Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 4, pp. 1127-1139, Warsaw 2017
DOI: 10.15632/jtam-pl.55.4.1127

A MULTI-SPRING MODEL FOR BUCKLING ANALYSIS OF CRACKED

TIMOSHENKO NANOBEAMS BASED ON MODIFIED COUPLE

STRESS THEORY

Majid Akbarzadeh Khorshidi, Mahmoud Shariati

Ferdowsi University of Mashhad, Department of Mechanical Engineering, Mashhad, Iran

e-mail: mshariati44@um.ac.ir

This paper develops a cracked nanobeam model and presents buckling analysis of this de-
velopedmodel based on amodified couple stress theory. The Timoshenko beam theory and
simply supported boundary conditions are considered. This nonclassical model contains a
material length scale parameter and can interpret the size effect. The cracked nanobeam is
modeled as two segments connected by two equivalent springs (longitudinal and rotational).
This model promotes discontinuity in rotation of the beam and additionally considers di-
scontinuity in longitudinal displacement due to presence of the crack. Therefore, this multi-
-spring model can consider coupled effects between the axial force and bending moment
at the cracked section. The generalized differential quadrature (GDQ) method is employed
to discretize the governing differential equations, boundary and continuity conditions. The
influences of crack location, crack severity, material length scale parameter and flexibility
constants of the presented spring model on the critical buckling load are studied.

Keywords: buckling, crack, modified couple stress theory, Timoshenko nanobeam, spring
model

1. Introduction

Many applicable structures have been used inmicro- and nano-scale dimensions. Size effects are
significant in themechanical behavior of these structures. Since the classical continuummecha-
nics cannot predict the size effect, some size-dependent continuum theories have been developed
to capture the size effect using some material length scale parameters. These parameters are
related to inherent properties of materials, which become considerable in small scale structu-
res. In view of the difficulties in determining these internal parameters, non-classical continuum
theories involving only one material length scale parameter are desirable. A modified couple
stress theory has been proposed byYang et al. (2012). This theory uses only onematerial length
scale parameter to capture the size-dependent behavior of structures and employs only the sym-
metric part of the couple stress tensor as a suitable measure of the continuum micro-rotation.
Many studies have been done on the static and dynamic behavior of nanostructures based on
the modified couple stress theory. Here we focuse on the static and dynamic behavior of the
micro/nanobeams specially on the buckling of the nanobeams.
Park and Gao (2006, 2008) formulated a modified couple stress based model for the Euler-

-Bernoulli beam and that model was extended by Ma et al. (2008) for the Timoshenko beam
using Hamilton’s principle. Also, a microstructure dependent non-classical Reddy-Levinson be-
am model was developed by Ma et al. (2010), and the difference between the results obtained
from the non-classical and the classical models was illustrated for static bending and free vi-
bration. Simsek (2010) proposed analytical and numerical solution procedures for vibration of
an embedded microbeam under action of a moving microparticle and studied the influences of
thematerial length scale parameter, Poisson’s ratio, velocity of themicroparticle and the elastic



1128 M. AkbarzadehKhorshidi, M. Shariati

medium constant on the maximum dynamic deflections of the microbeam. A size-dependent
nonlinear Euler-Bernoulli beam model was presented by Xia et al. (2010). They studied nonli-
near size-dependent static bending, postbuckling and the free vibration of beams, and Asghari
et al. (2010) developed their model for Timoshenko beam theory. Ke and Wang (2011) investi-
gated dynamic stability of FGM Timoshenko microbeams and showed the significance of the
size effect on dynamic stability. Buckling analysis of microbeamswith higher order theories and
general boundary conditions was investigated by Mohammad-Abadi and Daneshmehr (2014).
They revealed accuracy of theGDQmethod for thatmodified couple stress based size-dependent
buckling problem. Also, Mohammad-Abadi and Daneshmehr (2015) developed vibration ana-
lysis of composite laminated beams in order of microns using the GDQ method. Akbarzadeh
Khorshidi and Shariati (2016b) presented a comprehensive solution for free vibration of a shear
deformable S-FGM nanobeam by the GDQ method. Dehrouyeh-Semnani et al. (2015) inve-
stigated dynamic characteristics of axially moving Timoshenko microbeams using Hamilton’s
principle and Galerkin’s method. An exact solution for prediction of postbuckling behavior of
shear deformable nanobeamswas presented byAkbarzadehKhorshidi andShariati (2015). Also,
Akbarzadeh Khorshidi and Shariati (2016a) investigated the propagation of the stress wave in
a shear deformable nanobeam and evaluated the effects of shear deformation, material length
scale parameter and Poisson’s ratio on the phase velocity of it.

