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FORMA03 - Nonlinear TP2 static formation
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Organization (S): EDF-R & D/AMA
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
V6.03.114 document
FORMA03 - Travaux Pratiques of the formation
nonlinear statics: charge limit of a plate
perforated
Summary:
This test 2D in plane constraints quasi-static makes it possible to illustrate on a simple case the questions relative to
elastoplastic modeling; it highlights the effects of structure, of stress concentration, of
charge limit.
It is about a homogeneous rectangular plate, perforated in its center, consisted of an elastoplastic material
with isotropic work hardening, whose initial state is nonconstrained, which is subjected to a traction at its ends.
One is interested in the elastoplastic solution in load. More precisely, the analytical methods allow
to know a lower limit of the limiting load. By an elastoplastic calculation, one would like to find one
limit higher.
The objective of the test is to show the possibilities of modeling and postprocessing with STAT_NON_LINE.
Modeling A corresponds to the first loading, and comprises the commands of examination
useful for the TP
Modeling B clarifies the procedure to carry out calculation until the limiting load, and for
to model the discharge.
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Titrate:
FORMA03 - Nonlinear TP2 static formation
Date:
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Author (S):
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1
Problem of reference
1.1 Geometry
One models only one quarter of the plate because of symmetries.
P
G
F
G
F
H=150
With
a=10
With
B
D
B
D
L=100
1.2
Material properties
Elastoplastic behavior with isotropic work hardening given by the traction diagram.
Young modulus E = 1000 Mpa
NAKED Poisson's ratio = 0.3
Traction diagram (prolongation constant right)
Epsilon 0.004 0.006 0.009
0.02
Sigma 4.
5. 5.5 6.
1.3
Boundary conditions and loadings
Conditions of symmetry
The plate is blocked according to OX along the side AG and following OY along side data base
Loading in imposed constraint
It is subjected to a traction P following OY distributed on side FG.
Way of loading
One considers a monotonous way of loading, such as traction P grows since 1 MPa (
solution is then elastic) and until complete plasticization of the structure.
Handbook of Validation
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Titrate:
FORMA03 - Nonlinear TP2 static formation
Date:
16/07/03
Author (S):
Key J.M. PROIX
:
V6.03.114-B Page:
3/12
2
Reference solution
2.1 Solution
rubber band
In elasticity, for an infinite plate, comprising a hole of diameter, subjected to a loading P
according to y ad infinitum, the analytical solution in plane constraints and polar co-ordinates (R,) are [bib1]:
P
2
has
2
4
has
has
rr =
1
1 4
3
- - - + cos
2
2
R
R
R
2
4
P
has
has
=
1+
+ 1+
3
cos
2
2
R
R
2
4
P
has
has
R =
1+
2
-
3
sin
2
2
R
R
In particular, at the edge of the hole (R = has): = P ([1+ 2cos
2)]
2
4
P
has
has
And along axis X: = yy =
1+
+ 1+
3
2
R
R
Numerically, for P = 1 Mpa, and an infinite plate
Not CMP
MPa
WITH SIGXX
1
B SIGYY
3
For a plate of finished size, the abacuses [bib3] make it possible to obtain the coefficients of
stress concentration, and it is found that SIGYY are worth approximately 3.03 MPa.
2.2
Elastoplastic solution (load limits)
In elastoplasticity, by a static approach in plane constraints, one can obtain a terminal
lower of the load limits [bib2] for a tape of width 2L finished and infinite length,
comprising a hole of width 2a and subjected to an ad infinitum imposed constraint P:
P
lim = y [L - has]/L
Here one obtains as limits lower limiting load: -
lim
P = 5.4 MPa.
(One takes = 6 MPa, because the limiting load is identical between a perfect elastoplastic material and
y
a material, whose traction diagram presents a horizontal asymptote at 6 MPa).
2.3 References
bibliographical
[1]
Guide of Validation of Progiciels de Calcul de Structures SFM. Technical AFNOR.
[2]
Analyze limit of the fissured structures and criteria of resistance.
[3]
F. VOLDOIRE: Note EDF/DER/HI/74/95/26 1995
[4]
Stress concentration factors. R.E. PETERSON ED. J. WILEY P150
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/03/008/A

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FORMA03 - Nonlinear TP2 static formation
Date:
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V6.03.114-B Page:
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3 Modeling
With
3.1
Characteristics of modeling
Modeling C_PLAN. A quarter of the plate is modelled.
3.2
Characteristics of the grid
G
F
With
B
D
One uses a grid in QUAD8 (grid GIBI: forma03a. mgib). It comprises 186 QUAD8 and 617
nodes. The file forma03a. datg contains data GIBI to build this grid and also (in end
of file) commands GIBI for the examination of the results.
