The failure mechanisms of fasteners under multi-axial loading

Engineering Failure Analysis 105 (2019) 708–726

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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal

The failure mechanisms of fasteners under multi-axial loading ⁎

Zhenzhong Caoa, M.R.W. Brakeb, , Dingguo Zhanga a b

T

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China William Marsh Rice University, 6100 Main st, Houston, TX, USA

ABS TRA CT

Fasteners in assemblies are often subjected to multi-axial loading; however, the role of load angle and the assembly design/geometry on the failure of fasteners are not well understood. Existing experimental data often convolutes multiple sources of uncertainty. Thus, numerical modeling is needed to make accurate conclusions about the failure mechanisms of fasteners. This paper investigates the mechanics that controls the loading angle effects on the mechanical response of fasteners by using a simple finite element bolt model without threads. The material model of this model is calibrated against tensile testing data, which minimizes the contribution of the fixture to the bolt's response. The calibrated material model is then used to investigate the loading fixture effects (including the fixture gap, compliance, and double shear loading configuration) on the mechanical response of the fasteners. After identifying the fixture parameters that the fastener's response is most sensitive to, a model of a simplified fixture is validated against experimental results. This investigation reveals that the loading angle effects and most of the loading fixture effects on fasteners' mechanical response are attributed to the change of stress concentrations at the contact tips. The primary contributions of this paper are novel insights into the failure mechanisms of fasteners, and that a reduced model of a fastener can be used to predict its constitutive behavior accurately.

1. Introduction The primary function of a fastener in an assembly is to join two (or more) components and to support a load [1]. Fasteners in these assemblies are often under tension, shear or combined tension and shear loading conditions [2]. A prior experimental investigation [3] shows that the loading orientation has significant influences on the mechanical response of fasteners. As the test loading angle transitions from shear to tension, fasteners are able to support a higher load. The mechanics that govern the strength differences between the different loading conditions, however, are not well understood. These mechanics are influenced by the multiple interfaces in the fastener assemblies, which can augment the stiffness of and introduce uncertainty into the fasteners' mechanical response. Thus, the components that conduct loads to the fasteners must also be considered for investigating the failure of fasteners under combined shear and tension loading. Knowledge and understanding of the failure mechanisms for mechanical fasteners has always been germane to the aerospace industries [4–7], automotive [8–10], and manufacturing industries [11,12], among many others such as biomechanical applications [13–16]. Increasingly, though, with the advent of additively manufactured parts for the repair of damaged bones (whether from cancer, osteoarthritis, or breakage), understanding the performance of fasteners in non-tensile environments is paramount for ensuring successful surgeries and recoveries. A number of papers have numerically investigated the fasteners, such as locking bolt [17], T-stub joints [18], and tightening and loosening mechanism [19], however, the multi-axial loadings occur in these jointed structures are not further investigated in these papers. The causes of failure of threaded fasteners have been numerically studied [20,21], and fastener loosening has been parametrically studied [22], however, the multi-axial loading conditions are not investigated in these papers. One barrier to understanding how fasteners perform in multi-axial loading has been inconsistencies between experiments and the lack of an efficient numerical model to simulate the response of structures with multiple fasteners.

⁎

Corresponding author. E-mail address: [email protected] (M.R.W. Brake).

https://doi.org/10.1016/j.engfailanal.2019.06.100 Received 7 February 2019; Received in revised form 21 June 2019; Accepted 30 June 2019 Available online 15 July 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Failure Analysis 105 (2019) 708–726

