Bolted joints for small and medium reticulated timber ...73 Achie of Ciil and Mechanical Engineeing...

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Archives of Civil and Mechanical Engineering (2020) 20:73 https://doi.org/10.1007/s43452-020-00073-7

ORIGINAL ARTICLE

Bolted joints for small and medium reticulated timber domes: experimental study, numerical simulation, and design strength estimation

Zhan Shu1 · Zheng Li2 · Minjuan He2 · Fei Chen2 · Chao Hu2 · Feng Liang2,3

Received: 4 February 2020 / Revised: 24 April 2020 / Accepted: 28 May 2020 / Published online: 10 June 2020 © Wroclaw University of Science and Technology 2020

AbstractThis paper proposes a new type of bolted glulam joint for small-span and medium-span reticulated timber dome structures. The joint fastens the timber elements and the angled slotted-in steel plates together with steel bolts. Reasonably simplified experiments were designed and conducted to understand the mechanical properties of the proposed joint. Finite element models were also developed and calibrated with the tested results. A four-line model was provided to explain the mechanical properties of the joints, which were observed from the tests and simulations. To facilitate the future use of the proposed joint, theoretical derivations were provided to estimate its mechanical features. According to the estimation equations, bilinear moment–rotation curves could be easily obtained for the joints with different wood species, member sizes, joint designs, and/or bolt diameters. Finally, full-size structural models were created to investigate the static stability of K6 single-layered reticulated timber domes with the proposed joints. The influences on the ultimate structural stability capacity from the span, the rise-to-span ratio, the joint model (i.e., stiffness), the initial geometric imperfection introduced from the construction, and the load distribution were systematically investigated.

Keywords Reticulated dome · Timber structure · Spatial structure · Bolted joint · Stability analysis

1 Introduction

The single-layered reticulated domes/shells are commonly seen type of spatial structures. Due to the unique single-layered feature, some proper structural design and evaluation become necessary to avoid the damages to the joints and the members against special loads. Geometric and material nonlinearities shall be considered [1, 2]. Besides, similar to other spatial structures, the focal aspects of the single-layered reticulated domes are the stability analyses against static [3] and dynamic loads [4], the seismic performance [5], the progressive collapse mechanism [6], etc.

Meanwhile, domes built with different types of materials could provide various aesthetic perceptions. In the structural respect, many of existing studies investigate the performance of steel reticulated domes. Different types of joints were proposed and their semi-rigid characteristics were inves-tigated. For example, some of the existing joints are the TEMCOR joint [7], the socket joints [8], the welded hollow spherical joints [3], the bolted-arm joints [9, 10], the gear joints [11], and the welded grid shell joints [12]. Besides, aluminum dome joints were also investigated [13–15], try-ing to take advantage of the lightness of the material. Zhai et al. [16] investigated a liquefied natural gas tank structure with a steel panel covered by the reinforced concrete dome, which could be considered as a hybrid spatial structure. In addition, the dynamic behavior of a dome with substructure constructed underneath was investigated [17]. Considera-tion of the substructure’s vibration amplifications as well as the uncoordinated displacement of the sub-columns was proved necessary [18].

Wood is a unique construction material with a renewable characteristic that fosters the human–environment, in a way that it helps to reconstruct the environment contaminated

* Zheng Li [email protected]

1 Department of Civil Engineering, Shanghai University, Shanghai 200444, China

2 Department of Structural Engineering, Tongji University, Shanghai 200092, China

3 Tongji Architectural Design (Group) Co., Ltd., Shanghai 200092, China

http://orcid.org/0000-0002-1374-1698

http://orcid.org/0000-0003-1227-8168

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Archives of Civil and Mechanical Engineering (2020) 20:73

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by the heavy industry products such as steel or concrete [19, 20]. Consequently, the reticulated timber domes are also becoming more popular over the recent years [21, 22]. The world-famous timber domes include but are not limited to the Tacoma Dome in Tacoma, Washington, USA [23]; the Saldome 2 in Aargau, Swiss [24]; the wooden domes in Brindisi, Italy [25]; and the Odate dome in Odate, Akita, Japan [26]. Not only the architectural beauty of timber domes is widely appreciated by architects, their structural performances such as the lightness and ease of construc-tion are also favorable to structural engineers. Early studies started from multiple-layered timber spatial structures [27, 28]. Simplified material models such as the linear and the elasto-plastic constitutive models were developed to esti-mate the structural responses. Then, the estimated responses were compared with experiments and/or numerical models considering coupled nonlinearity. Later, the timber-framed

plywood panel dome structures were investigated and the structural performance under static loads was simulated [29]. Recently, a group of European scholars systematically studied the timber grid shells. Their work first explored the form finding and structural analysis algorithms [30], based on which a full-scale structure was designed and constructed [31].

