Thermal strength analysis of a steel bolted connection under bolt loss conditions

Steel bolted connections are used in many engineering, mechanical and building structures that can be subjected to a wide variety of thermal loads [11, 28]. As a result of these loads, the preload of bolts changes, which may be an unfavorable phenomenon [27, 51]. It has been observed that, in the case of moment connections, the tensile force in the bolts tends to decrease with increasing temperature. However, in the case of shear connections, the force in the bolts tends to increase with temperature [44]. At elevated temperatures, the mechanical properties of the bolts also drop significantly, which directly affects the safety and reliability of steel structures [3, 34, 53]. Many experimental studies with the use of testing machines have shown that temperature loads quickly lead to a significant decrease in the load capacity of bolted connections and affect their failure modes [22, 55, 62]. In order to prevent bolted connections from being failure by elevated temperatures, a series of experimental studies have been carried out, which have resulted in the development of special design rules, for example, for additional protection of bolts [7, 18, 23, 61]. In the papers written so far, the behaviour of bolted connections at elevated temperatures is sometimes analysed on the basis of the component method [2, 20, 46, 58, 63]. The still poor knowledge of some elements, especially the elements in shear, limits the wider application of this method. However, much more often it is modelled and described using the finite element method (FEM). Towarnicki and Grzejda [49] presented an analysis of the influence of the temperature load on the stress state in the joint separated from the bolted flange connection. They demonstrated the usefulness of using a simplified bolt model for modelling bolted connections in order to determine the forces acting on the bolt. The influence of the thread length on the failure of bolted connections at elevated temperatures was discussed by Shaheen et al. [45]. Schaumann and Kirsch [42] simulated a flush endplate connection at elevated temperatures taking into account nonlinearities, e.g. temperature dependent material. Lim and Young [25] investigated the influence of elevated temperatures on bolted moment-connections between cold-formed steel members and proposed some simple design rules that allow for this influence to be taken into account. Abid et al. [1] studied the strength and sealing performance of a preloaded gasketed bolted flange connection at combined pressure, axial and thermal loading to show its safe and unsafe operating limits. Similar thematic considerations were presented by Wang et al. [56]. The fracture behaviour of high-strength bolted steel connections The aim of the paper is to analyse an exemplary bolted connection under the conditions of loss of bearing capacity of some fasteners in the connection. The presented tests are numerical and concern an asymmetric bolted connection. The joined elements in the connection were modelled as 3D finite elements, and the fasteners were treated as hybrid models consisting of rigid heads and nuts, and flexible beams between them. The bolted connection was first preloaded according to six different tensioning orders using a standardised force. After all bolts were tensioned, the selected bolts were removed, simulating bolt damage. The connection was then tested for external loads with temperatures ranging from 20 to 600°C. The influence of loosening the connection on the load in the remaining bolts at the operational stage of the connection was investigated. The calculation results were compared with the test results for a healthy bolted connection. It was shown that the bearing capacity of the connection in the damaged state decreased by 13%. Highlights Abstract


Introduction
Steel bolted connections are used in many engineering, mechanical and building structures that can be subjected to a wide variety of thermal loads [11,28]. As a result of these loads, the preload of bolts changes, which may be an unfavorable phenomenon [27,51]. It has been observed that, in the case of moment connections, the tensile force in the bolts tends to decrease with increasing temperature. However, in the case of shear connections, the force in the bolts tends to increase with temperature [44]. At elevated temperatures, the mechanical properties of the bolts also drop significantly, which directly affects the safety and reliability of steel structures [3,34,53].
Many experimental studies with the use of testing machines have shown that temperature loads quickly lead to a significant decrease in the load capacity of bolted connections and affect their failure modes [22,55,62]. In order to prevent bolted connections from being failure by elevated temperatures, a series of experimental studies have been carried out, which have resulted in the development of special design rules, for example, for additional protection of bolts [7,18,23,61].
