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)
def __init__(self, n1, n2, material=None, properties=None):
BaseBeam.__init__(self)
self.nodeCount = 2
self.name = 'Structural 3D Truss'
self.dofSet = DofSet(Dx=True, Dy=True, Dz=True)
self.nodes = (n1, n2)
self.material = material
self.loads = List()
if properties is None:
properties = Properties()
self.properties = properties
def test_dofset_inplace(self):
dof1 = DofSet(Dx=True,Dy=True,Dz=True)
dof2 = DofSet(Dx=True,Dy=True,Dz=True,Rx=True,Ry=True,Rz=True)
dof1 &= dof2
self.assertTrue(dof1.count() == 3)
self.assertTrue(dof1.is3DSolid())
self.assertTrue(not dof1.is3DShell())
self.assertTrue(dof1.elementDimension() == 3)
dof1 = DofSet(Dx=True,Dy=True,Dz=True)
dof2 = DofSet(Dx=True,Dy=True,Dz=True,Rx=True,Ry=True,Rz=True)
dof1 |= dof2
self.assertTrue(dof1.count() == 6)
self.assertTrue(not dof1.is3DSolid())
self.assertTrue(dof1.is3DShell())
self.assertTrue(dof1.elementDimension() == 3)
def test_dofset_2d(self):
dof1 = DofSet(Dx=True,Dy=True,Dz=False)
self.assertTrue(dof1.count() == 2)
self.assertTrue(not dof1.is3DSolid())
self.assertTrue(not dof1.is3DShell())
self.assertTrue(dof1.elementDimension() == 2)
dof2 = DofSet(Dx=True,Dy=True,Dz=True,Rx=True,Ry=True,Rz=True)
dofa = dof1 & dof2
self.assertTrue(dofa.count() == 2)
self.assertTrue(not dofa.is3DSolid())
self.assertTrue(not dofa.is3DShell())
self.assertTrue(dofa.elementDimension() == 2)
dofo = dof1 | dof2
self.assertTrue(dofo.count() == 6)
self.assertTrue(not dofo.is3DSolid())
self.assertTrue(dofo.is3DShell())
self.assertTrue(dofo.elementDimension() == 3)
def __init__(self, n1, n2, material = None, properties = None):
BaseBeam.__init__(self)
self.nodeCount = 2
self.name = 'Structural 3D Truss'
self.dofSet = DofSet(Dx=True, Dy=True, Dz=True)
self.nodes = (n1,n2)
self.material = material
self.loads = List()
if properties is None:
properties = Properties()
self.properties = properties
def test_dofset_2d(self):
dof1 = DofSet(Dx=True, Dy=True, Dz=False)
self.assertTrue(dof1.count() == 2)
self.assertTrue(not dof1.is3DSolid())
self.assertTrue(not dof1.is3DShell())
self.assertTrue(dof1.elementDimension() == 2)
dof2 = DofSet(Dx=True, Dy=True, Dz=True, Rx=True, Ry=True, Rz=True)
dofa = dof1 & dof2
self.assertTrue(dofa.count() == 2)
self.assertTrue(not dofa.is3DSolid())
self.assertTrue(not dofa.is3DShell())
self.assertTrue(dofa.elementDimension() == 2)
dofo = dof1 | dof2
self.assertTrue(dofo.count() == 6)
self.assertTrue(not dofo.is3DSolid())
self.assertTrue(dofo.is3DShell())
self.assertTrue(dofo.elementDimension() == 3)
def test_dofset_dofPos(self):
dof = DofSet(Dx=True,Dy=True,Dz=True)
self.assertTrue(dof.dofPos(3) == 2)
dof = DofSet(Dx=False,Dy=False,Dz=False,Rx=True,Ry=True,Rz=True)
self.assertTrue(dof.dofPos(3) == 5)
def test_dofset_dofPos(self):
dof = DofSet(Dx=True, Dy=True, Dz=True)
self.assertTrue(dof.dofPos(3) == 2)
dof = DofSet(Dx=False, Dy=False, Dz=False, Rx=True, Ry=True, Rz=True)
