## Abstract

In designing a bolted joint consisting of dissimilar hollow cylinders, the load factor (the ratio of an increment in axial bolt force to an external tensile load) is important. However, no researches on the characteristics of bolted joints consisting of dissimilar joint members have been conducted. In this paper, the effects of load application positions, the ratio *b*/*a* of the outside diameter to the inside diameter of dissimilar hollow cylinders, Young's modulus ratio between the dissimilar hollow cylinders on the load factor, and a load when the interface separation starts are examined using finite element method (FEM) calculations. As a result, it is found that the values of the load factor decrease as the ratio *b*/*a* increases, and the load application position approaches the interfaces, while the value of the load factor is independent of the bolt preload. For verification of the FEM calculations, experiments to measure the load factor and a load when the interfaces start to separate were carried out. The FEM results are fairly coincident with the experimental results. It is found that the value of the load factor is less than 0.15 for steel and aluminum joints, it is less than 0.1 for steel and steel joints, and it is less than 0.2 for aluminum and aluminum joints, where the bolt material is steel in this study. Finally, using the obtained load factor, a new design method for bolted joints with dissimilar hollow cylinders is demonstrated for determining the nominal diameter of bolt, the bolt strength grade, a target tightening torque, and a target bolt preload. Additionally, in the design procedure, an issue if a washer should be inserted or not inserted is discussed as well as determination of bolt preload.

## 1 Introduction

Bolted joints are widely used in a lot of industrial fields such as mechanical structures, architecture, aerospace engineering, automobile vehicles, chemical and oil plants, and so on. However, a lot of bolted joints have been designed empirically and used in practice. Some accidents, therefore, are assumed to occur due to inadequate design for bolted joints in the world. In designing bolted joints, the nominal diameter of bolt, the bolt strength grade, a target bolt preload, and a tightening torque should be determined rationally. Therefore, it is necessary for mechanical engineers to know the mechanical behavior of bolted joints under external tensile loads, that is, to predict an increment in axial bolt force and a maximum bolt stress when the external tensile loads are applied to the bolted joints from bolt strength standpoint as well as to predict a force *F*_{c} which is eliminated from the interfaces of joint members from the joint integrity standpoint. The ratio of the increment *F*_{t} in axial bolt force to the external load *W* is defined as the load factor Φ (= *F*_{t}/*W*) and the force *F*_{c} is obtained as *F*_{c} = (1 − Φ)*W* using the load factor Φ. Some researches [1–4] were carried out to obtain the value of the load factor. Thum and Debus [2] and Bickford [4] proposed the formula Φ = *K*_{t}/(*K*_{t} + *K*_{c}), where *K*_{t} is the stiffness for bolt–nut system, and *K*_{c} is the compressive stiffness for joint members. This formula has been applied for a long time. However, it has been noticed that the load factors obtained from Thum's formula [2] are larger than the experimental values [5–7] even if the values of *K*_{c} [8–10] and *K*_{t} [11] can be calculated more precisely. Additionally, it is known that the effect of the load application position is not taken into account in Thum's formula [5–7].

Yoshimoto et al. [6] and Sawa and Omiya [7] proposed a new formula for obtaining the value of the load factor more precisely and theoretically by the equation Φ = *K*_{t}/(*K*_{t} + *K*_{c}) × (*K*_{c}/*K*_{pt}) for bolted joints where two similar hollow cylinders were clamped by a bolt and a nut [6,7]. In this formula, a new tensile stiffness *K*_{pt} is introduced, and the effect of the load application position is taken into account. The values of the load factor obtained from this formula are fairly coincident with the experimental results. Furthermore, this newly proposed formula was applied for analyzing the load factor of some bolted joints [12–14] using the theory of elasticity. The authors researched on the mechanical characteristics in bolted joints with similar material joint members of T-shape flanges [15] and circular flanges [16] using finite element method (FEM).

However, most previous papers mentioned above dealt with bolted joints with similar material joint members, and the material of the joint members is steel, while that of bolts is also steel. Recently, for lightening own weight of bolted joints, the materials of joint members have been used such as aluminum, polymer, and new material, while the bolt material used is steel. Also, it is difficult to apply the analytical methods mentioned above to the behavior of actual bolted joints. Additionally, joint members with dissimilar materials are clamped together such as aluminum and steel combination in practical bolted joints. Thus, it is necessary to research the characteristics of bolted joints with dissimilar material joint members under tensile loads in practice such as the load factor and the contact stress distribution at the interfaces which governs the clamping effect. As an external tensile load increases, the contact stress at the interfaces between the joint members decreases, and then a separation of the interfaces starts. When the interfaces start to separate, a possibility of bolt loosening occurs [17]. Bolted joints in practice of which the interfacial contact stress is reduced under external axial tensile loads are sometimes subjected to some unexpected external loads such as transverse loads, impact loads, and thermal loads. At that time, the possibility of the bolt loosening increases when the transverse loads as well as thermal loads are applied. So, it is desirable to increase the bolt preload for preventing the separation between the interfaces. As the bolt preload increases, the bearing stress of lower rigidity joint member should be less than the critical contact stress because the larger bearing stress leads to plastic deformation and a reduction in bolt preload (nonrotational loosening).

Furthermore, it is also necessary for mechanical engineers to establish a rational design method for determining the nominal diameter of bolt, the bolt strength grade (classification), a target bolt preload, and a target tightening torque for bolted joints with dissimilar material joint members. VDI 2230 demonstrates one design method of bolted joints with hollow cylinders [1,4] based on some important data. However, the similar joint members are dealt in VDI 2230 [1], and some issues remain to be examined. No research is conducted on the mechanical characteristics of bolted joints with dissimilar material joint members, and a rational design method mentioned is not established sufficiently.

The objectives of this paper are to examine the load factor and a load when the interfaces start to separate and to propose a new design method for bolted joints consisting of dissimilar hollow cylinders under tensile loading using FEM. The effects of a position where a load is applied to the outside diameter of hollow cylinders and the ratio of the outside to the inside diameters are examined on the values of the load factor. For verification of the FEM results, the experiments were carried out to measure the load factor and a load when the interfaces start to separate.

The materials of the joint members are chosen as combinations of steel and aluminum, aluminum and aluminum, and steel and steel, while bolt material is steel. The FEM results of the load factor and the load when the interfaces start to separate are compared with the measured results. Then, for safe and integral design of bolted joints, fundamental items to be considered are proposed. Using the obtained values of the load factor from the FEM calculations, a simple and rational design method for bolted joints with dissimilar hollow cylinders is proposed newly for satisfying the above fundamental items, that is, the method is demonstrated how to determine the nominal diameter of bolt, bolt strength grade, and a bolt preload taking account of the tightening coefficient *Q* and the equivalent bolt strength evaluation. Some calculation examples are demonstrated for determining the strength grade and the nominal diameter of bolt, and discussion is carried out on washer insertion and a determination of the higher bolt preload for practical use.

## 2 Finite Element Method Stress Analysis and the Load Factor of Bolted Joints Under Axial Tensile Loads

Figure 1(a) shows a bolted joint in which two dissimilar (materials are different, while the dimensions are the same) hollow cylinders are clamped with a bolt and a nut with an initial preload *F*_{f}. Figure 1(b) shows the case when an external tensile load *W* is applied to the outside diameter of the hollow cylinders. When the external tensile load *W* is applied to the joint, an increment *F*_{t} in axial bolt force occurs and a force *F*_{c} is eliminated from the interfaces, where *F*_{t} + *F*_{c} = *W*. The ratio *F*_{t} to the load *W* is called as the load factor Φ (= *F*_{t}/*W*). When the value of the load factor Φ is obtained, the force *F*_{c} is obtained from the equation *F*_{c} = (1 − Φ)*W* [2,6,7]. The inside diameter of the hollow cylinders is designated by 2*a* and the outside diameter by 2*b*. The thickness of the hollow cylinder is designated by *h* as shown in Fig. 1(a). Figure 2 shows an example of mesh divisions in FEM. The initial preload *F*_{f} is applied at the cross section of the bolt shank, and then the external tensile load *W* is applied to the outside diameter of the hollow cylinders. In the FEM calculations, axisymmetrical elements are used, and the employed FEM software is ansys (ver. 17). The threads of bolt and those of the nut are neglected and are replaced with the cylinders. The part of bolt threads is connected to the inside of the nut. The hexagon of bolt head and the nut are replaced with equivalent circles. The numbers of the elements are about 14,207–19,795, and those of the nodal points are about 14,666–20,000.

