- Email: [email protected]
Dynamic behaviour and seismic design method of a single-layer reticulated shell with semi-rigid joints
Thin-Walled Structures 119 (2017) 544–557
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Dynamic behaviour and seismic design method of a single-layer reticulated shell with semi-rigid joints
MARK
⁎
Huihuan Maa, , Zhiwei Shanb, Feng Fana a b
School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China School of Civil Engineering, The University of Hong Kong, Hong Kong, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Dynamic behaviour Seismic design Single-layer reticulated shell Semi-rigid joint Free vibration frequency Seismic internal force coefficient
Most of the existing studies on reticulated shells with a semi-rigid joint system have been focused on the mechanical properties under static loads. Taking material and geometric nonlinearities into account, the finite element analysis (FEA) model of a single-layer reticulated shell with semi-rigid joints was established using the software ABAQUS and then validated through comparison with the experimental result. Based on the bending stiffness of a bolt-column (BC) joint obtained through experiments, the dynamic behaviour and a seismic design method for single-layer reticulated shells with semi-rigid joints were investigated in this paper. First, analysis of the free vibration frequency of the single-layer latticed domes with semi-rigid bolt-column (BC) joints was conducted based on several different parameters, including joint stiffness, ratio of rise to span, initial geometric imperfection. Second, the seismic internal force coefficient of the members of the semi-rigidly jointed spherical single-layer reticulated shells of different parameters was studied in detail. Finally, the seismic internal force coefficients for spherical single-layer reticulated shells with semi-rigid joints under a common earthquake were derived.
1. Introduction Single-layer reticulated shells are well known to tend to be unstable under dynamic working actions. With large-span space structures, such as exhibition halls, stadiums and railway stations, being increasingly used in public facilities in recent years, reticulated shells have attracted more attention. The seismic performance of a reticulated dome not only is directly related to the safety of a building but also has a huge impact on the socio-economic development and the security of people. Therefore, researchers around the world have made great efforts to address the issues, such as the dynamic response and failure mechanism of these structures subjected to dynamic loads. However, most of the studies on the dynamic behaviour of a single-layer reticulated shell assume that the connections behave as perfect rigid joints, such as the dynamic performances and failure characteristics of single-layer reticulated domes subjected to an earthquake discussed in [1] as well as the study performed on the seismic performance of a single-layer reticulated dome [2,3]. However, the joints in most space structures are semi-rigid, for example the MERO joint systems [4–6], the aluminium alloy truss connector [7], the socket joint system [8] and the bolt-column joint system [9]. Single-layer reticulated shells with the above-described semi-rigid
⁎
Corresponding author. E-mail addresses: [email protected] (H. Ma), [email protected] (Z. Shan).
http://dx.doi.org/10.1016/j.tws.2017.07.003 Received 3 January 2017; Received in revised form 6 June 2017; Accepted 5 July 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
joints can provide a good solution for small- and medium-span space structures because of their rapid construction speed, high fabrication accuracy and beautiful appearance. Therefore, interest in single-layer spatial structures has grown significantly in the recent years. Observations from an earlier study [4,10] confirmed that connection stiffness had a significant effect on the load-displacement behaviour of a single-layer spatial structure. Aitziber et al. [11,12], Ma et al. [13,14], Fan et al. [15] and Kato et al. [16] verified that the rigidity of joints is an important factor that influences the behaviour of a single-layer latticed shell. However, those research studies primarily focus on the study of the mechanical properties of single-layer latticed shells with semi-rigid joint under static loads. Few investigations on the dynamic behaviour of a single-layer latticed shell with a semi-rigid joint have been conducted. Hence, it is worthwhile to explore dynamic mechanical performance of a singlelayer latticed shell with a semi-rigid joint. In this paper, finite element models of single-layer reticulated shells with the semi-rigid bolt-column joints are established. The rules of free vibration frequency are explored, considering the joint stiffness, ratio of rise to span, section of members, span and initial geometric imperfection. Under a common earthquake, seismic internal force coefficients of the single-layer latticed shell with different stiffness are derived for
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 1. Bolt-column joint.
seismic design.
Fig. 3. Moment -rotation curves of the bolt-column joints [17].
displacements at points P1-P3, which are obtained by the LVDTs. The moment-rotation curves of the bolt-column joints obtained from the experiments are plotted in Fig. 3. In the figure, k0 is the initial bending stiffness of the joints; ku is the post-limit stiffness of the joints, which is defined as 10% of the initial stiffness, k0; Mu is the plastic moment resistance of the joints, which corresponds to the point A of the regression line obtained for the post-limit (ku) stiffness. The bolt-column joints exhibit an adequate bending stiffness and bending capacity. The response of the connection exhibits linear behaviour in the early loading sequence and subsequent nonlinearity, which embodies typical elasto-plastic behaviour. The results indicate that the initial bending stiffness k0 of M27 BC joints is much higher than the initial bending stiffness of M24 BC joints. The plastic moment resistance Mu also improved when the diameters of the high-strength bolts increase. Therefore, the effect of the diameters of the bolts on the mechanical behaviour of the BC joint is significant.
2. The semi-rigid bolt-column (BC) joints The bolt-column joint is composed of a hollow cylinder, highstrength 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, which are welded at both ends of the members in the factory. At the construction site, the two high-strength bolts are used to connect the members to the column node without any welding work. All holes in the hollow column are tapped to accommodate the threaded part of the bolts. The two highstrength bolts are screwed into the hollow column node from the endcone part. One concave washer is employed at each bolted connection; they are placed at the outside of the column node. The washers smoothly transmit axial compressive or tensile force. The high competitiveness of the joining technologies is related to the high bending stiffness, easy assembly, machining reduction, and high speed of construction. As mentioned in [17], the BC joint system exhibits an adequate bending stiffness and bending capacity. The response of the connection embodies typical elasto-plastic behaviour under a bending moment. To obtain the whole moment-rotation curves of the joints, the experiment was performed on the joints with M24 (bolt diameter of 24 mm) and M27 (bolt diameter of 27 mm) high-strength bolts [17]. The test setup is shown in Fig. 2. The instrument arrangement and typology were considered to evaluate the main characteristics of the semi-rigid joints (stiffness, strength, rotation behaviour, and failure mode). The bottom plate of the specimen was employed to fix the specimen on the rigid frame; it is welded to the support beam to prevent slippage. The bending moment at the joint can be applied via the horizontal hydraulic jack. The connection rotation can be calculated based on the level
3. The numerical model of single-layer reticulated shells with semi-rigid joints Based on the moment-rotation curve of the bolt-column joints obtained from the experiment, the finite element models of single-layer reticulated shells with semi-rigid joint were built. 3.1. The member model The member model is the basic element in the semi-rigid jointed reticulated shells. The member model in this paper is established in the software ABAQUS as shown in Fig. 4. The member model in Fig. 4 consists of three main parts: joint Fig. 2. Picture of the test setup.
545
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
shown in Fig. 4. Because of the different mechanical performances of the different shells, the principle used to define the local coordinate systems is different. The x-axis of the local coordinate system is always oriented from Node 1 towards Node 2, as shown in Fig. 4, and the X-Y plane is defined by Nodes 1, 2 and O defined by the researcher according to the mechanical behaviour of the dome; for example, point O can be defined as the centre of the sphere for the single-layer spherical reticulated shells. According to the rule of right-handed coordinate system, the z-axis can be defined. According to the local right-handed coordinate system, the x-axis represents the torsional direction, whereas the y-axis and z-axis represent the bending directions.
Fig. 4. Member model and the local right-handed coordinate system.
3.2. Verification of the member model in a simple structure The FEA analysis of the latticed dome with semi-rigid joints begins with the verification of a simple semi-rigid structure, which is composed of two rectangle members, each of which has a section of 300 mm × 200 mm × 10 mm, as shown in Fig. 6. When the angle between the member and the horizontal changes from θ0 to θ, an exact relation of P-θ and Δ-θ can be given by the equation of equilibrium corresponding to the deformed configuration of the deformed simple structure as follows:
P= Fig. 5. Steel constitutive model for joint region and beam member.
Δ =
regions, connections and beam member. The beam and joint region in the model are both modelled using the fibre beam of B31 element in ABAQUS. B31 is a non-linear beam element that is suitable for analysing slender or thick beam structures. This element is based on Timoshenko beam theory. Shear deformation effects are included. The section of the joint region is normally greater than that of the beam because of its larger bending stiffness. The diameter of the joint region in the figure, D, represents the size of the centre part of semi-rigid joints. For example, in the bolt-column joints, D represents the outer diameter of the hollow cylinder; and in the bolt-ball joints, D represents the outer diameter of the ball node. Therefore, the length of the joint region could be defined according to the joint geometric parameters used in the structures, and the value may vary with different joint systems. The variable, L, represents the length of the beam member between two joints in single-layer reticulated structures. A two-linear kinematic hardening model was assumed for the stress-strain relationship for the joint region and beam members, as shown in Fig. 5. The Von Mises' yield criterion was adopted for all steel components along with the flow rule following yielding. At the connection point between the beam and the joint region, the translations in the nodal x, y and z directions are coupled. The connector element CONN3D2 in ABAQUS with nonlinear characteristics is used to simulate the bending and torsional stiffness of the semi-rigid joints by inputting real constants in the element. CONN3D2 is a multidirectional element with nonlinear generalized force-deflection capability that can be used in three-dimensional analysis. The connector elements have internal DOFs that do not exist at any node but are a part of the connector element itself. To distinguish the in-plane bending stiffness, out-plane bending stiffness and torsional stiffness, a local right-handed coordinate system according to every element must be assigned to the CONN3D2 element, as
2M cos θ0 ⎞ cos2 θ + 2EA ⎛1 − sin θ L/2 cos θ ⎠ ⎝ L (tg (θ0) − tg (θ)) 2
M = K × 2(θ0 − θ)
(1) (2) (3)
where L = 10 m is the span of the structure; θ (rad) is the angle between the member and the horizontal, and θ0 = 0.060 is its initial angle. The steel for the members and supporting frames was simplified into linear elastic materials. The Young's modulus is E = 206 GPa; A = 9.6 × 10−3 m2 is the area of the member section; M is the bending moment supplied by the CONN3D2 element; K is the bending-stiffness of the connection. The numerical analysis of this simple structure adopts the experimental moment-rotation curve of the bolt-column joint with M24 joints shown in Fig. 3. The P-Δ relationships obtained using the equation of equilibrium and the FEA analysis are shown in Fig. 7. The two P-Δ curves are extremely close to each other, which indicates that the actual bending stiffness of a bolt-column joint system can be accurately simulated using the connector element CONN3D2 in the analysis of the structure. 3.3. Verification for a single-layer cylindrical reticulated shell with semirigid joints Ma et al. [18–20] conducted a series of tests on the bolt-ball semirigid joint system and the single-layer cylindrical reticulated shell with semi-rigid joints. The bolt-ball semi-rigid joint system is shown in Fig. 8. The components of this system consist of a solid ball node, high strength bolts, sleeves, dowel pins, end-cones and tubular members. The end-cones with threaded holes are welded at both ends of the tubes. The bolts are screwed into the solid ball node from the end cones. The tensile force is transferred to the spherical ball through the interface between the end cone and bolt; the compressive force is transmitted to the spherical ball through the interface between the end cone and Fig. 6. Schematic diagram of the two-member structure. (a) Two-member model before deformation. (b) Two-member model after deformation.
