Uncertainty analysis of the coefficients of friction during the tightening process of bolted joints
© Seibel et al.; licensee Springer. 2014
Received: 18 February 2014
Accepted: 16 September 2014
Published: 3 October 2014
In this paper, we introduce closed-form symbolic expressions for the possibility distributions of the coefficients of friction during the tightening process of bolted joints. The parameters in the distribution functions are then identified by a standardized fastener testing system according to ISO 16047. An uncertainty analysis finally shows that the total amount of uncertainty in the coefficient of bearing friction is almost 40% larger than in the coefficient of thread friction. Furthermore, the real value of the coefficent of bearing friction is likely to be about 5% higher than the expected value.
Bolted joints are among the most used joints in mechanical engineering. But failures still occur during the tightening process and in operation. In order to improve the quality and reliability of bolted joints, a main goal is to achieve a proper clamping force. With a certain effort , it is possible to measure the clamping force directly. But most of the bolts and screws still are assembled by an indirect measurement of the clamping force .
where Tb denotes the bearing friction torque component and Tt the thread friction torque component. These components are in a direct relationship with the clamping force and the coefficients of friction of the particular material pairing. Hence, the information about the coefficients of friction has a great influence on the quality and reliability of bolted joint connections.
Suppliers of bolts and screws often give only parameter windows for the desired coefficients of friction. These intervals are rather general and may not be suitable for individual applications. Furthermore, no information about the distribution functions of the coefficients of friction is provided.
to meet the requirement (3).
The general disadvantage in working with probability distribution functions is that in order to construct a probabilistic model, a large amount of statistical data is needed, which, in reality, is not always the case. In contrast to that, when only little information about the uncertain parameters is available, possibility theory is most appropriate. For this reason, we use possibility theory in this paper to model parametric uncertainty.
The main contributions of this paper are the following. First, closed-form symbolic expressions for the possibility distributions of the coefficients of friction during the tightening process of bolted joints are introduced for the first time. The parameters in the distribution functions are then identified by a standardized fastener testing system according to ISO 16047. An uncertainty analysis finally compares the total amount of uncertainty in the coefficient of bearing friction with that in the coefficient of thread friction. Furthermore, the deviations of the real values of the coefficients of friction from the expected values are given.
of the model function f at the point , where d f represents the total change of f in a neighborhood of if all parameters are changed simultaneously. The change dx i of a single parameter x i contributes to the amount (d f) i to the total change d f.
In order to propagate the uncertainties through the computations, we introduce the following transformation:
Using the above transformation, we can formulate the following analytical approach :
In this section, we use the above analytical approach to derive closed-form symbolic expressions for the distribution functions of the coefficients of friction during the tightening process of bolted joints.
Coefficient of bearing friction
where Tb denotes the bearing friction torque component, F the clamping force, and rb the effective bearing radius.
During the tightening process, the clamping force F should be achieved exactly. Hence, it exhibits no uncertainty. The parameters Tb and rb, on the other hand, are assumed to be normally distributed.
Hence, the uncertainties of Tb and rb contribute to the same amount to the overall uncertainty of μb.
Coefficient of thread friction
where dt denotes the effective thread diameter, Tt the thread friction torque component, F the clamping force, and P the thread pitch. (Note that the possibility degree π, which is used throughout the paper, should not be confused with the mathematical constant π=3.14.).
Again, the clamping force F is assumed to exhibit no uncertainty, and the other parameters are assumed to be normally distributed.
Hence, the uncertainties of dt and Tt contribute to the nearly same amount to the overall uncertainty of μt, whereas the uncertainty of P has a negligible influence.
Materials and methods
Descriptions and materials of the test parts
DIN 6921 M8 × 50 10.9
DIN 934 M8 8.8
S235JRC+C (EN 1.0122)
For carrying out the experiments, we followed the instructions from ISO 16047 . A tightening torque was steadily applied until a clamping force of 16 kN was reached. Here, F, T, and Tb were measured, and Tt was computed according to Equation (1). The sample size was 32 bolts.
where do denotes the outer diameter and di the inner diameter of the bearing surface.
where d2 denotes the thread pitch diameter, d the nominal thread diameter, and D1 the minor nut thread diameter. The standard deviations are chosen such that the intervals [ di, do] and [ D1, d] correspond to and , respectively. (The factor two at results from the fact that the standard deviation of a diameter is twice as large as the standard deviation of the corresponding radius). Beyond the interval boundaries, the possibility values are smaller than 1%  and can be thus neglected. In fact, they are physically impossible.
meaning that in our application, the total amount of uncertainty in μb is almost 40% larger than in μt.
denotes the defuzzified value of π(x).
Both eccentricities are positive, that is, the real values of the coefficients of friction will be potentially higher than the expected values. More specifically, the real value of μb is likely to be about 5% and the real value of μt to be about 1% higher than the corresponding expected value.
We introduced closed-form symbolic expressions for the possibility distributions of the coefficients of friction during the tightening process of bolted joints. This relieves the engineer from the burden of propagating the uncertainties through the computations to obtain the uncertain output. The parameters in the distribution functions can be identified by a standardized fastener testing system according to ISO 16047 as has been demonstrated in this paper. An uncertainty analysis also revealed that the total amount of uncertainty in the coefficient of bearing friction is almost 40% larger than in the coefficient of thread friction. This finding suggests that in practice, it is more important to control the coefficient of bearing friction than the coefficient of thread friction. The uncertainty analysis also showed that the real value of the coefficient of bearing friction is likely to be about 5% and the real value of the coefficient of thread friction to be about 1% higher than the corresponding expected value.
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