Supercritical Pitchfork Bifurcation of Euler's Column¶
This notebook shows the reproduction of the supercritical pitchfork bifurcation of Euler's column with MSC Nastran SOL 106. Following the different buckling calculation methods demonstrated in our last notebook, now we want to test the nonlinear capabilities of SOL 106 by reproducing the nonlinear behavior of Euler's column. We first recall the analytical results for a 1 DOF system and then we move on to the numerical computations. Different methods and load case sequences are investigated to explore the branches of Euler's column supercritical pitchfork bifurcation.
1 DOF system ¶
Let's take the 1 DOF system represented below, composed by two inextensible and initially collinear rods connected by a linear torsional spring and subjected to the axial compressive load $P$.
The equilibrium equation is satisfied under two conditions:
$$\theta=0$$$$\frac{P}{P_c}=\frac{\theta}{\sin\theta}$$where $\theta=0$ is the trivial solution and $P_c=2k/L$ is the critical buckling load.
For $\theta=0$ the equilibrium is stable when $P<P_c$ and unstable when $P>P_c$, while for $P/P_c=\theta/\sin\theta$ the equilibrium is always stable.
If we add an imperfection to the system, such as the initial angle $\theta_0$ shown below, the equilibrium conditions change.
The symmetry of the problem is broken and the equilibrium is found for the following condition: $$\frac{P}{P_{cr}}=\frac{\theta-\theta_0}{\sin\theta}.$$
The equilibrium is stable for $\left(\theta-\theta_0\right)/\tan\theta<1$ and unstable for $\left(\theta-\theta_0\right)/\tan\theta>1$.
Let's plot the above expressions and observe the difference with respect to the system without imperfection.
import matplotlib.pyplot as plt # plotting package
import tol_colors as tc # package for colorblind-friendly colors
import numpy as np # package for matrix operations
# Create new figure with one subplot
plt.rc('axes', prop_cycle=plt.cycler('color', list(tc.tol_cset('bright')))) # set default color cycle to TOL bright for colorblind accessibility
colors = plt.rcParams['axes.prop_cycle'].by_key()['color'] # get list of default color cycle
plt.rcParams['figure.dpi'] = 120 # set default dpi of figures
fig, ax = plt.subplots()
# Plot trivial stable solution of perfect system
theta_trivial_stable = np.array([0., 0.]) # [deg]
load_trivial_stable = np.array([0., 1.])
stable_line = ax.plot(load_trivial_stable, theta_trivial_stable, color=colors[0], label="$\\theta_0=0$ deg, stable")
# Plot nontrivial stable solution of perfect system
theta_max = 120 # [deg]
theta_nontrivial = np.linspace(-theta_max, theta_max) # [deg]
load_nontrivial = np.deg2rad(theta_nontrivial)/np.sin(np.deg2rad(theta_nontrivial))
ax.plot(load_nontrivial, theta_nontrivial, color=colors[0])
# Plot stable and unstable solution of system with imperfections
unstable_color = colors[1] # set red as color for unstable solutions
del colors[1] # delete red from list of colors
imperfections = [1, 5, 10] # [deg]
for count, theta_0 in enumerate(imperfections): # iterate through initial angles theta_0
theta_negative = np.arange(-theta_max, 0) # array of negative angles
theta_positive = np.arange(theta_0, theta_max + 1) # array of positive angles
load_theta_negative = np.deg2rad(theta_negative - theta_0)/np.sin(np.deg2rad(theta_negative)) # array of loads corresponding to negative angles
load_theta_positive = np.deg2rad(theta_positive - theta_0)/np.sin(np.deg2rad(theta_positive)) # array of loads corresponding to positive angles
stability_theta_negative = np.deg2rad(theta_negative - theta_0)/np.tan(np.deg2rad(theta_negative)) # array of stability of negative angles
ax.plot(load_theta_negative[stability_theta_negative<1], theta_negative[stability_theta_negative<1],
color=colors[count + 1], label=f"$\\theta_0={theta_0:d}$ deg, stable") # plot stable solution of negative angles
ax.plot(load_theta_negative[stability_theta_negative>1], theta_negative[stability_theta_negative>1], '--',
color=unstable_color,) # plot unstable solution of negative angles
ax.plot(load_theta_positive, theta_positive, color=colors[count + 1]) # plot stable solution
# Plot trivial unstable solution of perfect system
theta_trivial_unstable = np.array([0., 0.]) # [deg]
load_trivial_unstable = np.array([1., np.max(load_nontrivial)])
ax.plot(load_trivial_unstable, theta_trivial_unstable, '--', color=unstable_color, label="unstable")
# Set plot appearance
ax.set_ylabel("$\\theta$, deg")
ax.set_xlabel("$P/P_c$")
ax.set_xlim([0, 2])
ax.legend()
ax.grid()
plt.show()
We observe the presence of a supercritical pitchfork bifurcation for the perfect system, while the imperfect system presents a so-called broken supercritical pitchfork. This broken bifurcation is characterized by a natural equilibrium path where the structure is always stable, and a complementary equilibrium path that is partly stable and partly unstable. The difference between the response of the perfect and the imperfect system grows as the imperfection becomes larger.
