NDT.net • April 2004 • Vol. 9 No.04


A. M. Abdelhay
Associate Professors, Production Engineering Department
Faculty of Engineering, Helwan University, Helwan, Cairo 11792-Egypt.

Corresponding Author Contact:
Email: Abdelhay1953@Yahoo.Com


The good mastership of working stresses level in mechanical components and structures is an important factor in engineering industries. Evaluation and monitoring of the stress state of these elements is a time consuming job, beside it involves quite tedious work. Ultrasonic nondestructive techniques offer the unique evaluation capability of establishing macro stresses along a finite path length, as well as gradients near to the surface. The principles of ultrasonic stress measurement of specific applications such as pressure vessels by employing the critically refracted longitudinal ultrasound waves are described. Also, the paper describes ultrasonic monitoring of longitudinal and hoop stress components in thin walled pressure vessel, and the determination of their acoustoelastic coefficients.

KEYWORDS: Biaxial stress state, Nondestructive stress evaluation, Ultrasonic technique. Acoustoelastic coefficients, Pressure vessels.


Various nondestructive techniques are available for the measurement of either applied stresses or residual stresses. The non-destructive test method based on Barkhausen emissions [1-4] from ferromagnetic materials is sensible to changes in microstructural characteristics and the stress state of the material and can be used to evaluate these materials characteristics. Silva and Mansur [4] studied the use of Barkhausen Noise Analysis in detection and measurement of residual stresses in ferromagnetic materials. Their results are compared with the ones obtained from the Hole-Drilling Strain-Gage Method. They found that for low values of the tension and compression stresses applied to special designed ASTM A 515 steel samples, the relation between the rms of the magnetic Barkhausen noise (MBN) value and stress magnitude is approximately linear. In all studied situations, the stress values determined by MBN measurements presented values smaller than those obtained by the Hole-Drilling Strain-Gage Method.

On the other hand, eddy current method [5,6] allows to evaluate the state of stress in ferromagnetic material. The given method is used for determining own stress as well as that formed in effect of outside load. The study by Dybiec et al. [5] used the eddy current method to evaluate the state of stress in ferromagnetic material. The given method is used for determining own stress as well as that formed in effect of outside load. The results of stress measurement originating from external loading have shown that within the range of tensile deformations the dependence of indications on the loading is linear, whereas after exceeding the proportionality limit the angle of curve inclination changes.

Among the various non-destructive evaluation (NDE) techniques, ultrasonic technique is a versatile tool for investigating the changes in microstructure [7,8], deformation process [9], stress measurements [10-14] and mechanical properties of different materials / components [15, 16]. It is due to the fact that, the ultrasonic waves are closely related with the elastic and inelastic properties of the materials.

Cost and portability are the two main factors limiting the extensive use of the non-destructive techniques. Ultrasonic instrumentation has an advantage in this aspect because it has the lowest cost among the previously cited methods. The increased utilization of this technique is also due to the availability of different frequency ranges and many modes of vibration of the ultrasonic waves to probe into the macro, micro and submicroic levels. Landa and Plesek [14] presented an excellent study for the evaluation of internal stresses using different ultrasonic techniques. They used compression tests of aluminum prismatic specimen for the evaluation of internal stresses. Resulting differences were less than 10%.

In the present paper a new experimental setup is used to study the principal stresses of a pressure vessel under biaxial stress state conditions by using the ultrasonic nondestructive evaluation technique, and also, to determine experimentally the required acoustoelastic coefficients (AEC) used for these stress evaluations.


