Design approach to bolted end-plate vertical-slits RWS connection

Extensive research has been carried out on steel moment frames to improve the cyclic performance of seismic resisting connections with reduced beam section (RBS). The RBS connections are conventionally known by the radial reduction of the beam flange. Where the contribution of the beam flange to the flexural resistance is greater than that of the beam web, some researchers have proposed reduced web section (RWS) connections, instead. The present study dedicates to the RWS connections with vertical-slits (VS), as a cost-effective alternative with multiple design parameters. This paper aims to obtain proper ranges for the geometric design parameters of the VS-RWS connection. In this order, two full-scale specimens of the bolted end-plate VS-RWS connection were experimentally tested under the SAC cyclic loading to evaluate the performance of connections, and then a parametric study was carried out using the verified numerical models. The parameters consist of the distance between the column face and the beginning of the reduced region, the length of the reduced region, as well as the depth and width of the vertical-slits. Based on the results, certain recommendations for the ranges of the geometric parameters of VS-RWS have been suggested. In order to obtain the story drift of the frame caused by the VS-RWS beam flexural deformation using the conjugate beam method, the original VS-RWS was replaced with an equivalent constant-cut reduced beam section (CC-RBS). At last, a simple design procedure for VS-RWS connections was provided according to AISC-358.


Introduction
Prior to 1994, the moment resisting frame system had been widely used in the construction industry in seismic zones. Due to the Northridge and Kobe earthquakes in 1994 and 1995, many traditional moment connections performed poorly in terms of providing the expected ductility, and the connections were subject to a brittle fracture. One of the ways to improve the performance of these connections is to weaken the beam section, based on the principle of the strong column-weak beam by reducing the beam flange or web to form a plastic hinge in the reduced region. This region, which acts as a fuse, creates some proper ductility to prevent the brittle damage to the connection. Since the web of the beam makes a small contribution to its flexural rigidity, the weakening of the beam web instead of the flange can be a good option for reducing the beam cross section as long as it provides adequate shear capacity.
In some types of reduced web section (RWS) connections, the beam web is weakened by creating a rectangular, circular, or vertical elliptical opening ( [1][2][3][4][5]). In these connections, a relatively large void is required to reduce the stresses over the beam-to-column CJP welding region, yet the large void causes the local buckling instability at the cutting edges of the beam web [1,4]. In order to overcome this instability, some researchers have proposed and investigated the accordion/tubular-web RWS connections in which the flat web is replaced by a corrugated-web [6][7][8][9]. Nadi et al. [8] showed that the Holed Tubular-Web (HTW) which is a kind of accordion-web, decreases the fatigue phenomenon caused by the stress concentration in the connection of the tube and beam. Besides, Zahraei et al. [6] demonstrated that the tubular-web RWS provides a better condition than the accordion-web RWS in terms of lateral-torsional buckling stability of the beam and low-cycle fatigue. In any case, although this type of connection behaves well in terms of plastic hinge formation in the reduced region, its construction is relatively difficult, costly, and time-consuming.
In addition to local instability, another drawback of the large openings is the early failure of the beam section due to the significant effect of the shear-flexure interaction in the T sections remaining at the top and bottom of the large opening. Therefore, in order to reduce the depth of the web perforation, Davarpanah et al. [10,11] introduced and studied the horizontal elliptical-shaped RWS connections. They suggested ranges for the purpose of choosing the geometric variables of the elliptical-cut and the reduced region to ensure that the formation of the plastic hinge is away from the column edge. Their research also illustrated that reducing the web instead of the flange reduces the lateral-torsional instability greatly. Some researchers turned toward the RWS connections with a multi-hole web instead of a single large-opening web [12][13][14]. Maleki and Tabbakhha [12] studied the reduced beam section (RWS) connections with the slotted beam web named slotted-web-reduced-flange (SWRF) connection. Hedayat and Celikag [13] proposed RWS connections with two parallel horizontal voids in the beam web and presented a step-by-step design process for this purpose.
In order to prevent the beam flange/web buckling, they had to add web stiffeners and two tubes in the center of voids. In a recent study, Nazaralizadeh et al. [14] have introduced a new type of the RWS connections with vertical-slits (VS) (Fig. 1) as an easy-to-build alternative that has multiple geometrical parameters for the control of the connection behavior. Their comparative study of the bolted end-plate VS-RWS connection with the conventional RBS and with various cut-shaped RWS connections exhibits good ductility and flexural strength, as well as low-localized stresses in the end-plate welding line of the proposed connection.
This research aims to perform experimental tests, numerical validation, and parametric study to obtain the knowledge required to control the performance of VS-RWS connections and to find the optimum ranges for the geometrical parameters of the reduced region using a design approach. This paper is structured as three following parts. The first part is dedicated to the experimental tests and performances of two full-scale VS-RWS connection specimens with different configurations of the reduced region. The second part is devoted to the development of a finite element model (FEM) for the VS-RWS connection to simulate the cyclic response of various VS-RWS connections with different geometrical parameters of the reduced region.
Based on the results, the optimum ranges for the RWS region parameters are recommended.
In the third part, the original VS-RWS is replaced with an equivalent constant-cut depth RBS (CC-RBS) to obtain the story drift of the frame caused by the VS beam flexural deformation using the conjugate beam method. Last of all, for the design purpose, a simple design procedure for VS-RWS connections according to the optimum ranges is provided for the reduced region.

