Parametric study of hexagonal castellated beams in post-tensioned self-centering steel connections

Abstract

The effects of important parameters (beam reinforcing plates, initial post-tensioning, and material properties of steel angles) on the behavior of hexagonal castellated beams in post-tensioned self-centering (PTSC) connections undergone cyclic loading up to 4% lateral drift have been investigated by finite element (FE) analysis using ABAQUS. The PTSC connection is comprised of bolted top and bottom angles as energy dissipaters and steel strands to provide self-centering capacity. The FE analysis has also been validated against the experimental test. The new formulations derived from analytical method has been proposed to predict bending moment of PTSC connections. The web-post buckling in hexagonal castellated beams has been identified as the dominant failure mode when excessive initial post-tensioning force is applied to reach greater bending moment resistance, so it is required to limit the highest initial post-tensioning force to prevent this failure. Furthermore, properties of steel material has been simulated using bilinear elastoplastic modeling with 1.5% strain-hardening which has perfectly matched with the real material of steel angles. It is recommended to avoid using steel angles with high yielding strength since they lead to the yielding of bolt shank. The necessity of reinforcing plates to prevent beam flange from local buckling has been reaffirmed.

Keywords

finite element analysis / hexagonal castellated beam / parametric study / post-tensioned self-centering steel connection / steel moment-resisting frame

Introduction

The steel structures are widely used in built-environment and infrastructures and bring many advantages such as high strength, excellent ductility, significant durability, rapid construction, and low waste. Among various types of steel constructions, steel moment-resisting frames (MRFs) have acceptable characterizations designed to resist earthquake loadings, large deformations and provide high level of seismic energy dissipation in the range of inelastic deformations. The beam-to-column connections have substantial impacts on the behavior of steel MRFs under earthquake and many investigations have been conducted to improve the connection properties. The 1994 Northridge earthquake leading to failure of many welded connections [¹] has contributed to extensive research in this field proposing different kind of modified connection details to prevent the damage of beam-to-column connections and provide energy dissipation through beam inelastic deformations with a distance from the connections such as using reduced beam section (RBS) [²], applying reinforcing cover plates [³], and haunches [⁴]. Although the features of modified connections eliminate the connection failure, the permanent beam damages lead to considerable cost and efforts to repair the steel structures after the earthquake. Therefore, it has been required to develop high-performance beam-to-column connections concentrating the damages to easy-replaceable elements.

The first type of post-tensioned self-centering (PTSC) steel connections comprised of bolted top and bottom angles as energy dissipation devices and high-strength steel strands to provide self-centering capability returning the MRFs to the initial position without residual drift before the earthquake has been originally introduced by Ricles et al. [⁵,⁶] which has many advantages such as the same initial stiffness as that of a fully-welded connection, eliminating the field welding, and easy-installation aspects requiring conventional material and skills. Some researchers have focused on this PTSC connection to comprehensively investigate its seismic performance. Garlock et al. [⁷] have studied the length of beam reinforcing plates required to resist beam flange buckling subjected to extreme loading and have suggested that the minimum length of reinforcing plate should limit the beam strains up to twice the yielding strain. Furthermore, step-by-step procedure has been provided to design the MRFs with the PTSC connections under earthquake [⁸]. The bolted top and bottom angles of such connections can reliably dissipate energy up to 5% lateral drift and higher thickness of angle increases the capacity of energy dissipation [⁹]. Further studies have been also carried out to reach more general conclusions regarding PTSC connections with bolted angles [¹⁰–¹²].

Christopoulos et al. [¹³] have proposed energy dissipating (ED) bars including confined steel bars in steel cylinders welded to the interior surface of beam flanges to eliminate the slab interferences. Faggiano et al. [¹⁴] have numerically studied the PTSC connections with ED steel bars by ABAQUS software. To eliminate the slab restraints, discontinuous slab has been proposed for the PTSC connections with ED bars [¹⁵]. The cyclic behavior of 1-story MRF with such PTSC connections have been analyzed [¹⁶]. Chou et al. [¹⁷] have used reduced flange plates (RFPs) bolted to steel beam flange and welded to concrete-filled tubular column as the energy dissipaters of the PTSC connections. A three-dimensional (3D) analytical method has been also proposed to estimate the beam compression force and the impact of column restraint on the behavior of MRFs with such PTSC connections [¹⁸]. Vasdravellis et al. [¹⁹] have experimentally evaluated the steel web hourglass-shaped cylindrical pins as energy dissipaters installed between the beam flanges of the PTSC connections. A simple nonlinear model of this PTSC connection has been proposed and calibrated against experiment showing high precision in simulating the behavior of this connection [²⁰]. The collapse risk analysis conducted for the MRFs comprised of such PTSC connections and viscous dampers near the fault regions has represented high reduction in risk of structural collapse [²¹]. Zhang et al. [²²] have introduced the U-shaped steel dampers of PTSC connections and have experimentally and numerically examined the behavior of this connection resulting in reliable level of energy dissipation under lateral cyclic loadings.

