Moment Resisting Connection with Curved Endplates: Behaviour Study

: This article provides a behaviour analysis of moment resisting joints with curved endplates. This is a new type of connection that can be used for joining steel beams to the circular hollow section (CHS) columns by means of bolts. Some researchers apply the Eurocode model without considering the differences in calculation schemes and assumptions, such as by using the general model of an equivalent T-stub in tension. Consequently, many of the existing behaviour studies are somewhat misleading, thus there is a need for further research. Apart from the absence of analytical methods that are devoted to predicting the initial stiffness and strength of the curved T-stub, other technical difﬁculties were encountered, such as gaps between the endplate and the column, as well as the initial pre-loading force of the bolts. In the previous studies, endplates were manufactured by rolling ﬂat plates to the precise curvature which resulted in ﬁrm contact. In contrast, in this study, endplates were manufactured from a standard CHS tube, which led to signiﬁcant initial gaps. Meanwhile, in terms of preloading force, it was found that it affected the moment resistance of the joint. This paper discusses problems associated with ongoing researches and presents experimental tests of the two connections. The obtained results were further used in the parametric ﬁnite element analysis (FEA) to determine the effect of the gaps and preloading force of the bolts on the moment resistance and initial rotational stiffness of the joint. The results indicate that the behaviour of curved plated connections is exceedingly complex and that the preloading force is the key factor, therefore, it should be controlled.


Introduction
Composite steel-concrete structures can be seen as a solution to the arising struggle with climate change [1][2][3][4] as they combine efficient use of materials with the robustness and ease of assembly, but their advantages are still not used widely enough. The main setback for composite structures is the lack of construction-wise and cost-effective joints. It is especially apparent in the design of frames with the circular concrete-filled steel tube (CFST) columns, which as a result of a superior confinement effect have greater compressive strength and deformation capacity than their square counterparts [5]. Such connections are considered non-straightforward, and because of this, they are not developed to the same degree as joints of square CFST columns. [6][7][8][9].
In recent years, a large-scale investigation of the new moment resisting joint with the curved endplates ( Figure 1) has been in progress. This includes experimental and numerical studies of the behaviour of the bolted endplates connection between a CFST column and a steel beam [10][11][12][13][14][15][16], as well as analysis of the connections between a composite beam and a stainless or carbon steel concrete-filled column [17][18][19][20][21][22]. There are also studies by Tao et al. [23] and Sheet et al. [24] on the connections with through-column bolts, which were either parallel to the beam, or passed through the column in an "X" shape. Lastly, there is the study by Oktavianus et al. [25], in which a diaphragm-like bolted endplate were either parallel to the beam, or passed through the column in an "X" shape. Lastly, there is the study by Oktavianus et al. [25], in which a diaphragm-like bolted endplate connection was analysed, but it would seem, that in case of this connection, unequal bolt force distribution should be anticipated, as it was reported by Yao et al. [26]. Based on the Theory of Shells, it is known that curved plates develop thrust to resist the applied loads. The same happens in curved endplates, in which bolts are not only in tension, but also in shear. The degree of tension/shear force vastly depends on the flexibility of the plate and support conditions of the bolts: if bolts could move easily, then thrust would not develop. Thrust force decreases the axial load of the bolts and increases their shear. Because of this, conventional prying models are not suitable for the general case of the curved T-stub. Moreover, this also applies to the use of the Eurocode [27] expressions for determination of the effective flange length, particularly in the case of estimation of the initial stiffness of the T-stub [28].
This paper investigates behaviour of new moment resisting connection between circular CFST and steel beam by means of experimental and numerical studies. Two connections with curved endplates of identical geometry were tested, in which bolts were snugtightened to establish firm contact between an endplate and the column, but in neither one firm contact was achieved. Further parametric study was utilised to explore the effects of gaps on stiffness, strength, and position of centre of compression in such joints. It showed that depending on the gap size and its position, not only the initial rotational stiffness, but also resistance of the joint is dependent on bolt preloading force. This finding led to the conclusion that without knowing the initial force in the bolts, it is impractical to calibrate the FEA model, as there are too many variables.