Test specimens
The beam and column profiles of the experimental test specimens were taken from the seventh floor of a 10-story library building in a high-seismic region. The structural building was analyzed and designed by ETABS [15] under the gravity and lateral loading. The firststory height was 4.5 m and the other stories were 3 m. The dead and live loads were 0.5 and 0.3 t/m 2 respectively. The lateral load resisting system was chosen as a special moment frame.
Based on the design, the IPE270 and IPB200 profiles with nominal yield and ultimate stresses of 240 and 370 MPa were employed for the beam and column, respectively. The lengths of the specimen beam and column in experimental tests were considered as 1460 and 1075 mm, respectively, according to the previous studies [16][17][18]. The connection details, including the end-plate, its bolts and holes, the web plates, the continuity plates, and the welds were designed according to the LRFD method. Two web plates were installed on either side of the column web, and the continuity plates were welded to the web plates and column flanges.
Thus, the panel zone could establish the strong-column-weak-beam strategy [17].
In this study, two full-scale bolted end-plate RWS connections with vertical-slits named VS-RWS1 and VS-RWS2 were built and tested. The details of the test specimens have been provided in Fig. 2. In this figure, "a", "b", "cʹ", and "g" are the distance between the column face and the beginning of the reduced region, the length of the reduced region, half of the vertical-slit depth, and the total cutting size along the beam length for a single cut or multiple discontinuous cuts, i.e. b ≥ g, respectively. The reduced region parameters, "a" and "b", for the case of VS-RWS1, as well as "b" for the case of VS-RWS2 were selected based on the criteria required by FEMA-350 [19] for conventional RBS connections (0.5bf ˂ a ˂ 0.75bf , The mechanical properties of materials were determined according to ASTM A370 [20] standard tensile tests [14] (Table 1). In many studies on connections [1,18,[21][22][23] the cyclic loading is based on the SAC [24] protocol; hence, in the present study, this protocol has been used [14]. The quasi-static cyclic load has been applied to the free-end of the beam by the displacement control method at a rate of 0.02 [16].   A local buckling occurred at the beam flange, small tears were observed at the strips between some of the slits, and therefore the flexural strength was reduced.

0.05
A slight deformation was seen around some slits. Also, a local buckling were occurred at the beam flange and web in the RWS region.
The buckling instability of the beam was increased, and some slit strips began to rupture.

0.06
The beam deformations were increased due to the local buckling, such that the out-of-plane deformation at the web around the upper and lower slits was clearly visible. The end-plate began a slight deformation.
The rupture in the slit strips was expanded toward the beam flanges.

0.07
Due to the large deformation of the beam, some slits were damaged. The rupture occurred in the upper and lower slits and extended obliquely toward the beam flange.
A full shear rupture occurred at the slit strips.

