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Parametric study and analytical characterization of the bolt–column (BC) joint for single-layer reticulated structures
Engineering Structures 123 (2016) 108–123
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Parametric study and analytical characterization of the bolt–column (BC) joint for single-layer reticulated structures Huihuan Ma ⇑, Shan Ren, Feng Fan School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China
a r t i c l e
i n f o
Article history: Received 20 January 2016 Revised 19 May 2016 Accepted 23 May 2016
Keywords: Single-layer latticed structure Parametric analysis Bolt–column joint Rotational capacity Failure mode Analytical model
a b s t r a c t The bolt–column (BC) joint, which can efficiently connect H, I and rectangular section members, is actually semi-rigid and of partial strength. The consideration of real semi-rigid behaviour can lead to a more accurate global analysis and, consequently, more optimized structures. In this paper a parametric analysis is conducted on the joint. A total of 341 models, divided into five series G1–G6, were analyzed. The main parameters affecting the rotational capacity and failure mode of the joint were considered in the finite element (FE) models. First, the optimal geometric parameters, such as the suitable thickness of the side plate and front plate, were obtained and discussed. Second, the mechanical behaviour of the joints under different static load conditions, including pure bending moment, eccentric pressure, eccentric tension, bending with a defined pressure and bending with a defined tension force, was investigated in detail. The theoretical equations and envelope curves for predicting the moment capacity of the joints under eccentric forces and bending with axial forces were established. Furthermore, based on the component methods in Eurocode 3 (EC3) (CEN, 2005) [1], a trilinear mathematical model for predicting the moment–rotation relationship of the joints was developed. The moment–rotation curves obtained by the mathematical model compare very satisfactorily with the experimental results. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The conventional design of steel structures is usually conducted under the assumption that joints are either fully rigid or ideally pinned. Currently, after a series of research, the concept that the actual joints exhibit behaviour that is intermediate between these two extreme cases—semi-rigid and nonlinear—has been accepted by the codes. This means that the effect of joint flexibility should be considered in the analysis and design procedures to facilitate numerical analysis of complete structures that incorporate this form of joint. Likewise, Eurocode 3 (EC3) [1] includes procedures and formulations to define both the stiffness and resistance of semi-rigid joints starting from their geometrical and mechanical properties. Normally, the moment–rotation curve, as the most important factor, can describe the rotational behaviour of the joint; the curve defines three main properties: rotational stiffness, moment resistance and rotation capacity. Many studies are aimed at obtaining moment–rotation curves of the joints so that they can be incorporated in structural analysis, such as the research for beam-to-column steel connections in [2–5] and semi-rigid joints
⇑ Corresponding author. http://dx.doi.org/10.1016/j.engstruct.2016.05.037 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.
in single-layer reticulated structures [6,7]. However, the failure mode of the joint is also a very important factor to evaluate the mechanical property of the joints. Whether the failure mode of a joint is sudden brittle failure or gradual plastic failure has an essential influence on the behaviour of entire structures. To date, few studies have focused on this issue for semi-rigid joints used in single-layer reticulated structures. Especially for a new joint system, determination of the optimal geometric parameters to avoid sudden brittle failure is of great significance. In addition, except for the investigations undertaken by Fan et al. [6,7] and Ueki et al. [8], most of the achievements regarding joints in spatial structures are related to the moment–rotation relationship of the joints under a pure bending moment or pure axial forces. However, in real spatial structures, the joints are mainly subject to combined bending moment and axial force, and the investigations in references [6–8] have shown that the axial forces have a significant effect on the mechanical behaviour of the joints, including the moment rotation curves and failure modes. Therefore, combinations of the different action types should be considered when investigating the mechanical characters of new joints. The actual working environment of a joint system is complex, and a further investigation into the effect of axial forces on the moment–rotation characteristic of a new joint
H. Ma et al. / Engineering Structures 123 (2016) 108–123
109
Nomenclature fy fu E M / t1 t2 Sj,inia,anal Sj,ini,exp Msup,anal Msup,exp d dc
gd
M dsup M 0sup
gdM
e Nt/Np Ntu =N pu
gt gp A1 A2
a
M Nsup
gNM Kj Mp Fb Fs L
yield strength of steel material tensile strength of steel material Young’s modulus of steel material bending moment joint rotation thickness of front plate thickness of side plate initial stiffness obtained numerically initial stiffness obtained experimentally plastic moment resistance obtained numerically plastic moment resistance obtained experimentally eccentricity of axial forces critical eccentricity absolute value of ratio of d to dc plastic moment resistance of joint under eccentric axial force plastic moment resistance of joint under pure bending ratio of Mdsup to M0sup natural logarithms axial tension/pressure force ultimate axial force of joint under pure axial tension or pressure ratio of Nt to Ntu ratio of Np to Npu total cross-sectional area of two side plates cross-sectional area of middle plate ratio of A1 to A2 plastic moment resistance of joint under bending with certain axial forces ratio of MNsup to M0sup initial stiffness obtained from the proposed model ultimate bending moment force in bolt tension/pressure force in side plates distance between shear force and end plate
is necessary to provide a better stiffness model that is more similar to practical circumstances. An experimental program was carried out to assess the real behaviour of the BC joint under bending [9]. Finite element models established through the ABAQUS finite element software program [10] with the same characteristics as those of the tests were developed and calibrated with the experimental results. The results demonstrate that the finite element model provides a reliable and robust tool to develop a parametric study, avoiding the need for additional and costly experiments. In this paper, a nonlinear parametric study is carried out. The moment–rotation curves and failure mode of the joints with varying side plate thickness, front plate thickness, and loading schemes are obtained and discussed in detail. The optimal geometric parameters of the joints are identified to achieve high stiffness and avoid the sudden brittle failure mode. The envelope curves for predicting the plastic moment resistance considering the affection of eccentricities and axial forces are established. Finally, an analytical model of the moment–rotation relationship is developed for design applications, and the analytical model is verified by the experimental results. 2. Finite element model The bolt–column (BC) joint is composed of a hollow column node, high-strength bolts, washers, and end-cone part, as shown in Fig. 1. It can be used to connect H, I or rectangular members in the real structures. The end-cone part consists of five plates:
d1 d2
DC DT kb kspt/kspp ksp As Lb tc Eb Lb tc tw tbh dc be n
q cp,k
m cp,M P P0 Wp he Mp
rcr b kp
v
distance between two high-strength bolts distance of centroid of side plate from compression zone to tension zone axial deformation of bolts deflection of side plates at compression or tension zone bolt stiffness stiffness of each side plate at tension/compression zone total stiffness of side plates section area of bolt shank bolt elongation thickness of column modulus of elasticity of bolts bolt elongation length thickness of column node washer thickness thickness of bolt head depth of side plate from front plate to end plate assumed effective width of side plates coefficient = 2.7 reduction factor q for side plate buckling influence coefficient of bolt pretension force on initial stiffness Poisson’s ratio for steel (=0.3) influence coefficient of bolt pretension force on ultimate plastic bending real preload force in bolt preload force in bolt according to code plastic inertia moment of side plate height of side plate ultimate plastic bending moment elastic critical buckling stress elastic buckling coefficient plate slenderness elastic constraint coefficient associated with middle plate
one front plate (t1), two side plates (t2), one middle plate (t3) and one end plate (t4). The cone parts are welded at both ends of the members in the factory. In the construction site, the two highstrength bolts are used to connect the members to the column node without any welding work. A total of 14 BC joint specimens (Fig. 2) were divided into 7 groups and tested to failure under monotonic loading. The threedimensional finite element model (Fig. 3) has the same configuration and dimensions as the tested specimens. The configurations of the specimens are shown in Fig. 4 and Table 1. The whole experimental joint, including the column node, the cone part, the bolts and the rectangular member, is modelled with 8-node solid elements C3D8R, available in the ABAQUS library [10]. The numerical models incorporated a number of detailed features such as contact phenomena, bolt slippage definition and bolt prestressing force application. A total of 14 tests, conducted based on varying pretension force, side plate thickness and bolt diameter, are used to validate the FE model. The geometric parameters, preload forces on the bolts and initial geometric imperfection of the side plates of the FE models are the same as the specimens in the tests. The comparison of the moment–rotation curves obtained through FE models and tests are shown in Fig. 5. From the curves, it can be observed that the curves obtained numerically and experimentally are similar. The good agreement with the experimental results confirms that the detailed FE models are able to simulate the mechanical behaviour
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
Fig. 1. The bolt–column joint.
Fig. 2. Specimen of the BC joint.
of BC joints with reasonable accuracy. The numerical curves of the specimens in group S1 and S2 are slightly higher than those experimentally obtained after the plastic flexural resistance. This variation can be attributed to the sudden buckling in the compression side of the thin side plates of the experimental specimens. The reliability of the FE models is also proved by the good agreement between the numerical failure mode and those observed after tests, as shown in Fig. 6. To provide a convenient way to analyze the M–u curves, the following characteristics are assessed, as shown in Fig. 5(a): Sj,ini is the initial stiffness; Sj,p-l is the postlimit stiffness, which is defined as 10% of the initial stiffness, Sj,ini; and Msup is the plastic moment resistance, which corresponds to point O of the regression line obtained for the post-limit (Sj,p-l) stiffness. 3. Nonlinear parametric analysis of the joints
Fig. 3. FE model of the BC joint.
It is well known that in real spatial reticulated structures, connections carry not only bending forces but also axial forces and that the ratios between the bending and axial forces in the joints vary. Hence, to reflect the mechanical performance of the joint in a real structure, the FE models were subjected to different load
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
Fig. 4. Detailed dimensions of the cone part.
Table 1 Geometric parameters of the specimens. Test ID
Beam section (mm)
Bolt diameter d (mm)
Bolt distance L1 (mm)
Front plate t1 (mm)
Side plates t2 (mm)
Middle plate t3 (mm)
End plate t4 (mm)
Prestressing forces P24 = 225 kN P27 = 290 kN
S1-A
h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 180 130 16 16
24
78
30
3.5
5.5
20
P24
24
78
30
5.5
5.5
20
1.25P24
24
78
30
5.5
5.5
20
P24
24
78
30
5.5
5.5
20
0.75P24
24
78
30
5.5
5.5
20
0.32P24
24
78
30
9.5
5.5
20
P24
27
87
30
7.5
7.5
20
P27
S2-A S2-B S2-C S2-D S3-A S4-A
The square box means rectangular steel tube.