For examining the effect of the ratio *b*/*a* of the outside diameter 2*b* to the inside diameter 2*a* on the load factor, the values of *b*/*a* are chosen as 1.6, 2.0, 3.0, 4.0, and 5.0, while the thickness *h* and the inside diameter 2*a* of the hollow cylinder (I) and (II) are held constant as *h* = 28 mm and 2*a* = 14 mm, respectively. The material of (I) is assumed as steel and that of (II) as aluminum. Young's modulus of steel is 207 GPa and that of aluminum as 68 GPa. Poisson's ratio is held constant as 0.3 for steel and aluminum. In the FEM calculations, the load factors of bolted joints with steel and steel joint members and aluminum and aluminum joint members are also analyzed. As the load *W* increases, the contact stress at the interfaces decreases, and then it becomes to be zero at the outside element of the interfaces. For examining the effect of the ratio *b*/*a* of the outside diameter 2*b* to the inside diameter 2*a* on the load factor, the values of *b*/*a* are chosen as 1.6, 2.0, 3.0, 4.0, and 5.0, while the thickness *h* and the inside diameter 2*a* of the hollow cylinder (I) and (II) are held constant as *h* = 28 mm and 2*a* = 14 mm, respectively. The material of (I) is assumed as steel and that of (II) as aluminum. Young's modulus of steel is 207 GPa and that of aluminum as 68 GPa. Poisson's ratio is held constant as 0.3 for steel and aluminum. In the FEM calculations, the load factors of bolted joints with steel and steel joint members and aluminum and aluminum joint members are also analyzed. As the load *W* increases, the contact stress at the interfaces decreases, and then it becomes to be zero at the outside element of the interfaces.

At that time, a load *W*_{s} is defined as the load when the interfaces start to separtate. In the FEM calculations, the minimum sizes of elements around the outside diameter of the interfaces in the joint members are chosen as 20–50 *μ*m for obtaining a precise load *W*_{s}.

## 3 Experimental Method

For verification of FEM results, the load factor and the load *W*_{s} when the interfaces start to separate are measured. Figure 3(a) shows the dimensions of hollow cylinders, and the materials are chosen as steel (carbon steel S55C, JIS) and aluminum (Al, A7570). The screw threads were manufactured at the outside circumference of hollow cylinders for applying an external load. The pitch diameters *d*_{2} of the screw threads of hollow cylinders are chosen as *d*_{2} = 27.7 mm for M30, *d*_{2} = 42.0 mm for M45, and *d*_{2} = 52.4 mm for M56, respectively. The outside diameter 2*b* is assumed to be the pitch diameter *d*_{2} as mentioned above. The hollow cylinder is named as (TS1) for the hollow cylinder with M30, (TS2) for the hollow cylinder with M45, and (TS3) for the hollow cylinder with M56. The inside diameter is held constant at 2*a* = 14 mm, and the thickness is also held constant at *h* = 28 mm. Figure 3(b) shows the dimensions of bolt used in the experiments. The nominal diameter of bolt was chosen as M12. Since two strain gauges were attached at the shank of the bolts, the bolt hole diameter 2*a* is determined as 14 mm mentioned above. The strain gauges attached were calibrated before the measurements. Figure 3(c) shows the position where a load is applied. For applying a tensile load to the joints, attachments (screw threads were manufactured at the inside circumference of the attachments) were joined with the threaded outside circumference of the joint members (hollow cylinders). The engagement in screw threads between the joint members and the attachments can be adjusted. The position of the load application is named as (WI) and (WII). The position (WI) means that the engagement of screw threads is from the upper surface to 0.5*h*. (WII) is from the upper surface to zero as shown in Fig. 3(c). Figure 3(d) shows assumption of the shear force distributions (=*W*_{y}) due to the external tensile load *W*. It is assumed as the uniform and the linear as shown in Fig. 3(d) in the FEM calculations. Figure 4(a) shows a schematic of experimental setup for applying tensile loads to the joints. Figure 4(b) shows a detail of bolted joint members and the attachments which are joined with the joint members. The experimental setup was installed on a material testing machine, and a compression load was applied to the setup. The compression load was converted to a tensile load through two U-shaped pedestals shown in Fig. 4. Two pedestals are assembled in different phase of 90 deg. Two attachments with the bolted joint members are fixed to the pedestals as shown in Fig. 4. Then, an external tensile load *W* is applied to the joint. The external load *W* was measured by a load cell which was attached on the setup, and a change in axial bolt force was measured from the strain gauges attached at the shank of bolt. The relationship between the load *W* and the increment *F*_{t} in the axial bolt force was measured. In addition, the load *W*_{s} when a separation of the interfaces occurs was also measured from the relationship between the load *W* and the increment *F*_{t} in the axial bolt force.

## 4 Finite Element Method Results and Comparison With Experimental Results

### 4.1 Comparison of the Load Factor and Load *W*_{s} Between the Finite Element Method Results and the Experimental Results.

In the experiments, screw threads were manufactured at the outside diameter of the hollow cylinder as shown in Fig. 3. So, when an external load is applied to the joint, the effect of the shear force distribution between the engaged threads is examined by changing the shear force distribution as the uniform and the linear (Fig. 3(d)). It was found that the difference of the values of the load factor is less than 2% between the uniform (Table 1, St–Al joint) and the linear shear force distributions for the case of *b*/*a* = 2.0 (St–Al joint) in this study. In addition, no difference in the value of *W*_{s} is found between the two cases. As a result for simplicity, the uniform shear force distribution is applied for obtaining the load factor in the FEM calculations.

Combination | St–Al | Al–Al | St–St | St–Al |
---|---|---|---|---|

Load position | (WII) | (WII) | (WII) | (WI) |

Φ (FEM) | 0.147 | 0.180 | 0.088 | 0.205 |

Φ (Expt.) | 0.149 | 0.187 | 0.097 | 0.192 |

W_{s} (kN) (FEM) | 12.9 | 14.9 | 7.88 | 15.8 |

W_{s} (kN) (Expt.) | 11.5 | 13.5 | 8.26 | 14.5 |

Combination | St–Al | Al–Al | St–St | St–Al |
---|---|---|---|---|

Load position | (WII) | (WII) | (WII) | (WI) |

Φ (FEM) | 0.147 | 0.180 | 0.088 | 0.205 |

Φ (Expt.) | 0.149 | 0.187 | 0.097 | 0.192 |

W_{s} (kN) (FEM) | 12.9 | 14.9 | 7.88 | 15.8 |

W_{s} (kN) (Expt.) | 11.5 | 13.5 | 8.26 | 14.5 |

Figure 5 shows comparison of the load factor and a load *W*_{s} when an interface separation occurs between the FEM results and the experimental results. Figure 5(a) is the case of steel and aluminum (St–Al) hollow cylinder joint. The bolt preload *F*_{f} is chosen as 0.5*σ*_{y}*A*_{s} (*F*_{f} = 26 kN) and 0.3*σ*_{y}*A*_{s} (*F*_{f} = 16 kN), where *σ*_{y} is the yield stress of bolt, and *A*_{s} is the stress area. The hollow cylinder is ((TS1) = M30) and the load application position is (WII) shown in Fig. 3(c). The abscissa is the load *W*, and the ordinate is the axial bolt force *F*_{f} + *F*_{t}. The solid lines show the experimental results, and the dashed lines show the FEM results. As the load *W* increases, the increment *F*_{t} in the axial bolt force increases linearly, and when a separation of the interfaces starts, *F*_{t} plots a curved line. The slope of the linear line means the load factor (= *F*_{t}/*W*). The load *W*_{s} when the interfaces start to separate is measured such that the linear line changes to be curved line in the relationship between the load *W* and the increment in the axial bolt force *F*_{t}, where the designation (*x*) is the value of *W*_{s}. The values of the load factor are obtained from FEM calculations as 0.147 for the bolt preload *F*_{f} of 0.3*σ*_{y}*A*_{s} and 0.147 for *F*_{f} = 0.5*σ*_{y}*A*_{s}, respectively. The experimental values of the load factor are obtained as 0.152 for the bolt preload *F*_{f} = 0.5*σ*_{y}*A*_{s} and as 0.146 for *F*_{f} =0.3*σ*_{y}*A*_{s}. The average of the two values of the measured load factor is 0.149. The FEM results of the load factor are fairly coincident with the experimental results. The difference in the slopes of the linear line between *F*_{f} = 0.3*σ*_{y}*A*_{s} and *F*_{f} = 0.5*σ*_{y}*A*_{s} is found to be extremely small as shown in Fig. 5(a). Thus, it can be said that the value of the load factor is independent of the bolt preload. In the case where the preload is *F*_{f} = 0.5*σ*_{y}*A*_{s} (*F*_{f} = 26 kN), the mean initial contact stress at the interfaces is 57.9 MPa, and it is 33.4 MPa after the tensile load *W* of 12.9 kN is applied, while the load factor Φ used is 0.147. The value of *W*_{s} is in a fairy good agreement between the FEM result and the measured result in Fig. 5(a).

