Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
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Theoretical and Applied Fracture Mechanics
journal homepage: www.elsevier.com/locate/tafmec
Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams,
using a 3D SGBEM–FEM Alternating Method
Longgang Tian a,b, Leiting Dong c,⇑, Sharada Bhavanam b, Nam Phan d, Satya N. Atluri b
a
Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China
Center for Aerospace Research & Education, University of California, Irvine, USA
c
Department of Engineering Mechanics, Hohai University, Nanjing, China
d
Structures Division, Naval Air Systems Command, Patuxent River, MD, USA
b
a r t i c l e
i n f o
Article history:
Available online 18 October 2014
Keywords:
SGBEM–FEM Alternating Method
I-beam
Stress intensity factor
Fatigue crack growth
a b s t r a c t
In the present paper, computations of mixed mode stress intensity factor (SIF) variations along the crack
front, and fatigue-crack-growth simulations, in cracked I-beams, considering different load cases and
initial crack configurations, are carried out by employing the three-dimensional SGBEM (Symmetric
Galerkin Boundary Element Method)–FEM (Finite Element Method) Alternating Method. For mode-I
cracks in the I-beam, the computed SIFs by using the SGBEM–FEM Alternating Method are in very good
agreement with available empirical solutions. The predicted fatigue life of cracked I-beams agrees well
with experimental observations in the open literature. For mixed-mode cracks in the web or in the flange
of the I-beams, no analytical or empirical solutions are available in the literature. Thus mixed-mode SIFs
for mixed-mode web and flange cracks are presented, and non-planar fatigue growth simulations are
given, as benchmark examples for future studies. Moreover, because very minimal efforts of preprocessing and very small computational burden are needed, the current SGBEM–FEM Alternating Method is
very suitable for fracture and fatigue analyses of 3D structures such as I-beams.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
The calculation of fracture mechanics parameters (such as the
Mode I, II and III stress intensity factors), for arbitrary surface
and embedded cracks in complex 3D structures, remains an important task for the structural integrity assessment and damage tolerance analyses [1]. The application of linear elastic fracture
mechanics to fracture and fatigue analyses of various types of
structures has been hindered by the lack of analytical solutions
of stress intensity factors, which characterize the magnitude of
the singular stress field near the crack-tip/crack-front. The strength
of a cracked structure and the fatigue crack growth rate under cyclic loading can be determined once the corresponding stress intensity factors are computed. A rational fatigue life estimation of the
cracked structure can be made based on these analyses, for the
design, maintenance and damage tolerance of civil, mechanical
and aerospace structures.
I-beams are widely used as typical structural components in
structural engineering. Since the I-beam has the characteristics of
both three-dimensional finite bodies as well as slender bars, it is
⇑ Corresponding author.
E-mail address: dong.leiting@gmail.com (L. Dong).
http://dx.doi.org/10.1016/j.tafmec.2014.10.002
0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
quite difficult to find the exact solution for stress intensity factors
for cracked I-beams by the existing classical analytical methods. By
applying the conservation laws and elementary beam theory, Kienzler and Hermann [2] obtained the approximate stress intensity
factors for cracked beams with different crack geometries and a
rectangular cross section. Hermann and Sosa [3] applied the
method in [2] to find the stress intensity factors of cracked pipes
under different loading conditions. The stress intensity factors
are normally expressed as a function of the finite-size correction
factor b in literature [2]. Gao and Hermann [4] applied asymptotic
techniques based on the method introduced in [2] to calculate a
better approximation for the correction factor b. Müller et al. [5]
extended the method of Kienzler and Hermann [2] to assess the
stress intensity factors of circumferentially cracked cylinders and
rectangular beams under different loading conditions. Dunn et al.
[6] calculated the stress intensity factors of I-beams under a bending moment, by extending the work of [2] along with dimensional
considerations and a finite element calibration.
Another approach for determining stress intensity factors of
engineering structures is by using the G⁄-integral, based on conservation laws and the concept of crack surface widening energy
release rate. Xie et al. [7–9] showed that the crack surface widening
energy release rate can be expressed by the G⁄-integral along with
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L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
elementary strength of materials theory for slender cracked structures. They applied the G⁄-integral to calculate the stress intensity
factors of thin-walled tubes with a cracked rectangular cross section, and they also applied this method to analyze homogeneous
and composite multi-channel beams [10]. Kishen and Kumar [11]
employed a cracked beam-column element, introduced by Tharp
[12], in a finite element simulation to study the fracture behavior
of cracked beam-columns with different load eccentricities.