conditions (as shown in Fig. 7), including pure bending, eccentric pressure, eccentric tension, bending with a defined pressure, and bending with a defined tension. The same boundary condition is used for the joints under different loading schemes: the half column node is fixed, and the forces act on the end of the member. The force Nt or Np in Fig. 7(c) can rotate with the model to maintain the direction of the axial forces pointing to the joint centre along the centreline of the members; hence, the secondary moment caused by the axial force can be eliminated during the loading. The stiffness, strength, rotation behaviour and failure mode of the joints under different static load conditions were obtained and are discussed later. The geometric characteristics and detailed loading conditions of joints during the parametric analysis are listed in Table 2. A total of 341 models, divided into six series G1–G6, have been analyzed. The models in series G1 were used to investigate the optimized thickness of the front plate and side plate; the models in series G2 were used to investigate the mechanical behaviour of the joints under shear force with different distances between the shear force and end plate; the models in series G3 and G4 were used to investigate the mechanical behaviour of the joints with different bolt diameters and side plate thicknesses under eccentric forces; the models in series G5 and G6 were used to investigate the mechanical behaviour of the joints with different bolt diameters and side plate thicknesses under bending with defined axial forces. Based on the parametric study, the following results are obtained:
– The complete moment–rotation curves of the joints (Figs. 9–12 and 16). – Stress distribution in the cone part and bolts of the joints (Figs. 9–11, 14 and 18). – Variation of initial stiffness and moment capacity of the joints (Figs. 13 and 17). – Relationship of moment capacity and eccentricity of axial forces (Fig. 15). – Relationship of moment capacity and axial forces (Fig. 19). Figs. 12–14 show the moment–rotation curves, stress distribution and variation of the initial stiffness and moment capacity, respectively, for the models for only one set of joints (t2 = 5.5 mm, M24 bolts) under different eccentric forces in series G3 and G4. For the other models in these two series, the effect of the eccentricity of the forces on the moment–rotation curves, stress distribution and change rule of the initial stiffness and moment capacity is similar. To obtain the envelope curve of the moment capacity of the joints under eccentric forces in Fig. 15, all moment–rotation curves of the joints in series G3 and G4 were obtained and analyzed. Figs. 16–18 also show the moment–rotation curves, stress distribution and change rule of initial stiffness and moment capacity, respectively, for the models for only one set of joints (t2 = 5.5 mm, M24 bolts) under bending with different defined axial forces in series G5 and G6. For the other models in these two series, the effect of the axial forces on the moment–rotation curves, stress distribution
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
40
40 Msup,exp: 28.77kN·m
Msup
S3-A 35
Sj,ini
Bending moment M (kN·m)
Bending moment M (kN·m)
35 30
Sj,p-l=0.1Sj,ini
O
Msup,anal: 28.06kN·m
25 20
Msup,exp: 17.52kN·m
15
Msup
S2-B : 17.58kN·m
anal
Msup,exp: 10.46kN·m
10
S1-A Msup,anal: 10.30kN·m
5
: Msup,exp
S4-A
30 Msup,anal:30.79kN·m
25 20
Msup,exp:17.52kN·m
S2-B 15
Msup,anal:17.59kN·m
10
test FEA
5
test FEA
: Msup,anal
Msup,exp:30.86kN·m
: Msup,exp 0 0.00
0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.01
0.02
: Msup,anal 0.03
0.04
0.05
(a) For different thicknesses of side plates
0.08
20 Msup,exp: 16.74kN·m
S2-C
Bending moment M (kN·m)
S2-A
16
Bending moment M (kN·m)
0.07
(b) For different bolt diameters
20
Msup,anal: 16.57kN·m 12
8
4 : Msup,exp 0 0.00
0.06
Rotation (rad)
Rotation (rad)
0.01
test FEA
: Msup,anal 0.02
16
Msup,exp:17.43kN·m Msup,anal:17.39kN·m
12
8
4 : Msup,exp
0.03
0.04
0 0.00
0.05
Rotation (rad)
0.01
test FEA
: Msup,anal 0.02
0.03
0.04
0.05
Rotation (rad)
20
Msup,exp: 17.28kN·m
Bending moment M (kN·m)
S2-D 16
Msup,anal:17.12kN·m 12
8
4
: Msup,exp
test FEA
: Msup,anal
0
0.00
0.01
0.02
0.03
0.04
0.05
Rotation (rad)
(c) For different pretension forces Fig. 5. Moment–rotation comparison of the BC joints.
and change rule of initial stiffness and moment capacity is similar. To obtain the envelope curve of the moment capacity of the joints under bending with different axial forces in Fig. 19, all moment– rotation curves of the joints in series G5 and G6 were obtained and analyzed. The stress–strain relationship for the different parts of the joints is modelled for nonlinear analysis as an elastoplastic material with a strain hardening and yielding plateau beyond the elastic phase. It is assigned a trilinear stress–strain law, symmetrical in tension and compression, as shown in Fig. 8. The steel used for the beam
member, cone part and washers is grade steel S235. The bolts are frictional high-strength bolts (class 10.9). The actual material properties of the steel were obtained from tensile tests on coupons and the bolt certificate of quality, as shown in Table 3. 3.1. BC joints under pure bending moment In this section, the 3D FE model is used for parametric studies in which different thicknesses of the front plate and side plate are considered under bending. Based on the parametric analysis, the
H. Ma et al. / Engineering Structures 123 (2016) 108–123
113
Fig. 6. Failure mode comparison of the BC joints.
Fig. 7. Static loading conditions.
whole moment–rotation curves and failure modes of the joints are obtained, and the appropriate thicknesses of front plate and side plate are discussed and suggested.
3.1.1. BC joints with different thicknesses of front plate The influence of the thickness of the front plate on the overall behaviour of the joints was analyzed. For the joint (G1-1 Table 2), four front plate thicknesses, 22, 26, 30 and 40 mm, were considered in the analysis. The moment–rotation curves and stress distribution at the cone part and bolts of the joints are shown in Fig. 9. The general law is that by increasing the thickness of the front plate, the rotational capacity will be correspondingly increased slightly under bending. Careful examination of the results shows similar failure modes for the joints with eccentric pressure and tension. The top and bottom areas of the side plates had yielded and formed into a plastic hinge before the tensional bolt yielded. From the stress distribution at
the cone and tensional bolt, it can be observed that the yield area of front plate decreases with increasing thickness of the front plate; the effect of the thickness of the front plate on the stress distribution of the bolts and side plates is small. The deformation of the bolt is linear in the joints with different thicknesses of the front plate; however, when t1 6 26 mm, a large area of the front plate finally yields, which will become worse when the joints are subjected to combined bending and axial force. Therefore, for BC joints with M24 high-strength bolts, the front plate thickness should not be thinner than 26 mm.
3.1.2. BC joints with different thicknesses of side plate The influence of the thickness of the side plate on the rotational capacity of joints was examined. Five thicknesses of the side plate were considered in the analysis—namely, 4, 6, 7, 10 and 12 mm. The other geometric dimensions of the joints are the same as G1-2 listed in Table 2. Fig. 10 shows the moment–rotation
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
600
1400 1200
500
0.0379
0.0352 1171.41 MPa
800 600 400
300
0.00146
303.93 MPa
200 100
200 0 0.00
455.58 MPa
400
0.00457 975 MPa
Stress (MPa)
Stress (MPa)
1000
0.02
0.04
0.06
0.08
0 0.00
0.02
0.04
0.06
0.08
Strain
Strain
(a) stress–strain curve for bolts
(b) stress–strain curve for column node, cone part and members
Fig. 8. Stress–strain behaviour of the joint components.
response and stress distribution at the cone part and bolts of the joints. It can be observed that the maximum moment and rotational capacity increase significantly with increasing thickness of side plates; the stress of the bolts increases with increasing thickness of side plates; the yield area of the side plate obviously decreases with increasing thickness of the side plate. The failure mode of the joints changes with increasing thickness of the side plate: (i) when the side plate thickness t2 6 7 mm, the top and bottom areas of the side plates yield, resulting in a plastic hinge before the tension bolts yield, and the joints fail; (ii) when t2 P 10 mm, the side plates cannot form a plastic hinge before the tensional bolts yield, and the joints fail because the tensional bolts yield. Apparently, failure mode (ii) should be particularly avoided in the design of real structures because it reduces the energydissipating capacity of the joints and may lead to a brittle failure of the joints and the whole structure. Therefore, to gain a greater moment–rotational capacity and better energy-dissipating capacity and avoid brittle failure, the thickness of the side plates should not be too large for the joints. 3.2. BC joints under shear force with different distance L To investigate the influence of shear force on the behaviour of the BC joints, four different distances between the shear forces and end plate were considered in the analysis. The shorter the distance L, the greater shear force at the connection when the bending moment is the same. The geometric dimensions of the joints in this section are the same as series G2 listed in Table 2. The moment–rotation curves and stress distribution at cone part and bolts of the joints are shown in Fig. 11. In the figure, the plastic moment resistance of the joint under shear decreased less than 4.87% compared with the joint under pure bending. A plastic hinge is formed at the cone part in all five joints. The stress distribution of the bolts and the yield area of the front plate increases with increasing distance between the shear force and the end plate. In total, the effect of shear on the behaviour of the BC joint is small. 3.3. BC joints under eccentric forces The geometric dimensions of the joints in this section are the same as series G2 and G3, listed in Table 2. During the loading process, there was bending moment owing to the eccentric pressure and tension. The bending and axial force increased together.
The moment–rotation curves of the joints (t2 = 5.5 mm, M24 bolts), subjected to different eccentric forces, are shown in Fig. 12. d = 0.02–1 m is the eccentric distance of the axial force, and the curve corresponding to d = 1 in the figure is the moment–rotation curve of the joints subjected to pure bending. Fig. 13 shows the change rule of the initial bending stiffness and plastic moment resistance of the joints (t2 = 5.5 mm, M24 bolts). In the figure, Sdj;ini =S0j;ini represents the ratio of the initial bending of the joints under axial forces with the eccentricity d to that of the joints under pure bending; M dsup =M 0sup represents the ratio of the plastic moment resistance of the joints under axial forces with the eccentricity d to that of the joints under pure bending. From the curves, it was found that the effect of the force eccentricities on the plastic moment resistance is similar, whereas the effect of the force eccentricities on the initial bending stiffness is different. As shown in Fig. 13, the curves can be described in two stages: – Stage-1: when d < 0.2 m, the eccentricity of the forces has an obvious effect on both the initial bending stiffness and moment capacity; the initial bending stiffness of the joints under the eccentric pressure force is stronger than that of the joints under pure bending, whereas the initial bending stiffness of the joints under eccentric tension force is smaller than that of the joints under pure bending; the plastic moment resistance of the joints under eccentric pressure and tension is smaller than that of the joints under pure bending, and it increases significantly when the eccentricity of the axial force increases from 0.02 m to 0.1 m. – Stage-2: when d P 0.2 m, the initial bending stiffness and plastic moment resistance of joints subjected to eccentric forces are almost the same as those of the joints under pure bending. Therefore, dc = 0.2 m is defined as the critical eccentricity. The mechanical behaviour of the joints under axial forces with eccentricity d P dc will be very close to the joints under pure bending. Another important effect of eccentricity is related to the failure modes of the joints. The stress distributions at the cone part and bolts of the joints (t2 = 5.5 mm, M24 bolts) are shown in Fig. 14. In the figures, it can be observed that (i) a plastic hinge is formed at the cone part of all joints; (ii) the neutral axis of the cone part in the failure modes deviates from the middle plate when the eccentric distance is less than 0.1 m; and the neutral axis of the
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H. Ma et al. / Engineering Structures 123 (2016) 108–123 Table 2 Detailed dimensions and loading conditions of the analysis models. Group
Parametric study (FEM) Beam section (mm)
Bolt diameter d (mm)
Bolt distance L1 (mm)
Front plate t1 (mm)
Side plates t2 (mm)
Middle plate t3 (mm)
End plate t4 (mm)
Preload P (kN)
Loading condition
G1-1
h 160 120 16 16
24
78
5.5
5.5
20
225
Fig. 5(a)
G1-2
h 160 120 16 16
24
78
22 26 30 40 30
4 6
5.5
20
225
Fig. 5(a)
G2
h 160 120 16 16
24
78
30
5.5
20
225
Fig. 5(b) L = 0.50 m L = 0.25 m L = 0.05 m L = 0.00 m
G3-1
h 160 120 16 16
24
78
30
3.5 5.5 9.5
5.5
20
225
G3-2
h 180 130 16 16
27
87
30
7.5
5.5
20
290
G4-1
h 160 120 16 16
24
78
30
5.5
20
225
G4-2
h 180 130 16 16
27
87
30
3.5 5.5 9.5 7.5
5.5
20
290
G5-1
h 160 120 16 16
24
78
30
3.5 5.5 9.5
5.5
20
225
G5-2
h 180 130 16 16
27
87
30
7.5
5.5
20
290
G6-1
h 160 120 16 16
24
78
30
3.5 5.5 9.5
5.5
20
225
G6-2
h 180 130 16 16
27
87
30
7.5
5.5
20
290
Fig. 5(c) dt = 0 dt = 0.01 m dt = 0.02 m ... dt = 0.1 m ... dt = 0.9 m dt = 1.0 m Fig. 5(d) dp = 0 dp = 0.01 m dp = 0.02 m ... dp = 0.1 m ... dp = 0.9 m dp = 1.0 m Fig. 5(e) gt = 0.05 gt = 0.10 gt = 0.15 ... gt = 0.85 gt = 0.90 gt = 0.95 Fig. 5(f) gp = 0.05 gp = 0.10 gp = 0.15 ... gp = 0.85 gp = 0.90 gp = 0.95
7 10 12 5.5
The square box means rectangular steel tube.