Setup of the numerical model ¶
Let's setup our numerical model for the analysis with Nastran. We consider the same column of our last notebook and we create a BDF object calling the function create_base_bdf from the column_utils module. Also in this case we are going to work with the millimeter - newton - megapascal - ton system of units.
from resources import column_utils # module with functions for working with Nastran models of Euler's column
E = 207000. # Young's modulus [MPa]
nu = 0.3 # Poisson's ratio
rho = 7.8e-4 # density [tons/mm^3]
d = 20 # diameter [mm]
l = 420 # length [mm]
no_elements = 420 # number of beam elements
bdf_input = column_utils.create_base_bdf(young_modulus=E, poisson_ratio=nu, density=rho, diameter=d, length=l,
no_elements=no_elements) # create base BDF object
subcase=0 already exists...skipping
Our BDF object includes nodes, beam elements, material properties, boundary conditions, unit compression force and some parameters to define the output files. We can check its content with the get_bdf_stats method.
print(bdf_input.get_bdf_stats())
---BDF Statistics--- SOL None bdf.loads[4]: 1 FORCE: 1 bdf.spcadds[3]: 1 SPCADD: 1 bdf.spcs[1]: 1 SPC1: 1 bdf.spcs[2]: 1 SPC1: 1 bdf.params: 0 PARAM : 1 bdf.nodes: 0 GRID : 421 bdf.elements: 0 CBEAM : 420 bdf.properties: 0 PBEAML : 1 bdf.materials: 0 MAT1 : 1
Before moving on to the definition of the load case sequence, we recall the linear buckling load calculated by SOL 105 in our last notebook. We will use this load in the remainder of this notebook to nondimensionalize the applied load.
sol_105_buckling_load = 90578. # [N]
Load case sequence definition ¶
To reproduce the supercritical pitchfork bifurcation of Euler's column we need to explore the associated paths of the equilibrium diagram. This means that we need to look for the stable path up to the buckling load, the unstable branch beyond the buckling load and one of the two symmetrical stable branches. In addition, we also want to visualize the broken pitchfork, in order to observe the influence of an initial imperfection or of an eccentricity in the load.
We are going to approach this problem by defining a load case sequence that will allow us to move along the different paths of the equilibrium diagram of Euler's column. In SOL 106 Nastran's subcases are executed in a sequential way, meaning that each subcase starts where the previous one ends. We define the following sequence of subcases to explore all the paths described earlier.
- Apply a compression load larger than the buckling load at the roller-supported end of the column. In this subcase we expect to end on the unstable branch of the bifurcation.
- Add a small transverse load at the middle of the column. This will break the symmetry of the problem and consequently also the supercritical pitchfork bifurcation. We expect the equilibrium configuration to move onto the broken supercritical pitchfork, resulting in a deflection of the column in the same direction of the applied transverse load.
- Remove the transverse load in order to restore the symmetry of the problem. As a consequence, we expect also the supercritical pitchfork bifurcation to be restored. Thus, the equilibrium configuration should move onto the stable branch of the supercritical pitchfork corresponding to the direction of the tranverse load that was just removed. We expect to observe a non-zero deflection of the column.
- Remove the compression load. We expect this to move the equilibrium configuration along the stable branch of the supercritical pitchfork until the applied load is equal to the buckling load and then back to the undeformed state of the column.
- Apply both the compression and the transverse load at the same time. This will let us visualize the equilibrium path of the broken supercritical pitchfork.
Let's start by defining the compression force and the transverse force, which we will later assign to the different load sets. We consider a compression force equal to twice the buckling load calculated by SOL 105, $P_x/P_\text{SOL 105}=2$. The FORCE card corresponding to the compression force has already been created by the create_base_bdf function, so we only need to change its magnitude.
compression_force_set_id = list(bdf_input.loads.keys())[0] # retrieve identification number of FORCE card representing the compression load
bdf_input.loads[compression_force_set_id][0].mag = sol_105_buckling_load*2 # change magnitude of the force to twice the buckling load calculated by SOL 105
Then we define the transverse force as a fraction of the buckling load, $P_y/P_\text{SOL 105}=1/100$. This time we need to create a new FORCE card.
transverse_force_set_id = compression_force_set_id + 1 # identification number of FORCE card representing the transverse force
middle_node_id = int(no_elements/2 + 1) # identification number of the node of application of the transverse force
transverse_force_magnitude = sol_105_buckling_load/100
transverse_force_direction = [0., 1., 0.]
bdf_input.add_force(sid=transverse_force_set_id, node=middle_node_id, mag=transverse_force_magnitude,
xyz=transverse_force_direction) # add FORCE card to the BDF object
FORCE 5 211 905.78 0. 1. 0.
Now we define the two load combinations that we need for our subcases: one with compression and transverse force and one with no applied force.
