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Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints
Thin-Walled Structures 86 (2015) 1–9
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints Huihuan Ma n,a, Feng Fan a, Peng Wen b, Hao Zhang c, Shizhao Shen a a
School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China No. 7 Jiananjie Beijing Economic-Technological Development Area, Beijing 100176, PR China c School of Civil Engineering, University of Sydney, Sydney 2006, NSW, Australia b
ar t ic l e i nf o
a b s t r a c t
Article history: Received 8 February 2014 Accepted 12 August 2014
Most existing studies on single-layer spatial structures with semi-rigid joints were focused on spherical domes. The present paper analyzed the squared plan-form single-layer structures, examining the influence of joint-rigidity on the mechanical performances of the structures. An experiment has been conducted on a 5 6 m single-layer cylindrical reticulated shell with semi-rigid bolt-ball joints. The mechanical performance of the latticed shell is investigated in detail. Finite element analysis (FEA) model of the latticed shell is established taking material and geometric nonlinearities into account. The results show that the behavior determined using the FEA models correlates well with the experimentally observed behavior for the single-layer structures with semi-rigid joints. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Single-layer spatial structures Semi-rigid joints Experiments Bending–stiffness FEA model
1. Introduction In the design and analysis of single-layer spatial structures, it is usually assumed that the connections behave either as pinned or rigid joints. However, the joints in realistic spatial structures are semi-rigid and their actual behavior does not conform to either of the two extremes. Joint systems which are frequently used in space frames include the MERO joint systems [1–3], the aluminum alloy truss connector [4], the ORTZ joint [5–7], the bolt-ball joint systems [8-11], the S14, D14 and D06 joints [12], the T.U.U.-S ball joint [13], the space-truss connector [14] and the socket joint system [15]. Recently, interest in single-layer spatial structures has grown significantly. Observations from an earlier study [1,16] confirmed that connection stiffness had a significant effect on the load– displacement behavior of a single-layer spatial structure. Aitziber et al. [5,6], Ma et al. [8], Fan et al. [9] and Kato et al. [17] verified that the rigidity of joints is an important factor that influences the behavior of a single-layer latticed dome. El-Sheikh [18] found that both the overall behavior and failure mode of a spatial structure are influenced by the bending stiffness of connections. Most previous research has focused on circular plan-form domes. Single-layer cylindrical reticulated shells (Fig. 1) with semirigid joints can provide a good solution for small and medium span
n
Corresponding author. E-mail addresses: [email protected] (H. Ma), [email protected] (F. Fan), [email protected] (P. Wen), [email protected] (H. Zhang), [email protected] (S. Shen). http://dx.doi.org/10.1016/j.tws.2014.08.006 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
space structures, because of their rapid construction speed, high fabrication accuracy and beautiful appearance. However, few such reticulated shells were used because their performance is complex and relevant studies are very limited. Hence, it is worthwhile exploring their mechanical performance and providing information that can be used by designers. In this paper, an experiment on a 5 m 6 m single-layer cylindrical reticulated shell with the boltball joints (which is widely used in spatial structures) was reported. The experiment recorded the complete load–displacement response, the stress development in the members, the nodal displacement and the buckling mode of the structure. FEA model of the cylindrical reticulated shell was also established and the results were compared with the test results.
2. Experimental study of the single-layer cylindrical reticulated shell 2.1. Description of the test structure A 3-D diagram of the single-layer cylindrical reticulated shell and overall layout of the tests structure are shown in Fig. 1(a) and (b). The bolt-ball joints were used to connect the members in the shell. The structural parameters are:
dimensions—5 m width and 6 m length; the height of the center point—1.25 m; all the tubular members are 76 mm diameter and 3.5 mm thick.
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
b
2
L
Fig. 1. Single-layer cylindrical reticulated shells. a) Layout of grid form, b) Supporting system without horizontal ribs and c) Supporting system with horizontal ribs.
Table 1 Details of the members used in the shell. Member type
Outer diameter thick (mm)
Length (m)
Quantity
Weight (t)
1 2 3 4 In total
76 3.5 76 3.5 76 3.5 76 3.5
0.962 1.000 1.388 2.000
12 6 36 18 72
0.050 0.027 0.249 0.195 0.520
Fig. 2. Illustrative diagram of the single-layer cylindrical reticulated dome. a) 3-D diagram of the structure and b) Overall layout of the test structure.
The details of the members are listed in Table 1. The shell was supported on its four edges: two arch supporting edges along the X-axis and two straight supporting edges along the Y-axis, as shown in Fig. 2(a). The structure was loaded at the central point by means of a hydraulic jack, as shown in Fig. 2(b). The structural members and joints were prefabricated in the factory, and assembled for testing in the Structural Laboratory in Harbin Institute of Technology. Pictures of the structure and its
supporting systems are shown in Fig. 3. The supporting nodes on the two straight edges were welded to the Ι-beams, and the H-beams were fixed to the floor by the anchor bolts, as shown in Fig. 3(e) and (d). The supporting nodes on the two arch edges were at different heights. Therefore, triangle steel frames with different height were used to support the nodes, and the frames were fixed to the floor by the anchor bolts. Angle bars were used to increase the stiffness of the supporting system, as shown in Figs. 3 and 4.