Figure 5(b) shows comparison of the load factor in the case where joint members are aluminum and aluminum (Al–Al), the dimensions of the hollow cylinder are (TSI = M30), the bolt preload is *F*_{f} = 0.3*σ*_{y}*A*_{s}, and the position of the load is (WII). It is observed that the load factor obtained from FEM (dashed line) is fairly agreed with the experimental result (solid line). The value of the load factor is obtained as Φ = 0.180 from the FEM and 0.187 from the experiments. These values are greater than those in the case where the joint members are steel and aluminum (St–Al) shown in Fig. 5(a). The value of *W*_{s} is also matched between the FEM and measured results in Fig. 5(b). Figure 5(c) shows comparison of the load factor and the load *W*_{s} for the joint with steel and steel (St–St) hollow cylinders (TS1), and the load application position is (WII). The value of the load factor obtained from the FEM calculations is 0.088, and it is 0.097 from the experiment. They are in a fairly good agreement. The FEM result of the load *W*_{s} is fairly matched with the measured result. When the interfaces are separated completely, the increment in the axial bolt force *F*_{t} equals to the load *W*, i.e., the force *F*_{t} plots the line with 45 deg.

Table 1 shows comparison of the values of the load factor and the load *W*_{s} between the FEM results and the experimental results for the bolt preload *F*_{f} = 16 kN. The value of the load factor for the joint (St–Al) where the load application position is (WI) shown in Fig. 3(c) is also described in Table 1. It is found that the value of the load factor for the joint with the load application position (WI) is larger than that of the load application position (WII) for the same material combination. It is therefore noticed that the value of the load factor is affected by the load application position. From comparison mentioned above, the value of the load factor is the largest for the aluminum and aluminum (Al–Al) joint, and the smallest is the steel and steel (St–St) joint under the same load application (WII) in this study. This is because that as Young's modulus of joint members decreases, the compressive stiffness *K*_{c} of joint members decreases. As a result, the value of the load factor increases.

In Table 1, it is also found that the load *W*_{s} increases as the value of the load factor increases. In addition, from the values of the load factor, the load *W*_{s} when the interfaces start to separate can be predicted. When the value of the load factor is the largest, the force *F*_{c} is the smallest from the equation *F*_{c} = (1 − Φ)*W*. Therefore, when the value of the load factor is the largest, the load *W*_{s} is the largest for Al–Al joint in this study. It can be concluded that as the value of the load factor increases, the load *W*_{s} increases.

### 4.2 The Effects of the Ratio *b*/*a* of the Outside Diameter 2*b* to the Inside Diameter 2*a* in Joint Members and the Load Application Position on the Load Factor.

The FEM calculations of the load factor and the load *W*_{s} were verified by the experimental results in Sec. 4.2. In this section, using the FEM calculations, the effects of load application positions and the ratio *b*/*a* are examined on the values of the load factor. Figure 6 shows three types of the load application positions in the FEM calculations, where the joint members are steel and aluminum (St–Al). The external tensile load *W* is applied to the width of 7 mm as uniform shear stress distribution at the outside circumference of the hollow cylinders in the FEM calculations (without engagement of screw threads) where the inside diameter 2*a* is held constant as 14 mm. In this paper, the load application position is named as (upper), (middle), and (lower) as shown in Fig. 6. Figure 7 shows the FEM results of the load factor. The abscissa is the ratio *b*/*a*, and the ordinate is the load factor Φ. It is found in Fig. 7 that the value of the load factor decreases as the value of *b*/*a* increases. Additionally, it is observed that the value of the load factor is the highest for the load application position is (upper), and it is the smallest for the (lower). When the value of *b*/*a* is around 4.0–5.0, the values of the load factor of the joints are almost zero for all load application positions, i.e., the increment in the axial force *F*_{t} decreases to be zero, while the force eliminated from the interfaces approaches *F*_{c} = *W*, namely, the interfaces are perfectly separated. The value of the load factor is affected substantially by the load application position.

### 4.3 The Load Factor of Joint Consisting of the Same Material Joint Members.

In special case, the effect of *b*/*a* on the load factor of joint consisting of the same material joint members is examined, that is, the material of the joint members is the case of steel and steel (St–St) and the case of aluminum and aluminum (Al–Al). Figure 8 shows the effect of the ratio *b*/*a* on the load factor for the joints with St–St and Al–Al hollow cylinders, while the bolt material is kept as steel. The abscissa indicates the ratio *b*/*a*, and the ordinate is the value of the load factor for the dimensions of the hollow cylinders shown in Fig. 6. The position of the load application is chosen as (upper) shown in Fig. 6. In Fig. 8, it is found that the value of the load factor for the joint with Al–Al hollow cylinders is larger than that for the joint with St–St hollow cylinders. Under the condition where the load application position is the same and the dimensions of the hollow cylinders are the same, the compressive stiffness *K*_{c} is smaller in the case of Al–Al hollow cylinders than that of St–St; thus, the load factor of Al–Al joint becomes larger. In the obtained results of the load factor (Figs. 7 and 8), it can be concluded that the value of the load factor for bolted hollow cylinder joints in this study (in the case where *b*/*a* is larger than 2.0) is less than 0.1 for St–St joint, less than 0.15 for St–Al joint, and less than 0.2 for Al–Al joint.

## 5 Discussion on the Load Factor of Special Bolted Joints

*F*

_{f}, and when a load

*W*is applied, an increment

*F*

_{t}in the axial bolt force occurs. The grip length is denoted by

*l*

_{f}(=2

*h*), and the load

*W*is applied to a position

*x*/2 from the interfaces of joint members as shown in Fig. 9. In addition, Young's modulus of upper joint member (hollow cylinder) is denoted by

*E*

_{1}and that of the lower joint member by

*E*

_{2}. The cross-sectional area of joint members is denoted by

*A*

_{f}. When the load

*W*is applied to the joint as shown in Fig. 9(a), the upper part of the hollow cylinder with the height of

*x*/2 is elongated by $\lambda \u20321$ and the lower part with the height of

*x*/2 by $\lambda \u20322$ as shown in Fig. 9(b). The total elongation

*λ*$\u2032$ between both the bearing surfaces in the joint members (thin hollow cylinders) is obtained by $\lambda \u20321+\lambda \u20322=((W\xb7(x/2))/Af)((1/E1)+(1/E2))\u2009$from Hooke's law. When the load

*W*is applied, the tensile stiffness

*K*

_{pt}of the joint members is defined by the following equation:

*W*is applied, an additional force

*F*

_{t}is applied to the both bearing surfaces as shown in Fig. 9(c). The shrinkages at both the bearing surfaces of hollow cylinders are denoted by$\u2009\lambda \u20331\u2009and\u2009\lambda \u20332$, respectively. The total shrinkage

*λ*$\u2033$ of the joint members due to the compressive force

*F*

_{t}is expressed by the following equation:

*K*

_{c}is defined as follows:

*K*

_{t}[11], the bolt elongation is expressed as

*F*

_{t}/

*K*

_{t}. The elongation

*λ*at the bearing surfaces in joint members is expressed by

*λ*=

*λ*$\u2032$ −

*λ*$\u2033$, and it is obtained by $(W/Kpt)\u2212(Ft/Kc)$. So, the following Eq. (4) is obtained from the equilibrium of the displacements at the bearing surfaces between the bolt (or nut) and the joint members:

*F*

_{t}to

*W*is obtained by the following equation:

*F*

_{t}to the load

*W*, which is called as the load factor Φ =

*F*

_{t}/

*W*, is obtained by the following equation:

*x*=

*l*

_{f}. This means that the external tensile load is applied to the bearing surfaces of the joint members. Practically, this case is not correct. The maximum value of the load factor Φ is obtained by the following Eq. (6). This equation corresponds to Thum's formula [2] for obtaining the load factor. So, the value of the load factor from Thum's formula [2] is not correct in practical use

Thus, the value of the load factor obtained from Thum's formula [2] is larger than that in practical bolted joints [5–7].

For the joint members shown in Fig. 6, when the outside diameter is chosen as that of the bearing surfaces of the bolt (M12), that is, 2*b* as 22.4 mm (= 1.6 × 2*a*) while the inside diameter 2*a* is 14 mm, the thickness of the joint member is chosen as 28 mm (*l*_{f} = 56 mm), and the material of upper hollow cylinder is chosen as aluminum and the lower one as steel (St–Al), the stiffness *K*_{c} is obtained as 439 N/$\mu m$, while the stiffness *K*_{t} for bolt–nut system is obtained as *K*_{t} = 272 N/$\mu m$ from Ref. [11]. So, the load factor Φ is obtained as Φ = 0.334 from Eq. (5) when the load is applied at (upper) shown in Fig. 6 where the value of *x* is *x* = 56 − 2 × 7/2 = 49 mm and the value of *x*/*l*_{f} is 0.875 in Eq. (5). When the positions are (middle) and (lower), the values of *x* are *x* = 28 mm and *x* = 7 mm, respectively. The values of the load factor are obtained as 0.191 and 0.0478 for (middle) and (lower), respectively. Also, the values of the load factor for St–St joint and Al–Al joint are obtained as 0.205 and 0.422, respectively, when the load is applied at (upper). The obtained values of the load factor are plotted by the designation X in Figs. 7 and 8, and they are in a fairy good agreement with the FEM results. As a result, it can be concluded that the maximum value of the load factor is predicted using Eq. (5).

