Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 3, pp. 813-822, Warsaw 2017
DOI: 10.15632/jtam-pl.55.3.813

EFFECT OF THE MULTIPLE DAMAGES AND TEMPERATURE CHANGES

ON THE NATURAL FREQUENCY

Sattar Mohammadi Esfarjani, Mehdi Salehi, Aazam Ghassemi

Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

e-mail: satar.iran@gmail.com (corresponding author); mehdi.salehi@pmc.iaun.ac.ir; a ghassemi@pmc.iaun.ac.ir

Damage detection based on structural dynamic characteristics, such as natural frequencies
andmode shapes, is an important area of research. Obtaining accurate structural dynamic
characteristics is perhaps the most challenging aspect. In particular, changes in environ-
mental temperature due to seasonal weather or radiation from sunshine leads to changes
in the dynamic characteristics of structures. An important conclusion is that changes in
the dynamic characteristics of a structure due to damage may be smaller than changes in
the dynamic characteristics due to variations in temperature. Also, damage can affect the
frequency response. This is the first study of evaluation of the effect of changes in tem-
perature and multiple damages on natural frequency at the same time. In this paper, the
simultaneous effect of the multiple defects and temperature on the natural frequencies of
6063 aluminum alloy beam are assessed numerically. ABAQUS finite element software is
used for the numerical analysis. The present paper aims to evaluate the temperature effect
and multiple damages on vibration responses. The variations in the frequency have been
analysed in simulation by using an aluminum specimen and obtaining impedance signatu-
res at temperatures ranging from −200◦C to 204◦C. The results show that an increase in
temperature leads to a decrease in structural frequency, and that a decrease in tempera-
ture leads to an increase in structural frequency. The evaluation of the effect of multiple
defects on natural frequency shows that when damages are created in the structure, there is
a significant decrease in the natural frequency responses of the 6063 aluminum alloy beam.
The results show that damage causes a decrease in the natural frequency of the structure.
This study highlights the importance of applying simulation methods to the evaluation of
the effect of changes in environmental temperature and multiple damages on the dynamic
characteristics such as natural frequencies andmode shapes, especially at the same time.

Keywords: structural healthmonitoring, damagedetection, vibration response, temperature,
natural frequency

1. Introduction

Many techniques have been proposed to detect and localize structural damage using changes in
dynamic characteristics of structures such as natural frequency and mode shape (Xu and Wu,
2007). Modal parameters such as natural frequencies and mode shapes are sensitive indicators
of structural damage.However, they are not only sensitive to damage, but also to environmental
conditions such as humidity, wind and, most important, temperature (Meruane and Heylen,
2012). In particular, changes in environmental temperature due to seasonal weather or radia-
tion from sunshine lead to changes in the dynamic characteristics of structures. An important
conclusion is that the changes in dynamic characteristics of the structure due to damagemay be
smaller than changes in the dynamic characteristics due to variations in temperature (Ralbovsky
et al., 2014). For large structures, such as long-span bridges, damage detection is affected by
environmental factors. Significant damage may cause very small changes in dynamic characte-
ristics, and these changes may go undetected due to changes in environmental and operational



814 S. Mohammadi Esfarjani et al.

conditions. These changes have an influence on the dynamic characteristics of the structure.
Damage induced dynamic characteristics may be completely masked by changes in dynamic
characteristics due to changes in environmental temperature (Xu andWu, 2007).

The effect of crack location in the modal frequency of a draft gear used in auto couplers
of freight railway wagon for various orientations was investigated by Harak et al. (2015). They
showed that the defect in consecutive pads causes more changed in frequency as compared to
a single defective pad. As far as the location of the defective pad is concerned, it is seen that
the draft gear frequency is more sensitive to defective pads located either near the housing ba-
se plate or top follower (Harak et al., 2015). The effects of crack ratios and positions on the
fundamental frequencies and buckling loads of slender cantilever Euler beams with a single-
edge crack are investigated byKaraag̃aç et al. (2009). Sayman et al. (2013) presented the effect
of interface crack on the free vibration response of a sandwich composite beam experimental-
ly and numerically. They showed that the natural frequency of the torsional mode decreases
as the crack length increases (Sayman et al., 2013). The effect of temperature on damage de-
tection results was detected early. Eigenfrequency changes caused by temperature effects on
different structures were described in the works of Peeters and DeRoeck (2001), Farrar et al.
(1997) or Ralbovsky et al. (2010). The effect of temperature was considered as an obstacle
for damage detection. Various methods for removing that effect were proposed, for example in
the works of DeRoeck et al. (2000), Hu et al. (2012) and many other scientists (Ralbovsky et
al., 2010). Variations in frequencies are caused mainly by a change in the modulus of a mate-
rial under different temperatures. Modal frequencies of steel structures, the aluminum beam,
and the RC structures decrease by about 0.02, 0.03, and 0.15%, respectively, when tempera-
ture increases by one degree Celsius, regardless of modes and structural types. Frequencies of
concrete structures are more sensitive to temperature changes than metallic structures (Xia
et al., 2012).

