# Assessment of the Durability of Threaded Joints

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Object

^{3}–5 × 10

^{3}load cycles during tightening and to overloads in cases of hydraulic testing, start-up, shutdown, full or partial cooling, change of efficiency and capacity, and in certain emergency and servicing of the equipment. Additional axial cyclic loading appears as a result of initialization, shutdown, change in operating mode, and lid strain. The latter also causes the bending moment. Depending on the function of the bolt and the stud joint, the tightening may reach $0.65{\sigma}_{y}$ in various units. The structure of critical threaded joints used in mineral grinding mills is shown in Figure 1.

## 3. Experimental Results and Their Analysis

#### 3.1. Experimental Procedures

#### 3.2. Influence of Tightening and Cyclic Bending

#### 3.3. Comparison of the Experimental Results to Benchmark Curves

_{adm}is determined at a certain amplitude of the stress cycle, and an admissible stress amplitude σ

_{a},

_{adm}is determined by providing a number of cycles. This is implemented using two methods: according to the calculated fatigue curves provided in the norms and according to the formulas, if the number of cycles does not exceed 10

^{6}. In the PNAE normative methodology, low-cycle durability is determined according to the first stage of failure, i.e., crack initiation in the stress concentration areas.

_{0}was determined on the basis of the two values calculated using these formulas and was considered to be a calculation result. According to the ASME code, in the case of the high-strength studs (700÷1150 MPa), the low-cycle durability is determined according to the calculated durability curves that are verified experimentally. For the threaded joints, the curves were calculated using the formula [39,40,41]:

^{8}, including an increase in concentration factors, shows that the permissible amplitude and number of cycles are subject to stricter regulation in the PNAE norms than in the ASME norms. The total tensile and bending stresses in the outlying layers of the bolt may sometimes exceed the yield strength. The resulting nonelastic strain conditions along the entire length of the bolt may lead to changes in the operating conditions of the joint. Nonplastic strain processes determine the failure, which may be a localized or general failure. In the first case, this is related to the accumulation of damage under the action of cyclic loads that lead to the propagation of the crack.

#### 3.4. Analysis of the Crack Propagation

#### 3.5. Loosening Analysis

_{t}

_{1}and the bending stresses were almost unchanged ${\sigma}_{b1}\approx {\sigma}_{b}$. As the crack spread, the tension decreased to σ

_{t}

_{2}just before the fracture and the bending stresses to ${\sigma}_{b2}$. If the stresses in the relative elasticity stage exceeded the yield stress ${\sigma}_{t}>{\sigma}_{y}$ (Figure 9d–f), then after 50 to 100 cycles the initial stress decreased to ${\sigma}_{t1}^{\prime},$ and the bending stresses were almost unchanged at ${\sigma}_{b1}^{\prime}$ ≈ ${\sigma}_{b}^{\prime}$. As the crack spread to the fracture, the stress decreased to ${\sigma}_{t2}^{\prime}$, and the bending stresses decreased to ${\sigma}_{b1}^{\prime}$.

_{.}In the general case, the relaxation of the nut–stud–nut system is made up of several components:

_{cr}is the depth of the crack, d is the diameter of the stud shank), was established, as shown in Figure 11.

**to 0.76σ**

_{y}**and remained stable for a long time. The stress magnitude as the crack started and spread is shown in curve 3. If the initial stress was 0.7σ**

_{y}**, the magnitude of the stress varied with the crack according to curve 4.**

_{y}## 4. Shakedown of the Threaded Joints

_{s}, at least at one point in the structure. If the stresses in the structure are proportionate to only one parameter, plastic strains with variable sign start developing, where the proportional strain interval exceeds 2σ

_{s}.

#### 4.1. Crack-Free Joints

- a given structure shakedown:$$\underset{0}{\overset{T}{\int}}dt{\displaystyle \underset{V}{\int}({\sigma}_{ij}-{\sigma}_{ij}^{(e)}){\epsilon}_{ij0}^{\u2033}dV}}>0$$
- a given structure that does not undergo shakedown, i.e., the structure must collapse eventually due to cyclic plastic deformations:$$\underset{0}{\overset{T}{\int}}dt{\displaystyle \underset{V}{\int}({\sigma}_{ij}-{\sigma}_{ij}^{(e)}){\epsilon}_{ij0}^{\u2033}dV}}<0$$

_{i}

_{0}. The respective fictive yield stresses ${\sigma}_{ij*}^{0}$ are determined by using the associated flow rule. According to the experimental data, crack initiation and propagation take place in the weakest link of the threaded joint, i.e., in the stud, after a certain number of cycles. Plastic strains occur in the outer layers of the stud before crack initiation, under the action of repetitive variable load. As a rod with circular cross-section, the stud is subjected to axial force F and symmetrically variable bending moment M (–M* < M

_{f}< M*). Figure 12 shows a cross-section of a stud and the stress distribution under M and F forces leading to yield stresses (${\sigma}_{y}$—yield stress). The tension zone A is equal to the compression zone C. Consequently, the longitudinal force F is equilibrated to the stresses acting in zone B. Bending moment M equilibrates with the stresses acting in zones B and C.