# Define load set with combined compression and transverse forces
combined_load_set_id = transverse_force_set_id + 1
bdf_input.add_load(combined_load_set_id, scale=1., scale_factors=[1., 1.], load_ids=[compression_force_set_id,
transverse_force_set_id])
# Define load set with no applied force
zero_load_set_id = combined_load_set_id + 1
bdf_input.add_load(zero_load_set_id, scale=1., scale_factors=[0.], load_ids=[compression_force_set_id])
LOAD 7 1. 0. 4
We finally define our subcases as explained earlier using the create_static_load_subcase function from the pynastran_utils module.
from resources import pynastran_utils # module with useful functions to work with pyNastran objects
pynastran_utils.create_static_load_subcase(bdf=bdf_input, subcase_id=1, load_set_id=compression_force_set_id) # first subcase
pynastran_utils.create_static_load_subcase(bdf=bdf_input, subcase_id=2, load_set_id=combined_load_set_id) # second subcase
pynastran_utils.create_static_load_subcase(bdf=bdf_input, subcase_id=3, load_set_id=compression_force_set_id) # third subcase
pynastran_utils.create_static_load_subcase(bdf=bdf_input, subcase_id=4, load_set_id=zero_load_set_id) # fourth subcase
pynastran_utils.create_static_load_subcase(bdf=bdf_input, subcase_id=5, load_set_id=combined_load_set_id) # fifth subcase
Load control method ¶
The first nonlinear iteration strategy that we want to test is the load control method. This method consists in dividing the total applied load into a number of load increments, and in converging to an equilibrium for each load increment by means of Newton iterations.
We choose SOL 106 as solution sequence and add the parameter PARAM,LGDISP,1 to consider large displacement effects. Then we define a NLPARM card with the same parameters of our last notebook. Finally, we select the NLPARM card in the case control deck as default for all subcases.
bdf_input.sol = 106 # assign solution sequence
bdf_input.add_param('LGDISP', [1]) # enable large displacement effects
nlparm_id = 1 # nlparm card id
bdf_input.add_nlparm(nlparm_id=nlparm_id, ninc=100, kmethod='ITER', kstep=-1, max_iter=25, conv='PUV', int_out='YES',
eps_p=1e-3, eps_u=1e-3, max_bisect=10) # add nlparm card
bdf_input.case_control_deck.subcases[0].add_integer_type('NLPARM', nlparm_id) # nlparm card default for all subcases
We can now run the analysis with Nastran! Let's define the name of the analysis directory and of the input file and let's call the function run_analysis from the pynastran_utils module.
import os # module for working with the operating system
analysis_directory_name = '02_Supercritical_Pitchfork_Bifurcation_of_Euler_Column'
analysis_directory_path = os.path.join(os.getcwd(), 'analyses', analysis_directory_name)
input_name = 'load_control_method'
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=bdf_input, filename=input_name, run_flag=False)
Nastran job load_control_method.bdf completed Wall time: 46.0 s
Once the analysis is done, we can read the op2 file and find the load and displacement history for all subcases using the function read_load_displacement_history_from_op2 from the pynastran_utils module. This function returns the history of load steps, overall applied loads, and displacements and rotations at the indicated node for each subcase of the analysis. We read the displacements and rotations at the hinge-supported node (node id 1) because we want to assess the equilibrium diagram of the column in terms of the rotation $\theta$ at that node, similarly to the 1-DOF system discussed earlier.
from pyNastran.op2.op2 import read_op2 # function to read op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None) # read op2 file without display of debug messages
displacement_node_id = 1 # id of the node for which the displacement history is to be extracted
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
Now we want to plot the results of our analysis in terms of equilibrium paths. For each converged iteration of each subcase we want to visualize:
- load step vs $\theta$;
- applied load $P_x$ vs $\theta$;
- applied load $P_y$ vs $\theta$.
The load step represents the progress of the nonlinear analysis towards the final applied load. It ranges from 0 to 1 for the first subcase, from 1 to 2 for the second subcase and so on.
In addition, we also want to print the value of $\theta$ at the end of each subcase. We define the function plot_2d_equilibrium_paths to do the plotting and print the value of $\theta$, and we call it to show the results of our analysis.