It should be noted that, all the above-mentioned analytical or
empirical solutions can only be used for analyzing symmetric
model-I cracks in I-beams. They are not valid for fatigue crack
growth simulations of mixed-mode cracks in I-beam, such as the
inclined cracks in the flanges. Numerical methods are necessary
for fracture and fatigue analyses of mixed-mode cracks in complex
3D structures.
In spite of its wide-spread popularity, the traditional Finite Element Method (FEM), with simple polynomial interpolations, is
unsuitable for modeling cracks and their propagations. This is partially due to the high-inefficiency of approximating stress & strainsingularities using polynomial FEM shape functions. In order to
overcome this difficulty, embedded-singularity elements by Tong
et al. [13], Atluri et al. [14], and singular quarter-point elements
by Henshell and Shaw [15] and Barsoum [16], were developed in
order to capture the crack-tip/crack-front singular field. Many such
related developments were summarized in the monograph by
Atluri [17] and they are now widely available in many commercial
FEM software. However, the need for constant re-meshing makes
the automatic fatigue-crack-growth analyses, using traditional
FEM, to be very difficult.
In a fundamentally different mathematical way, after the derivation of a complete analytical solution for an embedded elliptical
crack in an infinite body whose faces are subjected to arbitrary
tractions [19], the first paper on a highly-accurate Finite Element
(Schwartz–Neumann) Alternating Method (FEAM) was published
by Nishioka and Atluri [18]. The FEAM uses the Schwartz–Neumann alternation between a crude and simple finite element solution for an uncracked structure, and the analytical solution for the
crack embedded in an infinite body. Subsequent 2D and 3D variants of the Finite Element Alternating Methods were successfully
developed and applied to perform structural integrity and damage
tolerance analyses of many practical engineering structures [1].
Recently, the SGBEM (Symmetric Galerkin Boundary Element
Method)–FEM (Finite Element Method) Alternating Method, which
involves the alternation between the very crude FEM solution of
the uncracked structure, and an SGBEM solution for a small region
enveloping the arbitrary non-planar 3D crack, was developed for
arbitrary three-dimensional non-planar growth of embedded as
well as surface cracks by Nikishkov et al. [20], and Han and Atluri
[21]. An SGBEM ‘super element’ was also developed in Dong and
Aluri [22–25] for the direct coupling of SGBEM and FEM in fracture
and fatigue analysis of complex 2D & 3D solid structures and materials. The motivation for this series of works, by Atluri and many of
his collaborators since the 1980s, is to explore the advantageous
features of each computational method: model the complicated
uncracked structures with simple FEMs, and model the crack-singularities by mathematical methods such as complex variables,
special functions, boundary integral equations (BIEs), and by
SGBEMs.
In the present paper, by employing the SGBEM–FEM Alternating
Method, the stress intensity factors of I-beams with different crack
configurations are computed; fatigue-crack-growth process of the
I-beams subjected to cyclic loadings are examined. The stress
intensity factors of the crack-front are computed during each step
of crack increment. Crack growth rates are determined by the Paris
Law. The crack paths and number of loading cycles are predicted
for damage tolerance evaluation. The validations of SGBEM–FEM
Alternating Method are illustrated by the comparison of numerical
results with available empirical solutions as well as experimental
observations. For mode-I cracks in the I-beam, the computed SIFs
by using the SGBEM–FEM Alternating Method are in very good
agreement with empirical solutions. And the predicted fatigue life
of cracked I-beams agrees well with experimental observations in
the open literature. For inclined cracks in the web or in the flange
of the I-beams, no analytical or empirical solutions are available in
the literature. Thus mixed-mode SIFs of inclined web and flange
cracks are presented and non-planar fatigue growth simulations
are given, as benchmark examples for future investigators.
2. SGBEM–FEM Alternating Method: theory and formulation
The Symmetric Galerkin Boundary Element Method (SGBEM)
has several advantages over traditional and dual BEMs by Rizzo
[26], Hong and Chen [27], such as resulting in a symmetrical coefficient matrix of the system of equations, and the avoidance of the
need to treat sharp corners specially. Early derivations of SGBEMs
involve regularization of hyper-singular integrals, see the work by
Frangi and Novati [28]; Bonnet et al. [29]; Li et al. [30]; Frangi et al.