The present paper aims to evaluate the temperature effect and multiple damages on vibra-
tion responses. In this study, the frequency changes of a structure caused by multiple defects
and temperature changes in the 6063 aluminum alloy beam are studied. The thermally induced
changes in dynamic characteristics are compared with those due to damages to the 6063 alumi-
numalloy beam.The changes caused by temperature have been analyzed based on the following
aspects: (1) change of the passion rate of the aluminumalloy; (2) changes of the elasticmodulus
of the aluminumalloy. Guidelines to predict changes in the frequency andmode shape curvature
due to temperature changes in the aluminumalloy have been obtained,which is useful for design
purposes (Xu andWu, 2007). The remainder of the paper is organized as follows. In Section 2,
theory of the temperature effect on natural frequency is discussed. Section 3 is devoted to the
process of the evolution effect ofmultiple defects and temperature changes on structural natural
frequency responses. The verification and simulation of the method is discussed in Section 4.
Numerical methods and results for evaluation of the temperature effect and multiple damages
on vibration responses are presented in Section 5. The conclusion is reported in Section 6.

2. Theory

Letus consider a rectangular platewhich is subjected to an exponential temperaturedistribution
along the length, i.e. in the x-direction (Arun et al., 2014)

T =T0
e−ex

e−1
(2.1)

where T denotes the temperature excess above the reference temperature at any point at the
distance X = x/a and T0 denotes the temperature excess above the reference temperature at



Effect of the multiple damages and temperature changes... 815

the end, i.e. x= a orX =1. The temperature dependence of themodulus of elasticity for most
of engineering materials is given by Nowacki (1962), (Arun et al., 2014)

E(T)=E0(1−γT) (2.2)

where E0 is the value of Young’s modulus at the reference temperature, i.e. T = 0, and γ is
the slope of variation ofE with T (Arun et al., 2014). Taking as the reference temperature, the
temperature at the end of the plate, i.e. at X = 1, the modulus variation in view of (2.1) and
(2.2) becomes (Arun et al., 2014)

E(X) =E0
(

1−α
e−eX

e−1

)

(2.3)

where α = γT0(0−α < 1) is a constant known as the temperature constant. In the above
literature, most studies show that an increase in temperature leads to a decrease in structural
frequencies, whilemagnitude varies depending on structures, materials, and temperature range.
Variations in natural frequencies of structures with temperature are caused by changes inmate-
rial properties, in particular, themodulus of elasticity. To quantify the effect of temperature on
natural frequencies, a single-span or multi-span prismatic beam made of an isotropic material
is used as an example. Its undamped flexural vibration frequency of the order n is (Xia et al.,
2012; Blevins, 1979)

fn =
λ2
n

2πl2

√

EI

µ
(2.4)

where λn is a dimensionless parameter and is a function of boundary conditions, l is length of
the beam, µ is mass per unit length, E is the modulus of elasticity, and I is the moment of
inertia of cross-sectional area. It is assumed that variations in temperature do not affect mass
and boundary conditions, but only geometry of the structure and mechanical properties of the
material.

