FBE with uniaxial sections and asymmetric hysteretic curve

[Tas]

I am facing a difficulty in defining properly a Force Based Element by assigning Uniaxial sections at the I.P’s instead of Fiber sections in conjunction with a hysteretic material.
I am trying to define a Hysteretic material for the uniaxial section with an unsymmetric response curve (this kind of curve is used for RC beams that have different reinforcement at bottom and top fibers and also they have different effective compressive areas). I want to assign negative (or positive) values corresponding to tension of upper flange and positive values (or negative respectively) corresponding to tension of the lower flange of beam.
The problem is that I cannot find out how I suppose to assign properly this physical condition: the values of the one side of the hysteretic material curve to represent the bending that causes tension to the one flange of the beam and the values of the other side of the curve to represent the bending that causes tension to the opposite flange of the beam… I mean that I cannot find out which sign of values in the hysteretic material should represent properly the one or the other bending side…

Please take a look at the following script (i created it in order to test-investigate different kind of things) in which I want to define an asymmetric response curve to the hysteretic material of the beam and I can’t... (the results of the forces in the beam are not justified).


# 2D Frame
# Units Kg,KN,m,sec

file mkdir 2DM; # create data directory

# First steps of Modeling

model basic -ndm 2 -ndf 3

# Create nodes

node 1 0.0 0.0
node 2 0.0 2.0
node 3 2.0 2.0
node 4 2.0 0.0

fix 1 1 1 1; fix 4 1 1 1

### Masses - Materials - Elements ###

# Set parameters for masses
set P 50.0;
set g 9.81; # g
set m [expr $P/$g]; # nodal mass

mass 2 $m $m 0
mass 3 $m $m 0


# geometry parameters for Columns
set ACol 0.00538; # cross-sectional area, make stiff
set IzCol 0.000036922; # Column's moment of inertia

# geometry parameters for Beam
set ABeam 0.00284841; # cross-sectional area, make stiff
set IzBeam 0.0000194317; # Beam's moment of inertia

# geometry parameters for braces
set Abrace 0.005306; # cross-sectional area, make stiff
set Izbrace 0.0000256; # Brace's moment of inertia

# material and section parameters
set ColMatTagFlex 2; # assign a tag number to the column flexural behavior
set ColMatTagAxial 3; # assign a tag number to the column axial behavior
set BeamMatTagAxial 4; # assign a tag number to the beam axial behavior
set BeamMatTagFlex 5; # assign a tag number to the beam flexural behavior
set brace 6; # assign a tag number to the brace axial behavior
set fc 235000; # Steel Compressive-Tensive Strength (+Tension, -Compression)
set E 210000000; # Steel Young's modulus

# COLUMN section
# stiffness parameters
set EACol [expr $E*$ACol]; # EA, for axial force-strain relationship
set EICol [expr $E*$IzCol]; # EI, for flexural-elastic bending behavior

# BEAM section
# stiffness parameters
set EABeam [expr $E*$ABeam]; # EA, for axial-force-strain relationship
set EIBeam [expr $E*$IzBeam]; # EI, for flexural-elastic bending behavior

# BRACE section
# stiffness parameters
set EAbrace [expr $E*$Abrace]; # EA, for axial-force-strain relationship


## define response diagrams

# uniaxialMaterial Hysteretic $ColMatTagFlex 96.2 0.0124 96.2 0.124 -96.2 -0.0124 -96.2 -0.124 0.0 0.0 0.0 0.0 0.8;
# uniaxialMaterial Elastic $ColMatTagFlex $EICol;

uniaxialMaterial Hysteretic $BeamMatTagFlex 44.0 0.010783 48.0 0.10783 -22.0 -0.005392 -25.0 -0.05392 0.0 0.0 0.0 0.0 0.8;
# uniaxialMaterial Hysteretic $BeamMatTagFlex 44.0 0.010783 44.0 0.10783 -44.0 -0.010783 -44.0 -0.10783 0.0 0.0 0.0 0.0 0.8;
# uniaxialMaterial Elastic $BeamMatTagFlex $EIBeam;

# uniaxialMaterial Elastic $ColMatTagAxial $EACol; # this is not used as a material, represents an axial-force-strain response;
uniaxialMaterial Elastic $BeamMatTagAxial $EABeam; # this is not used as a material, represents is an axial-force-strain response;

# uniaxialMaterial Hysteretic $brace 1084.30 0.00275 1084.30 0.035 -248.50 -0.003153 -248.5 -0.03153 0.0 0.0 0.0 0.0 ;
# uniaxialMaterial Hysteretic $brace 204347.80 0.000973 204347.80 0.0117 -46828.40 -0.00111 -46828.40 -0.0111 0.0 0.0 0.0 0.0 ; O.K... ;
uniaxialMaterial Elastic $brace $EAbrace; # this is used when the element trussSection 4 1 3 103 is used;
# uniaxialMaterial Elastic $brace $E; # this is used when the element truss 4 1 3 $Abrace $brace is used;