[31]. A systematic procedure to develop weakly-singular symmetric Galerkin boundary integral equations was presented by Han
and Atluri [32,33]. The derivation of this simple formulation
involves only the non-hyper singular integral equations for tractions, based on the original work reported in Okada et al. [34,35].
It was used to analyze cracked 3D solids with surface flaws by
Nikishkov et al. [20] and Han and Atluri [21].
For a domain of interest as shown in Fig. 1, with source point x
and target point n, 3D weakly-singular symmetric Galerkin BIEs for
displacements and tractions are developed by Han and Atluri [32].
The displacement BIE is:
1
2
Z
v p ðxÞup ðxÞdSx ¼
Z
@X
v p ðxÞdSx
@X
þ
Z
Z
v p ðxÞdSx
@X
þ
Z
@X
t j ðnÞup
j ðx; nÞdSn
@X
Z
@X
v p ðxÞdSx
Z
@X
Di ðnÞuj ðnÞGp
ij ðx; nÞdSn
ni ðnÞuj ðnÞup
ij ðx; nÞdSn
ð1Þ
And the corresponding traction BIE is:
1
2
Z
@X
Z
Da ðxÞwb ðxÞdSx
tq ðnÞGq
ab ðx; nÞdSn
@X
@X
Z
Z
wb ðxÞdSx
na ðxÞt q ðnÞuq
ab ðx; nÞdSn
@X
@X
Z
Z
þ
Da ðxÞwb ðxÞdSx
Dp ðnÞuq ðnÞHabpq ðx; nÞdSn
wb ðxÞtb ðxÞdSx ¼
Z
@X
@X
ð2Þ
Fig. 1. A solution domain with source point x and target point n.
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L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
In Eqs. (1) and (2), Da is a surface tangential operator:
@
@ns
@
Da ðxÞ ¼ nr ðxÞersa
@xs
As shown in Fig. 2, By applying Eq. (1) to Su, where displacements are prescribed, and applying Eq. (2) to St, where tractions
are prescribed, a symmetric system of equations can be obtained:
Da ðnÞ ¼ nr ðnÞersa
ð3Þ
q
q
Kernels functions up
j ; Gab ; uab ; Habpq can be found in the paper by
Han and Atluri [32], which are all weakly-singular, making the
implementation of the current BIEs very simple.
Fig. 2. A defective solid with arbitrary cavities and cracks.
2
App
6
4 Aqp
Arp
Apq
Aqq
Arq
38 9 8 9
Apr <
>p>
< fp >
= >
=
7
Aqr 5 q ¼ f q
>
: >
: >
; >
;
Arr
r
fr
ð4Þ
where p, q, r denotes the unknown tractions at Su, unknown displacements at S0t , and unknown displacement discontinuities at Sc
respectively.
With the displacements and tractions being first determined at
the boundary and crack surface, the displacements, strains and
stress at any point in the domain can be computed using the
non-hyper singular BIEs in the paper by Okada et al. [34,35]. Therefore, the path-independent or domain-independent integrals can
be used to compute the stress intensity factors. Alternatively, with
the singular quarter-point boundary elements at the crack face,
stress intensity factors can also be directly computed using the displacement discontinuity at the crack-front elements, see the paper
by Nikishkov et al. [20] for detailed discussion. For fatigue growth
of cracks, there is no need to use any other special technique to
describe the crack surface, such as the Level Sets. The crack surface
is already efficiently described by boundary elements. In each fatigue step, a minimal effort is needed: one can simply extend the
crack by adding a layer of additional elements at the crack-tip/
crack-front. This greatly saves the computational time for fatigue-crack-growth analyses.
However, the SGBEM is unsuitable for carrying out large scale
simulation of complex structures. This is because of the fact that
the coefficient matrix for SGBEM is fully-populated. To further
explore the advantages of both FEM and SGBEM, Han and Atluri
[21] coupled FEM and SGBEM indirectly, using the Schwartz–Neumann Alternating Method. As shown in Fig. 3, simple FEM is used
Fig. 3. Superposition principle for SGBEM–FEM Alternating Method: (a) the uncracked body modeled by simple FEM, (b) the local domain containing cracks modeled by
SGBEM, (c) FEM model subjected to residual stresses, and (d) alternating solution for the original problem.
L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
to model the uncracked global structure, and SGBEM is used to
model the local cracked subdomain. By imposing residual stresses
at the global and the local boundaries in an alternating procedure,
the solution of the original problem is obtained by superposing the
solution of each individual sub-problem.