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
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Fig. 3. Picture of the single-layer cylindrical reticulated dome. a) Picture of the structure, b) Straight supporting edge, c) Supporting node on arch edges, d) End supporting node on straight edges and e) Middle supporting node on straight edges.
2.2. Loading system and procedure Obtain a complete load–displacement curve for the structure is the primary goal in this test. When the single-layer cylindrical reticulated shell loses its stability, the process is so rapid that it is difficult to record the displacement of the structure. To record the complete load– displacement curve successfully, the loading schemes in Fig. 5 were adopted. The ball node was connected to a load cell ③ through the crossed plate ②. The arc-shaped lib ④ used in the loading system can eliminate the secondary moment caused by the eccentricity of the tension force. The two hydraulic jacks ⑥ were connected to one oil pump through two distributor pipes, thus synchronizing the two horizontal load settings. The two screw jacks ⑦ were located besides the two hydraulic jacks for recording the instant response when the structure loses its stability. The loading procedure was as follows: (1) Initially, the structure was subject to a small load and unloaded in order to ensure the loading apparatus and the displacement gauges were working properly and to allow the different parts of the specimen to work as a whole. (2) With the two hydraulic jacks, the test was performed under load control conditions. (3) When the structure was going to attain its critical load, the test control was changed to displacement control using the two screw jacks until failure of the structure occurred.
During the test, the time for each load step was at least ten minutes to ensure the transformation was complete and stable.
Fig. 4. Supporting system of the test structure.
2.3. Strain and displacement monitoring system As shown in Fig. 6, at each load step the strains of the members and the displacement of the nodes were measured using strain gauges and displacement transducers. The strain gauges were located at the ends and the middle section of the members. For the central members around the loading node, four strain gauges were used at one section, and for other members two strain gauges were used at one section. The X, Y and Z direction displacements of the loading node 16 were monitored, and X and Z direction displacements of other inner nodes were monitored. The instrumentation setup for monitoring Z and X direction displacements of the nodes is shown in Fig. 7.
4
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
Fig. 5. Loading equipments. a) Loading system, b) Ball node and load cell and c) Arc-shaped lib.
Fig. 6. Layout of the strain gauges and displacement transducers.
2.4. Initial imperfection measurement
2.5. Mechanical properties
In real structures, initial imperfections are inevitable, e.g., bending defects in the members and geometrical imperfection of the nodes. The single-layer cylindrical reticulated shells are imperfection-sensitive structures; therefore, the initial bending defects and initial geometrical imperfection were measured before the beginning of the experiment. The bar numbers and their initial bending defect values are shown in Fig. 8 and Table 2. In the table, Vb/l means the ratio of initial bending defect value to the length of the corresponding bar. The biggest value occurred in bar c1, and the value was 1/1044. The initial geometrical imperfection values are shown Table 3. The biggest value occurred in node 15, and the value was 31 mm. The experimental model of the dome with initial geometrical imperfection is shown in Fig. 9.
The steel for the members and supporting frames was grade Q235 with a nominal yielding stress fy of 235 MPa. The material properties of the steel were obtained from tension coupon tests (as shown in Fig. 10), and the values are listed in Table 4. The average yield strength is fy ¼259 MPa, and the average ultimate tensile strength is fu ¼ 337 MPa.
3. Results and discussion Based on the commercial FE software ANSYS, the FE member model (shown in Fig. 11), which has been used for reticulated spherical domes [8,9], was used to establish a model of the cylindrical reticulated shell with semi-rigid joints. The member
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
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Fig. 7. Monitoring arrangement for displacements. a) Z direction and b) X direction.
Fig. 8. Measurement of the initial bending defect. a) Messurement process and b) The bar numbers.
Table 2 Values of the initial bending defect of bars in the shell. Bar number
Initial bending defect value (Vb/l)
Bar number
Initial bending defect value (Vb/l)
Bar number
Initial bending defect value (Vb/l)
a1 a2 a3 a4 a5 a6 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 e1 e2
1/2721 1/2985 1/1905 1/5300 1/2130 1/1428 1/5278 1/1252 1/1686 1/1663 1/1044 1/4762 1/2532 1/2556 1/6340 1/2083 1/2703 1/1667
d1 d2 d3 f1 f2 f3 f4 g1 g2 g3 g4 g5 g6 h1 h2 h3 i1 i2
1/1515 1/1493 1/1389 1/1510 1/1600 1/1822 1/1901 1/3280 1/1848 1/1872 1/3774 1/2439 1/2469 1/2262 1/3749 1/3124 1/1413 1/2273
j1 j2 j3 j4 k1 k2 k3 k4 k5 k6 e3 e4 e5 e6 i3 i4 i5 i6
1/1585 1/3632 1/3000 1/2464 1/2060 1/2262 1/3730 1/3804 1/1419 1/1942 1/3704 1/1429 1/3170 1/2248 1/2058 1/2023 1/2335 1/1990
6
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Table 3 Values of the initial geometrical imperfection in the shell. Node number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Standard coordinate values of nodes (m)
Measured coordinate values of nodes (m)
x
y
z
x
y
z
2.500 1.812 0.951 0.000 0.951 1.812 2.500 1.812 0.000 1.812 2.500 0.951 0.951 2.500 1.812 0.000 1.812 2.500 0.951 0.951 2.500 1.812 0.000 1.812 2.500 1.812 0.951 0.000 0.951 1.812 2.500
3.000 3.000 3.000 3.000 3.000 3.000 3.000 2.000 2.000 2.000 1.000 1.000 1.000 1.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 2.000 2.000 2.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000
1.250 0.578 0.148 0.000 0.148 0.578 1.250 0.578 0.000 0.578 1.250 0.148 0.148 1.250 0.578 0.000 0.578 1.250 0.148 0.148 1.250 0.578 0.000 0.578 1.250 0.578 0.148 0.000 0.148 0.578 1.250
2.51 1.813 0.954 0.000 0.945 1.814 2.483 1.802 0.01 1.818 2.51 0.973 0.948 2.483 1.784 0.003 1.813 2.526 0.977 0.959 2.481 1.801 0.001 1.797 2.519 1.805 0.953 0.000 0.936 1.807 2.491
3.001 2.999 3.007 3.000 2.991 2.986 2.984 1.995 2.001 1.994 1.001 1.018 1.007 0.984 0.009 0.004 0.005 0.986 0.992 0.985 1.017 1.999 1.99 2.005 3.000 3.007 3.001 2.993 2.994 3.006 3.017
1.25 0.58 0.158 0.000 0.147 0.575 1.248 0.574 0.009 0.572 1.25 0.156 0.151 1.248 0.569 0.011 0.576 1.255 0.153 0.146 1.249 0.569 0.014 0.581 1.248 0.571 0.145 0.000 0.149 0.575 1.245
Structure with initial imperfection Structure without initial imperfection Fig. 9. The experimental model of the dome with initial geometrical imperfection.
