## 1 Introduction

Yoshimoto et al.  and Sawa and Omiya  proposed a new formula for obtaining the value of the load factor more precisely and theoretically by the equation Φ = Kt/(Kt + Kc) × (Kc/Kpt) 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 Kpt 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  using the theory of elasticity. The authors researched on the mechanical characteristics in bolted joints with similar material joint members of T-shape flanges  and circular flanges  using finite element method (FEM).

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 , 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

For examining the effect of the ratio b/a of the outside diameter 2b to the inside diameter 2a 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 2a of the hollow cylinder (I) and (II) are held constant as h = 28 mm and 2a = 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 2b to the inside diameter 2a 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 2a of the hollow cylinder (I) and (II) are held constant as h = 28 mm and 2a = 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 Ws 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 μm for obtaining a precise load Ws.

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

### 4.1 Comparison of the Load Factor and Load Ws 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 Ws 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.

Table 1

Comparison of the load factor Φ and the load of Ws (Ff = 16 kN)

CombinationSt–AlAl–AlSt–StSt–Al
Φ (FEM)0.1470.1800.0880.205
Φ (Expt.)0.1490.1870.0970.192
Ws (kN) (FEM)12.914.97.8815.8
Ws (kN) (Expt.)11.513.58.2614.5
CombinationSt–AlAl–AlSt–StSt–Al
Φ (FEM)0.1470.1800.0880.205
Φ (Expt.)0.1490.1870.0970.192
Ws (kN) (FEM)12.914.97.8815.8
Ws (kN) (Expt.)11.513.58.2614.5

In Table 1, it is also found that the load Ws increases as the value of the load factor increases. In addition, from the values of the load factor, the load Ws when the interfaces start to separate can be predicted. When the value of the load factor is the largest, the force Fc is the smallest from the equation Fc = (1 − Φ)W. Therefore, when the value of the load factor is the largest, the load Ws 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 Ws increases.

### 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 Kc 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

Figure 9 shows a special case where the outside diameter of the hollow cylinders is equal to that of the bearing surfaces in the bolt head and the nut. The dissimilar two thin hollow cylinders are clamped with bolt preload Ff, and when a load W is applied, an increment Ft in the axial bolt force occurs. The grip length is denoted by lf (=2h), 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 E1 and that of the lower joint member by E2. The cross-sectional area of joint members is denoted by Af. 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 $λ′1$ and the lower part with the height of x/2 by $λ′2$ as shown in Fig. 9(b). The total elongation λ$′$ between both the bearing surfaces in the joint members (thin hollow cylinders) is obtained by $λ′1+λ′2=((W·(x/2))/Af)((1/E1)+(1/E2))$from Hooke's law. When the load W is applied, the tensile stiffness Kpt of the joint members is defined by the following equation:
$1Kpt=λ′W=(λ′1+λ′2)/W=x2Af(1E1+1E2)$
(1)
When the load W is applied, an additional force Ft 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$λ″1 and λ″2$, respectively. The total shrinkage λ$″$ of the joint members due to the compressive force Ft is expressed by the following equation:
$λ″=λ″1+λ″2=Ft· lf2Af(1E1+1E2)$
(2)
In addition, the compressive stiffness for joint members (thin hollow cylinder) Kc is defined as follows:
$1Kc= λ″/Ft=lf2Af(1E1+1E2)$
(3)
Using the stiffness for bolt–nut system Kt , the bolt elongation is expressed as Ft/Kt. The elongation λ at the bearing surfaces in joint members is expressed by λ = λ$′$ − λ$″$, and it is obtained by $(W/Kpt)−(Ft/Kc)$. 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:
$FtKt=WKpt−FtKc$
(4)
From Eq. (4), the ratio of Ft to W is obtained by the following equation:
$Φ=FtW=KtKt+KC(KcKPt)$
(4′)

In Eq. (4′), the ratio $Kc/KPt$ is changed with $(x/lf)$ using Eqs. (1) and (3).

Then, the ratio of Ft to the load W, which is called as the load factor Φ = Ft/W, is obtained by the following equation:
$Φ=FtW=KtKt+KC(xlf)$
(5)
In Eq. (5), the value of the load factor is the maximum when the load is applied to the upper part of the joint member (thin hollow cylinder), that is, x = lf. 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  for obtaining the load factor. So, the value of the load factor from Thum's formula  is not correct in practical use
$Φ=FtW=KtKt+KC$
(6)

Thus, the value of the load factor obtained from Thum's formula  is larger than that in practical bolted joints .

