5___AITI#8714__56-65


Advances in Technology Innovation, vol. 7, no. 1, 2022, pp. 56-65 

 

Effects of the Vibration Amplitude in Vibratory Stress Relief on  

the Fatigue Life of Structures 

Cuong Bui Manh
1,* 

, Duong Nguyen Van
2
, Si Do Van

1
, Manh Phan Van

1
, Van Thao Le

3
 

1
Department of Engineering Mechanics, Le Quy Don Technical University, Hanoi, Vietnam 

2
Department of Materials Science and Engineering, Le Quy Don Technical University, Hanoi, Vietnam 

3
Advanced Technology Center, Le Quy Don Technical University, Hanoi, Vietnam 

Received 16 October 2021; received in revised form 08 December 2021; accepted 09 December 2021 

DOI: https://doi.org/10.46604/aiti.2021.8714 

Abstract 

This research aims to investigate the effects of vibration amplitude in vibratory stress relief (VSR) on the 

fatigue strength of structures with residual stress. Experiments are carried out on specimens with residual stress 

generated by local heating. Flat specimens made of A36 steel are prepared to be suitable for setting up fatigue 

bending tests on a vibrating table. Several groups of samples are subjected to VSR at resonant frequencies with 

different acceleration amplitudes. The results show that VSR has an important influence on the residual stress and 

fatigue limit of steel specimens. The maximum residual stress in the samples is reduced about 73% when the 

amplitude of vibration acceleration is 57 m/s
2
. The VSR method can also improve the fatigue limit by up to 14% for 

steel samples with residual stress.  

 

Keywords: vibratory stress relief (VSR), residual stress, fatigue life, fatigue limit, vibration amplitude 
 

1. Introduction 

Residual stress is the internal stress existing between parts of workpieces in the absence of external loads. This stress is 

generated when the workpieces experience inhomogeneous deformation, e.g., heating, cooling, and plastic deformation [1-2]. 

Many technological processes, such as welding, quenching, and metal forming, can produce residual stress. Residual stress 

often leads to the loss in the geometry accuracy of workpieces after machining, and causes the degradation of corrosion 

resistance in various environments [3]. Residual stress also has effects on the mechanical properties of materials and therefore 

on the load-bearing capacity and life service of components. Especially, the tensile residual stress which usually appears in 

welding has negative effects on the fatigue limit and fatigue life of workpieces [4-5].  

Relieving residual stress is one of the major concerns in mechanical engineering processes. There are many relieving methods 

implemented in manufacturing: annealing, local heating, monotonic overloading, shot pinning, vibratory stress relief (VSR), etc. 

Among the mechanical methods for relieving residual stress, VSR is considered an effective, flexible, inexpensive, and 

eco-friendly solution [6-8]. In VSR, cyclic external loads are applied to the workpieces with residual stress for a certain length of 

time, and this process is believed to cause microplastic deformation in micro regions of the workpieces and lead to the relaxation of 

residual stress [8-9]. This technology has been successfully implemented in the manufacturing of many important workpieces, such 

as welded shafts, marine shafts, large rails, large surface plates from stainless steel, and thin parts from aluminum alloy [10-13]. 

There have been lots of publications investigating the VSR effects on the residual stress state and the mechanical 

properties of specimens [7-8, 12, 14-16]. However, the research examining the VSR effects on the fatigue characteristics of 

workpieces is still limited. Fang et al. [17] demonstrated that the fatigue life of welding steel samples could be improved up to 

                                                           
*
 
Corresponding author. E-mail address: buimanhcuongkck@lqdtu.edu.vn

 



Advances in Technology Innovation, vol. 7, no. 1, 2022, pp. 56-65 

 

25% by using VSR. Munsi et al. [18] also carried out a study on VSR for welded steel bars at non-resonant frequency with two 

load levels for a short interval of time, only 5 s. They found that applying VSR at high loads and a small number of cycles could 

reduce the residual stress and increase the fatigue life of structures up to 17%. Meanwhile, the thermal method could reduce the 

residual stress and reduce the fatigue life of structures up to 43%. Djuric et al. [19] studied the VSR effects on the properties of 

martensitic steel, and found an increase of fatigue damage due to the applied VSR treatment. In the work of Gao et al. [20], the 

fatigue characteristics of Ti-6Al-4V titanium alloys were determined after VSR at different vibration amplitudes. The fatigue 

limit of samples was found to be slightly reduced with the increasing VSR amplitude (several percent to 10%) while the 

residual stress could also be reduced up to 60%. Song and Zhang [21] applied VSR for 7075-T651 aluminum alloy, and found 

that the fatigue life of specimens could be enhanced to 6.3% with the help of VSR. Gao et al. [22] also conducted VSR for 

7075-T651 aluminum alloy, and concluded that low amplitude VSR could reduce the residual stress and improve the fatigue 

life of 7075-T651 aluminum alloy. 