model is composed of: (1) two ball nodes at both ends; (2) an elasto-plastic beam element; (3) three nonlinear spring elements at each end of the beam connecting the beam and ball nodes. Ks1, Ks2, Ks4 and Ks5 represent the bending stiffness of the two joints at each end about the Y and Z axes, and Ks3 and Ks6 represent the rotation stiffness of the joints about the axis X. Translations of Points a and c (b and d) are coupled in the nodal directions x, y, and z, which means the lengths of spring elements in the model are zero. The nonlinear spring element COMBIN39 in ANSYS with unidirectional freedom and nonlinear characteristics is used to represent the bending and torsional capacity of a bolt-ball joint by inputting the real constants of the element COMBIN39. Both tube and ball in the model are simulated by an elasto-plastic beam element, and the section of the ball is greater than that of the tube because of its bigger bending stiffness.
Imperfection value Vd (mm)
Vd/5000
9 2 13 0 11 14 23 12 13 10 9 30 8 23 31 12 5 29 28 17 25 14 17 16 18 12 4 7 16 8 19
1/552 1/2041 1/398 0 1/460 1/346 1/220 1/421 1/371 1/481 1/552 1/169 1/611 1/220 1/163 1/414 1/913 1/172 1/181 1/292 1/202 1/351 1/290 1/311 1/276 1/412 1/1336 1/714 1/309 1/598 1/257
The FEA model of the test structure was established based on the member model, the bending–rotation curve of the bolt-ball joint (shown in Fig. 12), and the values of the initial geometrical imperfection in the shell (listed in Table 3). The analysis was carried out for the test structure including the initial geometric imperfections. Both material and geometric nonlinearity were considered in the analysis. The experimental and FE-based load–deflection response curves of the test structure are compared in Fig. 13. The results show that the two curves agree reasonably well, indicating that the developed FEA model [8,9] has yielded a good estimation of the experimentally observed behavior. The critical load calculated using the FEA model has a relative error of 1.66%, compared with the experimental result. The test was carried out under force control in the elastic range of the shell behavior, and each load step was approximately 3 kN. When the load was 21.2 kN, the displacement of node 16 was about 45 mm, and the test control was changed to displacement control until failure of the structure occurred. In the process, the displacement of node 16 increased gradually, and the structure lost its stability at a critical load of 22.43 kN. The deformation patterns of the structure obtained from the test and FEA at the critical load are shown in Fig. 14. To provide a clear picture of the displacement development of the structure, the deformation patterns of the structure at three different moments (as shown in Fig. 13) are given in Fig. 15 based on the FEA model. The load–displacement curves of nodes 13 and 20 obtained through the FEA and the test are shown in Fig. 16, and the comparison of stress–displacement curves at member f3 near node 16 are shown in Fig. 17. From the two figures, it can be seen that the FEA results are close to the experimental results before the structure loses its stability.
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Fig. 10. Material test. a) Test specimens and b) Test picture.
30
Table 4 Mechanical properties of steel.
Moment 2
Coupon number
Yield strength fy (MPa)
Tensile strength fu (MPa)
Elastic modulus E (MPa)
1 2 3 4 Average value
260.48 254.69 267.42 254.69 259.32
347.01 334.28 339.05 327.11 336.86
1.61 105 1.58 105 1.66 105 1.50 105 1.59 105
25
Load (kN)
20
Moment 3 15 10
Test FEA with initial geometric imperfection
5
Moment 1 0 0.00
0.04
0.08
0.12
0.16
0.20
Displacement of node 16 (m) Fig. 13. Comparison of load–displacement curves at node 16.
Fig. 11. Member model.