## 6 Discussion on Design for Bolted Joints Consisting of Dissimilar Hollow Cylinders Under Tensile Loads

In an actual design of bolted joints, it is important how to determine the nominal diameter *d* of bolt, the bolt strength grade (classification), and the bolt preload under given conditions such as an external tensile load *W*, the material of joint members, required interfacial contact stress *σ*_{c}, and a tightening coefficient *Q* (scatter in preloads) [1,17]. In designing bolted joints, the following items should be satisfied for achieving the joint function and integrity.

An equivalent bolt stress

*σ*_{eqv}should be less than the yield stress*σ*_{y}of bolt (static strength)_{.}A bolt stress amplitude

*σ*_{a}under repeated external loading should be less than the bolt fatigue stress*σ*_{A}(fatigue strength).Never bolt loosening occurs.

The contact surfaces at the interfaces should be kept in contact for achieving the joint function, and the reduced contact stress of joint under external load should be larger than a required contact stress $\sigma c$ (the interfacial contact stress condition).

The bearing stress

*σ*_{w}should be less than the critical contact stress*σ*_{cri}(the bearing stress condition).

*τ*occurs at the engaged threads due to a torque

*T*

_{s}which is applied to the engaged threads. The torque

*T*

_{s}is obtained as Eq. (7) which is only applicable to threads with a 30 deg angle as ISO threads [1,4] for the highest initial preload $Ffmax\u2009$in scattered preloads, and the shear stress

*τ*is obtained by Eq. (8), where

*μ*

_{s}/cos30 deg is changed with 1.155

*μ*

_{s}in Eq. (7) which is applicable for the half thread angle with 30 deg, and where

*d*

_{s}is the diameter of a circle of which the area equals to the stress area

*A*

_{s}in Eq. (8),

*p*is the pitch of threads, and

*d*

_{2}is the pitch diameter

*σ*

_{bmax}in the axial direction is obtained by Eq. (9) taking account of a maximum bolt preload

*F*

_{fmax}in scattered bolt preloads and the increment in the axial bolt force

*F*

_{t}

*a*), i.e., Mises' equivalent stress

*b*) should be satisfied:

*A*

_{3}is the root area, and$\u2009\sigma A$ is the fatigue stress:

*σ*

_{w}between bolt head (or nut) and a joint member is increased. The bearing stress

*σ*

_{w}should be less than the critical bearing stress

*σ*

_{cri}[1] because the higher bolt preload leads to the plastic deformation at the bearing surfaces and the bolt loosening. This condition is expressed by the following Eq. (12), where

*A*

_{w}is the bearing area between bolt head (or nut) and a joint member:

*A*

_{f}is the interfacial cross section area between joint members:

*F*

_{fmax}in scattered preloads, the equivalent bolt stress

*σ*

_{eqv}must be less than the yield stress of bolt, and when it is tightened at the minimum preload

*F*

_{fmin}, the interfacial contact stress condition must be satisfied. The target bolt preload

*F*

_{ftarget}is obtained as the average of the maximum

*F*

_{fmax}and the minimum bolt preload

*F*

_{fmin}. In addition, a target tightening torque

*T*

_{ftarget}is obtained by the following Eq. (14

*a*) [1,4], where

*T*

_{w}is a torque applied to the bearing surfaces.

*T*

_{ftarget}is obtained from

*T*

_{s}+

*T*

_{w}, where

*T*

_{w}is obtained from Eq. (14

*b*):

where *D*_{w} is the equivalent diameter at the bearing surfaces, namely, $Dw=(D+dh)/2,$*D* is the outside diameter of washer, and *D* is changed with *d*_{w} of outside diameter at the bearing surfaces$\u2009$when the washer is not inserted. Additionally, *K* is the nut factor. When the friction coefficients $\mu s\u2009$for screw threads and $\mu w\u2009$for the bearing surfaces$\u2009$are provided, the target torque *T*_{ftarget} is calculated by the third equation in Eq. (14*a*).

Two examples for designing bolted joints are demonstrated and discussed in this study, that is, one is a problem how to determine the strength grade of bolt, namely, yield stress *σ*_{y} of bolt (Sec. 6.1), and the other is a problem how to determine the nominal diameter *d* of bolt (Sec. 6.2). In both examples, the materials of joint members are steel (I) and aluminum (II) where a bolt is steel; both the thickness of the hollow cylinders is chosen as 28 mm shown in Fig. 1. In addition, the maximum preload *F*_{fmax} is assumed as *F*_{fmax} = *βσ*_{y}*A*_{s} (*β* is chosen) in this study. Figure 10 shows the flowchart of calculation process for two problems in design of bolted joints.

### 6.1 The First Problem is to Estimate the Bolt Strength Grade Under the Condition That the Nominal Diameter *d* of Bolt is Given as *d* = 12 mm

#### 6.1.1 For Bolted Joint With St–Al Hollow Cylinders.

The conditions are provided as follows:

A repeated external load

*W*= 10 kN (0–10 kN).Required interfacial contact stress

*σ*_{c}= 10 MPa.^{1}The tightening coefficient

*Q*(=*F*_{fmax/}*F*_{fmin}) is chosen as*Q*= 2.0 [1,17].The bolt hole diameter

*d*_{h}is assumed as*d*_{h}= (2*a*)=1.1*d*(the first approximation).*F*_{fmax}is assumed to be*F*_{fmax}= $\beta \sigma yAs.$The outside diameter 2

*b*of hollow cylinders is 39 mm, and the inside diameter 2*a*is 13 mm (*b*/*a*= 3.0).The critical bearing stress for aluminum is 360 MPa (Al:A7075) and 700 MPa for steel [1].

- To satisfy the interfacial contact stress condition where the nominal diameter
*d*of bolt is 12 mm, the following Eq. (13′) is examined. The stress area*A*_{s}of threads is*A*_{s}= 84.3 mm^{2}for M12. In the following calculations, the value of*β*is chosen as 0.6. In addition, when*β*is 0.7 and 0.8, the calculations are also carried out for examining the target bolt preload and the target tightening torque. Additionally, the value of the load factor Φ is obtained as Φ = 0.06 for*b*/*a*= 3.0 from the case of (upper) in Fig. 7. Using the load factor Φ, the force*F*_{t}is obtained as*F*_{t}= 600 (N), and the value of*F*_{c}is obtained from the equation*F*_{c}= (1 − Φ) ×*W*= 9400 N. The minimum bolt preload*F*_{fmin}is obtained as follows:where$Ffmin\u2265Fc+\sigma cAf$(13′)*F*_{fmin}is obtained for*β*= 0.6 as follows:where$Ffmin=FfmaxQ=\beta \sigma yAs/Q=0.6\sigma y\xd784.32=25.29\sigma y$*σ*_{c}= 10 MPa and $Af=(\pi /4)(392\u2212132),$ and where the bolt hole diameter*d*_{h}(= 2*a*) is 13 mm. Thus, from Eq. (13′),*σ*_{y}is obtained by the following equation:As a result, a calculated yield stress$25.29\sigma y\u2009=\u20099400\u2009+\u200910,618\u2009=\u200920,018\u2009N$*σ*_{y}′ is obtained as 791 MPa. Then, actual yield stress $\sigma y\u2009$of bolt should be determined from the bolt strength grade, and the yield stress is$\u2009$determined as*σ*_{y}= 900 MPa from the strength grade 10.9. For*β*= 0.7 and 0.8, the values of calculated yield stress*σ*_{y}′ are obtained as 678 and 593 MPa, respectively. As a result, the yield stress*σ*_{y}is determined as 720 and 640 MPa for*β*= 0.7 and 0.8, respectively. Namely, the bolt strength grade is chosen as 9.8 for*β*= 0.7 and 8.8 for*β*= 0.8, respectively. Check the interfacial contact stress condition:

Using the newly determined yield stress

*σ*_{y}of bolt, the interfacial contact stress condition described in Eq. (13′) is examined. The minimum bolt preload*F*_{fmin}in the left term of Eq. (13′) is obtained as 22,761, 21,243, and 21,580 N for*β*= 0.6, 0.7, and 0.8, respectively, where the value of*F*_{c}+*σ*_{c}*A*_{f}in right term of Eq. (13′) is 20,018 N. So, the condition for Eq. (13′) is satisfied for all values of*β*.Check the bearing stress

*σ*_{w}:Using the determined yield stress$\u2009\sigma y$, the maximum bolt preload

*F*_{fmax}is calculated as$\u2009Ffmax=\beta \sigma yAs.$The maximum load at the bearing surfaces is obtained as

*F*_{fmax}+*F*_{t}, and the bearing stress*σ*_{w}is obtained using Eq. (12) where*A*_{w}(=74.1 mm^{2}) is the bearing area. Table 2 shows the calculated results of the bearing stress*σ*_{w}between the bearing surfaces. For all the values of*β*, the bearing stress*σ*_{w}exceeds the critical bearing stress of 360 MPa for aluminum material (A7075). So, a washer is needed for the bearing surface of the aluminum joint member. The dimensions of plain washer (steel) are as follows: the inside diameter*d*_{h}is13 mm, the outside diameter*D*is 24 mm, and the thickness is 2.5 mm [19,20]. The bearing stress $\sigma w\u2032$ between the washer and the joint member is also described in Table 2. It is noticed that the bearing stress $\u2009\sigma w\u2032\u2009$is less than the critical bearing stress of 360 MPa when the steel washer is inserted.Check the equivalent stress

