Equivalent modeling for a finite element analysis of the mechanical properties of a bolted joint with a superelastic shape memory alloy

Abstract

In this study, a beam element model with tension–compression asymmetry is constructed by using the improved Brinson model. A comparison between the calculated and experimental results verifies the computational effectiveness of the beam element. Then, a new mechanical property analysis model of a shape memory alloy (SMA) bolted joint based on the beam element model is proposed. Combined with a constructed convergence function, the mechanical equilibrium state of the SMA bolted joint under preload and external load is solved iteratively by the bisection method. Consequently, the stress–strain response curve of the SMA bolt rod and the root of the first thread, the deformation law of each thread and the separation conditions of the contact surface of members are analyzed by the proposed model. It is shown that the new proposed method can greatly improve the calculation efficiency with high calculation precision compared with the traditional calculation method.

Keywords

Shape memory alloys finite element modeling SMA bolted joint beam element

1. Introduction

As one of the most important connection forms in mechanical structures, the bolted joint is widely used in key component structures. Extensive research has shown that problems with bolted joints can become catastrophic in safety-critical applications (Jiang et al., 2013; Qin et al., 2016). In recent years, shape memory alloy (SMA), as a new functional material, has been widely used in vibration-resistant and self-centering connection intelligent structures (Wang et al., 2019), medical equipment and instruments (Machado et al., 2015) and aerospace applications (Bettini et al., 2021). It has been demonstrated that the design of a reasonable preloading force is important for the reliability and service life of bolted joints (Yu et al., 2015). Therefore, it is necessary to clarify the mechanical properties of SMA bolted joints under preloading forces and external loads.

Researchers have used the superior properties of SMAs, such as high damping (Helbert et al., 2021), durability and fatigue resistance, in bolted joint design. The vibration failure time of a Fe-Mn-Si SMA bolt is seven times that of a common bolt, and the anti-loosening effect is remarkable (Zhou et al., 2014). SMA corrugated gaskets (SMA-CGs) can adapt to fluctuating working conditions, which make them excellent sealing components (He et al., 2021). SMA bolted connections can provide reasonable damping for frame structures under explosion loads (Weli and Vigh, 2021).

In the field of civil engineering, a superelastic SMA is attractive for the design of seismic resistance structures because of its excellent self-centering and energy dissipation capabilities (Ghamari et al., 2020; Toghroli et al., 2020; Zheng et al., 2021). The SMA bolted extended end-plate has excellent recentering ability and moderate energy dissipation capacity (up to 17.5% equivalent viscous damping) (Fang et al., 2014). The composite slab systems equipped with superelastic SMA bolts exhibited satisfactory self-centering ability and ductility (Fang et al., 2017; Wang et al., 2017). A reasonable balance between structural performance and cost may be achieved by an appropriate arrangement of the SMA connections within the structure (Fang et al., 2018). SMA stiffened angle connections and SMA bolted end plate connections can significantly improve seismic resistance performance (Chowdhury et al., 2019). Experiments (Najmabad et al., 2021) have shown that the combination of energy dissipation and the superelastic behavior of SMA anchor bolts can significantly reduce the failure of a tank shell and damage to an anchoring system.

These studies have fully demonstrated the superior performance of SMA bolted joints. However, the existing finite element models for analyzing the mechanical properties of bolted joints adopt solid, plane or bar elements. Solid or plane element models have problems with complexity, low computational efficiency, and difficulty in convergence. A model based on the bar element is only able to consider the tension deformation of the bolt rod (Jiang et al., 2018). However, the deformation of a thread root can have a significant impact on the overall mechanical properties of a bolted joint (Jiang et al., 2019).

Therefore, it is necessary to establish a high-efficiency and high-precision calculation model to analyze the nonlinearity of material and structural deformations of superelastic SMA bolted joints. In this paper, a new mechanical property analysis model of SMA bolted joints based on beam elements is proposed. Combined with a constructed convergence function, the mechanical equilibrium state of different types of SMA bolted joints under preload and external load is solved iteratively by the bisection method. The novelty of this paper is that the proposed calculation model can address not only the shortcomings of high computational cost (CPU time) and poor convergence with continuum elements but also the deficiency of the bar element in not calculating the nonlinear bending deformation of threads.

2. Constitutive model of NiTi SMA

In recent years, many scholars have established constitutive models of NiTi SMAs. Among them, the phenomenological theory model has the advantages of simplicity and few introduced parameters. It is the best candidate integrated into structural calculation methods (such as the finite element method) to predict bolt rod and thread deformation (Kan et al., 2012).

2.1. Constitutive equations and internal variables

The SMA constitutive model used here is the macroscopic phenomenological constitutive model proposed by Brinson (1993), which can be expressed as:

σ - σ_{0} = E (ξ) ε - E (ξ_{0}) ε_{0} + Ω (ξ) ξ - Ω (ξ_{0}) ξ_{0} + Θ (T - T_{0})

(1)

where $σ$ and $ξ$ are the stress and martensite volume fraction of the SMA, respectively. $E$ is the Young’s modulus, and $Ω$ is the transition coefficient. $T$ is the temperature, and $Θ$ is the thermal elastic coefficient. The subscript “0” indicates the initial values.

