Prediction of the Bolt Fracture in Shear Using Finite Element Method

Structures 12 (2017) 188–210

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Prediction of the Bolt Fracture in Shear Using Finite Element Method Amir Ahmad Hedayat a b c

a,⁎,b

, Ehsan Ahmadi Afzadi

a,b

, Amin Iranpour

MARK

b,c

Department of Civil Engineering, Kerman-branch, Islamic Azad University, Kerman, Iran Pars Sustainable Seismic Structures Co., Kerman, Iran Department of Civil Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Bolt fracture Extended finite element method Bolt shear strength Finite element modeling

This study was aimed to propose appropriate failure criteria for bolt fracture prediction in shear when threads are excluded from the shear plane. In this regard, the finite element methods for such prediction were divided into two main categories and for each category the available methods were discussed. Then, using finite element modeling and available experimental results of previous researchers, three methods here referred to as MTD1, MTD2 and MTD3 were proposed. MTD1 and MTD2 were based on the monitoring of the level of stress and strain at the critical elements of the bolt shank, while MTD3 was based on the extended finite element method and consisted of two main steps including defining crack initiation and crack evolution. Analytical results indicated that MTD1 and MTD3 were reasonably acceptable for prediction of the bolt fracture in shear with negligible amount of error. Method MTD1 is a suitable method when a progressive collapse analysis is not of interest and only the capacity of the system at the onset of the first bolt fracture is required. However, MTD3 can be used in a progressive collapse analysis where the amount of reduction in a system strength by fracturing of each component is of interest. However, in comparison to MTD1, MTD3 is remarkably time consuming.

1. Introduction Generally, there are two main connection types in steel structures: welded and bolted connections. Although welded connections are more rigid than the bolted ones, the quality of such connections is questionable, especially when they are not shop fabricated. On the other hand, bolting of steel structures is a very rapid process that needs less skilled labor compared to the welding one. In addition, bolted connections are more ductile than the welded ones. All these give bolting distinct advantages over the welding from both economical and behavioral points of view. These features finally made the bolted connections more attractive for both researchers and designers. In order to investigate the behavior of newly proposed or existing bolted connections, it is essential to conduct experimental tests. However, such tests are usually expensive and time consuming. Hence, finite element method (FEM) has been an alternative and conventional method between researchers for preliminary investigation of the behavior of the bolted connections. However, a major concern regarding this method is how to capture the fracture of steel components especially high strength bolts. In this regard, several methods have been proposed based on FEM. These methods can be classified into two main categories. In the first category, the fracture initiation of a bolt can be estimated by monitoring the stress or strain level at the bolt shank. Such analyses

⁎

are generally simple and not time consuming. Despite the simplicity in the application of this method, it is not a suitable procedure when investigation of the progressive collapse is of interest. This is due to the fact that in this method the fracture is not automatically detected and propagated. It is worth mentioning that such simple methods are not only used to capture the bolt fracture [1,2], but also used by many researchers to detect the fracture of other steel components such as shear tab in simple connections, beam flange and web in welded connections and steel slit dampers in semi-rigid connections [3–9]. In 2013, Suleiman [1] investigated the behavior of extended shear tab connections. For such connections, he corresponded bolt yielding to the bolt shear failure by considering an elastic-perfectly plastic model for bolt's material. In this reference, bolt fracture was computationally determined by comparing von Mises stress contours against the engineering stress associated to the ultimate strength of bolt material (engineering value of Fu). It should be noted that, due to the big difference between the yield and ultimate strain of a ductile material, the true ultimate stress is remarkably greater than the engineering one. As a result, such comparison can lead to an early bolt's fracture estimation. Hence, defining such failure criteria for bolt might be questionable. In 2015, Abou-zidan and Liu [2] investigated the location of inflection point of extended shear tab connections using FEM. In order to capture the onset of bolt fracture, the fluctuation of shear stress along the

Corresponding author at: Department of Civil Engineering, Kerman Branch, Islamic Azad University, Kerman, P.O. Box 7635131167, Iran. E-mail address: [email protected] (A.A. Hedayat).

http://dx.doi.org/10.1016/j.istruc.2017.09.005 Received 20 January 2017; Received in revised form 6 September 2017; Accepted 22 September 2017 Available online 25 September 2017 2352-0124/ © 2017 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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simple but is limited to the LS-DYANA [21] software. It should be noted that, despite simplicity, element deletion method is severely mesh sensitive. Since ABAQUS software is a popular software among structural engineering researchers, it is worth finding a similar method in ABAQUS program. It seems that the extended finite element method (XFEM) in ABAQUS can be an appropriate alternative to the one implemented in LS-DYNA [21]. In addition, independency of mesh size gives the XFEM a remarkable advantage over element deletion method. Based on the discussions presented in the previous paragraphs, this study was aimed to modify the failure criteria presented in the first category based on the ones proposed by references [1,2] and proposing appropriate failure criteria pertaining to the second category based on the XFEM when bolts are subjected to shear. For this purpose, at first, using valuable experimental results of monotonically tested specimens presented in reference [22], finite element models for both A325 and A490 bolts were created and the analytical results were compared to the experimental ones to validate the finite element modeling. In the next step, mesh sensitivity was analyzed and an appropriate mesh size was determined to eliminate the effect of mesh size on the proposed failure criteria. Since, in this study, in addition to monotonically tested specimens, one specimen under cyclic loading was also used for FE validation, the calibration of material models under cyclic loading was also discussed in detail. At the end, the applicability of the proposed failure criteria was investigated through implementation of them into specimen 6U which is an extended shear tab connection, tested by Sherman and Ghorbanpoor [23] and a A490 high strength bolt tested in a double shear joint presented in reference [24].

centerline of the bolt was monitored. They assumed that the bolt shear rupture happens when the bolt shear stress decreases irreversibly. It is worth mentioning that this irreversible decrease may not necessarily be due to the bolt fracture. This also can happen due to encountering instability or failure of other connection components such as weld tearing or net section fracture of the shear tab. In addition, initial finite element investigations done by the authors indicated that this failure criterion is strongly dependent on the bolt's mesh size. While, such investigation was not considered in reference [2].This failure criterion is more discussed in the following sections. In recent years Hedayat et al. [3–7] and Saffari et al. [8] investigated the strength and ductility of postNorthridge connections. In these references, they assumed that the fracture of beam flanges initiates when the von Mises strain reaches the strain associated to the ultimate strength of the material at the whole width of the beam flange. In 2015, Hedayat [9] investigated the behavior of steel slit dampers. In this reference, the onset of fracture of slit damper was estimated by monitoring the values of plastic equivalent strain (PEEQ) and the fracture index (FI) at the corner of links. He assumed that the fracture of links initiates when simultaneously the values of PEEQ and FI reach 5 and 1.25 respectively. In the second category, fracture happens based on a predefined failure mechanism. A failure mechanism consists of two ingredients: a fracture (or damage) initiation criterion and a damage evolution law. In this case, fracture initiation is automatically detected and propagated so it is a suitable method for progressive collapse analysis. It should be noted that using this procedure is somehow complicated and the appropriate input parameters for such criteria are not always available. Furthermore, these analyses are always very time consuming and expensive. There are various types of fracture mechanisms and consequently fracture initiation models and evolutions. More discussion regarding these is presented in the following sections. For ductile materials, however, two most common fracture initiation models are ductile-damage [10–12] and shear-damage [11,13]. In 2012, Behan et al. [13] simulated high-resolution bolt models for different bolted connection tests. In this reference, they used an old version of ABAQUS program [14] and the used model for shear fracture initiation of bolts was shear damage. It should be noted that in this version of ABAQUS program, the shear damage initiation was controlled only by one parameter, shear plastic strain. For explicit analyses, the value used for this parameter was 0.24 mm/mm. It should be noted that in recent versions of ABAQUS (e.g., [15]), such formulation for this criterion is not supported anymore. Instead, the shear-damage criterion is defined as a function of three parameters, fracture strain, shear stress ratio and strain rate. In 2013, Pavlović et al. [11] investigated the behavior of bolted shear connectors based on experimental and analytical tests. They used ductile and shear damage criterions for bolts, while only ductile damage was utilized for steel sections. In ductile damage initiation criterion, the equivalent plastic strain at the onset of damage pl (ε D ) must be defined as a function of stress triaxiality (η) and strain rate. Based on the results presented in references [16,17], Pavlović et al. [11] proposed an exponential relationship between the equivalent plastic strain and stress triaxiality for bolts at the onset of damage

2. Finite element modeling 2.1. Material properties and boundary conditions In reference [22], four different bolt grades of six different diameters were tested in direct tension and shear with the threads excluded and not excluded from the shear plane. In this study, using this reference, six high strength bolts in shear with the threads excluded from the shear plane, corresponding to A325 and A490 bolts with diameters of 3/4, 7/8 and 1 in. were modeled using finite element program ABAQUS [15]. Fig. 1 shows the test machine used in this reference which was set in accordance to requirements given in ASTM F606-05 [25] for single shear testing. As it is shown in this figure, hardened steel of sufficient thickness was used for the shear fixture to prevent bearing failure. In this test setup, the bolt hole was 1.59 mm (1/ 16 in.) larger than the nominal diameter of the bolt. All finite element models were created using C3D8R element, which is a three-dimensional continuum (solid) 8-node linear brick elements

pl = 0.08e−1.5 (η − 3 ) ). In 2016, Seif et al. [18] developed a practical (ε D modeling approach to investigate the behavior of structural steel and structural bolts at elevated temperatures. They used element erosion to model material failure. Similar approaches were also used by Sadek et al. [19] and Main and Sadek [20] to investigate the behavior of moment and simple shear connections under column removal scenarios at ambient temperature respectively. In these studies, the criterion used for fracture initiation was as such, to initiate the fracture when the plastic equivalent strain (PEEQ) reaches a specific value, erosion strain. This method is based on element deletion method. For instance, in reference [18], the erosion strains corresponding to A325 and A490 bolts at 20 °C were 0.5 and 0.35 respectively. Compared to the ductile and shear damage criterions, usage of the mentioned procedure is fairly 1

Fig. 1. Test machine (shear fixture) used in reference [22].

