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Development of an Equivalent Numerical Model of a Bolted Point based on an Analytical Solution
Orwa Omran, Viet Dung Nguyen, H. Jaffal, P. Marchand, A. Bezza, P.
Coorevits
To cite this version:
Orwa Omran, Viet Dung Nguyen, H. Jaffal, P. Marchand, A. Bezza, et al.. Development of an Equiv- alent Numerical Model of a Bolted Point based on an Analytical Solution. The Ninth International Conference on Engineering Computational Technology„ Sep 2014, Naples, Italy. �hal-01711228�
Abstract
The study of mechanical structures using a three-dimensional finite element, taking into account the problem of non-linearities (behaviour and contact) increases the time of the calculation and requires significant computer memory. Likewise, the optimization of the assembly (the number and position of the points) leads to an important number of calculations. The aim of the work, presented in this paper, is to develop a methodology which allows us to simplify a bolted assembly while keeping a realist physical behavior and to reduce the time of calculation of the numerical simulations. This paper focuses in the comparison of numerical and experimental results in the traction between two steel sheets with one fixity.
Keywords: bolted assembly, equivalent element, connector, traction.
1 Introduction
Despite significant advances in mechanical assembly, bolting remains widespread because of its simplicity of realization. However, the disadvantage of this method by finite element modelling of great structures (airplane wing, train set ...) is the number of degrees of freedom and non-linearity of materials and contact behaviour.
Numerous studies have been conducted to study its behaviour and reduce the computation time.
Langrand [1] and Dang-Hoang [2] have studied the influence of edge effects, clearances and number of fixings on the global mechanical behaviour of the assembly of a mono-material. In [1], the author has showed that the state of the contact area (deformation, friction …) must be taken into account. The study [2]
showed that we can describe the behaviour of bolted assembly by introducing the preload to obtain the tightening torque. In the study [3], Berot has considered a simplification of a rivet by a third cylindrical body placed between two plates; this approach showed disadvantages because it does not represent the actual geometry of
Paper 58
Development of an Equivalent Numerical Model of a Bolted Point based on an Analytical Solution
O. Omran
1, V.-D. Nguyen
1, H. Jaffal
2, P. Marchand
2, A. Bezza
2and P. Coorevits
11
Eco-PRocédés, Optimisation et Aide à la Décision, IUT de l'Aisne University of Picardie Jules Verne, Saint-Quentin, France
2
CETIM, Senlis, France
Civil-Comp Press, 2014
Proceedings of the Ninth International Conference on Engineering Computational Technology, P. Iványi and B.H.V. Topping, (Editors), Civil-Comp Press, Stirlingshire, Scotland.
the assembly system (introduction of the distance between the plates, rivet head not taken into account,... ). In the second study of Berot, the rivet is replaced by a virtual model which presents an area joint to the two plates with a specific behaviour. The results are quite acceptable for the overall behaviour, but this method didn't describe the local behaviour of the bolted assembly; the author also noted that it is difficult to automate this method.
Another simple method consists of modelling the plates by finite element and the assembly point by an equivalent element. In the literature, these studies [1, 3-4] have showed that we can use kinematic constraints (relationship between the displacements of points or surfaces), non-linear elements (beam, spring or mixed elements) and hybrid formulations. Langrand has showed that the use of kinematic constraints is not appropriate to describe the nonlinear behaviour of the assembly [1]. Moreover, this approach depends on the compliance of meshing. The hybrid approach formulations are quite expensive in computation time.
In this paper, we propose a simple finite element model that consists of two thin sheets connected by a linear element which link the six degrees of freedom of each connecting node of element. The decoupling hypothesis has been proposed to reduce the number of stiffness to 6 instead of 36 (as theory says).
At first, an analytical approach is proposed by modelling the sheets by their mean planes separated by a distance h. This model is based on the classical hypothesis of strength of materials with a perfect adhesion of the contact surfaces. By using the equilibrium equations, the boundary conditions, and the behaviour law of sheets, we obtained analytical relationships that allow us to link the rigidities of connector to displacements and rotation of the assembly.
An experimental campaign has been realized by CETIM (partner of the project).
The experimental results allow us to identify the stiffness matrix and to validate our study.
2 Assembly model by an equivalent element
2.1 Description of the proposed model
In order to produce the behaviour of an assembly, it is necessary to define a reference solution on which identifies this equivalent mechanical behaviour. Figure 1 show two mechanical testing used to identify the mechanical behaviour of a structure.
a) b)
Figure 1: Basic solicitation (a-traction; b- bending)
In our study, the idea is to simplify the three-dimensional model (3D) by one dimensional beam model (1D); our approach consists of replacing the bolt by a simple connector that allows us to obtain a realist solution with a reasonable numerical calculation cost. The finite element model proposed is composed of two plates connected by an element which link the six degrees of freedom
ux u
y u
z
x
y
z of each node of bolted connecting. This model requires the determination of six rigidities corresponding to the six degrees of freedom according to x y
z
y
z
, , :
- 3 rigidities in traction K
x, K
y, K
z, - 3 rigidities in torsion C
x, C
y, C
z.
Theoretically, we needed to 36 rigidities to describe the stiffness matrix, so we have supposed the hypothesis of decoupling between the different types of load. The parameters those defining the mechanical behaviour (performance) of the connector can be identified by various mechanical testing: traction, bending, torsion of bolt.
Each test can identify two or three mechanical rigidities (Fig.2).