The Young’s modulus $E$ and the transition coefficient $Ω$ of the material in the Brinson model are functions of the martensite volume fraction $ξ$ , expressed as follows:

{\begin{matrix} E (ξ) = E_{A} + ξ (E_{M} - E_{A}) \\ Ω (ξ) = - ε_{L} E (ξ) \end{matrix}

(2)

where $E_{A}$ and $E_{M}$ are the Young’s modulus of the NiTi SMA in the austenite phase and martensite phase, respectively. $ε_{L}$ is the maximum residual strain.

The volume fraction of martensite in the Brinson model consists of a temperature-induced part ( $ξ_{T}$ ) and a stress-induced part ( $ξ_{S}$ ), which can be written as:

ξ = ξ_{S} + ξ_{T}

(3)

Since the object studied in this paper is at room temperature, the influence of the temperature-induced part $ξ_{T}$ on the martensite volume fraction $ξ$ is ignored.

According to the infinitesimal strain assumption, the total strain $ε$ can be divided into elastic strain $ε_{e}$ and inelastic strain $ε_{in}$ .

ε = ε_{e} + ε_{in}

(4)

According to the general plasticity, the inelastic strain of the SMA in equation (4) is equal to the transition strain in the process of martensitic phase transition.

ε_{in} = ε_{tr}

(5)

Substituting equation (5) into equation (4) gives:

ε = ε_{e} + ε_{tr}

(6)

It is assumed that the transition strain and the martensite volume fraction show a proportional relationship. Then, the following relationship can be obtained:

ξ = \frac{ε_{tr}}{ε_{L}}

(7)

Based on the Brinson model framework, the expression of the martensite volume fraction $ξ$ of the superelastic SMA beam element during transition can be written as:

(i) Transition to the martensite phase

if T > M_{s} and σ_{s}^{cr} + C_{M} (T - M_{s}) < σ < σ_{f}^{cr} + C_{M} (T - M_{s}) :

\begin{matrix} ξ_{A \to M} = \frac{1 - ξ_{0}}{2} \cos {\frac{π}{σ_{s}^{cr} - σ_{f}^{cr}} [σ - σ_{s}^{cr} - C_{M} (T - M_{S})]} \\ + \frac{1 + ξ_{0}}{2} \end{matrix}

(8)

(ii) Transition to the austenite phase

if T > A_{s} and C_{A} (T - A_{f}) < σ < C_{A} (T - A_{s})

ξ_{M \to A} = \frac{\underset{A \to M}{ξ^{f}}}{2} {\cos [\frac{π}{A_{f} - A_{s}} (T - A_{s} - \frac{σ}{C_{A}})] + 1}

(9)

where $σ_{s}^{cr}$ and $σ_{f}^{cr}$ are the starting and finishing stresses of martensite transition, respectively; $C_{A}$ and $C_{M}$ are the slopes for the relation between critical transition stress and temperature, respectively; $A_{s}$ and $A_{f}$ are the starting and finishing temperatures of austenite transition, respectively; $M_{s}$ and $M_{f}$ are the starting and finishing temperatures of martensite transition, respectively; and $ξ_{0}$ is the volume fraction of martensite at the beginning. $\underset{A \to M}{ξ^{f}}$ is the volume fraction of martensite at the end of martensitic transformation.

Now, considering that the research object has been at room temperature, equation (1) can be rewritten as follows:

σ - σ_{0} = E^{s} (ξ^{s}) ε - E^{s} (ξ_{0}) ε_{0} + Ω^{s} (ξ^{s}) ξ^{s} - Ω^{s} (ξ_{0}) ξ_{0}

(10)

The experimental results (De la Flor et al., 2011) showed that SMA has asymmetry under tensile and compressive loads. Poorasadion et al. (2014) proposed an improved Brinson model, which took into account the asymmetry of tension and compression, and its constitutive model and tangent stiffness matrix can be written as follows:

\begin{matrix} σ - σ_{0} = E^{s} (ξ^{s}) ε - E^{s} (ξ_{0}) ε_{0} + Ω^{s +} (ξ^{s}) ξ^{s +} \\ - Ω^{s +} (ξ_{0}) ξ_{0} + Ω^{s -} (ξ^{s}) ξ^{s -} - Ω^{s -} (ξ_{0}) ξ_{0} \end{matrix}

(11)

The tangent stiffness matrix is written as:

C = \frac{d σ}{d ε} = \frac{E^{s} (ξ^{s})}{1 - (ε - ε_{L}^{+} ξ^{s +} - ε_{L}^{-} ξ^{s -}) (E_{M}^{+} \frac{d ξ^{s +}}{d σ} + E_{M}^{-} \frac{d ξ^{s -}}{d σ} - E_{A} \frac{d ξ^{s}}{d σ}) - (Ω^{s +} (ξ^{s}) \frac{d ξ^{s +}}{d σ} + Ω^{s -} (ξ^{s}) \frac{d ξ^{s -}}{d σ})}

(12)

where the superscripts “+” and “–” in equations (11) and (12) represent tension and compression, respectively. In this paper, the above constitutive model and tangent stiffness matrix are adopted. Next, the finite element equation of the SMA beam element and its solution method are derived.

2.2. Finite element equation and solution method of SMA beam element

The beam element based on Euler–Bernoulli beam theory is shown in Figure 1. Each beam element consists of two nodes, and each node has three degrees of freedom $(d_{xi}, d_{yi}, ϕ_{zi})$ . The beam element can withstand axial, transverse and bending loads.