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There have been several experimental studies of fasteners subjected to different loading orientations [23–25]. These studies mainly concentrate on the strength of bolts under combined loadings, e.g., combined loading failure criteria [26], strength under combined loading at elevated loading rates [27], and post-fire effect on combined loading strength [28]. These studies, however, often unintentionally introduce uncertainty in the experimental data through the test fixture design. Further, there is a lack of investigation on how loading orientation affects fasteners' resistance strength, and also a limited understanding of how the fixtures, which conduct loads to the fasteners, affect the mechanical response of the components. A better understanding of the origin of the loading orientation and loading fixture effects will be helpful for the investigation of predicting the mechanical response of fasteners. It is crucial to choose a proper model for fasteners in numerical investigations [29]. Finite element analysis is a convenient method for investigating fastener failure [30–32]. However, computing numerous cases and large scale fastener assemblies with high fidelity and refined mesh finite element models are time-consuming. The problem is further exacerbated for complex structures where thousands of fasteners might be used (such as aerospace, engines, turbines, automobiles, etc.) [1]. The complex geometries and contacts between threads cause most of the computational burden in simulating fastener assemblies. Considering the limitation of computing capabilities, more efficient models are necessary for simulations [33]. To reduce the computational cost, modeling simplifications are often considered, e.g., using simple geometry models without threads [34] and employing coarser meshes. However, these simplifications often cause lower accuracy results. Thus, an efficient and accurate reduced order model is necessary for modeling fasteners. Creating an appropriate FE model to reproduce the real mechanical response of a fastener needs to consider various sources of uncertainties from the real model. A significant source of uncertainty in threaded fasteners is the initial pretension or preload [34–36]. Preload exists when a torque is applied to a fastener to fix it in place; many experiments on fastener properties, unfortunately, do not control for the preload and typically only have a small preload from the fastener being threaded into the test fixture. The existence of preload in assemblies has non-negligible effects on the measured mechanical response of a fastener, however. The loss of preload occurs before the yield of a fastener during the tests, and this loss of preload is hypothesized to be manifested as a sudden increase in the mechanical response curve of the fastener, which is absent in most of the experimental data analyzed in this paper. Since there is a lack of preload information pertaining to the available data that this paper uses, preload is not considered in the FE model in this study. The neglect of preload can lead to a possible source of error in the FE model. Another significant source of uncertainty in fasteners is measurement of the material properties. One challenge with measuring the properties of a fastener is that most, if not all, experiments create a joint/assembly when the fastener is placed in a test fixture [1]. The measurements, thus, are directly affected by the compliance of the fastener-fixture assembly. This highlights the limitations of many previous studies that attempted to analyze the fastener in isolation; without considering the effects of the fixture and the fastener-fixture assembly as a whole, it is unclear what is being modeled or measured. Thus, measurements and models of fasteners must take care to either account for the fixture's compliance in reporting the results or explicitly include the fixture and report the properties of the assembly. In what follows, the measured response of the fastener-fixture assembly is discussed and is referred to as the fastener for simplicity. For the purpose of both efficiency and accuracy, a feasible simplified model of a fastener needs to be created. Castelluccio and Brake [37] showed that a simple 1D bolt model without threads can be regularized to reproduce 3D models closely under tensile loading conditions. Further, [37] demonstrated a good agreement among 3D models and experiments for tensile testing by considering material property changes in manufacturing. A second result of [37] is that the high fidelity representation of the geometry (i.e., threads) is less important than accurately characterizing the material response. This finding led to a hypothesized approach in which the threads can be neglected, which is explored in the present work. That is, [37] suggests that a simple, 1D, smoothed model can reproduce most of the experiments. However, the study did not compare the 1D model with experiments directly, and it only investigated the tension loading case. To further demonstrate the feasibility of the 1D model for other loading orientations, validations against experiments are necessary for the 1D model. In this paper, a simple smoothed cylinder bolt model without threads (called the 3D model in this paper, and referred to as the “1D model” in [37]) is developed to study the influences of loading orientation and test fixture on the mechanical response of bolts. The material model of the fastener is calibrated against pure tension experiments (to correct for strain hardening behavior). This calibrated material model is used to investigate the influence of the fixture design in both shear testing and combined tension-shear testing experiments, where the fixture has a significant influence on the measured fastener's response. The model is then validated against experimental data by incorporating the key components of the fixture design into a simplified model of the fixture. A refined mesh 2D model without threads is also created to investigate the mechanism that governs the mechanical response changes under different loading orientations and test fixture gaps. Model validations against experiments are conducted to strengthen the results of these investigations. Compliance of the test fixture and the double shear loading condition are also studied with the validated model. 2. Sources of variability and error in modeling fasteners and fixtures The mechanical response of bolts in computational models is affected by many factors. A taxonomy was proposed in [37] that includes geometry, material, mechanics, and methodology in modeling the fasteners. Geometry variability was defined in terms of the geometry of the threads and the tradeoffs of using a 2D or 3D model. Material variability was based on the variability observed in the potential manufacturing methods for bolts. Lastly, the mechanics variability was defined in terms of the computational models. To systematically study the factors that can affect the mechanical response of fasteners, an expanded taxonomy is shown in Fig. 1, which considers both fastener and fixture variability. Here, three categories are considered for both the fastener and fixture: the geometric/fixture design, the materials, and the 709

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Fastener

Fixture

Geometry

Material

Mechanics

Fixture Design

e.g., shape, dimensions, tolerances

e.g., properties, anisotropy, inhomogeneity

e.g., friction, temperature, loading

e.g., geometry/gap, orientation, instrumentation points

Material e.g., properties, compliance

Mechanics e.g., loading, boundary, flexure

Mechanical response e.g., force, displacement, stress, strain, failure

Fig. 1. Most significant sources of sensitivity, error, and uncertainty for the mechanical response in modeling fasteners and fixtures.

mechanics. The present work is a sequel to [37], which investigated the contributions of the different sources of fastener variability to its constitutive response under normal loading. Thus the focus here is the influence of the fixture design and loading conditions (e.g. shear, mixed, etc.) on the constitutive response of the fastener. A better understanding of the fixture's role in the constitutive response of fasteners will enable improved fastener testing methods and will facilitate more accurate predictions of how a fastener will perform in an assembly with potentially different loading conditions. It is important to note that this paper does not consider the effects of the threads on the mechanical response of the fastener. In [37], it was found that the response of a fastener leading up to failure (i.e., the ultimate stress and strain) was more dependent upon the material model, modeling methodology, and loading mechanics than the precise representation of the geometry for predicting the bulk response of a fastener. Thus, without loss of generality, the fasteners in this paper are modeled without the threads in order to study the constitutive response of the fasteners approaching failure. While, this simplification does not capture the contribution and failure of the threads, [37] showed that a validated model can be developed without threads in order to predict the stress strain relationship for an entire fastener in an assembly. For the material model, variability in the elastic-plastic properties is considered, as the manufacturing procedures have significant influence on the yield strength, elastic modulus, and hardening range [38–40]. For the mechanics, this paper focuses on the effects of the loading orientation. In particular, as the loading orientation is changed from normal (0°) to shear (90°), the contributions of stress concentrations significantly change. Not considered in [37], the test fixture can have a significant (and often unintentional) effect on measurements of a fastener's constitutive properties. This effect can be observed as the high variability in measurements of the same fasteners in different experiments [4,5]. The geometric design of the fixture can introduce gaps that increase the fixture's gage length (and thus changing its effective stiffness), flanges that increase the system's compliance, and asymmetries that introduce moments to the fasteners. The boundary conditions of the text fixture have significant influence on the mechanical response of the fasteners. For example, fixtures with no guides can be unstable in the tests, so using fixture guides in the experiments maintains stable extension of the fastener in a single direction. Further, the fixture material choice is a non-negligible factor that can affect the mechanical response and system compliance too. Lastly, the response of a fastener in an assembly can deviate significantly from experiments due to the presence of bushings, shear plates, pinning, or other considerations not tested. In what follows, this paper is divided into two parts. First, in Section 3, an in depth analysis of the modeling considerations for a fastener under mixed loading conditions is performed, and the recommended model is then validated. Next, to gain insights into how these sources of fixture variability influence a fastener's response, section 4 investigates the effects of loading angle, fixture gaps, fixture compliance, and the double shear loading condition. 3. Modeling Due to the available experimental data for validation [3], #10-32 bolts are primarily considered in this investigation. A coarse schematic of the experiment's test fixture is shown in Fig. 2a. This test fixture geometry is simplified in the model as shown in Fig. 2b, which has two parts with a variable gap between them, and is based on the experiments of [3]. As mentioned in Section 2, a 3D finite element model without threads is used for modeling bolts (Fig. 3a). Both the bolt and the nut are considered to be one part to represent the threaded connection between them. Hexahedral solid elements are employed in meshing the models, and simulations are carried out using LS-DYNA. The boundary conditions of the FE models are defined on the two test fixtures. For pure tension (0°) loading conditions, velocity controlled loading (10 in/s, which is the loading rate applied in the tests) along the bolt axis is applied on the nodes of one fixture, and another fixture is constrained from displacing in any direction. In the experiments, force was measured using a load cell at the cross head of the test frame and a linear differential variable transducer was placed at the moving half of the fixture to measure the displacement. Fig. 3b shows where the load and displacement are measured in the FE model. The contact force between the moving half of the fixture and the bolt is obtained as the bolt load, since this force can accurately represent the load measured at the test 710

