Rigid Diaphragm Consequences
An unintended consequence of using the RigidDiaphragm constraint (or other constraint) on a beam column element is a result of the fact that it enforces a condition of zero axial strain condition on the element. For sections where the neutral axis does not shift as a consequence of bending in the beam this constraint has little effect on the axial load in the beam, e.g. elastic or steel sections. For those beams where the neutral axis does shift, i.e. Reinforced Concrete beams modeled using fiber sections under moment, the moments in an unconstrained beam will cause the beam to undergo axial deformation. To constrain this axial deformation requires axial forces, the magnitude of which can be significant.
Here is a stupid made up example that illustrates my point:
foreach rigidConstraint {no yes} {
puts "RIGID CONSTRAINT: $rigidConstraint"
foreach matType {steel concrete} {
foreach eleType {forceBeamColumn dispBeamColumn} {
wipe
# create model builder
model basic -ndm 2 -ndf 3
set width 360
set height 144
# create nodes
node 1 0.0 0.0
node 2 $width 0.0
node 3 0.0 $height
node 4 $width $height
# Fix supports at base of columns
# tag DX DY RZ
fix 1 1 1 1
fix 2 1 1 1
if {$rigidConstraint == "yes"} {
equalDOF 3 4 1
}
if {$matType == "concrete"} {
# 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
} else {
set fy 60.0; # Yield stress
set E 30000.0; # Young's modulus
# tag fy E0 b
uniaxialMaterial Steel01 1 $fy $E 0.01
uniaxialMaterial Steel01 2 $fy $E 0.01
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 elements
# ----------------------
geomTransf Linear 1
set np 5
# Create the coulumns using Beam-column elements
# e tag ndI ndJ nsecs secID transfTag
element $eleType 1 1 3 $np 1 1
element $eleType 2 2 4 $np 1 1
element $eleType 3 3 4 $np 1 1
# Define gravity loads
# --------------------
# Set a parameter for the axial load
set P 180; # 10% of axial capacity of columns
# Create a Plain load pattern with a Linear TimeSeries
pattern Plain 1 "Linear" {
# Create nodal loads at nodes 3 & 4
# nd FX FY MZ
load 3 0.0 [expr -$P] 0.0
load 4 0.0 [expr -$P] 0.0
}
system BandGeneral
constraints Transformation
numberer RCM
test NormDispIncr 1.0e-12 10 3
algorithm Newton
integrator LoadControl 0.1
analysis Static
analyze 10
loadConst -time 0.0
pattern Plain 2 "Linear" {
load 3 1.0 0.0 0.0
load 4 1.0 0.0 0.0
}
integrator DisplacementControl 3 1 0.1
analyze 10
set strains [eleResponse 3 basicDeformation]
set forces [eleResponse 3 forces]
puts "eleType: $eleType matType $matType axialForce [lindex $forces 0] axialDeformation: [lindex $strains 0]"
}
}
}and the results when this script is run are:
RIGID CONSTRAINT: no
eleType: forceBeamColumn matType steel axialForce -0.08875612010504939364 axialDeformation: 0.00000291960921383616
eleType: dispBeamColumn matType steel axialForce -0.00000000000010219973 axialDeformation: 0.00000000000000000000
eleType: forceBeamColumn matType concrete axialForce 1.25405861627505310629 axialDeformation: 0.12918648354852413362
eleType: dispBeamColumn matType concrete axialForce 3.61866611884458500015 axialDeformation: 0.17247077365271457072
RIGID CONSTRAINT: yes
eleType: forceBeamColumn matType steel axialForce -0.00000000000010287569 axialDeformation: 0.00000000000000000000
eleType: dispBeamColumn matType steel axialForce -0.00000000000013310834 axialDeformation: 0.00000000000000000000
eleType: forceBeamColumn matType concrete axialForce 186.25340185913745472135 axialDeformation: 0.00000000000000000000
eleType: dispBeamColumn matType concrete axialForce 163.64106186688189836786 axialDeformation: 0.00000000000000000000