FORMA03… Travaux Practice nonlinear static formation: charge limit dune plates perforated

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FORMA03 - Nonlinear TP2 static formation

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V6.03 booklet: Nonlinear statics of the plane systems

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EDF-R & D/AMA

Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
V6.03.114 document

FORMA03 - Work Practice formation
nonlinear statics: charge limit of a plate
perforated

Summary:

This test 2D in plane stresses 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 controls of examination
useful for the TP

Modeling B clarifies the procedure to carry out calculation until the limiting load, and for
to modelize the discharge.

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FORMA03 - Nonlinear TP2 static formation

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Problem of reference

1.1 Geometry

One modelizes only one quarter of the plate because of symmetries.

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 locked according to OX along the side AG and following OY along side data base

Loading in imposed stress
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.

With

L=100

H=150

a=10

With

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FORMA03 - Nonlinear TP2 static formation

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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 stresses and polar co-ordinates (R,

) is [bib1]:

cos

has

cos

has

sin

has

In particular, at the edge of the hole (R = has):

(

)

[

]

cos

And along axis X:

has

Numerically, for P = 1 Mpa, and an infinite plate

Not CMP

MPa

WITH SIGXX

 1

B SIGYY

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 stresses, 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 stress P:

[

]

has

lim

Here one obtains as limits lower limiting load:

lim

= 5.4 MPa.

(One takes

= 6 MPa, because the limiting load is identical between a perfect elastoplastic material and

a material, whose traction diagram presents a horizontal asymptote at 6 MPa).

2.3 References

bibliographical

[1]

Guide Validation of the Software packages of Structural analysis 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

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3 Modeling

With

3.1

Characteristics of modeling

Modeling C_PLAN. A quarter of the plate is modelized.

3.2

Characteristics of the mesh

With

One uses a mesh in QUAD8 (mesh GIBI: forma03a. mgib). It comprises 186 QUAD8 and 617
nodes. The file forma03a. datg contains data GIBI to build this mesh and also (in end
of file) controls GIBI for the examination of the results.

Note:

This mesh 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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3.3 Functionalities

tested

Controls

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

STAT_NON_LINE INCREMENT

SUBD_PAS

CALC_ELEM

OPTION

SIEF_ELNO_ELGA
VARI_ELNO_ELGA
EQUI_ELGA_SIGM

RECU_FONCTION

NOM_CHAM

DEPL,
SIEF_ELNO_ELGA

CURVED IMPR_COURBE

FUNCTION

IMPR_RESU RESU

FORMAT

CASTEM,

GMSH

IMPR_RESU RESU

VALE_MAX

YES

POST_RELEVE_T ACTION

IDENTIFY

POLAR

INTE_MAIL_2D

FORMULATE

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 controls 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 the 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 stresses:

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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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 controls 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

(

)

[

]

cos

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

allow 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 stresses 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.

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

Stress

SIYY

3.023 0.8

1. With

Stress

SIXX

 1. 1.02 2.1

5.4 G

Stress

SIYY

5.4

5.4026 0.5

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FORMA03 - Nonlinear TP2 static formation

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5 Modeling

5.1

Characteristics of modeling

This modeling allows, within the framework of Practical Work, to conclude calculation until
charge limit of the structure, thanks to the piloting of the loading and the discharge. The behavior is
elastoplastic with isotropic work hardening given by a traction diagram such as the stress
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 piloting of the stress
imposed by the displacement of a point.

5.2

Characteristics of the mesh

The mesh is identical to that of modeling A.

5.3 Functionalities

tested

Controls

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

STAT_NON_LINE INCREMENT

SUBD_PAS

STAT_NON_LINE PILOTING

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

FUNCTION

IMPR_RESU RESU

FORMAT

CASTEM

IMPR_RESU RESU

VALE_MAX

YES

STAT_NON_LINE NEWTON

PREDICTION

RUBBER BAND

CREA_MAILLAGE ECLA_PG

CREA_RESU ECLA_PG

RECU_FONCTION

NOM_PARA_RESU

ETA_PILOTAGE

CALC_NO

OPTION

FORC_NODA

POST_RELEVE_T ACTION

RESULTANT

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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 amendment 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 pitch 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 stress-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 an area 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 piloting of the loading imposed by
value of a displacement. It is thus proposed to modify the command file to use it
piloting. One will be able to use for example displacement U

point A to control the stress

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

(A) * 1. Thus time varies here between 0 and 2s (to represent one

displacement between 0 and 2mm).

One treats all the climb charges some in 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

PILOTING

is:

PILOTING: _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. U

not G or U

not

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
controls 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 stresses 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 control of lancerment of
Stanley.

5.4.3 Discharge: use of the elastic prediction and the elastic matrix

One will use initial calculation without piloting, 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

STAMP = “ELASTIC”

(by increasing the iteration count total of

Newton allowed).

To analyze the residual state: deformation, isovaleurs of stresses 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 the same control

STAT_NON_LINE

, (without piloting), it is possible, with search

linear and subdivision of the pitch 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 = “ELASTIC”

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.

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 piloting a value higher than this value of reference.

Moment Identification Reference Aster Difference

(%)

0.1 ETA_PILOTAGE 3.11

3.11

0.4 ETA_PILOTAGE 5.05

5.05

1. ETA_PILOTAGE 5.39

5.39

1.5 ETA_PILOTAGE 5.401

5.401

2. ETA_PILOTAGE 5.405

5.405

5.4 SIYY 5.4 5.4026 0.5

Examination: one notes in the file

RESULT

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.

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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 pitches to go until
P = 5.4 MPa, and 6 pitches 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 = “ELASTIC”

, one treats indeed

all calculation in 100s approximately (instead of 85s without linear search, by cumulating all the stages
the preceding ones, except piloting). One can in particular note that during the phase of discharge, the number
iterations is definitely weaker with piloting.

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FORMA03 - Nonlinear TP2 static formation

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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 piloting 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 piloting,

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 pitch of time,

linear search can be used to contribute to convergence, as well as the subdivision
automatic of the pitches of time.