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An innovative and inexpensive experimental setup for testing connections in gridshell structures
Engineering Structures 207 (2020) 110257
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
An innovative and inexpensive experimental setup for testing connections in gridshell structures Hamed Seifi, Anooshe Rezaee Javan, Xiaoshan Lin, Yi Min Xie
T
⁎
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne, Victoria 3001, Australia
ARTICLE INFO
ABSTRACT
Keywords: Gridshell structure Structural node Test rig Topology optimization
The design and manufacture of structural nodes for gridshell structures are complicated due to complex geometries and loading conditions. The validation of the design concepts of these nodes is even more challenging, because complex design loads are difficult to be applied in a laboratory test. In this paper, a testing rig is proposed and manufactured to test two different additively manufactured nodes. The two nodes are symmetrical and each connecting three members. These nodes are designed to sustain pure axial loads and pure out-of-plane bending moments, respectively. The results of the experiments show the importance of the bolt tolerance in the design of such a testing setup. Subsequently an innovative and inexpensive experimental setup is developed for testing nodes under dominant design loading conditions in gridshell structures. The proposed testing method is generalized, which can be applied to both individual and combined loading conditions, and the new testing rig can be easily fabricated at a low cost.
1. Introduction Structural node is one of the most critical components in a gridshell structure, which is responsible for transferring loads in the system. The node stiffness could significantly affect the behaviour of a gridshell structure. Many numerical and experimental studies have been reported on the investigation of the effects of mechanical properties or imperfection of nodes on the behaviour of gridshell structures [1–7]. Therefore, it is very important to evaluate the effectiveness of the node design concept. However, it is usually difficult to simulate the complex geometrical, topological and loading conditions of a node in experiment. In most cases, the node conditions are highly simplified in the experimental tests. In the past, several approaches have been used in the design of test setup for structural nodes. The first approach is to simplify the geometry, topology and loading conditions of the test specimen. In the experimental study carried out by López et al. [8], nodes were simplified as 2-way nodes for single layer structures, and only the out-of-plane bending moment was applied. They used the conventional four-point bending test setup with mobile supports to obtain bending behaviour and rotational stiffness of the joints. The second approach is to apply loads directly onto the prototype of a selected part of the structure in order to test the performance of the connecting nodes. Although, in this method, the node is subjected to the
⁎
internal loads from the connected members, these applied loads are usually different from those generated in the whole structure. Besides, other nodes and members in the structure might be damaged during the test of one specific node. This approach was used by Lopez et al. [1] in the investigation of the influence of joint rigidity on the global behaviour of single-layer latticed dome structure. In their study, two experimental tests were conducted. The first experiment was on a simple structure with only one free node, and the second was carried out on a complete single-layer dome with a span of 7 m. During the tests, the load was applied on the central node, and the vertical displacement was measured. The second approach was also employed in the study conducted by Ma et al. [9] to investigate the influence of joint rigidity on the mechanical performances of squared plan-form single-layer structures. In addition, Tian et al. [10] and Wei et al. [11] constructed a model for the substructure of a long-span single-layer spatial grid structure to investigate its anti-progressive collapse mechanism. Eight full-scale specimens were considered in their studies. Their designed test setup consisted of a self-balanced spatial support system and a vertical reaction frame used to apply an upward load. The third approach is designed to be capable of covering all dominant loading and geometrical conditions of a structural node. In the full scale laboratory test conducted in Zhejiang University [12], loading devices were aligned with the directions of the design loads. In this test
Corresponding author. E-mail address: [email protected] (Y.M. Xie).
https://doi.org/10.1016/j.engstruct.2020.110257 Received 27 October 2019; Received in revised form 17 January 2020; Accepted 17 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.
Engineering Structures 207 (2020) 110257
H. Seifi, et al.
bending node are formed in planes parallel to the planes of the applied loads. The distance between planar substructures in bending node is larger than that in axial node. Additionally, a truss-like structure is formed in the bending node to connect top and bottom planar structures, while the planar structures in the axial node are connected by the non-design domain (blue parts).
setup, a large-scale spherical frame was needed to support the loading devices pointing to the node from different directions in space. Although this test rig is versatile, it is extremely expensive to construct. In this study, an innovative test setup is designed to apply axial load, out-of-plane bending moment and in-plane bending moment based on a deterministic mechanism. To achieve this, three simple operations are proposed, including applying eccentricities, changing load directions, and combining vertical loads. The designed test rig is more flexible compared to the first two approaches and is cheaper and easier to set up compared to the full-scale loading device.
2.2. Manufacturing of node Additive manufacturing is an automatic manufacturing method which could eliminate or minimize human mistakes during manufacturing process. The challenges of combining BESO as a design tool and additive manufacturing as manufacturing method have been formerly investigated by many other researches [14–19]. In these studies, BESO was employed to optimise the design of a node so as to achieve satisfactory structural performance with minimum amount of material, which also enabled the additive manufacturing of structural nodes. In this study, both axial node and bending node designed in Section 2.1 are 3D printed in stainless steel using binder jet method, a simpler and more convenient manufacturing method for small size objects compared to other methods like casting. In the Binder Jet method, a binder is selectively deposited onto the powder bed, bonded metal particles all together to form a fragile and porous part one layer at a time. The process of separating the powder from the 3D printed porous model (by using shaking table) is called de-powdering step. Then a solid and strong part will be made by flowing bronze into the pores in an infiltration process [20]. To avoid breaking the porous model during depowdering step, six auxiliary members are added to the model to be 3D printed and cut off before test. The 3D printed nodes and auxiliary members are shown in Fig. 3. To obtain the properties of the printed stainless steel, a series of tensile tests are carried out on six 3D printed dog-bone samples and the test results are shown in Fig. 4. As can be seen, the tested modulus of elasticity, strength and elongation for the 3D printed material vary in different extent. This uncertainty in material properties may cause uncertain results in the structural tests. For numerical simulation purpose, the average of the test data is employed and simplified by a multilinear curve. Fig. 8 shows the averaged result from the tensile tests, and the simplified stress-strain curve used for non-linear simulations. The printing direction of dog-bone sample is also shown in Fig. 5.
