a simplified finite element riveted lap joint model in structural dynamic analysis

7/21/2019 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],

[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

186 Mechanical and Aerospace Engineering V



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



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.


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




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




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.


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




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

[1]. D.V. T. G. Pavan Kumar, S. Sathiya Naarayan, S. Kalyana Sundaram, S. Chandra: Furthernumerical and experimental failure studies on single and multi-row riveted lap joints,

Enginnering Failure Analysis Vol. 20 (2012), p. 9-24.

[2]. S. Sathiya Naarayan, D.V. T. G. Pavan Kumar, S. Chandra: Implication of unequal rivet load

distribution in the failures and damage tolerant design of metal and composite civil aircraft

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[3]. W. Huang, T-J. Huang, Y. Garbatov, C. Guedes Soares: Fatigue reliability assessment of riveted

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[5]. S. Xu, X. Deng: An evaluation of simplified finite element models for spot-welded joints, Finite

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[6]. M. I. Friswell, J. E.Mottershead, in: Finite Element Model Updating in Structural Dynamics,

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[7]. M. Dourado, J. Meireles, A. M. A. C. Rocha: Structural Dynamic Updating Using a Global

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a simplified finite element riveted lap joint model in structural dynamic analysis

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