Fig. 9. Moment–rotation curves and stress distribution of joints with different front plate thicknesses.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
Fig. 10. Moment–rotation curves and stress distribution of joints with different side plate thicknesses.
35
Bending moment M (kN·m)
30 25
Msup=21.897kN·m Msup=21.44kN·m
20 15 10
Msup=20.88kN·m Msup=21.21kN·m Msup=21.1kN·m
shear force with L=0.5m shear force with L=0.25m
5 0 0.00
pure bending
shear force with L=0.05m shear force with L=0.0m 0.02
0.04
Rotaion
0.06
rad
Fig. 11. Moment–rotation curves and stress distribution of joints under shear force with different distance L.
Fig. 12. Moment–rotation curves of one set of joints under eccentric forces.
side plates moves to the centre when the eccentric distance of the force increases. However, the effects of eccentric pressure and tension on the stress distribution of the front plate and bolts are quite distinct. When dt < 0.2, a large area of the bolts and front plate of the joint under eccentric tension force yields; when the joints are subjected to the eccentric pressure, the stress of the bolts and front plate is small and far below the yield stress.
For joints subjected to eccentric pressure and tension, designers are more concerned with the relationship between the ultimate moment resistance and eccentricity when the joints are broken. Hence, based on the foregoing parametric analysis, the non-dimensional relationship of the plastic moment resistance and the eccentricity of the pressure and tension forces for the BC joint have been established, as shown in Fig. 15. The theoretical
Ratio of initial stiffness/ultimate bending
H. Ma et al. / Engineering Structures 123 (2016) 108–123
1.3 1.2
Sj,ini/Sj,ini Eccentric pressure
1.1
Sj,ini/Sj,ini Eccentric tension
1.0
1 1 gdM ð1 þ 3e6gd Þ ¼ 1 4 sup
0
resistance of BC joints under different eccentricities of axial forces can be predicted by the designers.
Msup/Msup Eccentric pressure
0.8
0
Msup/Msup Eccentric tension
0.7
M24
0.6
t2=5.5mm
3.4. BC joints under bending with a defined axial force
The critical eccentricity: =0.2m c 0.0
ð1Þ
Md d where gdM ¼ Msup 0 ; gd ¼ d . Based on Eq. (1), the plastic moment c
0.9 0.96
0.4
envelope curve in the figure is approximated by the following equations:
1.03
0.5
117
0.2
0.4
0.6
Eccentric distances
0.8 t
/
p
1.0
1.2
(m)
Fig. 13. Change rule of initial stiffness and moment capacity of one set of joints under eccentric forces.
The geometric dimensions of the joints in this section are the same as series G4 and G5, listed in Table 2. Unlike the joints subjected to eccentric forces, the axial force in this section remains constant, and the direction of the axial forces always point to the joint centre along the centreline of the beam during the loading process; the bending increases stepwise until the joints fail. The moment–rotation curves and the stress distribution at the cone part of the joints are shown in Fig. 16. Fig. 17 shows the variation
(a) the stress cloud of the joint under eccentric tension
(b) the stress cloud under eccentric compression Fig. 14. Stress distribution of the joints under different eccentric forces.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
M
1.0 t
c
0.2m;
p
0.8
1.0; M t
1.0;
t2=5.5mm, M24
1.0;
1.0; p
M
t2=9.5mm, M24
0.2 m;
1.0;
t2=7.5mm, M27
0.2
t2=7.5mm, M27
c
1.0;
t2=5.5mm, M24
0.4
t2=9.5mm, M24
Theoretical curve
Theoretical curve
0.0
0.5
1.0
0.2 m ;
t2=3.5mm, M24
t2=3.5mm, M24
1.0; M
M
0.6
0.2 m ;
c
c
1.0;
Eccentric tension
0.5
1.0
Eccentric pressure
Fig. 15. Relationship of moment capacity and eccentricity of axial forces.
1.4
Sj,ini/Sj,ini Bending with axial pressure P
1.2
Sj,ini/Sj,ini Bending with axial tension t
Change slightly
1.0 t
/ p=0.1
0.2
0.8
0.3 0.4 0.5
0.6
t2=5.5mm, M24
0.4
0.6
0
Msup/Msup Bending with axial pressure p
0.2 0.0
Sharp drop
Ratio of initial stiffness/ultimate bending
Fig. 16. Moment–rotation curves of one set of joints under bending with different axial forces.
0
Msup/Msup Bending with axial tension t
0
150
0.7
300
Axial tension/pressure
450
/ t
600
(kN) p
Fig. 17. Variation of initial stiffness and moment capacity of one set of joints under bending with different axial forces.
of the initial bending stiffness and plastic moment resistance of the joints under bending with different axial forces. In the figure, N
ðt=pÞ Sj;ini =S0j;ini represents the ratio of the initial bending of the joints
under bending with axial forces Nt/Np to that of the joints under N
ðt=pÞ pure bending; Msup =M 0sup represents the ratio of the plastic moment resistance of the joints under bending with axial forces
Nt/Np to that of the joints under pure bending; gt ¼ N t =N tu and gp ¼ Np =Npu ; here, Ntu ¼ 797:4 kN and Npu ¼ 814:16 kN are the ultimate axial forces of the joints (t2 = 5.5 mm, M24 bolts) under pure axial tension and pressure, respectively. From the curves, it has been found that (i) the moment resistance decreases significantly with increasing axial pressure/tension force; however, when g(t/p) 6 0.3, the moment resistance of the joints under bending with axial forces is very close to that of the joints under pure bending; the difference in the curves of the joints under bending with axial pressure and tension forces narrows when gt/gp decreases; (ii) the axial pressure/tension force has a different effect on the initial bending stiffness of the joints; the initial stiffness of the joints under bending with axial tension force decreases significantly with increasing tension force, and the bending stiffness of all joints is smaller than that of the joints under pure bending; the initial stiffness of the joints under bending with axial pressure force remain stable and higher than that of the joints under pure bending when Np 6 366.21 kN (gp 6 0.5), and a sharp drop occurs in the curve when Np > 451.98 kN (gp > 0.6). The force transfer mechanisms of the BC joint under bending with axial pressure or tension are similar. From the stress distribution at the final step for different axial pressures, it can be found that the neutral axis of the middle plate moves from the centre to one side when gt/gp increases. The yield areas of the front plate and bolts increase with increasing tension force and decrease with increasing compression force. The stress distributions of the joints under bending with different axial pressure and tension forces are shown in Fig. 18.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
(a) the stress cloud of a joint under bending with a certain tension force
(b) the stress cloud of a joint under bending with a certain compression force Fig. 18. Stress distribution of the joints under bending with axial pressure and tension forces.
4
+2
2 (t / p )
N M
0.3
t
N M
1
p
4
1.0 t
0.3
N M
1.0
p N M
0.8
0.3
+2
2 (t / p )
N M
1
0 .3 1 .0
0.6
-1.0
-0.8
t2=3.5mm, M24 bolts
FEA;
t2=5.5mm, M24 bolts
FEA;
t2=9.5mm, M24 bolts
FEA;
t2=7.5mm, M27 bolts
FEA;
-0.6
-0.4
Bending with tension
0.4
Theoretical curve Theoretical curve
0.2
-0.2 t
Theoretical curve Theoretical curve 0.0
0.2
0.4
0.6
Bending with pressure
0.8
1.0
p
Fig. 19. Relationship of moment capacity and axial force.
The plastic neutral axis of the cone part in the failure modes deviates from middle plate to the tensional side when the axial forces increase. The stress distribution of the side plates of the joints in
which g(t/p) 6 0.3 is almost the same as that of the joint under pure bending. A plastic hinge is ultimately formed at the cone part of all joints. However, the effects of pressure and tension force on the
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
rotation curve of the joints is divided into three phases, as shown in Fig. 20:
Table 3 Material properties. Results of tensile coupon tests. Components
fy (N/mm2)
fu (N/mm2)
Young’s modulus E (N/mm2)
fy/fu
Plates in cone part
t2 = 3.5 mm t2 = 5.5 mm t2 = 7.5 mm t2 = 9.5 mm
312.65 303.93 286.57 297.58
470.81 455.58 419.95 413.87
207,053 207,848 202,476 207,882
0.67 0.67 0.68 0.66
Beam Bolts
t = 15.5 mm d = 24 mm
269.00 975.00
406.86 1171.41
205,569 213,302
0.66 0.83
– Phase I: M < 2/3 Mp, the curve has linear behaviour up to the moment value of 2/3 Mp, where Mp is the ultimate plastic moment of the joint; – Phase II: 2/3 Mp < M < Mp, the curve is nonlinear in this phase. – Phase III: M P Mp, in this phase, owing to different side plate thicknesses, two failure modes occurred in the test: strength failure and local buckling failure. According to the two failure modes, the curve in Phase III performs as a plastic hardening phase and plastic descent phase.
The square box means rectangular steel tube.
stress distribution of the bolts are quite distinct; compared with the joint under pure bending, the existence of tension force exacerbates the stress level in the bolts, whereas the existence of pressure force reduces the stress level in the front plate and bolts. To predict the bending capacity of the joints under bending with different axial pressure or tension intuitively, the envelope curve of the ultimate bending moment affected by different axial forces is shown in Fig. 19. According to the position of the neutral axis of the cone part in the failure modes, the envelope curve of the ultimate bending moment can be divided into two groups:
Based on the three phases, the bending moment M is expressed as
8 Kj/ for M 6 23 M p > > > > < Kj n / for 23 Mp < M < Mp M¼ 1:3MM > p > > > : M p K t / for M P M p
where the value of parameter n of the bolted end-plate connections is 2.7 according to Eurocode 3. The final stiffness, Kt, can be calculated using a reduced stiffness coefficient of 1/60 and expressed as K t ¼ K j =60.
