Fracture Behaviour and Design of Steel Tensile ...

International Journal of Steel Structures 14(4): 1-17 (2014) DOI 10.1007/s13296-014-

www.springer.com/journal/13296

Fracture Behaviour and Design of Steel Tensile Connections with Staggered Bolt Arrangements Feng Wei1,2, Cheng Fang3,4,*, Michael C H Yam5, and Yanyang Zhang5 1

State Key Lab of Subtropical Building Science, South China University of Technology, Guangzhou, China 2 Department of Civil Engineering, South China University of Technology, Guangzhou, China 3 Department of Structural Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China 4 School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom 5 Department of Building & Real Estate, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

Abstract This paper presents an experimental, numerical, and reliability study on the fracture behaviour of bolted steel tensile connections with various bolt arrangements. A total of 36 full-scale specimens were tested under tension until fracture. The main test parameters included the geometric pattern of the connections and the connection details such as pitch, gauge, edge distance and material. The test results show that the predictions of the ultimate capacity based on the s2/4g rule can be conservative, which may be mainly due to the ‘reinforcement’ effect of the biaxial stress state that exists between the bolt holes. However, for the specimens fabricated using higher strength steel with lower ductility, the conservatism is reduced substantially. Moreover, the test efficiency, which is defined as the ratio of the test ultimate capacity over the theoretical capacity of the gross section, was also examined. A higher test efficiency can be found when the hole spacing/edge distance is decreased/increased, although substantial increase in the edge distance can be ineffective. In addition, less ductile steel can lead to lower test efficiency. Furthermore, a numerical study was undertaken, where good agreements were observed between the numerical and the test results. With available test data, reliability analysis was finally performed to re-examine the rationale behind the resistance factors used in the current design codes. Keywords: staggered bolted connection, fracture, ultimate load capacity, tensile test, reliability analysis

1. Introduction Fracture of the cross-section over the net area is a common failure mode in structural steel members with holes (for bolts or rivets) under tension. For a tension member with a non-staggered pattern of hole arrangement, the net cross-section area for the calculation of ultimate member capacity can be defined as the gross area minus the area of holes along the transverse failure path. To obtain a more effective usable area in a connection, one may stagger the bolt/rivet layout, in which case tensile failure may develop along an alternate net area with a zigzag path. The influence of staggered fastener pattern on the net area computation has long been addressed using the ‘s2/4g rule’ (s=pitch, g=gauge) which was firstly Received December 1, 2014; accepted June 17, 2015; published online 00000 00, 2015 © KSSC and Springer 2015 *Corresponding author Tel: +86 21 65983894, Fax: E-mail: [email protected], [email protected]

proposed by Cochrane (1922) based on a maximum stress theory and has been adopted by most of the current design specifications (AISC 2005, AS4100 1998, BS5950 2000, CSA-S16-09 2009, Eurocode 3 2005). Using this rule, the net cross-section area is obtained by deducting from the gross area the areas of all holes along any possible failure path, and then by adding s2/4g for each gauge space, as expressed by: 2

s An = t ⎛ wg – nD + ∑ ------⎞ ⎝ 4g⎠

(1)

Thus the predicted ultimate capacity Ppred is given by: 2

s Ppred = fu A n = f u t ⎛ w g – nD + ∑ ------⎞ ⎝ 4g⎠

(2)

where An is the net area, t is the plate thickness, wg is the gross section width of the member, n is the number of holes intercepted by the failure path, D is the diameter of each hole, and fu is the ultimate tensile strength of the material.

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Feng Wei et al. / International Journal of Steel Structures

The first proposal of the s2/4g rule led to subsequent discussions by other researchers. Bijlaard (1940) restudied various plates with staggered holes and improved the net section calculation method based on an alternative maximum distortion energy theory. Wilson et al. (1952) reaffirmed that the s2/4g rule did produce reasonable approximations to the test results of riveted steel plate connections. Subsequently, McGuire (1968) showed that the predictions of the tensile capacity of riveted steel plate connections with staggered holes by the s2/4g rule and those predicted by the Bijlaard’s theory did not have significant differences. In addition, McGuire (1968) was concerned that the maximum stress theory, which the s2/ 4g rule was based on, might not be appropriate to predict the plastic flow of ductile materials. LaBoube (1996) found that the s2/4g rule was also appropriate to account for the effect of staggered holes for cold-formed steel connections. The cold formed steel tension members with two and three staggered bolts have also been recently studied by Fox and Schuster (2010), and it was found that the strength of the specimens are conservatively predicted by the current design code. Teh and Clements (2012) further proposed design equations based on 74 staggered cold-formed specimens. The fatigue behaviour of joints with staggered holes in steel bridges were investigated by Josi et al. (2004), and design recommendations were also proposed accordingly. Apart from tensile fracture of staggered bolted connections, block shear and shear lag effects could also cause fracture of tension members, and these were also widely investigated by many researchers (Clements and Teh, 2013; Fang et al., 2013). Although the s2/4g rule has been widely accepted in design practice, some other parameters, including net-togross area ratio, fabrication technology (punched or drilled holes), bearing action, and shear-lag, can also affect the behaviour of a tensile connection (Munse and Chesson 1963). Besides, the s2/4g rule was initially proposed in the early 20th century when mild steel (with relatively high ductility and low strength) was the most prevalent in use in structural steelworks (Wilson et al., 1952). Modern steel, however, can have higher strength and lower ductility, and thus the s2/4g rule may not be applicable in these conditions. Although Može et al. (2007) have investigated the net cross-section resistance of tension members made of high strength steel, only the failure patterns of one-hole or two non-staggered holes were investigated, whereas the influence of staggered bolt patterns was not considered. Moreover, most of the early test data (Wilson et al., 1952) were based on riveted connections, but the relevant information on modern hotrolled bolted steel connections with staggered holes was rare (although some later research interests have been given to cold-formed connections). In view of this, an experimental investigation of bolted steel plate tension members with staggered holes, fabricated using modern structural steels and bolted connections, is conducted and