3. Description of the process of the evolution effect of multiple defects and

temperature changes on structural natural frequency responses

In recent years, the use of simulation models to develop vibration based damage detection
techniqueshasbecomeverypopular, because it is a less expensive and time-consumingprocedure
than investigation of real structures or experimental models. Also, an experimental setup is also
a fairly difficult process. Since the Finite Element (FE) method has been widely accepted as
an analysis tool in Structural Health Monitoring (SHM), the above mentioned constraints can
be overcome by using a validated FE model to simulate the real structure (Moragaspitiya et
al., 2013). In this Section, a FE modeling method is used to obtain natural frequencies of the
structure.FEmodelsof the samplearedevelopedutilizing thecommercialFEpackageABAQUS.
As shown in Fig. 1, natural frequency responses of the structure are obtained fromFEmodeling
of the intact and damaged structure. Then, the natural frequency responses of the structure are
obtained for varying temperatures from−200◦C to 204◦C. This is illustrated in Fig. 2.

4. Numerical simulation

In this Section, evaluation of the effect of multiple defects on natural frequencies of the 6063
aluminum alloy beam is discussed. Then, the effect of temperature changes on the natural
frequencies of the 6063 aluminum alloy beam is investigated as well.



816 S. Mohammadi Esfarjani et al.

Fig. 1. Flowchart of the evolution effect of multiple defects on the natural frequency responses

Fig. 2. Flowchart of the evolution effect of temperature changes on the natural frequency responses of
the undamaged structure

4.1. Evaluation of the effect of multiple defects on natural frequencies

A 3-D FE model of the 6063 aluminum alloy beam has been created using ABAQUS (see
Fig. 3). The overall dimensions of the beam are 400mm×40mm×0.16mm. The material pro-
perties are listed in Table 1. The clamped end of the beam has no rotation or displacement in
any direction (see Fig. 4). Finally, themodel is analysed with a frequency step, and the natural
frequency responses for 10 modes are obtained from the results. As shown in Fig. 5, in order
to investigate the effect of multiple defects on natural frequencies, several damage scenarios are
simulated. Two types of damages are simulated (Table 2) and each is introduced using different
extents to investigate the effect of multiple defects on natural frequencies. Freemeshing is more
suitable on the damaged section, thus free meshing with triangular and tetrahedral elements is
utilized (see Fig. 6). The elements type, the total number of elements and nodes for the unda-
magedmodel and damage scenarios that are created in themodel are shown in Table 3. In each
step, the damaged model is analysed with a frequency step of the natural frequency responses



Effect of the multiple damages and temperature changes... 817

for 10 modes obtained from the results. Figure 7 shows the results of calculating the structural
response to various damage cases to investigate the effect of multiple defects on natural frequ-
encies. As shown in Fig. 7, when damages are created in the structure, the natural frequencies
responses of the 6063 aluminum alloy beam significantly decrease. The results show that the
damage decreases the natural frequency of the structure.

Table 1.Material properties of the 6063 aluminum alloy beam

Density Poisson’s ratio Modulus of elasticity
ρ [g/cc] ν [–] E [GPa]

2.7 0.33 68.9

Fig. 3. A FE model of the beam created using ABAQUS

Fig. 4. Boundary conditions

Fig. 5. Location of damages in the model

Fig. 6. Free meshing technique with linear tetrahedral and hexahedral elements

Table 2. Introduced damage types to investigate the effect of multiple defects on the natural
frequency

Damage Description
Dimensions Location
[mm] [mm]

#1 delamination in the beam 20×0.06×1 100-120

#2 delamination in the beam 2×0.02×0.5 180-182

#3 delamination in the beam 5×0.02×0.077 300-305

#4 transverse crack in the beam 0.02×0.17×5 200-200.02



818 S. Mohammadi Esfarjani et al.

Table 3. The type of elements, the total number of elements and nodes for undamaged model
and damage scenarios that were created in the model

Structure status
Total number Total number

Type of elements
of nodes of elements

Undmaged model 49323 32000
linear hexahedral elements

of type C3D8R

Damage scenario
49838 195122

linear tetrahedral elements
#1 of type C3D4

Damage scenario
98536 383130

linear tetrahedral elements
#2 of type C3D4

Damage scenario
147420 572781

linear tetrahedral elements
#3 of type C3D4

Damage scenario
196712 763944

linear tetrahedral elements
#4 of type C3D4

Fig. 7. Effect of multiple defects on natural frequency responses of the 6063 aluminum alloy beam

4.2. Investigation of the effect of temperature changes on natural frequencies

The investigation of the effect of temperature changes on natural frequencies is presented in
this Section. For thepurposeof this investigation, two types of scenarios are simulated to identify
the effect of temperature changes on the natural frequencies (Table 4). Thefirst type of scenario,