546
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 9. Moment-rotation curve of the M24 bolt-ball joint. Fig. 7. Comparison of the load-displacement curves of the intersection point.
stress-displacement curves at member f3 near node 16 is shown in Fig. 14. The two figures reveal that the FEA results are close to the experimental results before the structure loses its stability.
sleeve. Their experimental studies obtained the bending-rotation relationships of the joint, as shown in Fig. 9. In this paper, based on the moment-rotation curve of the M24 bolt-ball joint under bending moment, the FEA analysis of the single-layer cylindrical reticulated shell with bolt-ball joints is conducted. The FEA analysis calculates the full loaddeflection response of the dome in consideration of both material and geometric nonlinearities, which provides a relatively perfect picture regarding the nonlinear behaviour. The experimental picture and illustrative diagram of the single-layer latticed shell are shown in Fig. 10. The 5 m × 6 m single-layer cylindrical reticulated shell with a rise of 1.25 m is vertically supported on the eighteen outer nodes. The tubular members of 76 mm in diameter with a wall thickness of 3.5 mm (which are nominal values for the diameter and thickness) are used in the structure. Based on the proposed simulation method in Section 3.1, the FE member model (shown in Fig. 4) was used to establish a model of the cylindrical reticulated shell with bolt-ball joints, as shown in Fig. 11. All the geometric details and the material properties used in the FE model of the structure are the same as those in the experimental structure. The FE analysis takes both material and geometric nonlinearities into account. The average yield strength of the steel for the members and joint region was fy = 259 MPa, which was obtained from tension coupon tests. The Young's modulus is E = 206 GPa. The experimental and FE-based load-deflection response curves of the test structure are compared in Fig. 12. The results show that the two curves agree reasonably well, indicating that the developed FE model provides a good estimation of the experimentally observed behaviour. The critical load calculated using the FE model has a relative error of 2.23% compared with the experimental result. The load-displacement curves of nodes 13 and 20 obtained through the FEA and the test are shown in Fig. 13, and a comparison of the
4. Investigation on the natural vibration frequency of single-layer reticulated shells with bolt-column joints The FEA model of the single-layer reticulated shells was established based on the member model and the bending-rotation curves of the bolt-column joints obtained by the experiments shown in Fig. 3, as shown in Fig. 15. The reticulated shell with a span of 50 m and a rise of 10 m is vertically hinged supported on the outer nodes. The rectangular steel tube is used in the structure; the section of the radial and circumferential members, which are identified with red and blue lines in the figure, is 120 mm × 60 mm × 3.2 mm, and the section of the diagonal members, which are identified with black lines, are 120 mm × 60 mm × 2 mm; the size of the joint region used in the numerical analysis is the same as that of the experimental hollow cylinder, whose diameter is 200 mm; a roof load of 60 kg/m2 is used during the analysis. The steel for the members is grade Q235 with a nominal yielding stress fy of 235 MPa. All of the 400th order natural frequencies of the structures are shown in Fig. 16(a); the first 20th natural frequencies and 165th −185th natural frequencies are enlarged to be clearer, as shown in Fig. 16(b) and (c), respectively. It can be seen from the figures that the natural frequencies increase as the bending stiffness of the joints increases, and the difference increases as the mode order increases. The natural frequencies of the single-layer reticulated shells with M24 bolt-column joints, M27 boltcolumn joints and rigid joints can be divided into four stages: a) Basic frequency range: the first 1st and 2nd mode order; b) Low frequency range: 3rd–169th mode order; c) Skip range: 170th–175th mode order; Fig. 8. Bolt-ball joint system. (a) Picture of the joint specimen (a) Joint in section.
547
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 10. Illustrative diagram of the single-layer cylindrical reticulated dome. (a) Picture of the test (b) Layout of the test.
Fig. 11. FEA model of the single-layer cylindrical reticulated dome.
Fig. 13. Comparison of the load-displacement curves at nodes 13 and 20.
Fig. 12. Comparison of the load–displacement curves at loading node16.
d) High frequency range: 176th–400th mode order. There are two obvious skips that occur in the ‘critical sections’ in the natural frequencies, as indicated in Fig. 16(b) and (c). To investigate the causes of the skips, the graphs of the mode of vibration near the critical sections are shown in Fig. 17. From the figure, the 2nd-order vibration is the antisymmetric horizontal vibration of the whole structures; the 3rd-order vibration is the symmetrical vertical vibration of the whole structures; the 169th vibration is the local vibration in the structures; the 170th and 175th vibrations are the in-plane rotational vibration of the structures; 176th vibration is the member vibration in the structures. Therefore, it can be concluded that the skips in the natural frequencies are due to the sudden change of the mode of vibration.
Fig. 14. Comparison of the stress-displacement curves at member 13.
5. Effects of bending stiffness on the base frequency of the singlelayer reticulated shells A parametric analysis on the natural vibration of single-layer spherical reticulated shells (shown in Fig. 15) with BC joints is performed. The effects of bending stiffness on the base frequency of the single-layer reticulated shells are considered in detail. The parameters of single-layer reticulated shells, such as span, ratio of rise to span, roof load, geometric imperfections, are considered. Both geometric and 548
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
α=
kα k0
(4)
where k 0 is the initial bending stiffness of M24 BC joint; k α is the initial bending stiffness of the joints with the initial bending stiffness factor α.
• Different span and ratio of rise to span Considering the practical engineering situation, three spans and six rise-to-span ratios presented bellow were taken accounted into analysis: Span (L): 30 m, 40 m, and 50 m; Ratio of rise to span (f/L): 1/3, 1/4, 1/5, 1/6, 1/7, and1/8.
Fig. 15. The single-layer spherical reticulated shells. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
To assess the overall behaviour of a single-layer spherical reticulated shell with different ratios of rise to span, the pattern division number of a shell should allow for almost the same length (between 3 and 5 m) of members with different span and ratio of rise to span. Therefore, the number of rings for the shells with 30 m, 40 m and 50 m spans are 5, 6 and 7, respectively.
material nonlinearities are taken into consideration to determine the overall behaviour of the shells.
5.1. Scheme for parametric analysis
• Different bending stiffness
• Different roof loads
To fully investigate the influence of bending stiffness of joints on the natural vibration, 11 types of bending stiffness determined by the initial bending stiffness factor α, as shown in Fig. 18, are adopted based on the bending stiffness of M24 BC joint. In the figure, Mu is the plastic moment resistance of M24 BC joint; the initial bending stiffness factor, α, is defined as:
•
Different roof loads of 0.6 kN/m2, 1.2 kN/m2, and 1.8 kN/m2 are added in the models during the analysis; the load is distributed uniformly over the shell. Different geometric imperfections The base frequencies of single-layer shells with different initial imperfections are obtained through eigenvalue buckling analysis. The first eigenvalue buckling mode is used to simulate the distribution of the initial imperfection. A 40-m span dome with five initial
Fig. 16. Natural frequencies of single-layer spherical reticulated shells. (a) 400th order natural frequencies (b) The first 20th natural frequencies (c) 165th −185th natural frequencies.
549
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 17. Comparison of the typical modes of single-layer spherical reticulated shells with different joints. (a) Structures with M24 BC joints (b) Structures with M27 BC joints (c) Structures with rigid joints.
•
imperfection values, namely 40 mm (L/1000), 45 mm (L/900), 53 mm (L/750), 67 mm (L/600), 89 mm (L/450), and 133 mm (L/ 300), are analysed to investigate the variation trend of the base frequencies. Section of members
Each model contains two different members, as can be identified from Fig. 14, where the blue and red lines indicate the larger section 120 mm × 60 mm × 3.2 mm and the fine line the smaller section 120 mm × 60 mm × 2 mm;
• Diameter of joint region: 200 mm; • Supporting condition: the shell is supported by pins on the peripheral nodes.
Fig. 18. Moment-rotation curves defined by the initial stiffness factor α.
550
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 19. Base frequency of the shells with different initial bending stiffness coefficients. (a) With effects of different spans (b) With effects of different ratios of rise to span (c) With effects of roof loads.