from matplotlib.lines import Line2D # class defining the characters for the marker styles
markers = list(Line2D.markers.keys())[2:] # store list of marker characters to plot different marker for each subcase
def plot_2d_equilibrium_paths(disp, steps, loads):
# Create figure with 3 subplots
_, axs = plt.subplots(nrows=1, ncols=3, sharey='all')
# Iterate over the subcases
for subcase_id in disp:
# Convert rotation angle from radians to degrees
theta = np.rad2deg(disp[subcase_id][:,5])
# Print rotation at the end of subcase
print(f"Subcase {subcase_id:d}: theta = {theta[-1]:.2f} degrees")
# Plot load step vs rotation
axs[0].plot(steps[subcase_id], theta, markers[subcase_id - 1])
# Plot applied load along x vs rotation
axs[1].plot(-loads[subcase_id][:,0]/sol_105_buckling_load, theta, markers[subcase_id - 1]) # sign of load along x is inverted to show a positive load in the plot
# Plot applied load along y vs rotation
axs[2].plot(loads[subcase_id][:,1]/sol_105_buckling_load, theta, markers[subcase_id - 1],
label=f"Subcase {subcase_id:d}")
# Set plot appearance
axs[0].set_xlabel("Load step")
axs[0].set_ylabel('$\\theta$, deg')
axs[0].grid(visible=True)
axs[1].set_xlabel('$P_x/P_\mathrm{SOL\/105}$')
axs[1].grid(visible=True)
axs[2].set_xlabel('$P_y/P_\mathrm{SOL\/105}$')
axs[2].grid(visible=True)
axs[2].legend(loc="upper right", bbox_to_anchor=(2., 1.02))
plt.show()
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads) # call function to show results of the analysis
Subcase 1: theta = 0.00 degrees Subcase 2: theta = 124.04 degrees Subcase 3: theta = 124.08 degrees Subcase 4: theta = -0.00 degrees Subcase 5: theta = 124.04 degrees
As expected, the results show the presence of a supercritical pitchfork bifurcation. We can make the following observations.
- During the first subcase the rotation $\theta$ is always null. The equilibrium point moves along a stable equilibrium path for $P_x/P_\text{SOL 105}<1$ and then along the unstable branch of the bifurcation for $P_x/P_\text{SOL 105}>1$.
- In subcase 2, $\theta$ jumps from zero to about 124 degrees, with the equilibrium configuration moving from the unstable branch of the supercritical pitchfork bifurcation to the broken supercritical pitchfork.
- In the third subcase the equilibrium configuration moves from the broken supercritical pitchfork to the stable branch of the supercritical pitchfork bifurcation. Unexpectedly, at the end of the subcase $\theta$ is slightly smaller with respect to the value at the end of subcase 2. This is in contrast with the results of the 1 DOF system, where the imperfect sytem always shows a larger value of $\theta$ for the same applied load $P/P_c$ with respect to the perfect system.
- In subcase 4, the equilibrium configuration moves along the stable branch of the unbroken pitchfork as the column is unloaded and for $P_x/P_\text{SOL 105}<1$ it traverses a stable equilibrium path until the structures goes back to its undeformed state (null rotation).
- In subcase 5 we can observe the equilibrium path corresponding to the broken supercritical pitchfork. In particular, it is evident how the path of the broken supercritical pitchfork deviates from the one of the supercritical pitchfork bifurcation, especially close to $P_x/P_\text{SOL 105}=1$. As expected, the final value of $\theta$ corresponds to the one obtained at the end of the second subcase.
Except for the value of $\theta$ at the end of the third subcase, all the observations are in line with our expectations.
We can also visualize the equilibrium paths with a 3D plot, showing $\theta$, $P_x/P_\text{SOL 105}$ and $P_y/P_\text{SOL 105}$ at the same time. We define the function plot_3d_equilibrium_paths for such purpose and we call it to plot our results in 3D.
# Define function to plot 3D equilibrium paths
def plot_3d_equilibrium_paths(disp, loads):
# Create figure with three-dimensional axes
plt.figure(figsize=(8, 6))
ax_3d = plt.axes(projection='3d')
# Plot load applied along x vs rotation of pin-supported node vs applied along y
for subcase_id in disp:
ax_3d.plot3D(-loads[subcase_id][:,0]/sol_105_buckling_load, np.rad2deg(disp[subcase_id][:,5]),
loads[subcase_id][:,1]/sol_105_buckling_load, markers[subcase_id - 1],
label='Subcase {:d}'.format(subcase_id)) # sign of load along x is inverted to show a positive load in the plot
# Set plot appearance
ax_3d.set_xlabel("$P_x/P_\mathrm{SOL\/105}$")
ax_3d.locator_params(axis='x', nbins=5) # set number of bins along x-axis
ax_3d.set_ylabel("$\\theta$, deg")
ax_3d.yaxis.labelpad = 8 # increase distance of y-axis's label to axis
ax_3d.set_zlabel("$P_y/P_\mathrm{SOL\/105}$")
ax_3d.zaxis.labelpad = 8 # increase distance of z-axis's label to axis
ax_3d.legend(loc="upper left")
ax_3d.grid(visible=True)
ax_3d.set_box_aspect(aspect=None, zoom=0.85)
plt.show()
# Call function to show results of the analysis
plot_3d_equilibrium_paths(displacements, applied_loads)
Here we can recognize the equilibrium path of the supercritical pitchfork bifurcation on the $P_x/P_\text{SOL 105}-\theta$ plane and the jump of $\theta$ during subcase 2. We remind that the employed nonlinear method is based on load control, so the jump in subcase 2 suggests the presence of a limit point, which we should be able to observe using the arc-length method. Finally, the 3D view enables the observation of two features:
- the path of the broken pitchfork is actually three-dimensional when we also consider the transverse load as an independent parameter of the system;
- the supercritical pitchfork bifurcation and the broken supercritical pitchfork are connected by the equilibrium path found in subcase 3.
