Nonlinear Dynamic Analysis of Bolted Joints: Detailed and Equivalent Modelling

N. Jamia1, H. Jalali2, J. Taghipour1, M.I. Friswell1, H.H. Khodaparast1

1College of Engineering, Swansea University. Bay Campus, Fabian way,

Crymlyn Burrows, Swansea, SA1 8EN, U.K.

2Department of Mechanical Engineering

Arak University of Technology, Arak 38181-41167, Iran.

ABSTRACT A standard finite element analysis of individual components in aero engine and other systems shows a high accuracy compared to experimental measurements of the system response. However when it comes to assemblies, the conventional linear approaches fail to deliver good accuracy. This is due to the uncertain physical phenomena in the contact interface of the joints. A nonlinear contact problem is introduced by the joint and influences the overall dynamic behavior of the engine assembly. Therefore, the linear dynamic models must be coupled with nonlinear analysis of the assembly to investigate the accurate dynamics of the nonlinear system. Flanges are widely used joints that represent the main source of nonlinearities in assemblies. In this study, a finite element simulation of two bolted flanges is considered to identify the nonlinear behavior of the bolted flange joint caused by the presence of friction in contact interfaces. A detailed model of a bolted joint was built in ANSYS in order to evaluate the energy dissipated in a bolted joint and therefore provide accurate modeling of joint interfaces. Due to the high cost of the detailed model, an equivalent model was derived and predictions from this model are compared to the detailed model results in order to provide a robust model for designing bolted joints.

Keywords: Bolted flange, nonlinear analysis, detailed model, equivalent model

INTRODUCTION

Bolted joints are widely used due to their simplicity in assembling mechanical structures. However, despite their simplicity, their inherent dynamics is too complex to be easily analyzed. Bolted lap and flange joints show specific nonlinear behaviour particularly at higher excitation amplitudes resulting from nonlinear slip and slap mechanisms in their contact interface. These mechanisms manifest usually themselves as nonlinear stiffness and damping effects and the transfer of mechanical energy from lower to higher frequencies. Due to the micro-slip mechanism between the mating surfaces, a hysteresis phenomenon is present in bolted joint systems when a cyclic load is applied. The area inside a hysteresis loop represents the amount of dissipated energy in the contact interface in one cycle [1]. Many efforts have been made in the past to model the nonlinearities associated when hysteresis phenomena. It should be mentioned that any useful joint model must be capable of reproducing the properties of the contact interfaces in terms of energy dissipation and nonlinear stiffness. Moreover, the model parameters must be deducible from joint level experimental or very fine-scale finite element results and integration of the joint model into structural-level models must be also practical [2].

The first attempt to model hysteresis behaviour was undertaken by Timoshenko [3]. Later, Iwan [4-5] extended the approach suggested by Timoshenko which assumes that a general hysteretic system consists of a large number of elastoplastic elements having different yield levels. Gaul et al. [6] experimentally studied the nonlinear damping and response function characteristics of bolted joints. Lenz and Gaul [7] and Gaul and Lenz [8] studied the behaviour of a bolted joint when it is subjected to longitudinal and torsional forces. They used the Valanis model, well known from plasticity [9], as a reduced order lumped model and showed that this model is capable of describing the nonlinear dynamic behaviour of a bolted joint contact interface. Segalman [10] used the parallel series Iwan model in series with a soft element to represent respectively the contact patch and the rest of the joint and showed that this model is a good candidate for capturing the nonlinear physics involving in contact interfaces. Considering the softening effect at the contact interface of a bolted lap-joint, Ahmadian and Jalali [11] proposed a lumped joint model consisting of linear and cubic springs and linear damping elements. Yuan et al. [12] used the adjusted Iwan model to simulate the behaviour of a bolted joint.

In this paper, the microslip behavior at the contact pair interface between two flanges is analysed with two finite element models of a bolted joint; a detailed 3D FE model and a 1D equivalent model. The modelling details are described and a comparison of natural frequencies and micro slip behavior is performed between the two models. Then static and dynamic analyse is were simulated to generate the force-displacement curve for both cases and comparisons between the results of the two models are performed.

