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Arch Appl Mech (2013) 83:1087–1096
DOI 10.1007/s00419-013-0736-7
ORIGINAL
A. Ranjbaran ·H. Rousta ·M. O. Ranjbaran ·
M. A. Ranjbaran ·M. Hashemi ·M. T. Moravej
A necessary modification for the finite element analysis
of cracked members detection, construction, and justification
Received: 28 March 2012 / Accepted: 23 January 2013 / Published online: 5 February 2013
© Springer-Verlag Berlin Heidelberg 2013
Abstract The finite element analysis of cracked beam-like members is investigated. Through concise formu-
lation, computer implementation, and numerical experiment, the necessity for a paramount modification in
the governing differential equation is detected. In the course of numerical analysis, physical consideration,
and mathematical theory, the structure of the modification and its relation with the member geometry and
crack parameters is constructed and justified. Several new chain rules for use in the application of the method
of weighted residual to differential equations containing the derivatives of the Dirac’s delta distribution is
proposed. Through analysis of typical examples and comparison of the results with that of the others, the
accuracy, efficiency and robustness of the work are verified.
Keywords Ranjbaran factor ·Equivalent mass method ·Stiffness reduction method ·Cracked members ·
Modified method of weighted residual ·Modified finite element method ·New chain rules ·Dirac delta
1 Introduction
The influence of cracks on the dynamic characteristics of beam-like structures has been the subject of many
investigations in the last four decades [1]. Shifrin and Ruotolo [2] proposed a technique for calculating the
natural frequencies of a vibrating beam with an arbitrary finite number of transverse open cracks. Chondros
et al. [3] proposed a continuous cracked beam vibration theory for the lateral vibration of cracked Euler–Ber-
noulli beams with single or double-edge open cracks. They used the Hu–Washizu–Bar variational formulation
to develop the differential equation and the boundary conditions of the cracked beam as a one dimensional
continuum. Vibration of beams with multiple open cracks subjected to axial force is studied by Binici [4]. He
proposed a method to obtain the eigen-frequencies and eigen-modes of beams containing multiple cracks and
subjected to axial force. Wang and Qiao [5] presented a general solution for vibration of an Euler–Bernoulli
beam with arbitrary type and location of discontinuity. Unlike the commonly used approach in the literature,
the modal displacement of the whole beam is expressed by a single function using the Heaviside’s unit step
function to account for the discontinuities. The general modal displacement function is then solved by using
A. Ranjbaran (B
)·H. Rousta
Department of Civil Engineering, Shiraz University, Shiraz, Iran
E-mail: aranjbaran@yahoo.com; ranjbarn@shirazu.ac.ir
M. O. Ranjbaran
Department of Chemical Engineering, Sharif University, Tehran, Iran
E-mail: ranjbaran@che.sharif.ir; mranjbaran24@gmail.com
M. A. Ranjbaran ·M. Hashemi ·M. T. Moravej
Center of Excellence for Advanced Research, Shiraz, Iran
1088 A. Ranjbaran et al.
the Laplace transform. The presented solution reduced the complexity of the vibration of beams with arbitrary
discontinuities to the same order of the case without discontinuity. Caddemi and Calio [6] used the Dirac’s
distribution in the buckling analysis of an Euler–Bernoulli column. The influence of the axial load on the
vibration modes of beam-columns with different number and position of cracks, under different boundary
conditions, has been analyzed by Caddemi and Calio [7]. Moradi, et al. [8] applied the bees algorithm to the
problem of crack detection in beams. Their work contains both numerical and experimental studies. Ranjbaran
and co-workers [1,9–11] formulated the problem of cracked members in a special manner. The effect of crack
is defined by a new variable, and the derivative of the variable at the singular cracked point is derived (golden
derivative). Making use of the golden derivative an innovative method, the equivalent mass method, for the
analysis of free vibration of cracked members, is developed. The closed form and the finite element solutions
were derived. For the case of buckling, static and dynamic analysis of cracked members, the equivalent mass
method is not applicable. For these cases, the stiffness reduction method is proposed. The governing equation
is developed. Making use of the Laplace transform, the closed form solution is derived and verified. The
conventional method of weighted residual is used to derive the weak form equation. The derived weak form
equation is found to be in error, that is, the conventional methods are not applicable here. The aim is to develop
a procedure for modification of the governing differential equation to make it suitable for transformation to
the weak form and deriving a correct finite element equation.
