shm

Table of ContentsChapter 1 INTRODUCTION................................................................................................7

1.1 Background:................................................................................................................7

1.2 Effects of Structural Damage on Natural Frequency:.................................................8

1.3 Objective and Scope of the Project:............................................................................9

Chapter 2 STATE–OF–THE-ART IN STRUCTURAL HEALTH MONITORING.....10

2.1 Structural Health Monitoring (SHM): An Overview:...............................................10

2.2 Techniques of Health Monitoring:............................................................................11

2.2.1 Static Response Based Technique:.....................................................................11

2.2.2 Dynamic Response Based Techniques:.............................................................11

2.2.3 Local SHM Techniques:....................................................................................12

2.2.4 Global Vibration Techniques for SHM:.............................................................12

2.3 Techniques Using Smart Materials and Smart Structures Concept:.........................13

2.3.1 Smart Structures:................................................................................................14

2.3.2 Smart Materials:.................................................................................................14

2.3.3 Structural Health Monitoring Using PZT Patches.............................................15

Chapter 3 LITERATURE REVIEW...................................................................................19

3.1 Introduction:..............................................................................................................19

3.2 Damage Detection Methods for Beams:....................................................................19

3.2.1 Change in Curvature Mode Shape Method:.......................................................19

3.2.2 Change in Flexibility Method:...........................................................................22

3.2.3 Change in Stiffness Method:..............................................................................23

3.2.4 Improved Damage Detection Method Based On Modal Information................24

3.2.5 Damage Detection Using Strain Energy Theory:...............................................25

3.2.6 Damage Detection Using Higher Order Derivatives:........................................26

Chapter 4 EXPERIMENTAL WORK DONE ON STEEL BEAM...................................28

4.1 Introduction:..............................................................................................................28

4.2 Dimensions and Properties Steel Beam:....................................................................28

4.3 Experimental Setup:..................................................................................................29

4.4 Instruments Used:......................................................................................................30

4.4.1 Data acquisition systems:...................................................................................30

4.4.2 Force application systems:.................................................................................30

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4.4.3 Sensors:..............................................................................................................30

4.5 Mode Shape Based on Column of FRF (Single impact exitation) Method:..............31

4.5.1 Frequency Response Function:..........................................................................31

4.5.2 Procedure To Draw Mode Shape By Single Impact Exitation Method:............31

4.5.3 For Single Damage Criteria:..............................................................................32

4.5.4 For Multiple Damage Criteria:...........................................................................33

4.6 Experiments on Undamaged Beam:..........................................................................33

4.7 Mode Shapes of Undamaged Beam:.........................................................................35

4.8 Mode Shapes for Different Damage Conditions:......................................................36



4.9 Damage Index Plots:..................................................................................................44



4.10 Variation of Torque with Damage Index for Single Damage Criteria:.....................47

4.11 Damage Index Using 3rd Order Derivative:...............................................................47



Chapter 5 EXPERIMENTAL WORK DONE ON reinforced CONCRETE BEAM........52

5.1 Multiple Impact Excitation Method Procedure:........................................................52

5.1.1 Procedure to Draw Mode Shape using Multiple Impact Excitation Method:....52

5.2 Mode Shapes of Undamaged And Damaged Beam:.................................................55

5.3 Damage Index Plots Using Curvature Mode Shape Method:...................................58

5.4 Damage Index Using 3rd Order Derivative:...............................................................59

5.5 Roving of Accelerometer Approach:.........................................................................61

5.5.1 Procedure To Find Mode Shape:........................................................................61

Chapter 6 CONCLUSIONS AND FUTURE WORK........................................................62

6.1 Introduction:..............................................................................................................62

6.2 Research Conclusions and Contribution:..................................................................62

6.3 Limitations:................................................................................................................63

6.4 Scope for Future Work:.............................................................................................63

REFERENCES……………………………………………………………………………….64

APPENDIX…………………………………………………………………………………..66

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LIST OF FIGURES

Figure 2-1. Common smart materials and associated stimulus-response................................13

Figure 2-2. A piezoelectric plate under the action of stress and electric field 1D interaction (rama shanker et al., 2011).......................................................................................................16

Figure 3-1. Absolute difference between the curvature mode shapes of cantilever beam for 1st mode shape (pandey et al., 1991).............................................................................................20

Figure 4-1. Complete steel beam with dimensions (Abhishek, 2013).....................................28

Figure 4-2. PZT patch (Abhishek, 2013).................................................................................29

Figure 4-3. Showing the location of impact and PZT sensor (Abhishek, 2013)......................29

Figure 4-4. Images of Oscilloscope and QDA1008.................................................................30

Figure 4-5. Dytran sensor and force hammer made by steel and wood...................................30

Figure 4-6. Surface bonded PZT, Embedded Sensors and Accelerometers (Abhishek, 2013)30

Figure 4-7. Showing the plates connected by Bolts (Abhishek, 2011)....................................31

Figure 4-8. FFT of 1st sensor for Absolute Part (Amplitude(x-axis) vs Frequency(Y-axis)). 33

Figure 4-9. FFT of 1st sensor for Imaginary Part (Amplitude(x-axis) vs Frequency(Y-axis))..................................................................................................................................................34

Figure 4-10. FFT of 1st sensor for Real Part (Amplitude(x-axis) vs Frequency(Y-axis))......34

Figure 4-11. Mode Shape obtained from Absolute part of FRF.............................................35

Figure 4-12. Mode Shape obtained from Imaginart Part of FRF............................................35

Figure 4-13. Mode Shape obtained from Real Part of FRF.....................................................36

Figure 4-14. Mode shapes of Undamaged and Damaged states of beam obtained from Imaginary Part of FRF.............................................................................................................36

Figure 4-15. Mode shapes of Undamaged and Damaged states of beam obtained from Absolute Part of FRF...............................................................................................................38

Figure 4-16. Mode shapes of Undamaged and Damaged states of beam obtained from Real Part of FRF...............................................................................................................................39

Figure 4-17. Mode shapes of Undamaged and Damaged states of beam obtained from Imaginary Part of FRF.............................................................................................................41

Figure 4-18. Mode shapes of Undamaged and Damaged states of beam obtained from Absolute Part of FRF...............................................................................................................42

Figure 4-19. Mode shapes of Undamaged and Damaged states of beam for Real Part of FRF..................................................................................................................................................43

Figure 4-20. Plot for Imaginary Part of FRF............................................................................44

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Figure 4-21. Plot for Absolute Part of FRF..............................................................................44

Figure 4-22. Plot for Real Part of FRF.....................................................................................45




Figure 4-26. Variation of Damage Index with Torque Released for Imaginary Part of FRF. .47




Figure 4-30. Plot for imaginary Part of FRF............................................................................50



Figure 5-1. Undamaged Concrete beam on which experiments are conducted......................53

Figure 5-2. Damage1 condition of concrete beam...................................................................54

Figure 5-3. Damage2 and damage3 conditions of concrete beam...........................................54

Figure 5-4. Retofitting-1 of concrete beam..............................................................................54

Figure 5-5. Mode Shape obtained from absolute Part of FRF.................................................55

Figure 5-6. Mode Shape obtained from Imaginary Part of FRF..............................................56

Figure 5-7. Mode Shape obtained from Real Part of FRF.......................................................57

Figure 5-8. Plot For Absolute Part of FRF...............................................................................58

Figure 5-9. Plot for Imaginary Part of FRF..............................................................................58

Figure 5-10. Plot For Real Part of FRF....................................................................................59

Figure 5-11. Plot For Imaginary Part of FRF...........................................................................59

Figure 5-12. Plot for Asolute Part of FRF................................................................................60


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LIST OF TABLES

Table 4-1. Physical properties of beam....................................................................................28

Table 4-2. Relation of single damage with Torque..................................................................32

Table 4-3. Relation of Multiple damage with Torque..............................................................33

Table 4-4. Variation of Natural frequency, Damage location, and severity For Imaginary Part of FRF......................................................................................................................................37

Table 4-5. Variation of Natural frequency, Damage location, and severity For Absolute Part of FRF......................................................................................................................................38

Table 4-6. Variation of Natural frequency, Damage location, and severity For Real Part of FRF...........................................................................................................................................40




Table 4-10. Damage locations using 3rd order derivative for single damage location............49

Table 4-11. Damage locations using 3rd order derivative for multiple damage criteria.........51

Table 5-1. Properties of Concrete Beam..................................................................................52

Table 5-2. Description About Damage....................................................................................53




Table 5-6. Damage locations using 3rd order derivative for concrete beam...........................61

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LIST OF SYMBOLS

E Young’s Modulus

I Moment of Inertia

EI Flexural Rigidity

[K] Stiffness Matrix

[Ø] Mode Shape Matrix

[M] Mass Matrix

[Ω] Modal Stiffness Matrix

S Strain

d31 Piezoelectric Strain Coefficient

T Axial Tension

V Potential Difference

µ Poisons Ratio

v” Curvature At A Section

M Bending Moment at a Section

ω Natural Frequency

U Strain Energy

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CHAPTER 1 INTRODUCTION

1.1 Background:

Civil engineering structures form an important pillar of any modern nation’s economy, and

their poor performance can have a detrimental effect on the economic well-being (Aktan et

al., 1998). Therefore, intensive research is being focused on structural health monitoring

(SHM) worldwide. Structural health monitoring (SHM) denotes a reliable system with ability

to detect and interpret adverse changes in a structure due to damage or normal operations

(Kessler et al., 2002). It implies acquisition, validation and analysis of technical data to

facilitate life cycle management decisions. Damage is defined as changes to the material

and/or geometric properties of a structural system, including changes to the boundary

conditions and system connectivity, which adversely affect the system’s performance. The

SHM process involves the observation of a system over time using periodically sampled

dynamic response measurements from an array of sensors which are embedded or surface

attached the extraction of damage-sensitive features from these measurements, and the

statistical analysis of these features to determine the current state of system health. For long

term SHM, the output of this process is periodically updated information regarding the ability

of the structure to perform its intended function in light of the inevitable aging and

degradation resulting from operational environments. After extreme events, such as

earthquakes or blast loading, SHM is used for rapid condition screening and aims to provide,

in near real time, reliable information regarding the integrity of the structure.

