Modal analysis of cracked cantilever beam using ANSYS software

CONTENTS
Acknowledgement
Abstract
Chapter 1. Introduction
Chapter2. Literature Review
Chapter3. Theory
3.1 Vibrations
3.1.1Classification of vibration
3.1.2 Importance of vibration
3.2 Crack
3.2.1 Classification of Crack
3.3 Objectives
3.4 Methodology
Chapter4. Model geometry and validation
4.1 Approach of study
4.2 Concrete beam
4.2.1 Validation
4.2.2 Crack modeling
4.3 Steel beam
4.3.1 Validation
4.3.2 Crack modeling
Chapter5. Results and Discussion
5.1 Concrete beam
5.1.1 Effect of Crack Opening Size
5.1.2 Effect of Crack Depth
5.1.3 Parametric Study
5.2 Steel beam
5.2.1 Effect of Crack Opening Size
5.2.2 Effect of Crack Depth
5.2.3 Parametric Study

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 Modal analysis of cracked cantilever beam using ANSYS software

ABSTRACT
The presence of cracks causes changes in the physical properties of a structure which introduces
flexibility, and thus reducing the stiffness of the structure with an inherent reduction in modal natural
frequencies. Detection of cracks in engineering materials, structures and machines at the early stage is
an important issue of concern in the field of engineering. Cracks often occur first on the surface of
concrete structures under load and provide an indication for further degradation. Fatigue can have
significant influence on crack. It is therefore imperative to detect crack at the early stage to avoid
effects.
In this study of modal analysis, natural frequency and mode shapes of transverse vibration for
both un-cracked and cracked cantilever beam and steel beam has been extracted for first three modes.
The analysis has been extended to investigate the effect of crack opening size and mesh refinement. For
cracked beam, analysis is performed for various crack depth and crack location. As structural
discontinuity problems are difficult to solve analytically, leading commercial Finite Element Analysis
software –“ANSYS” is used to perform all the analysis computationally. In our study we observed that
natural frequency reduces with the presence of crack. The amount of reduction varies depending on
crack location, depth and crack opening size.

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 Modal analysis of cracked cantilever beam using ANSYS software

CHAPTER 1

INTRODUCTION

Any structure in presence of crack is susceptible to failure depending on the mode of vibration. Failure
is due to the resonance formed by the superposition of frequency of periodic force acting on structure
and the natural frequency of the structure. To be alert about resonance due to periodic load, it is
important to determine natural frequency. Besides this, information about the location and depth of
cracks in cracked steel beams and concrete beam can be obtained using this technique. Using vibration
analysis for early detection of cracks has gained popularity over the years and in the last decade
substantial progress has been made in that direction. Dynamic characteristics of damaged and
undamaged materials are very different. For this reason, material faults can be detected, especially in
steel beams, which are very important construction elements because of their wide spread usage
construction and machinery. Crack formation due to cycling loads leads to fatigue of the structure and to
discontinuities in the interior configuration. Cracks in vibrating components can initiate catastrophic
failures. Therefore, there is a need to understand the dynamics of cracked structures. When a structure
suffers from damage, its dynamic properties can change. Specifically, crack damage can cause a
stiffness reduction, with an inherent reduction in natural frequencies, an increase in modal damping, and
a change in the mode shapes. From these changes the crack position and magnitude can be identified.
Since the reduction in natural frequencies can be easily observed, most researchers use this feature.
Natural frequency of the concrete and steel beam is compared.

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 Modal analysis of cracked cantilever beam using ANSYS software

CHAPTER 2
LITERATURE REVIEW

B. Rajashekharam et al. “Damage detection in structural components using free vibration

analysis”,Eigen valve and Eigen vector are determined analytically using Euler Bernoulli approach and
finite element method (ANSYS).Free Vibration Analysis is carried out by inducing damage in the form
of stiffness reduction. Damage quantification is done by comparing Eigen valves and Eigen vectors
from modal analysis.