(i) When the axial forces are sufficiently small (g(t/p) 6 0.3), the plastic neutral axis is in the middle of the cone part. The formula of the envelope curve for the plastic moment resistance is as follows:
gNM ¼
M Nsup M 0sup
1:0
4.1. Initial bending stiffness Kj Eurocode 3 Part 1.8 [1] has adopted a component method for the design of joints. This method is based on dividing the joint into its basic components, and the design moment–rotation characteristic of a joint depends on the properties of its basic components. By assembling the contributions of individual components that represent the joint as a set of rigid and deformable elements, the entire behaviour of the joint may be determined. Based on the component method, the initial stiffness and ultimate capacity of each component is determined and assembled to form a spring model for some beam-to-column joints [11–14]. In the model, the joint components are treated as springs with predefined characteristics such as initial stiffness and ultimate capacity. In this paper, based on the structural characteristic of the BC joints under bending moment obtained by FE models, a spring-stiffness model for the joints is developed to predict the moment–rotation characteristic of the joints. The components involved in the BC joints under bending are assembled as series of springs according to the mechanical model
ð2Þ
(ii) When the axial forces are sufficiently large (g(t/p) > 0.3), the plastic neutral axis is not in the middle of the cone part. The formula of the envelope curve for the plastic moment resistance is as follows:
2 4 ða þ 2Þ gðt=pÞ a þ gNM ¼ 1
ð3Þ
where a ¼ A2 =A1 . 4. Analytical model for the joint under bending moment To establish the practical analytical model for the evaluation of the connection stiffness and capacity of the BC joints under a pure bending moment, the simplified trilinear model suggested in Eurocode 3 Part 1.8 [1] is considered. The elastoplastic moment–
M
M
Mp
3
Elastic phase Mp
0
-Kt
Mp
Kt 2
2
3
Mp
Elastic phase Nonlinear phase Plastic descent phase
Nonlinear phase Plastic hardening phase
Kj
Test Proposed model
(a) For the joints with strength failure mode
ð4Þ
Kj
Test Proposed model
(b) For the joints with local buckling failure mode
Fig. 20. Trilinear mathematical model of the M–u curve.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
illustrated in Fig. 21. For simplicity, in the model, the tension and compression zone comprise bolt stiffness Kb and side-plate stiffness Ksp. The bending moment M is equivalent to a pair of two forces; one of them pulls out from the joint along the tension bolt on one side, and the other pushes in the joint along the compression bolt on the opposite side. The global rotational stiffness of the joint Kj can then be determined at a given moment based on the assembled stiffness of all components from the following equations [15–17]:
M ¼ F b d1 ¼ F s d2
ð5Þ
DC DT þ d1 d2
ð6Þ
/¼
K j ¼ cp;k
cp;k F b d1 c d1 M ¼ p;k ¼ 1 / 2 F b þ F b d1 2 þ k d1d2 k d1 k d1 k d2 b
sp 2
b
ð7Þ
sp 2
where DC and DT sum up the contribution of both the axial deformation of the bolts and the deflection of the side plate; cp,k is an influence coefficient of bolt pretension force on initial stiffness and is given by
cp;k
p p0 ¼1þ 4p0
ð8Þ
At initial elastic stage a small deformation assumption is adopted. (1) Initial stiffness of bolts, kb The bolts are considered to be subjected to direct tensile force in isolation. High-strength bolts exhibit a brittle response up to failure, and the resistance of each bolt is given by F u ¼ 0:9f u As [18]. By applying the principles of Hooke’s law, the elastic stiffness of the bolt kb can be expressed as
kb ¼
1:6As E Lb
ð9Þ
(2) Side plate behaviour ksp The side plate can be divided into two zones: tension zone and compression zone. The stiffness of one-half of the side plate, kspt/ kspp, in the compression zone may be estimated by idealizing it as a plate of dimensions dc be subjected to a uniform compression force. By assuming that the plate obeys Hooke’s law, the elastic stiffness of the side plate can be expressed as
ksp ¼ 2kspt ¼ 2kspp ¼
ð11Þ
4.2. Ultimate plastic bending moment Mp The high-strength bolts and side plates are key components, and local buckling is an important issue for calculating the ultimate plastic bending moment when the side plate is thin. Consequently, based on Eurocode 3 Part 1-5 [1], the ultimate plastic bending moment, Mp, can be defined in the following form:
Mp ¼ qcp;M f y W p
ð12Þ t2 h2e 4
2 2t 2 be ;
where W p ¼ 2 ¼ the influence coefficient of the bolt pretension force on the ultimate plastic bending, cp,k, is given by
(
cp;M ¼
0 1 pp 4p
p P p0
1
p < p0
0
ð13Þ
According to Eurocode 3 Part 1-5 [1], the reduction factor q for side plate buckling can be obtained as follows:
8 < kp 0:2 kp > 0:673 k2p q¼ : 1 kp 6 0:673
The plate slenderness, kp , can be obtained by kp ¼
ð14Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi f y =rcr .
where rcr is the elastic critical plate buckling stress, based on the elastic stability theory; the critical buckling stress of the uniaxial compressional plate, rcr, can be calculated as follows:
2 t2 12ð1 m2 Þ be
The bolt elongation can be obtained from the following relationship based on Agerskov’s [19] recommendations:
rcr ¼ bv
t bh Lb ¼ t 1 þ tc þ tw þ 2
b ¼ 0:425 þ be =dc
ð10Þ
Et2 be dc
p2 E
(a) spring-model of the cone
(b) spring-model of the joint Fig. 21. General representation of the proposed model.
ð15Þ ð16Þ
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H. Ma et al. / Engineering Structures 123 (2016) 108–123 20
12
S1-A
S2-A
Bending moment M (kN·m)
Bending moment M (kN·m)
10 8 6 4
FEA Test Proposed model
2 0 0.00
0.01
0.02
0.03
0.04
0.05
16
12
8
0 0.00
0.06
test FEA Proposed momdel
4
0.01
0.02
0.03
0.04
20
Bending moment M (kN·m)
Bending Moment M (kN·m)
S2-C
16
12
8
FEA Test Proposed model
4
0.01
0.02
0.03
Rotation
0.04
0.05
12
8
test FEA Proposed model
4
0 0.00
0.06
0.01
0.02
0.03
0.04
0.05
0.06
Rotation (rad) 40
S2-D 16
12
8
test FEA Proposed model
4
0
0.01
0.02
0.03
0.04
0.05
S3-A
36
Bending Moment M (kN·m)
Bending moment M (kN·m)
16
(rad)
20
0.00
0.06
20
S2-B
0 0.00
0.05
Rotation (rad)
Rotation (rad)
0.06
Rotation (rad)
32 28 24 20 16 12
Test FEA Proposed model
8 4
0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Rotation
(rad)
40
S4-A
Bending moment M (kN·m)
36 32 28 24 20 16 12
Test FEA Proposed model
8 4 0 0.00
0.01
0.02
0.03
0.04
Rotation
0.05
0.06
0.07
0.08
(rad)
Fig. 22. The comparison of the moment–rotation relationship of BC joints.
where m = 0.3 is Poisson’s ratio; b is the elastic buckling coefficient; v is the elastic constraint coefficient associated with the middle plate, and v = 1.0 is used for the joints in this paper.
4.3. Validation of the proposed analytical model Fig. 22 presents a comparison of the M–u curves obtained from the tests, finite element analysis (FEA) and proposed analytical
H. Ma et al. / Engineering Structures 123 (2016) 108–123
123
model established by means of Eqs. (3)–(15). It can be observed from these figures that the equations presented above provide a good estimation of the M–u curves. The stiffness and bending capacity provided by the proposed analytical model are in very good agreement with the experimental results. It can be concluded that the analytical estimation can be a great help for a preliminary prediction of the initial stiffness and bending capacity of BC joints.
Further research should be conducted on the BC joints regarding the mechanical behaviour of the joints under torsion and out-of-plane loading conditions. Another important design consideration is related to the hysteretic behaviour of the joints under inor out-of-plane loads.
5. Conclusions
This research is supported by grants from the Natural Science Foundation of China (Grant No. 51308153); China Postdoctoral Science Foundation funded project (Grant No. 2013 M531020 and Grant No. 2014T70346); and The National Science Fund for Distinguished Young Scholars (Grant No. 51525802).
This paper addresses the FE model, parametric study and analytical model of a novel type of semi-rigid joint. First, in this paper, a parametric study including 341 models is carried out in which different parameters including front plate thickness, side plate thickness, and loading schemes are varied. The major conclusions related to the parametric analysis are summarized below: – The thickness of front plate has little effect on the moment–rotation curves and failure mode of the joints. The initial stiffness and bending capacity increase significantly with increasing thickness of side plates. However, when the thickness is too large (t2 > 7 mm for the joints in this paper), the tensional bolt in the joint will yield before the side plates, leading to brittle failure. To gain a greater bending–rotational capacity and avoid brittle failure, a reasonable thickness of the side plates should be chosen for the joints. – The critical eccentricity dc = 0.2 m was defined for the joints under eccentric pressure or tension. The effects of pressure and tension on the failure modes of the joints are quite distinct. A large area of the bolts and front plate yield when the joint is subjected to eccentric tension force with dt < 0.2; the stress of the bolts and front plate is small and far below the yield stress when the joints are subjected to eccentric pressure. Theoretical equations and envelope curves for predicting the plastic moment resistance of the joints under pressure or tension forces with different eccentric distances are proposed. – The initial bending stiffness and moment capacity decrease significantly with increasing axial tension force. The moment capacity decreases significantly with increasing axial pressure, whereas the initial bending stiffness increases at first and then decreases with a sharp drop Np > 451.98 kN (gp > 0.6). More attention should paid to the effect of pressure and tension force on the failure mode of the joints. Compared with the joint under pure bending, the existence of axial tension force exacerbates the stress level significantly in the bolts and front plate, whereas the existence of pressure force reduces the stress level in the bolts and front plate. The theoretical equations and envelope curves for predicting the plastic moment resistance of the joints under bending with different axial pressure or tension are established. Second, based on the structural characteristic of the BC joints, a spring-stiffness model for the joints was assessed, and a proposed analytical model was developed to predict the bending-rotation characteristic of the joints under pure bending. The results obtained by the analytical model were compared with the experimental results and FEA models, showing that the proposed analytical model predicts the behaviour of the BC joints with good accuracy.