The reticulated timber dome structures are advantageous as they are renewable, aesthetically attractive, and that they could be easily assembled. The metallic joints connecting the timber members are crucial components that have sig-nificant impacts on the overall structural performance. Based on different architectural designs, a few types of joints were developed for reticulated timber domes. Figure 1a is a ten-don embedded timber joint (referred as TET joint hereaf-ter) [32]. The Saldome 2 dome is a typical case using such design. Figure 1b is a steel plate splint joint (referred as SPS

Fig. 1 Joints for timber spatial structures, a tendon embedded timber joint [27]; steel plate splint joint; c slotted hole connection system [28]; d MERO joint


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joint hereafter) that was used in the timber dome in Brindisi, Italy. The TET and SPS joints provide some rotational stiff-ness. Hence, these joints are suitable for large-span timber domes. Nonetheless, their torsional resistances are relatively weak. Also notice that the timber elements for these joints are usually accurately bent into the proposed forms before the field assembling procedures. Figure 1c is a slotted hole connection system designed for timber grid shells [33, 34]. Figure 1d presents a type of steel MERO joint that could also be used for various timber structural systems. Furthermore, the cross-laminated timber (CLT) is used to construct some of the newest domes [35].

For steel structures, as the joint behaviors could be accu-rately estimated in most cases, the joint performance can be defined as rigid, semi-rigid, or flexible (close to pinned) following different definitions of semi-rigidity [36–39]. Comparatively, the reticulated timber spatial structures were traditionally designed assuming that the joints are ideally pinned or completely rigid, disregard of different joint con-figurations. However, for the joints of timber domes, such as the TET joints and the SPS joints, their actual behaviors are usually semi-rigid and do not conform to either of the two extremes. Focusing on this problem, the semi-rigidities of timber connections [40–43] were recently investigated.

This paper proposes a novel bolted connection for small-span and mid-span reticulated timber domes. The design of the connection considers an angle between the slotted-in bolted timber elements and the center steel portion, which slightly reduces the costs by removing the form curving pro-cedure of timber elements. With the proposed connection, small or medium reticulated timber domes could be designed and constructed easily with acceptable prices. First, two rea-sonably simplified sets of full-scale experimental tests were

designed and carried out to investigate the semi-rigid prop-erties of the proposed connection considering axial loads. Numerical simulations of the proposed connections were also provided. Then, a theoretical investigation on the semi-rigidity of the proposed connections was presented. A design formula with reasonable accuracy was provided for differ-ent joint configurations. Finally, based on the tested and simulated results, a few mid-span reticulated timber domes were hypothetically designed, and the stability of the sample domes was systematically investigated.

2 Joint design and experimental investigations

2.1 Bolted joints for reticulated timber domes

The proposed joint includes steel bolts connecting the tim-ber members with the angled slotted-in steel plates (referred as BASS joint hereafter), which is shown in Fig. 2a. For a K6 reticulated dome, the BASS joint contains 6 angled slotted-in steel plates (shown in Fig. 2b). To construct a reticulated dome, the steel plates are cut into the designed angles instead of forming the timber elements with different curvatures. Please notice that the BASS joints are currently proposed only for small-span or mid-span reticulated timber domes. With a reduced size of the dome, the aspect ratio and the length of the timber elements are smaller such that they could be experimentally tested in full size.

The dowel-type timber joints are usually not recom-mended for the single-layered spatial structures. For exam-ple, the current design practice tends to simplify the flexural behavior of the dowel-type joint as a pinned joint. Under

Fig. 2 3D view of the BASS joints designed for the K6 timber domes, a 3D view of the proposed joint, b joint details


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such assumptions, stability is difficult to achieve for single-layered reticulated timber structures. However, discovered in this study, the moment-resisting capacity of the BASS joints is large enough such that they could satisfy the struc-tural demands for the small-span or medium-span reticulated timber domes.

2.2 Material tests

Compressive strength and embedment strength of wood are two important material properties that have great impacts on the overall behavior of the bolted glulam joints. The glulam used in this study were made of Canadian spruce-pine-fir (SPF) lumber. The elastic modulus as well as the strength of the glulam was tested for both the cases parallel (//) and perpendicular (⊥) to the grain. The results from this section serve as an important reference for the subsequent studies.

While testing the compressive strength parallel to the grain (fc,//), the specimens were designed with a length of 100 mm and the cross section of 25 mm × 25 mm. The test method of the specimens was designed according to ASTM D143 [44]. The test was displacement controlled. The loads were applied at a speed of 0.3 mm/min. The test setup is presented in Fig. 3a. The compression of the material was measured over a 50-mm area in the center of the tested spec-imen. A total number of 30 specimens were prepared and

tested. Four failure modes were observed during the tests, which are shown in Fig. 3b. The four modes were: (1) the crushing failure, (2) the crush-shear combined failure, (3) the shear failure, and (4) the end zone failure. The first three modes were the most commonly observed modes during the tests. The stress–strain curves of the tested specimens are presented in Fig. 3c. Obvious damage could be captured at the strain level near 0.003. It could also be concluded that the specimens were not ductile since a few specimens failed immediately just after the damages occurred.

Different from the test of compression parallel to the grain, the specimens with the length of 150 mm and the cross section of 50 mm × 50 mm were designed for the test of the compressive strength perpendicular to the grain (fc,⊥). Following the test configuration shown in Fig. 3d, a total number of 30 specimens were prepared and tested. Most of the specimens failed with a similar mode, i.e., the crushing failure shown in Fig. 3e. Comparing Fig. 3c, f, it is obvi-ous that the performance perpendicular to the grain is more ductile than that of the performance parallel to the grain.

