# PressureDependMultiYield-quadUP element

<center>Solid-fluid fully coupled (u-p) plane-strain element: Inclined (4 degrees), saturated soil column with pressure dependent material subjected to 1D sinusoidal base shaking</center>

Input File

<syntaxhighlight lang="tcl">

  1. Created by Zhaohui Yang (zhyang@ucsd.edu)
  2. fully coupled, u-p formulation
  3. plane strain, shear-beam type mesh with single material,
  4. dynamic analysis, SI units (m, s, KN, ton)

wipe

  1. some user defined variables

set matOpt 1 ;# 1 = pressure depend;

# 2 = pressure independ;

set fmass 1 ;# fluid mass density set smass 2 ;# saturated soil mass density set G 6.e4 ; set B 2.4e5 ; set bulk 2.2e6 ;#fluid-solid combined bulk modulus set vperm 5.e-4 ;#vertical permeability (m/s) set hperm 5.e-4 ;#horizontal permeability (m/s) set accGravity 9.81 ;#acceleration of gravity set vperm [expr $vperm/$accGravity/$fmass] ;#actual value used in computation set hperm [expr $hperm/$accGravity/$fmass] ;#actual value used in computation set press 0. ;# isotropic consolidation pressure on quad element(s) set loadBias 0.07 ;# Static shear load, in percentage of gravity load (=sin(inclination angle))

set accMul 2. ;# acc. multiplier set accNam whatever.acc ;# file name for input acc. record set accDt 0.0166 ;# dt of input acc. record set period 1.0 ;# Period for applied Sine wave set deltaT 0.01 ;# time step for analysis set numSteps 2400 ;# number of time steps set gamma 0.6 ;# Newmark integration parameter

set massProportionalDamping 0. ; set InitStiffnessProportionalDamping 0.002;

set numXele 1 ;# number of elements in x (H) direction set numYele 10 ;# number of elements in y (V) direction set xSize 1.0 ;# x direction element size set ySize 1.0 ;# y direction element size

  1. BUILD MODEL
  1. create the ModelBuilder

model basic -ndm 2 -ndf 3

  1. define material and properties

switch $matOpt { 1 { nDMaterial PressureDependMultiYield 1 2 $smass $G $B 31.4 .1 80 0.5\ 26.5 0.1 .2 5 10 0.015 1. } 2 { nDMaterial PressureIndependMultiYield 2 2 1.8 4.e4 2.e5 40 .1 } }

set gravY -$accGravity ;#calc. gravity set gravX [expr -$gravY*$loadBias]

  1. define nodes

set numXnode [expr $numXele+1] set numYnode [expr $numYele+1]

for {set i 1} {$i <= $numXnode} {incr i 1} { for {set j 1} {$j <= $numYnode} {incr j 1} { set xdim [expr ($i-1)*$xSize] set ydim [expr ($j-1)*$ySize] set nodeNum [expr $i + ($j-1)*$numXnode] node $nodeNum $xdim $ydim } }

  1. define elements

for {set i 1} {$i <= $numXele} {incr i 1} { for {set j 1} {$j <= $numYele} {incr j 1} { set eleNum [expr $i + ($j-1)*$numXele] set n1 [expr $i + ($j-1)*$numXnode] set n2 [expr $i + ($j-1)*$numXnode + 1] set n4 [expr $i + $j*$numXnode + 1] set n3 [expr $i + $j*$numXnode]

  1. thick maTag bulk mDensity perm1 perm2 gravity press

element quadUP $eleNum $n1 $n2 $n4 $n3 1.0 $matOpt $bulk $fmass $hperm $vperm $gravX $gravY $press } }

  1. set material to elastic for gravity loading

updateMaterialStage -material $matOpt -stage 0

  1. fix the base, and free surface drainage

for {set i 1} {$i <= $numXnode} {incr i 1} { fix $i 1 1 0 set surfnode [expr ($numYnode-1)*$numXnode + $i] fix $surfnode 0 0 1 }

  1. tie all disp. DOFs at same level

for {set i 1} {$i < $numYnode} {incr i 1} { set nodeNum1 [expr $i*$numXnode + 1] for {set j 2} {$j <= $numXnode} {incr j 1} { set nodeNum2 [expr $i*$numXnode + $j] equalDOF $nodeNum1 $nodeNum2 1 2 } }

