Pull-through capacity of bolted thin steel plate

Abstract

The loading capacity in the axial direction of a bolted thin steel plate was investigated. A refined numerical model of bolt was first constructed and then validated using existing experiment results. Parametrical analysis was performed to reveal the influences of geometric parameters, including the effective depth of the cap nut, the yield strength of the steel plate, the preload of the bolt, and shear force, on the ultimate loading capacity. Then, an analytical method was proposed to predict the ultimate load of the bolted thin steel plate. Results derived using the numerical and analytical methods were compared and the results indicated that the analytical method can accurately predict the pull-through capacity of bolted thin steel plates. The work reported in this paper can provide a simplified calculation method for the loading capacity in the axial direction of a bolt.

Keywords

bolted thin steel plate / refined numerical model / loading capacity / nonlinear spring element / analytical method

Introduction

Methods for joining parts are always the key issues before the construction of steel building structures, and the bolt is one of the most representative fasteners that is widely used in the connections of steel structures. The reliability of bolted connections directly determines the safety of entire structures. Hence, investigations on the mechanical behavior of bolted connections have always been a popular topic among researchers.

The numerical method has been widely utilized to determine the ultimate loading capacity or potential failure modes of all types of materials [¹–⁶] and connections. The loading process of high-strength bolt is very complex due to the change of contact state. So, more iterations are needed to detect the change in contact state between steel plates and bolt shank. The refined model of bolts has always been established in the full numerical joints to accurately consider the influence of bolts [⁷–¹³]. To simplify the bond-slip behavior between the bolt and rock, a trilinear model was utilized by Ma et al. [¹⁴]. Hwang [¹⁵] simulated the installation process based on a refined numerical model of bolt connection. Zhao et al. [¹⁶] proposed a simplified slip model for high-strength bolted connections. The numerical method was validated to help deeply understand the mechanical behavior of bolted connections. Thus, this method was adopted in this research program.

Many researchers have conducted investigations on the loading capacity and failure of bolted connections when they were subjected to shear or tension force. Grimsmo et al. [¹⁷] investigated the causes of the thread failure of bolts under tension load and eventually put forward several suggestions to avoid thread failure. Hu et al. [¹⁸] studied the mechanical behavior of high-strength bolts which subjected to tensile load. The bolt capacities were observed to be closely associated with their failure mechanisms. Salih et al. [¹⁹] conducted investigations on the bearing behavior of stainless steel connections between thick and thin plates. The investigation showed that the deformation behavior of stainless steel connections differed from that of carbon steel connections, with stainless steel exhibiting pronounced strain hardening. Alkatan et al. [²⁰] investigated the equivalent axial stiffness of various components in bolted joints when subjected to axial load. Varsani et al. [²¹] examined the performance of shear connectors under combined shear and uplifted axial forces. The shear–tension interaction of welded headed studs has been investigated by many researchers [²²]. Lee et al. [²³] analyzed the failure location in a bolt through a wedge tension test.

The aforementioned investigations have focused on the failure of the bolt itself. Only a few studies on the pull-through failure of steel plates can be found in the existing literature. Turvey [²⁴] investigated the bolt pull-through stiffness and strength of a pultruded glass fiber-reinforced polymer plate. Banbury et al. [²⁵] investigated the fastener pull-through failure of composite laminates, whereas Draganić et al. [²⁶] studied bearing failure in steel single bolt lap connections. The bolted connections may be subjected to tension load besides the shear force. There have been many achievements about the mechanical behavior of bolts in shear condition. For bolted connections in tensioning condition, the attentions have always been focused on the bolt itself in existing researches. However, the failure may happen at the steel plates. For a bolted thin steel plate, the steel plate is likely to be pulled off the head of the bolt when it is directly subjected to tension load, as shown in Fig. 1. The formula predicting the pull through capacity of bolted thin steel plate cannot be found in existing literatures. Thus, investigating the load limit that corresponds to this failure mode is necessary. The loading capacity of a bolted thin steel plate was investigated based on this background. The cracks should be accurately simulated when failure occurred at the bolts due to its brittle characteristics. Many achievements have been achieved on the numerically modelling cracking [²⁷–⁴³]. However, the crack development of steel plates can be ignored due to the good ductility of low alloy steel, such as the Q345 steel.

