Compressive strength and behaviour of gusset plate connections with single-sided splice members

Journal of Constructional Steel Research 106 (2015) 166–183

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Journal of Constructional Steel Research

Compressive strength and behaviour of gusset plate connections with single-sided splice members Cheng Fang a,b,⁎, Michael C.H. Yam c, J.J. Roger Cheng d, Yanyang Zhang c a

Department of Structural Engineering, School of Civil Engineering, Tongji University, Shanghai 200092, China School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom Department of Building & Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China d Department of Civil & Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada b c

a r t i c l e

i n f o

Article history: Received 4 September 2014 Accepted 6 December 2014 Available online 31 December 2014 Keywords: Gusset plate connections Loading eccentricity Single-sided splice Full-scale tests FE study Design recommendation

a b s t r a c t Due to the ease of fabrication and construction, gusset plate connections with single-sided splice members are a popular connection type in building frames and light structures. However, this detail produces local out-of-plane eccentricity, which can be detrimental to the ultimate strength of the connections. This paper presents experimental, numerical, and analytical investigations of the compressive behaviour of eccentrically loaded gusset plate connections. Three full-scale tests were conducted, including two specimens with unstiffened splice members and the remaining one with stiffened splice members. The ultimate loads of the three specimens were found to be evidently less than the concentrically loaded specimen which has been earlier reported. Finite element (FE) models were then established, which were validated through comparisons against the test results. The sensitivity of the FE models to the initial imperfections was also studied. An extensive parametric study was subsequently performed, and the influences of varying gusset plate and splice member geometric configurations were discussed in detail. Three key failure modes were identified for the eccentrically loaded gusset plate connections, namely, Gusset–Splice Interactive Plastic Failure (G–SIPF), Splice Plastic Failure (SPF), and Gusset Plate Buckling (GPB). A set of design rules based on the nature of each failure mode was proposed accordingly, where good agreements were observed between the FE and the design results in terms of both the ultimate load and the governing failure mode. Worked examples were finally provided to clearly illustrate the proposed design procedure. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Gusset plates are commonly used in building frames to connect the bracing members to the framing elements; they are also widely employed in bridges to link the truss members at member conjunctions. There are various types of gusset plate connections, including double gusset plate, dual gusset plate, gusset plate with double-sided splice member, and gusset plate with single-sided splice member, as illustrated in Fig. 1. Prior to the 1990s, a considerable amount of research effort had been made to investigate the tensile behaviour of gusset plate connections, whereas only limited work on the compressive behaviour of gusset plate connections had been conducted [1–3]. More work on compressively loaded gusset plate connections was carried out since the early 1990s [4–8], and a literature review on the design of gusset plate connections in braced building frames was later reported by Chambers and Ernst [9]. In 2007, the collapse of a large steel truss bridge in Minnesota, where thirteen people were killed, shocked the general ⁎ Corresponding author at: Department of Structural Engineering, School of Civil Engineering, Tongji University, Shanghai 200092, China. Tel.: +86 21 65983894. E-mail addresses: [email protected], [email protected] (C. Fang).

http://dx.doi.org/10.1016/j.jcsr.2014.12.009 0143-974X/© 2014 Elsevier Ltd. All rights reserved.

public. The subsequent investigation deduced that the collapse was mainly attributed to overstressed and buckled gusset plates [10]. The catastrophic event highlighted the importance of an appropriate design of gusset plate connections, and since then a global attention on this issue was raised. More tests and numerical studies were carried out after the accident, where the connection details that are commonly used in both building frames and truss bridges have been extensively investigated [11–16]. However, most of the available research was focused on the connections with double/dual gusset plates or those with double-sided splice members (Fig. 1(a)), whereas the ones with single-sided splice members have received little attention. This gusset connection type is commonly considered in the situation that a tubular bracing member with a slotted-in splice plate is connected to the gusset plate, where the splice plate is normally shop welded into the slots of the tubular member and then field bolted to the gusset plate [17]. Alternatively, a tee can be welded to the end of the bracing member and then bolted to the gusset plate. A continuous tube-splice solution without slotted-in welding (also known as flattened end hollow section) is also commonly used. At present, these gusset plate connection types are still quite popular, especially in building frames and other lightly loaded structures, due

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in Fig. 1(b). The industry later warned that a sway failure mechanism is more common for bracing cleat connections, and thus the design method based on Kitipornchai et al. [19] may be unconservative. An advisory note was subsequently issued [20], and possible revisions of the design models for bracing cleat connections were attempted [21]. It should be noted that the behaviour of cleat plates is different from that of gusset plates because of the more complex geometric configurations and boundary conditions for the latter case. More recently, a FE parametric study on gusset plates under compression was performed by Crosti and Duthinh [22], where the influence of loading eccentricity, among other parameters, was discussed. It was assumed that the gusset plate was connected by a single-sided I-section, which was found to evidently decrease the loading capacity compared with the concentrically loaded case. Since only one comparison set was considered in Crosti and Duthinh [22], more data are required to confirm the effect of loading eccentricity. In view of the above, it can be seen that research works on gusset plate connections with single-sided splice members are still insufficient. To provide a better understanding of this gusset plate connection type, three full-scale tests were conducted in this study, where single-sided plate type (unstiffened) splice members were adopted in two specimens, and stiffened splice members were employed in the remaining one. A concentrically loaded case, which has been reported in Yam and Cheng [8], was also included for discussion as a basis for comparison. The general nonlinear FE program ABAQUS [23] was then used to simulate these specimens. Parametric studies were subsequently performed to take an in-depth look into the strength and behaviour of such connections, where a total of 48 models with varying gusset plate/splice member geometric details and material properties were analysed. Based on the parametric study, key failure modes of those gusset plate connections were identified, and design models were developed to predict their ultimate strengths and governing failure modes accordingly. Illustrative design examples were finally given in order to more clearly demonstrate the proposed design scheme. In the following discussions, the gusset plate connections with single-sided splice members are named as the ‘eccentrically loaded gusset plate connections’ for consistency of terminology. 2. Test programme 2.1. Test specimens

Fig. 1. Gusset plate connections: a) common connection types, and b) typical deformation modes.

to the ease of fabrication and construction [18]. However, the connection details produce local out-of-plane eccentricity to the gusset plate due to the offset of the line of action from the tubular bracing member to the gusset plate by the eccentrically connected splice member. Hu and Cheng [3] studied the elastic buckling strength of several gusset plate connection types. Two specimens with single-sided splice members were considered, and their main failure mode was yielding at the splice plate near the gusset-to-splice conjunction. Compared with the concentrically loaded specimens, a significant decrease of ultimate loads for the two eccentrically loaded specimens was reported. Kitipornchai et al. [19] studied the compressive behaviour of bracing members eccentrically connected to cleat plates. A plastic collapse approach was employed to evaluate the ultimate strength of the specimens, where a non-sway failure mechanism was assumed, as shown

Three eccentrically loaded gusset plate connections were tested, which were designated as EP1, EP2, and EP3. The concentrically loaded specimen was designated as GP1. A diagonal bracing angle of 45° was considered, and the gusset plate thickness and the bolting details were identical for all the four tests. For the three eccentrically loaded specimens, the splice plate thickness and the types of splice member were varied. The dimensions and arrangement of the splice members and the gusset plate are shown in Fig. 2(a) and Table 1. For the unstiffened splice members, two plate thicknesses were considered which were 9.5 mm (EP1) and 13.0 mm (EP2). For the stiffened splice members (EP3), a WT 125 × 22.5 tee-section plus a splice plate of 9.5 mm thick was employed. Eight ASTM A325-M22 bolts were used to connect the splice member to the gusset plate. For specimen GP1, which was concentrically loaded, two tee-sections and two splice plates (2 × WT 125 × 22.5 and 2 × Plate 870 × 148 × 13.0) were connected concentrically to the gusset plate. Further details of specimen GP1 can be found in Yam and Cheng [8]. The material properties of the main structural components were obtained based on tensile coupon tests and the results are summarised in Table 2. It should be noted that originally CSA G40.21 300W steels were ordered for all the plates and the tee-section (WT), but the yield strength for the 9.5 mm thick material is exceptionally high. It is believed that a higher grade of steel might have been provided by the supplier. Nevertheless, this set of material properties will be used in analysing the test results and for conducting the FE studies.

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Fig. 2. Test programme: a) details of test specimens and instrumentations, and b) test principle and setup.