In the papers written so far, the behaviour of bolted connections at elevated temperatures is sometimes analysed on the basis of the component method [2,20,46,58,63]. The still poor knowledge of some elements, especially the elements in shear, limits the wider application of this method. However, much more often it is modelled and described using the finite element method (FEM). Towarnicki and Grzejda [49] presented an analysis of the influence of the temperature load on the stress state in the joint separated from the bolted flange connection. They demonstrated the usefulness of using a simplified bolt model for modelling bolted connections in order to determine the forces acting on the bolt. The influence of the thread length on the failure of bolted connections at elevated temperatures was discussed by Shaheen et al. [45]. Schaumann and Kirsch [42] simulated a flush endplate connection at elevated temperatures taking into account nonlinearities, e.g. temperature dependent material. Lim and Young [25] investigated the influence of elevated temperatures on bolted moment-connections between cold-formed steel members and proposed some simple design rules that allow for this influence to be taken into account. Abid et al. [1] studied the strength and sealing performance of a preloaded gasketed bolted flange connection at combined pressure, axial and thermal loading to show its safe and unsafe operating limits. Similar thematic considerations were presented by Wang et al. [56]. The fracture behaviour of high-strength bolted steel connections The aim of the paper is to analyse an exemplary bolted connection under the conditions of loss of bearing capacity of some fasteners in the connection. The presented tests are numerical and concern an asymmetric bolted connection. The joined elements in the connection were modelled as 3D finite elements, and the fasteners were treated as hybrid models consisting of rigid heads and nuts, and flexible beams between them. The bolted connection was first preloaded according to six different tensioning orders using a standardised force. After all bolts were tensioned, the selected bolts were removed, simulating bolt damage. The connection was then tested for external loads with temperatures ranging from 20 to 600°C. The influence of loosening the connection on the load in the remaining bolts at the operational stage of the connection was investigated. The calculation results were compared with the test results for a healthy bolted connection. It was shown that the bearing capacity of the connection in the damaged state decreased by 13%.
Six different tensioning orders of the bolts were • tested.
The failure of selected bolts in a connection load-• ed with temperature was simulated.
The influence of tensioning order on the bolt forc-• es in a damaged connection was probed.
Thermal strength analysis of a steel bolted connection under bolt loss conditions at elevated temperatures and also loaded with shear forces was described by Cai et al. [5]. Li and Zhao [24] conducted an analytical tests of beam-to-column connections at elevated temperatures and the verification of these tests with the use of FEM.
There are also many papers on the determination of the behaviour of bolted connections under fire conditions [12,13,19,21,[37][38][39] and several on the behaviour of bolted connections at reduced temperatures [29,47,50] using FEM.
Only a few studies dealt with the issue of changing the stress distribution in a bolted connection subjected first to failure and then to temperature loads. Most often they concern the analysis of connections with properties changed by the earthquake, and therefore different from those designed for a given structure and then subjected to fire. Della Corte et al. [9] investigated the post-earthquake fire resistance of steel moment resisting frames, which confirmed that the fire resistance of structures can be significantly reduced depending on the damage caused by seismic action. Xu et al. [59] came to similar conclusions, but in the case of bolted end-plate connections with end-plate and web stiffeners. Tartaglia et al. [48] described the behaviour of seismically damaged extended stiffened end-plate connections at elevated temperature. They presented the results in terms of moment-rotation-temperature characteristics and pattern of plastic deformations, which indicate under what conditions the type of cyclic damage may affect the performance of the connection under high temperature. An example of a test stand and test procedure to analyse the behaviour of steel beam to column end-plate bolted connections under the post-earthquake fire action was published by Petrina [35].
Likewise, there are few papers on the effect of the number and arrangement of bolts in a bolted connection on its performance under the influence of high temperatures. Wang et al. [57] conducted an experimental study of the relative structural fire behaviour and robustness of different types of bolted steel connections in restrained steel frames. Wang and Wang [54], and Yahyai and Rezaeian [60] presented numerical analyses verified experimentally on essentially similar issues. However, they did not investigate the effect of bolts loss on the bolted connection behaviour.