In the local coordinate system, the axial displacement and transverse displacement of the neutral axis of the beam element are $u$ and $v$ , respectively.

[\begin{matrix} u \\ v \end{matrix}] = [\begin{matrix} N_{B 1} & 0 & 0 & N_{B 2} & 0 & 0 \\ 0 & N_{B 3} & N_{B 4} & 0 & N_{B 5} & N_{B 6} \end{matrix}] {d}

(13)

where $N_{Bi} (i = 1 ~ 6)$ are called the shape functions and ${d}$ is the nodal displacement vector. The shape functions $N_{Bi}$ are expressed as in Javanbakht and Öchsner (2018) and written as follows:

\begin{matrix} N_{B 1} = 1 - \frac{x}{L} & N_{B 2} = \frac{x}{L} \\ N_{B 3} = \frac{1}{L^{3}} (2 x^{3} - 3 x^{2} L + L^{3}) & N_{B 4} = \frac{1}{L^{3}} (x^{3} L - 2 x^{2} L^{2} + x L^{3}) \\ N_{B 5} = \frac{1}{L^{3}} (- 2 x^{3} + 3 x^{2} L) & N_{B 6} = \frac{1}{L^{3}} (x^{3} L - x^{2} L^{2}) \end{matrix}

(14)

where $L$ is the length of the beam element. In addition, ${d}$ is expressed as:

{d}^{T} = [\begin{matrix} d_{xi} & d_{yi} & ϕ_{zi} & d_{xj} & d_{yj} & ϕ_{zj} \end{matrix}]

(15)

According to the basic assumption of Euler–Bernoulli beam theory, the following equation can be obtained.

ε = \frac{du}{dx} - y \frac{d^{2} v}{d x^{2}}

(16)

Substituting equation (13) into equation (16) and writing it in matrix form gives:

\begin{matrix} ε = - [\begin{matrix} \frac{1}{L} y \frac{12 x - 6 L}{L^{3}} & y \frac{6 xL - 4 L^{2}}{L^{3}} - \frac{1}{L} & y \frac{- 12 x + 6 L}{L^{3}} & y \frac{6 xL - 2 L^{2}}{L^{3}} \end{matrix}] {d} \\ = B {d} \end{matrix}

(17)

where matrix $B$ is the element strain matrix. Based on the principle of virtual work, the stiffness matrix $K$ of the beam element in the local coordinate system can be obtained.

K = \int B^{T} CBdV

(18)

where $C$ is the tangent stiffness matrix and equation (12) has been given.

The solution of the beam element is carried out in the finite element (FEM) software and calculated by calling the FORTRAN subroutine. The flow chart of the SMA beam solution algorithm is shown in Figure A of Appendix A, where $Tol$ is the tolerance of the program convergence judgment, and $Tol = 10^{- 6}$ is used in this work.

According to the method of Poorasadion et al. (2014), the following equation can be obtained by shifting equation (11):

\begin{matrix} f (σ) = E^{s} (ξ^{s}) ε - E^{s} (ξ_{0}) ε_{0} + Ω^{s +} (ξ^{s}) ξ^{s +} - Ω^{s +} (ξ_{0}) ξ_{0} \\ + Ω^{s -} (ξ^{s}) ξ^{s -} - Ω^{s -} (ξ_{0}) ξ_{0} - σ + σ_{0} \end{matrix}

(19)

When the beam element is in the phase transition stage, its stress and strain satisfy the nonlinear equation (19), and we use the Newton–Raphson method to solve this.

2.3. Numerical simulation and model verification

In later work, the established beam element will be used to simulate the tension of the SMA bolt rod and the bending of the thread. Therefore, it is very important to evaluate the tensile and bending performance of the SMA beam element established above. In this section, the SMA beam element established in this work will be tested for tension and bending through several examples and compared with the experimental data obtained in the reference.

2.3.1. Tensile test

The Mehrabi et al. (2015) experiment performed uniaxial loading and unloading of a thin-walled NiTi tube of length $L = 14 mm$ and thickness $t = 0.3 mm$ at a room temperature of $23 ° C$ . The material parameters used in the experiment are shown in Table 1.

Table 1.

SMA material parameters of the Mehrabi experiment.

$E_{A} = 20 GPa$	$E_{M} = 13.3 GPa$	$ε_{L} = 3.8 %$
$σ_{s}^{AM} = 248 MPa$	$σ_{f}^{AM} = 328 MPa$	$σ_{s}^{MA} = 65.6 MPa$	$σ_{f}^{MA} = 229 MPa$

A circular tube is divided into 10 beam elements: the left end is fixed, and the right end is loaded and then unloaded. The stress–strain curve is shown in Figure 2 The present simulation and experimental data have relatively small errors, and the established beam element model can well simulate the uniaxial tensile performance of the SMA bar.

2.3.2. Three-point bending test

Gillet et al. (1998) conducted tensile and bending experiments on thin-walled $Cu - Al - Be$ alloy beams at a temperature of $T = 19.3 ° C$ . The SMA material parameters are shown in Table 2.

Table 2.

SMA material parameters.