Rojas et al. [²³] have applied brass-steel surfaces in friction devices set on the beam flanges that provide high level of energy dissipation and good ductility for the PTSC connections. Tsai et al. [²⁴] have proposed bolted web friction devices of PTSC connection providing stable energy dissipation characterizations under cyclic loading. Kim and Christopoulos [²⁵] have introduced a new frictional energy dissipater installed on the beam flanges of the PTSC connections. The step-by-step design method and ultimate capacity of this connection have been also evaluated [²⁶]. Another friction device set only at the beam bottom flange has been proposed as energy dissipater of PTSC connections by Wolski et al. [²⁷]. However, this kind of asymmetric energy dissipation device has led to greater plastic strains and beam local buckling as compared with the friction devices symmetrically installed at both beam flanges under extreme loadings [²⁸]. Lin et al. [²⁹] have studied web friction channel as energy dissipaters of PTSC connections based on the immediate occupancy performance and collapse prevention performance. Repairable limit state and re-centering limit state as new performance states have been defined for the MRFs with this kind of friction-damped PTSC connections [³⁰]. Zhang et al. [³¹] have investigated a friction device with different characteristics bolted to the beam web of the prefabricated PTSC connections comprised of beam, steel strands, and vertical plates. Herning et al. [³²] have used reliability-based approach to design the existing PTSC connections and have proposed modification factors to the current design procedures.

Abedi Sarvestani [³³] has investigated hexagonal castellated beams in the PTSC connections comprised of bolted top and bottom angles to reliably dissipate energy and high-strength steel strands to provide capability of self-centering behavior subjected to lateral cyclic loadings and has represented that the hexagonal castellated beams provide adequate resistance against shear failure, web-post buckling and vierendeel mechanism, higher bending strength, and lower weight in comparison with wide flange beams in the PTSC connections. Design guidelines have been developed for castellated beams subjected to different loading conditions [³⁴,³⁵]. Chung et al. [³⁶] have introduced the generalized moment-shear interactions to design castellated beams with different characteristics. The ultimate strength of castellated beams against web-post failure has been investigated [³⁷,³⁸]. Demirdjian [³⁹] has used the elastic and plastic modeling to examine the failure modes of castellated beams. Kerdal and Nethercot [⁴⁰] have evaluated the influence of steel beam web-opening on the lateral-torsional buckling (LTB) and have shown that castellated beams have similar strength against LTB as steel beams with solid webs. Distortional buckling of the perforated beams has also investigated by experimental tests [⁴¹]. Ellobody [⁴²] has analyzed the impact of beam length, section geometrics, and steel strength on the castellated beams resistance against the combined distortional buckling and LTB. Chung et al. [³⁶,⁴³] have analyzed the strength of perforated steel beams with different web-opening characteristics against vierendeel mechanism which is a beam failure mode related to four plastic hinges of top and bottom tee-sections of the steel beam with web-openings under the combined bending moment and vierendeel moments. Gholizadeh et al. [⁴⁴] have analyzed the strength of castellated beams against web-post buckling. Soltani et al. [⁴⁵] have proposed a numerical method to calculate the ultimate strength of castellated beams with different characterizations.

This investigation is based on the research conducted by Abedi Sarvestani [³³] on the behavior of hexagonal castellated beams in PTSC connections required a parametric study to obtain thorough conclusions regarding the influence of effective factors on the behavior of such unique system subjected to lateral cyclic loadings. The influence of the existence of beam flange reinforcing plates, the initial post-tensioning force, and the material of steel angles are investigated as the important factors in the study. In addition, an analytical method is introduced to predict the bending moment of PTSC connections with hexagonal castellated beams. The specimens are theoretically investigated and numerically analyzed by finite element (FE) software ABAQUS 6.11-PR3.

Behavior of hexagonal castellated beams in PTSC connections

The use of hexagonal castellated beams in the PTSC steel connections has been proposed by Abedi Sarvestani [³³]. The PTSC beam-to-column connection is based on that proposed by Garlock et al. [⁷], this type of PTSC connection includes the bolted top and bottom angles as energy dissipaters and 7-wire high-strength steel strands to provide the initial post-tensioning force compressing the beam flanges against the column flange. This compression results in the decompression moment resisting the initial bending moment prior to gap-opening of the PTSC connection at the beam-column interface. The steel strands provide self-centering capability for MRFs with PTSC connections and they are symmetrically arranged around the beam centroid. The theoretical formulations to design hexagonal castellated beams in PTSC connections can be found in the research conducted by Abedi Sarvestani [³³]. Figure 1 demonstrates the details of sample PTSC beam-to-column connection.

