200 lines
6.8 KiB
Python
200 lines
6.8 KiB
Python
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"""Simply Supported Beam with 2D Solid (Quad) Elements — OpenSees Ex 6.4.
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A 40" × 10" simply supported deep beam meshed with 16×4 plane-stress
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quadrilateral elements under two centre-span vertical loads (one at the
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top midspan, one at the bottom midspan, each 1 kip downward). The
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load is ramped up over 10 LoadControl steps, giving a final midspan
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deflection of ~0.394 in. A second case then *removes* the preload and
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lets the beam free-vibrate with 2% stiffness-proportional damping for
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1500 Newmark steps at dt = 0.5 s — reproducing the full Tcl flow.
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Matches the Tcl walkthrough at:
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https://opensees.berkeley.edu/wiki/index.php?title=Simply_supported_beam_modeled_with_two_dimensional_solid_elements
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Model (kip, in, sec):
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- Geometry: L = 40 in, H = 10 in, t = 1 in (plane stress)
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- Material: ElasticIsotropic — E = 1000 ksi, ν = 0.25, ρ = 3.0
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- Elements: 16 × 4 = 64 four-node quads (``quad`` variant,
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``PlaneStress2D`` formulation)
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- Supports: pin at node 1 (bottom-left), roller at node 17
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(bottom-right, Uy restrained only)
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- Loads: 2 × 1 kip ↓ at (20, 0) and (20, 10), ramped via
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LoadControl from 0 to 10 over 10 steps
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Analysis cases:
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1. ``Static`` — 10-step LoadControl, midspan Uy ≈ -0.394 in
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2. ``FreeVibration`` — Transient with preload = case 1, remove
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pattern 1, Newmark (γ=0.5, β=0.25), dt = 0.5 s, 1500 steps,
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βK auto-computed from 2% damping at the 1st mode.
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"""
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from __future__ import annotations
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from pathlib import Path
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from otko.core import (
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ElasticIsotropic,
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LinearTimeSeries,
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NodalLoad,
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Node,
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PlainLoadPattern,
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Project,
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ProjectMeta,
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QuadElement,
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StaticCase,
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TransientCase,
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UnitSystem,
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)
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from otko.services import load_project, save_project
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# Geometry (inches).
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L = 40.0
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H = 10.0
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THICKNESS = 1.0
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# Mesh — nx must be even so a node lands at the midspan.
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NX = 16
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NY = 4
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# Material (kip-in units).
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E = 1000.0
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NU = 0.25
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RHO = 3.0
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def _node_id(i: int, j: int) -> int:
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"""Row-major node id (matching the Tcl ``block2D`` numbering).
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i = 0..NX column index (along length), j = 0..NY row index (through depth).
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"""
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return j * (NX + 1) + i + 1
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def _mid_bottom_node_id() -> int:
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"""Node at (L/2, 0) — bottom flange midspan where one load attaches."""
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return _node_id(NX // 2, 0)
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def _mid_top_node_id() -> int:
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"""Node at (L/2, H) — top flange midspan where the other load attaches."""
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return _node_id(NX // 2, NY)
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def _right_bottom_node_id() -> int:
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"""Bottom-right corner — the roller support."""
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return _node_id(NX, 0)
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def build_beam_quad_2d() -> Project:
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dx = L / NX
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dy = H / NY
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nodes: list[Node] = []
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for j in range(NY + 1):
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for i in range(NX + 1):
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nid = _node_id(i, j)
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if nid == 1:
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restraint = (True, True, False, False, False, False) # pin
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elif nid == _right_bottom_node_id():
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restraint = (False, True, False, False, False, False) # roller (Uy only)
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else:
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restraint = (False,) * 6
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nodes.append(Node(
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id=nid, name=f"N{nid}",
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coords=(i * dx, j * dy, 0.0),
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restraint=restraint,
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))
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elements: list[QuadElement] = []
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eid = 1
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for j in range(NY):
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for i in range(NX):
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n1 = _node_id(i, j) # bottom-left
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n2 = _node_id(i + 1, j) # bottom-right
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n3 = _node_id(i + 1, j + 1) # top-right
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n4 = _node_id(i, j + 1) # top-left (counter-clockwise)
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elements.append(QuadElement(
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id=eid, name=f"Q{eid}",
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nodes=(n1, n2, n3, n4),
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thickness=THICKNESS,
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material_id=1,
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variant="quad",
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behaviour="PlaneStress2D",
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))
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eid += 1
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return Project(
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meta=ProjectMeta(
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name="Simply Supported Beam — Quad Elements (OpenSees Ex 6.4)",
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author="OpenSees Examples Manual",
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description=(
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f"{NX}×{NY} plane-stress quad mesh of a {L:.0f}×{H:.0f} deep "
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"beam with two midspan point loads, LoadControl 0→10 in 10 steps."
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),
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units=UnitSystem.US_IN_KIP,
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),
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ndm=2, ndf=2,
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nodes=nodes,
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materials=[ElasticIsotropic(id=1, name="Elastic", E=E, nu=NU, rho=RHO)],
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elements=elements,
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time_series=[LinearTimeSeries(id=1, name="Ramp")],
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load_patterns=[PlainLoadPattern(
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id=1, name="MidspanLoad",
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time_series_id=1,
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nodal_loads=[
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NodalLoad(node_id=_mid_bottom_node_id(),
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forces=(0.0, -1.0, 0.0, 0.0, 0.0, 0.0)),
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NodalLoad(node_id=_mid_top_node_id(),
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forces=(0.0, -1.0, 0.0, 0.0, 0.0, 0.0)),
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],
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)],
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analyses=[
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StaticCase(
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id=1, name="Static",
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pattern_ids=[1],
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n_steps=10, load_factor_increment=1.0,
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system="ProfileSPD", constraints="Plain",
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integrator="LoadControl", algorithm="Newton",
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test="EnergyIncr", tolerance=1e-12, max_iter=10,
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),
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# Free-vibration continuation: runs case 1 to completion,
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# drops the midspan load pattern, sets up 2% βK Rayleigh
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# damping from the 1st mode, then integrates 1500 steps of
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# Newmark (γ=0.5, β=0.25) at dt = 0.5 s.
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TransientCase(
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id=2, name="FreeVibration",
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preload_case_ids=[1],
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remove_patterns=[1],
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pattern_ids=[],
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dt=0.5, n_steps=1500,
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system="BandGeneral", constraints="Plain",
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integrator="Newmark", integrator_params=(0.5, 0.25),
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algorithm="Newton",
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test="EnergyIncr", tolerance=1e-12, max_iter=10,
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rayleigh_mode1_damping=0.02,
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),
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],
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)
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def main() -> None:
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project = build_beam_quad_2d()
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project.validate_references()
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print(f"Built '{project.meta.name}'")
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print(f" ndm={project.ndm}, ndf={project.ndf}, "
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f"units={project.meta.units.value}")
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print(f" {len(project.nodes)} nodes, {len(project.elements)} quads")
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print(f" Midspan loaded nodes: bottom={_mid_bottom_node_id()}, "
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f"top={_mid_top_node_id()}")
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out_path = Path(__file__).with_suffix(".osmodel")
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save_project(project, out_path)
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print(f"Saved -> {out_path}")
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restored = load_project(out_path)
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restored.validate_references()
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assert restored.model_dump(by_alias=True) == project.model_dump(by_alias=True)
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print("Round-trip OK.")
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if __name__ == "__main__":
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main()
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