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Table 1 Specimen geometric properties and test/FE results. Specimen

Gusset plate size (mm)

Splice member

PTest (kN)

PFE (kN) (no IMP)

PFE (kN) (IMP = 0.1 mm)

PFE (kN) (IMP = 0.5 mm)

PFE (kN) (IMP = 2.0 mm)

EP1 EP2 EP3 GP1

500 × 400 × 13.3 500 × 400 × 13.3 500 × 400 × 13.3 500 × 400 × 13.3

9.5 mm plate 13.0 mm plate WT125 × 22.5 + 9.5 mm plate 2 × WT125 × 22.5 + 2 × 13.0 mm plate

297.8 322.2 856.3 1869.1

292.1 310.6 852.6 1975.3

292.0 310.5 852.4 1791.8

288.9 308.4 847.6 1718.9

280.3 302.7 836.0 1598.7

Note: IMP represents ‘initial imperfection’.

2.2. Test setup, instrumentation and test procedure The test setup is shown schematically in Fig. 2(b). The original concept of the test frame was proposed by Hu and Cheng [3] and later adopted by Yam and Cheng [7]. The main idea of the testing frame is that the same out-of-plane displacement mode of the gusset plate can be achieved by allowing the beam and column base to sway out-ofplane instead of the bracing member as in a real frame. This test arrangement can well reflect the ‘sway’ deformation mode of the specimens, and is more compatible with common testing machines where the loading head is normally fixed horizontally. A more detailed description of the concept of the test frame can be found in [3,7,8]. Two W310 × 129 sections were used as the stub beam and column members, and the diagonal bracing member (W250 × 67) was fixed in place by lateral supports at the top and bottom of the bracing member. Fourteen ASTM A325-M22 bolts were used to connect the splice member to the diagonal bracing member. The stub beam and column were bolted to a distributing beam, which sat on rollers to allow for out-of-plane movements. Two channel sections were bolted to the distributing beam to provide lateral stability for the test frame. Linear Variable Displacement Transformers (LVDTs) were used to measure the out-of-plane displacement of the gusset plate free edges. The LVDTs were also located on the splice members, which were expected to be the critical region for these specimens as shown in Fig. 2(a). Two additional LVDTs were attached to the beam and column to record the out-of-plane displacement of the test frame. Strain gauges and rosettes were used to measure the strain distribution of the specimens, as shown in Fig. 2(a). All the strain gauges and rosettes were mounted on both sides of the specimen in order to confirm the readings and to detect possible bending behaviour. The load was applied to the specimens by a MTS testing machine, and stroke control was used for all the tests. A data acquisition system was used to collect all the test data and to record the MTS load and stroke. Whitewash was applied to all the specimens to detect material yielding. As specimens EP1 and EP2 were mainly used to examine the failure response of the splice plates (i.e. no or minor yielding was expected in the gusset plate), only one gusset plate (500 mm × 400 mm × 13.3 mm) was used to conduct all the three tests. In order to avoid excessive damage to the gusset plate, the specimens with the 9.5 mm thick splice plate was tested first, followed by that with the 13.0 mm thick splice plate, using the same gusset plate specimen. Any minor permanent deformation of the gusset plate after each test was corrected (i.e. pushed back to straight) before performing the next test. Since it was expected that using the tee-section splice member would result in a higher ultimate load, test EP3 was conducted last. Another gusset plate with the same

Table 2 Specimen material properties. Material

Static yield strength (MPa)

Ultimate strength (MPa)

Modulus of elasticity (MPa)

13.3 mm gusset plate 13.0 mm splice plate 9.5 mm splice plate 13.0 mm flange of tee-section splice

295 285 435 284

501 510 540 503

207,600 199,960 201,500 197,800

geometric dimensions was used for specimen GP1. It should be noted that since the ultimate failure mode was expected to be the combined bending and compression on the splice members for tests EP1 and EP2, any slight yielding developed in the gusset plate after test EP1 would not affect the test result for EP2 as failure of the specimens occurred in the splice member only (as will be discussed later). For specimen EP3, the reuse of the gusset plate would also have negligible influence on the ultimate load. This is because the gusset plate was pushed back to a straight position after each test, and therefore the gusset plate would behave elastically when reloaded. The only issue would be the reduced ductility of the steel material after being re-used; however, negligible influence on specimen EP3 was expected, as the failure mode was mainly governed by significant yielding in the splice member combined with yielding of the gusset plate (again this will be confirmed later), where material ductility was not a critical issue. In view of this, it was believed that repeatedly using one gusset plate for the three tests can well reflect the failure modes and ultimate loads of the specimens, and is therefore acceptable for the current study. 2.3. Test results and discussions In general, all the eccentrically loaded specimens failed by significant yielding in the splice member at the area of gusset-to-bracing conjunction as indicated by the strain gauge readings and the visible yield lines. Yielding at the gusset plate in the vicinity of the corner of the stub beam and column was also observed, and the significance of yielding varied with different specimens. For specimen EP3 in particular, extensive yielding was observed at the gusset plate underneath the splice member close to the beam boundary. The yield line originated from the end of the splice member and extended towards the ends of the free edges. For specimens EP1 and EP2, yield lines were also recorded near the beam boundary close to the short side free edge. Fig. 3(a) illustrates the yield lines observed in specimen EP1 on the gusset plate. The static load vs. vertical displacement and the static load vs. outof-plane displacement curves for the three specimens are shown in Fig. 4(a) and (b), respectively. The deflections started to be accumulated at the beginning of the loading stage. This was mainly due to the out-ofplane movement of the gusset plate caused by the loading eccentricity. For specimens EP1 and EP2, the vertical displacement at ultimate load was around 2 mm, while a 4 mm vertical displacement was achieved for specimen EP3 at ultimate load. The load vs. out-of-plane displacement curves showed that an evident amount of out-of-plane displacement prior to ultimate load could be sustained. Specimens EP1 and EP2 reached their ultimate loads at an out-of-plane displacement of about 20 mm. For specimen EP3, the ultimate load was attained at a reduced out-of-plane displacement of about 10 mm. This clearly illustrated the stiffening effect of using a stiffened splice member. Selected strain gauge readings are shown in Fig. 5. As the strain gauge results for specimens EP1 and EP2 were quite similar, the results for specimen EP2 were discussed as follows. For this specimen, which used a 13.3 mm-thick splice plate, yielding was first recorded at the gusset-to-splice conjunction at an applied load of 220 kN. The plot of strain gauges #19 and #20, which were located on two opposite sides of the splice plate at the conjunction, indicated a bending behaviour from the beginning of loading. The curve for strain gauge #24, which was located very close to the first row of bolts from the end of the

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(a)

(b) Fig. 3. Yielding patterns: a) general yielding behaviour of test specimen EP1, and b) yielding conditions of gusset plate at ultimate deformation for tests EP1 and EP2.

bracing member, showed a significant compressive strain at the ultimate load. At a load level of 260 kN, flaking of the whitewash was observed at the back side of the splice member at the conjunction, and

this echoed the corresponding strain gauge results (not shown in Fig. 5) that yielding occurred at the same load level. With the applied load further approaching the ultimate load, minor yield lines were

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Fig. 4. Load–deformation responses of test specimens: a) load–vertical displacement curves, and b) load–horizontal out-of-displacement displacement curves.

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The ultimate loads of the test specimens are given in Table 1. The effects of using single-sided splice members on the compressive strength of gusset plate connections were clearly demonstrated by comparing the test results of EP specimens to that of specimen GP1, where the ultimate loads of the former were significantly lower than that of the latter. The test results also showed that although the thickness of the splice plate for specimen EP2 was increased by approximately 40% compared with that for specimen EP1, the ultimate load was not significantly increased. The main reason was that the yield strength for the 13.0 mm thick splice plate in specimen EP2 (285 MPa) was significantly lower than that of the 9.5 mm thick splice plate in specimen EP1 (435 MPa). Concurrently, the increase of the splice plate thickness for EP2 also led to an increase of the loading eccentricity, which could further decrease the ultimate load. This indicated that the ultimate loads of the EP test specimens were determined by the combined effects of the loading eccentricity, splice member material strength, and splice member thickness. When a tee-section was applied as the additional splice member for specimen EP3, the ultimate load was increased to about 2.9 times of that of specimen EP1, noting that the tee-section provided an increase to the splice member in terms of both the cross-sectional strength and bending rigidity. However, the splice arrangement for EP3 also created a larger loading eccentricity. Again, the final increase of the ultimate load was a combined effect of the increasing of the splice stiffness (beneficial) and the increasing of eccentricity (detrimental). Nevertheless, the ultimate load of specimen EP3 was still much less than that of the concentrically loaded specimen GP1. 3. FE modelling of test specimens 3.1. Modelling strategy

Fig. 5. Strain gauge reading results.

found on the gusset plate right underneath the splice plate. The specimen finally failed showing significant yielding in the splice member near the first row of bolts from the end of the bracing member. For specimen EP3, yield lines were first observed at the gusset plate in the vicinity of the corner of the beam and column boundary at an applied load of around 550 kN. At this loading level, the strain gauges on the splice plate located at the gusset-to-splice conjunction also showed yielding, but no visible yield lines were detected. A plot of load versus strain readings recorded at the web of the tee-section and on the front side of the splice plate is shown in Fig. 5. Due to the significant bending behaviour, the strain reading of the web of the tee showed tension yielding, while compression yielding was shown for the strain gauge at the splice plate.