2. Design of test rig for structural nodes In this paper, an innovative test setup is designed to test nodes in gridshell structures, which are usually complex in terms of geometry, connectivity and load conditions. The combination of BESO design and additive manufacturing methods is chosen in this study. An automatic manufacturing method such as additive manufacturing (AM) can ease the manufacturing step of complex nodes, and to enable AM, an effective method is to use BESO in the design of nodes so that satisfactory structural performance with minimum material can be achieved. This combined method could be the trend of future construction industry. 2.1. Design of structural nodes In this study, structural nodes are designed using bi-directional evolutionary structural optimization (BESO) method which optimizes a structure by removing inefficient elements and adding elements where needed [13]. The highest structural stiffness can be achieved by using the smallest amount of material through BESO process. The process of adding and removing elements in BESO is applied to a predefined domain, called design domain. Other elements which are excluded from BESO process form non-design domain. In this paper, the BESO code for optimising the structural node is linked to ABAQUS as a structural analysis engine. BESO parameters, including evolutionary ratio, volume fraction and filter radius, are set to 2%, 95%, and 3 mm respectively. Laplacian smoothing algorithm is then applied to the optimised node to smooth rough surface caused by element removal. The geometry of the node, which is used as the initial model in BESO design, as well as its dimensions are shown in Fig. 1. In this model, the non-design domain consists of small rings around the bolt hole. Two symmetrical load cases applied in the design process include axial compressive loads and out-of-plane bending moment. Fig. 2 shows the rendered perspective views of the nodes designed for symmetrical bending moment and axial compressive forces. By comparing the BESO designs for axial forces (axial node) and out-of-plane bending moment (bending node), it can be clearly observed that the optimised planar topologies for both nodes are almost identical for top and bottom planes. In the axial node, loads are applied on the top and bottom planes, whereas the planar structures in the
2.3. Progressive design process of the test rig The design of test rig is evolved during a process in which the main idea of the final design is generated based on the necessities that the final design is capable of testing nodes under various loading conditions, such as bending, shear, axial and combined loads. Fig. 6 shows three test rigs generated during the design process. The first test rig is proposed for a symmetrical three-way node designed for resisting outof-plane bending moment. As shown in Fig. 6(a), the upper and lower
Fig. 1. Initial geometry of node used for BESO design (dimensions are in millimetres). 2
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Fig. 2. BESO designs for nodes under bending moment and axial forces.
Fig. 3. 3D printed axial node and bending node.
Fig. 4. Tensile test results for six 3D printed dog-bone samples.
parts of the test rig are designed to be clamped in the hydraulic jack. In this test rig, the out-of-plane bending moment is generated on each connecting member by applying two vertical forces in the opposite directions and with a certain eccentricity. This method may not be appropriate for full scale node test due to the limited space inside the testing machine. Besides, the test rig is customised for only one node under one loading case. In this design, the upper and lower parts of the test rig are complicated to be manufactured. In the following iterations shown in Fig. 6(b) and (c), the testing system is fixed onto the ground in order to eliminate the dependency on the size of testing machine. Also, to produce different bending moments, multiple bolt holes are considered in the connecting plate so that vertical loads can be applied in different locations. In the final design of the test rig (Fig. 6(d)), the distances between bolt holes and the dimensions of the connecting plates are determined
Fig. 5. Averaged result of tensile test and simplified stress-strain curve for numerical simulations. 3
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Fig. 6. Evolution of a test rig design for nodes in gridshell structures.
in a modular way in which different loading conditions are obtained by changing the configuration of plates and bolts. Configurations of the test rig for six different loading conditions, including bending, shear, compression, tension, and two combined loading conditions are shown in Fig. 7.
obtained from numerical models are shown in Figs. 9–11. The forcedisplacement curves measured at a reference point of each node are also shown in Figs. 9–11. As can be seen, the failure of the tension node occurs in the short members of the node as circled in Fig. 9. The failure in bending node only takes place in planar substructures in compression side of the node due to buckling, while the other side under combined bending and tension is not failed. Based on the predicted maximum loads that the structural node can sustain in different loading conditions, the loads required in the experimental tests can be calculated, considering the deterministic structural system of the test rig. Details of the applied loads and the dimensions of test rig in different tests are shown in Fig. 12.
2.4. Prediction of applied loads To pre-determine the load resisting capacities of the designed structural nodes, non-linear finite element analysis is carried out using Abaqus. To apply loads, the internal surfaces of the bolt holes are constrained to reference points. One set of the constrained joints and their respective reference points are shown in Fig. 8(a). The axial node is simulated under both tension and compression to evaluate the effect of buckling on the node behaviour when it is in compression. The loading condition is simulated by applying displacements or rotations to the reference points. In axial node, displacements in the direction of the connected beam members are applied. In bending node, out-of-plane rotations are applied to the reference points. Fig. 8(b) shows the applied displacements and rotations. The maximum forces applied to each bolt are 5kN and 2.63kN respectively for tension node and compression node, and the maximum bending moment applied to each branch of the bending node is 0.13kNm. The failure modes of tension, compression and bending nodes
2.5. Experiments In the experimental test of the axial node, the test rig is firstly assembled according to the configuration shown in Fig. 12 and then the node is placed in the designed location. It shall be mentioned that initial displacements and rotations of the rigid rotating plates (purple plates) are observed after setting up the test rig, as shown in Fig. 13. This is because of the difference between the diameters of bolt and its hole, leading to a movement of the bolt in the direction of the force being transferred by that bolt. The accumulation of these bolt movements may result in the rigid displacements and rotations in the rotating 4
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Fig. 7. Configurations of test rig for six different loading conditions.