Figure 2(a) represents the moment-relative rotation relationship at the beam-column interface (M – q_r) as the overall behavior of a PTSC steel connection with bolted top and bottom angles subjected to lateral cyclic loadings. The first event (point 1) shows when the decompression moment overcome by lateral loading resulting in gap-opening at the beam-column interface, the initial stiffness of PTSC connection up to decompression is the same as that of a fully-welded moment connection. By continuing lateral loading and subsequently gap-opening, the angles starts yielding at the second event (point 2). The stiffness of PTSC connection between the first and second events is associated with stiffness of post-tensioning steel strands and angles stiffness. The third event (point 3) demonstrates the angle yielding mechanism that are formed with three plastic hinges at the angle legs [⁴⁶] as illustrated in Fig. 2(b). The forth event (point 4) represents the unloading point prior to the strands yielding (point 4') due to the post-tensioning force. The connection stiffness from point 3 to point 4 is related to the elastic stiffness of strands and deformation-hardening of connection angles. After unloading, the energy is dissipated due to angles yielding up to the gap at the beam-column interface closes at the seventh event (point 7). This type of PTSC steel connection also demonstrates the same behavior when undergone the reverse lateral loading that is represented by points 0, and 8–14.

Analytical method formulations

A simplified analytical method is developed to investigate the behavior of PTSC connections with bolted top and bottom angles under cyclic loading. Once the moment at the beam-column interface overcomes the decompression resisting moment, the gap starts to open. Figure 3 represents the deformed shape of an interior PTSC steel connection following decompression.

The incident of gap-opening at the beam-column interface occurs when the decompression moment M_d,th is overcome [⁶]:

(1)M_{\mathrm{d,th}}=d_{\mathrm{c}}{\displaystyle \frac{T_{\mathrm{0}}}{\mathrm{2}}},

where T₀ and d_c ( = 2d₂) are respectively the sum of initial post-tensioning force and the distance between connection contact areas as shown in Fig. 4.

On the basis of the story elastic drift q_elastic, the relative rotation q_r at the story drift q is calculated as follows [⁶]:

(2){\theta }_{\mathrm{r}}=\theta -{\theta }_{\mathrm{elastic}}.

The elastic analysis of steel frame with rigid connections resulting in the elastic deformations in beams, columns, and panel zones is considered to calculate q_elastic [⁶]. Following the gap-opening, the elongation of high-strength steel strands leads to the increase in strands force and the shortening of beam. The sum of post-tensioning force in steel strands T_th can be calculated by following Eq [⁴⁷]:

(3)T_{\mathrm{th}}=T_{\mathrm{0}}+\mathrm{2}\left({\displaystyle \frac{K_{\mathrm{s}}{K}_{\mathrm{b}}}{K_{\mathrm{s}}+K_{\mathrm{b}}}}\right){\theta }_{\mathrm{r}}.

That K_s and K_b are respectively the strands axial stiffness and the beam axial stiffness.

The simplified model of steel angle after the gap-opening leading to the yielding mechanism with three plastic hinges is considered to develop the analytical method as illustrated in Fig. 5. The points of 1, 2, and 3 are provided to show the angle plastic hinges. The equilibrium of moments and forces should be satisfied at the points of angle plastic hinges in the column leg (12) and beam leg (23) as follows:

(4)M_{\mathrm{12}}^{\mathrm{u}}+M_{\mathrm{21}}^{\mathrm{u}}=T_{\mathrm{a}}^{\mathrm{u}}{G}_{\mathrm{2}}-V_{\mathrm{a}}^{\mathrm{u}}{\mathrm{\Delta }}_{\mathrm{gap}},

(5)M_{\mathrm{23}}^{\mathrm{u}}+M_{\mathrm{32}}^{\mathrm{u}}=V_{\mathrm{a}}^{\mathrm{u}}{G}_{\mathrm{1}},

where V_a^u is the shear force in unit length of angle, and T_a^u is the axial force in unit length of angle. G₁ is the distance between the beam bolt nut and angle fillet, and G₂ is the distance of column bolt nut and angle fillet. M₁₂^u and M₂₁^u are the moments in unit length of angle plastic hinges at 1 and 2 points with full-plastic state, respectively. Furthermore, M₂₃^u and M₃₂^u are the moments in unit length of angle plastic hinges at 2 and 3 points with full-plastic state, respectively. ∆_gap = q_rd₃ is the amount of gap-opening at the beam-column interface, d₃ is the distance between the centerline of tension angle leg and the center of rotation as illustrated in Fig. 4. V_a^u can be expressed based on Eq. (5) and the assumption that moments at points of angle plastic hinges reach the full-plastic moment capacity in unit length M_ap^u by:

(6)V_{\mathrm{a}}^{\mathrm{u}}={\displaystyle \frac{\mathrm{2}{M}_{\mathrm{ap}}^{\mathrm{u}}}{G_{\mathrm{1}}}}.