Bilinear curves
In this paper, the ductility ratio and effective stiffness have been calculated according to FEMA440 [25,26]. For this purpose, at first the backbone diagram was drawn using the envelope of the hysteresis curve. Then an equivalent bilinear curve was fitted to the backbone diagram (Fig. 5). Only the ascending segment of the hysteresis curve was used to obtain the backbone diagram and the decreasing portion was ignored ("m" is the slope of the "d" line and is positive). To equalize the two diagrams based on the concept of equivalent energy, the area underneath the bilinear diagram must be equal to that of the backbone diagram. This means that the area of the yellow and red hatched parts would be equal in Fig. 10. The point C in this figure is considered as a yield point. By using these diagrams, the yield flexural moment (My) and the yield rotation angle (θy), as well as the ultimate flexural moment (Mu) and the ultimate rotation angle (θu) of the tested connections are determined and indicated in Table 3. According to the available data, the effective stiffness of the connections as the line slope of the elastic zone in the bilinear diagram and the ductility ratio for the two connections are presented in Table 3. This table shows that both the ductility ratio and the effective stiffness of the VS-RWS1 are higher than those of the VS-RWS2.  Table 3. Ductility ratio and effective stiffness of tested specimens

Lateral-torsional buckling of beam
The lateral-torsional buckling phenomenon in the connections has a great effect on their cyclic performance; thus, this section is devoted to the investigation of this phenomenon in the proposed connection. One of the disadvantages of the common RBS connections is the large lateral deformation in the beam [27,28]. The experimental observations of the two tested connections showed that although the beam is weakened by vertical-slits, as can be seen in Fig. 6, at the end of the tests, the significant lateral-torsional buckling does not occur in the connections, which is one of the important advantages of connections of this type.

Stiffness degradation
Many structural steel components will exhibit some level of stiffness degradation when subjected to cyclic loading. To evaluate the stiffness degradation of connections under cyclic loading, the stiffness degradation coefficient (Sj) can be utilized as follows [22]: (1) where pj i is the maximum load applied at the beam tip in the i th cycle; uj i is the maximum rotation of the i th cycle; and "n" is the number of loading cycles. The diagram of Sj versus the rotation angle (θ) at different rotations is plotted in Fig. 7. According to this figure, by increasing the rotation angle, the stiffness degradation coefficient on the positive and negative sides is decreased. In addition, the flexural stiffness of VS-RWS2 is lower than that of VS-RWS1. This originates from the local buckling and deformation of the VS-RWS2 beam.

Numerical analysis and verification
The bolted end-plate RWS connections with vertical-slits were simulated in ABAQUS [29].
To achieve the accurate numerical results, it was necessary to include the slipping phenomenon in the modeling of the bolted junctions by introducing the contact interaction between the surfaces of the components. Fig. 8 shows the interacting surfaces and the meshing details of the connection model. The details of FEM modeling, including the element type, the geometric and material characteristics, the analysis type, the bolt pre-tension force, the type of contacts and the mesh size, were in accordance with Ref. [14].

Rotation angle (rad)
Interaction1 hinge in the predicted region, the proper bending capacity and the energy dissipation, the connection showed acceptable seismic behavior. Since the drop in the flexural strength of the VS-RWS1 connection at 0.04 rad is less than 20% of the cross-section plastic moment, this connection has met the requirements of the special bending frames in AISC-358 [30].
According to Fig. 9, the flexural capacity of the VS-RWS2 connection is equal to 104.2 kN, which is about 37% less than that of the VS-RWS1 connection. The bending moment of VS-RWS2 connection at 0.04 rad is less than 80% of the cross-section plastic moment.
Accordingly, it lacks the AISC-358 [30] criteria for being utilized in special moment frames.

Parametric study
To achieve the appropriate ranges for the reduced region parameters, including "a", "b", "cʹ" and "g", various numerical analyses were conducted. Each parameter is introduced as a percentage of the beam depth (db). It should be mentioned that the design codes have not already recommended any criterion for these parameters of the RWS connections [2].
According to Refs. [13,31], the plastic hinge should be formed at a distance approximately equal to the beam depth from the column face.
The numerical simulation comprises thirty models, which are categorized into five groups (Table 4). In Groups I, II and III, the parameters "a", "b", and "cʹ" are variable, respectively, and the other parameters are fixed constants like those of the VS-RWS connection. In Groups I, II and III, the variables "a", "b", and "cʹ" are assumed within the ranges of 25-65%db, 40-90%db and 10-35%db, respectively (in the VS-RBS1, a=37.5%db, b=80%db and cʹ=16.3%db).
Since the quantity of "cʹ" affects the plastic section modulus, this parameter is expected to have a significant effect on cyclic behavior of the VS-RWS connection. In Group IV, the parameters "a", "b" and "cʹ" are similar to the VS-RWS1 specimen, and the parameter "g" is variable within a range of 10-100%b (equal to 8-80%db). In order to consider the simultaneous effects of "g" and "cʹ", the Group V models were investigated. In this group, both the parameters "g" and "cʹ" are variable, and the parameters "a" and "b" are considered according to the VS-RWS1 specimen. To name the models, at first, the group number was used and then the variable parameter was introduced as a percentage of beam depth (db). For example, a model of the second group in which the parameter "b" is 70% of the beam depth is regarded as II (b=70%db). The Bolted ordinary rigid connection (ORC-B) is a non-reduced model as a reference; and the models in which "g" is equal to 100%b have a rectangular opening. To examine the effect of the mentioned parameters, the hysteresis curve, the PEEQ, the rupture index, the end-plate deformation, and the dissipated energy of the connection models were studied and compared. Hence, in the design approach, the proper ranges for the parameters would be recommended.

