Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method

Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

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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 xxxx 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: [email protected] (L. Dong).

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

http://dx.doi.org/10.1016/j.tafmec.2014.10.002 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: L. Tian et al., Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.002

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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 ðxÞt b ðxÞdSx ¼

Z

@X

@X

ð2Þ

Fig. 1. A solution domain with source point x and target point n.

Please cite this article in press as: L. Tian et al., Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.002

L. Tian et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

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.

3

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.

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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.

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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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

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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

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).

Please cite this article in press as: L. Tian et al., Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.002

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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).

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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.

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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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.

Please cite this article in press as: L. Tian et al., Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.002

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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.

Please cite this article in press as: L. Tian et al., Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.002

L. Tian et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

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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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Please cite this article in press as: L. Tian et al., Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.10.002