IMECE2009-10306

Proceedings of the ASME 2009 International Mechanical Engineering Congress & Exposition
IMECE2009
November 13-19, Lake Buena Vista, Florida, USA
IMECE2009-10306
NUMERICAL MODELING AND EXPERIMENTAL VALIDATION OF BOLTED LAP JOINTS WITH 1D
HYSTERESIS FINITE ELEMENTS
J. ABAD
Department of Mechanical Engineering
University of Zaragoza, 50018
Zaragoza, Spain
javabad@unizar.es
J. M. FRANCO
Department of Design and Manufacturing Engineering
University of Zaragoza, 50018
Zaragoza, Spain
jfranco@unizar.es
L. LEZAUN
Department of Mechanical Engineering
University of Zaragoza, 50018
Zaragoza, Spain
llezaun@unizar.es
F.J. MARTINEZ
Department of Mechanical Engineering
University of Zaragoza, 50018
Zaragoza, Spain
fjmargo@unizar.es
ABSTRACT
The work presented in this paper is part of a larger project
for the modeling of dynamic behavior in bolted joints, and it is
a further work on the adjustment of bolted joint 3D numerical
model
This work shows the study and conclusions of the
numerical modeling of a bolted lap joint by means of 1D
hysteresis finite element and its validation with dynamical
tests. The modeled joint is made up of two plates with a bolt,
nut and washer. The behavior curve of the hysteresis element
used was obtained by means of a 3D model of the joint, whose
parameters and validation were carried out from the results of
quasi-static laboratory tests. This procedure could be
advantageously extended to any other lap joint given that its
computational requirements are less than those required for a
detailed 3D modeling.
considering the contact among the joint elements. The only
practical way of considering the non-linear joint behavior in
the dynamic structure analysis is the use of simplified models
with freedom degrees which are consistent with the analysis
performed.
The Basic friction models used for modeling the behavior
in bolted joints are classified as phenomenological and
constitutive. Phenomenological models represent friction force
as a function of relative displacement. These models include
static friction described by signum-friction models [1], elasticslip models represented by a set of spring-slider elements in
parallel [2-6] (known as Jenkins- or Masing-element), the
LuGre [7] (Lund–Grenoble) model represented by elastic
bristles sliding over rigid bristles, and the Valanis [1,8] model,
which accommodates local microslip and macroslip in one
model. Constitutive models [9] establish relationships between
stress and displacement fields. They include joint description
by contact mechanics with statistical surface roughness
description, and fractal characterization of surface roughness
in joints.
The modeling of a bolted lap joint by means of 1D
hysteresis finite element, in which the joint force is defined
according to the relative edge displacement, is proposed in this
paper. The work is divided into two parts. In the first one a 3D
modeling of the bolted joint is made, and its numerical results
are experimentally validated by means of quasi-static trials
which have the purpose of defining the parameters of 1D
INTRODUCTION
The behavior of bolted joints in structures is responsible
for their dynamic behavior at a great extent. The joint
determines both stiffness and damping of the structure, and
thereby its response to a dynamic solicitation. The scale
difference between the global size of the structure and the
local effect of energy dissipation which is produced in the
joint, makes it practically impossible to carry out the
numerical analysis of the dynamic structure response,
1
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3D FEM MODEL OF JOINT
3D joint numerical model has been made by using
ANSYS 10 software, including the different existing contact
surfaces. The type of calculation made with numerical model
has been a non-linear analysis including three-steps. First of
all, the conditions of contour and contact are established, thus
stabilizing the model. Secondly, the load is applied to the bolt.
Thirdly, the displacement is imposed for the free edge, this
way reactions are achieved at the fixed edge. In the previous
steps, the number of sufficient sub-steps is defined in order to
both ensure the convergence of the analysis and reach a
sufficient number of intermediate results.
hysteresis finite element. In the second part a simplified
numerical model of the joint is made by means of 1D
hysteresis finite element and the results are validated by
comparing them with the experimental results of the dynamic
joint response.
DESCRIPTION OF THE JOINT ANALYZED
As shown in fig. 1, the joint analyzed in this paper
consists of the following components: two steel plates, two
washers, a nut and a bolt (M12). In fig. 1 the model contour
conditions are schematically shown: a fixed edge, and a free
edge to which a displacement is imposed.