Hysteresis curve
The moment-rotation curves of the models are drawn in Fig. 11. According to Fig. 11a,b, the connection models of Groups I and II are subject to strength degradation after the rotation angle of 0.05 rad due to the local buckling of the beam flange and web. Since the flexural strength of the connections in these groups at 0.04 rad is more than 80% of the plastic flexural capacity of the beam, these connections satisfy the seismic criteria of AISC-341 [32] standard for use in special moment frames. In Group III, the connection III(c=35% db) does not have the conditions for use in special moment frames because its flexural strength at 0.04 rad drops below 80% of the beam plastic flexural capacity (Fig. 11c). Hence, it is recommended that to achieve the proper cyclic performance of VS-RWS connections, the parameter "cʹ" does not exceed 30%db. By increasing the parameter "g" in Group IV, the strength degradation is intensified at a rotation angle of around 0.04 rad ; however, all the connections in this group meet the AISC-341 [32] requirement for use in special moment frames (Fig. 11d). In Group V, the connection V(cʹ=30%db, g =72%db) does not provide the conditions of AISC-341 [32] for application in special moment frames (Fig. 11e). Therefore, by examining the models of Groups III, IV, and V, it is recommended that the values of the parameters "cʹ" and "g" be limited to 25%db and 40%b (0.32db), respectively.

Equivalent plastic strain
The equivalent plastic strain (PEEQ) is an indicator that measures the local inelastic strain demand [33]. The PEEQ can be determined as follows [14]: where, ɛij is the plastic strain components.
To examine the strain concentration at various locations, the PEEQ distributions of the connection models are indicated in Fig. 13. Table 5  deformations of the connection models in the various groups. As can be seen, a significant strain concentration occurs at the slit tips. Due to the severe deformation and local buckling of the beam flange in the reduced region of the models II(b=40%db) and II(b =50%db), in Fig. 13b the lower and upper bounds for "b" are considered to be 60%db and 90%db, respectively The PEEQ distribution and the buckling of beam flange near the column face in models III(cʹ=10%db), IV(cʹ =10%b=8%db) and V(cʹ=10%, g=8%db) in Fig. 13c,d,e, indicate that a decrease in "cʹ" below 15%db, or a decrease in "g" below 20%b, increases the deformation of the beam flange at the column face. Therefore, the minimum values of "cʹ" and "g" are recommended 15%db and 20%b (0.16db), respectively. Thus, based on the results of the present and 5.1 sections, the ranges for parameters "cʹ" and "g" are presented as 15%db ≤ cʹ ≤ 25%db and 20%b ≤ g ≤ 40%b.  For ORC-B model, the strain concentration occurs at the column face, while for VS-RWS models, the strains are concentrated in a region far from the column face.
The torsional deformation of the beam is clearly observed in ORC-B model, though this phenomenon is negligible in the VS-RWS models.
Increasing the "a" causes the plastic hinge to move away from the column face.

b II
Increasing the "b" reduces the deformation of the flanges.
In the models of this group, the plastic hinges are formed in the flanges in line with the bottom slit, and the "b" has no significant effect on the location of plastic hinge.

