Bolting is a common connection method in engineering and construction

and the tightening torque is often used to control the preload of bolts on industrial sites to ensure the reliability. The torque coefficient is defined as the proportional factor between the tightening torque and the preload force of the bolt

which is essential to estimate the appropriate tightening torque range. At present

the research on torque coefficient is mainly focused on circular face bolts

and there is less research on other face bolts. Based on the force analysis of the bolt tightening process

a mechanical model of bolt preload is proposed

and an accurate calculation formula of the torque coefficient of arbitrary end face bolt is derived by using description of the internaland external contours of the bolt end face in polar coordinates. A simple torque coefficient measurement system is built to test and verify the torque coefficients of circular end face

hexagonal end face and T-shaped end face bolts. The testing results are consistent with the theoretical caleulation

which proves the accuracy of the theoretical caleulation formula of torque coefficient.