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Bolt

Nominal Diameter

Bolt

Nut

Fixture

Gap

(a) Geometry of the fixture in [3].

(b) Simplified geometric model.

Fig. 2. Model of the bolt and fixture considered in simulations.

Displacement Contact force

(a) Finite element model . (b) Measurement of load and displacement. Fig. 3. Finite element model and measurement of load and displacement.

frame in the experiments. The displacement at the center of the moving half of the fixture is considered to represent the measured displacement in the test, as this displacement can also represent the displacement of the bolt. A286 stainless steel and 302HQ stainless steel bolts, which were tested in [3] are considered in the simulations, and A286 stainless steel is also employed in modeling the fixtures. Simulations employ a simplified Johnson-Cook material model that considers strain rate effects [41]. Fracture and failure mechanisms are not considered in the simulations of this paper. Automatic surface to surface contacts are defined between bolts and fixtures, and a friction coefficient of 0.15 is used in the contacts. Nominal material properties adapted from [42] are listed in Table 1, as well as the geometric properties of the baseline fixture configuration listed in Table 2, based on [3]. 3.1. Mesh size sensitivity To investigate the influence of the mesh size on mechanical response, three different mesh size models are considered based on the analysis of [37]. The mesh sizes along the axis of the models are: 0.4 cm, 0.1 cm and 0.05 cm (Fig. 4). Mesh sensitivity studies are carried out under tension and shear loading conditions respectively. For tension loading condition, the force-displacement predictions of the simulation results are compared for the three models in Fig. 5a. The results indicate that the 0.4 cm mesh size model gives a lower force than the other two models after the bolt yields, and that the force levels of the 0.1 cm and 0.05 cm mesh size models are converged. For shear loading condition, Fig. 5b indicates that the 0.4 cm mesh size model cannot give a stable curve, and that the 0.1 cm mesh size model gives a little higher force than the fully converged results of the 0.05 cm and smaller mesh size models. However, this discrepancy is acceptable when considering that the 0.05 cm mesh size model needs about 20 times more than the 0.1 cm mesh size model in computing time. Thus, the 0.1 cm mesh size model is chosen in consideration of both accuracy and efficiency for the subsequent simulations. Table 1 Material property for A286 and 302HQ.

Elastic modulus Poisson ratio Yield stress Hardening modulus

A286

302HQ

200 GPa 0.28 827 MPa 1100 MPa

200 GPa 0.28 200 MPa 520 MPa

711

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Table 2 Geometric properties of the fixture from [3]. Bolt length

16 mm (5/8 in)

Bolt diameter Fixture flange thickness Nominal fixture gap Fixture width

4.8 mm (0.19 in) 5.1 mm~7.6 mm (0.2 in~0.3 in) 5.1 mm (0.2 in) 12.7 mm (0.5 in)

(1) 0.4 cm.

(2) 0.1 cm.

(3) 0.05 cm.

Fig. 4. Different bolt axial mesh size.

10

Force (kN)

Force (kN)

10

5

5

0.4 cm 0.1 cm 0.05 cm 0.02 cm

0.4cm 0.1cm 0.05cm 0

0 0

0.1

0.2

0

0.06

Displacement (cm)

Displacement (cm)

(a) Tension loading condition.

(b) Shear loading condition.

Fig. 5. Force-displacement for different mesh sizes.

Table 3 Material properties calibrated to the pure tension experiments.

Elastic modulus Poisson ratio Yield stress Hardening modulus

A286

302HQ

200GPa 0.28 864 MPa 232 MPa

140GPa 0.28 490 MPa 200 MPa

712

0.12

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18

Force (kN)

Force (kN)

24

12

9

0

0 0

0.1

0.2

0

0.1

Displacement (cm)

Displacement (cm)

(a) As-produced material properties.

(b) Strain hardened material properties.