The embedment strength (also referred to as the dowel-bearing strength) is needed to calculate the load-carrying capacity per shear plane of the bolted connections. A total number of 15 specimens were prepared and tested to study the embedment strength of the proposed joints. The speci-mens are illustrated in Fig. 4a. For the two group of tests

Fig. 3 Test setup, failure modes, and tested results of the compression test parallel to the grain


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(i.e., //and ⊥), the diameter of the fasteners (d), the size of the predrilled hole (d0), and the dimensions of the timber specimens are summarized in Table 1.

The testing method followed ASTM D5764 [45]. The loads were applied at the speed of 1 mm/min. When the specimens were loaded parallel to the grain, the failure mode included the embedment failure that arrived earlier and the splitting failure along the grain, as shown in Fig. 4b. When the specimens were loaded perpendicular to the grain, the specimens were undamaged until the occurrence of wood fiber fracture, as shown in Fig. 4c. The result of the embed-ment strength is now listed in Table 2.

The mean values of the test results are summarized and presented in Table 2. Besides the mean values, the standard deviations and the variable coefficients were also calculated

and summarized. It should be noted that for glulam, clear wood specimens which are smaller than one lamella cannot give enough information on the load-bearing capacity of the full-scale glulam members. As we have the grade of the glu-lam, the aim of the tests is to give a comparative information for the material properties of the material. It should be noted that the glulam used in this study was made according to the Chinese standard “Technical Code for Glued Laminated Timber Structures” [46], in which the strength grade classifi-cation of glulam is different from that in the European code [40]. In general, the strength of the glulam used in China is lower than the strength of the glulam used in Europe. Since the joint test results were related to the type and strength of the glulam, the results of the material tests are reported in this section to provide the readers with the tested material properties. Thus, if a different type of glulam is used by other researchers, a comparison can be made to have an in-depth understanding of the performance of the joints.

2.3 Mid‑span reticulated timber domes

Reticulated timber domes with rise-to-span ratios from 1/6 to 1/3 were hypothetically designed. The loads were determined by load specifications in China [47]. The dead

Fig. 4 Primary failure mode of the specimens in the dowel-bearing strength tests, a parallel to the grain; b perpendicular to the grain

Table 1 Dimensions of embedment strength specimens

Group Diameter of the fasteners

Size of the predrilled hole

Width Height Thickness

// 20 22 132 154 50⊥ 20 22 308 88 50

Table 2 Material and specimen properties

Formula Compression strength Embedment strength

Elastic modu-lus E (//)

Ultimate strength σu (//)

Elastic modu-lus E (⊥)

Yield strength σy (//)

Ultimate strength σu (//)

Yield strength σy (⊥)

Ultimate strength σu (⊥)

Mean X̄ =∑

Xi

n

14,269 MPa 33.82 MPa 246 MPa 24.00 MPa 27.07 MPa 11.39 MPa 16.40 MPa

Standard devia-tion S =

�

∑

(Xi−X̄)2

n−1

3521 MPa 3.51 MPa 75.66 MPa 4.37 MPa 1.54 MPa 0.72 MPa 1.38 MPa

Coefficient of variation

𝜈 =S

X̄× 100% 24.07% 10.39% 30.78% 18.19% 5.67% 6.33% 8.38%


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load was 0.96 kN/m2, and the live load was 0.5 kN/m2. The wind load form factors were different for the cases when the rise-to-span ratio was big or smaller than 1/4, as shown in Fig. 5a, b. The seismic design followed the requirements specified in [48]. The technical specification of space frame structures [49] was also used for the design of the dome. Meanwhile, as the specification is mainly focused on the steel space frame structures, the technical code of glued laminated timber structures was used to design the timber elements [46]. Regarding to the joint, two columns of bolts were used to keep the joint area small. The end and edge dis-tances were designed to satisfy the minimum code require-ments. At the structural level, the analysis and design were done using the software SAP2000.

2.4 Joint design

To use BASS joints, a K6 reticulated timber dome contains many stiffened hollow steel cylinders. As shown in Fig. 6, six timber elements are assembled around each of the stiff-ened hollow steel cylinder with the steel bolted connections.

Besides, straight timber elements could be used to avoid the form curving procedures for the glulam. An angle was designed for each of the timber element. Notice that the

angles for each of the joints might be different; thus, a quick analysis was performed to statistically collect all the angles for the joints in the single-layered reticulated dome with 6 nested rings. Domes with smaller rise-to-span ratios have smaller angles. Therefore, The domes with smallest (1/6) and largest (1/3) considered rise-to-span ratio was studied. As shown in Fig. 7 , no obvious patterns could be found for the distribution of the angles. However, most of the angles are bounded within the range between 4° and 8°, as shown in Fig. 7b.

