deformation. If no shear deflection
constant (ShearZ and ShearY) is set, the displacement considers
bending deformation only.
Shear deflection constants for other cross-sections can be found in
structural handbooks.
'''
Ke = np.zeros((12,12), dtype=float)
mat = self.material
prop = self.properties
L = self.length()
L2 = L*L
L3 = L2*L
E, G = mat["E"], mat["G"]
Ax, Iy, Iz = prop["Area"], prop["Iyy"], prop["Izz"]
J = prop.get("Ixx", 0.)
if J == 0.:
# Simplification only valid for circular sections
J = Iy + Iz
Asy = prop.get("ShearY", 0.)
Asz = prop.get("ShearZ", 0.)
if Asy != 0. and Asz != 0.:
Ksy = 12.*E*Iz / (G*Asy*L2)
Ksz = 12.*E*Iy / (G*Asz*L2)
else:
Ksy = Ksz = 0.
Ke[0,0] = Ke[6,6] = E*Ax / L
Ke[1,1] = Ke[7,7] = 12.*E*Iz / ( L3*(1.+Ksy) )
Ke[2,2] = Ke[8,8] = 12.*E*Iy / ( L3*(1.+Ksz) )
Ke[3,3] = Ke[9,9] = G*J / L
Ke[4,4] = Ke[10,10] = (4.+Ksz)*E*Iy / ( L*(1.+Ksz) )
Ke[5,5] = Ke[11,11] = (4.+Ksy)*E*Iz / ( L*(1.+Ksy) )
Ke[4,2] = Ke[2,4] = -6.*E*Iy / ( L2*(1.+Ksz) )
Ke[5,1] = Ke[1,5] = 6.*E*Iz / ( L2*(1.+Ksy) )
Ke[6,0] = Ke[0,6] = -Ke[0,0]
Ke[11,7] = Ke[7,11] = Ke[7,5] = Ke[5,7] = -Ke[5,1]
Ke[10,8] = Ke[8,10] = Ke[8,4] = Ke[4,8] = -Ke[4,2]
Ke[9,3] = Ke[3,9] = -Ke[3,3]
Ke[10,2] = Ke[2,10] = Ke[4,2]
Ke[11,1] = Ke[1,11] = Ke[5,1]
Ke[7,1] = Ke[1,7] = -Ke[1,1]
Ke[8,2] = Ke[2,8] = -Ke[2,2]
Ke[10,4] = Ke[4,10] = (2.-Ksz)*E*Iy / ( L*(1.+Ksz) )
Ke[11,5] = Ke[5,11] = (2.-Ksy)*E*Iz / ( L*(1.+Ksy) )
return Ke
def calcK(self):
'''
Eval element stiffness matrix in global coordinates.
'''
K = self.calcKe()
T = self.calcTe()
# Evaluate Transformation K = T^T * K * T
return np.dot(T.T,np.dot(K,T))
def calcStress(self):
raise NotImplementedError()
def calcNodalForces(self, res = None):
'''
Calculate corresponding global forces at nodes from
line loads
'''
res = self.calcLocalNodalForces(res)
Tl = self.calcT()
# Build the final transformation matrix - a 12x12 matrix
T = np.zeros((12,12), dtype=float)
for i in range(4):
for j in range(3):
for k in range(3):
T[j+3*i,k+3*i] = Tl[j,k]
res[:] = np.dot(T.T, res)
return res
def calcLocalNodalForces(self, res = None):
'''
Calculate corresponding end forces at nodes from
line loads in local directions
'''
if res is None:
res = np.zeros((self.dofSet.count() * self.nodeCount,), dtype = float)
L = self.length()
LL = L * L
load = self.calcLocalLoad()
load *= L
# Nx
res[0] += .5*load[0]
res[6] += .5*load[0]
# Vy
res[1] += .5*load[1]
res[7] += .5*load[1]
# VZ
res[2] += .5*load[2]
res[8] += .5*load[2]
# My
res[4] -= load[2] * L / 12.
res[10] += load[2] * L / 12.
# Mz
res[5] -= load[1] * L / 12.
res[11] += load[1] * L / 12.
return res
def calcSectionForces(self, dx = 1e9, sections=None):
# find number of divisions
try:
nx = int(self.length()/dx)
nx = min(max(1, nx), 100)
except ZeroDivisionError:
nx = 1
dx = self.length()
if sections is not None:
nx = sections - 1
dx = self.length() / (sections - 1)
res = np.zeros((nx + 1, 7), dtype=float)
L = self.length()
# Eval local coordinate transformation matrix
Tl = self.calcT()
# Build the final transformation matrix - a 12x12 matrix
T = np.zeros((12,12), dtype=float)
for i in range(4):
for j in range(3):
for k in range(3):
T[j+3*i,k+3*i] = Tl[j,k]
# nodes flipped?
n1, n2 = self.nodes
if n1.cz > n2.cz:
fac = -1.
else:
fac = 1.