A refined numerical model of a bolted steel plate connection was first constructed based on the general purpose finite element software, and parametrical analysis was performed to reveal the influences of geometric parameters, including the yield strength of the steel plate, the effective depth of the cap nut, the preload of the bolt, and shear force, on the ultimate loading capacity. The influence of uncertain input parameters on uncertain model outputs have been done by Vu-Bac et al. [⁴⁴] and Hamdia et al. [⁴⁵] based on the Matlab functions. The uncertainties of these parameters were not considered for simplicity. The loading capacity investigated in this study was the loading capacity in the axial direction of the bolt, as shown in Fig. 2. In addition, an analytical method was proposed to predict the ultimate load of the bolted thin steel plate. The results derived using the numerical and analytical methods were compared and were found to be in good agreement with each other. The work reported in this paper can provide a simplified calculation method for the pull through capacity bolted thin steel plate.

Establishment and validation of the numerical model

Establishment of the numerical model

The accuracy of the refined numerical model of high-strength bolted connections is first validated. The results are compared with experimental results that derived by Li et al. [⁴⁶]. The geometrical size of steel plates and bolts are depicted in Fig. 3. The total length of this connection is 515 mm and the width of the steel plates is 80 mm. the yield and ultimate strength of bolt are 940 MPa and 1,040 MPa. The pretightening force in the bolts is set to be 155 kN according to experiment conducted by Li et al. [⁴⁶]. The Q235B steel is adopted for the steel plates and the yield strength, elastic modulus (E_s), tangent moduli (E_t), Poisson’s ratio are 235 MPa, 210 GPa, 0.01 E_s and 0.3, respectively. The diameter of bolt hole is 21.5 mm and the diameter of bolt shank is 20 mm. The load-displacement curves derived by numerical model are compared with that derived in experiment to validate its accuracy.

The general-purpose commercial finite element software ANSYS is utilized to construct the numerical model of bolted connection. The solid element SOLID185 in ANSYS is utilized to discretize the bolts and steel plates. The SOLID185 element owns eight nodes and each node has three translational degrees in x-, y-, and z-directions. This element owns the ability to consider the plasticity occurred at bolt or steel plates. The contact element CONTA174 and TARGE170 is utilized to simulate the contact behavior between bolts and steel plates. The parameter F_kn which means the normal contact stiffness is set to 1. The CONTA174 and TARGE170 shared the same real constant and formed contact pair. Then the contact and sliding state can be detected. The PRETS179 is utilized to apply pretightening force in the bolt shank. Several iterations are also needed to reach the designed value of pretightening force and the pretension section should be first defined.

The bolt is centrally located in the bolt holes, and the gap between the bolt shank and the bolt hole is 0.75 mm. The analyzed bolted connection is shown in Fig. 4. The values of other parameters are set to the default values. The contact algorithm is set as the penalty function and the Lagrange multiplier.

Validation of the numerical model

Bolts subjected to shear force

The objective of this subsection is to validate the accuracy of refined numerical model established in this paper. Existing research on pulling off the head of a bolt is highly limited, and only a few experimental results can be found. The accurate modeling of the interaction between different parts of bolted connections is the most important issue. The results obtained by Li et al. [⁴⁶] were utilized to validate the contact behavior between different plates and a bolt. The load-displacement curve presented in Ref [⁴⁶]. is utilized to validate the load-displacement curves derived based on the FE model. All of the translational degrees of the nodes located at one side surface (left side) is constrained and the displacement is applied at another side surface (right side). The parameter D indicates the relative displacement of the two sides. The load-displacement curves derived based on numerical analysis and experiment are compared and shown in Fig. 5(a). It can be observed from the results that the results derived by numerical analysis are consistent with those derived in experiment. The sliding behavior can be observed which indicated the established FE model can exactly capture the sliding phenomenon of bolted connection. The analytical value of friction force is 151 KN and the friction force derived based on numerical analysis is 150 kN. The sliding can be observed after the external force exceeded the analytical friction force. The accuracy of numerical model established in this paper can be validated.

The time needed to complete the analysis once can significantly increase with the increase of element number, especially the contact element number. The mesh sensitivity analysis should be performed to determine the optimal element size. The element size is set to 1 mm and 5 mm to reveal the influence of element size on results, as shown in Fig. 5(b). It can be seen that the load corresponding to sliding phase derived based on E_size=1 mm was 161 kN and the force corresponding to E_size=5 mm is 141 kN, i.e., the element size can affect the force needed for sliding of bolted connection. But the ultimate loading capacity was not influenced by the element size. The objection of this work was the tensioning capacity of bolted steel plate. It can be concluded that the FE model can predict the loading capacity accurately. With the decrease of element size, the friction is closer to analytical value. The deformation and stress contour of steel plates is shown in Fig. 6.

