Based on the acoustoelasticity theory, a certain relationship exists between ultrasonic velocity and stress. By combining shear and longitudinal waves, this paper provides a nondestructive method of evaluating axial stress in a tightened bolt. For measuring the bolt axial stress in different situations, such as under low or high loads, this paper provides guidelines for calculating the stress for a given load factor. Experimental and calculated results were compared for three bolt test samples: an austenitic stainless steel bolt (A270) and lowcarbon steel 4.8 and 8.8 bolts. On average, the experimental results were in good agreement with those obtained through calculations, thus providing a nondestructive method for bolt stress measurements.
Bolts, a key component in equipment joints, are used to join, clamp, reinforce, and seal various parts and are widely used throughout industry (e.g., aerospace, marine, construction, chemical, and energy industries). Due to their importance, bolt axial stress has received much attention in an attempt to improve the bolt’s performance and useful life. However, the complexities of the bolt structure and threaded part deformation with axial loading have prevented accurate measurement of the axial stress. In recent decades, the research community has actively explored solutions to this problem [
Ultrasonic nondestructive testing with electromagnetic ultrasonic shock excitation shear waves has been used to measure the axial pretightening force of bolts, with some promising results; however, this approach fails to resolve the axial stress measurement when the bolt length is unknown [
From the sound theory of elasticity, the elastic wave propagation velocity in a solid material, the material itself (e.g., material density), and the secondorder elastic constants are closely related to the stress state of the material. As such, the stress medium elastic wave equation for the initial coordinate system can be expressed as follows:
Elastic sound is nonlinear. Assuming a plane ultrasound wave and the initial coordinates, (
Taking into account the stress state in which the speed of sound and the elastic waves in the material have a quantitative mathematical relationship, and given the independent Lagrange variables
The propagation of a longitudinal wave along the direction of stress is given by
The stress direction along the propagation direction, with the polarization direction perpendicular to the shear stress, is given as
Bolt axial stress measurement physical model.
Taking into account the measured temperature expansion of the overall impact of the bolt and the linear elastic deformation zone length for the bolt stress, the bolt length can be expressed as follows [
The relationship between the axial shear stress on the bolt and the longitudinal wave propagation velocity can be expressed as
The final expression for the stress
The experimental measurement system consisted of a symmetrical configuration of transverse and longitudinal wave transducers. The transverse and longitudinal wave transducer units operated independently and simultaneously produced transverse and longitudinal wave excitation signals (center frequency: 2.25 MHz). The broadband contact transducer produced a narrow pulse (diameter: 10 mm). The bolts whose models are M20 include A270, 4.8, and 8.8. In order to achieve stable coupling effects, nuts and flat tail had grinded the grips smooth, and the temperature is controlled at 26°C.
The system used an ultrasonic excitation pulse signal transceiver device. Oscilloscopes coupled to the transverse and longitudinal wave transducer units provided an accurate recording of the transit time. A stretching machine was used to quantify the tensile test results. The stress value was extrapolated from the time duration through the function relationship between the two. Due to the low yield strength of steel (maximum bolt strength: 320 MPa), the tensile test stress range for the A270 bolts was 0–360 MPa, using a step increment of 20 MPa; the axial stress range for the lowcarbon steel 4.8 and 8.8 bolts was 0–260 MPa, in 20 MPa step increments. Tables
Austenitic stainless steel bolt (A270) axial tensile stress measurement results.
Stress 





0  42.831  76.17  3.145160  
20  42.8415  76.204  3.146601  −2.53 × 10^{14} 
40  42.8605  76.2265  3.145552  −1.27 × 10^{14} 
60  42.88  76.247  3.14423  −8.57 × 10^{13} 
80  42.897  76.266  3.143192  −6.47 × 10^{13} 
100  42.9155  76.2835  3.14176  −5.22 × 10^{13} 
120  42.9325  76.3025  3.140722  −4.38 × 10^{13} 
140  42.9515  76.318  3.13902  −3.80 × 10^{13} 
160  42.9685  76.338  3.13808  −3.34 × 10^{13} 
180  42.988  76.357  3.13663  −3.00 × 10^{13} 
200  43.0044  76.3775  3.13583  −2.71 × 10^{13} 
220  43.0245  76.3965  3.13429  −2.49 × 10^{13} 
240  43.042  76.4175  3.13336  −2.30 × 10^{13} 
260  43.062  76.439  3.13207  −2.14 × 10^{13} 
280  43.081  76.462  3.131084202 

