The measurement of friction coefficients according to ISO 16047 [19] consists of using a specific test-bench defined by a controlled engine and 6 axis load-cell. After installing a specimen that represents the real assembly, one side of specimen will be locked (the nut, or the bolt head in our case). Then, a regular tightening rotation angle is applied to the second side. The used load-cell measures, the total applied torque “T”, the bearing surface torque of the turned side “T_b”, the thread torque “T_th” (useful torque + friction torque due to the movement between the screw thread and its nut thread), the clamped load “F₀”, the applied angle “ $\theta$” and the tightening time.

Measured torques are defined by equations given below:

$T=T_b+T_{th}=F_0\left(\frac{P}{2\pi }+0.577\times d_2\times {\mu }_t+\frac{D_b}{2}\times {\mu }_b\right)$(2)

where, P: thread Pitch.

d₂: pitch diameter of the bolt thread: $d_2=d-0.6495\times P$.

d: nominal thread diameter of the bolt.

For each measured point, friction coefficients could be obtained by using equations below:

Default values used for a non-qualified case is 0.15 ± 0.03 (e.g. Zinc Flake coating).

During tightening, the measured Torque-Clamp Force graph commonly shows a linear relationship between these factors. But the torque-angle graph shows three main areas:

1) A first non-linear area at the beginning of tightening. In fact, due to geometric defects and roughness, the contact between the parts at this stage is not perfect and not stable. At the moment, no analytical formula or numerical approach can be industrially used to determine this area. The end of this area defines the minimum level of the snug torque.

2) A second linear area shows a linear relationship between Torque-Angle and between Angle-Preload. This area can be represented by an analytical formula detailed in the paragraphs below. Most of the time, designers prefer to work in this area to prevent the assembly exceeding the yield point.

3) A third non-linear area, highly defined by clamped length, bolt design and their material properties (YLS, ULS, A%,…).

As mentioned above, for the snug torque, the designers and operators ignore the first non-linear area. For elastic tightening, this snug torque can be estimated by the torque that introduces a preload close to 30% of the final preload. And for a tightening at or after the yield point, this ratio can achieve 50% ( [1] , [3] and [12] ).

By using the standard ISO 16047 [19] , friction values, geometric data of assembly and tightening tool accuracy, the snug torque could be calculated by using Equation (2) as:

$T_{snug}=F_{snug}\left(\frac{P}{2\pi }+0.577\times d_2\times {\mu }_t+\frac{D_b}{2}\times {\mu }_b\right)$(2-b)

where: $F_{snug}$: introduced preload by applying the snug torque.

To calculate the required angle to introduce a target preload, the bolt and clamped parts can be represented by two elastic springs, which can be easily calculated using the listed references [1] - [9] or other equivalent references. Applying any additional preload “ΔF₀” will elongate the bolt and will compress clamped parts respectively by “ΔL_b” and “ΔL_P” given by Equation (3) and Equation (4). And due to the helicoidal shape of the threads, the relative rotation of the nut with respect to the screw “Δθ” will create a relative displacement of the nut “ΔL_sn”, calculated by two ways, Equation (5) and Equation (6). Finally, the relationship between the preload and the applied angle is detailed by Equation (7), which illustrates the importance of the thread pitch, the bolt resilience (stiffness), and the clamped parts in addition to the applied angle to calculate the preload. This formula is used in most academic and industrial applications.

$\Delta L_b=\frac{F_0}{K_b}=F_0\times {\delta }_b$ (3)

$\Delta L_P=\frac{F_0}{K_P}=F_0\times {\delta }_P$ (4)

$\Delta L_{sn}=\frac{P}{360}\Delta \theta$ (5)

$\Delta L_{sn}=\Delta L_b+\Delta L_P$ (6)

$\Delta F_0=\frac{P}{360\times \left({\delta }_b+{\delta }_p\right)}\Delta \theta$ (7)

$\begin{array}{c}{F}_0=\frac{T_{sung}}{\frac{P}{2\pi }+0.577\times d_2\times {\mu }_t+\frac{D_b}{2}\times {\mu }_b}\\ +\frac{\Delta \theta }{\frac{360\times \left({\delta }_b+{\delta }_p\right)}{P}+\left(\frac{P}{2\pi }+0.577\times d_2\times {\mu }_t\right)\times {\delta }_{b,t}}\end{array}$ (8)

where: $F_0$: Bolt Preload.

