def test_ueltruss2D():
"""Tests for 2-noded 2D truss uel"""
coord = np.array([[0, 0], [1, 0]])
params = 1.0, 1.0
stiff, mass = uel.ueltruss2D(coord, params)
stiff_ex = np.array([[1, 0, -1, 0], [0, 0, 0, 0], [-1, 0, 1, 0],
[0, 0, 0, 0]])
assert np.allclose(stiff, stiff_ex)
def test_ueltruss2D():
coord = np.array([
[0, 0],
[1, 0]])
stiff = uel.ueltruss2D(coord, 1.0 , 1.0)
stiff_ex = np.array([
[1, 0, -1, 0],
[0, 0, 0, 0],
[-1, 0, 1, 0],
[0, 0, 0, 0]])
np.allclose(stiff, stiff_ex)
def retriever(elements, mats, nodes, i, uel=None):
"""Computes the elemental stiffness matrix of element i
Parameters
----------
elements : ndarray
Array with the number for the nodes in each element.
mats : ndarray.
Array with the material profiles.
nodes : ndarray.
Array with the nodal numbers and coordinates.
i : int.
Identifier of the element to be assembled.
Returns
-------
kloc : ndarray (float)
Array with the local stiffness matrix.
ndof : int.
Number of degrees of fredom of the current element.
"""
IELCON = np.zeros([9], dtype=np.integer)
iet = elements[i, 1]
ndof, nnodes, ngpts = fem.eletype(iet)
elcoor = np.zeros([nnodes, 2])
im = np.int(elements[i, 2])
par0, par1 = mats[im, :]
for j in range(nnodes):
IELCON[j] = elements[i, j + 3]
elcoor[j, 0] = nodes[IELCON[j], 1]
elcoor[j, 1] = nodes[IELCON[j], 2]
if uel is None:
if iet == 1:
kloc = ue.uel4nquad(elcoor, par1, par0)
elif iet == 2:
kloc = ue.uel6ntrian(elcoor, par1, par0)
elif iet == 3:
kloc = ue.uel3ntrian(elcoor, par1, par0)
elif iet == 5:
kloc = ue.uelspring(elcoor, par1, par0)
elif iet == 6:
kloc = ue.ueltruss2D(elcoor, par1, par0)
elif iet == 7:
kloc = ue.uelbeam2DU(elcoor, par1, par0)
else:
kloc, ndof, iet = uel(elcoor, par1, par0)
return kloc, ndof, iet
def stress_truss(nodes, elements, mats, disp):
r"""Compute axial stresses in truss members
Parameters
----------
nodes : ndarray (float).
Array with nodes coordinates.
elements : ndarray (int)
Array with the node number for the nodes that correspond
to each element.
mats : ndarray (float)
Array with material profiles.
disp : ndarray (float)
Array with the displacements. This one contains both, the
computed and imposed values.
Returns
-------
stress : ndarray
Stresses for each member on the truss
Examples
--------
The following examples are taken from [1]_. In all the examples
:math:`A=1`, :math:`E=1`.