The use of non-destructive methods for detection and measurements of stresses has been increased in the last years, because they do not promote changes in the material under examination and do not interfere in its later use. The ultrasonic methods for detection and evaluation of applied and internal (residual) stresses are based on acoustoelastic effect, basically on stress-induced velocity variations of acoustic waves propagating in a deformed solid. The stress dependence of wave velocities is caused by a finite deformation of a body and, in addition, by a nonlinear stress-strain relationship of the material. From the theory of acoustoelasticity and for an isotropic material, the following linearization equation is cited [14],
i, j any two perpendicular wave propagation directions (i.e. axial and hoop)
Vij ultrasound wave velocity
Vo ultrasound velocity in Stress free media
bi acoustoelastic coefficient along the ith ultrasound wave propagation direction
si applied stress level along the ith - direction

To evaluate experimentally working stresses, one needs the value of the acoustoelastic coefficient ( bi ), which can be done through a calibration procedure. These coefficients are determined for a given material, by measuring the propagation speed (or propagation time) of ultrasonic waves versus stress to which the material is submitted. The value of this ratio can be obtained experimentally by calibration.

In the present study, one defines an acoustoelastic coefficient bi which represents the value of the curve of the relative speed variation (dV/Vo)versus stress ( s) . Eq. (1) can equivalently be rewritten in terms of times as follows :
t0 the ultrasonic flight time in stress free condition.
t the ultrasonic flight time in stressed condition

Therefore, for a distance ( d ) between the ultrasonic transmitter (T) and the ultrasonic receiver ( R ) is maintained constant (see Fig. 3), measurements are not carried out on the speed itself (Eq.1), but on ultrasonic travel-time ( i.e. time of flight ) Eq. (2); which is directly measurable.


A specially designed and fabricated experimental setup was used for generating a biaxial stress state conditions, and to measure the time of flight for a propagated ultrasound waves. Details of this setup is shown schematically in Fig. 1., and is highlighted as follow.

3.1 Test Specimen
In order to study the case of biaxial stress state; which is never studied before, a good candidate for this condition is a cylindrical pressure vessel. Therefore, test runs are carried on free surface of a pressure vessel; which is fabricated from thin-walled seamless tube of 500 mm long, 114 mm outer diameter and 5mm wall thickness, and is made from St 35. This pressure vessel is fitted with dial pressure gage (60 bar max. capacity). Measurement of working stresses (i.e. axial stress sa , and Hoop stress sh ) were done by measuring their corresponding strains (i.e. axial strain ea , and Hoop strain eh ). Two foil type strain gages (KFG-5-120C1-11L1M2R) were bonded using standard procedure. For such a purpose, two strain gages were bonded so that one is oriented parallel to the pressure vessel axis, and the second along its circumferential direction. Another two strain gages were bonded to a steel strip to serve as a temperature compensating gages for the active ones.

Fig 1: Schematic diagram for the experimental setup.

3.2 Angle Beam Transducers
According to Snell's law [ 17], the critically refracted longitudinal waves ( Lcr- waves) can be practically excited when the incident angle of the ultrasound beam is 28°, or slightly greater than it, say 30° , as shown in Fig. 2. Thus, specially designed perspex curved wedges were molded to a geometry suitable to adopt the curvature of the cylindrical surface of the pressure vessel, and with slanted faces of 30° normal to the pressure vessel's surface, in order to generate the desired Lcr- waves mode.

Fig 2: The critically refracted longitudinal waves (Lcr- waves ) excited by the angle beam transducer.

Measurements of the time of flight (t0 and t) are made using the transmission ultrasonic technique. Two separate and identical angle beam transducers were used as shown in Fig. 3. These contact type ultrasonic transducer are of 2.25 MHz frequency, and 12.5 mm contact diameter. Constant pressures were maintained between transducers and the pressure vessel and to conserve a constant distance (d) between the T and R transducers. White grease was used as a couplant between the different interfaces.

Fig 3: Arrangement for transmitter (T) and receiver ( R ) transducers for (a) axial and (b) hoop stress measurements.

3.2 Biaxial stress and Ultrasound Variable Measurements
Experimental work has been performed on the steel pressure vessel using a pressure range of 0 - 40 bar, with 5 bar increments. Measurements were carried out first along the axial direction (fig. 3a), then along the circumferential direction (Fig.3b), after replacing the curved transducer's wedges. For each case, and at the specified pressure, observed strains (i.e. axial strain ea , and Hoop strain eh ) were recorded with 0.15 µe SD (standard deviation), and ultrasound signal was acquired and stored for later analysis. Measured strains ( 1.0 µe accuracy) were used to calculate the principal working stresses (i.e. axial stress sa , and Hoop stress sh ) with 0.37 MPa. SD , as follows:
E modulus of elasticity of the steel pressure vessel,
n Poisson's ratio for the steel pressure vessel.

