Articles Yvonne Steige* Klaus Weynand DOI: 10.1002/stco.201510023 Design resistance of end plate splices with hollow sections This paper presents a design approach for calculating rectangular hollow section (RHS) splices (bolted end plate connections) under tension forces or bending moments in accordance with EN 1993-1-8. Based on models available in the literature, a Euro­ code-conform model is presented using the component method. The original model, based on experimental and numerical investigations, uses a three-dimensional yield line method to predict the tension resistance of bolted splices with hollow sections considering the joint as a whole. The adapted model is fully compatible with EN 1993-1-8. Moreover, the original model has been extended to predict also the design moment resistance of such RHS splices. 1 Background EN 1993 Part 1.8 contains application rules for evaluating the resistances of end plate connections with open sections (chapter 6) by means of the component method. Furthermore, the standard provides rules for calculating the design resistance of welded hollow section joints in lattice girders (chapter 7). However, there are no explicit application rules or design formulas for bolted end plate joints with hollow sections. Bolted end plate joints are normally used as chord splices in lattice girders under normal loading conditions. Typical bolt patterns in RHS splices are bolts on two opposite sides (Fig. 1a) or bolts placed on four sides of the hollow section (Fig. 1b). The hollow section is connected to the end plate by a one-sided fillet weld around the perimeter of the section. Full strength welds are recommended by Eurocode 3. The topic of the design resistance of two-sided end plate splices with RHS members is discussed in Packer et al. [5]. They present a design model using a modified T-stub model, which observed the yield line inside the RHS. This model is also introduced by CIDECT [6]. In [3] it is shown that the T-stub model in EN 1993-1-8 for this type of connection can be used under the following boundary conditions: the bolt positions have to be within Selected and reviewed by the Scientific Committee of the 13th Nordic Steel Construction Conference, 23 to 25 September 2015, Tampere, Finland * Corresponding author: yvonne.steige@kit.edu a) b) Fig. 1. RHS end plate splices: a) two-sided and b) foursided configuration the RHS dimension and be in the same position on both sides of the connection. For bolt positions outside the walls of the RHS, reference is made to [2]. For joints with bolts on all four sides of the connected hollow section, no information is available on how to determine the effective length of the effective T-stub for the “corner bolts” (i. e. bolt close to the corner of the RHS). The configuration with four-sided RHS splices is discussed by Kato and Mukai [4] and Willibald [9]. These two publications present three-dimensional yield line models for calculating the tension resistance of RHS splices. Based on the model of Willibald, the present paper proposes a formula for determining the effective length part for the corner bolts. Supplementing the effective length in EN 1993-1-8 with this newly developed effective length part, it is possible to calculate the design tension and moment resistance for an RHS splice. 