Dhiraj A et al. “Modal Analysis of cracked cantilever beam using ANSYS software”,This study
goal is to measure fracture size in damaged beams.(exp modal analysis). A defect free and fractured
beam was studied.

BalakrishnaAdavi et al. “Damage Localization of Cantilever Beam Based on Normalized Natural

Frequency Zones and Vibration Nodes”, in this proposed technique two approaches have been
followed to estimate the damage location. First approach is to identify the damage location in terms of
small zones with the help of Normalized Frequency information.Second approach is to identify the exact
damage location of the beam with vibration nodes.

Udyan Ghosh et al. “Modal Analysis of Cracked Cantilever Beam by Finite Element
Simulation”,The importance of natural frequency at different mode has been studied.

G. M. Owolabi et al. “Crack detection in beams using change in frequencies and amplitudes of
frequency response functions”,To detect the presence of crack in beams, and determine its location
and size based on experimental modal analysis results.

Aim and scope of present study

 The objective of this study is to analyze the vibration behavior of beams both experimentally and
using FEM software ANSYS subjected to single and multiple cracks under free and forced
vibration cases.
 Study of Natural frequency and mode shapes of beam without crack for Cantilever beam.

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 Study of Natural frequency and mode shapes of beam with crack at different locations.
 Study of Natural frequency and mode shapes of beam with different crack depths.

CHAPTER 3
THEORY
3.1 Vibrations
Vibrations are time dependent displacements of a particle or a system of particles w.r.t an equilibrium
position. If these displacements are repetitive and their repetitions are executed at equal interval of time
w.r.t equilibrium position the resulting motion is said to be periodic.

3.1.1Classification of vibration
Vibration can be classified in several ways. Some of the important classifications are as follows:

Free and forced vibration: If a system, after an internal disturbance, is left to vibrate on its own, the
ensuing vibration is known as free vibration. No external force acts on the system. The oscillation of the
simple pendulum is an example of free vibration.
If a system is subjected to an external force, the resulting vibration is known as forced vibration. The
oscillation that arises in machineries such as diesel engines is an example of forced vibration.
If the frequency of the external force coincides with one of the natural frequencies of the system, a
condition known as resonance occurs, and the system undergoes dangerously large oscillations.

Undamped and damped vibration: If no energy is lost or dissipated in friction or other resistance
during oscillation, the vibration is known as undamped vibration. If any energy lost in this way,
however, it is called damped vibration. In many physical systems, the amount of damping is so small
that it can be disregrated for most engineering purposes. However, consideration of damping becomes
extremely important in analyzing vibratory system near resonance.

Linear and nonlinear vibration: If all the basic components of vibratory system—the spring, the mass
and the damper—behave linearly, the resulting vibration is known as linear vibration. If, however, any
of the basic components behave nonlinearly, the vibration is called nonlinear vibration.

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3.1.2 Importance of vibration

 The measurement of the natural frequencies of the structure or machine is useful in selecting the
operational speed of nearby machinery to avoid resonant conditions.
 The theoretically computed vibration characteristics of a machine or structure may be different
from the actual values due to the assumptions made in the analysis.
 In many applications survivability of a structure or machine in a specified vibration environment
is to be determined. If the structure or machine can perform the expected task even after
completion of testing under the specified vibration environment, it is expected to survive the
specified conditions.
 Continuous systems are often approximated as multidegree of freedom systems for simplicity. If
the measured natural frequencies and mode shapes of a continuous system are comparable to the
computed natural frequencies and mode shapes of the multidegree of freedom model, then the
approximation will be proved to be a valid one.
 The measurement of the input and the resulting output vibration of a system help in identifying
the system interms of its mass stiffness and damp.

3.2 Crack:

A crack in a structural member introduces local flexibility that would affect vibration response of the
structure. This property may be used to detect existence of a crack together its location and depth in the
structural member. The presence of a crack in a structural member alters the local compliance that
would affect the vibration response under external loads.

