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##################################################################################################################
# Spring_PZ.tcl
#
# SubRoutine to construct a rotational spring with a trilinear hysteretic response representative of steel
# panel zone response with/without the consideration of composite action
#
# References:
#--------------
# Elkady, A. and D. G. Lignos (2014). "Modeling of the Composite Action in Fully Restrained Beam-to-Column
# Connections: Implications in the Seismic Design and Collapse Capacity of Steel Special Moment Frames."
# Earthquake Eng. & Structural Dynamics 43(13). DOI: 10.1002/eqe.2430.
#
# Skiadopoulos, A., Elkady, A. and D. G. Lignos (2020). "Proposed Panel Zone Model for Seismic Design of
# Steel Moment-Resisting Frames." ASCE Journal of Structural Engineering. DOI: 10.1061/(ASCE)ST.1943-541X.0002935.
#
##################################################################################################################
#
# Input Arguments:
#------------------
# SpringID Spring ID
# NodeI Node i ID
# NodeJ Node j ID
# E Young's Modulus
# mu Poisson's Ratio
# fy Yield Stress (Expected, measured or nominal)
# tw_Col Column Web Thickness
# tdp Doubler Plate(s) Total Thickness
# d_Col Column Depth
# d_Beam Beam Depth
# tf_Col Column Flange Thickness
# bf_Col Column Flange Width
# Ix_Col Column second-moment-of-interia about the strong axis
# n Axial load ratio (P/Py)
# trib Steel deck rib depth
# ts Concrete slab depth above the rib
# Response_ID ID for Panel Zone Response: 0 --> Interior Steel Panel Zone with Composite Action
# 1 --> Exterior Steel Panel Zone with Composite Action
# 2 --> Bare Steel Interior/Exterior Steel Panel Zone
# transfTag Geometric Transformation ID
#
# Written by: Dr. Ahmed Elkady, University of Southampton, UK
#
##################################################################################################################
proc Spring_PZ {SpringID NodeI NodeJ E mu fy tw_Col tdp d_Col d_Beam tf_Col bf_Col Ix_Col n trib ts Response_ID transfTag} {
set tpz [expr $tw_Col + $tdp]; # total PZ thickness
set G [expr $E/(2.0 * (1.0 + $mu))]; # Shear Modulus
# Beam's effective depth
if {$Response_ID==2} {
set d_BeamP $d_Beam;
} else {
set d_BeamP [expr $d_Beam + $trib + 0.5 * $ts]; # Effective Depth in Positive Moment
}
set d_BeamN $d_Beam; # Effective Depth in Negative Moment
# Stiffness Calculation
set Ks [expr $tpz * ($d_Col - $tf_Col) * $G]; # PZ Stiffness: Shear Contribution
set Kb [expr 12 * $E * ($Ix_Col + $tdp * pow(($d_Col - 2*$tf_Col),3)/12.) /pow($d_Beam,3) * $d_Beam]; # PZ Stiffness: Bending Contribution
set Ke [expr ($Ks * $Kb) / ($Ks + $Kb)]; # PZ Stiffness: Total
set Ksf [expr 2 * ($bf_Col * $tf_Col) * $G]; # Flange Stiffness: Shear Contribution
set Kbf [expr 2 * 12 * $E * $bf_Col * pow($tf_Col,3)/12. /pow($d_Beam,3) * $d_Beam]; # Flange Stiffness: Bending Contribution
set Kef [expr ($Ksf * $Kbf) / ($Ksf + $Kbf)]; # Flange Stiffness: Total
