a simplified finite element riveted lap joint model in structural dynamic analysis
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A Simplified Finite Element Riveted Lap Joint Model in Structural
Dynamic Analysis
Marco Daniel Malheiro Dourado1, a
, José Filipe Bizarro de Meireles1,b
1
Mechanical Engineering Department, Azurém Campus, 4800-058 Guimarães, [email protected],
Keywords: Riveted Lap Joint, Finite Element Model Updating, Modal Analysis, Optimization
Abstract. This paper proposes a simplified finite element model to represent a riveted lap joint in
structural dynamic analysis field. The rivet is modeled by spring-damper elements. Several numerical
models are studied with different quantities of rivets (1, 3 and 5) and spring-damper elements (4, 6, 8,
12, 16 and 20) per rivet. In parallel, samples of two aluminum material plates connected by different
quantities of rivets (1, 3 and 5) are built and tested in order to be known its modal characteristics –
natural frequencies and mode shapes. The purpose of the different settings is to get the best numericalriveted lap joint representation relatively to the experimental one. For this purpose a finite element
model updating methodology is used. An evaluation of the best numerical riveted lap joint is carried
out based on comparisons between the numerical model after updating and the experimental one. It is
shown that the riveted lap joints composed by eight and twelve spring-damper elements per rivet have
the best representation. A stiffness constant value k is obtained for the riveted lap joints in study.
Introduction
Riveted lap joints have wide application in various industrial sectors, mainly in the automotive and
aerospace industry. Such structures have thousands of riveted connections. It is not practical to model
the detailed rivet for numerical structural analysis. Furthermore, this implies a large number ofdegrees of freedom, increasing the computational cost analysis. Some works describing the fatigue
and static behaviour of riveted lap joints, use a simplified modeling for the joints [1, 2, 3, 4]. All of
them use spring elements or beam elements connecting two nodes to simulate the rivet, but the
stiffness constant value of the joint is not clearly explicit. To increase the accuracy of the numerical
riveted lap joint, the stiffness constant parameter must be defined. This modal parameter has large
influence in the resonance frequencies and mode shapes of the structure.
The work [5] describes an evaluation of simplified finite element models for spot-welded joints. In
this work four types of simplified models to simulate the spot-welded are evaluated by the authors: the
multiple rigid bar (MRB) model, the rigid bar-rigid shell (RB-RSH) model, the solid nugget (SN)
model and the rigid bar (RB) model. One of these simplified models is interesting for application inthis work, in particular the MRB model. In the MRB model, two sheets are connected with multiple
rigid bar elements to simulate the spot-welded. The other models are not interest for this work. The
RB–RSH and RB models with only one rigid bar do not provide sufficient stiffness to simulate the
dynamic behaviour of the riveted lap joint model. The SN model is composed by multiple solid
elements and therefore more complex and time consuming.
To determine the spring stiffness constant value of the rivets, an updating technique for improving
finite element models is used [6].
Experimental modal analysis is carried out in order to know the dynamic behaviour of the riveted
lap joint models, and will be described in section 3. A numerical riveted lap joint models are built in
ANSYS code, as described in section 4. An initial stiffness constant value is assigned to the
spring-damper elements. The damping is negligible and is not considered in the spring-damper element. With the updating software, described in [7], the spring stiffness constant value is estimated,
and the results are presented in section 5. The best numerical riveted lap joint representation of the
physical one is obtained, as concluded in section 6.
Advanced Materials Research Vol. 1016 (2014) pp 185-191© (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.1016.185
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Theoretical Problem
The finite element model updating methodology use an optimization technique to find the optimal
value of the stiffness constant k parameter of the spring-damper elements. The optimization problem
consists in the minimization of an objective function defined by a sum of three specific functions as
described below,
( ) ( ) ( ) ( )k k k k λ ϕ ϕ f f f f U C ++= (1)
The C f ϕ function represents the quantification of the difference between numerical and reference
correlated mode pairs. C N is the number of correlated mode pairs values, of the diagonal MAC
matrix, to sum. C f ϕ is given by,
( ) ( )
( )∑∑
=
=−=
C
C
N
i ii
N
i ii
C
MAC
MAC f
1
0
1
k
k k ϕ
(2)
where,
( )( )( )( )( )( )Num
j
T Num
j
Ref
i
T Ref
i
Num
j
T Ref
i
ij MAC ϕ ϕ ϕ ϕ
ϕ ϕ 2
= (3)
where, Ref
i ϕ is the th i reference mode shape andNum
j ϕ is the th j numerical mode shape [8].