The general nonlinear FE analysis package ABAQUS [23], which is capable of considering both material and geometric nonlinearities, was used to simulate the test specimens. 8-node linear brick elements with reduced integration and hourglass control, namely, C3D8R in ABAQUS nomenclature, were employed for the structural components, including the gusset plate, splice member, bracing member, supporting beam/column, and high-strength bolts. Two layers of elements were employed over the thickness of the gusset plate and splice members, and this meshing scheme has been found to be sufficient to capture the bending and instability behaviour of the plates [14]. ‘Hard contact’ behaviour with no penetration in the normal direction was assumed for all contact faces. A Coulomb friction model was used with a coefficient of friction of 0.2, which corresponds to the Class D slip factor for untreated hot roll steel [24]. ‘Tie’ interactions were employed to simulate all fillet welds. A rigid plate was created to simulate the loading head, and its reference point, which was located in the centroid, was used for applying the vertical load. The model was applied with boundary conditions reflecting the test setup: the base plate for the supporting beam and column was free to move out-of-plane, and the lateral translational degrees of freedom of the reference point of the ‘loading head plate’ were constrained. The out-of-plane movement of the entire bracing member was also prohibited in order to reflect the presence of the lateral supports. A representative finite element model (EP1) is shown Fig. 6(a). The material of steel was simulated by the isotropic hardening model with the von Mises yield criterion. The key material properties, e.g. Young's modulus, yield strength, and ultimate strength, were obtained from the tensile coupon test results. The engineering stress and strain relationships (trilinear) for the material were converted to true stress and strain. Two analysis steps were adopted for the FE study. The first step was eigenvalue analysis (buckling analysis) which was mainly used for the consideration of the initial imperfection in the nonlinear analysis. The second step was nonlinear analysis (Riks's arc-length analysis) which was used to obtain the nonlinear responses of the specimens under the applied load. The initial geometric imperfections were

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incorporated into the ABAQUS nonlinear analysis by automatically inputting an initial deformed configuration through the a priori definition of a buckling mode in the first step. Due to the lack of the data for the imperfection measurement, an initial imperfection amplitude of 0.1 mm using the first buckling mode was considered for all the models. The first buckling mode of model EP1 obtained from the eigenvalue analysis is shown in Fig. 6(a). A sensitivity study on the influence of imperfection amplitude was also performed, and is discussed in the following section. 3.2. Model validation The failure modes and the von Mises stress distributions of models EP1 and EP3 are shown in Fig. 6(b), noting that the behaviour of model EP2 is similar to that of model EP1. The FE results show that the specimens generally fail by significant yielding in the splice member, which is in line with the test observations. For model EP1, an evident bending deformation of the splice member is shown, and excessive yielding is featured in the splice member at the gusset-to-splice conjunction. When the ultimate load is achieved, minor yielding occurs in the gusset plate, as confirmed by the equivalent plastic strain (PEEQ) contour shown in Fig. 3(a). The pattern of this yield line is in good agreement with the test results. For model EP3, the bending action of the plate splice is less significant than that of EP1 due to the presence of the additional stiffened splice member (tee-section). However, significant yielding is still observed in the splice plate and the web of the teesection due to the out-of-plane bending action, and yielding of the gusset plate is also observed. For comparison, the behaviour of the concentrically loaded specimen GP1 was also studied by FE analysis, as shown in Fig. 6(c). A different governing failure mode, i.e. inelastic buckling of the gusset plate, is observed. No failure occurs in the splice members. The FE predicted failure mode for specimen GP1 is the same as that observed in the test, as detailed in Yam and Cheng [8]. As mentioned previously, it was assumed that no or minor yielding was expected in the gusset plate for specimens EP1 and EP2 which were mainly used to examine the failure response of the splice plates. Therefore, only one gusset plate was used to conduct all the three EP tests. The conditions of the gusset plate after tests EP1 and EP2 can be shown via the FE results. The equivalent plastic strain (PEEQ) contours of the gusset plate at ultimate out-of-plane displacement are given in Fig. 3(b). It should be noted that the ultimate out-of-plane displacement is the stage beyond the ultimate load, and at this displacement, the most extensive yielding occurred in the gusset plate for each test. The values of these displacements can be found according to Fig. 4(b). The PEEQ contours confirm that yielding could be developed in the gusset plate after tests EP1 and EP2, but the level of yielding was minor. According to the FE results, the level of equivalent plastic strain is generally less than 0.6% at ultimate displacement after test EP2. The load vs. deflection responses of the three specimens obtained from the test and FE analysis are compared in Fig. 4, where both vertical and horizontal out-of-plane displacements are shown. For the three models, the predicted ultimate loads are in good agreement with the test results. The load vs. vertical displacement curves are predicted reasonably by the FE models, although the initial stiffness of the FE curves is slightly higher than the test results. This could be due to the minor sources of vertical displacement (e.g. bolt slippage and possible deformation of the supporting members) that were not considered in the FE models. For the load vs. horizontal out-of-plane displacement response, the FE models also provide reasonable predictions, but the initial out-ofplane displacement seems to be slightly overestimated. This might be resulted from the omission of the possible restraining effects in the out-ofplane direction, e.g. slight friction caused by the roller supports. Nevertheless, the general trend of the load–deflection response is well captured. The strain gauge readings obtained from the tests are compared with the FE predictions as shown in Fig. 5. Good agreements are generally shown, especially in the elastic range. After the occurrence of yielding,

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a small level of discrepancy is observed, but the trend and the magnitude of the strains are still well predicted. It is noted that for specimen EP2, the test result of strain gauge #20 shows the development of tensile strain at the splice member surface (front side), but this is not reflected in the FE model. The test result of this strain gauge shows that the tensile strain caused by the bending action (due to the out-ofplane eccentricity) is larger than the compressive strain that is directly induced by the applied load; in other words, the algebraic sum of the strains caused by the two actions (bending and compression) is positive (tension). However, the FE result gives a small negative (compressive) strain developed at this location, indicating a less significant bending action. This discrepancy might be due to the initial local imperfections (those other than the first buckling mode) of the specimen which was difficult to be fully incorporated into the FE model. In spite of this, the bending action of the splice member is sufficiently reflected by the evident difference between the FE results of SG#19 and SG#20 locating at the two opposite sides. The ultimate loads of the specimens predicted by the FE analyses, PFE (initial imperfection, IMP = 0.1 mm), are summarised in Table 1. In order to examine their sensitivity to initial imperfections, three levels of imperfection amplitude were considered, namely, no imperfection, 0.5 mm, and 2.0 mm. It can be seen that the ultimate loads for the three eccentrically loaded gusset plate connection models are not very sensitive to initial imperfections as shown in the table. Compared with the ‘no imperfection’ cases, the decreasing rates of the ultimate loads are less than 5% when a maximum imperfection of 2.0 mm is considered. The non-sensitivity of the specimens to the initial imperfection is because of the presence of eccentricity which results in out-of-plane deformations at the very beginning of the loading process. As found in the load vs. out-of-plane displacement curves shown in Fig. 4(b), the out-of-plane displacement could achieve 20 mm at ultimate load; therefore, the initial imperfection plays a non-critical role. On the other hand, model GP1, which is concentrically loaded, is more sensitive to initial imperfections, as shown in Table 1. When an imperfection of 0.1 mm is considered, the ultimate load decreases from 1975.3 kN for the case of no imperfection to 1791.8 kN (decreasing rate = 9.3%), further increasing the initial imperfection to 2 mm leads to a ultimate load of 1598.7 kN (decreasing rate = 19.1% compared with the no imperfection case). The imperfection sensitive characteristic of model GP1 is due to the fact that the model is a symmetrical system, and buckling of the gusset plate is the governing failure mode. Comparing PFE with PTest, it is observed that assuming an imperfection of 0.1 mm can lead to reasonable predictions (slightly on the conservative side) of the ultimate loads for both GP and EP typed models. Therefore, an imperfection of 0.1 mm is consistently employed for the parametric studies discussed in the next section. 4. Parametric studies and discussions 4.1. Parameter matrix Following the validation study discussed above, a parametric study was performed to take an in-depth look into the compressive strength and behaviour of eccentrically loaded gusset plate connections. A total of 48 models were analysed, covering variations of gusset plate size, splice member configuration, and material property. In order to represent a common type of gusset plate connection used in modern construction, a square tubular bracing member (instead of the I-section used in the test programme) with a splice member welded to its end (via a cover plate) was considered, and the splice member was then bolted to the gusset plate. The FE modelling strategies were the same as those employed in the previous validation study. Detailed geometric configurations of the gusset plate connections are illustrated in Fig. 7(a). The parameter matrix was selected to represent typical configurations of gusset plate connections in practice. Two gusset plate sizes were considered, which are shown in Fig. 7(a). Three thicknesses were considered for each gusset plate type, namely, 4 mm, 8 mm, and