  1. GRAVITY APPLICATION (elastic behavior)
  1. create the SOE, ConstraintHandler, Integrator, Algorithm and Numberer

numberer RCM system ProfileSPD test NormDispIncr 1.0e-8 25 2 algorithm Newton constraints Penalty 1.e18 1.e18 integrator Newmark 1.5 1. analysis Transient

analyze 3 5.e3

  1. switch material stage from elastic (gravity) to plastic

switch $matOpt { 1 { updateMaterialStage -material $matOpt -stage 1 } 2 { updateMaterialStage -material $matOpt -stage 1 } } analyze 5 5.e3

  1. rezero time

wipeAnalysis setTime 0.0

  1. loadConst -time 0.0
  2. NOW APPLY LOADING SEQUENCE AND ANALYZE (plastic)
  1. base input motion

pattern UniformExcitation 1 1 -accel "Sine 0. 10. $period -factor $accMul"

  1. input motion through data file
  2. pattern UniformExcitation 1 1 -accel "Series -factor $accMul -filePath $accNam -dt $accDt"
  1. recorder for nodal variables along the vertical center line.

set nodeList {} for {set i 0} {$i < $numYnode} {incr i 1} { lappend nodeList [expr $numXnode/2 + $i*$numXnode] }

  1. define recorders for disp., excess pore pressure, and acc.
  2. Note: disp and acc outputs are relative to the base

eval "recorder Node -file disp -time -node $nodeList -dof 1 2 -dT $deltaT disp" eval "recorder Node -file pwp -time -node $nodeList -dof 3 -dT $deltaT vel" eval "recorder Node -file acc -time -node $nodeList -dof 1 2 -dT $deltaT accel"

  1. stress/strain output at Gauss point 1 of each element along center line

set name1 "stress"; set name2 "strain"; for {set i 1} {$i < $numYnode} {incr i 1} { set ele [expr $numXele-$numXele/2+($i-1)*$numXele] set name11 [join [list $name1 $i] {}] set name21 [join [list $name2 $i] {}] recorder Element -ele $ele -time -file $name11 -dT $deltaT material 1 stress recorder Element -ele $ele -time -file $name21 -dT $deltaT material 1 strain }

  1. analysis options

constraints Penalty 1.e18 1.e18 test NormDispIncr 1.e-5 25 0 numberer RCM algorithm Newton system ProfileSPD

  1. some mass proportional and initial-stiffness proportional damping

rayleigh $massProportionalDamping 0.0 $InitStiffnessProportionalDamping 0.0 integrator Newmark $gamma [expr pow($gamma+0.5, 2)/4] analysis VariableTransient

  1. analyze

set startT [clock seconds] analyze $numSteps $deltaT [expr $deltaT/100] $deltaT 15 set endT [clock seconds] puts "Execution time: [expr $endT-$startT] seconds."

wipe #flush ouput stream </syntaxhighlight>

MATLAB Plotting File

<syntaxhighlight lang="matlab"> clear all;

a1=load('acc'); d1=load('disp'); p1=load('pwp'); s1=load('stress1'); e1=load('strain1'); s5=load('stress5'); e5=load('strain5'); s9=load('stress9'); e9=load('strain9');

fs=[0.5, 0.2, 4, 6]; accMul = 2;

%integration point 1 p-q po=(s1(:,2)+s1(:,3)+s1(:,4))/3; for i=1:size(s1,1) qo(i)=(s1(i,2)-s1(i,3))^2 + (s1(i,3)-s1(i,4))^2 +(s1(i,2)-s1(i,4))^2 + 6.0* s1(i,5)^2; qo(i)=sign(s1(i,5))*1/3.0*qo(i)^0.5; end

figure(1); close 1; figure(1); %integration point 1 stress-strain subplot(2,1,1), plot(e1(:,4),s1(:,5),'r'); title ('shear stress \tau_x_y VS. shear strain \epsilon_x_y at 10 m depth'); xLabel('Shear strain \epsilon_x_y'); yLabel('Shear stress \tau_x_y (kPa)'); subplot(2,1,2), plot(-po,qo,'r'); title ('confinement p VS. deviatoric stress q at 10 m depth'); xLabel('confinement p (kPa)'); yLabel('q (kPa)'); set(gcf,'paperposition',fs); saveas(gcf,'SS_PQ_10m','jpg');