c III
The variation of "cʹ" has a considerable effect on plastic strain intensity.
As the slit length increases, a large plastic deformation at the slit tips and a severe local buckling in the beam web and flange occur, leading to a reduction in the flexural capacity of the connections.
In Models III(cʹ=30%db) and III(cʹ=35%db), the plastic deformations and the low-cycle fatigue may cause cracks and eventually a rupture at the slit tips.
In the model III(cʹ=10%db), the lateral-torsional deformation of the beam is observed.

d IV
The parameter "g" has a slight effect on plastic strain intensity.
The model IV(cʹ=10%b =8%db), unlike the other models, is subject to a severe torsional deformation in a region near the welding line of the beam to the end-plate. This occurrence might lead to brittle failure of the welds between beam and endplate, when the dynamic effects are considered.
The beams of the connection models have similar local buckling mode shapes and the plasticity occurs away from the column face.
The optimal value for parameter "g" is recommended about 30%b.

e V
The simultaneous increase of two parameters "cʹ" and "g", which means an increase in the cutting area, causes the plastic strains to move away from the column face.
In the model with cʹ=10%db, the beam torsional deformation is larger than that of the other models. Fig. 14 shows the comparison of the maximum PEEQ for all models at different rotation angles. Fig. 14a,b indicates that in Group I and II models, the parameters "a" and "b" have a slight effect on the maximum PEEQ value. According to Fig. 14c, the PEEQ is strongly increased by increasing the cutting depth, such that in models III(cʹ=30%db) and III(cʹ=35%db) the rupture in the slit strips due to the low-cycle fatigue or local buckling may lead to the connection failure. As can be seen in Fig. 14d,e for Groups IV and V, by increasing the cutting area, the maximum PEEQ increases. In addition, the maximum PEEQ is created at the tips of vertical-slits, while in the models with rectangular openings, it occurs in the corner of the rectangular opening with a smaller amount than that of the VS-RWS models.
In the ORC-B connection, the maximum PEEQ occurs at the column face and is less than the VS-RWS connections.  those of the VS-RWS models; therefore, creating a large opening in the beam web cannot be recommended. By examining the results obtained from the study of Groups III and V, the optimal value for the parameter "cʹ" is suggested to be 20%db. Increasing the "cʹ" to more than the optimal value improves the connection behavior in terms of removing the strain concentration from the column face and the lateral-torsional stability, while it drastically reduces the bending capacity (Fig. 11).

Rupture index
In this section, the rupture index (RI) is used as a criterion for assessing the fracture potential of various locations in a structure [17,34,35]. This index is obtained by Eq. (3) [14].
where εy and ɛf are the yield strain and fracture strain; α is a material constant; p and q are hydrostatic stress and equivalent stress (von-Mises stress), respectively.
The fracture of connections at the column face is an important failure mechanism that needs to be avoided. Also, the regions with the maximum PEEQ have the largest values of RI [35].
Hence, the maximum values of RI at the column face and at the locations of the maximum PEEQ are plotted in Fig. 16. According to Fig. 16a, the RI at the column face of all the VS-RWS connections is lower than that of the ORC-B connection. This means that the verticalslits reduce the potential of connections for the weld failure. Among the present studied connections, the V(cʹ=35%db, g=80%db) and III(cʹ=35%db) have the lowest RIs at the welding line; however, Fig. 16b shows that the mentioned connections have a high potential for the low-cycle fracture around the slits.   (  II  I  )   b   %d  cʹ=25  (  II  I  )   b   %d  cʹ=30  (  II  I  )   b   %d  cʹ=35  (  II  I   ) Fig. 18, which are a combination of Groups III and IV, confirm the effects of the two parameters "g" and "cʹ" on the out-of-plane deformation of the end-plate. Beam flange width

Group II Section (a-a)
Beam flange width

Group III Section (a-a)
Beam flange width

Energy dissipation
The area enclosed by the hysteresis curves of the connections represents the energy dissipated due to cyclic loading. Fig. 19 illustrates the energy dissipated by the modeled connections. As can be seen, the "a" and "b" parameters, in Groups I and II, have a different, yet insignificant effect on the energy dissipation. The highest and lowest energy dissipations are related to the models II(b=60%db) and I(a=55%db), respectively, with a roughly 6% difference. In Group III, increasing the "cʹ" from 10%db to 35%db has reduced the dissipated energy by about 30%, indicating that the "cʹ" is the most effective parameter on energy dissipation. Since the hysteresis curves of Group IV models are almost similar, as expected, the amounts of energy dissipation of connections in this group will be very close, and the "g" parameter will have a very small effect on the dissipated energy. Among the models, V(cʹ=35%db) with the largest slit depth and cutting area has the lowest energy dissipation and as mentioned in the previous sections, this connection has the lowest bending capacity.

