250 K.F. Chung and K.H. Ip Double bolted connection : 100-G550-B2- 48 • 95 -M12 (page 38 of Reference 7). Thickness, t = 0.99mm; Bolt diameter, d = 12mm, and Bolt spacing, Sp = 36mm. Figure 1 Geometry of a double bolted connection Figure 2 Proposed stress-strain curves for high strength low ductility cold-formed steel strips, FEA-pr and FEA-pr Finite Element Modelling of Double Bolted Connections 35 30 25 Z 9 -~ 20 "o m 15 O -J 10 5 0 J . oo ~ ;~~-' / FEA-pr L FEA-py I / i 0 0.5 1 1.5 2 2.5 3 Extension (mm) Figure 4(a) Load-extension curves for double bolted connections with different stress-strain curves (Bolts 1 and 2 in contact with CFS at the same time) 31.10kN 28.08kN 251 35- L 30 2 mm gap 25 Z 20 "O m 15 O _! 10 i r ~ Bolt 1 first in contact 5 ~ Bolt 2 first in contact Single bolt 0 i J 0 0.5 1 1.5 2 2.5 Extension (mm) Figure 4(b) Load-extension curves for double bolted connection with different deformation sequences 1 mm gap~ ~ FEA-pr 28.08kN 26.69kN 24.10kN 15.90kN 35 30 A 25 z 20 -o m 15 0 ,_1 10 5 0 j ] Sp=36m m Sp=48mm , 1 0 0.5 1 1.5 2 2.5 Extension (mm) Load-extension curves for double bolted connection with different bolt spacing, Sp (Bolts 1 and 2 in contact with CFS at the same time) o 31.82kN _ ~ 28.08kN Figure 4(c) 252 Figure 5 K.F. Chung and K.H. Ip Distribution of von Mises stress of the double bolted connection Figure 6 Deformed mesh of double bolted connection at 3mm extension (Failure mode - bearing failure of CFS strip) ANALYTICAL MODEL FOR EIGHT-BOLT RECTANGULAR HOLLOW SECTION BOLTED MOMENT END PLATE CONNECTIONS A. T. Wheeler l, M. J. Clarke 2 and G. J. Hancock 2 ~Department of Civic Engineering and Environment, The University of Western Sydney Nepean Kingswood, N.S.W., 2747, Australia 2Department of Civil Engineering, The University of Sydney, Sydney, N.S.W., 2006, Australia ABSTRACT The increase in the use of rectangular hollow sections in mainstream structures has highlighted the need for simple design methods for the production of economical connections. This paper presents a new model for the determination of the serviceability limit moment and the ultimate moment capacity of bolted moment end plate connections utilising rectangular hollow sections and eight bolts positioned in an approximately equidistant sense around the perimeter of the section. The model considers the combined effects of prying action due to flexible end plates, and the formation of yield lines in the end plate. Failure modes involving plate yielding, bolt fracture, punching shear and beam section capacity are considered. The model has been calibrated and validated using experimental data from an associated test program. The model constitutes a relatively simple method for predicting the serviceability limit moment and ultimate moment capacity of moment end plate connections utilising square and rectangular hollow sections and eight bolts. KEYWORDS Tubular, connections, moment end plate, structural design, prying, yield line. INTRODUCTION The use of moment end plate connections joining I-section members and their corresponding structural behaviour has been well documented (Murray, 1990). Contrastingly, research on end plate connections joining rectangular and square hollow sections has been limited and consequently few design models are available for routine use. Furthermore, documented studies have concentrated primarily upon pure tensile loading, or combined compression and bending, as in a column-to-column bolted flange splice connection (Packer et al., 1989; Kato and Mukai, 1991). The eight-bolt moment end plate connection described in this paper and depicted in Figure I has a similar layout to that used by Kato and Mukai (1991), and represents one of two fundamental bolting arrangements studied by Wheeler (1998). The other bolting arrangement utilises four bolts, with the corresponding design model described in Wheeler et al. (1998). 