{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import pyvista\npyvista.OFF_SCREEN = True\npyvista.set_jupyter_backend('static')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Cantilever under uniformly distributed load \u2014 Euler\u2013Bernoulli closed form\n\nFor a clamped-free prismatic cantilever of length $L$,\nflexural rigidity $EI$, and uniform transverse line load\n$w$ (force per unit length), the tip deflection is\n\n\\begin{align}\\delta_\\text{tip} = \\frac{w L^{4}}{8 E I},\n    \\qquad I = \\frac{w_b h^{3}}{12}\\end{align}\n\n(Timoshenko 1955 \u00a75.4, Cook 2002 \u00a72.4\u2013\u00a72.5).  The corresponding\nslope and curvature at the clamp are\n\n\\begin{align}\\theta_\\text{tip} = \\frac{w L^{3}}{6 E I}, \\qquad\n    \\kappa(0) = \\frac{M(0)}{E I} = \\frac{w L^{2}}{2 E I},\\end{align}\n\nso the maximum bending stress at the root extreme fibre is\n$\\sigma_\\text{max} = E\\, \\kappa(0)\\, c = w L^{2} c / (2 I)$.\n\n## References\n* Timoshenko, S. P.  *Strength of Materials, Part I: Elementary\n  Theory and Problems*, 3rd ed., Van Nostrand, 1955, \u00a75.4.\n* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002)\n  *Concepts and Applications of Finite Element Analysis*, 4th\n  ed., Wiley, \u00a72.4 (Hermite cubics) \u2014 the cantilever-UDL example\n  is the canonical test of consistent-load integration on the\n  beam element.\n* Gere, J. M. and Goodno, B. J. (2018) *Mechanics of Materials*,\n  9th ed., Cengage, \u00a79.3 Table 9-1 case 1.\n\n## Vendor cross-references\n\n==========================================  =====================  ==================================\nSource                                       Reported \u03b4_tip [m]     Problem ID / location\n==========================================  =====================  ==================================\nClosed form (Euler\u2013Bernoulli)                1.200 \u00d7 10\u207b\u00b3           Timoshenko *SoM Part I* \u00a75.4\nGere & Goodno (2018) \u00a79.3 Table 9-1 case 1   1.200 \u00d7 10\u207b\u00b3           UDL cantilever\nMAPDL Verification Manual                    1.20 \u00d7 10\u207b\u00b3            VM-2 *Beam stresses and deflections*\nAbaqus Verification Manual                   1.20 \u00d7 10\u207b\u00b3            AVM 1.5.x cantilever-with-UDL family\n==========================================  =====================  ==================================\n\n## Implementation note\nThe cantilever is meshed as a 3D HEX8 plate at slenderness\n$L/h = 20$ with three transverse layers; the\n``enhanced_strain`` formulation (Simo-Rifai 1990) is selected\nto skirt the shear-locking that plain bilinear hexes suffer on\nslender bending.  The UDL is distributed across the\ntop-surface nodes as a per-node ``F_y`` proportional to the\nnodal influence area.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "from __future__ import annotations\n\nimport numpy as np\nimport pyvista as pv\n\nimport femorph_solver\nfrom femorph_solver import ELEMENTS"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Geometry + material\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "L = 1.0\nWIDTH = 0.05\nHEIGHT = 0.05\nE = 2.0e11\nNU = 0.30\nRHO = 7850.0\n\n# Total load per unit length [N/m]; integrated over L gives the\n# total force for the equivalent point-load comparison.\nw = 1000.0\nW_total = w * L\n\nI = WIDTH * HEIGHT**3 / 12.0  # noqa: E741\n# Closed-form magnitude is wL^4 / (8EI); sign convention here is\n# negative because the applied UDL points in -y.\ndelta_published = -w * L**4 / (8.0 * E * I)\n\nprint(f\"problem: L={L} m, w={w} N/m (downward), EI={E * I:.4e} N\u00b7m\u00b2\")\nprint(f\"published \u03b4_tip = -w L^4 / (8 E I) = {delta_published:+.6e} m\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Build a HEX8 plate cantilever\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "NX, NY, NZ = 40, 3, 3\nxs = np.linspace(0.0, L, NX + 1)\nys = np.linspace(0.0, WIDTH, NY + 1)\nzs = np.linspace(0.0, HEIGHT, NZ + 1)\ngrid = pv.StructuredGrid(*np.meshgrid(xs, ys, zs, indexing=\"ij\")).cast_to_unstructured_grid()\n\nm = femorph_solver.Model.from_grid(grid)\nm.assign(\n    ELEMENTS.HEX8(integration=\"enhanced_strain\"),\n    material={\"EX\": E, \"PRXY\": NU, \"DENS\": RHO},\n)\n\npts = np.asarray(m.grid.points)\nclamped = np.where(pts[:, 0] < 1e-9)[0]\nm.fix(nodes=(clamped + 1).tolist(), dof=\"ALL\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Distribute the UDL as nodal F_y on the top surface\n\nA uniform line load ``w`` (force/length) on the beam axis maps\nto a per-node $F_y$ proportional to the nodal influence\narea on the top surface.  For a regular structured grid this is\nconstant per interior node and half at the edges.  Total\ndistributed force = ``w * L`` is invariant.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "top = np.where(pts[:, 2] > HEIGHT - 1e-9)[0]\n# Equal weights per top-surface node \u2014 coarse but acceptable on\n# a structured grid for this verification.\nfy_per_node = -W_total / len(top)\nfor n in top:\n    m.apply_force(int(n + 1), fy=fy_per_node)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Solve and compare\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "res = m.solve_static()\nu = np.asarray(res.displacement).reshape(-1, 3)\n\ntip = np.where(pts[:, 0] > L - 1e-9)[0]\ndelta_computed = float(u[tip, 1].mean())\n\nrel_err = (delta_computed - delta_published) / delta_published\nprint(f\"computed \u03b4_tip               = {delta_computed:+.6e} m\")\nprint(f\"relative error vs textbook   = {rel_err * 100:+.2f} %\")\n\n# 10 % tolerance at this mesh \u2014 HEX8 EAS recovers the\n# Euler-Bernoulli answer with ~5-8 % residual at L/h=20 on a\n# 40x3x3 grid.\nassert abs(rel_err) < 0.10, f\"cantilever UDL deflection failed: {rel_err:.2%}\""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Render the deflected shape\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "warped = m.grid.copy()\nwarped.points = m.grid.points + u\nwarped[\"|u|\"] = np.linalg.norm(u, axis=1)\n\nplotter = pv.Plotter(off_screen=True, window_size=(640, 240))\nplotter.add_mesh(m.grid, style=\"wireframe\", color=\"grey\", opacity=0.3)\nplotter.add_mesh(warped, scalars=\"|u|\", cmap=\"viridis\", show_edges=False)\nplotter.view_xy()\nplotter.camera.zoom(1.05)\nplotter.show()"
      ]
    }
  ],
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    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
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    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
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      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
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