5.2. Results and discussion of the parametric analysis Fig. 19 presents the curves of the base frequency of the semi-rigidly jointed spherical reticulated shells with different span, ratio of rise to span and roof load. The basic frequencies of the shells decrease with increasing span and roof load and increase with the increasing f/L ratio. To investigate the change law of the base frequency of shells with different bending stiffness, Fig. 20 presents the curves of the base frequency coefficient λα versus bending stiffness factor α of the joints. The base frequency coefficient λα in the figure is defined as:
λα =
Fig. 21. First-order mode of single-layer reticulated shells with semi-rigid joints.
fα fR
imperfections. For the single-layer reticulated shells with semi-rigid joints, the base frequency of the shells decreases obviously with the increasing initial geometric imperfection; when the ratio of rise to span f/L ≤ 1/5, the influence of initial geometric imperfection on base frequency gradually reduce with the increasing bending stiffness of the joints; the influence of initial geometric imperfection on base frequency is very small when the ratio of rise to span f/L = 1/3.
(5)
where fα is the natural frequency of the single-layer reticulated shells with the joints whose initial bending stiffness factor is α; fR is the natural frequency of the single-layer reticulated shells with rigid joints. The figures show that the base frequency coefficient of the singlelayer reticulated shells with different parameters increases when the bending stiffness of the joints in the structure increases, whereas the differences in base frequency due to the bending stiffness of the joints are small; the base frequency coefficient of the single-layer reticulated shells with semi-rigid joints slightly increases when the span and ratio of rise to span increases. The similar conclusion also was found by Fan et al. [22] during analysing the single-layer reticulated shells with boltball joints; and Chen et al. [23] during analysing the suspend-dome with semi-rigid joints. This indicates that for the first mode, axial deformation is still the dominant deformation. The first-order mode of single-layer reticulated shells with semirigid joints is horizontally anti-symmetric mode shape, as shown in Fig. 21. Fig. 22 presents the base frequencies versus the bending stiffness of the joints for the single-layer reticulated shells with different geometric
6. Investigation on the seismic response of single-layer reticulated shells with semi-rigid joints The time history analysis method based on the ABAQUS software is adopted to study seismic behaviour of single-layer spherical reticulated shells with semi-rigid joints under frequent earthquake load, including the El-Centro wave, the Taft wave, and another artificial earthquake waves. The peak accelerations of these three earthquake waves were adjusted to 70 cm/s2. Earthquake waves were imposed on the shells in three dimensions on the assumption that the ground motions on all supports of the structure are the same. The interval in the earthquake waves was set as 0.02 s, and duration of each of the earthquake waves was 20 s, according to Fan et al. [21].
Fig. 20. Base frequency coefficient of the shells with different initial bending stiffness coefficient. (a) With effects of different spans (b) With effects of different ratios of rise to span (c) With effects of roof loads.
551
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 22. Effect of bending stiffness of the joints on the base frequencies of the single-layer reticulated shells with different geometric imperfections. (a) f/L = 1/7 (b) f/L = 1/5 (c) f/L = 1/3.
class members due to action of earthquakes and dead load can be obtained by the maximal internal force (NDI max ) in the corresponding class members due to the action of the dead load multiplied by the relevant coefficient (ξI + 1). The relevant formula can be expressed as follows:
6.1. Definition of the seismic internal force coefficient For a latticed shell with many members, each steel member has a seismic internal force coefficient. To simplify the designing procedure, it is necessary to statistically summarize the seismic internal force coefficient of the shells. Therefore, according to the location of members in structure, members can be classified as circumferential members, radial members and diagonal members, as shown in Fig. 14. The seismic internal force coefficient of the three class members can be separately summarized. The statistical method based on the envelope of the maximal seismic response in [22] is used to obtain the seismic internal force coefficient for different members. The seismic internal force coefficient is defined as follows:
NEI max ξI = −1 NDI max
NEIDesign = NDI max × (ξI + 1)
where NEIDesign is the design axial force of the radial members (NERDesign), circumferential members (NECDesign) and diagonal members (NEDDesign) obtained using the method described in this paper. 6.2. Illustration with an example To verify the effectiveness and applicability of Eqs. (6) and (8) above, analyses of a single-layer reticulated shell with semi-rigid joints and the corresponding single-layer reticulated shell with rigid joints are conducted. The parameters for analysis are as follows:
(6)
where ξI is the seismic internal force coefficient for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) members; NEI max is the maximum axial force of one class members under the action of the earthquake waves and the dead load of single-layer reticulated shells with semi-rigid joints; NDI max is the maximum axial force of one class members under the dead load of the corresponding singlelayer reticulated shells with rigid joints. Similarly, seismic internal force coefficient for each member in every group can be defined by Eq. (7):
ξIi =
NEIi −1 NDIi
(8)
Joint systems: M24 BC joints and rigid joints; Span (L): 40 m; Section dimensions: 140mm × 100mm × 5mm Ratio of rise to span: 1/7; Roof Load: 1.8 kN/m2; Supporting condition: the shell is pin-supported on the peripheral nodes. After the analysis, based on Eq. (6), the seismic internal force coefficient for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) are shown in Fig. 23. The design axial forces of the radial members (NERDesign), circumferential members (NECDesign) and diagonal members (NEDDesign) obtained by Eq. (7) are shown in Fig. 23. In the figures, NEIi and ξIi are the actual seismic axial force and actual seismic internal force coefficient of each member. Fig. 24 reveals that the design seismic axial forces NERDesign, NECDesign and NEDDesign are the envelope value for each class members that can satisfy requirement
(7)
where ξIi is the seismic internal force coefficient for every member in 3 classified group. NEIi is the axial force of one member under the action of the earthquake loads and the dead load of single-layer reticulated shells with semi-rigid joints; NDIi is the axial force of the same member under the dead load. When structures are designed, the maximal internal force of one
Fig. 23. Seismic internal force coefficient of each class members. (a) Radial members (b) Diagonal members (c) Circumferential members.
552
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 24. Design axial forces of the members. (a) Radial members (b) Diagonal members (c) Circumferential members.
of the design.
ξ ′R = 0.475. The influence of bending moment on the seismic internal force coefficient for the three classes of member is little. Secondly, in the “Technical specification for latticed shells [24]”, the seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of single-layer reticulated shells are based on the Eq. (6) in the paper; which means the influence of the bending moments was not considered. To be consistent in the definition of the seismic internal force coefficient, Eq. (6) was also used to calculated the seismic internal force coefficients of singlelayer reticulated shells in this paper.
6.3. Influence of the bending moments on the seismic internal force coefficient Normally, in a single-layer reticulated shell, bending moments may be of relevance in calculating the seismic internal force coefficient. To investigate the Influence of the bending moments, the seismic internal force coefficient (ξ ′Ii ) for each member considering the influence of the bending moments were obtained. ξ ′Ii is calculated as below:
ξ ′Ii =
N M σEIi + σEIi −1 N M σDIi + σDIi
6.4. Seismic internal force coefficient of single-layer reticulated shells with semi-rigid joints
(9)
N where, σEIi is the axial stress of one member in single-layer reticulated shells with semi-rigid joints under the action of the earthquake loads M and the dead load; σEIi is the bending stress of one member in singlelayer reticulated shells with semi-rigid joints under the action of the N earthquake loads and the dead load; σDIi is the axial stress of the same M is the bending stress member in the structure under the dead load; σDIi of the same member in the structure under the dead load. The parameters for the reticulated shells are the same with Section 6.2 in the revised paper. The Comparison of seismic internal coefficient of each member with and without the influence of bending moment as shown in Fig. 25. It can be seen from the figure that the bending moment does have some influence on seismic internal force coefficient for each member. Based on the data in the figure, the seismic internal force coefficient for the radial members with and without the influence of bending moment are ξR = 0.212 and ξ ′R = 0.207 ; the seismic internal force coefficient for the circumferential members with and without the influence of bending moment are ξC = 0.700 and ξ ′C = 0.685; and the seismic internal force coefficient for the diagonal members with and without the influence of bending moment are ξD = 0.483 and
Parameter analysis of the seismic internal force coefficient is investigated to determine the change rule of seismic internal force coefficient with influence factors. Hence, the seismic internal force coefficient of single-layer reticulated shells with semi-rigid joints can be derived based on a statistical method. The parameters considered in the analysis are listed in Table 1. The change law of the seismic internal force coefficient of the radial members, diagonal members and circumferential members of the semirigidly jointed single-layer spherical reticulated shells with different parameters are shown in Figs. 26–28. it can be seen from the figures that the parameters, such as roof load, member section, span, ratio of rise to span and earthquake wave, have a significant effect on seismic internal force coefficient of the members. In general, the seismic internal force coefficients of the radial members and the diagonal members increase with bending stiffness increases, whereas the seismic internal force coefficients of the circumferential members first increase and then decrease with the increase of the initial bending stiffness ratio. Based on the 486 finite element models, the statistical values of seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of the shells with semi-rigid joints are obtained. During the analysis, the parameters of bending stiffness ratio, roof load, sectional dimension, span and ratio Table 1 Parameters of the shells for the parametric analysis. Parameters of the shells
Values
Span Bending stiffness coefficient Sectional dimensions
30 m, 40 m 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0 140 mm × 100 mm × 5 mm (Section 1) 150 mm × 100 mm × 5 mm (Section 2) 0.6 kN/m2, 1.2 kN/m2, 1.8 kN/m2 1/3, 1/5, 1/7 El-Centro, Taft, Artificial wave 200 mm pinned supported on the peripheral nodes
Roof load (RL) Ratio of rise to span Earthquake waves Diameter of the joint region Supporting condition
Fig. 25. Comparison of seismic internal coefficient of the members with and without the influence of bending moment.
553
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 26. Seismic internal force coefficient of the radial members for different parameters. (a) With different roof loads (b) With different member sections (c) With different f/L ratios (d) With different spans (e) With different earthquake waves.