It is well known that cracks increase flexibility of a structure. Therefore, the presence of a
crack leads to reduction of the stiffness of a structure. So, a simple and accuratemodel shouldbe
used to determine this reduction in stiffness. For this reason, the cracked beam ismodeled as two
segments connected by means of massless springs (Freund and Herrmann, 1976; Adams et al.,
1978).Rice andLevy (1972) viewed theplate as a strictly twodimensional continuumwitha local
reduction inbendingandextensional stiffness along the crack line.Thismodel,whichhasbecome
knownas the line springmodel for surfaceflaws,was alsodiscussed indetail byRice (1972).Most
of the studies which have been recently done utilized an equivalentmassless rotational spring at
the cracked section.Thus, at the cracked section, a discontinuity in rotation due to bendingmust
be considered. This model has been extensively used for vibration, buckling and postbuckling
analyses of cracked structures like cracked beams. Chaudhari and Maiti (2000) presented a
methodofmodelling for transverse vibrations of geometrically segmented slender cracked beams.
Lele and Maiti (2002) studied transverse vibration of short beams to detect the location of a
crack. Yang and Chen (2008) investigated free vibration and elastic buckling of beamsmade of
FGMs containing three open edge cracks. Ke et al. (2009) studied the postbuckling response of
edge cracked FG Timoshenko beams. Also, there are many other research works in the context
of cracked structures by using the equivalent springmodel, see for instance, Chati et al. (1997),
Krawczuk et al. (2003), El Bikiri et al. (2006), Kitipornchai et al. (2009), Yan et al. (2011). But,
thesemodels cannot satisfy themultiple discontinuities at the cracked section. Loya et al. (2006)
considered a rotational and an extensional spring at the cracked section for bending vibration of
a Timoshenko cracked beam. Thus, in addition to the discontinuity in rotation due to bending,
a discontinuity in the transverse deflection due to shearing has been also defined at the cracked
section. However, the contribution of the extensional spring to the strain energy of the system is
small in comparison with that of the rotational spring, nevertheless, this discontinuity has been
considered for coherencywith the general derivation of compliance for cracked beams (Okamura
et al., 1973; Tharp, 1987).

There are a few studies on cracked micro/nanostructures, which are mostly different types
of vibration analyses of cracked nanobeams based on a nonclassical continuum theory bymeans
of the rotational spring model. Flexural vibrations of cracked nanobeams based on the theory
of nonlocal elasticity applied to Euler-Bernouilli beamswas studied by Loya et al. (2009). They
proposed amodel containing a rotational and a longitudinal elastic spring at the cracked section
and defined compatibility relations of this model. This model presents a discontinuity in the



A multi-spring model for buckling analysis of cracked Timoshenko nanobeams... 1129

rotation and a discontinuity in the longitudinal displacement. But, they simplified their model
to one spring only (rotational spring) by some assumptions, and many authors followed them
for free transverse vibration analyses of cracked nanobeams (Hasheminejad et al., 2011; Torabi
and Nafar Dastgerdi, 2012; Hosseini-Hashemi et al., 2014). Of course, those assumptions can
be acceptable for free transverse vibration. Also, Hsu et al. (2011) investigated longitudinal
frequency of a cracked nanobeam by means of the longitudinal spring model. They expressed
continuity conditions as a relation between the longitudinal displacement and axial force, and
nelegted the discontinuity in rotation. Also, Loya et al. (2014) analysed torsional vibrations of a
nanorodwith a circumferential crack using a nonlocal elasticitymodel. They utilized a torsional
elastic spring at the cracked section to consider additional strain energy due to the presence of
the crack, so, a discontinuity in the torsion angle of the rod at the cracked section is introduced.
Therefore, an appropriatemodel can be selected to simulate the influence of the crack based on
the type of analysis and the applied loadings.
This papermakes one of the first attempts to investigate the buckling of cracked nanobeams