2 Basic theories
2.1 General consideration
A crack introduces a jump y(n−1)in the displacement of a cracked member as follows:
y(n−1)=cny(n),n=1,2(1)
where yis displacement, is the jump, (n)in the superscript denotes the order of derivative with respect to
x,cnis crack compliance, and n=1 is used for axial member and n=2 for flexural member. To prepare for
innovative formulation, the jump is defined by a new variable as follows:
y(n−1)
c=cny(n)H(x−xi)(2)
where His the Heaviside unit step function and xiis crack position. According to principles of fracture
mechanics, a crack introduces only one change in the continuity condition of the cracked member defined by
Eqs. (1)or(2). This leads to the conclusion that y(n)on both sides of the crack are equal or at the cracked
point y(n)is constant. As a result, the derivative of Eq. (2) is written as
y(n)
c=cny(n)δ(x−xi)(3)
where δ(x−xi)is the Dirac delta distribution. The derivative in Eq. (3) is called the golden derivative. It paved
the way for innovative formulation of cracked members. The governing equation for analysis of a beam-like
structure is defined in a general form as follows:
y(n)(n)
−g(x,y)=0(4)
For a cracked member, the Eqs. (3)and(4) are combined and a general equation for analysis of a cracked
member is obtained as follows:
y(n)−Rcncny(n)δ(x−xi)(n)
−g(x,y)=0(5)
where g(x,y)is a specified function and Rcn =1. In Eq. (5), the effect of the crack appeared as a reduction
in the stiffness term. In view of that, the method is called the stiffness reduction method. Typical functions are
defined as follows:
Cracked members detection, construction, and justification 1089
g(x,y)=(−1)nλ2n
ωy,−λ2
Py (6)
For two orders of differential equations and different forms of function g(x,y),theEq.(5) is solved and the
closed form solution is obtained by the Laplace transform. Comparison of the results with results of the others
verified the Eq. (5)[1,9–12].
The analysis of structures containing cracked members is best done by the finite element method. To do
that, the weak form equation corresponding to Eq. (5) should be derived. Making use of the method of weighted
residual [13], the weak form equation corresponding to Eq. (5) is derived as follows:
L
0
ψ(n)y(n)dx−Rcn
L
0
ψ(n)cnδ(x−xi)y(n)dx−G(ψ, y)=0(7)
where ψis a weight function and Rcn =1. Define the weight function ψand main parameter yin terms of
nodal values as
ψ=ψαNα,y=yβNβ,α,β=1,ned (8)
The subscripts denote nodal values. In this equation, αand βare degrees of freedom numbers, ned is number
of element’s degrees of freedom, and Nis the shape function. The Einstein summation convention is assumed.
In this convention, repeated index denotes summation. Substitution of Eq. (8) into the weak form Eq. (7) leads
to the finite element equation as follows:
L
0
N(n)
αN(n)
βdxy
β−Rcn
L
0
N(n)
αcnδ(x−xi)N(n)
βdxy
β−FNα,yβ=0(9)
The finite element Eq. (9) is implemented in a computer program. The program is used for numerical analyses.
The first try for analysis of a typical member failed. Closed insight into the program revealed that the Eq. (9)is
wrong. The parameter Rcn should be different from one. That is the conventional method of weighted residual
failed to apply to the proposed differential Eq. (5). A modification beyond the conventional method of weighted
residual is necessary. Extensive effort during more than two years leads to an innovative modification. The
modification is done by introduction of the parameter Rcn. The parameter is named as Ranjbaran factor. The
formulation for Ranjbaran factor is determined in three stages. The extensive numerical experiment is used to
define the correct formula in the first stage. In the second stage, the derived formula is verified by considering
the physics of the problems. Finally, the Ranjbaran factor formula is developed based on sound mathematical
principles in the third stage. The stages used for definition of the Ranjbaran factor are described in detail in
the following sections.
2.2 Numerical basis for Ranjbaran factor
The governing equation for free vibration analysis of cracked members is defined as
y(n)−Rcncnδ(x−xi)y(n)(n)
−λ2ny=0,n=1,2(10)
where (n)in the superscript denotes order of derivative with respect to x,cnis crack compliance, δ(x−xi)
is Dirac delta with center at cracked point xi,yis variable function, λis a working parameter, and Rcn is
Ranjbaran factor.
