class TestBeamFEAttachmentModes(unittest.TestCase): def setUp(self): x = array([0.0, 10.0, 22.0, 28.0]) EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]]) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0) self.fe.set_boundary_conditions('C', 'C') self.fe.set_dofs([False, False, True, False, True, False]) def test_simple_model(self): am = self.fe.attachment_modes() self.assertEqual(am.shape, (4 * 6, 2 * 2))
class TestBeamFE_Example62(unittest.TestCase): def setUp(self): x = linspace(0, 1, 16) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0) self.fe.set_boundary_conditions('C', 'F') self.fe.set_dofs([False, False, True, False, True, False]) def test_mode_frequencies(self): modal = self.fe.modal_matrices() assert_allclose(modal.w[:4], [3.5160, 22.0345, 61.6972, 120.9019], atol=1e-1)
def setUp(self): x = linspace(0, self.L, 15) fe = BeamFE(x, density=self.m, EA=0, EIy=self.EI, EIz=0) fe.set_boundary_conditions('C', 'C') fe.set_dofs([False, False, True, False, True, False]) beam = DistalModalElementFromFE('beam', fe, num_modes=1, damping=self.damping_coeff) system = System() system.add_leaf(beam) system.setup() system.update_kinematics() self.beam, self.system = beam, system
class TestBeamFE_Example62(unittest.TestCase): def setUp(self): x = linspace(0, 1, 16) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0) self.fe.set_boundary_conditions('C', 'F') self.fe.set_dofs([False, False, True, False, True, False]) def test_mode_frequencies(self): modal = self.fe.modal_matrices() assert_allclose(modal.w[:4], [3.5160, 22.0345, 61.6972, 120.9019], atol=1e-1)
class TestBeamFE_Example42(unittest.TestCase): def setUp(self): x = array([0.0, 4.0, 10.0]) EI = 144.0 # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0) self.fe.set_boundary_conditions('C', 'F') self.fe.set_dofs([False, False, True, False, True, False]) def test_stiffness(self): # Reddy1993, p 164 expected = array([ [27, 0, 0, 0, 0, 0], [-54, 144, 0, 0, 0, 0], [-27, 54, 27 + 8, 0, 0, 0], [-54, 72, 54 - 24, 144 + 96, 0, 0], [0, 0, -8, 24, 8, 0], [0, 0, -24, 48, 24, 96], ]) expected += expected.T - np.diag(expected.diagonal()) # make symmetric expected_II = expected[2:6, 2:6] expected_IB = expected[2:6, 0:2] assert_allclose(self.fe.K_II, expected_II) assert_allclose(self.fe.K_IB, expected_IB) def test_distributed_load(self): # Distributed force on second element: load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0]) Q = self.fe.distribute_load_on_element(1, load) Q = Q[[2, 4, 8, 10, 14, 16]] assert_allclose(Q, [0, 0, -90, 120, -210, -180]) def test_solution_without_spring(self): # Distributed force on first element: load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0]) QF = self.fe.distribute_load_on_element(1, load) # Solve static deflection deflections, reactions = self.fe.static_deflection(QF) # Expected values from Reddy expected_deflections = zeros_like(deflections) expected_deflections[[ 8, 10, 14, 16 ]] = array([-1.6, 0.72, -7.108, 0.99]) * 1e4 / 143.0 assert_allclose(deflections, expected_deflections, atol=1e1)
def setUp(self): x = linspace(0, self.L, 15) fe = BeamFE(x, density=self.m, EA=0, EIy=self.EI, EIz=0) fe.set_boundary_conditions('C', 'C') fe.set_dofs([False, False, True, False, True, False]) beam = DistalModalElementFromFE('beam', fe, num_modes=1, damping=self.damping_coeff) system = System() system.add_leaf(beam) system.setup() system.update_kinematics() self.beam, self.system = beam, system