189

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1200

Vertical displacement (dv) 1000 Stress (MPa)

(a) Point A

800 600 A325

400

A490

Critical shear plane

200 0 0

0.04

0.08 0.12 0.16 Strain (mm/mm)

0.2

0.24

Fig. 3. Engineering stress-strain curves for A325 and A490 bolts.

values to be utilized in the FE modeling. The engineering curves were mainly derived based on the numerous tensile tests on high strength bolts conducted by Moore [22]. In addition, in order to check the validity of the assumed curves, three key parameters (yield strength, ultimate strength and fracture strain) were compared to the data presented by other researchers. Regarding the yield strength of the bolts, by comparing the more recent experimental results presented by Kodur et al. [27] and Moore [22] to the old results presented by Kulak et al. [28], it was found that the yield strength of bolts have almost remained unchanged. As a result, in this study, the yield strength values proposed by Kulak et al. [28] were selected to construct the engineering stressstrain curves. As Fig. 3 indicates, the used values for the ultimate stress of materials for both A325 and A490 bolts are slightly greater than those presented in Table J3.2 of AISC360-10 [29] which are here summarized in Table 1. These higher values were chosen based on the experimental results presented by Moore [22]. Table 1 summarizes the minimum, maximum and average of ultimate strength of materials obtained by Moore [22]. In addition, experimental tests done by Liu and Astaneh-Asl [30] on simple shear connections approve the assumptions taken in this study to define stress-strain curves (see Table 1). Regarding the fracture strain of material, experimental tests done by Kodur et al. [27] at ambient temperature indicated a value of 0.21 mm/mm for A325 bolt which is close to the one proposed by Kulak et al. [28] (0.23 mm/mm). While for A490 bolt material, Kodur et al. [27] proposed the value of 0.16 mm/mm that is less than the one suggested by Kulak et al. [28] (0.20 mm/mm). In reference [22], as opposed to the references [27,28], elongation of bolts at failure was not obtained using a round specimen fabricated on basis of A370-05 [31] specifications, rather, elongation was calculated based on bolts tested in direct tension. In [22], after the tension bolts were tested, the thread length was measured and compared to the initial thread length. Due to the reduced area in the threads, the elongation occurred mostly in this region and not in the bolt shank. As a result, the elongation values reported by Moore [22] were different (smaller) than those proposed in references [27,28]. In reference [22], the minimum, average and maximum percent elongation for A325/F1852 bolts and A490/F2280 bolts were (0.58%, 9.83%, 21.98%) and (0.35%, 6.74%, 15.76%) respectively. As Fig. 3 indicates, the assumed strains at fracture of material for A325 and A490 bolts are 0.22 and 0.16 respectively which

Critical elements Hole border

(b)

Plate border

Fig. 2. Typical finite element mesh of test specimen.

with reduced integration and hourglass control. This element resolves shear locking problem which is a common problem associated to fully integrated elements. However, this technique can lead to numerical difficulties due to the very flexible nature of such elements. This finally may lead to generation of zero-energy singular modes that cause mesh instabilities and inaccurate results. Hour glassing control implemented into ABAQUS could avoid facing such problem. Fig. 2 shows the typical finite element model created in this study using solid element. Based on the report presented by Moore [22], the shear fixture consisted of three parts of different material properties: small insert, big insert and main body of the test machine (see Fig. 2b). As a result, in the finite element modeling, each part was modeled with its related material property and dimension. Small insert was made of 1045 steel, which has a yield strength of 393.3 MPa (57 ksi) and tensile strength of 660 MPa (95.65 ksi) [22,26]. While to make the big insert the D2 tool steel was used which has a yield strength equal to 1483.5 MPa (215 ksi) and a tensile strength equal to 1897.5 MPa (275 ksi) [22,26]. The main body of the test machine was made of a hardened steel. For modeling purpose, each material property was defined using a bilinear relationship made of true values of the mentioned yield and tensile strength. The inner and outer diameters of the small insert were equal to the bolt hole diameter and 38.1 mm (1.5 in.) respectively and the inner and outer diameter of the big insert were 38.1 mm (1.5 in.) and 76.2 mm (3 in.) respectively. Regarding the high strength A325 and A490 bolts, a multi linear relationship between stress and strain was defined. Fig. 3 shows these engineering stress-strain curves which were then converted to the true

Table 1 Ultimate stress of A325 and A490 bolts – MPa (ksi). Bolt

A325 A490

190

The present study

AISC 360-10 [29]

Moore [22] Min

Max

Average

998 (144.6) 1131 (164)

828 (120) 1035 (150)

838 (121.5) 1050 (152.1)

1078 (156.3) 1241 (179.8)

993 (143.9) 1135 (164.5)

Liu and Astaneh-Asl [30] 980 (142) –

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were mainly selected based on the experimental results of references [27,28]. The stress values of the mid-points of the stress-strain curves (between the yield and the fracture points) were selected based on the experimental results of Moore [22] corresponded to the bolts tested in direct tension. Material nonlinearity analyses were performed on the basis of von Mises yielding criteria. In this study, the hardening law, which describes how the yield surface expands or changes as the material deforms plastically, was determined by a trial and error method. In general, three kinds of hardening laws are represented in ABAQUS program [15]: isotropic hardening, kinematic hardening and combined (isotropic and kinematic) hardening. Generally, isotropic hardening is assumed for the monotonic analysis, whereas kinematic hardening is assumed for the cyclic analysis [4,8,9,32]. Hence, in this study, firstly, isotropic hardening was assumed for monotonic analysis, but comparison between the analytical and experimental results indicated that the usage of kinematic hardening law can lead to more accurate results than the usage of isotropic hardening law. Regarding the cyclic loading, comparison between the analytical and experimental results, showed that in compare to the kinematic hardening, combined hardening is a more appropriate hardening law which led to a more accurate analytical results. Hence, in this study for all FE models, kinematic hardening was assumed for the monotonic analysis, whereas where it was applicable (Section 3.1.1.1), a combined hardening law was assumed for the cyclic analysis. In addition, nonlinear geometry analyses were also considered through a large strain formulation to capture any local instability and large deformation effects. All analyses were conducted by applying a monotonic vertical displacement load to the top end of the left plate, point A shown in Fig. 2. Hereafter, the vertical displacement relevant to this point and its corresponding reaction force are referred as parameters dv and Rv respectively. In addition, the values of dv and Rv corresponding to the onset of fracture were denoted by parameters dvf and Rvf respectively. The used boundary conditions simulated those that were imposed by the test machine. In this respect, the bottom end of the right plate was restrained in all degrees of freedom and the horizontal displacements of the both left and right plates were restrained. Interaction between the two plates, bolt shank and the plates, and bolt head/nut and the plates were defined using a surface to surface formulation. In this respect, to prevent any penetration of slave nodes into the master surfaces, the hard-normal contact was used. With respect to the smooth surface between the plates of the test machine, a frictionless formulation was considered for tangential behavior. In reference [22], in order to determine the load rate for testing in shear, two extreme load rates, 0.25 in./min and 0.5 in./min, were evaluated on two 3/4-inch diameter A325 bolts with the threads excluded and not excluded from the shear plane. The results indicated that the load rate has not a remarkable effect on the strength of the fastener so a load rate of 0.5 in./min was used. Hence, it might be concluded that the inertial effects can be neglected. Therefore, in this study, a general static procedure in ABAQUS standard solver was adopted.

Table 2 Mesh sizes used for mesh sensitivity analysis. Mesh size (mm)

Case 1

Case 2

Case 3

Case 4

Case 5

Bolt

4 4 5 5

3 4 5 5

2 4 5 5

1 4 5 5

1 2 5 5

Plate

Shank Head & nut Hole border Plate border

300 250

Rv (KN)

200 150

Case 1 Case 2 Case 3 Case 4 Case 5

100 50 0 0

2

4 dv (mm)

6

8

Fig. 4. Effect of mesh size on the global response.

mesh size, the local response was also investigated on the basis of total shear stress calculated at the cross-sectional area of the bolt shank. This shear stress is named by parameter τbolt which was computed using three different methods: a) dividing the resultant shear sectional force at the bolt shank by the cross sectional area of that which was denoted by τbolt − a, b) dividing the value of parameter Rv by the cross sectional area of the bolt shank which was denoted by τbolt − b, c) by directly averaging of the resultant shear stress of the first row of elements on both sides of the critical shear plane (τbolt − c). Hereafter, these two rows of elements are referred as critical elements (see Fig. 2a). The level of convergence increases as the amount of discrepancy between these three curves decreases. Fig. 5 indicates that a mesh size of 1 mm for the bolt shank, 4 mm for the bolt head and nut and 5 mm for the both bolt hole and plate border (case 4) can lead to an acceptable level of convergence of these analytical results (i.e., τbolt − a ≅ τbolt − b ≅ τbolt − c). Hereafter, in all finite element models, except where XFEM was of interest, this mesh size (case 4) was used and shear stress at the bolt shank was calculated using the method-c (i.e., τbolt − c). In the case of using XFEM at which the prediction of the bolt fracture is less dependent to the local response, case 3 was used because in terms of the global response, both of the case 3 and case 4 showed reasonable convergence. It should be noted that the XFEM is time consuming so this selection, for complicated models with several bolts, would lead to a remarkable reduction in the analysis time while the analytical results will be still of high level of accuracy.

2.3. Simulation of bolts' tightening In finite element modeling using ABAQUS, tension in a tightened bolt is modeled by applying a bolt load to each bolt in the first step of the analysis. This load can be defined in terms of either a concentrated force or a prescribed change in the bolt's shank length (i.e., shortening the shank length). And this load is applied across the bolt cross-section surface which was previously specified. In later steps, however, the applied load can be modified to prevent further length changes so that the bolt acts as a standard, deformable component responding to other loadings on the assembly. This technique helps to avoid the problem with extensive elongation of the bolts under the loading. This modification can be done through changing the type of the loading procedure selected in the step one of the analysis to the “Fix the bolt at its current length” type in a subsequent analysis step. During the first step,

2.2. Mesh refinement study In this study, a mesh refinement study with different mesh sizes was conducted to determine the level of mesh refinement required to ensure convergence of the analytical results. This convergence was investigated from two different points of view, global and local responses. For the global response, Rv-dv curves corresponding to different mesh sizes were compared where Rv and dv were previously defined. Table 2 summarizes the different mesh sizes used for the bolt shank, the bolt head and nut, the bolt hole border and the plate border (see Fig. 2b). As Fig. 4 indicates, nearly identical global responses were achieved when the mesh size for different parts of the test machine was those that are identified by case numbers 3 and 4 in Table 2. In order to finalize the 191

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700 600

600

(a)

500

(MPa)

(MPa)

Fig. 5. Effect of mesh size on the local response for: a) case 3; b) case 4.

700

400

(b)

500 400

300

Method-a

300

Method-a

200

Method-b

200

Method-b

100

Method-c

100

Method-c

0

0 0

2

4 6 dv (mm)

8

0

2

4 6 dv (mm)

8

deformation of its shank becomes noticeable. As Fig. 6 indicates, the first method of pre-tensioning (i.e., applying a concentrated force without the modification) is unable to capture this lowering effect, while the other three methods reached to the same results and were able to capture the lowering effects. Among the last three methods, second method is the simples one since it does not need any trial and error procedure to find out the correct value of the bolt shortening length. It should be noted that, compared to the force-controlled methods (methods 1 and 2), the displacement-controlled methods (methods 3 and 4) are generally more stable and suitable for contact problems. In the other words, using the last two methods would lead to a more rapid convergence and consequently a significant reduction in the time of analysis especially in the case of having a complicated model. Hence in this study, where it is applicable, the fourth method (i.e., adjusting the bolt length with modification) was utilized for pretensioning of bolt.

all three translational degrees of freedom (DOF) at the section of pretension are restrained. These served as the artificial boundary condition to prevent the numerical singularity error which occurs as a result of rigid body motion. After the preloading and activating the contact properties, this artificial boundary condition is then removed [33]. Based on the above discussion, generally, a bolt can be tightened using four methods using ABAQUS [15]: applying a concentrated force or adjusting the bolt length with or without the modification. It is worth to mention that in some references, researchers did not apply the mentioned modification to fix the bolt at its current length [34,35]. In this study in order to realize the efficiency of each method, a 7/8 in A325 bolt was tightened using the four aforementioned methods and the results were compared. Amount of the shortening of the bolt's shank length (i.e., adjusted length) was obtained by a trial and error method. In this case the bolt was shortened by an arbitrary initial value and after the analysis, the bolt internal load was measured through the cutting section of the bolt shank. The initial value of the adjusted length was changed until achieving the correct pre-tensioning load. The initial value of the adjusted length, δi − adj, can be estimated using δi − adj = Pcode ∙ Lshank/(Abolt ∙ Es) where Pcode, Lshank, Abolt and Es are the minimum code-based pre-tensioning load, bolt's shank length, bolt's cross sectional area and the modulus of elasticity of the bolt material respectively. In addition, during the process of tightening of bolts, to avoid intrusion of bolt and steel plates, the contact model between the bolt head/nut and steel plates were defined as same as those explained in Section 2.1. Fig. 6 compares the application of the four mentioned pre-tensioning methods in ABAQUS software in terms of normalized pre-tensioning load and the vertical displacement of the nut in the direction of applied load. This figure is drawn for a 7/8 inch A325 bolt where the pre-tensioning load is normalized by the minimum pre-tensioning load presented in Table J3.1 of AISC360 [29]. Experimental tests conducted by Kulak [28] on pre-tensioned bolts have shown that a bolt starts to lose its pre-tensioning load as the shear

