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International Journal of Concrete Structures and Materials Slab Reinforcement Contributions to Negative Moment Strength of Reinforced Concrete T‑Beam with High Strength Steel at Exterior Beam‑Column Joints Zhamilya Mamesh1, Dilnura Sailauova1, Dichuan Zhang1*   , Hyunjin Ju2, Deuckhang Lee3 and Jong Kim1  Abstract  Previous studies have revealed that the contribution of slab reinforcement to the T-beam flexural strength in nega- tive moment regions are not negligible for the seismic capacity design. An effective slab width (i.e., effective width of flanged section) has been proposed, within which the slab reinforcement needs to be included in the calculation of the beam nominal flexural strength in negative moment regions. These studies mainly focused on the cases using normal-strength steel in moment resisting frames. However, recently high-strength steel has been widely used in rein- forced concrete moment resisting frames in high seismic regions to avoid congestion near beam-column joints. The use of high-strength steel may affect the beam stiffness due to the fact that it will require less amount of reinforce- ment, and result in a different normal stress distribution compared to the case with normal-strength steel. Therefore, this paper investigates the slab reinforcement contribution to the flexural strength of the reinforced concrete T-beam designed with high-strength steel in negative moment regions at exterior beam-column joints, for which nonlinear pushover analyses were conducted. Beam reinforcement grade was considered as a primary parameter with sev- eral other design variables including slab thickness, height, and span length of the beam. Analytical results show that the use of high-strength steel can result in a wider effective slab width than the case of normal-strength steel for calculating the beam nominal flexural strength under the negative moment. Based on these results, new design equations were proposed. Keywords  Effective slab width, High strength steel, Negative moment strength, Seismic design, Nonlinear finite element analysis 1  Introduction In high seismic regions, reinforced concrete (RC) moment resisting frames might have a congestion issue due to excessive amounts of reinforcement in beam-col- umn joints. Moreover, this congestion may result in high cost and low efficiency during construction (Kotsovos et al., 2013; Madias et al., 2017). One of the effective solu- tions to resolve this issue can be the use of high-strength steel (HSS) instead of normal-strength steel, particularly in beam-column joints of the moment-resisting frame to satisfy high seismic detailing requirements (NIST, 2014; Shahrooz et  al., 2011). For the same structural Journal information: ISSN 1976-0485 / eISSN 2234-1315. *Correspondence: Dichuan Zhang dichuan.zhang@nu.edu.kz 1 School of Engineering and Digital Sciences, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan 010000 2 Hankyong National University, 327, Jungang‑Ro, Anseong‑Si 17579, Gyeonggi‑Do, Korea 3 Chungbuk National University, 1 Chungdae‑Ro, Seowon‑Gu, Cheongju 28644, Chungbuk, Korea http://creativecommons.org/licenses/by/4.0/ http://crossmark.crossref.org/dialog/?doi=10.1186/s40069-023-00634-z&domain=pdf http://orcid.org/0000-0002-9253-4178 Page 2 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 component, application of HSS can significantly reduce the required amount of steel in comparison to normal- strength steel, thus preventing the reinforcement conges- tion and decreasing the material cost (Alavi-Dehkordi et al., 2019; Alaee & Li, 2017). Recently, ACI 318 (2019) has allowed using HSS in special moment frames, but with limited yield strength up to 550 MPa due to the serviceability issue. Despite the restrictions, several researchers have used HSS with the yield strength higher than 550 MPa in beam-column joints, thus showing that the crack width was within the allowable limits in struc- tures with HSS (Alaee & Li, 2017; Alavi-Dehkordi et al., 2019; Chang et al., 2014). The lateral-load resisting components (beams, col- umns, and beam-column joints) are the most critical elements in the structural design for the reinforced con- crete moment resisting frame  (Wang et  al., 2016). Cur- rent seismic design practices in Europe, North America, and Asian regions have commonly adopted a so-called “strong-column and weak-beam” concept to allow the yielding of beams under earthquakes. According to this criterion, the flexural strength of columns must be at least 20% greater than the actual flexural strength of the beams at the joint (ACI 318, 2019; Eurocode 8, 1996). There- fore, it is important to accurately estimate the flexural strength of the beam during the seismic design. The slab reinforcement contribution is not significant for the posi- tive moment since the slab is under compression. How- ever, for the negative moment, when the top of the beam is under tension, some portions of the slab reinforcement located close to the beam contribute to the moment resistance, and this contribution becomes less depending on the distance from the center of the beam due to the shear lag effect. Existing studies have proposed an effec- tive slab width, in which slab reinforcement would con- tribute to the flexural strength of the beam (Durrani & Zerbe, 1987; Pantazopoulou et  al., 1988; Qi et  al., 