Acknowledgements
References [1] CEN. Eurocode 3: design of steel structures. Part 1.8: design of joints (EN 19931-8:2005). CEN; 2005. [2] Citipitioglu AM, Haj-Ali RM, White DW. Refined 3D finite element modeling of partially-restrained connections including slip. J Constr Steel Res 2002;58:995–1013. [3] Maggi YI, Gonçalves RM, Leon RT, Ribeiro LFL. Parametric analysis of steel bolted end plate connections using finite element modeling. J Constr Steel Res 2005;61:689–708. [4] Mohamadi-shooreh MR, Mofid M. Parametric analysis on the initial stiffness of flush end-plate splice connections using FEM. J Constr Steel Res 2008;64:1129–41. [5] Shi Gang, Shi Yongjiu, Wang Yuanqing, Bradford Mark A. Numerical simulation of steel pretensioned bolted end-plate connections of different types and details. Eng Struct 2008;30:2677–86. [6] Fan Feng, Ma Huihuan, Chen Gengbo, Shen Shizhao. Experimental study of semi-rigid joint systems subjected to bending with and without axial force. J Constr Steel Res 2012;68(1):126–37. [7] Fan Feng, Ma Hui-huan, Cao Zheng-gang, Shen Shi-zhao. Numerical analyses of semi-rigid joints subjected to bending with and without axial force. J Constr Steel Res 2012;90(11):13–28. [8] Ueki T, Matsushita F, Shibata, Kato S. Design procedure for large single-layer latticed domes. In: Proceedings of the fourth international conference of space structures. UK: University of Surry; 1993. p. 237–46. [9] Ma Huihuan, Ren Shan, Fan Feng. Experimental and numerical research on a new semi-rigid joint for single-layer reticulated structures. Eng Struct 2015 [submitted for publication]. [10] ABAQUS/CAE. Hibbit, Karlsson & Sorensen Inc; 2005. [11] A1-Jabria KS, Burgessb IW, Plankc RJ. Spring-stiffness model for flexible endplate bare-steel joints in fire. J Constr Steel Res 2005;61:1672–91. [12] Gil Beatriz, Bijlaard Frans, Bayo Eduardo. T-stub behavior under out-of-plane bending. II: Parametric study and analytical characterization. Eng Struct 2015;98:241–50. [13] Block FM, Burgess IW, Davision JB, Plank RJ. A component approach to modelling steelwork connections in fire: behaviour of column webs in compression. In: Proc ASCE structures congress; 2004. p. 1–8. [14] Gil Beatriz, Bayo Eduardo. An alternative design for internal and external semirigid composite joints. Part II: finite element modelling and analytical study. Eng Struct 2008;30:232–46. [15] Lopez Aitziber, Puente Iñigo, Aizpurua Hodei. Experimental and analytical studies on the rotational stiffness of joints for single-layer structures. Eng Struct 2011;33(12):731–7. [16] Loureiro A, Moreno A, Gutiérrez R, Reinosa JM. Experimental and numerical analysis of three-dimensional semi-rigid steel joints under non-proportional loading. Eng Struct 2012;38:68–77. [17] Yang Huifeng, Liu Weiqing, Ren Xiao. A component method for momentresistant glulam beam–column connections with glued-in steel rods. Eng Struct 2016;115:42–54. [18] Simões da Silva L, Santiago Aldina, Vila Real Paulo. Post-limit stiffness and ductility of end-plate beam-to-column steel joints. Comput Struct 2002;80:515–31. [19] Agerskov H. High strength bolted connections subject to prying. J Struct Div ASCE 1976;102(ST1):161–75.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Parametric study and analytical characterization of the bolt–column (BC) joint for single-layer reticulated structures Huihuan Ma ⇑, Shan Ren, Feng Fan School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China
a r t i c l e
i n f o
Article history: Received 20 January 2016 Revised 19 May 2016 Accepted 23 May 2016
Keywords: Single-layer latticed structure Parametric analysis Bolt–column joint Rotational capacity Failure mode Analytical model
a b s t r a c t The bolt–column (BC) joint, which can efficiently connect H, I and rectangular section members, is actually semi-rigid and of partial strength. The consideration of real semi-rigid behaviour can lead to a more accurate global analysis and, consequently, more optimized structures. In this paper a parametric analysis is conducted on the joint. A total of 341 models, divided into five series G1–G6, were analyzed. The main parameters affecting the rotational capacity and failure mode of the joint were considered in the finite element (FE) models. First, the optimal geometric parameters, such as the suitable thickness of the side plate and front plate, were obtained and discussed. Second, the mechanical behaviour of the joints under different static load conditions, including pure bending moment, eccentric pressure, eccentric tension, bending with a defined pressure and bending with a defined tension force, was investigated in detail. The theoretical equations and envelope curves for predicting the moment capacity of the joints under eccentric forces and bending with axial forces were established. Furthermore, based on the component methods in Eurocode 3 (EC3) (CEN, 2005) [1], a trilinear mathematical model for predicting the moment–rotation relationship of the joints was developed. The moment–rotation curves obtained by the mathematical model compare very satisfactorily with the experimental results. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The conventional design of steel structures is usually conducted under the assumption that joints are either fully rigid or ideally pinned. Currently, after a series of research, the concept that the actual joints exhibit behaviour that is intermediate between these two extreme cases—semi-rigid and nonlinear—has been accepted by the codes. This means that the effect of joint flexibility should be considered in the analysis and design procedures to facilitate numerical analysis of complete structures that incorporate this form of joint. Likewise, Eurocode 3 (EC3) [1] includes procedures and formulations to define both the stiffness and resistance of semi-rigid joints starting from their geometrical and mechanical properties. Normally, the moment–rotation curve, as the most important factor, can describe the rotational behaviour of the joint; the curve defines three main properties: rotational stiffness, moment resistance and rotation capacity. Many studies are aimed at obtaining moment–rotation curves of the joints so that they can be incorporated in structural analysis, such as the research for beam-to-column steel connections in [2–5] and semi-rigid joints
⇑ Corresponding author. http://dx.doi.org/10.1016/j.engstruct.2016.05.037 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.
in single-layer reticulated structures [6,7]. However, the failure mode of the joint is also a very important factor to evaluate the mechanical property of the joints. Whether the failure mode of a joint is sudden brittle failure or gradual plastic failure has an essential influence on the behaviour of entire structures. To date, few studies have focused on this issue for semi-rigid joints used in single-layer reticulated structures. Especially for a new joint system, determination of the optimal geometric parameters to avoid sudden brittle failure is of great significance. In addition, except for the investigations undertaken by Fan et al. [6,7] and Ueki et al. [8], most of the achievements regarding joints in spatial structures are related to the moment–rotation relationship of the joints under a pure bending moment or pure axial forces. However, in real spatial structures, the joints are mainly subject to combined bending moment and axial force, and the investigations in references [6–8] have shown that the axial forces have a significant effect on the mechanical behaviour of the joints, including the moment rotation curves and failure modes. Therefore, combinations of the different action types should be considered when investigating the mechanical characters of new joints. The actual working environment of a joint system is complex, and a further investigation into the effect of axial forces on the moment–rotation characteristic of a new joint
H. Ma et al. / Engineering Structures 123 (2016) 108–123
109
Nomenclature fy fu E M / t1 t2 Sj,inia,anal Sj,ini,exp Msup,anal Msup,exp d dc
gd
M dsup M 0sup
gdM
e Nt/Np Ntu =N pu
gt gp A1 A2
a
M Nsup
gNM Kj Mp Fb Fs L
yield strength of steel material tensile strength of steel material Young’s modulus of steel material bending moment joint rotation thickness of front plate thickness of side plate initial stiffness obtained numerically initial stiffness obtained experimentally plastic moment resistance obtained numerically plastic moment resistance obtained experimentally eccentricity of axial forces critical eccentricity absolute value of ratio of d to dc plastic moment resistance of joint under eccentric axial force plastic moment resistance of joint under pure bending ratio of Mdsup to M0sup natural logarithms axial tension/pressure force ultimate axial force of joint under pure axial tension or pressure ratio of Nt to Ntu ratio of Np to Npu total cross-sectional area of two side plates cross-sectional area of middle plate ratio of A1 to A2 plastic moment resistance of joint under bending with certain axial forces ratio of MNsup to M0sup initial stiffness obtained from the proposed model ultimate bending moment force in bolt tension/pressure force in side plates distance between shear force and end plate
is necessary to provide a better stiffness model that is more similar to practical circumstances. An experimental program was carried out to assess the real behaviour of the BC joint under bending [9]. Finite element models established through the ABAQUS finite element software program [10] with the same characteristics as those of the tests were developed and calibrated with the experimental results. The results demonstrate that the finite element model provides a reliable and robust tool to develop a parametric study, avoiding the need for additional and costly experiments. In this paper, a nonlinear parametric study is carried out. The moment–rotation curves and failure mode of the joints with varying side plate thickness, front plate thickness, and loading schemes are obtained and discussed in detail. The optimal geometric parameters of the joints are identified to achieve high stiffness and avoid the sudden brittle failure mode. The envelope curves for predicting the plastic moment resistance considering the affection of eccentricities and axial forces are established. Finally, an analytical model of the moment–rotation relationship is developed for design applications, and the analytical model is verified by the experimental results. 2. Finite element model The bolt–column (BC) joint is composed of a hollow column node, high-strength bolts, washers, and end-cone part, as shown in Fig. 1. It can be used to connect H, I or rectangular members in the real structures. The end-cone part consists of five plates:
d1 d2
DC DT kb kspt/kspp ksp As Lb tc Eb Lb tc tw tbh dc be n
q cp,k
m cp,M P P0 Wp he Mp
rcr b kp
v
distance between two high-strength bolts distance of centroid of side plate from compression zone to tension zone axial deformation of bolts deflection of side plates at compression or tension zone bolt stiffness stiffness of each side plate at tension/compression zone total stiffness of side plates section area of bolt shank bolt elongation thickness of column modulus of elasticity of bolts bolt elongation length thickness of column node washer thickness thickness of bolt head depth of side plate from front plate to end plate assumed effective width of side plates coefficient = 2.7 reduction factor q for side plate buckling influence coefficient of bolt pretension force on initial stiffness Poisson’s ratio for steel (=0.3) influence coefficient of bolt pretension force on ultimate plastic bending real preload force in bolt preload force in bolt according to code plastic inertia moment of side plate height of side plate ultimate plastic bending moment elastic critical buckling stress elastic buckling coefficient plate slenderness elastic constraint coefficient associated with middle plate
one front plate (t1), two side plates (t2), one middle plate (t3) and one end plate (t4). The cone parts are welded at both ends of the members in the factory. In the construction site, the two highstrength bolts are used to connect the members to the column node without any welding work. A total of 14 BC joint specimens (Fig. 2) were divided into 7 groups and tested to failure under monotonic loading. The threedimensional finite element model (Fig. 3) has the same configuration and dimensions as the tested specimens. The configurations of the specimens are shown in Fig. 4 and Table 1. The whole experimental joint, including the column node, the cone part, the bolts and the rectangular member, is modelled with 8-node solid elements C3D8R, available in the ABAQUS library [10]. The numerical models incorporated a number of detailed features such as contact phenomena, bolt slippage definition and bolt prestressing force application. A total of 14 tests, conducted based on varying pretension force, side plate thickness and bolt diameter, are used to validate the FE model. The geometric parameters, preload forces on the bolts and initial geometric imperfection of the side plates of the FE models are the same as the specimens in the tests. The comparison of the moment–rotation curves obtained through FE models and tests are shown in Fig. 5. From the curves, it can be observed that the curves obtained numerically and experimentally are similar. The good agreement with the experimental results confirms that the detailed FE models are able to simulate the mechanical behaviour
110
H. Ma et al. / Engineering Structures 123 (2016) 108–123
Fig. 1. The bolt–column joint.
Fig. 2. Specimen of the BC joint.
of BC joints with reasonable accuracy. The numerical curves of the specimens in group S1 and S2 are slightly higher than those experimentally obtained after the plastic flexural resistance. This variation can be attributed to the sudden buckling in the compression side of the thin side plates of the experimental specimens. The reliability of the FE models is also proved by the good agreement between the numerical failure mode and those observed after tests, as shown in Fig. 6. To provide a convenient way to analyze the M–u curves, the following characteristics are assessed, as shown in Fig. 5(a): Sj,ini is the initial stiffness; Sj,p-l is the postlimit stiffness, which is defined as 10% of the initial stiffness, Sj,ini; and Msup is the plastic moment resistance, which corresponds to point O of the regression line obtained for the post-limit (Sj,p-l) stiffness. 3. Nonlinear parametric analysis of the joints
Fig. 3. FE model of the BC joint.
It is well known that in real spatial reticulated structures, connections carry not only bending forces but also axial forces and that the ratios between the bending and axial forces in the joints vary. Hence, to reflect the mechanical performance of the joint in a real structure, the FE models were subjected to different load
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
Fig. 4. Detailed dimensions of the cone part.