Bolts subjected to tension and shear forces

The experimental work conducted by Draganić et al. [²⁶] was also utilized to validate the accuracy of the FE model established in this study. Specimen 2-1-a-28 was adopted. The dimensions of this specimen are provided in Table 1, and the definition of each symbol is shown in Fig. 7. The steel plates were assembled using M16 bolts (Class 5.6). The M16 indicates the diameter of bolt shank is 16 mm. The nominal yield strength and tensile strength are 300 MPa and 500 MPa, respectively. The stress-strain curve of steel plates is shown in Fig. 8. The pretightening force was 12.5 kN, and the coefficient of friction was 0.2. The contact behavior between plates was defined as flexible-to-flexible, and the contact behavior between the plate and the bolt was defined as rigid-to-flexible. The bolt and the nuts were bound.

The comparison of the failure mode and the force-displacement curves is shown in Figs. 9 and 10. The results derived in this study agreed well with that derived by Draganić et al. [²⁶]. As shown in Fig. 11, the bolt was simultaneously subjected to tension and shear forces in the failure phase of the single-bolted connection. Hence, the established model achieves high accuracy in predicting the ultimate load for bolts subjected to tension and shear forces.

Ultimate loading capacity of bolted steel plate

The loading capacity of bolted steel plate can be directly affected by the material strength. The influence of yield strength and tangent moduli (E_t) of steel plate is investigated in this study. The bilinear isotropic hardening mode coupled with von Mises yield criteria in ANSYS is adopted. The initial slop of stress-strain curves equals to elastic modulus (E_s) of steel. The E_s is equal to 2.06 × 10¹¹ Pa. The tangent moduli (E_t) of steel adopted by the steel plate and the bolt were set as 0.007 E_s and 0, respectively, as shown in Fig. 12. The yield strength of the bolt was 960 MPa and remained unchanged. The diameter of bolt shank is indicated by d. The influences of several key parameters, including the yield strength of the steel plate (f_y), the thickness of the steel plate (t), the effective depth (w_e) of the cap nut, the preload in the bolt, and the shear force, and that of the geometric parameter, on the loading capacity were also investigated, as shown in Fig. 13. The boundary condition of the established FE model is shown in Fig. 14. The displacement load in z direction was applied at the end of thin steel plate (in red color). The translational degree in z direction of nodes located at the middle plane of base steel plate was fixed. All of the translational degrees of nodes located at the left side of the model were fixed to improve the convergence. Nearly all of the bolted steel plates were subjected to shear force (F_s) under actual condition. Hence, the influence of F_s was also investigated in section 4, as shown in Fig. 14.

The influence of element size on load-displacement curves was investigated to firstly determine the suitable element size for bolted plates subjected to both tension and shear force. The value of element size (E_size) is set to 3, 4, 5, 7 mm. Effective depth (we) was set as 5 mm. The preload of the bolt was 155 kN, and the thickness of the steel plate (t) was set as 4 mm. The value of F_s was set as 5kN. The load-displacement curves derived based on different element size were shown in Fig. 15. It can be observed that the value of E_size has great influence on the value of ultimate tension capacity of bolted steel plate. The values of ultimate tension capacity corresponding to E_size=7 mm, 5 mm, 4 mm and 3 mm were 53 kN, 36 kN, 33kN and 32kN. It can be seen that the loading capacity almost did not change any more when the E_size was less than 5 mm. Then, the value of E_size was set to 4 mm with considering the balance between accuracy and the need for computational power.

Influence of the yield strength of the steel plate

The influence of the yield strength of the steel plate on the loading capacity was first investigated. Effective depth (w_e) was set as 5 mm. The preload of the bolt was 155 kN, and the thickness of the steel plate (t) was set as 4 mm. The yield strength of the steel plate was set as 256, 300, and 345 MPa to investigate the influence of the yield strength of the bolted steel plate. The load displacement curves are shown in Fig. 16. The material of the bolt was set as ideal elastic (a) and elastic–plastic (b) to investigate the influence of bolt strength on the ultimate loading capacity of the bolted steel plate when yield strength was changed.

The entire loading process can be divided into three phases based on the load–displacement curves, namely, Phase I (bending of the steel plate), Phase II (anti-pull phase of the bolt), and Phase III (failure phase). Yield strength slightly influenced stiffness in Phase I and significantly influenced the ultimate loading capacity. The loading capacity can be slightly overestimated when the bolt is elastic, and ductility may also be overestimated.

