1 Introduction

In the project's design, one of the most important attention is the safety of structures. Progressive collapse is one of the failure forms of constructions that has excited much interest in current decades (Ch-Salmasi & Sheidaii, 2017). "Progressive collapse is defined as the spread of an initial local failure from element to element, resulting eventually in the collapse of an entire structure or a disproportionately large part of it" (ASCE, 2016). A list of different definitions for progressive collapse and a deep discussion on the nomenclature is reported by Kiakojouri et al. (2021).

The criteria checking of the structures against this phenomenon have gradually been inserted in the design codes. The concentration of engineering society stated in this event via the Ronan Point building's collapse (Ellingwood, 2006). More consideration of structural compliance to abnormal loads was attracted after the WTC's failure in 2001 (NIST, 2005, Kiakojouri et al., 2020). In the pushdown analysis, beam-column connections have a primary duty to resist gravitation loads in the removed column's region (Sadek et al., 2010). The evaluation of connections behavior corresponding to the forces transfer is essential in assessing structural resistance under a column removal (Brett & Lu, 2013).

Many experimental and theoretical investigations have been done to evaluate the beam-column connection performance in the progressive collapse of steel moment frames. Karns et al. (2009) studied the behavior of beam-column connections under the air blast loading. In their study, side-plate and typical welded connections were examined. Lee et al. (2010) examined the behavior of the tensile axial force-moment interaction of the welded moment frame under a column removal scenario. They proposed a parallel plastic hinge (PPH) model for the beams. Comparisons of analysis results demonstrated this method is reasonably reliable and exceptionally efficient. Sadek et al. (2011) investigated the performance of two kinds of steel moment connections against progressive collapse with experimental and numerical simulations. The ten-story structure models were special moment frames (SMF) and intermediate moment frames (IMF). The results displayed a similar failure mode in the structural systems. The collapse of models happened at the bottom beam flanges.

In another research, Sadek et al. (2013) and Lew et al. (2013) assessed two kinds of rigid connections under column removal. The results demonstrated that the connections withstand considerable axial force and resist against the employed vertical loads. Yang and Tan (2012a, 2012b, 2013) studied the progressive collapse performance of different bolted connections. Li et al., (2013, 2015) investigated the robustness and fracture mechanism of various kinds of rigid connections under column removal. They demonstrated that the models fundamentally maintained the applied force by the flexural action in the initial analysis steps.

Wang et al. (2016) evaluated the progressive collapse behavior of three connections with a theoretical and experimental assessment. These investigations concentrated on symmetrical-span structures. The results of the analysis showed that double full-span modeling could fully cover the failure progress of the connections. Zhong et al. (2017) assessed the behavior of three stiffness connections. The results of the analysis showed that with vertical-displacement increasing, the flexural mechanism of structures shifted towards the catenary action mode. Meng et al. (2018) assessed the welded unreinforced flanges–bolted web (WUF-B) connection's behavior under the column removal scenario. Three models were evaluated with the different span ratios of 1:0.6, 1:1.0, and 1:1.4. The results of the analysis showed that the unity span ratio displays better performance than other span ratios. The load transfer of structure with a span ratio of 1:1.0 is more desirable than the nonequal span ratios.

As mentioned, prior studies only concentrated on the performance of one or two of several conventional beam-column connections against progressive collapse. Moreover, the information gathered around general comparisons of the conventional connections' progressive collapse behavior is very poor. Hence, this investigation evaluated double full-span samples' behavior with various kinds of conventional beam-column connections under single-column removal. The modeling structures were analyzed by the nonlinear method. The load–displacement curves, fracture modes, Von-Mises stresses, and other theoretical outcomes were described for the different studied connections.

2 Finite Element Model Explanation

The rigid welded steel beam-column connections are one of the most common connections between rigid connections. In this study, the nonlinear finite element (FE) method is utilized to simulate and assess the behavior of welded beam-column connections against progressive collapse. The advanced commercial FE analysis package ABAQUS (2014) was utilized for this purpose. The design of the columns, beams, and beam-column connections was following the seismic design basis of AISC341 (2016a), AISC360 (2016b), and AISC358 (2016c) codes. Designing gravity and seismic loads were assessed based on criteria of ASCE7 (2016) code; the design acceleration parameters S1, at a period of 1 s, and SS, at short periods, were considered 0.38 g and 0.81 g, respectively. The occupancy of structures was considered commercial and is located in an area of moderate seismicity. The Poisson's ratio was considered 0.3, and grade S355 steel was used for the specimens' beam and column elements. The basic nonlinear material property of steel was simulated using the isotropic hardening model with the von Mises yield criterion. The fundamental material properties, including the modulus of elasticity, yield strength, and ultimate strength, were obtained from Wang et al. (2016) in which the tensile coupon tests according to the relevant specification (ASTM A370 2002) were carried out. Each set of results in the table was based on three coupons' average values for each part. The considered material properties for specimens have been summed in Table 1.