2.4. Validation of finite element modeling In order to verify the accuracy of the finite element modeling, the experimental results of 3/4, 7/8 and 1 inch A325 and A490 bolts presented in reference [22] were compared to the analytical results obtained from the finite element models in terms of the Rv-dv relationship. Since for each bolt size and type several bolts of different lots were tested, to validate the analytical results, the experimental curve corresponding to each bolt of a specific size and type was obtained by averaging of the all curves of different lots. This procedure was also adopted by other researchers (e.g. [18]). As a result, the analytical Rvdv curves should not be expected to perfectly match the experimental ones. Fig. 7 shows this comparison for 3/4, 7/8 and 1 inch A325 and A490 bolts. As this figure shows, generally, the analytical curves are in good agreement with the related experimental ones. In addition, Table A1 presented in Appendix A summarizes the strength ratios of the vaFEM

lidated FE models against all the available test data ( Rvtest ) at the onset Rv

Normalized pre-tensioning load

1.6

Fig. 6. Relationship between normalized pre-tensioning load and the vertical displacement of the nut in terms of different pre-tensioning methods.

Method1: Concentrated force without modification

1.4 Method2: Concentrated force with modification

1.2 Method3: Adjusting the bolt length without modification

1 Method4: Adjusting the bolt length with modification

0.8 0.6 0.4

Adjusted length=0.152 mm Applied load=176000 N

0.2 0 0

2

4 6 8 10 12 The vertical displacement of the nut (mm)

14

192

16

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300 250

250

(a)

200

Rv (KN)

Rv (KN)

Fig. 7. Comparison of analytical and experimental [22] Rvdv curves for a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

300

150 100

7/8 inch

150 100

Experimental result - A325

50

(b)

200

7/8 inch Experimental result - A490

50

Analytical result - A490

Analytical result - A325

0

0 0

2

4 dv (mm)

6

8

0

250

4 dv (mm)

(c)

8

(d)

Rv (KN)

200

150 100

3/4 inch Experimental result - A325

50

150 100 3/4 inch

50

Experimental result - A490

Analytical result - A325

Analytical result - A490

0

0 0

400 350 300 250 200 150 100 50 0

2

4 dv (mm)

6

0

8

(e)

1 inch Experimental result - A325 Analytical result - A325

0

2

4 dv (mm)

2

4

6

8

dv (mm)

Rv (KN)

Rv (KN)

6

250

200 Rv (KN)

2

6

8

400 350 300 250 200 150 100 50 0

(f)

1 inch Experimental result - A490 Analytical result - A490

0

2

4

6

8

10

dv (mm)

strains at the critical elements of a bolt shank as the value of parameter dv increases while, a capacity curve is a type of graph expressing the relationship between the stress and strain carrying capacity of the bolt material which was obtained based on theoretical formulas.

of the fracture for all bolts. As the results of this table indicate, these ratios are enough close to unity indicating a small dispersion between the Rvtest values at the onset of fracture and the accuracy of the finite element modeling to capture the ultimate strength of all bolts at the onset of fracture. However, as Fig. 7 shows, in terms of parameter dv, the analytical models could not provide a good prediction at the points of fracture. It is because of this fact that a failure mechanism has not incorporated in the finite element modeling yet. The methods to capture the fracture points and the efficiency of each method are completely discussed in the following sections.

3.1.1. First category on the basis of shear stress The proposed method is similar to the one suggested by Abou-zidan and Liu [2] but subjected to some modifications. Abou-zidan and Liu [2] assumed that the bolt shear rupture happens when the bolt shear stress measured at the critical elements of the bolt's shank (τbolt − c), demand values, decreases irreversibly. However, it can be shown that this irreversible decrease may not necessarily be due to the bolt fracture. For the sake of clarity, an interior single plate shear connection was firstly designed based on the procedure presented in the AISC manual [36] and recommendations given by Astaneh-Asl [37] as such among all possible limit states, the shear tab yielding and the bolt fracture were the first and the last limit states respectively. Since Astaneh-Asl [37] recommends that the weld fracture to be the last limit state, in this investigation, the limit state of weld fracture was ignored. Then, the shear tab of this specimen was intentionally weakened to ensure that the bolt fracture will never happen. This specimen is named here as specimen BF. The configuration of this specimen is similar to specimen 2A of reference [30] but its loading and boundary conditions were different from those used in specimen 2A. Since the behavior of specimen BF is somehow complicated (due to the interaction between connection parts), specimen 2A was firstly modeled and its analytical results were compared to the experimental one to ensure about the

3. Proposed methods for bolt fracture prediction in shear As mentioned in Section 1, methods to estimate the bolt failure can be classified into two main categories. 3.1. First category Generally, in the first category, the fracture initiation of a bolt can be estimated by monitoring either the stress or strain level at the bolt shank. In this section, firstly, the proposed criteria by previous researchers which are on the basis of either the shear [2] or the von Mises [1] stresses are discussed and then the accuracy of each method was investigated. Finally, to improve the accuracy of each criterion, a new procedure was proposed in which the onset of bolt fracture was estimated by intersecting the demand and capacity curves. Here, a demand curve is the graph depicting the variation of the induced stresses and 193

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Table 3 Geometrical and material properties of specimen 2A of reference [30] and specimen 6U of reference [23]. Specimen

2A

6U

Connection type

Interior

Exterior

W14 × 90 A572 3050 W24 × 55 A572 7924.8 457.2 × 127 × 9.525

W14 × 90 A572 2438.4 W24 × 146 A572 10,058.4 457.2 × 292.1 × 12.7

A36 6 22.225 (7/8)

A36 6 19.05 (3/4)

A325-N Pre-tensioned Cyclic

A325-X Snug tightened Monotonic

Column

Beam

Shear tab

Bolt

Loading type

Section Material Length (mm) Section Material Length (mm) Dimensions (mm) Material No. Diameter, mm (in.) Material Tightening type

accuracy of finite element modeling. A detailed explanation of the FE modeling of this specimen is explained in Section 3.1.1.1. Then, the validity of the method proposed by Abou-zidan and Liu [2] was investigated in Section 3.1.1.2. Finally, at Section 3.1.1.3 a method to improve the accuracy of the Abou-zidan and Liu method [2] was proposed. Fig. 8. The geometrical properties, applied boundary conditions and the associate finite element mesh for the specimens used in the flat coupon test (used in monotonic test) and the cyclic strain test.

3.1.1.1. FE modeling of specimen 2A of reference [30]. Table 3 summarizes all geometrical and material properties of specimen 2A. In FE analysis, both of material and geometrical nonlinearities were considered. As shown in Table 3, all beams and columns were made of A572 Gr. 50 steel and shear plates were selected from A36 steel. Hence, these material properties were calibrated against the monotonic and cyclic coupon tests results presented by Kaufmann et al. [38] through simulating of the coupon specimens. The objective of the investigation done by Kaufmann et al. [38] was to examine the basic mechanical properties of A572 Gr. 50, A913 Gr. 50 and A36 steel sections. In this reference the cyclic inelastic strain behavior of the materials under tension-compression loading was studied using standard strain controlled fatigue test methodologies [39]. The geometrical properties, applied boundary conditions and the associate finite element mesh for the specimens used in the flat coupon test (used in monotonic test) and the cyclic strain test are shown in Fig. 8. Fig. 9 shows the used plastic stress-strain curves for these materials utilized in FE modeling where the values beyond the onset of necking were determined by a trial and error procedure because of the complex triaxial state of stress after necking. FE models were created using C3D8R element. For instance, Fig. 10 shows the comparison of analytical and experimental results for A36 material. As this figure indicates, there is a good agreement between the analytical and experimental results. Since specimen 2A was subjected to cyclic loading, isotropic hardening parameters (Q∞ and b) of A36 and A572 Gr.50 materials were also needed. Q∞ is a saturation value (the asymptotic value in the regime of the maximum change in the size of the yield surface) and b is the saturation rate (the speed of stabilization or the rate at which the size of the yield surface changes as plastic straining develops). These parameters were obtained by using experimental results of the cyclic coupon specimens under reversed cyclic loading presented by Kaufmann et al. [38]. Calibrated isotropic hardening parameters (b, Q∞), for A36 and A572 Gr.50 materials are (10, 137 MPa) and (4, 83 MPa) respectively. Fig. 10b compares the analytical and experimental results of the stabilized cycle for the A36 material under 10 cycles at 8% strain range. As this figure shows, the analytical results are in good agreement with the experimental ones. Specimen 2A was an interior connection and was tested under a

700 600 Stress (MPa)

500 400 300 A36

200

A572-Gr.50

100 0 0

0.05

0.1 0.15 Strain (mm/mm)

0.2

0.25

Fig. 9. True plastic stress-strain curves used for A572 Gr. 50 and A36 steel materials.

cyclic loading. Since initial shear and rotation on the connection due to gravity loads would have a significant effect on the cyclic response of a shear tab connection [40], Judy Liu and Astaneh-Asl [30] simulated all these effects by applying the loads in three steps. In the first step, the beam ends were pushed down 9.525 mm (3/8 in.) and were kept constant in their positions during the test. In the next step, a 222.4 kN (50 kips) concentrated load was applied on each beam. These two load steps simulated the required shear and rotation demands on the connection due to the gravity loads. In the third step, the column tip was subjected to a cyclic loading. The location of applied loads and lateral braces are shown in Fig. 11a and the load history used in this test is shown in Fig. 11b. In FE modeling, the imposed boundary conditions related to this specimen were similar to those used in test setup of reference [30] which are shown in Fig. 11a. The mesh sizes varied from region to region. The size of mesh used for bolts and bolt holes was as same as the one presented in Section 2.2. Since yielding is expected in shear tab and beam web at the vicinity of the connection, a fine mesh size of maximum 5 mm was used for these regions. Shear tab, beam web and beam flanges were modeled with 194

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500

(a)

400

300 Stress (MPa)

Stress (MPa)

500 300 200

Experimental result

100 0

0.1

0.2 0.3 0.4 Strain (mm/mm)

100 -100 -300 Experimental result Analytical result

-500

Analytical result 0

Fig. 10. Comparison between experimental engineering stress-strain curves of [38] and analytical results of A36 steel material for the case of: a) monotonic loading; b) cyclic loading.