253 254 A.T. Wheeler et al. Figure 1" Typical eight-bolt end plate application and layout The theoretical model presented in this paper pertains to tubular eight-bolt end plate connections subjected to flexural loading. The model determines the yield moment of the connection using yield line analysis, and combines the yield line analysis with stub-tee analysis to predict the ultimate strength of the connection. Two additional failure modes observed in the experimental program, namely section capacity and punching shear, have also been included in the theoretical model. Full details of the derivation of the model are given in Wheeler (1998). The predictions of the model are compared with the results obtained from an associated experimental program (Wheeler et al., 1995). EXPERIMENTAL PROGRAM An experimental program in which ten eight-bolt connections were tested has been conducted at the University of Sydney (Wheeler et al., 1995). The connections were loaded in pure flexure by subjecting a beam, with a splice connection at mid-span, to four-point bending. As the sections were not susceptible to local buckling, the ultimate load of the specimen was limited to connection failure, which occurred due to tensile bolt fracture, excessive end plate deformations, section failure or punching shear failure. The experimental ultimate moment (Mcu) and the failure mode for each test are listed in Table 1. The end plate material properties of yield stress (fy) and ultimate tensile strength (fu), and the beam section dimensional details and measured ultimate moment capacity (Mus) are given in Table 2. The parameters varied in the experimental program are also given in Table 1 and include the plate size (Wp, Dp), the plate thickness (tp), the section shape, and the positions of the bolts with respect to the section flange and web (So and g). The bolt and nut assemblies were M20 structural grade 8.8 (Grade TABLE 1 END PLATE CONNECTION DETAILS AND TEST RESULTS Specimen No. Section Type SHS 2 RHS 3 SHS 4 SHS 5 RHS 6 RHS 7 SHS 8 SHS 9 RHS 10 RHS Pla~ Dimensions(mm) Mcu Wp Dp So ~ (kNm) 16 280 280 35 30 116.0 16 230 330 35 15 124.5 12 280 280 35 30 93.9 20 280 280 35 30 116.0 12 230 330 35 15 92.7 20 230 330 35 15 136.7 16 260 260 25 35 113.2 16 300 300 45 25 97.6 16 210 310 25 20 133.0 16 250 350 45 10 119.3 Failure Mode* Bolt Punching Bolt Bolt Punching Bolt Bolt Punching Punching Punching * Punching = Failure by section tearing away from plate at toe of weld (punching shear). Bolt = Failure by bolt fracture. Analytical Model for Bolted Moment End Plate Connections TABLE 2 END PLATE MATERIAL PROPERTIES AND BEAM SECTION DETAILS 255 End Plate Properties Beam Section Details tp (mm) fy (MPa) fu (MPa) Section Depth d (mm) Width b (mm) Thickness ts (mm) Mus (kNm) 12 354 499 SHS 151.0 150.9 9.0 119 16 349 482 RHS 199.5 101.5 9.1 138 20 351 496 8.8/T), with a measured yield strength and ultimate tensile strength of 195 kN and 230 kN, respectively. The connections were prefabricated using a combination fillet/butt weld joining the section to the end plate, with a nominal fillet leg length of 8 mm. YIELD LINE ANALYSIS The yield line analysis serves primarily to determine the failure mode of the end plate, with prying action of the bolts ignored. As a secondary function, the analysis provides an estimate of the yield moment of the connection (Mcy). To determine the critical yield line pattern, numerous plastic mechanisms were considered. Most of these entailed relatively complicated patterns and resulted in lengthy expressions for the collapse moment (Myl). The derivations of the collapse moments for the different mechanisms considered are given in Wheeler (1998). The three most critical end plate mechanisms are presented in Figure 2. For each test, the experimental yield moment (Mcy) and the corresponding calculated yield moments (My0 are presented in Table 3, with the critical mode highlighted. The yield mechanism termed "Mode 8" in fact corresponds to beam yield capacity, determined using the measured yield stress of the tubular section. Figure 2: End plate yield line mechanisms TABLE 3 THEORETICAL AND OBSERVED RESULTS FOR CONNECTION YIELD MOMENTS 256 A.T. Wheeler et al. It can be seen in Table 3 that the majority of the tests were govemed by section yielding (Mode 8). Additionally, the calculated yield moments for Modes 4 and 5 are virtually identical. CUMULATIVE MODIFIED STUB-TEE METHOD To consider both the combined effects of bolt prying and end plate yielding on the ultimate capacity of the connection, a modified version of the stub-tee analogy is employed. Stub-tee analogies have been used extensively to determine the strength of end plate connections in I-sections (Nair et al., 1974; Kennedy et al., 1981). Generally