Brick element Response

Supported Brick elements include most brick elements in OpenSees, including:

• ✅ stdBrick

• ✅ bbarBrick

• ✅ Brick20N

• ✅ SSPbrick

• ✅ FourNodeTetrahedron

• ✅ brickUP

• ✅ bbarBrickUP

• ✅ 20_8_BrickUP

• ✅ SSPbrickUP

• ✅ ……

import matplotlib.pyplot as plt
import numpy as np
import openseespy.opensees as ops

import opstool as opst

opst.load_ops_examples("Pier-Brick")
# or your model code here

# add gravity loads
ops.timeSeries("Linear", 1)
ops.pattern("Plain", 1, 1)
_ = opst.pre.gen_grav_load(direction="z", factor=-9.81)

# plot
opst.vis.pyvista.set_plot_props(notebook=True)
fig = opst.vis.pyvista.plot_model(show_nodal_loads=True)
fig.show(jupyter_backend="static")

Result Saving

Nsteps = 100

ops.system("BandGeneral")
# Create the constraint handler, the transformation method
ops.constraints("Transformation")
# Create the DOF numberer, the reverse Cuthill-McKee algorithm
ops.numberer("RCM")
# Create the convergence test, the norm of the residual with a tolerance of
# 1e-12 and a max number of iterations of 10
ops.test("NormDispIncr", 1.0e-12, 10, 3)
# Create the solution algorithm, a Newton-Raphson algorithm
ops.algorithm("Newton")
# Create the integration scheme, the LoadControl scheme using steps of 0.1
ops.integrator("LoadControl", 1 / Nsteps)
# Create the analysis object
ops.analysis("Static")

opstool allows us to save the data at each step of the analysis! First, we create a database class using opstool.post.CreateODB, and then, during each step of the analysis, we call its method .fetch_response_step to retrieve the data for the current step. Once all the analysis steps are completed, we use the .save_response method to save the data in one go.

compute_mechanical_measures is used to compute the mechanical measures, including various stress and strain measures.

project_gauss_to_nodes is used to project the Gauss point results to the nodes.

• “copy”: The response of each node is copied from the Gaussian point closest to it.

• “average”: The response of each node is equal to the weighted average of the responses of all Gaussian points of the element, with the weight being the integration point weight.

• “extrapolate”: The nodal responses are obtained by extrapolating the element shape functions.

ODB = opst.post.CreateODB(
    odb_tag=1,
    compute_mechanical_measures=True,
    project_gauss_to_nodes="copy",  # "extrapolate", "copy", "average"
    nd_material_type="brittle",
)
for i in range(Nsteps):
    # Perform the analysis step
    ops.analyze(1)
    # fetch the response step, every 10 steps for reducing the size of the ODB file
    if (i + 1) % 10 == 0:
        ODB.fetch_response_step()
    # ODB.fetch_response_step()   # or every step
ODB.save_response()  # save the response to a file

OPSTOOL ::  All responses data with _odb_tag = 1 saved in
g:\opstool\docs\src\post\.opstool.output/RespStepData-1.zarr!

Result Reading

The provided function opstool.post.get_element_responses() make it easy to read element responses.

ele_type="Solid" is used to specify extracting the response of solid elements.

all_resp = opst.post.get_element_responses(odb_tag=1, ele_type="Solid")

OPSTOOL ::  Loading Solid response data from g:\opstool\docs\src\post\.opstool.output/RespStepData-1.zarr ...

The result is an xarray DataSet object, and we can access the associated DataArray objects through .data_vars.

all_resp.data_vars

Data variables:
    StrainsAtNodes         (time, nodeTags, strainDOFs) float32 390kB 0.0 ......
    Strains                (time, eleTags, GaussPoints, strainDOFs) float32 2MB ...
    StrainsAtNodesErr      (time, nodeTags, strainDOFs) float32 390kB 0.0 ......
    StressAtNodesErr       (time, nodeTags, stressDOFs) float32 390kB 0.0 ......
    Stresses               (time, eleTags, GaussPoints, stressDOFs) float32 2MB ...
    StressMeasures         (time, eleTags, GaussPoints, measures) float32 1MB ...
    StressesAtNodes        (time, nodeTags, stressDOFs) float32 390kB 0.0 ......
    StressMeasuresAtNodes  (time, nodeTags, measures) float32 260kB 0.0 ... 1...