2.5 Test setup

The boundary conditions are very difficult to define espe-cially for a spatial dome structure test. Since the full-scale sphere test of the dome containing several rings was nearly impossible to realize in the laboratory, tests were designed to obtain the mechanical properties of the joint, mainly the properties of the joints under shear force, axial force, and moment. In fact, out of these properties, the shear and axial force capacities could be obtained from mul-tiple guidelines in the design codes. Moreover, it should be noted that the test setup used in this study might cause bending in the joints, which resulted in relatively

Fig. 5 Wind load form factors for timber domes, a rise-to-span ratio of 1/3; b rise-to-span ratio of 1/6

Fig. 6 Illustration of the BASS joints connecting with the center steel cylinder at an angle


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conservative load-resisting capacity of the joints. Thus, a detailed numerical model was developed and verified by the test results, and the model was further used to compre-hend the mechanical behavior of the timber dome under any loading scenarios.

To simplify the stiffened hollow steel cylinder of a K6 dome (shown in Fig. 8a), a steel box shown in Fig. 8b was designed. Each of the simplified specimens contains only two steel plates along a single direction. Besides, a steel plate was designed at the top of the steel box to apply the loads. The cross section of the timber elements is designed to be 150 mm × 200 mm to match the steel plates.

Besides, the test needed to consider the influences from the angled steel plates. Two different angles of the extreme cases (i.e., shown in Fig. 7) were considered, which were 4° and 8° according to the analysis above. The specimens were labeled J4 and J8 accordingly, whose designs are illustrated in Fig. 8c, d, respectively. The angle is the only difference between the J4 and J8 specimens.

There is a small gap between the timber beam and the steel cylinder, and the contact at the upper side contributes to the splitting failure of timber at the bottom. It should be noted that the bolts took shear load at the beginning, and the bolts could bring a little moment resistance at the beginning of the test due to the effect of the axial force.

The testing machine used in this study is shown in Fig. 9a. The loads were applied from a hydraulic actuator with a maximum loading capacity of 500 kN. The clear space between the upper and the bottom actuator was 1000 mm. The dimension of the testing device was 7 m (long) × 1.5 m (wide) × 4.8 m (tall).

In addition, simulations from SAP2000 showed that the axial forces could not be neglected for the K6 tim-ber domes. Therefore, a support (shown in Fig. 9b) was designed to consider the axial forces. The center connector and the support were also designed to avoid the out-of-plane buckling of the joints. The loads were applied above the center connector pointing downward. The specimens were pushed downward at the rate of 4 mm/min until the specimens failed. LVDTs were placed around the joint and near the supports to collect the displacement and the rota-tion of the specimens. As shown in Fig. 9c, LVTD-R1, R2, and R5 were used to measure the rotation of the tim-ber elements. LVTD-R3 and R4 were used to measure the rotation of the steel plates. LVDT-R6 was used to measure the vertical displacement at the beam ends. Finally, the specimens were placed onto the testing machine, as shown in Fig. 9d, e.

2.6 Test results

Three duplicates of specimens were fabricated in each test set. The six specimens were labeled as J4-1, J4-2, J4-3, J8-1, J8-2, and J8-3, respectively. The typical damages at the con-nection failure stage are shown in Fig. 10a, b. At this stage, the strength of the joint was significantly reduced by wood splitting. The damage modes were further studied by decom-posing the elements, as shown in Fig. 10c, d. The predrilled holes were deformed. Such embedment failure was ductile and desirable. The cracks in the wood were along the grain passing the horizontal predrilled holes. In addition, as the bolts were relatively large, they were only slightly bent after the tests. After one of the surfaces of the predrilled holes reached its shear capacity, cracks were highly likely to occur and propagate along the grain passing other predrilled holes.

The moment–rotation curves of the J4 and J8 joints are provided in Fig. 10e, f. While the rotation was smaller than 0.02 rad, the curves of J4 specimens were more likely to be concave and the curves of J8 specimens were more likely to be convex. Consequently, the initial stiffness (within 0.02 rad) for J8 specimens was higher than that of the J4 specimens. In addition, it is shown that most of the cracks occurred after the rotation of 0.035 rad, which led to brittle failures after the rotational deformation of the joints reached 0.04 rad. Finally, the behavior of J4 specimen could be esti-mated by an approach provided in the subsequent sections as given in Fig. 18.

Fig. 7 Statistical analysis of the angles of the single-layered reticu-lated dome with 6 nested rings (40 m span with 1/3 rise-to-span ratio), a angle analysis; b angle distribution


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3 Estimation of the mechanical properties of the proposed joint

3.1 Numerical simulation

In this section, a 3D finite element (FE) model was devel-oped using the simulation software Abaqus [50]. The model accuracy was validated and calibrated by the test data. The FE model could provide estimations of the rotational strength for different designs of the BASS joints. Concluded from the experimental results, the elastic performance of the BASS joints is usually decided by the embedment strength of the glulam. To consider the ductile performance at the local level, a common approach is to weaken the elastic modulus near the predrilled holes on the glulam elements [51, 52]. The elastic material properties are summarized in Table 3. In the model, a square area with the size of 40 mm × 40 mm around the predrilled hole was weakened for each bolt. The weakening of the material was mainly in the parallel to grain direction, which was calibrated according to the test data.