${\delta }_b$ and ${\delta }_P$: Axial resilience (stiffness) of bolt and clamped parts.

${\delta }_{b,t}$: Torsion resilience (stiffness) of bolt.

For very long clamping length, the bolt torsion angle can be significant. For this reason, we (Cetim) have improved the last formula as shown by Equation (8).

This program covers 37 assembly configurations. These configurations were defined with a group of French manufacturers (~60 companies). Figure 1 and Table 1 give a simplified view of the tested assembly configurations. Each tested assembly is defined by a screw or a stud (#1), a nut or a tapped part (#2), 0 to 2 washers (#3) that depend on the tested configuration, a set of clamped parts (#4) and a load sensor (#5) to measure the load in the bolt.

(a)

(b)

To reduce the number of assumptions regarding material properties and friction values, all materials of tested parts have been qualified with basic tensile tests according to ISO 898-part 1 [20] and part 2 (ISO 6892 [21] ) for material properties and according to ISO 16047 [19] for friction coefficients values (bearing surface friction coefficient “μ_b” and thread friction coefficient “μ_t”).

Table 1 details the experimental program in which we tested different types of fasteners (studs and screws), different product standards (normal, HR and HV bolts), different providers, different materials, different head shapes, different clamped lengths, …

To reduce the operator-effect during tightening, a specific tool has been designed to tighten the tested assemblies using the Torque-Clamp Force Test-bench (#A of Figure 1 ). This last one applies a regular angle and measures the applied torque. Additional accessories are used to measure screw elongation (LVDT) and bolt force (load sensor). For each configuration, 5 to 10 assemblies have been tested. All tests are performed by the same operator and by using the same equipment, except for M24 assemblies, which require other tools. Finally, all samples are tightened up to the bolt failure.

Experimental results are treated in two parts: qualification test-results and tightening test results. For the first part, Figure 2 shows the Stress-Strain graph (mean values) of two material specimens from the same batch. Figure 3 shows the linear relationship between the applied torque and the clamp force. It also shows the deviation of the preload due to friction deviation, for a target preload. All experimental results of those tests are summarized in Table 2 and Table 3 .

The second part of results (tightening tests) is presented and discussed in this section and in the validation section of this paper.

Using configuration 1 (clamped length “L_k” ~ 2d where “d” is the nominal diameter), Figure 4 shows the evolution of preload regarding bolt elongation. It shows a large plastique area for HR bolts. This area is mainly defined by the physical and geometrical properties of the tested bolt. We note also some deviations in the elastic area, which could be due to the quality of the contact between the bolt and used sensor LVDT.

Graph in Figure 5 gives an example of a bolt elongation (mean values) during the tightening operation. It shows a linear relationship between bolt elongation and applied angle, as well as non-linear behaviors at the start and end of the tightening tests. The first non-linearity is due to the contact between the sensor LVDT and the bolt, and the initial gap at the engaged threads. The second is due to the beginning of the thread stripping (for this case) or bolt yielding.

For the tested configuration shown in Figure 5 , by ignoring snugging angle (non-linear behavior), the theoretical rotation of the nut is estimated to

$\Delta \theta =\frac{\Delta L_b}{P}\times 360=\frac{3.3-0.6}{1.75}\times 360=555.4˚$ (according to Equation (5)). This rotation is significantly lower than the measured angle (662.3˚ − 40.6˚ = 621.7˚). This

difference is mainly due to the tool resilience (stiffness) and its accessories (sockets, extension…). It demonstrates that the real assembly is more flexible than the calculated one (theorical assembly). And it confirms the existence of an additional resilience ( Table 2 ). This conclusion is mentioned by Fukuoka [10] and Gold [11] . The first author improved the resilience formula by adding a non-linear contact resilience. The second gives other explanations based on the local stress and the elasto-plastic properties of the material, which will be investigated later. In fact, and according to two different papers of Fukuoka ( [10] and [20] ), we can see a good correlation between theorical approach and experimental results. But in the first paper, he considers the roughness effect on the additional contact resilience, and he ignores this factor in his second work (based on a specific axisymmetric FEA model).