Equivalent radial-cut and constant-cut for vertical-slits
In this paper, the equivalent radial-cut (RC) RBS and the equivalent constant-cut (CC) RBS of a VS-RWS, in terms of their flexural strain, are suggested via using the classical conjugate beam method in the closed form [36].
where V is the shear force at the inflection point of the beam; and Lb is the beam length.
The flexural strain of the beam flange is then achieved in the VS-section and in the fullsection by Eqs. (11) and (12), respectively.
By integrating the flexural strain over the length of the reduced region, the total elongation of the beam flange is obtained from Eq. (13). (13) Similarly, in the RC-RBS connection at the beam flange, the strain, and the elongation related to the reduced region is calculated as follows: where SRC(x) is the cross-section modulus of the beam in the RC-section; and bf(x) is the reduced flange width as a function of the distance from the column face (x) as Eq. (18). In this equation, the Taylor series expansion is used to simplify bf(x).
In the same way, in the CC-RBS connection with a constant cutting depth (ceq) at the beam flange, the strain and the elongation over the reduced region of the beam are achieved by Eqs. (19) to (22 where SCC is the cross-section modules of the beam with the constant-cut. Since the elongation of the beam flange is directly related to the flexural rotation of the beam, the validity of the elongation criterion for three connections is explained as Eq. (23).
where SRC is the cross-section modules of the RC-RBS beam in the center of the radial-cut. In order to obtain ∆CC-RBS in the CC-RBS connection, in addition to the presented method, it can be obtained by calculating the limit of ∆RC-RBS when "R" (the radius of the cut is depicted in Fig. 20) approaches infinity using the Maclaurin series as follows: In order to check the accuracy of the equivalent radial-cut and the equivalent constant-cut for the VS-RWS, a finite element analysis was conducted. In this numerical study, three connections (Table 6) were applied under the cyclic loading. The loading protocol, the beam and column profiles and the dimensions were considered to be the same as those in the tested specimens. All the equivalent parameters including "cʹ" and "ceq" implicitly satisfy Eq. (24). Table 6. Geometric specifications of VS-RWS, RC-RBS and CC-RBS The load-displacement curves of the beams in Fig. 22

Story drift due to beam flexural deformation
In regular moment frames under lateral loading, the inflection points are usually near the midspan of the beam and the mid-height of the column. The beam slope (θb) and the story drift (δb) in such frames can, therefore, be determined by analyzing a length of the beam between the middles of the two spans. Fig. 23 shows the deflected shape of the connection caused by only the beam flexural deformation.  Based on this method, a distributed load obtained by dividing the value of the real beam bending moment by the EI was placed on the conjugate beam. The obtained distributed load was then converted to the equivalent concentrated load (P) and was placed in the load area center ( x ). By writing equilibrium equations in the form of the free body diagram, the shear force on the conjugate beam was calculated, which is equal to the slope (θb) of the real beam.
The elastic load diagram of the equivalent beam with the constant-cut is illustrated in Fig. 24.
By solving the integrals of Eq. (26) and simplifying it, the value of δb is obtained as follows: (27) where Ip and tp are the moment of inertia and thickness of the end-plate, respectively. The story drift (δb) from Eq. (27) is obtained by the sum of three terms; the first, second and third terms are related to the end-plate, the reduced-section beam, and the full-section beam, respectively.

