Summary of the thesis of engineering and engineering engineering Mechanical: Modal analysis and Crack detection in stepped beams

The aim of this thesis is to study crack-induced change in natural frequencies and to develop a procedure for detecting cracks in stepped beams by measurement of natural frequencies.

ABSTRACT OF DOCTOR THESIS

IN MECHANICAL ENGINEERING AND ENGINEERING

MECHANICS

HANOI - 2018

The thesis has been completed at: Graduate University of Science and

Technology - Vietnam Academy of Science and Technology

Supervisions:

1: Prof.DrSc Nguyen Tien Khiem

2: Dr Tran Thanh Hai

Reviewer 1: Prof.DrSc Nguyen Van Khang

Reviewer 2: Prof.Dr Nguyen Manh Yen

Reviewer 3: Assoc.Prof.Dr Nguyen Đang To

Thesis is defended at Graduate University of Science and Technology

- Vietnam Academy of Science and Technology at…, on date…month…201

Hardcopy of the thesis be found at:

- Library of Graduate University of Science and Technology

- Vietnam National Library

INTRODUCTION

1 Necessarily of the thesis

Crack is a damage usually happened in structural members but dangerous for safety of structure if it is not early detected However, cracks are often difficult to identify by visual inspection as they occurred at the unfeasible locations Therefore, cracks could be indirectly detected from measured total dynamical characteristics of structures such as natural frequencies, mode shapes and frequency or time history response In order to identify location and size of a crack in a structure the problem of analysis of the crack’s effect on the dynamic properties is of great importance It could give also useful tool for crack localization and size evaluation On the other hand, beams are frequently used as structural member in the practice of structural engineering So, crack detection for beam-like structures gets to be an important problem Crack detection problem of beam with uniform cross section is thoroughly studied, but vibration of cracked beam with varying cross section presents a difficult problem It is because vibration

of such the structure is described by differential equations with varying coefficients that are nowadays not generally solved The beam with piecewise uniform beam, acknowledged as stepped beam is the simplest model of beam with varying cross section Although, vibration analysis and crack detection for stepped beam have been studied in some publications, developing more efficient methods for solving the problems of various types of stepped beams is really demanded

2 Objective of the thesis

The aim of this thesis is to study crack-induced change in natural frequencies and to develop a procedure for detecting cracks in stepped beams by measurement of natural frequencies

3 The research contents of the thesis

(1) Developing the Transfer Matrix Method (TMM) for modal analysis of stepped Euler – Bernoulli, Timosheko and FGM beams with arbitrary number of cracks

(2) Expanding the Rayleigh formula for computing natural frequencies of stepped beams with multiple cracks

(3) Employing the extended Rayleigh formula for developing an algorithm to detect unknown number of cracks in stepped beams by natural frequencies

(4) Experimental study of cracked stepped beams to

validate the developed theory

Thesis composes of Introduction, 4 Chapters and Conclusion Chapter 1 describes an overview on the subject literature; Chapter 2 – development of TMM; Chapter 3 – the Rayleigh method and Chapter 4 presents the experimental study

CHAPTER 1 OVERVIEW ON THE MODELS, METHODS AND PUBLISHED RESULTS

1.1 Model of cracked beams

1.1.1 On the beam theories

Consider a homogeneous beam with axial and flexural displacements ( , , )u x z t , ( , , ) w x z t at cross section x Based on

some assumptions the displacements can be represented as:

( , , ) ( , ) ( , ) ( ) ( , ); ( , , ) ( , ),

u x z t u x t zw x t  z x t w x z t w x t where u 0 (x, t), w 0 (x, t) are the displacements at the neutral axis,

(x,t) – shear slop, z is heigh from the neutral axis Function

(z), representing shear distribution can be chosen as follow:

(a) ( )z  0- for Euler-Bernoulli beam theory (the classical beam theory)

(b) ( )z z - for Timoshenko beam theory or the first order shear beam theory

   - the exponent shear beam theory

Recently, one of the composites is produced and called Functionally Graded Material (FGM), mechanical properties of

which are varying continuously along corrdinate z or x

Denoting elasticity modulus E, shear modulus G and material density , a model of the FGM is represented as

( )z b ( t b) ( )g z

where  b, t stand for the characteristics (E, , G) at the

bottom and top beam surfaces and function g(z) could be chosen

in the following forms:

a) P-FGM: g z( ) (zh 2) /hn - the power law material b) E-FGM: ( ) (1 2 / )z h, 0.5ln( / )