Two techniques in the field of SHM are generally available. Those are

1) Wave Propagation Based Techniques and

2) Vibration Based Techniques.

Broadly the literature for vibration based SHM can be divided into two aspects, the first

where in models are proposed for the damage to determine the dynamic characteristics, also

known as the direct problem, and the second, where in the dynamic characteristics are used to

determine damage characteristics, also known as the inverse problem. In the last ten to fifteen

years, SHM technologies have emerged creating an exciting new field within various

branches of engineering.

Health monitoring is a maturing concept in the manufacturing, automotive and aerospace

industries; there are a number of challenges for its effective applications on civil and defence

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infrastructure systems. While successful real-life studies on a new or an existing structure are

critical for transforming health monitoring from research to practice, laboratory benchmark

studies are also essential for addressing issues related to the main needs and challenges of

structural health monitoring.

A successful health monitoring system requires the recognition and integration of several

components. Identification of health and performance metric is the first component which is a

fundamental knowledge need and should dictate the technology involved.

PZT patches have been employed for the health monitoring of structures owing to their

simultaneous sensing/actuating capability. Structural health monitoring (SHM) has been

gaining more importance in civil engineering areas such as wind engineering and earthquake

engineering. However, only few structures such as historical buildings and few critical

bridges have been instrumented with structural monitoring system due to high cost of

installation, long and complicated installation of system of wires. This Thesis is aimed at

identifying damage location in 1D structure using a PZT patches, thus making it very

economically viable.

1.2 Effects of Structural Damage on Natural Frequency:

The presence of damage or deterioration in a structure causes changes in the natural

frequencies of the structure. The most useful damage location methods are probably those

using changes in the resonant frequencies because frequency measurements can be quickly

conducted and are often reliable. Abnormal loss of stiffness is inferred when measured

natural frequencies are substantially lower than expected. It would be necessary for a natural

frequency to change by about 5% for the damage to be detected with confidence. However,

significant frequency changes alone do not automatically imply the existence of damage since

frequency shifts (exceeding 5%) due to changes in ambient conditions which have been

measured for both concrete and steel bridges within a single day (Aktan et al., 1994).

At modal nodes (points of zero modal displacements), the stress is minimum for the

particular mode of vibration. Hence, the minimal change in particular modal frequency could

mean that the defect may be close to the modal node. From vibration tests on concrete portal

frames, it is reported that the degrees of freedom of reduction in natural frequency is

dependent on the position of the defect relative to the Mode Shape obtained from a particular

mode of vibration (Moradalizadeh et al., 1990), From these findings, we can say that damage

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detection using frequency measurements might be unreliable when the damage is located at

regions of low stress. Thus, a shift in natural frequencies alone might not provide sufficient

information for integrity monitoring, unless the damage is in an important load bearing

member. The natural frequency changes vary proportionally with the square root of the

stiffness change, thus underlining the need for relatively large stiffness changes before

significant frequency changes can be detected. The reduction in frequency becomes more

important when the crack is at regions of high curvature for the modes under consideration.

Results from some experimental and numerical studies have suggested that the lower

vibration modes would be best suited for damage detection. However Begg stated that modes

higher than first should be used in damage detection so as to improve the identification. The

increased sensitivity of the higher modes to local damage has been mentioned by. Since

higher modes are usually unavailable from the results of a full-scale modal survey, their use

in damage detection cannot be implemented in real practice.

1.3 Objective and Scope of the Project:

The primary objective to the thesis work is to locate the damage induced in a 1D structure

using global vibration technique with the aid of low-cost PZT patches. The scope and

objective can be summarised as under:

(a) To identify damage locations in 1D structures using Curvature Mode Shape Method,

Third Order Derivative methods.

(b) To determine Severity of damage (in the terms of ratio of EIdamaged to EIundamaged ) in

damaged structures.

(c) Developed automated software interphase for real time for Identification & location

of damage.

(d) To evaluate practical aspect of mode shape extraction namely, compare the mode

shape derived from absolute, imaginary and real parts of frequency response function.

(e) Retrofitting of damaged reinforced concrete beam to know improvement in gain in

strength of reinforced concrete beam.

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CHAPTER 2 STATE –OF–THE-ART IN STRUCTURAL HEALTH MONITORING

2.1 Structural Health Monitoring (SHM): An Overview:

In most developing countries increase in defence potential has flooded huge defence

infrastructure and equipment. Health of the defence equipment can be kept up to date by good

maintenance and periodical inspections. All in-service structures require some form of

maintenance for monitoring their integrity and good health condition. Breakdown of

equipment not only leads to immense economic loss but can also endanger defence of the

nation. Appropriate maintenance prolongs the lifespan of a structure and can be used to

prevent catastrophic failure. Current schedule driven inspection and maintenance techniques

can be time consuming, labour-intensive, and expensive. SHM, on the other hand, involves

autonomous in-service inspection of the structures. The first instances of SHM date back to

the late 1970s and early 1980s. The concept of SHM originally applied to aerospace and

mechanical systems, is now being extended to civil structures. Objectives of health

monitoring are damage identification and its location, to determine the severity of damage

and remaining useful life of structure. The concept of SHM originally applied to aerospace

and mechanical systems, is now being extended to civil structures.

Objectives of health monitoring are as follows.

a) To ascertain that damage has occurred or to identify damage

b) To locate the damage

c) To determine the severity of damage.

d) To determine the remaining useful life of the structure.

SHM consists of both passive and active sensing and monitoring. Passive sensing and

monitoring is used to identify the location and force–time–history of external sources, such as

impacts or acoustic emissions. Active sensing and monitoring is used to localize and

determine the magnitude of existing damages. An extensive literature review of damage

identification and health monitoring of structural and mechanical systems from changes in

their vibration characteristics is covered by Doebling (1996).

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2.2 Techniques of Health Monitoring:

1. Static Response Based Technique

2. Dynamic Response Based Techniques

3. Local SHM Techniques

4. Global Vibration Techniques For SHM

2.2.1 Static Response Based Technique:

According to Banan (1994), static forces are applied on structure and the corresponding

displacements are measured. It is not necessary to select the entire set of forces and

displacements. Any subset could be selected, but a number of load cases may be necessary in

order to obtain sufficient information for computation. Computational method based on least

scale error function between the model and actual measurement is used. The resulting

equations are to be solved to arrive at a set of structural parameters. Any change in the

parameters from the base line healthy structure is an indicator of damage. The main

shortcoming of this technique is that the measurement of displacements is not an easy task. It

requires establishment of frame of reference. Employing a member of load cases can be very

time consuming. Besides, the computational effort required by the method is enormous.

Another method has been proposed based on static strain method. The advantage of this

technique is strain measurement can be made accurately and easily compared to displace

measurement. Although the method has some advantages over the static displacement

method, its application on real life structures remains tedious.

2.2.2 Dynamic Response Based Techniques:

These techniques are subjected to low frequency vibrations, and dynamic response of the

structure are measured and analysed. By this analysis, a suitable set of parameters such as

modal frequencies, modal damping, and mode shapes associated with each mode are

identified. If structure undergoes damage, changes occur in structural parameters namely the

stiffness matrix, damping matrix and natural frequencies. In this method, the structure is

excited by appropriated means and the response data processed to obtain a quantitative index

or a set of indices representative of the condition of the structures.

These techniques are advantageous over static response since they are comparatively easier to

implement.

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A damage detection method based on this concept is the changes in the stiffness method. The

stiffness matrix is obtained from the mode shapes and the modal frequencies derived from the

measured dynamic response of the structure. The stiffness matrix [K] may be expressed in

terms of mode shape matrix [Ø], the mass matrix [M], and the modal stiffness matrix [Ω],

(diagonal ω2) as

[K]= [M][Ø][Ω][Ø]T[M] = [M](∑i=1

n

ω2 Øi ØiT)[M]

The method involves determining pre and post damage stiffness to identify and locate

damage.

2.2.3 Local SHM Techniques:

These techniques rely on the localized structural interrogation for detecting damages.

Ultrasonic techniques, acoustic emission, eddy currents, impact echo testing, magnetic field

analysis, penetration dye testing, and x-ray analysis are some of the techniques in this

category are

a. Ultra Sonic Technique

b. Acoustic Emission Technique

c. Eddy Currents Technique

d. Impact-Echo Technique

e. Magnetic Field Technique

f. Penetration Dye Test Technique

g. X-Ray Technique

2.2.4 Global Vibration Techniques for SHM:

These techniques involve acquiring the global vibration response of the structure, and look

for changes in the natural frequencies and the mode shapes arising out of any structural

damage. The early algorithms based on global dynamic techniques involved determining the

structural stiffness or the flexibility matrices from global vibration measurements (Pandey

and Biswas, 1994; Zimmerman and Kaouk, 1994). About the same time, the Damage index

method (Stubbs and Kim, 1994) based on the modal Strain energy was also proposed.