(b)

0.5

-700 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Strain (mm/mm)

Displacement (mm)

300 200 100 0 -100 -200 -300 0 10 20 30 40 50 60 70 80 90100 Number of cycles Fig. 11. a) Schematic view of the test setup used for specimen 2A [30]; b) load history used in reference [30].

load and displacement. As Fig. 12 shows, the stiffness in loading, unloading and reloading and the maximum load achieved in each cycle were well matched for the finite element model and the test specimen. Therefore, it might be concluded that the analytical results are in good agreement with experimental results.

three, two and two elements along their thickness respectively. The size of mesh was gradually increased up to 75 mm in the low strain regions. This finite element model was created using C3D8R element with reduced integration and hourglass control algorithms. Interactions between elements surfaces usually need special care in a connection due to complex numerical problems that can be faced. In this study, small surface-to-surface sliding was considered for all contacts. A “small sliding” option in ABAQUS meant that the contact between the master and slave nodes was defined in the initial stage and not redefined in the later stages of the analysis. Contacts were defined in the interfaces of the shear tab and the beam web, bolt heads/bolt nuts and shear tab/beam web, bolt shanks and bolt holes of the shear tab and the beam web, beam flanges and column flanges in the case of closing the gap between the beam end and the column flange. For normal behavior, “hard” contact pressure-over-closure relationship was defined and a separation after contact was allowed. Tangential behavior was specified in contact property, for which tangential behavior “Penalty” friction formulation was defined. In general, under cyclic loading, slip coefficient of the friction surfaces decreases as the number of loading cycles increases which is due to wear and tear. Hence, a coefficient of friction of 0.18 was used which was obtained by trial and error method. The welds which connect the shear tab to the column flange were assumed to be rigid and were modeled using the tie contact algorithm. Connection tests were generally considered quasi-static and consequently inertial effects can be assumed to be negligible. Hence, a general static procedure in ABAQUS standard solver was utilized. In order to consider out of plane deformations in the finite element models and to ensure that buckling occurs when the model becomes unstable; the imperfect model was analyzed under cyclic or monotonic loadings. In the present study, in order to determine the imperfect model, first the buckling mode shapes were computed in a separate buckling analysis and then were implemented to perturb the original perfect geometry of the model as it was done by many researchers [7,9,41]. In order to verify the accuracy of the modeling, the experimental results of specimens 2A were compared with the analytical results obtained from the finite element model. These results which are shown in Fig. 12 are drawn up to the achievement of the first fracture in the connection. The results are presented in terms of column tip horizontal

Load [KN]

3.1.1.2. Examination of the method proposed by Abou-zidan and Liu [2]. In specimen BF, the boundary conditions were set as such to create a pinned connection at both ends of column while a roller support was simulated at both beam ends. Then an incremental uniform gravity load on the beams was applied until the failure of specimen was reached. These applied boundary conditions were intended to simulate the realistic condition of an interior connection under uniform gravity load. Fig. 13a shows the finite element mesh of specimen BF along with the mentioned boundary conditions. Fig. 13b shows the variation of the total shear stress (τbolt − c) at the most stressed bolt versus the total shear force of the connection which here is referred to as demand curve. As mentioned before, based on the Abou-zidan and Liu method [2] any irreversible decreases in the demand curve can be considered as bolt fracture. As Fig. 13b shows, an irreversible reduction is obvious in this demand curve. However, as it is expected and clear from the analytical results, this reduction is not due to the bolt shear fracture, but it was because of the excessive deformation of the shear tab and local buckling

100 75 50 25 0 -25 -50 -75 -100 -250

Experimental result Analytical result

-150

-50

50

150

250

Displacement [mm] Fig. 12. Comparison between the analytical and experimental results of specimen 2A tested in reference [30].

195

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700

(b)

600

F'nv >> Fv,max

-c

(MPa)

500 400 300 200

Demand

100

Capacity (F'nv) 0 0

200 400 600 800 Total shear force of the connection (KN)

Fig. 13. Specimen BF: a) Finite element mesh and deformations; b) variation of τbolt − c in terms of shear force of the connection.

of the beam flange plates at the mid-span of beam where the moment demands are maximum (see Fig. 13a). Hence, a modification is needed to be applied on this method to improve its accuracy. Hereafter, this modified method is named as method MTD1 which is discussed in the following section.

ratio of the shear strength to ultimate tensile strength of a bolt is actually a random variable with a mean value of 0.62 and standard deviation of 0.03. In this study in order to propose a conservative failure criterion, by assuming that this random variable follows a normal distribution, this ratio was selected as such, to limit the probability of exceedance to 84%. This procedure yields to Fnv = 0.59 Fu (i.e.,

3.1.1.3. Proposed method MTD1. As concluded in the previous section, any reduction in the bolt shear stress may not be necessarily due to the bolt fracture and the method proposed by Abou-zidan and Liu [2] cannot be used alone. Rather, a controlling parameter which is directly correlated to the bolt material property is needed to be used in conjunction with the initial idea. In this study, modified nominal shear strength of bolt (F′nv) expressed by Eq. (C-J3-5a) of AISC 360 [29], denoted here as Eq. (1), was selected as controlling parameter. In this equation fv is required shear stress, ft is required tensile stress, Fnt is nominal tensile stress, Fnv is nominal shear stress, F′nv is nominal shear stress modified to include the effects of tensile stress, and ∅ is resistance factor. In this study, the resistance factor ∅ was taken as unity since the real behavior of the bolt was of interest.

Pr

2

2

⎛ ft ⎞ + ⎛ f v ⎞ = 1 ⎝ ∅Fnv ⎠ ⎝ ∅Fnt ⎠ ⎜

⎟

⎜

⎟

f F′nv = ∅Fnv 1 − ⎛ t ⎞ ⎝ ∅Fnt ⎠ ⎜

Fnv Fu

)

< 0.59 = 0.16). Fig. 13b compares the variation of the τbolt − c at

the most stressed bolt of specimen BF with the proposed controlling parameter, F′nv, which was denoted as capacity curve. As this figure shows, there is a remarkable difference between the maximum shear stress experienced by the bolt and the value of F′nv which clearly indicates that the bolt has not been fractured and something else has caused such irreversible reduction in the bolt shear stress. In other words, such controlling parameter is necessary to recognize the actual reason of the connection strength loss. In order to examine the applicability of the proposed method, using the test setup utilized by Moore et al. [22], 3/4, 7/8 and 1 inch A325 and A490 bolts were numerically subjected to incremental shear load until the achievement of bolt shank fracture. For all A325 and A490 bolts, Fig. 14 shows the variation of τbolt − c and the controlling parameter F′nv in terms of the value of the parameter dv. The former one was named as demand curve, while the later one was denoted as capacity curve. For 7/8 inch A325 and A490 bolts, this comparison was done for both snug tightened and fully pre-tensioned bolts. As Fig. 14 indicates, the capacity and the demand curves for both snug tightened and pretensioned bolts have intersected at a specific point which is failure

2

→

(

⎟

(1)

AISC 360 [29], states that Fnv is 0.625 times of the ultimate tensile strength of the bolt material (Fnv = 0.625Fu) when threads are excluded from the shear planes. However, experimental results on 142 A325 and A490 bolts presented by Kulak et al. [28] indicated that the 196

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(a)

700

600

600

(MPa)

700 500 400 300

Fig. 14. Demand and capacity curves in terms of τbolt − c and dv for: a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

(b)

800

-c

-c

(MPa)

800

500 400 300

Capacity (Pretensioned) Demand (Pretensioned) Capacity (Snug-tightened) Demand (Snug-tightened)

200 100 0 0

5

dv (mm)

10

Capacity (Pretensioned) Demand (Pretensioned) Capacity (Snug-tightened) Demand (Snug-tightened)

200 100 0 15

0

800

(c)

dv (mm)

10

15

(d)

800

600

600

(MPa)

500

(MPa)

500

300

-c

700

-c

700

5

300

400 200

200

Demand

100

400

Demand

100

Capacity

0

Capacity

0 0

800

5

dv (mm)

10

15

0

800

(e)

600

600

(MPa)

500

(MPa)

500

300

-c

300

400 200

Demand

100

Capacity

0

dv (mm)

10

15

(f)

700

-c

700

5

400 200

Demand

100

Capacity

0 0

5

dv (mm)

10

15

0

5

dv (mm)

10

15

MTD1 and also the one proposed by Abou-zidan and Liu [2] is strongly mesh sensitive. If in the finite element modeling this important point is not considered, these two curves will never intersect even if a bolt has definitely experienced a shear fracture. This behavior can be seen in the finite element analysis done by Abou-zidan and Liu [2] for an extended shear plate connection where the governing failure mode was bolt fracture. Fig. 15 shows the variation of shear stress at a fractured bolt drawn by Abou-zidan and Liu [2]. As this figure indicates for this fractured bolt, the demand and capacity curves did not intersect, which can be due to the usage of improper mesh size by Abou-zidan and Liu [2]. For this comparison, Fnv instead of F′nv was used, since other parameters to calculate F′nv were not reported by Abou-zidan and Liu [2]. Furthermore, as the deformation in the bolt releases its pre-tensioning load, there would not be a considerable difference between Fnv and F′nv at the onset of bolt fracture. Hence, based on this discussion, it can be concluded that the fracture of a bolt has happened when the demand (τbolt) and capacity (F′nv) curves have intersected, provided that, an appropriate mesh size and suitable pre-tensioning method is used.

point. Finite element models have shown that at this point the bolt shanks have experienced an excessive shear deformation, indicating a bolt shank fracture. Fig. 14 also indicates another important point regarding the behavior of pre-tensioned bolts. As Fig. 14a and b shows, the amount of F′nv increases as the shear deformation of the bolt shank increase. This behavior can be explained using Eq. (1). As this equation indicates, the amount of F′nv increases as the bolt tensile stress decreases. Since the method used for pre-tensioning of bolts are on the basis of the adjusting of bolt shank length (method-4 presented in Section 2.3), finite element models are able to automatically capture the reduction in bolt tensile stress and consequently increase the bolt shear strength (F′nv). For all 3/4, 7/8 and 1 inch A325 and A490 bolts, the obtained values of dvf (dv corresponding to the intersection point of the demand and capacity curves) and related experimental ones reported by Moore [22] along with the related errors are summarized in Table 4. As the results presented in this table indicate, the proposed method is generally conservative with a maximum and average error equal to 10% and 5% respectively. Abou-zidan and Liu [2] did not report the level of accuracy of their method; they just stated that their method is able to capture the bolt fracture. Using the original method proposed by Abouzidan and Liu [2], for all 3/4, 7/8 and 1 inch A325 and A490 bolts tested by Moore [22], the value of dvf was obtained and the related errors were calculated. These results are summarized in Table 4. As the results indicate, this method remarkably overestimates the value of dvf with maximum and average errors of 58% and 29% respectively. It should be noted that the implementation of the proposed method

3.1.2. First category on the basis of von Mises stress Suleiman [1] proposed that the bolt fracture can be predicted by comparing the von Mises stress contours against the ultimate engineering stress of the bolt material (Fu). The weak points of this procedure might be as follow: a) to have a more realistic comparison, instead of engineering ultimate stress, von Mises stress contours must be compared against true ones. b) This procedure is actually a visual 197

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Table 4 Comparison between the values of dvf (mm) obtained from different methods. Bolt type

A325

Diameter (inch)

3/4

7/8

1

3/4

7/8

1

Experimental value of dvf [22]

5.43

6.52

7.37

6.01

6.45

8.00

dvf predicted Error dvf predicted Error dvf predicted Error dvf predicted Error

5.60 3% 8.56 58% 3.50 − 36% 6.12 13%

6.37 − 2% 8.45 30% 3.83 − 41% 6.75 4%

7.30 − 1% 8.59 17% 4.64 − 37% 8.40 14%

5.39 − 10% 7.56 26% 3.33 − 45% 6.90 15%

5.79 − 10% 8.8 36% 4.06 − 37% 7.10 10%

7.20 −10% 8.54 7% 4.33 −46% 9.32 17%

using MTD1 method using Abou-zidan method [2] using Modified Suleiman method using MTD2 method

A490

700

(MPa)

500 400 300 200 Demand (Obtained by Abou-zidan and Liu [2]) Capacity (Fnv=0.59Fu)

0 0

100

200

300

400

−5% 29% −40% 12%

same elements, which is named here as demand curve. The failure point can be estimated by intersecting the demand and capacity curves. This method is hereafter referred to as method MTD2. In this method, to define the capacity curve, a relationship between plastic equivalent strain and von Mises stress based on the material property is needed. In this study, this relationship was defined based on the work done by Rice and Tracey [17] which is as given by Eq. (2), where ε f is the von Mises inelastic strain (equivalent plastic strain) at failure, εf is uniaxial plastic strain at fracture (≅ 0.21 and 0.16 for A325 and A490 bolts), σH is hydrostatic stress and σv is von Mises stress. In fact, this equation indicates that the presence of triaxial stresses, σ H σv , affects the plastic rupture strain. These demand and capacity curves for all A325 and A490 bolts along with their failure points are shown in Fig. 17. In order to determine the value of dv corresponding to these failure points (dvf), demand and capacity curves in terms of PEEQ were drawn against dv (see Fig. 18). As this figure shows, for 7/8 in A325 and A490 bolts the values of parameter dvf are 6.75 and 7.1 mm respectively which are enough close to the experimental values of dvf. Table 4 summarizes the predicted values of dvf obtained based on MTD2 method for all 3/4, 7/8 and 1-inch A325 and A490 bolts along with their related errors. As the results of this table indicate, this method is somehow non-conservative with a maximum and average error equal to 17% and 12% respectively.