presented in this paper with the main aims of (1) significantly enriching the physical test database; (2) examining the validity of the s2/4g rule for modern connections; (3) studying the influence of various parameters on the ultimate behaviour of tensile connections; and (4) proposing new design recommendations if necessary. A total of 36 full scale tests on bolted steel plate tension members with various test parameters were carried out, and the fracture modes observed in the tests and those predicted by the s2/4g rule were compared. As an important index which indicates the ability of the connection to develop the full strength of the gross section, the ‘test efficiencies’ of the specimens Etest, as expressed in Eq. (3), are also discussed (Ag =the gross section area of the specimen). Subsequently, finite element (FE) models are established to compare with selected test results, and the FE models are also used to explain some unexpected test behaviours. Finally, reliability analysis is performed to examine the rationale behind the current resistance factor used in design codes. Etest = P test ⁄ f u A g

(3)

2. Experimental Programme 2.1. Test specimens A total of 36 bolted steel plate members with three types of geometric pattern were tested under tension until fracture. Details of the specimens are shown in Fig. 1 and Table 1. The specimen designation used consists of several sets of letters and numbers, where ‘P’ represents bolt geometric pattern, ‘G’ represents gauge, ‘S’ represents pitch, ‘M’ represents material, and ‘E’ represents edge distance. For example, considering specimen P1G3S2M1, ‘P1’ (Pattern 1) represents a connection with two lines of bolt holes, ‘G3’ represents the gauge g=125 mm, S2 represents the pitch s=37.5 mm, and M1 represents the material grade S235. Details of the specimen designations are listed in the footnotes of Table 1 for easy reference. Among all specimens, twenty-eight of them were fabricated from 8 mm (nominal) thick steel plates conforming to the BS EN 10025-2 (2004) grade S235 steel (M1 material) and the remaining eight specimens were fabricated using a higher grade steel (M2 material). The stress-strain properties of the materials were obtained via coupon tests based on the ASTM A370 standard (2002), as illustrated in Fig. 2. It is shown that the grade S235 steel exhibits a typical stress-strain curve of low carbon steel with a yield plateau, and the higher grade steel exhibits a linear elastic region without the yield plateau and has a ductile inelastic behaviour. It is worth mentioning that grade S460 steel was originally ordered for the M2 material, but the actual property of the provided material was different. Nevertheless, the M2 material exhibited a higher tensile strength and lower ductility when comparing with those of the M1

Fracture Behaviour and Design of Steel Tensile Connections with Staggered Bolt Arrangements

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Figure 1. Geometric details of test specimens (dimensions in mm).

Figure 2. Stress-strain curves of M1 and M2 steels from coupon tests.

material. The average modulus of elasticity for M1 and M2 steels are 196.8 and 196.4 GPa, respectively. The

average static yield/ultimate strengths for the M1 and M2 steels are 245.1/361.2 and 265.2/492.6 MPa, respectively, where the average static yield strength of the M2 steel was obtained according to the 0.2% offset rule. The ultimate strains of the M1 and M2 steels vary between 18 and 22% and between 12.5 and 18%, respectively, with average values of around 20 and 15%, respectively. The specimens (main plates) were tested in bolted butt joints as shown in Fig. 3 and they were designed to fail at the net section. The lap plates were fabricated using 16 mm thick grade S235 steel plates with different lengths to meet the requirement of connection dimensions. For short connections (such as the ones examined in this study), as long as the main plate has sufficient ductility to allow load re-distribution of the bolts to occur, the shear loads on the bolts would distribute more or less evenly at ultimate load. A 170 mm clear spacing between the two inner edges of the lap plates was employed in order to fit the tension testing machine. The total length of the specimens

Figure 3. Test setup and strain gauge layout.

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Table 1. Details of test specimens and test/calculation results Width Thickness Edge Ultimate Test Final Number t distance Load, PTest Elongation of holes wg (mm) (mm) Le (mm) (kN) (mm) passed

Specimen code

Gauge g (mm)

Pitch s (mm)