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(a)

(b)

(c) Fig. 6. FE modelling of test specimens: a) typical mesh and eigenvalue analysis result, b) failure behaviour of eccentrically loaded connections, and c) failure behaviour of concentrically loaded connection.

12 mm. For each gusset plate, four different types of splice member were considered. The reference case was a plate type splice member with a thickness of 10 mm, and the other three cases used for comparisons were: 1) reduced gap length of the gusset-to-bracing member conjunction (designated as ‘L’ as shown in Fig. 7(a)); 2) double thickness of the splice member (20 mm); and 3) stiffened splice member with a 15 mm-thick stiffener. Through considering the above parameters, a practical range of gusset and splice sizes can be covered. For the material properties, two steel grades, S355 (M1) and S690 (M2), were considered, which represent the classes of normal-strength and highstrength steels, respectively. Tri-linear stress–strain relationships were assumed for the material models, as illustrated in Fig. 7(b). The assumed yield and ultimate strengths are typical for the considered steel grades

[25]. Based on the considered parameters, each model was designated with a code for easy identification, as shown in Table 3. For example, M1G500T4L70 represents: steel grades = S355 (M1), the length of the longer side of gusset plate = 500 mm (G500), gusset plate thickness = 4 mm (T4), and the gusset-to-bracing member conjunction gap = 70 mm (L70). For some models, the last letters ‘D’ and ‘S’ represent double splice thickness (20 mm) and stiffened splice member, respectively. 4.2. General results The ultimate load PFE, the out-of-plane displacement at ultimate load ΔFE, and the failure mode for all the FE models are summarised in

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Failure (G–SIPF). This failure mechanism is characterized by the formation of two evident yield lines, one developed in the splice member near the end of the bracing member, and the other one developed in the gusset plate near the bottom of the splice member, as shown in Fig. 7(c). When this failure mode happens, the gusset plate is mainly subjected to out-of-plane bending, and no obvious buckling behaviour is observed. Another failure mode observed in the models with unstiffened splice members is Splice Plastic Failure (SPF). This failure mechanism, which is only found in models with the 12 mm-thick gusset plate, is featured by two yield lines developed in the splice member, as shown in Fig. 7(c). It is worth mentioning that test specimens EP1 and EP2

Table 3. A considerable variation of ultimate load, ranging from 84.9 kN to 928.5 kN, is observed. In general, the ultimate load increases with increasing gusset/splice member thickness. The stiffeners also greatly increase the ultimate load of the models. The influences of various parameters on ultimate load are discussed later in detail. The out-ofplane displacement at ultimate load also varies significantly with different connection types, and it can be seen that the presence of splice stiffeners evidently reduces the out-of-plane displacement. Importantly, three distinct failure modes are observed for the considered models. For those with plate type (unstiffened) splice members, the most common failure mode is Gusset–Splice Interactive Plastic

(a)

(b)

Yield lines

175

Yield lines

Yield areas

(c) Fig. 7. Parametric study results: a) geometric configurations, b) material properties, and c) typical failure modes.

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Table 3 Summary of parametric study and design results. FE results

Design results

Model

PFE (kN)

ΔFE (mm)

Failure mode

PFHWA (kN)

PG–SIPF (kN)

PSPF (kN)

PGPB (kN)

Ratio of PFE/min (PG–SIPF, PSPF)

Ratio of PFE/min (PG–SIPF, PGPB)

Governing failure mode

M1G500T4L70 M1G500T4L30 M1G500T4L70D M1G500T4L70S M2G500T4L70 M2G500T4L30 M2G500T4L70D M2G500T4L70S M1G500T8L70 M1G500T8L30 M1G500T8L70D M1G500T8L70S M2G500T8L70 M2G500T8L30 M2G500T8L70D M2G500T8L70S M1G500T12L70 M1G500T12L30 M1G500T12L70D M1G500T12L70S M2G500T12L70 M2G500T12L30 M2G500T12L70D M2G500T12L70S M1G650T4L70 M1G650T4L30 M1G650T4L70D M1G650T4L70S M2G650T4L70 M2G650T4L30 M2G650T4L70D M2G650T4L70S M1G650T8L70 M1G650T8L30 M1G650T8L70D M1G650T8L70S M2G650T8L70 M2G650T8L30 M2G650T8L70D M2G650T8L70S M1G650T12L70 M1G650T12L30 M1G650T12L70D M1G650T12L70S M2G650T12L70 M2G650T12L30 M2G650T12L70D M2G650T12L70S

84.9 94.0 194.3 306.7 112.9 129.2 278.9 471.6 153.8 166.5 268.4 501.9 210.8 231.5 393.0 878.4 252.3 288.0 383.6 538.7 335.5 388.4 566.6 928.5 73.1 78.1 160.8 366.3 95.1 104.7 216.5 571.7 131.1 141.5 227.4 480.3 173.0 190.3 329.2 778.6 219.3 256.0 339.9 515.9 279.5 357.5 499.0 871.7

26.7 19.0 19.5 14.7 46.2 32.7 34.8 23.9 21.4 20.6 14.4 8.0 36.1 35.0 28.5 12.3 21.3 22.1 15.5 7.4 27.9 38.1 25.3 10.3 39.4 33.6 30.1 27.5 65.2 52.9 52.8 35.8 37.7 36.1 23.9 10.2 53.4 51.8 37.8 21.5 37.3 41.0 28.2 9.5 46.4 51.6 36.9 15.5

G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF

242.7 242.7 242.7 242.7 270.1 270.1 270.1 270.1 759.5 759.5 759.5 759.5 1282.4 1282.4 1282.4 1282.4 1236.9 1236.9 1236.9 1236.9 2259.9 2259.9 2259.9 2259.9 321.1 321.1 321.1 321.1 333.1 333.1 333.1 333.1 1115.7 1115.7 1115.7 1115.7 1822.7 1822.7 1822.7 1822.7 1853.5 1853.5 1853.5 1853.5 3335.2 3335.2 3335.2 3335.2

77.7 83.1 195.3 396.4 107.6 116.6 292.9 747.4 144.9 153.8 245.3 425.2 207.8 223.5 377.9 812.4 248.5 261.9 328.5 481.3 367.5 392.7 518.3 931.0 69.8 73.7 172.9 394.5 93.6 99.7 249.7 736.4 145.5 152.9 235.7 440.2 201.4 213.4 349.2 836.2 264.3 276.3 338.0 521.6 376.5 397.3 513.2 1007.2

226.5 270.0 669.0 – 320.4 398.1 1048.4 – 232.9 276.9 683.7 – 332.0 410.8 1079.1 – 240.1 284.2 699.0 – 344.4 424.4 1111.7 – 216.0 258.6 649.3 – 303.4 377.2 1007.8 – 222.7 265.4 664.3 – 314.6 389.7 1038.6 – 229.8 272.7 679.9 – 326.6 403.0 1071.3 –