Note:
This grid is sufficiently fine to have a good approximation of the solution in
elasticity: for example if one compares on the edge of the hole compared to the solution
analytical, one obtains:
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FORMA03 - Nonlinear TP2 static formation
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3.3 Functionalities
tested
Commands
AFFE_CHAR_MECA FACE_IMPO
AFFE_CHAR_MECA FORCE_CONTOUR
DEFI_MATERIAU TRACTION
STAT_NON_LINE COMP_INCR
RELATION
“VMIS_ISOT_TRAC”
STAT_NON_LINE NEWTON
REAC_ITER
1
STAT_NON_LINE INCREMENT
SUBD_PAS
4
CALC_ELEM
OPTION
SIEF_ELNO_ELGA
VARI_ELNO_ELGA
EQUI_ELGA_SIGM
RECU_FONCTION
NOM_CHAM
DEPL,
SIEF_ELNO_ELGA
CURVED IMPR_COURBE
FONCTION
IMPR_RESU RESU
FORMAT
CASTEM,
GMSH
IMPR_RESU RESU
VALE_MAX
OUI
POST_RELEVE_T ACTION
REPERE
POLAIRE
INTE_MAIL_2D
FORMULE
CALC_FONC_INTERP
3.4
Unfolding of the TP
This modeling (command files forma03a. COM) corresponds to a calculation in
elastoplasticity for P going up to 5.4 MPa. It also contains (in the command file
forma03a. com1, to use in continuation) the commands necessary to the examination (traced
graphic curves and postprocessing).
3.4.1 Results of calculation
Before the throw calculation itself, it is necessary to define a base, on the machine
of execution, which will contain all the results. One can launch provided calculation, to P = 5.4 MPa in
the goal post-to treat it. In card-indexing it result, one can observe how the maximum evolves/moves (on
all elements) of component VMIS of field EQUI_ELGA_SIGM, as well as the maximum of
internal variables.
One will be able to note on this level on the internal variables that:
· at moments 1 and 1.2, there is no plasticization,
· until moment 5.4, one is constantly in load.
and on the constraints:
· the maximum value of the criterion of Von Mises at the points of Gauss is always limited to 6 Mpa,
what shows that the solution checks the law of behavior well.
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FORMA03 - Nonlinear TP2 static formation
Date:
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3.4.2 Use of the tools for examination
One will be able to use in continuation the command file forma03a. com1, on the basis previously
created.
It is necessary to define the files following results:
· a file of the type “cast” if one wants effector a graphic examination with GIBI,
· a file of the type “pos” if one wants effector a graphic examination using GMSH,
· other files specific to the layouts of curves, (extension .dat for layout with xmgrace).
The commands necessary to the layout of curves are defined in the file forma03a. com1.
One finds there:
· evolution of the component Sigma-Theta at moment 1 at the edge of the hole, and the curve definite
by the analytical solution = P ([1+ 2cos
2)],
· evolution of the Syy component at the point G according to displacement at the same point, for
every calculated moment,
· evolution of the resultant of the nodal forces on edge FG, according to displacement with
not G.
These three calls to IMPR_COURBE make it possible to generate three files, which can be traced via one
spreadsheet, or a software of layout of curve like xmgrace.
To plot a curve using xmgrace, “grace should be launched”, then “dated/ASCII importation/” and
to select the file to be traced. For the moment, the file produced by Aster is not directly
readable: it is necessary to comment on the lines of text (character #) before the reading by xmgrace.
One can visualize with GIBI, or GMSH the deformation, the isovaleurs of constraints SYY and of
cumulated equivalent plastic deformation p. One will be able to note at moment 1. (elasticity)
stress concentration at the point B.A moment 5.4, one can notice on the isovaleurs of
cumulated plastic deformation, localization of the deformations in the vicinity of B.
One will be able to finally use interactive postprocessing Stanley to visualize fields or
curves. It is enough for that to remove the character comment (# in front of the ordering of launching of
Stanley.
3.4.3 Continuation of the TP
Then, to continue calculation, it is necessary to modify the way of applying the loading. This been the subject of
modeling B.
4
Results of modeling A
4.1 Values
tested
Value of the limiting loading. The value of reference corresponds on the lower terminal. Difficulties
of convergence occur as soon as one seeks to increase the loading beyond 5.4Mpa.
Moment
GROUP_NO
Identification Reference
Aster Difference
(%)
1. B
Constraint
SIYY
3.