Fig. 8. (a) Constraint definitions for bolt holes in axial node and bending node; and (b) symmetrical loading conditions.
plates, and hence change the force condition from what the mechanism is designed for. This defect is considered in the design of a generalised test setup in Section 3. Due to this initial movement, numerical simulation is conducted again, and the predicted results are found to be very different. The failure modes and the force-displacement and force-rotation curves of the compression node and bending node are shown in Fig. 14. As can be seen clearly from the curves, the applied load on each branch of the compression node increases rapidly from 3.7 kN at displacement of 3.0 mm (where all bolts become tight) to 8 kN at
displacement of 3.2 mm (where the node is failed). Similarly, the maximum applied moment on each branch of the bending node jumps from 0.04 kN m at rotation of 3.7° to 0.13 kN m at rotation of 4°. Both nodes are failed by bending in their members, which shows the deviation from the designed loading condition. Fig. 15 shows the failure of one axial node and one bending node due to applied loads in experiment. The failure of the compression nodes is due to the combination of axial force and compound bending, which matches very well with the failure mode obtained from the numerical simulation. As for the bending node, the top side (compression 5
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Fig. 11. Failure mode and force-displacement curve for node in out-of-plane bending.
side) is failed in combination of simple bending towards inside of the node and axial load, which is also in a good agreement with the predicted results. In the experimental tests, the maximum loads applied by the hydraulic jack on four axial nodes are 4.65 kN, 5.72 kN, 5.31 kN, 5.40 kN. Also, the maximum applied load for one bending node is 1.18 kN. The measured loads are equivalent to the axial loads of 6.35 kN, 7.81 kN, 7.25 kN and 7.38 kN to be applied on the axial node, and bending moment of 0.19 kN m to be applied on the bending node. The measured maximum loads for the compression nodes are 2–20% less than the predicted maximum axial load, while the peak bending moment recorded for bending node during the test is much higher than that obtained from numerical analysis. The differences between experimental results and numerical predictions might be explained by the following reasons. Firstly, the small-scale test increases the tolerances, and consequently the errors. The uncertainty of the material properties observed in the 3D printed dog-bone samples may also lead to inaccurate numerical predictions. Lastly, the deviation of the manufactured test rig from the ideal test condition has brought more uncertainty to the experimental results.
Fig. 9. Failure mode and force-displacement curve for node in tension.
3. Generalized concept for design of test setup At present, very limited testing facilities that currently exist can respond to the needs of the daily increasing gridshell market, and a versatile and easily accessible testing equipment is in high demand. Although the designed test rig is customised for some specific load cases and geometry conditions (three-way node under tension, compression, shear and bending), the rules applied in the design process are general. 3.1. Generalized concept
Fig. 10. Failure mode and force-displacement curve for node in compression.
In the test setup introduced in this section, three types of load are
6
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Fig. 12. Test setup for compression node and bending node.
Fig. 13. Initial rigid rotation and displacement in the assembled test setup for axial node.
applied to the node at the same time, including axial load, out-of-plane bending moment and in-plane bending moment. The idea of this setup is achieved by combining three simple operations. The first operation is to apply out-of-plane bending and in-plane bending moments by making use of the eccentricity of the axial load at each connecting face of the node (Fig. 16). The second operation is to change the direction of the applied loads (vertical loads) to the desired direction (design loads) by using a rotating part as a mechanism. This operation is schematically shown in Fig. 17. As can be seen in Fig. 17, in the proposed test rig, pin connections are used to support the rotating parts, which would generate minimal friction and reduce the influence on the rotational freedom. When design loads are applied, two pairs of reaction forces (dashed vectors) are generated in the opposite directions of the applied loads. These reaction forces are then combined and transferred to the ground through the pin support. As the structural system of the test rig is deterministic, the movement in each bolt can be obtained easily by solving the equilibrium equations. Therefore, the initial rigid movements of the plates induced by the looseness of the bolts can be cancelled by moving back the bolt holes in the directions of the loads. The amount this adjustment
shall be half of the difference between the diameter of the bolt and the diameter of the bolt hole. Lastly, the third operation is to combine all vertical loads into one resultant vertical load as the only controlling load in the test setup (Fig. 18). The location of the resultant of two loads can be calculated as:
D1 =
F2 × L F ×L , D2 = 1 F1 + F2 F1 + F2
(1a)
3.2. Design procedure To introduce the design procedures of the new test setup, the arbitrary geometry condition used for the nodes in the authors’ previous study [14] is used as an example herein. The geometrical parameters which are used in the node designs are listed in Table 1. Three angles α, β, γ and a distance d are used to define the direction and position of the connecting members, where α is the angle between the x axis and the projection of the member on the x-y plane, β is the angle between the member direction and the x-y plane, γ is the rotation angle of the members along its own centre line, and d is the distance from the centre point of the node to the closer end section of the 7
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Fig. 14. Failure mode and load-displacement curve of compression and bending node obtained from non-linear simulation.