set ay [expr (0.58 * $Kef / $Ke + 0.88) / (1 - $Kef / $Ke)];
set aw_eff_4gamma 1.10;
set aw_eff_6gamma 1.15;
set af_eff_4gamma [expr 0.93 * $Kef / $Ke + 0.015];
set af_eff_6gamma [expr 1.05 * $Kef / $Ke + 0.020];
set r {expr sqrt(1- pow($n,2))}; # Reduction factor accounting for axial load
set Vy [expr $r * 0.577 * $fy * $ay * ($d_Col - $tf_Col) * $tpz]; # Yield Shear Force
set Vp_4gamma [expr $r * 0.577 * $fy * ($aw_eff_4gamma * ($d_Col - $tf_Col) * $tpz + $af_eff_4gamma * ($bf_Col - $tw_Col) * 2*$tf_Col)]; # Plastic Shear Force @ 4 gammaY
set Vp_6gamma [expr $r * 0.577 * $fy * ($aw_eff_6gamma * ($d_Col - $tf_Col) * $tpz + $af_eff_6gamma * ($bf_Col - $tw_Col) * 2*$tf_Col)]; # Plastic Shear Force @ 6 gammaY
set gamma_y [expr $Vy/$Ke];
set gamma4_y [expr 4.0 * $gamma_y];
set gamma6_y [expr 6.0 * $gamma_y];
set My_P [expr $Vy * $d_BeamP];
set Mp_4gamma_P [expr $Vp_4gamma * $d_BeamP];
set Mp_6gamma_P [expr $Vp_6gamma * $d_BeamP];
set My_N [expr $Vy * $d_BeamN];
set Mp_4gamma_N [expr $Vp_4gamma * $d_BeamN];
set Mp_6gamma_N [expr $Vp_6gamma * $d_BeamN];
set Slope_4to6gamma_y_P [expr ($Mp_6gamma_P - $Mp_4gamma_P) / (2 * $gamma_y) ];
set Slope_4to6gamma_y_N [expr ($Mp_6gamma_N - $Mp_4gamma_N) / (2 * $gamma_y) ];
# Defining the 3 Points used to construct the trilinear backbone curve
set gamma1 $gamma_y;
set gamma2 $gamma4_y;
set gamma3 [expr 100 * $gamma_y];
set M1_P [expr $My_P];
set M2_P [expr $Mp_4gamma_P];
set M3_P [expr $Mp_4gamma_P + $Slope_4to6gamma_y_P * (100 * $gamma_y - $gamma4_y)];
set M1_N [expr $My_N];
set M2_N [expr $Mp_4gamma_N];
set M3_N [expr $Mp_4gamma_N + $Slope_4to6gamma_y_N * (100 * $gamma_y - $gamma4_y)];
set gammaU_P 0.3;
set gammaU_N -0.3;
set Dummy_ID [expr 12 * $SpringID];
# Hysteretic Material without pinching and damage
# uniaxialMaterial Hysteretic $matTag $s1p $e1p $s2p $e2p <$s3p $e3p> $s1n $e1n $s2n $e2n <$s3n $e3n> $pinchX $pinchY $damage1 $damage2
# Composite Interior Steel Panel Zone
if { $Response_ID == 0.0 } {
uniaxialMaterial Hysteretic $Dummy_ID $M1_P $gamma1 $M2_P $gamma2 $M3_P $gamma3 [expr -$M1_P] [expr -$gamma1] [expr -$M2_P] [expr -$gamma2] [expr -$M3_P] [expr -$gamma3] 0.25 0.75 0. 0. 0.;
uniaxialMaterial MinMax $SpringID $Dummy_ID -min $gammaU_N -max $gammaU_P;
}
# Composite Exterior Steel Panel Zone
if { $Response_ID == 1.0 } {
uniaxialMaterial Hysteretic $Dummy_ID $M1_P $gamma1 $M2_P $gamma2 $M3_P $gamma3 [expr -$M1_N] [expr -$gamma1] [expr -$M2_N] [expr -$gamma2] [expr -$M3_N] [expr -$gamma3] 0.25 0.75 0. 0. 0.;
uniaxialMaterial MinMax $SpringID $Dummy_ID -min $gammaU_N -max $gammaU_P;
}
# Bare Steel Interior/Exterior Steel Panel Zone
if { $Response_ID == 2.0 } {
uniaxialMaterial Hysteretic $Dummy_ID $M1_N $gamma1 $M2_N $gamma2 $M3_N $gamma3 [expr -$M1_N] [expr -$gamma1] [expr -$M2_N] [expr -$gamma2] [expr -$M3_N] [expr -$gamma3] 0.25 0.75 0. 0. 0.;
uniaxialMaterial MinMax $SpringID $Dummy_ID -min $gammaU_N -max $gammaU_P;
}
element zeroLength $SpringID $NodeI $NodeJ -mat $SpringID -dir 6;
}