based on a modified couple stress theory. This modified couple stress based Timoshenko beam
model contains a material length scale parameter and can interpret the size effect. The present
model considers coupled effects between the axial force andbendingmomentusing a longitudinal
and a rotational spring at the cracked section. Govrning equations are derived by using the
principle ofminimumpotential energy. TheGeneralizedDifferential Quadrature (GDQ)method
is employed to solve the governing differential equations. Then the critical buckling loads for
different crack locations, crack severities and ratios of the material length scale parameter to
thickness are computed, and the obtained results are compared with those corresponding to the
classical beammodel.

2. Formulation

2.1. Modified couple stress theory

In this Section, a modified couple stress theory (Yang et al., 2012) with only one material
length scale parameter is employed to capture the size effect. According to this theory, the strain
energy U in an isotropic linear elastic beam under an axial compressive load P at both ends is
defined as (Mohammad-Abadi and Daneshmehr, 2014)

U =
1

2

∫

Ω

(σijδεij +mijδχij) dv−
1

2

L
∫

0

P
(∂w

∂x

)2
dx i,j =1,2,3 (2.1)

where ε, σ, χ and m are the strain tensor, Cauchy stress tensor, symmetric curvature tensor
and the deviatoric part of the couple stress tensor, respectively. These tensors are defined as

εij =
1

2
(∇u+(∇u)T)=

1

2
(ui,j +uj,i) σij =λtr(εii)δij +2µεij

χij =
1

2
(∇θ+(∇θ)T)= 1

2
(θi,j +θj,i) mij =2ℓ

2µχij

(2.2)

where ℓ is the material length scale parameter which is mathematically the square root of
the ratio of the modulus of curvature to the modulus of shear and is physically a property
measuring the effect of couple stress (Park andGao, 2006;Ma et al., 2008). This parameter can
be determined from torsion tests of slim cylinders of different diameters or bending tests of thin
beams of different thickness. Also, λ and µ are Lame’s constants and are defined as

λ=
νE

(1+ν)(1−2ν)
µ=

E

2(1+ν)
(2.3)



1130 M. AkbarzadehKhorshidi, M. Shariati

whereE is Young’s modulus and ν is Poisson’s ratio. The rotation vector θ is given as

θ=
1

2
curl(u) (2.4)

2.2. Displacement field

Consider a beam of length L, width b and thickness h. The Timoshenko beam theory is
employed to describe the effect of shear deformation. According to the rectangular Cartesian
coordinate system shown in Fig. 1, where the x-axis is coincident with the centroidal axis of
the undeformed beam, the y-axis is the neutral axis and the z-axis is the symmetry axis, the
displacement field in a Timoshenko beam is described as

u1 =u(x,t)−zϕ(x,t) u2 =0 u3 =w(x,t) (2.5)

whereu1, u2 and u3 are, respectively, the x-, y- and z-components of the displacement vectoru,
u andw are, respectively, thex- and z-components of the displacement vector of a point located
on the beam axis, and ϕ is the angle of rotation of the cross-section.

Fig. 1. Beam configuration and coordinate system

From Eqs. (2.2)1 and (2.5), the nonzero strains are given as

εxx =
∂u1
∂x
=
∂u

∂x
−z
∂ϕ

∂x
εxz =

1

2

(∂u1
∂z
+
∂w

∂x

)

=
1

2

(∂w

∂x
−ϕ
)

(2.6)

and from Eqs. (2.2)2 and (2.5), the nonzero stresses are given as

σxx =(λ+2µ)εxx =(λ+2µ)
(∂u

∂x
−z
∂ϕ

∂x

)

σxz =2µεxz =µ
(∂w

∂x
−ϕ
)