%integration point 5 p-q po=(s5(:,2)+s5(:,3)+s5(:,4))/3; for i=1:size(s5,1) qo(i)=(s5(i,2)-s5(i,3))^2 + (s5(i,3)-s5(i,4))^2 +(s5(i,2)-s5(i,4))^2 + 6.0* s5(i,5)^2; qo(i)=sign(s5(i,5))*1/3.0*qo(i)^0.5; end

figure(5); close 5; figure(5); %integration point 5 stress-strain subplot(2,1,1), plot(e5(:,4),s5(:,5),'r'); title ('shear stress \tau_x_y VS. shear strain \epsilon_x_y at 6 m depth'); xLabel('Shear strain \epsilon_x_y'); yLabel('Shear stress \tau_x_y (kPa)'); subplot(2,1,2), plot(-po,qo,'r'); title ('confinement p VS. deviatoric stress q at 6 m depth'); xLabel('confinement p (kPa)'); yLabel('q (kPa)'); set(gcf,'paperposition',fs); saveas(gcf,'SS_PQ_6m','jpg');

%integration point 9 p-q po=(s9(:,2)+s9(:,3)+s9(:,4))/3; for i=1:size(s1,1) qo(i)=(s9(i,2)-s9(i,3))^2 + (s9(i,3)-s9(i,4))^2 +(s9(i,2)-s9(i,4))^2 + 6.0* s9(i,5)^2; qo(i)=sign(s9(i,5))*1/3.0*qo(i)^0.5; end

figure(6); close 6; figure(6); %integration point 9 stress-strain subplot(2,1,1), plot(e9(:,4),s9(:,5),'r'); title ('shear stress \tau_x_y VS. shear strain \epsilon_x_y at 2 m depth'); xLabel('Shear strain \epsilon_x_y'); yLabel('Shear stress \tau_x_y (kPa)'); subplot(2,1,2), plot(-po,qo,'r'); title ('confinement p VS. deviatoric stress q at 2 m depth'); xLabel('confinement p (kPa)'); yLabel('q (kPa)'); set(gcf,'paperposition',fs); saveas(gcf,'SS_PQ_2m','jpg');

figure(2); close 2; figure(2); %node 3 displacement relative to node 1 subplot(2,1,1),a=plot(d1(:,1),d1(:,8),'r'); hold on subplot(2,1,1),b=plot(d1(:,1),d1(:,14),'g'); hold on subplot(2,1,1),c=plot(d1(:,1),d1(:,22),'b'); title ('Lateral displacement wrt base'); xLabel('Time (s)'); yLabel('Displacement (m)'); legend([a,b,c],'8m depth','4m depth', 'Surface',2) set(gcf,'paperposition',fs); saveas(gcf,'Disp','jpg');

s=accMul*sin(0:pi/50:20*pi); s=[s';zeros(3000,1)]; s1=interp1(0:0.01:40,s,a1(:,1));

figure(3); close 3; figure(3); %node acceleration subplot(3,1,1),a=plot(a1(:,1),s1+a1(:,22),'r'); legend(a,'at surface',4); title ('Lateral acceleration'); xLabel('Time (s)'); yLabel('Acceleration (m/s^2)'); subplot(3,1,2),a=plot(a1(:,1),s1+a1(:,14),'r'); legend(a,'4 m depth',4); xLabel('Time (s)'); yLabel('Acceleration (m/s^2)'); subplot(3,1,3),a=plot(a1(:,1),s1+a1(:,8),'r'); legend(a,'8 m depth',4); xLabel('Time (s)'); yLabel('Acceleration (m/s^2)'); set(gcf,'paperposition',fs); saveas(gcf,'Acc','jpg');

figure(4); close 4; figure(4); subplot(3,1,1),a=plot(p1(:,1),p1(:,11),'r'); legend(a,'1 m depth',4); title ('Pore pressure'); xLabel('Time (s)'); yLabel('Pore pressure (kPa)'); subplot(3,1,2),a=plot(p1(:,1),p1(:,7),'r'); legend(a,'5 m depth',4); xLabel('Time (s)'); yLabel('Pressure (kPa)'); subplot(3,1,3),a=plot(p1(:,1),p1(:,2),'r'); legend(a,'10 m depth',4); xLabel('Time (s)'); yLabel('Pressure (kPa)'); set(gcf,'paperposition',fs); saveas(gcf,'EPWP','jpg'); </syntaxhighlight>

Displacement Output File

PD_Ex22Disp.jpg

Stress-Strain Output File (2 m depth)

PD_Ex22SS_PQ2m.jpg

Stress-Strain Output File (6 m depth)

PD_Ex22SS_PQ6m.jpg

Stress-Strain Output File (10 m depth)

PD_Ex22SS_PQ10m.jpg

Excess Pore Pressure Output File

PD_Ex22EPP.jpg

Acceleration Output File

PD_Ex22Accel.jpg

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