Arc-length control method ¶
Now we want to analyze the supercritical pitchfork bifurcation using the arc-length control method. This method allows both displacements and applied load to vary during each step of the nonlinear analysis, and in this way it makes possible to follow equilibrium paths also beyond limit points.
To use the arc-length method in SOL 106, we need to add a NLPCI card. We use a similar set of parameters employed in our last notebook. Here we increase the maximum allowable arc-length adjustment ratio to 1.01 and the maximum number of controlled increment steps in each subcase to 2000. Several tests have been performed on these two parameters as the last subcase struggled to converge with the original values, and the chosen values are the ones that allowed the analysis to converge.
bdf_input.add_nlpci(nlparm_id, Type='CRIS', minalr=.01, maxalr=1.01, desiter=5, mxinc=2000)
NLPCI 1 CRIS .01 1.01 0. 5 2000
Let's define the name of our input file and run the analysis.
input_name = 'arclength_method'
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=bdf_input, filename=input_name, run_flag=False)
Nastran job arclength_method.bdf completed Wall time: 150.0 s
We plot the results of our analysis in an analogous way as done for the load control method.
# Read op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None)
# Read load displacement history
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
# Plot equilibrium paths
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads)
Subcase 1: theta = 0.00 degrees Subcase 2: theta = -0.38 degrees Subcase 3: theta = -0.00 degrees Subcase 4: theta = -0.00 degrees Subcase 5: theta = 124.04 degrees
We can easily observe one main difference with respect to the results obtained with the load control method. After the first subcase, the arc-length solver does not succed to move from the unstable branch of the supercritical pirchfork bifurcation to the broken supercritical pitchfork. The solver appears to follow a path with negative $\theta$ for a positive transverse load, which means that the column is deflecting in the opposite direction with respect to the applied transverse load. This is a counterintuitive result that needs further investigation.
Also in this case we plot the 3D equilibrium paths to have another visualization of our results.
plot_3d_equilibrium_paths(displacements, applied_loads)
Considering the jump that we observed with the load control method in subcase 2, we would expect to observe a limit point with the arc-length method. However, we cannot see any evidence of such limit point, rather the structure follows an equilibrium path with slightly negative values of $\theta$. As a consequence, we need to investigate the equilibrium path around $P_x/P_\text{SOL 105}=2$ more in detail.
Investigation of equilibrium paths around $P_x/P_\text{SOL 105}=2$ ¶
We investigate the behavior of the equilibrium paths around $P_x/P_\text{SOL 105}=2$ by applying a large transverse force in one direction and then in the opposite one. For this purpose we need to make a deep copy of our BDF object and modify the subcases in the following way:
- Subcase 1 stays unchanged.
- In subcase 2 we keep the compression force and apply a large transverse force, setting $P_y/P_\text{SOL 105}=100$, with the aim of visualizing more extensively the shape of the equilibrium path.
- In subcase 3 we keep the compression force and reverse the transverse force, setting $P_y/P_\text{SOL 105}=-100$. This means that we apply an equal and opposite transverse force with respect to the one applied in subcase 2. In this way we study the other side of the equilibrium path.
Subcases 4 and 5 are removed.
# Increase maximum number of increments allowed within a subcase and create deep copy
px2_investigation_input = bdf_input.__deepcopy__({})
# Modify subcase 2
px2_investigation_input.loads[transverse_force_set_id][0].mag = sol_105_buckling_load*100 # change magnitude of transverse force
# Modify subcase 3
combined_load_opposite_direction_set_id = zero_load_set_id+1 # define identification number of load combination with reversed transverse force
scale_factors = [1., -1.] # define scale factors of the forces, -1 means transverse force applied in opposite direction w.r.t. subcase 2
px2_investigation_input.add_load(combined_load_opposite_direction_set_id, 1., scale_factors, [compression_force_set_id,
transverse_force_set_id]) # add new load combination with reversed transverse force
px2_investigation_input.subcases[3].params['LOAD'][0] = combined_load_opposite_direction_set_id # select load combination in subcase control deck
# Remove subcases 4 and 5
del px2_investigation_input.subcases[4]
del px2_investigation_input.subcases[5]
Let's run the analysis and visualize the results.
# Run analysis
input_name = 'arclength_method_px2_investigation' # input name
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=px2_investigation_input, filename=input_name, run_flag=False) # run analysis
# Read load and displacement history from op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None)
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
# Plot equilibrium paths
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads)
plot_3d_equilibrium_paths(displacements, applied_loads)
Nastran job arclength_method_px2_investigation.bdf completed Wall time: 84.0 s Subcase 1: theta = 0.00 degrees Subcase 2: theta = -256.42 degrees Subcase 3: theta = 256.43 degrees
The equilibrium path on the $P_x/P_\text{SOL 105}=2$ plane depicts a downward deflection of the column for an upward transverse force and vice versa. Once again, we cannot observe the presence of a limit point and the equilibrium path does not seem to justify the jump that we observed with the load control method.