Experimental Investigation
Two connections of identical geometry 2B and 3B were tested. For the tests, a 1000 t capacity press was used. Geometry of the joints is presented in Figure 1a. The characteristics of materials used for the tests were obtained experimentally and are presented in Tables 1 and 2. Loading was divided into two steps. In the first step, an initial 50 t compression force in the column was introduced and maintained, then, in the second step, equal and opposite moments were induced by manually controlled hydraulic cylinders ( Figure 1). The ends of the column were fixed; in contrast, the ends of the beams were not restrained. Based on the Theory of Shells, it is known that curved plates develop thrust to resist the applied loads. The same happens in curved endplates, in which bolts are not only in tension, but also in shear. The degree of tension/shear force vastly depends on the flexibility of the plate and support conditions of the bolts: if bolts could move easily, then thrust would not develop. Thrust force decreases the axial load of the bolts and increases their shear. Because of this, conventional prying models are not suitable for the general case of the curved T-stub. Moreover, this also applies to the use of the Eurocode [27] expressions for determination of the effective flange length, particularly in the case of estimation of the initial stiffness of the T-stub [28].
This paper investigates behaviour of new moment resisting connection between circular CFST and steel beam by means of experimental and numerical studies. Two connections with curved endplates of identical geometry were tested, in which bolts were snug-tightened to establish firm contact between an endplate and the column, but in neither one firm contact was achieved. Further parametric study was utilised to explore the effects of gaps on stiffness, strength, and position of centre of compression in such joints. It showed that depending on the gap size and its position, not only the initial rotational stiffness, but also resistance of the joint is dependent on bolt preloading force. This finding led to the conclusion that without knowing the initial force in the bolts, it is impractical to calibrate the FEA model, as there are too many variables.

Experimental Investigation
Two connections of identical geometry 2B and 3B were tested. For the tests, a 1000 t capacity press was used. Geometry of the joints is presented in Figure 1a. The characteristics of materials used for the tests were obtained experimentally and are presented in Tables 1 and 2. Loading was divided into two steps. In the first step, an initial 50 t compression force in the column was introduced and maintained, then, in the second step, equal and opposite moments were induced by manually controlled hydraulic cylinders ( Figure 1). The ends of the column were fixed; in contrast, the ends of the beams were not restrained. Vertical displacements of the ends of the beams were measured with two linear variable differential transformers (LVDT) per side to account for plausible torsion. These LVDTs were placed on the bottom flange near the loading gear. Lateral displacements of the beam ends were controlled with one LVDT, which was attached to the middle of the web. In addition to this, horizontal displacements of the endplates were measured by placing LVTDs in the middle of the bolt spacing: both in the compression and in the tension zones. For the redundancy of the measurements, the digital 2d image correlation technique was also tested, which proved to be accurate, but it did not show significant advantages in comparison to the conventional LVTDs.
Curved endplates were manufactured from a seamless CHS 219 × 10 pipe, which was cut into four equal pieces (the central angle of the endplate is equal to 90 deg.) The average throat thickness of the fillet weld joining the endplates with IPE 270 cantilevers was 4.44 mm. The endplates were connected to the cold formed CHS 219 × 6 column with four high-strength M12 × 35 HV bolts by using a regular wrench, and then the assembled columns were filled with the concrete.
Bolts were tightened by applying the full-strength torque to establish firm contact between the column and the endplate; however, no firm contact could be achieved. The bolt tightening force was not measured or controlled by any means. After tightening, the gaps at the ends of the plates were measured with the feeler gauges at three points (two corners and in the middle in accordance with Figure 1). The measurements are presented in Figure 2. Vertical displacements of the ends of the beams were measured with two linear variable differential transformers (LVDT) per side to account for plausible torsion. These LVDTs were placed on the bottom flange near the loading gear. Lateral displacements of the beam ends were controlled with one LVDT, which was attached to the middle of the web. In addition to this, horizontal displacements of the endplates were measured by placing LVTDs in the middle of the bolt spacing: both in the compression and in the tension zones. For the redundancy of the measurements, the digital 2d image correlation technique was also tested, which proved to be accurate, but it did not show significant advantages in comparison to the conventional LVTDs.
Curved endplates were manufactured from a seamless CHS 219 × 10 pipe, which was cut into four equal pieces (the central angle of the endplate is equal to 90 deg.) The average throat thickness of the fillet weld joining the endplates with IPE 270 cantilevers was 4.44 mm. The endplates were connected to the cold formed CHS 219 × 6 column with four high-strength M12 × 35 HV bolts by using a regular wrench, and then the assembled columns were filled with the concrete. Bolts were tightened by applying the full-strength torque to establish firm contact between the column and the endplate; however, no firm contact could be achieved. The bolt tightening force was not measured or controlled by any means. After tightening, the gaps at the ends of the plates were measured with the feeler gauges at three points (two corners and in the middle in accordance with Figure 1). The measurements are presented in Figure 2.