600

100

Avg. error

500

Total Shear Force (KN) Fig. 15. Comparison between the maximum experienced of τbolt and Fnv in the case of using an inappropriate bolt's mesh size.

inspection and it completely depends on the user's opinion to decide whether the bolt shank is fractured or it still can resist the load. Because Suleiman [1] did not clearly specify how much of the bolt shank must reach to Fu to be considered as a fractured shank. c) This method is a too conservative as it is explained in the following paragraph. In order to remedy the second weak point, a computational method is proposed. This method which is hereafter referred to as modified Suleiman method, is graphically shown in Fig. 16 for all A325 and A490 bolts tested by Moore [22]. As this figure shows, for these bolts, a demand curve which is the average of von Mises stresses for the critical elements of the bolt shank (see Fig. 2), σv − bolt, versus the value of parameter dv, were drawn along with the upper limits or capacity values. These upper limits or capacities were set to the engineering value of Fu because the von Mises stresses were obtained by averaging of the stress of the critical elements. As this figure indicates, the values of dv corresponding to the engineering value of Fu (dvf) for 7/8 inch A325 and A490 bolts are 3.83 and 4.06 mm respectively which are much < 6.52 and 6.45 mm that A325 and A490 bolts experienced during tests at the onset of fracture respectively. Table 4 summarizes the predicted values of dvf obtained based on modified Suleiman method for all 3/4, 7/8 and 1 inch A325 and A490 bolts along with their related errors. As the results of this table indicate, this method is inaccurate and too conservative with a maximum and average of error equal to 46% and 40% respectively. This inaccuracy comes from this fact that the ductile materials can undergo a significant inelastic deformation after reaching the ultimate stress of material. In other words, there is a significant difference between the strains corresponding to ultimate stress and the one associated to rupture stress while this point was ignored in this method. Hence, it seems that a more accurate criterion can be obtained if plastic strains are also involved in defining the failure criterion. As a result, in this study, the average of von Mises stresses of the critical elements (σv − bolt) were drawn against plastic equivalent strains (PEEQ) of the

1 3σ ε f = ε f exp ⎛ − H ⎞ 2σv ⎠ ⎝2 ⎜

⎟

(2)

3.2. Second category As mentioned in Section 1, in the second category, fracture happens based on a predefined failure mechanism. A failure mechanism consists of two ingredients: a fracture (or damage) initiation criterion and a damage evolution law. In this section, firstly, a comprehensive summary of different types of fracture mechanisms, fracture initiation models and evolutions are presented. However, among all possible methods of this category, the focus of this paper is on the procedure which is on the basis of the extended finite element method (XFEM) that is discussed in the following subsections. Hereafter, this method is referred to as MTD3 method. 3.2.1. Fracture mechanisms In general, metals and alloys fail due to one or a combination of the following three mechanisms [42]: a) Ductile fracture that occurs by a process known as microvoid coalescence. In such case, first, plastic strain causes small microvoids to form in the material, most often at sites of inclusions. As the process proceeds, these microvoids grow and begin to join together (coalesce). Final failure occurs when the walls of material between the growing voids finally break; b) shear fracture that is based on shear band localization and c) instability with localized necking that is followed by ductile or shear fracture inside the neck area. Under special conditions, however, cracks can form and propagate 198

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1400 1200

(b)

1200

1000

(MPa)

(MPa)

Fig. 16. Variation of von Mises stress along with upper limit for: a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/ 4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

1400

(a)

800 600

1000 800 600

400

400 Demand Capacity

200

Demand Capacity

200

0

0 0

5

10

15

0

5

dv (mm) 1400

1400

(c)

1000 800 600

1000 800 600

400

400 Demand Capacity

200

Demand Capacity

200

0

0

0

5

10

15

0

5

dv (mm) 1400

1200

10

15

dv (mm)

1400

(f)

1200

(e)

1000

(MPa)

(MPa)

15

(d)

1200

(MPa)

(MPa)

1200

10

dv (mm)

800 600

1000 800 600

400

400 Demand Capacity

200

200

0

Demand Capacity

0 0

5

10

15

0

5

dv (mm)

10

15

dv (mm)

generality, practitioners like it because location of possible fracture site can be determined simply by constructing color-coded plots of the equivalent strain and data can be easily found in handbook [53]. For the X-W fracture model, fracture is postulated to occur when the modified accumulated equivalent plastic strain reaches a limiting value. The ductile fracture model available in ABAQUS [15] which is a phenomenological model for predicting the onset of damage due to nucleation, growth, and coalescence of voids is on basis of the X-W model. This model assumes that the equivalent plastic strain at the onset of pl damage, ε D , is a function of stress triaxiality and equivalent plastic pl pl pl ̇ ̇ strain rate: ε D (η, ε ) , where η is the stress triaxiality, and ε is the equivalent plastic strain rate. The criterion for damage initiation is met when the condition presented in Eq. (3) is satisfied. Recent experimental results for aluminum alloys and other metals [55] reveal that, in addition to stress triaxiality and strain rate, ductile fracture can also depend on the third invariant of deviatoric stress, which is related to the Lode angle. ABAQUS [15] also considers this modification.

along grain boundaries. This rare fracture mechanism is known as intergranular fracture and as its name implies, occurs when the grain boundaries are the preferred fracture path in the material [43]. 3.2.2. Fracture initiation models So far, several fracture models (or damage criteria) are proposed by researchers to predict the fracture initiation of metals corresponding to the above-mentioned mechanisms. Examples of the most common fracture models regarding the predicting of the necking instability of sheet metal include forming limit diagram (FLD) [44]; forming limit stress diagram (FLSD) [45]; Muschenborn-Sonne forming limit diagram (MSFLD) [46]; and the Marciniak-Kuczyński (M-K) criterion [47]. In ABAQUS program [15], these criteria apply only to elements with a plane stress formulation (plane stress, shell, continuum shell, and membrane elements); ABAQUS ignores these criteria for other elements. Examples of the most common fracture models regarding the prediction of the ductile and shear fracture mechanisms include: Constant equivalent strain model [48]; X-W fracture model [49,50]; the shear fracture model [42]; the Wilkins (W) fracture model [51]; the Johnson-Cook (J-C) fracture model [52]; the maximum shear stress (MS) fracture model [53]; Cockcroft–Latham fracture model [54]; MAXE and QUADE fracture models [15]; MAXS and QUADS fracture models [15]; and MAXPS and MAXPE fracture models [15]. A brief explanation of some of these models is as follows: For the constant equivalent strain fracture model, fracture is assumed to occur in a material element when the plastic equivalent strains (PEEQ) reach a critical value. While this criterion lacks

∫

dε pl pl ε Dpl (η, ε ̇ )

=1 (3)

For the Wilkins fracture model, fracture is assumed to occur when the following integral exceeds a critical value Dc over a critical dimension Rc [51]. In Eq. (4), A = max(σ2 / σ1, σ2 / σ3) where σ1, σ2 and σ3 are principal stresses and parameters a, λ and μ are material constants.

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1400 1000

Failure Point: PEEQ=0.39 Mises Stress=1060.45 MPa

800 600

1000

Failure Point: PEEQ=0.28 Mises Stress=1242.19 MPa

800 600

400

400 Demand Capacity

200

Demand Capacity

200

0

0 0

0.5

1

1.5

2

0

0.5

1

PEEQ 1400

1400

(c) (MPa)

(MPa)

Failure Point: PEEQ=0.38 Mises Stress=1060.5 MPa

800 600

(d)

1000 Failure Point: PEEQ=0.36 Mises Stress=1242.8 MPa

800 600

400

400 Demand Capacity

200 0 0

0.5

1

1.5

Demand Capacity

200 0 2

0

0.5

1

PEEQ 1400

(e)

1000 Failure Point: PEEQ=0.37 Mises Stress=1057.8 MPa

800 600

2

(f)

1200

(MPa)

(MPa)

1200

1.5

PEEQ

1400

1000 Failure Point: PEEQ=0.36 Mises Stress=1238.2 MPa

800 600

400

400 Demand Capacity

200 0 0

0.5

1

1.5

Demand Capacity

200 0 2

0

0.5

PEEQ

εf

1 (2 − A)ηdε (1 − aσm) λ

1

ε Dpl = C1 + C2 exp(C3 η)

dε pl

2

assumed to occur when the maximum shear stress, τmax, reaches a critical value, τmax − f, (i.e., τmax = τmax − f). τmax can be obtained using Eq. (7) where σ1, σ2 and σ3 are the principal stresses. Eq. (7) is similar in form to the Tresca yield condition but in general τmax − f is larger than the yield stress in shear [53].

(4)

τmax = max

{ σ −2 σ , σ −2 σ , σ −2 σ } 1

2

2

3

3

1

(7)

For the maximum nominal stress (strain) model, MAXS (MAXE), fracture is assumed to initiate when the maximum nominal stress (strain) ratio (as defined in Eqs. (8) and (9)) reaches a value of one. For the quadratic nominal stress (strain) model, QUADS (QUADE), fracture is assumed to initiate when a quadratic interaction function involving the nominal stress (strain) ratios (as de fined in the Eqs. (10) and (11)) reaches a value of one [56]. In these equations, parameters tn0(εn0), ts0(εs0) and tt0(εt0) are maximum tolerable normal stress (strain), shear stress (strain) in the first direction and shear stress (strain) in the second direction, respectively. And parameters tn(εn), ts(εs), and tt(εt) are applied normal stress (strain), shear stress (strain) in the first direction and shear stress (strain) in the second direction, respectively. In ABAQUS [15], the MAXE, MAXS, QUADE and QUADS fracture initiation models can be used to predict fracture initiation in cohesive elements and XFEM enriched regions.