Table 1 Material properties of samples
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The fracture modeling of steel is a complicated issue, inconsistent approaches were normally employed in various previous investigations. In the current numerical study, ductile and shear damage in addition to elastic and plastic property were defined (Wang et al., 2016). For ductile damage fracture strain and stress triaxiality parameters were used. For shear damage fracture strain and shear stress ratio were defined.

These buildings were symmetrically designed with four spans with 4 and 7 m length in two main directions. The six stories were considered for the analytical models with 3.5 m height for each story. In this study, four kinds of rigid connections, including WUF-W, RBS, WFP, and I-W, were chosen, as shown in Fig. 1. The geometrical characteristics of various elements of four samples are exhibited in Table 2.

Fig. 1
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Details of beam-column connections for four specimens: a WUF-W; b RBS; c WFP; d I–W

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Table 2 Geometry and dimensions of samples (dimensions in mm)
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The performance of investigated specimens against progressive collapse was assessed by employing rising vertical displacement at the central damaged element's top node, considering the catenary action influences. A quasi-static analysis is carried out, and because the process of the experimental testing and numerical simulation is static, the strain rate influences are not considered (Qin et al., 2016). The samples' performance was assessed by increasing the gravity displacement to recognize the samples' behavior against progressive collapse. Throughout the models' analysis, the damaged column's vertical displacement increases until the specimens' ultimate capacity are achieved.

In the current study, the explicit time integration method was adopted to investigate specimens' progressive collapse behavior. The adopted time period for the "dynamic" analysis is 1 s. Automatic incrementation with global stable increment estimator is used in the analyses. Vertical downward displacement with the amplitude of 0.5 m is applied, however, the real amount of applied displacement is related to the onset of the failure in the models, which means, analyses continue until the occurrence of failure and separation in the connections, i.e., usually smaller than theoretical input. Abaqus predefined default mass scaling option is utilized.

After a middle column removed, the structure divided into the relevant and irrelevant spans according to the alternate path method in unified facilities criteria (UFC 2016) and general service administration (GSA 2013) guidelines as described in Fig. 2 (Barmaki et al., 2020). A double full-span assembles with two neighborhood beams created, as shown in Fig. 2a (Zhong et al., 2017). Figure 2b shows the standard finite element model of the specimens. In an actual frame construction with a lateral load-bearing system, the column's horizontal relative deformation in a given layer is generally most negligible, and the side columns' inflection points can be simplified to fixed hinge constraints. Therefore, the side columns' ends were constrained by horizontal and vertical hinges in the current study. In Fig. 2a, the side column length (Lc) is considered 350 cm. The axial restraints from the adjacent beams and diaphragms were considered, and out-of-plane movements of the affected beams were constrained by defining the appropriate axial constraints at the end nodes of the beam element, and the failed column can only move in the gravitational direction.

Fig. 2
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Beam-column assembly

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The failed column is removed from the analytical structural model, and the displacement-controlled pushdown analysis is applied to all specimens. The loading was implemented under displacement control using a sufficiently slow rate to eliminate inertial effects. In this research series, the vertical displacement is applied at the central damaged column's top node in the gravity direction, and the axial load is measured in the two adjacent beams. Displacements are continuously increased until the failure of elements occurs. A gravitational load combination of 1.2DL + 0.5LL, where DL and LL are the dead loads and live loads as defined in the UFC (2016) guidelines, was applied to all models. Displacements are continuously increased when the fracture of elements occurs. The concentrated vertical forces for every four connections were determined in each stage of analysis. Finally, the load factor, ultimate load per proposed UFC load, was assessed for each connection. The calculated value for the entire gravitational load of the UFC load combination is 188KN. Figure 3 presents the complete detail of the three-dimensional theoretical model and schematic shape of specimens.

Fig. 3
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Geometry and dimensions of numerical specimens (dimensions in mm)

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The details of the numerical models are presented herein. As proposed by Yang and Tan (2012a, 2012b) and Kiakojouri and Sheidaii (2018), the explicit solver can utilized for the progressive collapse analysis. When kinetic energy was maintained within 10% of internal energy (Barmaki et al., 2020) explicit solver can be used for static problems. Solid element type C3D8R, which are 8-node linear brick elements with reduced integration and hourglass control, was used to create all models' components. The mesh size of the beam, column element, and the areas nearby the beam-column connection considered 4 mm and 2 mm, respectively, as shown in Fig. 4. In this research, the "tie" contact interaction was utilized between the column and beam flange, because no weld fracture happened during the previous experimental studies (Wang et al., 2016).