265.0 265.0 265.0 265.0 441.5 441.5 441.5 441.5 776.6 776.6 776.6 776.6 1451.6 1451.6 1451.6 1451.6 1250.0 1250.0 1250.0 1250.0 2387.9 2387.9 2387.9 2387.9 358.2 358.2 358.2 358.2 574.2 574.2 574.2 574.2 1146.5 1146.5 1146.5 1146.5 2123.6 2123.6 2123.6 2123.6 1876.2 1876.2 1876.2 1876.2 3569.5 3569.5 3569.5 3569.5

1.09 1.13 1.00 – 1.05 1.11 0.95 – 1.06 1.08 1.09 – 1.01 1.04 1.04 – 1.05 1.10 1.16 – 0.97 0.99 1.09 – 1.05 1.06 0.93 – 1.02 1.05 0.87 – 0.90 0.93 0.97 – 0.86 0.90 0.95 – 0.96 0.94 1.01 – 0.86 0.90 0.97 – Mean = 1.00 CoV = 7.9%

– – – 1.15 – – – 1.07 – – – 1.18 – – – 1.08 – – – 1.12 – – – 1.00 – – – 1.02 – – – 1.00 – – – 1.09 – – – 0.93 – – – 0.99 – – – 0.87 Mean = 1.04 CoV = 8.6%

G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF GPB G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF G–SIPF SPF SPF G–SIPF G–SIPF SPF G–SIPF G–SIPF G–SIPF

Note: G–SIPF = Gusset–Splice Interactive Plastic Failure, SPF = Splice Plastic Failure, GPB = Gusset Plate Buckling.

also mainly failed in this manner. Both the G–SIPF and SPF modes are caused by the out-of-plane bending action induced by the loading eccentricity. When a relatively thick gusset plate is employed, the SPF mode tends to occur prior to the G–SIPF mode due to the weaker out-of-plane bending resistance of the splice member compared to that of the gusset plate. For the models with a stiffened splice member, G–SIPF is also a possible governing failure mode for those with 8 mm-thick and 12 mm-thick gusset plates, as shown in Fig. 7(c). Significant plastic deformation is developed over the stiffened splice member cross-section at the gusset-to-bracing member conjunction area. The unstiffened part of the stiffened splice member (i.e. the flange of the tee) experiences compressive yielding, whereas the tip of the stiffener experiences tensile yielding. This is in line with the stress distribution observed in test specimen EP3. In addition, a yield line is also exhibited in the gusset plate, indicating an out-of-plane bending action in the gusset plate as

well. When the thickness of the gusset plate is reduced to 4 mm, a Gusset Plate Buckling (GPB) failure mode is induced, as shown in Fig. 7(c). This failure mode has been commonly found in concentrically loaded gusset plate connections [8]. No yielding is observed in the stiffened splice member, whereas the gusset plate is buckled as indicated by the typical out-of-plane bowing of its two free edges. The area of yielding is widely spread in the gusset plate (particularly underneath the splice member) as shown by the PEEQ contour. Based on each observed failure mode discussed above, design recommendations can be proposed accordingly, which is further discussed in Section 5. 4.3. Influence of gusset plate thickness The ultimate loads Pu of the models with varied gusset plate thicknesses are shown in Fig. 8(a). The results are presented by normalised ultimate load ratios, i.e. the ratio of Pu of the models with 8 mm-thick

C. Fang et al. / Journal of Constructional Steel Research 106 (2015) 166–183

or 12 mm-thick gusset plates (PT8 or PT12) over that of the 4 mm-thick gusset plates (PT4). Since it is difficult to only discuss the influence of gusset plate thickness without simultaneously considering other parameters, a sub-categorisation using various symbols and zones is made along the abscissa in order to maximize the available information in the figure. In general, depending on different splice member types, doubling the gusset plate thickness can increase the ultimate load by 30% to 100%, and tripling the gusset plate thickness can achieve a 50% to 250% increase of the ultimate load. When the 20 mm-thick unstiffened splice member is considered, the increasing rate of the ultimate load due to the increase of gusset plate thickness becomes less evident. This implies that the splice member with increasing thicknesses starts to dominate the ultimate load, and thus the influence of gusset plate thickness becomes less obvious. A similar range of increasing rate is observed for the models with the stiffened splice members, but the increase of ultimate load when the gusset plate thickness increases from 8 mm to 12 mm is not very significant. Again, this is due to a more dominating role taken by the stiffened splice member in determining the ultimate load. Apart from the ultimate load, a clear trend in the failure mode is also observed with varying gusset plate thicknesses. When a thin gusset plate (4 mm) is considered, the governing failure modes are G–SIPF and GPB for the models with the unstiffened and stiffened splice members, respectively. As the gusset plate thickness increases to

177

8 mm, G–SIPF becomes the governing failure mode for both unstiffened and stiffened splice member cases. When the gusset plate thickness increases to 12 mm, the SPF mode occurs prior to the G–SIPF mode for the models with the 10 mm-thick unstiffened splice members and gusset-to-bracing gap length L = 70 mm. As discussed previously, this is due to the stronger out-of-plane bending resistance of the gusset plate which forces both yield lines to occur in the splice member. For the out-of-plane displacement at ultimate load, no obvious tendency is observed for the models with varying gusset plate thicknesses. 4.4. Influence of splice member thickness The ratios of the ultimate loads for the models with the 20 mm-thick splice members (P20 mm) over those with the 10 mm-thick splice members (P10 mm) are shown in Fig. 8(b). It can be seen that doubling the splice member thickness can lead to an increase of ultimate load by 50% to 150%. Due to the increased bending stiffness of the splice member, the out-of-plane displacement at ultimate load is decreased, as can be seen in Table 3. More obvious increase of ultimate load is observed in the specimens with a thin gusset plate. For the failure mode, SPF happens in the cases where the thickness of the splice member is smaller than that of the gusset plate. Increasing the splice thickness can avoid this failure mode, resulting in a G–SIPF failure mode instead. It is worth noting that even if the splice member

(a)

(b)

(c)

(d)

Fig. 8. Influences of varying parameters: a) influence of gusset plate thickness, b) influence of splice member thickness, c) influence of gusset-to-bracing member gap distance, and d) influence of splice member stiffener.

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thickness is five times that of the gusset plate thickness (e.g. for the case of 20 mm-thick splice and 4 mm-thickness gusset plate), G–SIPF still governs the failure mode with no obvious sign of Gusset Plate Buckling (GPB). This is because the out-of-plane deformation starts to be accumulated at the very beginning of the loading stage due to the initial loading eccentricity, and when the ultimate load is being approached, a considerable out-of-plane deformation has been developed, which leads to a bending type failure of the gusset plate (i.e. G–SIPF) rather than compressive buckling. This phenomenon also alerts that the current GPB-based design recommendations for concentrically loaded gusset plate connections may not be applicable to the eccentrically loaded cases.

4.5. Influence of gusset-to-bracing member gap distance The influence of gusset-to-bracing member gap distance L (shown in Fig. 7(a)) is studied by considering two values of L, namely, 70 mm and 30 mm. The ratios of the ultimate loads for the models with L = 30 mm (PL30) over those with L = 70 mm (PL70) are shown in Fig. 8(c). It can be seen that reducing this gap is beneficial for the ultimate load, and the increasing rate of ultimate load ranges from 6% to 21%. It is believed that the increase of ultimate load is due to the decreased distance between the two yield lines formed at the ultimate state, where for the G–SIPF failure mode as shown in Fig. 7(c), one yield line is developed in the splice member and the other one is developed in the gusset plate; for the SPF failure mode, both yield lines are developed in the splice member. For most models, the decreased distance between the two yield lines could decrease the out-of-plane displacement at ultimate load (as confirmed by Table 3), and thus a less significant out-of-plane bending effect is induced, leading to an increased ultimate load. However, an opposite trend is observed for the models with the 12 mm-thick gusset plate (i.e. decreasing the distance of L increases the out-ofplane displacement at ultimate load as shown in Table 3). This is due to the fact that the decrease of L changes the failure mode from SPF to G–SIPF, where the latter case enables a higher level of out-of-plane displacement sustained at ultimate load. Notwithstanding, the ultimate load is increased with the changing of the failure mode due to the greater bending resistance provided by the gusset plate.