Experimental Results
All joints failed in the tension zone either by punching shear (Figure 3a) or by fraction of the bolt (3B(L)). Non-fractured bolts from the tension zone were heavily bent (Figure 3b), fractured bolts did not exhibit necking. Due to flaking of the limewash near the weld, plastic deformations in the middle of the top flange were visible (Figure 4a). In contrast, the limewash layer was intact in the tension zone ( Figure 4b).

Experimental Results
All joints failed in the tension zone either by punching shear (Figure 3a) or by fraction of the bolt (3B(L)). Non-fractured bolts from the tension zone were heavily bent ( Figure  3b), fractured bolts did not exhibit necking. Due to flaking of the limewash near the weld, plastic deformations in the middle of the top flange were visible (Figure 4a). In contrast, the limewash layer was intact in the tension zone (Figure 4b).
The experiments showed that the average initial rotational stiffness of the joints was 17,381 kNm/rad with standard deviation of 3293 kNm/rad and the ultimate bending moment capacity of 50.48 kNm with deviation of 2.41 kNm. Moment-rotation curves are shown in Figure 5.

Experimental Results
All joints failed in the tension zone either by punching shear (Figure 3a) or by fraction of the bolt (3B(L)). Non-fractured bolts from the tension zone were heavily bent ( Figure  3b), fractured bolts did not exhibit necking. Due to flaking of the limewash near the weld, plastic deformations in the middle of the top flange were visible (Figure 4a). In contrast, the limewash layer was intact in the tension zone (Figure 4b).
The experiments showed that the average initial rotational stiffness of the joints was 17,381 kNm/rad with standard deviation of 3293 kNm/rad and the ultimate bending moment capacity of 50.48 kNm with deviation of 2.41 kNm. Moment-rotation curves are shown in Figure 5.   The experiments showed that the average initial rotational stiffness of the joints was 17,381 kNm/rad with standard deviation of 3293 kNm/rad and the ultimate bending moment capacity of 50.48 kNm with deviation of 2.41 kNm. Moment-rotation curves are shown in Figure 5.

Experimental Results
All joints failed in the tension zone either by punching shear (Figure 3a) or by fraction of the bolt (3B(L)). Non-fractured bolts from the tension zone were heavily bent ( Figure  3b), fractured bolts did not exhibit necking. Due to flaking of the limewash near the weld, plastic deformations in the middle of the top flange were visible (Figure 4a). In contrast, the limewash layer was intact in the tension zone (Figure 4b).
The experiments showed that the average initial rotational stiffness of the joints was 17,381 kNm/rad with standard deviation of 3293 kNm/rad and the ultimate bending moment capacity of 50.48 kNm with deviation of 2.41 kNm. Moment-rotation curves are shown in Figure 5.

Finite Element Analysis
Simulations were performed in the ANSYS Static Structural analysis system. With account of the symmetry, only a quarter of the model was analysed, which then was meshed with the higher-order tetrahedral ten-node SOLID187 elements ( Figure 6). After performing a mesh convergence study, different mesh sizes were used: 3 mm for bolts; 5 mm for extended parts of the endplate; 15 mm for the middle part of the endplate; 2.5 mm for fillet welds; 10 mm for the steel tube; 15 mm for the middle part of the concrete core. For the other parts, the default global mesh size of 83 mm was left. Additionally, mesh refinements were implemented near the wealds and in the tension zone of the steel tube. In these critical areas, mesh size was decreased in such a way that the endplate and the bolt hole would be meshed with four layers of elements along their thicknesses, which exceeds recommendations of Molitoris et al. [29]. Another plausibility of assessing the mesh quality is to directly compare FEA results with the experiments [30,31].