Let's try a different sequence of subcases to investigate the equilibrium paths on the $P_x/P_\text{SOL 105}=2$ plane. We create a deep copy of our original BDF object and we define the following subcases.
- Apply combined compression and transverse force, with $P_x/P_\text{SOL 105}=2$ and $P_y/P_\text{SOL 105}=1/100$. In this way we expect the equilibrium configuration to move along the broken supercritical pitchfork.
- Keep the compression load and reverse the transverse load, setting $P_y/P_\text{SOL 105}=-100$. The idea here is to observe whether we are able to move from one side of the broken supercritical pitchfork to the other side. We set a larger magnitude for the transverse load to overcome potential limit points in the equilibrium path.
Subcases 3, 4 and 5 are removed.
# Create deep copy of original BDF object
alternative_subcases_input = bdf_input.__deepcopy__({})
# Modify subcase 1
alternative_subcases_input.subcases[1].params['LOAD'][0] = combined_load_set_id
# Modify subcase 2
scale_factors = [1., -10000.] # define scale factors to apply a reversed transverse force of Py/pc = -100
alternative_subcases_input.add_load(combined_load_opposite_direction_set_id, 1., scale_factors,
[compression_force_set_id, transverse_force_set_id]) # add new load combination with reversed transverse force
alternative_subcases_input.subcases[2].params['LOAD'][0] = combined_load_opposite_direction_set_id # select load combination in subcase control deck
# Remove subcases 3 to 5
del alternative_subcases_input.subcases[3]
del alternative_subcases_input.subcases[4]
del alternative_subcases_input.subcases[5]
Once again, let's run the analysis and visualize the results.
# Run analysis
input_name = 'arclength_method_px2_investigation_alternative_subcases' # input name
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=alternative_subcases_input, filename=input_name, run_flag=False) # run analysis
# Read op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None)
# Read load displacement history
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
# Plot equilibrium paths
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads)
plot_3d_equilibrium_paths(displacements, applied_loads)
Nastran job arclength_method_px2_investigation_alternative_subcases.bdf completed Wall time: 1427.0 s Subcase 1: theta = 124.04 degrees Subcase 2: theta = 90.02 degrees
The analysis stops during subcase 2 because it exceeds the allowed maximum number of controlled iterations. We can observe negative load steps, indicating that the arc-length solver tries to seek new equilibrium points in the same loading direction of the previous subcase. As a consequence, the arc-length solver applies an increasing upward force instead of applying the prescribed downward force.
Since the arc-length method seems to encounter issues when the sign of the transverse force is inverted, we try to repeat the analysis substituting the arc-length method with the load control method in subcase 2. For this purpose we need to add another NLPARM card and assign its id to subcase 2.
nlparm2_id = 2
alternative_subcases_input.add_nlparm(nlparm2_id, ninc=100, kmethod='ITER', kstep=-1, max_iter=25, conv='PUV',
int_out='YES', eps_p=1e-3, eps_u=1e-3, max_bisect=10)
alternative_subcases_input.case_control_deck.subcases[2].add_integer_type('NLPARM', nlparm2_id) # select NLPARM card in subcase 2 control deck
We run the analysis and plot the results.
input_name = 'mixed_methods_px2_investigation_alternative_subcases' # input name
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=alternative_subcases_input, filename=input_name, run_flag=False) # run analysis
# Read op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None)
# Read load displacement history
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
# Plot equilibrium paths
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads)
plot_3d_equilibrium_paths(displacements, applied_loads)
Nastran job mixed_methods_px2_investigation_alternative_subcases.bdf completed Wall time: 171.0 s Subcase 1: theta = 124.04 degrees Subcase 2: theta = 267.67 degrees
Once again, the analysis stops during the second subcase becasue it exceeds the allowed maximum number of controlled iterations. However, this time we observe that the load control method is able to traverse the equilibrium path along the prescribed loading direction. The results indicate that for the investigated range of transverse loads the equilibrium path does not seem to point towards the other side of the broken supercritical pitchfork.
Reduced compression force ¶
At this point we want to assess how the equilibrium paths change across different $P_x/P_\text{SOL 105}=c$ planes, with $c$ constant. We use the same load case sequence defined for the alternative_subcase_input object, with the following values of $P_x/P_\text{SOL 105}$: $1.1$, $1.4$, $1.7$. For each value of $P_x/P_\text{SOL 105}$ we adjust the value of $P_y/P_\text{SOL 105}$ in the second subcase. The reason for this will be evident when we'll look at the results.