_{d}and m

_{d}, if the threaded joint is subject to axial force F and bending moment M. Limit point L coordinates m

_{lim.}, n

_{lim}. The inclination angle of the radius for similar cycles is

_{d}and P-m

_{d}are presented in Figure 14.

#### 4.2. Joints with a One-Sided Crack

_{cr}to the radius ratio of the cross-section k = h

_{cr}/r. By using those ratios according to Figure 16 and from Equation (35), the conditions of progressive plastic strain were obtained (Table 1).

_{d}and P-m

_{d}are presented in Figure 17.

#### 4.3. Joints with a Two-Sided Crack

_{cr}and the radius r. Crack depth 2h

_{cr}to cross-sectional diameter d = 2r ratio k = h

_{cr}/r. Using various values of the ratio, progressive plastic strain conditions were obtained (Table 2).

_{d}and P-m

_{d}are presented in Figure 19.

## 5. Discussion

## 6. Conclusions

- (1)
- Short cracks (up to 0.5 mm deep, up to 5 mm long) do not reduce the static strength of threaded connections. This is the size of the crack easily measurable by nondestructive inspection methods, which can be considered as the maximum allowed under service conditions.
- (2)
- The crack propagation characteristics are different in cyclically bending and tensile threaded joints. At a certain fixed nut-and-pin position (the bending plane is aligned with the most loaded position in the thread recess), the crack is one-sided. Otherwise, i.e., at any other position of the nut relative to the stud, the crack is two-sided. The adaptability of the crack decreases with the progression of a two-sided crack, i.e., the crack is more dangerous.
- (3)
- In prestressed and cyclically bending threaded joints, a relationship was found between the loosening and the parameters of the failure process, such as the shape and depth of the crack. The propagation of the crack is mainly due to the energy of the cyclic bending, so the stresses in the propagation plane of the crack increase as the stress is only slightly reduced. When the crack reaches a critical size, the rest of the crack breaks rapidly due to stress after a small pulse of cyclic bending energy.
- (4)
- In order to obtain a stable tension of the threaded joint and thus a stable clamping of the elements to be joined, it is necessary to apply a repeated (3–4 times) turn of the nut, irrespective of the method used to assemble the threaded joint. This prevents stress reduction during the assembly process.
- (5)
- Shakedown analysis of the threaded joints was performed. The safety margin of the progressive shape change in the experimentally tested joints was determined, and a statistical evaluation was performed. The safety factor ranged from 1.06 to 1.79 for the joints analyzed, with the probability varying from 1% to 99%.
- (6)
- Where the tightening was close to 0.8σ
_{y}, (σ_{y}—yield strength) and the bending was 0.4σ_{y}, the safety factor for the progressive change in shape was close to 1. These safety factors are considered to have insufficient reliability in the elastic–plastic area analyzed. - (7)
- The statistical investigation carried out as part of the study showed that to develop a reliable definition of the safety margin for progressive change in shape in threaded joints, an experimental curve of 50% must be developed, which, in the case analyzed, reflects a safety factor equal to 1.322.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | cross-section area of the stud (mm^{2}); |

${A}_{e}$$,C,{S}_{e}$ | parameters depending on the mechanical properties of steel and indicators of the load cycle (A_{e} = 0.149; C = 0.5; S_{e} = 0.45); |

${A}_{p}$ | area of fictive yield surface (mm^{2}); |

a, b, m, n | dimensionless load parameters; |

D | plastic energy dissipation; |

d | diameter of the stud (mm); |

E | elasticity modulus (MPa); |

${e}_{c}$ | material plasticity indicator determined by assessment of the variation in the cross-section area of the standard cylindrical specimen subjected to tension; |

$F,{F}_{t}$ | axial and tensile force (N); |

${F}_{y}$ | limiting axial force causing plastic deformation (N); |

${F}_{fd}$ | reaction force in the clamped parts; |

$F\left({\epsilon}_{ij0}^{\u2033}\right)$ | plastic energy dissipative function; |

${h}_{a}$, ${h}_{cr}$ | crack depth (mm); |

${I}_{x}$, ${I}_{x1}$ | moments of inertia of the cross-section (mm^{4}); |

${l}_{0}$ | effective length of the stud (distance between the cross-sections subjected to the largest loading in the nut–stud–nut assembly, mm); |