Therefore, T_a^u can be calculated based on substitution of V_a^u in Eq. (4) as follows:

(7)\mathrm{2}{M}_{\mathrm{ap}}^{\mathrm{u}}=T_{\mathrm{a}}^{\mathrm{u}}{G}_{\mathrm{2}}-\left({\displaystyle \frac{\mathrm{2}{M}_{\mathrm{ap}}^{\mathrm{u}}}{G_{\mathrm{1}}}}\right){\mathrm{\Delta }}_{\mathrm{gap}},

(8)T_{\mathrm{a}}^{\mathrm{u}}={\displaystyle \frac{\mathrm{2}{M}_{\mathrm{ap}}^{\mathrm{u}}}{G_{\mathrm{2}}}}\left(\mathrm{1}+{\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{gap}}}{G_{\mathrm{1}}}}\right).

The full-plastic moment capacity of angle M_ap = bM_ap^u in that b is the length of angle. As the angle is not expected to reach the full-plastic moment capacity under cyclic loading, the plastic moment capacity of angle will be estimated based on a modification factor r<1 depended on the geometry of angle as follows:

(9)M_{\mathrm{a}}^{\mathrm{T}}=M_{\mathrm{a}}^{\mathrm{C}}=\rho M_{\mathrm{ap}}.

That M_ɑ^T is the bending moment at plastic hinge of tension angle column leg, and M_ɑ^C is the bending moment at the plastic hinge of compression angle beam leg. The total angle axial force T_a can be calculated by following equation:

(10)T_{\mathrm{a}}=\rho {\displaystyle \frac{\mathrm{2}{M}_{\mathrm{ap}}}{G_{\mathrm{2}}}}\left(\mathrm{1}+{\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{gap}}}{G_{\mathrm{1}}}}\right).

The equilibrium of moments and forces at beam-column interface should be satisfied as represented in Fig. 4. Therefore, the theoretical bending moment of PTSC connection M_th is calculated as follows:

(11)M_{\mathrm{th}}=T_{\mathrm{a}}{d}_{\mathrm{1}}+T_{\mathrm{th}}{d}_{\mathrm{2}}+M_{\mathrm{a}}^{\mathrm{T}}+M_{\mathrm{a}}^{\mathrm{C}}.

In this study, the initial modification factor r = 0.63 derived from the experimental results of PTSC connections with bolted top and bottom angles carried out by Garlock et al. [⁷] has been applied to predict the behavior of hexagonal castellated beams in PTSC connections.

FE analysis of hexagonal castellated beams in PTSC connections

In this article, specimens of hexagonal castellated beams in PTSC steel connections considered in the research conducted by Abedi Sarvestani [³³] have been selected to develop the FE modeling for parametric study. The configurations of hexagonal castellated beams are exactly the same as that provided by Abedi Sarvestani [³³]. The castellated beam depth H is 1.5h in which h is the depth of original hot-rolled section and tanj = 2 that j is the angle of hexagonal opening. The length of hexagonal opening C is considered h/2; therefore, castellated beam pitch P is equal to H. Figure 6 demonstrates the characterizations of hexagonal castellated beams.

Abedi Sarvestani [³³] has studied the hexagonal castellated beams in PTSC connections and has shown that such castellated beams provide higher load-carrying capacity compared with the wide flange beams of higher weight. The detailed FE modeling of this parametric study is exactly the same as that of the reference research [³³]. The PTSC connections have been simulated as an interior connection of steel MRF as represented in the FE modeling setup of Fig. 7.

The design of beams with hexagonal web-openings is based on the beam resistance against the applied shear force in the hollow section under the standard cyclic loadings up to 4% lateral drift. Therefore, it has logically increased the depth of the beam in practice and also provide adequate strength against failure of web-post and vierendeel mechanism. The hexagonal castellated beams should meet the slenderness limits and bracing requirements based on AISC Seismic Provisions [⁴⁸]. Garlock et al. [⁷] have carried out the experimental tests on the PTSC connections and found that local buckling of beam flanges can be prevented by restricting the beam flange strains under twice the yield strain at the end of the reinforcing plates. In addition, the sum of contact force at the beam-column interface should be lower than the total force leading to beam flange yielding and web horizontal shear yielding to prevent beam horizontal shear yielding. Therefore, the length of reinforcing plates should be adequate to meet this criterion [⁸]. To prevent the beam yielding under the bearing stress at beam-column interface, the contact force should not exceed the total yielding force of the beam flange and reinforcing plate [⁸]. Also, the shim plates are used as construction fit-up between the end of the beam and column faces and prevent bearing of the beam web at beam-column interface.

In FE modeling, cyclic loadings performed by actuators in the experiment have been laterally applied at the top of the column with wide flange section of W14 × 398. The top and bottom angles of L203 × 203 × 19 with 406 mm width provide the energy dissipation during the lateral drift that are connected to the beam and column flanges with A490 high-strength steel bolts of 32 mm (\mathrm{1}{{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}}^{"}). Four steel bolts arranged in a single row with a distance of 137 mm from the angle bottom connect the angle leg to the column flange, also four bolts set in two rows provide the connection between the angle and the beam flange. The tension force of 454 kN (102 kips) has provided the minimum pretension force in the connection high-strength steel bolts based on AISC Specification [⁴⁹]. The holes with a diameter of 44 mm have been drilled in the column flanges to pass the 7-wire high-strength steel strands with the area of 140 mm² each and the ultimate force T_u, that are bundled together to form the 2–4 strands providing the post-tensioning force for the PTSC connections. The arrangement of strands have been symmetric to coincide the centroids of beam and strands.

