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Titrate:
Law of behavior of assembly ASSE_CORN
Date
:
14/04/05
Author (S):
G. DEVESA, J.L. FLEJOU, Key P. PENSERINI
:
R5.03.32-A Page
: 1/18
Organization (S): EDF-R & D/AMA, LME
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.32
Law of behavior of assembly ASSE_CORN
Summary:
This document describes the nonlinear behavior of the nonlinear assemblies of angles of pylons
modelled by discrete elements DIS_TR. This law of behavior is affected on the discrete elements
by means of relation ASSE_CORN called by the operators of resolution of nonlinear problems
STAT_NON_LINE [R5.03.01] or DYNA_NON_LINE [R5.05.05].
The law represents at the same time behavior in traction of the assembly and the relation moment-rotation around
the axis of the bolts perpendicular to the assembly. The other directions of loading present one
linear elastic behavior describes by traditional characteristics of rigidity.
One distinguishes in the law from behavior two phases associated with two mechanisms: the first
representing the friction and the slip of the bolts until the thrust, and the second representing
plasticization of the assembly until the ruin. The laws of the plastic type describing each one of these phases have
even pace and have a concavity at their connection which makes convergence problematic and
require a particular digital processing in the options of calculation to which the method appeals
iterative of Newton.
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Law of behavior of assembly ASSE_CORN
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Count
matters
1 Notations ................................................................................................................................................ 3
2 Physical model of the one-way behavior of the assembly ................................................... 5
3 Relation of behavior of the mechanisms ......................................................................................... 7
3.1 One-way behavior ......................................................................................................... 7
3.2 Incremental two-dimensional behavior .................................................................................... 8
4 Establishment in Code_Aster ......................................................................................................... 11
4.1 Formulation in sizes reduced in loading ....................................................................... 12
4.1.1 Operator Knr ........................................................................................................................ 12
4.1.2 Operator Kor ........................................................................................................................ 13
4.2 Formulation in sizes reduced in unloading .................................................................. 14
4.3 Tangent operators kN and KB ......................................................................................................... 14
4.4 Digital processing of connection enters the mechanisms of the law of assembly .............. 15
5 Variables and parameters of the law of behavior ............................................................................ 17
5.1 Variables of the law .......................................................................................................................... 17
5.2 parameters of the law ....................................................................................................................... 17
6 bibliographical References ............................................................................................................... 18
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Titrate:
Law of behavior of assembly ASSE_CORN
Date
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:
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1 Notations
SLF
Surface Limite de Frottement
M y
Moment in the assembly around the axis y
N1
Effort limits slip of the assembly on axis X
M 1
Moment limits slip of the assembly on the axis y
SLU
Surface Ultimate Limite
N2
Ultimate limiting effort of the assembly on axis X
M2
Ultimate limiting moment of the assembly on the axis y
NR
Limiting effort
M
Limiting moment
U 1
Displacement limits mechanism 1 on axis X
1
Rotation limits mechanism 1 on the axis y
U 2
Displacement limits mechanism 2 on axis X
2
Rotation limits mechanism 2 on the axis y
U
Displacement of the assembly on axis X
Rotation of the assembly on the axis y
N
Effort reduces N = Nx/NR
m
Moment reduces m = My/M
U R
Displacement reduces Ur = U/U
R
Reduced rotation R =/
U
Displacement limits on axis X
Rotation limits on the axis y
H (X)
Scalar function
has
Parameter of nonlinearity
D
Constant scalar
D
Vector reduced generalized displacement
F
Vector reduced generalized effort
p
Variable interns scalar
feq
Effort generalized equivalent reduces scalar
F
Surface loading
R (X)
Scalar function R (X) H 1
-
=
(X)
D
Vector generalized displacement
F
Vector generalized effort
[D]
Stamp displacement generalized limit
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[F]
Stamp effort generalized limit
+
X
Value of X at the moment T + dt
-
X
Value of X at the moment T
E
Eccentricity of loading E = M y/NR X
er
Reduced eccentricity of loading E = m N
R
/
Sign N
[]
Stamp
{}
Vector column
< >
Vector line
KB
Tangent operator at the moment T
KN
Tangent operator at the moment T + dt
Kor, Knr Opérateurs tangent reduced
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2 Physical model of the one-way behavior of
assembly
Assembly of an angle on the wing of another or a plate (bracket or splice plate) by
bolts is schematized by [Figure 2-a].