DETAILED FE MODELLING In this section a detailed three-dimensional finite element model of two bolted joints is developed. The FEM analysis is carried out using the commercial software ANSYS Workbench.

Fig. 1 Bolted flange design

A three bolted joint consisting of three M10 bolts, three nuts, three washers and two rectangular flanges is considered. The two flanges are connected through the three M10 bolts as shown in Fig. 1. Structural steel with the following parameters was assigned as the joint material; E = 200GPa, ν = 0.3 and ρ = 7.85g/cm3, where E is the Young’s modulus, ν is the Poisson’s ratio and ρ is the mass density. The different design parameters of the flange are given in Table 1.

Table 1 3-D model geometry parameters Description Measurement

Flange length 100 mm Flange height 200 mm

Flange thickness 7 mm Flange width 50 mm

Bolts M10 mm

Three types of contact embedded in ANSYS are used in the model; bonded, frictionless and frictional. Frictionless behavior allows the parts to slide relative to one another without any resistance. This type of behaviour was assigned to the contact between the bolt-shank and flanges holes and the bolt-shank to washer interface. The frictional contact provides shear forces between the parts in contact and was assigned between the contact bolt head to the top surface of the upper flange, the nut interface to the bottom surface of the lower flange and between the upper flange and the lower flange. Finally a bonded contact was assigned to the contact nut interface to the bolt. The contact interfaces of the bolted joint are modelled in ANSYS based on the type of the contact between parts. In the normal direction, ANSYS prevents the contacting bodies from interpenetration by enforcing contact compatibility using contact algorithms. For the case of nonlinear solid body contact, the Augmented Lagrange and Pure Penalty formulations are chosen for frictional or frictionless and bonded contacts respectively. These methods combine robustness and high accuracy of the nonlinear contact solution. These two formulations are a penalty-based contact formulation. Friction behaviour is considered following Coulomb’s law 𝐹!"#$%#!&"' ≤ µ𝐹#()*"' where µ is the coefficient of static friction. Once the tangential force 𝐹!"#$%#!&"' exceeds µ𝐹#()*"' sliding will occur. If µ is zero then the contact behaves as fricitonless. In this model, a coefficient of friction µ= 0.2 was assigned to the contacting area between the two flanges.

Regarding loads and boundary conditions, the lower side of the bottom flange was constrained and a bolt axial load equals to 5kN was applied to the three bolts. A pretension in the bolts was generated at the mid-plan of the bolt. To specify the pretension in the bolt, a local coordinate system was defined with the Z-axis along the bolt length as shown in Fig. 2(a). To provide high accuracy in the contact analysis, the model is meshed with a fine mesh in the contact area, bolts, nuts and washers. Tetrahedral elements were used in the mesh as shown in Fig. 2(b). The assembly has 16248 elements and 84317 nodes.

(a) (b) Fig. 2 (a) Pretension in the bolts (b) Mesh details of the model

Due to the presence of the bolt pretension, the simulation is performed over two load steps. During the first load step, the different assembly interferences are calculated to provide the prescribed preload and reaction forces into the bolts and mimic the assembly of the two flanges. Once the bolt pretension load is achieved, then calculated reaction force agrees with the applied preload in the bolts. During the second load step, the external load was applied to the bolted joint.

(a) (b)

Fig. 3 (a) Contact status (b) Contact pressure at the contact interface

The contact status shown in Fig. 3(a) shows the presence of sliding and sticking behaviour in the contacting area between the two flanges. Sliding is caused by the elements that slide on the surface of the contact; however sticking is caused by the contact elements that cannot move and therefore penetration occurs. The resulting contact pressure at the contact interface given in Fig. 3(b) shows a nearly uniform distribution.

To determine the natural frequencies and mode shapes of the system, an eigenfrequency study was performed using ANSYS Workbench. A fixed constraint was applied to the base of the lower flange. The first 6 natural frequencies and their corresponding mode shapes are given in Table 2 and Fig. 4, respectively.

