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Experimental investigation on the semi-rigid behaviour of aluminium alloy gusset joints
Thin-Walled Structures 87 (2015) 30–40
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Experimental investigation on the semi-rigid behaviour of aluminium alloy gusset joints Xiaonong Guo, Zhe Xiong n, Yongfeng Luo, Liqiu Qiu, Jia Liu Department of Building Engineering, Tongji University, Shanghai 200092, China
art ic l e i nf o
a b s t r a c t
Article history: Received 24 January 2014 Received in revised form 11 September 2014 Accepted 3 November 2014
Aluminium alloy gusset (AAG) joints are typical semi-rigid joints widely used in single-layer reticulated shells. Despite the semi-rigid behaviour of the AAG joints, structural analyses still show that dangerous situations can occur. To study the semi-rigid performance of the AAG joints, experiments on fourteen AAG joints are conducted. Initially, the failure modes of the AAG joints are summarised in terms of their collapse phenomena and the stress distributions of the plates are discussed based on the measured strain. Subsequently, the primary characteristics of the M-φ curves of the AAG joints are obtained. The bending stiffness properties of AAG joints are also investigated. The experimental results indicate the following: (1) the primary failure modes include member rupture, member buckling, block tearing of the top plates and local buckling in the bottom plates; (2) the moment-rotation relationship of the AAG joints exhibits a significant inelastic response; and (3) as the thickness of the gusset plate increases, the initial stiffness of the joint increases. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Aluminium alloy gusset joint Semi-rigid behavior Failure modes Bending stiffness
1. Introduction Compared with traditional construction material, single-layer reticulated shells made from an aluminium alloy offer a certain attractive appearance, transparency, material savings, lightness, ease of erection and favourable durability. Therefore, these types of single-layer reticulated shells have been widely used in spatial structures recently. As a typical semi-rigid joint system, aluminium alloy gusset (AAG) joints are widely applied in single-layer reticulated shells made from an aluminium alloy. However, in the design and analysis of single-layer reticulated shells over the years, the common practice has been to regard the joints as being ideally pinned or perfectly rigid [1]. Pinned joints make it difficult for single-layer reticulated shells to achieve the required stability, whereas rigid joints greatly improve the bearing capacity of single-layer reticulated shells. Consequently, it will strongly affect the safety and economy of structures if the AAG joints are assumed as pinned or rigid joints. Recent research has shown that the semi-rigid behaviour of joints plays a key role in the global buckling behaviour of singlelayer reticulated shells [2]. A large number of investigators have explored the influence of the semi-rigid behaviour of joints on the global buckling capacity of single-layer reticulated shells using numerical simulations, experimental programs and theoretical
n
Corresponding author. Tel.: þ 86 021 65980531. E-mail address: [email protected] (Z. Xiong).
http://dx.doi.org/10.1016/j.tws.2014.11.001 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
analyses. Lopez et al. [3] developed statistical regression techniques and nonlinear structural analysis models to investigate the effect of joint stiffness on the critical buckling load of reticulated domed-shaped structures. Subsequently, based on previous results, the authors proposed a new formula, which allows designers to rapidly estimate buckling loads for single-layer latticed domes with semi-rigid joints under symmetric loading conditions [4]. Kato et al. [5] established a nonlinear elastic–plastic hinge analysis formulated for three-dimensional beam-columns with elastic-plastic hinges located at both ends and at the midspan of each member. With the help of the proposed member model, reductions in collapse loads due to joint rigidity, geometric imperfections and member crookedness are discussed. Hiyama et al. [6,7] estimated the buckling behaviour of single-layer reticular domes made from an aluminium alloy composed of tubular truss members and ball joints via loading tests and numerical simulations on test structures scaled at a 1:5 ratio. The numerical results were in good agreement with the experimental results when the stiffness of the joints was 15.2 kN.m/rad. Fan et al. [8], Ma et al. [9] and Kitti [10] established finite element (FE) models in terms of single-layer latticed domes with semi-rigid joints and analyzed their buckling behaviour, where the bending stiffness, rise-to-span ratio, torsional stiffness, ball size, asymmetric load distribution, tube section, support condition and initial imperfections were considered. It should be noted that the geometric parameters, bending stiffness of the joint, rise-to-span ratio and tube section are the major factors that influence the critical load of these domes. The aforementioned studies showed
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that joint stiffness cannot be ignored in the analysis and design of single-layer reticulated shells, particularly for the bending stiffness of joints, which significantly affects their global buckling behavior. Thus far, extensive experimental studies and numerical simulations have been performed to investigate the stiffness behaviour of different joint systems in spatial structures, which include boltball joint system [11,12], welded hollow spherical joint system [13], tubular joint system [14–16] and so on. However, research on the semi-rigid behaviour of AAG joints is relatively limited. Comparing experimental results with FE results, Zeng et al. [17] found that to improve the accuracy of the numerical analysis in terms of single-layer reticulated shells, the stiffness of the AAG joints must be considered. To date, the behaviour of the AAG joints has primarily been investigated using the FE method. Based on the FE results of the AAG joints in practical engineering, Zou [18] highlighted that AAG joints are typical semi-rigid joints. The state of the research in the field of AAG joints is presently under development, where the primary limitations are concluded as follows: (1) research on the semi-rigid behaviour of AAG joints is currently at an early stage, and further study is required; (2) there is no specific design method to predict the bending stiffness of AAG joints; and (3) an experimental program on the semi-rigid characteristics of AAG joints must be performed. These research limitations greatly influence the widespread application of aluminium alloy single-layer reticulated shells with AAG joints. With the aim of resolving the aforementioned research limitations, this article is primarily focused on the semi-rigid behaviour of AAG joints. Tests on fourteen AAG joints are conducted to study their out-of-plane flexural capacity. Firstly, the experimental program is introduced. Secondly, the failure modes of the AAG joint specimens are summarised in terms of their collapse phenomena. Finally, the experimental results are discussed. In addition, both the stress distributions of the plates and the out-ofplane bending stiffness of the AAG joint specimens are evaluated.