Finite Element Analysis
Simulations were performed in the ANSYS Static Structural analysis system. With account of the symmetry, only a quarter of the model was analysed, which then was meshed with the higher-order tetrahedral ten-node SOLID187 elements ( Figure 6). After performing a mesh convergence study, different mesh sizes were used: 3 mm for bolts; 5 mm for extended parts of the endplate; 15 mm for the middle part of the endplate; 2.5 mm for fillet welds; 10 mm for the steel tube; 15 mm for the middle part of the concrete core. For the other parts, the default global mesh size of 83 mm was left. Additionally, mesh refinements were implemented near the wealds and in the tension zone of the steel tube. In these critical areas, mesh size was decreased in such a way that the endplate and the bolt hole would be meshed with four layers of elements along their thicknesses, which exceeds recommendations of Molitoris et al. [29]. Another plausibility of assessing the mesh quality is to directly compare FEA results with the experiments [30,31]. The material model for the concrete was assumed to be linear elastic. In contrast, bolts and beams were modelled with elastic perfectly plastic material. Moreover, for the column tube, the multi-linear isotropic plasticity model (Figure 7) was chosen to overcome problems with the convergence, which are related to the simulation of the punching shear fracture. Geometrical nonlinearities were not considered. The model was loaded in a similar sequence as the one in the experiment: the axial force in the column was introduced by displacement control loading during the first step, and a vertical force was applied to the end of the beam during the second step. In addition to this, the bolts were preloaded during the first step. The material model for the concrete was assumed to be linear elastic. In contrast, bolts and beams were modelled with elastic perfectly plastic material. Moreover, for the column tube, the multi-linear isotropic plasticity model (Figure 7) was chosen to overcome problems with the convergence, which are related to the simulation of the punching shear fracture. Geometrical nonlinearities were not considered. The model was loaded in a similar sequence as the one in the experiment: the axial force in the column was introduced by displacement control loading during the first step, and a vertical force was applied to the end of the beam during the second step. In addition to this, the bolts were preloaded during the first step.
Fourteen frictional contact pairs were introduced: between the endplate and the column; the bolt heads and the endplate; the nuts and the inner surface of the column; the bolt shanks and the bolt holes; the concrete and the inner surface of the column. Additionally, the nut of the bolt in tension zone was split into six surfaces ( Figure 8) to even out concentration of the force. Augmented Lagrange formulation was used for all contacts with stiffness update at each iteration. The initial contact interface treatment was set to the "adjust to touch" option and all friction coefficients were set to 0.2. Fourteen frictional contact pairs were introduced: between the endplate an umn; the bolt heads and the endplate; the nuts and the inner surface of the col bolt shanks and the bolt holes; the concrete and the inner surface of the column. A ally, the nut of the bolt in tension zone was split into six surfaces ( Figure 8) to concentration of the force. Augmented Lagrange formulation was used for al with stiffness update at each iteration. The initial contact interface treatment w the "adjust to touch" option and all friction coefficients were set to 0.2. Four models with the different gaps were additionally created in order to effect of the not firm contact. The endplates were modelled on a single Gaussi ture, as the main cause of such imperfection is the difference in radii of the su contact. This difference can be positive (gap in the middle), negative (gap at co absent (zero).

Results of the Finite Element Analysis
The parametric study has shown that connections with the gaps in the mid endplate have not only the lowest values of the initial rotational stiffness, bu lowest "ultimate" bending resistance. This may be caused by the prying action i zone and the warping of the endplate in compression zone. Warping occurred closure of the gap, when the corners of the endplate deformed outwards ( Figure   318  Fourteen frictional contact pairs were introduced: between th umn; the bolt heads and the endplate; the nuts and the inner sur bolt shanks and the bolt holes; the concrete and the inner surface o ally, the nut of the bolt in tension zone was split into six surface concentration of the force. Augmented Lagrange formulation w with stiffness update at each iteration. The initial contact interfa the "adjust to touch" option and all friction coefficients were set to Four models with the different gaps were additionally create effect of the not firm contact. The endplates were modelled on a ture, as the main cause of such imperfection is the difference in contact. This difference can be positive (gap in the middle), nega absent (zero).

Results of the Finite Element Analysis
The parametric study has shown that connections with the ga endplate have not only the lowest values of the initial rotationa lowest "ultimate" bending resistance. This may be caused by the p zone and the warping of the endplate in compression zone. War closure of the gap, when the corners of the endplate deformed ou Four models with the different gaps were additionally created in order to study the effect of the not firm contact. The endplates were modelled on a single Gaussian curvature, as the main cause of such imperfection is the difference in radii of the surfaces in contact. This difference can be positive (gap in the middle), negative (gap at corners), or absent (zero).