self.assertTrue(dof.dofPos(3) == 5)
def test_dofset_inplace(self):
dof1 = DofSet(Dx=True, Dy=True, Dz=True)
dof2 = DofSet(Dx=True, Dy=True, Dz=True, Rx=True, Ry=True, Rz=True)
dof1 &= dof2
self.assertTrue(dof1.count() == 3)
self.assertTrue(dof1.is3DSolid())
self.assertTrue(not dof1.is3DShell())
self.assertTrue(dof1.elementDimension() == 3)
dof1 = DofSet(Dx=True, Dy=True, Dz=True)
dof2 = DofSet(Dx=True, Dy=True, Dz=True, Rx=True, Ry=True, Rz=True)
dof1 |= dof2
self.assertTrue(dof1.count() == 6)
self.assertTrue(not dof1.is3DSolid())
self.assertTrue(dof1.is3DShell())
self.assertTrue(dof1.elementDimension() == 3)
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 Truss(BaseBeam):
'''
2-Node Structural 3D Truss
'''
def __init__(self, n1, n2, material = None, properties = None):
BaseBeam.__init__(self)
self.nodeCount = 2
self.name = 'Structural 3D Truss'
self.dofSet = DofSet(Dx=True, Dy=True, Dz=True)
self.nodes = (n1,n2)
self.material = material
self.loads = List()
if properties is None:
properties = Properties()
self.properties = properties
def __str__(self):
return '()'
def __repr__(self):
return '%s%s' % (self.__class__.__name__, str(self))
def validate(self):
'''
Validate truss
'''
if self.length() < TINY:
raise FEError('Truss length is to small')
# check material
for name in ('E',):
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',):
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 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)
load = self.calcLocalLoad()
# Nx
res[0] += .5*load[0]
res[3] += .5*load[0]
# Vy
res[1] += .5*load[1]
res[4] += .5*load[1]
# VZ
res[2] += .5*load[2]
res[5] += .5*load[2]
return res
def calcNodalForces(self, res = None):
'''
Calculate corresponding global forces at nodes from
line loads
'''
# calculate local forces
res = self.calcLocalNodalForces(res)
# Transforamtion matrix
T = self.calcT()
res[0:3] = np.dot(T.T, res[0:3])
res[3:6] = np.dot(T.T, res[3:6])
return res
def calcM(self, lumped = False):
'''
Eval elemenent consistent mass matrix in global coordinates
'''
mat = self.material
prop = self.properties
Ax, rho = prop["Area"], mat['Density']
n1, n2 = self.nodes
L = self.length()
# Eval direction cosines in order to transform stiffness matrix
# from local coords (link direction) to global coordinates
cxx = ( n2.cx - n1.cx ) / L
cxy = ( n2.cy - n1.cy ) / L
cxz = ( n2.cz - n1.cz ) / L
ll = sqrt(cxx**2 + cxy**2)
if ll != 0.:
cyx = -cxy/ll
cyy = cxx/ll
cyz = 0.
else:
cyx = 0.
cyy = 1.
cyz = 0.
czx = cxy*cyz - cxz*cyy;
czy = -cxx*cyz + cxz*cyx;
czz = cxx*cyy - cxy*cyx;
# Transforamtion matrix
T = np.zeros((6,6), dtype=float)
T[0,0] = cxx; T[0,1] = cxy; T[0,2] = cxz
T[1,0] = cyx; T[1,1] = cyy; T[1,2] = cyz
T[2,0] = czx; T[2,1] = czy; T[2,2] = czz
T[3,3] = cxx; T[3,4] = cxy; T[3,5] = cxz
T[4,3] = cyx; T[4,4] = cyy; T[4,5] = cyz
T[5,3] = czx; T[5,4] = czy; T[5,5] = czz
# Mass matrix
M = np.zeros((6,6), dtype=float)
if lumped:
M[0,0] = 1.; M[5,5] = 1.