Fig 4: Typical acquired ultrasound signal (test run no. a30), showing the readout of the flight time (dX), using WaveStar software Ver. 2.1.

The analysis of the ultrasonic signal ( Fig. 4) was concentrated on the travel time ( t ) between the initial pulse of the T-transducer and the received pulse by the R-transducer for the Lcr- waves. These flight time values are used to calculated the relative change in velocities for the Lcr- waves traveled between the fixed locations of the T and R transducers. Accuracy of time reading was 0.1 nS (nano seconds). The calculations were using the following formula:
where dV is the change in velocities (V -Vo) of the Lcr- waves along the specified direction ( i.e. axial or circumferential) .


Graphs of Fig. 5 show typical experimental results indicating the relationships between the measured principal stresses and the percent relative change in velocities of the Lcr- waves under biaxial stress state conditions. The stress - velocity change relationships were represented by linear relationships. Oscillations of experimental results along the ideal straight lines, can be attributed to such factors as micro-inhomogeneity of the pressure vessel's material, and temperature change resulting from the wave propagation.
Using the statistical linear regression, the acoustoelastic coefficient (AEC) for each stressed direction was found. As for the axial direction the AEC was found to be ba = - 0.0128 MPa-1 with correlation coefficient of 0.95, while for the hoop stresses or the circumferential direction it was bh = - 0.0140 MPa-1 with correlation coefficient of 0.93.

Fig 5: Relationships of the measured principal stresses; (a) Axial stress, (Two) Hoop stress and the percent relative change of the ultrasound velocity of the Lcr- waves.

The AEC value indicates the higher sensitivity of the relative change in ultrasound velocities along the circumferential direction compared to the axial one.
It is worth comparing these values of the AEC for the case of biaxial stress state, with those published for similar materials [12], but under uniaxial stress state, it is found that the former is lower. One can explain this phenomena by saying that there is some negative interactions between wave polarization along the principal stress directions; which leads to lower the acoustoelastic effects on the propagated Lcr- waves along that direction. But, wave polarization for the case of uniaxial stress states enhances and reinforces the acoustoelastic effects of wave propagation along the loading direction.


The effect of stresses on ultrasonic Lcr- waves velocity propagating along the two principal stress directions of an element under biaxial stress state was investigated experimentally. The ultrasonic modes considered were the critically refracted longitudinal ultrasonic waves polarized along the axial and circumferential directions of a pressure vessel. The experimental results indicated the existence of acoustoelasticity phenomena. The relative velocity changes of the Lcr- waves polarized axially and circumferentially were different from each other. The relative changes of the Lcr- waves velocities were given as functions of the working stresses. The acoustoelastic coefficients (AEC) were obtained from the relationships between the relative changes of the Lcr- waves velocities and the working stresses. The absolute values of the AEC of material under biaxial stress states were smaller than those under uniaxial stress state conditions.

The above mentioned results indicate that ultrasonic Lcr- waves exhibit more change in their velocities along the circumferential direction with a relatively greater sensitivity ( bh = - 0.0140 MPa-1 ) compared to the axial direction. This suggests the possible application of ultrasonic nondestructive technique for the evaluation of the stress state conditions of critically loaded engineering components such as pressure vessels. This kind of study can open the road to the concept of " Ultrasound Strain Gage" idea. Such non-contact type strain gage can avoid the limitations and difficulties of the ordinary electric wire type strain gages. To make this dream possible, systematic research on this area of stress induced ultrasonic velocity changes must be pursed to be reached for an integrated measurement system; which is simple and accurate for stress evaluation nondestructively.


The author wishes to express his gratitude to his colleague Prof. Dr. O. M. Dawood for his sincere help and objective discussions. Also, thanks are due to the lab. technician Mr. Farouk Abdelbaki for his assistance in molding the required transducer's perspex wedges used in this work.


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