E z E e    E E - the exponent law material

c) S-FGM: g z1 ( )   1 0.5 1 2 /  z hn,0 z h/ 2

d) g z2 ( )  1 2 /z hn/ 2,h/ 2 z 0 - Sigmoid law material

In this thesis only the FGM of power law is investigated

1.1.2 Crack model in homogeneous beams

Fig 1.2 edge crack model

Consider a homogeneous beam as shown in Fig 1.2 that

contains a crack with depth a at position e Based on the

fracture mechanics, Chondros, Dimagrogonas and Yao have proved that the crack can be represented by a rotational spring

of stiffness

where EI is bending stiffness, h is heigh of beam and function

Hence, compatibility conditions at the cracked sections are

For Timoshenko beam the conditions take the form ( 0, ) ( 0, );

1.1.3 Modeling crack in FGM beam

Crack in FGM beam can be modeled by a spring of stiffness calculated as

1.2 Vibration of cracked beams

1.2.1 Homogeneous beams

Consider an Euler-Bernoulli beam with n cracks at

positions 0 e1 e2  e nL and depth a j,j 1, 2, , nFree vibration of the beam is described by equation

for determining the unknown constants Hence, frequency equation can be obtained as

1.2.3 Conventional formulation of TMM

In this section, an Euler-Bernoulli homogeneous beam composed of uniform beam elements with the material and geometry constants:

It is well known that general solution of free vibration problem

in every beam segment is expressed in the form

with T being called transfer matrix of the beam Applying

boundary conditions for the latter connection allows one to get

1.2.4 Rayleigh method

For a standard beam flexural deflection in in vibration of frequency  is v x t( , )( )sinx twith function ( )x called mode shape of vibration In that case, potential and kinetic energies are

to the exact one Such calculating natural frequency from appropriately chosen shape function is acknowledged as Rayleigh method The Rayleigh formula was expanded for multiple cracked Euler-Bernoulli beam by N.T Khiem and T.T

Hai and applied for calculating natural frequencies of just uniform Euler-Bernoulli beams

1.3 Crack detection problem for beams

Contents of the crack detection problem is to localize and evaluate severity of crack based on the measured data gathered from testing on the structure of interest There are two appoaches to solve the problem: first approach is based only on measured data that are often response of the structure to a given load; the second one involves additionally a model of the structure with assumed cracks of unknown loacation and depth The crucial tool for the first approach is the method used for signal processing such as, for example, the Fourier or wavelet transform The second approach finds the way to connect the measured data with the structure model in form of diagnostic equations of unknown crack parameters The advantage of the model-based approach to crack detection is that enables to apply the latest achievements in both theoretical and numerical development of the system identification theory In this thesis, the model-based approach is applied and the crack parameters are determined from the equations connecting the measured and calculated natural frequencies

1.4 Overview on vibration of stepped beams

1.4.1 Spepped beams without cracks

Free vibration of stepped beams was studied by numerious authors such as Jang and Bert; Jaworski and Dowell, Cunha et al.; Kukla et al and Yang, The most important obtained results demonstrate that natural frequencies of stepped beam are

significantly affected by abrupt change in cross section area of stepped beams and the natural frequency variation is dependent also on the boundary conditions Sato studied an interesting problem that proposed to calculate natural frequency of beam with a groove in dependence on size of the groove Using a model of stepped beam and the Transfer Matrix Method combined with Finite Element Method the author demonstrated that (a) fundamental frequency of the structure increases with growing thickness and reducing length of the mid-step; (b) the mid-step could be modeled by a beam element, therefore, the TMM is reliably applicable for the stepped beam if ratio of its

length to the beam thickness (r=L2/h) is equals or greater 4.0

Comparing with experimental results the author concluded that error of the TMM may be up to 20% if the ratio is less than 0.2

1.4.2 Cracked stepped beams

Kukla studied a cracked onestep column with a crack at the step under compression loading Zheng et al calculted fundamental frequency of cracked Euler-Bernoulli stepped beam by using the Rayleigh method Li solved the problem of free vibration of stepped beam with multiple cracks and concentrated masses by using recurent connection between vibration mode of beam steps The crack detection problem for stepped beams was first solved by Tsai and Wang, then, it was studied by Nandwana and Maiti based on the so-called contour method for identification of single crack in three-step beam Zhang vet al solved the problem for multistep beam using wavelet analysis and TMM Besides, Maghsoodi et al have

proposed an explicit expression of natural frequencies of stepped beam through crack magnitudes based on the energy method and solved the problem of detecting cracks by measurements of natural frequencies The classical TMM was completely developed by Attar for both the forward and inverse problem of multistep beam with arbitrary number of cracks Neverthenless, the frequency equation used for solving the inverse problem is still very complicated so that cannot be usefully employed for the case of nember of cracks larger than

