Advanced Materials Research Vol.1016 (2014) pp 185-191 © (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.1016.185 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, Portugal a b
[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 numerical riveted 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 of degrees 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 in this 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. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 193.137.16.117-27/08/14,12:54:53) f (k ) = f ϕC (k ) + f ϕU (k ) + f λ (k ) (1) The f ϕC function represents the quantification of the difference between numerical and reference correlated mode pairs. k is the vector with the updating stiffness constant parameters used in the numerical . The f ϕU function represents the quantification of the difference between numerical and reference uncorrelated mode pairs. f λ is given by. f ϕU is given by. f ϕC is given by. N U is the number of uncorrelated mode pairs values. ∑ f ϕC (k ) = − Ni ∑i NC =1 MAC ii (k ) (2) MAC ii (k 0 ) =1 C where. λRef is the reference eigenvalue and λNum is the numerical eigenvalue. N ∑ij (ωi − ω j (k )) 2 λ f λ (k ) = =1 =1 ∑ (ω Nλ i =1 j =1 i ( )) − ωj k 0 2 (5) where. to sum. Ref ω i = λ 2π (6) is the reference frequency and. outside of the diagonal MAC matrix. Num ω j = λ 2π (7) is the numerical frequency. N C is the number of correlated mode pairs values. The optimization problem consists in the minimization of an objective function defined by a sum of three specific functions as described below. The denominator is used to obtain the normalized difference. of the diagonal MAC matrix. MAC ij = ((ϕ ((ϕ )ϕ Ref T i )ϕ ) )( ϕ ) ϕ Ref T i Ref i Num j 2 Num T j Num j ) (3) where. 5 convergence and to obtain only positive differences between the frequencies of the two models. N λ is the number of eigenvalues λ corresponding to the correlated mode pairs. Exponent 2 is used to accelerate the Eq. ϕ iRef is the i th reference mode shape and ϕ jNum is the j th numerical mode shape [8].186 Mechanical and Aerospace Engineering V 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. to sum. ∑j ∑ij NU NU =1 MAC ij (k ) 1 ≠1 N f ϕU (k ) = NU U 0 N U ∑j =1 ∑i =1 MAC ij k =1 ( ) (4) j ≠1 The f λ function represents the quantification of the difference between numerical and reference frequencies. using LMS SCADAS equipment for experimental modal analysis.. (b) sample subject to experimental modal analysis. are tested. λNUM = f (k 1 . and can be expressed as. . as shown in Figure 1b. N ∑ij (ωi − ω jfinal ) λ Average Difference = =1 =1 N λ ωi × 100 (10) where.Advanced Materials Research Vol. k 3 . The plates are connected by aluminum material rivets. three and five rivets. The tests are performed at room temperature. Updating parameters k are subject to lower and upper bounds inequality constraints defined as. about 20 ºC. ω jfinal is the numerical final frequency obtained after updating. k LB ≤ k ≤ k UB (9) The best updated values of the stiffness constant parameters k are obtained when objective function f is minimized. Then. It is means that the modes are correlated. Multiply by 100 to obtain the average percentage difference. (ϕ NUM ) . (c) location of the measured points. 1 (a) Sample schematic representation with joint of five rivets. the numerical riveted lap joint set evaluation is made by the average difference defined as. as shown in Figure 1c. the minimal objective function value is different for all studied riveted joint sets. P1 to P8. k p ) (8) where. suspending them in two points by a nylon yarn of sufficient length (350 mm) so as not to cause interference in the test. 1016 187 model updating. p is the number of updating stiffness constant parameters. with 3 mm diameter. with riveted joint of one. However.. Fig. The samples are tested in free-free boundary conditions. Experimental Procedure The samples for experimental modal analysis consisted of two connected aluminum material plates. Three samples. The tests are performed using an impact hammer to input the impact force in point P1. and therefore can not be considered as direct reference to evaluate the best riveted joint set. k 0 is the vector with the initial updating stiffness constant parameters.. Numerical mode shapes ϕ NUM and numerical eigenvalues λNUM are function of these updating parameters.. with 2 mm of tickness. as shown in Figure 1a. k 2 . The minimal objective function value only indicates that the optimal value of the stiffness constant parameters k was found. and the response measured with laser Doppler interferometer in eight points. 