2010; Zhen et al., 2009). However, the equations for the effec- tive slab width in the tension zone suggested in the exist- ing studies as well as those specified in code provisions are primarily for the structures configured with normal- strength steel. Since the use of HSS may affect the normal stress distribution in the beam and slab reinforcement, it is important to identify whether the effective slab width for the beams with normal-strength steel is applicable to the beams with HSS. However, only few studies have been conducted regarding the slab reinforcement contri- bution and effective slab width of the beam with HSS in the tension zone. For a better understanding of this phenomenon, this paper investigates the effect of beam reinforce- ment strength on the effective slab width for the exte- rior beam-column joints through analytical studies. Nonlinear static pushover analyses were conducted. To this end, a three-dimensional (3D) multi-truss model was adopted by applying the vertical displacement at the beam free end for an idealized beam-column-slab assembly. The beam reinforcement strength was set as the key influential factor, while several other parame- ters affecting the effective slab width were also consid- ered in the study including slab thickness, height, and span length of the beam. 2 � Background 2.1 � Effective Slab Width In the negative moment region, the imbalance of tension forces between mid-span and end support induces the shear flow at the web-flange joint, as shown in Fig. 1. This shear flow makes the normal stresses in the flange lag behind that of the web, thus causing Shear lag. Due to the uneven stress distribution, the concept of effective slab width has been introduced to simplify the beam section analysis (Moffatt & Dowling, 1978). The effective slab width (Beff) represents reduced slab width with a uniform stress distribution that replaces the actual stress distribu- tion to be equivalent to the total force. In seismic design, the “strong-column and weak-beam” principal has been implemented to make structural beams with high ductility weaker than columns with low ductility, so that the structure could have a ductile response rather than brittle collapses. According to Sec- tion Clause 18.7.3.2 in ACI 318-19, at a beam-column joint, it is required that the total moment capacity of the column should be 1.2 times greater than that of the beam to ensure “strong-column and weak-beam” design: where ∑ Mnc = the sum of the nominal moment capaci- ties of columns at the surface of the joint; ∑ Mnb = the sum of the nominal moment capacities of beams at the surface of the joint. Under negative moment, slab ten- sion reinforcement lying within the effective width (Beff) would contribute to nominal beam flexural strength, Mnb. A number of studies have suggested design equations for the effective slab width to evaluate the contribu- tion of slab reinforcement to the beam flexural strength (Durrani & Zerbe, 1987; Ning et  al., 2016; Pantazopou- lou, 1988; Qi et al., 2010; Zhen et al., 2009). Table 1 sum- marizes the suggested design equations in the existing studies and building codes with key parameters used in the equations. The parameters include beam height, slab thickness, and span length of beam. Moreover, all the equations are suggested for RC beams reinforced with normal-strength steel, which requires further studies on (1) ∑ Mnc ≥ 1.2 ∑ Mnb, Page 3 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 the effective slab width for RC beams configured with HSS. 2.2 � Review of RC Structures with High‑Strength Steel The use of HSS reinforcement can be an effective method in reducing the required amount of steel in RC members, thus resulting in cost savings in material, transportation, and placement (Alavi-Dehkordi et  al., 2019; Shahrooz et al., 2011). Different types of HSS have been developed, and those are currently available in international markets (NIST, 2014). Various grades of reinforcing steel have been adopted in ASTM specifications based on their production meth- ods. ASTM A706 includes deformed and plain low alloy steel reinforcing bars with limited mechanical proper- ties (ASTM, 2016a). Their Grade 60 and 80 reinforce- ments were the first HSS bars used for seismic purposes in moment-resisting frames and special structural walls Fig. 1  Shear flow occurred in the flange of the T beam under a negative moment: a Beam span; b Actual; c Assumed uniform stress distribution for design Table 1  Effective slab width in code provisions and existing studies b = web thickness, ts = slab thickness, h = beam height, d = beam effective depth, Lo = clear distance to the adjacent web, Ln = clear span, L = beam span Codes/existing studies Equations for Bef f Parameters Beam reinforcement ACI 318 (2019) b + Min (16ts, L0, Ln/4) b, ts, L0, Ln Normal-strength steel Eurocode 8 (1996) b + 4ts b, ts SNiP code (1997) Min (L/6, b + 12ts) b, ts, L Durrani and Zerbe (1987) b + 2 h b, h Pantazopoulou et al. (1988) b + 6d b, d Zhen et al. (2009) b + 4 h b, h Qi et al. (2010) b + Min (Max(L/5, 3 h), L0/2) b, h, L, L0 Ning et al. (2016) b + 5.4 h b, h Page 4 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 (NIST, 2014). ASTM A615 presents the deformed and plain carbon steel bars which include HSS up to Grade 100 (ASTM, 2016b). ASTM A1035 reinforcing steel pro- duced in Grades 100 and 120 is composed of