2. Experimental program 2.1. Specimens A series of tests on fourteen AAG joint specimens was performed to investigate their semi-rigid behaviour. The AAG joint system is commonly composed of six beam members tightly attached to top and bottom circular plates by means of bolted connections, as shown in Fig. 1(a). All the beam members have an I-shaped cross-section, and the angle between adjacent members is 60 degrees. The identification numbers of I-shaped members range from L1 to L6, as shown in Fig. 1(b). Hand tightened stainless
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steel M6 bolts in 6.5 mm drilled holes were used in all the specimens, as shown in Fig. 1(c). In addition, ten bolts were used to connect one side of the flanges to the gusset plate. In the AAG joint system, the force acting on the members can be transmitted to each other through the bolts and circular plates. It is worth noting that in practice, six I-shape beams meet at each joint. Therefore, the specimens designed in this article are similar to practical scenarios. All the I-shaped members have the same cross-sectional dimensions of 100 50 4 5 mm, which represents that the height of cross-section is 100 mm, the width of cross-section is 50 mm, the thickness of web is 4 mm and the thickness of each flange is 5 mm. The length of each I-shaped member is 890 mm, and the diameter of each circular plate is 280 mm. Detailed configurations of the test specimens are plotted in Fig. 2. Three primary parameters were varied. The first parameter was the plate thickness. Four specimens were classified as a thick plate joint system (plate thickness t Z5 mm), and ten specimens were classified as a thin plate joint system (plate thickness t r3 mm). The second parameter was the shear connector. The A series corresponded to a joint specimen without a shear connector, whereas the B series corresponded to a joint specimen with a type B shear connector, and the C series corresponded to a joint specimen with a type C shear connector, as illustrated in Fig. 3. The final parameter was the loading scheme. Three loading schemes, which include six loaded members, three loaded members and two loaded members, were applied to the specimens, as shown in Fig. 12. Detailed information of these specimens is presented in Table 1. 2.2. Materials Aluminium alloy 6063-T5 [19] was selected as the material for the extruded beam members and extruded plates. Austenitic stainless steel was the material chosen for the bolts (material grade of A2-70) [20]. Tensile tests were performed to investigate the actual mechanical properties of these components according to the Chinese mechanical testing standard [21]. Four tensile coupons were cut directly from the flange and the web of the I-shaped beam members, as shown in Fig. 4(a). Six tensile coupons were cut directly from the extruded plates, as shown in Fig. 4(b). In addition, the six tensile coupons, which had diameters of 6 mm, were the same material as that of the bolts, as shown in Fig. 4(c). All the aluminium alloy tensile coupons had the same dimensions described in Fig. 5. The mechanical properties of these components obtained from the tensile tests are listed in Table 2, where E is the elastic modulus, f0.2 is the nominal yield strength and fu is the ultimate tensile strength.
Fig. 1. AAG joint. (a) AAG joint specimen (b) Planar graph of the AAG joint (c) Cross-section of the AAG joint.
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Fig. 2. Detailed configurations of the test specimens. (a) Detailed configurations of the beam members (mm) (b) Detailed configurations of the gusset plates (mm).
Fig. 3. Types of shear connectors. (a) Specimen without a shear connector, (b) Specimen with a type B shear connector, (c) Specimen with a type C shear connector.
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2.3. Arrangement of measuring point For convenience in its description, the plate is divided into several parts, as shown in Fig. 6. Additionally, each part is represented by a symbol. The symbols and their interpretations are presented in Table 3. Three sets of measurement points were arranged in the AAG joint specimens as follows: (1) The unidirectional strain gauges placed on the beam members are illustrated in Fig. 7. Four unidirectional strain gauges (P1 P4) were located at the flanges of the beam members near the joint zone. These strain gauges were used to monitor the eccentric action during the loading process, and also used to calculate the bending moment and axial force acting at the joint. Four unidirectional strain gauges (P5 P8) were attached to the flanges of the beam members near the hinged support zone and were used to estimate the internal force of the support. (2) The strain gauges placed on the plates are plotted in Fig. 8. Although the arrangements of the strain gauges on the plates of the different joint specimens were slightly different, all of them obeyed the same principles: (i) for the free area of the plates, there are hoop strain gauges, radial strain gauges or strain-gauge rosettes, as shown in Fig. 8(a); (ii) for the connection area of the plates, there are radial strain gauges, as illustrated in Fig. 8(b); (iii) for the central area of the plates, there are hoop strain gauges and radial strain gauges, as shown in Fig. 8(c). In Fig. 8, “R83” and “R30” indicate that the distance between the strain gauge and the centre of plate are 83 mm and 30 mm, respectively. The strain gauge
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categories, which are denoted on the plates, are represented by symbols, as listed in Table 4. The interpretations of the symbols in Fig. 8 are listed in Table 5. In addition, to confirm the surface of the plates, the symbols Ut, Ub, Dt and Db are added. For example, for strain gauge “Db-f12H-d”, “Db”
Fig. 5. Dimensions of aluminium alloy coupons.
Table 2 Mechanical properties of the material. Components
E (MPa)
f0.2 (MPa)
fu (MPa)
Plates Beam members Bolts
69088 65364 206000
184.40 177.40 470
213.58 206.80 725
Table 1 Detailed specimen information. No.
Plate thickness (mm)
Shear connector
Number of loaded members
A1 A2 A3 A4 A5 A6 A7 B1 B2 B3 C1 C2 C3 C4
6.00 5.00 5.00 2.70 2.25 2.25 2.25 2.25 2.25 2.25 6.00 2.25 2.25 2.70
None None None None None None None B B B C C C C
Six members Six members Two members Six members Six members Three members Two members Six members Three members Two members Six members Six members Three members Two members Fig. 6. Partitions of the plates.
Fig. 4. Tensile coupons. (a) Coupons of the beam members (b) Coupons of the plates (c) Coupons of the bolts.
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represents that the strain gauge is placed at the outer surface of the bottom plates, “f12” means that the strain gauge is in the free area of the plates between member L1 and member L2 and “H-d” represents the radial strain gauge of the straingauge rosette. The strain gauges placed on the plates were used to determine the stress distributions of the plates. (3) Linear variable differential transducers (LVDTs) were used to measure the displacements of the specimens during the experimental process. All the specimens used the same arrangement of LVDTs, as illustrated in Fig. 9. Every LVDT was placed in the middle area of each member to measure its vertical displacement. The distance between the LVDT and the end of members was 350 mm. Corresponding to the identification numbers of the I-shaped members, the identification Table 3 Symbols of the plate parts. Symbols
Interpretations
Ut Ub Dt Db ck i fkj
Outer surface of the top plates Inner surface of the top plates Inner surface of the bottom plates Outer surface of the bottom plates Connection area of the plates for member Lk Central area of the plates Free area of the plates between member Lk and member Lj
numbers of the LVDTs range from ω1 to ω6. In addition, the identification number of the LVDT located in the middle of the joint plates is ω7.
2.4. Testing arrangements The objective of the experimental program is to provide insight into the out-of-plane bending capacity of AAG joints. To create the out-of-plane bending moment acting at the joints, the concentrated transverse load was applied within the middle area of the beam members by means of a vertical hydraulic jack. A selfbalancing reaction frame with central symmetry was designed for the experimental program, as shown in Fig. 10. Each beam member of the specimens was hinged to the reaction frame using a high strength M24 bolt, as shown in Fig. 11. These specimens were tested under three different loading schemes, including six loaded members, three loaded members and two loaded members. To avoid the out-of-plane torsion caused by unbalanced bending moment, a central symmetry loading scheme was adopted, as shown in Fig. 12. The distance between the hydraulic jack and the end of the members was 500 mm. Initially, a preload was performed to suppress slipping Table 4 The concise symbols of the categories of strain gauges. Symbols
Interpretations
D R H-d H-r H-s
Radial strain gauge Hoop strain gauge Radial strain gauge of strain-gauge rosette Hoop strain gauge of strain-gauge rosette Oblique strain gauge of strain-gauge rosette
Fig. 7. Unidirectional strain gauges placed on the beam members.
Fig. 8. Position of the strain gauges placed within the plates. (a) Strain gauges placed within the free area (mm) (b) Strain gauges placed within the connection area (mm), (c) Strain gauges placed within t the central area (mm).