Fig. 27. Seismic internal force coefficient of the diagonal members for different parameters. (a) With different roof loads (b) With different member sections (c) With different f/L ratios (d) With different spans (e) With different earthquake waves.
554
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 28. Seismic internal force coefficient of the circumferential members for different parameters. (a) With different roof loads (b) With different member sections (c) With different f/L ratios (d) With different spans (e) With different earthquake waves. Table 2 Seismic internal force coefficients of single-layer spherical reticulated shells with semi-rigid joints. Coefficients
ξR
ξD
ξC
f/L
1/3 1/5 1/7 1/3 1/5 1/7 1/3 1/5 1/7
Bending stiffness coefficients 0.05
0.1
0.2
0.4
0.6
0.8
1.0
2.0
Rigid [24]
0.417 0.537 0.434 1.214 1.008 0.748 0.984 0.877 0.681
0.407 0.484 0.391 1.192 0.946 0.638 0.976 0.900 0.679
0.403 0.438 0.368 1.164 0.842 0.514 0.973 0.905 0.668
0.404 0.411 0.340 1.145 0.790 0.515 0.958 0.911 0.649
0.383 0.383 0.326 1.144 0.766 0.516 0.936 0.955 0.634
0.361 0.373 0.325 1.135 0.739 0.506 0.913 0.825 0.636
0.332 0.355 0.323 1.128 0.722 0.501 0.903 0.759 0.633
0.331 0.327 0.323 1.098 0.704 0.501 0.913 0.668 0.624
0.320 0.320 0.320 0.640 0.560 0.500 0.760 0.640 0.580
7. Conclusions
of rise to span (as shown in Table 1) were considered. According to the statistical probability method, the seismic internal force coefficients according to different bending stiffness ratios are obtained with 95% insurance rate, as shown in Table 2. To make the comparison clearer, the ratio of the seismic internal force coefficients of single-layer reticulated shells with semi-rigid joints to that of single-layer reticulated shells with rigid joints are also presented in Fig. 29. The symbols ξRR , ξDR , and ξCR in the figure represent the seismic internal force coefficients for the radial members (ξRR ), diagonal members (ξDR ) and circumferential members (ξCR ) of single-layer reticulated shells with rigid joints. it can be seen from the comparison that: with the increasing of the joint bending stiffness, almost all the seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of single-layer reticulated shells with semi-rigid joints decrease, and the values are close to the values of the members in single-layer reticulated shells with rigid joints. especially in the singlelayer reticulated shells with the f/L = 1/7, the seismic internal force coefficients of circumferential members initially decrease until bending stiffness coefficient reaches 0.6 and then the seismic internal force coefficients of the three classed of members remain nearly stable.
The FE model of single-layer reticulated shells with semi-rigid joints established in ABAQUS with the beam element (B31) and the connector elements (CONN3D2) was validated through comparison with the experimental result. Based on the experimental moment-rotation curves of the M24 and M27 bolt-column joints, the natural vibration frequencies of singlelayer reticulated shells with semi-rigid joints were investigated in detail. The following phenomena were observed: i) the natural frequencies increase with the increasing of bending stiffness of the joints, and the difference increases as the modes order increases; ii) the natural frequencies of the single-layer spherical reticulated shells with boltcolumn joints can be divided into four stages: basic frequency range, low frequency range, skip range, and high frequency range; iii) the basic frequencies of semi-rigidly jointed single-layer reticulated shells decrease with increasing span and roof load and with decreasing f/L ratio, whereas, according to the base frequency coefficient λα (0.001 ≤ α ≤ 10.0), the effect of bending stiffness on the basic frequencies of shells without initial geometrical imperfections is small; iv) when the ratio of rise to span is f/L ≤ 1/5, the base frequency of shells decreases 555
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 29. Comparison of the seismic internal force coefficients of the three classes of members. (a) Radial members (b) Diagonal members (c) Circumferential Members.
References
obviously with the increasing initial geometric imperfection; the basic frequencies of shells with a certain initial geometric imperfection increases largely when the bending stiffness of joints increases, whereas when the ratio of rise to span f/L = 1/3 the influence of initial geometric imperfection on base frequency is very small; and the basic frequencies of shells with a certain initial geometric imperfection changes little with the increasing bending stiffness. The seismic internal force coefficient was defined for the singlelayer spherical reticulated shells with semi-rigid joints based on the statistical method. The affecting laws of rigidity of joints, span, ratio of rise to span, tube sections and roof load on the seismic internal force coefficients of the single-layer spherical reticulated shells with semirigid joints were obtained. Finally, under a common earthquake, the seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of the singlelayer latticed shell with different stiffness were derived for seismic design. The internal force coefficients in Table 2 are suitable for the single-layer spherical reticulated shells (as shown in Fig. 15) with semirigid joints under symmetrical load condition. More further studies should be carried out to extend the method to other single-layer reticulated shells, such as single-layer cylindrical reticulated shell, singlelayer elliptical paraboloid latticed shells, single-layer ribbed latticed shells, etc. The effect of the unsymmetrical load condition on the dynamic behaviour and internal force coefficients should also be investigated.
[1] Zhi-Wei Yu, Xu-Dong Zhi, Feng Fan, Chen Lu, Effect of substructures upon failure behavior of steel reticulated domes subjected to the severe earthquake, Thin-Walled Struct. 49 (2011) 1160–1170. [2] H. Dai, Z. Liu, W. Wang, Structural passive control on electromagnetic friction energy dissipation device, Thin-Walled Struct. 58 (2012) 1–8. [3] Xudong Zhi, Feng Fan, Shizhao Shen, Elasto-plastic instability of single-layer reticulated shells under dynamic actions, Thin-Walled Struct. 48 (2010) 837–845. [4] F.A. Fathelbab, The Effect of Joints on the Stability of Shallow Single Layer Lattice Domes, University of Cambridge, 1987. [5] H. Dai, Z. Zheng, W. Wang, A new fractional wavelet transform, Commun. Nonlinear Sci. Numer. Simul. 44 (2017) 19–36. [6] Reza Chenaghlou Mohammad, Semi-Rigidity of Connections in Space Structures, University of Surrey, 1997 Ph.D. Thesis. [7] Y. Hiyama, H. Takashima, T. Iijima, S. Kato, Buckling behaviour of aluminium ball jointed single layered reticular domes, Int. J. Space Struct. 15 (2) (2000) 81–94. [8] H. Dai, Z. Cao, A wavelet support vector machine-ased neural network metamodel for structural reliability assessment, Comput. Aided Civ. Infrastruct. Eng. 32 (4) (2017) 344–357. [9] H.H. Ma, S. Ren, F. Fan, Parametric study and analytical characterization of the bolt–column (BC) joint for single-layer reticulated structures, Eng. Struct. 123 (2016) 108–123. [10] T. See, Large Displacement Elastic Buckling Space Structures (Ph.D. Thesis), University of Cambridge, 1983. [11] L.ópez Aitziber, Puente Iñigo, Miguel A. Serna, Direct evaluation of the buckling loads of semi-rigidly jointed single-layer latticed domes under symmetric loading, Eng. Struct. 29 (2007) 101–109. [12] L.ópez Aitziber, Puente Iňigo, Miguel A. Serna, Numerical model and experimental tests on single-layer latticed domes with semi-rigid joints, Comput. Struct. 85 (2007) 360–374. [13] H.H. Ma, F. Fan, S.Z. Shen, Numerical parametric investigation of single-layer latticed domes with semi-rigid joints, J. Int Assoc. Shell Spat. Struct. 49 (2) (2008) 99–110. [14] H.H. Ma, F. Fan, J. Zhong, Stability analysis of single-layer elliptical paraboloid latticed shells with semi-rigid joints, Thin-Walled Struct. 72 (10) (2013) 128–138. [15] F. Fan, H.H. Ma, Z.G. Cao, S.Z. Shen, A new classification system for the joints used in lattice shells, Thin Walled Struct. 49 (12) (2011) 1544–1553. [16] S. Kato, I. Mutoh, M. Shomura, Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections, J. Constr. Steel Res. 48 (1998) 145–168. [17] H.H. Ma, S. Ren, F. Fan, Experimental and numerical research on a new semi-rigid
Acknowledgements This research is supported by grant of The Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration (Grant no. 2017D06), and The National Science Fund for Distinguished Young Scholars (Grant No. 51525802). 556
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
[21] F. Fan, S.Z. Shen, Study on the Dynamic Strength Failure of Reticulated Domes under Severe Earthquakes, International Association for Shell and Spatial Structures Symposium, Montpellier, France, 2004, pp. 140–141. [22] F. Fan, M.L. Wang, Z.G. Cao, Seismic behaviour and seismic design of dingle-layer reticulated shells with semi-rigid joint system, Adv. Struct. Eng. 15 (10) (2012) 1829–1841. [23] Z. Chen, H. Xu, Z. Zhao, Investigations on the mechanical behavior of suspenddome with semi-rigid joints, J. Constr. Steel Res. 122 (1) (2016) 14–24. [24] Ministry of Urban and Rural Construction of China, Technical Specification For Latticed Shells, JGJ 61-2003 (in Chinese), 2003.
joint for single-layer reticulated structures, Eng. Struct. 126 (11) (2016) 725–738. [18] F. Fan, H.H. Ma, G.B. Chen, S.Z. Shen, Experimental study of semi-rigid joint systems subjected to bending with and without axial force, J. Constr. Steel Struct. 68 (1) (2012) 126–137. [19] H.H. Ma, F. Fan, G.B. Chen, Z.G. Cao, S.Z. Shen, Numerical analyses of semi-rigid joints subjected to bending with and without axial force, J. Constr. Steel Res. 90 (2013) 13–28. [20] H.H. Ma, F. Fan, P. Wen, Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints, Thin-Walled Struct. 86 (2015) 1–9.
