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This example is of a reinforced concrete portal frame, as shown in figure~\ref{example3}, subject to gravity loads.
Files Required
Model
A nonlinear model of the portal frame shown in the figure is created. The model consists of four nodes, two nonlinear beam-column elements, 1 and 2, to model the columns and an elastic beam element, 3, to model the beam. For the column elements a section, identical to the section used in Example 2, is created using steel and concrete fibers.
A single load pattern with a linear time series, two vertical nodal loads acting at nodes 3 and 4, and single point constraints to constrain nodes 1 and 2 are created.
Analysis
The model contains material non-linearities, so a solution Algorithm of type Newton is used. The solution algorithm uses a ConvergenceTest which tests convergence of the equilibrium solution with the norm of the displacement increment vector. For this nonlinear problem, the gravity loads are applied incrementally until the full load is applied. To achieve this, a LoadControl integrator which advances the solution with an increment of 0.1 at each load step is used. The equations are formed using a banded storage scheme, so the System is BandGeneral. The equations are numbered using an RCM (reverse Cuthill-McKee) numberer. The constraints are enforced with a Plain constraint handler.
Once all the components of an analysis are defined, the Analysis object itself is created. For this problem a Static Analysis object is used. To achieve the full gravity load, 10 load steps are performed.
Output Specification
At end of analysis, the state at nodes 3 and 4 is output. The state of element 1 is also output.
OpenSees Script
# OpenSees Example 3.1
# OpenSees Primer
#
# Units: kips, in, sec
# ------------------------------
# Start of model generation
# ------------------------------
# Create ModelBuilder (with two-dimensions and 3 DOF/node)
model basic -ndm 2 -ndf 3
# Create nodes
# ------------
# Set parameters for overall model geometry
set width 360
set height 144
# Create nodes
# tag X Y
node 1 0.0 0.0
node 2 $width 0.0
node 3 0.0 $height
node 4 $width $height
# Fix supports at base of columns
# tag DX DY RZ
fix 1 1 1 1
fix 2 1 1 1
# Define materials for nonlinear columns
# ------------------------------------------
# CONCRETE tag f'c ec0 f'cu ecu
# Core concrete (confined)
uniaxialMaterial Concrete01 1 -6.0 -0.004 -5.0 -0.014
# Cover concrete (unconfined)
uniaxialMaterial Concrete01 2 -5.0 -0.002 0.0 -0.006
# STEEL
# Reinforcing steel
set fy 60.0; # Yield stress
set E 30000.0; # Young's modulus
# tag fy E0 b
uniaxialMaterial Steel01 3 $fy $E 0.01
# Define cross-section for nonlinear columns
# ------------------------------------------
# set some paramaters
set colWidth 15
set colDepth 24
set cover 1.5
set As 0.60; # area of no. 7 bars
# some variables derived from the parameters
set y1 [expr $colDepth/2.0]
set z1 [expr $colWidth/2.0]
section Fiber 1 {
# Create the concrete core fibers
patch rect 1 10 1 [expr $cover-$y1] [expr $cover-$z1] [expr $y1-$cover] [expr $z1-$cover]
# Create the concrete cover fibers (top, bottom, left, right)
patch rect 2 10 1 [expr -$y1] [expr $z1-$cover] $y1 $z1
patch rect 2 10 1 [expr -$y1] [expr -$z1] $y1 [expr $cover-$z1]
patch rect 2 2 1 [expr -$y1] [expr $cover-$z1] [expr $cover-$y1] [expr $z1-$cover]
patch rect 2 2 1 [expr $y1-$cover] [expr $cover-$z1] $y1 [expr $z1-$cover]
# Create the reinforcing fibers (left, middle, right)
layer straight 3 3 $As [expr $y1-$cover] [expr $z1-$cover] [expr $y1-$cover] [expr $cover-$z1]
layer straight 3 2 $As 0.0 [expr $z1-$cover] 0.0 [expr $cover-$z1]
layer straight 3 3 $As [expr $cover-$y1] [expr $z1-$cover] [expr $cover-$y1] [expr $cover-$z1]
}
# Define column elements
# ----------------------
# Geometry of column elements
# tag
geomTransf Linear 1
# Number of integration points along length of element
set np 5
# Create the coulumns using Beam-column elements
# tag ndI ndJ nsecs secID transfTag
element nonlinearBeamColumn 1 1 3 $np 1 1
element nonlinearBeamColumn 2 2 4 $np 1 1
# Define beam elment
# -----------------------------
# Geometry of column elements
# tag
geomTransf Linear 2
# Create the beam element
# tag ndI ndJ A E Iz transfTag
element elasticBeamColumn 3 3 4 360 4030 8640 2
# Define gravity loads
# --------------------
# Set a parameter for the axial load
set P 180; # 10% of axial capacity of columns
# Create a Plain load pattern with a Linear TimeSeries
pattern Plain 1 "Linear" {
# Create nodal loads at nodes 3 & 4
# nd FX FY MZ
load 3 0.0 [expr -$P] 0.0
load 4 0.0 [expr -$P] 0.0
}
# ------------------------------
# End of model generation
# ------------------------------
# ------------------------------
# Start of analysis generation
# ------------------------------
# Create the system of equation, a sparse solver with partial pivoting
system BandGeneral
# Create the constraint handler, the transformation method
constraints Transformation
# Create the DOF numberer, the reverse Cuthill-McKee algorithm
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
test NormDispIncr 1.0e-12 10 3
# Create the solution algorithm, a Newton-Raphson algorithm
algorithm Newton
# Create the integration scheme, the LoadControl scheme using steps of 0.1
integrator LoadControl 0.1
# Create the analysis object
analysis Static
# initialize in case we need to do an initial stiffness iteration
initialize
# ------------------------------
# End of analysis generation
# ------------------------------
# ------------------------------
# Start of recorder generation
# ------------------------------
# Create a recorder to monitor nodal displacements
recorder Node -file nodeGravity.out -time -node 3 4 -dof 1 2 3 disp
# --------------------------------
# End of recorder generation
# ---------------------------------
# ------------------------------
# Finally perform the analysis
# ------------------------------
# perform the gravity load analysis, requires 10 steps to reach the load level
analyze 10
# Print out the state of nodes 3 and 4
print node 3 4
# Print out the state of element 1
print ele 1 2
Results
Node: 3
Coordinates : 0 144
commitDisps: 1.7109e-18 -0.0183736 -2.81893e-20
unbalanced Load: 0 -180 0
ID : 3 4 5
Node: 4
Coordinates : 360 144
commitDisps: 1.71095e-18 -0.0183736 -2.79765e-20
unbalanced Load: 0 -180 0
ID : 0 1 2
Element: 1 Type: NLBeamColumn2d Connected Nodes: 1 3
Number of Sections: 5 Mass density: 0
End 1 Forces (P V M): 180 7.0121e-31 -8.88178e-16
End 2 Forces (P V M): -180 -7.0121e-31 8.88178e-16
Element: 2 Type: NLBeamColumn2d Connected Nodes: 2 4
Number of Sections: 5 Mass density: 0
End 1 Forces (P V M): 180 0 -8.88178e-16
End 2 Forces (P V M): -180 0 8.88178e-16
For the two nodes, displacements and loads are given. For the beam-column elements, the element end forces in the local system are provided.
The nodeGravity.out file contains ten lines, each line containing 7 entries. The first entry is time in the domain at end of the load step. The next 3 entries are the displacements at node 3, and the final 3 entries the displacements at node 4.