>>> import assemutil as ass
>>> import solidspy.solutil as sol
>>> def fem_sol(nodes, elements, mats, loads):
... DME , IBC , neq = ass.DME(nodes, elements)
... KG = ass.assembler(elements, mats, nodes, neq, DME)
... RHSG = ass.loadasem(loads, IBC, neq)
... UG = sol.static_sol(KG, RHSG)
... UC = complete_disp(IBC, nodes, UG)
... return UC
**Exercise 3.3-18**
The axial stresses in this example are
.. math::
[\sigma] = \left[\frac{1}{2},\frac{\sqrt{3}}{4},\frac{1}{4}\right]
>>> nodes = np.array([
... [0, 0.0, 0.0, 0, -1],
... [1, -1.0, 0.0, -1, -1],
... [2, -np.cos(np.pi/6), -np.sin(np.pi/6), -1, -1],
... [3, -np.cos(np.pi/3), -np.sin(np.pi/3), -1, -1]])
>>> mats = np.array([[1.0, 1.0]])
>>> elements = np.array([
... [0, 6, 0, 1, 0],
... [1, 6, 0, 2, 0],
... [2, 6, 0, 3, 0]])
>>> loads = np.array([[0, 1.0, 0]])
>>> disp = fem_sol(nodes, elements, mats, loads)
>>> stress = stress_truss(nodes, elements, mats, disp)
>>> stress_exact = np.array([1/2, np.sqrt(3)/4, 1/4])
>>> np.allclose(stress_exact, stress)
True
**Exercise 3.3-19**
The axial stresses in this example are
.. math::
[\sigma] = \left[\frac{1}{\sqrt{2}+2},
\frac{\sqrt{2}}{\sqrt{2}+1},
\frac{1}{\sqrt{2}+2}\right]
>>> nodes = np.array([
... [0, 0.0, 0.0, 0, 0],
... [1, -1.0, -1.0, -1, -1],
... [2, 0.0, -1.0, -1, -1],
... [3, 1.0, -1.0, -1, -1]])
>>> mats = np.array([[1.0, 1.0]])
>>> elements = np.array([
... [0, 6, 0, 1, 0],
... [1, 6, 0, 2, 0],
... [2, 6, 0, 3, 0]])
>>> loads = np.array([[0, 0, 1]])
>>> disp = fem_sol(nodes, elements, mats, loads)
>>> stress = stress_truss(nodes, elements, mats, disp)
>>> stress_exact = np.array([
... 1/(np.sqrt(2) + 2),
... np.sqrt(2)/(np.sqrt(2) + 1),
... 1/(np.sqrt(2) + 2)])
>>> np.allclose(stress_exact, stress)
True
**Exercise 3.3-22**
The axial stresses in this example are
.. math::
[\sigma] =\left[\frac{1}{3 \sqrt{2}},\frac{5}{12},
\frac{1}{2^{\frac{3}{2}}},
\frac{1}{12},
-\frac{1}{3 \sqrt{2}}\right]
>>> cathetus = np.cos(np.pi/4)
>>> nodes = np.array([
... [0, 0.0, 0.0, 0, 0],
... [1, -1.0, 0.0, -1, -1],
... [2, -cathetus, cathetus, -1, -1],
... [3, 0.0, 1.0, -1, -1],
... [4, cathetus, cathetus, -1, -1],
... [5, 1.0, 0.0, -1, -1]])
>>> mats = np.array([[1.0, 1.0]])
>>> elements = np.array([
... [0, 6, 0, 1, 0],
... [1, 6, 0, 2, 0],
... [2, 6, 0, 3, 0],
... [3, 6, 0, 4, 0],
... [4, 6, 0, 5, 0]])
>>> loads = np.array([[0, cathetus, -cathetus]])
>>> disp = fem_sol(nodes, elements, mats, loads)
>>> stress = stress_truss(nodes, elements, mats, disp)
>>> stress_exact = np.array([
... 1/(3*np.sqrt(2)),
... 5/12,
... 1/2**(3/2),
... 1/12,
... -1/(3*np.sqrt(2))])
>>> np.allclose(stress_exact, stress)
True
References
----------
.. [1] William Weaver and James Gere. Matrix Analysis
of Framed Structures. Third Edition, Van Nostrand
Reinhold, New York (1990).
"""
neles = elements.shape[0]
stress = np.zeros((neles))
for cont in range(neles):
ini = elements[cont, 3]
end = elements[cont, 4]
coords = nodes[[ini, end], 1:3]
tan_vec = coords[1, :] - coords[0, :]
length = np.linalg.norm(tan_vec)
mat_id = elements[cont, 2]
local_stiff = uel.ueltruss2D(coords, *mats[mat_id, :])
local_disp = np.hstack((disp[ini, :], disp[end, :]))
local_forces = local_stiff.dot(local_disp)
stress[cont] = local_forces[2:].dot(tan_vec) / (length *
mats[mat_id, 1])
return stress