557
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Dynamic behaviour and seismic design method of a single-layer reticulated shell with semi-rigid joints
MARK
⁎
Huihuan Maa, , Zhiwei Shanb, Feng Fana a b
School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China School of Civil Engineering, The University of Hong Kong, Hong Kong, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Dynamic behaviour Seismic design Single-layer reticulated shell Semi-rigid joint Free vibration frequency Seismic internal force coefficient
Most of the existing studies on reticulated shells with a semi-rigid joint system have been focused on the mechanical properties under static loads. Taking material and geometric nonlinearities into account, the finite element analysis (FEA) model of a single-layer reticulated shell with semi-rigid joints was established using the software ABAQUS and then validated through comparison with the experimental result. Based on the bending stiffness of a bolt-column (BC) joint obtained through experiments, the dynamic behaviour and a seismic design method for single-layer reticulated shells with semi-rigid joints were investigated in this paper. First, analysis of the free vibration frequency of the single-layer latticed domes with semi-rigid bolt-column (BC) joints was conducted based on several different parameters, including joint stiffness, ratio of rise to span, initial geometric imperfection. Second, the seismic internal force coefficient of the members of the semi-rigidly jointed spherical single-layer reticulated shells of different parameters was studied in detail. Finally, the seismic internal force coefficients for spherical single-layer reticulated shells with semi-rigid joints under a common earthquake were derived.
1. Introduction Single-layer reticulated shells are well known to tend to be unstable under dynamic working actions. With large-span space structures, such as exhibition halls, stadiums and railway stations, being increasingly used in public facilities in recent years, reticulated shells have attracted more attention. The seismic performance of a reticulated dome not only is directly related to the safety of a building but also has a huge impact on the socio-economic development and the security of people. Therefore, researchers around the world have made great efforts to address the issues, such as the dynamic response and failure mechanism of these structures subjected to dynamic loads. However, most of the studies on the dynamic behaviour of a single-layer reticulated shell assume that the connections behave as perfect rigid joints, such as the dynamic performances and failure characteristics of single-layer reticulated domes subjected to an earthquake discussed in [1] as well as the study performed on the seismic performance of a single-layer reticulated dome [2,3]. However, the joints in most space structures are semi-rigid, for example the MERO joint systems [4–6], the aluminium alloy truss connector [7], the socket joint system [8] and the bolt-column joint system [9]. Single-layer reticulated shells with the above-described semi-rigid
⁎
Corresponding author. E-mail addresses: [email protected] (H. Ma), [email protected] (Z. Shan).
http://dx.doi.org/10.1016/j.tws.2017.07.003 Received 3 January 2017; Received in revised form 6 June 2017; Accepted 5 July 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
joints can provide a good solution for small- and medium-span space structures because of their rapid construction speed, high fabrication accuracy and beautiful appearance. Therefore, interest in single-layer spatial structures has grown significantly in the recent years. Observations from an earlier study [4,10] confirmed that connection stiffness had a significant effect on the load-displacement behaviour of a single-layer spatial structure. Aitziber et al. [11,12], Ma et al. [13,14], Fan et al. [15] and Kato et al. [16] verified that the rigidity of joints is an important factor that influences the behaviour of a single-layer latticed shell. However, those research studies primarily focus on the study of the mechanical properties of single-layer latticed shells with semi-rigid joint under static loads. Few investigations on the dynamic behaviour of a single-layer latticed shell with a semi-rigid joint have been conducted. Hence, it is worthwhile to explore dynamic mechanical performance of a singlelayer latticed shell with a semi-rigid joint. In this paper, finite element models of single-layer reticulated shells with the semi-rigid bolt-column joints are established. The rules of free vibration frequency are explored, considering the joint stiffness, ratio of rise to span, section of members, span and initial geometric imperfection. Under a common earthquake, seismic internal force coefficients of the single-layer latticed shell with different stiffness are derived for
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 1. Bolt-column joint.
seismic design.
Fig. 3. Moment -rotation curves of the bolt-column joints [17].
displacements at points P1-P3, which are obtained by the LVDTs. The moment-rotation curves of the bolt-column joints obtained from the experiments are plotted in Fig. 3. In the figure, k0 is the initial bending stiffness of the joints; ku is the post-limit stiffness of the joints, which is defined as 10% of the initial stiffness, k0; Mu is the plastic moment resistance of the joints, which corresponds to the point A of the regression line obtained for the post-limit (ku) stiffness. The bolt-column joints exhibit an adequate bending stiffness and bending capacity. The response of the connection exhibits linear behaviour in the early loading sequence and subsequent nonlinearity, which embodies typical elasto-plastic behaviour. The results indicate that the initial bending stiffness k0 of M27 BC joints is much higher than the initial bending stiffness of M24 BC joints. The plastic moment resistance Mu also improved when the diameters of the high-strength bolts increase. Therefore, the effect of the diameters of the bolts on the mechanical behaviour of the BC joint is significant.
2. The semi-rigid bolt-column (BC) joints The bolt-column joint is composed of a hollow cylinder, highstrength 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, which are welded at both ends of the members in the factory. At the construction site, the two high-strength bolts are used to connect the members to the column node without any welding work. All holes in the hollow column are tapped to accommodate the threaded part of the bolts. The two highstrength bolts are screwed into the hollow column node from the endcone part. One concave washer is employed at each bolted connection; they are placed at the outside of the column node. The washers smoothly transmit axial compressive or tensile force. The high competitiveness of the joining technologies is related to the high bending stiffness, easy assembly, machining reduction, and high speed of construction. As mentioned in [17], the BC joint system exhibits an adequate bending stiffness and bending capacity. The response of the connection embodies typical elasto-plastic behaviour under a bending moment. To obtain the whole moment-rotation curves of the joints, the experiment was performed on the joints with M24 (bolt diameter of 24 mm) and M27 (bolt diameter of 27 mm) high-strength bolts [17]. The test setup is shown in Fig. 2. The instrument arrangement and typology were considered to evaluate the main characteristics of the semi-rigid joints (stiffness, strength, rotation behaviour, and failure mode). The bottom plate of the specimen was employed to fix the specimen on the rigid frame; it is welded to the support beam to prevent slippage. The bending moment at the joint can be applied via the horizontal hydraulic jack. The connection rotation can be calculated based on the level
3. The numerical model of single-layer reticulated shells with semi-rigid joints Based on the moment-rotation curve of the bolt-column joints obtained from the experiment, the finite element models of single-layer reticulated shells with semi-rigid joint were built. 3.1. The member model The member model is the basic element in the semi-rigid jointed reticulated shells. The member model in this paper is established in the software ABAQUS as shown in Fig. 4. The member model in Fig. 4 consists of three main parts: joint Fig. 2. Picture of the test setup.
545
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
shown in Fig. 4. Because of the different mechanical performances of the different shells, the principle used to define the local coordinate systems is different. The x-axis of the local coordinate system is always oriented from Node 1 towards Node 2, as shown in Fig. 4, and the X-Y plane is defined by Nodes 1, 2 and O defined by the researcher according to the mechanical behaviour of the dome; for example, point O can be defined as the centre of the sphere for the single-layer spherical reticulated shells. According to the rule of right-handed coordinate system, the z-axis can be defined. According to the local right-handed coordinate system, the x-axis represents the torsional direction, whereas the y-axis and z-axis represent the bending directions.
Fig. 4. Member model and the local right-handed coordinate system.
3.2. Verification of the member model in a simple structure The FEA analysis of the latticed dome with semi-rigid joints begins with the verification of a simple semi-rigid structure, which is composed of two rectangle members, each of which has a section of 300 mm × 200 mm × 10 mm, as shown in Fig. 6. When the angle between the member and the horizontal changes from θ0 to θ, an exact relation of P-θ and Δ-θ can be given by the equation of equilibrium corresponding to the deformed configuration of the deformed simple structure as follows:
P= Fig. 5. Steel constitutive model for joint region and beam member.
Δ =
regions, connections and beam member. The beam and joint region in the model are both modelled using the fibre beam of B31 element in ABAQUS. B31 is a non-linear beam element that is suitable for analysing slender or thick beam structures. This element is based on Timoshenko beam theory. Shear deformation effects are included. The section of the joint region is normally greater than that of the beam because of its larger bending stiffness. The diameter of the joint region in the figure, D, represents the size of the centre part of semi-rigid joints. For example, in the bolt-column joints, D represents the outer diameter of the hollow cylinder; and in the bolt-ball joints, D represents the outer diameter of the ball node. Therefore, the length of the joint region could be defined according to the joint geometric parameters used in the structures, and the value may vary with different joint systems. The variable, L, represents the length of the beam member between two joints in single-layer reticulated structures. A two-linear kinematic hardening model was assumed for the stress-strain relationship for the joint region and beam members, as shown in Fig. 5. The Von Mises' yield criterion was adopted for all steel components along with the flow rule following yielding. At the connection point between the beam and the joint region, the translations in the nodal x, y and z directions are coupled. The connector element CONN3D2 in ABAQUS with nonlinear characteristics is used to simulate the bending and torsional stiffness of the semi-rigid joints by inputting real constants in the element. CONN3D2 is a multidirectional element with nonlinear generalized force-deflection capability that can be used in three-dimensional analysis. The connector elements have internal DOFs that do not exist at any node but are a part of the connector element itself. To distinguish the in-plane bending stiffness, out-plane bending stiffness and torsional stiffness, a local right-handed coordinate system according to every element must be assigned to the CONN3D2 element, as
2M cos θ0 ⎞ cos2 θ + 2EA ⎛1 − sin θ L/2 cos θ ⎠ ⎝ L (tg (θ0) − tg (θ)) 2
M = K × 2(θ0 − θ)
(1) (2) (3)
where L = 10 m is the span of the structure; θ (rad) is the angle between the member and the horizontal, and θ0 = 0.060 is its initial angle. The steel for the members and supporting frames was simplified into linear elastic materials. The Young's modulus is E = 206 GPa; A = 9.6 × 10−3 m2 is the area of the member section; M is the bending moment supplied by the CONN3D2 element; K is the bending-stiffness of the connection. The numerical analysis of this simple structure adopts the experimental moment-rotation curve of the bolt-column joint with M24 joints shown in Fig. 3. The P-Δ relationships obtained using the equation of equilibrium and the FEA analysis are shown in Fig. 7. The two P-Δ curves are extremely close to each other, which indicates that the actual bending stiffness of a bolt-column joint system can be accurately simulated using the connector element CONN3D2 in the analysis of the structure. 3.3. Verification for a single-layer cylindrical reticulated shell with semirigid joints Ma et al. [18–20] conducted a series of tests on the bolt-ball semirigid joint system and the single-layer cylindrical reticulated shell with semi-rigid joints. The bolt-ball semi-rigid joint system is shown in Fig. 8. The components of this system consist of a solid ball node, high strength bolts, sleeves, dowel pins, end-cones and tubular members. The end-cones with threaded holes are welded at both ends of the tubes. The bolts are screwed into the solid ball node from the end cones. The tensile force is transferred to the spherical ball through the interface between the end cone and bolt; the compressive force is transmitted to the spherical ball through the interface between the end cone and Fig. 6. Schematic diagram of the two-member structure. (a) Two-member model before deformation. (b) Two-member model after deformation.