$E_{A} = 73.2 GPa$	$E_{M}^{+} = 10 GPa$	$E_{M}^{-} = 10 GPa$	$ε_{L}^{+} = 3.5 %$
$ε_{L}^{-} = - 3.0 %$
$σ_{s}^{AM +} = 309 MPa$	$σ_{f}^{AM +} = 969 MPa$	$σ_{s}^{AM -} = - 344 MPa$	$σ_{f}^{AM -} = - 689 MPa$
$σ_{s}^{MA +} = 317 MPa$	$σ_{f}^{MA} = 473 MPa$	$σ_{s}^{MA -} = - 251 MPa$	$σ_{f}^{MA -} = - 376 MPa$

Figure 3 shows a comparison between the stress–strain and load–displacement simulation results of the beam element and the experimental results of Gillet et al. In the three-point bending test, the span of the beam is $49.92 mm$ , and the two ends of the SMA beam are fixedly supported. A concentrated force of $26 N$ is loaded and unloaded in the middle of a thin-walled beam with a width of $2.9 mm$ and a thickness of $0.95 mm$ . In the present simulation, the beam is divided into 50 elements. It can be seen from the curves in Figure 3 that the overall simulation curves are very close to the experimental curves. The established beam element model can simulate the actual loading cases well.

3. Finite element modeling and solution method of a superelastic SMA bolted joint

In this section, the superelastic SMA beam element is used to establish a finite element model to simulate the preloading force and external load loading process of the bolted joint. For the above two processes, a convergence function is constructed, and the bisection method is proposed to iteratively solve the mechanical equilibrium state.

3.1. Finite element modeling of superelastic SMA bolted joint

The superelastic SMA bolted joint is simplified, and the main parts—the nuts, bolt rod and members—are simplified into beam elements. As shown in Figure 4, a superelastic SMA bolted joint finite element model based on beam elements is established. Since the stress distribution of the bolted joint is axisymmetric during the preloading force and external load loading process, according to research conclusions (Aljuboury et al., 2021; Shoji and Nagata, 2002), the three-dimensional finite element model of the bolted joint can be simplified to two-dimensions. The thread engagement of the bolt and nut is shown in Figure 5. Figure 5(a) shows the actual thread engagement section. The bolt thread and nut thread engagement section can be regarded as a trapezoid, where $P$ is the pitch of the screws, $H$ is the original triangular height and $H = \sqrt{3} P / 2$ , and $d 2$ is the pitch diameter of the bolt. The width of the fully engaged part of the bolt thread and the nut thread is $5 H / 8$ , while the height of the bolt thread and the nut thread are $P / 2$ at the pitch diameter line (chain line in Figure 5). As seen in Figure 5(a), the pitch diameter line is slightly close to the crest of the nut thread, and the distance is $2 H / 8$ .

In an experiment by Jiang et al. (2020), to avoid mutual rotation between the bolt and nut, the threads of the bolt and nut are bonded together with high-strength thread adhesive. In this work, the actual thread engagement is simplified, as shown in Figure 5(b). The bolt thread and the nut thread are coupled together in the y-direction degree of freedom at the pitch diameter line. The left side of the pitch diameter line is the bolt thread (width $2 H / 8$ ), and the right side of the pitch diameter line is the nut thread (width $3 H / 8$ ). In this work, a series of rectangular section beam elements are used to approximate the trapezoidal thread section.

3.2. Loading process of bolt external force load

The loading process of the external force load is divided into two stages: the loading stage of the preloading force and the loading stage of the external load, as shown in Figure 4(b) and (c), respectively.

3.2.1. Loading process of preloading force

In this research, a preset interference method that had successfully been used to simulate the preloading process of the bolted joint by Jiang et al. (2013, 2014, 2016) is adopted here to simulate the bolted joint preloading. The interference method means that when modeling a bolted joint, the preloading force is loaded by shortening the distance between the nut and the members in the model so that there is a certain hypothetical interference $Δ u_{0}$ between the contact surface of the nut and the members.

In the iterative calculation of the preloading force loaded to the superelastic SMA bolted joint, the tensile load and pressure load are applied to the nut and the members, respectively, to gradually decrease the interference $Δ u$ . Through calculation, $Δ u^{b}$ and $Δ u^{m}$ , the corresponding bolt preloading force $F_{b}$ and the compression force $F_{m}$ of the members are obtained. When the interference $Δ u = 0$ , the contact surface of the nut and the member fits into one surface. Meanwhile, the tensile force of the nut is equal to the compressive force of the members, that is, $F_{b} = F_{m} = F_{p}$ , which also means that the preloading process is finished.

The bisection method is used to iteratively solve the mechanical equilibrium state in this work. First, apply a $(1 + Tol) Δ u$ load to the nut, calculate the maximum force $F_{max}$ at which the nut is compressed to the lowest end, and then construct the following convergence function (20).

\begin{matrix} f (x_{p}) = | Δ u^{m} (x_{p}) | + | Δ u^{b} (- x_{p}) | - Δ u_{0}, \\ x_{p} \in (Tol \cdot F_{max}, (1 + Tol) \cdot F_{max}) \end{matrix}

(20)

where $f (x_{p})$ is continuous in the domain and $f (Tol \cdot F_{max}) \cdot f ((1 + Tol) \cdot F_{max}) < 0$ . The interval $(x_{p}^{a}, x_{p}^{b})$ where the zero point of $f (x_{p})$ is located can be continuously divided into two by the bisection method so that the two end points of the interval gradually approach the zero point. The iteration termination condition is $| f (x_{p}) | < Tol$ or $| x_{p}^{b} - x_{p}^{a} | < Tol$ .