the stub-tee utilises a simple rigid plastic analysis of an analogous beam that represents the one-dimensional behaviour of the end plate, with yield lines parallel to the axis of bending only. However, in the eight-bolt tubular end plate connections bending occurs about two axes, with the yield lines not necessarily being parallel to either axis of bending. The model presented in this paper is consequently termed the "cumulative modified stub-tee method", and is based on the analysis of analogous beams in both orthogonal directions. The principle of superposition is then used to obtain the resultant connection behaviour. Figure 3: Analogous beams for cumulative stub-tee model Simple representations of the analogous beams used in the cumulative modified stub-tee method are shown in Figure 3. The beam referred to as "in-plane bending" models the effect of the bolts below the flange of the section, with plastic hinges forming at points 1, 2 and 3 as shown in Figure 3a. The beam referred to as "out-of-plane bending", models the effect of the bolts lying on either side of the section webs. In this case, plastic hinges are assumed to form at points 4 and 5 on both sides of the hollow section, as indicated in Figure 3b. To simplify the problem, the bolts above the neutral axis are assumed to have a negligible effect on connection strength and are ignored. As defined by Kennedy et. al, (1981) the behaviour of the end plate may be defined as thick plate behaviour, intermediate plate behaviour and thin plate behaviour, depending on the thickness of the end plate (tp) and the magnitude of the applied load. In the cumulative stub-tee model, these categories are Analytical Model for Bolted Moment End Plate Connections 257 identified by the position and number of yield lines. Thick plate behaviour occurs when the connection fails due to bolt fracture, with a yield line forming only at point 1. Intermediate plate behaviour occurs when the bolts fracture after the formation of yield lines at points 1, 2 and 4 (i.e. plastic mechanism 5). Thin plate behaviour corresponds to the formation of yield lines at points 1, 2, 3, 4 and 5 in the end plate (i.e. plastic mechanism 2), without deformation of the bolts. To determine the moment capacity for the thick, intermediate and thin modes of behaviour, the analogous beams are analysed using statics as described by Wheeler (1998). The resulting capacities are given by Equations 1-3 following, in which it is assumed that the moment generated by the bending of the bolts is m b = ~Td~b3fyb/32 (where db = bolt diameter, fyb = bolt yield stress), and Mip is the plastic moment for the i th yield line. It is also assumed that the bolts below the flange reach their ultimate load, while those beside the webs of the section only reach a proportion (h) of their ultimate load based on their distance from the axis of rotation, h = (d- g)/(d +Soi). /Mlp +2.B~ "(d +Soi +h.d)l.(d_ts ) (l) M Cthick = d Mcint = (ap + Soi) +2. (ap+Soo) Jr d .(d-t~) (2) Mlp +M2p Msp +M2p +M b M3p +M2p +2.M b / Mcthi" = d +2. Soo + ~:So: .(d-t,) (3) / Since the yield lines invariably undergo significant rotations prior to the ultimate strength being reached, much of the material is stressed into the strain-hardening range. Consequently, the plastic moment Mip is defined in terms of a "design stress" (fp) rather than the yield stress (Packer et al., 1989). 1 2 fy + 2" fu (4) Mip = 4" tp " fp " I i f P : 3 The stub-tee analogy assumes that the yield lines form in a linear fashion, transversely across the end plate. However, the yield line analysis for the eight bolt end plates indicates that such patterns rarely occur in practice. To compensate for this inconsistency, "equivalent lengths" (for in-plane and out-of- plane bending) are determined for the yield lines such that the total amount of internal work involved in the mechanism remains unchanged. The equivalent lengths of the yield lines used for the cumulative stub-tee analysis depend on the assumed plastic collapse mechanism. Furthermore, these yield line lengths represent the cumulative length of the x or y components of several yield lines. Full details are given in Wheeler (1998). The theoretical connection capacities based on the cumulative modified stub- tee method are listed in Table 4 (presented later). PLASTIC SECTION