Table 1 represents the specimens of hexagonal castellated beams in PTSC connections considered in the study carried out by Abedi Sarvestani [³³]. This table includes the number of steel strands N_s, the total initial post-tensioning force T₀ normalized by the strands ultimate force T_u, and the length of reinforcing plates L_rp keeping the strain of beam flanges under twice the yielding strain.

Material properties

The material properties of steel components should be defined to develop FE modeling of hexagonal castellated beams in PTSC connections. The material type of all steel components except for high-strength steel bolts and strands is A572 Grade50 [⁵⁰]. The steel bolts and strands respectively have the material of A490 Grade50 [⁵¹] and A416 Grade270 [⁵²].

Contact conditions

To achieve the accurate results of FE analysis, the interaction of steel components in contact should be properly defined using the available modeling tools in ABAQUS 6.11-PR3 [⁵³]. The detailed contact conditions of FE modeling are as follows: the surface-to-surface contact with small relative sliding of contacted surfaces has been considered to define the two general contact types. The normal contact is implemented as a hard-contact defined for the behavior perpendicular to the surface, and the tangential contact defined for the behavior in the transverse direction to the surface is specified using the penalty method and the friction coefficient of 0.35 [⁴⁹,⁵³].

Boundary conditions and loading systems

To simulate the hinged support at the column base, the nodes of centerline of column bottom are constrained from displacement to apply the pin conditions. In addition, the roller supports at the beam ends are modeled by constraining the displacement of vertical direction [⁵³]. The beam out-of-plane displacements have been prevented by applying the lateral bracing at the locations based on AISC Seismic Provisions [⁴⁸] as represented in Fig. 7. Furthermore, the proper boundary condition has been used to model the anchorage systems at the end of beams [⁵³]. The minimum bolt pretension are applied to high-strength steel bolts as a boundary condition. The two parts of the bolt body split in the middle, perpendicular to the shank are pulled to each other to reach the minimum pretension force. Also, the high-strength steel strands are tensioned to the initial post-tensioning force as mentioned in Table 1. The standard cyclic loading based on AISC Seismic Provisions [⁴⁸] has been applied step-by-step with several cycles of the story drifts on the column flange as follows: the number of 6 cycles of 0.375% drift, 0.50% drift, and 0.75% drift, 4 cycles of 1% drift, and 2 cycles of 1.5% drift, 2% drift, 3% drift, and 4% drift.

Element types and meshing quality

In the FE modeling by ABAQUS 6.11-PR3, the elements should properly selected from various elements types with different characterizations to obtain the accurate results. 3D solid shape of C3D8R solid elements with 8-nodes have been used to model the steel components except for the high-strength steel strands which are simulated using 3D wire shape of B31 beam elements with 2-nodes [⁵³]. Figure 8 demonstrates the meshing of FE modeling of specimen HCB-05 by ABAQUS software.

Besides, the quality of meshing in FE analysis is as vital as the element types. To comply with this criterion, the key meshing metrics have been evaluated as shown in Table 2.

This table is comprised of the metrics of aspect ratio, skewness index, and orthogonality ratio in different steel components representing proper meshing quality to reach accurate results. The worst aspect ratio is limited to 4.482 found in the angle, the worst skewness index is 0.518 belonging to an element of W14 398, and the worst orthogonality ratio is restricted to 0.646 that is in the meshing of beam bolt.

Validation of FE analysis

The FE analysis of the hexagonal castellated beams in PTSC connections have been validated against the experimental test carried out by Garlock et al. [⁷]. In the validation process, one of the experimental specimens of PTSC connections with wide flange beam and bolted top and bottom angles (36s-20-P) [⁷] has been simulated by ABAQUS 6.11-PR3 and has been subjected to the standard cyclic loading up to 4% lateral drift. Figure 9 shows the lateral load P_lateral versus the displacements of column tip D for this specimen to provide the comparison between FE modeling and the experiment. This figure represents that the response of FE analysis has been well matched with the experimental results.

FE results of reference specimens

The results of FE analysis for the reference specimens with hexagonal castellated beams in PTSC connections mentioned in Table 1 represent that all steel components of the specimens except for the angles have remained elastic and have not suffered from any damages under standard cyclic loading up to 4% lateral drift. The criteria applied to calculate the beam reinforcing plates have been effective to prevent the beam flange buckling. In addition, the hexagonal castellated beams have shown adequate strength against the web shear buckling, web-post failure, and vierendeel mechanism under cyclic loadings. Furthermore, the high-strength steel strands remained elastic show that the initial post-tensioning force and number of strands have been accurately calculated to provide self-centering capability returning the MRFs with PTSC connections to the initial position prior to the lateral loadings. Table 3 shows the FE results of reference specimens subjected to lateral cyclic loadings. This table is comprised of the decompression moment M_d, the maximum bending moment at the beam-column interface M_max, the maximum post-tensioning force T_max, and the maximum relative rotation at the beam-column interface q_r,max.