We proceed with selecting again the arc-length method for the second subcase, modifying the applied loads, running the analysis and plotting the results for each value of $P_x/P_\text{SOL 105}$.
alternative_subcases_input.subcases[2].params['NLPARM'][0] = nlparm_id # select the NLPARM card associated to the arc-length method for subcase 2
compression_force_factors = [1.1, 1.4, 1.7] # define values of Px/pc
transverse_load_factors = [-1e1, -1e2, -5e2] # define scale factors of Py/pc
# Iterate through the different values of Px/pc and Py/pc
for force_factor, scale_factor in zip(compression_force_factors, transverse_load_factors):
# Print value of compression force
print("\033[1m" + f"Px/Psol105={force_factor:.1f}" + "\033[0m")
# Modify magnitude of compression force
alternative_subcases_input.loads[compression_force_set_id][0].mag = sol_105_buckling_load*force_factor
# Modify scale factor of transverse load in subcase 2
alternative_subcases_input.load_combinations[combined_load_opposite_direction_set_id][0].scale_factors[1] = scale_factor
# Define input name
input_name = f"arclength_method_px{force_factor:.1f}".replace('.','_')
# Run analysis
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=alternative_subcases_input, filename=input_name, run_flag=False)
# Read op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None)
# Read load displacement history
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
# Plot equilibrium paths
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads)
plot_3d_equilibrium_paths(displacements, applied_loads)
Px/Psol105=1.1
Nastran job arclength_method_px1_1.bdf completed
Wall time: 83.0 s
Subcase 1: theta = 50.66 degrees
Subcase 2: theta = -58.81 degrees
Px/Psol105=1.4
Nastran job arclength_method_px1_4.bdf completed
Wall time: 126.0 s
Subcase 1: theta = 90.37 degrees
Subcase 2: theta = -97.01 degrees
Px/Psol105=1.7
Nastran job arclength_method_px1_7.bdf completed
Wall time: 151.0 s
Subcase 1: theta = 110.76 degrees
Subcase 2: theta = -105.68 degrees
The results of these analyses show the presence of a limit point and consequently of a snap-through behavior when moving from one side to the other of the broken supercritical pitchfork. Subcase 1 brings the equilibrium configuration on the broken supercritical pitchfork corresponding to an upward deflection of the column (positive $\theta$). In subcase 2 the positive transverse force is gradually unloaded and the negative transverse force is loaded on the structure. As a consequence, during this process the transverse force becomes null and then negative with increasing magnitude, until a limit point is encountered. After this point the transverse force decreases in magnitude until it becomes null and changes sign. Once again, the force increases in magnitude until it encounters another limit point, this time for a positive force. Subsequently, the force decreases in magnitude, becomes null, changes its sign back to negative and finally achieves the prescribed value.
It should be noted that $P_y$ becomes null three times along the equilibrium path. Once for a positive value of $\theta$, once for $\theta=0$ and once for a negative value of $\theta$. This means that when moving from one side of the broken supercritical pitchfork to the other, the equilibrium path crosses the branches of the supercritical pitchfork bifurcation, where $P_y$ is null.
Furthermore, we can observe that the curve describing the equilibrium path connecting the two sides of the pitchfork changes with the compression load $P_x$. Not only the location of the limit points is different, but the overall shape of the curve changes. For example, we can notice that the slope of the $P_y/P_\text{SOL 105}-\theta$ curve at the start of subcase 2 is different in each case. This means that the level of compression of the column influences the equilibrium path connecting the two sides of the pitchfork. As a consequence, it is possible that for $P_x/P_\text{SOL 105}=2$ the equilibrium path becomes more complex and that we are not able to traverse it within the prescribed maximum number of iterations.
Finally, looking at the equilibrium point where $P_y=0$ and $\theta=0$, we notice the same seemingly nonphysical behavior that we observed in a previous analysis, that is to say downward deflection (negative $\theta$) for upward force (positive $P_y$) and vice versa. This can be explained by the fact that the points on that equilibrium path are all unstable, so when the column has a slight upward deflection, a downward force is needed to keep it in equilibrium and to prevent it from snapping to the stable branch of the supercritical pitchfork bifurcation. Analogously, when the column has a slight downward deflection, an equilibrium configuration can only be obtained with the application of an upward force that prevents the column from snapping further downward.
Final supercritical pitchfork bifurcation analysis ¶
We conclude this series of analyses with an investigation on both sides of the supercritical pitchfork bifurcation. We define the following subcases.
- Apply combined compression and transverse force, with $P_x/P_\text{SOL 105}=1.5$ and $P_y/P_\text{SOL 105}=1/100$, to move along the equilibrium path of the broken supercritical pitchfork.
- Remove the transverse load to move the equilibrium configuration onto the stable branch of the supercritical pitchfork bifurcation.
- Apply a downward transverse load, setting $P_y/P_\text{SOL 105}=-1$, to move to the other side of the broken supercritical pitchfork. The final equilibrium point should belong to the side of the broken supercritical pitchfork corresponding to a downward deflection of the column.
