A Dynamic Iwan Model to Describe the Impact Failure of Bolted Joints

Abstract

Due to the nonlinearity of the contact interface, as well as the material, jointed structures exhibit complex mechanical behaviors under impact loading. In order to accurately characterize the dynamic response of a joint, this work presents a nonlinear dynamic model (DICF model). First, the effects of loading velocity, preload and friction coefficient on the displacement–load curve are discussed based on a validated finite element model. Numerical simulation results show that the critical load and critical displacement are linearly related to the normalized logarithmic velocity and linearly related to the normalized preload and friction coefficient. Subsequently, a DICF model that consists of sliding, collision and failure is proposed. The constitutive relations of the model are derived, and dynamic correction functions are introduced to characterize the effects of velocity, preload and friction coefficient. A parameter identification method for the model is also provided. Finally, the DICF model is compared with the finite element simulation results, with an error of 0.43% for quasi-static conditions, a minimum error of 0.17% and a maximum error of −1.41% for impact conditions, in addition to significantly improved accuracy compared to the EC3 model, which indicates that it can effectively capture the behavior of bolted joints under impact loading conditions.

1. Introduction

Bolted joints have been widely used in various engineering structures due to their simplicity, cost effectiveness and ease of assembly [1]. The critical impact resistance provided by bolted joints is of vital importance when subjected to impact loading caused by explosions and collisions. In an assessment of the World Trade Center Collapse, the Institution of Structural Engineers stated that the ability of tall buildings to withstand impact loading is directly related to the strength, ductility and energy-absorbing capacity of the connections between the main structural elements [2]. However, the response of bolted structures under dynamic loadings is complex due to the presence of connection interfaces and material nonlinearities [3].

In recent years, researchers have conducted numerous experiments to discuss the response characteristics of bolted joints under impact loading. In a project on the safety of bolted joints in military vehicles, Horsfall et al. [4] carried out quasi-static, drop-tower and blast experiments on bolted joints, determining that the bolts exhibited ductile fracture under all of the above loading conditions. In another low-temperature impact experiment on M8.8 high-strength bolts, Gao et al. [5] pointed out that the bolts fractured in ductile form at both normal and high temperatures, and the failure mode changed from ductile fracture to brittle fracture at temperatures lower than −25.6 °C. In addition, Horsfall et al. [3] pointed out that compared to quasi-static loading, the strength of joints tend to increase under dynamic loading. Experiments carried out by Tyas et al. [6], Ribeiro et al. [7], Gao et al. [5] and Sanborn et al. [8,9,10] verified this conclusion. However, the effect of impact velocity on critical displacement is still debatable. In the SHPB impact tests carried out by Ren et al. [11] and Fransplass et al. [12], the critical displacements under impact loading increased compared to the quasi-static results. However, the results of the experiments of Ribeiro et al. [7] and Wanger et al. [13] were quite the opposite. Sanborn [8] tried to explain this in more detail; it was found that the critical load is mainly affected by bolt strength, while the critical displacement is mainly affected by the deformation of the connected parts. Song et al. [14], Brake et al. [15], Fransplass et al. [11] and Ren et al. [12] also discussed the effect of different loading directions on the failure mechanisms and found that the critical load under tensile loading was significantly higher than that under shear loading. In addition, Warren et al. [16] evaluated the residual load-carrying capacity of bolts under impact loading and found that impacts below the critical energy do not affect the strength of the joint but result in a reduction in ductility. Previous studies have demonstrated that the failure form of bolted joints under both quasi-static and impact loading is ductile fracture and that their strength is affected by the loading velocity, which is due to the strain-rate sensitivity of the materials.