# Transform displacements and rotations from global to local coordinates
d = np.zeros((12,), dtype=float)
d[:6] = n1.results
d[6:] = n2.results
if not n1.coordSys is None:
Tcs = n1.coordSys
d[:3] = np.dot(Tcs.T, d[:3])
d[3:6] = np.dot(Tcs.T, d[3:6])
if not n2.coordSys is None:
Tcs = n2.coordSys
d[6:9] = np.dot(Tcs.T, d[6:9])
d[9:12] = np.dot(Tcs.T, d[9:12])
u = np.dot(T, d)
# Stiffness matrix in local coordinates
Ke = self.calcKe()
# Calculate forces in local directions
forces = np.dot(Ke, u)
# Correct forces from eqivalent forces line loads
iforces = self.calcLocalNodalForces()
forces -= iforces
res[0,0] = 0.
res[0,1:] = forces[:6]
res[nx,0] = 1.
res[nx,1:] = -forces[6:]
if nx > 1:
# accumulate interior span loads
load = self.calcLocalLoad()
_dx = dx
x = _dx
for i in range(nx):
res[i + 1,0] = x/L # Position
res[i + 1,1] = res[i, 1] + load[0]*_dx # Axial force, Nx
res[i + 1,2] = res[i, 2] + load[1]*_dx # Sheare force Vy
res[i + 1,3] = res[i, 3] + load[2]*_dx # Sheare force Vz
res[i + 1,4] = res[i, 4] # Torsion, Txx
# correct last step if needed
if (i + 1)*_dx > L:
_dx = L - i*_dx
x += _dx
# trapezoidal integration of shear force for bending momemnt
_dx = dx
for i in range(nx):
res[i + 1,5] = res[i,5] + .5*(res[i + 1, 3] + res[i, 3])*_dx # Myy
res[i + 1,6] = res[i,6] + fac*.5*(res[i + 1, 2] + res[i, 2])*_dx # Mzz
# correct last step if needed
if (i + 1)*_dx > L:
_dx = L - i*_dx
return res
class Beam(BaseBeam):
'''
2-Node Structural 3D Beam
'''
def __init__(self, n1, n2, material=None, properties=None):
BaseBeam.__init__(self)
self.nodeCount = 2
self.name = 'Structural 3D Beam'
self.dofSet = DofSet()
self.dofSet.reset(True)
self.nodes = (n1, n2)
self.material = material
self.loads = List()
if properties is None:
properties = Properties()
self.properties = properties
def validate(self):
'''
Validate beam
'''
if self.length() < TINY:
raise FEError('Beam length is to small')
# check material
for name in ('E', 'G', 'nu'):
if name not in self.material:
raise FEError('Material definition not complete')
if self.material[name] < TINY:
raise FEError("Material parameter '%s' not correct" % name)
# check profile data
for name in ('Area', 'Iyy', 'Izz'):
if name not in self.properties:
raise FEError('Properties definition not complete')
if self.properties[name] < TINY:
raise FEError("Properties parameter '%s' not correct" % name)
def calcTe(self):
'''
Eval element tranformation matrix
'''
# Eval coordinate transformation matrix
Tl = self.calcT()
# Build the final transformation matrix - a 12x12 matrix
T = np.zeros((12, 12), dtype=float)
for i in range(4):
for j in range(3):
for k in range(3):
T[j + 3 * i, k + 3 * i] = Tl[j, k]
return T
def calcM(self, lumped=False):
'''
Eval elemenent consistent mass matrix in global coordinates
'''
mat = self.material
prop = self.properties
L = self.length()
LL = L * L
rho = mat["Density"]
Ax, Iy, Iz = prop["Area"], prop["Iyy"], prop["Izz"]
J = prop.get("Ixx", 0.)
if J == 0.:
# Simplification only valid for circular sections
J = Iy + Iz
M = np.zeros((12, 12), dtype=float)
t = rho * Ax * L
ry = rho * Iy
rz = rho * Iz
po = rho * J * L
if lumped:
t *= .5
ry *= .5
rz *= .5
po *= .5
M[0][0] = M[1][1] = M[2][2] = M[6][6] = M[7][7] = M[8][8] = t
M[3][3] = M[9][9] = po + ry + rz
M[4][4] = M[10][10] = po + ry + rz
M[5][5] = M[11][11] = po + ry + rz
M[3][4] = M[4][3] = M[9][10] = M[10][9] = po + ry + rz
M[3][5] = M[5][3] = M[9][11] = M[11][9] = po + ry + rz
M[4][5] = M[5][4] = M[10][11] = M[11][10] = po + ry + rz
else:
M[0][0] = M[6][6] = t / 3.