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and to allow the different components to settle and adjust. Subsequently, all the loads were increased simultaneously. The test was conducted under force control at the beginning of the loading process. When the behaviour of the specimens entered
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into non-linear region, the test control was changed to displacement control until failure occurred. The test procedure is illustrated in Fig. 13.
3. Experimental phenomena Table 5 Interpretations of the symbols in Fig. 8. Symbols Interpretations f61R f45H-s f45H-r f45H-d f23D c4D i34R i1D
Hoop strain gauge placed within the free area of the plates between member L6 and member L1 Oblique strain gauge of the strain-gauge rosette placed within the free area of the plates between member L4 and member L5 Hoop strain gauge of the strain-gauge rosette placed within the free area of the plates between member L4 and member L5 Radial strain gauge of the strain-gauge rosette placed within the free area of the plates between member L4 and member L5 Radial strain gauge placed within the free area of the plates between member L2 and member L3 Radial strain gauge placed within the connection area of the plates for member L4 Hoop strain gauge placed within the central area of the plates between member L3 and member L4 Radial strain gauge placed within the central area of the plates for member L1
To obtain the complete response of the AAG joint specimens, each load increment lasted for at least 5 min during the test. Both the failure modes and the measured ultimate loads Pu are summarised in Table 6. The failure phenomena of the specimens are shown in Fig. 14. It can be concluded that the primary failure modes of the specimens include member rupture, member buckling, block tearing of the top plates, buckling within the free area of the bottom plates and buckling within the central area of the bottom plates. For the thick plate specimens, their primary failure modes were member buckling and member rupture, whereas for the thin plate specimens, their primary collapse mechanisms involved buckling in the bottom plates caused by compressive force and block tearing of the top plates caused by tensile force. With respect to the specimens A1, A2 and A5, which only differ in plate thickness, the experimental results show that the bending capacity of the AAG joints is improved significantly with the increase of the plate thickness. Regarding the specimens A5, B1 and C2, which differ in shear connector types, the experimental results show that the shear connector enhances the bending capacity of the AAG joints, where the C shear connector is more efficient than that of the B shear connector. A possible reason for this result is that the shear connector improves the global behaviour of the AAG joints. For the specimens A5, A6 and A7, which differ in loading schemes, the ultimate loads of specimens A6 and A7 were much larger than that of specimen A5. This result implies that the AAG joints exhibit a good global performance in transmitting the force from loaded members to unloaded members.
Fig. 9. Arrangement of LVDTs (mm). (a) Planar graph, (b) Cross-section.
Fig. 10. Self-balancing reaction frame.
Fig. 11. Hinged support.
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Fig. 12. Three kinds of loading schemes. (a) Six loaded members, (b) Three loaded members, (c) Two loaded members (d) Cross-section.
Fig. 13. Test procedure.
4. Results and discussions 4.1. Load-strain curves of the plates The load-strain curves of the plates are plotted in Fig. 15, where it can be observed that the top plates are subjected to tensile force, whereas the bottom plates are subjected to compressive force. Fig. 15(a) exhibits the strain variations within the free area of specimen A2 measured by the strain-gauge rosettes during the loading process. The strain in the hoop direction is the largest, and
the strain in the radial direction is so small that it can be neglected. It is evident that the maximum principal stress direction is the hoop direction within the free area. Fig. 15(b) shows the load-strain curves within the free area of specimen A1 measured by the hydraulic jacks and hoop strain gauges. Fig. 15(a) and Fig. 15(b) indicate that when six members are loaded, the strains within the free area exhibit a good symmetric characteristic, which signifies that the specimens are predominantly subjected to a bending moment, whereas there is only a small axial force.
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Table 6 The failure modes and the measured ultimate loads. No. Failure modes
A1 A2 A3 A4 A5 A6 A7 B1 B2 B3 C1 C2 C3 C4
Pu (kN)
Rupture of member L1 10.146 Buckling of member L2 9.752 Buckling of member L6; rupture of member L1 24.976 Block tearing within the connection area c4, c5, c6 and the free area f45, f56 of the top plates 6.663 Block tearing within the connection area c4, c5 and the free area f45, f56 of the top plates 5.548 Block tearing within the connection area c4 of the top plates; buckling within the free area f61, f34 and the central area of the bottom plates 11.405 Buckling within the free area f56, f34 and the central area of the bottom plates 14.778 Block tearing within the connection area c4, c5 and the free area f45 of the top plates 5.924 Block tearing within the connection area c4, c5 and the free area f23, f34, f45 of the top plates; buckling within the free area f61 and the central area of the 12.200 bottom plates 14.487 Buckling within the free area f56, f12 and the central area of the bottom plates Rupture of member L3 and member L4 10.940 6.045 Block tearing within the connection area c1, c2, c6 and the free area f12, f61 of the top plates Block tearing within the connection area c1, c2, c5, c6 and the free area f12, f56, f61 of the top plates 10.239 Block tearing within the connection area c1, c6 and the free area f12, f61 of the top plates 16.991
Fig. 14. Failure phenomena of the specimens. (a) Member rupture, (b) Member buckling, (c) Block tearing of the plates, (d) Buckling within the free area of the plates, (e) Buckling within the central area of the plates.
Fig. 15(c) plots the load-strain curves within the connection area of specimen B1 measured by the hydraulic jacks and radial strain gauges. The radial strain in the connection area is extremely small. The load-strain curves within the central area of specimen A5 are illustrated in Fig. 15 (d). At the beginning of the loading process, the load-strain curves revealed a linear symmetric response. When the loads were greater than 3.5 kN, the top plates gradually became more flexible, and simultaneously, the bottom plates became more flexible suddenly, which implies a buckling failure in the bottom plates. 4.2. Derivation of the moment-rotation (M-φ) curves The mechanical characteristics of the AAG joints can be represented by moment-rotation curves that perform the relationship between the bending moment M and the corresponding rotation φ. To obtain M-φ curves, the measured strains of P1 P4
were used to calculate the bending moment M, and the vertical measured displacements of ω1 ω6 were adopted to estimate the rotation φ. The cross-section of the member is divided into fourteen parts, as shown in Fig. 16. Then the bending moment M could be derived from Eq. (1) as follows: 14
M ¼ ∑ σ j Aj Z j j¼1
ð1Þ
where Aj represents the area of part j; Zj is the distance between the geometric centre of part j and point O (geometric centre of the I-section) in the z direction; and σj is the stress of part j. Stress σj can be calculated in two steps. Firstly the strain of part j εj can be estimated by the measured strains of P1 P4 on the hypothesis of plane cross-section. Subsequently, stress σj can be obtained using the Ramburg-Osgood model [22], as described in Eq. (2). The constant n can be calculated according to the
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Fig. 16. Partitions of the cross-section.
Fig. 17. Deformation of the specimen model. (a) Specimen model, (b) Deformation of specimen.
10n ¼ f 0:2 ðMpaÞ
ð3Þ
The specimen deformation is shown in Fig. 17. The rotational deformation is the sum of the deformation of the I-shaped member and the rotational deformation of the joint. Thus, the total vertical displacement ω measured by the LVDTs (ω1 ω6) can be represented by Eq. (4): ω ¼ ω r þ ωb
ð4Þ
where ωr is the vertical displacement caused by the rotation of joint, and ωb is the vertical displacement caused by the deformation of the I-shaped member. The value of ωb can be calculated by the FE method. When the stiffness of the joint is infinite, then the total vertical displacement is caused by the deformation of the Ishaped member. An FE model was created using ABAQUS software to calculate the value of ωb, as plotted in Fig. 18. The rotation of the joint is calculated by Eq. (5): ω ω ω r ¼ arctan b ð5Þ ϕ ¼ arctan x x x where x is the distance between the position of the LVDT and the end of the I-shaped member.
