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 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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. REFERENCES [1] Göege D., Sinapius M., Füllekrug U., et al., 2005, “Detection and description of non-linear phenomena in experimental modal analysis via linearity plots”, International Journal of Non-Linear Mechanics, 40, pp. 27-48. [2] Song Y., Hartwigsen C.J., McFarland D.M., et al., 2004, “Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements”, Journal of Sound and Vibration, 273, pp. 249-276. [3] Oldfield M., Ouyang H., Mottershead J.E., 2005, “Simplified models of bolted joints under harmonic loading”, Computer & Structures, 84, pp. 25-33. 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. [4] Ouyang H., Oldfield M.J., Mottershead J.E., 2006, “Experimental and theoretical studies of a bolted joint excited by a torsional dynamic load”, International Journal of Mechanical Sciences, 48, pp. 1447-1455. 7 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use [5] Shiryayev O.V., Page S.M., Pettit C.L., et al., 2007, “Parameter estimation and investigation of a bolted joint model”, Journal of Sound and Vibration, 307, pp. 680-697. [6] Bauchau O.A. Ju C., 2006, “Modeling friction phenomena in flexible multibody dynamics”, Computer Methods in Applied Mechanics and Engineering, 195, pp. 6909-6924. [7] Chatterjee S., Chatterjee S., Singha T.K., 2005, “On the generation of steady motion using fast-vibration”, Journal of Sound and Vibration, 283, pp. 1187-1204. [8] Gaul L., Lenz J., 1997, “Nonlinear dynamics of structures assembled by bolted joints”, Acta Mechanica, 125, pp. 169181. [9] Hashiguchi K., Ozaki S., 2008, “Constitutive equation for friction with transition from static to kinetic friction and recovery of static friction”, International Journal of Plasticity, 24, pp. 2102-2124. [10] ANSYS Release 10.0 Reference: ANSYS Inc., 2005 Documentation. Element [11] Montgomery J. Methods for modeling bolts in the bolted joint. ANSYS User´s Conference 2002. [12] ANSYS Release 10.0 Documentation. Technology Guide: ANSYS Inc., 2005. Contact [13] OPTIMUS Release 5.2 Documentation. Theoretical Background. Noesis Solutions N.V. Inc., 2006. [14] ANSYS Release 10.0 Documentation. Teory Reference: ANSYS Inc., 2005. [15] ME`scopeVES Release 4.0 Documentation. Operating Manual: Vibrant Technology Inc., 2004. 8 Copyright © 2009 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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