M[1][1] = M[7][7] = 13. * t / 35. + 6. * rz / (5. * L)
M[2][2] = M[8][8] = 13. * t / 35. + 6. * ry / (5. * L)
M[3][3] = M[9][9] = po / 3.
M[4][4] = M[10][10] = t * LL / 105. + 2. * L * ry / 15.
M[5][5] = M[11][11] = t * LL / 105. + 2. * L * rz / 15.
M[4][2] = M[2][4] = -11. * t * L / 210. - ry / 10.
M[5][1] = M[1][5] = 11. * t * L / 210. + rz / 10.
M[6][0] = M[0][6] = t / 6.
M[7][5] = M[5][7] = 13. * t * L / 420. - rz / 10.
M[8][4] = M[4][8] = -13. * t * L / 420. + ry / 10.
M[9][3] = M[3][9] = po / 6.
M[10][2] = M[2][10] = 13. * t * L / 420. - ry / 10.
M[11][1] = M[1][11] = -13. * t * L / 420. + rz / 10.
M[10][8] = M[8][10] = 11. * t * L / 210. + ry / 10.
M[11][7] = M[7][11] = -11. * t * L / 210. - rz / 10.
M[7][1] = M[1][7] = 9. * t / 70. - 6. * rz / (5. * L)
M[8][2] = M[2][8] = 9. * t / 70. - 6. * ry / (5. * L)
M[10][4] = M[4][10] = -LL * t / 140. - ry * L / 30.
M[11][5] = M[5][11] = -LL * t / 140. - rz * L / 30.
T = self.calcTe()
# Evaluate Transformation M = T^T * M * T
return np.dot(T.T, np.dot(M, T))
def calcKe(self):
'''
Eval element stiffness matrix in local coordinates.
Includes transverse shear deformation. If no shear deflection
constant (ShearZ and ShearY) is set, the displacement considers
bending deformation only.
Shear deflection constants for other cross-sections can be found in
structural handbooks.
'''
Ke = np.zeros((12, 12), dtype=float)
mat = self.material
prop = self.properties
L = self.length()
L2 = L * L
L3 = L2 * L
E, G = mat["E"], mat["G"]
Ax, Iy, Iz = prop["Area"], prop["Iyy"], prop["Izz"]
J = prop.get("Ixx", 0.)
if J == 0.:
# Simplification only valid for circular sections
J = Iy + Iz
Asy = prop.get("ShearY", 0.)
Asz = prop.get("ShearZ", 0.)
if Asy != 0. and Asz != 0.:
Ksy = 12. * E * Iz / (G * Asy * L2)
Ksz = 12. * E * Iy / (G * Asz * L2)
else:
Ksy = Ksz = 0.
Ke[0, 0] = Ke[6, 6] = E * Ax / L
Ke[1, 1] = Ke[7, 7] = 12. * E * Iz / (L3 * (1. + Ksy))
Ke[2, 2] = Ke[8, 8] = 12. * E * Iy / (L3 * (1. + Ksz))
Ke[3, 3] = Ke[9, 9] = G * J / L
Ke[4, 4] = Ke[10, 10] = (4. + Ksz) * E * Iy / (L * (1. + Ksz))
Ke[5, 5] = Ke[11, 11] = (4. + Ksy) * E * Iz / (L * (1. + Ksy))
Ke[4, 2] = Ke[2, 4] = -6. * E * Iy / (L2 * (1. + Ksz))
Ke[5, 1] = Ke[1, 5] = 6. * E * Iz / (L2 * (1. + Ksy))
Ke[6, 0] = Ke[0, 6] = -Ke[0, 0]
Ke[11, 7] = Ke[7, 11] = Ke[7, 5] = Ke[5, 7] = -Ke[5, 1]
Ke[10, 8] = Ke[8, 10] = Ke[8, 4] = Ke[4, 8] = -Ke[4, 2]
Ke[9, 3] = Ke[3, 9] = -Ke[3, 3]
Ke[10, 2] = Ke[2, 10] = Ke[4, 2]
Ke[11, 1] = Ke[1, 11] = Ke[5, 1]
Ke[7, 1] = Ke[1, 7] = -Ke[1, 1]
Ke[8, 2] = Ke[2, 8] = -Ke[2, 2]
Ke[10, 4] = Ke[4, 10] = (2. - Ksz) * E * Iy / (L * (1. + Ksz))
Ke[11, 5] = Ke[5, 11] = (2. - Ksy) * E * Iz / (L * (1. + Ksy))
return Ke
def calcK(self):
'''
Eval element stiffness matrix in global coordinates.