def test_first_mode_frequency(self): # From Reddy1993, p. 160 x = np.linspace(0, 1, 16) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0) fe.set_boundary_conditions('C', 'F') fe.set_dofs([False, False, True, False, True, False]) element = ModalElementFromFE('elem', fe) Mmodal = element.mass_ee Kmodal = element.K w = np.sqrt(np.diag(Kmodal) / np.diag(Mmodal)) assert_aae(w[0], 3.5160, decimal=4)
def test_first_mode_frequency(self): # From Reddy1993, p. 160 x = np.linspace(0, 1, 16) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0) fe.set_boundary_conditions('C', 'F') fe.set_dofs([False, False, True, False, True, False]) element = ModalElementFromFE('elem', fe) Mmodal = element.mass_ee Kmodal = element.K w = np.sqrt(np.diag(Kmodal) / np.diag(Mmodal)) assert_aae(w[0], 3.5160, decimal=4)
class TestBeamFEAttachmentModes(unittest.TestCase): def setUp(self): x = array([0.0, 10.0, 22.0, 28.0]) EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]]) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0) self.fe.set_boundary_conditions('C', 'C') self.fe.set_dofs([False, False, True, False, True, False]) def test_simple_model(self): am = self.fe.attachment_modes() self.assertEqual(am.shape, (4*6, 2*2))
class TestBeamFE_Example42(unittest.TestCase): def setUp(self): x = array([0.0, 4.0, 10.0]) EI = 144.0 # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0) self.fe.set_boundary_conditions('C', 'F') self.fe.set_dofs([False, False, True, False, True, False]) def test_stiffness(self): # Reddy1993, p 164 expected = array([ [27, 0, 0, 0, 0, 0], [-54, 144, 0, 0, 0, 0], [-27, 54, 27+8, 0, 0, 0], [-54, 72, 54-24, 144+96, 0, 0], [0, 0, -8, 24, 8, 0], [0, 0, -24, 48, 24, 96], ]) expected += expected.T - np.diag(expected.diagonal()) # make symmetric expected_II = expected[2:6, 2:6] expected_IB = expected[2:6, 0:2] assert_allclose(self.fe.K_II, expected_II) assert_allclose(self.fe.K_IB, expected_IB) def test_distributed_load(self): # Distributed force on second element: load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0]) Q = self.fe.distribute_load_on_element(1, load) Q = Q[[2, 4, 8, 10, 14, 16]] assert_allclose(Q, [0, 0, -90, 120, -210, -180]) def test_solution_without_spring(self): # Distributed force on first element: load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0]) QF = self.fe.distribute_load_on_element(1, load) # Solve static deflection deflections, reactions = self.fe.static_deflection(QF) # Expected values from Reddy expected_deflections = zeros_like(deflections) expected_deflections[[8, 10, 14, 16]] = array( [-1.6, 0.72, -7.108, 0.99]) * 1e4 / 143.0 assert_allclose(deflections, expected_deflections, atol=1e1)
def test_static_deflection(self): x = array([0.0, 4.0, 10.0]) EI = 144.0 # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. fe = BeamFE(x, density=10, EA=0, EIy=EI, EIz=0) fe.set_boundary_conditions('C', 'F') fe.set_dofs([False, False, True, False, True, False]) element = ModalElementFromFE('elem', fe) # Distributed load, linearly interpolated load = np.zeros((3, 3)) load[-1, 2] = -100 # Load in z direction at tip element.apply_distributed_loading(load) defl = -element.applied_stress / np.diag(element.K) # Check against directly calculating static deflection from FE Q = fe.distribute_load(interleave(load, 6)) defl_fe, reactions_fe = fe.static_deflection(Q) assert_aae(dot(element.shapes, defl), defl_fe, decimal=2)