CAPACITY The plastic section capacity of the tubular member may also govem the ultimate moment that the connection can attain. For compact cross-sections, design specifications generally define the plastic section capacity as the yield stress (fy) times the plastic section modulus (S). Although appropriate for design, this method of calculating the section plastic capacity does not usually reflect the experimentally measured ultimate moment as the cold working of the section produces significant strain hardening of the material. A more accurate method to predict the experimental plastic section capacity is to use the design stress (fp) as defined in Equation 4, fumishing M s = S.fp (5) 258 A.T. Wheeler et al. PUNCHING SHEAR Punching shear failure (tearing of the end plate) occurs when the concentrated loads transferred from the section to the end plate exceed the shear capacity of the end plate over a localised region. To model punching shear failure, a simple approach is used in which it is assumed that shear failure planes are defined by the geometry of the connection. It is also assumed that the punching shear capacity of the end plate is not affected by any concomitant bending moment. The connection is considered to have failed in punching shear when the load in the tensile flange and adjacent regions of the section (Figure 4) exceed the shear capacity of a predefined "nominal shear length" of the end plate. The nominal shear length is the length around the perimeter of the section that is assumed to fail as a result of the section pulling out from the end plate. As shown in Figure 4, the nominal shear length is divided into two regions, corresponding to flange failure (/sf) and web failure (lsw). Figure 4: Punching shear failure regions In Figure 4, s denotes the fillet weld leg length, dbh is the diameter of the bolt head, and it is assumed that the tubular section has an extemal comer radius of 2.5 times the wall thickness. Using the von Mises yield criterion, the moment capacity of the connection with respect to punching shear failure is given by Mp s :~3.tp.(Isf.(d-ts)+Isw.(d-g)) (6) The theoretical capacities of the connections tested in the experimental program with respect to the punching shear are shown in Table 4. GENERALISED CONNECTION MODEL The model described in this paper identifies three modes of failure, namely connection capacity (cumulative modified stub tee model), plastic section capacity, and punching shear. The computed capacities for each mode of failure are presented in Table 4, with the critical one highlighted. Failure modes determined using the cumulative modified stub-tee model may be govemed by bolt capacity or end plate capacity. Bolt capacity (fracture of bolts) is associated with either thick or intermediate plate behaviour, while plate capacity occurs with thin plate behaviour and is independent of the bolt loads. The results shown in Table 4 indicate that for the ten experimental tests carried out, four of these were limited in strength by punching shear and a further four were govemed by plastic section capacity. Only two tests were govemed by failure of the bolts according to the stub tee model. While the ultimate failure mode of the specimens was generally punching shear, bolt failure or section failure, substantial yielding in the end plates was observed in the experimental program. The failure criteria and failure loads for the standard SHS tests (Tests 1, 3, 4) and the RHS tests (Tests 2, 5, 6) are presented in Figures 4 and 5, respectively. Analytical Model for Bolted Moment End Plate Connections TABLE 4 THEORETICAL AND OBSERVED ULTIMATE CONNECTION MOMENTS 259 Figure 5: Failure criteria for SHS connections (So = 35 mm) Figure 6" Failure criteria for RHS connections (So = 35 mm) . for this inconsistency, "equivalent lengths" (for in- plane and out-of- plane bending) are determined for the yield lines such that the total amount of internal work involved in the. ~:So: .(d-t,) (3) / Since the yield lines invariably undergo significant rotations prior to the ultimate strength being reached, much of the material is stressed into the strain-hardening range Proposed stress-strain curves for high strength low ductility cold-formed steel strips, FEA-pr and FEA-pr Finite Element Modelling of Double Bolted Connections 35 30 25 Z 9 -~ 20 "o