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, 2b as 22.4 mm (= 1.6 × 2a) while the inside diameter 2a is 14 mm, the thickness of the joint member is chosen as 28 mm (lf = 56 mm), and the material of upper hollow cylinder is chosen as aluminum and the lower one as steel (St–Al), the stiffness Kc is obtained as 439 N/$μm$, while the stiffness Kt for bolt–nut system is obtained as Kt = 272 N/$μm$ from Ref. . 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/lf 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 σc, 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.

1. An equivalent bolt stress σeqv should be less than the yield stress σy of bolt (static strength).

2. A bolt stress amplitude σa under repeated external loading should be less than the bolt fatigue stress σA (fatigue strength).

3. Never bolt loosening occurs.

4. 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 $σc$ (the interfacial contact stress condition).

5. The bearing stress σw should be less than the critical contact stress σcri (the bearing stress condition).

For satisfying item (3), the higher bolt preload is desirable for preventing bolt loosening; however, at the same time, the bearing stress condition (item (5)) should be satisfied. Item (4) is for keeping joint function and integrity. In an initial bolt tightening using a wrench, the shear stress τ occurs at the engaged threads due to a torque Ts which is applied to the engaged threads. The torque Ts 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 $Ffmax$in scattered preloads, and the shear stress τ is obtained by Eq. (8), where μs/cos30 deg is changed with 1.155μs in Eq. (7) which is applicable for the half thread angle with 30 deg, and where ds is the diameter of a circle of which the area equals to the stress area As in Eq. (8), p is the pitch of threads, and d2 is the pitch diameter
$Ts=Ffmax2(Pπ+1.155μsd2)$
(7)
$τ=16Tsπds3$
(8)
The maximum bolt stress σbmax in the axial direction is obtained by Eq. (9) taking account of a maximum bolt preload Ffmax in scattered bolt preloads and the increment in the axial bolt force Ft
$σbmax=Ffmax+FtAs$
(9)
Thus, the equivalent bolt stress is obtained by Eq. (10a), i.e., Mises' equivalent stress
$σeqv=σbmax2+3τ2$
(10a)
For satisfying the above item (1), the following Eq. (10b) should be satisfied:
$σeqv≤σy$
(10b)
In addition, for satisfying above item (2), a stress amplitude of bolt $σa$ should be satisfied with the following Eq. (11), where $αA$ is the stress concentration factor at the first thread root , A3 is the root area, and$σA$ is the fatigue stress:
$σa=Ft2A3αA≤σA$
(11)
Concerning item (3) and item (5), the higher bolt preloads are required for preventing bolt loosening. As the bolt preload increases, the bearing stress σw between bolt head (or nut) and a joint member is increased. The bearing stress σw should be less than the critical bearing stress σcri  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 Aw is the bearing area between bolt head (or nut) and a joint member:
$σw= (Ffmax+Ft)/Aw<σcri$
(12)
For satisfying the above item (4), the following Eq. (13) should be satisfied, where $Ffmin=Ffmax/Q$, and Af is the interfacial cross section area between joint members:
$Ffmin−Fc≥σcAf$
(13)
Taking account of the scattered bolt preloads, when the bolt preload is tightened at the maximum preload of Ffmax in scattered preloads, the equivalent bolt stress σeqv must be less than the yield stress of bolt, and when it is tightened at the minimum preload Ffmin, the interfacial contact stress condition must be satisfied. The target bolt preload Fftarget is obtained as the average of the maximum Ffmax and the minimum bolt preload Ffmin. In addition, a target tightening torque Tftarget is obtained by the following Eq. (14a) [1,4], where Tw is a torque applied to the bearing surfaces. Tftarget is obtained from Ts + Tw, where Tw is obtained from Eq. (14b):
$Tftarget= Ts+Tw=K×Fftarge×d = Fftarget2(Pπ+1.155μsd2+μwDw)$
(14a)
$Tw=Fftarget2μwDw$
(14b)

where Dw 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 dw of outside diameter at the bearing surfaceswhen the washer is not inserted. Additionally, K is the nut factor. When the friction coefficients $μs$for screw threads and $μw$for the bearing surfacesare provided, the target torque Tftarget is calculated by the third equation in Eq. (14a).