Fig. 15. Testing of axial and bending nodes.
member. The width and the height of all connecting members are 80 mm and 180 mm respectively. The geometry of the node is shown in Fig. 19. It should be noted that the test rig design procedures discussed here are for testing nodes under individual loading case. Similar procedures can be applied to the design for combined loading case. The first step is to calculate the eccentricities and apply the axial loads at the eccentric centres of the connecting faces (Fig. 19). In this example, constant eccentricities of 30 mm are applied in both in-plane and out-of-plane directions. In the second step, eye-connection plates are designed to be vertically mounted on the connecting members as shown in Fig. 20. As discussed before, the positions of bolt holes on these plates should be determined with the consideration of the bolt looseness, so that after initial movement, the bolts should be placed in the right positions as
they are designed for. As can be seen in Fig. 20, the vertical plates are parallel to the connected members’ centre-lines. In the third step, the rigid rotating plates are designed to transform the applied vertical load (Vi) to the design load (Fi) as shown in Fig. 21. The last step is to design a set of balanced beams for combining vertical loads to obtain one resultant load (F) as shown in Fig. 22. 4. Discussion and conclusions In this study, an innovative test rig is designed to test two different types of nodes. By slightly adjusting the setup, the rig can be used to test structural nodes under various loading conditions. As the looseness of the bolts in the test rig may lead to changes in the designed loads, the initial movements of the connecting plates should be considered in the prediction of testing load. The results of the experiments are compared 8
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Fig. 16. Eccentricities in out-of-plane bending and in-plane bending moments.
Fig. 17. Schematic of applying loads using a rotating part.
Table 1 Geometrical design parameters of case study node. Member
L1 L2 L3 L4 L5 L6
Rotation angles (degrees) α
β
γ
Distance of the member from the centre (mm) d
0 50 130 175 220 290
6 8 1 7 2 10
8 4 7 9 4 3
195.712 199.272 198.155 208.345 199.771 173.877
third operation is to combine vertical loads from different branches of the node into one vertical load to be applied to the system. The new experimental setup developed in this study is general, inexpensive and able to simulate the dominant loading conditions of the nodes in gridshell structures. It is worth to mention that the design of the test setup in this paper depends only on the geometrical, topological and loading condition of the node. The internal structure of the node can be designed and manufactured using any method and any material.
Fig. 18. Combination of vertical loads into one resultant load.
to the results of non-linear finite element simulations of the real test conditions. The comparison shows good agreement in failure mode of the nodes, which confirms the concept of the proposed test setup. Differences are found between the predicted and observed maximum loads, which could be attributed to uncertainties of testing conditions. To have a more generalized test rig which is applicable to both individual and combined loading scenarios, an innovative design concept for test setup is proposed, which is based on three operations. The first operation is to apply suitable eccentricities to the axial design loads to introduce bending moments. The second operation is to change the direction of the vertical load in each branch to generate axial load. The
CRediT authorship contribution statement Hamed Seifi: Conceptualization, Investigation, Methodology, Validation, Writing - original draft. Anooshe Rezaee Javan: Writing original draft. Xiaoshan Lin: Supervision, Writing - review & editing. Yi Min Xie: Supervision, Conceptualization, Methodology. 9
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Fig. 19. Application of eccentric axial loads.
Fig. 20. Eye-connection plates.
Fig. 21. Rigid rotating part for changing direction and magnitude of applied vertical load. 10
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Fig. 22. Applying axial loads and bending moments through single resultant force.
Declaration of Competing Interest
2007;29(1):101–9. [7] 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. [8] Lopez A, Puente I, Aizpurua H. Experimental and analytical studies on the rotational stiffness of joints for single-layer structures. Eng Struct 2011;33(3):731–7. [9] Ma HH, Fan F, Wen P, Zhang H, Shen SZ. Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints. Thin Wall Struct 2015;86:1–9. [10] Tian L, Wei J, Hao J. Anti-progressive collapse mechanism of long-span single-layer spatial grid structures. J Constr Steel Res 2018;144:270–82. [11] Wei JP, Tian LM, Hao JP. Improving the progressive collapse resistance of long-span single-layer spatial grid structures. Constr Build Mater 2018;171:96–108. [12] Chen M, Xing D, Xao Y, Su L, Dong S, Wang D, et al. Experimental research and finite element analysis on joints of Sun Valley steel structure for the Expo Axis project. J Build Struct 2010;31(5):34–41. [13] Huang X, Xie YM. Evolutionary topology optimization of continuum structures: methods and applications. Chichester: Wiley; 2010. [14] Seifi H, Rezaee Javan A, Xu S, Zhao Y, Xie YM. Design optimization and additive manufacturing of nodes in gridshell structures. Eng Struct 2018;160:161–70. [15] van der Linden LPL. Innovative joints for gridshells Master thesis Delft University of Technology; 2015 [16] O’Donnell J, Seifi H, Sitler B, Williams N, Crolla K, Xie YM. Smart nodes pavilion: bi-directional evolutionary structural optimization and additive manufacturing. Proceedings of the international association for shell and spatial structures symposium. 2015. [17] Seifi H, Xie YM, O’Donnell J, Williams N. Design and fabrication of structural connections using bi-directional evolutionary structural optimization and additive manufacturing. In: 2nd Australian conference for computational mechanics 2015; 571–76. [18] Williams N, Prohasky D, Burry J, Crolla K, Leary M, Brandt M, et al. Challenges of scale modelling material behaviour of additive-manufactured nodes. In: 5th designing modelling symposium 2015; 45–51. [19] Galjaard S, Hofman S, Ren S. New opportunities to optimize structural designs in metal by using additive manufacturing. Adv Archit Geometry 2014:79–93. [20] Varotsis AB. Introduction to binder jetting 3D printing. Available from: https:// www.3dhubs.com/knowledge-base/introduction-binder-jetting-3d-printing.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This project was supported by the National Natural Science Foundation of China (51778283) and the Australian Research Council (DP200102190). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2020.110257. References [1] López A, Puente I, Serna MA. Numerical model and experimental tests on singlelayer latticed domes with semi-rigid joints. Comput Struct 2007;85(7–8):360–74. [2] Fathelbab FA. The effect of joints on the stability of shallow single layer lattice domes PhD thesis University of Cambridge; 1987 [3] See T, McConnel RE. Large displacement elastic buckling of space structures. Struct Eng 1986;112(5):1052–69. [4] Fan F, Ma HH, Cao Z, Shen SZ. A new classification system for the joints used in lattice shells. Thin Wall Struct 2011;49(12):1544–53. [5] Kato S, Mutoh I, Shomura M. Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections. J Constr Steel Res 1998;48(2–3):145–68. [6] López A, Puente I, Serna MA. Direct evaluation of the buckling loads of semi-rigidly jointed single-layer latticed domes under symmetric loading. Eng Struct