4.6. Influence of splice member stiffener The ratios of the ultimate loads for the models with stiffened splice members (PS) over those with unstiffened splice members (PUS, where PUS = P10 mm) are shown in Fig. 8(d). The results show a significant increase of ultimate load due to the presence of the stiffeners, and the ultimate loads of the specimens with stiffener can be up to six times those without stiffener. The highest ultimate load increasing rate is found in the models with the 4 mm-thick gusset plates, in which case the governing failure mode changes from G–SIPF for the models without splice member stiffeners to GPB when the stiffeners are applied. This is because the stiffeners tend to decrease the out-ofplane displacement (as confirmed in Table 3), and as a result, the bending type failure (i.e. G–SIPF) is significantly delayed, which makes the compressive buckling of the gusset plate occur first. For the remaining cases, although a similar failure mode (i.e. G–SIPF) is observed for the models with and without splice member stiffeners, the ultimate load is greatly increased due to the increased bending resistance of the stiffened splice member. Moreover, the influence of the stiffeners on ultimate load becomes less significant when thicker gusset plates are employed (the reason is similar to that discussed in Subsection 4.3). Notwithstanding, the increasing rate is at least over 100% for all the considered cases, thus demonstrating the effectiveness of splice member stiffeners in increasing the ultimate load of eccentrically loaded gusset plate connections.

5. Design considerations 5.1. Current design guidelines After the failure of the truss bridge in Minnesota, the Federal Highway Administration (FHWA) has issued a consistent design guideline [26] for gusset plate connections based on the existing test database and past design practice [16]. It is assumed in the FHWA guideline that the compressive strength of a gusset plate is treated as that of an equivalent column with the cross-section of gusset plate thickness × Whitmore effective width. The Whitmore effective width [27] is determined by the intersections of the last bolt line with two other lines originating from the ends of the first row of bolts and extending towards to the last bolt line at an angle of 30°, as illustrated in Fig. 1(a). Based on the work of Thornton [28], the length of the column L is determined as the average of three lengths L1, L2, and L3. The Euler buckling strength fE of the column is expressed as: fE ¼ E


πr 2 KL

ð1Þ

where E = Young's modulus, r = radius of gyration of the equivalent column, and K = effective length factor which is taken as 0.65. Using the ratio of the yield strength fy over the Euler buckling strength fE (as expressed by λ = fy/fE), the unfactored design compressive strength fd can be obtained from:
 λ for λ ≤ 2:25 ðinelastic bucklingÞ f d ¼ f y 0:66

ð2Þ

f d ¼ 0:88f E for λ N 2:25 ðelastic bucklingÞ:

ð3Þ

The ultimate loads predicted by the FHWA guideline, PFHWA, for all the FE models are listed in Table 3. A high level of discrepancy is shown when comparing PFE with PFHWA, which indicates that the current FHWA guideline is not suited to the case of eccentrically loaded gusset plate connections. The main reason is that the FHWA guideline is based on the bucking response (either elastic or inelastic) of gusset plate connections, but this type of failure is not common for the connections considered in this study. Therefore, a more rational design method, which can sufficiently reflect the actual failure mechanism of those gusset plate connections, is required. 5.2. Proposed design approaches Based on the three observed failure modes, the following preliminary design approaches are proposed accordingly. In the following discussion, the eccentricity e for the unstiffened splice member cases is defined as the distance between the centroids of the gusset plate and the splice member, i.e. e = (tG + tS) ∕ 2, where tG and tS are the thicknesses of gusset plate and splice, respectively. For the stiffened splice member cases, the eccentricity e is the distance between the centroids of the gusset plate and the tee. The definition of eccentricity e is illustrated in Fig. 7(a). • Design for Gusset–Splice Interactive Plastic Failure (G–SIPF) It is indicated from the experimental and FE studies that the eccentrically loaded gusset plate connections are very susceptible to the G–SIPF failure mode, cases which are featured by the formation of two clear yield lines acting like plastic hinges. Since a failure mechanism was developed by these plastic hinges, considering a rigid-plastic analogy of the connections may provide a good direction for developing new design approaches. It is noted that Kitipornchai et al. [19] also employed rigid plastic collapse analysis to evaluate the ultimate strength of the eccentrically loaded cleat plates.

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179

moment capacity of the gusset plate. Compared with the actual hinge, the simplified hinge tends to be conservative because a decreased plastic hinge length is considered. With this rigid-plastic failure mechanism identified, an equilibrium approach can be used to derive the equation of the rigid-plastic unloading line of the gusset plate connection. It is assumed that full plastic moment capacities were achieved in both plastic hinges. The plastic moment capacity may need to be reduced due to the interaction between the axial load and applied moment. Therefore, in order to achieve an equilibrium state of the free body diagram, the sum of the reduced plastic moment for the gusset plate MpG − r and the reduced plastic moment for the splice member MpS − r should be

A simplified plastic collapse model is considered in this study to predict the ultimate load of the eccentrically loaded gusset plate connections failed by the G–SIPF mode. The free body diagram of the mechanism is illustrated in Fig. 9(a). The two plastic hinges are assumed to be located at the splice plate and the gusset plate, as mentioned above. For simplicity of design, the plastic hinge in the gusset plate is assumed to be a horizontal straight line crossing the gusset plate beneath the bottom end of the splice member (marked as ‘hinge for design’). Although the actual pattern of the plastic hinge observed in the test specimens and FE models (marked as ‘actual hinge’) can be more complex, the assumed plastic hinge offers a simple evaluation of the plastic

(a)

(b)

(c) Fig. 9. Design considerations: a) free body diagram for G–SIPF and SPF, b) comparisons between design and FE results, and c) given properties for illustrative design examples.

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balanced by the moment caused by the applied load P multiplied by the initial eccentricity e plus the out-of-plane deflection Δ (see Fig. 9(a)), as expressed by:

  f y 0:75 θu ¼ ð0:034−0:0012eÞ for stiffened cases: 355

P ðe þ ΔÞ ¼ MpS−r þ M pG−r :

Eqs. (9) and (10) are developed to be applicable to the unstiffened and stiffened splice member cases failed by G–SIPF. Employing the two equations, and considering Δu = θu × Lsg, Eq. (7) can be finally used to predict the ultimate load Pu, noting that for the stiffened splice member cases the calculation of MpS, PyS and e should be based on a tee section.

ð4Þ

In order to consider the reduced plastic moment capacity MpS − r and MpG − r, the simplified axial force and moment interaction equation proposed in AISC [29] is employed, as given in Eq. (5). This equation assumes that the reduced plastic moment capacity Mp − r may be obtained by multiplying the plastic moment capacity Mp by a reduction factor related to P and Py, where P is the applied axial load and Py is the axial yield load of the considered member. Mp−r

! 9 P 1:0− ¼ M p ≤Mp 8 Py

ð5Þ

Therefore, Eq. (4) can be re-expressed by: ! ! 9 P 9 P 1:0− 1:0− P ðe þ ΔÞ ¼ M pS þ M pG 8 P yS 8 P yG

 9
MpS þ MpG 8 P¼ : 9 MpS 9 M pG eþΔþ þ 8 P yS 8 P yG

• Design for Splice Plastic Failure (SPF) The same design concept can be used for the SPF failure mode, as shown in Fig. 9(a). In this case, the distance Lsg should be replaced by Ls, where Ls is the distance between the bottom end of the bracing member and the top bolt row of the gusset plate (near which the two yield lines are developed). With the free body diagram identified in Fig. 9(a), the same equilibrium approach as the G–SIPF mode can be used to derive the equation of the rigid-plastic unloading line of the gusset plate connection, as given by:

ð6Þ

where MpS and MpG are the plastic moment capacities of the crosssections of the splice member and the gusset plate, respectively. Rearranging the above equation, the applied load can be given by:

ð7Þ

This rigid-plastic equation provides an unloading line (with increasing Δ) which represents the changes of the plastic collapse load due to the change of the deformed state [30]. The ultimate load can be evaluated by determining the out-of-plane displacement at ultimate load Δu and then solve for P from Eq. (7). In this study, an analytical equation, which is based on curve fitting of the parametric study results, is developed for calculating Δu. The first step for calculating Δu is to choose an appropriate equation format with a few number of constants (the constants will be determined later via curve fitting). It can be observed from Table 3 that Δu (symbol ΔFE is used in Table 3) increases with increasing distances of the two plastic hinges Lsg, and thus it may be reasonable to assume that Δu can be expressed in terms of Δu = θu × Lsg, where θu is the inclined angle at ultimate load and Lsg is the distance between the two plastic hinges, as shown in Fig. 9(a). According to the FE results (checking the trend of ΔFE/Lsg), θu tends to decrease with increasing eccentricity. A further decrease of θu is observed when the stiffeners are considered. The FE results also clearly show that θu for the models using the higher steel grade (S690) is evidently larger than those using S355 steel. In view of the above, the inclined angle θu may be obtained using the following form of equation:

ð10Þ

P¼

9 M 4 pS : 9 MpS Δþ 4 P yS

ð11Þ

As G–SIPF and SPF share a similar failure mechanism (i.e. out-ofplane bending), the form of Eq. (9) can also be used to consider the inclined angle at ultimate load for the SPF mode:     f y 0:75 Lsg 0:25 θu ¼ ð0:056−0:0012eÞ : 355 Ls

ð12Þ

Similarly, the ultimate load Pu of the connections with the SPF failure mode can be obtained by incorporating Δu = θu × Ls for Δ in Eq. (11). It should be noted that although the gusset plate behaviour is not directly relevant to the SPF failure mode (which only involves the behaviour of the splice plate), it is indicated in Table 3 that a larger value of Lsg of the gusset plate can lead to a decreased ultimate load (e.g. comparing G500 and G650 connections failed by SPF). Therefore, the additional factor (Lsg/Ls)0.25 in Eq. (12) is used to consider this effect. As the SPF failure mode is not likely to happen in stiffened splice members, Eqs. (11) and (12) are only used for the connections with unstiffened splice members. • Design for Gusset Plate Buckling (GPB)

where the previously discussed trends related to eccentricity e and material properties can be well reflected in this equation. The constants C1, C2, and C3 are then determined in order to provide a good fit to the FE results. Through trial-and-error, and making the average FE-topredicted ratio close to unity, it is found that the following equation can provide good agreements with the FE results in terms of ultimate load:

Finally, it is noted that there are four models (with stiffened splice members) failed by the GPB mode. Their ultimate loads PFHWA predicted by the FHWA guideline, which is based on the previously mentioned column buckling approach, are found to be quite conservative, especially for the models with S690 steel. The conservatism of the column buckling design philosophy has also been reported in other investigations on concentrically loaded gusset plate connections [5,8]. This suggests that the current guideline for predicting the GPB strength of gusset plate connections, especially for the case of high strength steel, may need to be re-visited. Currently, there is no research data on eccentrically loaded gusset plate connections using high strength steel, a preliminary design approach is proposed in this study based on the limited FE results. According to Table 3, the mean PFE/PFHWA ratio for the two S355 models failed by GPB is approximately 1.20, while that for the two S690 models is around 1.73. Considering this, a modified effective length factor K is proposed, as given by:

  f y 0:75 θu ¼ ð0:056−0:0012eÞ for unstiffened cases 355

!0:3 355 K ¼ 0:6 : fy

  f y C3 θu ¼ ðC 1 −C 2 eÞ 355

ð8Þ

ð9Þ

ð13Þ

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Compared with the recommendations in FHWA (2009), the basic value of K is slightly decreased from 0.65 to 0.6 in order to better fit the FE results for normal steel conditions. The effect of increased steel grade is also considered by introducing a fy related factor. Eq. (13) can be preliminarily used to overcome the conservatism of the current FHWA guideline; however, it needs to be further validated when more test or numerical data are available, but this is beyond the scope of this paper. The investigation of the buckling behaviour of a high strength steel gusset plate is currently underway, which will be reported in future publications.

5.4. Illustrative design examples

5.3. Discussion on design results

5.4.1.1. Check for S–GIPF. The initial eccentricity can be obtained as: e = (tS + tG)/2 = (10 + 4)/2 = 7 mm, and for the gusset plate and splice member:

Employing the above design approaches, the design ultimate loads based on the failure modes of G–SIPF, SPF, and GPB are listed in Table 3, as denoted by PG–SIPF, PSPF, and PGPB, respectively. Since the gusset plate connections with unstiffened and stiffened splice members are found to behave differently, the following design schemes for the two cases are recommended: • Gusset plate connections with unstiffened splice members — only the G– SIPF and SPF failure modes need to be checked, and the lower value of PG–SIPF and PSPF governs the design ultimate load. The GPB mode is unlikely to occur in this case, and thus it can be neglected. • Gusset plate connections with stiffened splice members — only the G– SIPF and GPB failure modes need to be checked, and the lower value of PG–SIPF and PGPB governs the design ultimate load. The SPF mode is unlikely to occur in this case, and thus it can be neglected.

To illustrate the proposed design schemes, two models, M1G500T4L70 and M1G500T4L70S, were selected as design examples, representing the unstiffened and stiffened splice member cases, respectively. The key dimensions and the required geometric properties of the models are given in Fig. 9(c). 5.4.1. Design Example 1 — gusset plate connections with unstiffened splice member

M pG ¼ Z pG f y ¼ 1896  355 ¼ 673; 080 N  mm; P yG ¼ AG f y ¼ 1896  355 ¼ 673; 080 N M pS ¼ Z pS f y ¼ 3750  355 ¼ 1; 331; 250 N  mm; P yS ¼ AS f y ¼ 1500  355 ¼ 532; 500 N: The out-of-plane displacement at ultimate load is:   f y 0:75 Δu ¼ Lsg ð0:056−0:0012eÞ 355   355 0:75 ¼ 380  ð0:056−0:0012  7Þ ¼ 18:1 mm 355 and the ultimate load is:

Based on the above design strategy, the predicted ultimate loads for all the considered FE models are obtained, as given in Table 3. For the models with unstiffened splice members, the FE-to-predicted ratios (i.e. PFE/min (PSPF, PG–SIPF)) vary from 0.86 to 1.16. The mean value of the FE-to-predicted ratios is 1.00, and the coefficient of variation (CoV) is 7.9%. Similar results are observed for the models with stiffened splice members, where the FE-to-predicted ratios (i.e. PFE/min (PG–SIPF, PGPB)) range from 0.87 to 1.18. The mean value of the FE-to-predicted ratios is 1.04, and the CoV is 8.6%. In order to more clearly show the comparisons between the FE and design results, the FE-to-predicted ratios for all the considered models are reproduced in Fig. 9(b) along with the 10% and 15% FE-to-predicted discrepancy lines. It is shown that the discrepancies are within 10% for most models, although the maximum observed discrepancy is 18% (on the conservative side). Importantly, as shown in Table 3, the predicted design failure modes agree very well with the FE results for the considered models, and this further validates the rationale behind the proposed design approaches in terms of both ultimate load and failure mode. The only exception is model M1G650T12L30, where the predicted failure mode is SPF but the FE result shows a G–SIPF mode. However, it should be noted that the predicted PG–SIPF and PSPF are quite close for this model. Finally, the proposed design method is used to predict the ultimate loads Pu of the three test specimens. The predicted failure mode for both specimens EP1 and EP2 is SPF, which agrees with the test results. The predicted ultimate loads are 249.3 kN and 327.5 kN, respectively, as compared with the test and FE results in Fig. 4. The test-topredicted ratios for the two specimens are 1.18 and 0.98, respectively. While the predictions of Pu for specimens EP1 and EP2 are quite straightforward, simplifications are required in order to apply the proposed method to specimen EP3 because of its special splice-bracing member combination. In order to use the current design method, it has to be assumed that the total thickness of the flange of the stiffened splice member is the sum of the thicknesses of the tee, gap plate, and splice plate. Adopting this simplification, the predicted ultimate load is 791.9 kN with a test-to-predicted ratio of 1.08. The predicted failure mode is G–SIPF, which agrees well with the test observation.