Stresses and Strains refer to the stress and strain at the Gauss points. Stress and strain consist of six components aligned with the global coordinate system, as well as additional stress measures:

print(all_resp.stressDOFs.data)
print(all_resp.strainDOFs.data)
print(all_resp.measures.data)

['sigma11' 'sigma22' 'sigma33' 'sigma12' 'sigma23' 'sigma13']
['eps11' 'eps22' 'eps33' 'eps12' 'eps23' 'eps13']
['p1' 'p2' 'p3' 'tau_max']

Although we analyzed 100 steps, we saved the data every 10 steps, so we only have data for 10 steps, and the time corresponds accordingly.

print(all_resp.time.data)

[0.  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. ]

all_resp.attrs  # attributes

{'sigma11, sigma22, sigma33': 'Normal stress (strain) along x, y, z.',
 'sigma12, sigma23, sigma13': 'Shear stress (strain).',
 'para#i': 'The additional output of stress, which is useful for some elements, such as * eta_r * for some u-p elements. eta_r--Ratio between the shear (deviatoric) stress and peak shear strength at the current confinement.',
 'p1, p2, p3': 'Principal stresses (strains).',
 'sigma_vm': 'Von Mises stress.',
 'tau_max': 'Maximum shear stress.',
 'p_mean': 'Hydrostatic or confining stress.',
 'q_triaxial': 'Deviatoric stress in triaxial test: q_tri = p1 - p3',
 'q_cs': 'Deviatoric stress in critical state soil mechanics, q_cs = √(3J₂), where J2 = 1/6 * [ (p1-p2)^2 + (p2-p3)^2 + (p3-p1)^2 ]',
 'q_oct': 'Deviatoric stress in octahedral shear stress, τ_oct = √(2/3) * √(J2)'}

Below, we retrieve the stress and strain data, which is a 4D array. The dimensions are, in order, (‘time’, ‘eleTags’, ‘GaussPoints’, ‘DOFs’), and we can conveniently retrieve data based on these dimensions and their coordinates.

Gauss points results

stresses = all_resp["Stresses"]
strains = all_resp["Strains"]
stress_measures = all_resp["StressMeasures"]
print(stresses.dims)
print(strains.dims)
print(stress_measures.dims)

('time', 'eleTags', 'GaussPoints', 'stressDOFs')
('time', 'eleTags', 'GaussPoints', 'strainDOFs')
('time', 'eleTags', 'GaussPoints', 'measures')

stresses2 = stress_measures.sel(eleTags=1, measures="tau_max")
gauss_points = stresses2.coords["GaussPoints"].data

for gp_no in gauss_points:
    s = stresses2.sel(GaussPoints=gp_no)
    time = s.coords["time"].data
    plt.plot(time, s, label=f"GP{gp_no}")
plt.title("Ele 1 Stress-time curve: max shear stress")
plt.xlabel("Time")
plt.ylabel("Stress")
plt.legend()
plt.show()

We can also compute averages along a specific dimension. For example, below, we calculate the average stress at the Gauss points:

stresses2 = stress_measures.sel(measures="tau_max")
stresses3 = stresses2.mean(dim="GaussPoints")

for eletag in np.arange(1, 11):
    s = stresses3.sel(eleTags=eletag)
    plt.plot(time, s, label=f"Ele {eletag}")
plt.title("Mean stress-time curve: max shear stress")
plt.xlabel("Time")
plt.ylabel("Stress")
plt.legend()
plt.show()

Results at nodes

stress_measures = all_resp["StressMeasuresAtNodes"]
stresses = stress_measures.sel(measures="tau_max")

print(stresses.dims)

('time', 'nodeTags')

plt.plot(stresses.time, stresses.sel(nodeTags=10))
plt.title("Node 10: tau_max vs time")
plt.show()