It should be noted that the modeling method was firstly proposed by Foschi and Bonac [53] and was verified and used by other researchers [51, 52]. The method includes a

wood foundation which is a prescribed zone surrounding a bolt. The foundation material parameters were determined through bolt-embedment tests, and then the parameters were assigned to the prescribed foundation zone to account for the wood crushing (embedment) behavior. However, such modeling method has its limitations since a hypothetical weakening zone near the bolt holes was assumed, and it is essentially a phenomenon-based model which aims to pro-vide a relatively accurate estimation for the load-carrying capacity of the dowel-type joints.

The nonlinear behavior of the glulam elements was assumed to follow a tri-linear model as shown in Fig. 11a. Three points were, respectively, defined to mark the elastic (stress and strain with subscript e), the plastic (stress and strain with subscript p), and the ultimate (stress and strain with subscript u) characteristics of the material. The areas near the predrilled holes were also weakened to consider plasticity. The tri-linear material properties are listed in Table 4.

Furthermore, Hill yield criterion was applied to consider the anisotropic plastic deformation of wood. For the steel members, a typical bilinear model (shown in Fig. 11b) is used with the mechanical properties listed in Table 5.

Fig. 8 Simplification and design of the tested specimens, a stiffened hollow steel cylinder; b steel box; c specimen configuration of J4; d speci-men configuration of J8


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The global views of the J4 and J8 models are shown in Fig. 12a. The numerical model ignored the imperfection of the wood and the influence from the threads of the bolts. The contact surfaces between the nuts and the wood were slightly increased as the washers were not included in the model. Friction contact was defined for the model. Hard contacts were defined between the bolts and the steel holes. All the other contacts were soft contacts, setting the surfaces of the

steel elements as main surfaces and the surfaces of the wood elements as the slave surfaces. Reference points (RP) were created for the model to apply the loads and constrains. The boundary condition at the end of the beam was set as pinned connection. No cracks were predefined in the model. The mesh details are illustrated in Fig. 12b.

Both J4 and J8 models were simulated with the mod-eling details provided above. The loads of the simulated

Fig. 9 Test setup, a testing machine; b configuration of the test specimen; c locations of LVDT; d setup of J4 type joint; e setup of J8 type joint


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Fig. 10 Experimental phenomena and tested results, a failure mode of specimen J4; b failure mode of specimen J8; c deformations in the structural members of specimen J4; d deformations in the structural

members of specimen J8; e moment–rotation curves of specimen J4; f moment–rotation curves of specimen J8

Table 3 Elastic material properties of the joint and the weakened material properties near the predrilled holes

a 1 in the subscript stands for the // direction, and 2 and 3 stand for ⊥ direction

Elastic modulus (MPa) Shear modulus (MPa) Poisson’s ratio

E1a E2 E3 G12 G13 G23 ν12 ν13 ν23

Joint (global) 14,270 832 832 900 900 90 0.37 0.37 0.38Hole (local) 760 125 125 134 134 45 0.37 0.37 0.38


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Fig. 11 Demonstration of the nonlinear material models in Abaqus model, a tri-linear material properties for timber members; b bilinear material properties for steel members

Table 4 Tri-linear material properties for the glulam elements

Stress (MPa) Strain

σe σp σu εe εp εu

// (global) 28.4 34 30 0.002 0.005 0.01// (local) 25.4 33.2 26 0.04 0.18 0.5⊥ (local) 4.3 14 18 0.24 0.64 0.94

Table 5 Bilinear material properties for the steel members

Items E (MPa) Poisson’s ratio fy (MPa) fu (MPa) εu

Q235B steel plates 2.06 × 105 0.3 270 550 0.2Grade 8.8 high strength bolts 2.06 × 105 0.3 640 800 0.1

Fig. 12 Finite element simulation of the glulam bolted connections, a overview of J4 and J8 models; b model meshing


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specimens were also applied on the created reference points. Take J8 as an example, the component-level stress is illus-trated in Fig. 13a, b. For the timber elements, stress concen-tration was observed near the predrilled holes. Meanwhile, the bolts were under bending, which led to higher stresses in the center between the steel plates and the timber mem-bers. The model predictions of moment–rotation curves for J4 and J8 joints are presented in Fig. 13c, d. It could be concluded that the simulation generally captures the trend of the moment–rotation behavior of the joint. In addition, as no cracks were predefined in the model, the simulations were stopped before 0.04 rad to avoid the over-estimation of the moment resistance.

3.2 Evaluation of semi‑rigid flexural behavior

To systematically evaluate the semi-rigidity of the BASS joints, two moment–rotation models (shown in Fig. 14a) were derived based on the test and simulation results. The first model is a four-line model, which describes the moment–rotation behavior of the BASS joints by capturing the underlying mechanics of the components. Meanwhile, the second model is a bilinear model, which describes the

joint behavior by a simplified approach. The bilinear model connects the original point on the moment–rotation curves with the ultimate point (point C in Fig. 14a). To validate the accuracy of the simplified bilinear model, a quick analysis at the system level was performed. Specifically, the overall buckling forces of two reticulated timber domes (span 30 m and span 40 m, rise-to-span ratio 1/5) were investigated. A quick comparison of the ultimate buckling capacities is rep-resented in Table 6. It is shown that the differences between the two models were no more than 6.2%, which proves the validity of the bilinear simplification. The details of the sta-bility analysis will be presented in the last section of this paper.