2

1.5 Formulation of problem for the thesis

Based on the overview there will be formulated subjects for the thesis as follow:

(1) Further developing the TMM for modal analysis of stepped Euler – Bernoulli; Timoshenko and FGM beams; (2) Extending the Rayleigh formula for calculating natural frequencies of stepped beam with multiple cracks;

(3) Using the established Rayleigh formula to propose an algorithm for multi-crack detection in stepped beam from natural frequencies;

(4) Overall, carrying out an experimental study on cracked stepped beam to validate the developed theories

CHAPTER 2 THE TRANSFER MATRIX METHOD FOR VIBRATION ANALYSIS OF STEPPED BEAMS WITH

MULTIPLE CRACKS 2.1 Stepped Euler-Bernoulli beam with multiple cracks

2.1.1 General solution for uniform homogeneous

Euler-Bernoulli beam element is

2.1.2 The transfer matrix

Using the solution for mode shape, transfer matrix for the beam with cracks is conducted in the form

Fig 2.1 Two models of stepped beam used in numerical

analysis

Fig 2.2 Effect of crack position and depth on three lowest natural frequencies of B1S (right) and B2S (left)

Notice: Observing graphs given in Fig 2.2 allows the following

remerks to be made: (1) Likely to the uniform beams, there exist on stepped beam positions crack occurred at which does

30% 10%

30% 10%

not change some natural frequencies; (2) Change in natural frequencies due to crack undergoes a jump for crack passing across step; (3) Increasing depth leads to reduced natural frequencies, but reduction of natural frequencies is different for

cracks at different steps

2.2 Stepped Timoshenko beam with multiple cracks

2.2.1 General solution

W ( , )x C coshk x C sinhk x C cosk x C sink x;

0 ( , )x r C1 1 sinhk x1 r C1 2 coshk x1 r C2 3 sink x2 r C2 4 cosk x2

stepped are L=0.5m;E=210Gpa;  7860 kg m / 3;b=12mm;

h 1 =20mm; h 2 =16 mm

0

0 0

Note that variation of natural frequencies caused by crack at the step also has a jump; crack near the clamped end is more affecting on natural frequencies than that located near the free end

Fig 2.6 First natural frequency ratio of cantilever stepped

beam versus crack position

Fig 2.7 Second natural frequency ratio of cantilever stepped

beam versus crack position

2.3 Stepped FGM beams with cracks

Clamped Beam, L1=L2=L3=1;a/h=5,10,20,30,40%

Vi tri vet nut

S1 S1

1.01 Clamped Beam, L1=L2=L3=1;a/h=5,10,20,30,40%

Vi tri vet nut

S2

S0 S1

S0 S2

S1

S1: h1=h3=0.1;h2=0.2 S2: h1=h3=0.1;h2=0.05 S0: h1=h2=h3=0.1

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 Clamped Beam, L1=L2=L3=1;a/h=5,10,20,30,40%

Vi tri vet nut

S1: h1=h3=0.1;h2=0.2 S2: h1=h3=0.1;h2=0.05

S1 S2

S0

S1 S0

S1

S0 S1

S1

S2

S0

S0 S1 S2

x x

Φ

11 22

Notice: Numerical analysis has shown that material gradient

and thickness variation of beam are significantly affecting the crack-induced change in natural frequencies For instance, abrupt change in beam thickness produces a jump in variation

of natural frequencies versus crack position that is continuous for uniform beam

Concluding remarks for Chapter 2

Results obtained in this Chapter are as follow:

 TMM has been developed for modal analysis of stepped Euler-Bernoulli; Timoshenko and FGM beams with arbitrary number of transverse open cracks;

 The developed TMM allows studying effect of change in cross section area on natural frequencies in combination with cracks and material parameters of beams

CHAPTER 3 RAYLEIGH METHOD IN ANALYSIS AND CRACK DETECTION FOR STEPPED BEAMS 3.1 Rayleigh formula

2 1

.( )

Fig 3.3 Effect of crack position on natural frequencies of

two-step beam with simply supported ends

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.75

0.8 0.85 0.9 0.95 1

Table 3.5 Result of crack detection in cantilever beam

 Numerical results showed that natural frequencies of stepped beam can be more easily calculated by Rayleigh qoutient in comparison with the TMM

 The Rayleigh quotient is applied also for deriving an efficient peocedure to detect multiple cracks in stepped beam by natural frequencies