M *= rho * Ax * L / 2.
else:
M[0,0] = 2.; M[1,1] = 2.; M[2,2] = 2.
M[3,3] = 2.; M[4,4] = 2.; M[5,5] = 2.
M[3,0] = 1.; M[4,1] = 1.; M[5,2] = 1.
M[0,3] = 1.; M[1,4] = 1.; M[2,5] = 1.
M *= rho * Ax * L / 6.
# 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
'''
mat = self.material
prop = self.properties
L = self.length()
Ax, E = prop["Area"], mat["E"]
# Stiffness matrix
Ke = np.zeros((2,2), dtype=float)
AxEoverL = Ax*E/L
Ke[0,0] = AxEoverL; Ke[0,1] = -AxEoverL
Ke[1,0] =-AxEoverL; Ke[1,1] = AxEoverL
return Ke
def calcK(self):
'''
Eval element stiffness matrix in global coordinates.
'''
L = self.length()
n1, n2 = self.nodes
# Transforamtion matrix
T = np.zeros((2,6), dtype=float)
# Eval direction cosines in order to transform stiffness matrix
# from local coords (link direction) to global coordinates
cx = ( n2.cx - n1.cx ) / L
cy = ( n2.cy - n1.cy ) / L
cz = ( n2.cz - n1.cz ) / L
T[0,0] = cx; T[0,1] = cy; T[0,2] = cz
T[1,3] = cx; T[1,4] = cy; T[1,5] = cz
# Stiffness matrix
K = self.calcKe()
# Evaluate Transformation K = T^T * K * T
return np.dot(T.T,np.dot(K,T))
def calcStress(self):
raise NotImplementedError()
def calcSectionForces(self, dx = 1e9):
'''
Eval beam section forces in nodes
'''
res = np.zeros((2,7), dtype=float)
prop = self.properties
mat = self.material
n1, n2 = self.nodes
L = self.length()
E = mat['E']
G = mat['G']
Ax = prop['Area']
# Eval coordinate transformation matrix
T = self.calcT()
# Transform displacements from global to local coordinates
u1 = np.dot(T, n1.results[:3])
u2 = np.dot(T, n2.results[:3])
# Axial force, Nx
res[0,0] = 0.
res[0,1] = (-Ax*E/L)*(u2[0] - u1[0])
res[1,0] = 1.
res[1,1] = -res[0,1]
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
class Truss(BaseBeam):
'''
2-Node Structural 3D Truss
'''
def __init__(self, n1, n2, material=None, properties=None):
BaseBeam.__init__(self)
self.nodeCount = 2
self.name = 'Structural 3D Truss'
self.dofSet = DofSet(Dx=True, Dy=True, Dz=True)
self.nodes = (n1, n2)
self.material = material
self.loads = List()
if properties is None:
properties = Properties()
self.properties = properties
def __str__(self):
return '()'
def __repr__(self):
return '%s%s' % (self.__class__.__name__, str(self))
def validate(self):
'''
Validate truss
'''
if self.length() < TINY:
raise FEError('Truss length is to small')
# check material
for name in ('E', ):
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', ):
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 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)
load = self.calcLocalLoad()
# Nx
res[0] += .5 * load[0]
res[3] += .5 * load[0]
# Vy
res[1] += .5 * load[1]
res[4] += .5 * load[1]
# VZ
res[2] += .5 * load[2]
res[5] += .5 * load[2]
return res
def calcNodalForces(self, res=None):
'''
Calculate corresponding global forces at nodes from
line loads
'''
# calculate local forces
res = self.calcLocalNodalForces(res)
# Transforamtion matrix
T = self.calcT()
res[0:3] = np.dot(T.T, res[0:3])
res[3:6] = np.dot(T.T, res[3:6])
return res
def calcM(self, lumped=False):
'''
Eval elemenent consistent mass matrix in global coordinates
'''
mat = self.material
prop = self.properties
Ax, rho = prop["Area"], mat['Density']
n1, n2 = self.nodes
L = self.length()
# Eval direction cosines in order to transform stiffness matrix
# from local coords (link direction) to global coordinates
cxx = (n2.cx - n1.cx) / L
cxy = (n2.cy - n1.cy) / L
cxz = (n2.cz - n1.cz) / L
ll = sqrt(cxx**2 + cxy**2)
if ll != 0.:
cyx = -cxy / ll
cyy = cxx / ll
cyz = 0.