3.2.1 Classification of Crack

Based on geometries, cracks can be broadly classified as follows::

 Transverse crack: These are cracks perpendicular to beam axis. These are the most commonand

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 Modal analysis of cracked cantilever beam using ANSYS software

most serious as they reduce the cross section as by weakens the beam. They introduce alocal
flexibility in the stiffness of the beam due to strain energy concentration in the vicinity
orcracktip.
 Longitudinal cracks: These are cracks parallel to beam axis. They are not that common but they
pose danger when the tensile load is applied at right angles to the crack direction i.e.
perpendicular to beam axis.

 Open cracks: These cracks always remain open. They are more correctly called “notches”.
Open cracks are easy to do in laboratory environment and hence most experimental work is
focused on this type of crack.

 Breathingcrack: These are cracks those open when the affected part of material is subjected to
tensile stress and close when the stress is reversed. The component is most influenced when
under tension. The breathing of crack results in non‐linearity in the vibration behavior of the
beam. Most theoretical research efforts are concentrated on “transverse breathing” cracks due to
their direct practical relevance

 Slant cracks: These are cracks at an angle to the beam axis, but are not very common. There
effect on lateral vibration is less than that of transverse cracks of comparable severity.

 Surface cracks: These are the cracks that open on the surface. They can normally be detected by
dye‐penetrates or visual inspection.

 Subsurface cracks: Cracks that do not show on the surface are called subsurface cracks. Special
techniques such as ultrasonic, magnetic particle, radiography or shaft voltage dropare needed to
detect them. Surface cracks have a greater effect than subsurface cracks in the vibration behavior
of shafts.

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3.3 Objectives
 Study of Natural frequency and mode shapes of beam without crack for Cantilever beam and
Steel beam.
 Study of Natural frequency and mode shapes of beam with crack at different locations.
 Study of Natural frequency and mode shapes of beam with different crack depths.
 Based on the frequency Data identifying crack depth and crack location usingFrequency Plot
Contours.

3.4 Methodology
 Modeling of structural element using ANSYS.
 Using modal analysis, study of natural frequency and mode shapes for uncracked beam.
 Modeling the cracks at different locations and using modal analysis, study of natural frequency
and mode shapes for cracked beam.
 Using the frequency data and by plotting the Frequency Contours. Finding the crack depth and
crack location for different boundary conditions.

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CHAPTER 4
MODEL GEOMETRY AND VALIDATION

4.1 Approach of study

Natural frequency is the frequency at which a system or structure vibrates when subjected to an initial
excitation in the absence of any driving or damping force. To determine natural frequency, thus free un-
damped vibration is considered. For any cracked structure analysis, the study of resonance is important
as it affects the structure in number of ways. When the frequency of applied load becomes equal to
associated natural frequency, the structure vibrates theoretically at infinite amplitude leading to failure.
To be alert about structural failure due to periodic load, thus it is important to determine resonant
frequency.

4.2 CONCRETE BEAM:

Modal analysis is performed to determine frequency of vibration for different mode shapes. A system
may undergo vibration with different mode shapes depending upon the constraints imposed on it.
For modal analysis, a steel cantilever beam of length 2.2m, width 0.5m and depth 0.25m is considered.
Table 1 represents model dimensions and properties of the material used. It is assumed that crack have
uniform depth across the width of the cantilever beam. An open edge crack, perpendicular to the
longitudinal axis, is present in the cantilever beam. For the study on the behavior of crack the crack
opening size of 2 mm is considered.
Crack is positioned at 0.55m, 1.1m, and 1.65m from the fixed end for the analysis. For every
crack position, crack of varying depth such as 0.05m, 0.075m, 0.1m, 0.125m and 0.15m are taken.
Considering these factors, the effect of crack on a cantilever beam is investigated. For all cases, data of
first three modes of vibration are taken under consideration.