546
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 9. Moment-rotation curve of the M24 bolt-ball joint. Fig. 7. Comparison of the load-displacement curves of the intersection point.
stress-displacement curves at member f3 near node 16 is shown in Fig. 14. The two figures reveal that the FEA results are close to the experimental results before the structure loses its stability.
sleeve. Their experimental studies obtained the bending-rotation relationships of the joint, as shown in Fig. 9. In this paper, based on the moment-rotation curve of the M24 bolt-ball joint under bending moment, the FEA analysis of the single-layer cylindrical reticulated shell with bolt-ball joints is conducted. The FEA analysis calculates the full loaddeflection response of the dome in consideration of both material and geometric nonlinearities, which provides a relatively perfect picture regarding the nonlinear behaviour. The experimental picture and illustrative diagram of the single-layer latticed shell are shown in Fig. 10. The 5 m × 6 m single-layer cylindrical reticulated shell with a rise of 1.25 m is vertically supported on the eighteen outer nodes. The tubular members of 76 mm in diameter with a wall thickness of 3.5 mm (which are nominal values for the diameter and thickness) are used in the structure. Based on the proposed simulation method in Section 3.1, the FE member model (shown in Fig. 4) was used to establish a model of the cylindrical reticulated shell with bolt-ball joints, as shown in Fig. 11. All the geometric details and the material properties used in the FE model of the structure are the same as those in the experimental structure. The FE analysis takes both material and geometric nonlinearities into account. The average yield strength of the steel for the members and joint region was fy = 259 MPa, which was obtained from tension coupon tests. The Young's modulus is E = 206 GPa. The experimental and FE-based load-deflection response curves of the test structure are compared in Fig. 12. The results show that the two curves agree reasonably well, indicating that the developed FE model provides a good estimation of the experimentally observed behaviour. The critical load calculated using the FE model has a relative error of 2.23% compared with the experimental result. The load-displacement curves of nodes 13 and 20 obtained through the FEA and the test are shown in Fig. 13, and a comparison of the
4. Investigation on the natural vibration frequency of single-layer reticulated shells with bolt-column joints The FEA model of the single-layer reticulated shells was established based on the member model and the bending-rotation curves of the bolt-column joints obtained by the experiments shown in Fig. 3, as shown in Fig. 15. The reticulated shell with a span of 50 m and a rise of 10 m is vertically hinged supported on the outer nodes. The rectangular steel tube is used in the structure; the section of the radial and circumferential members, which are identified with red and blue lines in the figure, is 120 mm × 60 mm × 3.2 mm, and the section of the diagonal members, which are identified with black lines, are 120 mm × 60 mm × 2 mm; the size of the joint region used in the numerical analysis is the same as that of the experimental hollow cylinder, whose diameter is 200 mm; a roof load of 60 kg/m2 is used during the analysis. The steel for the members is grade Q235 with a nominal yielding stress fy of 235 MPa. All of the 400th order natural frequencies of the structures are shown in Fig. 16(a); the first 20th natural frequencies and 165th −185th natural frequencies are enlarged to be clearer, as shown in Fig. 16(b) and (c), respectively. It can be seen from the figures that the natural frequencies increase as the bending stiffness of the joints increases, and the difference increases as the mode order increases. The natural frequencies of the single-layer reticulated shells with M24 bolt-column joints, M27 boltcolumn joints and rigid joints can be divided into four stages: a) Basic frequency range: the first 1st and 2nd mode order; b) Low frequency range: 3rd–169th mode order; c) Skip range: 170th–175th mode order; Fig. 8. Bolt-ball joint system. (a) Picture of the joint specimen (a) Joint in section.
547
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 10. Illustrative diagram of the single-layer cylindrical reticulated dome. (a) Picture of the test (b) Layout of the test.
Fig. 11. FEA model of the single-layer cylindrical reticulated dome.
Fig. 13. Comparison of the load-displacement curves at nodes 13 and 20.
Fig. 12. Comparison of the load–displacement curves at loading node16.
d) High frequency range: 176th–400th mode order. There are two obvious skips that occur in the ‘critical sections’ in the natural frequencies, as indicated in Fig. 16(b) and (c). To investigate the causes of the skips, the graphs of the mode of vibration near the critical sections are shown in Fig. 17. From the figure, the 2nd-order vibration is the antisymmetric horizontal vibration of the whole structures; the 3rd-order vibration is the symmetrical vertical vibration of the whole structures; the 169th vibration is the local vibration in the structures; the 170th and 175th vibrations are the in-plane rotational vibration of the structures; 176th vibration is the member vibration in the structures. Therefore, it can be concluded that the skips in the natural frequencies are due to the sudden change of the mode of vibration.
Fig. 14. Comparison of the stress-displacement curves at member 13.
5. Effects of bending stiffness on the base frequency of the singlelayer reticulated shells A parametric analysis on the natural vibration of single-layer spherical reticulated shells (shown in Fig. 15) with BC joints is performed. The effects of bending stiffness on the base frequency of the single-layer reticulated shells are considered in detail. The parameters of single-layer reticulated shells, such as span, ratio of rise to span, roof load, geometric imperfections, are considered. Both geometric and 548
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
α=
kα k0
(4)
where k 0 is the initial bending stiffness of M24 BC joint; k α is the initial bending stiffness of the joints with the initial bending stiffness factor α.
• Different span and ratio of rise to span Considering the practical engineering situation, three spans and six rise-to-span ratios presented bellow were taken accounted into analysis: Span (L): 30 m, 40 m, and 50 m; Ratio of rise to span (f/L): 1/3, 1/4, 1/5, 1/6, 1/7, and1/8.
Fig. 15. The single-layer spherical reticulated shells. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
To assess the overall behaviour of a single-layer spherical reticulated shell with different ratios of rise to span, the pattern division number of a shell should allow for almost the same length (between 3 and 5 m) of members with different span and ratio of rise to span. Therefore, the number of rings for the shells with 30 m, 40 m and 50 m spans are 5, 6 and 7, respectively.
material nonlinearities are taken into consideration to determine the overall behaviour of the shells.
5.1. Scheme for parametric analysis
• Different bending stiffness
• Different roof loads
To fully investigate the influence of bending stiffness of joints on the natural vibration, 11 types of bending stiffness determined by the initial bending stiffness factor α, as shown in Fig. 18, are adopted based on the bending stiffness of M24 BC joint. In the figure, Mu is the plastic moment resistance of M24 BC joint; the initial bending stiffness factor, α, is defined as:
•
Different roof loads of 0.6 kN/m2, 1.2 kN/m2, and 1.8 kN/m2 are added in the models during the analysis; the load is distributed uniformly over the shell. Different geometric imperfections The base frequencies of single-layer shells with different initial imperfections are obtained through eigenvalue buckling analysis. The first eigenvalue buckling mode is used to simulate the distribution of the initial imperfection. A 40-m span dome with five initial
Fig. 16. Natural frequencies of single-layer spherical reticulated shells. (a) 400th order natural frequencies (b) The first 20th natural frequencies (c) 165th −185th natural frequencies.
549
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 17. Comparison of the typical modes of single-layer spherical reticulated shells with different joints. (a) Structures with M24 BC joints (b) Structures with M27 BC joints (c) Structures with rigid joints.
•
imperfection values, namely 40 mm (L/1000), 45 mm (L/900), 53 mm (L/750), 67 mm (L/600), 89 mm (L/450), and 133 mm (L/ 300), are analysed to investigate the variation trend of the base frequencies. Section of members
Each model contains two different members, as can be identified from Fig. 14, where the blue and red lines indicate the larger section 120 mm × 60 mm × 3.2 mm and the fine line the smaller section 120 mm × 60 mm × 2 mm;
• Diameter of joint region: 200 mm; • Supporting condition: the shell is supported by pins on the peripheral nodes.
Fig. 18. Moment-rotation curves defined by the initial stiffness factor α.
550
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 19. Base frequency of the shells with different initial bending stiffness coefficients. (a) With effects of different spans (b) With effects of different ratios of rise to span (c) With effects of roof loads.