else:
cyx = 0.
cyy = 1.
cyz = 0.
czx = cxy * cyz - cxz * cyy
czy = -cxx * cyz + cxz * cyx
czz = cxx * cyy - cxy * cyx
# Transforamtion matrix
T = np.zeros((6, 6), dtype=float)
T[0, 0] = cxx
T[0, 1] = cxy
T[0, 2] = cxz
T[1, 0] = cyx
T[1, 1] = cyy
T[1, 2] = cyz
T[2, 0] = czx
T[2, 1] = czy
T[2, 2] = czz
T[3, 3] = cxx
T[3, 4] = cxy
T[3, 5] = cxz
T[4, 3] = cyx
T[4, 4] = cyy
T[4, 5] = cyz
T[5, 3] = czx
T[5, 4] = czy
T[5, 5] = czz
# Mass matrix
M = np.zeros((6, 6), dtype=float)
if lumped:
M[0, 0] = 1.
M[5, 5] = 1.
M *= rho * Ax * L / 2.
else:
M[0, 0] = 2.
M[1, 1] = 2.
M[2, 2] = 2.
M[3, 3] = 2.
M[4, 4] = 2.
M[5, 5] = 2.
M[3, 0] = 1.
M[4, 1] = 1.
M[5, 2] = 1.
M[0, 3] = 1.
M[1, 4] = 1.
M[2, 5] = 1.
M *= rho * Ax * L / 6.
# 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
'''
mat = self.material
prop = self.properties
L = self.length()
Ax, E = prop["Area"], mat["E"]
# Stiffness matrix
Ke = np.zeros((2, 2), dtype=float)
AxEoverL = Ax * E / L
Ke[0, 0] = AxEoverL
Ke[0, 1] = -AxEoverL
Ke[1, 0] = -AxEoverL
Ke[1, 1] = AxEoverL
return Ke
def calcK(self):
'''
Eval element stiffness matrix in global coordinates.
'''
L = self.length()
n1, n2 = self.nodes
# Transforamtion matrix
T = np.zeros((2, 6), dtype=float)
# Eval direction cosines in order to transform stiffness matrix
# from local coords (link direction) to global coordinates
cx = (n2.cx - n1.cx) / L
cy = (n2.cy - n1.cy) / L
cz = (n2.cz - n1.cz) / L
T[0, 0] = cx
T[0, 1] = cy
T[0, 2] = cz
T[1, 3] = cx
T[1, 4] = cy
T[1, 5] = cz
# Stiffness matrix
K = self.calcKe()
# Evaluate Transformation K = T^T * K * T
return np.dot(T.T, np.dot(K, T))
def calcStress(self):
raise NotImplementedError()
def calcSectionForces(self, dx=1e9):
'''
Eval beam section forces in nodes
'''
res = np.zeros((2, 7), dtype=float)
prop = self.properties
mat = self.material
n1, n2 = self.nodes
L = self.length()
E = mat['E']
G = mat['G']
Ax = prop['Area']
# Eval coordinate transformation matrix
T = self.calcT()
# Transform displacements from global to local coordinates
u1 = np.dot(T, n1.results[:3])
u2 = np.dot(T, n2.results[:3])
# Axial force, Nx
res[0, 0] = 0.
res[0, 1] = (-Ax * E / L) * (u2[0] - u1[0])
res[1, 0] = 1.
res[1, 1] = -res[0, 1]
return res