With the help of method of weighted residual, the weak form equation corresponding to Eq. (10) is written as
L
0
ψ(n)y(n)dx−Rcn
L
0
ψ(n)cnδ(x−xi)y(n)dx−λ2n
L
0
ψydx=0,n=1,2 (11)
1090 A. Ranjbaran et al.
Tabl e 1 Ranjbaran factor for 2nd and 4th order equation
Length, L Rc1Length, L Rc2
Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Column 7 Column 8
0.5 0.200 1/5 0.5/(0.5+2) 0.50 0.111111 2/18 0.50/(0.50+4)
1.0 0.333 2/6 1.0/(1.0+2) 0.75 0.157574 3/19 0.75/(0.75+4)
1.5 0.428 3/7 1.5/(1.5+2) 1.00 0.199999 4/20 1.00/(1.00+4)
2.0 0.500 4/8 2.0/(2.0+2) 1.25 0.239315 5/21 1.25/(1.25+4)
2.5 0.555 5/9 2.5/(2.5+2) 1.50 0.276189 6/22 1.50/(1.50+4)
3.0 0.600 6/10 3.0/(3.0+2) 1.75 0.311111 7/23 1.75/(1.75+4)
3.5 0.636 7/11 3.5/(3.5+2) 2.00 0.344444 8/24 2.00/(2.00+4)
4.0 0.666 8/12 4.0/(4.0+2) 2.25 0.376471 9/25 2.25/(2.25+4)
2.50 0.407406 10/26 2.50/(2.50+4)
where (Rcn =1)and ψis a weight function. The finite element equation is
L
0
N(n)
αN(n)
βdxy
β−Rcn
L
0
N(n)
αcnδ(x−xi)N(n)
βdxyβ−λ2n
L
0
NαNβdxy
β=0(12)
The finite element equation is implemented in a computer program. The program is used for numerical analysis
of typical problems. The first try failed. Close insight into the program revealed that Rcn should be different
from one. The work is continued for two orders n=1&2. For n=1 the compliance c1=2 and for n=2the
compliance c2=1 are used. Through extensive numerical analyses, the correct values for Rcn are determined.
The results are shown in Table 1.
The results for n=1 is shown in columns 1 to 4. Corresponding to each length in column 1, the correct
factor is shown in column 2. The factor in column 2 is determined in decimal form. The decimal values are
written as fraction in column 3. The fractions have a regular form. The fractions in column 3 are written
heroically in terms of the length in column 1 as in column 4. Based on the fractions in column 4, the following
relation is proposed for the Ranjbaran factor
Rc1=L/(L+2)(13)
The value of c1is changed, and the corresponding results are obtained. A closed insight into the results revealed
the following equation
Rc1=L/(L+c1)(14)
The simple and beautiful Eq. (14) is the result of more than 2,000 finite element analyses and more than 12
months engagement with the problem.
The results for n=2 is shown in columns 5–8. Corresponding to each length in column 5, the correct factor
is shown in column 6. The factor in column 6 is determined in decimal form. The decimal values are written
as fraction in column 7. The fractions have a regular form. The fractions in column 7 are written heroically in
terms of the length in column 5 as in column 8. Based on the fractions in column 8, the following relation is
proposed for the Ranjbaran factor
Rc2=L/(L+4)(15)
The value of c2is changed, and the corresponding results are obtained. A close insight into the results revealed
the following equation
Rc2=L/(L+4c2)(16)
The simple and beautiful Eq. (16) is the result of more than 2,000 finite element analyses and more than 12
months engagement with the problem.
Based on the results obtained, the weak form equation is written as
L
0
ψ(n)y(n)dx−Rcn
L
0
ψ(n)cnδ(x−xi)y(n)dx−λ2n
L
0
ψydx=0,n=1,2 (17)
Cracked members detection, construction, and justification 1091
e
k
eq
c
k
(a)Serial model
e
fc
f
(b) Parallel model
Fig. 1 Models for cracked element
where the Ranjbaran factor is defined as
Rcn =L/L+n2cn(18)
2.3 Physical basis for Ranjbaran factor
The finite element equation for free vibration of cracked members as in Eq. (12) is considered.