Since the interference $Δ u_{0}$ is assumed in advance, the preloading force $F_{p}$ calculated according to $Δ u_{0}$ is not necessarily the aimed preloading force. Therefore, it is necessary to continuously adjust the interference by the trial-and-error method to obtain the targeted bolt preloading force. The specific calculation process is shown in Appendix Figure B.

3.2.2. Loading process of the external load

The second stage is the external load loading stage. In the initial preload stage, the members are in a compression state. Due to the effect of the external load $F_{e}$ , the compression extent on the side of the bearing contact of the nut increases, and one on the side of the bearing contact of another member is released. The external load $F_{e} = K_{m} u_{e}$ , where $K_{m}$ is the stiffness of the members.

The specific iterative process is described as follows. The external load $F_{e}$ is loaded on the intermediate node between the two elements of the members. At this time, element 2 is tensioned, $Δ F_{n}^{m} = Δ F_{n}^{e}$ , and the distance between the two nodes of element 1 remains unchanged. Therefore, the distance $Δ u^{tr}$ appears between the member and the nut. Since the contact surface between the nut and the member is close (nodes 2 and 2′), it is necessary to solve a new equilibrium state so that $Δ u^{tr} = 0$ . The calculation under an external load is similar to the loading process of preloading. In this work, the bisection method is still used to solve, and the following function (21) is constructed.

\begin{matrix} f (x_{e}) = | Δ u^{m} (x_{e}) | + | Δ u^{b} (- x_{e}) | - Δ u^{tr}, \\ x_{e} \in (Tol \cdot F_{e}, (1 + Tol) \cdot F_{e}) \end{matrix}

(21)

When the iteration converges, the nut contact surface and the member contact surface fit into one surface, and the bolted joint reaches a new equilibrium state.

According to the descriptions above, under the effect of the external load $F_{e}$ , there exist two contact statuses between the contact surfaces of members.

The external load $F_{e}$ is less than the initial bolt preloading force $F_{p}$ , and the system satisfies the force equilibrium relationship $F_{b} = F_{m} + F_{e}$ .

The external load $F_{e}$ is greater than the initial bolt preloading force $F_{p}$ , and the system satisfies the force equilibrium relationship $F_{b} = F_{e}$ . Meanwhile, the constraint state of node 3 will be released, and element 2 will not be in an internal force state.

4. Results and discussion

Based on a finite element model of the superelastic SMA bolted joint, this section analyzes and discusses the mechanical properties of the superelastic SMA bolted joint under preload and external load. M6 and M10 SMA bolts are selected with a pitch of $1 mm$ . The material parameters of the SMA bolted joint are shown in Table 3, and the temperature is $25^{°} C$ . Motosh and Nassar (Motosh, 1976; Nassar and Abboud, 2009) developed an analytical model to describe the member stiffness precisely, which is adopted here to calculate the stiffness value as $4 \times 10^{8} N \cdot m^{- 1}$ with a dimensional height of $35 mm$ . The elastic modulus is $70 GPa$ , and the Poisson’s ratio is 0.28 for the selected aluminum. The material of the nut is the same as the members.

Table 3.

SMA material parameters.

$E_{A} = 48 GPa$	$E_{M} = 35 GPa$	$v_{A} = v_{M} = 0.3$	$ε_{L} = 6.3 %$
$σ_{s}^{AM} = 285 MPa$	$σ_{f}^{AM} = 458 MPa$	$σ_{s}^{MA} = 345 MPa$	$σ_{f}^{MA} = 164 MPa$

Figure 6 shows a simulation of the preload loading process of the M10 bolted joint under different preloading forces $P_{0}$ and different preload interferences $u_{0}$ . The node deformation of each threaded beam element, the stress–strain response of the beam element node at the root of the first thread with the largest stress and the center node of the bolt rod, and the force–deformation response of the bolt rod are solved and analyzed.

According to Figure 6, each thread of the SMA bolt is divided into 11 beam elements and 12 nodes, and the positions of node Nos. 1–12 from the thread root to the pitch diameter line are sorted from small to large according to the $x$ coordinate value. The peak deformation of the thread under the three preloads are $0.01205 mm, 0.01462 mm, 0.01563 mm$ , respectively. According to the curve fitting of the node deformation of each thread, the deformation curve of the threads is in the shape of a parabola. From left to right, the slope of the deformation curve increases with increasing $x$ coordinate value. From top to bottom, the clearance between threads gradually decreases at the same position.

When $P_{0} = 14.7 kN, u_{0} = 0.25 mm$ , the bolt rod has not undergone nonlinear deformation, while the stress and strain at the root of the first thread are large because the pitch selected during modeling is $1 mm$ , which is a fine thread. With the increase in preloading force and preload interference, the center of the bolt rod gradually begins to undergo nonlinear deformation. As seen in Figure 6(b) and (c), after the phase transition, the stress increment of the bolt rod is small, but the strain increment is large, and the force and deformation of the bolt rod also show this law, which is in line with the characteristics of SMA material.

Figure 7 shows the mechanical response of M6 under different preloading forces. For the M6 bolted joint, a small preloading force can make the thread root and bolt rod enter the stage of nonlinear deformation because the stiffness of the M6 bolts is relatively small.