Influence of the thickness of the steel plate

The influence of steel plate thickness was investigated in this section. Effective depth (w_e) was set as 5 mm, and the preload of the bolt was 155 kN. The thickness of the steel plate (t) was set as 3, 4, 5, 6, 7, and 8 mm. Yield strength (f_y) was set as 235 MPa, as shown in Fig. 12.

The results derived corresponding to all conditions are shown in Fig. 17. The material of the bolt was set as ideal elastic (a) and elastic–plastic (b) to reveal the effect of bolt strength on the ultimate loading capacity of the bolted steel plate. A change in the ultimate loading capacity with t is shown in Fig. 18, and the failure mode when t = 3 mm is shown in Fig. 19. The results indicated that loading capacity increased linearly with an increase in steel plate thickness. Loading capacity can be overestimated when the bolt seems elastic.

Influence of the effective depth and thickness of the cap nut

The effective depth (w_e) of the cap nut (Fig. 13) is a key parameter that influences loading capacity. Its influence was analyzed in this section. The thickness of the steel plate (t) was set as 4 mm, and yield strength (f_y) was set as 235 MPa. The bolt diameter (d) was 20 mm, and the preload of the bolt was set as 155 kN. The material of the bolt was set as ideal elastic (a) and elastic–plastic (b) to investigate the influence of bolt strength on the ultimate loading capacity of the bolted steel plate. The values of w_e were set as 4, 5, 6, 7, 10, and 15 mm. The results derived under different conditions are presented in Fig. 20. In general, loading capacity increases with an increase in w_e. The initial stiffness in the axial direction of the bolt was also improved. The ultimate loading capacity was improved from 30 kN to 53 kN when the w_e values increased from 4 mm to 15 mm. The changing tendency of loading capacity with w_e is shown in Fig. 21. The material constitutive model of the bolt exerted nearly no influence on loading capacity.

The failure modes that correspond to different w_e values are shown in Fig. 22. Failure was concluded to be caused by the yielding of the steel plate when w_e = 10 mm and by the penetration of the bolt cap when w_e = 4 mm.

The influence of bolt cap thickness (t_c) was also investigated in this section. The results are shown in Fig. 23. The ultimate loading capacity was slightly improved when t_c was increased from 6 mm to 18 mm based on the results. Loading capacity nearly remained unchanged when t_c≥12 mm. Hence, the influence of t_c can be disregarded when thickness is larger than 1.5 t.

Influence of the preload in the bolt

The influence of the preload in the bolt was investigated in this section. The effective depth (w_e) and thickness (t) of the steel plate were set as 5 mm and 4 mm, respectively. Yield strength (f_y) was set as 235 MPa, and bolt diameter (d) was 20 mm. The preload of the bolt was set as 50, 100, and 155 kN. The results are presented in Fig. 24. The material of the bolt was set as ideal elastic (a) and elastic-plastic (b) to investigate the influence of bolt strength on the ultimate loading capacity of the bolted steel plate. The results showed that the ultimate loads derived under the two conditions were nearly the same. The results also indicated that preload exerted nearly no influence on the ultimate loading capacity. The ductility of the bolt connection was overestimated when the bolt seemed elastic.

Influence of shear force

Nearly all of the bolted steel plates were subjected to shear force (F_s) under actual condition. Hence, the influence of F_s on loading capacity in the axial direction was analyzed. The effective depth (w_e) and thickness (t) of the steel plate were set as 5 mm and 4 mm, respectively, and yield strength (f_y) was set as 235 MPa. Bolt diameter (d) was 20 mm, and the preload of the bolt was set as 155 kN. The value of F_s was set as 0, 5, 10, and 20 kN. The yield strength of the steel plate at the net section is 67 kN. The load displacement curves derived under the four conditions are shown in Fig. 25. The results indicated that stiffness was improved when F_s was increased. However, loading capacity remained unchanged; that is, loading capacity was not influenced by F_s.

Analytical analysis

The entire loading process can be divided into three phases according to the numerical results presented in the preceding section. That is, Phase I (bending of the steel plate), Phase II (anti-pull phase of the bolt), and Phase III (failure phase). The details of the failure of the bolted steel plate are presented in Fig. 26. D indicates the vertical displacement at the end of the steel plate.

Establishing a numerical model of bolted connections will consume considerable time and effort. Moreover, conducting an analysis is time-consuming because refined numerical models are constantly required. An analytical method was proposed in this section to simplify analysis.

The research focused on Phases I and II because the failure phase was insignificant in the actual project.

