Linear static analysis

A Shell object describes a plate or a cylindrical shell through its attributes: the geometry, the laminate, the number of terms m and n of the approximation functions and the model, by default 'plate_clpt_donnell' for plates and 'cylshell_clpt_sanders' for cylindrical shells, see panels.models for the other models. The boundary conditions are controlled by the flags x1u, x1ur, x2u, …, y2wr, where 0 constrains and 1 releases the translation (e.g. x1u) or the rotation (e.g. x1ur) of each displacement component at each edge, see Bardell’s functions. The models based on shear deformation theories have the rotations as independent fields, with the analogous flags x1phix, x1phixr, …, y2phiyr.

The loads are added with Shell.add_point_load(), Shell.add_distr_load_fixed_x() or Shell.add_distr_load_fixed_y(), and the linear system \([K_C]\{c\} = \{F_{ext}\}\) is solved with structsolve. The displacement, strain and stress fields are calculated from the Ritz constants c with Shell.uvw(), Shell.strain() and Shell.stress(). The code below is extracted from one of the panels unit tests:

def test_panel_field_outputs():
    m = 7
    n = 6
    for model in ['plate_clpt_donnell',
                  'cylshell_clpt_donnell',
                  'cylshell_clpt_sanders',
                  'plate_fsdt_donnell',
                  'plate_tsdt_donnell']:
        print('Testing model %s' % model)
        s = Shell()
        s.model = model
        s.x1u = 1
        s.y1u = 1
        s.y2u = 1
        s.x1v = 0
        s.x2v = 0
        s.y1v = 0
        s.y2v = 0

        s.a = 2.
        s.b = 1.
        s.r = 1.e5
        s.stack = [0, -45, +45, 90, +45, -45, 0, 0]
        s.plyt = 1e-3*0.125
        s.laminaprop = (142.5e9, 8.7e9, 0.28, 5.1e9, 5.1e9, 5.1e9)
        s.nx = m
        s.ny = n
        s.m = m
        s.n = n

        P = 1000.
        Nxx = P/s.b
        s.add_distr_load_fixed_x(0, funcx=lambda y: Nxx, funcy=None, funcz=None, cte=False)
        increments, cs = static(s.calc_kC(), s.calc_fext(), silent=True)
        _, fields = s.uvw(cs[0])
        _, fields = s.stress(cs[0])
        _, fields = s.strain(cs[0])
        assert np.isclose(fields.get('w').max(), 0.00014, rtol=0.05)
        assert np.isclose(fields.get('Nxx').min(), -1017.4018, rtol=0.001)
        assert np.isclose(fields.get('eyy').min(), -4.1451e-07, rtol=0.001)