chromium alloy with low carbon content (NIST, 2014). Fig. 2a pre- sents the typical stress–strain curves for different grades of HSS from various studies (Barbosa et al., 2016; CRSI, 2016; El-Hacha & Rizkalla, 2002; Huq, 2018). As can be seen, both A706 and A615 have clearly defined yielding points (from Grade 60–100), while ASTM A1035 rein- forcing steel shows no clear yielding point (from Grade 100–120). Fig.  2b converts the stress–strain curves into the force–displacement curves by imposing the same yield force and gauge length for different steel grades. Since the yield strength for different steel grades is dif- ferent, the cross-section area does also vary. As seen in Fig.  2b, the high-strength steel shows a lower stiffness than the normal-strength steel. There have been numerous studies on the seismic behavior of HSS in RC structures including structural walls, beams, columns, and beam-column joints (Alaee and Li, 2017; Alavi-Dehkordi et  al., 2019; Cheng et  al., 2016; Li et  al., 2018). Also, some studies were reported on the beam-column joints in both interior and exterior cases (Alaee and Li, 2017; Alavi-Dehkordi et  al., 2019; Chang et  al., 2014). The yield strength of reinforcement in these studies ranged from 420 to 1034 MPa. Addition- ally, It has been discovered that the use of HSS as a beam longitudinal reinforcement with decreased cross-sec- tional area resulted in the reduction of energy dissipation capacity and stiffness of beam-column joints. Besides, beam-column assemblies configured with a lower rein- forcement ratio of HSS can obtain a comparable load-car- rying capacity to those with normal-strength steel (Alaee & Li, 2017; Alavi-Dehkordi et al., 2019). Aoyama (2001) studied slab effect on flexural behavior of beams using HSS for slab reinforcement. The assembly consisted of beam and slab with 689 MPa and 1034 MPa for beam and slab reinforcements, respectively. It has been observed that high-strength slab reinforcement contributes to the beam flexural strength in the negative moment regions (Aoyama, 2001). However, his study did not evaluate the effective width of the slab for the beam with HSS. Fig. 2  a Stress–strain curves; b force–displacement curves of different reinforcement grades (*without clear yielding points) Table 2  Study parameters Beam reinforcement Type Slab thickness, ts (mm) Beam height/Beam span, h/L (mm/mm) Grade Yield strength (MPa) Reinforcement ratio, ρ (%) 60 420 1.5 Type I (with a clear yield- ing point) 100 500/5000 80 593 1 150 600/6000 100 690 0.85 Type II (without a clear yielding point) 200 700/7000 120 830 0.7 Page 5 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 3 � Description of Analitical Study 3.1 � Study Parameters Several parameters were considered in this study to investigate their effects on the normal stress distribu- tion and the slab effective width, as presented in Table 2. The main study parameter is the beam reinforcement grade, i.e., the yield strength, which ranged from 420 to 830  MPa. To keep a similar strength for the beam, the total yielding force ( T = fyAs ) was set to be constant for the beam with different steel grades, which subsequently resulted in different beam reinforcement ratios (ρ). Also, two types of beam reinforcement grades (Type I and Type II) were used in this study. Grades 60, 80, and 100, which have a clear yielding point, are applied to Type I, while Grades 100 and 120 without a clear yielding point are considered as Type II reinforcement. Other parameters considered in this study are slab thickness (t), height (h), and span length of beam (L) because most of the studies included these parameters in their design equations for the effective slab width. Typically, the height of beam varies with the span length. Therefore, in this study, those parameters were chosen to be varied proportional, as summarized in Table 2. The other study parameters were kept as constants to represent typical dimensions of beam-column-slab assemblage at an exterior joint. Dimensions of trans- verse beams are taken the same as that of the longitudinal beam under study. The beam width is set as 50% of the beam height. The square column with a length of 3 m and the same height as the beam height was used. The slab reinforcement is kept as Grade 60 and #3 rebar with a cross-sectional area of 71 mm2 with a constant spacing of 150 mm. Compressive and tensile strength of unconfined concrete are 41 MPa and 4 MPa, respectively. 3.2 � Model Development The 3D multi-truss modeling approach was adopted to con- struct an exterior beam-column-slab assemblage, rather than a 3D model with solid elements. The multi-truss model has less numbers of degree of freedoms as compared to the solid model, which is able to save the computational cost. The solid model approximates the discrete cracking behavior with continuous plasticity. It tends to have a con- centrated deformation in the cracking region which causes deformation incompatibility along the height of the beam and the width of the slab. On the other hand, the multi- truss model is able to capture the discrete cracking behavior through the deterioration of individual nonlinear truss ele- ment and the elastic beam element inside the truss assem- blage is able to maintain the deformation compatibility in the cracking zone. The 3D multi-truss modeling approach was used to model reinforced concrete shear walls under seismic loading and showed good agreement with the test results (Panagiotou et al. 2012). This approach has also used in the modeling of precast concrete connectors (Zhang et al., 2016), and the models were verified by structural tests (Zhang et al., 2011).  