Analytical results in Phase I

The initial bending stiffness S_b and yielding moment of the steel plate M_y can be derived using Eqs. (1) and (2):

(1)S_{\mathrm{b}}={\displaystyle \frac{Ew{t}^{\mathrm{3}}}{\mathrm{12}}},

(2)M_{\mathrm{y}}={\displaystyle \frac{w{t}^{\mathrm{2}}{f}_{\mathrm{y}}}{\mathrm{6}}},

where E is the elasticity modulus of steel, f_y is the yield stress of the steel plate, w is the width of steel plate, and t is the thickness of the steel plate.

In the ultimate bending moment, the steel plate is assumed as M_u when its thickness is t. M_u can be derived through Eq. (3):

(3)M_{\mathrm{u}}=f_{\mathrm{y}}\times {\displaystyle \frac{w{t}^{\mathrm{2}}}{\mathrm{4}}}.

In the ultimate bending moment, the steel plate was subjected to moment in Phase I, which can be presented by F_t and F_s, as shown in Eq. (4):

(4)M_{}=F_{\mathrm{t}}\times \sqrt{l^{\mathrm{2}}-D^{\mathrm{2}}}-F_{\mathrm{s}}\times D,

where l is the length of the steel plate between the loading point and the bolt side (Fig. 26), and l can be quantified through Eq. (5):

(5)l^{\mathrm{2}}={(l_{\mathrm{o}}+{\displaystyle \frac{\sqrt{F_{\mathrm{t}}{}^{\mathrm{2}}+F_{\mathrm{s}}{}^{\mathrm{2}}}}{E{A}_{\mathrm{e}}}}\times l_{\mathrm{o}})}^{\mathrm{2}}=(\mathrm{1}+\mathrm{2}\times {\displaystyle \frac{\sqrt{F_{\mathrm{t}}{}^{\mathrm{2}}+F_{\mathrm{s}}{}^{\mathrm{2}}}}{E{A}_{\mathrm{e}}}}+{\displaystyle \frac{F_{\mathrm{t}}{}^{\mathrm{2}}+F_{\mathrm{s}}{}^{\mathrm{2}}}{E^{\mathrm{2}}{A}_{\mathrm{e}}{}^{\mathrm{2}}}})\times l_{\mathrm{o}}{}^{\mathrm{2}},

(6)A_{\mathrm{e}}=(d+w_{\mathrm{e}}\times \mathrm{2})\times t,

where A_e is the effective area of the steel plate, and d is the diameter of bolt shank.

Given that {\displaystyle \frac{F_{\mathrm{t}}{}^{\mathrm{2}}+F_{\mathrm{s}}{}^{\mathrm{2}}}{E^{\mathrm{2}}{A}_{\mathrm{e}}{}^{\mathrm{2}}}} is considerably less than {\displaystyle \frac{\sqrt{F_{\mathrm{t}}{}^{\mathrm{2}}+F_{\mathrm{s}}{}^{\mathrm{2}}}}{E{A}_{\mathrm{e}}}}, the third term in Eq. (5) can be disregarded; hence, Eq. (5) can be transformed into Eq. (7):

(7)l^{\mathrm{2}}\approx (\mathrm{1}+\mathrm{2}\times {\displaystyle \frac{\sqrt{F_{\mathrm{t}}{}^{\mathrm{2}}+F_{\mathrm{s}}{}^{\mathrm{2}}}}{E{A}_{\mathrm{e}}}})\times l_{\mathrm{o}}{}^{\mathrm{2}}.

Then, the tension load F_t can be represented by the ultimate bending moment M_u, shear load F_s, and vertical load.

(8)F_{\mathrm{t}}={\displaystyle \frac{M_{\mathrm{u}}+F_{\mathrm{s}}\times D}{\sqrt{l^{\mathrm{2}}-D^{\mathrm{2}}}}}={\displaystyle \frac{M_{\mathrm{u}}+F_{\mathrm{s}}\times D}{\sqrt{(\mathrm{1}+{\displaystyle \frac{\mathrm{2}\times \sqrt{F_{\mathrm{s}}{}^{\mathrm{2}}+F_{\mathrm{t}}{}^{\mathrm{2}}}}{E{A}_{\mathrm{e}}}})\times l_{\mathrm{o}}{}^{\mathrm{2}}-D^{\mathrm{2}}}}}

In Phase I, the tension load F_t is generally considerably less than the shear load F_s. Thus, Eq. (8) can be deduced as

(9)F_{\mathrm{t}}={\displaystyle \frac{M_{\mathrm{u}}+F_{\mathrm{s}}\times D}{\sqrt{l^{\mathrm{2}}-D^{\mathrm{2}}}}}={\displaystyle \frac{M_{\mathrm{u}}+F_{\mathrm{s}}\times D}{\sqrt{(\mathrm{1}+{\displaystyle \frac{\mathrm{2}\times F_{\mathrm{s}}}{E{A}_{\mathrm{e}}}})\times l_{\mathrm{o}}{}^{\mathrm{2}}-D^{\mathrm{2}}}}}(D\le l_{\mathrm{o}}).