As shown in Figures 6 and 7, when comparing the stress–strain response curves of bolts M6 and M10 during preloading, it can be seen that the stress–strain response at the root of thread M10 is much larger than the center of the bolt rod. Moreover, the deformation of the M10 thread is larger, which is mainly because the M10 bolt has a fine thread, and the received force is greater. This also shows that when the bolt rod is subjected to a certain force, the thread of the M10 bolts is more prone to cracks at the root.

Figure 8 shows the law of mechanical response of the M6 SMA bolted joint under different external loads with an identical bolt preloading force of $2.1 kN$ . The external load $u_{e}$ is $0.01127 mm, 0.01227 mm, 0.01267 mm, 0.01307 mm$ . With increasing external load, the stress and strain of the first thread root and bolt rod increase and begin to show superelasticity. After unloading, the stress of the thread root and bolt rod can return to the stress level at the beginning of deformation. At the same time, the dissipated energy of the SMA bolt (area around the stress–strain curve in the loading and unloading cycle) also increases with increasing external load. These characteristics can not only significantly improve the vibration-resistant performance of the structure but also reduce the recovery work after strong vibrations to a certain extent, such as earthquakes, which are not possessed by common bolted joints.

Using the same method, the external load loading process of the M6 bolted joint with a preloading force of $1.7 kN$ is also simulated. Figure 9 shows the node deformation of each thread and stress–strain response curve of the bolt under an external load $u_{e} = 0.01138 mm$ . Figures 8 and 9 show that the dissipated energy of SMA bolts increases with increasing bolt preloading force. The relatively high preloading force of SMA bolts will improve the vibration-resistant capacity of SMA bolted joint damping structures, which can provide a useful reference for engineers.

To further verify the effectiveness of the proposed bolted joint model, as shown in Figure 10, the simulation results are compared with the experimental results of Jiang et al. (2020). The superelastic SMA bolted joint model can well analyze the nonlinear deformation and superelastic recovery of SMA bolts. The simulation results show the stress–strain response of the bolt rod in Figures 8(c) and 9. It can be seen in Figure 10 that under the same preloads $2.1 kN$ and $1.7 kN$ , the stress value of the simulation in the initial stage is higher than that of the experiment. This is because the experimental curves are the stress response of the first cycle in the third stage of loading cases 2 and 3, respectively, and the preloading force of the bolted joint was attenuated in this stage.

Figure 11 shows the stress–strain response curves of the bolt rod and the root of the first thread and the force–deformation responses of M6 bolt rod when the preloading force of M6 is $P_{0} = 5.73 kN$ and the external loads are $u_{e} = 0.01960 mm, u_{e} = 0.02060 mm, u_{e} = 0.02110 mm, u_{e} = 0.02160 mm$ , respectively. When the external load $u_{e} > 0.02060 mm$ , the bolt rod shows obvious superelastic hysteresis characteristics, which is more obvious when the external load is increased. Similarly, it can also be obtained that when the preloading force of M10 is $P_{0} = 16.53 kN$ and the external load is $u_{e} > 0.07111 mm$ , the bolt rod shows obvious superelastic hysteresis characteristics.

Figure 12 shows the change in the internal force of elements M6 and M10 with the load step under an external load. The preloading forces of bolts M6 and M10 are $P_{0}^{M 6} = 5.73 kN$ and $P_{0}^{M 10} = 16.53 kN$ , respectively. The three curves numbered 1, 2, and 3 in the figure represent the three elements in Figure 4(a). The forces of bolt element 3 and member element 1 are equal in magnitude and opposite in direction and are a pair of forces and reaction forces. Because the SMA bolt quickly enters the phase transition stage when the external load is loaded, it produces great deformation under a relatively small external load. Therefore, the internal force of element 2 is rapidly attenuated, and the contact surface of the connected member is separated. After separation, the internal force of element 2 is zero. From the comparison of slopes a, b, and c, it can be seen that slope b and slope c reflect the change in the internal force of the NiTi SMA bolt with the load step, and the rate of change is obviously smaller than slope a and has nothing to do with the evolution of slope a. During the unloading of the external load, the internal force of element 2 also increases rapidly. Combining the curve in Figure 12 with the stress–strain curve under preload and external load, it can be determined that when the external load seems to be greater than a certain value (e.g. $u_{e}^{M 6} = 0.02060 mm, u_{e}^{M 10} = 0.07111 mm$ ), the contact surface of the members is separated. Jiang et al. (2018) showed that this separation phenomenon of contact surfaces will lead to a significant increase in the attenuation rate of the bolt clamping force.

The above calculations are based on SMA beam elements. To better reflect the advantages of beam elements in nonlinear calculation efficiency, in this work, a two-dimensional 4-node plane element is used to establish the M6 bolted joint model, and the simulation of the preload process was carried out under $P_{0} = 6.0 kN, u_{0} = 0.3 mm$ . The solution result of the plane element is shown in Figure 13. The node deformation of each threaded, the stress–strain response of the plane element node at the root of the first thread with the largest stress and the center node of the bolt rod, and the force-deformation response of the bolt rod are solved. Compared with the solution result of the beam element in Figure 7, it can be seen that the error of the peak deformation of the first thread and the strain at the root of the first thread are the largest. The peak deformations of the first thread of the plane element and beam element are $0.0071$ and $0.0093 mm$ , respectively, and the strains at the root of the first thread are 0.96% and 1.63%, respectively. The main reason for these errors is that the beam element considers the geometric nonlinearity and SMA material nonlinearity, which is more in line with the actual working conditions. In addition, the deformation trend of each thread is the same from left to right, and the stress and strain values of the bolt rod obtained by the two elements is the same. In general, the numerical results of the two elements are relatively close.