4.3. M-φ curves of the specimens with six loaded members According to the aforementioned derivation, the M-φ curves can be obtained. The M-φ curves of the specimens with six loaded members are shown in Fig. 19. To obtain their initial bending stiffness, linear regression analysis is used in the elastic phases. The values of the initial bending stiffness are listed in Table 7. According to Fig. 19 and Table 7, the bending stiffness characteristics of the AAG joints are discussed as follows: Fig. 15. Load-strain curves of the plates. (a) Load-strain curves within the free area of the specimen A2 plates, (b) Load-strain curves within the free area of the specimen A1 plates, (c) Load-strain curves within the connection area of the specimen B1 plates, (d) Load-strain curves within the central area of the specimen A5 plates.
Steinhardt suggestion [23], as shown in Eq. (3). εj ¼
σj σj n þ 0:002 E f 0:2
ð2Þ
(1) During the loading process, the bending stiffness characteristics of all the specimens underwent at least two phases. In the first phase, the bending stiffness was responsible for the elastic behaviour (Mo1 kN.m). When the bending moment reached eighty percent of the ultimate bending moment, the bending stiffness decreased rapidly. This reflected the second phase of the bending stiffness characteristics. In addition, it is worth noting that for the thick plate specimens A1 and C1, the bending stiffness clearly increased after the second phase.
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Fig. 18. FE model with a rigid joint.
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5 mm to 6 mm. For the thin plate specimens A4 and A5, the initial bending stiffness improved by 15.7% when the plate thickness increased from 2.25 mm to 2.7 mm. (3) The shear connector can effectively improve the initial bending stiffness of AAG joints, particularly in the case of thin plate specimens. Regarding the specimens with a 6 mm thickness plate, with the help of C shear connector, the initial bending stiffness was 14.5% greater than those without a shear connector. Regarding the specimens with a 2.25 mm thickness plate, using a type B shear connector, the initial bending stiffness increased 21.5%, whereas the initial bending stiffness
Fig. 19. M-φ curves for the specimens with six loaded members. (a) M-φ curves for the thick plate specimens, (b) M-φ curves for the thin plate specimens.
Table 7 Initial bending stiffness of the specimens (kN.m/rad). No.
Initial bending stiffness
No.
Initial bending stiffness
A1 A2 A4 A5
441 385 376 325
B1 C1 C2
395 505 475
(2) It is found that the plate thickness has an important effect on the initial bending stiffness of the AAG joints. For the thick plate specimens A1 and A2, the initial bending stiffness improved by 12.7% when the plate thickness increased from
Fig. 20. M-φ curves for specimens with three loaded members. (a) M-φ curves of specimen A6, (b) M-φ curves of specimen B2, (c) M-φ curves of specimen C3.
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increased 46.1% using a type C shear connector. The primary reason for this may be because the global behaviour of the thick plate specimens is better than that of the thin plate specimens such that the shear connector improves the global behaviour of the thin plate specimens more efficiently.
4.4. M-φ curves of the specimens with three loaded members The M-φ curves of the specimens with three loaded members are shown in Fig. 20. Each specimen has two types of M-φ curves, one that corresponds to the loaded member and the other that corresponds to the unloaded member. During the initial phase, the bending characteristics of the two types of M-φ curves were the same. However, the bending characteristic of the M-φ curve of the unloaded member became flexible after the initial phase. A possible reason for this may be that the deformation of the unloaded members is affected by the one of the loaded members. This phenomenon implies the interaction of the members on the bending stiffness of the AAG joints. 5. Conclusions and prospects In this article, the semi-rigid behaviour of the AAG joints has been investigated via an experimental study on fourteen AAG joint specimens. The following are the main conclusions: (1) The primary failure modes of the AAG joint specimens include member rupture, member buckling, block tearing of the top plates, buckling within the free area of the bottom plates and buckling within the central area of the bottom plates. For the thick plate specimens, their primary failure modes are member buckling and member rupture. For the thin plate specimens, their primary collapse mechanisms involve buckling of the bottom plates caused by compressive force and block tearing of the top plates caused by tensile force. (2) It is found from the strain distributions of the plates that the maximum principal stress direction is in the hoop direction. Under the loading scheme with six loaded members, the strains are central symmetric. (3) It can be observed from the M-φ curves that the AAG joints are typical semi-rigid joints. The joint specimens reveal a nonlinear bending stiffness behaviour response as the bending stiffness decreases during the second phase. Both the plate thickness and the shear connector have great effects on the initial bending stiffness of the AAG joints. The thicker the plate is, the stiffer the AAG joint is. The shear connector enhances the initial bending stiffness of the AAG joints by improving their global behaviour. (4) An interaction phenomenon exists with the bending stiffness of the AAG joints between loaded members and unloaded members.
Acknowledgement The authors gratefully acknowledge the financial support provided by the Natural Science Foundation of China under Grant No. 50908168 and No. 51478335.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.tws.2014.11.001.