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 σy 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 Ffmax is assumed as Ffmax = βσyAs (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:

1. A repeated external load W = 10 kN (0–10 kN).

2. Required interfacial contact stress σc = 10 MPa.1

3. The tightening coefficient Q (= Ffmax/Ffmin) is chosen as Q = 2.0 [1,17].

4. The bolt hole diameter dh is assumed as dh = (2a)=1.1d (the first approximation).

5. Ffmax is assumed to be Ffmax = $βσyAs.$

6. The outside diameter 2b of hollow cylinders is 39 mm, and the inside diameter 2a is 13 mm (b/a = 3.0).

7. The critical bearing stress for aluminum is 360 MPa (Al:A7075) and 700 MPa for steel .

8. 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 As of threads is As = 84.3 mm2 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 Ft is obtained as Ft = 600 (N), and the value of Fc is obtained from the equation Fc = (1 − Φ) × W = 9400 N. The minimum bolt preload Ffmin is obtained as follows:
$Ffmin≥Fc+σcAf$
(13′)
where Ffmin is obtained for β = 0.6 as follows:
$Ffmin=FfmaxQ=βσyAs/Q=0.6σy×84.32=25.29σy$
where σc = 10 MPa and $Af=(π/4)(392−132),$ and where the bolt hole diameter dh (= 2a) is 13 mm. Thus, from Eq. (13′), σy is obtained by the following equation:
$25.29σy = 9400 + 10,618 = 20,018 N$
As a result, a calculated yield stress σy′ is obtained as 791 MPa. Then, actual yield stress $σy$of bolt should be determined from the bolt strength grade, and the yield stress isdetermined as σy = 900 MPa from the strength grade 10.9. For β = 0.7 and 0.8, the values of calculated yield stress σy′ are obtained as 678 and 593 MPa, respectively. As a result, the yield stress σy 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.
9. Check the interfacial contact stress condition:

Using the newly determined yield stress σy of bolt, the interfacial contact stress condition described in Eq. (13′) is examined. The minimum bolt preload Ffmin 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 Fc + σcAf in right term of Eq. (13′) is 20,018 N. So, the condition for Eq. (13′) is satisfied for all values of β.

10. Check the bearing stress σw:

Using the determined yield stress$σy$, the maximum bolt preload Ffmax is calculated as$Ffmax=βσyAs.$

The maximum load at the bearing surfaces is obtained as Ffmax + Ft, and the bearing stress σw is obtained using Eq. (12) where Aw (=74.1 mm2) is the bearing area. Table 2 shows the calculated results of the bearing stress σw between the bearing surfaces. For all the values of β, the bearing stress σw 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 dh is13 mm, the outside diameter D is 24 mm, and the thickness is 2.5 mm [19,20]. The bearing stress $σw′$ between the washer and the joint member is also described in Table 2. It is noticed that the bearing stress $σw′$is less than the critical bearing stress of 360 MPa when the steel washer is inserted.

11. Check the equivalent stress σeqv of bolt:

Using Eq. (7), the tightening torque Ts applied to the engaged threads is obtained, and the shear stress τ is also obtained using Eq. (8) where the friction coefficient μs is chosen as 0.1 under lubricant. Additionally, the maximum bolt stress in the axial direction σbmax is obtained from Eq. (9). Thus, Mises' equivalent stress σeqv 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 σy is determined from the strength grade for the values of β = 0.6, 0.7, and 0.8, respectively. The value of Ffmax is obtained for β = 0.6, 0.7, and 0.8 (see Table 3). It is found that the highest Ffmax is obtained when β is 0.6 among three values of β. For all the values of β, the values of σeqv are less than each yield stress σy. So, the condition for bolt strength is satisfied.

12. Check the bolt fatigue strength:

Using Eq. (11), the stress amplitude of bolt is obtained by the following equation, where the stress concentration factor αA is assumed to be 4.2 from Ref. , the root area of threads A3 is 76.2 mm2, and fatigue strength σA is around 50 MPa [1,21]:
$σa=6002×76.24.2=16.5 MPa≤50 MPa$
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.
Table 2

The calculated results of the bearing stresses σw and σw′ for each β

βFfmax+Ft (N)σw (MPa)σw′ (MPa)
0.645,922622144
0.742,887581134
0.843,561590136
βFfmax+Ft (N)σw (MPa)σw′ (MPa)
0.645,922622144
0.742,887581134
0.843,561590136
Table 3