M | variable bending moment (N * mm); |

${M}_{f},{M}^{*}$ | threshold bending moment (N * mm); |

${M}_{y}$ | limiting bending moment causing plastic deformation (N*mm); |

${m}_{0}$ | number of specimens in the selected interval; |

N | number of cycles; |

${N}_{adm}$ | admissible number of cycles; |

${N}_{f}$ | number of cycles to failure; |

${n}_{0}$ | number of tested specimens; |

${n}_{N}$ | safety factor of strength by the number of cycles; |

${n}_{\sigma}$ | safety factor of strength by stress; |

P | random value probability; |

P(x) | probability density function; |

r | stud radius (mm); |

${r}_{a}$ | cycle asymmetry coefficient; |

${r}_{c}$ | thread root radius (mm); |

T, t | time (s); |

${u}_{i0}$ | plastic displacement components (mm); |

$\mathrm{\Delta}{u}_{i0}$ | plastic displacement increment components (mm); |

$\mathrm{\Delta}{u}_{z0}$ | elongation of the stud (mm); |

V | volume (mm^{3}); |

${x}_{i}^{0},{p}_{i}^{0},{p}_{z}^{0}$ | volumetric and superficial force components; |

${x}_{i}$ | random measure; |

${x}_{0}$ | mean arithmetic value; |

${x}_{1}$$,{y}_{1}$ | coordinates of the neutral axis (mm); |

${y}_{0}$ | coordinate of the crack (mm); |

Greek symbols | |

${\alpha}_{0}$ | angle to crack (°); |

${\alpha}_{1}$ | angle to neutral axis (°); |

${\epsilon}_{ij0}^{\u2033}$ | $\mathrm{plastic}\mathrm{strain}\mathrm{components}\mathrm{corresponding}\mathrm{to}{u}_{ij0}$; |

$\mathrm{\Delta}{\epsilon}_{ij0}^{\u2033},\mathrm{\Delta}{\epsilon}_{z0}^{\u2033}$ | $\mathrm{plastic}\mathrm{strain}\mathrm{increment}\mathrm{components}\mathrm{corresponding}\mathrm{to}\mathrm{\Delta}{u}_{ij0}$; |

ε | stud turns strain (%); |

${\epsilon}_{f}$ | failure strain (%); |

${\epsilon}_{d}$ | clamped parts strain (%); |

η | safety factor for the progressive shape change; |

${\sigma}_{0}$ | mean square deviation; |

${\sigma}_{0.2}$ | elastic limit or yield strength (MPa), the stress at which 0.2% plastic strain occurs; |

${\sigma}_{-1}$ | endurance limit of the material (MPa); |

*σ_{a, adm} | amplitude of the relative local tensile stress cycle (MPa); |

${\sigma}_{b}$$,{\sigma}_{b1}$$,{\sigma}_{b2}$$,{\sigma}_{b}^{\prime}$$,{\sigma}_{b1}^{\prime}$$,{\sigma}_{b2}^{\prime}$ | bending stress (MPa); |

${\sigma}_{ij}$ | stress of the fictive yield surface (MPa); |

${\sigma}_{ij}^{0}$ | minimum stress of the fictive yield surface (MPa); |

${\sigma}_{ij*}^{0}$ | $\mathrm{stress}\mathrm{of}\mathrm{the}\mathrm{fictive}\mathrm{yield}\mathrm{surface}\mathrm{related}\mathrm{to}\mathrm{the}$$\mathrm{\Delta}{\epsilon}_{ij0}^{\u2033}$ associated flow rule (MPa); |

${\sigma}_{ij}^{\left(e\right)},{\sigma}_{z}^{\left(e\right)}$ | stress components caused by external loads on a perfectly elastic material (MPa); |

${\sigma}_{z}^{0}$ | variable stress component (MPa); |

${\sigma}_{f}$ | failure stress (MPa); |

${\sigma}_{max}$$,{\sigma}_{min}$ | maximum and minimum cycle stress (MPa); |

${\sigma}_{y}$ | yield stress (MPa); |

${\sigma}_{S}$ | total bending and tensile stress (MPa); |

${\sigma}_{t},{\sigma}_{tmax}$ | tension stress (MPa); |

${\sigma}_{td}$ | tension stress reduction due to contact between intermediate parts (MPa); |

${\sigma}_{tb}$ | tension stress reduction due to shear, thread surface contact and crack propagation (MPa); |

${\sigma}_{tn}$ | tension stress reduction due to shear of the first turns of the nut (MPa); |

${\sigma}_{t}^{\supset}$ | tension stress reduction due to self-rotation of the nut (MPa); |

${\sigma}_{u}$ | ultimate tensile stress (MPa); |

ψ | percent area reduction (%). |

Abbreviations | |

PNAE norms | Regularities and Norms in Nuclear Power Engineering; |

ASME norms | The American Society of Mechanical Engineers. Boiler and Pressure Vessel Code, an internationally recognized code. |

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**Figure 1.**Fragment of the mineral grinding mill: (

**a**) stress concentration areas in the pressure vessel stud joints; (

**b**): 1—in the stud (bolt)–nut joint; 2—in the free-threaded region; 3—in the transitional threaded region; 4—in the smooth region (due to unevenness occurring during surface processing); 5—in the stud-body joint.

