Cantilever with a composite laminate

A 100 mm x 50 mm x 1 mm cantilever plate, clamped at one end and loaded by transverse forces at the two tip corners. The plate is made of a linear-elastic cross-ply laminate.

The Shell measures ply angles from a global orientation (projected onto each element), so we can sweep that reference direction to rotate the laminate's reference direction and study how the tip deflection depends on the fiber angle.

Material and laminate

import math

import matplotlib.pyplot as plt
import torch

from torchfem import Laminate, Shell
from torchfem.materials import OrthotropicElasticityPlaneStress
from torchfem.mesh import rect_tri

torch.set_default_dtype(torch.float64)

# Unidirectional GFRP
gfrp = OrthotropicElasticityPlaneStress(
    E_1=54000.0,
    E_2=9400.0,
    nu_12=0.33,
    G_12=5500.0,
    G_13=5500.0,
    G_23=3000.0,
    rho=1.8e-9,
)

# Cross-ply laminate
layup = Laminate(
    materials=[gfrp] * 4,
    thicknesses=[0.25] * 4,
    angles=[0.0, torch.pi / 2, torch.pi / 2, 0.0],
)

layup.plot()

Model setup

A reusable helper builds the clamped, tip-loaded cantilever for a given global reference orientation.

nodes, elements = rect_tri(10, 5, Lx=100.0, Ly=50.0, variant="zigzag")
nodes = torch.hstack([nodes, torch.zeros((nodes.size(0), 1))])

left = nodes[:, 0] < 0.01
corner_1 = (nodes[:, 0] > 99.9) & (nodes[:, 1] < 0.01)
corner_2 = (nodes[:, 0] > 99.9) & (nodes[:, 1] > 49.99)


def cantilever(orientation):
    plate = Shell(nodes, elements, layup, orientation=orientation)
    plate.constraints[left] = True
    plate.forces[corner_1, 2] = -2.0
    plate.forces[corner_2, 2] = -2.0
    return plate

Sweep the fiber orientation

Rotate the global reference direction (the 0° ply direction) from 0° to 180° and record the maximum tip deflection for each angle.

angles = torch.linspace(0.0, 180.0, 19)
deflections = []
for angle in angles:
    # Build orientation vector from angle in degrees
    theta = math.radians(angle.item())
    orientation_vector = torch.tensor([math.cos(theta), math.sin(theta), 0.0])
    # Build cantilever problem and solve
    plate = cantilever(orientation_vector)
    u, _, _, _, _ = plate.solve()
    deflections.append(u[:, 2].abs().max().item())
deflections = torch.tensor(deflections)

Visualization

The deflection is minimal if the 0° outer plies are oriented with the plate. The maximum deflection does not occur at 90°, because a slight off-angle introduces twisting which increases the maximum deflection.

# ABAQUS reference deflection (with S3 elements)
abq_half = [6.60, 8.09, 10.66, 13.85, 17.06, 19.68, 21.16, 21.60, 21.50, 21.13]
abq_deflection = abq_half + abq_half[-2::-1]

plt.plot(angles, abq_deflection, ".-", color="gray", label="ABAQUS")
plt.plot(angles, deflections, ".-", color="deeppink", label="torch-fem")
plt.xlabel("Reference orientation")
plt.ylabel("Maximum tip deflection")
plt.grid(True)
plt.legend()
plt.xticks(range(0, 181, 30), [f"{t}°" for t in range(0, 181, 30)])
plt.show()