Proposed design procedure
In order to design the common RBS connections, a specified procedure and suitable criteria have been presented in Chapter 5 of AISC-358 [30]. Nevertheless, no suitable criteria have been provided for VS-RWS connections. In this section, according to the parametric study of the previous sections, as well as to the AISC-358 [30], a step-by-step procedure for the design of VS-RWS connections is proposed as follows: Step 1. According to the following limits, consider the trial value for the location and length of the beam web reduction: Step 2. Choose the trial value for the cutting depth (2cʹ) based on the following range: For the first attempt, it is recommended that you assume "cʹ" and g to be equal to the optimal values of 0.2db and 0.3b, respectively.
Step 3. By using Eq. (24), obtain Seq and "ceq" for the equivalent RC-RBS or for the equivalent CC-RBS. Seq is the reduced-section modulus (Seq=SRC for RC-RBS; Seq=SCC for CC-RBS).
Step 4. Compute the plastic reduced-section modulus of about x-axis (Zeq, mm 3 ), and the probable maximum moment (Mpr) at the equivalent RC-RBS or CC-RBS.
where based on the AISC-358 [30] seismic provisions, Cpr and Ry are the factors which account for the peak strength and the ratio of the expected yield stress to the specified minimum yield stress, respectively.
Step 5. Obtain the shear force (Veq) in the center of the radial-cut for RC-RBS, or at the beginning of the constant-cut for CC-RBS at each end of the beam by using the free-body diagram of the beam (Fig. 25), including Mpr and gravity loads (Vgravity) acting on the beam as Eq. (34).
Step 6. Compute Mf, the probable maximum moment at the column face from a free-body diagram of the segment of the beam between the center of the radial-cut and the column face where Sh=a+b/2 for RC-RBS, and Sh=a for CC-RBS.
Step 7. Obtain the plastic moment of the beam (Mpe) based on the expected yield stress as follows: Step 8. Check the flexural strength of the beam at the column face through Eq. (37). In this equation, φd is the resistance factor for ductile limit states (φd =1).
Step 10. Determine the beam-required shear strength (Veq) using Eq. (34), and then obtain the nominal shear strength as follows [37]: where, Fy and Aw are the specified minimum yield stress of the type of steel being used (MPa) and the area of the beam web (mm 2 ), respectively. Check the shear strength of the beam through Eq. (39).
In LRFD method, Cv1 and ϕv for the webs of the rolled I-shaped members with h/tw ≤ 2.24(E/Fy) 0.5 are equal to one, where E, h and tw are the modulus of elasticity of steel (=200 GPa), the clear distance between flanges less the fillet at each flange (mm), and the web thickness, respectively.
Step 11. Establish preliminary values for the end-plate geometry, namely the horizontal and vertical distances between bolts (mm), thickness of end-plate (mm) etc, and bolt grade.
Step 12. Determine the required bolt diameter (db,req) using the following expressions: where Fnt is the nominal tensile strength of the bolt from the AISC-358 [30] specification (MPa); h0 and h1 are the distances from centerline of the beam compression flange to the centerline of the i th tension bolt row and the distance from centerline of compression flange to tension-side outer bolt row (mm), respectively. ϕn is equal to 0.90.
Step 13. Determine the required dimensions of the end-plate based on the section 6.8 of AISC-358 [30]. Step 14. Design the flange-to-end-plate and web-to-end-plate welds using the requirements of the section 6.7.6 of AISC-358 [30]. end-plate and to transfer them around the slits. This will prevent a brittle fracture or damage at the column face.
 As compared with the ordinary rigid connection, the vertical-slits improve the lateraltorsional stability behavior of VS-RWS connections.
 According to the conducted experimental tests, the depth and area of the vertical-slits play an important role in the local instability and the beginning of slit-rupture, which leads to the strength degradation in the cyclic behavior.
 The depth of the vertical-slit is an effective parameter in the behavior of the VS-RWS connections, such that the increase of the slit depth from cʹ=10%db to cʹ=35% db can reduce the amounts of the flexural capacity and the energy dissipation by about 40% and 25%, respectively.
 Considering (cʹ > 0.25db) reduces the flexural capacity and energy dissipation, and (cʹ < 0.15db) causes the local or torsional buckling, the increase of stress/strain concentration at the column face, as well as the out-of-plane deformation of the endplate.
 Based on the parametric study, the optimal values for "cʹ" and "g" parameters in this type of connection are about 20%db and 30%b, respectively.
 Considering the proper geometric parameters of the reduced region, the VS-RWS connections are capable of being used in special moment frames based on AISC-358.
 This paper calculates the story drift of the VS-RWS frame caused by the flexural deformation of the beam by presenting an equivalent constant-cut or radial-cut RBS connection using the conjugate beam method. By applying the equivalent RBS connection, a step-by-step design procedure has been proposed for VS-RWS connections.

Recommendations for further research
The authors recommend further research on the bolted or welded VS-RWS connections with different beam profiles, or with the non-equal depth of vertical-slits for various parameters, such as the number of slits and the beam length.