P S–GIPF

 9
MpS þ MpG 8 ¼ 9 MpS 9 M pG eþΔþ þ 8 P yS 8 P yG 9 ð1; 331; 250 þ 673; 080Þ 8 ¼ 77; 654 Ν ¼ 77:7 kN: ¼ 9 1; 331; 250 9 673; 080 þ 7 þ 18:1 þ 8 532; 500 8 673; 080

5.4.1.2. Check for SPF.     f y 0:75 Lsg 0:25 Δu ¼ Ls ð0:056−0:0012eÞ 355 Ls     355 0:75 380 0:25 ¼ 120  ð0:056−0:0012  7Þ ¼ 7:6 mm 355 120 9 9 MpS 1; 331; 250 4 ¼ 226; 489 N ¼ 226:5 kN: ¼ P SPC ¼ 4 9 1; 331; 250 9 MpS 7:6 þ Δþ 4 532; 500 4 P yS Therefore, the ultimate strength P = min (PS–GIPF, PSPC) = min (77.7 kN, 226.5 kN) = 77.7 kN, which is governed by S–GIPF. 5.4.2. Design Example 2 — gusset plate connections with stiffened splice member 5.4.2.1. Check for S-GIPF. As already obtained from Design Example 1, MpG = 673,080 N·mm, PyG = 673,080 N. For the stiffened splice: MpS ¼ Z pS f y ¼ 44; 000  355 ¼ 15; 620; 000 N  mm; P yS ¼ AS f y ¼ 2550  355 ¼ 905; 250 N:

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It is calculated that the eccentricity e = 23.5, and therefore   f y 0:75 Δu ¼ Lsg ð0:034−0:0012eÞ 355   355 0:75 ¼ 380  ð0:034−0:0012  23:5Þ ¼ 2:2 mm 355  9
MpS þ MpG 8 P S–GIPF ¼ 9 MpS 9 M pG eþΔþ þ 8 P yS 8 P yG 9 ð15; 620; 000 þ 673; 080Þ 8 ¼ 9 15; 620; 000 9 673; 080 þ 23:5 þ 2:2 þ 8 905; 250 8 673; 080 ¼ 396; 432 N ¼ 396:4 kN:

5.4.2.2. Check for GPB. L ¼ ðL1 þ L2 þ L3 Þ=3 ¼ ð203:4 þ 216:6 þ 62:0Þ=3 ¼ 160:7 mm; K
0:3 ¼ 0:6 ¼ 0:6 355=f y the Euler buckling strength f E ¼ E

 πr 2 KL


pffiffiffiffi  π4= 12 2 ¼ 205; 000 0:6160:7 ¼

289:9 MPa , λ = fy/fE = 355/289.9 = 1.225 b 2.25, and thus fd = fy(0.66λ) = 355(0.661.225) = 213.4 MPa, P GPB ¼ f d t G W e ¼ 213:4  4  310:5 ¼ 265; 043 N ¼ 265:0 kN: Therefore, the ultimate strength P = min (PS–GIPF, PGPB) = min (396.4 kN, 265.0 kN) = 265.0 kN, which is governed by GPB. 6. Summary and conclusions This paper reports a comprehensive study on the compressive strength and behaviour of eccentrically loaded gusset plate connections. Three tests have been conducted to observe the basic failure characteristics, and the test results indicated that the loading eccentricity significantly reduced the compressive strength. A stiffened splice member led to a better performance but its ultimate load was still much less than that of the concentrically loaded specimen. The basic failure mode of the test specimens was featured by evident yielding at the splice member near the area of the gusset-to-bracing member conjunction. Varied significance of yielding was also observed in the gusset plate. These yield regions introduced a collapse mechanism to the connections at the ultimate state. The test specimens were then simulated using the general nonlinear FE analysis package ABAQUS [23]. The predictions of the load–deflection behaviour of the specimens were, in general, in good agreement with the test results. For the strain gauge readings, good agreements were also shown, although some minor discrepancies were observed for some cases. It was believed that the discrepancy was due to the local imperfections and other minor test uncertainties that were not reflected in the FE model. The FE study also showed that the ultimate loads for the eccentrically loaded gusset plate connection models were not sensitive to initial imperfections, but the concentrically loaded connection was more sensitive to initial imperfections. A parametric study was performed following the validation study. Three typical failure modes have been identified for the eccentrically loaded gusset plate connections, namely, Gusset–Splice Interactive Plastic Failure (G–SIPF), Splice Plastic Failure (SPF), and Gusset Plate Buckling (GPB). The parametric study also showed that adding a stiffener on the splice member can be very effective in increasing the ultimate load (by up to 500% compared with the unstiffened case). Increasing the gusset plate and splice member thicknesses were also effective, where 30% to 100% and 50% to 150% increases of the ultimate load were observed when doubling the gusset

plate and splice member thicknesses, respectively. Another beneficial method was to reduce the gap between the end of the bracing member and the gusset plate, and the increasing rate of ultimate load varied from 6% to 21% when the gap decreased from 70 mm to 30 mm. Importantly, it was found that the FHWA guideline was not able to predict the ultimate loads of the connections since the equations in the guidelines were based on Gusset Plate Buckling. Based on the findings from the test programme and the subsequent parametric study, a set of design schemes have been proposed, which were then validated through comparisons with the FE results. Good correlations were observed between the FE and design results in terms of both ultimate load and governing failure mode. While the maximum discrepancy of ultimate load was 18% (on the conservative side), the discrepancies for most cases were within 10%. The mean values of the FE-topredicted ratios for the unstiffened and stiffened splice cases were 1.00 and 1.04, respectively, and the corresponding CoVs were 7.9% and 8.6%, respectively. Design examples were subsequently provided to illustrate the proposed design procedure. Finally, it is worth noting that based on the considered gusset plate types, the current design scheme may be limited to the cases of rectangular gusset plate. For those beyond this limit, further studies may be required. Acknowledgements Support of this work has been provided by the Natural Sciences and Engineering Research Council of Canada under grant No. A4727. References [1] Brown VLS. Stability of gusseted connections in steel structures. University of Delaware; 1988. [2] Yamamoto K, Akiyama N, Okumura T. Buckling strengths of gusseted truss joints. J Struct Eng 1988;114:575–90. [3] Hu SZ, Cheng JJR. Compressive behavior of gusset plate connections 1987. Structural engineering report no. 153; 1987. [4] Cheng JJR, Yam MCH, Hu SZ. Elastic buckling strength of gusset plate connections. J Struct Eng 1994;120:538–59. [5] Gross JL. Experimental study of gusseted connections. Eng J 1990;27:89–97. [6] Chakrabarti SK, Richard RM. Inelastic buckling of gusset plates. Struct Eng Rev 1990; 12–29. [7] Yam MCH, Cheng JJR. Experimental investigation of the compressive behavior of gusset plate connections 1994. Structural engineering report no. 194; 1994. [8] Yam MCH, Cheng JJR. Behavior and design of gusset plate connections in compression. J Constr Steel Res 2002;58:1143–59. [9] Chambers J, Ernst CJ. Brace frame gusset plate research—phase I literature review. American Institute of Steel Construction; 2005. [10] Berman JW, Wang BS, Olson A, Roeder CW, Lehman DE. Simple check for yielding in truss bridge gusset plate connections. Structures Congress 2011ASCE; 2011 1027–35. [11] Lehman DE, Roeder CW, Herman D, Johnson S, Kotulka B. Improved seismic performance of gusset plate connections. J Struct Eng 2008;134:890–901. [12] Martinez-Saucedo G, Packer JA, Christopoulos C. Gusset plate connections to circular hollow section braces under inelastic cyclic loading. J Struct Eng 2008;134:1252–8. [13] Chou CC, Chen PJ. Compressive behavior of central gusset plate connections for a buckling-restrained braced frame. J Constr Steel Res 2009;65:1138–48. [14] Liao M, Okazaki T, Ballarini R, Schultz AE, Galambos TV. Nonlinear finite-element analysis of critical gusset plates in the I-35W bridge in Minnesota. J Struct Eng 2011;137:59–68. [15] Chou CC, Liou GS, Yu JC. Compressive behavior of dual-gusset-plate connections for buckling-restrained braced frames. J Constr Steel Res 2012;76:54–67. [16] Higgins C, Hafner A, Turan OT, Schumacher T. Experimental tests of truss bridge gusset plate connections with sway-buckling response. J Bridge Eng 2013;18: 980–91. [17] Packer JA, Henderson JE. Hollow structural section connections and trusses: a design guide. Willowdale, Ontario, Canada: Canadian Institute of Steel Construction; 1992. [18] Gebremeskel A. Lapped joints in compression. Steel construction. South Afr Inst Steel Constr 2013;37:38–9. [19] Kitipornchai S, Al-Bermani FGA, Murray NR. Eccentrically connected cleat plates in compression. J Struct Eng 1993;119:767–81. [20] Munter S. Advisory note, design method for eccentrically loaded cleats not to be used. Steel construction. Australian Steel Institute; 2005. p. 16. [21] Khoo XE, Perera M, Albermani F. Design of eccentrically connected cleat plates in connections. Int J of Adv Steel Constr 2010;6:678–87. [22] Crosti C, Duthinh D. Instability of steel gusset plates in compression. Struct Infrastruct Eng 2014;10(8):1038–48. [23] Simulia DCS. ABAQUS analysis user's manual. Providence, RI, USA: Dassault Systèmes Simulia Corp; 2010.

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