Appear 2-a: locates local connection; axis X is confused with the axis of the bar
and centers it is confused there with the axis of the bolts
The one-way behavior of the assembly is modelled for the loading in traction or in
inflection.
The modeling selected of the one-way behavior in loading of the assembly subjected to one
normal effort or a moment around is represented there by [Figure 2-b].
Normal effort
Moment/y
SLU
SLU
mechanism 1
mechanism 2
mechanism 1
mechanism 2
SLF
SLF
U
butted
butted
1
Displacement
Rotation
1
0
0
0
U
0
2
2
Appear 2-b: mechanisms of assembly in normal effort and moment
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One distinguishes two phases of the behavior associated with two mechanisms:
· mechanism 1: friction and slip until the thrust (beginning of the shearing of
bolts).
· mechanism 2: plasticization of the assembly until the ruin by shearing of the bolts
or tearing of the grips.
The limiting surface of friction (SLF) is the curve corresponding to the appearance of the slip in
space NR X M Y. friction is described by the law of Coulomb.
Ultimate limiting surface (SLU) is the curve corresponding to the ruin of the assembly in space
NR X M Y. the ruin can be due, according to the design of the assembly, with the shearing of the bolts
or with the tearing of the grips.
Tests on the same geometry but with tightening torques of the different bolts
show that the tangent stiffness of mechanism 2 at the point of thrust decreases when the SLF
bring closer the SLU.
This justifies the physical modeling retained for the assembly of the two mechanisms [Figure 2-b].
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:
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3
Relation of behavior of the mechanisms
The behavior of mechanisms 1 and 2 is similar. It is nonlinear between a behavior
initial tangent rigid and an asymptotic limiting behavior.
It is described by two essential parameters: the parameter of nonlinearity and the parameter surface
limit.
The thrust (mechanism 1) or ruins it (mechanism 2) are described by an associated kinematic criterion.
3.1 Behavior
one-way
We said to [§2] that one-way behaviors in normal effort and moment around
are similar [Figure 2-b] there.
They can be described consequently relation if the adimensional sizes are used:
NR
M
·
X
y
reduced forces: N =
and
m =
NR
M
U
· reduced displacements: U R =
and R =
U
[Figure 3.1-a] represents in adimensional form the one-way behavior.
Analytically, it can be written (it is a choice):
U
R = H (N) or
R = H (m)
+
1
1
with H (X)
has
X
=
has
D 1 - X
a+1
N
D =
has
1 - N
A is the scalar parameter of nonlinearity. N and A are identified on the one-way tests. N
who takes into account the variability of the tests generally takes value 0.95.
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N
m
N = Nx/NR
1
m = My/M
N m
Ur = U/U
R =/
criterion
kinematics
Ur
1
R
Appear 3.1-a: relation of behavior of assembly
It is noticed that H (N) = 1 or H (m) = 1, i.e.: U R =1 or R = 1, or: U = U or
=.
The one-way kinematic criterion is thus checked for N = N or m = Mr.