3.0 2.5
M (kN⋅m)
2.0 1.5 1.0 0.5 Test 0.0 0.00
0.02
0.04
0.06
0.08
0.10
Φ (rad)
obtained by the FEA, as shown in Fig. 18. It can be seen from the figures that: (i) the buckling mode obtained from the FEA model is double symmetric, while the experimental buckling mode is not; (ii) In the FEA buckling mode, the upward displacement of the nodes on the 2 and 20 axis is bigger than the nodes on the 1 and 10 axis, which is opposite to the situation in the experimental buckling mode; (iii) The displacement trends of the nodes on the 1, 10 , 2 and 20 axis are all upward and they are the same in the two buckling modes; (iv) The displacement trends of the nodes besides the loading node on the 3 axis are opposite in the two buckling modes. There are some differences between the two buckling modes, but overall, the displacement trends of most nodes in the structure, except the nodes on the 3 axis, are the same. Finally, Fig. 19 compares the complete load–displacement curve from the test with the FEA results assuming rigid joints, semi-rigid joints, and pin joints, respectively. As expected, the experimental results are between the results of rigid joints and pin joints FE model, and closest to the semi-rigid joint model. Therefore, in the design of cylindrical reticulated shells, the hypothesis of pinned joints leads to structures with low capacity in terms of stability and resistance, and obviously, they cannot be regarded as rigidly jointed structures either. The bending stiffness should be considered in the FEA model for analyzing single-layer spatial structures.
Fig. 12. Moment–rotation curves of the joint.
4. Conclusions During the experiment, the node coordinates were recorded at two moments: before and after the buckling load. Then the first order buckling mode was obtained and compared with that
A series of experimental tests have been conducted on a semi-rigid bolt-ball joint and a 5 m 6 m single-layer cylindrical reticulated shell.
8
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
Fig. 14. The deformation of the structure at the critical load. a) Test and b) FEA.
Fig. 15. The deformation of the structure.
30
Displacement of node 16 (mm)
200
Load (kN)
25 20 15 10
Node13(FEA) Node13(Test) Node20(FEA) Node20(Test)
5
5
10
15
20
25
120 80 40 0
0 0
160
Axial stress (Test) Axial stress (FEA) Bending stress (Test) Bending stress (FEA)
30
Displacement (mm) Fig. 16. Comparison of load–displacement curves.
The experimental investigation on the structure presented in this paper provides an initial insight into the mechanical performance of single-layer cylindrical reticulated shells with semi-rigid joints. The complete load–deflection response curve and the buckling mode of the test structure were obtained. FEA models were also established to carry out numerical analysis.
0
-50
-100
-150
-200
Stress at member f3 (MPa) Fig. 17. Comparison of stress–displacement curves.
Through comparison of the FEA results and the corresponding test results, it is shown that the proposed FEA model can be used to effectively predict the mechanical behavior of the reticulated shell including its buckling load. Furthermore, the work in this paper verified the FEA model of the single-layer spatial structures with semi-rigid joints used in [8,9], and it also verified that the
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
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Fig. 18. Comparisons of the buckling modes. a)Test (1:40) and b)FEA (1:40).
70
Test FEA (Rigid) FEA (Semi-rigid) FEA (Pinned)
60
Load (kN)
50 40 30 20 10 0 0.00
0.03
0.06
0.09
0.12
0.15
0.18
Displacement of node 16 (m) Fig. 19. Comparisons of load–displacement curves.
FEA model is valid for both the single-layer reticulated spherical domes and non-spherical domes. Acknowledgement This research is supported by grants from the Natural Science Foundation of China under Grant no. 51308153, China Postdoctoral Science Foundation funded project no. 2013M531020, Project HIT. NSRIF2014 Supported by the Fundamental Research Funds for the Central Universities Grant No. HIT. NSRIF. 2014099, and the International Program Development Fund from the University of Sydney. These supports are gratefully acknowledged. References [1] Fathelbab FA., The effect of joints on the stability of shallow single layer lattice domes. PhD thesis. University of Cambridge, September; 1987.
[2] Fathelbab FA., Tangent stiffness matrix for space frame members with both member and joint imperfections. In: Space structure 4. editors G.A.R. Parke, C.M. Howard. conf. Guildford, Sept, London; 1993, pp. 1323–1333. [3] Mohammad Reza Chenaghlou, Semi-rigidity of connections in space structures. PhD thesis. University of Surrey, October; 1997. [4] Hiyama Y, Takashima H, Iijima T, Kato S. Buckling behaviour of aluminium ball jointed single layered reticular domes. Int J Space Struct 2000;15(2):81–94. [5] López 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 2007;29:101–9. [6] López Aitziber, Puente Iňigo, Miguel A, Serna. Numerical model and experimental tests on single-layer latticed domes with semi-rigid joints. Comput Struct 2007;85:360–74. [7] Loureiro A, Goni R, Bayo E., A one-step method for buckling analysis of single layer latticed structures with semi-rigid connections. In: Proceedings of the fifth international conference of space structures, Thomas Telford. London; 2002, pp. 1481–1490. [8] Ma HH, Fan F, Shen SZ. Numerical parametric investigation of single-layer latticed domes with semi-rigid joints. J Int Assoc Shell Spatial Struct 2008;49(2):99–110. [9] Fan F, Ma HH, Cao ZG, Shen SZ. A new classification system for the joints used in lattice shells. Thin Walled Struct 2011;49(12):1544–53. [10] Lee SL, See T, Swaddiwudhipong S. Development and testing of a universal space frame connector. Int J Space Struct 1990;5(2):138 (pp130). [11] Swaddiwudhipong S, Koh CG, Lee SL. Development and experimental investigation of a space frame connector. Int J Space Struct 1994;9(2):99–106. [12] Shibata R, Kato S, Yamada S. Experimental study on the ultimate strength of single-layer reticular domes. In: Proceedings of the fourth international conference of space structures, University of Surry, UK; 1993, pp. 387–395. [13] Ueki T, Matsushita. F., Shibata and Kato S., Design procedure for large singlelayer latticed domes. In: Proceedings of the fourth international conference of space structures, University of Surry, UK; 1993, pp. 237–246. [14] Fülöp Attila, Iványi Miklós. Experimentally analyzed stability and ductility behaviour of a space-truss roof system. Thin Walled Struct 2004;42:309–20. [15] Ma HH, Fan F, Ke J, Shen SZ. Experimental research on semi-rigid joint used in grid structures and reticulated domes. J Build Struct 2010;11(31):1–8. [16] See T., Large displacement elastic buckling space structures. PhD thesis. University of Cambridge, September; 1983. [17] Kato S, Mutoh I, Shomura M. Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections. J Constr Steel Res 1998;48:145–68. [18] EI-Sheikh AI. Numerical analysis of space trusses with flexible member-end joints. Int J Space Struct 1993;8(3):189–97.