# section Aggregator 101 $ColMatTagAxial P $ColMatTagFlex Mz ; # combine axial and flexural behavior of Columns into one section (no P-M interaction here);
section Aggregator 102 $BeamMatTagAxial P $BeamMatTagFlex Mz ; # combine axial and flexural behavior of Beam into one section (no P-M interaction here);

section Uniaxial 103 $brace P

# define geometric transformation: performs a linear geometric transformation
set ColTransfTag 1; # associate a tag to column transformation
geomTransf Linear $ColTransfTag ;


## element connectivity:
set numIntgrPts 5; # number of integration points for force-based element
# element nonlinearBeamColumn 1 1 2 $numIntgrPts 101 $ColTransfTag;
# element nonlinearBeamColumn 3 4 3 $numIntgrPts 101 $ColTransfTag;
element nonlinearBeamColumn 2 2 3 $numIntgrPts 102 $ColTransfTag;

element elasticBeamColumn 1 1 2 $ACol $E $IzCol $ColTransfTag;
element elasticBeamColumn 3 4 3 $ACol $E $IzCol $ColTransfTag;
# element elasticBeamColumn 2 2 3 $ABeam $E $IzBeam $ColTransfTag;


# element truss 4 1 3 $Abrace $brace
# element truss 5 2 4 $Abrace $brace
element trussSection 4 1 3 103
element trussSection 5 2 4 103

### Define RECORDERS -------------------------------------------------------------
recorder Node -xml 2DM/DFree2.out -time -node 2 -dof 1 2 3 disp; # displacements of free nodes
recorder Node -xml 2DM/DFree3.out -time -node 3 -dof 1 2 3 disp; # displacements of free nodes
recorder Node -xml 2DM/RBase1.out -time -node 1 -dof 1 2 3 reaction; # support reaction
recorder Node -xml 2DM/RBase4.out -time -node 4 -dof 1 2 3 reaction; # support reaction
recorder Drift -xml 2DM/Drift2.out -time -iNode 1 -jNode 2 -dof 1 -perpDirn 2 ; # lateral drift
recorder Drift -xml 2DM/Drift3.out -time -iNode 4 -jNode 3 -dof 1 -perpDirn 2 ; # lateral drift
recorder Element -xml 2DM/Beam.out -time -ele 2 localForce; # element forces - Beam
recorder Element -xml 2DM/BeamForceSec1.out -time -ele 2 section 1 forces ; # section results - Beam
recorder Element -xml 2DM/BeamForceSec3.out -time -ele 2 section 3 forces ; # section results - Beam
recorder Element -xml 2DM/BeamForceSec5.out -time -ele 2 section 5 forces ; # section results - Beam
recorder Element -xml 2DM/Beamdefsec1.out -time -ele 2 section 1 deformation; # section deformation node i - Beam
recorder Element -xml 2DM/Beamdefsec5.out -time -ele 2 section 5 deformation; # section deformation node j - Beam
recorder Element -xml 2DM/Col1.out -time -ele 1 localForce; # element forces - Column 1
recorder Element -xml 2DM/Col3.out -time -ele 3 localForce; # element forces - Column 3
recorder Element -xml 2DM/BeamDef.out -time -ele 2 basicDeformation; # element deformation
recorder Element -xml 2DM/Inflcol.out -time -ele 1 inflectionPoint; # inflection Point
recorder Element -xml 2DM/Brace4.out -time -ele 4 axialForce; # element forces - brace4
recorder Element -xml 2DM/Brace5.out -time -ele 5 axialForce; # element forces - brace5



# Create a Plain load pattern with a Linear TimeSeries
set P 50.0;
timeSeries Linear 1
pattern Plain 1 1 {load 2 [expr -100.0*$P] [expr -$P] 0.0; load 3 0.0 [expr -$P] 0.0}
# pattern Plain 1 1 {load 2 [expr 55.0*$P] [expr -$P] 0.0; load 3 0.0 [expr -$P] 0.0}
# pattern Plain 2 1 {eleLoad -ele 2 -type -beamUniform -85.0};

### Apply Gravity loads ###
constraints Plain; # how it handles boundary conditions
numberer Plain; # renumber dof's to minimize band-width (optimization), if you want to
system BandGeneral; # how to store and solve the system of equations in the analysis
test NormDispIncr 1.0e-8 10 ; # determine if convergence has been achieved at the end of an iteration step
algorithm Newton; # use Newton's solution algorithm: updates tangent stiffness at every iteration
integrator LoadControl 0.1; # determine the next time step for an analysis
analysis Static; # define type of analysis static or transient
analyze 10; # apply gravity

# maintain constant gravity loads and reset time to zero
loadConst -time 0.0

puts "eigen values at start: [eigen 2]"