The number of substeps specified in the two element solvers is 10, and the maximum number of substeps is 300. At the same time, record the CPU time of the solution, the number of elements and the number of nodes that the solution is divided into. The solution information of the two elements is shown in Table 4.

Table 4.

Beam element and plane element solution information.

Element type	Number of elements	Number of nodes	CPU solution time
Beam element	119	130	23.718750
Plane element	24,667	25,107	93.453123

It can be seen in Table 4 that more than 20,000 elements need to be divided when using plane elements to solve this problem. Where the bolt and nut engage and where the nut contacts the members, contact elements are needed to set contact. The modeling process is more complicated, and convergence is difficult. However, when solving with beam elements, only more than 100 elements need to be divided. Furthermore, the simplified calculation method proposed in this work in which the bolt threads and nut threads are coupled together in the y-direction degree of freedom at the pitch diameter line can avoid setting the contact elements and improve the convergence of the algorithm. In addition, it can be seen from the CPU time that the time taken to solve the beam element is much less than that of the plane element. The above comparison shows the advantages of the new proposed model in computational efficiency and convergence.

In addition, in the system-level analysis, the SMA connection modeling can adopt the multi-spring approach (Fang et al., 2018). The method of using beam element to simulate SMA bolted joint proposed in this paper can also be applied to system-level analysis. The advantage of using the approach proposed in this paper is that it can calculate the tensile deformation, stress and strain of the bolt rod under load, and the deformation of the bolt and nut thread bending caused by the tension of the bolt rod. Of course, compared with the solid element, the method proposed in this paper may have some limitations in calculation accuracy, but it also has higher calculation efficiency. What’s more, compared with the bolt preloading model of one-dimensional bar element, the calculation accuracy is also improved. The concrete application of beam element in SMA connection modeling in system-level analysis is worth exploring in the future.

5. Conclusions

In this paper, a finite element model for the analysis of the mechanical properties of bolted joints based on superelastic SMA beam elements is established, and the stress–strain response and nonlinear deformation law of different types of bolts under preload and external load are discussed. The main conclusions are as follows.

The deformation curve of each thread has a parabolic shape, the slope of the deformation curve from the root of the thread to the middle diameter line gradually increases, and the clearance between each thread gradually decreases.

Due to the fine thread, the stress–strain level at the root of the M10 bolt thread is higher, which is significantly higher than the stress state of the bolt rod, and the M10 bolt thread is more prone to cracks at the root.

When the preloading force is constant and the external load increases to a certain value (e.g. $u_{e}^{M 6} = 0.02060 mm, u_{e}^{M 10} = 0.07111 mm$ ), the contact surface of the members is separated, and the bolt rod shows obvious superelastic hysteresis characteristics.

Compared with the traditional calculation methods, the calculation efficiency of the new proposed method is improved by four times, and the calculation accuracy is relatively high.

Footnotes

Appendix A

Appendix B

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by [National Natural Science Foundation of China] grant number [51775406 and 52175246] and [Natural Science Foundation of Shaanxi Province] grant number [2022JQ-561].

ORCID iDs

Xingkun Dong

Xiangjun Jiang

References

Aljuboury

Rizvi

Grove

, et al. (2021) A numerical investigation of the sealing performance and the strength of a raised face metallic bolted flange joint. International Journal of Pressure Vessels and Piping 189: 104255.

Bettini

Rigamonti

Sala

(2021) Shape memory alloy engineering. In: Concilio

Antonucci

Auricchio

, et al. (eds) SMA for Composite Aerospace Structures. Oxford: Butterworth-Heinemann, pp.561–590.

Brinson

(1993) One-dimensional constitutive behavior of shape memory alloys: Thermomechanical derivation with non-constant material functions and redefined martensite internal variable. Journal of Intelligent Material Systems and Structures 4(2): 229–242.

Chowdhury

Rahmzadeh

Alam

(2019) Improving the seismic performance of post-tensioned self-centering connections using SMA angles or end plates with SMA bolts. Smart Materials and Structures 28(7): 075044.

De la Flor

Urbina

Ferrando

(2011) Asymmetrical bending model for NiTi shape memory wires: Numerical simulations and experimental analysis. Strain 47(3): 255–267.

Fang

Yam

MCH

Lam

ACC

, et al. (2014) Cyclic performance of extended end-plate connections equipped with shape memory alloy bolts. Journal of Constructional Steel Research 94: 122–136.

Fang

Wang

, et al. (2017) Self-centring behaviour of steel and steel-concrete composite connections equipped with NiTi SMA bolts. Engineering Structures 150: 390–408.

Fang

Wang

Ricles

, et al. (2018) Application of an innovative SMA ring spring system for self-centering steel frames subject to seismic conditions. Journal of Structural Engineering 144(8): 04018114.

Ghamari

Amini

De Crescenzo

, et al. (2020) Investigating the behaviour of steel end-plate connections with shape memory alloy bolts. Proceedings of the Institution of Civil Engineers-Structures and Buildings. London: Institution of Civil Engineers-Structures and Buildings, pp.1–16.

