.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "gallery/verification/example_verify_cc_beam_central_load.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_gallery_verification_example_verify_cc_beam_central_load.py: Clamped-clamped beam under a central point load ================================================ Statically-indeterminate first-order beam with both ends fully clamped, carrying a single transverse point load :math:`P` at midspan. Symmetry plus compatibility at the clamps gives the elementary closed form (Roark & Young 1989, Table 8 case 5; Timoshenko 1955 §40): .. math:: R_\mathrm{left} = R_\mathrm{right} = \tfrac{P}{2}, \qquad M_\mathrm{end} = \pm \tfrac{P L}{8}, with the mid-span deflection (the indeterminate clamping moments stiffen the beam by a factor of 4 vs. the simply-supported case) .. math:: \delta_\mathrm{mid} = \frac{P\, L^{3}}{192\, E I}, and the mid-span bending moment :math:`M_\mathrm{mid} = +PL/8` (numerically equal to the end-moment magnitude because the moment diagram is anti-symmetric about midspan). Implementation -------------- A 20-element BEAM2 (Hermite-cubic Bernoulli) line spans the beam. Both end nodes are clamped (all 6 DOFs); a single :math:`-P\,\hat y` force is applied at the midspan node. The Hermite cubic basis recovers Euler-Bernoulli kinematics exactly under nodal point loads, so every node hits the analytical value to machine precision and the clamping reactions / moments come out cleanly. References ---------- * Timoshenko, S. P. (1955) *Strength of Materials, Part I*, 3rd ed., Van Nostrand, §40 (statically indeterminate beams). * Roark, R. J. and Young, W. C. (1989) *Roark's Formulas for Stress and Strain*, 6th ed., McGraw-Hill, Table 8 case 5 (clamped-clamped beam, central point load). * Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) *Concepts and Applications of Finite Element Analysis*, 4th ed., Wiley, §2.5. .. GENERATED FROM PYTHON SOURCE LINES 52-62 .. code-block:: Python from __future__ import annotations import numpy as np import pyvista as pv from vtkmodules.util.vtkConstants import VTK_LINE import femorph_solver from femorph_solver import ELEMENTS .. GENERATED FROM PYTHON SOURCE LINES 63-65 Problem data ------------ .. GENERATED FROM PYTHON SOURCE LINES 65-90 .. code-block:: Python E = 2.0e11 # Pa (steel) NU = 0.30 RHO = 7850.0 # kg/m^3 b = 0.050 # square cross-section side [m] A = b * b I = b**4 / 12.0 # noqa: E741 J = 2.0 * I L = 1.0 # span [m] P = 5.0e3 # central point load magnitude [N], applied downward (-y) N_ELEM = 20 # even so a node lands exactly at midspan # Closed-form quantities ----------------------------------------------- R_published = P / 2.0 M_end_published = P * L / 8.0 delta_mid_published = P * L**3 / (192.0 * E * I) print(f"P = {P:.1f} N, L = {L:.2f} m, EI = {E * I:.3e} N m^2") print(f"R_L = R_R = P/2 = {R_published:.4f} N") print(f"|M_end| = P L / 8 = {M_end_published:.4f} N m (clamping moment)") print(f"δ_mid = P L^3/(192 EI) = {delta_mid_published:.4e} m") print("stiffness ratio C-C / S-S = 4x (192 / 48 = 4)") .. rst-class:: sphx-glr-script-out .. code-block:: none P = 5000.0 N, L = 1.00 m, EI = 1.042e+05 N m^2 R_L = R_R = P/2 = 2500.0000 N |M_end| = P L / 8 = 625.0000 N m (clamping moment) δ_mid = P L^3/(192 EI) = 2.5000e-04 m stiffness ratio C-C / S-S = 4x (192 / 48 = 4) .. GENERATED FROM PYTHON SOURCE LINES 91-93 Build a 20-element BEAM2 line ----------------------------- .. GENERATED FROM PYTHON SOURCE LINES 93-112 .. code-block:: Python points = np.array( [[i * L / N_ELEM, 0.0, 0.0] for i in range(N_ELEM + 1)], dtype=np.float64, ) cells_list: list[int] = [] for i in range(N_ELEM): cells_list.extend([2, i, i + 1]) cells = np.asarray(cells_list, dtype=np.int64) cell_types = np.full(N_ELEM, VTK_LINE, dtype=np.uint8) grid = pv.UnstructuredGrid(cells, cell_types, points) m = femorph_solver.Model.from_grid(grid) m.assign( ELEMENTS.BEAM2, material={"EX": E, "PRXY": NU, "DENS": RHO}, real=(A, I, I, J), ) .. GENERATED FROM PYTHON SOURCE LINES 113-115 Boundary conditions: both ends clamped -------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 115-120 .. code-block:: Python right = N_ELEM + 1 m.fix(nodes=[1], dof="ALL") m.fix(nodes=[right], dof="ALL") .. GENERATED FROM PYTHON SOURCE LINES 121-123 Apply the central point load ---------------------------- .. GENERATED FROM PYTHON SOURCE LINES 123-127 .. code-block:: Python mid_node = N_ELEM // 2 + 1 m.apply_force(mid_node, fy=-P) .. GENERATED FROM PYTHON SOURCE LINES 128-130 Static solve + reaction extraction ---------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 130-164 .. code-block:: Python res = m.solve() dof = m.dof_map() def _react(node: int, dof_idx: int) -> float: rows = np.where((dof[:, 0] == node) & (dof[:, 1] == dof_idx))[0] return float(res.reaction[rows[0]]) if len(rows) else 0.0 R_left = _react(1, 1) R_right = _react(right, 1) M_left = _react(1, 5) # ROTZ reaction = clamp moment M_right = _react(right, 5) print() print("reactions (femorph-solver) → (analytical)") print(f" R_left_UY = {R_left:+.4f} → {+R_published:+.4f} N") print(f" R_right_UY = {R_right:+.4f} → {+R_published:+.4f} N") print(f" M_left_RZ = {M_left:+.4f} → {+M_end_published:+.4f} N m") print(f" M_right_RZ = {M_right:+.4f} → {-M_end_published:+.4f} N m") # Vertical equilibrium and exact reaction match assert np.isclose(R_left + R_right, P, rtol=1e-12), "vertical equilibrium failed" assert np.isclose(R_left, R_published, rtol=1e-12), "left reaction off" assert np.isclose(R_right, R_published, rtol=1e-12), "right reaction off" # Clamp moment-reactions are equal in magnitude, opposite in sign — both # act to resist the downward load at midspan. M_left = +PL/8 (CCW about # +z) lifts the right end against the load; M_right = -PL/8 (CW) # mirrors it on the other side. assert np.isclose(M_left, +M_end_published, rtol=1e-12), "left clamp moment off" assert np.isclose(M_right, -M_end_published, rtol=1e-12), "right clamp moment off" assert np.isclose(M_left + M_right, 0.0, atol=1e-9), "moment anti-symmetry violated" .. rst-class:: sphx-glr-script-out .. code-block:: none reactions (femorph-solver) → (analytical) R_left_UY = +2500.0000 → +2500.0000 N R_right_UY = +2500.0000 → +2500.0000 N M_left_RZ = +625.0000 → +625.0000 N m M_right_RZ = -625.0000 → -625.0000 N m .. GENERATED FROM PYTHON SOURCE LINES 165-167 Mid-span deflection ------------------- .. GENERATED FROM PYTHON SOURCE LINES 167-176 .. code-block:: Python mid_uy = float(res.displacement[np.where((dof[:, 0] == mid_node) & (dof[:, 1] == 1))[0][0]]) err_mid = (mid_uy - (-delta_mid_published)) / (-delta_mid_published) print() print(f"δ_mid computed = {mid_uy:+.4e} m") print(f"δ_mid published = {-delta_mid_published:+.4e} m") print(f"relative error = {err_mid * 100:+.6f} %") assert abs(err_mid) < 1e-8, "mid-span deflection error too large for Bernoulli BEAM2" .. rst-class:: sphx-glr-script-out .. code-block:: none δ_mid computed = -2.5000e-04 m δ_mid published = -2.5000e-04 m relative error = -0.000000 % .. GENERATED FROM PYTHON SOURCE LINES 177-179 Render the deflected line ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 179-210 .. code-block:: Python pts = np.asarray(grid.points) disp_y = np.zeros(N_ELEM + 1) for i in range(N_ELEM + 1): rows = np.where((dof[:, 0] == i + 1) & (dof[:, 1] == 1))[0] if len(rows): disp_y[i] = float(res.displacement[rows[0]]) warped = grid.copy() warped.points = pts + np.column_stack([np.zeros(N_ELEM + 1), 200.0 * disp_y, np.zeros(N_ELEM + 1)]) warped["uy"] = disp_y plotter = pv.Plotter(off_screen=True, window_size=(720, 280)) plotter.add_mesh(grid, color="grey", line_width=2, opacity=0.5) plotter.add_mesh(warped, scalars="uy", line_width=5, cmap="viridis") plotter.add_points( np.array([[0.0, 0.0, 0.0], [L, 0.0, 0.0]]), render_points_as_spheres=True, point_size=18, color="black", label="clamps", ) plotter.add_points( np.array([[0.5 * L, 0.0, 0.0]]), render_points_as_spheres=True, point_size=14, color="#d62728", label=f"P = {P:.0f} N", ) plotter.view_xy() plotter.camera.zoom(1.05) plotter.show() .. image-sg:: /gallery/verification/images/sphx_glr_example_verify_cc_beam_central_load_001.png :alt: example verify cc beam central load :srcset: /gallery/verification/images/sphx_glr_example_verify_cc_beam_central_load_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.183 seconds) .. _sphx_glr_download_gallery_verification_example_verify_cc_beam_central_load.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: example_verify_cc_beam_central_load.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: example_verify_cc_beam_central_load.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: example_verify_cc_beam_central_load.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_