The analytical model has been developed by using the commercial finite-element (FE) analysis platform ABAQUS, as shown in Fig. 3. The model consists of two types of the elements considering the characteristics of components: (1) linearly elastic beam elements, which were used to model the transverse beam, column, and vertical elements of beam and slab; (2) nonlinear truss elements for the concrete beam and slab as well as the longitudinal reinforcement as shown in Fig.  3. The assemblage represents a cut of the exterior joint from inflection points in the moment resisting frame. The slab is modeled with a layer of horizontal nonlinear trusses placed at the center of the slab thickness. The lon- gitudinal beam is modeled with a layer of vertical nonlinear trusses placed at the center of the beam width. These lay- ers of trusses were divided into 40 mm by 40 mm subsets to represent the solid concrete. The subset consists of horizon- tal trusses, diagonal trusses, and elastic beams. Beam and slab reinforcement are modeled as nonlinear truss elements and embedded in the truss layers. The transverse beam and column are modeled as 3D elastic beam elements and are rigidly connected to the slab nodes and beam nodes, respec- tively. Pinned boundary conditions (i.e., UX = UY = UZ = 0, where U denotes the translation, and X, Y, and Z indicate each direction in three dimensional space) were applied at the top and bottom nodes of the column. Beam nodes were constrained in the Z-axis direction (UZ = 0) to prevent out- of-plane movement. The length of each beam and truss element is set to be 40 mm based on a preliminary mesh sensitivity study (Mamesh, 2022). Nonlinear pushover anal- yses were conducted by applying an incremental downward vertical displacement (∆) at the beam end nodes. The concrete material is simulated with the concrete damaged plasticity model (CDP), which is implemented in ABAQUS software (2010) with isotropic damage. The concrete model is able to capture brittle tensile cracking and compressive crushing behavior. Fig.  4a, b present the compressive and tensile stress–strain relationship of the confined and unconfined concrete based on mate- rial properties obtained from tests conducted by Durrani and Zerbe’s (1987), respectively. The confined concrete behavior was simulated by using Mander’s model (1988), and it was applied to the inner horizontal trusses of the beam, as shown in Fig. 4c. The concrete was modeled as multi-truss elements to achieve an equivalent strength and stiffness for a solid concrete block. As shown in Fig. 5, solid elements were converted into the subset of truss elements, where its size is 40 mm (be = Le), by matching their global strength and stiffness for tensile, compressive, and shear behaviors. Page 6 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 Horizontal trusses were used to model the tension and compression behavior of concrete, while diagonal trusses (θ = 45°) were used to model the coupled shear and axial behavior of concrete. Elastic vertical beams were used to keep deformation compatibility after the truss element cracks or crushes. By assigning concrete strength (i.e., tension and compression) to the truss element, the cross- sectional area and Young’s modulus of each truss ele- ment were calibrated to achieve the same global strength and stiffness as the solid concrete. The calculated values for beams and slabs are presented in Table  3, and the detailed calculation and calibration process for the multi- truss element can be found in elsewhere (Mamesh, 2022). For the longitudinal reinforcement, the same mechani- cal properties of reinforcing bar shown in Fig.  3 were directly assigned to the truss element. The bonding between reinforcement and concrete was assumed to be rigid (i.e., no bond slip condition). 3.3 � Model Validation An experimental study conducted by Durrani and Zerbe (1987) was used to validate the analytical model. The study investigates the exterior beam-column-slab assem- blage subjected to cyclic loadings. The column has a square section with a height of 305  mm. The longitudi- nal beam with a section of 381 × 254 mm has a length of 1.676 m. The slab thickness and width are 102 mm and 1778  mm, respectively. A transverse beam with a sec- tion of 381 × 254  mm was provided. The dimensions of the assemblage and reinforcement detailing are given in Fig.  6. The longitudinal beam and column ends were provided with hinged connections. Concrete strength on the day of the test was 40.68 MPa, and the specimen was configured with Grade 60 reinforcement. During the test, cyclic vertical displacements were applied at the end of the longitudinal beam. One cycle per amplitude was executed with an increasing amplitude of 12.7 mm. The maximum amplitude is 101.6 mm at the 8th cycle, which is corresponding to a 6% inter-story drift. The beam and slab reinforcement were instrumented with strain gauges to measure the strain/stress at the maximum displace- ment of each loading amplitude. More information for the tests can be found in Durrani and Zerbe (1987). Fig. 3  Multi-truss model of beam-column-slab