Although the lower natural frequencies (and the corresponding mode shapes) of the structure

can be determined easily, it is generally observed that they change by very small amounts till

the damage reaches moderate to severe magnitudes (Pandey and Biswas, 1994; Farrar and

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Jauregui, 1998). This is because damage, especially at the initial stage, is a local

phenomenon, whereas the first few natural frequencies and the corresponding mode shapes

are global parameters of the structure. These and other limitations of the global dynamic

techniques have been well acknowledged (Catbas et al., 2007). In addition, the sensors

(generally accelerometers and PZT’s) and the data acquisition systems used by the global

dynamic techniques are usually fragile and exorbitant from cost point of view. These

difficulties associated with the global dynamic techniques can be addressed by using the

electromechanical impedance (EMI) technique alongside the global dynamic techniques,

sharing the same set of PZT patches as sensors.

2.3 Techniques Using Smart Materials and Smart Structures Concept:

Smart materials are materials which have ability to change their physical properties such as

shape, stiffness, viscosity, etc. in a specific manner according to certain specific type of

stimulus input.

Examples of smart materials are electro-strictive materials, magneto-strictive materials, shape

memory alloys, magneto- or electro-rheological fluids, polymer gels, and piezoelectric

materials, optical fibres. These are explained in detailed manner later.

Figure 2-1. Common smart materials and associated stimulus-response.(Bhalla,2004)

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2.3.1 Smart Structures:

The new field ‘smart materials and smart structures’ refers to structures that can assess own

health, perform self-repair or can make critical adjustments in their behaviours as conditions

change. Smart materials deals with structures which are ‘adaptable’ in some sense, perhaps

like as biological system. The design of smart structures involves more challenges because

the structural behaviour is not fixed but depends upon environment.

According to Ahmad (1988), a system is termed as “smart” if it is capable of recognizing an

external stimulus and responding to it within a given time in predetermined manner. In

addition, it is supposed to have the capability of identifying its status and may optimally

adapt its function to external stimuli or may give appropriate signal to the user. Smart

structures that can monitor their own conditions, detect impending failure, control, or heal

damage and adapt to changing environment. Because of their inherent capability of detecting

the any change in structure, smart materials, systems and structures are being used for SHM

and NDE from past two decades.

Research on smart civil structural system is focused on two areas. They are

1. Control of structural response to external loading, such as wind and earth quake.

2. Detection of damage and flaw in the system, and its severity.

2.3.2 Smart Materials:

There are a number of types of smart material, some of which are already common. Some

examples are as following:

1. Piezoelectric materials are materials that produce a voltage when stress is applied.

Since this effect also applies in the reverse manner, a voltage across the sample will produce

stress within the sample. Suitably designed structures made from these materials can

therefore be made that bend, expand or contract when a voltage is applied.

2. Shape-memory alloys and shape-memory polymers are materials in which large

deformation can be induced and recovered through temperature changes or stress changes

(pseudo elasticity). The large deformation results due to martensitic phase change.

3. Magneto-strictive materials exhibit change in shape under the influence of magnetic

field and also exhibit change in their magnetization under the influence of mechanical stress.

4. Magnetic shape memory alloys are materials that change their shape in response to a

significant change in the magnetic field.

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http://en.wikipedia.org/wiki/Magnetic_shape_memory

http://en.wikipedia.org/wiki/Magnetostriction

http://en.wikipedia.org/wiki/Shape-memory_polymer

http://en.wikipedia.org/wiki/Shape-memory_alloy

http://en.wikipedia.org/wiki/Piezoelectricity

5. PH-sensitive polymers are materials that change in volume when the pH of the

surrounding medium changes.

6. Temperature-responsive polymers are materials which undergo changes upon

temperature.

7. Halo chromic materials are commonly used materials that change their colour as a

result of changing acidity. One suggested application is for paints that can change colour to

indicate corrosion in the metal underneath them.

8. Chromogenic systems change colour in response to electrical, optical or thermal

changes. These include electro chromic materials, which change their colour or opacity on

the application of a voltage (e.g., liquid crystal displays), thermo chromic materials change in

colour depending on their temperature, and photochromic materials, which change colour in

response to light—for example, light sensitive sunglasses that darken when exposed to bright

sunlight.

9. Ferro fluid

10. Photomechanical materials change shape under exposure to light.

11. polymorph mold under hot water

12. Self-healing materials have the intrinsic ability to repair damage due to normal usage,

thus expanding the material's lifetime

13. Dielectric elastomers (DEs) are smart material systems which produce large strains

(up to 300%) under the influence of an external electric field.

14. Magneto caloric materials are compounds that undergo a reversible change in

temperature upon exposure to a changing magnetic field.

15. Thermoelectric materials are used to build devices that convert temperature

differences into electricity and vice-versa.

2.3.3 Structural Health Monitoring Using PZT Patches

Piezoelectric Materials as Dynamic Strain Sensors:

The phenomenon of piezoelectricity occurs in certain classes of non-Centro-symmetric

crystals, such as quartz, in which electric dipoles (and hence surface charges) are generated

because of mechanical deformations. The same crystals also exhibit the converse effect; that

is, they undergo mechanical deformations when subjected to electric fields.

The constitutive relations for piezoelectric materials for one-dimensional (1D) interaction,

such as for a piezoelectric plate shown in Figure 1, are (Ikeda, 1990):

D3=❑33T E3+d31T 1(1)

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http://en.wikipedia.org/wiki/Thermoelectric_effect

http://en.wikipedia.org/wiki/Thermoelectric_effect

http://en.wikipedia.org/wiki/Thermoelectric_materials

http://en.wikipedia.org/w/index.php?title=Magnetocaloric_materials&action=edit&redlink=1

http://en.wikipedia.org/wiki/Dielectric_elastomers

http://en.wikipedia.org/wiki/Self-healing_material

http://en.wikipedia.org/wiki/Polymorph

http://en.wikipedia.org/wiki/Photomechanical_effect

http://en.wikipedia.org/wiki/Ferrofluid

http://en.wikipedia.org/wiki/Sunglasses

http://en.wikipedia.org/wiki/Photochromic

http://en.wikipedia.org/wiki/Thermochromic

http://en.wikipedia.org/wiki/Liquid_crystal_display

http://en.wikipedia.org/wiki/Electrochromic

http://en.wikipedia.org/wiki/Chromism

http://en.wikipedia.org/wiki/Corrosion

http://en.wikipedia.org/wiki/Halochromism

http://en.wikipedia.org/wiki/Temperature-responsive_polymers

http://en.wikipedia.org/wiki/PH-sensitive_polymers

S1=T 1

Y E +d31 E3(2)

Where S1 is the strain in direction ‘‘1,’’ D3 the electric displacement over the PZT patch, d31

the piezoelectric strain coefficient, and T1 the axial stress in direction ‘‘1.’’

Y E= y E(1+ηj)

is the complex Young’s modulus of elasticity of the PZT patch at constant electric field and

ε 33T =e33

T (1−δj)

is the complex electric permittivity (in direction ‘‘3’’) at constant stress, with j=√−1

In these expressions, η and δ denote the mechanical loss factor and the dielectric loss factor

of the PZT material, respectively.

Eq. (1) is used in sensing applications and Eq. (2) in actuation applications of the

piezoelectric materials.

Figure 2-2. A piezoelectric plate under the action of stress and electric field 1D interaction (Shanker et al., 2011).

If a PZT patch surface bonded on a structure is desired to be used as a sensor only (with no

external electric field across its terminals, i.e. E3= 0), its governing sensing equation [Eq. (1)]

can be reduced to:

D3=d31Y E S1(3)

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Where YE S1 been substituted for T1 making use of the Hooke’s law. From the theory of

parallel place capacitors, the charge density can also be expressed as:

D3=❑33

T Vh

(4)

where V is the potential difference across the terminals of the PZT patch of thickness h.

Using Eq. (3) and Eq. (4), the voltage generated across the terminals of the PZT patch can be

expressed in terms of the strain in the patch (and hence on the surface of the structure it is

bonded to) as:

S1=( ❑33T

d31h Y E )V=K p V (5)

The output voltage can be easily measured by an oscilloscope (with or without conditioning

circuit) or directly using the modern digital multimeters, such as Agilent 34411A (Bhalla et

al., 2009).

Eq. (5) is applicable to PZT patches bonded to beams, where the structure–PZT interaction is

1D in nature. For a PZT patch bonded to plates, where the interaction is 2D in nature, the

governing equations [Eq. (1) and Eq. (2)] will take following form:

D3=ε33T E33+d31(T 1+T 2) (6)

S1=T 1−µT 2

Y E +d31 E3 (7)

S2=T 2−µT 1

Y E +d31 E3 (8)

Where S1 and S2 are the strains along the two principal direction, respectively, and µ is the

Poisson’s ratio. Making use of the above equations for the case of sensor (E3 = 0), yields:

D3=d31Y E(S1+S2)

(1−µ) (9)

Similar to the 1D case, following relation can be derived for the voltage output across a PZT

patch bonded to a plate like structure in terms of the two principal strains:

V=d31Y E h(S1+S2)

ε33T (1−µ)

=K p' (S1+S2) (10)

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Hence, for the 2D case, the voltage measured across the PZT sensor is proportional to the

sum of the strains in the PZT patch (and hence on the surface of the structure) along the two

principal directions.