The great flexibility of this SGBEM–FEM Alternating Method is
obvious. The SGBEM mesh of the cracked sub-domain is totally
independent of the crude FEM mesh of the uncracked global structure. Because the SGBEM is used to capture the stress singularity at
crack-tips/crack fronts, a very coarse mesh can be used for FEM
model of the uncracked global structure. Since FEM is used to
model the uncracked global structure, large-scale structures can
be efficiently modeled.
In this study, the SGBEM–FEM Alternating Method is used to
simulate the fatigue growth of both mode-I and mixed-mode
cracks in I-beams. The detailed discussions of numerical simulations and their comparison to available empirical solutions and
experimental results are given in the next section.
3. Numerical examples
In this section, SGBEM–FEM Alternating Method is used to analyze mixed-mode fracture and fatigue growth of cracks in I-beams.
Comparisons are made between the results obtained by SGBEM–
FEM Alternating Method, and other empirical solutions as well as
experimental data reported in the literature. All the numerical simulations are carried out on a PC with an Intel Core i7 CPU.
3.1. An I-beam with mode-I crack subjected to bending moment and
axial tensile load
A simply-supported I-beam with a crack in its middle span is
considered as the first example. The crack cuts through the bottom
flange and the crack-front is located in the web of the I-beam. The
I-beam is subjected to a uniformly-distributed surface pressure qt
on its top flange (which results in a bending moment Mc on the
crack surface) and a uniformly-distributed axial tensile stress qa
on its cross section of the free roller side (which brings an axial
tensile load N to the crack surface).
Fig. 4 shows a half of the cracked I-beam, where Yc is the difference between the position of the neutral axis in the uncracked and
cracked cross sections, M is the bending moment, Q is the transverse shear force and N is the axial force. Subscript ‘c’ denotes
the cracked cross section of the I-beam and ‘uc’ denotes the
uncracked cross section of the I-beam. Geometrical parameters bf
is the width of the flange, tf is the thickness of the flange, h is the
height of the web, tw is the thickness of the web, a is the crack
length, and L is the span of the I-beam.
191
As shown in Fig. 5, in order to obtain the approximate solution
of SIFs of the cracked I-beam specimen, the surface crack is often
approximated as a semi-elliptical hole. By employing the classical
bending theory of beams and G⁄-integral, Ghafoori and Motavalli
[36] acquired the empirical relation for the plane strain SIFs of
the cracked I-beam as:
"
KI ¼
!
#12
N2 ðMc Y c NÞ2
p
2
2
þ N ðk1 þ k2 Þ þ Mc ðg1 þ g2 Þ
A
I
tw ð1 m2 Þ
ð5Þ
where A is the area of the uncracked cross section, I is the moment
of the inertial of the uncracked cross section, v is Poisson’s ratio, and
k1 ; k2 ; g1 ; g2 are the coefficients related with elliptic integrals.
This problem is solved in this study using SGBEM–FEM
Alternating Method. Geometrical parameters are taken
as: bf = 64 mm; tf = 10 mm; h = 100 mm; tw = 8 mm; L = 400 mm. A
uniformly-distributed pressure qt = 10.0 N/mm2 is applied to the
upper surface of the I-beam specimen, and a uniformly-distributed
axial tensile stress qa = 5.0 N/mm2 is applied to cross-section.
Young’s modulus E = 200 GPa and Poisson’s ratio m = 0.30 are considered. And the independent meshes for the uncracked structure
and the crack are given in Fig. 6.
The computed stress intensity factors are compared with the
empirical solution of Eq. (5) in Table 1 and Fig. 7, considering the
crack depth a = 70 mm. Note that the stress intensity factors of
every node at the crack-front are computed and listed, and for this
specific case, the empirical solution of the stress intensity factor is
14,720. The comparison of the normalized stress intensity factors
versus normalized crack length between SGBEM–FEM Alternating
Method and empirical solution is made In Fig. 8. As can be seen,
the SGBEM–FEM Alternating Method produces almost the same
result as [36] for the stress intensity factor at every value of the
crack length even with a very coarse mesh. Since the computation
burden for the SGBEM–FEM Alternating Method is very small, and
the independent meshing of the structure and the crack takes only
minimal human labor, it is very convenient to simulate the entire
fatigue crack growth process up to failure, for the damage tolerance analyses of cracked I-beams.