Meshing and types of element used
The meshing used in the model is shown in fig. 3. A
convergence study has been made to determine the optimum
mesh density at the lowest computational cost. The material is
defined for all components according to bilinear elastic-plastic
behavior, with a Poisson coefficient = 0.3 and a longitudinal
elasticity module E = 206 GPa. The characteristic material
parameters for each component are shown in Tab. 1.
Bolt
Fixed displacement
Displacement
imposed
Plate
Plate
Nut
Fig. 1: Bolted joint analyzed
The dimensions of plates are shown in fig. 2, and the rest
of the components have standardized dimensions (Tab. 1).
10
15
10
Fig. 3: Meshing of FEM Model
R5
All the model components have been meshed by using the
type of element SOLID185, 3D element of 8 nodes and 3
freedom degrees per node (displacements in nodal directions
x,y,z) [10]. This element has plasticity, large deflection, and
large strain capabilities.
Contacts have been defined by means of ANSYS contact
assistant by using elements of surface-surface contact type
designated as CONTA174 and TARGE170, which allow
several friction models to be defined. Bolt preload is applied
by means of element type PRETS179, which serves to define a
pre-stressed section in previously meshed structures [11]. This
element has only one movement freedom degree according to
the direction of preload application. The generation of contacts
has been carried out in ASYS by means of its tool ‘Contact
Manager’ [12], contact-type and target-type surfaces being
defined. The contact pairs on contact surfaces established
among the joint components shown in Tab. 2 have been
defined by means of a deformable target-type surface, which
allows a relative sliding with other contact-type surfaces. The
friction model used applies Coulomb law, establishing a
difference between two types of behavior. One of them is
sticking-type and the other one is sliding-type. During the
sticking-type behavior the shear force is transmitted without a
sliding between surfaces, while not exceeding a shear force
limit value (equation 1). By exceeding this limit, sliding-type
behavior is produced between both surfaces.
R5
13
40
65
25
65
160
Fig. 2: Plates geometry (dimensions in mm)
Component
Designation
Material
Standard
Plate
Bolt
Nut
Washer
S355
DIN 631 8.8
DIN 934 6.8
DIN 126 6.8
EN-10025
DIN-ISO 898
DIN-ISO 898
DIN-ISO 898
Yielding
stress
[MPa]
355
640
480
480
Tensile
strength
[MPa]
470
800
640
640
Max.
Elongation
[%]
17
12
8
8
Tab. 1: Mechanical properties of the materials used for the different
components of the joint
2
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Target surface
Contact surface
Fixed Plate
Washer
Washer
Washer
Inner surfaces of washers
Bolt hole of plates
Free Plate
Plates
Nut
Head of Bolt
Bolt Grip
Bolt Grip
Number of
Contact Pairs
1
2
1
1
2
2
Bolt
preload (N)
20000
30000
Tab. 2: Contact Pairs definition
 lim    P  b
(1)
40000
The limit shear force value is defined according to eq. (1),
where  is friction coefficient, P is the contact pressure, and b
is the contact cohesion which provides sliding resistance when
there is no contact pressure.
50000
Numerical results
With the purpose of characterizing the bolted joint
behavior, the hysteresis cycles of reaction force vs.
displacement (see fig. 4) have been determined for several
values of pre-load and displacement amplitude. Twenty
analyses have been made with four preload values (20kN,
30kN, 40kN and 50kN) and five displacement amplitudes
(0.05mm, 0.10mm, 0.15mm, 0.20mm and 0.25mm).
The displacement has been applied according to a triangular
wave with a period of 2.5s. The parameters of both dissipated
energy per cycles and of equivalent stiffness have been
calculated for each cycle obtained, and these values are shown
in Tab. 3. The dissipated energy per cycle is determined by
calculating the area within the hysteresis cycle and the
equivalent stiffness as the quotient between the maximum
force reached in the hysteresis cycle and the displacement it is
produced at.
