2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The constituent material verifies the vonMises criterion, with the threshold being
. The structure is subjected to pressure at the horizontal edges
and vertical
with
(
in 2D,
in 3D). In 2D plane, we consider two ways of doing this: on the one hand by amplifying the two pressures together, on the other hand by amplifying only the horizontal pressure, and by leaving the vertical pressure constant. In axisymmetry, the solid is subject to internal pressure alone
. We obtain the exact limit load and that by the regularization method [R7.07.01] in this loading direction, for the vonMises criterion, with the threshold
.
2.2. Case plan#
The structure is subjected to pressures on the horizontal edges:
and vertical:
, with:
, and a block is exerted by
. Two ways of controlling the load are considered:
cas1: both horizontal and vertical pressures are set by
,
cas2: the horizontal pressure is set by
, while the vertical pressure is constant
, with
.
2.2.1. Boundary analysis solution#
The solution is homogeneous (biaxial stresses)
:
,
, plane deformations
). We obtain [bib2] the limit load in these loading directions, for the vonMises criterion, in plane deformations, with the threshold
:
case 1:
eq 2.2.1-1
case 2:
Eq 2.2.1-2
We check that if we take
in case 2, we then find case 1.
2.2.2. Regularized limit analysis solution#
The solution is homogeneous. The plane deformations are necessarily of the form:
eq 2.2.2-1
By Norton-Hoff’s law, the coefficient
given, we obtain the deviatory constraints:
eq 2.2.2-2
The normalization (unit power) of the load for which the limit load parameter is sought, cf. [R7.07.01] [§1.2], leads to:
case 1:
Eq 2.2.2-3
case 2:
Eq 2.2.2-4
The terms of the suite
Approximations by excess of the limit load in these two load settings are then:
case 1:
eq 2.2.2-5
case 2:
eq 2.2.2-6
The invariance as a function of
observed here (which is a particular case) results from the fact that we are in an isostatic situation. In case 1, we can also use the series of approximations by default to the limit load
:
case 1:
eq 2.2.2-7
So we get the limit load
Exact when
.
In case2, where the vertical pressure is constant, the power of this « permanent » load in the displacement solution is:
case 2:
eq 2.2.2-8
2.3. Axisymmetric case#
In axisymmetric 2D, the same geometry is considered, but the solid, on which a complete axial locking is imposed, is only subjected to pressure on the internal wall:
set by
.
2.3.1. Boundary analysis solution#
We obtain [bib2] the limit load in this loading direction, for the vonMises criterion, in axisymmetry and zero axial deformations, with the threshold
:
eq 2.3.1-1
2.3.2. Regularized limit analysis solution#
The solution is homogeneous. Since the displacement is only radial, the isochoric deformations are necessarily of the form:
eq 2.3.2-1
By Norton-Hoff’s law, the coefficient
given, we obtain the deviatory constraints:
eq 2.3.2-2
The axial and radial equilibrium equations lead to the determination of the mean stress:
Eq 2.3.2-3
where
is a constant, which is calculated from the condition at the zero pressure limits on the outer wall. The components of the constraints are then obtained:
eq 2.3.2-4
The normalization (unit power) of the load for which the limit load parameter is sought, cf. [R7.07.01] [§1.2], leads to:
.
The terms of the suite
Approximations by excess of the limit load for this load are then:
eq 2.3.2-5
The terms of the suite
Some approximations by default to the load limit for this load are:
eq 2.3.2-6
En
, we find:
, that is, the same value as
and
.
2.4. Three-dimensional case#
In 3D, we consider the same geometry, but the solid, of unit thickness, is free in the antiplane direction.
. The solid is subjected to pressures on the horizontal walls:
and vertical:
, with:
. Both horizontal and vertical pressures are set by
.
2.4.1. Boundary analysis solution#
The solution is homogeneous (biaxial stresses)
:
,
,
, deformations
). We obtain the limit load in this loading direction [bib2], for the vonMises criterion, with the threshold
:
eq 2.4.1-1
2.4.2. Regularized limit analysis solution#
The solution is homogeneous. Isochoric deformations are necessarily of the form:
eq 2.4.2-1
By Norton-Hoff’s law, the coefficient
given, we obtain the deviatory constraints:
;
eq 2.4.2-2
We deduce from
:
. Hence the constraints:
.
The balance of the solid requires that
. The parameter is deduced
.
The normalization (unit power) of the load for which the limit load parameter is sought, cf. [R7.07.01] [§1.2], leads to:
Eq 2.4.2-3
The terms of the suite
upper limit load limits in this charging case are thus identical to:
Eq 2.4.2-4
2.5. Benchmark results#
Modeling |
sightings |
|
|
Power |
|
A (case 1) |
2D map |
|
|
|
|
Avis (cas2) |
2D map |
|
|
nought |
|
B |
3D |
|
|
|
|
C |
2D AXIS |
|
|
|
|
Modeling |
sightings |
|
|
) |
) |
Power |
A |
2D map |
11.547 |
11.547 |
9.6225 |
11.5458 |
0 |
Abis |
2D map |
14.6837 |
14.6837 |
nought |
nought |
0.25 |
B ( ) |
3D |
13.8675 |
13.8675 |
11.5562 |
13.8661 |
0 |
C |
2D AXIS |
12.6857 |
12.6857 |
8.5545 |
12.6830 |
0 |
Note:
Code_Aster calculates the opposite value of the power of the « permanent » load in the displacement solution
.
2.6. Bibliographical references#
VOLDOIRE, E. LORENTZ, J.M. PROIX, E. VISSE: Limit load calculation by the Norton-Hoff-Friaâ method. [R7.07.01].
VOLDOIRE, Fracture calculation and structural limit analysis, note EDF HI-74/93/082.