Table 1 Geometric parameters of the specimens. Test ID
Beam section (mm)
Bolt diameter d (mm)
Bolt distance L1 (mm)
Front plate t1 (mm)
Side plates t2 (mm)
Middle plate t3 (mm)
End plate t4 (mm)
Prestressing forces P24 = 225 kN P27 = 290 kN
S1-A
h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 160 120 16 16 h 180 130 16 16
24
78
30
3.5
5.5
20
P24
24
78
30
5.5
5.5
20
1.25P24
24
78
30
5.5
5.5
20
P24
24
78
30
5.5
5.5
20
0.75P24
24
78
30
5.5
5.5
20
0.32P24
24
78
30
9.5
5.5
20
P24
27
87
30
7.5
7.5
20
P27
S2-A S2-B S2-C S2-D S3-A S4-A
The square box means rectangular steel tube.
conditions (as shown in Fig. 7), including pure bending, eccentric pressure, eccentric tension, bending with a defined pressure, and bending with a defined tension. The same boundary condition is used for the joints under different loading schemes: the half column node is fixed, and the forces act on the end of the member. The force Nt or Np in Fig. 7(c) can rotate with the model to maintain the direction of the axial forces pointing to the joint centre along the centreline of the members; hence, the secondary moment caused by the axial force can be eliminated during the loading. The stiffness, strength, rotation behaviour and failure mode of the joints under different static load conditions were obtained and are discussed later. The geometric characteristics and detailed loading conditions of joints during the parametric analysis are listed in Table 2. A total of 341 models, divided into six series G1–G6, have been analyzed. The models in series G1 were used to investigate the optimized thickness of the front plate and side plate; the models in series G2 were used to investigate the mechanical behaviour of the joints under shear force with different distances between the shear force and end plate; the models in series G3 and G4 were used to investigate the mechanical behaviour of the joints with different bolt diameters and side plate thicknesses under eccentric forces; the models in series G5 and G6 were used to investigate the mechanical behaviour of the joints with different bolt diameters and side plate thicknesses under bending with defined axial forces. Based on the parametric study, the following results are obtained:
– The complete moment–rotation curves of the joints (Figs. 9–12 and 16). – Stress distribution in the cone part and bolts of the joints (Figs. 9–11, 14 and 18). – Variation of initial stiffness and moment capacity of the joints (Figs. 13 and 17). – Relationship of moment capacity and eccentricity of axial forces (Fig. 15). – Relationship of moment capacity and axial forces (Fig. 19). Figs. 12–14 show the moment–rotation curves, stress distribution and variation of the initial stiffness and moment capacity, respectively, for the models for only one set of joints (t2 = 5.5 mm, M24 bolts) under different eccentric forces in series G3 and G4. For the other models in these two series, the effect of the eccentricity of the forces on the moment–rotation curves, stress distribution and change rule of the initial stiffness and moment capacity is similar. To obtain the envelope curve of the moment capacity of the joints under eccentric forces in Fig. 15, all moment–rotation curves of the joints in series G3 and G4 were obtained and analyzed. Figs. 16–18 also show the moment–rotation curves, stress distribution and change rule of initial stiffness and moment capacity, respectively, for the models for only one set of joints (t2 = 5.5 mm, M24 bolts) under bending with different defined axial forces in series G5 and G6. For the other models in these two series, the effect of the axial forces on the moment–rotation curves, stress distribution
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
40
40 Msup,exp: 28.77kN·m
Msup
S3-A 35
Sj,ini
Bending moment M (kN·m)
Bending moment M (kN·m)
35 30
Sj,p-l=0.1Sj,ini
O
Msup,anal: 28.06kN·m
25 20
Msup,exp: 17.52kN·m
15
Msup
S2-B : 17.58kN·m
anal
Msup,exp: 10.46kN·m
10
S1-A Msup,anal: 10.30kN·m
5
: Msup,exp
S4-A
30 Msup,anal:30.79kN·m
25 20
Msup,exp:17.52kN·m
S2-B 15
Msup,anal:17.59kN·m
10
test FEA
5
test FEA
: Msup,anal
Msup,exp:30.86kN·m
: Msup,exp 0 0.00
0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.01
0.02
: Msup,anal 0.03
0.04
0.05
(a) For different thicknesses of side plates
0.08
20 Msup,exp: 16.74kN·m
S2-C
Bending moment M (kN·m)
S2-A
16
Bending moment M (kN·m)
0.07
(b) For different bolt diameters
20
Msup,anal: 16.57kN·m 12
8
4 : Msup,exp 0 0.00
0.06
Rotation (rad)
Rotation (rad)
0.01
test FEA
: Msup,anal 0.02
16
Msup,exp:17.43kN·m Msup,anal:17.39kN·m
12
8
4 : Msup,exp
0.03
0.04
0 0.00
0.05
Rotation (rad)
0.01
test FEA
: Msup,anal 0.02
0.03
0.04
0.05
Rotation (rad)
20
Msup,exp: 17.28kN·m
Bending moment M (kN·m)
S2-D 16
Msup,anal:17.12kN·m 12
8
4
: Msup,exp
test FEA
: Msup,anal
0
0.00
0.01
0.02
0.03
0.04
0.05
Rotation (rad)
(c) For different pretension forces Fig. 5. Moment–rotation comparison of the BC joints.
and change rule of initial stiffness and moment capacity is similar. To obtain the envelope curve of the moment capacity of the joints under bending with different axial forces in Fig. 19, all moment– rotation curves of the joints in series G5 and G6 were obtained and analyzed. The stress–strain relationship for the different parts of the joints is modelled for nonlinear analysis as an elastoplastic material with a strain hardening and yielding plateau beyond the elastic phase. It is assigned a trilinear stress–strain law, symmetrical in tension and compression, as shown in Fig. 8. The steel used for the beam
member, cone part and washers is grade steel S235. The bolts are frictional high-strength bolts (class 10.9). The actual material properties of the steel were obtained from tensile tests on coupons and the bolt certificate of quality, as shown in Table 3. 3.1. BC joints under pure bending moment In this section, the 3D FE model is used for parametric studies in which different thicknesses of the front plate and side plate are considered under bending. Based on the parametric analysis, the
H. Ma et al. / Engineering Structures 123 (2016) 108–123
113
Fig. 6. Failure mode comparison of the BC joints.
Fig. 7. Static loading conditions.
whole moment–rotation curves and failure modes of the joints are obtained, and the appropriate thicknesses of front plate and side plate are discussed and suggested.
3.1.1. BC joints with different thicknesses of front plate The influence of the thickness of the front plate on the overall behaviour of the joints was analyzed. For the joint (G1-1 Table 2), four front plate thicknesses, 22, 26, 30 and 40 mm, were considered in the analysis. The moment–rotation curves and stress distribution at the cone part and bolts of the joints are shown in Fig. 9. The general law is that by increasing the thickness of the front plate, the rotational capacity will be correspondingly increased slightly under bending. Careful examination of the results shows similar failure modes for the joints with eccentric pressure and tension. The top and bottom areas of the side plates had yielded and formed into a plastic hinge before the tensional bolt yielded. From the stress distribution at
the cone and tensional bolt, it can be observed that the yield area of front plate decreases with increasing thickness of the front plate; the effect of the thickness of the front plate on the stress distribution of the bolts and side plates is small. The deformation of the bolt is linear in the joints with different thicknesses of the front plate; however, when t1 6 26 mm, a large area of the front plate finally yields, which will become worse when the joints are subjected to combined bending and axial force. Therefore, for BC joints with M24 high-strength bolts, the front plate thickness should not be thinner than 26 mm.
3.1.2. BC joints with different thicknesses of side plate The influence of the thickness of the side plate on the rotational capacity of joints was examined. Five thicknesses of the side plate were considered in the analysis—namely, 4, 6, 7, 10 and 12 mm. The other geometric dimensions of the joints are the same as G1-2 listed in Table 2. Fig. 10 shows the moment–rotation
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
600
1400 1200
500
0.0379
0.0352 1171.41 MPa
800 600 400
300
0.00146
303.93 MPa
200 100
200 0 0.00
455.58 MPa
400
0.00457 975 MPa
Stress (MPa)
Stress (MPa)
1000
0.02
0.04
0.06
0.08
0 0.00
0.02
0.04
0.06
0.08
Strain
Strain
(a) stress–strain curve for bolts
(b) stress–strain curve for column node, cone part and members
Fig. 8. Stress–strain behaviour of the joint components.
response and stress distribution at the cone part and bolts of the joints. It can be observed that the maximum moment and rotational capacity increase significantly with increasing thickness of side plates; the stress of the bolts increases with increasing thickness of side plates; the yield area of the side plate obviously decreases with increasing thickness of the side plate. The failure mode of the joints changes with increasing thickness of the side plate: (i) when the side plate thickness t2 6 7 mm, the top and bottom areas of the side plates yield, resulting in a plastic hinge before the tension bolts yield, and the joints fail; (ii) when t2 P 10 mm, the side plates cannot form a plastic hinge before the tensional bolts yield, and the joints fail because the tensional bolts yield. Apparently, failure mode (ii) should be particularly avoided in the design of real structures because it reduces the energydissipating capacity of the joints and may lead to a brittle failure of the joints and the whole structure. Therefore, to gain a greater moment–rotational capacity and better energy-dissipating capacity and avoid brittle failure, the thickness of the side plates should not be too large for the joints. 3.2. BC joints under shear force with different distance L To investigate the influence of shear force on the behaviour of the BC joints, four different distances between the shear forces and end plate were considered in the analysis. The shorter the distance L, the greater shear force at the connection when the bending moment is the same. The geometric dimensions of the joints in this section are the same as series G2 listed in Table 2. The moment–rotation curves and stress distribution at cone part and bolts of the joints are shown in Fig. 11. In the figure, the plastic moment resistance of the joint under shear decreased less than 4.87% compared with the joint under pure bending. A plastic hinge is formed at the cone part in all five joints. The stress distribution of the bolts and the yield area of the front plate increases with increasing distance between the shear force and the end plate. In total, the effect of shear on the behaviour of the BC joint is small. 3.3. BC joints under eccentric forces The geometric dimensions of the joints in this section are the same as series G2 and G3, listed in Table 2. During the loading process, there was bending moment owing to the eccentric pressure and tension. The bending and axial force increased together.