For the four-line model, four phases were observed, as shown in Fig. 14b. The first phase is a short phase called steel plate squeezing phase. In this phase, most of the bolts are not contacting the predrilled holes in the timber mem-bers. The moment resistance is provided by the squeez-ing between the steel plate and the inner surface of the slot in the timber members. The second phase is the bolt slipping phase during which the bolts start to slip and hit the surfaces of the predrilled holes successively. When all the bolts hit the inner surfaces of the predrilled holes, the

Fig. 13 Simulated results from the FEM analyses, compared with the tested results, a stress distribution in the timber members; b stress status in the bolts; c predicted moment–rotation curve of J4; d predicted moment–rotation curve of J8


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third phase, namely the dowel-bearing phase, arrives. The behavior of the BASS joints in the third phase is ductile and desirable. The last phase is the failure phase when some large cracks start to occur. Figure 14c shows the dif-ference of the four-line curves for the J4 and J8 specimens. Such difference is purely caused by the effect of the angle (referred as the arch effect hereafter). Larger angles lead to more obvious arch effects, which increase the initial stiffness and provide greater moment resistance during the bolt slipping phase. Nonetheless, the arch effects become smaller near the failure phase.

On the other hand, the bilinear model is used as a basic model to develop the hand calculation method avail-able in the following section to estimate the semi-rigid joint properties. With the proposed hand calculation, the moment–rotation curves of the BASS joints could be roughly estimated without doing new experiments or simulations.

3.3 Theoretical analysis of the load‑resisting capacity

This section provides an estimation methodology for the load-carrying capacities of the BASS joints. For a slotted-in glulam bolted connection used for timber frames, its moment capacity could be roughly estimated from a latest work [43]. For the BASS joints, however, such estimation approach could not be directly applied due to larger axial force and the arch effect. This part of the paper proposes a modified approach that is more accurate for the BASS joints.

A couple of assumptions were made: (1) The deformation within the timber elements are small such that the relative locations of the holes do not change; (2) the center steel connector and the steel plates suffer no damage; (3) the rota-tional strength is influenced by the embedment strength of each fastener. The direction of the dowel-bearing force is

Fig. 14 Illustration of the four-line model and the bilinear model, a four-line model and bilinear model; b four phases of the four-line model; c difference of the four-line curves for the J4 and J8 specimens

Table 6 Difference between the bilinear model and the four-line model in terms of the ultimate buckling capacities

Domes With bilinear model (kN/m2)

With four-line model (kN/m2)

Difference (kN/m2)

Percentage

L = 30 m, fH/L = 1/5 33.36 31.41 1.95 6.20L = 40 m, fH/L = 1/5 16.79 17.63 − 0.84 − 4.80


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perpendicular to the line between the fastener and the center of rotation which is referred to as the CoR hereafter.

Multiple existing timber design codes provide the strength estimation of the embedment strength and the load-carrying capacity per shear plane. Four well-known codes to esti-mate the embedment strength of glulam bolted connections were systematically compared in a recent work [43]. This study adopts the approach from Eurocode 5 as it provides the closest estimation compared with the results of the material tests. From the Eurocode, the embedment strength could be computed according to Eq. (1):

where fe,0 and fe,90 are the embedment strength values paral-lel and perpendicular to the grain, unit in MPa; d is the diam-eter of the bolt, unit in mm; ρ is the density of the lumber, unit in kg/m3; θ is the angle between the dowel force and the grain. In addition, out of all the damage modes given in the code, three main ductile yield modes were identified for the BASS joints. As shown in Fig. 15, the three modes are: (Mode I) the yield of the wood fibers in contact with the fastener in the side timber member; (Mode II) the yield of fastener in bending at one plastic hinge point per shear plane, and bearing-dominated yield of wood fibers in contact with the fastener in the side member; and (Mode III) the yield of fastener in bending at two plastic hinge point per shear plane, with limited localized crushing of wood fibers near the shear planes.

Then, modified from the Eurocode 5, the load-carrying capacities per shear plane corresponding to the three modes could be calculated following Eq. (2):

(1)

fe,0 = 0.082(1 − 0.01d)�

fe,90 =fe,0

k90

fe,� =fe,0

k90 sin2� + cos2 �

k90 = 1.35 + 0.015dwhere Fv,b is the characteristic value of the load-carrying capacities per shear plane, which is close to the tested aver-age; fes and ts are the embedment strength and the thick-ness of the side (timber) element; My,b is the yield bending strength of the fastener, unit in kN∙m; fa,b is the characteristic withdrawal capacity of the fastener; fu is the ultimate tensile strength of the fastener, unit in MPa; kmod is the duration and saturation factor, which could be found in Table 3.1 from Eurocode 5; γm is the partial factor for material prop-erties and resistances, which are provided in Table 2.3 from Eurocode 5. To obtain the design value (F), the character-istic value shall be divided by a factor (i.e., γm/kmod = 1.56 for this case). Then, the design moment capacity (Md) of the joint is the factored summation of the moments provided from each of the n dowels, which is expressed by Eq. (3):

where Φan is the coefficient considering the angle of the joints and Φax is the coefficient considering the axial forces. It is observed from the test that the angle of the joint could slightly increase the moment capacity. Calibrated from the tested values, Φan is 1.135 and 1.24 for the J4 joints and the J8 joints, respectively. The coefficient for other BASS joints with the angles between 4° and 8° could be linearly interpolated.