Fig. 4
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Schematic shape of the finite element models

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The failure of mild steel when loaded in quasi-static tensile loads is characterized by a ductile fracture mechanism (Ribeiro et al., 2016). The fracture is called ductile when it results from void nucleation followed by their growth and coalescence. The stress–strain curve with consideration of failure mode of the progressive collapse was defined for all models is shown in Fig. 5. After the D = 0 point, the dashed curve represents a generic material response without any damage, while the solid line corresponds to the material softening. In this figure, σu and \({\bar{{\varepsilon }}}_{0}^{\mathrm{pl}}\) are the ultimate strength and equivalent plastic strain at the onset of damage, and \({\bar{{\varepsilon }}}_{\mathrm{f}}^{\mathrm{pl}}\) is the equivalent plastic strain at failure. In this study's analysis, when the elements failed, these elements were removed from all over the sample model's meshing.

Fig. 5
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True stress–strain curve with progressive damage (Barmaki et al., 2020)

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3 Verification Results

In the paper of Wang et al. (2016), the behavior of beam-to-SHS column connections has been investigated under the column removal by full-scale experimental tests and numerical simulation. A sample structure with three different types of connections was analyzed in their research. The three tested samples included I-W, I-WB, and ST-WB connections. Finally, the vertical force–displacement, axial force–displacement curves, and strain in separate points of the considered cross-section were extracted, and the results were analyzed. In the current study, to validate the simulations' results, Wang's et al. (2016) model was modeled in the ABAQUS with the I-W connection's complete details, according to Fig. 6. Finally, the performance of the I-W connection was calculated against progressive collapse via the dynamic explicit analysis.

Fig. 6
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Details of specimen with I–W connection

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The vertical displacement-force diagram, alongside the I-W connection failure in the main structure, examined by Wang et al., is illustrated in Fig. 7 in a remodeled arrangement by the author that verified the current study's results. In Fig. 7a the analysis continues to much higher load compared to the experimental results, such a differences in comparison of experimental and numerical results is very common. In the experimental research reported by Wang et al., limitation of the experimental setup prevents the development of complete curve, however, in the current study, which is numerical, there is no such a limitation and the complete route can be monitored. The fracture mode of the samples is also an exemplified validation of current study results, According to Fig. 7c. It should be noted that slight differences between two results is due to the differences in mesh size, otherwise, the fracture modes are reasonably similar.

Fig. 7
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Diagrams and FE results for specimen with I–W connection

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4 Analysis Results

In the present research, numerical simulations were conducted, and the local failures of whole samples in the different beam-column connection areas against progressive collapse are investigated. A displacement-controlled analysis with the step-by-step increase of the imposed loads and a nonlinear quasi-static method was carried out to evaluate various connections' performance. Based on the previous research (Lee et al., 2010; Li et al., 2013; Sadek et al., 2010, 2011; Wang et al., 2016; Yang & Tan, 2013), the FE model was used to model and analyze the sample models' susceptibility against the progressive collapse. The sample model's analysis results with a WUF-W connection are presented with full details in the following. Then, the comparison curves of results for different investigated connections have been demonstrated and discussed.

The load–displacement curves, fracture modes and Von-Mises stresses are used for comparison. In this section, vertical load refer to the vertical concentrated load (corresponds to the imposed displacement) that applied at the central damaged column's top node in the gravity direction, and the axial force is the measured value in the two adjacent beams. It should be noticed that, while displacement control approach is adopted, internal forces developed within the subframe can be calculated using equilibrium equations based on the techniques reported by Wang et al (2016), such techniques is adopted and used to convert the applied displacements to associated forces.

4.1 Results of the Sample Structure with WUF-W Connection

For the nonlinear quasi-static analysis with the WUF-W connection, the vertical displacement-load curve is plotted in Fig. 8. Then, the diagrams of axial load–displacement, load factor-displacement, stress conditions, and fracture modes /sequence are demonstrated in Fig. 9 to discover the structure's susceptibility with the WUF-W connection. The vertical displacement of the failed column is the same displacement of these diagrams. In this research, the load factor was described as the gravity load ratio applied in each step of the nonlinear analysis to the whole gravity load specified in the UFC guideline. Therefore, whenever the load applied to the sample structure is equal to the total value of the progressive collapse load, the load factor equals 1. In fact, if the displacement-load factor diagram for a structure under the gravity load of the progressive collapse can grow increasingly to the load factor of the unity, the structure can withstand the entire collapse load and will be resistant against the progressive collapse. Otherwise, the structure will not have the required strength against progressive collapse.