2 Resistance model The design resistance of end plate connections with open sections subjected to a bending moment can be calculated according to EN 1993-1-8 based on the component method. This method could be applied to end plate connections with RHS members as well. The basic components of the joint depending on the loading of the joint are shown in Table 1. The formulas for beam web in tension and beam web in compression are also listed in Table 1. The component resistance of the end plate in bending is calculated with the T-stub model. EN 1993-1-8 considers three failure modes, which are shown in Table 2. The relevant resistance results from the minimum of the three resistances. Modes 1 and 2 are calculated with © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · Steel Construction 8 (2015), No. 3 187 Y. Steige/K. Weynand · Design resistance of end plate splices with hollow sections Table 1. Joint components dependent on internal forces Component Tension resistance Moment resistance Bolts in tension Ft,Rd relevant relevant End plate in bending FT,Rd relevant relevant Beam web in tension Ft,bw,Rd = beff,t,wb ⋅ t b ⋅ fy / γ M0 relevant relevant Beam web in compression Fc,bw,Rd = Mc,Rd /(h b − t b ) not relevant relevant Table 2. T-stub resistance Failure mode T-stub resistance 1 Complete yielding of flange FT,1,Rd = 4 ⋅ Mpl,1,Rd /m 2 Bolt failure with yielding of flange FT,2,Rd = 2 ⋅ Mpl,2,Rd + n ⋅ 3 Bolt failure FT,3,Rd = ( the plastic moment of the T-stub flange, which depends on the effective lengths (Eq. (1) and Eq. (2)). Mpl,1,Rd = 0.25 · ∑ 1eff,1· t 2p · fyp / γ M0 (1) Mpl,2,Rd = 0.25 ⋅ ∑ leff,2 ⋅ t p (2) 2 ⋅ fyp / γ M0 The effective lengths of two-sided splices can be calculated with Table 3 in accordance with EN 1993-1-8. For four-sided connections, the effective lengths in Table 4 have to be added to those of Table 3 for outer bolt rows. Note that the effective length depends on the number of bolts per side ns. ∑ Ft,Rd ) / ( m + n ) ∑ Ft,Rd The derivation of leff,i is based on the three-dimensional yield line model of Willibald [9], which is shown in Fig. 2. She specifies a formula for the total design resistance of the joint, which is used here to calculate an effective length for the corner bolts (see the highlighted part in Fig 2). Setting this design resistance equal to the resistance of a half T-Stub produces the estimated effective length in Eq. (3). leff,i = 1 4 ⋅ mi ⋅ x i + 4 ⋅ ei ⋅ s1i 2 ⋅ m ⋅ e 2 + 2 ⋅ m ⋅ x 2 + 2 ⋅ s ⋅ m ⋅ x + 4 ⋅ e ⋅ s 2 ... i i i i 1i i i i 1i ... + mi ⋅ 2 ⋅ (mi + ei )2 + (mo + eo )2 x i + ei + 2 ⋅ ei ⋅ mi 2 ( ) (3) Table 3. Effective lengths for two-sided end plate splices Bolt row considered individually Bolt row considered as part of a group of bolt rows Non-circular patterns leff,nc Circular patterns leff,cp Non-circular patterns leff,nc Circular patterns leff,cp 2 ⋅ m + 0.625 ⋅ e + e1 π ⋅ m + 2 ⋅ e1 2 ⋅ m + 0.625 ⋅ e + 0.5 ⋅ p π⋅m + p 4 ⋅ m + 1.25 ⋅ e 2⋅π⋅m e1 + 0.5 ⋅ p 2 ⋅ e1 + p Inner bolt row 4 ⋅ m + 1.25 ⋅ e 2⋅π⋅m p 2⋅p Mode 1 leff,1 = leff,nc ≤ leff,cp ∑ leff,1 = ∑ leff,nc ≤ ∑ leff,cp Mode 2 leff,2 = leff,nc ∑ leff,2 = ∑ leff,nc Bolt row location Outer bolt row 188 Steel Construction 8 (2015), No. 3 Y. Steige/K. Weynand · Design resistance of end plate splices with hollow sections Table 4. Effective lengths for four-sided end plate splices for outer bolt rows Bolt row considered individually Bolt row considered as part of a group Non-circular patterns leff,nc Circular patterns leff,cp Non-circular patterns leff,nc Circular patterns leff,cp ns = 1 2 ⋅ leff,i not relevant not relevant not relevant ns > 1 leff,i + 2 ⋅ m + 0.625 ⋅ e not relevant leff,i + 0.5 ⋅ p not relevant Bolt row location Outer bolt row Fig. 3. Modified T-stub This model does not take into account the area between the two webs of the hollow section. As described in EN 1993-1-8, one bolt row includes two bolts. The resistances of the individual connection sides are then added. Fig. 2. Three-dimensional yield line model of Willibald [9] with dimensions of Eq. (3) The dimensions in Eq. (3) are illustrated in Fig. 2. Index i represents the bolt dimensions corresponding to the T-stub and index o is for the opposite side. Length xi in Eq. (3) results from the minimum of the resistance of the yield line model and can be calculated with Eq. (4). 