0.2

Fig. 6. Force-displacement for 0° (tension) loading condition and experiment. Shown are experimental (dashed) and simulated (solid) responses for 302 HQ (red, square) and A286 (blue, circle). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.2. Material property corrected for matching experiments The results of [37] indicate that discrepancies between the models and experiments can be attributed to work hardening and microstructural changes that occur during the manufacturing process. To account for this discrepancy, the material properties are calibrated with the pure tension loading data, with the final values used for all subsequent simulations listed in Table 3. Fig. 6 shows the comparison of the force-displacement relationship between the simplified models (Fig. 4) and the experimental data [3] for both A286 steel and 302HQ steel under pure tension loading for both the reported and calibrated material properties. The tensile experiments are used for calibration of the material properties as this configuration minimizes the contribution of the fixture effects to the measured response of the fastener (as opposed to the shear testing condition for instance). Fig. 6a is the comparisons before the calibration of material properties, and some differences are found between the simulations and experiments. For A286 steel, the force-displacement curve predicted with the reported material properties matches the experiments in the elastic regime, but not the plastic regime. For 302HQ, qualitative differences between the experiments and the simulations using the reported material properties are observed in both the elastic and plastic regimes. These differences should be attributed to the lack of consideration of work hardening during the manufacturing process. For the purpose of the matching to the experiments, calibration for the material properties is necessary. Decreases in hardening modulus for both materials and an increase in the yield stress for 302HQ steel (only a slight adjustment for A286 steel in the yield stress) would improve the matching of the curves between the simulations and the experiments. These modifications to the material properties of A286 are well within the range of properties commonly reported for this material [40], and highlight the need for accurately characterizing materials as there is significant variability observed in the same material from two different manufacturers. A decrease in elastic modulus for 302HQ steel, by contrast, would be necessary to compensate for the prior deformation of the threads in the real bolt model. A new set of simulations with the calibrated material properties (Table 3) are presented in Fig. 6b. The simulation results are in good agreement with the experimental results up until the peak stress, the response after which is dominated by fracture nucleated near the threads in the experiments, which is not considered here. This suggests that the cylinder bolt model without threads can reproduce the experimental mechanical response under pure tension loading by considering the prior work hardening in the material properties. The efficacy of this approach is assessed for mixed loading conditions in Section 4.

3.3. Consideration for mixed loading and pure shear loading The test fixture in [3] shows that a spacer was placed between the head mount and the thread mount to maintain the gage length, and a fixture guide was placed to maintain the fixture loading orientation for mixed loading and pure shear loading. These boundary conditions have non-negligible influences on the mechanical response of the bolts, so the spacer and fixture guide should be taken into account for simulations to see what influences they have on the mechanical response and also to have proper boundary conditions for the models. Note that this is often neglected in other studies of fasteners [26,37]. As the threads are not considered in this paper, the gage length will be represented by the gap between the two fixtures. The fixture guide will be simplified as a constraint on the moving half of the fixture. These boundary conditions in the models without threads do not affect the mechanical response of pure tension loading condition since there is no contact between the fixtures and the bolt shank under tension loading. These loading conditions are implemented via boundary conditions, as shown in Fig. 7. Two sets of boundary conditions are 713

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Velocity

Velocity

Velocity

Y

BC2

BC2 BC1

X

BC1

BC1

Z

Fig. 7. Three different fixture conditions (All DOFs to be constrained for BC1, X and Z displacements to be constrained for BC2, all loading directions are along Y axis).

defined for these models: all DOFs of the fixture are constrained on the specified nodes for boundary condition 1 (BC1), and the displacements (excluding the loading direction) of the fixture are constrained on the nodes for boundary condition 2 (BC2). Velocity controlled loadings are placed on the moving fixture for all models. The boundary conditions for each of the models are defined as: Model 1: BC1 is placed on the fixed half of the fixture, no constraints on the moving half of the fixture and no gap between the two fixtures. Model 2: BC1 is placed on the fixed half of the fixture, BC2 is placed on the moving half of the fixture and no gap between the two fixtures. Model 3: BC1 is placed on the fixed half of the fixture, BC2 is placed on the moving half of the fixture and 0.4 cm gap between the two fixtures is considered. 302HQ steel bolts with a loading angle of 60° are used to perform the simulations for these models. The force-displacement curves of these models are compared against the experimental data, to determine the most appropriate 3D model. Fig. 8 shows that Model 3 has the highest qualitative agreement with the experimental data of the three models. These results also indicate that both the gage length and fixture guide have significant influence on the mechanical response, which are further studied in Section 4. Discrepancies between model 3 and the experiment at the onset of yield (around 4.5 kN in this configuration) can be attributable to the neglect of preload in the FE model, since the loss of preload during test can leads to a kink in the mechanical response curve, as well as material defects and epistemic uncertainty.

3.4. Model validation To validate the fidelity of Model 3 under various loading angles, simulations for 0°, 45°, 60°, 75° and 90° loading angles are conducted for A286 steel and 302HQ steel in order to compare them with the experiments from [3]. Material properties are taken from the corrected material properties from Section 3.2. Comparisons are presented in Fig. 9, and the legend of the figure is shown in Table 4. Simulations are in good agreement with the

Force (kN)

9.0

4.5

0 0

0.1 Displacement (cm)

0.2

Fig. 8. Force-displacement curves for the three boundary condition models (model 1: black, square; model 2: red, circle; model 3: green, up triangle) and experiment (orange, solid line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 714

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11.0

Force (kN)

Force (kN)

18

9

0

5.5

0 0

0.05 Displacement (cm)

0.10

0

(a) A286.

0.1 Displacement (cm)

0.2

(b) 302HQ.