It may be noted that it is implicit that strains developed in the PZT are same as that on the

surface of the host structure. However, as a PZT patch is bonded using a finitely thick

adhesive layer, there is a differential strain between the patch and the host structure, as, given

by (Bhalla and Soh, 2004):

Γ2=(Gs

Y p ts t p

+3Gs w p

Y b wbt b t p

) (11)

where Gs is the shear modulus of the bond layer, Yp the Young’s modulus of the PZT patch,

Yb the Young’s modulus of the structure (beam), wp the width of the patch, wb the width of

the beam, tp the thickness of the patch and tb the depth of the beam.

If, however, the stiffness of the adhesive layer is sufficiently high and its thickness is small,

such that Ƭ>30 cm-1, the shear lag effect can be neglected.

18 | P a g e

CHAPTER 3 LITERATURE REVIEW

3.1 Introduction:

Damage detection in various structures is a very live subject and has very wide ranging

application. Extensive research has been taken in this area, thereby contributing rich source

of literature. Described below is some of the key literature which is relevant to the present

thesis work.

3.2 Damage Detection Methods for Beams:

3.2.1 Change in Curvature Mode Shape Method:

According to Pandey and Biswas (1990), any crack or localized damage in a structure

reduces the stiffness and increases the damping in the structure. Reduction in stiffness is

associated with decreases in the natural frequencies and modification of the modes of

vibration of the structure. Many researchers have used one or more of the above

characteristics to detect and locate a crack. Most of the emphasis has been on using the

decrease in frequency or the increase in damping to detect the crack. Very little work has

been done on using the changes in the mode shapes to detect the crack.

Yuen in his paper showed for a cantilever beam that there is a systematic change in the first

mode shape with respect to the damage location. He used finite element analysis to obtain the

natural frequencies and the mode shapes of the damaged structure. To detect damage location

curvature mode shape is used. The difference in the curvature mode shapes between the intact

and the damaged case is utilized to detect the location of the crack. The changes in the

curvature mode shapes are shown to be localized in the region of damage compared to the

changes in the displacement mode shapes.

According to Pandey et al., (1994) curvature mode shapes are related to the flexural stiffness

of beam cross-sections. Curvature at a point is given by

v”= M/(EI) ( 12)

Where v” is the curvature at a section, M is the bending moment at a section, E is the

modulus of elasticity and I is the second moment of the cross-sectional area.

If a crack or other damage is introduced in a structure, it reduces the stiffness (EI) of the

structure at the cracked section or in the damaged region, which increases the magnitude of

curvature at that section of the structure. The changes in the curvature are local in nature and

hence can be used to detect and locate a crack or damage in the structure. The change in

19 | P a g e

curvature increases with reduction in the value of (EI), and therefore, the amount of damage

can be obtained from the magnitude of change in curvature.

Figure 3-3. Absolute difference between the curvature mode shapes of cantilever beam for 1st mode shape (pandey et al., 1991)

3.2.1.1 Severity of Damage:

According to Rama shanker et al., (2011) Denoting the amplitude of the motion by the

generalized coordinate Z (t), the displacement at any point in a structure can be expressed as:

y ( x , t )=Ø (x )∗Z (t )

Where Ø(x) is the mode shape function. Thus, the harmonic variation of the generalized

coordinate at a given time in free vibration can be expressed as:

y ( x , t )=Ø (x )∗Zo sin (ωt )(12)

This Eq. (12) expresses the assumption that the shape of the vibrating beam does not change

with time, only the amplitude of motion varies, as it varies harmonically in a free-vibration

condition. At the point of maximum displacement:

y ( x , t )=Z0sin (ωt )

The strain energy of this flexural system is given by:

Umax=0.5∫0

L

EI ( x ) ¿¿(x))2dx

20 | P a g e

Where L is the length of the beam, E the Young’s modulus of the material of the structure,

and I is the moment of inertia. Thus, substituting the mode shape function and letting the

displacement amplitude take its maximum value leads to:

Umax=0.5∗Z02∫

0

L

EI ( x ) ¿¿(x))2dx

Where Ø ' ' ( x )denotes the second derivative of a particular mode shape, that is the curvature

mode shape. In general, the total energy of the system is distributed among various mode

shapes. If the structure is damaged, it is assumed that there is no significant change in the

distribution of the energy in various modes if the excitation force and the boundary

conditions remain unchanged. If there occurs any damage/crack in structure in any element,

the value of EI will reduce more at that element compared with other parts. This change

reflects the severity of damage. In the present approach, only average value of ‘‘EI’’ of whole

structure has been determined. If any element of structure is damaged, the tendency of that

element is to undergo greater displacement compared with the undamaged case. However,

because of the inertia and the adhesive forces, the neighbouring elements, which are less

damaged, oppose it and undergo somewhat greater displacement to balance the equilibrium.

Hence, the damage affects the entire mode shape of the structure. As the severity of the

damage increases, the strain at the damaged section increases further. By conservation of

energy for any mode:

(U damaged)undamaged=(U ¿¿max)damaged ¿

From above equations we can conclude that

EI (x)damaged

EI (x )undamaged

=η=(Z0

2 ) undamaged(∫0

L

Ø ' ' ( x ) dx)undamaged

(Z02) damaged(∫

0

L

Ø ' ' ( x ) dx)damaged

Where η represents the ratio of the current stiffness of structure to the undamaged stiffness.

In PZT sensors and the measurement approach described herein, is obtained directly from

measurement. Clearly, Ø’’ (x) this circumvents the necessity of numerical differentiation,

which would otherwise have to be performed if any other sensor types, say accelerometers

were used.

21 | P a g e

We know that Integration of [Ø’’ (x)] 2 with respect to x gives the area of square of the mode

shape. Hence, the damage severity can be determined in terms of the original stiffness after

drawing the experimental mode shape of structure and the computing area of the square of

the mode shape and the maximum relative amplitude Z0 of the mode shapes.

3.2.2 Change in Flexibility Method:

According to Pandey and Biswas (1991), this flexibility method does not required the

development of analytical model of the structure to be investigated. All predictions to the

state of damage should be made from experimental data collected from the structure. Also the

measurement of only few lower natural frequencies are sufficient to detect the damage of

structure. Unless the stiffness method, flexibility method will converges rapidly with

increasing frequency. The presence of a crack or a localized damage in a structure reduces the

stiffness of the Structure. Since flexibility is the inverse of stiffness, reduction in stiffness will

produce an increase in the flexibility of the structure. With mode shapes normalized to unit

mass,

ØT MØ=1

The flexibility matrix, F, can be obtained from the modal data as

K=MØΩ ØT M=M (∑i=1

n

ωi2 Ø i Ø i

T )M(13)

F=Ø Ω−1 ØT=∑i=1

n1

ω2 Ø i Ø iT(14)

Where K is the stiffness matrix, M is the mass matrix, Ø = [Ø1, Ø2, Ø3, Ø4, Ø5, Ø6 ............Øn],

is the mode shape matrix, Øi is the ith mode shape, Ω=diag (ωi2) is the modal stiffness matrix,

ωi is the ith modal frequency, n is the number of degrees of freedom, and F is the flexibility

matrix. The mode shapes and natural frequencies are obtained from the analytical or

experimental data.

From the Eq. (14) one can see that the modal contribution to the flexibility matrix decreases

as the frequency increase, i.e the flexibility matrix will converge rapidly with increasing

values of frequency, when compare to stiffness matrix. Therefore from few lower modes a

good estimate of flexibility matrix will obtained.

22 | P a g e

From the pre and post –damage flexibility matrices, a measure of the flexibility change

caused by the damage can be obtained from the difference of the respective matrices as

[ Δ F ]=[F ]−[F ]¿

Where [ΔF] represents the change in flexibility matrix. For each column of this matrix δ j=

max ׀δ ij׀ , i =1…n. The column of the flexibility matrix corresponding to the largest change

is indicative of the degree of freedom where the damage is located.

3.2.3 Change in Stiffness Method:

Zimmerman and Kaouk (1994), have developed a damage detection method based on

changes in the stiffness matrix that is derived from measured modal data. The eigenvalue

problem of an undamaged, undamped structure is

¿ (15)

The eigenvalue problem of the damaged structure is formulated by first replacing the pre-

damaged eigenvectors and eigenvalues with a set of post-damaged modal parameters and

second, subtracting the perturbations in the mass and stiffness matrices caused by damage

from the original matrices. Letting ΔMd and ΔKd represents the perturbations to the original

mass and stiffness matrices, the Eigen value Eq. (15) becomes to Eq. (16)

λ i¿[ M – Δ M d]+[ K – Δ K d]¿ {Ψ i }¿={0}(16)

Two forms of a damage vector, {Di} for the ith mode are then obtained by separating the

terms containing the original matrices from those containing the perturbation matrices.