### Perform PushOver Analysis ###
# source "Pushover_1.tcl"
# source "PushoverAnalysis.tcl"



### Perform an eigenvalue analysis ###

#puts "eigen values at end: [eigen 2]"

# Print state of node 3
#print node 3

[Tas]

In order to make my question more specific:
For the selected load pattern the results of local forces of the beam don’t correspond to the previously defined hysteretic material curve that it is assigned to the sections and then to the FBE that simulates the beam. At both ends of the beam the moments are positive and it seems that the negative values of the hysteretic curve aren’t activated! But this is not true: at the two ends of the beam the moments, caused by the specific lateral loading, act in a different way: at the one end the acting moment causes tension to the upper flange and at the other one the acting moment causes tension to the lower flange!
So in one of the beam’s ends the moment shouldn’t exceed the limiting value of 25.0 that is assigned at the corresponding hysteretic material (look at the values in the last step of analysis !)


<ElementOutput eleType="ForceBeamColumn2d" eleTag="2" node1="2" node2="3">
<ResponseType>N_1</ResponseType>
<ResponseType>V_1</ResponseType>
<ResponseType>M_1</ResponseType>
<ResponseType>N_2</ResponseType>
<ResponseType>V_2</ResponseType>
<ResponseType>M_2</ResponseType>
</ElementOutput>
<Data>
0.1 -200.703 5.53158 6.34439 200.703 -5.53158 4.71878
0.2 -401.407 11.0632 12.6888 401.407 -11.0632 9.43755
0.3 -602.11 16.5947 19.0332 602.11 -16.5947 14.1563
0.4 -803.398 20.1011 22.3791 803.398 -20.1011 17.823
0.5 -1005.2 21.8284 23.0913 1005.2 -21.8284 20.5655
0.6 -1207 23.5557 23.8035 1207 -23.5557 23.3079
0.7 -1408.8 25.283 24.5156 1408.8 -25.283 26.0504
0.8 -1610.6 27.0104 25.2278 1610.6 -27.0104 28.7929
0.9 -1812.4 28.7377 25.94 1812.4 -28.7377 31.5354
1 -2014.2 30.465 26.6521 2014.2 -30.465 34.2779
</Data>
</OpenSees>

[fmk]

25 is not the limiting stress (force in your case) for the material as you have defined it! it would be if you were to add e3, s3 points that have same stress but larger absolute strain values than the e2,s2 points

ps. setting the limits will result in a 0 tangent which is going to cause you problems. see if you can figure out why from what is printed to the screen and then see if you can come up with an alternative.


pps. i suggest you use OpenSeesTk and play with the tool strain.tcl in the following directory (you also need the other file in the directory) if you want to see the behaviour of the materials.
http://opensees.berkeley.edu/WebSVN/lis ... SCRIPTS%2F

[Tas]

Thank you for the reply Frank.

After some hours that I spent by making tests and in order to realize what happens, I made the following conclusions:
1. There is no matter if you don't assign the values e3p, s3p, e3n and s3n, it won't cause any problem.
2. There is no any limiting stress because of what it says as a note under the description of the MaterialUniaxial Hysteretic command in OpenWiki: "In cases $s3p > $s2p and abs($s3n) > abs($s2n), the envelope of the hysteretic material after $e3p or $e3n follows the slope defined by 2nd and 3rd point of the envelope"... so the exceedance of the last point (e2p or e3p, if the last one is used in the command) it won't result in a 0 tangent (maybe the number of the I.P's should be increased in order to achieve convergence).
3. The sign of the values of this uniaxial material (or maybe of anyone else, i tried and verified the same things with ElasticMultiLinear Material too) always correspond to the sign of section forces (that are the same, of course, with the signs of section deformations). The positive sign of section’s moment (and so, the sign that correspond to the positive values of the hysteretic material curve) is always according to the moment that causes tension to the lower flange of the beam and the negative values correspond to the moment that causes tension to the upper flange of the beam.

Hope this helps…
(if anyone has any different opinion or experience on what I am writing, you are welcome…)

[vesna]

I just want to add a comment on sign for the moment in a cross section. It depends on the local coordinate system that you set for your section. If local y is pointing up and local z to the left, positive moment will put bottom of the section in tension. However, if local y is pointing down and local z to the right, positive moment will put top of the section in tension.

[Tas]

Thank you for your reply dear Vesna,

Well.., in my "tests" this is not confirmed: when i changed the orientation of local coordinates of the beam (i.e when in the "element conectivity" i assigned for element 2 as starting node -i- the one with number 3 and for ending node -j- the one with number 2) the results were the same.
Positive section moments cause always tension in bottom flange.

[vesna]

The sign of the moment depends on orientation of local z axis, not on local x.

[Tas]

O.k thanks. So, in 2D frames the convention of moment signs allways follows the way i discribed while for 3D frames it depends on the definition of the vector vecxz...