assemblage Page 7 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 Fig.  7 shows the load (reaction at the beam free end) vs. drift ratio (vertical displacement normalized by the length of the beam) response of the multi-truss model in comparison with the experimental data. The multi- truss model shows similar behavior as observed from the experiment. The discrepancy in overall behavior might be Fig. 4  Modeling of concrete: a compression behavior, b tension behavior, and c truss model Fig. 5  Conversion from solid to truss elements Table 3  Input parameters of truss elements for concrete model Elements Beam Slab Properties Horizontal truss Diagonal truss Horizontal truss Diagonal truss Cross-sectional area (mm2) 3923 1298 1800 196 Young’s modulus (GPa) 29.5 132.55 25.72 355.01 Page 8 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 attributed to the fact that a cyclic loading was applied in the test while a monotonic pushover was applied in the model. Fig. 8 demonstrates the strain distribution in slab rein- forcement for the multi-truss model compared to experi- mental results at different drift ratios. The simulation reasonably captures the general trend of experimental results. 4 � Analytical Results 4.1 � Effects of Beam Reinforcement Grade This section aims to evaluate the effects of beam rein- forcement grade on the slab reinforcement contribu- tion to the beam flexural strength for the cases with ts = 150  mm, h = 500  mm, and L = 5000  mm, where ts Fig. 6  Reinforcement detailing and dimensions of the experimental set-up, modified from (Durrani & Zerbe, 1987) Fig. 7  Load vs. drift ratio curves of simulation and experiment Fig. 8  Strain distribution in slab reinforcement at different drift ratios Page 9 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 is the slab thickness, h is the beam height, and L is the length of a beam member. Fig.  9a, b shows the load vs. drift ratio curves for the model with Type I and Type II condition of the beam reinforcement grades, respectively. It highlights the points at the key events, such as yielding of beam rein- forcement and the unconfined concrete strain reaches the ultimate strain, εcu = 0.003. The load-drift ratio responses response indicates that the strength of the assemblage can still increase after the unconfined concrete strain reaches the ultimate strain due to the fact that the con- fined concrete can still carry the compression force. The load-drift ratio responses are similar for all cases due to the beam was designed with the same strength regard- less of the grade of the steel; however, the beam with a higher steel grade shows a lower initial stiffness and delayed reaching of ultimate strain at a larger drift ratio. This is because the HSS reinforcement has a smaller total cross-sectional area than the normal-strength steel does for achieving a similar total tensile strength. Although the beam flexural stiffness is not significantly affected by the steel grade (see Fig. 9a), this small influence can play a role in changing the stress distribution of the slab rein- forcement which can be observed in Fig. 10. Fig. 9  Load vs. drift ratio response for the model with (a) Type I and (b) Type II beam reinforcement Fig. 10  Stress and strain distribution in the reinforcement along the slab when the unconfined concrete reaches ultimate compression strain: a Strain and (b) Normalized stress (*Type II reinforcement) Page 10 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 Fig. 10a, b shows distributions of strain and correspond- ing normalized stress (by the yield stress) in reinforce- ment along the width of the slab, respectively, at the point when unconfined concrete reaches ultimate compression strain. As shown in Fig. 10, the strain and stress in the slab reinforcement gradually reduce as a considered location moves from the center of the beam to the edge of slab due to the shear lag effect. The unconfined concrete reaching its ultimate compression strain occurs at larger drifts for HSS cases due to their higher flexibility as shown in Fig. 9. Therefore, the cases with HSS have a higher strain at the same location in the slab reinforcement, as compared to the case with normal-strength steel. As a consequence, when the beam reinforcement grade increases, the stress in the slab reinforcement increases and more slab reinforce- ments can be yielded. This trend indicates a wider effective width for the cases with HSS at the point when the uncon- fined concrete reaches its ultimate compression strain. The contribution of the slab reinforcement to the beam flexural strength can be estimated by the summation of the tensile forces in the slab reinforcement at each side of the beam. The effective width (Beff) can then be calculated to have an equivalent total tensile force when the reinforce- ment within this width is considered to be yielded, as shown in Fig. 11. The amount of slab reinforcement within the effective slab width can be determined by dividing the total tension force ( Fn , summation of the tensile forces in all slab rein- forcement) by the yield tensile force in a single slab rebar ( Abfy ), which gives the number of slab reinforcement ( N  ) yielded in an average sense, as follows: where Ab and fy are the cross-sectional area and yield strength of single slab rebar, respectively, and σi is the actual stress in each slab