3.023 0.8
1. With
Constraint
SIXX
1. 1.02 2.1
5.4 G
Constraint
SIYY
5.4
5.4026 0.5
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FORMA03 - Nonlinear TP2 static formation
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5 Modeling
B
5.1
Characteristics of modeling
This modeling allows, within the framework of Travaux Pratiques, to conclude calculation until
charge limit of the structure, thanks to the control of the loading and the discharge. The behavior is
elastoplastic with isotropic work hardening given by a traction diagram such as the constraint
uniaxial tends towards a constant value (6 MPa). There is thus a limiting load for this structure
whose lower limit is known: Plim > 5.4 MPa [bib2].
The plate is subjected to a traction P following OY on side FG. Therefore, if P largely exceeds 5.4
MPa, it has there no more solution. Numerically, there are thus concern! It is what is observed: for
P=5.4 Mpa, convergence is very slow. If calculation were continued, there would be no more convergence with
beyond 5.41 Mpa.
The best solution if one wishes to calculate the load limits (by the resolution of a problem
elastoplastic incremental - other methods exist like the use of material
incompressible with direct methods of analysis limits) is to use the control of the constraint
imposed by the displacement of a point.
5.2
Characteristics of the grid
The grid is identical to that of modeling A.
5.3 Functionalities
tested
Commands
AFFE_CHAR_MECA FACE_IMPO
AFFE_CHAR_MECA FORCE_CONTOUR
DEFI_MATERIAU TRACTION
STAT_NON_LINE COMP_INCR
RELATION
“VMIS_ISOT_TRAC”
STAT_NON_LINE NEWTON
REAC_ITER
1
STAT_NON_LINE INCREMENT
SUBD_PAS
4
STAT_NON_LINE CONTROL
TYPE
DDL_IMPO
CALC_ELEM
OPTION
SIEF_ELNO_ELGA
VARI_ELNO_ELGA
EQUI_ELGA_SIGM
EPSI_ELNO_DEPL
CALC_NO
OPTION
SIEF_NOEU_ELGA
RECU_FONCTION
NOM_CHAM
DEPL
CURVED IMPR_COURBE
FONCTION
IMPR_RESU RESU
FORMAT
CASTEM
IMPR_RESU RESU
VALE_MAX
OUI
STAT_NON_LINE NEWTON
PREDICTION
ELASTIQUE
CREA_MAILLAGE ECLA_PG
CREA_RESU ECLA_PG
RECU_FONCTION
NOM_PARA_RESU
ETA_PILOTAGE
CALC_NO
OPTION
FORC_NODA
POST_RELEVE_T ACTION
RESULTANTE
DY
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FORMA03 - Nonlinear TP2 static formation
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5.4
Calculation until the limiting load
5.4.1 Test of increase in the loading to P = 5.4 MPa
One wishes to continue calculation with an aim of considering a limit upper of the limiting load, in
trying to go beyond 5.4 MPa (limits lower analytical), without modification of the loading. One
note whereas:
· calculation is possible up to 5.41 Mpa, beyond that, it does not converge more, even by subdividing it
no time,
· convergence is increasingly slow.
The cause of difficult convergence is well the proximity of the limiting load. This is why it is necessary
to subdivide the step of time. One can realize it by the value of the loading (5.4 Mpa =
theoretical value of the lower limit) and by the curve constraint-displacement in top of the structure
: one can note that for P=5.4 MPa the limiting load is not completely reached (not
of horizontal asymptote) but that one approaches some.
The isovaleurs of p show a zone of concentration of plastic deformation (comparable to one
line of slip) tilted of 53° approximately compared to the vertical, energy of the point B at the flat rim
This corresponds rather well to the theory which says that the lines of slip are tilted of
54,44° [bib2]. There is here of course an approximation of the line of slip which is in theory
of null thickness.
5.4.2 Calculation with loading controlled by a displacement
A good way of carrying out this type of calculation is to use the control of the loading imposed by
value of a displacement. It is thus proposed to modify the command file to use it
control. One will be able to use for example displacement UY of point A to control constraint YY
imposed on FG.
One will increase it up to 2 mm for example. One will take a coefficient equal to 1. One will thus be used
fictitious time T such as t= UY (A) * 1. Thus time varies here between 0 and 2s (to represent one
displacement between 0 and 2mm).
One treats all the rise charges some in a new STAT_NON_LINE. One can cut out the interval
(0. 2.) in 10 or 20 increment for example.
The syntax of the key word factor PILOTAGE is:
PILOTAGE: _F (GROUP_NO = A, STANDARD = “DDL_IMPO”, NOM_CMP = “DY”,
COEF_MULT = 1.) )
Attention with the type of loading CH2 (force distributed): to use “FIXE_PILO”.
Note:
One could have controlled by the displacement of other points, e.g. Uy of the point G or UX of the point
D.
To observe in the file “message” the value of the parameter: “ETA_PILOTAGE”.
