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Spring_IMK.tcl
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##################################################################################################################
# Spring_IMK.tcl
#
# SubRoutine to construct a rotational spring representing the moment-rotation behaviour of steel beam-columns
# and beams that are part of fully-restrained beam-to-column connections.
#
# The subroutine also considers modeling uncertainty based on the logarithmic standard deviations specified by the user.
#
# References:
#--------------
# Lignos, D. G. and H. Krawinkler (2011). "Deterioration Modeling of Steel Components in Support of Collapse
# Prediction of Steel Moment Frames under Earthquake Loading." Journal of Structural Engineering 137(11).
#
# 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).
#
# Lignos, D. G., et al. (2019). "Proposed Updates to the ASCE 41 Nonlinear Modeling Parameters for Wide-Flange
# Steel Columns in Support of Performance-based Seismic Engineering." Journal of Structural Engineering 145(9).
#
##################################################################################################################
#
# Input Arguments:
#------------------
# SpringID Spring ID
# NodeI Node i ID
# NodeJ Node j ID
# E Young's modulus
# Fy Yield stress
# Ix Moment of inertia of section
# d Section depth
# htw Web slenderness ratio
# bftf Flange slenderness ratio
# L Member Length
# Ls Shear Span
# Lb Unbraced length
# My Effective Yield Moment
# PgPye Axial load ratio due to gravity
# CompositeFlag FLAG for Composite Action Consideration: 0 --> Ignore Composite Effect
# 1 --> Consider Composite Effect
# ConnectionType Type of Connection: 0 --> Reduced Beam Section
# 1 --> Non-Reduced Beam Section
# 2 --> Column Section
# Units Unsed Units: 1 --> millimeters and MPa
# 2 --> inches and ksi
#
# Written by: Dr. Ahmed Elkady, University of Southampton, UK
#
##################################################################################################################
proc Spring_IMK {SpringID NodeI NodeJ E Fy Ix d htw bftf ry L Ls Lb My PgPye CompositeFlag ConnectionType Units} {
set n 10.0;
if {$Units == 1} {
set c1 1.0;
set c2 1.0;
set c3 25.4;
set c4 1000.0;
} else {
set c1 25.4;
set c2 6.895;
set c3 1.0;
set c4 1.0;
}
set K [expr ($n+1.0) * 6 * $E * $Ix / $L];
#######################################################################################################
#######################################################################################################
#######################################################################################################
#######################################################################################################
if {$ConnectionType == 0} {
# Rotational capacities calculated using Lignos and Krawinkler (2009) RBS equations
set theta_p [expr 0.19 * pow(($htw),-0.314) * pow(($bftf),-0.100) * pow(($Lb/$ry),-0.185) * pow(($Ls/$d),0.113) * pow(($c1 * $d/533),-0.760) * pow(($c2 * $Fy* $c4/355),-0.070)];
set theta_pc [expr 9.52 * pow(($htw),-0.513) * pow(($bftf),-0.863) * pow(($Lb/$ry),-0.108) * pow(($c2 * $Fy* $c4/355),-0.360)];
set Lmda [expr 585 * pow(($htw),-1.140) * pow(($bftf),-0.632) * pow(($Lb/$ry),-0.205) * pow(($c2 * $Fy* $c4/355),-0.391)];
# FOR BARE STEEL BEAM
if {$CompositeFlag == 0} {
set MyPMy 1.0;
set MyNMy 1.0;
set McMyP 1.1;
set McMyN 1.1;
# Corrected rotations to account for elastic deformations
set theta_y [expr $My/(6 * $E * $Ix / $L)];
set theta_p [expr $theta_p - ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_pc [expr $theta_pc + $theta_y + ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_p_P $theta_p;
set theta_p_N $theta_p;
set theta_pc_P $theta_pc;