1 xi = −2 ⋅ ei ⋅ s1i + 2 ⋅ mi 4 ⋅ ei 2 ⋅ s1i 2 + 4 ⋅ mi ⋅ ei ⋅ s1i 2 + 4 ⋅ mi 2 ⋅ ei 2 + 4 ⋅ ei ⋅ mi 3 ... ... + 2 ⋅ mi ⋅ ei ⋅ −s1i + mi ⋅ 2 ⋅ (mi + ei )2 + mo + eo ( ) ( ) 1 2 2 (4) Formula Eq. (3) is not very practical for hand calculation; therefore, the next step would be to find a simplification. 2.1 Tension resistance The design tension resistance results from the end plate in bending FT,Rd and the beam web in tension Ft,bw,Rd components. The formula for Ft,bw,Rd is listed in Table 1. Dimension beff,t,wb results from the relevant effective length. That means: if the connection collapses in mode 1, then beff,t,wb = leff,1, in other cases beff,t,wb = leff,2. The resistances of the individual connection sides have to be calculated (height and/or width) for the determination of FT,Rd and Ft,bw,Rd. Therefore, a modified T-stub model as shown in Fig. 3 is introduced. 2.2 Moment resistance The determination of the design moment resistance of the joint Mj,Rd is explained here for bending about the y-axis. The calculation for the z-direction can be derived in the same way. The design moment resistance of the joint Mj,Rd results from the effective tension resistance Ftr,Rd of the individual bolt rows r multiplied by the relevant lever arm, i.e. the distance of the bolt row from the centre of compression hr (see Eq. (5)). M j,Rd = ∑ h r ⋅ Ftr,Rd (5) It is assumed that the centre of compression lies in the beam flange of the hollow section (see Fig 4). Fig. 4 shows the design resistance of the joint components. To perform the calculation it is necessary to distinguish between the two different bolt patterns, i.e. two- and four-sided. The design moment resistance of two-sided connections can be calculated according to EN 1993-1-8. For four-sided connections, the first row, here called the external row, has to be considered separately. In EN 1993-1-8 one bolt row consists of two bolts, but in the case of an RHS splice, the external row can also have just one or even more than two bolts. Therefore, the resistance of the external row is calculated with a rotated T-stub as it is already presented in EN 1993-1-8 for the external part of an end plate, but taking into account the fact that the number of bolts, here n, is not set to a certain value. The effective length of the external row can be calculated with Table 5. The effective resistance of the individual bolt rows is the minimum of the end plate in bending and beam web in tension components. Steel Construction 8 (2015), No. 3 189 Y. Steige/K. Weynand · Design resistance of end plate splices with hollow sections a) b) Fig. 4. Design resistances of joint components for a) two-sided and b) four-sided connection Fc,bw,Rd < Table 5. Effective length for an external bolt row Non-circular patterns leff,nc Circular patterns leff,cp Smallest of: Smallest of: leff,i + (n − 1) ⋅ (2 ⋅ m x + 0.625 ⋅ e x ) n ⋅ ( π ⋅ m) leff,i + (n − 1) ⋅ 0.5 ⋅ p2 π ⋅ m + (n − 1) ⋅ p2 n ⋅ (2 ⋅ m x + 0.625 ⋅ e x ) e1,x + (n − 1) ⋅ p2 e1,x + (n − 1) ⋅ 0.5 ⋅ p2 not relevant 2 ⋅ m x + 0.625 ⋅ e x + (n − 1) ⋅ 0.5 ⋅ p2 not relevant leff,1 = leff,nc ≤ leff,cp leff,2 = leff,nc ∑ Ft,Rd (8) 2.3 Worked example (6) End plate Bolts p1 = p2 = 100 mm d = 16 mm e1 = e1,x = 125 s1 = 50 mm a = 6 mm d0 = 18 mm –– m = mx = 30 – 0.8 · √ 2 · 6 = 23.22 mm Ft,Rd = 113 kN e = 45 mm n = 29.03 mm (n ≤ 1.25 m) fyp = 235 N/mm2 tp = 12 mm Steel Construction 8 (2015), No. 3 3.The effective resistance of one bolt row is > 1.9 times the tension design resistance of one bolt. After the reduction in the effective (i.e. reduced if necessary) resistances, the design moment resistance can be determined using Eq. (5). 