P1G1S1M1 P1G1S2M1 P1G1S3M1 P1G1S4M1

75 75 75 75

0 37.5 60 90

194 195 195 195

7.6 7.6 7.7 7.7

59.5 60 60 60

418 430 451 457

28.0 36.7 36.8 32.8


100 100 100 100

0 37.5 60 90

195 195 195 194

7.6 7.7 7.7 7.6

47.5 47.5 47.5 47

433 441 438 443


125 125 125 125

0 37.5 60 90

195 194 195 194

7.7 7.6 7.6 7.6

35 34.5 35 34.5

P2G1S1M1E1 P2G1S2M1E1 P2G1S3M1E1 P2G1S4M1E1

75 75 75 75

0 37.5 60 90

220 220 218 220

7.6 7.7 7.7 7.7


75 75 75 75

0 37.5 60 90

269 268 270 270


75 75 75 75

0 37.5 60 90


75 75 75 75

P1G1S1M2 P1G1S2M2 P1G1S3M2 P1G1S4M2 P2G1S1M2E2 P2G1S2M2E2 P2G1S3M2E2 P2G1S4M2E2

Ppred P test ----------(kN) P pred

Ptest -------Ag fu

P pred ---------Ag fu

2 2 2 1

384 401 423 466

1.09 1.07 1.07 0.98

0.79 0.80 0.84 0.84

0.72 0.75 0.78 0.86

29.3 26.4 28.7 39.3

2 2 1* 2

387 403 417 439

1.12 1.09 1.05 1.01

0.81 0.81 0.81 0.83

0.72 0.74 0.77 0.83

426 419 417 421

28.5 24.6 24.1 25.4

2 2 1* 2

392 393 406 428

1.09 1.07 1.03 0.98

0.79 0.78 0.78 0.79

0.72 0.74 0.76 0.81

35 35 34 35

430 470 479 473

16.1 21.8 27.6 28.9

3 3 2 2

381 411 446 460

1.13 1.14 1.07 1.03

0.71 0.77 0.79 0.78

0.63 0.67 0.74 0.75

7.7 7.6 7.6 7.6

59.5 59 60 60

573 614 650 658

24.4 25.7 38.3 37.9

3 3 2 2

519 540 582 595

1.10 1.14 1.12 1.10

0.77 0.83 0.88 0.88

0.70 0.73 0.79 0.80

319 319 320 320

7.7 7.7 7.6 7.7

84.5 84.5 85 85

698 749 797 801

23.9 18.9 40.5 42.2

3 3 2 2

658 684 726 737

1.06 1.09 1.10 1.09

0.79 0.85 0.90 0.90

0.75 0.78 0.82 0.83

0 37.5 60 90

320 320 320 319

7.7 7.6 7.6 7.7

47.5 47.5 47.5 47

665 698 729 760

22.2 22.8 30.6 35.1

4 4 4 2

586 609 646 736

1.13 1.15 1.13 1.03

0.75 0.79 0.83 0.86

0.66 0.69 0.74 0.83

75 75 75 75

0 37.5 60 90

195 195 196 196

7.7 7.5 7.7 7.6

60 60 60.5 60.5

567 573 576 580

18.7 12.4 16.9 15.7

2 2 1 1

531 541 583 630

1.07 1.06 0.99 0.92

0.77 0.79 0.78 0.79

0.72 0.75 0.79 0.86

75 75 75 75

0 37.5 60 90

270 270 271 270

7.7 7.7 7.7 7.6

60 60 60.5 60

762 806 843 816

13.6 11.4 16.5 18.2

3 3 2 2

715 751 807 804

1.07 1.07 1.04 1.01

0.75 0.79 0.82 0.81

0.70 0.73 0.79 0.80

Mean=1.07 COV=5% Notes: P1, P2 and P3 represent two rows, three rows and four rows of bolts, respectively; G1, G2 and G3 represent a gauge of 75, 100, and 125 mm, respectively; S1, S2, S3 and S4 represent a pitch of 0, 37.5, 60, and 90 mm, respectively; E1, E2 and E3 represent an edge distance of 35, 60, and 85 mm, respectively; M1 and M2 represent grade S235 steel and a higher grade steel with a tensile strength of 492.6 MPa, respectively. *featured by combined fractural paths

varied from 695 to 1010 mm to allow for varied pitch. An end distance of 75 mm along the loading direction was adopted for all specimens.

2.2. Test setup and test procedure A SATEC universal testing machine with a tensile capacity of 2000 kN was employed to conduct the tests. ASTM


A490M M24 bolts (AISC 2005) were used to connect the main plates (test specimens), the lap plates, and the loading plates to the end fixtures, as shown in Fig. 3. The diameter of the bolt hole was 27 mm and the bolts were installed in a “snug-tight” manner. A built-in load cell of the testing machine was used to record the quasi-static load applied onto the specimens. A cable transducer was installed to measure the elongation of the specimens between two small steel plates which were attached to the connection region by welding. Strain gauges were mounted on the specimens to measure the deformation of the specimens near critical cross sections. In general, similar test procedures were employed for all specimens. Initially, the specimens were aligned in the testing machine and subsequently a small preload was applied to counteract the weight of the upper cross-head and the end fixtures and thus the majority of the bolts were in contact with the bolt holes. This procedure reduced major slippages of the connections during testing. Subsequently, load was applied incrementally and a

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smaller load increment was used to capture the nonlinear behaviour of the specimens in the inelastic stage of the tests. A stroke control was used during testing in order to hold the applied loads at regular intervals in the inelastic stage allowing for proper stress redistribution in the specimens. At the end of each interval, a set of static readings were recorded. Readings of the load cells, strain gauges and cable transducers were taken continually during the loading process.

3. Test Results 3.1. General The tests were terminated when the net sections of the specimens fractured. The observed fracture paths are shown in Fig. 4 through Fig. 6 for the specimens with the geometric pattern type P1, P2, and P3, respectively. For comparisons, the failure paths of the specimens predicted by the s2/4g rule as expressed in Eq. (1) are also given (shown as white dash lines). The test ultimate tensile

Figure 4. Test and predicted failure paths of P1 specimens.