In complex engineering dynamics analysis, direct numerical simulations using refined models are often difficult to carry out due to the limitation of arithmetic power. Therefore, constructing a theoretical model describing the mechanical behavior of the joint can effectively reduce the size of the problem solution. Researchers have proposed a large number of contact models, such as the Valanis model [17,18], the LuGre model [19,20,21], the Bouc–Wen model [22,23,24] and the Iwan model [25,26,27,28,29,30,31,32] et al. Among them, the Iwan model is widely used because of the clarity of the analytical functions and the simplicity of parameter identification. The Iwan model is composed of a large number of Jenkins cells, which consist of a spring and a slider, with the slip of the slider indicating the cell yields. This model was first used to describe the elastic plasticity of materials, and later, researchers applied it to describe connected structures. Song et al. [25] and Li et al. [26] added a linear spring to the original Iwan model to describe the residual stiffness of the bolted interface during the macroscopic sliding phase, expressed as two Dirac functions added to the density function. Segalman et al. [27] investigated the energy dissipation of joints under microslip and found that the energy dissipation was power related to the load amplitude, so the density function was assumed to be non-uniformly distributed, proposing a four-parameter Iwan model. Wang et al. [28] replaced the Dirac function with a peak function and proposed an improved four-parameter Iwan model in order to characterize the smooth transition between the microslip and macroslip. In other works, Brake et al. [29] and Shen et al. [30] considered the collision of bolts and holes, also known as the pinning phenomenon, and proposed modified Iwan models. Ranjan et al. [31] corrected the contact stiffness in Brake’s model by introducing boundary effects of finite-length bolts. In addition, in order to consider the effect of normal behavior on the tangential friction behavior, Li et al. [32] proposed an improved Iwan model considering normal load variation. In existing works, investigations of Iwan models concentrated on micro-and macroslip, without describing bolt deformation and the effects of impact velocity.

It can be concluded from the open literature that the velocity has a significant effect on both the strength and ductility of bolts, but the existing Iwan model is unable to describe the dynamic response under impact loading. To overcome the mentioned limitations, this paper considers the shear failure of bolts and introduces dynamic correction functions to establish a dynamic Iwan model (DICF model). Section 2 presents a refined finite element model and comparisons with the experimental results. In Section 3, the effects of loading velocity, preload and friction coefficient are discussed. In Section 4, a dynamic Iwan model is proposed by introducing the collision and failure stages into the classic Iwan model. The constitutive relationship of the model is derived, and a method for parameter identification is also provided. Section 5 presents a comparison of the proposed model with the FEM simulation results. Section 6 presents the conclusion of this paper.

2. Numerical Model of Bolted Joints

Lapped joints are very common in architectural and mechanical structures. When subjected to transverse shear load, the bolt interface first slips; then, the bolt makes contact with the hole; finally, the bolt fractures. This work focuses on the shear damage of bolts under impact loading. The bolted joint chosen for this research consists of four S8.8 M20 bolts and three plates. The thicknesses of the plates are 30 mm and 10 mm, and the assembly clearance between the bolts and holes (${\delta }_0$) is 1 mm. The test is performed using a hydraulic testing machine with a total capacity of 1000 kN. One end of the specimen is fixed, and a tensile displacement load is applied to the other end, with a displacement load rate of 1 mm/min [33]. According to EC3-1-8 [34], the standard value of pre-tension force corresponding to an S8.8 M20 bolt is F = 120 kN. The coefficient of friction is μ = 0.1. The tested components and finite element model are shown in Figure 1, with all components discretized by eight-node hexahedral elements. Mesh verification analysis was performed to obtain a reasonable mesh, providing reliable results in less computational time. A fine mesh was created at the region around the bolt, and according to the mesh verification results, the number of circumferential element divisions of the threads is 60, with a smallest element sizes of 0.14 mm and a total of 708,072 elements in the whole model. The material of the connected plates is Q355B alloy, with 45 steel for the bolt. Considering the effect of material strain rate during impact loading, the J-C model [35] was chosen for all material properties, the constitutive equation of which is expressed as

$$\sigma =(A+B{\epsilon }^n)(1+C\ln {\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}})[1-{({\displaystyle \frac{T-T_r}{T_m-T_r}})}^m]$$

(1)

$$\epsilon =[D_1+D_2\exp (D_3{\displaystyle \frac{\sigma }{{\sigma }_0}})](1+D_4\ln {\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}})[1-D_5({\displaystyle \frac{T-T_r}{T_m-T_r}})].$$

(2)

Equation (1) is the J-C plasticity model, and Equation (2) is the J-C damage model, where σ is the flow stress; A is the yield stress under quasi-static conditions; B and n are strain-hardening parameters; m controls the temperature dependence; C is the strain-rate dependence; $\epsilon$ is the equivalent plastic strain; $\dot{\epsilon }$ is the plastic strain rate; $\dot{{\epsilon }_0}$ is the reference strain rate; T is the absolute temperature; suffixes r and m indicate room and melting temperature, respectively; and D₁, D₂, D₃, D₄ and D₅ are experimentally determined material damage parameters. The detailed material parameters are shown in

Table 1. Material parameters of the components [36,37,38].