Calculated results of equivalent stress of bolt for each β

βFfmax (N)Ts (N·mm)τ (MPa)σbmax (MPa)σeqv (MPa)Judge (MPa)
0.645,52241,220188544603<900
0.742,48738,471176508592<720
0.843,16139,082179516601<640
βFfmax (N)Ts (N·mm)τ (MPa)σbmax (MPa)σeqv (MPa)Judge (MPa)
0.645,52241,220188544603<900
0.742,48738,471176508592<720
0.843,16139,082179516601<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 σc 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 σc at the interfaces should be reduced as σc = 3 MPa because the maximum bolt preload Ffmax 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 σy′ 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 (σy = 640 MPa). Next, the bearing stress is examined. From Eq. (12), σw is obtained as 358 MPa for β = 0.48 and σw = 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 σeqv is obtained as 364 MPa for β = 0.48, and this value is less than the yield stress of 640 MPa. The stress amplitude σa 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 σc = 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 σc 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 σy′ 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 σy 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 σy, the interfacial contact stress condition is examined. As a result, for all the values of β, Eq. (13′) is satisfied. Then, the bearing stress σw and the equivalent stress σeqv of bolt are examined. Table 4 shows the calculated results of σw and σeqv. All values of σw are less than 700 MPa (σwcri). The equivalent stresses σeqv are less than the yield stress σy. It is shown that the results of σw and σeqv 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. (10a). Table 5 shows the target preload Fftarget, target torque Tftarget, and the nut factor K for each value of β, where the diameter at bearing surface dw is chosen as 16.63 mm for M12 in Eq. (14b). The target bolt preload Fftarget is obtained by the average of Ffmin and Ffmax, and the target torque Tftarget is obtained using Eq. (14a). The friction coefficients μs and μw 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.

Table 4

Calculated results of σw and σeqv for each value of β

βFfmax + Ft (N)σw (MPa)σcri (MPa)σbmax (MPa)Ts (N·mm)σeqv (MPa)σy (MPa)Judge
0.742,78757770050738,471591720OK
0.843,46158670051539,082602640OK
0.8546,15962270054741,525638640OK
βFfmax + Ft (N)σw (MPa)σcri (MPa)σbmax (MPa)Ts (N·mm)σeqv (MPa)σy (MPa)Judge
0.742,78757770050738,471591720OK
0.843,46158670051539,082602640OK
0.8546,15962270054741,525638640OK
Table 5

Target bolt preload, target torque, and nut factor K for each value of β

βFftarget (N)Tftarget (N·mm) (lub)K (lub)Tftarget (N·mm) (no-lub)K (no-lub)
0.731,86552,4560.13787,3360.228
0.832,37053,2890.13788,7200.228
0.8534,39456,6200.13794,2680.228
βFftarget (N)Tftarget (N·mm) (lub)K (lub)Tftarget (N·mm) (no-lub)K (no-lub)
0.731,86552,4560.13787,3360.228
0.832,37053,2890.13788,7200.228
0.8534,39456,6200.13794,2680.228

### 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 (σy = 900 MPa).

Condition:

1. Strength grade: 10.9 (σy = 900 MPa).

2. The outside diameter 2b 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).

3. Repeated external tensile load W: 10 × 103 N (0–10 kN).

4. Tightening coefficient Q (Ffmax/Ffmin) is 2.0.

5. Required interfacial contact stress between joint members σc = 10 MPa.

6. Bolt hole diameter dh is assumed as 1.1d (first approximation).

The maximum preload Ffmax and the minimum preload Ffmin in scattered bolt preloads are described by the following Eqs. (14c) and (14d), where the stress area As is assumed as $(π/4)d2 (the first approximation)$. Additionally, β is assumed as 0.6, 0.7, and 0.8:
$Ffmax=βσyπ4d2=706.8βd2$
(14c)
$Ffmin=βσyπ4d2/Q=353.4βd2$
(14d)
In Fig. 7, the load factor Φ is assumed as 0.13 for St–Al joint (upper) when the ratio b/a is 2.5 (approximation). So, the additional bolt force Ft is 1300 N, and the force Fc is 8700 N. The interface area Af in the joint members is obtained as follows, where the bolt hole diameter is assumed as 1.1d:
$Af=π4(31.52−(1.1d)2)$
(15)
For satisfying the interfacial contact stress condition, the nominal diameter d is expressed by the following Eq. (16) using Eq. (13′) with Eqs. (14d) and (15):
$(353.4β+9.5)d2=16,493$
(16)
From Eq. (16), the calculated nominal diameter 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 As = 58.0 mm2, and the bolt hole diameter dh (2a) 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 Ft is 600 N, and the force Fc is 9400 N. The interfacial contact stress condition Eq. (13′) is changed with the following Eq. (13″), where As is 58.0 mm2 for d = 10 mm (M10) and Q = 2.0:
$βσyAsQ>Fc+σcAf$
(13″)

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 dh = 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 Af is changed as $Af=646 mm2$. The thread pitch p and the pitch diameter d2 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 Ts is obtained using Eq. (7), and the shear stress τ due to the torque Ts is obtained using Eq. (8). The equivalent stress σeqv is obtained from Eq. (10a). The bearing stress σw is obtained from Eq. (12).

