212 N. Jamia et al. nonlinear generic element formulation. In [14], Ahmadian and Jalali proposed a nonlinear mathematical model for bolted lap-joints to predict the micro-slip/slap behaviour in the joint contact interface. Noel et al. [15] introduced a nonlinear statespace identification approach to estimate the hysteresis dynamics. Miller and Quinn [16] developed a two-sided interface model using a series-series Iwan model and an elastic chain to predict the dynamic behaviour of structures with frictional joints. They also developed a reduced-order model of the presented approach in order to reduce the computational costs of the modelling. Jalali et al. [18] showed the application of the force-state mapping technique in the identification of nonlinear lap-jointed structures. Thin-layer element theory is another method introduced by Desai et al. [19] to model the contact interface of jointed structures. By carrying out a parametric study, they illustrated that the ratio of the thickness of the thin-layer element to the mean dimension of the adjacent elements should be between 0.01 and 0.1. Jalali et al. [20] exploited thin-layer element theory to model the dynamics of bolted lap-joints experiencing micro-slip/slap behaviour. Ehrlich et al. [21] reduced the computational costs and calculation time required for model updating and uncertainty analysis using thin-layer element theory, by proposing a reduced-order model of the thin-layer element theory. Wang et al. [22] developed a parametric modelling approach, based on the finite element method and thin-layer element theory, to estimate the parameters of jointed structures. Iranzad and Ahmadian [23] modelled the dynamics of a lap-type mechanical joint using thin-layer element theory and utilizing experimentally measured data. Zhan et al. [24] focused on modelling and estimating the parameters of the dynamics of bolted joint structures. To this end, they utilized thin-layer elements to model the joint contact interface and exploited three-dimensional brick elements to model the substructures of the joint. This paper investigates modelling the dynamics of a bolted lap-joint utilizing a detailed three-dimensional model and a two-dimensional equivalent beam model. In both models, a modified thin-layer element approach is used to model the joint contact interface where the material properties of the thin layer are considered to be variable over the contact interface. The variable material properties resemble the distribution of the normal contact pressure in the contact interface. Impact modal testing was employed to obtain the measured natural frequencies of the lap-joint. A 3D detailed model and a 2D equivalent model are created using the thin-layer approach. An identification of the model parameters is employed using the experimental modal properties of the lap-joint obtained by modal testing. 2 Experimental Analysis of Lap-Joint In this section, modal testing is performed on a lap-joint structure in order to obtain the undamped natural frequencies, which will be used for the identification of the joint parameters in the next sections. Two different beams with two different lengths joined by two M8 bolts were designed and fabricated in order to compose a lap-joint structure. The design and dimension of the two beams are shown in Fig. 1. The beams are made of structural steel with the following material properties, E=208GPa, ρ =7860kg/m3 andυ=0.3, where E is the Young’s modulus, ρ is the density and υis Poisson’s ratio. In order to obtain the natural frequencies of the lap-joint, an impact hammer experiment was performed. In this experiment, a 4-channel (DAQ) system, an impact hammer and data acquisition software were used. In the modal testing procedure, care has been taken to keep the uncertainties to a minimum. Two identical M8 bolts and nuts are used to join the two beams and a KTC Digital Ratchet Torque Wrench was used to tighten the bolts to a specific level corresponding to a torque level equal to 23 Nm. Free-free boundary conditions were used for the lap-joint structure by suspending it using flexible strings as shown in Fig. 2. Fig. 1 The lap-joint: (a) design, (b) dimensions
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