def test_static_deflection(self): x = array([0.0, 4.0, 10.0]) EI = 144.0 # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. fe = BeamFE(x, density=10, EA=0, EIy=EI, EIz=0) fe.set_boundary_conditions('C', 'F') fe.set_dofs([False, False, True, False, True, False]) element = ModalElementFromFE('elem', fe) # Distributed load, linearly interpolated load = np.zeros((3, 3)) load[-1, 2] = -100 # Load in z direction at tip element.apply_distributed_loading(load) defl = -element.applied_stress / np.diag(element.K) # Check against directly calculating static deflection from FE Q = fe.distribute_load(interleave(load, 6)) defl_fe, reactions_fe = fe.static_deflection(Q) assert_aae(dot(element.shapes, defl), defl_fe, decimal=2)
def test_rigid_element_reference_loading(self): # Make an element with no modes (rigid) fe = BeamFE(np.linspace(0, 1, 11), density=1, EA=0, EIy=1, EIz=0) fe.set_boundary_conditions('C', 'F') fe.set_dofs([False, False, True, False, True, False]) element = ModalElementFromFE('elem', fe, 0) # Distributed load, linearly interpolated load = np.zeros((11, 3)) load[:, 2] = -100 # Uniform load in z direction element.apply_distributed_loading(load) assert_aae(element.applied_stress, []) assert_aae(element.applied_forces[0:6], [0, 0, -100 * 1, 0, 50, 0]) assert_aae(element.applied_forces[6:12], 0) # Distributed load, triangle at tip load = np.zeros((11, 3)) load[-1, 2] = -100 # tapered load in z direction element.applied_forces[:] = 0 # reset element.apply_distributed_loading(load) F = -100 * 0.1 / 2 assert_aae(element.applied_forces[0:6], [0, 0, F, 0, -F * (0.9 + 0.1 * 2 / 3), 0]) assert_aae(element.applied_forces[6:12], 0)
def test_rigid_element_reference_loading(self): # Make an element with no modes (rigid) fe = BeamFE(np.linspace(0, 1, 11), density=1, EA=0, EIy=1, EIz=0) fe.set_boundary_conditions('C', 'F') fe.set_dofs([False, False, True, False, True, False]) element = ModalElementFromFE('elem', fe, 0) # Distributed load, linearly interpolated load = np.zeros((11, 3)) load[:, 2] = -100 # Uniform load in z direction element.apply_distributed_loading(load) assert_aae(element.applied_stress, []) assert_aae(element.applied_forces[0:6], [0, 0, -100 * 1, 0, 50, 0]) assert_aae(element.applied_forces[6:12], 0) # Distributed load, triangle at tip load = np.zeros((11, 3)) load[-1, 2] = -100 # tapered load in z direction element.applied_forces[:] = 0 # reset element.apply_distributed_loading(load) F = -100 * 0.1 / 2 assert_aae(element.applied_forces[0:6], [0, 0, F, 0, -F * (0.9 + 0.1 * 2 / 3), 0]) assert_aae(element.applied_forces[6:12], 0)
class TestBeamFE_Example41(unittest.TestCase): def setUp(self): x = array([0.0, 10.0, 22.0, 28.0]) EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]]) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0) self.fe.set_boundary_conditions('P', 'C') self.fe.set_dofs([False, False, True, False, True, False]) def test_stiffness(self): # Reddy1993, pp 161-162 expected = 1e7 * array([ [0.024, -0.12, -0.024, -0.12, 0, 0, 0, 0], [0, 0.80, 0.12, 0.40, 0, 0, 0, 0], [0, 0, 0.0309, 0.0783, -0.00694, -0.04167, 0, 0], [0, 0, 0, 1.133, 0.0417, 0.167, 0, 0], [0, 0, 0, 0, 0.0625, -0.125, -0.0556, -0.167], [0, 0, 0, 0, 0, 1, 0.1667, 0.333], [0, 0, 0, 0, 0, 0, 0.0556, 0.1667], [0, 0, 0, 0, 0, 0, 0, 0.6667], ]) expected += expected.T - np.diag(expected.diagonal()) # make symmetric expected_II = expected[1:6, 1:6] expected_IB = expected[1:6, [0, 6, 7]] assert_allclose(expected_II, self.fe.K_II, atol=1e-2 * 1e7) assert_allclose(expected_IB, self.fe.K_IB, atol=1e-3 * 1e7) def test_distributed_load(self): # Distributed force on first element: load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0]) Q = self.fe.distribute_load_on_element(0, load) Q = Q[[2, 4, 8, 10, 14, 16, 20, 22]] assert_allclose(Q / 1e3, [-12, 20, -12, -20, 0, 0, 0, 0]) def test_solution(self): # Distributed force on first element: load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0]) QF = self.fe.distribute_load_on_element(0, load) # Nodal forces: Q = np.zeros(self.fe.K.shape[0]) Q[14] = -10000 # z force at 3rd node # Solve static deflection deflections, reactions = self.fe.static_deflection(QF + Q) # Expected values from Reddy expected_deflections = zeros_like(deflections) expected_deflections[[4, 8, 10, 14, 16]] = [0.03856, -0.2808, 0.01214, -0.1103, -0.02752] assert_allclose(deflections, expected_deflections, atol=1e-4) expected_reactions = zeros_like(reactions) expected_reactions[[2, 20, 22]] = [18565.54, 15434.46, 92164.83] assert_allclose(reactions, expected_reactions, atol=1e-2) def test_load_matrix_gives_same_result_as_method(self): # Distributed force on all elements load = np.zeros(4 * 6) load[2::6] = [350, 324, 654, 54] # Z component Q1 = self.fe.distribute_load(load) Q2 = dot(self.fe.F, load) assert_allclose(Q1, Q2)
class TestBeamFE_Example41(unittest.TestCase): def setUp(self): x = array([0.0, 10.0, 22.0, 28.0]) EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]]) # Using the z axis as the transverse direction gives the same # sign convention as Reddy uses in 2D, namely that rotations # are positive clockwise. self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0) self.fe.set_boundary_conditions('P', 'C') self.fe.set_dofs([False, False, True, False, True, False]) def test_stiffness(self): # Reddy1993, pp 161-162 expected = 1e7 * array([ [0.024, -0.12, -0.024, -0.12, 0, 0, 0, 0], [0, 0.80, 0.12, 0.40, 0, 0, 0, 0], [0, 0, 0.0309, 0.0783, -0.00694, -0.04167, 0, 0], [0, 0, 0, 1.133, 0.0417, 0.167, 0, 0], [0, 0, 0, 0, 0.0625, -0.125, -0.0556, -0.167], [0, 0, 0, 0, 0, 1, 0.1667, 0.333], [0, 0, 0, 0, 0, 0, 0.0556, 0.1667], [0, 0, 0, 0, 0, 0, 0, 0.6667], ]) expected += expected.T - np.diag(expected.diagonal()) # make symmetric expected_II = expected[1:6, 1:6] expected_IB = expected[1:6, [0, 6, 7]] assert_allclose(expected_II, self.fe.K_II, atol=1e-2 * 1e7) assert_allclose(expected_IB, self.fe.K_IB, atol=1e-3 * 1e7) def test_distributed_load(self): # Distributed force on first element: load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0]) Q = self.fe.distribute_load_on_element(0, load) Q = Q[[2, 4, 8, 10, 14, 16, 20, 22]] assert_allclose(Q / 1e3, [-12, 20, -12, -20, 0, 0, 0, 0]) def test_solution(self): # Distributed force on first element: load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0]) QF = self.fe.distribute_load_on_element(0, load) # Nodal forces: Q = np.zeros(self.fe.K.shape[0]) Q[14] = -10000 # z force at 3rd node # Solve static deflection deflections, reactions = self.fe.static_deflection(QF + Q) # Expected values from Reddy expected_deflections = zeros_like(deflections) expected_deflections[[4, 8, 10, 14, 16]] = [ 0.03856, -0.2808, 0.01214, -0.1103, -0.02752 ] assert_allclose(deflections, expected_deflections, atol=1e-4) expected_reactions = zeros_like(reactions) expected_reactions[[2, 20, 22]] = [18565.54, 15434.46, 92164.83] assert_allclose(reactions, expected_reactions, atol=1e-2) def test_load_matrix_gives_same_result_as_method(self): # Distributed force on all elements load = np.zeros(4 * 6) load[2::6] = [350, 324, 654, 54] # Z component Q1 = self.fe.distribute_load(load) Q2 = dot(self.fe.F, load) assert_allclose(Q1, Q2)