In this study, VSR is conducted on specimens of ASTM A36 steel, which is a popular material for weld structures. The 

residual stress is created by local heating, which is similar to the welding process and different from the pre-strain method in 

recent literature [19, 21-22]. Fatigue bending tests are conducted on a vibrating table with flat specimens, and the fatigue limit is 

determined by an improved staircase method. Herein, this study focuses on the effects of VSR amplitude on the fatigue properties 

of A36 steel with residual stress, demonstrating the VSR ability of reducing the residual stress and improving the fatigue limit. 

2. Materials and Experimental Methods  

2.1.   Test specimens 

In this work, fifty-five identical flat test specimens, which are designed according to ASTM E466 for bending fatigue tests, 

are extracted from a large carbon steel panel (Fig. 1). The specimens’ thickness is 6 mm. The end part of each specimen is with 

a threaded hole M5 (Fig. 1(a)) to attach additional weight, which creates an addition flat bending load and increases the stress 

at the A-A cross-section during the vibration test. Tables 1 and 2 present the mechanical properties (e.g., density, modulus, and 

yield strength (YS)) and the chemical compositions of the carbon steel specimens. 

 
(a) Geometry and dimensions of specimens 

 

(b) Fabricated specimens 

Fig. 1 Design and preparation of specimens 

 

Table 1 Mechanical properties of low carbon steel 

Density 
 

(kg/m
3
) 

Modulus of  

elasticity (GPa) 
Poisson’s ratio 

Yield strength 

(MPa) 

Ultimate tensile  

strength (MPa) 

7850 200 0.3 296 440 

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Table 2 Chemical compositions of specimens 

Element C Si Mn P S Cr Mo Ni Al 

wt. % 0.1452 0.0087 0.4483 0.0139 0.0028 0.0054 0.0030 0.0091 0.0723 

Element Co Cu Ti V Nb W Pb Fe  

wt. % 0.0080 0.0118 < 0.0010 < 0.0020 0.0041 0.0470 < 0.0100 99.183  
 

2.2.   Generation of the residual and dynamic stress in specimens 

First, for each steel specimen, high residual stress is induced by spot heating in the transition zone (Fig. 2(a)). The 

displacement velocity of the heat source and the heating time are controlled so that the temperature at the center of the heating 

zone is around 1000°C. The heat source is moved along line A-A (Fig. 1(a)) at a speed of 2 mm/s. After the local heating, each 

sample is rapidly cooled to room temperature by quickly putting it into a water tank. To monitor the dynamic strain and stress, 

an HBM strain gage (type 1-LM11-3/350GE) is bonded onto the middle of line A-A. The strain gauge is attached to the surface 

of specimens in the region of interest (RoI) (Fig. 2(b)). 

The dynamic stress is generated and controlled by varying the exciting acceleration amplitude and maintaining the 

constant frequency of a shaker. Therefore, it is necessary to survey the relationship between the exciting acceleration amplitude 

of the samples and the dynamic stress generated at the cross-section (A-A) before conducting VSR and fatigue tests.  

An LDS electrodynamic shaker (model V830-335T) with an LDS LASER USB Vibration Controller (model LAS200, 

S/N 10870124) is used for generating the vibration of the samples. An LMS noise and vibration test system (LMS SCADAS 

mobile system) is used for signal acquisition, monitoring the displacement at the end of the samples and the dynamic strain in 

RoI, as shown in Fig. 3(a).  

  

(a) Location of local heating to create residual stress (b) Strain gauge attachment on a specimen 

Fig. 2 Generating residual stress and attaching a strain gage to monitor the dynamic strain and stress 

 

 
(a) The LMS noise and vibration test system and the experimental setup 

 
(b) Natural frequency of a sample with residual stress 

Fig. 3 Experiment for monitoring the displacement and the dynamic strain in RoI  

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The data acquired by the LMS SCADAS mobile system enables studying and establishing the relationship of the exciting 

frequency and acceleration amplitude with the dynamic strain and stress. The sine vibration experiment is set at 30 m/s
2
 with a 

frequency range from 260 to 300 Hz. A sweep rate is set at 0.25 oct/min to exactly determine the resonance frequency of test 

specimens. The specimens are heated locally to generate the residual stress. The transmissibility function is used for the 

resonance search, and is computed based on the ratio of the dynamic strain and exciting acceleration, as shown in Fig. 3(b). Fig. 

