Add example (plastic zone)

Author: mkraska

NonLinear/Sections/Plastic/Biegung.inp

@@ -0,0 +1,77 @@
+*include, input=all.msh
+*include, input=x0.nam
+*include, input=fixy.nam
+*include, input=xL.nam
+*include, input=xLsquare.nam
+*include, input=xLcircle.nam
+*include, input=xLhcircle.nam
+*include, input=xLIprofile.nam
+*include, input=disp.nam
+** constraints
+*boundary
+Nx0,1
+Nfixy,2
+NxL,3
+
+** **mean displacement of block
+** *node, nset=Nref1
+** 1, 0, 0, 0
+** *element, elset=Ecoup1,type=dcoup3d
+** 1,1
+** *distributing coupling, elset=Ecoup1
+** Nx0, 1
+** ** mean displacement of cylinder
+** *node, nset=Nref2
+** 2, 1, 0, 0
+** *element, elset=Ecoup2,type=dcoup3d
+** 2,1
+** *distributing coupling, elset=Ecoup2
+** Nnmid_cylinder, 1
+** ** mean displacement of hollow cylinder
+** *node, nset=Nref3
+** 3, 0, 0, 0
+** *element, elset=Ecoup3,type=dcoup3d
+** 3,1
+** *distributing coupling, elset=Ecoup3
+** Nnmid_hcylinder, 1
+** **mean displacement
+** *node, nset=Nref4
+** 4, 0, 0, 0
+** *element, elset=Ecoup4,type=dcoup3d
+** 4,1
+** *distributing coupling, elset=Ecoup4
+** Nnmid_Ibeam, 1
+
+** material definition
+*material, name=steel
+*elastic
+210000,0.3
+*plastic
+200,0
+**1000,1
+*density
+7.85e-9
+** material assignment to bodies
+*solid section, elset=Eall, material=steel
+*step,nlgeom
+*static
+0.05,1,0.00000001,0.05
+*boundary
+Ndisp,3,3,20
+** Nref1,3,3,-3
+** Nref2,3,3,-3
+** Nref3,3,3,-3
+** Nref4,3,3,-3
+*node file
+U
+*el file
+S,PE
+*node print, nset=Nxlsquare,totals=only
+RF
+*node print, nset=Nxlcircle,totals=only
+RF
+*node print, nset=Nxlhcircle,totals=only
+RF
+*node print, nset=Nxliprofile,totals=only
+RF
+*end step

NonLinear/Sections/Plastic/Biegung.png

@@ -0,0 +1,57 @@
+# Properties of Cross Sections (elasto-plastic)
+Tested with CGX/CCX 2.10
+
++ Simply supported beam with a prescribed displacement in the center
++ Parametric section shapes of identical area
++ Symmetry reduction
++ Elasto-plastic material without hardening (ideal plastic)
++ Objective:
+  + Evolution of the plastic zone
+  + Force-displacement curves (TBD)
+  + Ultimate bending load - plastic section modulus (TBD)
+
+
+File                           | Contents    
+ :-------------                | :-------------
+ [par.pre.fbl](par.pre.fbl)    | CGX script, pre-processing, parametrized with param.py
+ [movie.fbl](movie.fbl)        | CGX script, movie generation
+ [Biegung.inp](Biegung.inp)    | CCX input
+
+
+
+## Pre-Processing
+
+The problem is parametrized using [param.py](../../../Scripts/param.py).
+
+The model contains several simply supported beams of equal section area but different section shape.
+
+Objective is to visualize the evolution of the plastic zone and to determine the relative ultimate strength (full plastic section) with respect to the square section.
+
+Starting from a given cross section area, the half span of the beam and a thickness, four section shapes of identical cross section area are generated.
+
+The beams are supported vertically at all nodes at x=Length and z=0 and have a symmetry boundary at x=0.
+
+One row of nodes per beam is fixed in y-direction. This is a bit over-constrained, but prevents lateral bending, which might happen due to the developing plastic hinges if just a single node per beam is constrained in y-direction.
+
+The load is applied as a prescribed displacement of the nodes at x=0 (center between the supports) and at z=0 (through the section centroid).
+```
+> param.py par.pre.fbl
+> cgx -b pre.fbl
+```
+<img src="mesh.png" width="400" title="Mesh density is biased towards the center section at x=0">
+<img src="sets.png" width="400" title="">
+
+## Solving
+The ideal-plastic material can create convergence problems. These are mitigated by load application via prescribed displacement. A maximum increment size of 5% of the step is set for smooth force-displacement curves and movies.
+```
+> ccx Biegung
+```
+<img src="Biegung.png" >
+## Post-Processing
+```
+> cgx -b movie.fbl
+```
+The plastic zone is marked using a two-step color scheme with the plastic proof strain of 0.2% marking the limit between the two colors. The right movie shows the longitudinal (bending) stress with color bar limits 5% above the yield limit.
+
+<img src="PE.gif" width="400" title="Plastic zone (plastic strain > 0.2%)">
+<img src="SXX.gif" width="400" title="Bending stress">

NonLinear/Sections/Plastic/PE.gif

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+read Biegung.frd new
+
+view disp
+view elem
+rot -y
+rot l 110
+rot u 20
+
+seta base all
+copy base new mir x
+comp new do
+ds 2 e 1
+plot fv base
+hcpy png
+sys mv hcpy_1.png S11-1.png
+
+rot -y
+rot l 135
+rot u 20
+ds -1 e 1
+plot fv base
+mm 210
+steps 21
+zoom 3
+hcpy png
+sys mv hcpy_1.png PE.png
+
+ds -0 e 1
+min 0 e l
+max 0.004 e l
+steps 2
+plot fv all
+
+frame
+zoom 7
+tra r 20
+plot fv base
+scal d
+anim real
+movi frames auto
+ds 3 ah 1
+sys mv movie.gif PE.gif
+
+mm 210 e l
+steps 21
+movi frames auto
+ds 2 ah 1
+sys mv movie.gif SXX.gif