*σ*_{eqv}of bolt:Using Eq. (7), the tightening torque

*T*_{s}applied to the engaged threads is obtained, and the shear stress*τ*is also obtained using Eq. (8) where the friction coefficient*μ*_{s}is chosen as 0.1 under lubricant. Additionally, the maximum bolt stress in the axial direction*σ*_{bmax}is obtained from Eq. (9). Thus, Mises' equivalent stress*σ*_{eqv}is obtained, where the shear stress*τ*is calculated using Eqs. (7) and (8). Table 3 shows the calculated results. As mentioned above, the yield stress σ_{y}is determined from the strength grade for the values of*β*= 0.6, 0.7, and 0.8, respectively. The value of*F*_{fmax}is obtained for*β*= 0.6, 0.7, and 0.8 (see Table 3). It is found that the highest*F*_{fmax}is obtained when*β*is 0.6 among three values of*β*. For all the values of*β*, the values of*σ*_{eqv}are less than each yield stress*σ*_{y}. So, the condition for bolt strength is satisfied.Check the bolt fatigue strength:

Using Eq. (11), the stress amplitude of bolt is obtained by the following equation, where the stress concentration factor*α*_{A}is assumed to be 4.2 from Ref. [19], the root area of threads*A*_{3}is 76.2 mm^{2}, and fatigue strength*σ*_{A}is around 50 MPa [1,21]:As a result, the stress amplitude of bolt is less than the fatigue strength. It is shown that the condition for fatigue strength of bolt is satisfied. In this study, the strength grade less than 12.9 is used because a possibility of delayed fracture still remains for higher strength grade than 12.9.$\sigma a=6002\xd776.24.2=16.5\u2009MPa\u226450\u2009MPa$

β | F_{fmax}+F_{t} (N) | σ_{w} (MPa) | σ_{w}′ (MPa) |
---|---|---|---|

0.6 | 45,922 | 622 | 144 |

0.7 | 42,887 | 581 | 134 |

0.8 | 43,561 | 590 | 136 |

β | F_{fmax}+F_{t} (N) | σ_{w} (MPa) | σ_{w}′ (MPa) |
---|---|---|---|

0.6 | 45,922 | 622 | 144 |

0.7 | 42,887 | 581 | 134 |

0.8 | 43,561 | 590 | 136 |

β | F_{fmax} (N) | T_{s} (N·mm) | τ (MPa) | σ_{bmax} (MPa) | σ_{eqv} (MPa) | Judge (MPa) |
---|---|---|---|---|---|---|

0.6 | 45,522 | 41,220 | 188 | 544 | 603 | <900 |

0.7 | 42,487 | 38,471 | 176 | 508 | 592 | <720 |

0.8 | 43,161 | 39,082 | 179 | 516 | 601 | <640 |

β | F_{fmax} (N) | T_{s} (N·mm) | τ (MPa) | σ_{bmax} (MPa) | σ_{eqv} (MPa) | Judge (MPa) |
---|---|---|---|---|---|---|

0.6 | 45,522 | 41,220 | 188 | 544 | 603 | <900 |

0.7 | 42,487 | 38,471 | 176 | 508 | 592 | <720 |

0.8 | 43,161 | 39,082 | 179 | 516 | 601 | <640 |

#### 6.1.2 The Conditions for Bolted Joint With St–Al Hollow Cylinders Without Washer.

In Sec. 6.1.1, it is shown for each *β* that a washer is needed to insert for the bearing surfaces of aluminum joint member, where the required interfacial contact stress *σ*_{c} is 10 MPa. However, it is better for engineer not to insert a washer in bolted joint. For achieving the bolted joint with St–Al dissimilar joint members without a washer, the required contact stress *σ*_{c} at the interfaces should be reduced as *σ*_{c} = 3 MPa because the maximum bolt preload *F*_{fmax} should be reduced, while the other conditions are the same as Sec. 6.1.1. Additionally, the value of *β* is chosen as 0.48 and 050. Then, for both *β* = 0.48 and 0.50, the yield stress is calculated using Eq. (13′). The calculated yield stress *σ*_{y}′ is obtained as 597 MPa for *β* = 0.48 and 542 MPa for *β* = 0.50. So, the strength grade of bolt is determined as 8.8 (*σ*_{y} = 640 MPa). Next, the bearing stress is examined. From Eq. (12), *σ*_{w} is obtained as 358 MPa for *β* = 0.48 and *σ*_{w} = 372 MPa for *β* = 0.50, while the critical bearing stress is 360 MPa. So, the cases of *β* = 0.48 are available.

In addition, the equivalent stress *σ*_{eqv} is obtained as 364 MPa for *β* = 0.48, and this value is less than the yield stress of 640 MPa. The stress amplitude *σ*_{a} is the same value mentioned above. Thus, the conditions of the equivalent stress and the fatigue strength of bolt are both satisfied. As a result, for achieving the bolted joint without washer, the conditions are as follows: (1) the required interfacial contact stress should be reduced as *σ*_{c} = 3 MPa, and (2) *β* should be reduced to 0.48, namely, the maximum bolt preload should be reduced for satisfying the bearing stress between the bolt head (or nut) and aluminum joint member as well as the required interfacial contact stress. Additionally, it is shown that the conditions for the equivalent stress and fatigue strength of bolt are satisfied as well as the interfacial contact stress condition and the bearing stress condition.

#### 6.1.3 The Case of Bolted Joint With Similar St–St Joint Members.

In this case where the required interfacial contact stress *σ*_{c} is 10 MPa and the critical bearing stress is 700 MPa for St–St joint members, the value of the load factor is obtained as Φ = 0.03 from Fig. 8. The values of *β* are assumed to be 0.7, 0.8, and 0.85. According to Eq. (13′), the calculated yield stress *σ*_{y}′ is obtained as 688, 602, and 567 MPa, so the bolt strength grade is chosen as 9.8, 8.8, and 8.8 for *β* = 0.7, 0.8, and 0.85, respectively, namely, the yield stress *σ*_{y} is determined as 720 MPa (strength grade 9.8) for *β* = 0.7 and 640 MPa (strength grade 8.8) for *β* = 0.8 and 0.85.

Using the determined yield stress *σ*_{y}, the interfacial contact stress condition is examined. As a result, for all the values of *β*, Eq. (13′) is satisfied. Then, the bearing stress *σ*_{w} and the equivalent stress *σ*_{eqv} of bolt are examined. Table 4 shows the calculated results of *σ*_{w} and *σ*_{eqv}. All values of *σ*_{w} are less than 700 MPa (*σ*_{wcri}). The equivalent stresses *σ*_{eqv} are less than the yield stress *σ*_{y}. It is shown that the results of *σ*_{w} and *σ*_{eqv} are satisfied with the conditions. Additionally, the bolt fatigue condition is also satisfied. When the joint members are both steel, no washers are needed. When the value of *β* is 0.9, it is impossible to satisfy the equivalent stress condition because of the effect of the shear stress *τ* in Eq. (10*a*). Table 5 shows the target preload *F*_{ftarget}, target torque *T*_{ftarget}, and the nut factor *K* for each value of *β*, where the diameter at bearing surface *d*_{w} is chosen as 16.63 mm for M12 in Eq. (14*b*). The target bolt preload *F*_{ftarget} is obtained by the average of *F*_{fmin} and *F*_{fmax}, and the target torque *T*_{ftarget} is obtained using Eq. (14*a*). The friction coefficients *μ*_{s} and *μ*_{w} are chosen as 0.1 when the lubricant is applied (lub), and they are chosen as 0.18 when it is not applied (nonlub). The nut factor *K* is around 0.137 in the case of the lubricant, and it is 0.228 in the case of nonlubricant.

