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EXAMPLE 2 -- Moment-Curvature Analysis of a RC Section

This next example covers the moment-curvature analysis of a reinforced concrete section. The zero-length element with a fiber discretization of the cross section is used in the model. In addition, Tcl language features such as variable and command substitution, expression evaluation, and procedures are demonstrated.

In this example, a moment-curvature analysis of the fiber section is undertaken. Figure~\ref{rcsection4} shows the fiber discretization for the section.

Files Required

Model

The model consists of two nodes and a ZeroLengthSection element. A depiction of the element geometry is shown in the figure . The drawing on the left of the figure shows an edge view of the element where the local z-axis, as seen on the right side of the figure and in figure~\ref{rcsection0}, is coming out of the page. Node 1 is completely restrained, while the applied loads act on node 2. A compressive axial load, P, of 180 kips is applied to the section during the moment-curvature analysis.

Geometry of Zero-Length Element

For the zero length element, a section discretized by concrete and steel is created to represent the resultant behavior. UniaxialMaterial objects are created to define the fiber stress-strain relationships: confined concrete in the column core, unconfined concrete in the column cover, and reinforcing steel.

The dimensions of the fiber section are shown in the figure. The section depth is 24 inches, the width is 15 inches, and there are 1.5 inches of cover around the entire section. Strong axis bending is about the section z-axis. In fact, the section z-axis is the strong axis of bending for all fiber sections in planar problems. The section is separated into confined and unconfined concrete regions, for which separate fiber discretizations will be generated. Reinforcing steel bars will be placed around the boundary of the confined and unconfined regions. The fiber discretization for the section is shown in the figure.

Dimensions of RC Section

Fiber-Section Discretization

Analysis

The section analysis is performed by the Tcl procedure MomentCurvature defined in the file MomentCurvature.tcl. The arguments to the procedure are the tag of the section to be analyzed, the axial load applied to the section, the maximum curvature, and the number of displacement increments to reach the maximum curvature.

Output Specification

The output for the moment-curvature analysis will be the section forces and deformations, stored in the file section1.out. In addition, an estimate of the section yield curvature is printed to the screen.

OpenSees Script

In the script below variables, are set and can then be used with the syntax of \$variable. Expressions can be evaluated, although the Tcl syntax at first appears cumbersome. An expression is given by an expr command enclosed in square brackets []'s. Typically, the result of an expression is then set to another variable. A simple example to add 2.0 to a parameter is shown below:

set v 3.0

set sum [expr $v + 2.0]

puts $sum; # print the sum

Comments with # can appear on the same line as a command, as long as the command is terminated with a semi-colon.

# OpenSees Example 2.1

# OpenSees Primer

#

# Units: kips, in, sec

# Define model builder

# --------------------

model BasicBuilder -ndm 2 -ndf 3

# 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 parameters

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 co

The Tcl procedure to perform the moment-curvature analysis follows. In this procedure, the nodes are defined to be at the same geometric location and the ZeroLengthSection element is used. A single load step is performed for the axial load, then the integrator is changed to DisplacementControl to impose nodal displacements, which map directly to section deformations. A reference moment of 1.0 is defined in a Linear time series. For this reference moment, the DisplacementControl integrator will determine the load factor needed to apply the imposed displacement. A node recorder is defined to track the moment-curvature results. The load factor is the moment, and the nodal rotation is in fact the curvature of the element with zero thickness.

# proc MomentCurvature

# Arguments

# secTag -- tag identifying section to be analyzed

# axialLoad -- axial load applied to section (negative is compression)

# maxK -- maximum curvature reached during analysis

# numIncr -- number of increments used to reach maxK (default 100)

#

# Sets up a recorder which writes moment-curvature results to file

# section$secTag.out ... the moment is in column 1, and curvature in column 2

proc MomentCurvature {secTag axialLoad maxK {numIncr 100} } {

# Define two nodes at (0,0)

node 1 0.0 0.0

node 2 0.0 0.0

# Fix all degrees of freedom except axial and bending at node 2

fix 1 1 1 1

fix 2 0 1 0

# Define element

# tag ndI ndJ secTag

element zeroLengthSection 1 1 2 $secTag

# Create recorder

recorder Node -file section$secTag.out -time -node 2 -dof 3 disp

# Define constant axial load

pattern Plain 1 "Constant" {

load 2 $axialLoad 0.0 0.0

}

# Define analysis parameters

integrat

Results

Estimated yield curvature: 0.000126984126984

The file section1.out contains for each committed state a line with the load factor and the rotation at node 3. This can be used to plot the moment-curvature relationship as shown in figure~\ref{momcurv}.

FIGURE HERE

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