(5)

The shear fracture model is a phenomenological model for predicting the onset of damage due to shear band localization. This model which is also available in ABAQUS [15], assumes that the equivalent plastic strain at the onset of damage, ε Spl , is a function of the shear stress pl ratio and the equivalent plastic strain rate: ε Spl (θs , ε ̇ ) . Here θs = (1 − ksη)/ϕs where ϕs is the ratio of the maximum shear stress (τmax) and the von Mises stress (σv), η is the stress triaxiality and ks is a material parameter. The criterion for fracture initiation is met when the condition presented in Eq. (6) is satisfied: pl ε Spl (θs , ε ̇ )

1.5

PEEQ

The Johnson-Cook fracture initiation model [52] which is also available in ABAQUS [15] is a special case of the ductile fracture initiation model and is for predicting the onset of damage due to nucleation, growth, and coalescence of voids in ductile metals. The model assumes that the equivalent plastic strain at the onset of damage (for constant strain rate and temperature) is a monotonic function of the stress triaxiality (see Eq. (5)). In Eq. (5), the constants C1, C2 and C3 are failure parameters and can be determined mainly from tensile tests with high triaxiality and in some cases from a shear test.

∫

2

1200

1000

∫0

1.5

PEEQ

1200

Dc =

Fig. 17. Demand and capacity curves in terms of σv − bolt and PEEQ for: a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

(b)

1200

(MPa)

(MPa)

1400

(a)

1200

=1

〈t 〉 t t max ⎧ n0 , 0s , 0t ⎫ = 1 ⎨ ⎭ ⎩ tn ts tt ⎬

(6)

For the maximum shear stress (MS) fracture model, fracture is 200

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1.2

(a)

1

(b)

1 0.8

PEEQ

0.8

PEEQ

Fig. 18. Demand and capacity curves in terms of PEEQ and dv for: a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/ 4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

1.2

0.6 0.4

0.6 0.4

Capacity Demand

0.2

0.2

0

Capacity Demand

0 0

5

10

15

0

5

dv (mm) 1.2

(c)

(d)

1 0.8

PEEQ

0.8

PEEQ

15

1.2

1

0.6 0.4

0.6 0.4

Capacity Demand

0.2 0 0

5

10

Capacity Demand

0.2 0 15

0

5

dv (mm)

10

15

dv (mm)

1.2

1.2

(e)

1

(f)

1 0.8

PEEQ

0.8

PEEQ

10

dv (mm)

0.6 0.4

0.6 0.4

Capacity

0.2

0.2

Demand

0 0

5

10

Capacity Demand

0

15

0

5

dv (mm)

〈ε 〉 ε ε max ⎧ n0 , 0s , 0t ⎫ = 1 ⎨ ⎭ ⎩ εn εs εt ⎬ 2

2

2

2

(10)

2

⎧ 〈εn 〉 ⎫ + ⎧ εs ⎫ + ⎧ εt ⎫ = 1 0 0 0 ⎨ ⎨ ⎨ ⎩ εn ⎬ ⎭ ⎩ εs ⎬ ⎭ ⎩ εt ⎬ ⎭

(11)

For the maximum principal stress (strain) model, MAXPS (MAXPE), fracture is assumed to initiate when the maximum principal stress (strain) ratio reaches a value of one (i.e.,

15

assumption that when a crack extends by a small amount, the energy dissipated in the process is equal to the work required to close the crack to its original length [58]. This approach can be computationally effective when sufficiently refined meshes are used, and when all the elements at the crack tip have the same dimensions in the crack growth direction [58]. In addition, in this method, care must be taken in selecting the element size at the crack tip to simulate discontinuity (delamination). The XFEM which was originally proposed by Dolbow [59] is an extension of the conventional finite element method based on the concept of partition of unity. This method allows the presence of discontinuities in an element by enriching degrees of freedom with special displacement functions and does not require the mesh to match the geometry of the discontinuities. In addition, this method is a very attractive and effective way to simulate initiation and propagation of a discrete crack along an arbitrary, solution-dependent path without the requirement of remeshing in the bulk materials [15]. In this method, unlike other traditional methods, it is not necessary to pre-model a crack tip singularity or its direction of crack growth. The minimum required is to define a part or area of a part as an enriched zone. This will be the zone where a crack can initiate and grow. This method can determine where a crack will initiate and where it will propagate without any extra modeling effort. All these have made the XFEM distinguished from other approaches such as boundary element methods [60], remeshing methods [61,62], and element deletion methods [63]. To define a crack using the XFEM, two main steps must be taken;

(9)

⎧ 〈t n〉 ⎫ + ⎧ t s ⎫ + ⎧ tt ⎫ = 1 0 ⎨ ⎨ t s0 ⎬ ⎨ t t0 ⎬ ⎩ tn ⎬ ⎭ ⎩ ⎭ ⎩ ⎭ 2

10

dv (mm)

{

〈σmax 〉 σ0max

} = 1; { } = 1). 〈εmax 〉 ε0max

Parameter σmax0(εmax0) represents the maximum allowable principal stress (strain) and σmax(εmax) is the principal stress (strain) demand in the enriched elements. In Eqs. (8) through (11), the Macaulay bracket, 〈 〉, denotes that the compressive stress (strain) does not contribute to damage initiation. In ABAQUS [15], the MAXPE and MAXPS fracture initiation models can be used only to predict fracture initiation in the XFEM enriched regions. 3.2.3. Extended finite element method (XFEM) In general, in finite element modeling, methods to simulate crack growth can be classified into two main groups: traditional and recent methods. Virtual crack closure technique (VCCT) [57] and extended finite element method (XFEM) are examples of the traditional and the modern methods respectively. The VCCT method is based on the

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t

the crack initiation to the bolt material property and also to consider the interaction of the bolt internal forces (horizontal and vertical shear and tensile force), crack initiation was initially defined based on QUADS. However, analytical results indicated that usage of QUADS as crack initiation criterion may result in inaccurate fracture time prediction because beyond the yield point, material in different strains can experience same stresses so all of these points may meet the stress criterion. As a result, in this study, QUADE was chosen as appropriate criterion for crack initiation (i.e., Eq. (11)). The numerators of Eq. (11) can be considered as demand values, while the denominators can be seen as capacity values. These capacities must be as such, to initiate the crack correctly. Such values can be defined as a multiple of bolt material properties that are discussed in Section 3.2.5. Once the damage initiation criterion is met, damage can occur according to a pre-defined damage evolution law. The damage evolution law defines the post damage-initiation material behavior which characterizes the rate of reduction in the material stiffness when the initiation criterion is satisfied. There are two methods to the definition of the evolution of damage. The first method involves specifying the effective displacement at complete failure, δf, relative to the effective displacement at the initiation of damage, δ0. In the second method, damage evolution can be defined based on the energy that is dissipated as a result of the damage process, also called the fracture energy, GC. In ABAQUS, fracture energy can be specified as a material property and ABAQUS ensures that the area under the linear damaged response shown in Fig. 19 is equal to the fracture energy. In the current study, the energy-based damage evolution criterion was used since it led to a more rapid convergence. In addition, the fracture energy was assumed to be mode-independent where the fracture energy, GC, was estimated using Eq. (12) [64]. In this equation Kc is fracture toughness and Es is modulus of elasticity of the bolt material. The value of Kc was estimated from the work done by Weaver et al. [65] with data on 300C maraging steel. The material properties (e.g., yield and tensile strength) of this steel are similar to those of high strength bolts. In this reference, the values provided for the fracture toughness were between 44 and 110 MPa m . Using Eq. (12), this range of fracture toughness yields to a range of fracture energy, GC, between 8.9 and 55.76 N·mm/mm2. Fig. 20 shows the effect of parameter GC on the response of a high strength bolt under shear force. As this figure indicates and as it is expected, this parameter had no any effect on the onset of fracture initiation, but increase in this parameter caused an increase in the amount of displacement at the complete failure. In this study, the value of Kc was assumed to be 60 MPa m which finally led to GC = 16.59 N ∙ mm/mm2. As mentioned before, in this study a linear softening behavior (see Fig. 19) was used. However, in ABAQUS program an exponential softening behavior also can be used. Fig. 20

Damage initiation

Traction

Linear softening Failure

Separation Fig. 19. Traction-separation model.

defining crack initiation and crack evolution. 3.2.4. Crack initiation and evolution models used in the XFEM In this study, the XFEM capability in ABAQUS was used to model cracks. ABAQUS provides two approaches for studying crack initiation and propagation using XFEM: a) Linear elastic fracture mechanics (LEFM) and b) traction-separation cohesive behavior. The LEFM approach uses the modified VCCT to calculate the strain energy release rate at the crack tip. The LEFM approach is more appropriate for brittle fracture problems [15]. The second approach is a very general interaction modeling capability, which can be used for modeling brittle or ductile fracture [15]. As a result, in this study, the second approach, XFEM-based Traction-separation cohesive behavior, was used for modeling of damage. Fig. 19 shows the traction-separation model used in this study with a linear softening behavior along the failure me→ chanism. In this figure, the nominal traction stress vector, t , consists of three components in three-dimensional problems, tn is the elemental normal stress and ts and tt are the two shear stress components perpendicular to the normal stress. The corresponding separation vector is → denoted by δ (δn, δs and δt). T0 denotes the maximum traction and δf is the separation at which the final failure occurs. Among all fracture initiation models, mentioned in Section 3.2.2, six built-in fracture initiation models are available in ABAQUS using XFEM. These fracture initiation models are MAXS, MAXPS, QUADS, MAXE, MAXPE and QUADE. In this study, in order to directly correlate

Fig. 20. Rv-dv curves for a typical high strength bolt with different values of damage initiation and evolution parameters.

Rv (KN)

200 180

Fracture energy=5, Softening=Linear

160

Fracture Energy = 10, Softening=Linear

140

Fracture Energy = 16.59, Softening=Linear

120

Fracture Energy = 30, Softenin:=Linear

100

Fracture Energy = 50, Softening=Linear

80

Fracture Energy = 75, Softening=Linear

60

Fracture Energy = 16.59, Softening=Exponential

40

Reduced damage initiation parameters

20

Increased damage initiation parameters

0 0

1

2

3

4

5

6

7

dv (mm) 202

8

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Eq. (2) would be 1.65 times of εf. Furthermore, ε f can be obtained using Eq. (16) [66]. By knowing that, in the shear case, all strain components (including normal strains, εii, and shear strains, γij) except γ12 is zero and with respect to the fact that in the plastic case, the effective Poisson's ratio, ν, is 0.5, ε f in Eq. (16) would be 0.577 times of γ12. By equating Eqs. (2) and (16) and with respect to the fact that the difference between the total and plastic strain is negligible, α is approximately 2.86.

compares the response of the fractured bolt under both softening behaviors for a given value of fracture energy. As this figure indicates, by changing the softening behavior from linear to exponential, the onset of the complete bolt fracture was postponed significantly. For the linear softening response shown in Fig. 19, ABAQUS calculate the damage variable, D, as expressed by Eq. 13. In this equation, Teff0 is the effective traction at damage initiation and δmax is the maximum value of the effective displacement attained during the loading history.