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MATERIAL PROPERTIES

Table 1: Model Geometry and properties

Property Value
Length (L) 2.2 m
Width (w) 0.15 m
Depth (t) 0.25 m
Material concrete
Elastic Modulus (E) 25x109 N/m2
Density 2500kg/m3
Poisson’s Ratio 0.2

Fig. 1: Boundary Condition: Cantilever Beam

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Fig. 2: Meshing of Cantilever Beam

4.2.1 Validation

In the initial stage, natural frequency of un-cracked cantilever beam has been analyzed using ANSYS.
In ANSYS the model has been analyzed considering different mesh element. The natural frequencies
(ω) as shown in fig. 3 and table 2 of mode-1, mode-2 and mode-3 are found as 173.2cycles/sec,
1063.9/sec, 2084.3 cycles/sec.

Simulation set up for concrete Beam

Table 2: natural frequency Modes Frequencies(Hz) of undamaged beam

Mode 1 173.2

Mode 2 1063.9

Mode 3 2084.3

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Fig. 3: (a) mode 1

Fig. 3(a) mode 1

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Fig. 3(b) mode 2

Fig. 3(c)
Mode 3

4.2.2 Crack modeling

Presence of crack causes a

complex geometrical property which is difficult to study. So finite element method (FEM) is used and
for finite element analysis (FEA), ANSYS (workbench 19.2) has been used. For a certain crack depth,
analyzing the results a V-shaped notched crack of opening 2 mm is finally taken for all the models with
crack.

Fig. 4: Cracked cantilever beam

As presence of crack creates complexities regarding meshing, for analysis in ANSYS, special
distribution of mesh element is considered by performing local meshing. The tip of the crack requires
special attention. For a V-shaped notch crack, at crack location partition is created.

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Table 3: Crack orientation

Crack depth (m), a Location ( m),c

0.05 0.55
0.075 1.1
0.1 1.65
0.125
0.15

Fig. 4: crack orientation

4.3 STEEL BEAM:

For modal analysis, a steel cantilever beam of length 3m, width 0.25m and depth 0.2m is considered.
Table 4 represents model dimensions and properties of the material used. It is assumed that crack have
uniform depth across the width of the cantilever beam. An open edge crack, perpendicular to the
longitudinal axis, is present in the cantilever beam. For the study on the behavior of crack the crack
opening size of 2 mm is considered.
Crack is positioned at 0.75m, 1.5m, and 2.25m from the fixed end for the analysis. For every
crack position, crack of varying depth such as 0.05m, 0.075m, 0.1m, 0.125m and 0.15m are taken.
Considering these factors, the effect of crack on a cantilever beam is investigated. For all cases, data of

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first three modes of vibration are taken under consideration.

Table 4: Model Geometry and properties

Property Value
Length (L) 3.00 m
Width (w) 0.25 m
Depth (t) 0.20 m
Material Mild Steel
Elastic Modulus (E) 210 × 109 N/m2
Density 7860 kg/m3
Poisson’s Ratio 0.3

Fig. 5: Boundary Condition: Cantilever Beam

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Fig. 6: Meshing of Cantilever Beam

4.3.1 Validation
In the initial stage, natural frequency of un-cracked cantilever beam has been analyzed using ANSYS.
In ANSYS the model has been analyzed considering different mesh element. The natural frequencies
(ω) as shown in fig. 7 and table 5 of mode-1, mode-2 and mode-3 are found as 18.113cycles/sec,
111.24/sec, 234.36 cycles/sec.
Simulation set up for Steel Beam
Table 5: natural frequency of undamaged beam

Fig. 7: (a) mode 1

Fig. 7(a) Mode 1

Modes Frequencies(Hz)

Mode 1 18.113

Mode 2 111.24

Mode 3 234.36

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(b) Mode 2

(c) Mode 3

4.3.2 Crack modeling

Presence of crack causes a complex geometrical property which is difficult to study. So finite element
method (FEM) is used and for finite element analysis (FEA), ANSYS (workbench 19.2) has been used.
For a certain crack depth, analyzing the results a V-shaped notched crack of opening 2 mm is finally
taken for all the models with crack.