2.The resistance of the beam web in compression is lower than the sum of the bolt row resistances. 190 (7) Ft,i,Rd ≥ 1.9 ⋅ Ft,Rd These calculated resistances have to be reduced if at least one of the following conditions is satisfied: 1.The resistance of the bolt group is smaller than the sum of the individual bolt row resistances. Ft,Group,Rd < ∑ Ft,Rd The following example illustrates the calculation approach for design tension and moment resistance. The connection is a symmetrical end plate splice with RHS 200×10 in grade S355 and 8 No. M16 grade 10.9 bolts. 2.3.1 Calculation of design tension resistance The first step requires the calculation of the resistance of the end plate in bending component. Therefore, it is necessary to determine the effective lengths leff,1 and leff,2. The connection has two outer bolt rows on both sides, which is why only this type is calculated and listed in Table 6. Beam RHS 200 × 10 fyb = 355 N/mm2 Wpf,y = 5.309 · 105 mm3 Safety factor gM0 = 1.0 Fig. 5. Dimensions of RHS bolted end plate connection of calculation example Y. Steige/K. Weynand · Design resistance of end plate splices with hollow sections Table 6. Effective length of the individual outer bolt row of the connection example Bolt row location leff,cp [mm] Bolt row location 2 ⋅ m + 0.625 ⋅ e + e1 = 199.6 π ⋅ m + 2 ⋅ e1 = 323 Outer bolt row 4 ⋅ m + 1.25 ⋅ e = 149.1 2 ⋅ π ⋅ m = 146 leff,nc [mm] Outer bolt row leff,i + 2 ⋅ m + 0.625 ⋅ e = 154 leff,cp [mm] leff,i + 0.5 ⋅ p = 130 2 ⋅ m + 0.625 ⋅ e + 0.5 ⋅ p = 125 π ⋅ m + p = 173 e1 + 0.5 ⋅ p = 175 e1 + p = 225 leff,1 = 125 (leff,nc = 125 ≤ leff,cp = 173) leff,2 = leff,nc = 149,1 leff,2 = leff,nc = 125 Mode 1: Mode 2: 2 ⋅ 2 115 000 + 29.03 ⋅ 4 ⋅ 113 ⋅ 103 = 332.09 kN 103 ⋅ (23.22 + 29.03) FT,2,Rd = (16) 4 ⋅ 1 235 160 = 213 kN 103 ⋅ 23.22 (9) Mode 3: FT,3,Rd = 4 ⋅ 113 = 452 kN Mode 2: FT,2,Rd = leff,nc [mm] leff,1 = 146 (leff,nc = 149,1 > leff,cp = 146) Afterwards, the resistances of the three failure modes are calculated for one individual bolt row: FT,1,Rd = Table 7. Effective length of the outer bolt row as part of a group of the connection example 2 ⋅ 1 261 639.8 + 29.03 ⋅ 2 ⋅ 113 ⋅ 103 = 173.9 kN 103 ⋅ (23.22 + 29.03) (10) (17) where: Mpl,1,Rd = Mpl,2,Rd = 0.25 ⋅ 2 ⋅ 125 ⋅ 122 ⋅ 235 /1.0 = (18) = 2 115 000 Nmm Mode 3: FT,3,Rd = 2 ⋅ 113 = 226 kN (11) where: Mpl,1,Rd = 0.25 ⋅ 146 ⋅ 122 ⋅ 235/1.0 = 1 235 160 Nmm (12) Mpl,2,Rd = 0.25 ⋅ 149.1 ⋅ 122 ⋅ 235/1.0 = (13) = 1 261 639.8 Nmm The relevant resistance of an outer bolt row is 173.9 kN and thus the total resistance of one connection side is FT,s,Rd = 2 ·173.9 = 347.8 kN. The resistance of the beam web in tension component is determined with beff,t,wb = leff,2: ( ) Ft,bw,Rd = 149.1 ⋅ 10 ⋅ 355/ 103 ⋅ 1.0 = 529.4 kN Two bolt rows as part of a group have a resistance of 332.09 kN and this result is less than the resistance of the two bolt rows considered individually and therefore relevant. The resistance of the beam web in tension component is determined with beff,t,wb = leff,2: ( ) Ft,bw,Group,Rd = 2 ⋅ 125 ⋅ 10 ⋅ 355 / 103 ⋅ 1.0 = 887.7 kN (19) As