Table 2 Simulated natural frequencies (Hz) - 𝜔! 𝜔" 𝜔# 𝜔$ 𝜔% 𝜔&

Simulated 34.69 181.18 245.88 419.95 590.04 760.2

The interest in this paper is the bending modes, and their corresponding natural frequencies will be compared to the equivalent model given in the next section.

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

Fig. 4 The first six modes of the bolted flanges model

A static load was applied to the lower face of the upper flange, close to the contact interface between the two flanges. This will help to excite the nonlinear behaviour of the joint interface and creates a sliding behavior in the contact interface. A parametric study was performed where the amplitude of the static load was varied and the relative displacement at a point of the contact interface between the two flanges was monitored. Figure 5(a) shows the behaviour of the relative displacement at the contact interface between the two flanges under the action of the static external load. The graph shows a linear behavior while the force is between -3kN and 3kN and beyond that range, a nonlinear behaviour occurs. This static study gave an approximation of the value of static force starting from which the structure starts to behave nonlinearly.

(a) (b) Fig. 5. Force-displacement curve obtained from (a) static loading (b) dynamic loading

A dynamic analysis was performed where a periodic force is applied to the bolted joint. This force is defined as 𝐹(𝑡) = |𝐹|sin(𝜔𝑡) where |𝐹 |= 4 kN and 𝜔=15 Hz. A simulation was performed for a period of 4s with a time step equal to 0.0025s and the relative displacement of a point from the contact area of the bolted flanges is calculated for each time step. Figure 5(b) shows the curve of the external load versus the relative displacement at the interface contact between the two flanges for a single cycle of loading-unloading. It can be observed that the curve shows a hysteresis loop which has a parallelogram shape where two stages can be identified. The first stage where the variation of the relative displacement is very slow relatively with the variation of the applied load. This behavior corresponds to sticking. Increasing the applied load leads to gross slip and a microslip occurs during the transition between the stick and gross slip. This loop is used to quantify the energy dissipated in the joint contact due to the damping of the joint. The detailed model has a very high cost in terms of computational time, and therefore the numerical limitations in terms of mesh and time step are the reason behind the noise in the data shown in Figure 5(b). Therefore, a 1D equivalent model was performed in the next section.

EQUIVALENT BEAM MODELLING An effective approach to construct an equivalent model for the bolted flange structure considered in previous section is through using beam elements to model the beam sections and using appropriate joint models to represent the effects of slip and slap mechanisms in the contact interface. A schematic of FE model of the structure is shown in Fig. 6. DOF constraints (or MPCs) are used to relate the movement of the end nodes in modelling two perpendicular beam sections. Beam elements are two-noded with {𝑢, 𝑣, 𝜃+} as the DOFs at each node. Joint elements are used to model the effect of contact interface dynamics in the FE model. Because of the two slap and slip mechanisms which are likely to develop in normal, i.e. x, and tangential, i.e. y, directions at the contact interface, the following models are proposed to consider the effects of the slip and slap mechanisms.

Fig. 6 FE modelling of the bolt jointed flange

Micro/macro slip can initiate in the tangential direction at the contact interface if loading and bolt preload conditions are enough to initiate them. Prior to micro-slip the behaviour of the contact interface is linear under small loading amplitudes. It is assumed in this paper that macro-slip will not happen in the contact interface and the focus is to model the micro-slip phenomenon. The Jenkin’s element is used to model the micro-slip in the tangential direction of the bolt jointed flange as shown in Fig. 7.