20
Disipated energy
per cycle (Nm)
0.124
0.878
1.753
2.714
4.275
0.061
0.802
1.936
3.204
4.762
0.038
0.596
2.007
3.546
5.285
0.026
0.398
1.989
3.817
5.770
Stifness (N/m)
9.538E+07
6.168E+07
4.934E+07
4.105E+07
3.332E+07
1.043E+08
8.016E+07
6.319E+07
5.422E+07
4.777E+07
1.103E+08
9.765E+07
7.550E+07
6.435E+07
5.703E+07
1.158E+08
1.106E+08
8.668E+07
7.293E+07
6.432E+07
Tab. 3: Characteristic parameters of bolted joint obtained with
numerical model.
E
E
Bolt preload = 50000N
(a)
15
Reaction force (kN )
Displacement
imposed (mm)
 0.05
 0.10
 0.15
 0.20
 0.25
 0.05
 0.10
 0.15
 0.20
 0.25
 0.05
 0.10
 0.15
 0.20
 0.25
 0.05
 0.10
 0.15
 0.20
 0.25
10
5
0
-5
-10
K
-15
-20
-0.3
K
-0.2
-0.1
0
0.1
0.2
0.3
Displacement imposed (mm)
Fig. 4: Hysteresis cycles obtained with ANSYS
The results obtained have allowed the response surface
(RSM) to be defined by making an adjustment per minima
squared. In fig. 5 the RSM obtained by means of OPTIMUS
software [13] for the parameters of stiffness and dissipated
energy per cycle are shown according to the preload applied to
the bolt (F) and the displacement imposed (X).
(b)
Fig. 5: RSM for the characteristic joint parameters: (a) E: Dissipated
energy, (b) K: Stiffness
(X: displacement) (F: bolt preload)
3
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Model Validation
The 3D model used has been experimentally validated by
reproducing the analysis made with ANSYS. The experiments
were carried out in a universal testing machine, 8032 model of
INSTRON (fig. 6.a) with dynamic control, hydraulic grip jaws
and a maximum force of 100kN. The machine control
software records the force applied and the moving grip jaw
displacement. The application of tightening torque on bolt was
made by means of a dynamometric spanner of 30-160 Nm
range. The preload force on the bolt was measured with an
ALD-W-200 loading cell model of A.L. Trademark of Design
Inc., 0-10000 daN range and 2.14 mV/V sensitivity, connected
to a portable bridge of Wheaststone model P3 of Vishay
Micro-Measurements Trademark (fig. 6.b)
15
Bolt preload = 35981N
Reaction force (kN)
10
5
0
-5
-10
-15
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Displacement imposed (mm)
Fig. 7: Hysteresis cycles experimentally obtained
The purpose of these tests is to verify the results obtained
by means of FEM analysis using the RSM previously
determined. These results are shown in Tab. 4.
1D FEM MODEL OF JOINT
With the purpose of determining the dynamic response of
the joint numerically, this one has been simplified to the
model in fig.8. The reduced model size will allow a high
number of its temporal response items to be obtained at a low
computational cost.
1D hysteresis finite element
The joint model has been implemented by means of
ANSYS 10, using two types of elements in the model. The
former MASS21 type element (1 node element), has been used
to define lumped masses: m1 y m2. The latter, COMBIN39
type element, has been used to define the non-linear and
hysteretic behavior of the joint. This COMBIN39 type element
requires the definition of the force-displacement curve by the
input of discrete points of force versus displacement.
Unloading along the line parallel to the slope at the origin of
the curve allows modeling of hysteretic effects [14] as
illustrated in fig. 9.
(b)
(a)
Fig. 6: (a) Assembly in testing machine, (b) Instrumentation
The machine control has been performed in
displacements, carrying a total of 12 tests. Four imposed
displacement amplitudes (0.10mm, 0.14mm, 0.18mm and
0.22mm) and three preloads in the bolt (26928N, 35981N
and 42016N) have been defined according to a triangular wave
with a period of 2.5s. From the hysteresis cycles obtained
(fig.7) the dissipated energy parameters per cycle and
equivalent stiffness has been determined as it is made in
numerical model.
