def str_el3(coord, ul):
"""Compute the strains at each element integration point
This one is used for 3-noded triangular elements.
Parameters
----------
coord : ndarray
Coordinates of the nodes of the element (nn, 2).
ul : ndarray
Array with displacements for the element.
Returns
-------
epsGT : ndarray
Strain components for the Gauss points.
xl : ndarray
Configuration of the Gauss points after deformation.
"""
epsl = np.zeros([3])
epsG = np.zeros([3, 3])
xl = np.zeros([3, 2])
XW, XP = gau.gpoints3()
for i in range(3):
ri = XP[i, 0]
si = XP[i, 1]
ddet, B = stdm3NT(ri, si, coord)
epsl = np.dot(B, ul)
epsG[:, i] = epsl
N = sha3(ri, si)
xl[i, 0] = sum(N[0, 2*i]*coord[i, 0] for i in range(3))
xl[i, 1] = sum(N[0, 2*i]*coord[i, 1] for i in range(3))
return epsG.T, xl
def uel3ntrian(coord, par):
"""Triangular element with 3 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (3, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (6, 6).
Examples
--------
>>> coord = np.array([
... [0, 0],
... [1, 0],
... [0, 1]])
>>> stiff = uel3ntrian(coord, 1/3, 8/3)
>>> stiff_ex = 1/2 * np.array([
... [4, 2, -3, -1, -1, -1],
... [2, 4, -1, -1, -1, -3],
... [-3, -1, 3, 0, 0, 1],
... [-1, -1, 0, 1, 1, 0],
... [-1, -1, 0, 1, 1, 0],
... [-1, -3, 1, 0, 0, 3]])
>>> np.allclose(stiff, stiff_ex)
True
"""
Emod = par[0]
enu = par[1]
rho = par[2]
calpha = par[3]
cbeta = par[4]
kl = np.zeros([6, 6])
ml = np.zeros([6, 6])
cl = np.zeros([6, 6])
C = fem.umat(enu, Emod, rho)
XW, XP = gau.gpoints3()
ngpts = 3
for i in range(ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm3NT(ri, si, coord)
N = fem.sha3(ri, si)
kl = kl + 0.5 * np.dot(np.dot(B.T, C), B) * alf * ddet
ml = ml + 0.5 * rho * np.dot(N.T, N) * alf * ddet
cl = calpha * kl + cbeta * ml
return kl, ml, cl
def uel3ntrian(coord, enu, Emod):
"""Triangular element with 3 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (3, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (6, 6).
Examples
--------
>>> coord = Matrix([
... [0, 0],
... [1, 0],
... [0, 1]])
>>> stiff = uel3ntrian(coord, S(1)/3, S(8)/3)
>>> stiff_ex = 1/2 * Matrix([
... [4, 2, -3, -1, -1, -1],
... [2, 4, -1, -1, -1, -3],
... [-3, -1, 3, 0, 0, 1],
... [-1, -1, 0, 1, 1, 0],
... [-1, -1, 0, 1, 1, 0],
... [-1, -3, 1, 0, 0, 3]])
>>> (stiff - stiff_ex).norm()/stiff_ex.norm() < 1e-6
True
"""
kl = np.zeros([6, 6])
C = fem.umat(enu, Emod)
XW, XP = gau.gpoints3()
ngpts = 3
for i in range(ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm3NT(ri, si, coord)
kl = kl + 0.5*B.T*C*B*alf*ddet
return kl
def str_el3(coord, ul):
"""Compute the strains at each element integration point
This one is used for 3-noded triangular elements.
Parameters
----------
coord : ndarray
Coordinates of the nodes of the element (nn, 2).
ul : ndarray
Array with displacements for the element.
Returns
-------
epsGT : ndarray
Strain components for the Gauss points.
xl : ndarray
Configuration of the Gauss points after deformation.
"""
epsl = np.zeros([3])
epsG = np.zeros([3, 3])
epsGT = np.zeros([3, 3])
xl = np.zeros([3, 2])
x, y = symbols('x y')
XW, XP = gau.gpoints3()
for i in range(3):
ri = XP[i, 0]
si = XP[i, 1]
ddet, B = stdm3NT(ri, si, coord)
epsl = B*ul
epsG[:, i] = epsl[:]
N = sha3(ri, si)
NN = N.subs([(x, ri), (y, si)])
xl[i, 0] = sum(NN[0, 2*i]*coord[i, 0] for i in range(3))
xl[i, 1] = sum(NN[0, 2*i]*coord[i, 1] for i in range(3))
epsGT = epsG.T
return epsGT, xl