The U f ϕ function represents the quantification of the difference between numerical and reference
uncorrelated mode pairs. U N is the number of uncorrelated mode pairs values, outside of the diagonal
MAC matrix, to sum. U f ϕ is given by,
( )( )
( )∑ ∑∑ ∑
=≠
=
=≠
=
=
U U
U U
N
j
N
j i ij
N
j
N
j i ij
U
U
MAC
MAC
N f
111
0
111
1
k
k k ϕ
(4)
The λ f function represents the quantification of the difference between numerical and reference
frequencies. λ N is the number of eigenvalues λ corresponding to the correlated mode pairs. λ f is
given by,
( )( )( )
( )( )∑
∑
=
=
=
=
−
−
=λ
λ
ω ω
ω ω
λ N
j
i j i
N
j i j i
f
1
1
20
11
2
k
k
k (5)
where,
π λ ω
2
Ref
i = (6)
is the reference frequency and,
π λ ω
2
Num
j = (7)
is the numerical frequency. Ref λ is the reference eigenvalue and Num λ is the numerical eigenvalue.
Exponent 2 is used to accelerate the Eq. 5 convergence and to obtain only positive differences between the frequencies of the two models. The denominator is used to obtain the normalized
difference. k is the vector with the updating stiffness constant parameters used in the numerical
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model updating. Numerical mode shapes NUM ϕ and numerical eigenvalues NUM λ are function of
these updating parameters, and can be expressed as,
( ) ( ) p
NUM NUM k k k k f ,...,,,, 321=λ ϕ (8)
where, p is the number of updating stiffness constant parameters. 0k is the vector with the initial
updating stiffness constant parameters. Updating parameters k are subject to lower and upper bounds
inequality constraints defined as,
UBLB k k k ≤≤ (9)
The best updated values of the stiffness constant parameters k are obtained when objective
function f is minimized. It is means that the modes are correlated. However, the minimal objective
function value is different for all studied riveted joint sets, and therefore can not be considered as
direct reference to evaluate the best riveted joint set. The minimal objective function value only
indicates that the optimal value of the stiffness constant parameters k was found. Then, the numerical
riveted lap joint set evaluation is made by the average difference defined as,
( )100DifferenceAverage
11
×
−
=∑ =
=
i
N
j i
final j i
N ω
ω ω
λ
λ
(10)
where, final
j ω is the numerical final frequency obtained after updating. Multiply by 100 to obtain the
average percentage difference.
Experimental Procedure
The samples for experimental modal analysis consisted of two connected aluminum material plates, with
2 mm of tickness, as shown in Figure 1a. The plates are connected by aluminum material rivets, with
3 mm diameter. Three samples, with riveted joint of one, three and five rivets, are tested. The tests are performed at room temperature, about 20 ºC, using LMS SCADAS equipment for experimental
modal analysis. The samples are tested in free-free boundary conditions, suspending them in two
points by a nylon yarn of sufficient length (350 mm) so as not to cause interference in the test, as
shown in Figure 1b. The tests are performed using an impact hammer to input the impact force in
point P1, and the response measured with laser Doppler interferometer in eight points, P1 to P8, as
shown in Figure 1c.
Fig. 1 (a) Sample schematic representation with joint of five rivets; (b) sample subject to
experimental modal analysis; (c) location of the measured points.
Advanced Materials Research Vol. 1016 187
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The selected eight points are the minimum to represent the first eight mode shapes of the sample.