Hence,

{Di } = (λ i¿ [M ] +[K ]){Ψ i }¿ = (λ i

¿ [ΔM ] +[ΔK ]){Ψ i }¿

To simplify the investigation, damage is considered to alter only the stiffness of the structure

of the structure (i.e. ΔMd = [0]). Therefore, the damage vector reduces to

{Di}= [ΔKd]{Ψ i }¿

In a similar manner as the modal-based flexibility matrices previously defined, the stiffness

matrices, before and after damage, can be approximated from incomplete mass-normalized

modal data as

[ K ]=∑ωi2 Ø i Ø i

T

23 | P a g e

[ K ]¿=∑ ωi¿ 2 Ø ¿

i Ø i¿T

3.2.4 Improved Damage Detection Method Based On Modal Information

According to Kim (2001), for a linear, undamaged, skeletal structure with elements and N

nodes, the modal Stiffness of the arbitrary structure is given by

K i=Ø iT∗C∗Ø i

Where Øi is the ith modal vector and C is the system stiffness matrix. The contribution of j th

member to ith modal stiffness, is given by

K ij=Ø iT∗C j∗Ø i

K i¿=Ø i

¿T C ¿ Ø i¿

Where is Cj the contribution of Jth member to the system stiffness matrix. Then, the fraction of

modal energy (i.e., the undamaged modal sensitivity) of the i th mode and the jth member is

defined as

F ij=K ij

K i

Let the corresponding modal parameters in equations associated with a subsequently

damaged structure be characterized by asterisks. Then for the damaged structure, the

damaged sensitivity of the ith mode and jth member is defined as

F ij¿=

K ij¿

K i¿

Where Kij* and Ki

* are given by

K ij¿=Ø i

¿T C j¿ Ø i

¿

Where

C j=E j C jo∧C j¿=E j

¿C jo

Where the scalars Ej and Ej* are parameters representing material stiffness properties of

undamaged and damaged jth members, respectively. The matrix Cjo involves only geometric

quantities (and possibly terms containing Poisson's ratio) and it can represent beam or plate

element. On making the assumption that modal sensitivities for the i th mode and jth location is

the same for both damaged and undamaged structure (i.e Fij*=Fij), then the above equations

can be rearranged to get

F ij¿

F ij

=K ij

¿ K i

K i¿ K ij

=1

24 | P a g e

Thus we can define the damage index as follows

β j=E j

E j¿=

∑i=1

nm

γij¿ K i

∑i=1

nm

γij K i¿

Where

γ i , j=Ø iT C Ø i∧γi , j

¿ =Ø iT C Ø i

¿

3.2.5 Damage Detection Using Strain Energy Theory:

The strain energy of a Bernoulli’s Euler beam is given by

U=0.5∫0

l

EI ( d2 wd x2 )dx

Where EI is the Flexural rigidity of the beam. For a particular mode shape, Ψ i (x), the energy

associated with that mode shape is

U i=0.5∫0

l

EI ( d2Ψ i

d x2 )2

dx

If the beam is subdivided into Nd divisions as shown in below Figure, then the energy

associated with each sub-region j due to the ith mode is given by

U ij=0.5∫a j

a j+1

EI j( d2Ψ i

d x2 )2

dx

The fractional energy is therefore

F ij=U ij

U i

∧∑j=1

Nd

F ij=1

Similar quantities can be defined for a damaged structure and are given as

U i¿=0.5∫

0

l

EI¿ ( d2Ψ ¿i

d x2 )2

dx

U ij¿=0.5∫

a j

a j+1

EI j¿ ( d2Ψ i

¿

d x2 )2

dx

F ij¿=

U ij¿

U i¿ ∧∑

j=1

Nd

Fij=∑j=1

N d

F ij¿=¿1¿

25 | P a g e

By choosing the sub-regions to be relatively small, the Flexural rigidity for the jth Sub-

region, EIj is roughly constant and Fij* becomes

F ij¿=(EI )j

¿∫a j

a j+1 ( d2Ψ i¿

d x2 )2

dxU i

¿

If one assumes that the damage is primarily located at a single sub-region then the fractional

energy will remain relatively constant in undamaged sub-regions and F*ij= Fij. For a single

damaged location at sub-region j=k one can finds

(EI )k ∫a j

a j+1 ( d2Ψ i¿

d x2 )2

dxU i

¿ =(EI )k¿ ∫

ak

ak+1 ( d2 Ψ i¿

d x2 )2

dxU i

¿ (17)

By assuming that EI is essentially constant over the length of the beam for both the

undamaged and damaged modes, Eq. (17) can be rearranged to give an indication of the

change in the flexural rigidity of the sub-region:

( EI )kEIk

¿ ={∫

a j

aj+1

( d2Ψ i¿

d x2 )2

dx|∫0

l

( d2Ψ i¿

d x2 )2

dx }}{∫

a j

a j+1 ( d2Ψ i

d x2 )2

dx|∫0

l

( d2Ψ i

d x2 )2

dx }=

f ik¿

f ik

In order to use all the measured modes, m, in the calculation, the damage index for sub-

region k is defined to be

βk=∑i=1

m

f ik¿

∑i=1

m

f ik

3.2.6 Damage Detection Using Higher Order Derivatives:

The concept of using fourth derivatives of mode shapes in damage detection of beams was

first used by Whalen (2006), who used the Euler-Bernoulli beam model. They assumed that

the shearing deformations, rotational inertia and axial effects to be negligible. They assumed

that the stiffness EI of the beam can vary with position x, the governing equation of motion

for undamped free vibration of the beam can be expressed as Eq. (18):

26 | P a g e

d2

dx2 [EI ( x ) d2 ydx2 ]+ ρA ( x ) d2 y

dt 2 =0 (18)

Where ρA(x) is the linear mass density of the structure. With standard separation of variables

argument and solving for the fourth derivative we got:

Ψ n(4)=

ωn2 ρA ( x )EI (x )

Ψ n (x )−2EI ( 1) (x )EI ( x )

Ψ n(3 ) ( x )− EI ( 2) ( x )

EI ( x )Ψ n

(2)(x )

Where ωn(x) is the natural frequency of vibration for nth mode shape. He showed that if

damage causes a change in EI(x), the terms involving derivatives of EI(x) can have large

values due to the localized nature of this change. Therefore, if there is damage in a structure,

large discontinuities in the magnitude of the fourth derivative of mode shape will increase

sharply at the location of damage. Hence, it is suitable for locating damage in beams.

27 | P a g e

CHAPTER 4 EXPERIMENTAL WORK DONE ON STEEL BEAM

4.1 Introduction:

This study is aimed for the occurrence of damage, location of damage and severity of

damage. The specimen prepared in such a way that (using nut-bolt system) repeatability of

the damaged can occur in a beam, and both can be assess by system (i.e. damaged as well as

normal) using Global Vibration Technique. Result of single hit approach method and

multiple hit approach method are shown here. Other method to create mode shape without

force quantification is also given in this chapter.

4.2 Dimensions and Properties Steel Beam:

Structure consists of three steel beam pieces each of 0.8 m. Steel beam is welded to steel

plate. The steel plates are connected to each other by nut-bolt system. Compete Steel beam

structure is shown below

Figure 4-4. Complete steel beam with dimensions (Abhishek, 2013)

Table 4-1. Physical properties of beam

Properties Values

Section area 1900mm2

Depth of section 150 mm

Width of flange 80 mm

Thickness of flange 7.6 mm

Thickness of web 4.8 mm

Plastic modulus of section 110500mm3

28 | P a g e

PZT Patches:

Piezoceramics are the most popular amongst the smart materials. Typical PZT of size

10*10*0.3 mm is used in experiment is shown in below figure. Piezoceramics materials have

the advantages of being lightweight, low-cost, and easy-to-implement and offer the sensing

and actuation capabilities that can be utilized for passive and active vibration control.

Figure 4-5. PZT patch (Abhishek, 2013)

4.3 Experimental Setup:

The steel beam of length of 2.4 m is supported on four wheels. The wheels are welded on the

plates of size 150 mm X 150 mm and then plate is welded with the beam at both end of the

beam specimen and shown in below figure

Figure 4-6. Showing the location of impact and PZT sensor (Abhishek, 2013)

After the fabrication of steel beam, PZT patch is attached to beam with adhesive araldite as

Two PZT sensors are attracted to the beam on the same vertical line passing through the plate

of symmetry (vertical plane in along the length of beam).The PZT patch is soldered with wire

and connector is used to facilitate ease in working with QDA 1008 system.

29 | P a g e

4.4 Instruments Used:

4.4.1 Data acquisition systems:

Oscilloscope (TDS2004B) manufactured by Techtronix inc, QDA1008 manufactured by

Quazar Tech 2013, were used to take response of structure under vibration. Figures are

shown below respectively

Figure 4-7. Images of Oscilloscope and QDA1008

4.4.2 Force application systems:

A force sensor (1051V4, Dytran), A SSD hammer which is prepared in lab using four PZT

patches were used to apply force and vibrate the structure by hitting.

Figure 4-8. Dytran sensor and force hammer made by steel and wood.

4.4.3 Sensors:

In steel beam surface bonded PZT sensors were applied and in concrete beam lab made

embedded concrete vibration sensor were used which are shown below. In few experiments

accelerometers (PCB Piezotronics inc.) as shown in below Fig were used.

Figure 4-9. Surface bonded PZT, Embedded Sensors and Accelerometers (Abhishek, 2013)

30 | P a g e

4.5 Mode Shape Based on Column of FRF (Single impact exitation) Method:

4.5.1 Frequency Response Function:

Frequency response is the quantitative measure of the output spectrum of a system or device

in response to a stimulus, and is used to characterize the dynamics of the system. It is a

measure of magnitude and phase of the output as a function of frequency, in comparison to

the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a

linear system will respond at that same frequency with a certain magnitude and a certain

phase angle relative to the input.

From Peter Availtable lessons we can conclude that, any Row or any Column in a matrix

FRF matrix of a beam is sufficient to draw the mode shape of a beam.

If we draw mode shape by taking any Column, it is called as Mode Shape Based on Column

of FRF (Single impact exitation), and if we consider any Row in to consideration it is caled

Multiple Impact Exitation Method.

4.5.2 Procedure To Draw Mode Shape By Single Impact Exitation Method:

1. Connect all sensors of the beam to QDA1008, which should be connected to computer

using data cable

2. Take the responses of beam from different points when hit at any selected point of

beam

3. The resposes which we had taken are in the Time domain format, which we need to

convert it in to frequency domain using FFT(Fast Fourier transform)

4. Corresponding to the natural frequency of beam we need to take Amplitudes of the

beam.