300  43.1025  76.486  3.129780475 

320  43.124  76.512  3.128663993 

340  43.148  76.5355  3.127547129 

360  43.1755  76.5565  3.12635963 

Explanation:
Stress [MPa] bolt axial stress.
Tensile stress measurement results for a lowcarbon 4.8 steel bolt.
Stress 





0  46.3535  84.2395  3.27610  
20  46.369  84.256  3.27509  −5.27 × 10^{14} 
40  46.3865  84.274  3.273907  −2.65 × 10^{14} 
60  46.405  84.29  3.272383  −1.78 × 10^{13} 
80  46.424  84.3155  3.27161  −1.34 × 10^{13} 
100  46.44  84.3375  3.271006  −1.07 × 10^{13} 
120  46.458  84.358  3.26995  −8.96 × 10^{13} 
140  46.4755  84.382  3.26929  −7.70 × 10^{13} 
160  46.4945  84.3985  3.26774  −6.78 × 10^{13} 
180  46.513  84.417  3.26644  −6.05 × 10^{13} 
200  46.533  84.432  3.26460  −5.49 × 10^{13} 
220  46.555  84.456  3.2632  −5.01 × 10^{13} 
240  46.575  84.4745  3.26171  −4.68 × 10^{13} 
260  46.6  84.5  3.26001  −4.39 × 10^{13} 
Axial tensile stress measurement results for a lowcarbon 8.8 steel bolt.
Stress 





0  46.402  84.484  3.28979  
20  46.412  84.4885  3.2886  −4.99 × 10^{14} 
40  46.427  84.495  3.286792  −2.51 × 10^{14} 
60  46.4405  84.505  3.285529  −1.69 × 10^{14} 
80  46.456  84.514  3.283863  −1.27 × 10^{14} 
100  46.47  84.524  3.282523  −1.02 × 10^{14} 
120  46.487  84.533  3.280624  −8.59 × 10^{13} 
140  46.5025  84.543  3.27905  −7.41 × 10^{13} 
160  46.519  84.552  3.277234  −6.53 × 10^{13} 
180  46.5335  84.562  3.275821  −5.84 × 10^{13} 
200  46.549  84.572  3.274252  −5.29 × 10^{13} 
220  46.57  84.582  3.27182  −4.85 × 10^{13} 
240  46.5825  84.59  3.270553  −4.47 × 10^{13} 
260  46.6  84.6  3.268675  −4.16 × 10^{13} 
The
The fitting curve
The comparison result of an A270 austenitic stainless steel bolt.
The fitting curve
The comparison result of a 4.8 lowcarbon steel bolt.
The fitting curve
The comparison result of a 8.8 lowcarbon steel bolt.
The distribution curves shown in Figures
Matlab least squares fit of the stress measurement coefficients,
Material  Coefficient  

0–100 MPa  100–360 MPa or 100–260 MPa  



 
A270  3.7835 × 10^{16}  1.1885 × 10^{17}  2.813 × 10^{15}  8.7741 × 10^{15} 
4.8  5.5036 × 10^{16}  1.7992 × 10^{17}  4.2938 × 10^{15}  1.396 × 10^{16} 
8.8  3.3484 × 10^{16}  1.0982 × 10^{17}  3.543 × 10^{15}  1.1544 × 10^{16} 
In this paper, an ultrasonic technique involving the combination of transverse and longitudinal waves provided a fast and effective way to measure bolt axial stress in three bolt materials: an austenitic stainless steel bolt (A270) and lowcarbon steel 4.8 and 8.8 bolts. In the measurement process, the effect of temperature and elastic deformation was eliminated allowing the axial length of the bolt and the plug axial stress to be measured. The measurements were performed under various load conditions. The stress measurement coefficients, determined from least squares fitting of the stress data, were classified into two stress ranges, to provide a more accurate reading of the bolt axial stress. The average error of this new method is less than 20 MPa and can be applied to various industries, including aerospace, marine, turbines, bridge construction, chemical equipment, and energy systems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project is supported by the National Natural Science Foundation of China (Grant no. 51305028) and Beijing Higher Education Young Elite Teacher Work.