set theta_pc_N $theta_pc;
set theta_u 0.2;
set D_P 1.0;
set D_N 1.0;
set Res_P 0.4;
set Res_N 0.4;
set c 1.0;
}
# FOR COMPOSITE BEAM
if {$CompositeFlag != 0} {
set MyPMy 1.35;
set MyNMy 1.25;
set McMyP 1.30;
set McMyN 1.05;
# Corrected rotations to account for elastic deformations
set theta_y [expr $My/(6 * $E * $Ix / $L)];
set theta_p_p [expr $theta_p - ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_p_n [expr $theta_p - ($McMyN-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_pc_p [expr $theta_pc + $theta_y + ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_pc_n [expr $theta_pc + $theta_y + ($McMyN-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_p_P [expr 1.80*$theta_p_p];
set theta_p_N [expr 0.95*$theta_p_n];
set theta_pc_P [expr 1.35*$theta_pc_p];
set theta_pc_N [expr 0.95*$theta_pc_n];
set theta_u 0.2;
set D_P 1.15;
set D_N 1.0;
set Res_P 0.3;
set Res_N 0.2;
set c 1.0;
}
}
#######################################################################################################
#######################################################################################################
#######################################################################################################
#######################################################################################################
if {$ConnectionType == 1} {
# Rotational capacities calculated using Lignos and Krawinkler (2009) other-than-RBS equations
if {$d > [expr $c3*21.0]} {
set theta_p [expr 0.318 * pow(($htw),-0.550) * pow(($bftf),-0.345) * pow(($Lb/$ry),-0.023) * pow(($Ls/$d),0.090) * pow(($c1 * $d/533),-0.330) * pow(($c2 * $Fy* $c4/355),-0.130)];
set theta_pc [expr 7.500 * pow(($htw),-0.610) * pow(($bftf),-0.710) * pow(($Lb/$ry),-0.110) * pow(($c1 * $d/533),-0.161) * pow(($c2 * $Fy* $c4/355),-0.320)];
set Lmda [expr 536 * pow(($htw),-1.260) * pow(($bftf),-0.525) * pow(($Lb/$ry),-0.130) * pow(($c2 * $Fy* $c4/355),-0.291)];
} else {
set theta_p [expr 0.0865 * pow(($htw),-0.360) * pow(($bftf),-0.140) * pow(($Ls/$d),0.340) * pow(($c1 * $d/533),-0.721) * pow(($c2 * $Fy* $c4/355),-0.230)];
set theta_pc [expr 5.6300 * pow(($htw),-0.565) * pow(($bftf),-0.800) * pow(($c1 * $d/533),-0.280) * pow(($c2 * $Fy* $c4/355),-0.430)];
set Lmda [expr 495 * pow(($htw),-1.340) * pow(($bftf),-0.595) * pow(($c2 * $Fy* $c4/355),-0.360)];
}
# FOR BARE STEEL BEAM
if {$CompositeFlag == 0} {
set MyPMy 1.0;
set MyNMy 1.0;
set McMyP 1.1;
set McMyN 1.1;
# Corrected rotations to account for elastic deformations
set theta_y [expr $My/(6 * $E * $Ix / $L)];
set theta_p_P $theta_p;
set theta_p_N $theta_p;
set theta_pc_P $theta_pc;
set theta_pc_N $theta_pc;
set theta_u 0.2;
set D_P 1.0;
set D_N 1.0;
set Res_P 0.4;
set Res_N 0.4;
set c 1.0;
}
# FOR COMPOSITE BEAM
if {$CompositeFlag != 0} {
set MyPMy 1.35;
set MyNMy 1.25;
set McMyP 1.30;
set McMyN 1.05;
# Corrected rotations to account for elastic deformations
set theta_y [expr $My/(6 * $E * $Ix / $L)];
set theta_p_p [expr $theta_p - ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_p_n [expr $theta_p - ($McMyN-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_pc_p [expr $theta_pc + $theta_y + ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_pc_n [expr $theta_pc + $theta_y + ($McMyN-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_p_P [expr 1.80*$theta_p_p];
set theta_p_N [expr 0.95*$theta_p_n];
set theta_pc_P [expr 1.35*$theta_pc_p];
set theta_pc_N [expr 0.95*$theta_pc_n];
set theta_u 0.2;
set D_P 1.15;
set D_N 1.00;
set Res_P 0.3;
set Res_N 0.2;
set c 1.0;
}
}
#######################################################################################################
#######################################################################################################