3.2
Incremental two-dimensional behavior
The coupling in extreme cases is defined by limiting surface:
2
2
NR X
M y
+
= 1
NR
M
The one-way behavior in reduced variables is described by the relation of [§3.1]:
D = H (F)
U R
where
D is the vector reduced displacements
R
N
F is the vector reduced forces
m
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Into two-dimensional behavior, the isotropy is translated by a model with a variable interns scalar
p such as:
p = H (feq)
loading
in
where feq is the equivalent reduced force (scalar).
feq is defined such as:
F = feq F *
where
NR X
F is the current point of loading M
y
*
NR X
F * is the limiting loading associated F
*
M y
The expression of feq results from the expression of limiting surface. Membership of F * with
surface limit is written:
2
2
*
*
NR X
M y
+
= 1
NR
M
By the definition of feq, one can write:
2
2
NR
M
X
y
+
= 1
feqN
feqM
i.e. according to reduced forces N and m:
2
2
N
m
+
= 1
feq
feq
from where
2
2
feq = N + m
One defines then the surface of loading F, homothetic on limiting surface, by:
F: feq - R (p) = 0
R (p) H 1
where
-
=
(p)
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For a formalism similar to that of plasticity with isotropic work hardening [bib2], one obtains the relation
of behavior continues expressed in reduced sizes:
·
·
·
F
D = p F = p
F
feq
·p = 0
if feq - R (p) < 0
·
·
p = h' (feq) F eq if feq - R (p) = 0
The relation of behavior of the rigid type - plastic without elasticity is written finally:
·
· = p
1
D
[D] [F] - F
feq
U
Nx
where D =
and F =
M
y
[
0
0
D] U
=
NR
and [F]
=
0
0 M
The relation of incremental behavior in reduced sizes is obtained by integration of the relation
continue between T (variable -) and T + dt (variable +).
In loading, p
check F = 0 with T + dt:
feq+ = R (p + p
)
éq
2.2-1
By introducing the relation of behavior,
+
F
D = p
éq
2.2-2
+
feq
one deduces the value from p
,
2
2
p
= D.D =
U
R +
R
and one calculates the value of
+
feq by [éq 2.2-1]. The relation of behavior [éq 2.2-2] gives them
reduced efforts:
U
+
N
R
=
R (p + p
)
p
+
m
R
=
R (p + p
)
p
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In unloading, p
= 0 and one have by [éq 2.2-2]:
D = 0
4
Establishment in Code_Aster
The relation of behavior ASSE_CORN is assigned to discrete elements of modeling
DIS_TR with 2 confused nodes. This relation is called by the operators of resolution of
nonlinear problems STAT_NON_LINE [R5.03.01] or DYNA_NON_LINE [R5.05.05].
The local axes of these elements X, y, Z are defined as on [Figure 2-a].
The integration of this relation of behavior of the assemblies in operator STAT_NON_LINE
of Code_Aster the formulation of the tangent operators requires KB and kN [bib3].
· KB is tangent rigidity at the beginning of the step of time, urgent T.
· KN is tangent rigidity at the end of the step of time, urgent T + dt.
The illustration of the operators KB and kN is given by [Figure 4-a].
Appear 4-a: definition of the operators KB and kN
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4.1
Formulation in sizes reduced in loading
4.1.1 Operator
Knr
We saw with [§3.2] that the relation of behavior is written:
+
D
F
=
R (p + p
)
p
2
2
with
p
= D.D =
U
R +
R
The Knr operator is defined by:
F
K
= I
NR
1 I, J 2
D J
It is written:
p
[Id] - {}
p
D · <
>+
D
K
J
=
R +
NR
2
(p) +
p
{
+
'
D}
p
R
+
(p)
· <
>
D
p
J
Calculation gives then:
p
+
Ur
<
> =
R
;
D J
p
p
U 2r
Ur
R
{D}
p + p
p
· <
> =
D
2
J
Ur R
R
p
p
1 x2
and with A = 1
has
one: H (X) =
D 1 - X
1
2
1
2
R (p) = H (p) =
- D p + D p + 4 D p
2
1
1 -
2
R' (p)
D [R (p)]
=
=
h' [R (p)] R (p) [2 - R (p)]
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4.1.2 Operator
Kor
For the elastoplastic behaviors, the operator KB with T = 0 is equal to the rigidity of the structure
rubber band. In our case, the tangent initial behavior is rigid. The Kor operator is defined then
by the passage in extreme cases when p tends towards 0 of the Knr operator. One obtains:
R' (p)
R (p)
=
p0
p
R (p)
from where Kor =
[Id]
p0
p
However R (p) < 1
p
and if one supposes that the user gives, for the first step of loading, of
values such as
4
p 10
>
, one can retain in practice:
4
10
0
K
=
T
however =0
4
0
10
These remarks are illustrated by [Figure 4.1.2-a].