- Remove the transverse load to move the equilibrium configuration onto the other stable branch of the supercritical pitchfork bifurcation. In this subcase we want to verify that the final rotation $\theta$ is equal and opposite to the one at the end of subcase 2.
- Remove the compression load to walk the supercritical pitchfork bifurcation back to the undeformed configuration of the structure. In this subcase we use the load control method because we have observed the arc-length method to have issues with unloading the compression force.
- Apply a compression force combined with a downward transverse force, setting $P_x/P_\text{SOL 105}=1.5$ and $P_y/P_\text{SOL 105}=-1$. In this way we check that the final value of $\theta$ is equal to the one obtained at the end of subcase 3. Also in this subcase we use the load control method, this time because the arc-length method appeared to have issues finding solutions in the opposite loading direction of the previous subcase.
# Modify magnitude of compression force
alternative_subcases_input.loads[compression_force_set_id][0].mag = sol_105_buckling_load*1.5
# Modify subcase 2
alternative_subcases_input.subcases[2].params['LOAD'][0] = compression_force_set_id
# Create subcase 3
alternative_subcases_input.load_combinations[combined_load_opposite_direction_set_id][0].scale_factors[1] = -1e2 # modify scale factor of reversed transverse force to apply Py/pc = -1
pynastran_utils.create_static_load_subcase(bdf=alternative_subcases_input, subcase_id=3,
load_set_id=combined_load_opposite_direction_set_id)
# Create subcase 4
pynastran_utils.create_static_load_subcase(alternative_subcases_input, subcase_id=4,
load_set_id=compression_force_set_id)
# Create subcase 5
pynastran_utils.create_static_load_subcase(alternative_subcases_input, subcase_id=5, load_set_id=zero_load_set_id)
alternative_subcases_input.case_control_deck.subcases[5].add_integer_type('NLPARM', nlparm2_id) # set load control method as nonlinear solution method
# Create subcase 6
pynastran_utils.create_static_load_subcase(alternative_subcases_input, subcase_id=6,
load_set_id=combined_load_opposite_direction_set_id)
alternative_subcases_input.case_control_deck.subcases[6].add_integer_type('NLPARM', nlparm2_id) # set load control method as nonlinear solution method
We run the analysis, read the output file and plot the results.
input_name = 'mixed_methods_px1_5'
pynastran_utils.run_analysis(
directory_path=analysis_directory_path, bdf=alternative_subcases_input, filename=input_name, run_flag=False)
# Read op2 file
op2_filepath = os.path.join(analysis_directory_path, input_name + '.op2')
op2_output = read_op2(op2_filename=op2_filepath, debug=None)
# Read load displacement history
load_steps, applied_loads, displacements = pynastran_utils.read_load_displacement_history_from_op2(
op2=op2_output, node_ids=[displacement_node_id]) # read load and displacement history
displacements = displacements[displacement_node_id] # extract displacement history of the node
# Plot equilibrium paths
plot_2d_equilibrium_paths(displacements, load_steps, applied_loads)
plot_3d_equilibrium_paths(displacements, applied_loads)
Nastran job mixed_methods_px1_5.bdf completed Wall time: 178.0 s Subcase 1: theta = 98.34 degrees Subcase 2: theta = 98.28 degrees Subcase 3: theta = -101.55 degrees Subcase 4: theta = -98.28 degrees Subcase 5: theta = 0.00 degrees Subcase 6: theta = -101.55 degrees
We can make the following observations.
- In subcase 1 the broken supercritical pitchfork is on the side corresponding to an upward deflection of the column, so in the same direction as the applied transverse load.
- Subcase 3 highlights the snap-through behavior of the column when the equilibrium configuration moves from one side to the other of the broken supercritical pitchfork.
- The rotation $\theta$ at the end of subcase 2 is equal and opposite to the one at the end of subcase 4.
- In subcase 5 the stable branch of the supercritical pitchfork bifurcation is on the side corresponding to a downward deflection of the column, so in the same direction as the applied transverse load.
- In subcase 6 the broken supercritical pitchfork is again on the side corresponding to a downward deflection of the column, as the transverse force is applied in that direction. We can notice that the curve in the $P_x/P_\text{SOL 105}-\theta$ plot is more distant from the bifurcation point with respect to the broken supercritical pitchfork curve described by subcase 1. This is caused by the different magnitude of the transverse load.
- The rotation $\theta$ at the end of subcase 3 is equal to the one at the end of subcase 6.
All these results are in line with our expectations.
Conclusion ¶
In this notebook we have successfully used the arc-length method available in MSC Nastran SOL 106 to reproduce the supercritical pitchfork bifurcation of Euler's column. We can draw the following final remarks regarding the use of the arc-length method in Nastran:
- the order of subcases matters to get to a specific point of the equilibrium diagram;
- the arc-length method has issues when the subcase applies a load in the opposite direction with respect to the one of the previous subcase. In these situations, the load control method should be used.
In the next notebook we will test further the nonlinear capabilities of SOL 106 investigating the equilibrium diagram of a thin plate under uniaxial compression.