(2.7)

Using Eqs. (2.4) and (2.5), the nonzero component of the rotation vector is obtained as

θy =−
1

2

(

ϕ+
∂w

∂x

)

(2.8)

and from Eqs. (2.2)3,4 and (2.8), we have

χxy =−
1

4

(∂ϕ

∂x
+
∂2w

∂x2

)

mxy =−
1

2
ℓ2µ
(∂ϕ

∂x
+
∂2w

∂x2

)

(2.9)

2.3. Governing equations

Substituting Eqs. (2.6), (2.7) and (2.9) into Eq. (2.1), the total strain energy of the Ti-
moshenko nanobeam is achieved. Then, using the principle of minimum potential energy and



A multi-spring model for buckling analysis of cracked Timoshenko nanobeams... 1131

fundamental lemma of the calculus of variation, the governing equations of the Timoshenko
nanobeam in terms of displacements are obtained as

(λ+2µ)A
∂2u

∂x2
=0

KsµA
(∂2w

∂x2
−
∂ϕ

∂x

)

−
1

4
ℓ2µA

(∂4w

∂x4
+
∂3ϕ

∂x3

)

−P
∂2w

∂x2
=0

KsµA
(∂w

∂x
−ϕ
)

+
1

4
ℓ2µA

(∂3w

∂x3
+
∂2ϕ

∂x2

)

+(λ+2µ)I
∂2ϕ

∂x2
=0

(2.10)

whereA and I are, respectively, the beam cross-sectional area and the secondmoment of cross-
-sectional area. Also,Ks is the Timoshenko shear coefficient which is introduced as a correction
factor to account for the non-uniformity of the shear strain over the beam cross-section (Park
and Gao, 2006) and is defined with Poisson’s ratio asKs =(5+5ν)/(6+5ν).

Note that all of the body forces and body couples are neglected in this study. Therefore, the
governing equations presented in Eqs. (2.10) are identical to those given by Mohammad-Abadi
and Daneshmehr (2014).

the boundary conditions for a simply supported beam are stated as

u=w=Y =M =0 at x=0,L (2.11)

where Y is the couple moment, which is a resultant of the couple stress component mxy. Also,
M is the conventional bendingmoment. These moments are defined as

Y =

∫

A

mxy dA=
1

2
ℓ2µA

(∂2w

∂x2
+
∂ϕ

∂x

)

M =

∫

A

σxxz dA=(λ+2µ)I
∂ϕ

∂x
(2.12)

So, the boundary conditions defined in Eq. (2.11), can be writen as

u=w=
∂ϕ

∂x
=
∂2w

∂x2
=0 at x=0,L (2.13)

2.4. Cracked nanobeam model

Consider a crack at a distance Lc(e = Lc/L) at the left end of the beam (Fig. 2). It is
assumed that the crack is perpendicular to the beam surface and always remains open. As
shown in Fig. 2, the cracked beam ismodeled as two segments connected by twomassless elastic
springs (longitudinal and rotational) (Loya et al., 2009). This modeling promotes flexibility at
the cracked section and introduces discontinuity in rotation and discontinuity in longitudinal
displacementof thebeam,whichareproportional to coupledeffects of theaxial force andbending
moment. Therefore, the continuity conditions are developed for the Timoshenko nanobeam as
(at x=Lc)

w1 =w2 N1 =N2 M1 =M2

Q1−Pϕ1 =Q2−Pϕ2 ϕ2−ϕ1 ∝ f(N,M) u2−u1 ∝ g(N,M)
(2.14)

where subscripts 1 and 2 refer to the left segment and right segment of the beamdivided by the
crack. Also,N andQ are, respectively, the axial force and the shear force, which are defined as

N =

∫

A

σxx dA=(λ+2µ)A
∂u

∂x
Q=

∫

A

σxz dA=KsµA
(∂w

∂x
−ϕ
)

(2.15)



1132 M. AkbarzadehKhorshidi, M. Shariati

Using Eqs. (2.12) and (2.15), the continuity conditions can be written as (at x=Lc)

w1 =w2
∂u1
∂x
=
∂u2
∂x

∂ϕ1
∂x
=
∂ϕ2
∂x

KsµA
(∂w1
∂x
−
∂w2
∂x
+ϕ2−ϕ1

)