β | F_{fmax} + F_{t} (N) | σ_{w} (MPa) | σ_{cri} (MPa) | σ_{bmax} (MPa) | T_{s} (N·mm) | σ_{eqv} (MPa) | σ_{y} (MPa) | Judge |
---|---|---|---|---|---|---|---|---|

0.7 | 42,787 | 577 | 700 | 507 | 38,471 | 591 | 720 | OK |

0.8 | 43,461 | 586 | 700 | 515 | 39,082 | 602 | 640 | OK |

0.85 | 46,159 | 622 | 700 | 547 | 41,525 | 638 | 640 | OK |

β | F_{fmax} + F_{t} (N) | σ_{w} (MPa) | σ_{cri} (MPa) | σ_{bmax} (MPa) | T_{s} (N·mm) | σ_{eqv} (MPa) | σ_{y} (MPa) | Judge |
---|---|---|---|---|---|---|---|---|

0.7 | 42,787 | 577 | 700 | 507 | 38,471 | 591 | 720 | OK |

0.8 | 43,461 | 586 | 700 | 515 | 39,082 | 602 | 640 | OK |

0.85 | 46,159 | 622 | 700 | 547 | 41,525 | 638 | 640 | OK |

β | F_{ftarget} (N) | T_{ftarget} (N·mm) (lub) | K (lub) | T_{ftarget} (N·mm) (no-lub) | K (no-lub) |
---|---|---|---|---|---|

0.7 | 31,865 | 52,456 | 0.137 | 87,336 | 0.228 |

0.8 | 32,370 | 53,289 | 0.137 | 88,720 | 0.228 |

0.85 | 34,394 | 56,620 | 0.137 | 94,268 | 0.228 |

β | F_{ftarget} (N) | T_{ftarget} (N·mm) (lub) | K (lub) | T_{ftarget} (N·mm) (no-lub) | K (no-lub) |
---|---|---|---|---|---|

0.7 | 31,865 | 52,456 | 0.137 | 87,336 | 0.228 |

0.8 | 32,370 | 53,289 | 0.137 | 88,720 | 0.228 |

0.85 | 34,394 | 56,620 | 0.137 | 94,268 | 0.228 |

### 6.2 The Second Problem to Determine the Nominal Diameter *d* of Bolt

#### 6.2.1 A Problem to Determine the Nominal Diameter in Bolted Joint With St–Al Hollow Cylinders.

In this section, a method for determining the nominal diameter *d* in St–Al hollow cylinder joint is discussed when the bolt strength grade is given as 10.9 (*σ*_{y} = 900 MPa).

Condition:

Strength grade: 10.9 (

*σ*_{y}= 900 MPa).The outside diameter 2

*b*of hollow cylinders is fixed as 31.5 mm, and the height of a hollow cylinder as 28 mm (the grip length is 56 mm).Repeated external tensile load

*W*: 10 × 10^{3}N (0–10 kN).Tightening coefficient

*Q*(*F*_{fmax}/*F*_{fmin}) is 2.0.Required interfacial contact stress between joint members

*σ*_{c}= 10 MPa.Bolt hole diameter

*d*_{h}is assumed as 1.1*d*(first approximation).

*F*

_{fmax}and the minimum preload

*F*

_{fmin}in scattered bolt preloads are described by the following Eqs. (14

*c*) and (14

*d*), where the stress area

*A*

_{s}is assumed as $(\pi /4)d2\u2009(the\u2009first\u2009approximation)$. Additionally,

*β*is assumed as 0.6, 0.7, and 0.8:

*b*/

*a*is 2.5 (approximation). So, the additional bolt force

*F*

_{t}is 1300 N, and the force

*F*

_{c}is 8700 N. The interface area

*A*

_{f}in the joint members is obtained as follows, where the bolt hole diameter is assumed as 1.1

*d*:

*d*′ is obtained as 8.6 mm for

*β*= 0.6, 8.0 mm for

*β*= 0.7, and 7.5 mm for

*β*= 0.8, where a solution of Eq. (16) is denoted by

*d*′. So, the nominal diameter

*d*, which is specified in the standards, is determined from the safety standpoint as 10 mm for all values of

*β*. The stress area for

*d*= 10 mm is

*A*

_{s}= 58.0 mm

^{2}, and the bolt hole diameter

*d*

_{h}(2

*a*) is 10.5 mm. Then, the value of the load factor Φ is obtained as Φ = 0.06 in Fig. 7 for the value of

*b*/

*a*(= 31.5/10.5 = 3.0). Using the new value of the load factor, the force

*F*

_{t}is 600 N, and the force

*F*

_{c}is 9400 N. The interfacial contact stress condition Eq. (13′) is changed with the following Eq. (13″), where

*A*

_{s}is 58.0 mm

^{2}for

*d*= 10 mm (M10) and

*Q*= 2.0:

For each value of *β*, the interfacial contact stress condition is checked using Eq. (13″). However, when the value of *β* is 0.6, the above condition is not satisfied, while it is satisfied for *β* = 0.7 and 0.8. So, the nominal diameter *d* should be changed from *d* = 10 mm to *d* = 12 mm for *β* = 0.6. Then, the bolt hole of hollow cylinders is changed as *d*_{h} = 13 mm for *d* = 12 mm. The load factor Φ is also changed, and it is obtained as Φ = 0.12 in Fig. 7 when the ratio is *b*/*a* = 31.5/13 = 2.4 where the interface area *A*_{f} is changed as $Af=646\u2009mm2$. The thread pitch *p* and the pitch diameter *d*_{2} are 1.75 mm and 10.863 mm for *d* = 12 mm and 1.5 mm and 8.59 mm for *d* = 10 mm, respectively. Tightening torque *T*_{s} is obtained using Eq. (7), and the shear stress *τ* due to the torque *T*_{s} is obtained using Eq. (8). The equivalent stress *σ*_{eqv} is obtained from Eq. (10*a*). The bearing stress *σ*_{w} is obtained from Eq. (12).

Table 6 shows the calculated results of the equivalent stress *σ*_{eqv} and the bearing stress *σ*_{w}. Since the values of *σ*_{eqv} are smaller than the yield stress of 900 MPa, the bolt strength condition is satisfied. All values of *σ*_{w} for each value of *β* are larger than the critical contact stress *σ*_{cri} = 360 MPa for aluminum (A7075). It is necessary to insert a plain washer at the bearing surface of aluminum joint member. The outside diameter *D* of washer is chosen as 24 mm for *d* = 12 and 21 mm for *d* = 10 mm, and the inside diameter *d*_{h} is 13 mm for *d* = 12 mm and 10.5 mm for *d* = 10 mm, respectively. When the washer is inserted at the aluminum joint member, the bearing stress *σ*_{w}′ between the washer and the joint member is obtained. It is found that the bearing stress condition is satisfied, i.e., the value of *σ*_{w}′ is less than the critical value of 360 MPa.

β | d (mm) | A_{s} (mm^{2}) | d_{s} (mm) | F_{fmax} (N) | F_{t} (N) | σ_{bmax} (MPa) | T_{s} (N·mm) | τ (MPa) | σ_{eqv} (MPa) | σ_{w} (MPa) | σ_{w}′ (MPa) |
---|---|---|---|---|---|---|---|---|---|---|---|

0.6 | 12 | 84.3 | 10.36 | 45,522 | 1200 | 558 | 41,220 | 188 | 642 | 630 | 146 |

0.7 | 10 | 58.0 | 8.59 | 36,540 | 600 | 640 | 27,769 | 223 | 747 | 508 | 143 |

0.8 | 10 | 58.0 | 8.59 | 41,760 | 600 | 730 | 31,736 | 255 | 852 | 579 | 163 |

β | d (mm) | A_{s} (mm^{2}) | d_{s} (mm) | F_{fmax} (N) | F_{t} (N) | σ_{bmax} (MPa) | T_{s} (N·mm) | τ (MPa) | σ_{eqv} (MPa) | σ_{w} (MPa) | σ_{w}′ (MPa) |
---|---|---|---|---|---|---|---|---|---|---|---|

0.6 | 12 | 84.3 | 10.36 | 45,522 | 1200 | 558 | 41,220 | 188 | 642 | 630 | 146 |

0.7 | 10 | 58.0 | 8.59 | 36,540 | 600 | 640 | 27,769 | 223 | 747 | 508 | 143 |

0.8 | 10 | 58.0 | 8.59 | 41,760 | 600 | 730 | 31,736 | 255 | 852 | 579 | 163 |

Finally, the fatigue strength of bolt is examined. Using Eq. (11), the bolt stress amplitude $\sigma a$ is calculated. For *β* = 0.6, *F*_{t} is 1200 N, *A*_{3} is 76.2 mm^{2}, and the stress concentration factor is assumed as *α*_{A} = 4.2 [19]. The value of $\sigma a$ is obtained as 19.2 MPa [1,21]. In addition, for *β* = 0.7 and 0.8, $\sigma a$ is obtained as 12.0 MPa, where *A*_{3} is 52.3 mm^{2}, *F*_{t} is 300 N, and *α*_{A} is the same as 4.2, while $\sigma A$ is greater than 50 MPa for *d* = 12 mm and the strength grade 10.9, and where $\sigma A$ is greater than 55 MPa for *d* = 10 mm and 10.9 grade [1,21]. So, it is found that the fatigue strength condition is satisfied sufficiently.

As a result, the nominal diameter *d* is determined as 12 mm, 10 mm, and 10 mm for *β* = 0.6, 0.7, and 0.8, respectively, while all four conditions are also satisfied. However, washers should be inserted at the bearing surfaces of the aluminum joint member. In assembling the bolted joints, the target bolt preload *F*_{ftarget} is obtained using the average of the maximum $Ffmax\u2009$and the minimum $Ffmin\u2009$bolt preloads as follows: $Fftarget=(Ffmax\u2009+\u2009Ffmin)/2$. The target tightening torque *T*_{ftarget} is obtained as *T*_{ftarget} = *T*_{s} + *T*_{w}, shown in Eq. (14*a*), where the values of *D*_{w} in Eq. (14*b*) are 18.5 mm for *d* = 12 mm (*β* = 0.6) and 15.75 mm for *d* = 10 mm (*β* = 0.7 and 0.8), respectively.