Fig. 8
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Vertical load–displacement curve of WUF-W sample

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Fig. 9
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Step by step collapse results of the of WUF-W sample

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According to Fig. 8, the displacement smaller than 2.02 cm (A1 point), which corresponds to the vertical load of 117.62 kN, the linear performance is shown in the sample structure with the WUF-W connection. At this point, the load factor is 0.66. This point illustrates the elastic flexural response in the initial steps of analysis. From point A1, the specimen enters the nonlinear range of the material. As shown in Fig. 9a, in the displacement of 6.8 cm at the A2 point, and after the force reaches load factor of 1.02, the local buckling appears in the beam top-flange, probably due to the combination of bending and minor compression. This minor compression force results from the compressive arching effect. As shown in Fig. 8 at point A3, up to the vertical load of 219.81 kN (load factor of 1.23), the structure could withstand the equivalent load of the progressive collapse. The displacement of this point is 13.52 cm. A flexural fracture of the bottom flange has occurred at a distance of 2.5 cm from the column. After this point, a sudden drop in the strength has been observed. Considering that the maximum load factor tolerated is higher than 1, the WUF-W connection has higher strength. The column starts to bend by increasing the displacement, and cracks are formed near the access hole. According to Fig. 9c, at point A4, these cracks immediately propagate in the beam web direction and the shear tab near the weld, resulting in a drop in axial compressive strength. Continuing the analysis process of the displacement controlled, the growth of cracks and joining of the cavities and the shear fracture caused by the shear band localization, the shear tab failed, causing a total collapse in the connection by a 41.9 cm displacement and a load factor of 0.74 according to Fig. 9d at the point A5.

At the initiation of the analysis process, a minor compressive force was observed in the beam. With increasing displacement and the continuation of the analysis process, the axial compressive force was reduced, and the axial tensile force replaced it. The axial force–displacement curve and the relevant P/Py-displacement at a distance of 2.5 cm from the column are shown in Fig. 10 in which Py = Aσy is the yielding axial force of the beam. At a displacement of 39.9 cm, the maximum axial force developed at the connection is about 0.7Py.

Fig. 10
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axial force–displacement curve for the WUF-W connection

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This indicates the conversion of the structure behavior from bending to the tensile mode in large displacements, and as a result, catenary action is observed in the sample. In other words, during the displacement increasing from 25.8 to 40.6, and the Growth of failure in the shear tab and pre-failure in the entire sample, the axial force has been increasing as shown in Fig. 10, which has occurred after the failure of the beam bottom-flange and the failure of the shear tab.

Based on the results of the analysis, it is inferred that the collapse of the WUF-W sample occurs under the progressive collapse load in the connection during the large displacements; however, such a force is not usually considered in the design of the connection under the customary gravitational and seismic loads. It should be noted that up to this load level, the column element has not involved the plastic behavior and has remained in the elastic domain, which indicates initiation and propagation of the failure in the beam elements and their connections.

4.2 Resistance of Various Connections Against Progressive Collapse

The sample model with three kinds of the prequalified fully restrained beam-column connection was created as finite elements and analyzed under single-column removal. To specify the capacity of studied connections against progressive collapse, the results of all connections in the framework of the vertical displacement–force, axial force–displacement, load factor–displacement, and stress-displacement curves are drawn in this section. Figure 11 demonstrates the comparison results of vertical displacement-force and load factor–displacement curves of various connections. Suppose the ductility coefficient is defined as the ratio of the displacement corresponding to the maximum load factor per the specimen's yield limit (Barmaki et al., 2020). The WFP connection has a load factor of 1.55 and a ductility coefficient of 11.86. In that case, this connection has the highest strength and ductility response compared to the other connections. The RBS connection with the load factor of 1.06 showed undesirable strength behavior while the ductility in this connection approximately equals the WFP. The results summary of four connection samples against progressive collapse is given in Table 3.

Fig. 11
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Vertical load–displacement and load factor-displacement curve of the various connections

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Table 3 Results of investigated connections against progressive collapse
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The axial force–displacement curves (and the ratio of the axial force of the considered cross-section on the yielding axial force (P/Py)) of the beam-column connection are compared in Fig. 12 for the four various types of connections. In all samples, in small displacements, the axial force is very low, i.e., an average of 150 kN, and with the displacement increasing to 20 cm, the amount of axial force increases in all diagrams with a high slope and actually, catenary action is observed in all models. The maximum axial forces generated in various connections are in the range of 600–1050 kN, that is, (0.4–0.7)Py, which are very large tensile axial forces that appear in the connections under progressive collapse. Since the connections' design is typically performed only for the bending and shear forces resulting from gravitational and seismic loads, in the building's design exposed to the abnormal loads, this important axial force in the connections could be the origin of the collapse. Therefore, this load should be considered in the systematic design process of these elements.