L
0
N(n)
αN(n)
βdxy
β−Rcn
L
0
N(n)
αcnδ(x−xi)N(n)
βdxyβ−λ2n
L
0
NαNβdxy
β=0(12)
The finite element equation is implemented in a computer program. The program is used for numerical analysis
of typical problems. The first try failed. Close insight into the program revealed that Rcn should be different
from one. The problem is solved by considering the physics of the problem as follows.
A cracked member is modeled by two springs in series, Fig. 1a. The flexibility of the cracked member fce
is defined in terms of the flexibility of the intact member feplus the flexibility of the crack fcas
fce =fe+fc(19)
As a result, the stiffness of the cracked member kce is defined as
kce =1/(fe+fc)(20)
In the finite Eq. (12), the cracked member is modeled as two parallel springs, Fig. 1b one for the member and
one for the crack. The stiffness kce of the parallel system is defined as
kce =ke−keq
c(21)
where keis intact member stiffness, and keq
cis crack equivalent stiffness. The stiffness of two systems in Eqs.
(20)and(21) should be equal. As a result, the equivalent stiffness of the crack is obtained as
keq
c=fc/fe(fe+fc)(22)
For the problems under consideration fe=L/n2and fc=cn. As a result, Eq. (22) is written as
keq
c=cnn4/LL+n2cn(23)
The following equality is observed for the shape functions
L2
L
0
N(n)
αδN(n)
βdx=n4(24)
Multiply both side of Eq. (24)bycn/LL+n2cnto obtain
keq
c=L/L+n2cn
L
0
N(n)
αcnδN(n)
βdx(25)
Comparison of Eq. (25) with the second term in Eq. (12) leads to a definition for the Ranjbaran factor as
Rcn =L/L+n2cn(26)
The formulation in this section is considered as a physical basis for the Ranjbaran factor.
1092 A. Ranjbaran et al.
2.4 Mathematical basis for Ranjbaran factor
The governing equation for free vibration analysis of cracked member is defined as
y(n)−Rcncnδ(x−xi)y(n)(n)
−λ2ny=0,n=1,2 (27)
where (n)in the superscript denotes order of derivative with respect to x,cnis crack compliance, δ(x−xi)
is the Dirac delta with center at cracked point xi,yis variable function, λis a working parameter, and Rcn is
Ranjbaran factor.
The ntimes integration of Eq. (27) is written as
y(n)−Rcncnδ(x−xi)y(n)−z(x)=0,z=λ2n
n
ydx(28)
Multiply both sides of Eq. (28)byδ(x−xi)to obtain
δ(x−xi)y(n)−Rcncnδ(x−xi)δ(x−xi)y(n)−δ(x−xi)z(x)=0 (29)
Substitute for (δ(x−xi)δ(x−xi)=Anδ(x−xi)) [14] to obtain
δ(x−xi)y(n)=δ(x−xi)z(x)/(1−RcncnAn)(30)
Insert Eq. (30) into Eq. (28) to obtain the integrated governing equation as
y(n)−(1+Rcncnδ(x−xi)/(1−Rcn cnAn)) z(x)=0 (31)
As an alternative procedure, the integrated governing equation for cracked members is obtained by making
use of the golden derivative concept as follows. The governing equation for the intact part is
y(n)(n)
−λ2ny=0(32)
Equation (32)isntimes integrated as
y(n)−z(x)=0,z=λ2n
n
ydx(33)
The jump at a cracked point is defined as
y(n−1)
c=cny(n)H(x−xi)(34)
The derivative, golden derivative, of the jump is defined as
y(n)
c=cny(n)δ(x−xi)(35)
The integrated equation for the cracked member with the use of golden derivative is
y(n)−y(n)
c−z(x)=0 (36)
Or after substitution from Eq. (35)as
y(n)−(1+cnδ(x−xi)) z(x)=0 (37)
Equation (37) is an equivalent of Eq. (31).Theyaretwosidesofacoin.Asaresult
1=Rcn/(1−Rcn cnAn)(38)
Solution of Eq. (38)for Rcn leads to
Rcn =1/(1+cnAn)(39)
Substitution for An=n2/L[15] leads to the predefined value for Ranjbaran factor as follows
Rcn =L/L+n2cn(40)
The Ranjbaran factor was based on sound mathematical basis in this section.
Cracked members detection, construction, and justification 1093
0.8
0.85
0.9
0.95
1
1.05
0123456
RW
a(mm)
RE1
RS1
RT1
Fig. 2 Variation of the first frequency versus crack depth
2.5 The modified chain rules
Based on the formulation in previous sections, the following chain rules are proposed.