There are also several papers on the evaluation of the reliability of bolted connections, or larger structures with these connections. However, they do not take into account the influence of temperature on reliability, but rather relate to the assessment of fatigue reliability [26,30,41].
According to the conducted review of the state of knowledge, few scientists have so far dealt with the health assessment of the bolted connections under conditions of elevated temperatures and after the failure occurred earlier in the connection. Therefore, this theme was taken up by the author of the featured paper. The paper concerns a bolted connection, some experimental tests of which have been described in [14][15][16]. The novelty in relation to the above-cited articles is that in the current paper an asymmetrical connection was modelled under bolt loss conditions. The failure of the connection was simulated by removing the selected bolts from the connection, similarly to the explicit finite element analysis using element erosion, in which the elements are removed from the analysis after meeting certain failure criteria [43]. The calculations were made for an exemplary bolted connection using the finite element system called Midas NFX 2020 R2, and their results are the courses of force changes in bolts remaining in the connection, which was pretensioned and then subjected to variable temperature loads.

Material and methods
The subject of research and analysis is the bolted connection presented in Fig. 1a. The tested connection is made of a pair of plates joined with i fasteners shown in Fig. 1b (for i = 1, 2, …, 7). The fasteners are of the M10x1.25 type and consist of a bolt and a nut as one solid [52]. The joined plates are welded to the top and bottom bases. The thickness of the joined plates and the bases is equal to 28 mm. The connection is sloped to the horizontal at an angle of 60 deg (for comparison, see [8]). The total heigh of the structure is approximately 266 mm. The plates and bases are performed of 1.0577 steel. After machining, the bolts and nuts have been tempered to achieve the characteristics for the class of mechanical property 8.8 and 8, respectively.
The contact surface area between the joined plates and the assumed numbering of the bolts are presented in Fig. 1c. This area fits in a circle with a radius of 90 mm, and its outline does not exceed 90 cm 2 .
Using the Midas NFX 2020 R2 finite element system tools, a model of the bolted connection was made, shown in Fig. 2. The joined plates were divided into 3D finite elements, while the fasteners were modelled as flexible beams with rigid heads and nuts (for comparison, see [32,33]). All connection parts were assigned the properties of isotropic linear steel materials. The constitutive relationships in this case can be described by Hooke's law [10].
General surface-to-surface contact elements were applied between the joined plates. They allow for non-linear analysis taking into consideration the possibility of separating the joined plates in the vertical direction and the occurrence of sliding in the horizontal direction. The following values of the contact layer parameters were adopted: normal stiffness ratio equal to 10, • tangential stiffness ratio equal to 1, • static friction factor equal to 0.14 [17]. Welded type contact elements were used between the plates and the bases, preventing the elements from moving relative to each other in any direction. The bolted connection model was created with a total of 78750 elements and 133162 nodes. The maximum size of the side length of a finite element in the mesh does not exceed 10 mm. The mesh has been significantly densified at the point of contact between the joined plates and at the point of contact between the fasteners and the joined plates (for comparison, see [4]). The model was constrained by taking away all degrees of freedom on the underside of the bottom base. The preload of the bolts was applied via the "pretension" function built into the Midas NFX 2020 R2 system. A non-linear static analysis was chosen as the type of FE-analysis. The mathematical description of the temperature distribution in this case can be done using the Fourier-Kirchhoff partial differential equation for stationary isotropic bodies [40].
The research was divided into the following stages: Pretensioning the bolted connection according to one of the six 1.
adopted tensioning orders presented in Table 1 with a pretension F p equal to 22 kN. Performing calculations of forces in the bolts in the connec-2.
tion subjected to thermal loads in the range from 20 to 600°C (temperature boundary conditions were assigned to all model nodes as uniform prescribed nodal temperature [6]). Entering the failure states according to the diagram shown in 3. Table 2 (after each way of tensioning the bolted connection). Designation of changes in the value of forces in the bolts re-4.
maining in the connection. Duplication the above steps for all connection tensioning or-5. ders.