Ft,bw,Group,Rd > FT,Group,Rd, it is not relevant. Therefore, the total design tension resistance of the bolted end plate connection is NRd = 2 ⋅ 332.09 = 664.2 kN (20) 2.3.2 Calculation of design moment resistance (14) Afterwards, the design moment resistance about the y-axis for the connection shown in Fig. 5 is calculated. Here, Ft,bw,Rd > FT,s,Rd and, consequently, is not relevant. In the next step the resistance of the two bolt rows has to be checked as a group. The effective lengths of this step are given in Table 7. Bolt row considered as part of a group of two bolts: Beam web in compression The section class for an RHS 200×10 in moment loading is section class 1. Thus, the resistance of the beam web in compression is calculated with Eq. (21) and the associated moment capacity of the beam web is determined using Eq. (22). Mode 1: FT,1,Rd = 4 ⋅ 2 115 000 = 364.34 kN 103 ⋅ 23.22 (15) Fc,bw,Rd = Mc,Rd hb − t b = 188.5 ⋅ 103 = 991.9 kN (200 − 10) Steel Construction 8 (2015), No. 3 (21) 191 Y. Steige/K. Weynand · Design resistance of end plate splices with hollow sections Table 8. Effective length of the external bolt row (row 1) leff,nc [mm] leff,cp [mm] Smallest of: Smallest of: leff,i + (n − 1) ⋅ (2 ⋅ m x + 0.625 ⋅ e x ) = 154.1 n ⋅ ( π ⋅ m) = 145.8 leff,i + (n − 1) ⋅ 0.5 ⋅ p2 = 129.5 π ⋅ m + (n − 1) ⋅ p2 = 172.9 n ⋅ (2 ⋅ m x + 0.625 ⋅ e x ) = 149.1 2 ⋅ e1,x + (n − 1) ⋅ p2 = 350 e1,x + (n − 1) ⋅ 0.5 ⋅ p2 = 175 not relevant ( ) 2 · m x + 0.625 · e x + n − 1 · 0.5 · p2 = 124.5 not relevant leff,1 = leff,nc = 124.5 ≤ leff,cp = 145.8 leff,2 = leff,nc where: Mc,Rd = Wpl,y ⋅ fyb / γ M0 = 5.309 ⋅ 10−1 ⋅ 355/1.0 = (22) = 188.5 kNm End plate in bending and beam web in tension First, the resistance of the individual bolt rows has to be calculated. For the first row, the effective lengths are listed in Table 8. The resistances of the individual bolt rows and the associated effective bolt row resistances are combined in Table 9. Bolt rows 2 and 3 can be part of a group and are determined as described in section 2.3.1. The capacity of this group is calculated using Eqs. (15) to (18). As the resistance of the group is less than the sum of the resistances of the individual bolt rows, the resistance of bolt row 3 must be reduced to Ft,3,Rd = 332.09 – 173.9 = 158 kN. Total design moment resistance A reduction caused by beam web in compression and elastic distribution of the bolts is not necessary. Thus, the total design moment resistance of the joint is M j,Rd = 0.225 ⋅ 166 + 0.145 ⋅ 173.9 + 0.045 ⋅ 158 = (23) = 69.7 kNm 3 Verification The model for calculating the design tension resistance presented in this paper was verified by the results of exper- Table 9. Effective resistances of individual bolt rows Bolt row leff,r,1 leff,r,2 FT,1,r,Rd FT,2,r,Rd FT,3,r,Rd beff,t,r,wb Ft,bw,r,Rd Ft,r,Rd hr r [mm] [mm] [kN] [kN] [kN] [mm] [kN] [kN] [mm] 1 124.5 124.5 181.6 166 226.1 124.5 884.3 166 225 2 145.8 149.1 212.6 173.9 226.1 149.1 529.3 173.9 145 3 145.8 149.1 212.6 173.9 226.1 149.1 529.3 158 45 Table 10. Comparison of tension resistances from experimental results and model 192 Test NRk [kN] Nexp [kN] 1 664.6 847 2 758.3 3 627.0 4 Nexp Nexp Test NRk [kN] Nexp [kN] 1.27 10 770.1 946 1.23 955 1.26 11 736.7 843 1.14 792 1.26 12 878 946 1.08 718.0 910 1.27 13 694.1 881 1.27 5 1005.6 1108 1.10 14 836.6 1019 1.22 6 1214.4 1162 0.96 15 919.2 1030 1.12 7 956.4 1240 1.30 16 1093.2 1153 1.05 8 1130.1 1190 1.05 17 1016.1 1105 1.09 9 797.5 903 1.13 18 1174.7 1240 1.06 Steel Construction 8 (2015), No. 3 NRk NRk Y. Steige/K. Weynand · Design resistance of end plate splices