Fig. 9. Constitutive response of the fasteners as the orientation angle changes from pure tension to pure shear. The legend is described in Table 4. Table 4 The legend for Fig. 9. Legend

Name Experiment under 0° loading angle Experiment under 45° loading angle Experiment under 60° loading angle Experiment under 75° loading angle Experiment under 90° loading angle Simulation of model 3 under 0° loading angle Simulation of model 3 under 45° loading angle Simulation of model 3 under 60° loading angle Simulation of model 3 under 75° loading angle Simulation of model 3 under 90° loading angle

experiments among most loading angles in the overall mechanical behavior. The simulations capture the experiments at the elastic range, onset of plastic deformation and the plastic range. There are, however, several differences exist between the simulations and experiments. The simulation curves of 45°and 60° consistently do not fit well compared to the other loading angles for A286 steel. This is hypothesized to be due to the slip between the fixture and the bolt without threads. The maximum load carrying capacity of both the 45°and 60° loading angles for A286 steel is over estimated, which may be due to an improper friction coefficient between the fixture and the bolt. The experiments show more compliance than simulations when bolts begin to yield. This is attributed to the threads being partly involved in the resisting of yield in the experiments while only the bolt shank resists the yield in the simulations. The comparison of simulations and experiments suggests that the simple smoothed bolt model without threads can reproduce the mechanical response of the experiments under various loading angles. This has significant ramifications for models of large assemblies with many fasteners as simple bolt models without threads can replace complex bolt models with threads in these conditions to reduce the computational time significantly. 3.5. Validation of the whole fixture model In order to reduce computational time, the fixtures are simplified to capture the main features that affect the fastener's mechanical response. To further verify the accuracy of the simplified model, a numerical study of the whole fixture model (model with key components of the fixture design) is carried out to compare with the simplified model, and then the whole fixture model is validated against experimental data. Simulations of the whole fixture model for a 302HQ steel bolt in Fig. 10 are conducted under tension and shear loading, with force-displacement results compared with the simplified fixture model in Fig. 11. From the comparison, the model with the key features of the fixture design's results are almost the same as the simplified fixture model's results. To validate the whole fixture model against experimental data and to further verify the convergence of the 0.05 cm mesh size under various loading angles, simulations of the whole fixture model are carried out for 0.1 cm size model and 0.05 cm size model. Fig. 12 shows the force-displacement comparison of the whole fixture model and the experiment for 302HQ steel bolt under various 715

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Fig. 10. Model with the key features of the fixture design.

6

Force (kN)

Force (kN)

10

5

0 0.0

3

0 0.1 Displacement (cm)

0.2

0

0.1 Displacement (cm)

0.2

Fig. 11. Force-displacement for 302HQ under 0° (tension) and 90° loading conditions. Shown are whole fixture model (red, circle) and simplified fixture model (blue, square). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

11.0

Force (kN)

Force (kN)

11.0

5.5

5.5

0

0 0

0.1 Displacement (cm)

0.2

0

0.1 Displacement (cm)

0.2

(2) 0.05 cm mesh size model

(1) 0.1 cm mesh size model

Fig. 12. Force-displacement comparison of the model with the key features of the fixture design and the experiment for 302HQ under various loading angles. The legend is described in Table 4.

loading angles. The simulation results of 0.1 cm mesh size model and 0.05 cm mesh size model are both in good agreement with the experiment, and the coincidence of the 0.1 cm model is almost the same as the simplified fixture model in Section 3.4. The 0.05 cm mesh size model reproduces the experimental data slightly more accurately than the 0.1 cm mesh size model under two of the loading angles, but not enough to warrant its use instead of the 0.1 cm mesh size model in what follows.

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9.0

Force (kN)

Force (kN)

10

5

0

4.5

0 0

0.1 Displacement (cm)

0.2

0

(a) 45°.

0.2

(b) 60°. 9.0

Force (kN)

7.0

Force (kN)

0.1 Displacement (cm)

3.5

0

4.5

0 0

0.05 Displacement (cm)

0.10

(c) 75°.

0

0.075 Displacement (cm)

0.15

(d) 90°.

Fig. 13. Force-displacement for different gaps (0 cm: black, square; 0.2 cm: red, circle; 0.4 cm green triangle) comparison with experiment (blue, solid line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

This comparisons indicate that the 0.1 cm mesh size whole fixture model is accurate enough for convergence in reproducing the mechanical response of the bolts in the experiment. The model with the key features of the fixture design and the simplified fixture model are both able to reproduce the force-displacement behavior of the bolts in the experiment. In consideration of reducing computing time, the simplified fixture model is used in the following investigation. 4. Influence of the fixture design Following the validation in the previous section, the model is now used to investigate the influence of the fixture design. Specially, in this section, the gage length, fixture compliance, and single versus double shear conditions are investigated. These investigations can be used to understand how the fixture itself affects the perceived constitutive property of a fastener, and what design features are more important for accurately characterizing the fastener's response itself for future modeling purposes. 4.1. Influence of the gap From the comparison of Model 2 and Model 3, it is evident that changing the gage length leads to different mechanical responses. In order to determine how the test fixture gaps affect the mechanical response, simulations of different fixture gaps have been conducted for various loading angles by using the validated 3D model. 302HQ steel material properties are used in the simulations, BC1 is placed on the fixed half of the fixture and BC2 is placed on the moving half of the fixture. Fig. 13 presents the force-displacement comparison of the 3D model simulation results and the experiment results. It is obvious 717

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10 0° Maximum tension force (kN)

45°

60° e Lin

5

of

Eq

l ua

Te

n

n sio

an

h dS

ea

r

75°

90°

0 0

5

10

Maximum shear force (kN) Fig. 14. The maximum shear force and maximum tension force in the 302HQ steel fastener for each orientation.