Appear 4.1.2-a: operator KB in T = 0
At the moment T running, the Kor operator is equal to the Knr operator of the preceding step defined by
[§4.1.1].
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4.2
Formulation in sizes reduced in unloading
To avoid numerical problems, one describes the behavior (rigid) in unloading by:
Kor = Knr = K
T
however =0
4.3
Tangent operators kN and KB
· Tangent operator kN is written:
F
K =
I
N
1 I, J 6
D J
with
F
NR
1
X
N
NR
=
=
×
D
U
1
U
R U
F
NR
1
X
N
NR
=
=
×
D
5
R
F
M
5
y
m
M
=
=
×
D
U
1
U
R U
F
M
5
y
m
M
=
=
×
D
5
R
F
2 = Ky
D
2
F
3 = Kz
D
3
F
4 = KRx
D
4
F
6 = KRz
D
6
The other values are null.
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· The tangent operator KB, with T = 0, is written:
4 NR
10
O
U
K y
K
KB =
Z
K X-ray
4 M
O
10
K Rz
4.4 Digital processing of connection enters the mechanisms of
law of assembly
During the resolution of each step of loading by the iterative method of Newton, one
must calculate with each iteration the tangent with the curve of balance force-displacement of the law of
behavior. The problem is that connection enters the mechanisms of the law of assembly, on
the law of behavior, has a concavity (cf [Figure 2-b]) which returns convergence
problems when, during a step of loading, one passes from one mechanism to the other.
In the subroutine TE0041 which calculates, for each increment of load, the elementary matrix of
tangential rigidity of a discrete finite element with 2 nodes having of the degrees of freedom in translation
and in rotation, it proved to be necessary to converge, to calculate a directed secant stiffness of the state
initial of null effort and displacement towards the state, at the end of the step of loading, consisted the effort
imposed and displacement corresponding on the curve of balance of the law of behavior. It was necessary
for that, which was unusual on the level of this option, to know the number of iteration interns
numerical process calculating the step of loading, then to consider the effort imposed on the element with the end of
this step.
Indeed, if one notes
+
F effort imposed on the level of an element (a priori unknown since one does not know
that assembled efforts),
+
U displacement corresponding on the curve of balance, and for
iteration I, values respective U (I), F (I), Ks (I) of displacement, the effort and the matrix
secant acting as tangent matrix calculated at the end of the iteration, one knows only in
input of the above mentioned subroutine
I
U, and values at the beginning of the step of load F (0) and U (0), because one
the values with the preceding iteration I did not store - 1. In the expression of the residue calculated in end
+
of iteration I - 1: F - F (I -)
1 = Ks (I -)
1.(U (I) - U (I -)
1, one thus knows nothing any more but U (I) with
iteration I, except in the particular case I = 1 where one a:
F + - F (0) = K
-
S (0).(U ()
1 U (0)
+
F there is the only unknown value at the beginning and results from the others. One also deduces it
displacement
+
U at the end of the step according to the relation of balance:
·
p. [N, m]
·
·
= R (p)
.U
+
R, R, from where secant stiffness K S ()
+
1 = F/U.