Comparison of analytical predictions and FE results

Table 4 provides the comparison between the analytical responses of reference specimens with the FE results to assess the accuracy of the introduced analytical method. It is concluded that the analytical formulations precisely predict the behavior of the hexagonal castellated beams in PTSC steel connections subjected to cyclic loading up to 4% lateral drift. It shows the negligible difference between maximum bending moments predicted from the analytical method and that obtained from FE analysis.

The average modification factor r of the reference specimens from FE results is 0.61 that is very close to the initial amount of r = 0.63 based on the analytical method to design the specimens. This average can be implemented to design the hexagonal castellated beams in PTSC connections comprised of similar angle geometry. Therefore, the proposed analytical method can be reliably used in design of PTSC connections with hexagonal castellated beams. Correspondingly, post-tensioning force and decompression moment from theoretical predictions have agreed well with that of FE results.

Parametric studies

The parametric studies have been carried out to evaluate how different factors are influential on the response of hexagonal castellated beams in PTSC connections subjected to lateral cyclic loadings. The crucial factors considered in the study are: the existence of beam reinforcing plates, initial post-tensioning force of steel strands, and the material properties of steel angles as energy dissipation devices. Three out of five original specimens investigated by Abedi Sarvestani [³³] have been selected as reference specimens for this parametric research.

Effect of beam reinforcing plates

As one of the parametric factors, the influence of reinforcing plates on the behavior of hexagonal castellated beams in the PTSC connections has been examined. The specimen HCB-01 (-RP) without the reinforcing plates based on the similar details of specimen HCB-01 has been considered to provide this assessment. The beam local buckling of HCB-01 (-RP) without reinforcing plates has initiated at the first half-cycle of 2.4% drift. Figure 10 demonstrates the displacement contour and beam local buckling in deformed shape of the beam specimen of HCB-01 (-RP).

On the other hand, HCB-01 has not suffered from any beam damages up to 4% lateral drift since the reinforcing plates provide adequate strength against the beam flange buckling. The lateral load-displacement response (P_lateral– D) of these specimens compared in Fig. 11 represents that while the elastic post-tensioning strands have provided self-centering capability for both specimens, HCB-01 (-RP) has returned to its initial position with residual drift due to the beam local buckling under the cyclic loading up to 4% drift which indicates the behavior of the hexagonal castellated beams without the reinforcing plates in the PTSC connections is controlled by the beam local buckling. HCB-01 (-RP) has shown lower lateral load resistance in comparison with HCB-01.

Table 5 provides the summary of FE results of parametric studies. It shows that HCB-01 (-RP) has lower M_d and higher q_r,max as compared to HCB-01. As a result of beam flange buckling of HCB-01 (-RP), the maximum moment M_max and maximum post-tensioning force T_max have respectively decreased by 9.7% and 6.2% in comparison with HCB-01. As represented in Table 5, HCB-01 (-RP) has shown lower initial stiffness K_i by 14.9%. Although, the sum of energy dissipation SE_d of HCB-01 (-RP) was greater than HCB-01 due to the beam flange yielding, the energy dissipated by angles prior to the beam buckling has been 3.7% lower.

Effect of initial post-tensioning force

The initial post-tensioning force has a pivotal influence on the self-centering behavior and load-carrying capacity of the PTSC connections with hexagonal castellated beams. Based on the reference details of HCB-03, three different specimens including lower post-tensioning force, HCB-03 (PT₁), and higher post-tensioning force, HCB-03 (PT₂) and HCB-03 (PT₃), have been selected to evaluate this parameter on the response of specimens subjected to lateral cyclic loading. Figure 12 shows the comparative lateral load-displacement response (P_lateral– D) of the specimens under cyclic loadings. The result of FE analysis have represented that the different initial post-tensioning forces have led to different decompression behavior, relative rotation, and maximum moment capacity of the specimens.

The specimen of HCB-03 (PT₁) with less post-tensioning force has shown lower decompression moment resulting in greater gap-opening and relative rotation at the beam-column interface as shown in Table 5. The maximum moment of this specimen has also reduced to 4025.07 kN.m which is 3.1% less than that of HCB-03. The specimen of HCB-03 (PT₂) with 11.9% increase in initial post-tensioning force has enhanced 4.1% of moment capacity that is 4327.23 kN.m and has provided 7.3% higher level of energy dissipation due to greater load-carrying capacity of the specimen. The initial post-tensioning force should be restricted to prevent the failure modes of hexagonal castellated beams in the PTSC connection including web-post failure, vierendeel mechanism, and beam flange buckling. In the FE analysis of HCB-03 (PT₃) with the highest post-tensioning force, the relative rotation has reduced due to the greater decompression moment. Furthermore, the hexagonal castellated beam of this specimen has suffered from the web-post buckling at the first half-cyclic of 3.65% drift indicating the hexagonal castellated beams in PTSC connections are more vulnerable to this failure mode than other beam damages. Figure 13 illustrates the displacement contour and beam web-post buckling of HCB-03 (PT₃) at 3.65% drift.