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capacities and the fracture elongations of the specimens are given in Table 1. Two typical failure modes were observed, namely, the transverse net section fracture and the staggered net section fracture. For some specimens, the capacity calculated along the transverse and staggered net section fracture path is equal, so either failure pattern is possible. It is of interest to observe that transverse net section fracture (rather than staggered net section fracture)

always occurred in these specimens, and this might be ‘triggered’ by minor stress concentration of the main plate near the first bolt line. Through comparisons with the predicted fracture path according to the s2/4g rule, most of the specimens follow the predicted failure patterns, except for seven specimens which were found to fail differently. For these seven specimens, three belong to the P1 geometric pattern group (two bolts along the


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Figure 7. Typical load-elongation curves for different pattern types.

transverse direction), and the remaining four specimens belong to the P2 geometric pattern group (three bolts along the transverse direction). The observed failure modes for the group P3 specimens (four bolts along the transverse direction) correlate well with those predicted by the s2/4g rule. For the three P1 geometric pattern specimens (P1G1S3M2, P1G2S3M1, and P1G3S3M1) with a pitch of 60 mm, fracture occurs along the transverse net section path rather than the expected staggered path. It is observed that for two cases out of the three (i.e. P1G2S3M1 and P1G3S3M1), the fracture path again follows the staggered path as predicted when a larger pitch (90 mm) is considered (P1G2S4M1 and P1G3S4M1). It is noteworthy that for

specimens P1G2S3M1 and P1G3S3M1, the failure mode features a combined transverse/staggered fracture path, since the edge of the second row in the staggered path is also fractured. This implies that the unexpected failure path in the three specimens may not be attributed to the inaccuracy of the s2/4g rule, but could be due to the slight variation of the material properties near the fracture section. For the remaining four specimens (P2G1S3M1E1, P2G1S3M1E2, P2G1S3M1E3, and P2G1S3M2E2) with the P2 geometric pattern, fracture occurred along the transverse net section prior to the expected occurrence of staggered fracture, thus potentially indicating the conservatism of the s2/4g rule in predicting the ultimate strength of the

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Figure 8. Strain-load responses for varied geometric patterns.

staggered net section. However, considering the fact that the tensile capacities calculated for the transverse net section of these specimens are only around 1-2% larger than those of the staggered net section, it may be possible for these specimens to fail in either the transverse net section or the staggered net section. Therefore, the s2/4g rule can still be generally considered as a reliable method for predicting the failure path. The possible failure path featuring both transverse and staggered net section fractural behaviours are further discussed in Section 5 where numerical studies are employed to show the fracture pattern.

3.2. Load-elongation response Typical applied load versus elongation curves for the specimens with various geometric patterns fabricated with M1 and M2 materials are shown in Fig. 7, where the elongations were obtained from the cable transducer’s readings as mentioned above. Similar responses were observed for the other specimens with similar bolt arrangements. In general, the elastic deformations of the specimens were relatively small for both M1 and M2 specimens, while obvious inelastic deformations could be developed subsequently. These deformations allow stress redistribution in the critical net sections of the specimens


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Figure 9. Elastic strain distributions at typical sections for varied geometric patterns.

to occur prior to fracture. The final elongations of the specimens were found to depend on the material ductility and the specimen length. Specifically, the load elongation curves for the specimens fabricated with the M1 steel exhibited more ductile behaviour, where a considerable load-elongation plateau was observed after an obvious yield point (at around 70 to 80% of the ultimate loads). For the specimens with the M2 steel, linear load-elongation behaviour was maintained until the load level achieves 50 to 60% of the ultimate loads, and subsequently, less elongation was developed before final fracture. Therefore, incomplete stress redistribution might have occurred for M2 specimens. In addition, it is found that for most specimens, a larger pitch can enable considerable elongation before failure. This can be due to a larger overall length of the specimen with a larger pitch.

3.3. Strain distribution The strain gauges were mainly mounted on the specimens along the loading direction to monitor the strain development over the transverse net section and the middle-length section, as illustrated in Fig. 3. For the specimens with staggered geometric patterns, strain gauge rosettes were mounted in the inclined path, and principal strains were obtained from the readings of the rosettes. Typical load versus strain curves of the specimens with three different pattern types (P1, P2, and P3) and two different values of pitch (0 and 90 mm) are shown in Fig. 8. The strains were recorded along the mid-length gross section (Section AA) and along the transverse net section (Section B-B) as shown in the figure. The ultimate tensile capacities (Ptest) of these specimens are also included for comparison. It is found that yielding of the net section occurred first, at

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which much less strains were developed along the middle gross section. With an increasing applied load, yielding of the gross section along the mid-length section occurred prior to fracture. This implies that gross section yielding was developed throughout the plates, thus offering a certain level of ductility before fracture. Figure 9 gives the elastic strain distributions of the specimens along the considered two sections. Plastic strain distributions are not shown because the readings from the strain gauges may not be reliable when excessively large strains were achieved. As observed in Fig. 9, the elastic strain distributions are relatively even along the mid-length gross sections, while uneven strains are found along the net section where larger strains are developed near the bolt holes, which indicates a strain concentration effect. For example, under a tensile load of 290 kN, the readings of the strain gauges 6 and 7 in the specimen P1G1S4M1 are larger than the readings from the strain gauges 4 and 5. This shows that material yielding was initiated near the bolt hole area of the net section, and subsequently it propagated towards the plate edge. Although the final fracture path of the specimen was not visible, plate fracture might also follow the same path. Similar phenomena were also observed in specimen P2G1S4M1E2 and P3G1S4M1. So generally speaking, the strain gauge readings clearly show the strain concentration effects along the net sections at the elastic stage, and this effect can affect the ultimate tensile capacity of the specimens, as discussed in more detail in the next section.