The data is collected in the time domain (amplitude vs. time) and processed in the LMS modal
analysis software to convert to the Frequency Response Function (FRF) domain. The resonance
frequencies and amplitudes at each point are obtained from the adjusted FRF curve.
Numerical Models
Numerical models are built using the commercial finite element ANSYS code, with same geometric
and mechanical properties of the experimental models, as presented in Table 1.
Table 1: Mechanical properties.
Property Symbol Units Value
Young Modulus E x [Pa] 66.870x109
Young Modulus E [Pa] 71.030x109
Poisson Ratio υ xy - 0.31
Density ρ [kg/m3] 2707
The plates are modeled with shell (shell 63) elements. The rivet is built with spring-damper (combination 14) elements, which can combine the stiffness constant and damping coefficient. It is
considered negligible the damping coefficient. Instead the stiffness is variable. The rivet mass is
punctual and divided by the nodes that interconnect the two plates through of the combination
elements, but not variable. The rivet mass is modeled with mass (mass 21) elements. The initial
numerical rivet properties are presented in Table 2.
Table 2: Initial numerical rivet properties.
Property Symbol Units Value
Stiffness Constant per combination element k [N/m] 5x107
Rivet Mass m [kg] 2.5x10-4
Figure 2a shows an example of a numerical rivet representation, composed by eight combination
elements and eight mass elements per plate. Figure 2b shows an example of a numerical model of two
plates connected by five rivets.
Fig. 2 (a) Example of a numerical rivet representation and (b) example of a numerical riveted joint
with five rivets.A finite element model updating methodology is used to find the optimal value of the stiffness
constant parameter k of the combination elements, and consequently of the numerical riveted lap
joint. It is expected that this optimal value can vary according to the quantity of rivets and
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combination elements per rivet. For this purpose eighteen different settings of numerical riveted lap
joints are analyzed: riveted lap joints with three different quantities of rivets (1, 3 and 5), vs. six
different quantities of combination elements (4, 6, 8, 12, 16 and 20) per rivet.
Results and Discussion
The results of studied riveted lap joint sets are presented in this section. The graphs of the Figure 3
show the relationship between the average percentage difference, obtained by application of Eq. 10,
after updating between resonance frequencies of the numerical and experimental model, and the
quantity of combination elements. The riveted lap joints with eight combination elements per rivet are
the best numerical representation relatively to the experimental model. The average percentage
difference after updating is lower relatively to the other quantities of combination elements. Rivets
with twelve combination elements also present the same behaviour for the models with three (Figure
3b) and five (Figure 3c) rivets. However, its higher complexity relatively to the rivet with eight
combination elements, does not justify its implementation. The graphs are shown also that the
increase of rivets in the joint reduce the average percentage difference. This means that it may be
possible to reduce the quantities of combination elements per rivet.
Fig. 3 Percentage difference after updating between numerical and experimental model with revited
joint of (a) one rivet; (b) trhee rivets and (c) five rivets.
Tables 3, 4 and 5 show the ressonance frequencies and MAC values, before and after updating, for
the models with one, three and five rivets, respectevely. Only shown the results for the models with
eight combination elements per rivet.
Table 3: Frequencies and mode shapes evolution for the numerical model with one rivet and eightcombination elements.
ModeExp. Freq.
[Hz]
Num. initial
Freq.
[Hz]
Difference
before
Updating
[%]
Num. final
Freq.
[Hz]
Difference
after
Updating
[%]
Initial
MAC
Final
MAC
1 101.90 104.44 2.49 102.31 0.41 1 1
2 325.37 322.34 0.93 322.32 0.94 1 1
3 444.19 389.43 12.33 380.11 14.43 1 1
4 564.80 572.16 1.30 564.12 0.12 1 1
5 856.42 852.69 0.44 852.64 0.44 1 1
6 1057.65
1039.53
1.71
1039.34
1.73
1 1
7 1363.77 1210.99 11.20 1182.92 13.26 1 1
8 1402.80 1420.80 1.28 1402.80 0.00 1 1
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Table 4: Frequencies and mode shapes evolution for the numerical model with three rivets and eight
combination elements.