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:
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The problem is that in this first iteration, displacement U ()
1 imposed is different from
final displacement to calculate
+
U balances some with +
F from now on known (with the test of balance close to
no the preceding loading). Effort calculated at the end of this iteration F ()
1 must thus be also
different from
+
F and such as F ()
1 = Ks ()
1 .U ()
1 so that starting from the couple U ()
1 and F ()
1, one points
with the secant K ()
1
S
on the couple
+
U and +
F. One thus obtains at the beginning of iteration 2 one
displacement U (2) very near to +
U and one can then calculate by the relation of balance F (2) very
near also to
+
F as well as the secant stiffness Ks (2) = F (2)/U (2).
If one converged exactly with the preceding step of load, 2 internal iterations are enough to converge
exactly, if not one needs some additional iterations to satisfy the test of balance on
residue.
The method known as of “directed secant” is schematized on [Figure 4.3-a] where one has them
following correspondences:
U I = U (I)
K (U I
T
) = Ks (I)
for a law of behavior LLC (U I) = F (I).
F (U1)
Elementary loop of Newton:
Inputs: U0 Ui+1 Sig- VAr Mater->LC iter = i+1
If iter = 1:
One estimates F+ = LLC (U0) + Kt (U0) (U1 U0)
One estimates U+ = LC-1 (F+)
K (U1) = F+/U+, F (U1) = K (U1) * U1
F+
Outputs: K (U1) F (U1) Sig+ Var+
If iter > 1:
Outputs: Kt (Ui+1) LLC (Ui+1) Sig+ Var+
F
0
0
U0
U+
U1
Appear 4.3-a: method of directed secant
One thus sees now why it was necessary in the option calculated by the above mentioned subroutine to know it
number of iteration interns I in order to distinguish the particular case I = 1.
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:
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5
Variables and parameters of the law of behavior
5.1
Variables of the law
The law of behavior comprises 4 internal variables per point of calculation of which 3 only are
active:
· V1 is displacement reduces equivalent p maximum reached out of mechanism 1,
· V2 is displacement reduces equivalent p maximum reached out of mechanism 2,
· V3 is an indicator which is worth 1 or 2 according to whether one is respectively on surface limits
mechanism 1 or 2, and 0 if one is under this limiting surface (after discharge for example),
· V4 is inactive for the moment (thus remains to 0).
5.2
parameters of the law
The parameters of the law of behavior entered like data under key word ASSE_CORN of
command DEFI_MATERIAU [U4.43.01]:
· NU_: one enters behind this key word the value of the N1 parameter of mechanism 1,
· MU_1: one enters behind this key word the value of the parameter M 1 of mechanism 1,
· DXU_1: one enters behind this key word the value of the parameter U 1 of mechanism 1,
· DRYU_1: one enters behind this key word the value of parameter 1 of mechanism 1,
· C_1: one enters behind this key word the value common to parameters N and m of mechanism 1,
· NU_2: one enters behind this key word the value of the parameter N2 of mechanism 2,
· MU_2: one enters behind this key word the value of the parameter m2 of mechanism 2,
· DXU_2: one enters behind this key word the value of the parameter U 2 of mechanism 2,
· DRYU_2: one enters behind this key word the value of parameter 2 of mechanism 2,
· C_2: one enters behind this key word the value common to parameters N and m of mechanism 2,
· KY, KZ, KRX, KRZ take the values of the characteristics of linear behavior in
local directions y, Z, X-ray, Rz respectively.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Law of behavior of assembly ASSE_CORN
Date
:
14/04/05
Author (S):
G. DEVESA, J.L. FLEJOU, Key P. PENSERINI
:
R5.03.32-A Page
: 18/18
6 References
bibliographical
[1]
P. PENSERINI: “Modeling of the assemblies bolted in the webmasts”
Note EDF/R & D HM-77/93/287
[2]
P. PENSERINI: “Characterization and modeling of the behavior of the connections structure
metal-foundation “Thèse of doctorate of Université Paris 6, 1991
[3]
J.P. LEFEBVRE, P. MIALON: “Quasi-static nonlinear Algorithm of Code_Aster”
Note EDF/R & D HI-75/7832
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Outline document