The F_t values that correspond to the yield bending moment M_y and the ultimate bending moment M_u are 1.8 kN and 2.8 kN, respectively, when the shear load F_s is zero.

Analytical results in Phase II

The tension load F_t increased rapidly in the beginning of Phase II, and the peak value achieved a slight increase in vertical displacement (D). The peak value remained nearly the same during Phase II. The stress distribution mode of the steel plate during Phase II is shown in Fig. 27. It can be observed that stress distribution is generally axisymmetric about the symmetry axis of bolted steel plate. The error is caused by the meshing method and the error in numerical calculation. The yielding line indicated that the ultimate bending moment was reached in the width direction. From the stress distribution mode at the end of the steel plate, the load was concluded to suffer in the middle part of the steel plate and the width of this part was nearly equal to the diameter of the cap nut. Therefore, only this part suffered from the load, and it was a safer computational method. The ultimate load F_t can be derived using Eq. (10). The corresponding displacement can be derived as Eq. (11). The displacement of the steel plate includes two parts: l₀ and l₁. l₀ denotes the original length of the steel plate before deformation, whereas l₁ indicates the deformation caused by an external force, as shown in Fig. 28. The resultant force of F_s and F_t can be expressed as \sqrt{F_s{}^{\mathrm{2}}+F_t{}^{\mathrm{2}}}. The fact that the loading process was reached before the ultimate loading capacity is a concern of researchers and engineers; hence, the analytical method for Phase III was not considered in this study.

(10)F_{\mathrm{t}}=f_{\mathrm{y}}\times (\mathrm{d}+w_{\mathrm{e}}\times \mathrm{2})\times t�

(11)D=l_{\mathrm{o}}+l_{\mathrm{1}}=l_{\mathrm{o}}+{\displaystyle \frac{\sqrt{F_{\mathrm{s}}{}^{\mathrm{2}}+F_{\mathrm{t}}{}^{\mathrm{2}}}}{E{A}_{\mathrm{e}}}}{l}_{\mathrm{o}}.

Comparison of the results derived using different methods

The analytical method presented in the preceding section was adopted to analyze the mechanical behavior of the bolted steel plate, and the results derived were compared with those derived using the numerical method. From the results obtained in Section 4.2, the ultimate load capacity is positively and linearly related to the yield strength of the material adopted by the steel plate. The results derived using different methods were compared, as shown in Fig. 29, at different thickness values of the steel plate (t). The preload of the bolt was 155 kN in this section, and the yield strength (f_y) was set as 235 MPa.

The results showed that the data derived through the proposed analytical method agreed well with those derived using the numerical method. The ultimate bending moment and ultimate tension force can be predicted exactly using the proposed analytical method when the thickness of the steel plate changed from 3 mm to 8 mm. The ultimate loading capacity predicted using the analytical method was slightly larger than that derived using the numerical method, but it became slightly smaller when the thickness of the steel plate was 8 mm because the integrity of the steel plate was improved with an increase in thickness.

In addition, the load-displacement curves derived through the analytical method agreed well with that derived using the numerical method during the entire loading process when t = 3 and 4 mm. However, errors may occur after the ultimate bending moment M_u and before the ultimate loading capacity were reached when t has a large value. The ultimate loading capacity can still be predicted exactly.

From the results derived in Section 4.3, the diameter of the cap nut significantly influenced the ultimate loading capacity. Hence, the analytical method was utilized to investigate the influence of cap nut diameter on the ultimate loading capacity. The thickness of the steel plate (t) was set as 4 mm, and yield strength (f_y) was set as 235 MPa. The results derived using different methods were compared and shown in Fig. 30. The results derived using the proposed analytical method agreed well with that derived using the numerical methods when the cap nut diameter increased from 28 mm to 50 mm. This finding indicated that the proposed analytical method was suitable for a bolt with various cap nut diameters. The ultimate tension force can be predicted exactly, which validates the accuracy of Eq. (10).