Fig. 8: Cracked cantilever beam

As presence of crack creates complexities regarding meshing, for analysis in ANSYS, special
distribution of mesh element is considered by performing local meshing. The tip of the crack requires
special attention. For a V-shaped notch crack, at crack location partition is created.

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 Modal analysis of cracked cantilever beam using ANSYS software

Table 6: Crack orientation

Crack depth (m), a Location ( m),c

0.05 0.75
0.075 1.5
0.1 2.25
0.125
0.15

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Fig. 9: crack orientation

CHAPTER 5

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RESULTS AND DISCUSSION

5.1 Concrete beam
5.2 Steel beam
5.2.1 Effect of Crack location
Effect of crack location for specified crack depth has been represented in Table 7 and Figure 9(a), 9(b)
and 9(c). These show that frequency varies for a certain crack depth depending on the crack location and
variation pattern is different for different mode shapes. For mode-1, frequency increases as the crack
moves away from the fixed end. In other words, reduction in frequency is less for crack located near
free end. For mode-2, frequency starts to decrease and then increase again. Frequency pattern is
somewhat repetitive pattern of decrease-increase-decrease for mode-3.

Table 7: Frequency at different crack location for crack depth d=0.1m

Frequencies(Hz)
Crack depth (m) Mode 1 Mode 2 Mode 3
0.05 17.953 107.29 232.89
0.075 17.728 102.38 231.75
0.1 17.291 94.631 229.4
0.125 16.463 84.047 225.87
0.15 14.659 70.49 220.29

Fig. 10(a) Frequency vs. crack location for Crack depth 0.1m atMode-1

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Fig. 10(b) Frequency vs. crack location forCrack depth 0.1m at Mode-2

Fig. 10(c) Frequency vs. crack location for Crack depth 0.1m at Mode-3

5.2.2 Effect of Crack Depth

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Effect of crack depth for specified crack location has been shown through Table 8 and table 9 from
which it is obvious that frequency decreases as the crack depth increases.

Table 8: Frequency at different crack depth for crack location 1.0 m

Frequencies(Hz)

Location(m) Mode 1 Mode 2 Mode 3

15.551 110.56 226.22
0.75
17.291 94.631 229.4
1.5
18.066 103.6 233.1
2.25

5.2.3 Parametric Study

Frequency ratio for varying crack depth and for varying crack location is calculated and shown
graphically using non-dimensional parameters.
 Frequency ratio (normalized natural frequency) is the ratio between the frequency of cracked
beam (ωc) and that of uncracked beam (ω).
 Crack depth ratio (d/t) is the ratio between the actual crack depth and depth of the beam
 While crack location ratio (x/L) is obtained dividing distance of crack from fixed end by the
length of the beam.

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Table 9(a): Normalized Natural Frequencies of cantilever beam for various crack location
(Mode 1)

Mode 1
a/h vs Location 0.75 1.5 2.25
(from Fixed end)

0.25 0.970 0.991 0.999

0.375 0.927 0.979 0.998
0.5 0.859 0.955 0.997
0.625 0.750 0.909 0.992
0.75 0.584 0.809 0.980

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Fig. 11(a): Mode 1

Table 9(b): Normalized Natural Frequencies of cantilever beam for various crack location
(Mode 2)

Mode 2
a/h vs Location 0.75 1.5 2.25
(from Fixed end)

0.25 0.999 0.964 0.988

0.375 0.997 0.920 0.968
0.5 0.994 0.851 0.931
0.625 0.989 0.756 0.862
0.75 0.982 0.634 0.721

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Fig. 11(b): Mode 2

Table 9(c): Normalized Natural Frequencies of cantilever beam for various crack location
(Mode 3)

Mode 3
a/h vs Location 0.75 1.5 2.25
(from Fixed end)

0.25 0.992 0.994 0.999

0.375 0.981 0.989 0.997
0.5 0.965 0.979 0.995
0.625 0.941 0.964 0.968
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0.75 0.907 0.940 0.838
 Modal analysis of cracked cantilever beam using ANSYS software

Fig. 11(c): Mode 3

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