#######################################################################################################
#######################################################################################################
if {$ConnectionType == 2} {
# Rotational capacities calculated using Lignos et al. (2019) column regression equations for monotonic
set theta_p [expr 294 * pow(($htw),-1.700) * pow(($Lb/$ry),-0.700) * pow((1-$PgPye),1.600)];
set theta_pc [expr 90 * pow(($htw),-0.800) * pow(($Lb/$ry),-0.800) * pow((1-$PgPye),2.500)];
if {$theta_p > 0.20} {set theta_p 0.2}
if {$theta_pc > 0.30} {set theta_pc 0.3}
if {$PgPye <= 0.35} {
set Lmda [expr 25500 * pow(($htw),-2.140) * pow(($Lb/$ry),-0.530) * pow((1-$PgPye),4.920)];
} else {
set Lmda [expr 268000* pow(($htw),-2.300) * pow(($Lb/$ry),-1.300) * pow((1-$PgPye),1.190)];
}
if {$PgPye <= 0.2} {
set My [expr (1.15/1.1)*$My*(1-$PgPye/2)];
} else {
set My [expr (1.15/1.1)*$My*(9/8)*(1-$PgPye)];
}
set McMy [expr 12.5 * pow(($htw),-0.200) * pow(($Lb/$ry),-0.400) * pow((1-$PgPye),0.400)];
if {$McMy < 1.0} {set McMy 1.0}
if {$McMy > 1.3} {set McMy 1.3}
set MyPMy 1.0;
set MyNMy 1.0;
set McMyP $McMy;
set McMyN $McMy;
# Corrected rotations to account for elastic deformations
set theta_y [expr $My/(6 * $E * $Ix / $L)];
set theta_p [expr $theta_p - ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_pc [expr $theta_pc + $theta_y + ($McMyP-1.0)*$My/(6 * $E * $Ix / $L)];
set theta_p_P $theta_p;
set theta_p_N $theta_p;
set theta_pc_P $theta_pc;
set theta_pc_N $theta_pc;
set theta_u 0.15;
set D_P 1.0;
set D_N 1.0;
set Res_P [expr 0.5-0.4*$PgPye];
set Res_N [expr 0.5-0.4*$PgPye];
set c 1.0;
}
#######################################################################################################
#######################################################################################################
#######################################################################################################
#######################################################################################################
set My_P [expr $MyPMy * $My];
set My_N [expr $MyNMy * $My];
# # Bilin material model
#set My_P [expr $MyPMy * $My];
#set My_N [expr -$MyNMy * $My];
#set as_mem_p [expr ($McMyP-1.)*$My_P/($theta_p_P * 6.*$E * $Ix/$L)];
#set as_mem_n [expr -($McMyN-1.)*$My_N/($theta_p_N * 6.*$E * $Ix/$L)];
#set SH_mod_P [expr ($as_mem_p)/(1.0+$n*(1.0-$as_mem_p))];
#set SH_mod_N [expr ($as_mem_n)/(1.0+$n*(1.0-$as_mem_n))];
#uniaxialMaterial Bilin $SpringID $K $SH_mod_P $SH_mod_N $My_P $My_N $L_S $L_C $L_A $L_K $c_S $c_C $c_A $c_K $theta_p_P $theta_p_N $theta_pc_P $theta_pc_N $Res_P $Res_N $theta_u $theta_u $D_P $D_N
##################################################################################################################
#Random generation of backbone parameters based on assigned uncertainty
##################################################################################################################
global Sigma_IMKcol Sigma_IMKbeam; global xRandom;
if {$ConnectionType == 2} {
set SigmaX [lindex $Sigma_IMKcol 0]; Generate_lognrmrand $K $SigmaX; set K $xRandom;
set SigmaX [lindex $Sigma_IMKcol 1]; Generate_lognrmrand $My_P $SigmaX; set My_P $xRandom;
set My_N $xRandom;
set SigmaX [lindex $Sigma_IMKcol 2]; Generate_lognrmrand $McMyP $SigmaX; set McMyP [expr max(1.01,$xRandom)];
set McMyN [expr max(1.01,$xRandom)];
set SigmaX [lindex $Sigma_IMKcol 3]; Generate_lognrmrand $Res_P $SigmaX; set Res_P $xRandom;
set Res_N $xRandom;
set SigmaX [lindex $Sigma_IMKcol 4]; Generate_lognrmrand $theta_p_P $SigmaX; set theta_p_P $xRandom;
set theta_p_N $xRandom;
set SigmaX [lindex $Sigma_IMKcol 5]; Generate_lognrmrand $theta_pc_P $SigmaX; set theta_pc_P $xRandom;
set theta_pc_N $xRandom;
set SigmaX [lindex $Sigma_IMKcol 6]; Generate_lognrmrand $theta_u $SigmaX; set theta_u $xRandom;
