Force vs. Displacement formulations

An investigation of various frame formulations

Investigate the two most commonly used OpenSees elements for modeling beam-column elements: the force  and cubic displacement  formulations.

Although the parameters for defining these two elements are the same, a beam-column member needs to be modeled differently using these two elements to achieve a comparable level of accuracy. The intent of this example is to show how to properly model inelastic beam-columns with both force and displacement formulations.

Model Generation  

• The column is modeled using either force-based or displacement-based elements.
• The column height H is 96 inches

import xara

# Create a 2D model with 3 DOFs per node
model = xara.Model(ndm=2, ndf=3)

H = 96.0 * inch  # Column height

# Define nodes
model.node(1, (0.0, 0.0))
for i in range(ne):
    model.node(i + 2, (0.0, (i + 1) * H/ne))

Boundary Conditions  

Node 1 is fixed in both horizontal and vertical directions.

model.fix(1, (1, 1, 1))

Materials  

Longitudinal reinforcement  

• Bar area barArea is 0.31 square inches (for bar #5).
• Bar diameter db is 0.625 inches.
• Yield strength fy of longitudinal bars is 70 ksi.
• Modulus of elasticity Es of steel is 29,000 ksi.
• Tangent at initial strain hardening (Esf) is calibrated from coupon tests.

Transverse reinforcement  

• Spiral diameter dh is 0.25 inches.
• Number of hoops NoHoops is 1.
• Area of transverse reinforcement bar (Asp1) is 0.0491 square inches.
• Centerline distance between spirals stran is 1.25 inches.
• Yield strength of the hoop (fyh) is 96.6 ksi.

Unconfined concrete  

• Compressive strength $f'_c$ (fc) is 4.4 ksi.
• Strain corresponding to fc is eps0.
• Ultimate strain for unconfined concrete is epss.
• Elastic modulus Ec is calculated based on ACI 318.

fc = 4.4 * ksi  # Compressive strength of plain concrete
eps0 = 0.002    # Strain corresponding to fc'
epss = 0.005    # Ultimate strain for unconfined concrete
Ec = 57000.0 * sqrt(fc * 1000.0) / 1000.0

Confined concrete  

Compressive strength and strain are determined using Mander’s equations (Eq. 6 in Mander, 1988):

Section  

• The diameter D is 24 inches.
• The clear cover of concrete clearCover is 0.75 inches.

D = 24.0 * inch  # Column diameter
clearCover = 0.75 * inch  # Clear cover of concrete

theta = 360.0 / numBars
model.section("fiberSec", secnTag, GJ=1e8)
# Core patch
model.patch("circ", coreTag, nfCoreT, nfCoreR, 0, 0, 0, ri, 0.0, 360.0, section=secnTag)
# Cover patch
model.patch("circ", coverTag, nfCoverT, nfCoverR, 0, 0, ri, ro, 0.0, 360.0, section=secnTag)
# Reinforcing bars
model.layer("circ", steelTag, numBars, barArea, 0, 0, rl, theta / 2.0, 360.0 - theta / 2.0, section=secnTag)

Elements  

Coordinate transformation is applied using the Corotational method.

# Define coordinate transformation
transfTag = 1
model.geomTransf("Corotational", transfTag)

for i in range(ne):
    if eleType == 1:
        model.element("forceBeamColumn", i + 1, (i+1, i+2), nIP, secTag, transfTag)
    elif eleType == 2:
        model.element("dispBeamColumn", i + 1, (i+1, i+2), nIP, secTag, transfTag)

Analysis  

Gravity  

Cycling  

# Define recorders for displacement and force
model.recorder("Node", "disp", "-time", file="out/Disp.out", node=ne+1, dof=1)
model.recorder("Node", "reaction", "-time", file="out/Force.out", node=1, dof=1)

Resources  

The slides from the original presentation can be downloaded from here and video of the seminar can be found here: http://www.youtube.com/watch?v=yk-1k2aF53E 

• Supporting files to be stored in the same folder with the main file:
  • leh415.txt : experimental force-displacement response