Graphs of Figure 6 and Figure 7 illustrate the importance of the resilience (stiffness) of the bolt and clamped parts, defined mainly by the length and the Young modulus. It shows that the higher the clamped length ratio (L_k/d), the more the bolt has to be turned to install required preload. This reduces the sensitivity of the angle factor on the applied preload. In fact, for a stiff assembly (L_K close to 2d), any error of ±10 degrees on applied angle introduces a preload deviation of ±10 kN against ~±5 kN in the case of a highly elastic assembly (L_k = 6d).

The graphs in Figure 8 show the evolution of applied torque and introduced load as a function of applied angle. From those graphs we can note:

The graphs of Figure 9 show a limited effect of the thread pitch on the applied angle. In fact, the use of a fine pitch increases the required angle to introduce a similar load compared to a coarse pitch. This effect may be more significant in the case of a stiff assembly. Indeed, for a clamped length L_k = 6d, to introduce 39.6 kN, using a fine pitch (1.5 mm) increases the required angle by only 4˚ (36˚ instead of 32˚).

The graphs in Figure 10 show the same behavior of two similar assemblies where only the nut has been modified. It demonstrates that for a snug torque higher than the locking torque, the combined tightening can ignore the locking torque effect. But for higher locking torque, this observation may be wrong.

Finally, using test-results of the Torque-Clamp Force qualification according to ISO 16047 (Bolt HR 10.9 M12x40), we can note a significant evolution of the basic relationship between bolt preload and Torques (bearing surface torque, thread torque and total torque) after the yielding of the bolt as shown in Figure 11 . According to these graphs, the linear relationship between torques and bolt preload is only valid before the yield point of the bolt. This means the friction coefficient at the bearing surface ( Figure 11(c) ) and on the threads ( Figure 11(d) ) decreases after bolt yielding. In this paper, this evolution will be ignored by the proposed model (below) when extended to plastic tightening. And it will be developed in future work.

Figure 12 illustrates the difference between the measured angle and the effective angle, which generates a relative rotation between the nut and the bolt. We can note that the measured angle is the sum of 3 different rotations.

$\Delta \theta =\Delta {\theta }_{tool}+\Delta {\theta }_{b,Torsion}+\Delta {\theta }_u$ (9)

1) $\Delta {\theta }_{tool}$: Torsion rotation (angle) of the tool and its accessories. Some automatic screw drivers compensate for this quantity. At the end of this paper, we propose a simplified way to qualify the tool stiffness/resilience.

$\Delta {\theta }_{tool}=\frac{\Delta T}{K_{tool}}=\Delta F_0\times \left(\frac{P}{2\pi }+0.577\times d_2\times {\mu }_t+\frac{D_b}{2}\times {\mu }_b\right)\times {\delta }_{tool}$ (10)

2) $\Delta {\theta }_{b,Torsion}$: Torsion rotation (angle) of the bolt. The torsion resilience/stiffness of the bolt can be calculated in the same way as axial and bending resilience.

$\Delta {\theta }_{b,Torsion}=\frac{T_{thread}}{K_{b,Torsion}}=\Delta F_0\times \left(\frac{P}{2\pi }+0.577\times d_2\times {\mu }_t\right)\times {\delta }_{b,t}$ (11)

3) $\Delta {\theta }_u$, useful angle

$\Delta {\theta }_u=\frac{\Delta F_0}{K_{\theta }}=\Delta F_0\times \frac{360\times \left({\delta }_b+{\delta }_p\right)}{P}$ (12)

With Equations (8) to (12), we can deduce the final formula (13).