with hollow sections Tests 1 to 4 Tests 5 to 14 Tests 15 to 18 Fig. 6. Bolt patterns of tested end plate splices [9] iments on bolted end plate connections in axial tension, which are presented in [9]. Fig. 6 shows the bolt pattern of the test specimens. The geometrical and material properties of the test specimens are given in detail in [9]. Table 10 lists the characteristic tension resistances NRk calculated with the model presented here along with the test capacities. These are compared with the tension resistance Nexp results from the experimental tests. The fourth and eight columns show the relations between them. All relations except one show that the tension resistances calculated using the model presented in this paper are smaller than the experimental results. That means the model gives a safe prediction of the resistance. Nevertheless, a statistical processing according to the specific safety arrangement would be needed. The variation of the moment resistance is not specified. 4 Conclusion The model presented here provides a calculation method for bolted end plate joints with RHS members according to EN 1993-1-8. The resistance of the end plate in bending component is calculated with the T-stub model with two bolts. It would be useful to apply a half T-stub for bolted joints with RHS members so that the individual sides of the connection can be represented. This approach would also correspond to the component method and could thus be applied more easily to other connection types. References [1] EN 1993-1-8:2005, Eurocode 3: Design of steel structures – Part 1-8: Design of joints. CEN, 2005. [2] Heinisuo, M., Ronni, H., Perttola, H., Aalto, A., Tiainen, T.: End and base plate joints with corner bolts for rectangular tubular member. Journal of Constructional Steel Research, vol. 75, 2012, pp. 85–92. [3] Karlsen, F. T., Aalberg, A.: Bolted RHS end-plate joints in axial tension. Nordic Steel Construction Conference, Norway, 2012. [4] Kato, B., Mukai, A.: Bolted tension flange joining square hollow section. Journal of Constructional Steel Research, vol. 5, No. 3, 1985, pp. 163–177. [5] Packer, J. A., Bruno, L., Birkemoe, P. C.: Limit analysis of bolted RHS flange plate joints. Journal of Structural Engineering, vol. 115, No. 9, 1989. [6] Packer, J. A., Wardenier, J., Kurobane, Y., Dutta, D., Yeomans, N.: Design guide for rectangular hollow section (RHS) joints under predominantly static loading. CIDECT, 2009. [7] Steige, Y.: Entwicklung von Bemessungsalgorithmen für Stöße von Hohlprofilen im Stahlhochbau. Diploma thesis, Karlsruhe Institute of Technology in cooperation with Feldmann + Weynand GmbH Aachen, 2014 (unpublished). [8] Wheeler, A., Clarke, M., Hancock, G. J.: Design Model for Bolted Moment End Plate Connections Joining Rectangular Hollow Sections Using Eight Bolts. Research report No. R827, University of Sydney, 1980. [9] Willibald, S.: Bolted Connections for Rectangular Hollow Sections under Tension Loading. Dissertation, University Karlsruhe, 2003. [10] Willibald, S., Packer, J. A., Puthli, R. S.: Experimental study of bolted HSS flange-plate connections in axial tension. Journal of Structural Engineering, American Society of Civil Engineers, vol. 128, No. 3, 2002, pp. 328–336. Keywords: Rectangular hollow section; bolted end plate splices; steel joints Authors: Dipl.-Ing. Yvonne Steige KIT Campus Süd Holzbau und Baukonstruktionen R.-Baumeister Platz 1, 76049 Karlsruhe yvonne.steige@kit.edu Dr.-Ing. Klaus Weynand Feldmann + Weynand GmbH Pascalstr. 61, 52076 Aachen k.weynand@fw-ing.de Steel Construction 8 (2015), No. 3 193