3.2. Fatigue crack growth in a four-point bending cracked I-beam
specimen
A steel I-beam of type S355J0 (ST 52-3) is cyclically-loaded in a
four-point bending test scheme. Fig. 9 shows a schematic view of
the notched I-beam and the relevant geometrical dimensions.
The maximum amplitude of the load is 10.0 KN and the stress ratio
is 0.1. The crack is under a mode-I fatigue loading and eventually
Fig. 4. Schematic view of the cracked I-beam.
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L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
Fig. 5. The crack approximated as a semi-elliptical hole through the I-beam.
breaks through the cross-section of the I-beam specimen. Details of
the experimental configuration and results can be found in Ghafoori and Motavalli [36].
Fig. 6. Meshes for the cracked I-beam: (a) the uncracked global structure is
modeled with Finite Elements (10-noded tetrahedron element); and (b) the crack
surface is modeled independently with Boundary Elements (8-noded quadrilateral
element).
The fatigue-crack-growth and fatigue life of this cracked I-beam
are estimated by employing SGBEM–FEM Alternating Method, and
the meshes of the numerical model are similar to those in Fig. 6.
Paris Law [37] is used here to calculate the crack-growth-rate,
according to Paris Law:
Fig. 7. Normalized stress intensity factors at crack-front when crack length is
70 mm.
Table 1
The computed stress intensity factors for the cracked I-beam, using SGBEM–FEM Alternating Method (crack length a = 70 mm) at various nodes along the crack front.
KI
Difference (%)
KI
Difference (%)
KI
Difference (%)
KI
Difference (%)
Empirical
14522.2
14782.6
14859.4
14935.3
1.34
0.42
0.94
1.46
14977.8
15020.1
15030.5
15040.8
1.75
2.04
2.11
2.18
15029.9
15018.9
14980.9
14942.6
2.10
2.03
1.77
1.51
14856.3
14769.2
14598.8
14429.5
0.92
0.33
0.82
1.97
14720.0
L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
da
¼ C ðDK Þn
dN
ð6Þ
where C = 7.3790 108, n = 1.1877. In Eq. (6), da/dN is in mm/
pffiffiffiffiffiffiffiffiffi
cycle, and DK is in MPa mm.
The material parameters of S355J0 are: Young’s modulus
E = 203 GPa, Poisson’s ratio m = 0.30.
The process of fatigue-crack-growth for the cracked I-beam is
divided into 15 analysis steps. In each fatigue analysis step, the SIFs
along the crack-front are computed by the SGBEM–FEM Alternating Method. Then the crack-growth is modeled by adding a layer
of additional elements at the crack-front, in the direction determined by the Eshelby-force vector [38], with the size of surface
advance of the segment of the crack-front being determined by
Paris Law.
Based on Paris Law, the fatigue life of the cracked I-beam is:
N¼
Z
af
a0
1
da
C ðDK Þn
193
material. The crack length versus fatigue load cycles computed
by SGBEM–FEM Alternating Method is shown in Fig. 11, which also
shows good agreement with the experimental observations.
It should be noted that, some recent studies consider the plasticity induced crack closure to account for the effect of plate thickness on crack growth rates [39]. In this paper, simple Paris Law is
used to predict the fatigue growth rate. However, other models
which account for the effect of the plate thickness can also be
incorporated in the framework of the current SGBEM–FEM Alternating method, which will be our future study. Moreover, some
3D effects of fracture mechanics are also neglected in this study,
such as 3D corner singularity, and mode II and III coupling
[40,41], which are expected to have an insignificant effect for the
damage tolerance of the cracked I-beam structures.
3.3. An I-beam with an inclined crack in the web
ð7Þ
Thus, at each fatigue-growth-step, the maximum value of DK along
crack-front is computed by the SGBEM–FEM Alternating Method.
And Eq. (7) is numerically evaluated using Trapezoidal rule.
The growth of the fatigue crack up to failure is depicted in
Fig. 10, and the final crack length reaches 70.36 mm while the
stress intensity factor reaches the fracture toughness value of the
Fig. 8. Normalized stress intensity factor versus normalized crack length of the Ibeam under a bending moment and axial tensile load.
In this section, the same I-beam structure and bending/axial
loads are considered, as discussed in Section 3.1. However, in
Fig. 10. Growth of the crack in the I-beam: 2D view (a) initial crack length
a0 = 12.2 mm, (b) a = 19.4 mm, (c) a = 30.7 mm, (d) a = 39.8 mm (e) a = 48.7 mm, (f)
a = 55.7 mm, (g) a = 65.6 mm, and (h) a = 70.4 mm, failure).