Results and discussion
The obtained courses of the bolt forces variation for individual types of pretensioning of the connection are qualitatively very similar. Since their quantitative comparison also gives a high similarity, this paper presents a set of diagrams for an example type of pretensioning of the connection, which is the queue No. 3, marked with bold letters in Table 1.
The graphical presentation of the calculation results is shown in Fig. 3. The values of the operating forces in the bolts remaining in the connection after the introduction of two successive states of failure and those obtained for the healthy connection were compared. The operating forces were related to the preload of the bolts to improve the readability of the graphs.
The forces in the bolts tend to increase with temperature, which is inherent in shear connections [44]. In general, increments in operating bolt forces due to temperature decrease as the number of bolts removed from the connection increases. However, due to the asymmetry in the arrangement of bolts in the connection, there is no single relationship to determine the magnitude of the force decrease in bolts compared to the healthy connection, as a function of the number of bolts lost. Its search may be the subject of a separate publication, which would take into account other types of connections, with a different number of bolts. The force increments in bolts for individual load conditions of the connection are summarised in Table 3. They are expressed using the following Z 1 indicator: where: T bi F is the operating force in the i-th bolt at the end of the connection heating process as a function of the type of tensioning, and p bi F is the preload of the i-th bolt.
In most cases, the force increments in bolts under the influence of temperature acting on the damaged connection are smaller than for the healthy connection. The exceptions are the increments in force in bolt No. 2 in the first stage of failure, when they are greater than corresponding increments for the healthy connection. The force in this bolt then increases by about 20% for all tensioning cases. This does not exceed the permissible value of the preload according to Eurocode 3 [36], which for the M10x1.25 bolt in the class 8.8 is 31 kN. On the other hand, it may pose a threat due to the high pressure on the contact surface of the nut and the joined plate.
The bolt tensioning method has a slight effect on the Z 1 indicator values.
The last comparisons of the calculation results presented in Table  3 were aimed at determining the value of the temperature T at which the operating forces in the bolt No. 2 in the first state of connection failure reach the values corresponding to the final load condition in the healthy connection. The results of these analyses are summarised in Table 4.
The bearing capacity of bolt No. 2 in the failure state No. 1 decreased by approx. 13%. The bolt tensioning method has a slight effect on the value of the bearing capacity of the connection.
The total bolts tension at the end of the connection loading process was compared on the basis of the proposed Z 2 indicator described by the formula: where: h t F is the total force in the bolts in the considered distribution of forces at the end of the process of loading the healthy connection, and d t F is an analogous force in the case of the damaged connection.
The greatest drops in total force do not exceed 2% for failure state No. 1 and 5% for failure state No. 2, for all methods of tensioning the bolted connection (Table 5). The bolt tensioning method has a slight effect on the differences in these drops.
In the literature cited in the introduction, the differences in the displacements of the joined elements as a result of the removal of selected bolts were studied rather than the differences in the values of forces in the bolts not removed from the connection. In this sense, the presented research and their results supplement the state of knowledge on the behaviour of damaged bolted connections at elevated temperatures.

Conclusions
The paper presents a method of modelling preloaded asymmetric bolted connections in conditions of failure and load with increased temperature. This method can be used in assessing the health of bolted connections. The results of the study lead to the following conclusions: The increments in operating forces in a damaged connection 1.
subjected to elevated temperature may be greater than in the case of a healthy connection. In extreme cases, the differences in the values of the operating forces can be as high as 20%.
The bearing capacity of a damaged connection may be reduced 2.
by approx. 13%. The method of tensioning the connection has a slight effect on 3.
the magnitude of increments in the values of bolt forces and the bearing capacity of the connection in its failure state. The described tests can be extended to determine the universal 4.
load-strength interference [31] allowing the assesment of the