(a) (b)

Fig. 7 (a) Jenkin's element used to model the slip mechanism (b) Corresponding force-displacement diagram

In Fig. 7 𝑘! is the tangential stiffness, 𝜇 is the friction coefficient and 𝐹, is the pointwise normal force applied to the contact interface due to bolt clamping force. The force displacement relationship for the Jenkins element can be expressed as,

𝑘6 7 1 −1−1 1 : ;

𝑣&𝑣-< = 𝐹6 ;−11 <,𝐹& = −𝐹- = 𝐹6 (1)

where,

𝑘6 = =𝑘! ,ifsliderwasnotslidinginthepreviousiteration0,ifsliderwasslidinginthepreviousiteration (2)

𝐹6 = =±𝜇𝐹,,𝑖𝑓𝑠𝑙𝑖𝑑𝑒𝑟𝑖𝑠𝑠𝑙𝑖𝑑𝑖𝑛𝑔𝑘!𝑢.,𝑖𝑓𝑠𝑙𝑖𝑑𝑒𝑟𝑖𝑠𝑛𝑜𝑡𝑠𝑙𝑖𝑑𝑖𝑛𝑔𝑏𝑢𝑡ℎ𝑎𝑠𝑠𝑙𝑖𝑑𝑏𝑒𝑓𝑜𝑟𝑒

(3)

𝑢. = 𝑢& − 𝑢- − 𝑢/ (4)

The slip element shown in Fig. 7 is used between each two adjacent i and j nodes in the bolted section of the structure as shown in Fig. 6. In using the slip elements, it is assumed that all slip elements have the same tangential stiffness𝑘!. Due to a variable distribution of normal pressure in the contact interface, different 𝐹, are considered for different slip elements used in the contact interface. The distribution of pointwise normal forces 𝐹, is similar to the normal pressure distribution in the contact interface. It is assumed that in the simulated structure in the previous sections, and due to the location of the applied load, the slap mechanism is not activated. Therefore, the behaviour of the contact interface in the normal direction is considered linear and a linear stiffness 𝑘# is used in the joint model to represent the normal stiffness of the contact interface. The normal and tangential stiffness coefficients can be identified from simulated linear dynamic properties of the structure such as natural frequencies presented in previous sections. In next section identification of the linear and nonlinear joint model parameters is considered.

JOINT MODEL IDENTIFICATION RESULTS

The first three bending natural frequencies presented in Table 2 were used and the contact stiffness coefficients in the tangential and normal directions were identified. Table 3 shows the simulated and updated natural frequencies and their differences. The results presented in this Table show that the equivalent beam model is capable of accurately reproducing the linear behaviour of the detailed model presented in the previous sections:

Table 3 Comparison of the simulated and updated bending natural frequencies (Hz), 𝑘' = 1341063(()) and 𝑘* = 62698201((

))

- 𝜔! 𝜔" 𝜔% Simulated 34.69 181.18 590.04 Updated 34.92 180.97 589.87

Error 0.66 -0.11 -0.03

Next identification of the distribution of the normal forces 𝐹, is considered. In Fig. 8(a) an identified distribution of the normal forces and in Fig. 8(b) the force-displacement curve are compared for the equivalent model and the detailed model.

(a) (b)

Fig. 8 (a) An initial identification for the distribution of 𝐹+ (b) Comparison between force-displacement diagrams

It is worth noting that since the linear model presented in Table 3 is an accurate model, it is possible to use an identification approach to match the force-displacement diagram from the detailed and equivalent models as shown in Fig. 8(b). Next, the prediction of the hysteresis loops is considered by using the identified equivalent beam model. The dynamic load applied to detailed model and described in previous section is applied to the equivalent beam model and the response is calculated. By using the calculated displacements, the hysteresis loop is constructed and compared to the hysteresis loop from the detailed model in Fig 9.

Fig. 9 Comparison of the hysteresis loops from the detailed (solid line) and equivalent (circles) modelling methods

CONCLUSION In this paper, a detailed model of bolted flanges was constructed to capture the friction behaviour in the contact interface due to micro-slip phenomena. Static and dynamic loads were applied to the joint and the force-displacement curves and hysteresis loops were obtained for the contact interface. Since detailed modelling is often not possible in real engineering applications, an equivalent model was constructed by using beam elements and a proposed joint element. The linear and nonlinear parameters of the equivalent model were obtained by using the results from the detailed modelling approach.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council through the award of the Programme Grant “Digital Twins for Improved Dynamic Design”, grant number EP/R006768.

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