One obtains in theory a good approximation of the limiting load (by higher value) than one
will be able to compare on the analytical lower terminal (5.4).
One will be able to plot, in continuation, the curve forces resultant-displacement in G according to time. (them
commands necessary are in the file forma03a. com1). One should find on this curve
value of the load limits given by “ETA_PILOTAGE”.
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FORMA03 - Nonlinear TP2 static formation
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One can visualize with GIBI, or GMSH the deformation, the isovaleurs of constraints SYY and of
cumulated equivalent plastic deformation p. A moment 2, one can notice on the isovaleurs of
cumulated plastic deformation, localization of the deformations in the vicinity of B.
One will be able to finally use interactive postprocessing Stanley to visualize fields or
curves. It is enough for that to remove the character comment (# in front of the command of lancerment of
Stanley.
5.4.3 Discharge: use of the elastic prediction and the elastic matrix
One will use initial calculation without control, in continuation since P = 5.4 MPa. Several solutions are
possible:
· To try to discharge with tangent matrix. Conclusion?
· To test the elastic prediction for the phase of discharge.
· To test MATRICE = “ELASTIQUE” (by increasing the iteration count total of
Newton allowed).
To analyze the residual state: deformation, isovaleurs of constraints and variables internal.
5.4.4 Use of linear search
If one wants to calculate all the way of loading (P growing up to 5.4 MPa, then decreasing
up to 0) in same command STAT_NON_LINE, (without control), it is possible, with search
linear and subdivision of the step of time (and stamps tangent reactualized).
One will be able to put 10 iterations of linear search, and to facilitate the phase of discharge, one will be able
to put PREDICTION = “ELASTIQUE”.
One will be able to compare the solution obtained with that obtained previously (they must be
identical!) and to compare time CPU and the iteration count of each solution.
6
Results of modeling B
6.1 Values
tested
Value of the limiting loading. The value of reference corresponds on the lower terminal. Grace is obtained
with control a value higher than this value of reference.
Moment Identification Reference Aster Difference
(%)
0.1 ETA_PILOTAGE 3.11
3.11
0
0.4 ETA_PILOTAGE 5.05
5.05
0
1. ETA_PILOTAGE 5.39
5.39
0
1.5 ETA_PILOTAGE 5.401
5.401
0
2. ETA_PILOTAGE 5.405
5.405
0
5.4 SIYY 5.4 5.4026 0.5
Examination: one notes in file RESULTAT that the limiting load is almost reached:
parameter ETA_PILOTAGE tends towards a constant value: 5.405, which is very close to the terminal
lower theoretical (5.4): This load pattern thus makes it possible to correctly treat this case which
could be treated by a loading in conventional force.
Handbook of Validation
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FORMA03 - Nonlinear TP2 static formation
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Discharge:
The way of loading goes up to 5.4 MPa then totals 0. One discretizes in 27 steps to go until
P = 5.4 MPa, and 6 steps to total 0. The consistent tangent matrix (FULL_MECA) does not allow
to converge at the time of the discharge. To treat the discharge, two methods can be
used:
1) to calculate the discharge with elastic prediction (but reactualization of the tangent matrix with
run of the iterations). One can note that that makes it possible to cross the phase of beginning of
discharge. But that does not make it possible to calculate all the discharge (until a loading
no one).
2) to use the elastic matrix. This method functions but convergence is slow. It is necessary
simply to think of increasing the maximum iteration count (ITER_GLOB_MAXI).
Curve forces resulting - displacement: there is a state of residual stresses no one everywhere safe in
some areas (around the point B in particular), but of the nonnull residual deformations.
6.1.1 Use of linear search
With 10 iterations of linear search, and PREDICTION = “ELASTIQUE”, one treats indeed
all calculation in 100s approximately (instead of 85s without linear search, by cumulating all the stages
the preceding ones, except control). One can in particular note that during the phase of discharge, the number
iterations is definitely weaker with control.
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Titrate:
FORMA03 - Nonlinear TP2 static formation
Date:
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7
Summary of the results
This test makes it possible to go up how to carry out the calculation of an elastoplastic structure and sound
examination, and in particular to highlight the benefit to use control for one
problem of limiting load.
One can retain of this test some ideas:
· even apart from a perfect elastoplastic behavior, it can exist a limiting load:
it is the case with all the real traction diagrams. It is then necessary to adapt the method of
resolution with the mechanical solution and for example to use control,
· cutting in small increments of load is often necessary to integrate
correctly the relation of behavior. That can also contribute to convergence, it is thus
advised to use the automatic recutting of the step of time,
· linear search can be used to contribute to convergence, as well as the subdivision
automatic of the steps of time.
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Titrate:
FORMA03 - Nonlinear TP2 static formation
Date:
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