4. Discussion of Test Results 4.1. Effect of pitch Pitch is considered not having a significant effect on the ultimate strength of steel connections with non-staggered patterns (Fisher and Rumpf, 1965; Sterling and Fisher, 1966). However, for those connections with staggered bolt hole patterns, pitch can obviously affect the stress distribution in the steel plates, thus influencing the ultimate strength. Fig. 10(a) shows the tendency of the ultimate loads of the specimens with the increase of pitch. It can be seen that the ultimate tensile capacity of the specimens generally increases with increasing pitch as shown in the figure. However, when the pitch spacing achieves a certain level, i.e. 90 mm (S4) the rate of increase of the ultimate tensile capacity decreases. This conforms to the behaviour predicted by the s2/4g rule: firstly, failure of the connection with a non-staggered bolt arrangement occurs along the transverse net section; secondly, the fracture path transfers from the transverse net section to the staggered net section with an increased pitch spacing; and finally, the fracture path returns to the transverse net section when the pitch distance is increased beyond the limited value, and therefore a further increase of pitch spacing will not enhance the ultimate capacity. For four of the specimen groups (i.e. P1G2S(1-4)M1,

P1G3S(1-4)M1, P2G1S(1-4)M1E1, and P2G1S(1-4)M2E2), the ultimate tensile capacity did not always increase with increasing pitch. This can be explained by two reasons. Firstly, a longer distance along the staggered failure path due to the increasing pitch can lead to a more nonuniform stress distribution with a high stress level localized near the bolt holes (Gaylord et al., 1992). Hence, early fracture of the material adjacent to the bolt holes may occur prior to the completion of stress redistribution, thus leading to a reduced ultimate capacity. Secondly, the benefit of ‘reinforcement’ effect or bi-axial stress effect, which has been shown to increase the ultimate capacity of tension members (Munse and Chesson, 1963), is decreased as the increase of the failure path length due to the increased pitch. The ‘reinforcement’ effect is due to the influence of the metal on the parallel gross sections that constrains the ‘area reduction’ effect of the adjacent net section metal between holes (Munse and Chesson, 1963). Therefore, the expected increase of the ultimate capacity as the increase of pitch may be counteracted by the detrimental effect of non-uniform strain distributions as well as the reduced beneficial effect of ‘reinforcement’ constraints.

4.2. Effect of gauge The variation of ultimate tensile capacity of the specimens with respect to gauge is shown in Fig. 10(b). A generally decreasing trend is observed for the ultimate capacities of the specimens P1G(2-3)S1M1, P1G(2-3)S2M1, P1G(13)S3M1 and P1G(1-3)S4M1 with the increase of gauge while the width is kept unchanged. This phenomenon is again due to the ‘reinforcement’ effect as discussed previously, where it is generally shown in (Munse and Chesson 1963) that more significant ‘reinforcement’ effect can be induced for specimens with smaller gauge. Two exceptions, however, are found on the P1S(1-2)M1 specimens when their gauge is increased from 75 mm (G1) to 100 mm (G2), as shown in Fig. 10(b). A possible reason can be attributed to the influence of edge distance (which is discussed in detail in the next section). Since the same section width is employed for all the considered specimens with the P1 pattern, increasing the gauge means decreasing the edge distance, and vice versa. For the specimens with a small gauge and large edge distance, the benefit of the ‘reinforcement effect’ due to the small gauge may be counteracted by the non-uniform stress distribution along the relatively wide edge section. The non-uniform stress can cause an early initiation of fracture near the bolt hole prior to a complete stress redistribution, which decreases the ultimate tensile capacity. In this case, it is possible that increasing the gauge distance with the overall width unchanged can increase the ultimate tensile capacity. 4.3. Effect of edge distance To study the influence of edge distance on the ultimate


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Figure 10. Influence of test parameters.

tensile behaviour of the steel plate specimens, three values of edge distance (i.e. e=35, 60, or 85 mm) were considered in the test programme while the gauge was kept the same (g=75 mm). It seems that the edge distance has no obvious influence on the failure path, as observed in Fig. 5. The effect of edge distance on the ultimate load of the specimens is shown in Fig. 10(c), where it is clearly found that the ultimate tensile capacity of the specimens increases with increasing edge distance. This is not surprising, since a larger edge distance (with the same

gauge distance) means a larger specimen width, and as a reason, a larger net section to sustain the tensile load. While the ultimate tensile capacity of the specimens, Ptest, is a reasonable parameter directly indicating the strength of the considered in tension members, the test efficiency, which is defined as the ratio of test ultimate load to the tensile capacity of the gross section, is another important parameter indicating the cross-section utilization efficiency of a tension member. It is worth mentioning that the influences of the previously discussed pitch/

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Figure 11. Comparisons of efficiencies by test and s2/4g rule.

gauge on the test efficiency were not highlighted because the effects of them on the ultimate tensile capacity and on the test efficiency are the same. This is, however, not the case for the specimens with varied edge distances (and varied materials as discussed later). The calculated test efficiencies for all specimens are listed in Table 1, where the value varies from 0.71 to 0.90 which is within the extreme limits of 0.65 to 0.90 as suggested by McGuire (1968). The influence of edge distance on the test efficiency is shown in Fig. 10(d). An obvious increasing trend is observed (around 10% increment) for the test efficiency as the edge distance increases from 35 to 60 mm, but the rate of increase is significantly decreased (only 2% increment) when the edge distance is further increased from 60 to 85 mm. The decreased rate can be attributed to the more significant nonlinear stress distribution along the excessively large edge distance, which leads to the decreased ultimate capacity of the specimens.