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints Huihuan Ma n,a, Feng Fan a, Peng Wen b, Hao Zhang c, Shizhao Shen a a
School of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China No. 7 Jiananjie Beijing Economic-Technological Development Area, Beijing 100176, PR China c School of Civil Engineering, University of Sydney, Sydney 2006, NSW, Australia b
ar t ic l e i nf o
a b s t r a c t
Article history: Received 8 February 2014 Accepted 12 August 2014
Most existing studies on single-layer spatial structures with semi-rigid joints were focused on spherical domes. The present paper analyzed the squared plan-form single-layer structures, examining the influence of joint-rigidity on the mechanical performances of the structures. An experiment has been conducted on a 5 6 m single-layer cylindrical reticulated shell with semi-rigid bolt-ball joints. The mechanical performance of the latticed shell is investigated in detail. Finite element analysis (FEA) model of the latticed shell is established taking material and geometric nonlinearities into account. The results show that the behavior determined using the FEA models correlates well with the experimentally observed behavior for the single-layer structures with semi-rigid joints. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Single-layer spatial structures Semi-rigid joints Experiments Bending–stiffness FEA model
1. Introduction In the design and analysis of single-layer spatial structures, it is usually assumed that the connections behave either as pinned or rigid joints. However, the joints in realistic spatial structures are semi-rigid and their actual behavior does not conform to either of the two extremes. Joint systems which are frequently used in space frames include the MERO joint systems [1–3], the aluminum alloy truss connector [4], the ORTZ joint [5–7], the bolt-ball joint systems [8-11], the S14, D14 and D06 joints [12], the T.U.U.-S ball joint [13], the space-truss connector [14] and the socket joint system [15]. Recently, interest in single-layer spatial structures has grown significantly. Observations from an earlier study [1,16] confirmed that connection stiffness had a significant effect on the load– displacement behavior of a single-layer spatial structure. Aitziber et al. [5,6], Ma et al. [8], Fan et al. [9] and Kato et al. [17] verified that the rigidity of joints is an important factor that influences the behavior of a single-layer latticed dome. El-Sheikh [18] found that both the overall behavior and failure mode of a spatial structure are influenced by the bending stiffness of connections. Most previous research has focused on circular plan-form domes. Single-layer cylindrical reticulated shells (Fig. 1) with semirigid joints can provide a good solution for small and medium span
n
Corresponding author. E-mail addresses: [email protected] (H. Ma), [email protected] (F. Fan), [email protected] (P. Wen), [email protected] (H. Zhang), [email protected] (S. Shen). http://dx.doi.org/10.1016/j.tws.2014.08.006 0263-8231/& 2014 Elsevier Ltd. All rights reserved.
space structures, because of their rapid construction speed, high fabrication accuracy and beautiful appearance. However, few such reticulated shells were used because their performance is complex and relevant studies are very limited. Hence, it is worthwhile exploring their mechanical performance and providing information that can be used by designers. In this paper, an experiment on a 5 m 6 m single-layer cylindrical reticulated shell with the boltball joints (which is widely used in spatial structures) was reported. The experiment recorded the complete load–displacement response, the stress development in the members, the nodal displacement and the buckling mode of the structure. FEA model of the cylindrical reticulated shell was also established and the results were compared with the test results.
2. Experimental study of the single-layer cylindrical reticulated shell 2.1. Description of the test structure A 3-D diagram of the single-layer cylindrical reticulated shell and overall layout of the tests structure are shown in Fig. 1(a) and (b). The bolt-ball joints were used to connect the members in the shell. The structural parameters are:
dimensions—5 m width and 6 m length; the height of the center point—1.25 m; all the tubular members are 76 mm diameter and 3.5 mm thick.
H. Ma et al. / Thin-Walled Structures 86 (2015) 1–9
b
2
L
Fig. 1. Single-layer cylindrical reticulated shells. a) Layout of grid form, b) Supporting system without horizontal ribs and c) Supporting system with horizontal ribs.