that the fixture gap has more influence for larger loading angles in the force-displacement curves than for smaller angles. The forcedisplacement curves for three different gaps are quite close for the 45° loading angle, while the discrepancies among them are larger as the loading angle becomes larger. In particular the 75° and 90° tests show that the gap influences both the elastic regime and the peak force in the plastic regime. For reference, the experimental results shown in Fig. 13 reported a gap of 0.2 in (0.51 cm). Fig. 13 also shows that the smaller gap model presents a higher apparent elastic modulus for all loading angles. This should be attributed to the stiffness that relates to the fixture gap. From [43], the stiffness of the fastener can be expressed as:

kb =

Ad At E Ad lt + At ld

(1)

where the cross-sectional area of the unthreaded portion Ad, the cross-sectional area of the threaded portion At, the elastic modulus E, and the length of the threaded portion in grip lt are constants, and ld is the length of the test fixture gap. When ld decreases, the bolt stiffness kb increases, which is why the smaller gap model presents a higher apparent elastic modulus. The force-displacement curves in Fig. 13 show that there is a transition in the yield behavior as the loading angle increases: for the loading angles of 45° and 60°, the yield force level increases as the fixture gap increases. In contrast, the yield force level decreases as the fixture gap increases further to the 75° and 90° loading angles. This transition between 60° and 75° can be partially explained by Fig. 14: for loading angles up to 60°, the model's response is dominated by a tensile component. As the loading angle is increased to 75° and higher, the response becomes dominated by a shear component. This is somewhat unexpected as this transition would be expected to occur at a loading angle of 45°. While these results come from a simplified model, they agree with the experiments (as shown in Fig. 13). To understand these mechanisms better, these two regimes (termed the tension dominated and shear dominated regimes) are further investigated in what follows. 4.1.1. Stresses in the tension dominated regime When bolts are under smaller loading angles, the contribution of tensile stresses is more than shear stresses, thus the yield mechanism of the bolt is in the tension dominated regime. For the yield mechanism of the tension dominated regime, the bolt stiffness constant can be expressed in terms of kb and the spring rate of the C assembly members km, which is a constant for a given fixture design [43].

C=

kb . kb + k m

(2)

This gives the tensile stress in the bolt from [43] as

σb =

Fb CP + Fi kb P F = = + i (kb + km ) At At At At

(3)

with constant resultant bolt load Fb, external tensile load per bolt P, and preload Fi. Smaller fixture gaps increase kb, so that C also increases. This leads to the increase of the bolt stress σb, which indicates that the bolt will yield earlier. Thus, the smaller fixture gap models have a higher apparent elastic modulus and a lower yield force level, Even though these results come from a model based on pure tension loading, they still provide insights into the present problem for small loading angles. Further, the trends observed for the intermediate loading angles (45° and 60°) are similar to this analysis, suggesting a possible explanation for the increased stiffness and lower yield force level. 718

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Fig. 15. 2D model with 0.005 cm element size.

4.1.2. Stresses in the shear dominated and mixed loading regimes For loading angles greater than 60°, the yield mechanism is in the shear dominated regime. For the purpose of investigating the mechanism that causes the strength differences as both the loading angles and the fixture gap are varied, the stress distribution inside the bolts is investigated. However, the meshes of the 3D models in this paper are too coarse to look into the details of the stress distribution. The computing time of the refined mesh 3D models (i.e., the 0.05 cm element size model in Section 3.1) is much more than the 3D models (0.1 cm element size model) used in this paper. Furthermore, the details are not enough even using the 0.05 cm mesh 3D model and the computational cost of the more refined mesh 3D model is not acceptable. Hence, a much more refined mesh 2D model has been employed in this section to conduct the simulations for the details of stress distribution. In order to investigate the stress state inside the bolts to determine how the loading angles and fixture gaps affect the mechanical responses, two dimensional model with highly refined meshes (0.005 cm element size) for all loading angles and fixture gaps are created (Fig. 15). In this model, velocity controlled loading is placed on the moving fixture, BC1 is placed on the fixed half of the fixture, BC2 is placed on the moving half of the fixture and 0.4 cm gap between the two fixtures is considered for model validation. The 302HQ steel bolt with a loading angle of 60° is used for the 2D model validation. Fig. 16 shows the force-displacement curves comparison of the 2D model against the experimental data. It is observed from the comparison that the force-displacement curve of 2D model is more compliant than the experimental data among the elastic range and yields at a lower force level. However, the 2D model matches well with the experiment up to the peak force. In a real fastener, the internal stress state is a combination of the stress concentrations due to each thread, the bulk stress state from the far field loading, and the stress concentrations due to the sharp edge interactions between the fixture and the bolt. In the subsequent analysis, it is assumed that the dominant terms are due to the stress concentrations between the fixture and bolt, and not the additional effects from the stress concentrations associated with the bolt threads. This rationale is partially justified by the agreement exhibited in Fig. 16 (i.e. the main trends are represented, though there is some acceptable discrepancy between the experiment and 2D model). Further, the analysis of [37] indicated that the peak stresses in tensile loading were located in the first thread of a fastener, which has an analogue in the 2D model as the sharp edge contact between the fixture and the fastener. Therefore, a stress analysis investigation employing this 2D model is used to gain insights into how the loading angle and gap length combine to affect the stresses within a smoothed model of fastener. This analysis is expected to capture the bulk response due to the far field loading (i.e. how the shear and tensile loading change with orientation angle) and the response of the primary stress concentration

Force (kN)

9.0

4.5

0 0

0.1 Displacement (cm)

0.2

Fig. 16. Force-displacement curves of 2D model (blue, down triangle) and experiment (green, solid line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 719

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Fig. 17. 2D model von Mises stress distribution for different conditions.