$\begin{array}{l}{F}_0=\frac{T_{sung}}{\frac{P}{2\pi }+0.577\cdot d_2\cdot {\mu }_t+r_b\cdot {\mu }_b}\\ +\frac{\Delta \theta }{\frac{360\cdot \left({\delta }_b+{\delta }_p\right)}{P}+\left(\frac{P}{2\pi }+0.577\cdot d_2\cdot {\mu }_t\right)\cdot {\delta }_{b,t}+\left(\frac{P}{2\pi }+0.577\cdot d_2\cdot {\mu }_t+\frac{D_b}{2}\cdot {\mu }_b\right)\cdot {\delta }_{tool}}\end{array}$ (13)

This final formula shows all the factors influencing the relation between the bolt preload, snug torque and the applied angle ( $\Delta \theta$). It adds the effect of friction, and additional factors that may be ignored in some cases.

Based on the results of Gold [11] and of Fukuoka [22] , we have performed an axisymmetric FEA model ( Figure 13 ). We were interested in the evolution of part compression under a tightening load of 59.6 kN. The graph of Figure 14 shows the evolution of the axial displacement of the parts at the hole from the head (axial position Z = 22) to the nut (axial position Z = 62). It shows that a significant portion (more than 50%) of part compression is obtained with two local areas located at the interface in contact with the fastener up to a depth of 6 mm.

According to VDI 2230 approach, the part compression is detailed in Table 4 .

Using the preload and the FEA displacement at the hole edge, the local resilience is calculated at 1.29e−6 mm/N. Using the preload and the difference between the mean displacement under the bearing surfaces (bolt head and under the nut), the mean FEA resilience is equal to 9.36e−7 mm/N, that is closer to the analytical calculated resilience.

Due to the difference between analytical and FEA results, we can assume that the part resilience to be used during tightening is different to that used for load factor calculations. According to Alkatan [7] and Massol [8] , the part resilience calculation is based on elastic energy of the part. This resilience gives only a mean part-compression. With regard to the surface contact between bolt head and part, we can assume that this contact is a Herz contact. As a consequence, the full parts resilience during tightening can be calculated by Equation (14).

Figure 14 and Figure 15 demonstrate that the compression of the clamped parts is defined by a mixture of Hertzian and normal elongations.

${\delta }_P={\displaystyle \sum {\delta }_{P,Elastique\left(NFE,VDI,\cdots \right)}}+{\displaystyle \sum {\delta }_{P,Hertz}}$(14)

${\delta }_{P,Hertz}=\frac{1-{\nu }^2}{D_{w,eq}\times E_{eq}}$(15)

Where equivalent Young modulus

$E_{eq}=\frac{1}{\frac{1}{E_{P,i}}+\frac{1}{E_{P,i+1}}}$(16)

And Equivalent diameter:

$D_{w,eq}=\sqrt{d_w^2-d_h^2}$(17)

Finally, the simplified method consists of calculating the parts resilience according to the basic approach (VDI 2230 [1] , Alkatan [7] , Massol [8] , Bickford [23] or Rasmussen [24] ...) and adding Hertz resilience for the smallest contact surfaces.

Today, maximum snug torque is defined by experience or by practical rules. These rules specify 25% to 30% of maximal target preload should be introduced by the snug torque.

In this section, we determine the maximal snug torque to achieve any final tightening factor (“α” as defined by VDI 2230 [1] and Equation (18)). At first, we suppose the maximal preload “ $F_0^{+}$” and tightening factor “α” are known (e.g. $F_0^{+}=80\%\text{YLS}$ and α = 1.4), then we calculate the required snug torque.

The deviation of the final preload (introduced by angle and by the snug torque) and of snug preload can be calculated respectively by Equation (19) and Equation (21). The combination of Equation (18) and Equation (21) defines the maximum preload to be introduced by the snug torque (Equation (22)). Finally, the snug torque (Equation (23)) can be calculated using any formula describing the relationship between torque and preload (ISO 16047 or equivalent).