Table 1 Details of the members used in the shell. Member type
Outer diameter thick (mm)
Length (m)
Quantity
Weight (t)
1 2 3 4 In total
76 3.5 76 3.5 76 3.5 76 3.5
0.962 1.000 1.388 2.000
12 6 36 18 72
0.050 0.027 0.249 0.195 0.520
Fig. 2. Illustrative diagram of the single-layer cylindrical reticulated dome. a) 3-D diagram of the structure and b) Overall layout of the test structure.
The details of the members are listed in Table 1. The shell was supported on its four edges: two arch supporting edges along the X-axis and two straight supporting edges along the Y-axis, as shown in Fig. 2(a). The structure was loaded at the central point by means of a hydraulic jack, as shown in Fig. 2(b). The structural members and joints were prefabricated in the factory, and assembled for testing in the Structural Laboratory in Harbin Institute of Technology. Pictures of the structure and its
supporting systems are shown in Fig. 3. The supporting nodes on the two straight edges were welded to the Ι-beams, and the H-beams were fixed to the floor by the anchor bolts, as shown in Fig. 3(e) and (d). The supporting nodes on the two arch edges were at different heights. Therefore, triangle steel frames with different height were used to support the nodes, and the frames were fixed to the floor by the anchor bolts. Angle bars were used to increase the stiffness of the supporting system, as shown in Figs. 3 and 4.
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Fig. 3. Picture of the single-layer cylindrical reticulated dome. a) Picture of the structure, b) Straight supporting edge, c) Supporting node on arch edges, d) End supporting node on straight edges and e) Middle supporting node on straight edges.
2.2. Loading system and procedure Obtain a complete load–displacement curve for the structure is the primary goal in this test. When the single-layer cylindrical reticulated shell loses its stability, the process is so rapid that it is difficult to record the displacement of the structure. To record the complete load– displacement curve successfully, the loading schemes in Fig. 5 were adopted. The ball node was connected to a load cell ③ through the crossed plate ②. The arc-shaped lib ④ used in the loading system can eliminate the secondary moment caused by the eccentricity of the tension force. The two hydraulic jacks ⑥ were connected to one oil pump through two distributor pipes, thus synchronizing the two horizontal load settings. The two screw jacks ⑦ were located besides the two hydraulic jacks for recording the instant response when the structure loses its stability. The loading procedure was as follows: (1) Initially, the structure was subject to a small load and unloaded in order to ensure the loading apparatus and the displacement gauges were working properly and to allow the different parts of the specimen to work as a whole. (2) With the two hydraulic jacks, the test was performed under load control conditions. (3) When the structure was going to attain its critical load, the test control was changed to displacement control using the two screw jacks until failure of the structure occurred.
During the test, the time for each load step was at least ten minutes to ensure the transformation was complete and stable.
Fig. 4. Supporting system of the test structure.
2.3. Strain and displacement monitoring system As shown in Fig. 6, at each load step the strains of the members and the displacement of the nodes were measured using strain gauges and displacement transducers. The strain gauges were located at the ends and the middle section of the members. For the central members around the loading node, four strain gauges were used at one section, and for other members two strain gauges were used at one section. The X, Y and Z direction displacements of the loading node 16 were monitored, and X and Z direction displacements of other inner nodes were monitored. The instrumentation setup for monitoring Z and X direction displacements of the nodes is shown in Fig. 7.
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Fig. 5. Loading equipments. a) Loading system, b) Ball node and load cell and c) Arc-shaped lib.
Fig. 6. Layout of the strain gauges and displacement transducers.
2.4. Initial imperfection measurement
2.5. Mechanical properties
In real structures, initial imperfections are inevitable, e.g., bending defects in the members and geometrical imperfection of the nodes. The single-layer cylindrical reticulated shells are imperfection-sensitive structures; therefore, the initial bending defects and initial geometrical imperfection were measured before the beginning of the experiment. The bar numbers and their initial bending defect values are shown in Fig. 8 and Table 2. In the table, Vb/l means the ratio of initial bending defect value to the length of the corresponding bar. The biggest value occurred in bar c1, and the value was 1/1044. The initial geometrical imperfection values are shown Table 3. The biggest value occurred in node 15, and the value was 31 mm. The experimental model of the dome with initial geometrical imperfection is shown in Fig. 9.
The steel for the members and supporting frames was grade Q235 with a nominal yielding stress fy of 235 MPa. The material properties of the steel were obtained from tension coupon tests (as shown in Fig. 10), and the values are listed in Table 4. The average yield strength is fy ¼259 MPa, and the average ultimate tensile strength is fu ¼ 337 MPa.
3. Results and discussion Based on the commercial FE software ANSYS, the FE member model (shown in Fig. 11), which has been used for reticulated spherical domes [8,9], was used to establish a model of the cylindrical reticulated shell with semi-rigid joints. The member
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Fig. 7. Monitoring arrangement for displacements. a) Z direction and b) X direction.
Fig. 8. Measurement of the initial bending defect. a) Messurement process and b) The bar numbers.