Table 6 shows the calculated results of the equivalent stress σeqv and the bearing stress σw. Since the values of σeqv are smaller than the yield stress of 900 MPa, the bolt strength condition is satisfied. All values of σw for each value of β are larger than the critical contact stress σcri = 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 dh 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 σw′ between the washer and the joint member is obtained. It is found that the bearing stress condition is satisfied, i.e., the value of σw′ is less than the critical value of 360 MPa.

Table 6

The equivalent stress σeqv and the bearing stress σw and σw′ for each β

βd (mm)As (mm2)ds (mm)Ffmax (N)Ft (N)σbmax (MPa)Ts (N·mm)τ (MPa)σeqv (MPa)σw (MPa)σw′ (MPa)
0.61284.310.3645,522120055841,220188642630146
0.71058.08.5936,54060064027,769223747508143
0.81058.08.5941,76060073031,736255852579163
βd (mm)As (mm2)ds (mm)Ffmax (N)Ft (N)σbmax (MPa)Ts (N·mm)τ (MPa)σeqv (MPa)σw (MPa)σw′ (MPa)
0.61284.310.3645,522120055841,220188642630146
0.71058.08.5936,54060064027,769223747508143
0.81058.08.5941,76060073031,736255852579163

Finally, the fatigue strength of bolt is examined. Using Eq. (11), the bolt stress amplitude $σa$ is calculated. For β = 0.6, Ft is 1200 N, A3 is 76.2 mm2, and the stress concentration factor is assumed as αA = 4.2 . The value of $σa$ is obtained as 19.2 MPa [1,21]. In addition, for β = 0.7 and 0.8, $σa$ is obtained as 12.0 MPa, where A3 is 52.3 mm2, Ft is 300 N, and αA is the same as 4.2, while $σA$ is greater than 50 MPa for d = 12 mm and the strength grade 10.9, and where $σA$ 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 Fftarget is obtained using the average of the maximum $Ffmax$and the minimum $Ffmin$bolt preloads as follows: $Fftarget=(Ffmax + Ffmin)/2$. The target tightening torque Tftarget is obtained as Tftarget = Ts + Tw, shown in Eq. (14a), where the values of Dw in Eq. (14b) 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 Fftarget and target torque Tftarget 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 μs and μw 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.

Table 7

Target torque Tftarget and the nut factor K for the cases with lubricant and without lubricant (washers are inserted)

βd (mm)Fftarget (N)$μs=μw$Tftarget (N·m)K (with-lubricant)Ftarget (N)$μs=μw$Ttarget (N·m)K (nonlubricant)
0.61234,1410.162,5060.15256,8440.18104,9050.256
0.71027,4050.142,4080.15438,8450.1871,1000.259
0.81031,3200.148,4660.15444,3950.1881,2580.259
βd (mm)Fftarget (N)$μs=μw$Tftarget (N·m)K (with-lubricant)Ftarget (N)$μs=μw$Ttarget (N·m)K (nonlubricant)
0.61234,1410.162,5060.15256,8440.18104,9050.256
0.71027,4050.142,4080.15438,8450.1871,1000.259
0.81031,3200.148,4660.15444,3950.1881,2580.259

#### 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 Ffmax, while the interfacial contact stress σc condition (Eq. (13)), the bearing stress σw condition (Eq. (12)), and the equivalent stress condition σeqv (Eqs. (10a) and (10b)) 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 (σy = 480 MPa) and the nominal diameter d as d = 14 mm. The bolt hole diameter dh (=2a) is chosen as dh = 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 Ft is 1600 N and the force Fc 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.

1. The case of St–St joint members: The provided conditions are the same as mentioned before. The strength grade of bolt is 10.9 (σy = 900 MPa), the required interfacial contact stress σc = 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 σeqv, and the bolt fatigue strength are all satisfied. In addition, the bearing stress σw 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 σw 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.

