Steel Shear Link

This example compares the shear response of the MultiaxialFiber section with the NDFiber section of OpenSees.

It is demonstrated that NDFiber produces a poorer representation of the link’s shear capacity.

Modeling

Material

Begin by defining a xara.MultiaxialMaterial using the PlasticJ2 formulation.

#
# Hjelmstad's shear link #4
#
import xara
import veux

import xara.units.iks as units
from xara.helpers import find_node


verbose = False
Save = False


Fy = 35.0*units.ksi
E  = 30e3*units.ksi
G  = 12e3*units.ksi

Aspect = 1.5

material = xara.MultiaxialMaterial(
    type="PlasticJ2",
    E =  E,
    G =  G,
    Fy = Fy,
    Hiso = 0.002*E,
    Hkin = 0.002*E
)

Next, a shape object is created. This will be used to generate fibers with appropriate shear properties, like the shear correction factor \(\kappa\) used by NDFiber and the warping shapes \(\varphi\) used by MultiaxialFiber.

from xsection.library import from_aisc
shape = from_aisc("W18x40",
                units=units,
                mesh_scale=1,
                material=material,
                fillet=True,
                mesh_type="T3",
                mesher="gmsh")


L = shape.d*Aspect

Finally, the xara.FrameSection objects are created with the two formulations of interest.

sections = [
    xara.FrameSection("NDFiber", shape),
    xara.FrameSection("MultiaxialFiber", shape)
]

Model

def create_model(length, section, material):
    model = xara.Model(ndm=3, ndf=6)
    
    model.node(1, (     0, 0, 0))
    model.node(2, (length, 0, 0))

    model.fix(1, (1,1,1,  1,1,1))
    model.fix(2, (1,1,0,  0,1,1))

    model.material(material)
    model.section(section)

    model.geomTransf("Linear",  1, (0, 0, 1))
    model.element("ForceFrame", 1, (1,2), 
                  section=section, shear=1, transform=1)

    return model

Analysis

def analyze(model, ctrl_node, plots=(), verbose=False):

    # Loading
    step = 100
    a = 0.1


    series = xara.TimeSeries(values=[0,  0.05,   0.5, -0.5, -0.05],
                             time=  [0,   a,   1,    4,   5 ])

    # Create the control load
    ctrl = xara.NodalValue(model, node=ctrl_node, dof=3, value=1.0)
    pattern = xara.StaticPattern(ctrl, series=series)

    model.pattern(pattern)

    analysis = xara.StaticAnalysis(model, 
                                   integrator=xara.LoadFactorControl("Proportional", 1/step),
                                   system="BandGeneral",
                                   constraints="Transformation", 
                                   test=("Energy", 1e-18, 20, 2 if verbose else 0))

    for _ in range(step*5):
        if model.state.time >= 5.0:
            break

        if analysis.analyze(1) != 0:
            print(f"Failed at time = {model.state.time}")
            return

        for plot in plots:
            plot.update(model)

import matplotlib.pyplot as plt
from post import PlotResponse
from xara.post import FiberStress

##
plot_1 = PlotResponse()


_,plot_stress = plt.subplots(1,2)
plot_stress[0].figure.subplots_adjust(wspace=0.8)

##

for i,section in enumerate(sections):


    print(f"Running {section.type}")

    model = create_model(L, section, material)


    plot_1.reset(model,
                node=find_node(model, x=L), 
                dof=3,
                label=section.type
    )


    analyze(model,
            find_node(model, x=L),
            verbose=verbose,
            plots=[plot_1]
    )

    plot_1.draw()

    # Plot stress distribution at the end of the analysis
    artist = veux.ShapeArtist(shape, ax=plot_stress[i])
    artist.draw_surfaces(
        field=FiberStress(model, shape, section=1, stress="svm", element=1),
        cbar_label="Von Mises Stress (ksi)",
    )


plot_1.finish()

Running NDFiber
Running MultiaxialFiber