Table 2 Values of the initial bending defect of bars in the shell. Bar number
Initial bending defect value (Vb/l)
Bar number
Initial bending defect value (Vb/l)
Bar number
Initial bending defect value (Vb/l)
a1 a2 a3 a4 a5 a6 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 e1 e2
1/2721 1/2985 1/1905 1/5300 1/2130 1/1428 1/5278 1/1252 1/1686 1/1663 1/1044 1/4762 1/2532 1/2556 1/6340 1/2083 1/2703 1/1667
d1 d2 d3 f1 f2 f3 f4 g1 g2 g3 g4 g5 g6 h1 h2 h3 i1 i2
1/1515 1/1493 1/1389 1/1510 1/1600 1/1822 1/1901 1/3280 1/1848 1/1872 1/3774 1/2439 1/2469 1/2262 1/3749 1/3124 1/1413 1/2273
j1 j2 j3 j4 k1 k2 k3 k4 k5 k6 e3 e4 e5 e6 i3 i4 i5 i6
1/1585 1/3632 1/3000 1/2464 1/2060 1/2262 1/3730 1/3804 1/1419 1/1942 1/3704 1/1429 1/3170 1/2248 1/2058 1/2023 1/2335 1/1990
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Table 3 Values of the initial geometrical imperfection in the shell. Node number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Standard coordinate values of nodes (m)
Measured coordinate values of nodes (m)
x
y
z
x
y
z
2.500 1.812 0.951 0.000 0.951 1.812 2.500 1.812 0.000 1.812 2.500 0.951 0.951 2.500 1.812 0.000 1.812 2.500 0.951 0.951 2.500 1.812 0.000 1.812 2.500 1.812 0.951 0.000 0.951 1.812 2.500
3.000 3.000 3.000 3.000 3.000 3.000 3.000 2.000 2.000 2.000 1.000 1.000 1.000 1.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 2.000 2.000 2.000 3.000 3.000 3.000 3.000 3.000 3.000 3.000
1.250 0.578 0.148 0.000 0.148 0.578 1.250 0.578 0.000 0.578 1.250 0.148 0.148 1.250 0.578 0.000 0.578 1.250 0.148 0.148 1.250 0.578 0.000 0.578 1.250 0.578 0.148 0.000 0.148 0.578 1.250
2.51 1.813 0.954 0.000 0.945 1.814 2.483 1.802 0.01 1.818 2.51 0.973 0.948 2.483 1.784 0.003 1.813 2.526 0.977 0.959 2.481 1.801 0.001 1.797 2.519 1.805 0.953 0.000 0.936 1.807 2.491
3.001 2.999 3.007 3.000 2.991 2.986 2.984 1.995 2.001 1.994 1.001 1.018 1.007 0.984 0.009 0.004 0.005 0.986 0.992 0.985 1.017 1.999 1.99 2.005 3.000 3.007 3.001 2.993 2.994 3.006 3.017
1.25 0.58 0.158 0.000 0.147 0.575 1.248 0.574 0.009 0.572 1.25 0.156 0.151 1.248 0.569 0.011 0.576 1.255 0.153 0.146 1.249 0.569 0.014 0.581 1.248 0.571 0.145 0.000 0.149 0.575 1.245
Structure with initial imperfection Structure without initial imperfection Fig. 9. The experimental model of the dome with initial geometrical imperfection.
model is composed of: (1) two ball nodes at both ends; (2) an elasto-plastic beam element; (3) three nonlinear spring elements at each end of the beam connecting the beam and ball nodes. Ks1, Ks2, Ks4 and Ks5 represent the bending stiffness of the two joints at each end about the Y and Z axes, and Ks3 and Ks6 represent the rotation stiffness of the joints about the axis X. Translations of Points a and c (b and d) are coupled in the nodal directions x, y, and z, which means the lengths of spring elements in the model are zero. The nonlinear spring element COMBIN39 in ANSYS with unidirectional freedom and nonlinear characteristics is used to represent the bending and torsional capacity of a bolt-ball joint by inputting the real constants of the element COMBIN39. Both tube and ball in the model are simulated by an elasto-plastic beam element, and the section of the ball is greater than that of the tube because of its bigger bending stiffness.
Imperfection value Vd (mm)
Vd/5000
9 2 13 0 11 14 23 12 13 10 9 30 8 23 31 12 5 29 28 17 25 14 17 16 18 12 4 7 16 8 19
1/552 1/2041 1/398 0 1/460 1/346 1/220 1/421 1/371 1/481 1/552 1/169 1/611 1/220 1/163 1/414 1/913 1/172 1/181 1/292 1/202 1/351 1/290 1/311 1/276 1/412 1/1336 1/714 1/309 1/598 1/257
The FEA model of the test structure was established based on the member model, the bending–rotation curve of the bolt-ball joint (shown in Fig. 12), and the values of the initial geometrical imperfection in the shell (listed in Table 3). The analysis was carried out for the test structure including the initial geometric imperfections. Both material and geometric nonlinearity were considered in the analysis. The experimental and FE-based load–deflection response curves of the test structure are compared in Fig. 13. The results show that the two curves agree reasonably well, indicating that the developed FEA model [8,9] has yielded a good estimation of the experimentally observed behavior. The critical load calculated using the FEA model has a relative error of 1.66%, compared with the experimental result. The test was carried out under force control in the elastic range of the shell behavior, and each load step was approximately 3 kN. When the load was 21.2 kN, the displacement of node 16 was about 45 mm, and the test control was changed to displacement control until failure of the structure occurred. In the process, the displacement of node 16 increased gradually, and the structure lost its stability at a critical load of 22.43 kN. The deformation patterns of the structure obtained from the test and FEA at the critical load are shown in Fig. 14. To provide a clear picture of the displacement development of the structure, the deformation patterns of the structure at three different moments (as shown in Fig. 13) are given in Fig. 15 based on the FEA model. The load–displacement curves of nodes 13 and 20 obtained through the FEA and the test are shown in Fig. 16, and the comparison of stress–displacement curves at member f3 near node 16 are shown in Fig. 17. From the two figures, it can be seen that the FEA results are close to the experimental results before the structure loses its stability.