from the fastener-fixture interaction modeled, but not the other stress concentrations due to the engagement of every thread. For the 2D model stress analysis, simulations have been conducted for 0 cm, 0.2 cm and 0.4 cm gaps under 0°, 15°, 30°, 45°, 60°, 75° and 90° loading angles, and 302HQ material properties. The von Mises stress distributions as gage length and loading angle are varied are shown in Fig. 17. The maximum stress distribution along the contact tip diagonal line (defined in Fig. 18a) is normalized with respect to the line length in Fig. 18(b-d). The increase of the fixture gap effect on the mechanical response of the bolt as the loading angle increases can be explained by Fig. 17 and Fig. 18. When the loading angle is small, the contact effects between the fixture and bolt shank are negligible, and the stress distributions do not have any significant differences between the different fixture gaps. As the loading angle increases, the contacts between the fixture and the bolt shank are more severe due to a stress concentration that increases with the fixture gap, similar as to what is seen in sharp edge contacts [44–47]. This phenomenon is observed in Fig. 18(b-d). Since the stress of the bolts determines the onset of yield, the increased stress at larger loading angle leads to earlier failures/lower max loads. The stress distributions can also explain the shear dominated regime that controls the yield mechanism for larger loading angle conditions. Fig. 18 shows that the stress profiles among the three different fixture gaps at 90° loading angle are much larger than the stress profiles under smaller loading angles. Further, larger fixture gap models have much higher maximum stresses at the two contact tips for the 90° loading angle, even though the smaller gap models have higher stress distributions overall. The higher maximum stress at the contact tips results in the earlier yield, so that larger fixture gaps have lower yield force level under larger loading angles. The yield mechanism for mixed loading angles is the combination of tensile stresses and shear stresses. For the smaller loading angles, the contribution of tensile stresses dominates more than the shear stress, the smaller fixture gaps have a lower yield force level; by contrast, for the larger loading angles, the larger fixture gaps have a lower yield force level because the contribution of shear stresses dominates over tensile stresses. The origin of larger loading angles having lower yield force level in the experiments can also be explained by the stress distribution of the bolts. When the loading angle increases, the stress concentrations at the two contact tips are more severe. Higher stress concentrations result in earlier yield, so larger loading angles lead to a lower yield force level. 4.2. Compliance of the test fixture Another variable that can affect the mechanical response of the bolts under various loading angles is the compliance of the fixture. To find out how the fixture stiffness affects the mechanical response, a new set of simulations are employed using the 3D model. Simulation results for bolts made of 302HQ steel with a 0 cm gap and a 0.4 cm gap are shown in Figs. 19 and 20 respectively as the fixture's elastic modulus is varied from 140 GPa (the same as the bolts) to 1200 GPa. The force-displacement curves are compared under the same loading angle for 0°, 45°, 75° and 90°. Figs. 19 and 20 show that the fixture compliance has negligible influence on the 0° loading angle (as there are no contacts between the fixture and the bolt shank), and that the influence increases as the loading angle increases. The stress concentrations at the contacts between the bolt shank and test fixtures become significant as the loading angle increases, leading to the compliance of the fixtures having more of an effect. The higher fixture elastic modulus (Efixture) leads to stiffer curves at the elastic regime for all loading angles (though this is not 720

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1250

σ max (Mpa)

1000

L x

750

500 0

0.5

1.0

X/L

(b) 0°.

1250

1250

1000

1000 σ max (Mpa)

σ max (Mpa)

(a) Contact tip diagonal line.

750

500

750

500 0

0.5

1.0

X/L

0

0.5

1.0

X/L

(c) 45°.

(d) 90°.

Fig. 18. Von-Mises Stress along the diagonal for different gaps (0 cm: black, square; 0.2 cm: red, circle; 0.4 cm blue triangle). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

appreciable for small loading angles). There is a transition in the yield behavior as the loading angle increases: for smaller loading angles, a higher Efixture has a lower yield force at the onset of yield; for larger loading angles, a higher Efixture has a higher yield force at the onset of yield. This transition can be explained by the tension and shear dominated regimes respectively. For smaller loading angles, the stress state inside the bolt is dominated by the tension regime. Larger fixture stiffnesses have larger assembly stiffnesses from [43,48]:

1 1 1 = + km kb kf

(4)

where kf is the fixture stiffness. Thus, this leads to a steeper slope for the elastic regime. The fixture, however, is not appreciably deforming compared to the fastener in these simulations, which means that the yield displacement is constant as the fastener is being stretched the same amount for all fixture compliances. Post yield, because the simulations are displacement controlled, there are lower stresses for the same displacement, according to Eq. (3). This behavior is evident in Fig. 19 and Fig. 20: larger fixture compliances result in stiffer elastic regimes for the tension dominated loading cases, all simulations yield at the same displacement level, and in the plastic regime the higher fixture stiffnesses lead to lower stresses and thus lower force levels. For larger loading angles, the stress state inside the bolt is dominated by the shear regime. From the stress distribution of 90° loading angle for different Efixture in Fig. 21, the higher Efixture results in lower stress in the contact areas of the bolt. Thus, it has a higher yield force at the onset of yield. The force-displacement curves converge when Efixture increases to a high value, so increasing Efixture has no influence for the mechanical response when Efixture is approximately twice Ebolt. From all of the force-displacement curve comparisons in Fig. 19 and Fig. 20, it is observed that as Efixture changes, there is no appreciable difference in the response for the fully plastic regime of the 721

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10

0° E=140 E=170 E=200 E=230 E=260 E=300 E=1200

5

45°

Force (kN)

Force (kN)

10

E=140 E=170 E=200 E=230 E=260 E=300 E=1200

5

0

0

0

0.06 Displacement (cm)

0.12

0

0.06 Displacement (cm)

0.12

8

75° E=140 E=170 E=200 E=230 E=260 E=300 E=1200

4

Force (kN)

Force (kN)

8

90° E=140 E=170 E=200 E=230 E=260 E=300 E=1200

4

0

0 0

0.06 Displacement (cm)

0

0.12

0.06 Displacement (cm)

0.12

Fig. 19. Simulation results for 0 cm gap as the fixture modulus is varied, as indicated by the legend.

fastener. This suggests that the compliance of the test fixture has an influence on the elastic range (as the slope of the elastic regime is observed to change with Efixture) and the onset of the yield range (in terms of displacement), but does not affect the ultimate failure of the fastener.