Results of the Finite Element Analysis
The parametric study has shown that connections with the gaps in the middle of the endplate have not only the lowest values of the initial rotational stiffness, but also the lowest "ultimate" bending resistance. This may be caused by the prying action in tension zone and the warping of the endplate in compression zone. Warping occurred with the closure of the gap, when the corners of the endplate deformed outwards (Figure 9c,d).
In contrast, in the connections with the firm contact and connections with the gaps at the corners, prying did not occur due to clamping force in the initial loading stage and limited contact area in its final stages. Moreover, warping of the endplate in compression zone was also limited. This is evident from the formation of the plastic hinges, which were forming near the weld line (Figure 9a,b). This resulted in higher values of initial stiffness and bending resistance. In contrast, in the connections with the firm contact and connections with the the corners, prying did not occur due to clamping force in the initial loading sta limited contact area in its final stages. Moreover, warping of the endplate in comp zone was also limited. This is evident from the formation of the plastic hinges, whic forming near the weld line (Figure 9a,b). This resulted in higher values of initial s and bending resistance.
A further sensitivity study, which was based on Wang and Zhang's [13] obse that the increase in preloading force can increase the moment resistance of curve plated connection, not only confirmed it, but also demonstrated that imperfectio augment this phenomenon ( Figure 10). A further sensitivity study, which was based on Wang and Zhang's [13] observation that the increase in preloading force can increase the moment resistance of curved endplated connection, not only confirmed it, but also demonstrated that imperfections can augment this phenomenon (Figure 10). In contrast, in the connections with the firm contact and connections with the gaps at the corners, prying did not occur due to clamping force in the initial loading stage and limited contact area in its final stages. Moreover, warping of the endplate in compression zone was also limited. This is evident from the formation of the plastic hinges, which were forming near the weld line (Figure 9a,b). This resulted in higher values of initial stiffness and bending resistance.
A further sensitivity study, which was based on Wang and Zhang's [13] observation that the increase in preloading force can increase the moment resistance of curved endplated connection, not only confirmed it, but also demonstrated that imperfections can augment this phenomenon ( Figure 10). This is remarkable because it is well known that the initial preloading force has no effect on the ultimate capacity of the flat T-stubs [32]. Moreover, this cannot be simply This is remarkable because it is well known that the initial preloading force has no effect on the ultimate capacity of the flat T-stubs [32]. Moreover, this cannot be simply explained by the change of the centre of compression, as it was found that the lever arm of the internal forces does not depend on the preloading (Figure 11).
. Sci. 2021, 11, x FOR PEER REVIEW explained by the change of the centre of compression, as it was found that the leve of the internal forces does not depend on the preloading (Figure 11).
The position of the compression centre is mainly defined by the gap location (Figur Due to the rotation of the joint, it was not constant and moved outwards. This resulted connection with the gaps at the corners' internal lever arm being increased by 14%. In connections, the increase was smaller.  Regardless of the preloading force and the gaps, failure mode of the numeric s lation was the punching shear of the bolts (Figure 13a). It can be seen from Figure 13 bolts also experienced bending, as von Mises stress values in the half of the shank rea the ultimate stress. The position of the compression centre is mainly defined by the gap location ( Figure 12). Due to the rotation of the joint, it was not constant and moved outwards. This resulted in the connection with the gaps at the corners' internal lever arm being increased by 14%. In other connections, the increase was smaller.
l. Sci. 2021, 11, x FOR PEER REVIEW 8 o explained by the change of the centre of compression, as it was found that the lever a of the internal forces does not depend on the preloading ( Figure 11). The position of the compression centre is mainly defined by the gap location ( Figure 1 Due to the rotation of the joint, it was not constant and moved outwards. This resulted in connection with the gaps at the corners' internal lever arm being increased by 14%. In ot connections, the increase was smaller.  Regardless of the preloading force and the gaps, failure mode of the numeric sim lation was the punching shear of the bolts (Figure 13a). It can be seen from Figure 13b t bolts also experienced bending, as von Mises stress values in the half of the shank reach the ultimate stress. Regardless of the preloading force and the gaps, failure mode of the numeric simulation was the punching shear of the bolts (Figure 13a). It can be seen from Figure 13b that bolts also experienced bending, as von Mises stress values in the half of the shank reached the ultimate stress.
Regardless of the preloading force and the gaps, failure mode of the nume lation was the punching shear of the bolts (Figure 13a). It can be seen from Figure bolts also experienced bending, as von Mises stress values in the half of the shank the ultimate stress.   Figure 14 shows that compression and tension zones of the endplate behaved differently during the experiments as well as in the FEA. When the middle of the endplate was in firm contact with the column, it increased the stiffness of the compression zone and lowered the stiffness of the tension zone. Such contact significantly reduces the warping of the endplate in the compression zone and does not allow the prying action to occur (Figure 15a). Furthermore, in the tension zone, connections with the gaps in the middle have a higher initial stiffness than firm contact. It is plausibly due to the prying action, which causes favourable deformations, increasing the clamping (Figure 15b).