Fig. 9. Schematic view of the cracked I-beam and relevant dimensions (initial crack length a0 = 12.2 mm).
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L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
current case, a mixed-mode inclined crack is considered instead of
a mode-I vertical crack. Fig. 12 shows a schematic view of the initial cracks with different lengths and inclination angles, where a
80
Experimental result [36]
Present numerical result
70
Crack length
a (mm)
60
50
40
30
20
10
100
101
102
103
104
105
106
log(N) (cycles)
Fig. 11. Crack length versus number of load cycles for the I-beam (stress ratio
R = 0.1).
Fig. 13. Normalized stress intensity factors versus crack inclination angle (initial
crack length a0 = 40 mm).
Fig. 12. A schematic view of the crack for the I-beam with different lengths and
inclination angles.
Table 2
The configurations of the cracks for the I-beam and their labels.
Crack length a (mm)
Fig. 14. Normalized stress intensity factors versus crack inclination angle (initial
crack length a0 = 60 mm).
Inclination h ()
40
60
80
30
45
60
C1
C4
C7
C2
C5
C8
C3
C6
C9
Table 3
The computed stress intensity factors for the cracked I-beams with different crack
configurations.
Crack label
KI
KII
KIII
C1
C2
C3
C4
C5
C6
C7
C8
C9
3138.29
4541.45
5959.65
3381.71
5530.70
8139.04
3312.22
7025.69
12258.59
2149.56
2093.26
1745.98
2326.39
2522.13
2353.40
2747.07
3282.32
3482.24
30.56
52.54
39.37
77.79
89.84
9.02
325.61
137.35
75.59
Fig. 15. Normalized stress intensity factors versus crack inclination angle (initial
crack length a0 = 80 mm).
L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
represents the crack length, and h represents the inclination angle.
The detailed initial lengths and inclination angles of each crack are
listed in Table 2.
195
The computed SIFs for each case by SGBEM–FEM Alternating
Method are listed in Table 3, and the normalized SIFs versus crack
inclination angles for different initial crack lengths are plotted in
Figs. 13–15. It is shown that the stress intensity factors vary significantly with respect to the crack inclination angle: when the crack
inclination angle is rather small, both mode I and mode II SIFs are
comparable, but mode I SIFs increase rapidly while crack inclination angle is increased. Since analytical or empirical SIFs for these
mixed-mode cracks are not available in the open literature, SIFs
obtained by SGBEM–FEM Alternating Method can be used as
benchmarks in the future studies of cracked I-beams by other
investigators.
Fatigue-crack-growth of the cracked I-beam with initial crack
length a0 = 40 mm and crack inclination angle h = 45° is also simulated using SGBEM–FEM Alternating Method. Paris Law is used
here with material constants C = 7.3790 108, n = 1.1877, and
the stress ratio is R = 0.1. The fatigue life and fatigue-crack-growth
paths are shown in Figs. 16 and 17.
3.4. An I-beam with a slant edge crack in the flange subjected to
torsional forces
Fig. 16. Fatigue crack length versus number of load cycles for the I-beam (initial
crack length a0 = 40 mm).
In this section, a cantilever I-beam with a slant through-thethickness edge crack on its bottom flange is considered. The I-beam
Fig. 17. Growth of the inclined crack in the I-beam: 3D view, (a) initial crack length a0 = 40 mm, (b) crack length a = 51.78 mm, (c) a = 69.40 mm, (d) final crack, a = 94.89 mm,
and (e) 2D view, the final crack in the I-beam.
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L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
is subjected to shear forces with opposite directions on the cross
section of its top and bottom flange, which leads to a resultant torsional moment. A slant edge crack (a = 45°) which cuts through the
bottom flange has been introduced to the I-beam. Material properties are (Young’s modulus E = 206.8 GPa; Poisson’s ratio m = 0.29).
All the relevant dimensions are shown in Fig. 18.
Fig. 18. The I-beam with slant edge crack: geometry and load.
Table 4
The difference in the computed stress intensity factors between different meshes
(a = 10 mm).
Mesh
KI
KII
Keff
Fig.
Fig.
Fig.
Fig.