3(b) shows that the resonant frequency of the test specimens with residual stress is about 283.59 Hz. After the controller finds 

the resonant frequencies in the sweep range, the sine dwell tests are performed. The controller runs a single sine tone at the 

resonant frequency instead of sweeping through the frequency range, with the acceleration amplitudes of 60 m/s
2
, 90 m/s

2
, 120 

m/s
2
, 150 m/s

2
, and 180 m/s

2
, respectively. Subsequently, the dynamic stress is calculated from the dynamic strain by Hook’s 

law, as shown in Eq. (1).  

σ = E × ε (1) 

where σ is the stress (N/m2), ε is the strain (m/m), and E is the modulus of elasticity (N/m2). 

2.3.   VSR tests 

VSR tests are conducted on the LDS electrodynamic shaker in a scheme like the one in dynamic stress tests. 56 

specimens are randomly selected and divided into 7 groups denoted by A, B, C, D, E, F, and G. Each group consists of 8 test 

specimens. Three test specimens in group A are randomly selected to measure the residual stress before VSR, and the 

remaining five samples of group A are used for determining the fatigue limit. The VSR processes are conducted for the 

specimens in 6 groups (B, C, D, E, F, and G) using the LDS electrodynamic shaker with the acceleration amplitudes of 29 

m/s
2
, 43 m/s

2
, 57 m/s

2
, 71 m/s

2
, 99 m/s

2
, and 156 m/s

2
, respectively. Each sample in these 6 groups is stimulated by the 

shaker within 10 minutes. The exciting frequency is set at 283.59 Hz according to the resonant frequency of the test 

specimens. After the VSR processes, three samples in each group (B, C, D, E, F, and G) are randomly selected to measure 

and check the residual stress, and the remaining five test specimens are used for determining the fatigue limit. The diagram 

of using different test specimen groups is shown in Fig. 4. 

 

 

56 test 

samples 

Group A: 

Without conducting VSR, 3 of 8 test samples in the group  

are used for measuring the residual stress. 

Group B:  

VSR is conducted with the exciting acceleration amplitude  

of 29 m/s
2
. 

Group C:  

VSR is conducted with the exciting acceleration amplitude  

of 43 m/s
2
. 

Group D:  

VSR is conducted with the exciting acceleration amplitude  

of 57 m/s
2
. 

Group E:  

VSR is conducted with the exciting acceleration amplitude  

of 71 m/s
2
. 

Group F:  

VSR is conducted with the exciting acceleration amplitude  

of 99 m/s
2
. 

Group G:  

VSR is conducted with the exciting acceleration amplitude  

of 156 m/s
2
. 

Each group 

contains 8 test 

samples. 

After VSR, 3 of 8 test samples in each group  

are used for measuring the residual stress. 

The other 5 samples are used for 

determining the fatigue limit. 

Fig. 4 Diagram of different specimen groups 

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2.4.   Measuring residual stress 

Three test specimens of group A are used for measuring the residual stress; the measured value is considered the residual 

stress before VSR. For the other 6 groups (B, C, D, E, F, and G), after VSR is conducted, three samples are randomly chosen 

from each group for the measurement of residual stress. The residual stress on the samples before and after VSR is then 

compared to each other to assess the ability of reducing the residual stress in different loading modes.  

The measurement of residual stress is performed according to ASTM E837-13a [23], using a RS200 system (Vishay 

Group, USA) and an EA-06-062RE-120 strain gauge rosette. Fig. 5 shows the residual stress measuring system marked RS200. 

When measuring the residual stress, the electrical resistance-strain gauge rosettes are attached to the surface of specimens in 

the same position as in the measurement of dynamic strains (Fig. 3(b)). The residual stress on the surface layer is determined by 

ASTM E837-13a, as described in the work of Gao et al. [16]. 

 

Fig. 5 Specimens and the residual stress measuring system marked RS200 

2.5.   Fatigue test 

The fatigue tests with a flat bending load are also conducted on the LDS electrodynamic shaker, as shown in Fig. 6. In 

each group, 5 test samples are used to determine the fatigue limit. The fatigue test parameters are as follows: the frequency of 

fatigue vibration test at the resonance frequency of samples is 283.59 Hz, the stress ratio is -1, the ambient temperature is 27°C, 

and the humidity is 60%. 