4.4. Effect of material Two grades of material (M1 and M2) with different levels of tensile strength and ductility were employed in the tests. The tensile strength and the ductility of the M1 material were 361.2 MPa and 20%, respectively. For the M2 material, a tensile strength of 492.6 MPa and a ductility of 15% were recorded. The influence of material on the ultimate tensile capacity of the specimens is shown in Fig. 10(e), where as expected, the loads of M2 specimens are larger than those of M1 ones. However, with an increase of about 36% of the material tensile strength, the ultimate load of M2 specimens is only 23% higher than that of the M1 specimens on average, thus indicating a potential lower efficiency of specimen with the M2 steel. This can be confirmed in Fig. 10(f), which shows that employing the M2 steel with a relatively higher strength at the cost of a lower ductility (compared with M1) indeed decreases the test efficiency. This is due to the fact that the lower ductility of the M2 steel tends to prohibit

an adequate redistribution of stress along the critical net section prior to the fracture of the material in the vicinity of the bolt holes. The effect of material can be also clearly shown in Figs. 4 and 5, where a less obvious necking effect is featured for M2 specimens upon fracture, this again being due to the insufficient stress distribution caused by the lower ductility M2 steel. Finally, it seems that the material ductility affects less the test efficiency for the specimens with smaller pitch, where it is found in Fig. 10(f) that the decreasing rate of the test efficiency is slightly less for the specimens P1G1S1M(1-2), P1G1S2M(12), and P2G1S1M(1-2)E2 than the other specimen groups. This may be explained by the fact that the distance between bolt holes is relatively closer when a smaller pitch is employed, and the smaller distance may encourage more stress redistribution. In other words, the influence of material ductility on the test efficiency may be decreased for the specimens with a smaller distance between bolt holes (e.g. a smaller pitch) where early fracture (due to stress concentration) can be alleviated.

4.5. Comparison of test results with s2/4g rule In general, the predictions of ultimate tensile capacity of the specimens using the s2/4g rule, Ppred, tend to be conservative except for specimens P1G1S4M1, P1G3S4M1, and P1G1S(3-4)M2. As shown in Table 1, the test-topredicted ratio of ultimate capacity Ptest/Ppred varies between 0.92 and 1.15 with an average ratio and a coefficient of variation (COV) of 1.07 and 5.0%, respectively. As discussed previously, the conservative prediction may be due to the ‘reinforcement’ effect of the biaxial stress state in the area between the bolt holes, where the presence of bolt holes in a steel plate restricts a free lateral contraction of the material in the net section when subjected to tension. The test efficiencies of the specimens are plotted against the efficiencies predicted by the s2/4g rule, as shown in Fig. 11(a). For comparison purpose, the test results of riveted truss type connections with staggered holes reported by


Wilson et al. (1952) are also included in the figure. It is observed that most of the test data in the current study are generally above the 45o line, thus indicating that the efficiencies evaluated by the s2/4g rule (theoretical efficiencies) are conservative. In addition, the test data are seen to lie well within or close to the ±10% lines with a only few scattered ones beyond the +10% line (but within the +15% line). This shows that the prediction by the s2/ 4g rule correlates reasonably well with the test results. Moreover, Fig. 11(a) shows that employing M2 steel with a higher strength but lower ductility can lead to a slightly lower ratio of the test efficiency to the theoretical efficiency. This is consistent with the discussion in section 4.4, where reduced test efficiencies are found in the specimens with less ductile material M2. Another interesting finding is that the ultimate tensile capacity predicted by the s2/4g rule is more conservative for the specimens with smaller pitch, while the s2/4g rule tends to give less conservative results (more close to test results) with the increase of pitch, as illustrated in Fig. 11(b). Again, this is due to the less significant ‘reinforcement’ effects when the bolt spacing is increased.

5. Numerical Analysis While experimental studies can be the most realistic and straightforward way to examine the performance of steel plates with staggered geometric patterns, the finite element (FE) method can also be employed as a reliable approach offering supplementary analysis to explain some of the observations from the tests. In this section, preliminary results of FE analysis are reported, where 3D nonlinear FE models for selected test specimens were established and validated via the corresponding test results. The general FE program ABAQUS (2011), which is capable of simulating both material nonlinearity and geometric nonlinearity, was employed. Solid elements

13

C3D8R were used to model all connection components, including the specimen (main plate), lap plates, and bolts. The contacts between plates (or between bolts and bolt holes) were simulated via the ‘Hard contact’ behaviour with no penetration in the normal direction. A coefficient of friction of 0.2 was used, which corresponds to the Class D slip factor for untreated hot-rolled steel (Eurocode 3 2005). The general meshing sizes for the specimen plate and bolts were approximately 3 and 4 mm, respectively. For the lap plate, the meshing size was around 20 mm but with refined meshing sizes for the areas adjacent to the bolt holes. The head of the bolt was modelled by a cylinder with a diameter of 40 mm. To increase the efficiency of numerical analysis, only a quarter of the connection region was modelled due to the symmetry of the specimens, as shown in Fig. 12. The boundary conditions of the models were applied to reflect the actual conditions of the test setup. The translational degrees of freedom along the longitudinal direction at the mid-length of the specimen were fixed, and only one of the double lap plates was modelled. Only half of the thickness of the specimen was considered, and the out-of-plane displacement of the surface nodes was constrained to consider symmetry. The loading edge of the lap plate (loading plate) was only allowed to move longitudinally with no lateral movements. Quasi-static uniform displacements were applied onto the edge of the lap plate to simulate the applied loading. For material modelling, the values of true stress σtrue and true plastic strain ε ptrue based on the engineering stresses σnom and engineering strains εnom obtained from the tension coupon test results were input into the ABAQUS model, as expressed by: σtrue = σnom ( 1 + εnom )

(4)

σtrue⎞ p εtrue = ln ( 1 + εnom ) – ⎛ --------⎝ E ⎠

(5)

Figure 12. Finite element model of specimen.

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Figure 13. Comparisons of test and FE results.