Figure 2: Proposed model of assembly
An analytical approach is proposed by two beams represented by their mean planes separated by a distance h. This model is based on the classical hypothesis of strength of materials with a perfect adhesion of the contact surfaces. We have used the equations of equilibrium to obtain analytical relationships that allow us to identify the rigidities of connector proposed. The structure is fixed at one end A and blocked at the other end B by a cross-head which subjected to a tensile force.
yB
B
M
Z , are the reactions of the cross-head on the specimen. It is therefore a problem of tensile-bending in the plan ( x z
, ) (cf. figure 3).
Figure 3: Analytical model of traction 1D Figure 4: Perfect adhesion of surfaces
For the plate model of the structure, we have supposed the hypothesis of Love- Kirchhoff. The sections planes remain plane and normal to the deformed longitudinal axis during the mechanical testing. As a result of this approach, the displacement fields vary linearly according to the thickness of the plat. So the relation of displacement expressed by:
z u h y
u h x
u h z
u U
z u h y
u h x
u U h z u
U
z x
y y
x J
z x
y y
x I I
. ).
2 . ( ).
2 . ( 2 .
. ).
2 . (
).
2 . ( 2 .
3 3
3 3
3 3
2 2
2 2
2 2
3 2
(1)
For the beam model of the structure, we have supposed the hypothesis of Euler- Bernoulli. The relations between load and displacement are expressed as flowing depending on the equations of equilibrium:
x y
x y
K L F h L
ES
= FL L
u . ( ) ( ) 2
) 2 2
( 2 3
(2)
3
2 1 3 8 3 2 ) 1
1
( K L
EI FhL
FhL L EI
z G
y
Y
(3)
y Gy
t
y
C
FL EI
L
FL
2 ) 3 (
2
2(4)
S is the section,
Gy
I is the quadratic moment according to y
and L
tthe total length of the structure.
Based on the analytical solution (2-4) and experimental results
(F,ux(2L)), ( , ( ))
2
L
F
y,
(F,y3(L)), we can identify the rigidities
y z x
, K , C
K .
2.2 Stiffness matrix of connector
Taking into account the equation (1), the potential energy is written:
2
22 2
2 2
2 2 3
3
2 3 2
3 2
3 2 3 2
3 2 3
2 1 2
1
2 1 2
. 1 2 2
. 1 2 2
1
z z z y
y y
x x x z
z z x
x y y y y
y x x x
C C
C u
u h K
u u h K
u u K W
(5)
In based on the principle of minimizing the potential energy, we obtain the stiffness
matrix of connector:
z z
x y x
x y x
y x y
x y
z z
y y
y y
x x
x x
z z
x y x
x y x
x y
y x y
z z
y y
y y
x x
x x
C C
K h h C
h K K h C
K
K h h C
K h C
K
K K
K h h K
K K
K h h K
K K
C C
K h h C
h K K h C
K
h C h K
K h C
K
K K
K h h K
K K
K h h K
K K
0 0
0 0 0 0
0 0
0 0
4 0 . 0
0 2 0
4 0 0
0 2 0
0 4 0
. 2 0
0 0 0
2 0 0
0 0
0 0
0 0 0
0 0
0
0 2 0
0 0
0 2 0
0 0
2 0 0
0 0 2 0
0 0
0
0 0
0 0 0 0
0 0
0 0
4 0 0
0 2 0
4 0 . 0
0 2 0
0 0
2 0 0
0 4 0
. 2 0
0
0 0
0 0
0 0 0
0 0
0
0 2 0
0 0
0 2 0
0 0
2 0 0
0 0 2 0
0 0
0
2 2
2
2 2
2
(6)
3 Validation of connector model in traction
3.1 Tensile test
A tensile test was carried out to characterize the behaviour of the assembly. The experimental results allowed us to identify K
x, K
z, in the analytical solution.
The mechanical properties of the structure are described in table 1, which the length of the plat is
L, width
b, thickness
ei, and Young's modulus
Ei. We propose, at first study that the material is the same for two plates ant they have the same dimension: E
1 E
2 E and e
1 e
2 e ; the assemblies of structure (plate/bolt) used are made of steel type S235 and M8x30 for bolt.
L (mm) b (mm) e (mm) E (MPa)
135 40 8 205 000
Table 1: Mechanical and geometrical properties.
Figure (5) represents this overall behaviour of a typical curve of the load applied according to displacement. We note the existence of 5 phases:
phase 1 : elastic behaviour corresponds to the kinetic friction sheet-sheet, with the frictional force F
nestimated by Coulomb's friction law F
n . F
t(where
is the coefficient of friction, F
tclamping force)
phase 2 : non-linear behaviour related to relative displacement bolt/sheet
which leads to the sliding sheet,
phase 3 : slip phase consist of three steps : loss adhesion between the head of screw and the upper sheet, then slip of the upper sheet corresponding to the clearance sheet/bolt, at last slip of the upper sheet/bolt corresponding to the clearance bolt/sheet.
phase 4 : elasto-plastic behaviour of the structure,
phase 5 : rupture of the assembly.
Figure 5: Traction force as a function of the displacement
3.2 Numerical model (two dimensional-2D)
The structure had been modelled by two thin sheets with elements DKT linked by a connector (equivalent element). The proposed connector consists of two circular areas (blue area) having the same diameter of the bolt (It is necessary to note that the point connection has not given satisfactory results in 2D model). Each pair of nodes of two circular areas is linked by the stiffness matrix:
zone connector the
at nodes of number n
unitary matrix
Stiffness I
matrix Stiffness I
I
I
in t
i t
: : :
1
Figure 6: Numerical model 2D Figure 7: Measurement points
3.3 Comparison of experimental-numerical results
These equations (2, 3, and 4) allowed us to identify stiffness matrix as following
rad m N C
m N K
m N K
y z x
/ . 10 05 . 4
/ 10 06 . 3
/ 10 56 . 3
3 6 9