Shear force may exist in every bolted steel plate in an actual project. Thus, the analytical method was utilized to investigate the influence of shear force (F_s) on the ultimate loading capacity, and the results were compared with those derived in Section 4.5, as shown in Fig. 31. The effective depth (w_e) and thickness (t) of the steel plate were set as 5 mm and 4 mm, respectively, and yield strength (f_y) was set as 235 MPa. Shear force was changed from 0 to 30 kN. The results derived using the analytical and numerical methods were highly consistent with each other. This finding indicated that the proposed analytical method maintained high accuracy when it was adopted to compute the ultimate loading capacity under shear force.

The proposed analytical method was utilized to investigate the influence of yield strength (f_y) on the ultimate loading capacity, and the results were compared with that derived in Section 4.1 (Fig. 32). The yield strength of the steel plate was set to 256, 300, and 345 MPa. Accuracy was maintained when yield strength obtained different values.

Conclusions

A refined numerical model of bolted steel plate connection was established based on the FE code, and parametrical analysis was conducted to investigate the influences of geometric parameters, including the yield strength of the steel plate, the effective depth of the cap nut, the preload of the bolt, and shear force, on the ultimate loading capacity. The loading capacity investigated in this study indicated the loading capacity in the axial direction of the bolt. In addition, an analytical method was proposed to predict the ultimate load of the bolted thin steel plate. The conclusions can be summarized as follows.

1) The ultimate loading capacity increased linearly with an increase in yield strength. Loading capacity can be slightly overestimated when the bolt seemed elastic, and ductility may also be overestimated. The results indicated that loading capacity increased linearly with an increase in steel plate thickness.

2) Loading capacity generally increased with an increase in w_e. The initial stiffness in the axial direction of the bolt was also improved. Meanwhile, the ultimate loading capacity was improved from 30 to 53 kN when the w_e value increased from 4 to 15 mm. The material constitutive model of the bolt exerted nearly no influence on loading capacity. The influence of t_c can be disregarded when thickness was larger than 1.5t.

3) The preload exerted nearly no influence on the ultimate loading capacity. The ductility of the bolt connection was overestimated when the bolt seemed elastic. Stiffness was improved when F_s was increased. However, loading capacity was not changed. That is, loading capacity was not influenced by F_s.

4) The analytical method reported in this paper can provide a simplified calculation method for loading capacity in the axial direction of a bolt.

References

[1] Rabczuk T, Ren H, Zhuang X. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers, Materials and Continua, 2019, 59(1): 31–55

[2] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[3] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[4] Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[5] Areias P, Reinoso J, Camanho P P, César de Sá J, Rabczuk T. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360

[6] Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41

[7] Kim T S, Kuwamura H. Finite element modeling of bolted connections in thin-walled stainless steel plates under static shear. Thin-walled Structures, 2007, 45(4): 407–421

[8] Abdelfattah F A, Ali abdo A A H. Reinforced connection for upgrading strength and efficiency of single angle tension members. International Journal of Steel Structures, 2016, 16(2): 637–645

[9] Ashakul A, Khampa K. Effect of plate properties on shear strength of bolt group in single plate connection. Steel and Composite Structures, 2014, 16(6): 611–637

[10] Sabuwala T, Linzell D, Krauthammer T. Finite element analysis of steel beam to column connections subjected to blast loads. International Journal of Impact Engineering, 2005, 31(7): 861–876

[11] Rahbari R, Tyas A, Buick Davison J, Stoddart E P. Web shear failure of angle-cleat connections loaded at high rates. Journal of Constructional Steel Research, 2014, 103: 37–48

[12] Grimsmo E L, Clausen A H, Aalberg A, Langseth M. A numerical study of beam-to-column joints subjected to impact. Engineering Structures, 2016, 120: 103–115

[13] Al-Rifaie A, Guan Z W, Jones S W, Wang Q. Lateral impact response of end-plate beam-column connections. Engineering Structures, 2017, 151: 221–234

[14] Ma S, Zhao Z, Nie W, Gui Y. A numerical model of fully grouted bolts considering the tri-linear shear bond-slip model. Tunnelling and Underground Space Technology, 2016, 54: 73–80

[15] Hwang H Y. Bolted joint torque setting using numerical simulation and experiments. Journal of Mechanical Science and Technology, 2013, 27(5): 1361–1371

[16] Zhao Z, Liang B, Liu H, Li Y. Simplified numerical model for high-strength bolted connections. Engineering Structures, 2018, 164: 119–127