5. Plot those corresponding Amplitudes vs sensor position which we will get mode

shape of beam.

Figure 4-10. Showing the plates connected by Bolts (Abhishek, 2011)

31 | P a g e

http://en.wikipedia.org/wiki/Frequency_spectrum

Here, at support points, voltage value is assumed zero here. Rest seven locations are PZT

sensor position. Position 4 is impact position. For all seven sensors responses are

collected and they are converted in to frequency domain using FFT. From that we had

drawn the Mode shape of beam.

The above procedure is repeated for Undamaged and Damaged case of steel beam, where

damage were induced in connection between plate 3 and plate 4 by loosening the bottom

2 bolts with gradually decreasing torque. We created 12 damage conditions (single

damage location), and 5 multiple damage conditions. The results are shown below.

4.5.3 For Single Damage Criteria:

For Undamaged case all the bolts are tightened with Torque of 30Nm

B34 = Bottom Bolts of connection between plates of 3 and 4.

Table 4-2. Relation of single damage with Torque

S.N

O.

Condition of Beam Torque Released on B34

1 Undamaged state 0

2 Damage -1 5

3 Damage -2 10

4 Damage -3 15

5 Damage -4 20

6 Damage -5 25

7 Damage -6 30

8 Damage -7 35

9 Damage -8 40

10 Damage -9 45

11 Damage -10 50

12 Damage -11 55

13 Damage -12 60

32 | P a g e

4.5.4 For Multiple Damage Criteria:

In this case 5 multiple Damage locations are created. Here the bottom bolts of plates of

connection between 3 and 4(B34) are completely loosened (i.e Total Torque released is

60Nm), and another progressive Damage were created on Bottom bolts of connection

between Plates 1 and 2. (B12)

Table 4-3. Relation of Multiple damage with Torque

S.NO. Condition of

Beam

Torque

Released on B34

Torque

Released on B12

1 Undamaged state 0 0

2 Damage -1 60 5

3 Damage -2 60 10

4 Damage -3 60 30

5 Damage -4 60 35

6 Damage -5 60 60

4.6 Experiments on Undamaged Beam:

Experiment was done on undamaged beam i.e. when bolts between sections of beam were

fully tight with 30Nm on each bolt. Impact was done on midpoint and readings were taken on

all seven points. Figure 4.8, figure 4.9, figure 4.10, shows the FFT of 1st sensor for Absolute

part, imaginary part and real part of Undamaged beam from MATLAB, corresponding to

Natural Frequency of 35.6, 35.2, 35.6 respectively

Figure 4-11. FFT of 1st sensor for Absolute Part (Amplitude(x-axis) vs Frequency(Y-axis))

33 | P a g e

Figure 4-12. FFT of 1st sensor for Imaginary Part (Amplitude(x-axis) vs Frequency(Y-axis))

Figure 4-13. FFT of 1st sensor for Real Part (Amplitude(x-axis) vs Frequency(Y-axis))

34 | P a g e

4.7 Mode Shapes of Undamaged Beam:

Figure 4-14. Mode Shape obtained from Absolute part of FRF

Figure 4-15. Mode Shape obtained from Imaginart Part of FRF

35 | P a g e

Figure 4-16. Mode Shape obtained from Real Part of FRF

4.8 Mode Shapes for Different Damage Conditions:


Figure 4-17. Mode shapes of Undamaged and Damaged states of beam obtained from Imaginary Part of FRF

36 | P a g e

Table 4-4. Variation of Natural frequency, Damage location, and severity For Imaginary Part of FRF

S.NO. Damage

States

Natural

Frequency(Ω)

Damage Location

Observed

Severity Of

Damage

1 Undamaged

state

35.2 - -

2 Damage-1 34.4 6 0.9834

3 Damage-2 34.4 5 0.9296

4 Damage-3 34 6 0.9345

5 Damage-4 34 6 0.8547

6 Damage-5 32.4 2 0.8023

7 Damage-6 30.4 6 0.7953

8 Damage-7 29.6 6 0.7665

9 Damage-8 27.6 2 0.7719

10 Damage-9 22 2 0.7554

11 Damage-10 21.6 6 0.7012

12 Damage-11 21.6 6 0.6749

13 Damage-12 20 6 0.7192

37 | P a g e

Figure 4-18. Mode shapes of Undamaged and Damaged states of beam obtained from Absolute Part of FRF

Table 4-5. Variation of Natural frequency, Damage location, and severity For Absolute Part of FRF

S.NO. Damage

states

Natural

frequency(ω)

Damage Location

observed

Severity of

Damage

1 Undamaged

state

35.6 - -

2 Damage-1 34.6 5 0.9582

3 Damage-2 34 5 0.9470

4 Damage-3 33.6 6 0.8843

5 Damage-4 33.6 2 0.9186

38 | P a g e

6 Damage-5 32.8 2 0.8837

7 Damage-6 30.4 2 0.8935

8 Damage-7 28.8 2 0.8913

9 Damage-8 28.4 2 0.8825

10 Damage-9 21.6 2 0.8430

11 Damage-10 22.4 2 0.8159

12 Damage-11 22.4 2 0.6799

13 Damage-12 19.6 6 0.7148

Figure 4-19. Mode shapes of Undamaged and Damaged states of beam obtained from Real Part of FRF

39 | P a g e

Table 4-6. Variation of Natural frequency, Damage location, and severity For Real Part of FRF

S.NO. Condition of

beam

Natural

frequency(ω)

Damage Location

observed

Severity of

Damage

1 Undamaged state 35.6 - -

2 Damage-1 34.5 3 0.9879

3 Damage-2 34.8 3 0.9706

4 Damage-3 32.4 4 0.8915

5 Damage-4 32.4 2 0.8772

6 Damage-5 32.8 3 0.8468

7 Damage-6 30.8 3 0.7937

8 Damage-7 29.2 6 0.7867

9 Damage-8 28 6 0.7268

10 Damage-9 21.6 3 0.7315

11 Damage-10 22.8 2 0.7231

12 Damage-11 22.8 2 0.4600

13 Damage-12 18.4 4 0.8123

40 | P a g e


Figure 4-20. Mode shapes of Undamaged and Damaged states of beam obtained from Imaginary Part of FRF


S.NO. Damage states Natural

frequency(ω)

Damage Location

observed

Severity of

Damage

1 Undamaged

state

35.2 - -

2 Damage-1 22 2 0.8592

3 Damage-2 21.2 2 0.9408

4 Damage-3 21.2 6 0.7998

5 Damage-4 20.8 6 0.7344

6 Damage-5 12.4 7 0.7211

41 | P a g e

Figure 4-21. Mode shapes of Undamaged and Damaged states of beam obtained from Absolute Part of FRF


S.NO. Condition of

beam

Natural

frequency(ω)

Damage Location

observed

Severity of

Damage

1 Undamaged

state

35.6 - -

2 Damage-1 22.8 2 0.8928

3 Damage-2 22.8 2 0.7453

4 Damage-3 21.2 3 0.8211

5 Damage-4 21.2 6 0.7912

6 Damage-5 12 2 0.7523

42 | P a g e

Figure 4-22. Mode shapes of Undamaged and Damaged states of beam for Real Part of FRF


S.NO. Condition of

beam

Natural

frequency(ω)

Damage Location

observed

Severity of

Damage

1 Undamaged

state

35.6 - -

2 Damage-1 22.4 5 0.8590

3 Damage-2 22 7 0.8122

4 Damage-3 22 4 0.8335

5 Damage-4 21.2 5 0.7655

6 Damage-5 12.4 6 0.7264

43 | P a g e

4.9 Damage Index Plots:


1 2 3 4 5 6 7 80.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

Damage Index vs Beam Elements

damage 1damage 2damage 3damage 4damage 5damage 6damage 7damage 8damage 9damage 10damage 11damage 12

Figure 4-23. Plot for Imaginary Part of FRF

1 2 3 4 5 6 7 80.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23



Figure 4-24. Plot for Absolute Part of FRF

44 | P a g e

1 2 3 4 5 6 7 80.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31



Figure 4-25. Plot for Real Part of FRF


1 2 3 4 5 6 7 80.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2


damage 1damage 2damage 3damage 4damage 5


45 | P a g e

1 2 3 4 5 6 7 80.1

0.12

0.14

0.16

0.18

0.2

0.22




1 2 3 4 5 6 7 80.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5




46 | P a g e

4.10 Variation of Torque with Damage Index for Single Damage Criteria:

0 10 20 30 40 50 60 700.1

0.15

0.2

0.25

0.3

Torque Relaeased(Nm)

Dam

age

Inde

x

Figure 4-29. Variation of Damage Index with Torque Released for Imaginary Part of FRF

4.11 Damage Index Using 3rd Order Derivative:

Here we investigated whether 3rd order derivative is good indicator for detecting damage

location or not? Central difference method is used to find out the points from 2 nd order (i.e.

curvature mode shape).

According to central difference method,

( d3 yd3 x

)i

=( d2 yd2 x

)i+1

−( d2 yd2 x

)i−1

2h

Where h= difference between i and i+1 or i and i-1 points.

For all points of undamaged and damaged cases of steel and concrete beams we should do

central difference. Difference between damaged and undamaged cases are calculated at all

nodal points. From this we will calculate the damage index of elements, which is the average

of surrounding nodal points.