Component	Material	E (GPa)	ν	Ρ (kg/m3)	A (MPa)
Plates	Q355B steel	206	0.28	7850	339.45
Bolt and Nut	45 steel	206	0.28	7850	714
B (MPa)	C	n	m	Tm (°C)	Tr (°C)
620	0.045	0.412	0.66	1800	293
563	0.037	0.52	0.7	1808	293
$\dot{\epsilon }$ (s−1)	D1	D2	D3	D4	D5
1.33 × 10−3	0.81	6.05	−7.09	−0.003	2.0
8.33 × 10−4	0.10	0.76	1.57	0.005	−0.84

.

As can be seen in Figure 2, in the experiment, the failure of a bolted joint subjected to transverse loads involves three stages. First, sliding occurs, and the sliding distance is the same as the assembly clearance. Secondly, collision occurs, during which the stiffness is gradually reduced. Finally, failure occurs. The failure mode was bolt shear fracture with the fracture surface located at the contact interface of the connected plates. In addition, according to EC-1993-1-8(EC-3) [38], the theoretical critical load of this lapped joint is 255.6 kN, which was calculated by FEM simulation as 256.4 kN with an error of 0.3%, and the experimental result was 257.6 kN with an error of 0.8%. The bolt deformation and critical load of the simulation results are in good agreement with the experimental results.

3. Parametric Analysis

In this section, the effects of impact velocity, preload force and friction coefficient on the impact response are discussed, and the results of the variable analysis of the three parameters are shown in Figure 3.

The plastic parameters of metallic materials are significantly influenced by the strain rate, so it makes sense to discuss the effect of impact velocity. In this paper, five working conditions from 1 mm/min(quasi-static) to 1000 mm/s are discussed, and the calculation results are similar to Sanborn’s summary [6]; as the velocity increases, the critical load of the bolted joint shows an increasing tendency, while the fracture displacement decreases. From the J-C model, it can be seen that metallic materials are subjected to both strain hardening and strain-rate hardening, and the stress and failure strain are linearly related to the logarithm of the normalized strain rate.

For bolted joints, the preload is an important parameter, and according to the available experimental data, the loss of preload in bolted joints during long-term service seems to be inevitable. In addition to affecting the magnitude of the interfacial friction, the preload also affects the stress distribution inside the entire bolted joint. In this paper, five preload conditions of 120 kN, 90 kN, 60 kN, 30 kN and 0 kN are discussed, and it is found that the critical load, as well as the fracture displacement, increases slightly as the preload decreases. The preload increases the tensile stresses inside the bolt, which is reflected in the J-C damage model as an effect of the stress triaxiality, leading to an increase in plastic strain accumulation.

Similar to the preload, the friction coefficient also affects the interfacial friction. In this paper, six cases of friction coefficient of 0.10, 0.12, 0.14, 0.16, 0.18 and 0.20 are discussed, and it can be seen that the larger the friction coefficient, the more the critical load and critical displacement increase.

Table 2. Comparison of the numerical analysis results with the EC-3 prediction results.

Velocity (mm/s)	Preload (kN)	Friction Coefficient	Critical Load (kN)
			Simulation	EC-3	Error (%)
QS	120	0.10	256.4	255.6	−0.31
1	120	0.10	289.6	255.6	−11.74
10	120	0.10	312.8	255.6	−18.29
100	120	0.10	327.4	255.6	−21.93
1000	120	0.10	346.2	255.6	−26.17
100	90	0.10	337.1	255.6	−24.18
100	60	0.10	338.8	255.6	−24.56
100	30	0.10	338.3	255.6	−24.45
100	0	0.10	341.3	255.6	−25.11
100	120	0.12	331.7	255.6	−22.94
100	120	0.14	332.3	255.6	−23.08
100	120	0.16	334.9	255.6	−23.68
100	120	0.18	335.4	255.6	−23.79
100	120	0.20	335.6	255.6	−23.84

compares the critical loads at different velocities, preload forces and friction coefficients with the EC-3 prediction results, and it can be seen that EC-3 agrees with the quasi-static results but with a significant error in the impact results.