**Figure 3.**Strains in individual turns of the stud M52×4:

`○`—σ

_{t}= (0.6 ± 0.35)σ

_{y}; □—σ

_{t}= (0.6 ± 0.45)σ

_{y}.

**Figure 4.**Variation of the stress of the threaded joints M52×4:

**Δ**—measurement limits;

**—measurement limits upon crack initiation;**

`▲`**●**—crack initiation; ✕—testing stopped.

**Figure 5.**Variation of the stresses in the M52×4 stud depending on the number of cycles and coefficient of asymmetry; the empty symbols refer to maximum cycle stresses and the filled symbols represent the minimum stresses.

**Figure 6.**Comparison of the experimental results with the fatigue curves according to the PNAE (straight line) and ASME (dotted line) norms: ●—experimental data; a—maximum nominal stresses $0.95{\sigma}_{y}$; b—maximum nominal stresses $0.6{\sigma}_{y}$.

**Figure 7.**(

**a**)—one-sided fracture of the stud: 1—crack area; 2—normal propagation area of the crack; 3—accelerated propagation area of the crack; 4—brittle failure area; 5—trace of sample stoppage; (

**b**)—two-sided fracture of the stud; (

**c**)—the start of crack propagation in the root of the stud thread profile.

**Figure 8.**Tension stress changes (decreasing) during assembly of the threaded joint: (

**a**)—initial tightening cycle; (

**b**)—repeated tightening cycles.

**Figure 9.**Loading diagrams of the threaded joints in the bending plane: when σ

_{t}< σ

_{y}(

**a**–

**c**); when σ

_{t}> σ

_{y}(

**d**–

**f**).

**Figure 10.**Loading diagrams of threaded joints in the plane perpendicular to bending: (

**a**)—initial tightening cycle; (

**b**)—at the start of cyclic loading; (

**c**)—before fracture.

**Figure 13.**Position of the threaded joints in the shakedown diagram: 1—variable flow condition; 2—progressive shape change condition; 3—collapse range; 4—elastic range; •—experimental data.

**Figure 14.**Probabilistic relationships of the parameters m

_{d}(1) and n

_{d}(2) and their 95% probabilistic intervals for crack-free threaded joints.

**Figure 17.**Probabilistic relationships of the parameters m

_{d}(1) and n

_{d}(2) and their 95% probabilistic intervals for crack-free threaded joints.

**Figure 19.**Probabilistic relationships of the parameters m

_{d}(1) and n

_{d}(2) and their 95% probabilistic intervals for threaded joints with a two-sided crack.

Relative Crack Depth k = h _{cr}/r | Conditions of Progressive Plastic Fracture |
---|---|

k = 0 | $n+0.72m=1$ |

k = 0.05 | $1.014n+0.735m=1$ |

k = 0.10 | $1.04n+0.768m=1$ |

k = 0.15 | $1.074n+0.808m=1$ |

k = 0.20 | $1.117n+0.852m=1$ |

Relative Crack Depth k = h _{cr}/r | Conditions of Progressive Plastic Fracture |
---|---|

k = 0 | $n+0.72m=1$ |

k = 0.05 | $1.013n+0.745m=1$ |

k = 0.10 | $1.039n+0.791m=1$ |

k = 0.15 | $1.074n+0.854m=1$ |

k = 0.20 | $1.116n+0.932m=1$ |

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## Share and Cite

**MDPI and ACS Style**

Bazaras, Ž.; Leonavičius, M.; Lukoševičius, V.; Raslavičius, L.
Assessment of the Durability of Threaded Joints. *Appl. Sci.* **2021**, *11*, 12162.
https://doi.org/10.3390/app112412162

**AMA Style**

Bazaras Ž, Leonavičius M, Lukoševičius V, Raslavičius L.
Assessment of the Durability of Threaded Joints. *Applied Sciences*. 2021; 11(24):12162.
https://doi.org/10.3390/app112412162

**Chicago/Turabian Style**

Bazaras, Žilvinas, Mindaugas Leonavičius, Vaidas Lukoševičius, and Laurencas Raslavičius.
2021. "Assessment of the Durability of Threaded Joints" *Applied Sciences* 11, no. 24: 12162.
https://doi.org/10.3390/app112412162