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Fig. 10. Material test. a) Test specimens and b) Test picture.
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Table 4 Mechanical properties of steel.
Moment 2
Coupon number
Yield strength fy (MPa)
Tensile strength fu (MPa)
Elastic modulus E (MPa)
1 2 3 4 Average value
260.48 254.69 267.42 254.69 259.32
347.01 334.28 339.05 327.11 336.86
1.61 105 1.58 105 1.66 105 1.50 105 1.59 105
25
Load (kN)
20
Moment 3 15 10
Test FEA with initial geometric imperfection
5
Moment 1 0 0.00
0.04
0.08
0.12
0.16
0.20
Displacement of node 16 (m) Fig. 13. Comparison of load–displacement curves at node 16.
Fig. 11. Member model.
3.0 2.5
M (kN⋅m)
2.0 1.5 1.0 0.5 Test 0.0 0.00
0.02
0.04
0.06
0.08
0.10
Φ (rad)
obtained by the FEA, as shown in Fig. 18. It can be seen from the figures that: (i) the buckling mode obtained from the FEA model is double symmetric, while the experimental buckling mode is not; (ii) In the FEA buckling mode, the upward displacement of the nodes on the 2 and 20 axis is bigger than the nodes on the 1 and 10 axis, which is opposite to the situation in the experimental buckling mode; (iii) The displacement trends of the nodes on the 1, 10 , 2 and 20 axis are all upward and they are the same in the two buckling modes; (iv) The displacement trends of the nodes besides the loading node on the 3 axis are opposite in the two buckling modes. There are some differences between the two buckling modes, but overall, the displacement trends of most nodes in the structure, except the nodes on the 3 axis, are the same. Finally, Fig. 19 compares the complete load–displacement curve from the test with the FEA results assuming rigid joints, semi-rigid joints, and pin joints, respectively. As expected, the experimental results are between the results of rigid joints and pin joints FE model, and closest to the semi-rigid joint model. Therefore, in the design of cylindrical reticulated shells, the hypothesis of pinned joints leads to structures with low capacity in terms of stability and resistance, and obviously, they cannot be regarded as rigidly jointed structures either. The bending stiffness should be considered in the FEA model for analyzing single-layer spatial structures.
Fig. 12. Moment–rotation curves of the joint.
4. Conclusions During the experiment, the node coordinates were recorded at two moments: before and after the buckling load. Then the first order buckling mode was obtained and compared with that
A series of experimental tests have been conducted on a semi-rigid bolt-ball joint and a 5 m 6 m single-layer cylindrical reticulated shell.
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Fig. 14. The deformation of the structure at the critical load. a) Test and b) FEA.
Fig. 15. The deformation of the structure.
30
Displacement of node 16 (mm)
200
Load (kN)
25 20 15 10
Node13(FEA) Node13(Test) Node20(FEA) Node20(Test)
5
5
10
15
20
25
120 80 40 0
0 0
160
Axial stress (Test) Axial stress (FEA) Bending stress (Test) Bending stress (FEA)
30
Displacement (mm) Fig. 16. Comparison of load–displacement curves.
The experimental investigation on the structure presented in this paper provides an initial insight into the mechanical performance of single-layer cylindrical reticulated shells with semi-rigid joints. The complete load–deflection response curve and the buckling mode of the test structure were obtained. FEA models were also established to carry out numerical analysis.
0
-50
-100
-150
-200
Stress at member f3 (MPa) Fig. 17. Comparison of stress–displacement curves.
Through comparison of the FEA results and the corresponding test results, it is shown that the proposed FEA model can be used to effectively predict the mechanical behavior of the reticulated shell including its buckling load. Furthermore, the work in this paper verified the FEA model of the single-layer spatial structures with semi-rigid joints used in [8,9], and it also verified that the
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Fig. 18. Comparisons of the buckling modes. a)Test (1:40) and b)FEA (1:40).
70
Test FEA (Rigid) FEA (Semi-rigid) FEA (Pinned)
60
Load (kN)
50 40 30 20 10 0 0.00
0.03
0.06
0.09
0.12
0.15
0.18
Displacement of node 16 (m) Fig. 19. Comparisons of load–displacement curves.
FEA model is valid for both the single-layer reticulated spherical domes and non-spherical domes. Acknowledgement This research is supported by grants from the Natural Science Foundation of China under Grant no. 51308153, China Postdoctoral Science Foundation funded project no. 2013M531020, Project HIT. NSRIF2014 Supported by the Fundamental Research Funds for the Central Universities Grant No. HIT. NSRIF. 2014099, and the International Program Development Fund from the University of Sydney. These supports are gratefully acknowledged. References [1] Fathelbab FA., The effect of joints on the stability of shallow single layer lattice domes. PhD thesis. University of Cambridge, September; 1987.
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