# Dependence of the Preload on the Tightening Torque for Hydraulic Plugs
## 1 Introduction  
In hydraulics, threaded plugs are used to close various manufacturing holes and other fluid channels and thus prevent external leakage of the fluid [1; 2; 3]. To ensure sufficient sealing force (clamping force), the plugs must be preloaded like bolts. Preloaded bolted connections are used in engineering applications (e.g., in [4; 5]) to ensure sufficient clamping force and/or to achieve a higher fatigue strength [5] of the structure and are discussed in publications in the field of machine elements, e.g., in [6; 7; 8]. Preload is normally achieved by tightening the bolt or nut of the bolted connection. Given the relationship between the tightening torque and the preload, the preload in the bolt can be controlled with a torque wrench. In general, only 10% to 20% of the applied torque generates the preload in the bolt, while the rest of the torque is needed to overcome the friction in the bolted joint [9; 10]. Therefore, friction plays an essential role in preload formation and its variations have a major influence on the preload [11]. For example, a 5% increase in the underhead or thread friction can reduce the preload by half [9]. For an accurate description of the preload-torque relationship in bolted connections, it is crucial to know the exact tribological conditions in the bolted connections [12; 13] in addition to their geometric and material properties and the possible deformation mechanisms involved. An incorrect evaluation of the preload-torque relationship can lead to excessive or insufficient preload, where both may lead to failure of the bolted connection [11] and eventually to failure of the entire engineering structure [14; 15]. This is especially important in applications where human safety is involved [16]. In general, the tightening torque must overcome the underhead friction between the bolt head and the connected parts, as well as the thread friction between the male and female threads [17], while the rest of the tightening torque generates the preload in the bolted connection [18]. The preload-torque relationship in preloaded bolts can be described by a constant \(K\), known as the torque coefficient or the nut factor [8]:  
\[T=K\cdot F_{V}\cdot d \tag{1}\]  
where \(T\) is the tightening torque [N m], \(F_{V}\) the preload [N] and \(d\) the nominal thread diameter [m]. [PERSON] [8] provides some mean values of the torque coefficient for various combinations of joint materials and surface conditions. An approximate value of 0.2 has been given by various authors for typical unlubricated mid-size steel fasteners, while a value of 0.3 has been used by different authors for blind bolted connections [19]. A more advanced model of the preload-torque relationship was introduced by [PERSON] [17]. This model considers the torque necessary for raising the bolt threads with respect to the nut threads under preload (\(T_{p}\), i.e., the stretch component), the torque to overcome the friction between the threads (\(T_{l}\)) and the torque to overcome the underhead friction (\(T_{u}\)):  
\[T=T_{p}+T_{t}+T_{u} \tag{2}\]  
This preload-torque relationship has been implemented in some standards [20; 21]. In VDI 2230 [20], the preload-torque relationship is specified in an extended form for threads with a 60\({}^{\circ}\) thread angle as follows:  
\[T=F_{V}\cdot[0.16\cdot p+0.58\cdot\mu_{t}\cdot d_{2}+0.5\cdot\mu_{u}\cdot d_{u}] \tag{3}\]  
where \(p\) is the thread pitch [m], \(\mu_{t}\) the coefficient of friction between the male and female threads, \(d_{2}\) the mean thread diameter (\(d_{2}=d-0.6459\cdot p\)) [m], \(\mu_{u}\) the coefficient of friction between the bolt head and the flange surface and \(d_{u}\) the mean underhead diameter [m]. The underhead friction coefficients \(\mu_{u}\) given in the literature for bolted connections vary from 0.2 to 0.45 [19] and the thread friction coefficients from vary 0.025 to 0.1 [22] and also vary depending on each application.