=P(ϕ1−ϕ2)

ϕ2−ϕ1 =KMM
∂ϕ

∂x
+KMN

∂u

∂x
u2−u1 =KNN

∂u

∂x
+KNM

∂ϕ

∂x

(2.16)

where KMM, KMN, KNM and KNN are the flexibility constants and in nondimensional forms
they are given as k1 =KMM/L, k2 =KMN/L, k3 =KNN/L, k4 =KNM/L.

Then, governing equations (2.10) must be applied for each segment (r=1,2) of the cracked
beam, so, we have

(λ+2µ)A
∂2ur
∂x2r
=0

KsµA
(∂2wr
∂x2r
−
∂ϕr
∂xr

)

−
1

4
ℓ2µA

(∂4wr
∂x4r
+
∂3ϕr
∂x3r

)

−P
∂2wr
∂x2r
=0

KsµA
(∂wr
∂xr
−ϕr

)

+
1

4
ℓ2µA

(∂3wr
∂x3r
+
∂2ϕr
∂x2r

)

+(λ+2µ)I
∂2ϕr
∂x2r
=0

(2.17)

where 0¬x1 ¬L1 andL1 ¬x2 ¬L.

Fig. 2. Model of the cracked beam

3. Solution procedure

TheGDQmethod (Shu, 1991; Bellman et al., 1972; Shu andDu, 1997) is employed to discretize
the governing equations, boundary conditions and continuity conditions, therefore, the differen-
tial equations are simplified to algebraic equations. In thismethod, the region is discretized into
several sample points. The sample points are obtained as

xi =
1

2

(

1− cos
(i−1)π

N−1

)

i=1,2, . . . ,N (3.1)

According to the GDQmethod (to find details, see Akbarzadeh Khorshidi and Shariati, 2016b;
Shu and Du, 1997), governing equations (2.17) are discretized as

(λ+2µ)A
N
∑

j=1

c
(2)
ij ur(xj)= 0 (3.2)



A multi-spring model for buckling analysis of cracked Timoshenko nanobeams... 1133

and

KsµA
(

N
∑

j=1

c
(2)
ij wr(xj)−

N
∑

j=1

c
(1)
ij ϕr(xj)

)

−
1

4
ℓ2µA

(

N
∑

j=1

c
(4)
ij wr(xj)+

N
∑

j=1

c
(3)
ij ϕr(xj)

)

−P
(

N
∑

j=1

c
(2)
ij wr(xj)

)

=0

KsµA
(

N
∑

j=1

c
(1)
ij wr(xj)−ϕr(xj)

)

+
1

4
ℓ2µA

(

N
∑

j=1

c
(3)
ij wr(xj)+

N
∑

j=1

c
(2)
ij ϕr(xj)

)

+(λ+2µ)I
(

N
∑

j=1

c
(2)
ij ϕr(xj)

)

=0

(3.3)

where i,j =1, . . . ,N and c
(n)
ij is the weighting coefficient which can be found from the recursive

formula (Akbarzadeh Khorshidi and Shariati, 2016b; Shu andDu, 1997).
The discretized boundary conditions are expressed as

u1(x1)=w1(x1)=
N
∑

j=1

c
(1)
1j ϕ1(xj)=

N
∑

j=1

c
(2)
1j w1(xj)= 0

u2(xN)=w2(xN)=
N
∑

j=1

c
(1)
Nj
ϕ2(xj)=

N
∑

j=1

c
(2)
Nj
w2(xj)= 0

(3.4)

also, continuity conditions (2.16) are discretized as

w1(xN)=w2(x1)
N
∑

j=1

c
(1)
Nju1(xj)=

N
∑

j=1

c
(1)
1j u2(xj)

N
∑

j=1

c
(1)
Nj
ϕ1(xj)=

N
∑

j=1

c
(1)
1j ϕ2(xj)