L
0
ψc0δ(x−xi)y(0)(0)
dx=Rc0
L
0
ψ(0)c0δ(x−xi)y(0)dx(41)
and
L
0
ψc1δ(x−xi)y(1)(1)
dx=−Rc1
L
0
ψ(1)c1δ(x−xi)y(1)dx(42)
and
L
0
ψc2δ(x−xi)y(2)(2)
dx=+Rc2
L
0
ψ(2)c2δ(x−xi)y(2)dx(43)
where the Ranjbaran factor is defined as
Rcn =L/L+n2cn,n=0,1,2(44)
These equations should be considered as new adds to the literature.
The materials presented in the previous sections recommend a necessary modification into the method of
weighted residual for the governing differential equations containing the derivatives of the Dirac’s delta dis-
tribution. Using the modified chain rules in Eqs. (41)–(44), the modified weak form equation will be obtained.
For the problems under consideration, only the finite element equations based on the modified weak form lead
to correct results. It must be noted that the proposed modification is only necessary for solution by the so-called
direct integration methods. In the case of transformation methods such as Laplace transform, no modification
is required.
3 Verification
The presented formulation is implemented in a computer program. The program is used for analysis of the
examples. The results are compared with that of the others and or known solutions. Excellent agreement of
the results verified the work. In the Figures, the frequencies are denoted by the ratio of the cracked frequency
over the intact frequency. Moreover, RW is frequency ratio, RE and RS are the frequency ratios computed by
the equivalent mass method and the stiffness reduction methods, respectively, RT is tested frequency ratio, RS
is the crack position to member length ratio, a is the crack depth in millimeter, ERF is the difference between
the test and stiffness reduction results in percent, and the number in front of them denote frequency number.
1094 A. Ranjbaran et al.
0.8
0.85
0.9
0.95
1
1.05
0123456
RW
a(mm)
RE2
RS2
RT2
Fig. 3 Variation of the second frequency versus crack depth
0.8
0.85
0.9
0.95
1
1.05
0123456
RW
a(mm)
RE3
RS3
RT3
Fig. 4 Variation of the third frequency versus crack de
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
012345
Difference %
a(mm)
ERF1
ERF2
ERF3
Fig. 5 The percent difference between the analysis and test results
Example 1 The cantilever steel beam considered is 400 mm long with a 14mm ×14 mm cross-sectional area.
Modulus of elasticity, density and Poisson’s ratio are 181 GPa,7,800 kg/m3,and0.3, respectively. The cracks
with different depths varying from 1 to 5 mm by a step of 1 mm were introduced in each beam. The location
of cracks are 133 and 240 mm from the fixed support.
Solution The beam is analyzed. The results are as follows. The beam is tested in the Moradi et al. [8]. The
results of the present work for the first three frequency ratios are compared with the test results in Figs. 2–4,
respectively. The difference between the analysis and test results is shown in Fig. 5. Excellent agreement of
the results verified the work.
Cracked members detection, construction, and justification 1095
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.2 0.4 0.6 0.8 1 1.2
RW
RX
RS1
RS2
RS3
RE1
RE2
RE3
Fig. 6 The comparison between results of two methods
Example 2 The beam in example 1is analyzed by the equivalent mass method and the stiffness reduction
method. The crack depth is 7 mm. The crack is positioned at 0–400 mm by a step of 40mm.
Solution The results for the first three frequency ratios are compared to Fig. 6. The accurate agreement of the
results verified the accuracy, efficiency, and robustness of the work.
4 Conclusions
The following conclusions regarding the analysis of cracked members are obtained from this study:
1. The Governing differential equation containing the derivatives of the Dirac delta distributions should be
modified before deriving their corresponding weak form equations.
2. The necessary modification is done by introduction of the Ranjbaran factor into the differential equation.
3. The logical definition for Ranjbaran factor is based on three different alternative methods making use of
numerical, physical, and mathematical theories. The full agreement between the results from the three
alternative methods verified the work.
4. Based on the present work, three new chain rules for use in the method of weighted residual are proposed.
5. Through analysis of typical examples and comparison of the results with that of the others, the accuracy
and efficiency of the work is verified.
6. The present formulation paved the way for the simple, accurate and robust static dynamic, and buckling
analysis of frames containing cracked members.
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