Bolt preload (N)
26928
35981
42016
Displacement imposed (mm)
 0.098
 0.138
 0.179
 0.219
 0.098
 0.137
 0.178
 0.217
 0.099
 0.137
 0.179
 0.217
Dissipated energy per cycle (Nm)
Test
RSM
Error %
prediction
0.953
0.826
-13.3
1.857
1.655
-10.9
3.075
2.570
-16.4
4.319
3.617
-16.3
0.702
0.619
-11.8
1.740
1.620
-6.9
2.981
2.744
-8.0
4.538
3.963
-12.7
0.496
0.525
5.8
1.618
1.615
-0.1
3.018
2.890
-4.2
4.607
4.204
-8.7
Stiffness (N/m)
Test
7.321E+07
6.113E+07
5.284E+07
4.565E+07
9.571E+07
7.922E+07
6.723E+07
5.843E+07
1.078E+08
8.758E+07
7.464E+07
6.578E+07
RSM
prediction
7.504E+07
6.116E+07
5.318E+07
4.788E+07
9.224E+07
7.508E+07
6.470E+07
5.797E+07
1.008E+08
8.245E+07
7.070E+07
6.335E+07
Error %
2.5
0.0
0.7
4.9
-3.6
-5.2
-3.8
-0.8
-6.5
-5.9
-5.3
-3.7
Tab. 4: Comparison of experimental and RSM prediction of dissipated energy per cycle and stiffness
4
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Dynamic test of the joint
In order to validate the results of the simplified model, a
dynamic test of the bolted joint has been carried out, and fig.
11 illustrates the test setup. A 1 kg mass has been fixed on one
edge of the joint, whereas a shaker applies a vibratory
movement on the base of the other edge. The shaker control
generates an electric signal which feeds the shaker, using the
control accelerometer to adjust the amplitude and form of
vibratory signal wanted to be applied to the joint. The
acceleration of base (a1) and in mass (a2) is measured by
means of piezoelectric accelerometers. Preload force is
measured using the load cell previously indicated. Figure 12 is
a photograph of the test setup.
The bolted joint has been subjected to a sinusoidal
vibration for each applied preload, (353N, 873N, 1344N,
2080N, 3571N, 8554N) making a sweep around the frequency
corresponding to longitudinal vibration mode. The joint has
been subjected to a random type vibration in order to
determine its resonant frequency with each preload. In fig. 13
transmissibility functions between acceleration signals: a2 and
a1, are shown, which indicate the variation of this frequency
with preload.
a2(t)
MASS21
m2
a2(t)
Node 2
K non-lineal
COMBIN39
a1(t)
a1(t)
Node 1
m1
MASS21
(a)
(b)
Fig. 8: (a) Simplified diagram of bolted joint, (b) FEM 1D model
Force
Lumped mass accelerometer
(Mod. 4506B B&K)
Lumped mass
Bolt preload
Measurement
Displacement
Front-end:
3560C B&K
Control accelerometer
(Mod. 4507B B&K)
Base accelerometer
(Mod. 4517 B&K)
Software:
PULSE 9.0
ME´Scope 4.0
Shaker Control
LDS-Dactron
Fig. 9: Hysteretic behavior of COMBIN39 element
Amplifier
PA100E
The curve of hysteretic joint behavior depends on the applied
preload. These curves have been determined from the previous
3D model of the bolted joint, defining preload and applied
displacement. Fig. 10 shows the six force-displacement curves
which will define COMBIN39 type element.
Shaker
LDS V406
Fig. 11: Illustration of the test setup
200
175
Force [N]
150
125
100
75
50
25
1.6E-06
1.4E-06
1.2E-06
1.0E-06
8.0E-07
6.0E-07
4.0E-07
2.0E-07
0.0E+00
0
Displacement [m]
Fig. 10: Force-displacement curve obtained with 3D FEM model for
several preloads on the joint bolt
Fig. 12: Photograph of the test setup
5
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1.E+03
1.E+03
Bolt preload (N)
3571N
2080N
353
1.E+02
8554N
Transisibility (a2/a1)
Transmissibility (a2/a1)
1344N
873N
353N
1.E+01
1.E+00
1000
1100
1200
1300
1400
1500
873
1.E+02
1344
2080
3571
8554
1.E+01
1.E+00
1200
1600
1225
1250
1275
1300
1325
1350
1375
1400
Frequency (Hz)
Frequency (Hz)
Fig. 13: Transmissibility functions for several preloads on the joint
bolt
Fig. 15: Transmissibility functions for several preloads on the joint
bolt
Numerical results
The dynamic tests the bolted joints have been subjected to
have been reproduced in ANSYS, defining the characteristic
curve in COMBIN39 element for each preload in the bolt. A
vibration of variable frequency, between 1000Hz and 1500Hz,
and 1m/s2 amplitude has been defined in the node 1 (base),
and the response has been registered in the node 2 (lumped
mass). A transient analysis has been made in one second
defining a time step size of 0.02ms (fig 14).