47 | P a g e


1 2 3 4 5 6 7 80.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Damage Index vs Beam element



1 2 3 4 5 6 7 80.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7




48 | P a g e

1 2 3 4 5 6 7 80.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1




Table 4-10. Damage locations using 3rd order derivative for single damage location

S.NO. Damage

states

Damage locations

Imaginary Absolute Real

1 Damage-1 3 3 7

2 Damage-2 4 4 6

3 Damage-3 4 4 7

4 Damage-4 1 1 7

5 Damage-5 1 2 3

6 Damage-6 1 2 8

7 Damage-7 8 4 1

8 Damage-8 5 2 5

9 Damage-9 2 6 5

49 | P a g e

10 Damage-10 8 3 1

11 Damage-11 2 3 6

12 Damage-12 2 1 5


1 2 3 4 5 6 7 80.3

0.35

0.4

0.45

0.5

0.55

0.6



Figure 4-33. Plot for imaginary Part of FRF

1 2 3 4 5 6 7 80.3

0.32

0.34

0.36

0.38

0.4

0.42




50 | P a g e

1 2 3 4 5 6 7 80.3

0.4

0.5

0.6

0.7

0.8

0.9




Table 4-11. Damage locations using 3rd order derivative for multiple damage criteria

S.NO. Damage

states

Damage locations


1 Damage-1 1 1 8

2 Damage-2 1 2 3

3 Damage-3 8 8 5

4 Damage-4 8 5 3

5 Damage-5 5 5 6

51 | P a g e

CHAPTER 5 EXPERIMENTAL WORK DONE ON REINFORCED CONCRETE BEAM

5.1 Multiple Impact Excitation Method Procedure:

In Multiple Impact Excitation Method only one sensor was used and hitting was done at

different selected locations of structure

5.1.1 Procedure to Draw Mode Shape using Multiple Impact Excitation Method:

1. Connect the sensor of the beam to QDA1008, which should be connected to

computer using data cable.

2. Take the responses of beam from that sensor for different hitting locations which are

selected on the beam.

3. The resposes which we had taken are in the Time domain format, which we need to

convert it in to frequency domain using FFT(Fast Fourier transform)

4. Response of force need to convert in to frequency domain using FFT.

5. Frequency Responce function(each element FFT of responce/ each element of FFT of

force) should caluclate.

6. Corresponding to the natural frequency of beam we need to take Amplitudes of the

beam.

7. Plot those corresponding Amplitudes vs sensor position which we will get mode

shape of beam.

Experiments were carried on Reinforced Concrete Beam using Multiple Impact Exitation

Method

Table 5-12. Properties of Concrete Beam

S.NO. Properties Description

1 Length 4.0m

2 Cross- section 160x200mm

3 Sensors type used Endedded vibro sensors

4 Number of sensors used 19

5 Location of sensors Top fiber of beam

6 Grade of concrete used M40

7 Supplements used 30% fly ash

52 | P a g e

Table 5-13. Description About Damage

S.NO. Condition of Beam Description

1 Damage -1 Chipping of concrete between 7th and

8th sensors(4th element of Beam) from bottom

to mid-section of RCC beam

2 Damage -2 Damage -1 + Cutting of bottom one

reinforcement bar out of two bars

3 Damage -3 Damage -2 + Cutting of another

bottom reinforcement bar

4 Retrofitting-1 Welding of two Reinforcement bar

Figure 5-36. Undamaged Concrete beam on which experiments are conducted

53 | P a g e

Figure 5-37. Damage1 condition of concrete beam

Figure 5-38. Damage2 and damage3 conditions of concrete beam

54 | P a g e

Figure 5-39. Retofitting-1 of concrete beam

5.2 Mode Shapes of Undamaged And Damaged Beam:

0 2 4 6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

Undamaged damage 1damage 2damage 3Retroffiting 1

Length of Beam (m)

Am

plit

ude

Figure 5-40. Mode Shape obtained from absolute Part of FRF


S.NO. Condition of

beam

Natural

frequency(ω)

Damage

Location

observed

Severity of

Damage

1 Undamaged

state

22 - -

2 Damage-1 18 4 1.0298

55 | P a g e

3 Damage-2 16.8 6 0.8305

4 Damage-3 10.4 4 0.6917

5 Retrofitting-1 18 - 0.9211

0 2 4 6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

undamageddamage 1damage 2damage 3Retrofitting 1

Length of Beam (m)

Am

plit

ud

e

Figure 5-41. Mode Shape obtained from Imaginary Part of FRF


S.NO. Condition of

beam

Natural

frequency(ω)

Damage

Location

observed

Severity of

Damage

1 Undamaged

state

22 - -

2 Damage-1 18.4 4 0.8626

3 Damage-2 16.8 3 0.7547

4 Damage-3 10.4 4 0.8450

56 | P a g e

5 Retrofitting-1 18 - 0.8669

0 2 4 6 8 10 12 14 16 18 20 220

0.2

0.4

0.6

0.8

1

1.2

1.4

undamaged damage 1damage 2damage 3Retroffiting 1

Length of Beam (m)

Am

plit

ude

Figure 5-42. Mode Shape obtained from Real Part of FRF


S.NO. Damage

states

Natural

frequency(ω)

Damage

Location

observed

Severity of

Damage

1 Undamaged

state

22 - -

2 Damage-1 18 3 0.6093

3 Damage-2 16.8 4 1.0127

4 Damage-3 11.2 2 0.9100

57 | P a g e

5 Retrofitting-1 18.4 - 0.8429

5.3 Damage Index Plots Using Curvature Mode Shape Method:

1 2 3 4 5 6 7 8 9 100.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Damage Index vs element No.

damage 1damage 2damage 3

Figure 5-43. Plot For Absolute Part of FRF

58 | P a g e

1 2 3 4 5 6 7 8 9 100.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22




1 2 3 4 5 6 7 8 9 100.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5



Figure 5-45. Plot For Real Part of FRF

5.4 Damage Index Using 3rd Order Derivative:

59 | P a g e

1 2 3 4 5 6 7 8 9 100.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29



Figure 5-46. Plot For Imaginary Part of FRF

1 2 3 4 5 6 7 8 9 100.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29



Figure 5-47. Plot for Asolute Part of FRF

60 | P a g e

1 2 3 4 5 6 7 8 9 100.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5




Table 5-17. Damage locations using 3rd order derivative for concrete beam

S.NO. Damage

states

Damage locations


1 Damage-1 5 6 3

2 Damage-2 1 4 4

3 Damage-3 4 7 1

5.5 Roving of Accelerometer Approach:

In Real life it is very difficult to measure Force (as we had done in Multiple Impact Exitation

Method), i.e for example to measure the force in the bridge where vehicles are moving, there

the New approach is introduced i.e roving of Accelerometer or Reference Accelerometer

approach.

61 | P a g e

5.5.1 Procedure To Find Mode Shape:

1. In this approach we need teo Accelerometers, One Accelerometer called as Reference

Accelerometer, and other as Moving Accelerometer.

2. Reference accelerometer was fixed at some position of Beam, and other moving

accelerometer should be shifted to next nodal point after every reading.

3. Both accelerometer was connected to QDA1008 through PCB electronics, and impact,

was done on a fixed point, by using a hammer.

4. For getting mode shape FFT of moving accelerometer at corresponding Natural

frequency is normalized with FFT of readings of reference accelerometer which is

same as FRF process.

62 | P a g e

CHAPTER 6 CONCLUSIONS AND FUTURE WORK

6.1 Introduction:

This chapter deals with the conclusions and the scope of future work. The whole experiments

are conducted on steel and concrete beams of properties shown in previous chapters.

Curvature mode shape and Third order derivative methods are used for Damage detection on

steel and concrete beams. Conclusions of work done and scope of future work are shown

below.

6.2 Research Conclusions and Contribution:

Major research contribution and conclusions can be summarized as follows.

1. Curvature mode shape (2nd order derivative) based response of surface bonded PZT

patches can successfully identify the location of damage in one dimensional structure

for single and multiple damage conditions which was proven by experiments.

2. Third order derivative was unable to identify the location of damage in one

dimensional structure for single and multiple damage criteria.

3. For single damage criteria, Imaginary part of FRF (Frequency Response Function)

gives better results than real part and absolute part of FRF for damage detection.

4. For single damage conditions, Imaginary part of FRF and Absolute part of FRF are

able to detect damage location when compared to Real part of FRF for higher

intensity damages.

5. Curvature mode shape method is unable to detect exact damage locations for multiple

damage criteria.

6. Third order derivative also unable to detect exact damage locations for multiple

damage criteria.

7. Severity of damage [in terms of Ratio of (EI)damaged to (EI)damaged ] was calculated for all

damages

8. Successful Development of automated software interphase for instantaneous

Identification & location of damage.

9. Ratio of (EI)damaged to (EI)undamaged was increased after retrofitting of damage, which

should be true and the same was proven from experiments also.

63 | P a g e

6.3 Limitations:

1. The experimentation on the test models took very long time and we need to repeat the

experiments 2 to 3 times for accurate results.

2. Due to complex system of wires, we should be very careful while doing the experiments

6.4 Scope for Future Work:

1. Need to extend these damage location methods for 2D structures like plates.

2. Need to Develop automated software interphase for instantaneous Identification & location

of damage in 2D structures also.

3. To find out mode shape using mass normalised mode shape.

64 | P a g e

REFERENCES

1. Bhalla, S. (2001), “Smart System Based Automated Health Monitoring of Structure”,

Master Thesis, Nanyang Technological University, Singapore.