Effect of material properties of steel angles

The inelastic behavior of steel material has substantial effect on the energy dissipation capacity of the bolted angles in PTSC connections with hexagonal castellated beams. To study the impact of steel cyclic strain-hardening law and yield strength on steel angles as energy dissipation devices, three bilinear elastoplastic material models (M_1, M_2, and M₃) have been proposed for the angle steel type of A572 Grade50 as shown in Fig. 14. Each material has been applied to the same detail as that of HCB-05 to examine the steel material influence on the specimens namely HCB-05 (M₁), HCB-05 (M₂), and HCB-05 (M₃).

Table 5 represents the FE response of specimens with different material modellings subjected to cyclic loading up to 4% lateral drift. The first specimen of HCB-05 (M₁) with elastic-perfectly plastic model of angle steel material has shown a decrease in moment capacity by 5.4% and similar decompression moment and relative rotation as compared to the reference specimen of HCB-05. The material model M₂ with 1.5% strain-hardening for the steel angles of HCB-05 (M₂) has provided similar moment, post-tensioning force, and gap-opening at the beam-column interface as that of HCB-05. Figure 15 shows the stress contour of HCB-05 (M₂) at 4% lateral drift.

The material of M₃ assumed to have higher yielding strength and 1.5% strain-hardening has been assigned to the last specimen of HCB-05 (M₃) which represented higher moment capacity by 2.2% as compared with HCB-05. The higher yielding strength for the angle material of this specimen has led to greater tensile force on the steel bolts of the angle column leg and gradually the yielding of column bolt shank have been initiated at the first cycle of 3.9% lateral drift. Figure 16 represents the comparative lateral load-displacement response (P_lateral– D) of the specimens with different material models subjected to cyclic loadings. The lower strain-hardening of M₁ has led to lower stiffness of HCB-05 (M₁) after the gap-opening. Also, this specimen has provided lower level of energy dissipation by 11.5% compared with HCB-05. On the other hand, HCB-05 (M₃) has shown 1.6% higher stiffness and 6.7% greater energy dissipation as compared with that of HCB-05. As seen from Fig. 16, the material model of M₂ assigned to HCB-05 (M₂) has provided the same stiffness and energy dissipation as that of HCB-05. From the detailed behavior of HCB-05 (M₂) summarized in Table 5 and Fig. 16, it is concluded that the simple bilinear model of M₂ is perfectly matched with the real steel material of bolted angles as energy dissipaters in PTSC connections with hexagonal castellated beams.

Sensitivity analysis as future work

The changes in values of input parameters have led to different outputs of numerical modeling in this research and it is required to follow a reliable process to calculate the influences of such varying parameters. Sensitivity analysis (SA) is one of the greatest methods to reach this aim [⁵⁴]. Therefore, different SA methods based on a stochastic modeling will be developed to quantify the effects of selected input parameters on the behavior of hexagonal castellated beams in PTSC connections. The input parameters for future work will be the reinforcing plate length of beam flanges, post-tensioning force, variations in steel material properties including the yielding strength and strain-hardening schemes. In this section, a brief description of different methods and recent developments in SA methods is provided.

A stochastic model is required to investigate the quantitative results. For this purpose, Latin Hypercube Sampling is one of the reliable sampling strategy to approximate the stochastic characteristics [⁵⁵–⁵⁶]. SA methods provide useful procedures to determine how the values of outputs are susceptible to the changes in values of input parameters [⁵⁷]. Following generating random samples of input parameters, scatter plots can be adopted to visualize the relations between input variables and output responses. In addition, partial derivatives can be used to determine the sensitivity measures [⁵⁷]. Also, SA can be carried out using the elementary effects method [⁵⁷]. First-order sensitivity index and total-effect sensitivity index are useful to compute the impacts of input parameters on output results [⁵⁷–⁵⁸]. To decrease the computational costs, response surface (RS) methods, called surrogate models, are adopted as the approximation of mechanical models to perform SA [⁵⁹]. The regression models can be developed using the polynomial regression method [⁶⁰] as well as the moving least squares (MLS) regression method [⁶¹]. The details of different SA methods are available in recent research as mentioned below.