5x10-4 Figure 2a shows an example of a numerical rivet representation. Fig. A finite element model updating methodology is used to find the optimal value of the stiffness constant parameter k of the combination elements. The resonance frequencies and amplitudes at each point are obtained from the adjusted FRF curve. The rivet is built with spring-damper (combination 14) elements. with same geometric and mechanical properties of the experimental models.870x109 71. but not variable. Figure 2b shows an example of a numerical model of two plates connected by five rivets. as presented in Table 1.030x109 0. and consequently of the numerical riveted lap joint. It is considered negligible the damping coefficient. Property Young Modulus Young Modulus Poisson Ratio Density Symbol Ex Ey υxy ρ Units [Pa] [Pa] [kg/m3] Value 66. Instead the stiffness is variable. Numerical Models Numerical models are built using the commercial finite element ANSYS code. composed by eight combination elements and eight mass elements per plate. The data is collected in the time domain (amplitude vs. 2 (a) Example of a numerical rivet representation and (b) example of a numerical riveted joint with five rivets.188 Mechanical and Aerospace Engineering V The selected eight points are the minimum to represent the first eight mode shapes of the sample. Property Stiffness Constant per combination element Rivet Mass Symbol k m Units [N/m] [kg] Value 5x107 2.31 2707 The plates are modeled with shell (shell 63) elements. It is expected that this optimal value can vary according to the quantity of rivets and . The initial numerical rivet properties are presented in Table 2. The rivet mass is punctual and divided by the nodes that interconnect the two plates through of the combination elements. Table 1: Mechanical properties. The rivet mass is modeled with mass (mass 21) elements. Table 2: Initial numerical rivet properties. which can combine the stiffness constant and damping coefficient. time) and processed in the LMS modal analysis software to convert to the Frequency Response Function (FRF) domain. 44 322. Freq. 6. six different quantities of combination elements (4. 8. [Hz] Num.71 11. final Freq.43 572.16 852.69 1039. 4 and 5 show the ressonance frequencies and MAC values.49 0. initial Freq. after updating between resonance frequencies of the numerical and experimental model. Tables 3.77 1402. Fig. 10. Results and Discussion The results of studied riveted lap joint sets are presented in this section.99 1420.80 104.53 1210. For this purpose eighteen different settings of numerical riveted lap joints are analyzed: riveted lap joints with three different quantities of rivets (1.11 564. The graphs of the Figure 3 show the relationship between the average percentage difference.92 1402.34 1182.31 322. [Hz] 102.80 856. The average percentage difference after updating is lower relatively to the other quantities of combination elements.19 564. its higher complexity relatively to the rivet with eight combination elements.94 14. Mode Exp. Table 3: Frequencies and mode shapes evolution for the numerical model with one rivet and eight combination elements. [Hz] 1 2 3 4 5 6 7 8 101.41 0.44 1.12 852.93 12.Advanced Materials Research Vol.20 1. Only shown the results for the models with eight combination elements per rivet. obtained by application of Eq. vs. The riveted lap joints with eight combination elements per rivet are the best numerical representation relatively to the experimental model. 3 Percentage difference after updating between numerical and experimental model with revited joint of (a) one rivet.00 Initial MAC Final MAC 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .34 389.73 13. 16 and 20) per rivet. Rivets with twelve combination elements also present the same behaviour for the models with three (Figure 3b) and five (Figure 3c) rivets.80 Difference after Updating [%] 0. (b) trhee rivets and (c) five rivets.37 444.32 380. The graphs are shown also that the increase of rivets in the joint reduce the average percentage difference. does not justify its implementation.65 1363. 3 and 5).33 1.42 1057.90 325. 12. 1016 189 combination elements per rivet. However. and the quantity of combination elements. three and five rivets.28 Num. respectevely.44 1.80 Difference before Updating [%] 2.26 0. before and after updating. for the models with one.43 0. This means that it may be possible to reduce the quantities of combination elements per rivet.30 0.64 1039.12 0. The evaluation is based on the comparison between the modal characteristics of the numerical and experimental model.5x107 324. presented in Table 6.80 0.99 1512.44 0. [Hz] Num.12 446.72 606.26 325.26 0.60 849.31 1400.17 0.67 846.6x107 646.24 1400. for all cases.01 0. reveals. Only the stiffness constant k of the spring is updated in the finite element model updating software.31 Initial MAC Final MAC 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The average percentage difference.92 1061.35 Num.26 0.88 112.50 Difference after Updating [%] 0.18 0. [Hz] 112.18 0.59 1390.44 0.90 1509.12 446.10 1061. initial Freq.22 0.23 0.81 0.75 325. Different quantities of combination elements per rivet are evaluated.94 1511.71 0. simplified numerical riveted lap joint