Ejemplo n.º 1
0
def str_el4(coord, ul):
    """Compute the strains at each element integration point

    This one is used for 4-noded quadrilateral elements.

    Parameters
    ----------
    coord : ndarray
      Coordinates of the nodes of the element (4, 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, 4])
    xl = np.zeros([4, 2])
    XW, XP = gau.gpoints2x2()
    for i in range(4):
        ri = XP[i, 0]
        si = XP[i, 1]
        ddet, B = stdm4NQ(ri, si, coord)
        epsl = np.dot(B, ul)
        epsG[:, i] = epsl[:]
        N = sha4(ri, si)
        xl[i, 0] = sum(N[0, 2*i]*coord[i, 0] for i in range(4))
        xl[i, 1] = sum(N[0, 2*i]*coord[i, 1] for i in range(4))
    return epsG.T, xl
Ejemplo n.º 2
0
def uel4nquad(coord, par):
    """Quadrilateral element with 4 nodes

    Parameters
    ----------
    coord : ndarray
      Coordinates for the nodes of the element (4, 2).
    enu : float
      Poisson coefficient (-1, 0.5).
    Emod : float
      Young modulus (>0).

    Returns
    -------
    kl : ndarray
      Local stiffness matrix for the element (8, 8).

    Examples
    --------

    >>> coord = np.array([[-1, -1], [1, -1], [1, 1], [-1, 1]])
    >>> stiff = uel4nquad(coord, 1/3, 8/3)
    >>> stiff_ex = 1/6 * np.array([
    ...             [ 8,  3, -5,  0, -4, -3,  1,  0],
    ...             [ 3,  8,  0,  1, -3, -4,  0, -5],
    ...             [-5,  0,  8, -3,  1,  0, -4,  3],
    ...             [ 0,  1, -3,  8,  0, -5,  3, -4],
    ...             [-4, -3,  1,  0,  8,  3, -5,  0],
    ...             [-3, -4,  0, -5,  3,  8,  0,  1],
    ...             [ 1,  0, -4,  3, -5,  0,  8, -3],
    ...             [ 0, -5,  3, -4,  0,  1, -3,  8]])
    >>> np.allclose(stiff, stiff_ex)
    True

    """
    Emod = par[0]
    enu = par[1]
    rho = par[2]
    calpha = par[3]
    cbeta = par[4]
    kl = np.zeros([8, 8])
    ml = np.zeros([8, 8])
    cl = np.zeros([8, 8])
    C = fem.umat(enu, Emod, rho)
    XW, XP = gau.gpoints2x2()
    ngpts = 4
    for i in range(0, ngpts):
        ri = XP[i, 0]
        si = XP[i, 1]
        alf = XW[i]
        ddet, B = fem.stdm4NQ(ri, si, coord)
        N = fem.sha4(ri, si)
        kl = kl + np.dot(np.dot(B.T, C), B) * alf * ddet
        ml = ml + rho * np.dot(N.T, N) * alf * ddet
    cl = calpha * kl + cbeta * ml
    return kl, ml, cl
Ejemplo n.º 3
0
def uel4nquad(coord, enu, Emod):
    """Quadrilateral element with 4 nodes

    Parameters
    ----------
    coord : ndarray
      Coordinates for the nodes of the element (4, 2).
    enu : float
      Poisson coefficient (-1, 0.5).
    Emod : float
      Young modulus (>0).

    Returns
    -------
    kl : ndarray
      Local stiffness matrix for the element (8, 8).

    Examples
    --------
    
    >>> coord = Matrix([[-1, -1], [1, -1], [1, 1], [-1, 1]])
    >>> stiff = uel4nquad(coord, S(1)/3, S(8)/3)
    >>> stiff_ex = 1/6 * Matrix([
    ...             [ 8,  3, -5,  0, -4, -3,  1,  0],
    ...             [ 3,  8,  0,  1, -3, -4,  0, -5],
    ...             [-5,  0,  8, -3,  1,  0, -4,  3],
    ...             [ 0,  1, -3,  8,  0, -5,  3, -4],
    ...             [-4, -3,  1,  0,  8,  3, -5,  0],
    ...             [-3, -4,  0, -5,  3,  8,  0,  1],
    ...             [ 1,  0, -4,  3, -5,  0,  8, -3],
    ...             [ 0, -5,  3, -4,  0,  1, -3,  8]])
    >>> (stiff - stiff_ex).norm()/stiff_ex.norm() < 1e-6
    True

    """
    kl = np.zeros([8, 8])
    C = fem.umat(enu, Emod)
    XW, XP = gau.gpoints2x2()
    ngpts = 4
    for i in range(0, ngpts):
        ri = XP[i, 0]
        si = XP[i, 1]
        alf = XW[i]
        ddet, B = fem.stdm4NQ(ri, si, coord)
        kl = kl + B.T*C*B*alf*ddet
    return kl
Ejemplo n.º 4
0
def str_el4(coord, ul):
    """Compute the strains at each element integration point

    This one is used for 4-noded quadrilateral elements.

    Parameters
    ----------
    coord : ndarray
      Coordinates of the nodes of the element (4, 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, 4])
    epsGT = np.zeros([4, 3])
    xl = np.zeros([4, 2])
    x, y = symbols('x y')

    XW, XP = gau.gpoints2x2()
    for i in range(4):
        ri = XP[i, 0]
        si = XP[i, 1]
        ddet, B = stdm4NQ(ri, si, coord)
        epsl = B*ul
        epsG[:, i] = epsl[:]
        N = sha4(ri, si)
        NN = N.subs([(x, ri), (y, si)])
        xl[i, 0] = sum(NN[0, 2*i]*coord[i, 0] for i in range(4))
        xl[i, 1] = sum(NN[0, 2*i]*coord[i, 1] for i in range(4))
    epsGT = epsG.T
    return epsGT, xl