Vu-Bac et al. [⁶²] have carried out a SA based on RS models using polynomial regression and MLS approach and determined elementary effects on the mechanical model to evaluate the effects of input parameters on the material properties of amorphous polyethylene on the basis of molecular dynamics simulations. Vu-Bac et al. [⁶³] have investigated the influence of uncertain input variables on interfacial shear stress of polymeric nanocomposites (PNCs) in context of local and global SA. Surrogate models have developed using polynomial, MLS, and hybrid regressions, also elementary effects method have used on the mechanical model which requires less samples than the other methods. Vu-Bac et al. [⁶⁴] have proposed a unified framework using SA methods to determine the impact of a variety of input parameters on mechanical properties of PNCs. In addition, Several SA methods have been considered to quantify the influences of correlated input parameters on the mechanical properties of PNCs in which a hierarchical multiscale modeling has been employed in context of global SA [⁶⁵]. Vu-Bac et al. [⁶⁶] have utilized Matlab functions to provide a software-based SA toolbox comprising sampling model, surrogate models, and SA method to determine how much uncertain outputs are influenced by uncertain input variables.

Conclusions

The FE analysis of the hexagonal castellated beams in PTSC connections has been developed by 3D modeling of ABAQUS 6.11-PR3 software. The details of PTSC connection has been selected from that proposed by Garlock et al. [⁷]. The PTSC connection is comprised of bolted top and bottom angles as energy dissipaters and high-strength steel strands to provide post-tensioning force and self-centering capability reverting the MRFs with such connections to its initial position prior to the earthquake. A simplified analytical method has been proposed to predict the behavior of PTSC connections with hexagonal castellated beams and has been compared with the results obtained from FE analysis. The FE modeling has been also validated against the experimental test conducted by Garlock et al. [⁷], and the validation process has shown that the FE response has highly matched with the experiment. As a consequence, the FE modeling can be reliably applied to study the cyclic behavior of hexagonal castellated beams in PTSC connections. In addition, three different hexagonal castellated beams in PTSC connections investigated by Abedi Sarvestani [³³] have been considered as the reference specimens to examine the effects of the important parametric factors on the behavior of such systems under cyclic loading up to 4% lateral drift. The parametric factors evaluated in this research include the existence of beam flange reinforcing plates, three different initial post-tensioning force, and three different modellings for the steel material of the angles as energy dissipation devices. Therefore, seven specimens have been analyzed and compared with their reference specimens. The results of this research have been summarized as the following comments:

1) The proposed analytical method provides a simple and reliable model to design the MRFs with hexagonal castellated beams in PTSC connections. It accurately predicts the bending moment of PTSC connections with hexagonal castellated beams under cyclic loading. The analytical predictions have shown good agreement with the responses of validated FE analysis. The average amount of modification factor r = 0.61 derived from the FE results can be applied to predict the behavior of hexagonal castellated beams in PTSC connections with the same angle specifications as energy dissipation devices.

2) The existence of reinforcing plates are essential to prevent the local buckling of hexagonal castellated beams in PTSC connections. The specimen with reinforcing plates has represented effective self-centering capability without residual drift, and has resulted in 10.8% greater moment capacity and 17.5% higher initial stiffness as compared with that without reinforcing plates.

3) The parametric studies have shown the initial post-tensioning force has substantial effect on the behavior of hexagonal castellated beams in PTSC connections. The increase in post-tensioning force has boosted the moment capacity and energy dissipation by 4.1% and 7.3%, respectively, as compared with the reference specimen HCB-03. Also, the increased post-tensioning force has resulted in the higher initial stiffness. On the other hand, the hexagonal castellated beam of specimen HCB-03 (PT₃) with much higher post-tensioning force has suffered from the web-post buckling before 4% lateral drift. Although, the higher initial post-tensioning force can enhance the moment capacity, energy dissipation, and self-centering behavior of hexagonal castellated beams in PTSC connections, it should be checked to prevent the web-post buckling of hexagonal castellated beams as the dominant failure.

4) The material strain-hardening of steel angles has considerable influence on the energy dissipation and maximum moment resisted by hexagonal castellated beams in PTSC connections. The bilinear elastoplastic modeling with 1.5% strain-hardening of steel material (M₂) for the bolted angles has provided the same cyclic behavior as that of validated reference specimen HCB-05. However, the elastic-perfectly plastic modeling has led to 11.5% lower energy dissipation and 5.4% lower moment capacity. As a result, the proposed steel modeling M₂ is an acceptable stress-strain relationship used for simulating the material cyclic behavior of steel angles as energy dissipation devices.

5) The yielding strength has been another effective factor on the cumulative energy dissipation, maximum moment resistance, and self-centering behavior of the hexagonal castellated beams in PTSC connections. The FE results have shown that the greater yielding strength of steel material of the bolted angles has led to yielding of bolt shank of angle column leg since the high yielding strength of steel angles has contributed to higher force of column bolts. Consequently, the steel angles with the material of high yielding strength are not suitable to use for hexagonal castellated beams with PTSC connections.

This research will be extended to determine the sensitivity of selected input parameters. Several SA methods will be considered to quantify the influence of input parameters on the behavior of hexagonal castellated beams in PTSC connections. The input parameters will be the length of reinforcing plate, post-tensioning force, and variations in steel material characteristics (yielding strength and strain-hardening models).

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