The other coefficient, Φax, accounts for the moment increase due to the effect of axial force. Notice that for the BASS joints, the axial force keeps changing as the deforma-tion becomes larger. The coefficient is calibrated from the simulated results. If the axial forces are not available, this study recommends a default Φax value of 1.3 for the BASS joints. With the axial force obtained from the FEM analysis, the coefficient could be calculated according to Eq. (4):

(2)

Fv,b = min

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

festsd (I)

min

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

festsd

⎛

⎜

⎜

⎝

�

2+4My,b

fesdt2s

− 1

⎞

⎟

⎟

⎠

+fa,b

4

1.25 ⋅ festsd

⎛

⎜

⎜

⎝

�

2+4My,b

fesdt2s

− 1

⎞

⎟

⎟

⎠

(II)

min

⎧

⎪

⎨

⎪

⎩

2.3�

My,bfesd +fa,b

4

1.25 ⋅ 2.3 ⋅�

My,bfesd

(III)

My,b = 0.3fud2.6

F =k mod

�m

Fv,b =

0.8

1.25Fv,b

(3)Md = �an�ax

n∑

i=1

Fi⋅ l

i

t1 t1tsteel

(a) Failure mode I (b) Failure mode II (c) Failure mode III

Fig. 15 Three ductile failure modes identified for the BASS joints


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where a1 is the maximum axial force distributed to a sin-gle hole on the timber element. In this case, the axial force could be assumed to be evenly distributed. F is the load-carrying capacities defined in Eq. (2). The coefficient is set to 1 when the axial force is smaller than 20% of the load-carrying capacities. Take the J4 specimen as an example: the peak axial forces reached 78 kN according to the simulated results. Then, a1 = 78 kN/4 = 19.5 kN and Φax = 1.43.

The moment capacity of the joints (Mu) usually refers to the mean moment strength (Mave), which could be obtained by multiplying the factor γm/kmod, as expressed by Eq. (5):

The dowel load of the ith bolt (Fi) and the corresponding moment arm (li) are critical components of the provided for-mula to accurately decide the moment capacity of the joint. Under complex loads, the dowel load is the vector sum of the moment, the shear, and the axial forces. The shear and axial forces are assumed to be evenly resisted by the n fasteners (as shown in Fig. 16a).

When the spacing between the bolts is the same, the CoR is assumed to be located along the angled vertical line between the left most column and right most column of bolts. The CoR is not at the geometric centroid of the bolts as the dowel forces for different rows are not the same. For the tested case, the CoR is located below the geometric centroid toward the tension side of the glulam member. The precise location of the CoR is relatively hard to be hand calculated. In this study, an assumption is made such that lAC:lCB = 2:1, where point A is the center of the outmost row in the compression side and point B is the center of the out-most row in the tension side. For simplicity, no matter how many horizontal rows of bolts were designed for a joint, this

(4)𝛷ax=

{

1 a1∕F ≤ 20%

1 + 0.3 ⋅ (a1∕F) a1∕F > 20%

(5)Mave = �an�ax

�m

k mod

n∑

i=1

Fi⋅ l

i

ratio stays fixed. With such assumption, the moment from the shear forces is zero about the CoR.

Now that the moment arms are calculated, the next step is to compute the dowel force of the ith bolt. From the simu-lated model of the joints, the axial forces are large and could not be neglected due to the arch effect. All the dowels in compression reach their capacities (i.e., Fc = F) at the ulti-mate stage, which is shown in Fig. 16. In addition, the dowel forces for the bottom horizontal rows might be different. The dowel forces are larger when they are located further from the CoR. The dowel forces are assumed as zero if the connecting line of a layer of bolts passes through the CoR. For the outmost layer in tension, the dowel forces usually do not reach their capacities. In addition, the maximum dowel forces in the tension side (Ft) are usually different for dif-ferent joint designs. According to the simulated results for the investigated scenarios in this study, their values could be estimated by Eq. (6):

The dowel forces could be considered in the directions perpendicular to their moment arms, which are shown in Fig. 16a. If there are more than two horizontal rows of dow-els, the dowel loads of the middle layers shall be linearly interpolated. Notice that the assumptions about the locations of CoR and the dowel forces were carefully calibrated from multiple FEM simulated results. Nonetheless, these assump-tions might not be precise for all the sizes and designs of BASS joints. Future studies are encouraged to further elabo-rate these variables toward higher accuracy.