$\alpha =\frac{F_0^{+}}{F_0^{-}}$(18)

$\Delta F_0=F_0^{+}-F_0^{-}=\left(1-\frac{1}{\alpha }\right)\times F_0^{+}$(19)

${\alpha }_{sung}=\frac{F_{0,snug}^{+}}{F_{0,snug}^{-}}$(20)

$\Delta F_{0,snug}=F_{0,sung}^{+}-F_{0,sung}^{-}\le \Delta F_0$(21)

$F_{0,snug}^{+}\le \frac{\left(1-\frac{1}{\alpha }\right)}{\left(1-\frac{1}{{\alpha }_{sung}}\right)}\times F_0^{+}$(22)

$T_{sung}=\frac{F_{0,sung}^{+}}{1+\frac{\Delta T}{T}}\times \left(\frac{P}{2\pi }+0.577\times d_2\times {\mu }_{th}+\frac{D_b}{2}\times {\mu }_b\right)$(23)

Equation (22) demonstrates the importance of the snug torque accuracy, on the final preload accuracy. For industrial applications, the snug-tightening factor can be estimated at 2 ( $1.5<{\alpha }_{sung}<2.5$) and the final tightening factor for combined tightening is estimated between 1.2 and 1.4. In this case, the snug torque/preload should not exceed 33% to 57% of the final Torque or Preload.

According to the graph in Figure 16 (Equation (22)), an accurate tightening using combined tightening (α < 1.1), requires a very low snug ratio. However, using a low snug ratio could cause the tightening preload to fall within the non-linear range of the angle-preload relationship, which is another source of final preload-deviation. In other case, for a higher level of the tightening factor, the snug-Preload could be closed to the target preload (e.g. ${\alpha }_{sung}=\alpha =1.6$). In this case, combined tightening offers no advantage.

To measure the stiffness of the tightening tool and its accessories (socket, extension, adapters…), several experimental ways can be used. Below, we propose a simplified method which can be used. It consists of recording the torque and the angle during the bolt tightening at any torque-level (close to the bolt’s maximum capacity) and during the untightening step, as shown in Figure 17 .

The torsion stiffness of the tool used to tighten the bolt is determined by the “torque-angle” slope during the untightening (or re-tightening) operation before any relative rotation between the nut and the screw. In our case, the stiffness of the tightening tool, as shown in Figure 17 (during the untightening step), is equal to 24 Nm/˚.

This stiffness should be ignored when using a servo-controlled screwdriver, which anticipates this additional rotation due to its accessories.

The tightening after the yield point consists of tightening bolts beyond their elastic capacities. This leads to strain hardening of the screw material (on its critical/threaded area). Friedrich [12] presented a simplified approach to define the angle and the snug torque for this kind of tightening. The tightening after the yield point allows for a significant deviation in the applied angle and reduces the effect of any inaccuracy in the snug torque.

In our study, we adjusted the existing approach using the real S-S curve of the bolt.

This extension is described by the calculation steps detailed below:

1) We transform the body of the fastener into a set of cylinders or shanks (threaded and unthreaded shanks).

2) For any target preload “F₀”, we calculate the relative nut-bolt rotation as described below:

a) We calculate the equivalent (Von Mises) stress (combination of axial and torsional stress) in any critical shank.

b) Using Neuber correction and bolt stress-strain curve, we calculate the real stress and strain in the critical shank.

c) From the calculated “real stress”, we calculate the residual preload.

d) From the calculated strain, we calculate the shank elongation.

e) We perform the same calculation for all other shanks using the residual preload.

f) The total bolt elongation can be calculated by summing all shank-elongations.

g) Using the parts resilience and the residual preload, we calculate the clamped-parts elongation.

h) Finaly, the nut rotation “ ${\theta }_{F_0}$” can be calculated using the bolt and parts elongations, torsion angle of the bolt-body and the additional angle introduced by tightening-tool-accessories.

2) With the snug torque “ $T_{snug}$”, we calculate the snug preload by using the ISO 16047 formula (formula 2 [19] ).

3) We perform the same calculation for the snug preload “ $T_{snug}$” (step 2) to calculate the snug angle “ ${\theta }_{snug}$”.

4) Finally, the final combined tightening will be defined by “ $T_{snug}+\text{Δ}\theta$”, where the final angle to be applied is defined by Equation (24).

$\Delta \theta ={\theta }_{F_0}-{\theta }_{sung}$ (24)

This extension will be developed in another paper discussing the fatigue behavior of yielded assemblies, where S. Gold [11] demonstrated the performance of these assemblies in this case.