Table 4. Introduced damage types to investigate the effect of temperature changes on the
natural frequency

Structure status Description

First type of scenario: intact structure,
undmagedmodel #1 investigate at−200◦C to 204◦C

Second type of scenario: damaged structure with transverse crack,
damaged model #2 investigate at−200◦C to 204◦C

the intact model, is analysed with some frequency step, and the natural frequency responses for
10modesare obtained.For the second typeof scenario, a transverse crack is created on themodel
(see Fig. 8). The size of the transverse crack is 5mm×0.17mm×0.02mm, and is located at a
distance of 200mm to 200.02mm from the fixed end. Then, the damagedmodel is analysed and
the natural frequency responses for 10modes are obtained. Tables 5 and 6 show respectively the



Effect of the multiple damages and temperature changes... 819

results of calculation of the undamaged anddamaged structural response to various temperature
cases presenting the effect of temperature on thenatural frequency.Figures 9 and10 respectively
show the effect of temperature changes on natural frequency responses pertaining to undamaged
and damaged structures. The results show that as temperature of the structure increases, the
natural frequencies responses of the structure decrease.

Fig. 8. Transverse crack created in the model

Table 5. Effect of temperature changes on the natural frequency responses of the undamaged
structure

Natural frequencies ω [rad/s] and corresponding temperatures t [◦C]
Mode −200◦C −129◦C −73◦C 21◦C 93◦C 149◦C 204◦C

ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s]

1 0.00747343 0.00742029 0.00731643 0.00719205 0.0069961 0.00684871 0.00665982

2 0.0468166 0.0464838 0.0458331 0.045054 0.0438264 0.042903 0.0417199

3 0.131184 0.130251 0.128428 0.126245 0.122805 0.120218 0.116903

4 0.143114 0.142097 0.140108 0.137726 0.133974 0.131151 0.127534

5 0.212142 0.210634 0.207686 0.204156 0.198593 0.194408 0.189047

6 0.257473 0.255643 0.252064 0.247780 0.241028 0.235950 0.229443

7 0.426512 0.423480 0.417552 0.410455 0.399271 0.390858 0.380080

8 0.431727 0.428658 0.422657 0.415473 0.404153 0.395637 0.384727

9 0.638622 0.634082 0.625206 0.614579 0.597833 0.585236 0.569098

10 0.727410 0.722239 0.712130 0.700025 0.680951 0.666603 0.648221

Table 6. Effect of temperature changes on the natural frequency responses of the damaged
structure

Natural frequencies ω [rad/s] and corresponding temperatures t [◦C]
Mode −200◦C −129◦C −73◦C 21◦C 93◦C 149◦C 204◦C

ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s] ω [rad/s]

1 0.0111536 0.0110743 0.0109193 0.0107337 0.0104412 0.0102212 0.00993934

2 0.0699099 0.0694129 0.0684413 0.0672779 0.0654448 0.0640658 0.0622991

3 0.195750 0.194358 0.191637 0.188380 0.183247 0.179386 0.174439

4 0.212484 0.210973 0.208020 0.204484 0.198912 0.194721 0.189352

5 0.224307 0.222713 0.219595 0.215862 0.209981 0.205556 0.199888

6 0.383982 0.381252 0.375916 0.369526 0.359457 0.351883 0.342180

7 0.634711 0.630199 0.621377 0.610815 0.594172 0.581652 0.565613

8 0.676172 0.671365 0.661967 0.650715 0.632985 0.619647 0.602560

9 0.948804 0.942059 0.928872 0.913084 0.888204 0.869489 0.845512

10 1.13783 1.12974 1.11393 1.09499 1.06516 1.04271 1.01396



820 S. Mohammadi Esfarjani et al.

Fig. 9. Effect of temperature changes on the natural frequency for the undamaged structure

Fig. 10. Effect of temperature changes on the natural frequency for the damaged structure

5. Conclusion

This paper reviewsmultiple defects and the temperature effect on variations inmodal properties
of the structure. This is the first study of evaluation of the effect of temperature changes and
multiple damages on the natural frequency at the same time. The variations in the frequency
are analysed in simulation by using an aluminum specimen obtaining impedance signatures at
temperatures ranging from−200◦C to 204◦C.The results show that an increase in temperature