The material properties for the high-strength bolts were adopted from relevant test data (Popov and Takhirov, 2002; Kulak et al., 1987), where the yield strength and the ultimate tensile strength were taken as 890 and 938 MPa, respectively. To simulate the fracture of plates, ABAQUS offers a capability for modelling progressive damage and failure in ductile metals, where two basic parameters are required, namely, a damage initiation criterion and a damage evolution response with a choice of element removal. The damage initiation criterion describes the maximum strain which initiates a decreasing branch of the stress-strain relationship (decreasing of material stiffness), and this strain can be obtained directly from the coupon test (20 and 15% for M1 and M2 steel, respectively). The damage evolution law describes the rate of degradation of the material stiffness once the corresponding initiation criterion has been reached. In the current model, a linear damage evolution law was considered, and a strain of around 2% beyond the damage initiation strain was employed to allow the damage evolution (from the initiation of damage to a complete loss of the element). The load-elongation responses and fracture patterns of typical specimens (P2G1S2E2M1-2 and P2G1S4E2M12) obtained from the test and FE analysis are shown in

Fig. 13(a), where sound comparisons of ultimate capacity are observed. The discrepancy of the prediction on fracture elongation is probably attributed to the employment of a typical material damage model (e.g. using a typical fracture strain as obtained from the coupon tests) which can be slightly different from the actual material properties of the considered specimen. In addition, a symmetric modelling strategy is considered in the current model (half-length and half-thickness of the specimen), while a perfectly symmetrical specimen with identical fracture characteristic on the two half-lengths is unlikely in reality. Such sources of discrepancy are, however, unimportant when the FE models are mainly considered for parametric studies. Furthermore, the FE model could be used to explain the difference of the failure paths observed in the test and predicted by the s2/4g rule. Considering the specimen P2G1S3E2M2 for example, the fracture path was observed along the transverse net section rather than the staggered one predicted by the s2/4g rule, as shown in Fig. 13(b). As discussed previously, the ultimate tensile capacities for the two failure paths predicted by the s2/4g rule is close; therefore, both failure paths could be potentially possible. This is confirmed by the FE results as shown in Fig. 13(b), where it is found that the failure path predicted


by the ABAQUS model is very close to the test result, and the failure path features a combination of the net section fracture and the staggered section fracture. The contour shown in the FE model is a ‘damage initiation index’ (ABAQUS 2011), where a value of 1.0 means the initiation of material damage. Generally speaking, the sound correlation observed in the current FE results shows the great potential of employing finite element models for further investigations, which is beneficial for a clearer understanding of the influence of various factors, and finally towards a more accurate prediction of the fracture response of bolted connections with stagger. This section presents a preliminary FE analysis for the specimens, while more detailed work is currently undertaken by the authors and will be reported in due course.

6. Reliability Analysis and Design Recommendations The s2/4g rule to calculate the ultimate capacity of tension members with staggered bolted connections was mainly based on earlier test results of rivet connections, while the information on modern steel plates with staggered bolted connections is rare. With the available test data reported in this paper, it is beneficial to re-examine the reliability of the s2/4g rule used in current design specifications. The consideration of reliability for a design equation is normally reflected by a resistance factor, φ, which is applied to the nominal design resistance to achieve a certain level of safety margin, e.g. φ =0.75 in AISC (2005) and CSA-S16-09 (2009) for tension members governed by the rupture of the effective net section. The following equation proposed by Fisher et al. (1978) can be used to calculate the resistance factor: φ = CρR exp ( –βαR V R )

(6)

where β is the safety index, of which a higher value indicates a stricter failure probability control (greater safety margin). Although there is still argument on the selection of appropriate values of safety index β, it is commonly considered to have a higher safety index for connections than for other structural members such as beams (Schmidt and Bartlett, 2002b). Structural members (e.g. beams) are usually assigned a safety index of about 3.0, while connections have typically been assigned a value of approximately 4.5. For this study, the specimens should fall into the category of ‘connections’, and therefore β =4.5 was employed. It should be noted that although the specimens exhibited a certain level of ductility, the stress concentration/shear lag effect (due to the presence of bolt holes) can cause earlier fracture of the connections, thus causing less ductility than normal beams; αR is a separation factor taken as 0.55 as proposed by Fisher et al. (1978); the constant C is a correction factor that considers the interdependence of the resistance factor and

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the load factors. Approximated values of C are given by Yam et al. (2011): 2

C = 0.008β – 0.1584β + 1.4056

(7)

ρR is the bias coefficient for the resistance; and VR is the associated coefficient of variation for the resistance, as expressed below: ρR = ρG ρM ρP 2

(8) 2

2

VR = VG + VM + VP

(9)

As indicated in Eqs. (8) and (9), three types of parameters are required to obtain the bias coefficient for the resistance ρR as well as the associated coefficient of variation VR. The three parameters are ‘Geometry’, ‘Material’, and ‘Professional’. The geometry factor ρG is the mean measured-to-nominal ratio of structural member geometric sizes (e.g. plate thickness). Kennedy and Gad Aly (1980) proposed 0.994 for the geometry factor ρG and 0.033 for its coefficient of variation VG through an intensive investigation on the North American steel manufactory markets involving varied suppliers, and these values are used in the current study. The material factor ρM is the mean measured-to-nominal ratio of material strength. In this study, ultimate tensile strength (instead of yield strength) should be considered since the main failure mode is the fracture of net sections. For material in fracture, Schmidt and Bartlett (2002a, 2002b) presented the relevant data for the tensile strength of hotrolled members from the raw data provided by Dexter et al. (2000), who conducted an intensive investigation (including field measurements and shop measurements from varied suppliers) on the American and Canadian steel manufactory markets. The weighted average bias coefficient ρM and the coefficient of variation VM were concluded as 1.130 and 0.044, respectively. These values were used in this study. The professional factor ρP is the mean test-to-predicted ratio of member capacity from available test data, e.g. the test-to-predicted ratios listed in Table 1 for the current study. For the reported 36 tests, staggered fracture modes were found in thirteen specimens, and the remaining twenty-three specimens were fractured along the transverse net section. Therefore, the test data can be divided into two groups, representing the two fracture patterns separately. For the first group, i.e. staggered fracture pattern group, the professional factor ρP and the associated coefficient of variation VP are 1.082 and 0.047, respectively; for the second group, i.e. transverse fracture pattern group, the professional factor ρP and the associated coefficient of variation VP are 1.062 and 0.05, respectively. It is worth mentioning that the above discussed geometry and material factors and their coefficients of variation are based on the survey of North American markets, therefore, the resistance factor used in AISC (2005) and CSA-S1609 (2009) is mainly discussed herein. Considering the