NonLinear/Sections/Plastic/README.md

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+# <area=300>
+# <length=400>
+# <thickness = 4>
+# <dist = 40>
+# <divx = 20>
+
+asgn n 005
+asgn e 005
+
+# square
+# <l1=area**0.5>
+seto square
+  pnt ! 0 0 0
+  swep all new tra 0 <l1> 0 10
+  swep all new tra 0 0 <l1> 10
+  move square tra 0 <-l1/2.> <-l1/2.>
+setc
+seta fixy D001
+seto block
+swep square xLsquare tra <length> 0 0 <divx> a
+setc block
+
+
+# circle
+# <radius=(area/pi)**0.5>
+seto circle
+ pnt pc 0 0 0
+ pnt py 0 <radius> 0
+ pnt pz 0 0 <radius>
+ line l1 pc py 6
+ line l2 pc pz 6
+ line l3 py pz pc 8
+ surf s1 l1 l2 l3
+ copy circle new mir y
+ copy circle new mir z
+ move circle tra 0 <dist> 0
+setc
+seta fixy p pc
+seto cylinder
+swep circle xLcircle tra <length> 0 0 <divx> a
+setc cylinder
+
+# Hollow circle
+# <ri=area/(2.*pi*thickness)-thickness*0.5>
+seto hcircle
+ pnt pri 0 0 <ri> 0
+ swep hcircle new tra 0 0 <thickness> 2
+ swep hcircle new rot x 90 10
+ copy hcircle new rot x 90
+ copy hcircle new rot x 180
+ move hcircle tra 0 <2.*dist> 0
+setc
+seta fixy p pri
+seto hcylinder
+swep hcircle xLhcircle tra <length> 0 0 <divx> a
+setc hcylinder
+
+# I Profile made of three identical rectangles of given thickness
+# <l1=area/(3.*thickness)>
+seto Iprofile
+  pnt pI0 0 0 0
+  swep Iprofile pI1 tra 0 0 <l1/2.> 0
+  swep Iprofile new tra 0 <thickness> 0
+  seto f0
+    swep pI1 new tra 0 0 <thickness>
+  setc f0
+  swep f0 f1 tra 0 <thickness> 0
+  swep f1 new tra 0 <(l1-thickness)/2.> 0
+  swep f0 new tra 0 <-(l1-thickness)/2.> 0
+  copy Iprofile new mir z
+setc Iprofile
+move Iprofile tra 0 <3.*dist> 0
+seta fixy p pI0
+seto Ibeam
+swep Iprofile xLIprofile tra <length> 0 0 <divx> a
+setc Ibeam
+
+# set the bias to the longitudinal lines
+# enquiring lines is only possible at their end points
+seta lines l all
+enq lines l0 rec 0 _ _ 0.1
+enq lines ll rec <length> _ _ 0.1
+seti lb l l0 ll
+bia lb 10
+
+#Meshing
+elty all he8i
+mesh all
+merg n all
+send all abq
+
+# # choose nodes of the middle length:
+# # --> square / block:
+# seta nblock n block
+# enq nblock nmid_block rec <length/2> _ _ 0.1
+# # --> circle / cylinder:
+# seta ncylinder n cylinder
+# enq ncylinder nmid_cylinder rec <length/2> _ _ 0.1
+# # --> hcircle / hcylinder:
+# seta nhcylinder n hcylinder
+# enq nhcylinder nmid_hcylinder rec <length/2> _ _ 0.1
+# # --> Iprofile / Ibeam:
+# seta nIbeam n Ibeam
+# enq nIbeam nmid_Ibeam rec <length/2> _ _ 0.1
+
+# Mesh image
+rot -y
+rot l 110
+rot u 20
+frame
+zoom 2.5
+tra l <dist>
+view elem
+seta ! all
+hcpy png
+sys mv hcpy_1.png mesh.png
+
+# sets
+seta nodes n all
+enq nodes xL rec <length> _ 0 0.01 a
+send xL abq nam
+
+enq nodes x0 rec 0 _ _ 1 a
+send x0 abq nam
+
+send fixy abq nam
+
+enq x0 disp rec 0 _ 0 0.1
+send disp abq nam
+
+send xLsquare abq nam
+send xLcircle abq nam
+send xLhcircle abq nam
+send xLIprofile abq nam
+
+text x0 (red), y0 (green), disp (turq), xL (blue)
+rot x
+rot c -90
+rot l 5
+rot u 10
+frame
+zoom 2.5
+tra <dist> r
+plot n x0 r
+plus n fixy g
+plus n disp t
+plus n xL b
+hcpy png
+sys mv hcpy_2.png sets.png
+
+
+# send nmid_block abq nam
+# send nmid_cylinder abq nam
+# send nmid_hcylinder abq nam
+# send nmid_Ibeam abq nam
+
+volu block
+volu cylinder
+volu hcylinder
+volu Ibeam

NonLinear/Sections/Plastic/SXX.gif

@@ -161,3 +161,8 @@ This page uses animated gifs. In Firefox, you might want to install the plugin [
   width="200"
   title="Buckling and modal analysis of a pre-loaded system">
 ](Linear/L-Plate/)
+[<img
+  src="NonLinear/Sections/Plastic/PE.gif"
+  width="200"
+  title="Different section shapes of identical area. Plastic zone">
+](NonLinear/Sections/Plastic)