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Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
An innovative and inexpensive experimental setup for testing connections in gridshell structures Hamed Seifi, Anooshe Rezaee Javan, Xiaoshan Lin, Yi Min Xie
T
⁎
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne, Victoria 3001, Australia
ARTICLE INFO
ABSTRACT
Keywords: Gridshell structure Structural node Test rig Topology optimization
The design and manufacture of structural nodes for gridshell structures are complicated due to complex geometries and loading conditions. The validation of the design concepts of these nodes is even more challenging, because complex design loads are difficult to be applied in a laboratory test. In this paper, a testing rig is proposed and manufactured to test two different additively manufactured nodes. The two nodes are symmetrical and each connecting three members. These nodes are designed to sustain pure axial loads and pure out-of-plane bending moments, respectively. The results of the experiments show the importance of the bolt tolerance in the design of such a testing setup. Subsequently an innovative and inexpensive experimental setup is developed for testing nodes under dominant design loading conditions in gridshell structures. The proposed testing method is generalized, which can be applied to both individual and combined loading conditions, and the new testing rig can be easily fabricated at a low cost.
1. Introduction Structural node is one of the most critical components in a gridshell structure, which is responsible for transferring loads in the system. The node stiffness could significantly affect the behaviour of a gridshell structure. Many numerical and experimental studies have been reported on the investigation of the effects of mechanical properties or imperfection of nodes on the behaviour of gridshell structures [1–7]. Therefore, it is very important to evaluate the effectiveness of the node design concept. However, it is usually difficult to simulate the complex geometrical, topological and loading conditions of a node in experiment. In most cases, the node conditions are highly simplified in the experimental tests. In the past, several approaches have been used in the design of test setup for structural nodes. The first approach is to simplify the geometry, topology and loading conditions of the test specimen. In the experimental study carried out by López et al. [8], nodes were simplified as 2-way nodes for single layer structures, and only the out-of-plane bending moment was applied. They used the conventional four-point bending test setup with mobile supports to obtain bending behaviour and rotational stiffness of the joints. The second approach is to apply loads directly onto the prototype of a selected part of the structure in order to test the performance of the connecting nodes. Although, in this method, the node is subjected to the
⁎
internal loads from the connected members, these applied loads are usually different from those generated in the whole structure. Besides, other nodes and members in the structure might be damaged during the test of one specific node. This approach was used by Lopez et al. [1] in the investigation of the influence of joint rigidity on the global behaviour of single-layer latticed dome structure. In their study, two experimental tests were conducted. The first experiment was on a simple structure with only one free node, and the second was carried out on a complete single-layer dome with a span of 7 m. During the tests, the load was applied on the central node, and the vertical displacement was measured. The second approach was also employed in the study conducted by Ma et al. [9] to investigate the influence of joint rigidity on the mechanical performances of squared plan-form single-layer structures. In addition, Tian et al. [10] and Wei et al. [11] constructed a model for the substructure of a long-span single-layer spatial grid structure to investigate its anti-progressive collapse mechanism. Eight full-scale specimens were considered in their studies. Their designed test setup consisted of a self-balanced spatial support system and a vertical reaction frame used to apply an upward load. The third approach is designed to be capable of covering all dominant loading and geometrical conditions of a structural node. In the full scale laboratory test conducted in Zhejiang University [12], loading devices were aligned with the directions of the design loads. In this test
Corresponding author. E-mail address: [email protected] (Y.M. Xie).
https://doi.org/10.1016/j.engstruct.2020.110257 Received 27 October 2019; Received in revised form 17 January 2020; Accepted 17 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.
Engineering Structures 207 (2020) 110257
H. Seifi, et al.
bending node are formed in planes parallel to the planes of the applied loads. The distance between planar substructures in bending node is larger than that in axial node. Additionally, a truss-like structure is formed in the bending node to connect top and bottom planar structures, while the planar structures in the axial node are connected by the non-design domain (blue parts).
setup, a large-scale spherical frame was needed to support the loading devices pointing to the node from different directions in space. Although this test rig is versatile, it is extremely expensive to construct. In this study, an innovative test setup is designed to apply axial load, out-of-plane bending moment and in-plane bending moment based on a deterministic mechanism. To achieve this, three simple operations are proposed, including applying eccentricities, changing load directions, and combining vertical loads. The designed test rig is more flexible compared to the first two approaches and is cheaper and easier to set up compared to the full-scale loading device.