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target safety index of 4.5, the calculated resistance factors based on Eq. (6) for the two groups are 0.87 and 0.85, respectively, which are larger than 0.75 as proposed in AISC (2005) and CSA-S16-09 (2009). This indicates that the current design codes are conservative. In order to ensure a safety index of 4.5 and at the same time to maintain the resistance factor 0.75 employed in the two codes, Eq.(1) can be slightly modified as expressed below: 2

s A n = 1.16t ⎛ wg – nD + ∑ ------⎞ for staggered fracture path ⎝ 4g⎠ (10) A n = 1.13t ( wg – n'D ) for transverse fracture path (11) where n is the bolt hole number along staggered fracture path, n' is the bolt hole number along transverse fracture path. Through applying the correction factors which are larger than 1.0, the modified Eqs. (10) and (11) may lead to more realistic design and thus can result in material saving. Due the limited number of specimens considered in this study, these correction factors can be seen as preliminary values for design use. In order to obtain more reliable values, more extensive database may be required, including more test data and FE analysis results.

7. Conclusions This paper presents an experimental investigation on the structural performance of tension steel plate members using modern structural steels and high strength bolts with staggered geometric patterns, bearing the main aim of reconfirming the reliability of the widely used s2/4g rule in most of the current design specifications for predicting the ultimate strength and failure mode of tension members. A total of 36 full scale tests on the connection specimens with and without staggered holes were carried out, and various test parameters have been considered, including geometric pattern, pitch, gauge, edge distance, and steel grade. The main conclusions and comments are noted as follows: (1) Regarding the fracture path prediction, the test results show that the fracture paths predicted by the s2/4g rule agreed reasonably well with the test results, although different fracture paths were observed in seven specimens out of 36 in total. The pitch for the seven specimens was 60 mm (S3), and a common failure mode was found which occurred along the transverse net section rather than the staggered path as predicted by the s2/4g rule. This phenomenon indicates that the s2/4g rule may be conservative for predicting the staggered net section capacity of the specimens. In other words, the actual staggered net section capacity may be higher than the s2/ 4g rule prediction, thus fracture finally occurred alternatively along the transverse net section. (2) The ultimate capacities of both M1 and M2 steel specimens can be well predicted by the conventional

calculation method based on the s2/4g rule. The ratio of the test ultimate capacity Ptest to the predicted capacity Ppred ranged between 0.92 and 1.15 with an average ratio of 1.07. For most cases, the s2/4g rule was found to be conservative. The main reason for the conservatism is due to the ‘reinforcement’ effect on the specimens with relatively closely spaced bolt holes (e.g. small gauge spacing). However, this ‘reinforcement’ effect can be reduced, as observed in some specimens with larger spaced bolt holes or with lower ductility steel. This behaviour can be explained by the fact that a more significant nonlinear stress distribution (mainly due to the stress concentration near the bolt holes) combined with a lower material ductility can lead to a premature plate fracture around the bolt hole before the completion of stress redistribution, thus counteracting the ‘reinforcement’ effect. (3) The test efficiencies for the specimens ranged between 0.71 and 0.90. Varied values of pitch, gauge, edge distance, and different material properties were found to affect the test efficiency. In general, increasing the pitch or decreasing the gauge could improve the ultimate capacity/test efficiency, but the beneficial effect could be counteracted due to detrimental factors such as stress concentration. When the edge distance was increased (the gauge was kept unchanged), an increase of test efficiency was observed initially, but the increasing rate tended to diminish as the edge distance was further increased. Furthermore, increasing the material tensile strength of the test specimens (but decreasing the ductility) could decrease the test efficiency, although the ultimate tensile capacities of these specimens were improved. This observation is again due to the stress concentration near the bolt hole which cannot be sufficiently redistributed upon the initiation of fracture near the hole, thus degrading the efficiency of the specimen. (4) The results of preliminary finite element analysis on selected test specimens were presented. In general, the ultimate tensile capacity and the failure path of the specimens were well predicted, although some discrepancies on the prediction of fracture elongation were observed. The FE models were also used to explain the difference of the failure paths observed in the test and predicted by the s2/ 4g rule. Further finite element analysis for all the specimens is on-going, and a parametric study will be conducted to further investigate the tensile strength and behaviour of steel plates with staggered holes. (5) Finally, employing the test data reported in this study, the current recommendations of the resistance factor in the North American codes were re-examined. It was concluded that when a target safety index of 4.5 was considered, the resistance factor of 0.75 was conservative. Increasing the resistance factor up to around 0.85 might lead to more reasonable and economical design. In view of this, two correction factors were proposed for the cases of staggered fracture mode and transverse fracture mode. Using this strategy may lead to more realistic design if the resistance factor of 0.75 is maintained.


Acknowledgments The financial support provided by The Hong Kong Polytechnic University central research grant (G-U082) is greatly acknowledged. The authors wish to acknowledge the contributions of Dr. Sun Wei Wei to this study. The assistance provided by K. I. Hoi and O. K. Lou for conducting the tests is also appreciated.

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