References [1] Lopez A, Puente I, Aizpurua H. Experimental and analytical studies on the rotational stiffness of joints for single-layer structures. Eng Struct 2011;33:731–7. [2] Lopez A, Puente I, Serna MA. Numerical model and experimental tests on singlelayer latticed domes with semi-rigid joints. Comp Struct 2007;85:360–74. [3] Lopez A, Gerald K. The effect of joint stiffness on the buckling behaviour of reticulated dome structures. Asia-Pacific Conference on Shell and Spatial Structures 1996:564–70 (CCES-IASS, Beijing). [4] Lopez A, Puente I, Miguel A. Direct evaluation of the buckling loads of semirigidly jointed single-layer latticed domes under symmetric loading. Eng Struct 2007;29:101–9. [5] Kato S, Mutoh I, Shomura M. Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections. J Const Steel Res 1998;48:145–68. [6] Hiyama Y, Ishikawa K, Kato S, et al. Experiments and analysis of the postbuckling behaviours of aluminium alloy double layer space grids applying ball joints. Struct Eng Mech 2000;9(3):289–304. [7] Hiyama Y, Takashima H, Iijima T, et al. Buckling behaviour of aluminium ball jointed single layered reticular domes. Int J Space Struct 2000;15(2):81–94. [8] Fan F, Ma HH, Cao ZZ, et al. Direct estimation of critical load for single-layer reticulated domes with semi-rigid joints. Int J Space Struct 2010;25(1):15–24. [9] Ma HH, Fan F, Shen SZ. Numerical parametric investigation of single-layer latticed domes with semi-rigid Joints. J Int Assoc Shell Spatial Struct 2008;49 (2):99–110. [10] Kitti G. Effect of connection rigidity on the behaviour of single-layer steel grid shells. Conference of Junior Researchers in Civil Engineering 2012:58–65 (Budapest, Hungary). [11] Fan F, Ma HH, Chen GB, et al. Experimental study of semi-rigid joint systems subjected to bending with and without axial force. J Constr Steel Res 2012;68 (1):126–37. [12] Ma HH, Fan F, Chen Gengbo, et al. Numerical analyses of semi-rigid joint systems subjected to bending with and without axial force. J Constr Steel Res 2013;90:13–28. [13] Han QH, Liu XL. Ultimate bearing capacity of the welded hollow spherical joints in spatial reticulated structures. Eng Struct 2004;26(1):73–82. [14] Hyde TH, Leen SB. Prediction of elastic–plastic displacements of tubular joints under combined loading using an energy-based approach. J Strain Anal 1997;32(6):435–54. [15] Leen SB, Hyde TH. On the prediction of elastic-plastic generalized loaddisplacement responses for tubular joints. J Strain Anal 2000;35(3):205–19. [16] ZHAO QIU G Z. J C. Analysis and calculation of axial stiffness of tubular X joints under compression on braces. J Shanghai Jiaotong Univ 2009;14 (4):410–7. [17] Zeng YZ, Qian RJ, Wang RP, et al. Test research on the aluminous-alloy dome. Spatial Struct 2000;6(4):47–52 (In Chinese). [18] Zou L. Structure analysis study on aluminium alloy dome of Chong Qing Kong Gang Stadium. Master. Thesis, College of Civil Engineering of Chongqing University 2009 (In Chinese). [19] GB 50429-2007. Code for design of aluminium structures; 2007.(In Chinese). [20] GB/T 3098.6-2000. Mechanical properties of fasteners: boles, screws and studs made of stainless-steel; 2001. (In Chinses). [21] GB/T 228-2002. Metallic materials-tensile testing at ambient temperature; 2002 (In Chinese). [22] Ramberg W., Osgood W.R. Description of stress-stain curves by three parameters. Washington DC; NACA TN-902; 1943. [23] SteinHardt O. Aluminium constructions in civil engineering. Aluminium 1971;47:131–9.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Experimental investigation on the semi-rigid behaviour of aluminium alloy gusset joints Xiaonong Guo, Zhe Xiong n, Yongfeng Luo, Liqiu Qiu, Jia Liu Department of Building Engineering, Tongji University, Shanghai 200092, China
art ic l e i nf o
a b s t r a c t
Article history: Received 24 January 2014 Received in revised form 11 September 2014 Accepted 3 November 2014
Aluminium alloy gusset (AAG) joints are typical semi-rigid joints widely used in single-layer reticulated shells. Despite the semi-rigid behaviour of the AAG joints, structural analyses still show that dangerous situations can occur. To study the semi-rigid performance of the AAG joints, experiments on fourteen AAG joints are conducted. Initially, the failure modes of the AAG joints are summarised in terms of their collapse phenomena and the stress distributions of the plates are discussed based on the measured strain. Subsequently, the primary characteristics of the M-φ curves of the AAG joints are obtained. The bending stiffness properties of AAG joints are also investigated. The experimental results indicate the following: (1) the primary failure modes include member rupture, member buckling, block tearing of the top plates and local buckling in the bottom plates; (2) the moment-rotation relationship of the AAG joints exhibits a significant inelastic response; and (3) as the thickness of the gusset plate increases, the initial stiffness of the joint increases. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Aluminium alloy gusset joint Semi-rigid behavior Failure modes Bending stiffness
1. Introduction Compared with traditional construction material, single-layer reticulated shells made from an aluminium alloy offer a certain attractive appearance, transparency, material savings, lightness, ease of erection and favourable durability. Therefore, these types of single-layer reticulated shells have been widely used in spatial structures recently. As a typical semi-rigid joint system, aluminium alloy gusset (AAG) joints are widely applied in single-layer reticulated shells made from an aluminium alloy. However, in the design and analysis of single-layer reticulated shells over the years, the common practice has been to regard the joints as being ideally pinned or perfectly rigid [1]. Pinned joints make it difficult for single-layer reticulated shells to achieve the required stability, whereas rigid joints greatly improve the bearing capacity of single-layer reticulated shells. Consequently, it will strongly affect the safety and economy of structures if the AAG joints are assumed as pinned or rigid joints. Recent research has shown that the semi-rigid behaviour of joints plays a key role in the global buckling behaviour of singlelayer reticulated shells [2]. A large number of investigators have explored the influence of the semi-rigid behaviour of joints on the global buckling capacity of single-layer reticulated shells using numerical simulations, experimental programs and theoretical
n
Corresponding author. Tel.: þ 86 021 65980531. E-mail address: [email protected] (Z. Xiong).
http://dx.doi.org/10.1016/j.tws.2014.11.001 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
analyses. Lopez et al. [3] developed statistical regression techniques and nonlinear structural analysis models to investigate the effect of joint stiffness on the critical buckling load of reticulated domed-shaped structures. Subsequently, based on previous results, the authors proposed a new formula, which allows designers to rapidly estimate buckling loads for single-layer latticed domes with semi-rigid joints under symmetric loading conditions [4]. Kato et al. [5] established a nonlinear elastic–plastic hinge analysis formulated for three-dimensional beam-columns with elastic-plastic hinges located at both ends and at the midspan of each member. With the help of the proposed member model, reductions in collapse loads due to joint rigidity, geometric imperfections and member crookedness are discussed. Hiyama et al. [6,7] estimated the buckling behaviour of single-layer reticular domes made from an aluminium alloy composed of tubular truss members and ball joints via loading tests and numerical simulations on test structures scaled at a 1:5 ratio. The numerical results were in good agreement with the experimental results when the stiffness of the joints was 15.2 kN.m/rad. Fan et al. [8], Ma et al. [9] and Kitti [10] established finite element (FE) models in terms of single-layer latticed domes with semi-rigid joints and analyzed their buckling behaviour, where the bending stiffness, rise-to-span ratio, torsional stiffness, ball size, asymmetric load distribution, tube section, support condition and initial imperfections were considered. It should be noted that the geometric parameters, bending stiffness of the joint, rise-to-span ratio and tube section are the major factors that influence the critical load of these domes. The aforementioned studies showed
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that joint stiffness cannot be ignored in the analysis and design of single-layer reticulated shells, particularly for the bending stiffness of joints, which significantly affects their global buckling behavior. Thus far, extensive experimental studies and numerical simulations have been performed to investigate the stiffness behaviour of different joint systems in spatial structures, which include boltball joint system [11,12], welded hollow spherical joint system [13], tubular joint system [14–16] and so on. However, research on the semi-rigid behaviour of AAG joints is relatively limited. Comparing experimental results with FE results, Zeng et al. [17] found that to improve the accuracy of the numerical analysis in terms of single-layer reticulated shells, the stiffness of the AAG joints must be considered. To date, the behaviour of the AAG joints has primarily been investigated using the FE method. Based on the FE results of the AAG joints in practical engineering, Zou [18] highlighted that AAG joints are typical semi-rigid joints. The state of the research in the field of AAG joints is presently under development, where the primary limitations are concluded as follows: (1) research on the semi-rigid behaviour of AAG joints is currently at an early stage, and further study is required; (2) there is no specific design method to predict the bending stiffness of AAG joints; and (3) an experimental program on the semi-rigid characteristics of AAG joints must be performed. These research limitations greatly influence the widespread application of aluminium alloy single-layer reticulated shells with AAG joints. With the aim of resolving the aforementioned research limitations, this article is primarily focused on the semi-rigid behaviour of AAG joints. Tests on fourteen AAG joints are conducted to study their out-of-plane flexural capacity. Firstly, the experimental program is introduced. Secondly, the failure modes of the AAG joint specimens are summarised in terms of their collapse phenomena. Finally, the experimental results are discussed. In addition, both the stress distributions of the plates and the out-ofplane bending stiffness of the AAG joint specimens are evaluated.