2.2. Manufacturing of node Additive manufacturing is an automatic manufacturing method which could eliminate or minimize human mistakes during manufacturing process. The challenges of combining BESO as a design tool and additive manufacturing as manufacturing method have been formerly investigated by many other researches [14–19]. In these studies, BESO was employed to optimise the design of a node so as to achieve satisfactory structural performance with minimum amount of material, which also enabled the additive manufacturing of structural nodes. In this study, both axial node and bending node designed in Section 2.1 are 3D printed in stainless steel using binder jet method, a simpler and more convenient manufacturing method for small size objects compared to other methods like casting. In the Binder Jet method, a binder is selectively deposited onto the powder bed, bonded metal particles all together to form a fragile and porous part one layer at a time. The process of separating the powder from the 3D printed porous model (by using shaking table) is called de-powdering step. Then a solid and strong part will be made by flowing bronze into the pores in an infiltration process [20]. To avoid breaking the porous model during depowdering step, six auxiliary members are added to the model to be 3D printed and cut off before test. The 3D printed nodes and auxiliary members are shown in Fig. 3. To obtain the properties of the printed stainless steel, a series of tensile tests are carried out on six 3D printed dog-bone samples and the test results are shown in Fig. 4. As can be seen, the tested modulus of elasticity, strength and elongation for the 3D printed material vary in different extent. This uncertainty in material properties may cause uncertain results in the structural tests. For numerical simulation purpose, the average of the test data is employed and simplified by a multilinear curve. Fig. 8 shows the averaged result from the tensile tests, and the simplified stress-strain curve used for non-linear simulations. The printing direction of dog-bone sample is also shown in Fig. 5.
2. Design of test rig for structural nodes In this paper, an innovative test setup is designed to test nodes in gridshell structures, which are usually complex in terms of geometry, connectivity and load conditions. The combination of BESO design and additive manufacturing methods is chosen in this study. An automatic manufacturing method such as additive manufacturing (AM) can ease the manufacturing step of complex nodes, and to enable AM, an effective method is to use BESO in the design of nodes so that satisfactory structural performance with minimum material can be achieved. This combined method could be the trend of future construction industry. 2.1. Design of structural nodes In this study, structural nodes are designed using bi-directional evolutionary structural optimization (BESO) method which optimizes a structure by removing inefficient elements and adding elements where needed [13]. The highest structural stiffness can be achieved by using the smallest amount of material through BESO process. The process of adding and removing elements in BESO is applied to a predefined domain, called design domain. Other elements which are excluded from BESO process form non-design domain. In this paper, the BESO code for optimising the structural node is linked to ABAQUS as a structural analysis engine. BESO parameters, including evolutionary ratio, volume fraction and filter radius, are set to 2%, 95%, and 3 mm respectively. Laplacian smoothing algorithm is then applied to the optimised node to smooth rough surface caused by element removal. The geometry of the node, which is used as the initial model in BESO design, as well as its dimensions are shown in Fig. 1. In this model, the non-design domain consists of small rings around the bolt hole. Two symmetrical load cases applied in the design process include axial compressive loads and out-of-plane bending moment. Fig. 2 shows the rendered perspective views of the nodes designed for symmetrical bending moment and axial compressive forces. By comparing the BESO designs for axial forces (axial node) and out-of-plane bending moment (bending node), it can be clearly observed that the optimised planar topologies for both nodes are almost identical for top and bottom planes. In the axial node, loads are applied on the top and bottom planes, whereas the planar structures in the
2.3. Progressive design process of the test rig The design of test rig is evolved during a process in which the main idea of the final design is generated based on the necessities that the final design is capable of testing nodes under various loading conditions, such as bending, shear, axial and combined loads. Fig. 6 shows three test rigs generated during the design process. The first test rig is proposed for a symmetrical three-way node designed for resisting outof-plane bending moment. As shown in Fig. 6(a), the upper and lower
Fig. 1. Initial geometry of node used for BESO design (dimensions are in millimetres). 2
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Fig. 2. BESO designs for nodes under bending moment and axial forces.
Fig. 3. 3D printed axial node and bending node.
Fig. 4. Tensile test results for six 3D printed dog-bone samples.
parts of the test rig are designed to be clamped in the hydraulic jack. In this test rig, the out-of-plane bending moment is generated on each connecting member by applying two vertical forces in the opposite directions and with a certain eccentricity. This method may not be appropriate for full scale node test due to the limited space inside the testing machine. Besides, the test rig is customised for only one node under one loading case. In this design, the upper and lower parts of the test rig are complicated to be manufactured. In the following iterations shown in Fig. 6(b) and (c), the testing system is fixed onto the ground in order to eliminate the dependency on the size of testing machine. Also, to produce different bending moments, multiple bolt holes are considered in the connecting plate so that vertical loads can be applied in different locations. In the final design of the test rig (Fig. 6(d)), the distances between bolt holes and the dimensions of the connecting plates are determined
Fig. 5. Averaged result of tensile test and simplified stress-strain curve for numerical simulations. 3
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Fig. 6. Evolution of a test rig design for nodes in gridshell structures.
in a modular way in which different loading conditions are obtained by changing the configuration of plates and bolts. Configurations of the test rig for six different loading conditions, including bending, shear, compression, tension, and two combined loading conditions are shown in Fig. 7.
obtained from numerical models are shown in Figs. 9–11. The forcedisplacement curves measured at a reference point of each node are also shown in Figs. 9–11. As can be seen, the failure of the tension node occurs in the short members of the node as circled in Fig. 9. The failure in bending node only takes place in planar substructures in compression side of the node due to buckling, while the other side under combined bending and tension is not failed. Based on the predicted maximum loads that the structural node can sustain in different loading conditions, the loads required in the experimental tests can be calculated, considering the deterministic structural system of the test rig. Details of the applied loads and the dimensions of test rig in different tests are shown in Fig. 12.
