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Deflection_FEM_Timoshenko.py
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Deflection_FEM_Timoshenko.py
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pip install numpy matplotlib
import numpy as np
import matplotlib.pyplot as plt
from scipy.linalg import eigh
def form_stiffness_mass_timoshenko_beam(GDof, numberElements, elementNodes, numberNodes, xx, C, P, rho, I, thickness):
stiffness = np.zeros((GDof, GDof))
mass = np.zeros((GDof, GDof))
force = np.zeros(GDof)
gauss_locations_bending = [0.577350269189626, -0.577350269189626]
gauss_weights_bending = [1, 1]
for e in range(numberElements):
indices = elementNodes[e, :]
elementDof = np.concatenate([indices, indices + numberNodes])
length_element = xx[indices[1]] - xx[indices[0]]
detJacobian = length_element / 2
invJacobian = 1.0 / detJacobian
for q in range(len(gauss_weights_bending)):
pt = gauss_locations_bending[q]
shape, naturalDerivatives = shape_function_L2(pt)
Xderivatives = naturalDerivatives * invJacobian
B = np.zeros((2, 4))
B[0, 2:] = Xderivatives
stiffness[np.ix_(elementDof, elementDof)] += B.T @ (C[0, 0] * B) * gauss_weights_bending[q] * detJacobian
force[indices] += shape * P * gauss_weights_bending[q] * detJacobian
# Mass matrix contributions
N = np.zeros((4, 4))
N[:2, :2] = np.outer(shape, shape)
N[2:, 2:] = np.outer(shape, shape)
mass[np.ix_(elementDof, elementDof)] += rho * A * N * gauss_weights_bending[q] * detJacobian
gauss_location_shear = 0.0
gauss_weight_shear = 2.0
for e in range(numberElements):
indices = elementNodes[e, :]
elementDof = np.concatenate([indices, indices + numberNodes])
length_element = xx[indices[1]] - xx[indices[0]]
detJacobian = length_element / 2
pt = gauss_location_shear
shape, naturalDerivatives = shape_function_L2(pt)
Xderivatives = naturalDerivatives * invJacobian
B = np.zeros((2, 4))
B[1, :2] = Xderivatives
B[1, 2:] = shape
stiffness[np.ix_(elementDof, elementDof)] += B.T @ (C[1, 1] * B) * gauss_weight_shear * detJacobian
return stiffness, force, mass
def shape_function_L2(xi):
shape = np.array([(1 - xi) / 2, (1 + xi) / 2])
naturalDerivatives = np.array([-0.5, 0.5])
return shape, naturalDerivatives
def solution(GDof, prescribedDof, stiffness, force):
activeDof = np.setdiff1d(np.arange(GDof), prescribedDof)
K_active = stiffness[np.ix_(activeDof, activeDof)]
F_active = force[activeDof]
U_active = np.linalg.solve(K_active, F_active)
displacements = np.zeros(GDof)
displacements[activeDof] = U_active
return displacements
def solution_modal(prescribedDof, K, M, num_modes):
activeDof = np.setdiff1d(np.arange(len(K)), prescribedDof)
K_active = K[np.ix_(activeDof, activeDof)]
M_active = M[np.ix_(activeDof, activeDof)]
eigvals, eigvecs = eigh(K_active, M_active, subset_by_index=[0, num_modes-1])
eigvals = np.sqrt(np.real(eigvals))
eigvecs_full = np.zeros((len(K), num_modes))
eigvecs_full[activeDof, :] = eigvecs[:, :num_modes]
return eigvals[:num_modes], eigvecs_full
def output_displacements_reactions(displacements, stiffness, GDof, prescribedDof):
print("Displacements:")
for i in range(GDof):
print(f"{i + 1}: {displacements[i]}")
F = stiffness @ displacements
reactions = F[prescribedDof]
print("Reactions:")
for i, r in zip(prescribedDof, reactions):
print(f"{i + 1}: {r}")
def plot_displacements(nodeCoordinates, displacements):
plt.figure()
plt.plot(nodeCoordinates, displacements)
plt.xlabel('Node')
plt.ylabel('Displacement')
plt.title('Displacement of nodes')
plt.grid(True)
plt.show()
def plot_forces(nodeCoordinates, forces):
plt.figure()
plt.plot(nodeCoordinates, forces)
plt.xlabel('Node')
plt.ylabel('Force')
plt.title('Forces at nodes')
plt.grid(True)
plt.show()
def plot_mode_shapes(nodeCoordinates, mode_shapes, num_modes):
plt.figure()
for i in range(num_modes):
plt.subplot(num_modes, 1, i + 1)
plt.plot(nodeCoordinates, mode_shapes[:, i])
plt.grid(True)
plt.ylabel(f'Mode {i + 1}')
plt.xlabel('Node')
plt.suptitle('Mode Shapes')
plt.show()
def get_boundary_conditions(boundary_type, numberNodes):
if boundary_type == 'c-c':
fixedNodeW = [0, numberNodes - 1]
fixedNodeTX = fixedNodeW
elif boundary_type == 'c-s':
fixedNodeW = [0]
fixedNodeTX = [0, numberNodes - 1]
elif boundary_type == 's-s':
fixedNodeW = [0, numberNodes - 1]
fixedNodeTX = []
elif boundary_type == 'c-f':
fixedNodeW = [0]
fixedNodeTX = [0]
else:
raise ValueError("Invalid boundary condition type")
prescribedDof = fixedNodeW + [node + numberNodes for node in fixedNodeTX]
return prescribedDof
# Constants
E = 2.11e11
poisson = 0.30
rho = 7850
L = 1
b = 1
h = 0.1
I = b * h**3 / 12
kapa = 5 / 6
A = b * h
P = -1 # Uniform pressure
G = E / (2 * (1 + poisson))
# Constitutive matrix
C = np.array([[E * I, 0], [0, kapa * h * G]])
# Mesh
numberElements = 100
nodeCoordinates = np.linspace(0, L, numberElements + 1)
elementNodes = np.vstack([np.arange(numberElements), np.arange(1, numberElements + 1)]).T
# Generation of coordinates and connectivities
numberNodes = len(nodeCoordinates)
GDof = 2 * numberNodes
# Compute stiffness matrix and force vector
stiffness, force, mass = form_stiffness_mass_timoshenko_beam(GDof, numberElements, elementNodes, numberNodes, nodeCoordinates, C, P, rho, I, h)
# Choose boundary conditions
boundary_type = 'c-c' # Change this to 'c-c', 'c-s', 's-s', or 'c-f' for different boundary conditions
prescribedDof = get_boundary_conditions(boundary_type, numberNodes)
# Solution for static analysis
displacements = solution(GDof, prescribedDof, stiffness, force)
# Output displacements/reactions
output_displacements_reactions(displacements, stiffness, GDof, prescribedDof)
# Max displacement
U = displacements[:numberNodes]
max_displacement = np.min(U)
print(f"Max displacement: {max_displacement}")
# Plot displacements
plot_displacements(nodeCoordinates, displacements[:numberNodes])
# Plot forces
F = stiffness @ displacements
plot_forces(nodeCoordinates, F[:numberNodes])
# Normal modes analysis
num_modes = 8
eigenvalues, mode_shapes = solution_modal(prescribedDof, stiffness, mass, num_modes)
# Print natural frequencies
frequencies = eigenvalues / (2 * np.pi)
print("Natural frequencies (Hz):")
print(frequencies)
# Plot mode shapes
plot_mode_shapes(nodeCoordinates, mode_shapes[:numberNodes, :], num_modes)