Advances in Materials Science and Engineering

Volume 2015, Article ID 134090, 6 pages

http://dx.doi.org/10.1155/2015/134090

## Characteristic Evaluation on Bolt Stress by Ultrasonic Nondestructive Methods

^{1}School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China^{2}Key Laboratory of Fundamental Science for National Defense for Advanced Machining Technology, Beijing 100081, China^{3}BMW China Services Ltd., Beijing 101312, China

Received 15 May 2015; Revised 9 July 2015; Accepted 26 July 2015

Academic Editor: Rui Vilar

Copyright © 2015 Qinxue Pan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 (A2-70) and low-carbon 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.

#### 1. Introduction

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 [1]. Williams et al. gave the ultrasonic analysis of bolt preload [2], Johnson et al. used time-of-flight measurements of both shear and longitudinal waves propagating along the axis of a bolt along with the material parameters to calculate the stress acting on the bolt [3], and Koshti developed the application to make ultrasonic measurement of preload in sleeve bolts [4]. At present, a torque wrench is commonly used to determine the degree of bolt axial stress in the industrial field. However, due to the reverse torque in the nut of the thread friction pair, as well as the elastic deformation, the torque used to produce the bolt axial stress is actually <20% of the torque wrench numerical value [5]. Ultrasonic nondestructive testing of the vertical screw bolt axial stress (e.g., the pulse echo reflection method, transit time method, and phase method) cannot measure the axial stress when the bolts are installed or buried to a depth at which the original length of the anchor bolt is unknown [6, 7].

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 [8]. According to the acoustoelasticity theory, the ultrasonic velocity and the magnitude of the stress in isotropic metal materials are related in such a way that the effect of the bolt axial length and temperature change are eliminated through simultaneous solving of the transverse and longitudinal wave equations [9–13]. However, in practice, the relationship between the acoustic time difference and the stress magnitude does not have a simple linear form under extremely low or high loading conditions. Thus, a more detailed stress measurement model is required to accurately resolve the bolt axial stress under various loading conditions.

#### 2. Theory

From the sound theory of elasticity, the elastic wave propagation velocity in a solid material, the material itself (e.g., material density), and the second-order 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:where is a parameter of the Kronecker delta function, is the material density, is the distance of propagation, is the vector particle of a point, is the equivalent stiffness, and is the Cauchy stress.

Elastic sound is nonlinear. Assuming a plane ultrasound wave and the initial coordinates, (1) can be simplified to the following form:

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 , , and , and the particle displacement described by , , and , we have the following:The stress in (3) is given below:where are the Jacobian matrix elements, are the strain matrix elements, and is the elastic potential energy.

is determined using , , and . According to the acoustoelasticity theory, the transverse and longitudinal ultrasonic wave propagation direction and the polarization direction are related to the stress direction as follows [14]:(1)The propagation of a longitudinal wave along the direction of stress is given by(2)The stress direction along the propagation direction, with the polarization direction perpendicular to the shear stress, is given aswhere for is the direction of wave propagation, is the wave polarization direction, and is the load direction. In (5) and (6), the constant is given by is the density and and and , , and are second- and third-order elastic constants of the material, respectively. For the purposes of a semi-infinite plate,According to the basic assumptions of elasticity, before the axial stress, less than the yield limit of the bolt, the material can be considered as a complete bolt steel elastomer. Thus, when measuring the temperature constant at room temperature, the second bolt material and the third-order elastic constants do not contribute to increases or changes in the axial stress. Under these conditions, the only factor that has an effect on the axial bolt stress and bolt length measurement is the measurement temperature. Considering that the effective stress which can tighten the nut bolt deputy district is not wrapped in the entire area, as shown in Figure 1, we can create a physical model of the axial stress during fastening of the bolt [8, 9]. The head of the bolt is placed in transverse and longitudinal wave transducers. The pulse echo method is used to measure the horizontal and vertical wave crossings in the screw. The bolt axial stress and length measurement can be obtained indirectly from these measurements.