ModeExp. Freq.
[Hz]
Num. initial
Freq.
[Hz]
Difference
before
Updating
[%]
Num. final
Freq.
[Hz]
Difference
after
Updating
[%]
Initial
MAC
Final
MAC
1 112.13 112.26 0.12 112.13 0.00 1 1
2 325.71 325.12 0.18 325.12 0.18 1 1
3 448.72 446.75 0.44 446.73 0.44 1 1
4 606.86 606.20 0.11 605.60 0.21 1 1
5 851.80 849.10 0.32 849.06 0.32 1 1
6 1058.86 1061.24 0.23 1061.23 0.22 1 1
7 1389.75 1400.94 0.81 1400.90 0.80 1 1
8 1508.88 1511.15 0.15 1509.50 0.04 1 1
Table 5: Frequencies and mode shapes evolution for the numerical model with five rivets and eight
combination elements.
ModeExp. Freq.
[Hz]
Num. initial
Freq.
[Hz]
Difference
before
Updating
[%]
Num. final
Freq.
[Hz]
Difference
after
Updating
[%]
Initial
MAC
Final
MAC
1 112.75 112.78 0.03 112.75 0.00 1 1
2 325.68 325.13 0.17 325.12 0.17 1 1
3 449.07 446.76 0.51 446.72 0.52 1 1
4 607.59 607.85 0.04 607.67 0.01 1 1
5 849.17 846.98 0.26 846.92 0.26 1 1
6 1058.59 1061.31 0.26 1061.30 0.26 1 1
7 1390.99 1400.98 0.72 1400.88 0.71 1 1
8 1512.36 1517.62 0.35 1517.06 0.31 1 1
The average percentage difference, obtained by aplication of Eq. 10, is: 3.92% for the model with
one rivet (Figure 3a), 0.28% for the model with three rivets (Figure 3b) and 0.28% for the model with
five rivets (Figure 3c). By other hand, the value 1 for the initial and final MAC, reveals, for all cases,
that the modes are correlated. This fact indicates that the obtained stiffness constant values, presented
in Table 6, for the combination elements are obtained with high reliability and accuracy.
Table 6: Final stiffness constant values for the rivets with eight combination elements.
Property Symbol Units Value
1 rivet 3 rivets 5 rivets
Stiffness Constant per combination element k [N/m] 2.2x107 27.0x107 40.5x107
Stiffness Constant per rivet 17.3x107 215.6x107 324.2x107
Stiffness Constant of the joint 646.8 x107 1621 x10
7
The presented results justify the choice by the numerical riveted joint model with eight
combination elements per rivet.
Conclusions
In this paper, simplified numerical riveted lap joint models are evaluated. The evaluation is based on
the comparison between the modal characteristics of the numerical and experimental model. Three
specimens of experimental riveted lap joint models are subject to experimental modal analysis to
collect data. Eighteen numerical riveted lap joint models are built in ANSYS code. The rivet is
modeled with spring-damper (combination 14) elements. Only the stiffness constant k of the spring is
updated in the finite element model updating software. Different quantities of combination elements
per rivet are evaluated. Based on the results after updating, conclusions can be made. The riveted
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joints with eight combination elements per rivet have the lower dynamic behaviour difference
between numerical and experimental model. For the joints with three and five rivets, the set with
twelve combination elements per rivet have the same behaviour that the rivet with eight combination
elements. But its greater complexity does not justify its implementation. The results of the evaluations
allow conclude that, the stiffness constant values are obtained with good accuracy (Table 6), and the
numerical model with eight combination elements is reliably to represent the experimental riveted joint model with one, three and five rivets. With this study, we make an important contribution to the
designers in modeling riveted lap joints for structural finite element analysis. For the future it is
important understand how evolves the average percentage difference with the increase of rivets in the
lap joint. It may be possible to reduce the quantity of spring-damper elements per rivet in the
numerical model representation. Finally an equation can be developed in order to calculate the value
of the stiffness constant k of the combination element, according with the characteristics of the
physical riveted lap joint.
References
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