set SigmaX [lindex $Sigma_IMKcol 7]; Generate_lognrmrand $Lmda $SigmaX; set Lmda $xRandom;
#set SigmaX [lindex $Sigma_IMKcol 8]; Generate_lognrmrand $c $SigmaX; set c $xRandom;
}
if {$ConnectionType != 2 && $CompositeFlag == 0} {
set SigmaX [lindex $Sigma_IMKbeam 0]; Generate_lognrmrand $K $SigmaX; set K $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 1]; Generate_lognrmrand $My_P $SigmaX; set My_P $xRandom;
set My_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 2]; Generate_lognrmrand $McMyP $SigmaX; set McMyP [expr max(1.01,$xRandom)];
set McMyN [expr max(1.01,$xRandom)];
set SigmaX [lindex $Sigma_IMKbeam 3]; Generate_lognrmrand $Res_P $SigmaX; set Res_P $xRandom;
set Res_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 4]; Generate_lognrmrand $theta_p_P $SigmaX; set theta_p_P $xRandom;
set theta_p_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 5]; Generate_lognrmrand $theta_pc_P $SigmaX; set theta_pc_P $xRandom;
set theta_pc_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 6]; Generate_lognrmrand $theta_u $SigmaX; set theta_u $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 7]; Generate_lognrmrand $Lmda $SigmaX; set Lmda $xRandom;
#set SigmaX [lindex $Sigma_IMKbeam 8]; Generate_lognrmrand $c $SigmaX; set c $xRandom;
}
if {$ConnectionType != 2 && $CompositeFlag == 1} {
set SigmaX [lindex $Sigma_IMKbeam 0]; Generate_lognrmrand $K $SigmaX; set K $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 1]; Generate_lognrmrand $My_P $SigmaX; set My_P $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 1]; Generate_lognrmrand $My_N $SigmaX; set My_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 2]; Generate_lognrmrand $McMyP $SigmaX; set McMyP [expr max(1.01,$xRandom)];
set SigmaX [lindex $Sigma_IMKbeam 2]; Generate_lognrmrand $McMyN $SigmaX; set McMyN [expr max(1.01,$xRandom)];
set SigmaX [lindex $Sigma_IMKbeam 3]; Generate_lognrmrand $Res_P $SigmaX; set Res_P $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 3]; Generate_lognrmrand $Res_N $SigmaX; set Res_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 4]; Generate_lognrmrand $theta_p_P $SigmaX; set theta_p_P $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 4]; Generate_lognrmrand $theta_p_N $SigmaX; set theta_p_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 5]; Generate_lognrmrand $theta_pc_P $SigmaX; set theta_pc_P $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 5]; Generate_lognrmrand $theta_pc_N $SigmaX; set theta_pc_N $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 6]; Generate_lognrmrand $theta_u $SigmaX; set theta_u $xRandom;
set SigmaX [lindex $Sigma_IMKbeam 7]; Generate_lognrmrand $Lmda $SigmaX; set Lmda $xRandom;
#set SigmaX [lindex $Sigma_IMKbeam 8]; Generate_lognrmrand $c $SigmaX; set c $xRandom;
}
##################################################################################################################
##################################################################################################################
##################################################################################################################
# Cyclic deterioration parameters
if {$ConnectionType == 2} {
set L_S $Lmda; set L_C [expr 0.9*$Lmda]; set L_A $Lmda; set L_K [expr 0.9*$Lmda];
} else {
set L_S $Lmda; set L_C $Lmda; set L_A $Lmda; set L_K $Lmda;
}
set c_S $c; set c_C $c; set c_A $c; set c_K $c;
# IMKBilin material model (This is the updated version of the Bilin model)
uniaxialMaterial IMKBilin $SpringID $K $theta_p_P $theta_pc_P $theta_u $My_P $McMyP $Res_P $theta_p_N $theta_pc_N $theta_u $My_N $McMyN $Res_N $L_S $L_C $L_K $c_S $c_C $c_K $D_P $D_N;
element zeroLength $SpringID $NodeI $NodeJ -mat 99 99 $SpringID -dir 1 2 6 -doRayleigh 1;
}