4.3. Double shear loading condition The last analysis of this paper focuses on the double shear loading condition, which is common in fastener assemblies. To investigate how the response of the bolt is changed by the double shear condition, a new model for the double shear loading condition is created based on the 3D mesh model: an extra fixture is installed between the two parts of the test fixture for the double shear loading condition. This is different from the single shear (90°) loading condition, which has been investigated in previous sections. The double shear loading condition model is shown in Fig. 22a: a velocity controlled loading is placed on the middle fixture, and BC1 is set on both of the outer fixtures. The 302HQ material is chosen for the bolts and A286 material is chosen for the fixtures, using the material properties listed in Table 3. To investigate the double shear loading condition, simulations have been done for the middle fixture thickness of 0.1 cm, 0.2 cm, 0.3 cm, 0.4 cm and 0.5 cm. The force-displacement curves of the double shear models and single shear models (0 cm gap and 0.4 cm gap) are compared in Fig. 22b. The yield force level and the stiffness of elastic range are higher in the double shear loading models than the single shear models. These two loading conditions are inherently different test configurations, so they are not directly comparable. However, these simulations show that the resistibility of double shear loading is generally higher than single shear loading. Another phenomenon can be observed from the double shear loading is that the yield force level and stiffness of the elastic range increase as the middle fixture thickness increases up to 0.3 cm (in contrast, the yield force level and stiffness of the elastic range decrease as the gap length increases in single shear loading conditions). For thicknesses above 0.3 cm, the force-displacement curves appear to converge. To explain this phenomenon, the von Mises stress distributions of the bolts' cross section for double shear loading conditions are shown in Fig. 23. For larger thicknesses of the middle fixture (0.3 cm, 0.4 cm and 0.5 cm), the stress concentrations are 722

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10

10

Force (kN)

Force (kN)

0° E=140 E=170 E=200 E=230 E=260 E=300 E=1200

5

0

E=140 E=170 E=200 E=230 E=260 E=300 E=1200

5

0 0

0.06 Displacement (cm)

0.12

0

7.0

0.06 Displacement (cm)

0.12

7.0

75°

Force (kN)

Force (kN)

45°

E=140 E=170 E=200 E=230 E=260 E=300 E=1200

3.5

0

90° E=140 E=170 E=200 E=230 E=260 E=300 E=1200

3.5

0 0

0.06 Displacement (cm)

0.12

0

0.06 Displacement (cm)

0.12

Fig. 20. Simulation results for 0.4 cm gap as the fixture modulus is varied, as indicated by the legend.

separated into two sections. These stress states have no essential differences, so the force-displacement curves of these thicknesses are very similar. When the thickness decreases to 0.2 cm and 0.1 cm, the stress concentrations combine to result in a much higher stress state, so the higher stress leads to earlier yield. 5. Conclusions In this work, the influences of loading angle, fixture boundary condition, fixture gap, fixture compliance, and double shear loading conditions on the force-displacement response of bolts have been investigated by using a simple smoothed model without threads. The finite element model in this study is validated by experiments from the literature. The results reveal the mechanics that govern the influence of the loading angle and the test fixture. From this study, it is concluded that stress concentrations at the contact tips strongly influence the force-displacement curves when loading angle is close to 90°. This effect explains most of the phenomena that happen in the conditions of different loading angles, different fixture gaps, different fixture compliances and double shear loading. Specific conclusions are: 1. A simple smoothed bolt model without threads can reproduce most of the force-displacement response of the experiments under all loading angles. This simple model significantly reduces computational cost compared to models with threads, and is accurate enough for most design analyses. 2. Larger loading angles are observed to result in lower yield force levels. The simplified analysis and parameter studies indicates that as the loading angle increases to pure shear, the magnitude of the stress concentration increases, resulting in earlier failure. 3. The fixture gap's influence on the force-displacement response can be explained by the tension and shear dominated regimes. The stiffness that determines the yield force level is affected by the fixture gap in the tension dominated regime. In the shear dominated regime, the stress concentration at the sharp corner between the bolt shank and bolt head significantly affects the yield force level and becomes more severe for larger fixture gaps. 4. The compliance of the test fixture has no appreciable influence on the yield force level of the force-displacement response; it only affects the compliance at the onset of yield range. 723

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Fig. 21. Stress distribution for different elastic modulus (all stresses are plotted on the same scale).

Velocity

15

Force (kN)

10 0.1cm 0.2cm 0.3cm 0.4cm 0.5cm 0cm gap 0.4cm gap

5

BC1

BC1

0 0

(a) Double shear loading condition model.

0.01

0.02 0.03 Displacement (cm)

0.04

0.05

(b) Force-displacement curves comparison.

Fig. 22. Double shear loading model and force-displacement comparison of double shear loadings against single shear loadings (solid and dashed lines without markers, as points of reference).

5. For the double shear condition, the yield force level decreases as the middle fixture's thickness decreases due to the overlap of peak stresses in the superposition. The findings of this paper suggest that a simplified fastener model without threads is feasible in simulating large scale fastener assemblies and complex structures with fasteners to reduce the computing time while maintaining acceptable accuracy. The study of 724

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Fig. 23. Von Mises stress distributions of the bolts cross section for double shear loading conditions. Shown are the results for 0.1 cm (upper left), 0.2 cm (upper middle), 0.3 cm (upper right), 0.4 cm (bottom left), and 0.5 cm (bottom right).

the fixture effects suggests that the fixture gap significantly affects the mechanical response of the fastener in the test. This highlights the importance of carefully designing the fixtures used for testing the mechanical strength of fasteners. The yield behavior of the fastener and peak force (as a good indication for the onset of failure) is investigated in this study, however, the fracture failure of the fastener is not considered in this paper. The fastener is subjected to fracture failure when the load exceeds the fastener's maximum load carrying capacity. However, this part of work is more complicated than the prediction of the yield behavior, and warrants further investigation in future work. Declaration of Competing Interest None. Acknowledgments The authors are grateful for the support from the National Science Foundation of China (Grant Numbers 11772158, 11502113), and the Fundamental Research Funds for Central Universities (Grant Number 30917011103). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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