models are evaluated.15 Num.86 1389. Three specimens of experimental riveted lap joint models are subject to experimental modal analysis to collect data.11 0.75 325. Mode Exp. [Hz] 1 2 3 4 5 6 7 8 112. Property Stiffness Constant per combination element Stiffness Constant per rivet Stiffness Constant of the joint Symbol k Units [N/m] 1 rivet 2. the value 1 for the initial and final MAC. Eighteen numerical riveted lap joint models are built in ANSYS code.13 325.72 0. is: 3.73 605.80 1058. final Freq.72 607. Based on the results after updating.28% for the model with five rivets (Figure 3c).12 446.3x107 Value 3 rivets 27.20 849.03 0.190 Mechanical and Aerospace Engineering V Table 4: Frequencies and mode shapes evolution for the numerical model with three rivets and eight combination elements.98 1517.86 851. The riveted .71 448.51 0. Mode Exp.06 1061.26 0.07 607.98 1061.15 Difference before Updating [%] 0.17 0.04 Initial MAC Final MAC 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Table 5: Frequencies and mode shapes evolution for the numerical model with five rivets and eight combination elements. Conclusions In this paper. Freq. initial Freq.78 325.26 0.06 Difference after Updating [%] 0.36 112. By other hand. 0. Table 6: Final stiffness constant values for the rivets with eight combination elements. [Hz] 1 2 3 4 5 6 7 8 112.76 607.13 325.17 1058.2x107 17.21 0.88 1517.28% for the model with three rivets (Figure 3b) and 0. Freq.85 846.12 0.00 0. obtained by aplication of Eq.30 1400.00 0.0x107 215. conclusions can be made.32 0. that the modes are correlated.75 606. This fact indicates that the obtained stiffness constant values. [Hz] Num.75 1508.2x107 1621 x107 The presented results justify the choice by the numerical riveted joint model with eight combination elements per rivet. The rivet is modeled with spring-damper (combination 14) elements. final Freq.13 446. [Hz] 112.59 849. for the combination elements are obtained with high reliability and accuracy.52 0.92% for the model with one rivet (Figure 3a).68 449.04 0.62 Difference before Updating [%] 0.23 1400. 10.32 0.8 x107 5 rivets 40. V. Guimarães (2013). . C. 25 (2003). L. Y. G. [4]. 1013–1026. [5]. Sathiya Naarayan. 54-61. D. References [1]. W.V. [7]. M. S. 43 (2012). 20 (2012). S. T. Friswell. in: Finite Element Model Updating in Structural Dynamics. For the joints with three and five rivets. p. Guedes Soares: Fatigue reliability assessment of riveted lap joint of aircraft structures. Dourado. Huang. Allemang. 9-24. S. M. Finite Elements in Analysis and Design Vol. 5th Int. It may be possible to reduce the quantity of spring-damper elements per rivet in the numerical model representation. Pavan Kumar. S. The results of the evaluations allow conclude that. I. Finally an equation can be developed in order to calculate the value of the stiffness constant k of the combination element. [6]. X. M. For the future it is important understand how evolves the average percentage difference with the increase of rivets in the lap joint. Dordrecht (2000). and the numerical model with eight combination elements is reliably to represent the experimental riveted joint model with one. 1-8. D. S. three and five rivets. A. Chandra: Implication of unequal rivet load distribution in the failures and damage tolerant design of metal and composite civil aircraft riveted lap joints. 40 (2004). But its greater complexity does not justify its implementation. 16 (2009). 1st Int. Enginnering Failure Analysis Vol. Huang. Kluwer Academic Publishers.Mottershead. D. Modal Analysis Conference & Exhibit. Florida. we make an important contribution to the designers in modeling riveted lap joints for structural finite element analysis. [2]. International Journal of Fatigue Vol. Holiday Inn (1982). J. p. T. 1175–1194. G. A. T-J. p. [8]. With this study. 110-116. J. Brown: A Correlation Coefficient for Modal Vector Analysis. the set with twelve combination elements per rivet have the same behaviour that the rivet with eight combination elements. the stiffness constant values are obtained with good accuracy (Table 6). p. according with the characteristics of the physical riveted lap joint. Sathiya Naarayan. C. E. 1016 191 joints with eight combination elements per rivet have the lower dynamic behaviour difference between numerical and experimental model. Xu. M. Rocha: Structural Dynamic Updating Using a Global Optimization Methodology. International Journal of Fatigue Vol. Urban: Analysis of the fatigue life of riveted sheet metal helicopter airframe joints. in: Proc. Meireles.Advanced Materials Research Vol. S. in: Proc. 2255-2273. Garbatov. R. Kalyana Sundaram. J. [3]. Pavan Kumar. Enginnering Failure Analysis Vol. Chandra: Further numerical and experimental failure studies on single and multi-row riveted lap joints. p. p. Operational Modal Analysis Conference.R. Deng: An evaluation of simplified finite element models for spot-welded joints. p.