reinforcement within one side overhanging as the stress/strain distribution is symmetric on both sides. The number of bars ( N  ) is then multiplied by slab reinforcement spacing ( s ) to find overhanging effective width ( Beff ′ ), as follows: On this basis, the effective slab width ( Beff  ) can be computed by adding the beam width ( b ) to the twice of the overhanging width, as follows: (2)N = Fn Abfy = Ab ∑ σi Abfy = ∑ σi fy , (3)Beff ′ = N × s. Fig. 11  Calculation of effective slab width in negative moment regions Fig. 12  Effective slab width at different drift ratios Page 11 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 Fig.  12 shows the effective slab width calculated for the cases with different beam reinforcement grades using Eq. (4) at different drift ratios and the point when concrete reaches ultimate compression strain. It can be observed that the effective slab width increases as the drift ratio increases due to the stress/strain penetration into the slab. At large drift ratios from 1.5% to 2.5%, the effective slab width is similar regardless of the grades of the beam reinforcement. However, at lower drift ratios (< 1%), the effective slab width increases with the increase of the beam reinforcement grade, particularly at a point when the unconfined concrete reaches ultimate com- pression strain. Since the nominal moment strength is typically defined at a point when the concrete reaches the ultimate compression stain, the effective slab width cal- culated for this point will be presented and discussed in the following sections. 4.2 � Effect of Slab Thickness In this section, various slab thicknesses are chosen from 100 to 200 mm to evaluate its effect on the effective slab width calculated for the point when the unconfined con- crete reaches ultimate compression strain. This point was also used for calculation of the nominal flexural strength according to ACI 318 (2019). Different beam reinforce- ment grades were also used, namely Grade 60 and Grade 80 for Type I, and Grade 120 for Type II. The height and span length of the beam member were kept as 500 mm and 5000 mm, respectively. (4)Beff = b+ 2Beff ′ = b+ 2 ∑ σi fy s. Fig.  13 shows the effect of the slab thickness on the effective slab width for the cases with different beam reinforcement grades. Increase in slab thickness showed no significant effect on Beff. This is because the Beff is calculated at the tension side, where the concrete has cracked, and its influence is limited. From Fig. 13a, it can be also confirmed that the Beff increases with increase of the beam reinforcement grade. This increasing trend is similar among the different slab thicknesses, as observed in Fig.  13b, where the Beff was normalized by the case with Grade 60 beam reinforcement. Therefore, the effect of slab thickness can be ignored for calculating effective slab width with HSS. 4.3 � Effect of Height and Span Length of Beam In this section, the dimensional properties (i.e., height and span length) of the beam varied from 500 to 700 mm and 5000 to 7000  mm for the height and span length, respectively, to examine their effects on the effective slab width calculated for the point when the unconfined con- crete reaches ultimate compression strain. Note that the ratio between the beam height and span length was set to be the same, and cases with different beam reinforcement grades were also considered. In addition, the slab thick- ness was kept as 150 mm. Fig. 14 shows the effect of the height and span length of the beam on the effective slab width for the cases with different beam reinforcement grades. As shown in Fig.  14a, the higher the height/span length of the beam is, the slightly higher Beff is. This trend is consistent with the studies in the literature (refer to Table 1). Fig. 14a also confirms that the Beff increases with the increase of the beam reinforcement grade. This increase trend is almost Fig. 13  Effect of the slab thickness on Beff: a Absolute values and (b) Normalized values Page 12 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 the same among the different heights/span lengths of the beam as observed in Fig.  14b, where the Beff was normalized by the case with Grade 60 beam reinforce- ment. Therefore, the effect of the height/span length of the beam on the slab effective width can be considered independently. 5 � Design Recommendations The effective slab width calculated at the point, when the unconfined concrete reaches ultimate compression strain, for different cases is compared with the values determined based on the design equations from the code provisions including ACI 318 (2019), Eurocode 8 (1996), and SNiP code (1997) as well as existing studies pre- sented in Table 1. Through this comparison, it is possible to modify the proposed equations from the literature to include the effect of the beam reinforcement grade. The modified expression can then be used for calculating the nominal beam flexural strength with the HSS. Fig. 15 shows the comparison of the effective slab width at the point, when the unconfined concrete reaches ulti- mate compression strain, among the current study, code provisions, and existing studies. All the cases considered in this study are plotted in Fig.  15. The code provisions generally have a lower effective slab width, and it means those code models underestimated the slab reinforce- ment contribution to the nominal beam flexural strength. The results obtained from the multi-truss approach show a similar higher Beff as the studies in the literature, and it appeared that current model is able to capture the effect of the beam reinforcement grade. The Beff obtained from the multi-truss model is quite similar with those estimated from Zhen et  al. (2009) for the cases with the Grade 60 beam reinforcement, as shown in Fig. 15. Therefore, this study modifies the Zhen et  al.’s equation to include the effect of the beam rein- forcement grades. If Zhen et al.’s equation is taken as the Beff for normal-strength steel ( Beff ,G60 ), a modification coefficient of β can be introduced, as follows: Fig.  16 shows a comparison of Beff among the values calculated by using the proposed Eq.  (6), Zhen et  al.’s original equation, and simulation results from this paper. As presented in the Fig.  16, the proposed equation can reasonably capture the effect of the beam reinforcement grade on the slab effective width in a simple manner. In general, the proposed equation shows a slightly wider Beff than the simulation results, which can be more conserva- tive to satisfy the “strong column, weak beam” criterion. 6 � Conclusions This study investigates the slab contribution to the nega- tive moment strength in the tension zone for T-beam configured with HSS in the exterior beam-column joints. The nonlinear pushover analyses were conducted on a multi-truss model of beam-column-slab assemblages to identify the effect of the beam reinforcement grade on the stress and strain distribution along the width of the slab. Moreover, the effects of other influential factors (5)Beff ,G60 = b+ 4h for Grade 60 ( fy = 420MPa ) . (6)Beff = β × B eff ,G60 . (7)β = ( fy 420 )0.15 . Fig. 14  Effect of the height/span length of the beam on Beff: a Absolute values and (b) Normalized values Page 13 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 on the effective slab width were also considered, which includes the slab thickness, height, and span length of the beam. On this basis, the following conclusions can be drawn: • The slab reinforcement can contribute to the beam flexural strength in the negative moment region. The contribution reduces as moving far from the center of the beam due to the shear lag effect, though. • When the grade of the beam reinforcement increases, the behavior of beam-column-slab assem- blage becomes more flexible under lateral loads. This higher flexibility results in a larger drift ratio at the point when the concrete in the beam reaches ulti- mate compression strain. The larger drift ratio at this point increases the stress/strain in the slab reinforce- ment, resulting in a wider effective slab width for the cases with HSS. • The effective slab width increases with the increase of the drift ratio since the strain penetrates into more slab reinforcement. At large drift ratios, the effective slab width is insensitive to the beam rein- forcement grade. However, at low drift ratios, this width increases as the beam reinforcement grade increases, particularly at the point where the uncon- fined concrete reaches ultimate compression strain. • The effective slab width for the negative moment is insensitive to the slab thickness since the concrete has cracked. On the other hand, the height/span length of the beam influences the effective slab width. However, its effect is independent of the one from the beam reinforcement grade. • In general, design code provisions tend to have a lower effective slab width and underestimate the contribu- tion of the slab reinforcement. The simulation results obtained from this paper as well as those from existing studies show a much higher effective slab width. The results from this paper indicate an effect of the beam reinforcement grade on the effective slab width. • Based on the parametric study results, a modifica- tion coefficient was derived to incorporate the effect of the beam reinforcement grade on the effective slab width. This coefficient was applied to the design equation developed by Zhen et  al. (2009) which shows a similar result for the case with normal- strength reinforcement. Fig. 15  Comparison of the effective slab width at the concrete crushing point among this paper, code provisions, and existing studies Page 14 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 • The proposed simple expression with the modifica- tion coefficient can reasonably capture the effective slab width obtained from the simulations, and it can be applied for the reinforced concrete T-beam con- figured with HSS in negative moment regions to esti- mate the beam nominal flexural strength with contri- butions of slab reinforcement. This study was limited to the investigation of exterior beam-column joints. For the interior joint, the effec- tive slab thickness might be different due to the effect of moment on the other side of the column. Therefore, this developed equation (Eq. 5) might only be applicable to the exterior beam-column joint and further valida- tion is needed for the interior beam-column joint design. At the same time, the developed equation (Eq.  5) was purely from numerical simulations, and it might require validation from a further experimental study on the high- strength steel before applying it for the seismic design in the construction practices. Abbreviations A � Area of slab reinforcement AD � Area of diagonal truss element AH � Area of horizontal truss element b � Longitudinal beam width bc � Column width Beff � Effective slab width Beff’ � Overhanging effective slab width d � Effective beam depth Ec � Young’s modulus of the concrete ED � Young’s modulus of diagonal truss element EH � Young’s modulus of horizontal truss element fc’ � Compressive strength of concrete ft � Tensile strength of concrete FT ,d � Tension forces assumed to be developed along the diagonal truss bar FC ,d � Compression forces assumed to be developed along the diagonal truss bar FV ,d � Vertical components of the forces along the diagonal truss fy, σy � Yield strength of reinforcement Gc � Shear modulus of the concrete h � Longitudinal beam height Ilong � Moment of inertia of the longitudinal beam Itrans � Moment of inertia of the transverse beam KC/T � Total stiffness of the solid element KC/T ,hor � Shear stiffness of horizontal element KC/T ,diag � Shear stiffness of diagonal element L � Beam span L0 � Clear distance to the adjacent web Fig. 16  Effective slab wdith from the proposed equation: (a) h/L = 500/5000 mm; (b) h/L = 600/6000; (c) h/L = 700/7000 Page 15 of 16Mamesh et al. Int J Concr Struct Mater (2023) 17:69 LD � Length of diagonal truss element LD,v � Length of vertical component of elongated/shortened diagonal truss Le � Length of solid element LH � Length of horizontal truss element Mnb � The nominal moment capacity of beams calculated at the surface of the joint Mnc � The nominal moment capacity of columns calculated at the surface of the joint N � Number of bars within the effective slab width s � Slab reinforcement spacing ts � Slab thickness β � Adjustment coefficient for Beff to include the effect of HSS δ, Δ � Vertical displacement ν � Poisson’s ratio of the concrete, εcu � Ultimate strain of concrete εd � Strain of diagonal truss element θ � Angle between horizontal and diagonal truss elements ρ � Beam reinforcement ratio σC/T � Tension/compression stress developing in solid element σi � Stress in each slab reinforcement τ � Shear stress across the solid element surface Acknowledgements Nazarbayev University funded this research under Faculty Development Com- petitive Research Grant No. 021220FD2151. The fourth author also would like to acknowledge the support from the National Research Foundation of Korea (KRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C2093437). Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Nazarbayev University. Author contributions ZM: formal analysis, investigation, visualization, writing—original draft. DS: data curation, investigation, validation. DZ: conceptualization, funding acquisition, investigation, methodology, supervision, writing—review and editing. HJ: funding acquisition, investigation, methodology, supervision, writing—review and editing. DL: conceptualization, methodology, resources, writing—review and editing. JK: funding acquisition, project administration, resources, supervision. Funding Nazarbayev University: Research Grant No. 021220FD2151. National Research Foundation of Korea (KRF): 2021R1C1C2093437. Availability of data and materials Not applicable. Declarations Ethics approval and consent to participate We confirm that we have given due consideration to the protection of intel- lectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. Consent for publication We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. Competing interests Not applicable. Received: 27 January 2023 Accepted: 14 August 2023 References ABAQUS (2006) ABAQUS Analysis User’s Manual (Version 6.6). Providence, RI: Dassault Systems SIMULIA Corporation. 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Experimental evalua- tion of pretopped precast diaphragm critical flexure joint under seismic demands. Journal of Structural Engineering, 137(10), 1063–1074. Zhang, D., Fleischman, R., Naito, C. J., & Zhang, Z. (2016). Development of dia- phragm connector elements for three-dimensional nonlinear dynamic analysis of precast concrete structures. Advances in Structural Engineering, 19(2), 187–202. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in pub- lished maps and institutional affiliations. Zhamilya Mamesh  Master Student, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan, 010000. Dilnura Sailauova  Master Student, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan, 010000. Dichuan Zhang  Associate Professor, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan, 010000. Hyunjin Ju  is a Assistant Professor, Hankyong National University, 327, Jungang-ro, Anseong-si, Gyeonggi-do, 17579, Korea. Deuckhang Lee  is a Associate Professor, Chungbuk National Uni- versity, 1 Chungdae-ro, Seowon-gu, Cheongju, Chungbuk 28644, Korea. Jong Kim  is a Professor, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan, 010000. Slab Reinforcement Contributions to Negative Moment Strength of Reinforced Concrete T-Beam with High Strength Steel at Exterior Beam-Column Joints Abstract 1 Introduction 2 Background 2.1 Effective Slab Width 2.2 Review of RC Structures with High-Strength Steel 3 Description of Analitical Study 3.1 Study Parameters 3.2 Model Development 3.3 Model Validation 4 Analytical Results 4.1 Effects of Beam Reinforcement Grade 4.2 Effect of Slab Thickness 4.3 Effect of Height and Span Length of Beam 5 Design Recommendations 6 Conclusions Acknowledgements References