Effect of the multiple damages and temperature changes... 821

leads to a decrease in the structural frequency, and that a decrease in temperature leads to an
increase in the structural frequency. Therefore, temperature effects are a critical problem for
structural health monitoring based on vibration responses, especially in detecting low damage
levels. Efficient compensatory methods for temperature effects remain to be developed. The
evaluation of the effect of multiple defects on the natural frequency shows that when damages
are created in the structure, there is a significant decrease in natural frequency responses of
the 6063 aluminum alloy beam. The results show that damage causes a decrease in the natural
frequency of the structure. This study highlights the importance of application of simulation
methods to the evaluation the effect of changes in environmental temperature and multiple
damages on dynamic characteristics, such as natural frequencies and mode shapes, especially
taking place at the same time.

References

1. ArunK.,Gupta,MamtaJ., 2014,Exponential temperature effect on frequencies of a rectangular
plate of non-linear varying thickness: a quinitic spline technique,Journal of Theoretical andApplied
Mechanics, 52, 1, 15-24

2. Blevins R.D., 1979, Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold,
NewYork

3. DeRoeckG., Peeters B.,Maeck J., 2000,Dynamicmonitoring of civil engineering structures.
Computational methods for shell and spatial structures, IASS-IACM, Athens, Greece

4. Farrar C.R., Doebling S.W., Cornwell P.J., Straser E.G., 1997, Variability of modal
parameters measured on the Alamosa Canyon Bridge, Proceedings of 15th International Modal
Analysis Conference, Orlando, USA, 1997, 257-263

5. Harak S.S, Sharma S.C., Shukla S., Gupta P., Kumar S., Harsha S.P., 2015, Effect of
multiple location defects on the dynamics of draft gear used in freight railwaywagon, International
Journal of Vehicle Structures and Systems, 7, 3, 107-113

6. HuW.H.,MoutinhoC.,De SaCaetanoE.,Magalhães F., CunhaA.A.M.F, 2012,Conti-
nuousdynamicmonitoringof a lively footbridge for serviceabilityassessmentanddamagedetection,
Mechanical Systems and Signal Processing, 33, 38-55

7. Karaag̃aç C., Özturk H., Sabuncu M., 2009, Free vibration and lateral buckling of a canti-
lever slender beamwith an edge crack: experimental and numerical studies, Journal of Sound and
Vibration, 326, 235-250

8. Meruane V., Heylen W., 2012, Structural damage assessment under varying temperature con-
ditions, Structural Health Monitoring, 11, 3, 345-357

9. Moragaspitiya P.H.N., Thambiratnam D.P., Perera N.J., Chan T.H.T., 2013, Develop-
ment of a vibration based method to update axial shortening of vertical load bearing elements in
reinforced concrete buildings,Engineering Structures, 46, 49-61

10. Peeters B., DeRoeck G., 2001, One-year monitoring of the Z24-Bridge: environmental effects
versus damage events,Earthquake Engineering and Structural Dynamics, 30, 149-171

11. Ralbovsky M., Deix S., Flesch R., 2010, Frequency changes in frequency-based damage iden-
tification, Structure and Infrastructure Engineering, 6, 611-619

12. Ralbovsky M., Santos J., Kwapisz M., Dallinger S., Catarino J.M., 2014, Damage de-
tection based on structural response to temperature changes and model updating, 7th European
Workshop on Structural Health Monitoring, July 8-11, La Cité, Nantes, France

13. Sayman O.M., Toygar E., Kiral Z., Kiral B.G., 2013, Effect of the root crack on natural
frequency of sandwitch composite beams, Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi,
19, 7, 298-302



822 S. Mohammadi Esfarjani et al.

14. Xia Y., Chen B., Weng S., Ni Y.-Q., Xu Y.-L., 2012, Temperature effect on vibration pro-
perties of civil structures: a literature review and case studies, Journal of Civil Structural Health
Monitoring, 2, 29-46

15. Xu Z.-D., Wu Z., 2007, Simulation of the effect of temperature variation on damage de-
tection in a long-span cable-stayed bridge, Structural Health Monitoring, 6, 177-189, DOI:
10.1177/1475921707081107

Manuscript received November 3, 2016; accepted for print January 18, 2017