The fatigue limit can be determined by using the staircase method [24-25] (Fig. 7). The test is performed for the first 

specimen at a stress level around the fatigue limit. If the result of the above test is “non-fracture”, the second specimen is tested at 

the stress level with an increment of d. On the other hand, if the result is “fracture”, the second specimen is tested at the stress level 

with a decrement of d. The fatigue test is repeated until a given number of specimens is used, then the fatigue limit is calculated. 

This fatigue test method has been widely accepted to examine the fatigue limit of metals. There were also some modified staircase 

procedures to reduce the number of test specimens and hence lower the cost and running time of this test [26-27].  

 

Fig. 6 A Specimen and strain gauge in the fatigue test on the LDS electrodynamic shaker 
 

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In this study, a modified staircase method proposed by International Council for Combustion Engines is implemented for 

bending fatigue tests with a limited number of specimens [28]. Herein, the first specimen is subjected to a stress level that is 

most likely well below the average fatigue strength. When this specimen survives 5 × 106 cycles, this same specimen is 

subjected to a stress level one increment above the previous. This is continued with the same specimen until failure. Then, the 

number of cycles is recorded, and the next specimen is subjected to a stress that is at least 2 increments below the level where 

the previous specimen fails. The first stress amplitude level is determined, depending on the tensile strength of the material and 

according to the following empirical expression [29]: 

σ1 = 0.68 × (0.55 − 0.0001 × σu) × σu (2) 

where σu is the material tensile strength. When σu = 440 MPa, the first stress amplitude level σ1 is about 152 MPa. 

The next stress level applied on a specimen is determined by the results obtained in the preceding test. If the preceding 

specimen, i, fails at the level σi, the next stress level will be σi + 1 = σi – 2d. If the preceding specimen is not broken, σi + 1 = σi + 

d, where d is a predefined difference in applied stress levels. In this study, d is determined as d = 0.068 × σ1 ≈ 10.4 MPa. The 

fatigue testing results of five samples in group C is illustrated in Fig. 8. There are 5 specimens that undergo the bending tests. The 

first specimen is broken in the third test at a stress level of σ2. Therefore, the second specimen is firstly tested at a stress level σ0 = σ3 − 

2d, and then it is tested at elevated stress levels until it is broken at the fourth test with a stress level of σ3. Subsequently, the third 

specimen is tested, and the tests are repeated until the fifth specimen is broken.  

Based on the test results as shown in Fig. 8, the fatigue limit is calculated by Eq. (2) [24]: 

0

1

2
R

A
d

F
= + ±  

 
σ σ  (3) 

where σ0 is the lowest stress level (i = 0) in which the less frequent event run-out appears; F and A are the parameters calculated 

by Eqs. (4) and (5).  

F i= ∑  (4) 

i
A f= ∑  (5) 

where i is the number of stress levels, and fi is the number of run-outs obtained at the stress level i. 

Determining the load amplitude  

Fatigue test 

Samples are broken or not 

at the load circle 5 × 106 
Decreasing the stress amplitude  

by 2d = 20.8 MPa  

Determining the fatigue limit 

of sample groups 

Yes No 

If all 5 samples are tested  

Increasing the stress amplitude  

by d = 10.4 MPa  

Start 

Fig. 7 Diagram illustrating the process of determining the fatigue limit 

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Fig. 8 Log sheet of a modified staircase test 

3. Results and Discussion 

3.1.   Dynamic stress 

The correlation between the exciting acceleration amplitude and the strain amplitude on the surface of test specimens in 

RoI is shown in Fig. 9. With each case of the exciting acceleration amplitude, there is a corresponding graph that shows the 

change of strain over time. It can be seen that, in the sweep sine mode, the largest strain response is achieved when the exciting 

frequency of the LDS shaker coincides with the natural frequency of the samples. 

Fig. 10 shows the dynamic stress at different exciting acceleration amplitudes. This dynamic stress-vibration acceleration 

correlation can be approximately estimated by Eq. (6): 

σ = 2.1049 × A + 47.953 (6) 

where A is the amplitude of vibration acceleration (m/s
2
); σ is the principal stress on the surface of test specimens in RoI (MPa). 

It is revealed that the dynamic stress linearly increases with the amplitude of vibration acceleration. 