4. Dynamic Iwan Model

4.1. Analytical Development

Brake et al. [16] proposed a modified Iwan model (RIPP model) considering collision, and an illustrative drawing of the constitutive force as a function of displacement is shown in Figure 4a, which is expressed in the following form:

$$F_{\mathrm{RIPP}}=F_{\mathrm{PIN}}+F_{\mathrm{S}}$$

(3)

$$F_{\mathrm{S}}={\displaystyle {\int }_0^{x}f\rho (\phi )}\mathrm{d}f+{\displaystyle {\int }_x^{\infty }x\rho (\phi )}\mathrm{d}f$$

(4)

$$F_{\mathrm{PIN}}={\displaystyle \frac{\pi }{4}}{E}^{*}Ld$$

(5)

$$\rho (\phi )=R{\phi }^{\chi }[H(\phi )-H(\phi -{\phi }_2)]+K\delta (\phi -{\phi }_2)$$

(6)

where F_RIPP is the constitutive force of the RIPP model, F_PIN describes the collision and F_S describes the slide. In F_S is the classical Iwan model, and F_PIN is calculated by Hertz contact stiffness. Ρ(φ) is the Iwan density function, E is the contact stiffness, L is the contact length and d is the contact depth. In ρ(φ), H(φ) is the Heaviside function, δ(φ) is the Dirac function, ${\phi }_{\mathrm{MAX}}$ is the starting point of macroscopic slip, K is the tangential stiffness at the moment before macroscopic slip, and R and χ are parameters describing the power function. The RIPP model can describe the elastic collision of the bolt with the hole but cannot describe plastic deformation or even fracture failure under large sliding.

This paper proposes the DICF model, and the relationship between the constitutive force and the displacement is shown in Figure 4b, which was constructed with reference to the J-C model (Equations (1) and (2)), which means that multiple correction terms are used to modify the model under the reference operating conditions. The sliding part of the DICF model is still described by the classic Iwan model (Equation (4)), while the collision part is corrected with normalized velocity, normalized preload force and the normalized friction coefficient. The general form is expressed as

$$F_{\mathrm{MAX}}(v,\mu ,p)=F_{\mathrm{MAX}}(v_0,{\mu }_0,p_0)f_1(v*)f_2(\mu *)f_3(P*)$$

(7)

$$D_{\mathrm{MAX}}(v,\mu ,p)=D_{\mathrm{MAX}}(v_0,{\mu }_0,p_0)f_4(v*)f_5(\mu *)f_6(P*)$$

(8)

$$v*={\displaystyle \frac{v}{v_0}},\mu *={\displaystyle \frac{\mu }{{\mu }_0}},P*={\displaystyle \frac{P}{P_0}},$$

(9)

where v is the loading velocity, μ is the friction coefficient, P is the preload force, the suffix 0 indicates the reference state, and f₁ to f₆ are correction functions. ${\delta }_0$ is the assembly clearance, and the complete expression of the DICF model is

$$F_{D\mathrm{ICF}}(x)=\left\{\begin{array}{cc}{F}_{\mathrm{S}}(x)& x\le {\delta }_0\\ F_{\mathrm{PIN}}(x-{\delta }_0)& {\delta }_0<x\le D_{\mathrm{MAX}}\\ 0& D_{\mathrm{MAX}}\le x\end{array}\right..$$

(10)

As shown in Figure 4b, $D_{\mathrm{MAX}}$ is the critical displacement, corresponding to the displacement of the critical load ($F_{\mathrm{MAX}}$).

4.2. Parameter Identification

4.2.1. Sliding

Segalman et al. [13] and Li et al. [15] have provided detailed summaries for the parameter identification of the Iwan model (F_s). Displacement–load curves with different preload forces and friction coefficients were extracted based on a finite element simulation and fitted based on the Iwan model, whose corresponding Iwan destiny functions are power functions and truncated by the Dirac function, as shown in Figure 5a. The corresponding stiffness curves are shown in Figure 5b. The displacement–load curve is differentiated to the first order to obtain the displacement stiffness curve and make the following assumptions: $0 < x\le {\phi }_1$, no yielding of the Jenkins elements, constant stiffness; ${\phi }_1 < x\le {\phi }_{\mathrm{MAX}}$, partial yielding of the Jenkins elements and a gradual decrease in stiffness;${ \phi }_{\mathrm{MAX}}\le x$, complete yielding of the Jenkins elements and a reduction in stiffness to 0. A comparison between the Iwan model and the FEM results of the sliding stage is shown in Figure 6, and the Iwan model and the FEM model are in good agreement.

4.2.2. Collision

The impact velocity, friction coefficient and preload force in the DICF model are not coupled to each other, so their effects on the load curve can be extracted separately, normalized and fitted to obtain the expressions for f₁ to f₆. The selected normalized base parameters are v₀ = 100 mm/s, μ = 0.1 and P = 120 kN; the curve under the reference condition is fitted with the asymptotic function (Boxlucas model); and the fitting result is shown in Figure 7. The general form of the Boxlucas model [39] is

$$F(x)=a\left(1-b^x\right).$$

(11)

Equation (11) is a kind of asymptotic function, where parameter a controls the critical load and b controls the shape of the curve.