2. Bhalla, S. (2004), “A Mechanical Impedance Approach for Structural Identification,

Health Monitoring and Non-Destructive Evaluation using Piezo-Impedance

Transducers”, PhD Thesis, Nanyang Technological University, Singapore.

3. Bhalla, S., Vittal, A. P. R and Veljkovic, M. (2012), “Piezo-Impedance Transducers

for Residual Fatigue Life Assessment of Bolted Steel Joints”, Structural Health

Monitoring, An International Journal, Vol. 11, No 6 (Nov), pp. 733-750.

4. Pandey, A. K. and Biswas, M. (1994), “Damage Detection in Structures Using

Changes in Flexibility”, Journal of Sound and Vibration, Vol. 169, No. 1, pp. 3 – 17.

5. Pandey, A. K., Biswas, M. and Samman, M. M. (1991), “Damage Detection from

Changes in Curvature Mode Shapes”, Journal of Sound and Vibration, Vol. 145, No.

2, pp. 321 – 332.

6. Shanker, R., Bhalla, S. and Gupta, A. (2010), “Integration of Electro-mechanical

Impedance and Global Dynamic Technique for Improved Structural health

Monitoring”, Journal of Intelligent Material Systems and Structures, Vol. 21, No. 2

(Feb), pp.285-295.

7. Mohamed Abdel-Basset Abdo (2012), “Damage detection in plate-like structures

using High-Order mode shape”, International Journal of Civil and Structural

Engineering Volume 2, No 3.

8. Moatasem M. Fayyadh and H. Abdul Razak (2011) “Detection of damage location

using mode shape deviation: Numerical Study”, International Journal of the Physical

Sciences Vol. 6(24), pp. 5688-5698, 16 October.

9. Mark J. Schulz, Ahmad S. Naser, P. Frank Pai and Jaycee Chung (1998), “Locating

Structural Damage Using Frequency Response Reference Functions”, Journal of

Intelligent Material Systems and Structures.

10. Esfandiari and Amirkabir (2011), “Structural Damage Identification of Plate

Structures based on Frequency Response Function and Natural Frequencies”, Journal

of Structural Engineering and Geotechnics.

11. Stubbs N. and Kim J. T. (1994), “Field Verification of a Non-destructive Damage

Localization and Severity Estimation Algorithm”, Journal of Vibration and Acoustics,

Vol. 116, pp. 222 – 231.

65 | P a g e

12. Peter Avitable (2008), “A Simple Non-Mathematical Presentation on Modal Analysis

and Controls Laboratory Mechanical Engg”.

13. Zhou Z, Wenger and Sparling (2007), “Vibration based detection of small scale

damage on bridge desk”, Journal of structural engineering 133: page no.-1257-1267.

14. T. Siebel, A. Friedmann, M. Koch and D. Mayer (1990) “Assessment of mode shape-

based damage detection methods under real operational conditions”, Journal of

Intelligent Material Systems and Structures.

15. M. Salehi1, S. Ziaei rad1, M. Ghayour1 and M.A. Vazir (1991), “A Non model-based

damage detection technique using dynamically measured flexibility matrix”, Journal

of Intelligent Material Systems and Structures.

66 | P a g e

APPENDIX

MATLAB CODE FOR DAMAGE DETECTION USING SINGLE HIT AND

MULTIPLE IMPACT EXITATION METHOD

1. For Single impact excitation Applicable To QDA:

clear;

clc;

display('Hello');

l=input('Enter the length of the beam (in meters) = ');

N = input('Number of Sensors = ');

udbl=N;

display('enter the time step');

t=input('');

display('enter NUMBER OF READINGS in time domain of exel file');

rows=input('');

display('enter 1 for absolute');

display('enter 2 for imaginary');

display('enter 3 for real');

ss=input('');

display('Type of method used:');

display('For single hit approach enter = 1');

display('For multiple hit approach enter = 2');

final=[];

file=input('Enter the excel file name of the undamaged beam -> ','s');

arr=xlsread(file);

display('Hello');

arr=xlsread(file);

m=2;

for i=2:N+1;

final(i)=excel(arr,i,ss,rows,t);

j=input('To proceed further for getting the plot of next sensor enter 1 else 2 ->');

if j ==2

67 | P a g e

break

else

display('The plot for sensor');

display(i)

end

end

maxud=max(final);

for i=2:N+1;

final(i)=(final(i)/maxud);

end

display('enter 1 if sensors are placed equally');

display('enter 2 if sensors are not placed equally');

bpl=input('');

x1=[];

if bpl==1

x1(1)=0;

for i=1:N+1

x1(i+1)=x1(i)+l/(N+1);

end

else

x1(1)=0;

display('enter the distance from left support of the beam to first sensor : ');

x1(2)=input('');

for i=1:N-1

fprintf('enter the distance from sensor %d to sensor %d : ',i,i+1);

ip=input('');

x1(i+2)=x1(i+1)+ip;

end

x1(N+2)=l;

end

final(N+2)=0;

final

x1

plot(x1,final);

68 | P a g e

display('This is the plot of first Mode Shape obtained from the undamaged beam');

c=input('To proceed further enter 1,to end process enter 2 ->');

if c==1

fnl=[];

dc=0;

xc=[];

fx=[];

check=1;

while check==1

dc=dc+1;

display('enter the no of sensors');

N=input('');

file1=input('Enter the excel file name of the damaged beam -> ','s');

display('Hello');

arr1=xlsread(file1);

if arr1(32,1)==0

cv(file1);

arr1=xlsread(file1);

end

m=2

for i=2:N+1;

final1(i)=excel(arr1,i,ss,rows,t);

j=input('To proceed further for getting the plot of next sensor enter 1 else 2 ->');

if j ==2

break

else

display('The plot for sensor');

display(i)

end

end

maxd=max(final1);

for i=2:N+1;

final1(i)= final1(i)/maxd;

end

69 | P a g e

display('enter 1 if sensors are placed equally');

display('enter 2 if sensors are not placed equally');

bpl=input('');

x=[];

if bpl==1

x(1)=0;

for i=1:N+1

x(i+1)=x(i)+l/(N+1);

end

else

x(1)=0;

display('enter the distance from left support of the beam to first sensor : ');

x(2)=input('');

for i=1:N-1

fprintf('enter the distance from sensor %d to sensor %d : ',i,i+1);

ip=input('');

x(i+2)=x(i+1)+ip;

end

x(N+2)=l;

end

final1(N+2)=0;

final1

x

plot(x,final1);

display('This is the plot of first Mode Shape obtained from the damaged beam');

final2=abs(final-final1);

for i=1:(N+1)

final3(i)=(final2(i+1)+final2(i))/2;

end

hold off;

b=input('To continue and see the bar graph enter 1 ->');

if b==1

bar(final3);

end

70 | P a g e

for itr=1:N+1

if (final3(itr)==max(final3))

str = ['The damaged element is ',num2str(itr)];

disp(str);

end

end

q=input('To see the beam and different elements enter 1 ->');

if q==1

i=imread('beam2.jpg');

image(i);

end

fnl=[fnl final1];

fx=[fx x];

xc=[xc N+2];

display('enter 1 to continue for next damage else 2');

check=input('');

end

figure;

hold on;

plot(x1,final,'r');

temp=1;

for loop=1:dc

plot(fx(temp:(temp+xc(loop)-1)),fnl(temp:(temp+xc(loop)-1)));

temp=temp+xc(loop);

end

end

opt=input('for severity enter 1 else 2');

if opt==1

sum1=final(1)+final(end);

for loop=2:2:(size(final,2)-1)

sum1=sum1+4*final(loop);

end



71 | P a g e

end

sum1=(l/(udbl+1))*sum1/3;

temp=1;

for lop=1:dc

final1=[];

final1=fnl(temp:(temp+xc(lop)-1));

sum2=final1(1)+final1(end);

for loop=2:2:(size(final1,2)-1)

sum2=sum2+4*final1(loop);

end



end

sum2=(l/(xc(lop)-1))*sum2/3;

%disp(sum2);

severity=(sum1/sum2)^2;

fprintf('severity of damage of %d in terms of EI = ',lop);

disp(severity);

temp=temp+xc(lop);

end

end

display('Process end');

Funtmp:

function [oup] = funtmp(val,opt)

if opt==1

oup=abs(val);

else if opt==2

oup=imag(val);

else if opt==3

oup=real(val);

end

end

72 | P a g e

end

end

detmax:

function [B2]= detmax(arr,f,rows)

arr1=[];

var = input('Enter the X-coordinate of the peak you want -> ');

display('The natural frequency is:');

display(var);

B2=[];

for i=1:rows;

if(f(i)== var);

B2=arr(i);

end

end

display('The y ordinate is ->');

display(B2);

end

excel:

function [temp]= excel(arr,x,ss,rows,t)

arr1 = [];

arr1=arr(1:rows,x);

display('This is the plot of Voltage v/s Time');

plot(arr1);

i=input('To continue enter 1 -> ');

if i==1

vf=funtmp(fft(arr1),ss);

R = rows ;

f=(0:R-1)/(t*R);

f=f';

B=vf(1:rows);

display('This is the plot of Voltage (in frequency domain) v/s frequency');

73 | P a g e

plot(f,B);

temp=detmax(B,f,rows);

end

end

temp1:



end



end

sum1=(0.26/3)*sum1;

disp(sum1);

sum2=final1(1)+final1(end);



end



end

sum2=(0.26/3)*sum2;

disp(sum2);

severity=(sum2/sum1)^2;

disp('severity of damage = ');

disp(severity);

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