2. 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 σc, the bolt equivalent stress σeqv, 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 σw between bolt head (nut) and aluminum joint member and σw′ 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. (10a)) 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 σw 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.

Table 8

Effect of β on the nominal diameter d and insertion of washer (Y: washer is needed)

No.βCombination of materialsd (mm)Strength gradeσw (MPa)Washerσw′ (MPa)
10.6St–Al1210.9623Y144
20.7St–Al1010.9503Y142
30.8St–Al1010.9575Y162
40.5St–Al148.8336No
50.7St–Al146.8352No
60.8St–St1010.9572No
70.85St–St1010.9608No
80.7Al–Al146.8356No
90.5Al–Al148.8340No
No.βCombination of materialsd (mm)Strength gradeσw (MPa)Washerσw′ (MPa)
10.6St–Al1210.9623Y144
20.7St–Al1010.9503Y142
30.8St–Al1010.9575Y162
40.5St–Al148.8336No
50.7St–Al146.8352No
60.8St–St1010.9572No
70.85St–St1010.9608No
80.7Al–Al146.8356No
90.5Al–Al148.8340No

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 σc is incorporated in the present design method .

## 7 Conclusions

In this paper, the objectives are to examine the load factor and a load Ws 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 Ws 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.

1. 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.

2. 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 Kc decreases as Young's modulus decreases.

3. A load Ws when the interfaces start to separate is examined. It is found that the load Ws decreases as the ratio b/a increases. It is also observed that the load Ws increases as the value of the load factor increases, namely, the value of Ws of Al–Al joint is larger than that of St–Al and St–St joints.

4. 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 Ws are fairly coincident with the experimental results.

5. 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.

6. Five important items for safety design of bolted joints are proposed. In a new simple design method, the interfacial contact stress (required contact stress σc) 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 (=Ffmax/Ffmin), 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.

7. 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

Af =

cross-sectional area at the interfaces between joint members (mm2)

As =

stress area at engaged threads in bolt (mm2)

Aw =

bearing area between bolt head (or nut) and joint member (mm2)

A3 =

d =

nominal diameter of bolt (mm)

D =

outside diameter of washer (mm)

d′ =

calculated nominal diameter (mm)

dh =

bolt hole diameter (= 2a) (mm)

ds =

diameter of circle of which area is the same as the stress area (mm)

dw =

outside diameter of bearing surfaces (mm)

d2 =

pitch diameter (mm)

Dw =

equivalent diameter at the bearing surfaces (mm)

Eb =

Young's modulus of bolt (MPa)

Ei =

Young's modulus of joint member (i = 1, 2) (MPa)

Fbmax =

maximum bolt force (= Ffmax + Ft) (N)

Fc =

force eliminated from the interfaces in joint (N)

Ff =

bolt preload in initial clamping (N)

Ffmax =

maximum bolt force among scattered preloads (N)

Ffmin =

minimum bolt force among scattered preloads (N)

Fftarget =

Ft =

increment in axial bolt force (N)

K =

nut factor (K-factor)

Kc =

compressive stiffness for joint members (N/μm)

Kpt =

tensile stiffness for joint members (N/μm)

Kt =

stiffness for bolt–nut system (N/μm)

p =

Q =

tightening coefficient (= Ffmax/Ffmin)

Tftarget =

target torque in initial clamping (N·mm)

Ts =

torsional torque applied to engaged screw threads (N·mm)

Tw =

torsional torque applied to the bearing surfaces (N·mm)

W =

Ws =

external tensile load when the interfaces start to separate (N)

2a =

inside diameter of hollow cylinder (mm)

2b =

outside diameter of hollow cylinder (mm)

$μs$ =

$μw$ =

friction coefficient between bearing surfaces

νi =

Poisson's ratio of joint member (i = 1, 2)

σa =

stress amplitude in bolt (MPa)

σA =

fatigue strength of bolt (MPa)

σbmax =

maximum bolt stress in axial direction (MPa)

σc =

required interfacial contact stress between joint members (MPa)

σeqv =

equivalent stress (= $σ2+3.0τ2$ ) (MPa)

σy =

yield stress (strength) of bolts (MPa)

σy′ =

yield stress in calculating process in Eq. (13′) (MPa)

$τ$ =

shear stress due to torsional load Ts (MPa)

Φ =

## Footnotes

1

σc is determined such that when a soft gasket is used in a joint, a minimum gasket stress for sealing is around 10 MPa.

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