Model validation
In fig. 16 and fig. 17 numerical and experimental results
corresponding to dynamic joint response are compared. The
parameters used for validation are resonant frequency (fig 16)
and damping ratio (fig 17).
Resonant frequency (Hz)
1360
Bolt preload (N)
200
8554
150
3571
2080
Response a2 (m/s2)
100
1344
873
50
353
1340
1320
1300
1280
1260
Test
1240
1D FEM
1220
1200
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Bolt preload (N)
-50
Fig. 16: Comparison of experimental and 1D FEM model results of
resonant frequency
-100
-150
1.2
-200
0.2
0.4
0.6
0.8
1
Damping ratio  (%)
0
Time (s)
Fig. 14: Time response of acceleration along the sweep sine for
several preloads on the joint bolt
The results obtained in the dynamic calculation with
ANSYS have been exported for their treatment in ME´Scope
software [15]. Transmissibility functions have been
determined with the registered data of acceleration in the two
nodes of joint 1D numerical model by means of the software
just mentioned
1
0.8
Test
1D FEM
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Bolt preload (N)
Fig. 17: Comparison of experimental and 1D FEM model results of
damping ratio
6
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A good correlation is observed between numerical and
experimental results.
1350
Resonant frequency (Hz)
1330
Extend results
Once analyzed 1D model, the study is extended by
applying different excitation levels to the joint: a1 = 0.5m/s2,
1m/s2, 2m/s2, 4m/s2 and 6m/s2 (fig 18), determining the values
of resonant frequency and damping ratio. Figure 19.a shows
frequency variation, where a stiffness increase with preload
and a decrease with the amplitude of the exciting vibration are
observed. Figure 19.b shows how damping increases with
vibration amplitude, and how it decreases with bolt preload.
400
353
1270
873
1250
1344
1230
2080
3571
1210
8554
1190
0
1
2
3
4
5
6
7
2
Excitation a1 (m/s )
(a)
6
4
5
2
200
4.5
1
100
4
0.5
Damping ratio  (%)
2
Bolt preload (N)
1290
Excitation a1 (m/s2)
Bolt preload: 2080N
300
Response a2 (m/s )
1310
0
-100
-200
-300
3.5
Bolt preload (N)
3
353
2.5
873
2
1344
1.5
2080
1
3571
0.5
8554
0
-400
0.0
0.2
0.4
0.6
0.8
0
1.0
1
2
3
4
5
6
7
2
Excitation a1 (m/s )
Time (s)
(b)
Fig. 18: Time response of acceleration a2 along the sweep sine for
several excitation levels.
Fig. 19: Influence of preload and vibration amplitude on joint
behavior; (a) Resonant frequency, (b) Damping ratio
CONCLUSIONS
In this work the use of a 1D hysteresis finite element has
been presented for the numerical modeling of a bolted lap joint
behavior. The results of numerical simulations of joint 3D
model have been used to define 1D element. The model has
been validated by means of dynamic joint tests, thus obtaining
a good correlation in results.
The analysis of vibratory response of the bolted joint has
allowed verifying its non-linear behavior, as well as the
influence that preload and vibration amplitude have on it. The
inclusion of 1D hysteresis finite element in the structure model
will allow dynamic analysis to be carried out considering the
non-linear behavior of the joint with a reasonable
computational cost.
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ACKNOWLEDGMENTS
This work has been funded with project BIA200615266-C02-02 of the National R & D 2004-07, Ministry of
Science and Innovation of the Government of Spain, and cofinanced with FEDER funds.
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Element
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Contact
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