KsµA
(

N
∑

j=1

c
(1)
Njw1(xj)−

N
∑

j=1

c
(1)
1j w2(xj)+ϕ2(x1)−ϕ1(xN)

)

=P
(

ϕ1(xN)−ϕ2(x1)
)

ϕ2(x1)−ϕ1(xN)=KMM

N
∑

j=1

c
(1)
Njϕ1(xj)+KMN

N
∑

j=1

c
(1)
Nju1(xj)

u2(x1)−u1(xN)=KNM

N
∑

j=1

c
(1)
Nj
u1(xj)+KNN

N
∑

j=1

c
(1)
Nj
ϕ1(xj)

(3.5)

Using discretized equations (3.2)-(3.5), a set of algebraic equations is established, and by solving
the eigenvalue problem, the critical buckling load of cracked nanobeams is obtained.

4. Results and discussion

This Section presents the influence of crack location, crack severity and material length scale
parameter to thickness ratio on the nondimensional critical buckling load of simply supported
cracked nanobeams. The nondimensional critical buckling load is defined as P =PcrL

2/EI. To
validate the presented results, the critical buckling load of an intactmicrobram is comparedwith
the obtained results by Mohammad-Abadi and Daneshmehr (2014). An excellent agreement is
shown in Table 1.



1134 M. AkbarzadehKhorshidi, M. Shariati

Table1.Nondimensional critical buckling loadP of intactmicrobeams (E =1.44GPa,L=20h,
ℓ=17.6µm, b=2h, ν =0.38)

h [µm]
17.6 52.8 88 123.2

Ref. [25] 60.3571 22.9608 19.9491 19.1188

Present paper 60.4853 22.9962 19.9795 19.1479

Ref. [25] –Mohammad-Abadi and Daneshmehr (2014)

To present a parametric study, a Timoshenko beam with L = 20h, b = 2h and ν = 0.38
with simply supported boundary conditions and an open edge crack is considered. The effects
of the flexibility constants k1, k2, k3 and k4 are shown in Tables 2-5. These tables present
nondimensional critical buckling loads fordifferentflexibility constants.Asmentioned, this study
introduces local flexibility bymeans of four constants which relate discontinuities of the cracked
section with the axial force and bending moment. Therefore, different conditions of these four
constants state different crack severities. It is found that the flexibility constant related to the
bending moment k1 is more effective on the buckling behavior of the cracked nanobeam than
the other constants. In fact, the discontinuity in rotation due to bending has a great effect on
the local flexibility introduced by the crack. However, the other constants k2, k3 and k4 have a
scant influence on the local flexibility, but this influence becomes larger when the crack severity
is increased (Tables 3-5). Also, the crossover flexibility constants caused by the coupled effects
between the axial force and bendingmoment (k2 and k4) have similar influences on the critical
buckling load (Tables 3 and 5).

Table 2.Effect of k1 on the nondimensional critical buckling load P (ℓ/h=1 and e=0.5)

k1 k2 = k3 = k4 =0 k2 = k3 = k4 =0.5 k2 = k3 = k4 =1 k2 = k3 = k4 =2

0 60.4853 59.4491 58.4024 55.5903

0.25 40.3538 39.7141 39.0139 37.1375

0.5 29.0547 28.5334 28.0332 26.6845

1 18.8317 18.1148 17.7901 16.9438

2 10.7610 9.8532 9.6804 9.2153

Table 3.Effect of k2 on the nondimensional critical buckling load P (ℓ/h=1 and e=0.5)

k2 k1 = k3 = k4 =0 k1 = k3 = k4 =0.5 k1 = k3 = k4 =1 k1 = k3 = k4 =2

0 60.4853 28.3580 18.3580 9.9372

0.25 60.4853 28.6881 18.2012 9.8551

0.5 60.4853 28.5334 18.0786 9.7668

1 60.4852 28.2029 17.7901 9.5905

2 60.4852 27.5220 17.1818 9.2153

Figure 3 shows the effect of the material length scale parameter to thickness ratio ℓ/h on
the critical buckling load of Timoshenko cracked nanobeams. It is observed that the critical
buckling load increases as ℓ/h increases, and this increase is decreased by the growing severity of
the crack.Note that ℓ/h=0virtually indicates the results obtained fromthe classical continuum
theory which neglects the couple stress effect.