2.4. Prediction of applied loads To pre-determine the load resisting capacities of the designed structural nodes, non-linear finite element analysis is carried out using Abaqus. To apply loads, the internal surfaces of the bolt holes are constrained to reference points. One set of the constrained joints and their respective reference points are shown in Fig. 8(a). The axial node is simulated under both tension and compression to evaluate the effect of buckling on the node behaviour when it is in compression. The loading condition is simulated by applying displacements or rotations to the reference points. In axial node, displacements in the direction of the connected beam members are applied. In bending node, out-of-plane rotations are applied to the reference points. Fig. 8(b) shows the applied displacements and rotations. The maximum forces applied to each bolt are 5kN and 2.63kN respectively for tension node and compression node, and the maximum bending moment applied to each branch of the bending node is 0.13kNm. The failure modes of tension, compression and bending nodes
2.5. Experiments In the experimental test of the axial node, the test rig is firstly assembled according to the configuration shown in Fig. 12 and then the node is placed in the designed location. It shall be mentioned that initial displacements and rotations of the rigid rotating plates (purple plates) are observed after setting up the test rig, as shown in Fig. 13. This is because of the difference between the diameters of bolt and its hole, leading to a movement of the bolt in the direction of the force being transferred by that bolt. The accumulation of these bolt movements may result in the rigid displacements and rotations in the rotating 4
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Fig. 7. Configurations of test rig for six different loading conditions.
Fig. 8. (a) Constraint definitions for bolt holes in axial node and bending node; and (b) symmetrical loading conditions.
plates, and hence change the force condition from what the mechanism is designed for. This defect is considered in the design of a generalised test setup in Section 3. Due to this initial movement, numerical simulation is conducted again, and the predicted results are found to be very different. The failure modes and the force-displacement and force-rotation curves of the compression node and bending node are shown in Fig. 14. As can be seen clearly from the curves, the applied load on each branch of the compression node increases rapidly from 3.7 kN at displacement of 3.0 mm (where all bolts become tight) to 8 kN at
displacement of 3.2 mm (where the node is failed). Similarly, the maximum applied moment on each branch of the bending node jumps from 0.04 kN m at rotation of 3.7° to 0.13 kN m at rotation of 4°. Both nodes are failed by bending in their members, which shows the deviation from the designed loading condition. Fig. 15 shows the failure of one axial node and one bending node due to applied loads in experiment. The failure of the compression nodes is due to the combination of axial force and compound bending, which matches very well with the failure mode obtained from the numerical simulation. As for the bending node, the top side (compression 5
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Fig. 11. Failure mode and force-displacement curve for node in out-of-plane bending.
side) is failed in combination of simple bending towards inside of the node and axial load, which is also in a good agreement with the predicted results. In the experimental tests, the maximum loads applied by the hydraulic jack on four axial nodes are 4.65 kN, 5.72 kN, 5.31 kN, 5.40 kN. Also, the maximum applied load for one bending node is 1.18 kN. The measured loads are equivalent to the axial loads of 6.35 kN, 7.81 kN, 7.25 kN and 7.38 kN to be applied on the axial node, and bending moment of 0.19 kN m to be applied on the bending node. The measured maximum loads for the compression nodes are 2–20% less than the predicted maximum axial load, while the peak bending moment recorded for bending node during the test is much higher than that obtained from numerical analysis. The differences between experimental results and numerical predictions might be explained by the following reasons. Firstly, the small-scale test increases the tolerances, and consequently the errors. The uncertainty of the material properties observed in the 3D printed dog-bone samples may also lead to inaccurate numerical predictions. Lastly, the deviation of the manufactured test rig from the ideal test condition has brought more uncertainty to the experimental results.
Fig. 9. Failure mode and force-displacement curve for node in tension.
3. Generalized concept for design of test setup At present, very limited testing facilities that currently exist can respond to the needs of the daily increasing gridshell market, and a versatile and easily accessible testing equipment is in high demand. Although the designed test rig is customised for some specific load cases and geometry conditions (three-way node under tension, compression, shear and bending), the rules applied in the design process are general. 3.1. Generalized concept
Fig. 10. Failure mode and force-displacement curve for node in compression.
In the test setup introduced in this section, three types of load are
6
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Fig. 12. Test setup for compression node and bending node.
Fig. 13. Initial rigid rotation and displacement in the assembled test setup for axial node.