The moment–rotation curves of the joints (t2 = 5.5 mm, M24 bolts), subjected to different eccentric forces, are shown in Fig. 12. d = 0.02–1 m is the eccentric distance of the axial force, and the curve corresponding to d = 1 in the figure is the moment–rotation curve of the joints subjected to pure bending. Fig. 13 shows the change rule of the initial bending stiffness and plastic moment resistance of the joints (t2 = 5.5 mm, M24 bolts). In the figure, Sdj;ini =S0j;ini represents the ratio of the initial bending of the joints under axial forces with the eccentricity d to that of the joints under pure bending; M dsup =M 0sup represents the ratio of the plastic moment resistance of the joints under axial forces with the eccentricity d to that of the joints under pure bending. From the curves, it was found that the effect of the force eccentricities on the plastic moment resistance is similar, whereas the effect of the force eccentricities on the initial bending stiffness is different. As shown in Fig. 13, the curves can be described in two stages: – Stage-1: when d < 0.2 m, the eccentricity of the forces has an obvious effect on both the initial bending stiffness and moment capacity; the initial bending stiffness of the joints under the eccentric pressure force is stronger than that of the joints under pure bending, whereas the initial bending stiffness of the joints under eccentric tension force is smaller than that of the joints under pure bending; the plastic moment resistance of the joints under eccentric pressure and tension is smaller than that of the joints under pure bending, and it increases significantly when the eccentricity of the axial force increases from 0.02 m to 0.1 m. – Stage-2: when d P 0.2 m, the initial bending stiffness and plastic moment resistance of joints subjected to eccentric forces are almost the same as those of the joints under pure bending. Therefore, dc = 0.2 m is defined as the critical eccentricity. The mechanical behaviour of the joints under axial forces with eccentricity d P dc will be very close to the joints under pure bending. Another important effect of eccentricity is related to the failure modes of the joints. The stress distributions at the cone part and bolts of the joints (t2 = 5.5 mm, M24 bolts) are shown in Fig. 14. In the figures, it can be observed that (i) a plastic hinge is formed at the cone part of all joints; (ii) the neutral axis of the cone part in the failure modes deviates from the middle plate when the eccentric distance is less than 0.1 m; and the neutral axis of the
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H. Ma et al. / Engineering Structures 123 (2016) 108–123 Table 2 Detailed dimensions and loading conditions of the analysis models. Group
Parametric study (FEM) Beam section (mm)
Bolt diameter d (mm)
Bolt distance L1 (mm)
Front plate t1 (mm)
Side plates t2 (mm)
Middle plate t3 (mm)
End plate t4 (mm)
Preload P (kN)
Loading condition
G1-1
h 160 120 16 16
24
78
5.5
5.5
20
225
Fig. 5(a)
G1-2
h 160 120 16 16
24
78
22 26 30 40 30
4 6
5.5
20
225
Fig. 5(a)
G2
h 160 120 16 16
24
78
30
5.5
20
225
Fig. 5(b) L = 0.50 m L = 0.25 m L = 0.05 m L = 0.00 m
G3-1
h 160 120 16 16
24
78
30
3.5 5.5 9.5
5.5
20
225
G3-2
h 180 130 16 16
27
87
30
7.5
5.5
20
290
G4-1
h 160 120 16 16
24
78
30
5.5
20
225
G4-2
h 180 130 16 16
27
87
30
3.5 5.5 9.5 7.5
5.5
20
290
G5-1
h 160 120 16 16
24
78
30
3.5 5.5 9.5
5.5
20
225
G5-2
h 180 130 16 16
27
87
30
7.5
5.5
20
290
G6-1
h 160 120 16 16
24
78
30
3.5 5.5 9.5
5.5
20
225
G6-2
h 180 130 16 16
27
87
30
7.5
5.5
20
290
Fig. 5(c) dt = 0 dt = 0.01 m dt = 0.02 m ... dt = 0.1 m ... dt = 0.9 m dt = 1.0 m Fig. 5(d) dp = 0 dp = 0.01 m dp = 0.02 m ... dp = 0.1 m ... dp = 0.9 m dp = 1.0 m Fig. 5(e) gt = 0.05 gt = 0.10 gt = 0.15 ... gt = 0.85 gt = 0.90 gt = 0.95 Fig. 5(f) gp = 0.05 gp = 0.10 gp = 0.15 ... gp = 0.85 gp = 0.90 gp = 0.95
7 10 12 5.5
The square box means rectangular steel tube.
Fig. 9. Moment–rotation curves and stress distribution of joints with different front plate thicknesses.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
Fig. 10. Moment–rotation curves and stress distribution of joints with different side plate thicknesses.
35
Bending moment M (kN·m)
30 25
Msup=21.897kN·m Msup=21.44kN·m
20 15 10
Msup=20.88kN·m Msup=21.21kN·m Msup=21.1kN·m
shear force with L=0.5m shear force with L=0.25m
5 0 0.00
pure bending
shear force with L=0.05m shear force with L=0.0m 0.02
0.04
Rotaion
0.06
rad
Fig. 11. Moment–rotation curves and stress distribution of joints under shear force with different distance L.
Fig. 12. Moment–rotation curves of one set of joints under eccentric forces.
side plates moves to the centre when the eccentric distance of the force increases. However, the effects of eccentric pressure and tension on the stress distribution of the front plate and bolts are quite distinct. When dt < 0.2, a large area of the bolts and front plate of the joint under eccentric tension force yields; when the joints are subjected to the eccentric pressure, the stress of the bolts and front plate is small and far below the yield stress.
For joints subjected to eccentric pressure and tension, designers are more concerned with the relationship between the ultimate moment resistance and eccentricity when the joints are broken. Hence, based on the foregoing parametric analysis, the non-dimensional relationship of the plastic moment resistance and the eccentricity of the pressure and tension forces for the BC joint have been established, as shown in Fig. 15. The theoretical
Ratio of initial stiffness/ultimate bending
H. Ma et al. / Engineering Structures 123 (2016) 108–123
1.3 1.2
Sj,ini/Sj,ini Eccentric pressure
1.1
Sj,ini/Sj,ini Eccentric tension
1.0
1 1 gdM ð1 þ 3e6gd Þ ¼ 1 4 sup
0
resistance of BC joints under different eccentricities of axial forces can be predicted by the designers.
Msup/Msup Eccentric pressure
0.8
0
Msup/Msup Eccentric tension
0.7
M24
0.6
t2=5.5mm
3.4. BC joints under bending with a defined axial force
The critical eccentricity: =0.2m c 0.0
ð1Þ
Md d where gdM ¼ Msup 0 ; gd ¼ d . Based on Eq. (1), the plastic moment c
0.9 0.96
0.4
envelope curve in the figure is approximated by the following equations:
1.03
0.5
117
0.2
0.4
0.6
Eccentric distances
0.8 t
/
p
1.0
1.2
(m)
Fig. 13. Change rule of initial stiffness and moment capacity of one set of joints under eccentric forces.
The geometric dimensions of the joints in this section are the same as series G4 and G5, listed in Table 2. Unlike the joints subjected to eccentric forces, the axial force in this section remains constant, and the direction of the axial forces always point to the joint centre along the centreline of the beam during the loading process; the bending increases stepwise until the joints fail. The moment–rotation curves and the stress distribution at the cone part of the joints are shown in Fig. 16. Fig. 17 shows the variation
(a) the stress cloud of the joint under eccentric tension
(b) the stress cloud under eccentric compression Fig. 14. Stress distribution of the joints under different eccentric forces.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
M
1.0 t
c
0.2m;
p
0.8
1.0; M t
1.0;
t2=5.5mm, M24
1.0;
1.0; p
M
t2=9.5mm, M24
0.2 m;
1.0;
t2=7.5mm, M27
0.2
t2=7.5mm, M27
c
1.0;
t2=5.5mm, M24
0.4
t2=9.5mm, M24
Theoretical curve
Theoretical curve
0.0
0.5
1.0
0.2 m ;
t2=3.5mm, M24
t2=3.5mm, M24
1.0; M
M
0.6
0.2 m ;
c
c
1.0;
Eccentric tension
0.5
1.0
Eccentric pressure
Fig. 15. Relationship of moment capacity and eccentricity of axial forces.
1.4
Sj,ini/Sj,ini Bending with axial pressure P
1.2
Sj,ini/Sj,ini Bending with axial tension t
Change slightly
1.0 t
/ p=0.1
0.2
0.8
0.3 0.4 0.5
0.6
t2=5.5mm, M24
0.4
0.6
0
Msup/Msup Bending with axial pressure p
0.2 0.0
Sharp drop
Ratio of initial stiffness/ultimate bending
Fig. 16. Moment–rotation curves of one set of joints under bending with different axial forces.
0
Msup/Msup Bending with axial tension t
0
150
0.7
300
Axial tension/pressure
450
/ t
600
(kN) p
Fig. 17. Variation of initial stiffness and moment capacity of one set of joints under bending with different axial forces.
of the initial bending stiffness and plastic moment resistance of the joints under bending with different axial forces. In the figure, N
ðt=pÞ Sj;ini =S0j;ini represents the ratio of the initial bending of the joints
under bending with axial forces Nt/Np to that of the joints under N
ðt=pÞ pure bending; Msup =M 0sup represents the ratio of the plastic moment resistance of the joints under bending with axial forces
Nt/Np to that of the joints under pure bending; gt ¼ N t =N tu and gp ¼ Np =Npu ; here, Ntu ¼ 797:4 kN and Npu ¼ 814:16 kN are the ultimate axial forces of the joints (t2 = 5.5 mm, M24 bolts) under pure axial tension and pressure, respectively. From the curves, it has been found that (i) the moment resistance decreases significantly with increasing axial pressure/tension force; however, when g(t/p) 6 0.3, the moment resistance of the joints under bending with axial forces is very close to that of the joints under pure bending; the difference in the curves of the joints under bending with axial pressure and tension forces narrows when gt/gp decreases; (ii) the axial pressure/tension force has a different effect on the initial bending stiffness of the joints; the initial stiffness of the joints under bending with axial tension force decreases significantly with increasing tension force, and the bending stiffness of all joints is smaller than that of the joints under pure bending; the initial stiffness of the joints under bending with axial pressure force remain stable and higher than that of the joints under pure bending when Np 6 366.21 kN (gp 6 0.5), and a sharp drop occurs in the curve when Np > 451.98 kN (gp > 0.6). The force transfer mechanisms of the BC joint under bending with axial pressure or tension are similar. From the stress distribution at the final step for different axial pressures, it can be found that the neutral axis of the middle plate moves from the centre to one side when gt/gp increases. The yield areas of the front plate and bolts increase with increasing tension force and decrease with increasing compression force. The stress distributions of the joints under bending with different axial pressure and tension forces are shown in Fig. 18.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
(a) the stress cloud of a joint under bending with a certain tension force
(b) the stress cloud of a joint under bending with a certain compression force Fig. 18. Stress distribution of the joints under bending with axial pressure and tension forces.
4
+2
2 (t / p )
N M
0.3
t
N M
1
p
4
1.0 t
0.3
N M
1.0
p N M
0.8
0.3
+2
2 (t / p )
N M
1
0 .3 1 .0
0.6
-1.0
-0.8
t2=3.5mm, M24 bolts
FEA;
t2=5.5mm, M24 bolts
FEA;
t2=9.5mm, M24 bolts
FEA;
t2=7.5mm, M27 bolts
FEA;
-0.6
-0.4
Bending with tension
0.4
Theoretical curve Theoretical curve
0.2
-0.2 t
Theoretical curve Theoretical curve 0.0
0.2
0.4
0.6
Bending with pressure
0.8
1.0
p
Fig. 19. Relationship of moment capacity and axial force.
The plastic neutral axis of the cone part in the failure modes deviates from middle plate to the tensional side when the axial forces increase. The stress distribution of the side plates of the joints in
which g(t/p) 6 0.3 is almost the same as that of the joint under pure bending. A plastic hinge is ultimately formed at the cone part of all joints. However, the effects of pressure and tension force on the
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
rotation curve of the joints is divided into three phases, as shown in Fig. 20:
Table 3 Material properties. Results of tensile coupon tests. Components
fy (N/mm2)
fu (N/mm2)
Young’s modulus E (N/mm2)
fy/fu
Plates in cone part
t2 = 3.5 mm t2 = 5.5 mm t2 = 7.5 mm t2 = 9.5 mm
312.65 303.93 286.57 297.58
470.81 455.58 419.95 413.87
207,053 207,848 202,476 207,882
0.67 0.67 0.68 0.66
Beam Bolts
t = 15.5 mm d = 24 mm
269.00 975.00
406.86 1171.41
205,569 213,302
0.66 0.83
– Phase I: M < 2/3 Mp, the curve has linear behaviour up to the moment value of 2/3 Mp, where Mp is the ultimate plastic moment of the joint; – Phase II: 2/3 Mp < M < Mp, the curve is nonlinear in this phase. – Phase III: M P Mp, in this phase, owing to different side plate thicknesses, two failure modes occurred in the test: strength failure and local buckling failure. According to the two failure modes, the curve in Phase III performs as a plastic hardening phase and plastic descent phase.