GC = K c 2 Es

D=

(12)

δ f (δmax − δ0) ; δ f = 2GC T 0eff δmax (δ f − δ0)

εf =

(16)

(13)

β is a constant value and it is expected to be mainly affected by the bolt material properties. Since the left side of Eq. (14) reflects the total strain and its right side reflects the plastic strain, β must be slightly greater than unity. But as mentioned before, in this study, in order to have a conservative criterion, the bolt fracture prediction was based on a lower bound of parameter dvf, so, the value of parameter β will not be as expected. As a result, this parameter for different bolt materials and diameters was determined by trial and error method as such the analytical onset of bolt fracture was as same as the experimental one (i.e., dvfXFEM = dvf90%). Related values of parameter β are summarized in Table 5. With respect to the explanations provided in this section, a summary of parameters of the material failure models (damage initiation and evolution) used in ABAQUS for A325 and A490 bolts are as those summarized in Table 6, where the fracture initiation of bolts was controlled using Eq. (11) with α equal to 2.86 and proposed values of parameters β presented in Table 5. It is expected that by using the stress-strain curves presented in Fig. 3 (to obtain the plasticity model) and applying the fracture initiation and evolution models (summarize in Table 6) the response of a high strength bolt under shear to be appropriately predictable. Fig. 22 compares the experimental and analytical results of all A325 and A490 bolts under pure shear by implementing the proposed values summarized in Table 6 based on XFEM along with the shape of fractured bolts. As these figures indicate, the proposed procedures could appropriately predict the fracture initiation and evolution.

3.2.5. Determination of the crack initiation parameters In this study, the appropriate values for the denominators of Eq. (11) (capacity values) were determined based on the experimental results presented by Moore [22]. The experimental results indicated that for a given material property and bolt diameter, the shear displacement corresponding to the onset of fracture, dvf, was not a constant value. In other words, this parameter can be considered as a random variable. As a result, these capacity values which control the onset of the fracture initiation are also random variables. Fig. 20 compares the effect of these parameters on the response of a bolt subjected to shear. As this figure indicates, by increasing the values of damage initiation parameters (for a given value of parameter GC and given softening behavior) the onset of damage would be postponed. Table 5 summarizes the mean and standard deviation, SDTEV, for parameter dvf for different bolt materials and diameters. Table A2 of Appendix A lists all the values of parameter dvf that have led to these statistical quantities. These values are obtained from the experimental results presented in reference [22]. Fig. 21 shows the cumulative distribution function (CDF) of random variable dvf for all A325 and A490 bolts. All these data were first plotted on normal probability paper and results indicated that a normal probability distribution could fit the data remarkably well. In the current study, in order to achieve a conservative criterion for bolt fracture prediction, lower bound of dvf parameters with probability of exceedance of 90% was calculated on the basis of the experimental results of Moore [22]. This parameter is denoted by dvf90%. As it is shown in Fig. 21a, this value for 7/8 inch A325 bolt is 5.36 mm. All dvf90% values for different bolt diameters of A325 and A490 bolts are summarized in Table 5. In this study, the denominators of Eq. (11) (capacity values) were determined as such, to initiate the fracture of bolts when the value of parameter dv reaches its related lower bound value, dvf90%. To reach this goal, capacity values were defined by Eqs. (14) and (15).

ε 0n = β ε f

(14)

εs0 = εt0 = α ε 0n

(15)

1 3 2 2 (ε11 − ε22 )2 + (ε22 − ε33)2 + (ε33 − ε11)2 + (γ12 + γ 223 + γ31 ) 2 2 (1 + υ)

4. Implementing the proposed methods to capture bolt fracture In order to check the applicability of the proposed bolt fracture methods, an extended shear tab connection, specimen 6U, which was experimentally tested by Sherman and Ghorbanpoor [23] and a A490 high strength bolt tested in a double shear joint presented in reference [24] were modeled using finite element method. 4.1. Specimen 6U of reference [23] In order to develop a design procedure for extended shear tabs, a research program that consisted of 31 full-scale tests was performed by Sherman and Ghorbanpoor [23]. Among these specimens, specimen 6U was selected since it experienced bolt fracture. The geometric and material properties of this specimen are summarized in Table 3. Fig. 23

In these equations, α is a parameter that correlates the capacity values of shear strain to that of tension and it was determined as follow: Under pure shear the triaxial stress, σ H σ , is zero and consequently ε f in

Table 5 Bolt fracture parameters obtained based on reference [22]. Bolt type

A325

Diameter mm (in.)

19 (3/4)

22 (7/8)

25 (1)

19 (3/4)

22 (7/8)

25 (1)

5.43 0.43 4.87 0.57

6.52 0.90 5.36 0.57

7.37 1.07 5.99 0.57

6.01 0.48 5.38 0.78

6.45 0.66 5.60 0.63

8.00 0.99 6.72 0.81

dvf (mm)

β

Mean STDEV 90% probability 90% probability

A490

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1

0.8

0.8

CDF

CDF

1

0.6 0.4 0.2

(a)

Normal fit Experiment

0.6 0.4

Normal fit Experiment

0.2

0

(b)

0 3

4

5

6

7

8

9

3

4

5

dvf (mm)

7

8

1.2 Normal fit Experiment

1

1

Normal fit

0.8

CDF

0.8

CDF

6

dvf (mm)

1.2

0.6 0.4

Experiment

0.6 0.4

(c)

0.2

(d)

0.2

0

0 3

4

5

6

7

3

4

5

dvf (mm)

6

7

dvf (mm)

1.2

1.2 Normal fit

1

Normal fit

1

Experiment

Experiment

0.8

CDF

0.8

CDF

Fig. 21. Cumulative distribution function for: a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

1.2

1.2

0.6

0.6 0.4

0.4 0.2

0.2

(e)

(f)

0

0 3

4

5

6

7

8

9

10 11 12 13

3

4

5

6

7

8

9

10

11

12

dvf (mm)

dvf (mm)

Table 6 Summary of material failure models of bolts used in ABAQUS. Material failure model

A325-bolt

A490-bolt

Damage initiation

Type: Quade

Type: Quade

Bolt diameter (inch) Nominal strain normal-only mode Nominal strain shear-only mode first direction Nominal strain shear-only mode second direction

3/4, 7/8 and 1 0.120 0.3432 0.3432

3/4 0.1250 0.3575 0.3575

7/8 0.1000 0.2860 0.2860

1 0.1300 0.3718 0.3718

Material failure model

A325-bolt

A490-bolt

Damage evolution

Type: Energy

Type: Energy

Softening Degradation Mixed mode behavior Mode mix ratio Fracture energy, N·mm/mm2

Linear Maximum Mode-Independent Energy 16.59

Linear Maximum Mode-Independent Energy 16.59

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300

300

250

250

200

200

Rv (KN)

Rv (KN)

A.A. Hedayat et al.

150 100

Fig. 22. Comparison of experimental [22] and analytical results, fracture initiation based on QUADE and dvf90%, for: a) 7/8 inch A325 bolt; b) 7/8 inch A490 bolt; c) 3/4 inch A325 bolt; d) 3/4 inch A490 bolt; e) 1 inch A325 bolt; f) 1 inch A490 bolt.

150 100

50

Experimental result

(a)

Analytical result

0 0

50

1

2

3

4

Experimental result

(b)

Analytical result

0 5

6

0

1

2

250

250

200

200

150 100 Experimental result

(c)

50

Analytical result 1

2

3

4

Analytical result

0 5

0

6

1

2

350 300

250

250

Rv (KN)

Rv (KN)

3

4

200 150

150

(f)

Experimental result

50

0

Analytical result

0 1

2

3

4

6

200

100

Experimental result Analytical result

0

5

dv (mm)

300

(e)

6

Experimental result

(d)

350

50

5

100

dv (mm)

100

4

150

50

0 0

3

dv (mm)

Rv (KN)

Rv (KN)

dv (mm)

5

6

7

8

0

1

2

dv (mm)

3

4

5

6

7

8

dv (mm)

both of material and geometrical nonlinearities were considered. For the former one, depends on the type of material used in experimental test for each component, the associated calibrated material properties presented in Sections 2.1 and 3.1.1.1 were assigned. However, since this specimen was subjected to monotonic loading, isotropic hardening parameters (Q∞ and b) were excluded from the material definition.

shows the test setup of specimen 6U. This specimen which was an exterior connection was tested under monotonic loading by applying a concentrated load on the beam which was located 88.5 in. away from the bolt line. Other boundary conditions including the location of lateral braces are shown in Fig. 23. In FE modeling, the imposed boundary conditions were similar to those used in the test setup. In FE analysis,

Fig. 23. Schematic view of specimen 6U and its related the test setup used in reference [23].

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Fig. 24. Finite element mesh of specimen 6U tested by Sherman and Ghorbanpoor [23].

Fig. 26a shows the compression jig setup used in this test along with its boundary conditions and Fig. 26b shows its finite element mesh. The bolt holes in the plates of test jig were 1.59 mm (1/16 in.) larger than the nominal bolt diameter. The compression test jig shown in Fig. 26a was composed of two 25.4 mm (1 inch) lap plates connected to two 25.4 mm (1 inch) main plates by a single test bolt. Figs. 3 and 26c show the engineering stress-strain curves for the A490 and A440 materials respectively, which were then converted to the true stress-strain values to be utilized in FE modeling. The used element type and mesh sizes, interaction properties and analysis type were similar to those explained in Section 3.1.1.1. Fig. 27 compares the experimental and analytical results of this specimen. The results are presented in terms of shear stress in the bolt shank (τbolt − b) and vertical displacement at the point of applied load (dv). As this figure shows, the stiffness and the maximum load achieved by finite element model were well matched with the experimental ones. Table 7 summarizes the predicted value of dv at fracture point, dvf, and the amount of error corresponding to each method. Experimental value of dvf reported by Wallaert and Fisher [24] was 4.24 mm.

700 600

Shear (KN)

500 400 300 200 Experimental result 100

Analytical result

0 0

5

10

15

20

Displacement (mm) Fig. 25. Comparison between the analytical and experimental [23] results of specimen 6U.

Table 7 Value of dvfc corresponding to different methods.

4.3. Comparison of the accuracy of methods MTD1-MTD3

Specimen

Method

MTD1

MTD2

MTD3

6U of reference [23]

dvfc (mm) Error (%) dvf (mm) Error (%)

15.43 0.5 4.02 5.2 2.9

18.90 23.1 4.75 12.0 17.6

15.62 1.8 4.13 2.6 2.2

A490 bolt in compression jig of reference [24] Average error (%)

As the results of Table 7 indicate, all three methods were able to predict the bolt fracture but with different levels of accuracy. Based on these results, among these three methods, the first and the second accurate methods are MTD3 and MTD1 with the amount of average error equal to 2.2% and 2.9% respectively. However, unlike the MTD3 method which is based on the XFEM, MTD1 is not a suitable method when a progressive collapse analysis is of interest. Figs. 25 and 27 also show the shape of the fractured bolts at the end of test using MTD3. Although this method is time consuming, as these figures indicate, fracture of all bolts was captured appropriately. As it was expected (on the basis of the results obtained in Section 3), the least accurate method was MTD2, with 17.6% of non-conservative error. Hence, based on the results presented in this study and also in comparison to the other two methods (MTD1 and MTD3), it can be concluded that this non-conservative method may not be a suitable method for bolt fracture prediction in shear.

The used element type and mesh sizes, interaction properties and analysis type were similar to those used for specimen 2A which were explained in Section 3.1.1.1. Fig. 24 shows the finite element mesh of this model and Fig. 25 compares the experimental and analytical results in terms of the curves of the shear and vertical displacement of the connection. As this figure shows, there is a good agreement between analytical and experimental results. As mentioned before, this connection failed due to bolt fracture. In this finite element model, bolt fracture was evaluated using the three methods presented in Section 3. Table 7 summarizes the predicted value of connection displacement at the failure time, dvfc, and the amount of error corresponding to each method. Experimental value of the connection displacement at the failure time reported by Sherman and Ghorbanpoor [23] is 15.35 mm.