Table 7 shows the target preload *F*_{ftarget} and target torque *T*_{ftarget} for each value of *β* and in the cases of with-lubricant and nonlubricant (washers are inserted). When the lubricant is applied, the friction coefficients *μ*_{s} and *μ*_{w} are chosen as 0.1. They are chosen as 0.18 in the case of nonlubricant. These data are available in tightening works. It is found that the nut factor *K* (with-lubricant) is around 0.15 when the lubricant is applied, and it (nonlubricant) is around 0.26 when the lubricant is not applied.

β | d (mm) | F_{ftarget} (N) | $\mu s=\mu w$ | T_{ftarget} (N·m) | K (with-lubricant) | F_{target} (N) | $\mu s=\mu w$ | T_{target} (N·m) | K (nonlubricant) |
---|---|---|---|---|---|---|---|---|---|

0.6 | 12 | 34,141 | 0.1 | 62,506 | 0.152 | 56,844 | 0.18 | 104,905 | 0.256 |

0.7 | 10 | 27,405 | 0.1 | 42,408 | 0.154 | 38,845 | 0.18 | 71,100 | 0.259 |

0.8 | 10 | 31,320 | 0.1 | 48,466 | 0.154 | 44,395 | 0.18 | 81,258 | 0.259 |

β | d (mm) | F_{ftarget} (N) | $\mu s=\mu w$ | T_{ftarget} (N·m) | K (with-lubricant) | F_{target} (N) | $\mu s=\mu w$ | T_{target} (N·m) | K (nonlubricant) |
---|---|---|---|---|---|---|---|---|---|

0.6 | 12 | 34,141 | 0.1 | 62,506 | 0.152 | 56,844 | 0.18 | 104,905 | 0.256 |

0.7 | 10 | 27,405 | 0.1 | 42,408 | 0.154 | 38,845 | 0.18 | 71,100 | 0.259 |

0.8 | 10 | 31,320 | 0.1 | 48,466 | 0.154 | 44,395 | 0.18 | 81,258 | 0.259 |

#### 6.2.2 A Problem to Determine the Nominal Diameter d in Bolted Joint With St–Al Hollow Cylinders Without Washer.

For achieving a bolted joint without washer, it is necessary to decrease the maximum bolt preload *F*_{fmax}, while the interfacial contact stress *σ*_{c} condition (Eq. (13)), the bearing stress *σ*_{w} condition (Eq. (12)), and the equivalent stress condition *σ*_{eqv} (Eqs. (10*a*) and (10*b*)) as well as the bolt fatigue condition (Eq. (11)) should be satisfied. So, it is better to increase the value of *β* and the nominal diameter *d*, while the yield stress should be decreased. The method for determining the nominal diameter is the same procedure mentioned above, while the provided conditions are the same. Here, the strength grade is chosen as 6.8 (*σ*_{y} = 480 MPa) and the nominal diameter *d* as *d* = 14 mm. The bolt hole diameter *d*_{h} (=2*a*) is chosen as *d*_{h} = 15 mm. The value of the load factor is obtained as Φ = 0.16 for *b*/*a* = 31.5/15 = 2.1 (Fig. 7), and the additional bolt force *F*_{t} is 1600 N and the force *F*_{c} is 8400 N. As a result, for satisfying all the conditions, it is found that when the nominal diameter *d* = 14 mm, the value of *β* is 0.5, and the strength grade is 8.8 in St–Al joint without washer, the bearing stress is obtained as 336 MPa (<360 MPa). Additionally, when the value of *β* is determined as 0.7 and the strength grade as 6.8 for *d* = 14 mm, and when a washer is not inserted, the bearing stress is obtained as 352 MPa. This value is less than the critical bearing stress of 360 MPa. Thus, the bearing stress condition is satisfied as well as the other conditions.

#### 6.2.3 Determination of Nominal Diameter for the Bolted Joint With Similar Joint Members.

The case of St–St joint members: The provided conditions are the same as mentioned before. The strength grade of bolt is 10.9 (

*σ*_{y}= 900 MPa), the required interfacial contact stress*σ*_{c}= 10 MPa, and the tightening coefficient*Q*= 2.0. The calculation method is the same as mentioned above. The value of the load factor is determined as Φ = 0.03 when the ratio of*b*/*a*= 31.5/10.5 = 3.0 for St–St joint in Fig. 8. As a result, the nominal diameter*d*is obtained as*d*= 10 mm for*β*= 0.7, 0.8, and 0.85, while the interfacial contact stress condition, the equivalent stress*σ*_{eqv}, and the bolt fatigue strength are all satisfied. In addition, the bearing stress*σ*_{w}is obtained as 501, 572, and 608 MPa for*β*= 0.7, 0.8, and 0.85, respectively. It is found that the values of*σ*_{w}are less than the critical value of 700 MPa. So, it is obvious that no washers are needed in the bolted joints with St–St joint members. The higher value of*β*is desirable for increasing the bolt preload.The case of Al–Al joint members: When the joint members are both aluminum, the bearing stress condition is critical. The bearing stress and other three conditions for

*β*, the nominal diameter*d*, and the strength grade should be examined for the joint without washer because a bolted joint without a washer is better in assembling work. The calculation method is the same as mentioned above. The value of the load factor is determined as Φ = 0.2 in Fig. 8 when the ratio*b*/*a*= 31.5/15.0 = 2.1.

The value of *β* is chosen as 0.6 and 0.5, and the bolt strength grade is determined as 6.8 and 8.8, respectively, while the nominal diameter *d* of bolt is chosen as *d* = 14 mm. Then, the bearing stress is obtained as 301 and 334 MPa for *β* = 0.6 and 0.5, respectively. The bearing stress condition is satisfied. In addition, the interfacial contact stress condition *σ*_{c}, the bolt equivalent stress *σ*_{eqv}, and the bolt fatigue strength are all satisfied. As a result, the value of *β*, the nominal diameter *d*, and strength grade are determined for the bolted joint without washer.

Table 8 shows the obtained results of the nominal diameter *d*, the strength grade of bolt, and a necessity of washer insertion for bolted joints with combination of joint material. In addition, the bearing stress *σ*_{w} between bolt head (nut) and aluminum joint member and *σ*_{w}′ between a washer and the aluminum joint member are shown. For No. 1–No. 3, washers are needed for St–Al joints in Table 8. For No. 4 and No. 5, when the nominal diameter *d* is chosen as 14 mm (M14), the strength grade is reduced as 8.8 and 6.8, and the value of *β* is 0.5 and 0.7, respectively. In the cases, no washers are needed. Furthermore, for the case where no washer is needed, it is shown that when the value of *β* is increased, the strength grade of bolt should be decreased as seen in the case of No. 5. On the contrary, when the strength grade is increased as seen in No. 4, the value of *β* should be reduced. So, no washer is needed. For No. 6 and No. 7 (St–St joints), no washer is needed. Additionally, the nominal diameter *d* can be reduced in comparison with St–Al joints (No. 4 and No. 5), and the value of *β* can be increased to *β* = 0.85. However, the value of *β* cannot be increased to *β* = 0.9 because the bolt strength condition (Eq. (10*a*)) cannot be satisfied due to the shear stress *τ*. For No. 8 and No. 9, the nominal diameter *d* is increased by 14 mm (M14), so no washers are needed. The bearing stress *σ*_{w} for No. 8 and No. 9 (Al–Al) is a little bit larger than that of No. 4 and No. 5 (St–Al) because the values of the load factor for No. 8 and No. 9 are larger than those of No. 4 and No. 5. A bolted joint without washers can be achieved when the nominal diameter *d* is increased by 14 mm, where the value of *β* and the strength grade should be changed appropriately.

No. | β | Combination of materials | d (mm) | Strength grade | σ_{w} (MPa) | Washer | σ_{w}′ (MPa) |
---|---|---|---|---|---|---|---|

1 | 0.6 | St–Al | 12 | 10.9 | 623 | Y | 144 |

2 | 0.7 | St–Al | 10 | 10.9 | 503 | Y | 142 |

3 | 0.8 | St–Al | 10 | 10.9 | 575 | Y | 162 |

4 | 0.5 | St–Al | 14 | 8.8 | 336 | No | |

5 | 0.7 | St–Al | 14 | 6.8 | 352 | No | |

6 | 0.8 | St–St | 10 | 10.9 | 572 | No | |

7 | 0.85 | St–St | 10 | 10.9 | 608 | No | |

8 | 0.7 | Al–Al | 14 | 6.8 | 356 | No | |

9 | 0.5 | Al–Al | 14 | 8.8 | 340 | No |

No. | β | Combination of materials | d (mm) | Strength grade | σ_{w} (MPa) | Washer | σ_{w}′ (MPa) |
---|---|---|---|---|---|---|---|

1 | 0.6 | St–Al | 12 | 10.9 | 623 | Y | 144 |

2 | 0.7 | St–Al | 10 | 10.9 | 503 | Y | 142 |

3 | 0.8 | St–Al | 10 | 10.9 | 575 | Y | 162 |

4 | 0.5 | St–Al | 14 | 8.8 | 336 | No | |

5 | 0.7 | St–Al | 14 | 6.8 | 352 | No | |

6 | 0.8 | St–St | 10 | 10.9 | 572 | No | |

7 | 0.85 | St–St | 10 | 10.9 | 608 | No | |

8 | 0.7 | Al–Al | 14 | 6.8 | 356 | No | |

9 | 0.5 | Al–Al | 14 | 8.8 | 340 | No |

The above mentioned design method can be applied to a sealing design of bolted gasketed connections with dissimilar (or similar) flanges because the required contact stress *σ*_{c} is incorporated in the present design method [22].