'''
K = self.calcKe()
T = self.calcTe()
# Evaluate Transformation K = T^T * K * T
return np.dot(T.T, np.dot(K, T))
def calcStress(self):
raise NotImplementedError()
def calcNodalForces(self, res=None):
'''
Calculate corresponding global forces at nodes from
line loads
'''
res = self.calcLocalNodalForces(res)
Tl = self.calcT()
# Build the final transformation matrix - a 12x12 matrix
T = np.zeros((12, 12), dtype=float)
for i in range(4):
for j in range(3):
for k in range(3):
T[j + 3 * i, k + 3 * i] = Tl[j, k]
res[:] = np.dot(T.T, res)
return res
def calcLocalNodalForces(self, res=None):
'''
Calculate corresponding end forces at nodes from
line loads in local directions
'''
if res is None:
res = np.zeros((self.dofSet.count() * self.nodeCount, ),
dtype=float)
L = self.length()
LL = L * L
load = self.calcLocalLoad()
load *= L
# Nx
res[0] += .5 * load[0]
res[6] += .5 * load[0]
# Vy
res[1] += .5 * load[1]
res[7] += .5 * load[1]
# VZ
res[2] += .5 * load[2]
res[8] += .5 * load[2]
# My
res[4] -= load[2] * L / 12.
res[10] += load[2] * L / 12.
# Mz
res[5] -= load[1] * L / 12.
res[11] += load[1] * L / 12.
return res
def calcSectionForces(self, dx=1e9):
# find number of divisions
try:
nx = int(self.length() / dx)
nx = min(max(1, nx), 100)
except ZeroDivisionError:
nx = 1
dx = self.length()
res = np.zeros((nx + 1, 7), dtype=float)
L = self.length()
# Eval local coordinate transformation matrix
Tl = self.calcT()
# Build the final transformation matrix - a 12x12 matrix
T = np.zeros((12, 12), dtype=float)
for i in range(4):
for j in range(3):
for k in range(3):
T[j + 3 * i, k + 3 * i] = Tl[j, k]
# nodes flipped?
n1, n2 = self.nodes
if n1.cz > n2.cz:
fac = -1.
else:
fac = 1.
# Transform displacements and rotations from global to local coordinates
d = np.zeros((12, ), dtype=float)
d[:6] = n1.results
d[6:] = n2.results
if not n1.coordSys is None:
Tcs = n1.coordSys
d[:3] = np.dot(Tcs.T, d[:3])
d[3:6] = np.dot(Tcs.T, d[3:6])
if not n2.coordSys is None:
Tcs = n2.coordSys
d[6:9] = np.dot(Tcs.T, d[6:9])
d[9:12] = np.dot(Tcs.T, d[9:12])
u = np.dot(T, d)
# Stiffness matrix in local coordinates
Ke = self.calcKe()
# Calculate forces in local directions
forces = np.dot(Ke, u)
# Correct forces from eqivalent forces line loads
iforces = self.calcLocalNodalForces()
forces -= iforces
res[0, 0] = 0.
res[0, 1:] = forces[:6]
res[nx, 0] = 1.
res[nx, 1:] = -forces[6:]
if nx > 1:
# accumulate interior span loads
load = self.calcLocalLoad()
_dx = dx
x = _dx
for i in range(nx):
res[i + 1, 0] = x / L # Position
res[i + 1, 1] = res[i, 1] + load[0] * _dx # Axial force, Nx
res[i + 1, 2] = res[i, 2] + load[1] * _dx # Sheare force Vy
res[i + 1, 3] = res[i, 3] + load[2] * _dx # Sheare force Vz
res[i + 1, 4] = res[i, 4] # Torsion, Txx
# correct last step if needed
if (i + 1) * _dx > L:
_dx = L - i * _dx
x += _dx
# trapezoidal integration of shear force for bending momemnt
_dx = dx
for i in range(nx):
res[i + 1,
5] = res[i,
5] + .5 * (res[i + 1, 3] + res[i, 3]) * _dx # Myy
res[i + 1, 6] = res[i, 6] + fac * .5 * (res[i + 1, 2] +
res[i, 2]) * _dx # Mzz
# correct last step if needed
if (i + 1) * _dx > L:
_dx = L - i * _dx
return res