The effects of crack location on the critical buckling load of simply supported nanobeams
for different crack severities and ℓ/h ratios are respectively demonstrated in Figs. 4 and 5. The
critical buckling load is sensitive to crack location and is decreased when the crack gets near to



A multi-spring model for buckling analysis of cracked Timoshenko nanobeams... 1135

Table 4.Effect of k3 on the nondimensional critical buckling load P (ℓ/h=1 and e=0.5)

k3 k1 = k2 = k4 =0 k1 = k2 = k4 =0.5 k1 = k2 = k4 =1 k1 = k2 = k4 =2

0 60.4853 28.6977 18.4197 11.1871

0.25 60.4853 28.6128 18.2461 10.5983

0.5 60.4853 28.5334 18.0981 10.1732

1 60.4852 28.4527 17.7901 9.6471

2 60.4852 27.3721 17.5985 9.2153

Table 5.Effect of k4 on the nondimensional critical buckling load P (ℓ/h=1 and e=0.5)

k4 k1 = k2 = k3 =0 k1 = k2 = k3 =0.5 k1 = k2 = k3 =1 k1 = k2 = k3 =2

0 60.4853 28.8544 18.3579 9.9384

0.25 60.4853 28.6879 18.2010 9.8552

0.5 60.4853 28.5334 18.0787 9.7673

1 60.4852 28.2045 17.7901 9.5911

2 60.4851 27.5220 17.1817 9.2153

Fig. 3. Effects of ℓ/h ratio and crack severity on the nondimensional critical buckling load
(k2 = k3 = k4 =1 and e=0.5)

beam midpoint. Also, the influence of the crack tends to be very small as the crack gets closer
to the beam ends. By increasing the crack severity, the effect of crack location is increased; in
other words, the crack severity ismore effective when the crack is located at themidpoint of the
beam (e=0.5). Figure 5 indicates that the effect of crack location increases as the ℓ/h increases.
Also, it is found that the ℓ/h ratio at different crack locations has different effects on the critical
buckling load of cracked nanobeams, but these effects get similar when the crack gets near to
the midpoint.



1136 M. AkbarzadehKhorshidi, M. Shariati

Fig. 4. Effects of crack location and crack severity on the nondimensional critical buckling load
(k2 = k3 = k4 =1 and ℓ/h=1)

Fig. 5. Effects of crack location and ℓ/h ratio on the nondimensional critical buckling load
(k1 = k2 = k3 = k4 =1)



A multi-spring model for buckling analysis of cracked Timoshenko nanobeams... 1137

5. Conclusion

The buckling behavior of cracked nanobeams is studied within the framework of Timoshenko
beam theory anda modified couple stress theory. A material length scale parameter is used to
capture the size effect. The crack is modeled as two massless elastic springs (rotational and
longitudinal) which promote discontinuities at the cracked section. This study employs four
flexibility constants which introduce crack severity by two discontinuities which are proportional
to the bending moment and axial force transmitted through the cracked section. The principle
of minimum potential energy is employed to derive the governing differential equations which
are discretized using theGDQmethod. The effects of crack severity, crack location andmaterial
length scale parameter to thickness ratio are studied.
The obtained results show that the first flexibility constant which connects the discontinuity

in rotation with the bending moment makes the main part of the crack severity. The material
length scaleparameter to thickness ratio ℓ/h considerablyaffects thevalueof the critical buckling
loadwhich increases as the ℓ/h increases.The influenceof ℓ/h is stronger at lower crack severities,
inageneralmanner, reductionof the crack effects (crack severity and location) leads toadecrease
in the influence of ℓ/h. The critical buckling loads are significantly affected by crack location
and are continuously decreasedwhen the crack gets near to themidpoint of the nanobeam.Also,
the crack location is more effective at high crack severities and ℓ/h ratios.

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Manuscript received February 6, 2017; accepted for print April 14, 2017