Comparison of the FEA Results with the Experiment
i. 2021, 11, x FOR PEER REVIEW Figure 14 shows that compression and tension zones of the endplate behaved ently during the experiments as well as in the FEA. When the middle of the endpla in firm contact with the column, it increased the stiffness of the compression zo lowered the stiffness of the tension zone. Such contact significantly reduces the w of the endplate in the compression zone and does not allow the prying action to (Figure 15a). Furthermore, in the tension zone, connections with the gaps in the m have a higher initial stiffness than firm contact. It is plausibly due to the prying which causes favourable deformations, increasing the clamping (Figure 15b).

Comparison of the FEA Results with the Experiment
The shape of the graphs also suggests that variation of imperfections took pla diminishment of the gaps in the corners was led by the closing of the gap in the m Furthermore, higher experimental strength of the tension zone ( Figure 14) i parison to the FEA was probably caused by the adopted engineering stress-strain ( Figure 7). Nevertheless, even with the engineering stress-strain curve, the graphs merical simulations are a bit higher than the experimentally obtained ones (Figure the true stress-strain curve was adopted, it would increase the moment resistance connection even more, so one would need to decrease the preloading force in th Therefore, as long as the experimental bolt force is unknown, the FEA becomes a fitting problem-picking the proper combination of material model, imperfection, c definition, and preloading force.    The shape of the graphs also suggests that variation of imperfections took place, i.e., diminishment of the gaps in the corners was led by the closing of the gap in the middle. Furthermore, higher experimental strength of the tension zone ( Figure 14) in comparison to the FEA was probably caused by the adopted engineering stress-strain curve ( Figure 7). Nevertheless, even with the engineering stress-strain curve, the graphs of numerical simulations are a bit higher than the experimentally obtained ones ( Figure 16). If the true stress-strain curve was adopted, it would increase the moment resistance of the connection even more, so one would need to decrease the preloading force in the FEA. Therefore, as long as the experimental bolt force is unknown, the FEA becomes a curve-fitting problem-picking the proper combination of material model, imperfection, contact definition, and preloading force.

Summary and Conclusions
The article presents the results of an experimental investigation of the joints between a steel beam and a circular CFST column, as well as the FEA parametric study, in which effects of the gaps and preloading force of the bolts were analysed. Performed experi mental investigations and FEA results are in good agreement and correlate. Therefore based on the obtained results, the research conclusions are drawn.

Summary and Conclusions
The article presents the results of an experimental investigation of the joints between a steel beam and a circular CFST column, as well as the FEA parametric study, in which effects of the gaps and preloading force of the bolts were analysed. Performed experimental investigations and FEA results are in good agreement and correlate. Therefore, based on the obtained results, the research conclusions are drawn.
The behaviour of the joints with the curved T-stubs is exceedingly complex and heavily dependent on imperfections as well as on the preloading force of the bolts. Gaps in the middle of the endplate in compression zone can decrease the initial rotational stiffness and plastic bending resistance of the joint, although gaps at the corners can increase the initial rotational stiffness. In both cases, the magnitude of the change in stiffness and resistance heavily depends on the bolt preloading force.
As the initial preloading force of the bolt can increase the ultimate bending resistance of the joint, it should be properly controlled and measured. Without such control the FEA becomes a curve-fitting problem and exhibits four parameters: material model (σ-ε); contact definition; initial imperfections; and bolt preload. Yet, solving the problem in this way is somewhat meaningless since it cannot be used to predict the results.