Fig. 12
figure 12

Axial force and P/Py—displacement curve of the different connections

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The other three connections' performance in the most critical analysis steps, namely, yield load (corresponding to the initiation of yielding in the connection), ultimate load, and the connection's fracture, is shown in Fig. 13. As clearly demonstrated in Fig. 13, in all connections, the bottom flange is damaged from the lower side due to the combined forces of the bending moment and the axial tensile force. After the fracture of the bottom flange, the beam web and connected plates are damaged in the process of analysis. Also, in the WFP connection, the beam top-flange at the ending area of the connection zone undergoes high stress, but the bottom flange and then the shear tab is exposed to fracture in this connection.

Fig. 13
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Connections performance in various analysis steps

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In Fig. 14, the Von-Mises stress-displacement diagrams of the specimens in the beam bottom-flange, web, and top-flange at a distance of 2.5 cm (0.083db) from the beam-column connection are compared for the four types of connections. As shown in Fig. 14a, the beam top-flange is under compressive stress in all connections before the bottom flange fracture. With the displacement increasing and the catenary action's appearance, tensile stresses appeared in the beam flange, and the web in the considered section undergoes tensile stresses. As shown in Fig. 13, in the WFP connection, due to the existence of top and bottom plates in the flanges, the stress is higher outside the connection at the plates' end than other areas of the connection. So, the stress in the web near the connection area will be lower than the other regions, i.e., 240 MPa, as shown in Fig. 14b, and the RBS connection sustains maximum tensile stress of 573 MPa. The beam bottom-flange, which in all connections with the increasing displacement of the fracture initiated from this region, is under tensile stress at all analysis steps. Between all the studied connections, the maximum stresses were observed in the RBS connection. The WFP connection has the best performance due to removing critical stresses from the connection area and the inducement of the lowest compressive and tensile stresses in this critical area. This is despite the fact that compared to the other connections, the RBS connection has shown the weakest performance during the progressive collapse under excessive stresses and early fracture, and the vertical load. The beam bottom-flange has undergone tensile stresses at all stages of the analysis, as shown in Fig. 14c.

Fig. 14
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Stress–displacement curve of the different connections

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5 Conclusion

When the progressive collapse occurs, the buildings' beam-column connections should be strong enough and be able to make a "bridge" over between the beams at the two sides of the column. Since the prequalified connections contain a large number of the connections used in the steel structures with lateral load resisting system of the moment frames, it is necessary to accurately analyze the collapse process and the retrofitting method of these connections. In this study, to obtain the structures' performance, a numerical analysis of the finite element was done on the sample structure with four different types of connections under the progressive collapse. The samples' modeling was carried out in three columns and two beams (double full-span) with various connections. The nonlinear quasi-static analysis was performed for the numerical simulation. Comparisons between the numerical analysis and the experimental results were performed to validate the models created in this study. The results showed a high consistency between the finite elements and experimental tested results.

The results of the analyses demonstrated that each of the four investigated connections showed sufficient strength under the middle column removal against progressive collapse. The WFP connections exhibited the best behavior due to the presence of top and bottom plates in the top and bottom flanges, the removal of critical stresses from the connection area, and the inducement of the minimum compressive and tensile stresses critical area. The RBS connections showed the weakest behavior compared to other connections under the progressive collapse due to absorbing excessive stresses, fast fracture and tolerating minimum vertical loads.

In all investigated connections, under the internal moments and tensile axial force resulting from the progressive collapse, the initial damage began to propagate from the beam flange in the lower area of connection. The fracture grew with the displacement increasing in the beam-connected plates. The plastic strain accumulation occurred around the access hole in most studied connections, and the failure was initiated from these areas. In the large displacements, samples' behavior changed from bending to tensile mode with the appearance of catenary action, and the considerable tensile axial force at this level led to a general failure in the connection area. This occurs in the structure after the bottom beam flange's failure and the fracture beginning of the beam web and shear tab.

In general, the connections are only designed for an internal moment and shear forces resulting from gravitational and seismic loads. In the building's design exposed to the abnormal loads, the critical axial force is generated in the connections that could be the collapse's origin. Therefore, this load should be considered in the systematic design process of these elements. Determining the exact amount of this axial force depends on considering various parameters such as connection type, amount and type of loadings, span length, steel type, etc. The study of each of these parameters has been postponed to future research.