The fitted curves of the normalized parameters are shown in Figure 8. It can be seen that the normalized critical load is linearly related to the log-normalized velocity, the normalized preload force and the normalized friction coefficient; the normalized critical displacement is linearly related to the log-normalized velocity, the normalized preload force and the normalized friction coefficient. The specific expressions of Equation (7) and Equation (8) in the DICF model can then be expressed as

$$F_{\mathrm{MAX}}(x*,v,\mu ,P)=F_{\mathrm{MAX}}(x*,v_0,{\mu }_0,P_0)\left(1+A_{v1}\ln (v*)\right)\left(1+A_{\mu 1}\mu *\right)\left(1+A_{P1}P*\right)$$

(12)

$$D_{\mathrm{MAX}}(x*,v,\mu ,P)=D_{\mathrm{MAX}}(x*,v_0,{\mu }_0,P_0)\left(1+A_{v2}\ln (v*)\right)\left(1+A_{\mu 2}\mu *\right)\left(1+A_{P2}P*\right)$$

(13)

5. Results and Discussions

Figure 9 illustrates the displacement–load curves of the DICF model for different loading velocities, preload forces and friction coefficients. Comparison of Figure 9 and Figure 3 indicates that the DICF model can describe the effects of the above variables on the critical loads and critical displacements.

Table 3. Comparison of the numerical analysis results with the DICF model prediction results.

Velocity (mm/s)	Preload (kN)	Friction Coefficient	Critical Load (kN)
			Simulation	DICF Model	Error (%)
QS	120	0.10	256.4	257.5	0.43
1	120	0.10	289.6	290.1	0.17
10	120	0.10	312.8	308.4	−1.41
100	120	0.10	327.4	326.6	−0.24
1000	120	0.10	346.2	344.7	−0.43
100	90	0.10	337.1	331.3	−1.72
100	60	0.10	338.8	336.0	−0.83
100	30	0.10	338.3	340.5	0.65
100	0	0.10	341.3	345.0	1.08
100	120	0.12	331.7	329.2	−0.75
100	120	0.14	332.3	331.7	−0.18
100	120	0.16	334.9	334.2	−0.21
100	120	0.18	335.4	336.7	0.38
100	120	0.20	335.6	339.1	1.04

compares the critical load results from the DICF model and numerical calculations. The maximum error is −1.41% (v = 10 mm/s, P = 120 kN, μ = 0.1), and the minimum error is 0.17% (v = 1 mm/s, P = 120 kN, μ = 0.1). Compared to the EC-3 predictions (Table 2, maximum error of −26.17%), the DICF model is more accurate.

Compared to the quasi-static experiments, the strength of the bolted joints subjected to impact loading increases significantly, which is due to the strain-rate sensitivity of the material. For steel, the yield strength increases significantly with increasing strain rate. In addition, increasing the preload and friction coefficient both lead to an increase in friction, which results in a tendency for the critical load and critical displacement to increase. However, when preload is applied, the stiffness of the bolt increases, making it more difficult to absorb and buffer the energy from the external impact and resulting in a reduction in its ductility.

6. Conclusions

This paper presents simulation and modeling investigation of bolted joints subjected to impact loading. The effects of loading velocity, preload and friction coefficient on the strength and ductility of the joint are discussed. Finally, a dynamic Iwan model is proposed to describe the impact failure. The main conclusions are outlined as follows:

• The response of bolted joints under impact loading consists of three main stages, namely slip, collision and bolt shear failure, and the developed fine finite element model can accurately simulate the form of damage, including the shear fracture of the bolt and the prediction of the critical load.
• Numerical simulation results show that increases in velocity and friction coefficient lead to an increase in critical load, whereas preload leads to a decrease. In addition, critical load and critical displacement are linearly related to the normalized logarithmic velocity and the normalized preload and friction coefficient.
• A dynamic Iwan model (DICF model) is proposed to describe the dynamic response of bolted joints under impact loading, the constitutive relationship is derived and a parameter identification procedure is proposed. The parameters can be obtained from experimental or numerical simulation results to accurately predict the force–displacement relationship for different velocities, preloads and friction coefficients.
• Calculation of the refined finite element model is inefficient due to the contact interfaces and the complexity of the threads. In this study, a model is presented to describe the force–displacement relationship of bolted joints under impact loading more conveniently, which can be used in future studies to characterize the impact response of large structures containing bolted joints.