5.2. Results and discussion of the parametric analysis Fig. 19 presents the curves of the base frequency of the semi-rigidly jointed spherical reticulated shells with different span, ratio of rise to span and roof load. The basic frequencies of the shells decrease with increasing span and roof load and increase with the increasing f/L ratio. To investigate the change law of the base frequency of shells with different bending stiffness, Fig. 20 presents the curves of the base frequency coefficient λα versus bending stiffness factor α of the joints. The base frequency coefficient λα in the figure is defined as:
λα =
Fig. 21. First-order mode of single-layer reticulated shells with semi-rigid joints.
fα fR
imperfections. For the single-layer reticulated shells with semi-rigid joints, the base frequency of the shells decreases obviously with the increasing initial geometric imperfection; when the ratio of rise to span f/L ≤ 1/5, the influence of initial geometric imperfection on base frequency gradually reduce with the increasing bending stiffness of the joints; the influence of initial geometric imperfection on base frequency is very small when the ratio of rise to span f/L = 1/3.
(5)
where fα is the natural frequency of the single-layer reticulated shells with the joints whose initial bending stiffness factor is α; fR is the natural frequency of the single-layer reticulated shells with rigid joints. The figures show that the base frequency coefficient of the singlelayer reticulated shells with different parameters increases when the bending stiffness of the joints in the structure increases, whereas the differences in base frequency due to the bending stiffness of the joints are small; the base frequency coefficient of the single-layer reticulated shells with semi-rigid joints slightly increases when the span and ratio of rise to span increases. The similar conclusion also was found by Fan et al. [22] during analysing the single-layer reticulated shells with boltball joints; and Chen et al. [23] during analysing the suspend-dome with semi-rigid joints. This indicates that for the first mode, axial deformation is still the dominant deformation. The first-order mode of single-layer reticulated shells with semirigid joints is horizontally anti-symmetric mode shape, as shown in Fig. 21. Fig. 22 presents the base frequencies versus the bending stiffness of the joints for the single-layer reticulated shells with different geometric
6. Investigation on the seismic response of single-layer reticulated shells with semi-rigid joints The time history analysis method based on the ABAQUS software is adopted to study seismic behaviour of single-layer spherical reticulated shells with semi-rigid joints under frequent earthquake load, including the El-Centro wave, the Taft wave, and another artificial earthquake waves. The peak accelerations of these three earthquake waves were adjusted to 70 cm/s2. Earthquake waves were imposed on the shells in three dimensions on the assumption that the ground motions on all supports of the structure are the same. The interval in the earthquake waves was set as 0.02 s, and duration of each of the earthquake waves was 20 s, according to Fan et al. [21].
Fig. 20. Base frequency coefficient of the shells with different initial bending stiffness coefficient. (a) With effects of different spans (b) With effects of different ratios of rise to span (c) With effects of roof loads.
551
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 22. Effect of bending stiffness of the joints on the base frequencies of the single-layer reticulated shells with different geometric imperfections. (a) f/L = 1/7 (b) f/L = 1/5 (c) f/L = 1/3.
class members due to action of earthquakes and dead load can be obtained by the maximal internal force (NDI max ) in the corresponding class members due to the action of the dead load multiplied by the relevant coefficient (ξI + 1). The relevant formula can be expressed as follows:
6.1. Definition of the seismic internal force coefficient For a latticed shell with many members, each steel member has a seismic internal force coefficient. To simplify the designing procedure, it is necessary to statistically summarize the seismic internal force coefficient of the shells. Therefore, according to the location of members in structure, members can be classified as circumferential members, radial members and diagonal members, as shown in Fig. 14. The seismic internal force coefficient of the three class members can be separately summarized. The statistical method based on the envelope of the maximal seismic response in [22] is used to obtain the seismic internal force coefficient for different members. The seismic internal force coefficient is defined as follows:
NEI max ξI = −1 NDI max
NEIDesign = NDI max × (ξI + 1)
where NEIDesign is the design axial force of the radial members (NERDesign), circumferential members (NECDesign) and diagonal members (NEDDesign) obtained using the method described in this paper. 6.2. Illustration with an example To verify the effectiveness and applicability of Eqs. (6) and (8) above, analyses of a single-layer reticulated shell with semi-rigid joints and the corresponding single-layer reticulated shell with rigid joints are conducted. The parameters for analysis are as follows:
(6)
where ξI is the seismic internal force coefficient for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) members; NEI max is the maximum axial force of one class members under the action of the earthquake waves and the dead load of single-layer reticulated shells with semi-rigid joints; NDI max is the maximum axial force of one class members under the dead load of the corresponding singlelayer reticulated shells with rigid joints. Similarly, seismic internal force coefficient for each member in every group can be defined by Eq. (7):
ξIi =
NEIi −1 NDIi
(8)
Joint systems: M24 BC joints and rigid joints; Span (L): 40 m; Section dimensions: 140mm × 100mm × 5mm Ratio of rise to span: 1/7; Roof Load: 1.8 kN/m2; Supporting condition: the shell is pin-supported on the peripheral nodes. After the analysis, based on Eq. (6), the seismic internal force coefficient for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) are shown in Fig. 23. The design axial forces of the radial members (NERDesign), circumferential members (NECDesign) and diagonal members (NEDDesign) obtained by Eq. (7) are shown in Fig. 23. In the figures, NEIi and ξIi are the actual seismic axial force and actual seismic internal force coefficient of each member. Fig. 24 reveals that the design seismic axial forces NERDesign, NECDesign and NEDDesign are the envelope value for each class members that can satisfy requirement
(7)
where ξIi is the seismic internal force coefficient for every member in 3 classified group. NEIi is the axial force of one member under the action of the earthquake loads and the dead load of single-layer reticulated shells with semi-rigid joints; NDIi is the axial force of the same member under the dead load. When structures are designed, the maximal internal force of one
Fig. 23. Seismic internal force coefficient of each class members. (a) Radial members (b) Diagonal members (c) Circumferential members.
552
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 24. Design axial forces of the members. (a) Radial members (b) Diagonal members (c) Circumferential members.
of the design.
ξ ′R = 0.475. The influence of bending moment on the seismic internal force coefficient for the three classes of member is little. Secondly, in the “Technical specification for latticed shells [24]”, the seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of single-layer reticulated shells are based on the Eq. (6) in the paper; which means the influence of the bending moments was not considered. To be consistent in the definition of the seismic internal force coefficient, Eq. (6) was also used to calculated the seismic internal force coefficients of singlelayer reticulated shells in this paper.
6.3. Influence of the bending moments on the seismic internal force coefficient Normally, in a single-layer reticulated shell, bending moments may be of relevance in calculating the seismic internal force coefficient. To investigate the Influence of the bending moments, the seismic internal force coefficient (ξ ′Ii ) for each member considering the influence of the bending moments were obtained. ξ ′Ii is calculated as below:
ξ ′Ii =
N M σEIi + σEIi −1 N M σDIi + σDIi
6.4. Seismic internal force coefficient of single-layer reticulated shells with semi-rigid joints
(9)
N where, σEIi is the axial stress of one member in single-layer reticulated shells with semi-rigid joints under the action of the earthquake loads M and the dead load; σEIi is the bending stress of one member in singlelayer reticulated shells with semi-rigid joints under the action of the N earthquake loads and the dead load; σDIi is the axial stress of the same M is the bending stress member in the structure under the dead load; σDIi of the same member in the structure under the dead load. The parameters for the reticulated shells are the same with Section 6.2 in the revised paper. The Comparison of seismic internal coefficient of each member with and without the influence of bending moment as shown in Fig. 25. It can be seen from the figure that the bending moment does have some influence on seismic internal force coefficient for each member. Based on the data in the figure, the seismic internal force coefficient for the radial members with and without the influence of bending moment are ξR = 0.212 and ξ ′R = 0.207 ; the seismic internal force coefficient for the circumferential members with and without the influence of bending moment are ξC = 0.700 and ξ ′C = 0.685; and the seismic internal force coefficient for the diagonal members with and without the influence of bending moment are ξD = 0.483 and
Parameter analysis of the seismic internal force coefficient is investigated to determine the change rule of seismic internal force coefficient with influence factors. Hence, the seismic internal force coefficient of single-layer reticulated shells with semi-rigid joints can be derived based on a statistical method. The parameters considered in the analysis are listed in Table 1. The change law of the seismic internal force coefficient of the radial members, diagonal members and circumferential members of the semirigidly jointed single-layer spherical reticulated shells with different parameters are shown in Figs. 26–28. it can be seen from the figures that the parameters, such as roof load, member section, span, ratio of rise to span and earthquake wave, have a significant effect on seismic internal force coefficient of the members. In general, the seismic internal force coefficients of the radial members and the diagonal members increase with bending stiffness increases, whereas the seismic internal force coefficients of the circumferential members first increase and then decrease with the increase of the initial bending stiffness ratio. Based on the 486 finite element models, the statistical values of seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of the shells with semi-rigid joints are obtained. During the analysis, the parameters of bending stiffness ratio, roof load, sectional dimension, span and ratio Table 1 Parameters of the shells for the parametric analysis. Parameters of the shells
Values
Span Bending stiffness coefficient Sectional dimensions
30 m, 40 m 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0 140 mm × 100 mm × 5 mm (Section 1) 150 mm × 100 mm × 5 mm (Section 2) 0.6 kN/m2, 1.2 kN/m2, 1.8 kN/m2 1/3, 1/5, 1/7 El-Centro, Taft, Artificial wave 200 mm pinned supported on the peripheral nodes
Roof load (RL) Ratio of rise to span Earthquake waves Diameter of the joint region Supporting condition
Fig. 25. Comparison of seismic internal coefficient of the members with and without the influence of bending moment.
553
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 26. Seismic internal force coefficient of the radial members for different parameters. (a) With different roof loads (b) With different member sections (c) With different f/L ratios (d) With different spans (e) With different earthquake waves.
Fig. 27. Seismic internal force coefficient of the diagonal members for different parameters. (a) With different roof loads (b) With different member sections (c) With different f/L ratios (d) With different spans (e) With different earthquake waves.