[17] Grimsmo E L, Aalberg A, Langseth M, Clausen A H. Failure modes of bolt and nut assemblies under tensile loading. Journal of Constructional Steel Research, 2016, 126: 15–25

[18] Hu Y, Shen L, Nie S, Yang B, Sha W. FE simulation and experimental tests of high-strength structural bolts under tension. Journal of Constructional Steel Research, 2016, 126: 174–186

[19] Salih E L, Gardner L, Nethercot D A. Bearing failure in stainless steel bolted connections. Engineering Structures, 2011, 33(2): 549–562

[20] Alkatan F, Stephan P, Daidie A, Guillot J. Equivalent axial stiffness of various components in bolted joints subjected to axial loading. Finite Elements in Analysis and Design, 2007, 43(8): 589–598

[21] Varsani H, Tan E L, Singh B. Behaviour of innovative demountable shear connectors subjected to combined shear and axial tension. Ce/papers, 2017, 1(2–3): 1948–1955

[22] Saari W K, Hajjar J F, Schultz A E, Shield C K. Behavior of shear studs in steel frames with reinforced concrete infill walls. Journal of Constructional Steel Research, 2004, 60(10): 1453–1480

[23] Lee H C, Jin Y G, Hwang S K, Jung K H, Im Y T. Wedge tension test of a high-strength bolt of fully pearlitic high-carbon steel. Journal of Materials Processing Technology, 2011, 211(6): 1044–1050

[24] Turvey G J. Experimental evaluation of bolt pull-through in pultruded glass-fibre-reinforced polymer plate. Structures and Buildings, 2011, 164(5): 307–319

[25] Banbury A, Kelly D W, Jain L K. A study of fastener pull-through failure of composite laminates. Part 2: Failure prediction. Composite Structures, 1999, 45(4): 255–270

[26] Draganić H, Dokšanović T, Markulak D. Investigation of bearing failure in steel single bolt lap connections. Journal of Constructional Steel Research, 2014, 98: 59–72

[27] Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

[28] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[29] Zhang Y, Lackner R, Zeiml M, Mang H A. Strong discontinuity embedded approach with standard SOS formulation: Element formulation, energy-based crack-tracking strategy, and validations. Computer Methods in Applied Mechanics and Engineering, 2015, 287: 335–366

[30] Zhang Y, Zhuang X. Cracking elements method for dynamic brittle fracture. Theoretical and Applied Fracture Mechanics, 2019, 102: 1–9

[31] Zhang Y, Zhuang X. Cracking elements: A self-propagating strong discontinuity embedded approach for quasi-brittle fracture. Finite Elements in Analysis and Design, 2018, 144: 84–100

[32] Zhang Y, Zhuang X. A softening-healing law for self-healing quasi-brittle materials: Analyzing with strong discontinuity embedded approach. Engineering Fracture Mechanics, 2018, 192: 290–306

[33] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[34] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[35] Rabczuk T, Ren H. A peridynamics formulation for quasi-static fracture and contact in rock. Engineering Geology, 2017, 225: 42–48

[36] Han F, Lubineau G, Azdoud Y, Askari A. A morphing approach to couple state-based peridynamics with classical continuum mechanics. Computer Methods in Applied Mechanics and Engineering, 2016, 301: 336–358

[37] Wu J Y, Qiu J F, Nguyen V P, Mandal T K, Zhuang L J. Computational modeling of localized failure in solids: XFEM vs PF-CZM. Computer Methods in Applied Mechanics and Engineering, 2018, 345: 618–643

[38] Ren H L, Zhuang X Y, Anitescu C, Rabczuk T. An explicit phase field method for brittle dynamic fracture. Computers & Structures, 2019, 217: 45–56

[39] Wu J Y, Nguyen V P. A length scale insensitive phase-field damage model for brittle fracture. Journal of the Mechanics and Physics of Solids, 2018, 119: 20–42

[40] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[41] Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[42] Zhang Y. Multi-slicing strategy for the three-dimensional discontinuity layout optimization (3D DLO). International Journal for Numerical and Analytical Methods in Geomechanics, 2017, 41(4): 488–507

[43] Zhang Y, Zhuang X, Lackner R. Stability analysis of shotcrete supported crown of NATM tunnels with discontinuity layout optimization. International Journal for Numerical and Analytical Methods in Geomechanics, 2018, 42(11): 1199–1216

[44] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[45] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[46] Li Q, Gu Q, Su M, Chen A. Experiment of high-strength bolted connection behavior. Journal of Xi’an University of Science and Technology, 2003, 23(3): 322–327

RIGHTS & PERMISSIONS

Higher Education Press