2. Experimental program 2.1. Specimens A series of tests on fourteen AAG joint specimens was performed to investigate their semi-rigid behaviour. The AAG joint system is commonly composed of six beam members tightly attached to top and bottom circular plates by means of bolted connections, as shown in Fig. 1(a). All the beam members have an I-shaped cross-section, and the angle between adjacent members is 60 degrees. The identification numbers of I-shaped members range from L1 to L6, as shown in Fig. 1(b). Hand tightened stainless
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steel M6 bolts in 6.5 mm drilled holes were used in all the specimens, as shown in Fig. 1(c). In addition, ten bolts were used to connect one side of the flanges to the gusset plate. In the AAG joint system, the force acting on the members can be transmitted to each other through the bolts and circular plates. It is worth noting that in practice, six I-shape beams meet at each joint. Therefore, the specimens designed in this article are similar to practical scenarios. All the I-shaped members have the same cross-sectional dimensions of 100 50 4 5 mm, which represents that the height of cross-section is 100 mm, the width of cross-section is 50 mm, the thickness of web is 4 mm and the thickness of each flange is 5 mm. The length of each I-shaped member is 890 mm, and the diameter of each circular plate is 280 mm. Detailed configurations of the test specimens are plotted in Fig. 2. Three primary parameters were varied. The first parameter was the plate thickness. Four specimens were classified as a thick plate joint system (plate thickness t Z5 mm), and ten specimens were classified as a thin plate joint system (plate thickness t r3 mm). The second parameter was the shear connector. The A series corresponded to a joint specimen without a shear connector, whereas the B series corresponded to a joint specimen with a type B shear connector, and the C series corresponded to a joint specimen with a type C shear connector, as illustrated in Fig. 3. The final parameter was the loading scheme. Three loading schemes, which include six loaded members, three loaded members and two loaded members, were applied to the specimens, as shown in Fig. 12. Detailed information of these specimens is presented in Table 1. 2.2. Materials Aluminium alloy 6063-T5 [19] was selected as the material for the extruded beam members and extruded plates. Austenitic stainless steel was the material chosen for the bolts (material grade of A2-70) [20]. Tensile tests were performed to investigate the actual mechanical properties of these components according to the Chinese mechanical testing standard [21]. Four tensile coupons were cut directly from the flange and the web of the I-shaped beam members, as shown in Fig. 4(a). Six tensile coupons were cut directly from the extruded plates, as shown in Fig. 4(b). In addition, the six tensile coupons, which had diameters of 6 mm, were the same material as that of the bolts, as shown in Fig. 4(c). All the aluminium alloy tensile coupons had the same dimensions described in Fig. 5. The mechanical properties of these components obtained from the tensile tests are listed in Table 2, where E is the elastic modulus, f0.2 is the nominal yield strength and fu is the ultimate tensile strength.
Fig. 1. AAG joint. (a) AAG joint specimen (b) Planar graph of the AAG joint (c) Cross-section of the AAG joint.
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Fig. 2. Detailed configurations of the test specimens. (a) Detailed configurations of the beam members (mm) (b) Detailed configurations of the gusset plates (mm).
Fig. 3. Types of shear connectors. (a) Specimen without a shear connector, (b) Specimen with a type B shear connector, (c) Specimen with a type C shear connector.
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2.3. Arrangement of measuring point For convenience in its description, the plate is divided into several parts, as shown in Fig. 6. Additionally, each part is represented by a symbol. The symbols and their interpretations are presented in Table 3. Three sets of measurement points were arranged in the AAG joint specimens as follows: (1) The unidirectional strain gauges placed on the beam members are illustrated in Fig. 7. Four unidirectional strain gauges (P1 P4) were located at the flanges of the beam members near the joint zone. These strain gauges were used to monitor the eccentric action during the loading process, and also used to calculate the bending moment and axial force acting at the joint. Four unidirectional strain gauges (P5 P8) were attached to the flanges of the beam members near the hinged support zone and were used to estimate the internal force of the support. (2) The strain gauges placed on the plates are plotted in Fig. 8. Although the arrangements of the strain gauges on the plates of the different joint specimens were slightly different, all of them obeyed the same principles: (i) for the free area of the plates, there are hoop strain gauges, radial strain gauges or strain-gauge rosettes, as shown in Fig. 8(a); (ii) for the connection area of the plates, there are radial strain gauges, as illustrated in Fig. 8(b); (iii) for the central area of the plates, there are hoop strain gauges and radial strain gauges, as shown in Fig. 8(c). In Fig. 8, “R83” and “R30” indicate that the distance between the strain gauge and the centre of plate are 83 mm and 30 mm, respectively. The strain gauge
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categories, which are denoted on the plates, are represented by symbols, as listed in Table 4. The interpretations of the symbols in Fig. 8 are listed in Table 5. In addition, to confirm the surface of the plates, the symbols Ut, Ub, Dt and Db are added. For example, for strain gauge “Db-f12H-d”, “Db”
Fig. 5. Dimensions of aluminium alloy coupons.
Table 2 Mechanical properties of the material. Components
E (MPa)
f0.2 (MPa)
fu (MPa)
Plates Beam members Bolts
69088 65364 206000
184.40 177.40 470
213.58 206.80 725
Table 1 Detailed specimen information. No.
Plate thickness (mm)
Shear connector
Number of loaded members
A1 A2 A3 A4 A5 A6 A7 B1 B2 B3 C1 C2 C3 C4
6.00 5.00 5.00 2.70 2.25 2.25 2.25 2.25 2.25 2.25 6.00 2.25 2.25 2.70
None None None None None None None B B B C C C C
Six members Six members Two members Six members Six members Three members Two members Six members Three members Two members Six members Six members Three members Two members Fig. 6. Partitions of the plates.
Fig. 4. Tensile coupons. (a) Coupons of the beam members (b) Coupons of the plates (c) Coupons of the bolts.
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represents that the strain gauge is placed at the outer surface of the bottom plates, “f12” means that the strain gauge is in the free area of the plates between member L1 and member L2 and “H-d” represents the radial strain gauge of the straingauge rosette. The strain gauges placed on the plates were used to determine the stress distributions of the plates. (3) Linear variable differential transducers (LVDTs) were used to measure the displacements of the specimens during the experimental process. All the specimens used the same arrangement of LVDTs, as illustrated in Fig. 9. Every LVDT was placed in the middle area of each member to measure its vertical displacement. The distance between the LVDT and the end of members was 350 mm. Corresponding to the identification numbers of the I-shaped members, the identification Table 3 Symbols of the plate parts. Symbols
Interpretations
Ut Ub Dt Db ck i fkj
Outer surface of the top plates Inner surface of the top plates Inner surface of the bottom plates Outer surface of the bottom plates Connection area of the plates for member Lk Central area of the plates Free area of the plates between member Lk and member Lj
numbers of the LVDTs range from ω1 to ω6. In addition, the identification number of the LVDT located in the middle of the joint plates is ω7.