10796.11
10618.79
11098.95
10907.66
5964.51
5927.99
6031.65
6004.52
14119.99
13938.81
14433.79
14248.38
13(a) and (c)
13(a) and (d)
13(b) and (c)
13(b) and (d)
Fig. 20. I-beam with a slant edge crack in the flange: normalized stress intensity
factors versus crack length.
Fig. 19. Meshes for the I-beam with a slant edge crack: the uncracked global structure is modeled with Finite Elements, (a) coarse mesh with 544 10-noded tetrahedron
element, (b) dense mesh with 2660 10-noded tetrahedron element; the crack surface is modeled independently with Boundary Elements, (c) coarse mesh with 25 8-noded
quadrilateral elements, and (d) dense mesh with 100 8-noded quadrilateral elements.
L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
197
The SGBEM–FEM Alternating Method is employed here to solve
this problem, with two different FEM meshes for the uncracked
global structure, as well as two different BEM meshes for the crack
surface. The computed SIFs are given in Table 4, demonstrating
that the computed SIFs are insensitive to FEM and SGBEM meshes.
Moreover, as shown in Fig. 19, the meshes for the SGBEM–FEM
Alternating Method consist of up to a few thousands of finite elements and a few hundreds boundary elements. On the other hand,
for pure finite-element-based methods, it is typical to use more
than a million degrees of freedom to model a cracked 3D complex
structure. Thus the current SGBEM–FEM Alternating Method
greatly reduces the burden of computation as well as the
human-labor for preprocessing, compared to FEM-based numerical
methods, by several orders of magnitude.
Different crack lengths are also considered for this problem
(a = 10, 20, 30, 40 mm). The computed SIFs by the SGBEM–FEM
Alternating Method are shown in Fig. 20, which clearly indicates
that it is a mixed-mode crack for which mode I and mode II SIFs
are dominant compared to the mode III SIF.
Fatigue-crack-growth analyses of the I-beam with initial
crack length a0 = 10 mm and a0 = 40 mm are simulated using
SGBEM–FEM Alternating Method. Paris Law is used here with
C = 3.6 1010, n = 3.0.
Fig. 21. Crack length versus number of load cycles for the I-beam (initial crack
length a0 = 10 mm).
Fig. 23. Crack length versus number of load cycles for the I-beam (initial crack
length a0 = 40 mm).
Fig. 22. Growth of the crack in the I-beam: 3D view, (a) initial crack length a0 = 10 mm, (b) crack length a = 41.75 mm, (c) a = 64.41 mm, (d) final crack, a = 82.90 mm, (e) final
crack surface, and (f) 2D view, final crack with the I-beam.
198
L. Tian et al. / Theoretical and Applied Fracture Mechanics 74 (2014) 188–199
Fig. 24. Growth of the crack in the I-beam: 3D view, (a) initial crack length a0 = 40 mm, (b) crack length a = 57.06 mm, (c) a = 73.61 mm, (d) final crack, a = 90.63 mm, (e) final
crack surface, and (f) 2D view, final crack with the I-beam.
The maximum magnitude of the cyclic load is F = 100 N/mm.
And the stress ratio is R = 0.1. For the I-beam with an initial crack
length a0 = 10 mm, the fatigue life and fatigue crack paths are
shown in Figs. 21 and 22; for the I-beam with an initial crack
length a0 = 40 mm, the fatigue life and fatigue crack paths are
shown in Figs. 23 and 24.
4. Conclusions
In this paper, several numerical models of cracked I-beams are
considered, for which the stress intensity factor analyses and fatigue-crack-growth simulations under different load conditions are
carried out by employing the 3D SGBEM–FEM Alternating Method.
For mode-I cracks in the I-beam, the computed SIFs by using the
SGBEM–FEM Alternating Method are in very good agreement with
empirical solutions, and the predicted fatigue life of cracked Ibeams agrees well with experimental observations. For inclined
cracks in the web or in the flange of the I-beams, no analytical or
empirical solutions are available in the literature. Thus mixedmode SIFs of inclined web and flange cracks are presented and
non-planar fatigue growth simulations are given, as benchmark
examples for future studies. Because very minimal efforts of preprocessing and very small computational burden are needed, the
current SGBEM–FEM Alternating Method is very suitable for fracture and fatigue analyses of 3D structures such as I-beams.
Acknowledgements
The first author gratefully acknowledges the financial support
of China Scholarship Council (Grant No. 201306260034); National
Basic Research Program of China (973 Program: 2011CB013800);
and New Century Excellent Talents Project in China (NCET-120415).
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