## 7 Conclusions

In this paper, the objectives are to examine the load factor and a load *W*_{s} when the interfaces start to separate and to propose a new simple design method for bolted joints consisting of dissimilar hollow cylinders (steel and aluminum). The load factor and the load *W*_{s} are calculated using FEM, and the obtained results are verified by the experiments. The obtained value of the load factor is compared with that of the joints with similar hollow cylinders (St–St and Al–Al). In addition, important five items for safety and integrity design of bolted joints are described, and a new simple method for determining the nominal diameter of bolt, the bolt strength grade, target bolt preload, and target bolt tightening torque taking account of the torque coefficient *Q* is demonstrated using the obtained values of the load factor. The results obtained are as follows.

The values of the load factor in bolted joint consisting of steel and aluminum (St–Al) joint members are calculated using FEM. In addition, the effect of a position where a load is applied on the value of the load factor is examined. It is found that the value of the load factor increases as the position of the load application (upper) increases toward the bearing surfaces at the outside diameter of the hollow cylinders. The effect of the ratio of

*b*/*a*(the ratio of the outside diameter to the inside diameter in the joint members) is also examined, and it is found that the load factor decreases as the ratio*b*/*a*increases. The value of the load factor is found to be less than 0.15 (in the case where the ratio of*b*/*a*is greater than 2.5) in the present cases.The values of the load factor for bolted joints consisting of steel and aluminum joint members are compared with those of the joints with St–St combination and Al–Al combination. The value of the load factor of the joint consisting of St–St joint members is less than 0.1, and that of the joint consisting of Al–Al joint members is less than 0.2 (in the case where the ratio of

*b*/*a*is greater than 2.5). As a result, it is observed that the value of the load factor is the largest for the Al–Al joint, and the smallest is the St–St joint. This is because the compressive stiffness*K*_{c}decreases as Young's modulus decreases.A load

*W*_{s}when the interfaces start to separate is examined. It is found that the load*W*_{s}decreases as the ratio*b*/*a*increases. It is also observed that the load*W*_{s}increases as the value of the load factor increases, namely, the value of*W*_{s}of Al–Al joint is larger than that of St–Al and St–St joints.The FEM results of the load factor are observed to be in a fairly agreement with the experimental results. Furthermore, the values of the load

*W*_{s}are fairly coincident with the experimental results.In the special case where the outside diameter of the joint members is equal to that of the bearing surfaces, a simple calculation method for obtaining the load factor is demonstrated. The calculated values of the load factor are fairly matched with the FEM results. When the values of obtained load factors are used, the bolt strength and the bearing stress conditions are evaluated in the sufficiently safety side. However, the interfacial stress condition is not evaluated in the safety side because the value of load factor is too large.

Five important items for safety design of bolted joints are proposed. In a new simple design method, the interfacial contact stress (required contact stress

*σ*_{c}) should be kept under external load, and a bolt preload should be increased as possible as we can for preventing the bolt loosening. Using the obtained load factor for bolted joints consisting of St–Al joint members, a new design method for determining the nominal diameter of bolt and the bolt strength grade is demonstrated. In the present design method, the tightening coefficient*Q*(=*F*_{fmax}/*F*_{fmin}), the bolt strength criteria, and the critical bearing stress are taken into consideration. As the first problem, the numerical calculation example is shown for determining the strength grade of bolt under a given nominal diameter of bolt. Additionally, the nut factor*K*for the joint with lubricant and without lubricant is shown for practical use. As the bolt preload increases, the bearing stress increases. So, a method whether a washer should be inserted at the bearing surface of lower rigidity joint member or not is shown by reducing the value of*β*and strength grade of bolt.As the second problem, a design method for determining the nominal diameter of bolt under a given strength grade is demonstrated. Numerical calculations are shown, and a method for determining a relationship between the nominal diameter

*d*and the strength grade is shown. Furthermore, a method for determining a bolt preload is shown when no washer is needed for bolted joint consisting of St–Al joint. This is useful in practical tightening work. In addition, a difference in the determination of the nominal diameter of bolt*d*is demonstrated between St–Al joint and St–St joint. As a result, in the design for St–Al joint, it is noticed that steel washers should be inserted, while they are not needed for steel–steel joint. However, a method is demonstrated in the case where no washer is needed in St–Al joints by increasing the bolt nominal diameter. Additionally, it is also shown that when the value of*β*is increased, the strength grade of bolt should be decreased.

In this study, the behaviors and a design method of bolted joints are described. In practice, bolted joints are subjected to eccentric loads, and then a bending moment is applied in bolts. In near future, the authors will research and propose a design method for bolted joints with dissimilar joint members under eccentric loadings such as bolted T-shape flange and circular flange joints.

## Acknowledgment

The authors would like to express the thanks to Mr. Komei Kobayashi of Hard Lock Industry for his useful assistance in our study and to Professor Emeritus Toshiyuki Sawa of Hiroshima University in Japan for his suggestions.

## Nomenclature

*A*_{f}=cross-sectional area at the interfaces between joint members (mm

^{2})*A*_{s}=stress area at engaged threads in bolt (mm

^{2})*A*_{w}=bearing area between bolt head (or nut) and joint member (mm

^{2})*A*_{3}=root area in threads (mm

^{2})*d*=nominal diameter of bolt (mm)

*D*=outside diameter of washer (mm)

*d′*=calculated nominal diameter (mm)

*d*_{h}=bolt hole diameter (= 2

*a*) (mm)*d*_{s}=diameter of circle of which area is the same as the stress area (mm)

*d*_{w}=outside diameter of bearing surfaces (mm)

*d*_{2}=pitch diameter (mm)

*D*_{w}=equivalent diameter at the bearing surfaces (mm)

*E*_{b}=Young's modulus of bolt (MPa)

*E*=_{i}Young's modulus of joint member (

*i*= 1, 2) (MPa)*F*_{bmax}=maximum bolt force (=

*F*_{fmax}+*F*_{t}) (N)*F*_{c}=force eliminated from the interfaces in joint (N)

*F*_{f}=bolt preload in initial clamping (N)

*F*_{fmax}=maximum bolt force among scattered preloads (N)

*F*_{fmin}=minimum bolt force among scattered preloads (N)

*F*_{ftarget}=target bolt preload (N)

*F*_{t}=increment in axial bolt force (N)

*K*=nut factor (

*K*-factor)*K*_{c}=compressive stiffness for joint members (N/

*μ*m)*K*_{pt}=tensile stiffness for joint members (N/

*μ*m)*K*_{t}=stiffness for bolt–nut system (N/

*μ*m)*p*=pitch of screw threads (mm)

*Q*=tightening coefficient (=

*F*_{fmax}/*F*_{fmin})*T*_{ftarget}=target torque in initial clamping (N·mm)

*T*_{s}=torsional torque applied to engaged screw threads (N·mm)

*T*_{w}=torsional torque applied to the bearing surfaces (N·mm)

*W*=external tensile load (N)

*W*_{s}=external tensile load when the interfaces start to separate (N)

- 2
*a*=inside diameter of hollow cylinder (mm)

- 2
*b*=outside diameter of hollow cylinder (mm)

- $\mu s$ =
friction coefficient between engaged threads

- $\mu w$ =
friction coefficient between bearing surfaces

*ν*=_{i}Poisson's ratio of joint member (

*i*= 1, 2)*σ*_{a}=stress amplitude in bolt (MPa)

*σ*_{A}=fatigue strength of bolt (MPa)

*σ*_{bmax}=maximum bolt stress in axial direction (MPa)

*σ*_{c}=required interfacial contact stress between joint members (MPa)

*σ*_{eqv}=equivalent stress (= $\sigma 2+3.0\tau 2$ ) (MPa)

*σ*_{y}=yield stress (strength) of bolts (MPa)

*σ*_{y}′ =yield stress in calculating process in Eq. (13′) (MPa)

- $\tau $ =
shear stress due to torsional load

*T*_{s}(MPa) - Φ =
load factor (=

*F*_{t}/*W*)

## Footnotes

*σ*_{c} is determined such that when a soft gasket is used in a joint, a minimum gasket stress for sealing is around 10 MPa.

## References

*Steifigkeit und Dauerhaltbarkeit von Schraubenverbindungen)*,