The square box means rectangular steel tube.
stress distribution of the bolts are quite distinct; compared with the joint under pure bending, the existence of tension force exacerbates the stress level in the bolts, whereas the existence of pressure force reduces the stress level in the front plate and bolts. To predict the bending capacity of the joints under bending with different axial pressure or tension intuitively, the envelope curve of the ultimate bending moment affected by different axial forces is shown in Fig. 19. According to the position of the neutral axis of the cone part in the failure modes, the envelope curve of the ultimate bending moment can be divided into two groups:
Based on the three phases, the bending moment M is expressed as
8 Kj/ for M 6 23 M p > > > > < Kj n / for 23 Mp < M < Mp M¼ 1:3MM > p > > > : M p K t / for M P M p
where the value of parameter n of the bolted end-plate connections is 2.7 according to Eurocode 3. The final stiffness, Kt, can be calculated using a reduced stiffness coefficient of 1/60 and expressed as K t ¼ K j =60.
(i) When the axial forces are sufficiently small (g(t/p) 6 0.3), the plastic neutral axis is in the middle of the cone part. The formula of the envelope curve for the plastic moment resistance is as follows:
gNM ¼
M Nsup M 0sup
1:0
4.1. Initial bending stiffness Kj Eurocode 3 Part 1.8 [1] has adopted a component method for the design of joints. This method is based on dividing the joint into its basic components, and the design moment–rotation characteristic of a joint depends on the properties of its basic components. By assembling the contributions of individual components that represent the joint as a set of rigid and deformable elements, the entire behaviour of the joint may be determined. Based on the component method, the initial stiffness and ultimate capacity of each component is determined and assembled to form a spring model for some beam-to-column joints [11–14]. In the model, the joint components are treated as springs with predefined characteristics such as initial stiffness and ultimate capacity. In this paper, based on the structural characteristic of the BC joints under bending moment obtained by FE models, a spring-stiffness model for the joints is developed to predict the moment–rotation characteristic of the joints. The components involved in the BC joints under bending are assembled as series of springs according to the mechanical model
ð2Þ
(ii) When the axial forces are sufficiently large (g(t/p) > 0.3), the plastic neutral axis is not in the middle of the cone part. The formula of the envelope curve for the plastic moment resistance is as follows:
2 4 ða þ 2Þ gðt=pÞ a þ gNM ¼ 1
ð3Þ
where a ¼ A2 =A1 . 4. Analytical model for the joint under bending moment To establish the practical analytical model for the evaluation of the connection stiffness and capacity of the BC joints under a pure bending moment, the simplified trilinear model suggested in Eurocode 3 Part 1.8 [1] is considered. The elastoplastic moment–
M
M
Mp
3
Elastic phase Mp
0
-Kt
Mp
Kt 2
2
3
Mp
Elastic phase Nonlinear phase Plastic descent phase
Nonlinear phase Plastic hardening phase
Kj
Test Proposed model
(a) For the joints with strength failure mode
ð4Þ
Kj
Test Proposed model
(b) For the joints with local buckling failure mode
Fig. 20. Trilinear mathematical model of the M–u curve.
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H. Ma et al. / Engineering Structures 123 (2016) 108–123
illustrated in Fig. 21. For simplicity, in the model, the tension and compression zone comprise bolt stiffness Kb and side-plate stiffness Ksp. The bending moment M is equivalent to a pair of two forces; one of them pulls out from the joint along the tension bolt on one side, and the other pushes in the joint along the compression bolt on the opposite side. The global rotational stiffness of the joint Kj can then be determined at a given moment based on the assembled stiffness of all components from the following equations [15–17]:
M ¼ F b d1 ¼ F s d2
ð5Þ
DC DT þ d1 d2
ð6Þ
/¼
K j ¼ cp;k
cp;k F b d1 c d1 M ¼ p;k ¼ 1 / 2 F b þ F b d1 2 þ k d1d2 k d1 k d1 k d2 b
sp 2
b
ð7Þ
sp 2
where DC and DT sum up the contribution of both the axial deformation of the bolts and the deflection of the side plate; cp,k is an influence coefficient of bolt pretension force on initial stiffness and is given by
cp;k
p p0 ¼1þ 4p0
ð8Þ
At initial elastic stage a small deformation assumption is adopted. (1) Initial stiffness of bolts, kb The bolts are considered to be subjected to direct tensile force in isolation. High-strength bolts exhibit a brittle response up to failure, and the resistance of each bolt is given by F u ¼ 0:9f u As [18]. By applying the principles of Hooke’s law, the elastic stiffness of the bolt kb can be expressed as
kb ¼
1:6As E Lb
ð9Þ
(2) Side plate behaviour ksp The side plate can be divided into two zones: tension zone and compression zone. The stiffness of one-half of the side plate, kspt/ kspp, in the compression zone may be estimated by idealizing it as a plate of dimensions dc be subjected to a uniform compression force. By assuming that the plate obeys Hooke’s law, the elastic stiffness of the side plate can be expressed as
ksp ¼ 2kspt ¼ 2kspp ¼
ð11Þ
4.2. Ultimate plastic bending moment Mp The high-strength bolts and side plates are key components, and local buckling is an important issue for calculating the ultimate plastic bending moment when the side plate is thin. Consequently, based on Eurocode 3 Part 1-5 [1], the ultimate plastic bending moment, Mp, can be defined in the following form:
Mp ¼ qcp;M f y W p
ð12Þ t2 h2e 4
2 2t 2 be ;
where W p ¼ 2 ¼ the influence coefficient of the bolt pretension force on the ultimate plastic bending, cp,k, is given by
(
cp;M ¼
0 1 pp 4p
p P p0
1
p < p0
0
ð13Þ
According to Eurocode 3 Part 1-5 [1], the reduction factor q for side plate buckling can be obtained as follows:
8 < kp 0:2 kp > 0:673 k2p q¼ : 1 kp 6 0:673
The plate slenderness, kp , can be obtained by kp ¼
ð14Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi f y =rcr .
where rcr is the elastic critical plate buckling stress, based on the elastic stability theory; the critical buckling stress of the uniaxial compressional plate, rcr, can be calculated as follows:
2 t2 12ð1 m2 Þ be
The bolt elongation can be obtained from the following relationship based on Agerskov’s [19] recommendations:
rcr ¼ bv
t bh Lb ¼ t 1 þ tc þ tw þ 2
b ¼ 0:425 þ be =dc
ð10Þ
Et2 be dc
p2 E
(a) spring-model of the cone
(b) spring-model of the joint Fig. 21. General representation of the proposed model.
ð15Þ ð16Þ
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H. Ma et al. / Engineering Structures 123 (2016) 108–123 20
12
S1-A
S2-A
Bending moment M (kN·m)
Bending moment M (kN·m)
10 8 6 4
FEA Test Proposed model
2 0 0.00
0.01
0.02
0.03
0.04
0.05
16
12
8
0 0.00
0.06
test FEA Proposed momdel
4
0.01
0.02
0.03
0.04
20
Bending moment M (kN·m)
Bending Moment M (kN·m)
S2-C
16
12
8
FEA Test Proposed model
4
0.01
0.02
0.03
Rotation
0.04
0.05
12
8
test FEA Proposed model
4
0 0.00
0.06
0.01
0.02
0.03
0.04
0.05
0.06
Rotation (rad) 40
S2-D 16
12
8
test FEA Proposed model
4
0
0.01
0.02
0.03
0.04
0.05
S3-A
36
Bending Moment M (kN·m)
Bending moment M (kN·m)
16
(rad)
20
0.00
0.06
20
S2-B
0 0.00
0.05
Rotation (rad)
Rotation (rad)
0.06
Rotation (rad)
32 28 24 20 16 12
Test FEA Proposed model
8 4
0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Rotation
(rad)
40
S4-A
Bending moment M (kN·m)
36 32 28 24 20 16 12
Test FEA Proposed model
8 4 0 0.00
0.01
0.02
0.03
0.04
Rotation
0.05
0.06
0.07
0.08
(rad)
Fig. 22. The comparison of the moment–rotation relationship of BC joints.
where m = 0.3 is Poisson’s ratio; b is the elastic buckling coefficient; v is the elastic constraint coefficient associated with the middle plate, and v = 1.0 is used for the joints in this paper.
4.3. Validation of the proposed analytical model Fig. 22 presents a comparison of the M–u curves obtained from the tests, finite element analysis (FEA) and proposed analytical
H. Ma et al. / Engineering Structures 123 (2016) 108–123
123
model established by means of Eqs. (3)–(15). It can be observed from these figures that the equations presented above provide a good estimation of the M–u curves. The stiffness and bending capacity provided by the proposed analytical model are in very good agreement with the experimental results. It can be concluded that the analytical estimation can be a great help for a preliminary prediction of the initial stiffness and bending capacity of BC joints.
Further research should be conducted on the BC joints regarding the mechanical behaviour of the joints under torsion and out-of-plane loading conditions. Another important design consideration is related to the hysteretic behaviour of the joints under inor out-of-plane loads.
5. Conclusions
This research is supported by grants from the Natural Science Foundation of China (Grant No. 51308153); China Postdoctoral Science Foundation funded project (Grant No. 2013 M531020 and Grant No. 2014T70346); and The National Science Fund for Distinguished Young Scholars (Grant No. 51525802).
This paper addresses the FE model, parametric study and analytical model of a novel type of semi-rigid joint. First, in this paper, a parametric study including 341 models is carried out in which different parameters including front plate thickness, side plate thickness, and loading schemes are varied. The major conclusions related to the parametric analysis are summarized below: – The thickness of front plate has little effect on the moment–rotation curves and failure mode of the joints. The initial stiffness and bending capacity increase significantly with increasing thickness of side plates. However, when the thickness is too large (t2 > 7 mm for the joints in this paper), the tensional bolt in the joint will yield before the side plates, leading to brittle failure. To gain a greater bending–rotational capacity and avoid brittle failure, a reasonable thickness of the side plates should be chosen for the joints. – The critical eccentricity dc = 0.2 m was defined for the joints under eccentric pressure or tension. The effects of pressure and tension on the failure modes of the joints are quite distinct. A large area of the bolts and front plate yield when the joint is subjected to eccentric tension force with dt < 0.2; the stress of the bolts and front plate is small and far below the yield stress when the joints are subjected to eccentric pressure. Theoretical equations and envelope curves for predicting the plastic moment resistance of the joints under pressure or tension forces with different eccentric distances are proposed. – The initial bending stiffness and moment capacity decrease significantly with increasing axial tension force. The moment capacity decreases significantly with increasing axial pressure, whereas the initial bending stiffness increases at first and then decreases with a sharp drop Np > 451.98 kN (gp > 0.6). More attention should paid to the effect of pressure and tension force on the failure mode of the joints. Compared with the joint under pure bending, the existence of axial tension force exacerbates the stress level significantly in the bolts and front plate, whereas the existence of pressure force reduces the stress level in the bolts and front plate. The theoretical equations and envelope curves for predicting the plastic moment resistance of the joints under bending with different axial pressure or tension are established. Second, based on the structural characteristic of the BC joints, a spring-stiffness model for the joints was assessed, and a proposed analytical model was developed to predict the bending-rotation characteristic of the joints under pure bending. The results obtained by the analytical model were compared with the experimental results and FEA models, showing that the proposed analytical model predicts the behaviour of the BC joints with good accuracy.
Acknowledgements
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