  

Fig. 9 Correlation between strain and amplitude of  

vibration acceleration 

Fig. 10 Dependence of stress on exciting acceleration 

 

3.2.   Residual stress and fatigue limit 

The results of measuring residual stress according to ASTM E837-13a include a variety of stress components. To compare 

and match with the results of dynamic stress measurement, the residual stress components will be chosen as the first principal 

stress. The residual stress of the samples after VSR is shown in Fig. 11. It is noted that the samples without VSR are considered 

the ones with the vibration amplitude of zero. 

It is clear that the residual stress will be reduced when the amplitude of exciting acceleration increases. According to the 

graph, the residual stress in the samples without using VSR is 216 MPa. In all the cases using VSR, the residual stress is 

reduced by more than 36%. In particular, the maximum residual stress reduction can be attained by up to 73% when the 

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vibration amplitude reaches 57 m/s
2
. This is corresponding to the total stress exceeding 30% of the YS of the material. In 

contrast, when the amplitude of vibration acceleration exceeds 57 m/s
2
, the residual stress will increase as the vibration 

amplitude increases. This may be explained by the intensive deformation of the samples. 

The process of determining the fatigue limit of sample groups is conducted according to the modified staircase method 

and is shown in Fig. 7. The experimental data is processed, and the final results of the fatigue limit are given in Fig. 12. The 

experimental results of the fatigue limit demonstrate that the fatigue limit of samples is generally increased with the VSR 

treatment. The fatigue limit of test specimens increases as the vibration acceleration amplitude increases from 0 to 57 m/s
2
, and 

it reaches a peak of 14% when the vibratory exciting acceleration amplitude is 57 m/s
2
. After that, if the vibratory exciting 

acceleration amplitude continues increasing, the fatigue limit will gradually decrease. The result is in good agreement with the 

studies on tensile fatigue characteristics after VSR [19, 21]. It is obvious from the tests that, if the amplitude of the vibration 

acceleration is very high, the fatigue limit of specimens will be decreased to the value even lower than the value of the samples 

without VSR. In general, if the vibration amplitude in VSR is too high or the VSR time is too long, micro voids will be formed 

and specimens’ fatigue limit will be lowered. 

  
Fig. 11 Residual stress in post-vibrating samples with  

different vibration acceleration amplitudes 

Fig. 12 Fatigue limit of sample groups according to vibration  

acceleration amplitudes  
 

4. Conclusions 

In this study, the VSR effects on the residual stress and fatigue limit of samples are investigated. The key findings of this 

research are described as follows:  

(1)  The dynamic stress generated through VSR has a linear relationship with the amplitude of vibratory exciting acceleration. 

At the amplitude of vibratory exciting acceleration lower than 57 m/s
2
, the residual stress is reduced with the increase in the 

amplitude of vibratory exciting acceleration. The maximum principal residual stress is reduced about 73% when the 

amplitude of vibratory exciting acceleration is equal to 57 m/s
2
. As the amplitude of vibratory exciting acceleration 

exceeds 57 m/s
2
, the residual stress will increase. However, it is generally still lower than those in the cases without using 

VSR. 

(2)  VSR also has an effect on the fatigue limit of the samples made from low-carbon steel with residual stress. VSR can 

improve the fatigue limit by up to 14% for the samples under residual stress. However, if the vibrating samples are with 

very high amplitude of acceleration, their fatigue limit will be reduced to the values that are lower than those without VSR. 

(3)  When the vibrating test samples are with moderate acceleration amplitude, VSR is able to both reduce the residual stress 

and increase the fatigue limit. If the amplitude of vibration acceleration is very high, this effect will be possibly reversed. 

The residual stress will be improved less, and the fatigue limit can be lower than the cases without VSR. 

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Symbols and Abbreviated Terms 

Symbol Unit Definition 

VSR - Vibratory stress relief 

YS MPa Yield strength 

σ MPa Stress 
ε m/m Strain 
E MPa Modulus of elasticity 

σu MPa Material tensile strength 

σ1 MPa The first stress amplitude level 

σi MPa The i
th

 stress level 

σ0 MPa The lowest stress level (i = 0) 

σR MPa Fatigue limit 

d MPa Predefined difference in applied stress levels 

A m/s
2
 Amplitude of vibration acceleration 

 

Conflicts of Interest 

The authors declare no conflict of interest. 

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[28] International Council on Combustion Engines Working Group 4 (CIMAC WG4), “IACS UR M53, Appendix IV, 

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[29] Strength Calculation and Testing: Methods of Fatigue Strength Behavior Calculation, GOST 25.504-82, 1982. (In 

Russian) 

 

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