10.

Gillet

Patoor

Berveiller

(1998) Calculation of pseudoelastic elements using a non-symmetrical thermomechanical transformation criterion and associated rule. Journal of Intelligent Material Systems and Structures 9(5): 366–378.

11.

Zhu

, et al. (2021) Influence of structural parameters of shape memory alloy corrugated gaskets on the contact pressure of bolted flange joints. Advances in Materials Science and Engineering 2021: 1–19.

12.

Helbert

Volkov

Evard

, et al. (2021) On the understanding of damping capacity in SMA: From the material thermomechanical behaviour to the structure response. Journal of Intelligent Material Systems and Structures 32(11): 1167–1184.

13.

Javanbakht

Öchsner

(2018) Computational statics revision course. In: Javanbakht

Öchsner

(eds) Euler–Bernoulli Beams and Frames. Cham: Springer. pp.63–144.

14.

Jiang

Chen

, et al. (2020) Experimental investigation of self-loosening behavior of bolt joints with superelastic shape-memory alloy by macroscopic-mechanical response and microscopic evolution. Journal of Engineering Mechanics 146(7): 06020004.

15.

Jiang

Huang

Wang

, et al. (2018) Finite element analysis for the self-loosening behavior of the bolted joint with a superelastic shape memory alloy. Materials 11(9): 1592.

16.

Jiang

Wang

, et al. (2019) Self-loosening behavior of bolt in curvic coupling due to materials ratcheting at thread root. Advances in Mechanical Engineering 11(5): 841137.

17.

Jiang

Zhu

Hong

, et al. (2013) Investigation into the loosening mechanism of bolt in curvic coupling subjected to transverse loading. Engineering Failure Analysis 32: 360–373.

18.

Jiang

Zhu

Hong

, et al. (2014) Development and validation of analytical model for stiffness analysis of curvic coupling in tightening. Journal of Aerospace Engineering 27(4): 04014012.

19.

Jiang

Zhu

Hong

, et al. (2016) Stiffness analysis of curvic coupling in tightening by considering the different bolt structures. Journal of Aerospace Engineering 29(3): 04015076.

20.

Kan

Kang

Guo

(2012) Finite element implementation of a super-elastic constitutive model for transformation ratchetting of NiTi alloy. International Journal of Computational Methods 09(1): 1240022.

21.

Machado

Louche

Alonso

, et al. (2015) Superelastic cellular NiTi tube-based materials: Fabrication, experiments and modeling. Materials & Design 65: 212–220.

22.

Mehrabi

Andani

Kadkhodaei

, et al. (2015) Experimental study of NiTi thin-walled tubes under uniaxial tension, torsion, proportional and non-proportional loadings. Experimental Mechanics 55(6): 1151–1164.

23.

Motosh

(1976) Determination of joint stiffness in bolted connections. Journal of Engineering for Industry 98: 858–861.

24.

Najmabad

Razzaghi

Alahi

(2021) Retrofitting of a damaged liquid storage tank using steel and SMA anchor bolts. International Journal of Pressure Vessels and Piping 194: 104530.

25.

Nassar

Abboud

(2009) An improved stiffness model for bolted joints. Journal of Mechanical Design 131(12): 1990.

26.

Poorasadion

Arghavani

Naghdabadi

, et al. (2014) An improvement on the Brinson model for shape memory alloys with application to two-dimensional beam element. Journal of Intelligent Material Systems and Structures 25(15): 1905–1920.

27.

Qin

Han

Chu

(2016) Bolt loosening at rotating joint interface and its influence on rotor dynamics. Engineering Failure Analysis 59: 456–466.

28.

Shoji

Nagata

(2002) Analysis of Gasketed Flanges With Ordinary Elements Using Apdl Control. Narashino, Japan: Toyo Engineering Corporation.

29.

Toghroli

Nasirianfar

Shariati

, et al. (2020) Analysis of extended end plate connection equipped with SMA bolts using component method. Steel and Composite Structures: An International Journal 36(2): 213–228.

30.

Wang

Jiang

Wang

(2019) Numerical simulation and behavior insights of steel columns with SMA bolts towards earthquake resilience. Journal of Constructional Steel Research 161: 285–295.

31.

Wang

Fang

Liu

(2017) Self-centering beam-to-column connections with combined superelastic SMA Bolts and steel angles. Journal of Structural Engineering 143(2): 04016175.

32.

Weli

Vigh

(2021) Blast Performance Evaluation of steel moment-resisting frame equipped with smart bolted connection. Journal of Performance of Constructed Facilities 35(5): 04021050.

33.

Zhou

Wang

(2015) Finite element analysis of relationship between tightening torque and initial load of bolted connections. Advances in Mechanical Engineering 7(5): 15588477.

34.

Zheng

Fang

Liang

, et al. (2021) An innovative seismic-resilient bridge with shape memory alloy-washer-based footing rocking RC piers. Journal of Intelligent Material Systems and Structures 32(5): 549–567.

35.

Zhou

Lin

Liu

(2014) Advanced materials research. In: Choi

S-B

Kim

Y-H

(eds) Study on Fe-Mn-Si Shape Memory Alloy Anti-Loosening Bolt, vol. 900. Stafa-Zurich: Trans Tech Publications, pp.78–82.