2.4. Testing arrangements The objective of the experimental program is to provide insight into the out-of-plane bending capacity of AAG joints. To create the out-of-plane bending moment acting at the joints, the concentrated transverse load was applied within the middle area of the beam members by means of a vertical hydraulic jack. A selfbalancing reaction frame with central symmetry was designed for the experimental program, as shown in Fig. 10. Each beam member of the specimens was hinged to the reaction frame using a high strength M24 bolt, as shown in Fig. 11. These specimens were tested under three different loading schemes, including six loaded members, three loaded members and two loaded members. To avoid the out-of-plane torsion caused by unbalanced bending moment, a central symmetry loading scheme was adopted, as shown in Fig. 12. The distance between the hydraulic jack and the end of the members was 500 mm. Initially, a preload was performed to suppress slipping Table 4 The concise symbols of the categories of strain gauges. Symbols
Interpretations
D R H-d H-r H-s
Radial strain gauge Hoop strain gauge Radial strain gauge of strain-gauge rosette Hoop strain gauge of strain-gauge rosette Oblique strain gauge of strain-gauge rosette
Fig. 7. Unidirectional strain gauges placed on the beam members.
Fig. 8. Position of the strain gauges placed within the plates. (a) Strain gauges placed within the free area (mm) (b) Strain gauges placed within the connection area (mm), (c) Strain gauges placed within t the central area (mm).
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and to allow the different components to settle and adjust. Subsequently, all the loads were increased simultaneously. The test was conducted under force control at the beginning of the loading process. When the behaviour of the specimens entered
35
into non-linear region, the test control was changed to displacement control until failure occurred. The test procedure is illustrated in Fig. 13.
3. Experimental phenomena Table 5 Interpretations of the symbols in Fig. 8. Symbols Interpretations f61R f45H-s f45H-r f45H-d f23D c4D i34R i1D
Hoop strain gauge placed within the free area of the plates between member L6 and member L1 Oblique strain gauge of the strain-gauge rosette placed within the free area of the plates between member L4 and member L5 Hoop strain gauge of the strain-gauge rosette placed within the free area of the plates between member L4 and member L5 Radial strain gauge of the strain-gauge rosette placed within the free area of the plates between member L4 and member L5 Radial strain gauge placed within the free area of the plates between member L2 and member L3 Radial strain gauge placed within the connection area of the plates for member L4 Hoop strain gauge placed within the central area of the plates between member L3 and member L4 Radial strain gauge placed within the central area of the plates for member L1
To obtain the complete response of the AAG joint specimens, each load increment lasted for at least 5 min during the test. Both the failure modes and the measured ultimate loads Pu are summarised in Table 6. The failure phenomena of the specimens are shown in Fig. 14. It can be concluded that the primary failure modes of the specimens include member rupture, member buckling, block tearing of the top plates, buckling within the free area of the bottom plates and buckling within the central area of the bottom plates. For the thick plate specimens, their primary failure modes were member buckling and member rupture, whereas for the thin plate specimens, their primary collapse mechanisms involved buckling in the bottom plates caused by compressive force and block tearing of the top plates caused by tensile force. With respect to the specimens A1, A2 and A5, which only differ in plate thickness, the experimental results show that the bending capacity of the AAG joints is improved significantly with the increase of the plate thickness. Regarding the specimens A5, B1 and C2, which differ in shear connector types, the experimental results show that the shear connector enhances the bending capacity of the AAG joints, where the C shear connector is more efficient than that of the B shear connector. A possible reason for this result is that the shear connector improves the global behaviour of the AAG joints. For the specimens A5, A6 and A7, which differ in loading schemes, the ultimate loads of specimens A6 and A7 were much larger than that of specimen A5. This result implies that the AAG joints exhibit a good global performance in transmitting the force from loaded members to unloaded members.
Fig. 9. Arrangement of LVDTs (mm). (a) Planar graph, (b) Cross-section.
Fig. 10. Self-balancing reaction frame.
Fig. 11. Hinged support.
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Fig. 12. Three kinds of loading schemes. (a) Six loaded members, (b) Three loaded members, (c) Two loaded members (d) Cross-section.
Fig. 13. Test procedure.
4. Results and discussions 4.1. Load-strain curves of the plates The load-strain curves of the plates are plotted in Fig. 15, where it can be observed that the top plates are subjected to tensile force, whereas the bottom plates are subjected to compressive force. Fig. 15(a) exhibits the strain variations within the free area of specimen A2 measured by the strain-gauge rosettes during the loading process. The strain in the hoop direction is the largest, and
the strain in the radial direction is so small that it can be neglected. It is evident that the maximum principal stress direction is the hoop direction within the free area. Fig. 15(b) shows the load-strain curves within the free area of specimen A1 measured by the hydraulic jacks and hoop strain gauges. Fig. 15(a) and Fig. 15(b) indicate that when six members are loaded, the strains within the free area exhibit a good symmetric characteristic, which signifies that the specimens are predominantly subjected to a bending moment, whereas there is only a small axial force.
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Table 6 The failure modes and the measured ultimate loads. No. Failure modes
A1 A2 A3 A4 A5 A6 A7 B1 B2 B3 C1 C2 C3 C4
Pu (kN)
Rupture of member L1 10.146 Buckling of member L2 9.752 Buckling of member L6; rupture of member L1 24.976 Block tearing within the connection area c4, c5, c6 and the free area f45, f56 of the top plates 6.663 Block tearing within the connection area c4, c5 and the free area f45, f56 of the top plates 5.548 Block tearing within the connection area c4 of the top plates; buckling within the free area f61, f34 and the central area of the bottom plates 11.405 Buckling within the free area f56, f34 and the central area of the bottom plates 14.778 Block tearing within the connection area c4, c5 and the free area f45 of the top plates 5.924 Block tearing within the connection area c4, c5 and the free area f23, f34, f45 of the top plates; buckling within the free area f61 and the central area of the 12.200 bottom plates 14.487 Buckling within the free area f56, f12 and the central area of the bottom plates Rupture of member L3 and member L4 10.940 6.045 Block tearing within the connection area c1, c2, c6 and the free area f12, f61 of the top plates Block tearing within the connection area c1, c2, c5, c6 and the free area f12, f56, f61 of the top plates 10.239 Block tearing within the connection area c1, c6 and the free area f12, f61 of the top plates 16.991
Fig. 14. Failure phenomena of the specimens. (a) Member rupture, (b) Member buckling, (c) Block tearing of the plates, (d) Buckling within the free area of the plates, (e) Buckling within the central area of the plates.