5. Summary and conclusion This study was aimed to propose appropriate failure criteria for bolt fracture prediction in shear when threads are excluded from the shear plane. In this regard, the finite element methods for such prediction were divided into two main categories and for each category the available methods were discussed. Then using finite element modeling and valuable experimental results presented by Moore [22], three methods, here referred to as methods MTD1, MTD2 and MTD3 were proposed. Method MTD1 belongs to the first category and is a modified version of the Abou-zidan and Liu [2] method. Abou-zidan and Liu [2]

4.2. A490 high strength bolt tested in a double shear joint presented in reference [24] In the experimental work done by Wallaert and Fisher [24], 75 bolts made of different materials were tested in jigs made of A440 and constructional alloy steel. Among these specimens, the one consisted of a 22.23 mm (7/8 inch) A490 bolt was used for this investigation. 206

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Stress (MPa)

(b) (a)

Fig. 26. a) Schematic view of compression jig setup [24]; b) finite element mesh; c) engineering stressstrain curve for A440 material [67].

600 500 400 300 200 100 0

(c)

(MPa)

0

0.05

0.1 0.15 0.2 Strain (mm/mm)

•

800 700 600 500 400 300 200 100 0

•

Experimental result e Analytical result

0

1

2 dv (mm)

3

4

•

5

Fig. 27. Comparison between the analytical and experimental results of A490 bolt tested in compression test jig of reference [24].

assumed that the bolt shear rupture happens when the bolt shear stress decreases irreversibly. Based on MTD1 prediction method the fracture of a bolt happens when the curves of F′nv and shear stress of the bolt shank has intersected, provided that, an appropriate mesh size and suitable pre-tensioning method is used. Method MTD2 also belongs to the first category and is a quantitative version of the Suleiman [1] method. Based on this prediction method, the failure point can be estimated by intersecting the demand and capacity curves. In this method, demand and capacity curves were defined as a function of plastic equivalent strain and von Mises stress. Method MTD3 pertains to the second category and is based on the XFEM. To define a crack using this method, two main steps must be taken including defining crack initiation and crack evolution. In this method, crack initiation was defined based on quadratic nominal strain (QUADE) and crack evolution law was defined in terms of fracture energy. Furthermore, appropriate parameters related to crack initiation and evolution were proposed. The level of accuracy of each method was evaluated using several experimental results of individual bolts presented by Moore [22], experimental results of a bolted extended shear tab connection tested by Sherman and Ghorbanpoor [23] and a A490 high strength bolt tested in a double shear joint presented in reference [24] in which the failure mode was bolt fracture. Following conclusions can be drawn from this finite element study:

• •

•

• In bolted connections, an appropriate mesh size to achieve an acceptable level of convergence might be 1 mm for the bolt shank,

207

0.25

4 mm for the bolt head and nut and 5 mm for the bolt hole. The appropriate method for tightening of a bolt is to shorten its shank length rather than applying a pre-tensioning load to the cross section of the bolt shank. By adjusting the bolt shank length, a finite element model is able to automatically capture the reduction in bolt tensile stress and consequently to increase the bolt shear strength (F′nv). Any reduction in the bolt shear stress may not be necessarily due to the bolt fracture, indicating that the use of Abou-zidan and Liu method without any controlling parameter, may incorrectly lead to a bolt fracture prediction. By comparing the analytical and experimental results of the individual bolts, it was found that: a) method MTD1 is generally conservative with a maximum and average error of 10% and 5% respectively; b) Abou-zidan and Liu method remarkably overestimates the value of dvf with maximum and average errors of 58% and 29% respectively; c) the modified Suleiman's method is too conservative with a maximum and average of error equal to 46% and 40% respectively. And d) method MTD2 is somehow non-conservative with a maximum and average of error 17% and 12% respectively. For method MTD3, the use of quadratic nominal stress (QUADS) as crack initiation criterion can lead to an inaccurate prediction of the bolt fracture time. By comparing the analytical and experimental results of a bolted extended shear tab connection and a A490 high strength bolt tested in a double shear joint it was found that among three proposed methods MTD1, MTD2 and MTD3: a) the first and the second accurate methods are MTD3 and MTD1 with the amount of average error equal to 2.2% and 2.9% respectively; b) the least accurate method is MTD2, with 17.6% of non-conservative error, indicating the unsuitability of this method for bolt fracture prediction in shear. Finally, based on the results presented in this study it can be concluded that both of the methods MTD1 and MTD3 are reasonably acceptable for prediction of the bolt fracture in shear. Method MTD1 is a suitable method when a progressive collapse analysis is not of interest and only the capacity of the system at the onset of the first bolt fracture is required. While, MTD3 method can be used in a progressive collapse analysis where the amount of reduction in a system strength by fracturing of each component is of interest. However, in comparison to the method MTD1, MTD3 method is remarkably time consuming.

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A.A. Hedayat et al.

Appendix A. The database used to evaluate the accuracy of finite element modeling and to calculate the statistical quantities of parameter dvf Table A1 Comparison of the strength ratios (

Lot no.

1

2

3

4

5

6

7

8

9

10

Minimum Maximum Average

Test no.

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5

RFEM v ) Rtest v

of the bolts at the onset of fracture.

A325-bolt

A490-bolt

3/4 (in.)

7/8 (in.)

1 (in.)

3/4 (in.)

7/8 (in.)

1 (in.)

0.99 0.97 0.97 0.98 0.98 0.93 0.96 0.95 0.96 0.96 0.96 0.94 0.94 0.94 0.93 – 0.93 0.93 0.92 0.94 0.92 0.91 0.91 0.93 0.93 0.95 0.93 0.95 0.93 0.94 0.93 0.91 0.91 0.92 0.91 0.91 0.91 0.90 0.90 0.91 0.90 – 0.91 0.91 0.91 0.90 0.90 – 0.88 0.88 0.89 0.88 0.90 0.88 0.99 0.93

0.94 0.96 0.96 0.95 0.95 0.96 0.97 0.96 0.97 0.96 0.94 0.95 0.95 0.95 0.94 – 0.96 0.95 0.95 0.92 0.96 0.91 0.91 0.90 0.90 0.92 0.92 0.93 0.93 0.94 0.92 0.92 0.92 0.92 0.92 0.91 0.91 0.89 0.91 0.89 0.89 0.87 0.88 0.87 0.87 0.88 0.88 0.87 – – – – – 0.87 0.97 0.92

0.97 0.97 0.98 0.98 0.97 0.98 0.99 0.97 0.98 0.98 0.97 0.98 0.97 0.97 0.99 – 1.01 1.02 1.01 1.02 1.03 1.00 0.98 1.00 1.03 0.98 0.96 0.95 0.96 0.96 0.96 0.97 0.97 0.99 0.96 0.99 0.96 0.95 0.96 0.96 0.96 – 0.91 0.91 0.93 0.91 0.92 – – – – – – 0.91 1.03 0.97

0.99 1.01 1.01 1.01 1.02 0.98 0.99 0.99 0.99 0.98 1.01 1.01 1.02 1.01 1.01 – 1.05 1.07 1.06 1.08 1.06 0.98 0.99 1.01 1.01 0.99 0.96 0.96 0.95 0.92 0.94 1.01 1.01 1.01 1.01 1.04 0.97 0.95 0.94 0.96 0.99 – 0.99 1.01 1.00 1.03 1.01 – – – – – – 0.92 1.08 1.00

0.95 0.93 0.97 0.94 0.98 0.96 0.96 0.97 0.96 0.96 1.00 0.99 0.99 0.99 1.00 – 1.02 1.02 1.02 1.05 1.04 0.98 0.97 0.97 0.97 0.98 0.91 0.91 0.93 0.92 0.92 0.94 0.94 0.95 0.94 0.93 1.01 1.01 0.99 0.91 0.99 – 0.93 0.94 0.93 0.95 0.94 – 0.96 0.96 0.95 0.98 0.96 0.91 1.05 0.97

0.95 0.94 0.93 0.94 0.94 0.98 0.97 0.98 0.97 0.97 1.08 1.08 1.04 1.04 1.05 1.04 0.98 0.97 0.99 0.99 0.99 1.01 1.01 1.01 0.97 1.02 0.99 0.99 0.99 1.00 0.99 0.96 0.93 0.93 0.93 0.94 0.95 0.94 0.98 0.93 0.94 – 1.00 0.96 0.96 0.95 0.95 0.95 – – – – – 0.93 1.08 0.98

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Table A2 Experimental values of dvf (mm) [22].

Lot no.

1

2

3

4

5

6

7

8

9

10

Test no.

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5

A325

A490

3/4

7/8

1

3/4

7/8

1

5.59 5.59 5.59 6.10 6.10 5.59 5.59 5.84 5.84 5.84 5.59 5.59 5.59 5.84 5.84 – 4.57 5.08 5.08 5.33 5.33 4.83 5.08 5.33 5.84 5.84 5.08 5.08 5.08 5.59 5.59 5.08 5.08 5.59 5.59 6.10 4.57 5.08 5.33 5.59 5.59 – 4.57 5.08 5.08 5.59 5.59

6.10 6.10 6.10 6.86 7.11 6.35 6.60 6.86 7.11 7.11 6.10 6.60 6.86 6.86 6.86 – 6.10 6.10 6.35 6.35 6.86 7.62 7.62 7.62 7.87 7.87 6.60 6.60 6.60 6.86 6.86 5.59 5.84 5.84 6.10 6.10 4.83 5.08 6.10 7.62 8.13 8.38 4.83 5.08 5.59 5.84 5.84 6.10 – – – – –

7.62 8.13 8.38 8.38 8.64 7.11 7.37 7.37 7.37 7.37 7.11 7.37 7.37 7.37 7.62 – 6.10 6.10 6.10 6.35 6.35 8.89 8.89 9.40 9.40 9.40 6.35 6.35 6.60 6.60 6.60 6.60 6.60 6.60 7.11 7.11 6.35 6.60 6.86 6.86 7.11 – 6.60 7.87 7.87 8.64 8.89

5.59 5.59 5.84 6.10 6.60 6.10 6.60 6.60 6.86 6.86 5.84 5.84 6.35 6.35 6.60 – 6.35 6.35 6.35 6.60 6.86 6.10 6.10 6.10 6.10 6.35 5.59 5.59 5.59 5.84 6.10 5.08 5.59 5.84 5.84 5.84 5.33 5.33 5.59 5.84 5.84 – 5.59 5.59 5.59 5.84 5.84

5.08 5.33 5.59 5.84 6.10 5.59 6.35 6.35 6.35 6.60 6.35 6.60 6.60 6.86 7.11 – 5.33 5.59 5.59 5.84 6.10 6.86 7.37 7.37 7.37 7.62 5.84 6.60 6.60 6.60 7.11 6.60 6.60 6.86 6.86 7.11 5.59 6.10 6.60 6.35 6.86 – 6.60 6.86 7.11 7.37 7.37

– – – – –

– – – – –

6.10 6.10 6.10 6.35 6.60

6.60 6.60 7.11 7.11 7.87 5.84 7.62 7.62 7.87 7.87 6.60 7.62 9.14 9.65 9.91 9.91 6.86 7.37 7.87 8.38 8.38 8.13 8.64 9.14 9.14 9.40 8.13 8.38 8.38 8.38 8.38 7.87 8.13 8.38 8.38 8.89 7.62 7.62 8.13 8.38 8.89 – 6.10 6.86 7.62 7.62 7.62 7.87 – – – – –

5.08 5.08 5.33 5.59 5.84

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