Engineering, 2013, 5, 56-60

http://dx.doi.org/10.4236/eng.2013.51009 Published Online January 2013 (http://www.scirp.org/journal/eng)

Analyzing Some Behavior of a Beam with Different Crack

Positions Transversely inside It

Behrooz Yazdizadeh

Department of Mechanics, Kerman Branch, Islamic Azad University, Kerman, Iran

Email: behyazd@iran.ir

Received October 4, 2012; revised November 6, 2012; accepted November 25, 2012

ABSTRACT

Vertical displacement, critical Euler buckling load and vibration behavior of a cracked beam are considered in this re-

search. The crack inside the beam is placed in different positions and results compared for each crack position. On first

Eigenvalue of free vibration results, there is a border that first Eigenvalue of free vibration does not change if center of

crack is located on that border, and after that border, the first Eigenvalue of free vibration is increased that is a counter-

example relation of critical Euler buckling load and first Eigenvalue of free vibration.

Keywords: Crack; Mesh; Vibration; Euler Buckling; Displacement

1. Introduction

Fracture mechanics, was originated by Wieghardt and

Inglis [1]. Both independently showed that cavities and

flaws in continuum materials act as stress concentrators

which, in the limit of sharp edges (cracks), produce infi-

nite stress at the tip [2]. A fairly thorough description of

the approaches for solving the crack problems is made by

many researchers [3-6]. These were the first attempts to

bring closer the theories of fracture mechanics (FM) and

continuum mechanics (CM). About the same time, the

Finite Element Method (FEM) and digital computers

dashed into the engineering community as a gifted means

for quantifying solutions in structural and solid mechan-

ics. Naturally, fracture mechanic researchers implement-

ed their FE methods, while continuum mechanic re-

searchers implemented theirs [7].

Over the years, the finite element technique has been

so well established that today it is considered one of the

best methods for solving a wide variety of practical

problems efficiently and is more and more feasible, pro-

vided that finite element (FE) models can be shown to be

correct [8]. It can be used for the solution of any problem

simply by changing the input data [9].

A lot of effort has been spent in the last 30 years to

investigate and treatment the observed drawbacks of (FE)

method [10] and Finite element analysis is one of the

usual ways to solve crack problems.

In this research, some beam characteristic behavior

comparing with each other in different crack Positions

inside the beam. These comparing consist of maximum

vertical displacement under simple bending, first mode

first Eigenvalue of free vibration and critical Euler buck-

ling load.

Critical Euler buckling load for single beam-columns

can be evaluated from analytical expressions. These so-

lutions, as given for various boundary conditions, follow

the first analytical method given by Euler [11] for pre-

dicting the reduced strengths of slender columns. The

finite elements method can be easily implemented for

beam elements without cracks since the stiffness and

generalized geometrical stiffness matrices of a non-

cracked beam are already commonly known (for example

in [12]). However, the situation essentially changes if the

structural elements are transversely cracked. Two dimen-

sion (2D) or three dimension (3D) finite element ana-

lyzes must be implemented in order to achieve a com-

plete model of the structure. When studying the elastic

Euler buckling load of a column, it is necessary to deter-

mine the maximum load at which the structure remains in

equilibrium at the deformed position.

Stress intensity factor, normal, and shear stresses (at the

tip of the crack) are calculated for simple cracked beam

under vertical pressure before [13].

Here we examine changes of some parameters of a

beam with different crack position inside of it for con-

clusion that whether crack position has an important role

to change the vertical displacement, first Eigenvalue of

free vibration and critical Euler buckling load or not.

2. Material and Methods

Consider a beam as shown in Figure 1. This beam

loaded with a vertical load at end of it then calculating

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