Fig. 15(c) plots the load-strain curves within the connection area of specimen B1 measured by the hydraulic jacks and radial strain gauges. The radial strain in the connection area is extremely small. The load-strain curves within the central area of specimen A5 are illustrated in Fig. 15 (d). At the beginning of the loading process, the load-strain curves revealed a linear symmetric response. When the loads were greater than 3.5 kN, the top plates gradually became more flexible, and simultaneously, the bottom plates became more flexible suddenly, which implies a buckling failure in the bottom plates. 4.2. Derivation of the moment-rotation (M-φ) curves The mechanical characteristics of the AAG joints can be represented by moment-rotation curves that perform the relationship between the bending moment M and the corresponding rotation φ. To obtain M-φ curves, the measured strains of P1 P4
were used to calculate the bending moment M, and the vertical measured displacements of ω1 ω6 were adopted to estimate the rotation φ. The cross-section of the member is divided into fourteen parts, as shown in Fig. 16. Then the bending moment M could be derived from Eq. (1) as follows: 14
M ¼ ∑ σ j Aj Z j j¼1
ð1Þ
where Aj represents the area of part j; Zj is the distance between the geometric centre of part j and point O (geometric centre of the I-section) in the z direction; and σj is the stress of part j. Stress σj can be calculated in two steps. Firstly the strain of part j εj can be estimated by the measured strains of P1 P4 on the hypothesis of plane cross-section. Subsequently, stress σj can be obtained using the Ramburg-Osgood model [22], as described in Eq. (2). The constant n can be calculated according to the
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Fig. 16. Partitions of the cross-section.
Fig. 17. Deformation of the specimen model. (a) Specimen model, (b) Deformation of specimen.
10n ¼ f 0:2 ðMpaÞ
ð3Þ
The specimen deformation is shown in Fig. 17. The rotational deformation is the sum of the deformation of the I-shaped member and the rotational deformation of the joint. Thus, the total vertical displacement ω measured by the LVDTs (ω1 ω6) can be represented by Eq. (4): ω ¼ ω r þ ωb
ð4Þ
where ωr is the vertical displacement caused by the rotation of joint, and ωb is the vertical displacement caused by the deformation of the I-shaped member. The value of ωb can be calculated by the FE method. When the stiffness of the joint is infinite, then the total vertical displacement is caused by the deformation of the Ishaped member. An FE model was created using ABAQUS software to calculate the value of ωb, as plotted in Fig. 18. The rotation of the joint is calculated by Eq. (5): ω ω ω r ¼ arctan b ð5Þ ϕ ¼ arctan x x x where x is the distance between the position of the LVDT and the end of the I-shaped member.
4.3. M-φ curves of the specimens with six loaded members According to the aforementioned derivation, the M-φ curves can be obtained. The M-φ curves of the specimens with six loaded members are shown in Fig. 19. To obtain their initial bending stiffness, linear regression analysis is used in the elastic phases. The values of the initial bending stiffness are listed in Table 7. According to Fig. 19 and Table 7, the bending stiffness characteristics of the AAG joints are discussed as follows: Fig. 15. Load-strain curves of the plates. (a) Load-strain curves within the free area of the specimen A2 plates, (b) Load-strain curves within the free area of the specimen A1 plates, (c) Load-strain curves within the connection area of the specimen B1 plates, (d) Load-strain curves within the central area of the specimen A5 plates.
Steinhardt suggestion [23], as shown in Eq. (3). εj ¼
σj σj n þ 0:002 E f 0:2
ð2Þ
(1) During the loading process, the bending stiffness characteristics of all the specimens underwent at least two phases. In the first phase, the bending stiffness was responsible for the elastic behaviour (Mo1 kN.m). When the bending moment reached eighty percent of the ultimate bending moment, the bending stiffness decreased rapidly. This reflected the second phase of the bending stiffness characteristics. In addition, it is worth noting that for the thick plate specimens A1 and C1, the bending stiffness clearly increased after the second phase.
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Fig. 18. FE model with a rigid joint.
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5 mm to 6 mm. For the thin plate specimens A4 and A5, the initial bending stiffness improved by 15.7% when the plate thickness increased from 2.25 mm to 2.7 mm. (3) The shear connector can effectively improve the initial bending stiffness of AAG joints, particularly in the case of thin plate specimens. Regarding the specimens with a 6 mm thickness plate, with the help of C shear connector, the initial bending stiffness was 14.5% greater than those without a shear connector. Regarding the specimens with a 2.25 mm thickness plate, using a type B shear connector, the initial bending stiffness increased 21.5%, whereas the initial bending stiffness
Fig. 19. M-φ curves for the specimens with six loaded members. (a) M-φ curves for the thick plate specimens, (b) M-φ curves for the thin plate specimens.
Table 7 Initial bending stiffness of the specimens (kN.m/rad). No.
Initial bending stiffness
No.
Initial bending stiffness
A1 A2 A4 A5
441 385 376 325
B1 C1 C2
395 505 475
(2) It is found that the plate thickness has an important effect on the initial bending stiffness of the AAG joints. For the thick plate specimens A1 and A2, the initial bending stiffness improved by 12.7% when the plate thickness increased from
Fig. 20. M-φ curves for specimens with three loaded members. (a) M-φ curves of specimen A6, (b) M-φ curves of specimen B2, (c) M-φ curves of specimen C3.
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increased 46.1% using a type C shear connector. The primary reason for this may be because the global behaviour of the thick plate specimens is better than that of the thin plate specimens such that the shear connector improves the global behaviour of the thin plate specimens more efficiently.
4.4. M-φ curves of the specimens with three loaded members The M-φ curves of the specimens with three loaded members are shown in Fig. 20. Each specimen has two types of M-φ curves, one that corresponds to the loaded member and the other that corresponds to the unloaded member. During the initial phase, the bending characteristics of the two types of M-φ curves were the same. However, the bending characteristic of the M-φ curve of the unloaded member became flexible after the initial phase. A possible reason for this may be that the deformation of the unloaded members is affected by the one of the loaded members. This phenomenon implies the interaction of the members on the bending stiffness of the AAG joints. 5. Conclusions and prospects In this article, the semi-rigid behaviour of the AAG joints has been investigated via an experimental study on fourteen AAG joint specimens. The following are the main conclusions: (1) The primary failure modes of the AAG joint specimens include member rupture, member buckling, block tearing of the top plates, buckling within the free area of the bottom plates and buckling within the central area of the bottom plates. For the thick plate specimens, their primary failure modes are member buckling and member rupture. For the thin plate specimens, their primary collapse mechanisms involve buckling of the bottom plates caused by compressive force and block tearing of the top plates caused by tensile force. (2) It is found from the strain distributions of the plates that the maximum principal stress direction is in the hoop direction. Under the loading scheme with six loaded members, the strains are central symmetric. (3) It can be observed from the M-φ curves that the AAG joints are typical semi-rigid joints. The joint specimens reveal a nonlinear bending stiffness behaviour response as the bending stiffness decreases during the second phase. Both the plate thickness and the shear connector have great effects on the initial bending stiffness of the AAG joints. The thicker the plate is, the stiffer the AAG joint is. The shear connector enhances the initial bending stiffness of the AAG joints by improving their global behaviour. (4) An interaction phenomenon exists with the bending stiffness of the AAG joints between loaded members and unloaded members.
Acknowledgement The authors gratefully acknowledge the financial support provided by the Natural Science Foundation of China under Grant No. 50908168 and No. 51478335.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.tws.2014.11.001.
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