applied to the node at the same time, including axial load, out-of-plane bending moment and in-plane bending moment. The idea of this setup is achieved by combining three simple operations. The first operation is to apply out-of-plane bending and in-plane bending moments by making use of the eccentricity of the axial load at each connecting face of the node (Fig. 16). The second operation is to change the direction of the applied loads (vertical loads) to the desired direction (design loads) by using a rotating part as a mechanism. This operation is schematically shown in Fig. 17. As can be seen in Fig. 17, in the proposed test rig, pin connections are used to support the rotating parts, which would generate minimal friction and reduce the influence on the rotational freedom. When design loads are applied, two pairs of reaction forces (dashed vectors) are generated in the opposite directions of the applied loads. These reaction forces are then combined and transferred to the ground through the pin support. As the structural system of the test rig is deterministic, the movement in each bolt can be obtained easily by solving the equilibrium equations. Therefore, the initial rigid movements of the plates induced by the looseness of the bolts can be cancelled by moving back the bolt holes in the directions of the loads. The amount this adjustment
shall be half of the difference between the diameter of the bolt and the diameter of the bolt hole. Lastly, the third operation is to combine all vertical loads into one resultant vertical load as the only controlling load in the test setup (Fig. 18). The location of the resultant of two loads can be calculated as:
D1 =
F2 × L F ×L , D2 = 1 F1 + F2 F1 + F2
(1a)
3.2. Design procedure To introduce the design procedures of the new test setup, the arbitrary geometry condition used for the nodes in the authors’ previous study [14] is used as an example herein. The geometrical parameters which are used in the node designs are listed in Table 1. Three angles α, β, γ and a distance d are used to define the direction and position of the connecting members, where α is the angle between the x axis and the projection of the member on the x-y plane, β is the angle between the member direction and the x-y plane, γ is the rotation angle of the members along its own centre line, and d is the distance from the centre point of the node to the closer end section of the 7
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Fig. 14. Failure mode and load-displacement curve of compression and bending node obtained from non-linear simulation.
Fig. 15. Testing of axial and bending nodes.
member. The width and the height of all connecting members are 80 mm and 180 mm respectively. The geometry of the node is shown in Fig. 19. It should be noted that the test rig design procedures discussed here are for testing nodes under individual loading case. Similar procedures can be applied to the design for combined loading case. The first step is to calculate the eccentricities and apply the axial loads at the eccentric centres of the connecting faces (Fig. 19). In this example, constant eccentricities of 30 mm are applied in both in-plane and out-of-plane directions. In the second step, eye-connection plates are designed to be vertically mounted on the connecting members as shown in Fig. 20. As discussed before, the positions of bolt holes on these plates should be determined with the consideration of the bolt looseness, so that after initial movement, the bolts should be placed in the right positions as
they are designed for. As can be seen in Fig. 20, the vertical plates are parallel to the connected members’ centre-lines. In the third step, the rigid rotating plates are designed to transform the applied vertical load (Vi) to the design load (Fi) as shown in Fig. 21. The last step is to design a set of balanced beams for combining vertical loads to obtain one resultant load (F) as shown in Fig. 22. 4. Discussion and conclusions In this study, an innovative test rig is designed to test two different types of nodes. By slightly adjusting the setup, the rig can be used to test structural nodes under various loading conditions. As the looseness of the bolts in the test rig may lead to changes in the designed loads, the initial movements of the connecting plates should be considered in the prediction of testing load. The results of the experiments are compared 8
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Fig. 16. Eccentricities in out-of-plane bending and in-plane bending moments.
Fig. 17. Schematic of applying loads using a rotating part.
Table 1 Geometrical design parameters of case study node. Member
L1 L2 L3 L4 L5 L6
Rotation angles (degrees) α
β
γ
Distance of the member from the centre (mm) d
0 50 130 175 220 290
6 8 1 7 2 10
8 4 7 9 4 3
195.712 199.272 198.155 208.345 199.771 173.877
third operation is to combine vertical loads from different branches of the node into one vertical load to be applied to the system. The new experimental setup developed in this study is general, inexpensive and able to simulate the dominant loading conditions of the nodes in gridshell structures. It is worth to mention that the design of the test setup in this paper depends only on the geometrical, topological and loading condition of the node. The internal structure of the node can be designed and manufactured using any method and any material.
Fig. 18. Combination of vertical loads into one resultant load.
to the results of non-linear finite element simulations of the real test conditions. The comparison shows good agreement in failure mode of the nodes, which confirms the concept of the proposed test setup. Differences are found between the predicted and observed maximum loads, which could be attributed to uncertainties of testing conditions. To have a more generalized test rig which is applicable to both individual and combined loading scenarios, an innovative design concept for test setup is proposed, which is based on three operations. The first operation is to apply suitable eccentricities to the axial design loads to introduce bending moments. The second operation is to change the direction of the vertical load in each branch to generate axial load. The
CRediT authorship contribution statement Hamed Seifi: Conceptualization, Investigation, Methodology, Validation, Writing - original draft. Anooshe Rezaee Javan: Writing original draft. Xiaoshan Lin: Supervision, Writing - review & editing. Yi Min Xie: Supervision, Conceptualization, Methodology. 9
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Fig. 19. Application of eccentric axial loads.
Fig. 20. Eye-connection plates.
Fig. 21. Rigid rotating part for changing direction and magnitude of applied vertical load. 10
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Fig. 22. Applying axial loads and bending moments through single resultant force.
Declaration of Competing Interest
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This project was supported by the National Natural Science Foundation of China (51778283) and the Australian Research Council (DP200102190). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2020.110257. References [1] López A, Puente I, Serna MA. Numerical model and experimental tests on singlelayer latticed domes with semi-rigid joints. Comput Struct 2007;85(7–8):360–74. [2] Fathelbab FA. The effect of joints on the stability of shallow single layer lattice domes PhD thesis University of Cambridge; 1987 [3] See T, McConnel RE. Large displacement elastic buckling of space structures. Struct Eng 1986;112(5):1052–69. [4] Fan F, Ma HH, Cao Z, Shen SZ. A new classification system for the joints used in lattice shells. Thin Wall Struct 2011;49(12):1544–53. [5] Kato S, Mutoh I, Shomura M. Collapse of semi-rigidly jointed reticulated domes with initial geometric imperfections. J Constr Steel Res 1998;48(2–3):145–68. [6] López A, Puente I, Serna MA. Direct evaluation of the buckling loads of semi-rigidly jointed single-layer latticed domes under symmetric loading. Eng Struct
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