554
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 28. Seismic internal force coefficient of the circumferential members for different parameters. (a) With different roof loads (b) With different member sections (c) With different f/L ratios (d) With different spans (e) With different earthquake waves. Table 2 Seismic internal force coefficients of single-layer spherical reticulated shells with semi-rigid joints. Coefficients
ξR
ξD
ξC
f/L
1/3 1/5 1/7 1/3 1/5 1/7 1/3 1/5 1/7
Bending stiffness coefficients 0.05
0.1
0.2
0.4
0.6
0.8
1.0
2.0
Rigid [24]
0.417 0.537 0.434 1.214 1.008 0.748 0.984 0.877 0.681
0.407 0.484 0.391 1.192 0.946 0.638 0.976 0.900 0.679
0.403 0.438 0.368 1.164 0.842 0.514 0.973 0.905 0.668
0.404 0.411 0.340 1.145 0.790 0.515 0.958 0.911 0.649
0.383 0.383 0.326 1.144 0.766 0.516 0.936 0.955 0.634
0.361 0.373 0.325 1.135 0.739 0.506 0.913 0.825 0.636
0.332 0.355 0.323 1.128 0.722 0.501 0.903 0.759 0.633
0.331 0.327 0.323 1.098 0.704 0.501 0.913 0.668 0.624
0.320 0.320 0.320 0.640 0.560 0.500 0.760 0.640 0.580
7. Conclusions
of rise to span (as shown in Table 1) were considered. According to the statistical probability method, the seismic internal force coefficients according to different bending stiffness ratios are obtained with 95% insurance rate, as shown in Table 2. To make the comparison clearer, the ratio of the seismic internal force coefficients of single-layer reticulated shells with semi-rigid joints to that of single-layer reticulated shells with rigid joints are also presented in Fig. 29. The symbols ξRR , ξDR , and ξCR in the figure represent the seismic internal force coefficients for the radial members (ξRR ), diagonal members (ξDR ) and circumferential members (ξCR ) of single-layer reticulated shells with rigid joints. it can be seen from the comparison that: with the increasing of the joint bending stiffness, almost all the seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of single-layer reticulated shells with semi-rigid joints decrease, and the values are close to the values of the members in single-layer reticulated shells with rigid joints. especially in the singlelayer reticulated shells with the f/L = 1/7, the seismic internal force coefficients of circumferential members initially decrease until bending stiffness coefficient reaches 0.6 and then the seismic internal force coefficients of the three classed of members remain nearly stable.
The FE model of single-layer reticulated shells with semi-rigid joints established in ABAQUS with the beam element (B31) and the connector elements (CONN3D2) was validated through comparison with the experimental result. Based on the experimental moment-rotation curves of the M24 and M27 bolt-column joints, the natural vibration frequencies of singlelayer reticulated shells with semi-rigid joints were investigated in detail. The following phenomena were observed: i) the natural frequencies increase with the increasing of bending stiffness of the joints, and the difference increases as the modes order increases; ii) the natural frequencies of the single-layer spherical reticulated shells with boltcolumn joints can be divided into four stages: basic frequency range, low frequency range, skip range, and high frequency range; iii) the basic frequencies of semi-rigidly jointed single-layer reticulated shells decrease with increasing span and roof load and with decreasing f/L ratio, whereas, according to the base frequency coefficient λα (0.001 ≤ α ≤ 10.0), the effect of bending stiffness on the basic frequencies of shells without initial geometrical imperfections is small; iv) when the ratio of rise to span is f/L ≤ 1/5, the base frequency of shells decreases 555
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
Fig. 29. Comparison of the seismic internal force coefficients of the three classes of members. (a) Radial members (b) Diagonal members (c) Circumferential Members.
References
obviously with the increasing initial geometric imperfection; the basic frequencies of shells with a certain initial geometric imperfection increases largely when the bending stiffness of joints increases, whereas when the ratio of rise to span f/L = 1/3 the influence of initial geometric imperfection on base frequency is very small; and the basic frequencies of shells with a certain initial geometric imperfection changes little with the increasing bending stiffness. The seismic internal force coefficient was defined for the singlelayer spherical reticulated shells with semi-rigid joints based on the statistical method. The affecting laws of rigidity of joints, span, ratio of rise to span, tube sections and roof load on the seismic internal force coefficients of the single-layer spherical reticulated shells with semirigid joints were obtained. Finally, under a common earthquake, the seismic internal force coefficients for the radial members (ξR), circumferential members (ξC) and diagonal members (ξD) of the singlelayer latticed shell with different stiffness were derived for seismic design. The internal force coefficients in Table 2 are suitable for the single-layer spherical reticulated shells (as shown in Fig. 15) with semirigid joints under symmetrical load condition. More further studies should be carried out to extend the method to other single-layer reticulated shells, such as single-layer cylindrical reticulated shell, singlelayer elliptical paraboloid latticed shells, single-layer ribbed latticed shells, etc. The effect of the unsymmetrical load condition on the dynamic behaviour and internal force coefficients should also be investigated.
[1] Zhi-Wei Yu, Xu-Dong Zhi, Feng Fan, Chen Lu, Effect of substructures upon failure behavior of steel reticulated domes subjected to the severe earthquake, Thin-Walled Struct. 49 (2011) 1160–1170. [2] H. Dai, Z. Liu, W. Wang, Structural passive control on electromagnetic friction energy dissipation device, Thin-Walled Struct. 58 (2012) 1–8. [3] Xudong Zhi, Feng Fan, Shizhao Shen, Elasto-plastic instability of single-layer reticulated shells under dynamic actions, Thin-Walled Struct. 48 (2010) 837–845. [4] F.A. Fathelbab, The Effect of Joints on the Stability of Shallow Single Layer Lattice Domes, University of Cambridge, 1987. [5] H. Dai, Z. Zheng, W. Wang, A new fractional wavelet transform, Commun. Nonlinear Sci. Numer. Simul. 44 (2017) 19–36. [6] Reza Chenaghlou Mohammad, Semi-Rigidity of Connections in Space Structures, University of Surrey, 1997 Ph.D. Thesis. [7] Y. Hiyama, H. Takashima, T. Iijima, S. Kato, Buckling behaviour of aluminium ball jointed single layered reticular domes, Int. J. Space Struct. 15 (2) (2000) 81–94. [8] H. Dai, Z. Cao, A wavelet support vector machine-ased neural network metamodel for structural reliability assessment, Comput. Aided Civ. Infrastruct. Eng. 32 (4) (2017) 344–357. [9] H.H. Ma, S. Ren, F. Fan, Parametric study and analytical characterization of the bolt–column (BC) joint for single-layer reticulated structures, Eng. Struct. 123 (2016) 108–123. [10] T. See, Large Displacement Elastic Buckling Space Structures (Ph.D. Thesis), University of Cambridge, 1983. [11] L.ópez Aitziber, Puente Iñigo, Miguel A. Serna, Direct evaluation of the buckling loads of semi-rigidly jointed single-layer latticed domes under symmetric loading, Eng. Struct. 29 (2007) 101–109. [12] L.ópez Aitziber, Puente Iňigo, Miguel A. Serna, Numerical model and experimental tests on single-layer latticed domes with semi-rigid joints, Comput. Struct. 85 (2007) 360–374. [13] H.H. Ma, F. Fan, S.Z. Shen, Numerical parametric investigation of single-layer latticed domes with semi-rigid joints, J. Int Assoc. Shell Spat. Struct. 49 (2) (2008) 99–110. [14] H.H. Ma, F. Fan, J. Zhong, Stability analysis of single-layer elliptical paraboloid latticed shells with semi-rigid joints, Thin-Walled Struct. 72 (10) (2013) 128–138. [15] F. Fan, H.H. Ma, Z.G. Cao, S.Z. Shen, A new classification system for the joints used in lattice shells, Thin Walled Struct. 49 (12) (2011) 1544–1553. [16] S. Kato, I. Mutoh, M. Shomura, Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections, J. Constr. Steel Res. 48 (1998) 145–168. [17] H.H. Ma, S. Ren, F. Fan, Experimental and numerical research on a new semi-rigid
Acknowledgements This research is supported by grant of The Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration (Grant no. 2017D06), and The National Science Fund for Distinguished Young Scholars (Grant No. 51525802). 556
Thin-Walled Structures 119 (2017) 544–557
H. Ma et al.
[21] F. Fan, S.Z. Shen, Study on the Dynamic Strength Failure of Reticulated Domes under Severe Earthquakes, International Association for Shell and Spatial Structures Symposium, Montpellier, France, 2004, pp. 140–141. [22] F. Fan, M.L. Wang, Z.G. Cao, Seismic behaviour and seismic design of dingle-layer reticulated shells with semi-rigid joint system, Adv. Struct. Eng. 15 (10) (2012) 1829–1841. [23] Z. Chen, H. Xu, Z. Zhao, Investigations on the mechanical behavior of suspenddome with semi-rigid joints, J. Constr. Steel Res. 122 (1) (2016) 14–24. [24] Ministry of Urban and Rural Construction of China, Technical Specification For Latticed Shells, JGJ 61-2003 (in Chinese), 2003.
joint for single-layer reticulated structures, Eng. Struct. 126 (11) (2016) 725–738. [18] F. Fan, H.H. Ma, G.B. Chen, S.Z. Shen, Experimental study of semi-rigid joint systems subjected to bending with and without axial force, J. Constr. Steel Struct. 68 (1) (2012) 126–137. [19] H.H. Ma, F. Fan, G.B. Chen, Z.G. Cao, S.Z. Shen, Numerical analyses of semi-rigid joints subjected to bending with and without axial force, J. Constr. Steel Res. 90 (2013) 13–28. [20] H.H. Ma, F. Fan, P. Wen, Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints, Thin-Walled Struct. 86 (2015) 1–9.
557