Finally, to fully define the bilinear curve of the force–dis-placement curve of the BASS joints, the rotational deformation at the ultimate stage (θu) is needed in addition to the ultimate capacity (Mu). Furthermore, the ultimate rotation of a glulam

(6)Ft=

⎧

⎪

⎨

⎪

⎩

0.40F 2 horizontal rows

0.65F 3 horizontal rows

0.90F 4 or more horizontal rows

Fig. 16 Demonstration of the internal forces during hand calculations of BASS joints


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bolted connections has never been clearly defined from the existing studies. Shu et al. [43] estimated that the rotation is usually between 0.04 and 0.05 rad while considering slight axial forces of the timber members. In this study, θu is assumed to be relevant to three parameters, i.e., the diameter of the bolts (d), the width of the BASS joint (i.e., tw in this case), and the axial force coefficient Φax. θu becomes larger when the joint section is wider and/or the bolt diameters are smaller. Larger axial forces make the timber elements more likely to split, which also reduces the ultimate rotation. The ultimate rota-tion (in the unit of rad) could be roughly estimated by Eq. (7):

(7)𝜃u = 𝛽 ×

(

0.07

𝛷ax

− 0.0005 × d

)

,

𝛽 =

{

1.0 tw ≤ 200mm

1.1 tw> 200mm

3.4 Application of the proposed analytical method

To better present the proposed estimation methodology, the moment capacities of three BASS joints were calculated and presented in this section. Different section sizes, bolt diam-eters, and bolt arrangements were designed for three BASS joint specimens (i.e., specimens S1, S2, and S3). The dowel forces were first calculated to obtain the moment capacities. Then, the ultimate rotational deformations were estimated for the three BASS joints. As shown in Fig. 17, the first specimen is same as the tested J4 joint. Besides, the diam-eters of the bolts for the three joints are 20 mm, 24 mm, and 10 mm, respectively. The tensile strength of the bolts (fu,k) is 600 MPa. The specific gravity (ρk) is 400 kg/m3. Fur-thermore, the size of steel predrilled hole (dm) = [23 mm, 24 mm, 14 mm] and the size of timber predrilled hole (ds) = [22 mm, 24 mm, 12 mm] for the three designs.

Then, the calculated values for each of the bolts are presented in Table 7. In the table, Fa is the force adjust-ing matrix for each of the bolts that was described above;

(a) (b) (c)

Fig. 17 Design of the three BASS joints for the example of the implementation of the proposed hand calculation method, a specimen S1; b specimen S2; c specimen S3

Table 7 Variable matrices for the three examples of the BASS joints

Fa Fa*Fv,b (kN) ri (mm) Mi= Fa*Fv,b*ri (kN m)

Φan Φax

S1 1 1 13.6 13.6 88.0 88.0 2.4 2.4 1.135 1.430.4 0.4 4.9 4.9 74.9 74.9 0.7 0.7

S2 1 1 23.4 23.4 174.6 174.6 8.2 8.2 1.24 1.30.5 0.5 10.1 10.1 106.3 106.3 2.1 2.10 0 0.0 0.0 70.0 70.0 0.0 0.00.9 0.9 18.2 18.2 106.3 106.3 3.9 3.9

S3 1 1 1 11.0 11.7 11.0 160.9 133.3 160.9 3.5 3.1 3.5 1.24 1.30.25 0.25 0.25 2.3 2.9 2.3 96.0 33.3 96.0 0.4 0.2 0.40.65 0.65 0.65 6.4 7.6 6.4 112.0 66.7 112.0 1.3 0.9 1.3


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Fv,b could be obtained following Eq. (2); ri are the distances from each of the bolts to the CoR; Mi is the contribution of moment from each of the bolts.

Finally, the estimation of the moment–rotation curves of the three examples of BASS joints is presented in Fig. 18. In the figure, the expected average moment–rotation curves for S1 (J4), S2, and S3 are provided. The design values from the proposed theoretical analysis method are also marked. For specimen 1 (J4), the tested curve is also provided to show the accuracy of the provided method.

3.5 Performance along the weak axis

Compared with the TET and SPS joints, the slotted-in steel plates could provide the stiffness along the weak axis, which makes the domes more reliable against lateral loads such as strong winds and earthquakes. This is another advantage of the BASS joints. This section provides some preliminary FE simulation results of the moment–rotation behavior about the weak axis of the beam. The moment resistance over the weak axis of the beam is mainly contributed from the inter-action between the steel plate and the timber members. As shown in Fig. 19, as no bolts were placed in this direction, the performance of the joints is close to a linear line. There-fore, the moment resistance over the weak axis of the beam could be represented by a linear line if it is needed for the future design.

4 Stability analysis of reticulated timber domes with the proposed joint

Full-size structural models were created using ANSYS [54]. The models were used to investigate the static stabil-ity of K6 single-layered reticulated timber domes. With the nonlinear spring element COMBIN39, the force–displace-ment correlations over all the degree of freedoms could be

accurately established at the joint level. In addition, by using the ANSYS Parametric Design Language (APDL), a system-atic analysis was performed to investigate the influences of several key parameters on the stability of the domes.

4.1 Modeling of the single‑layered reticulated timber dome

To investigate the stability of the small-span and mid-span reticulated timber domes geared with BASS joints, a few key parameters were considered. They were: the span (L) and the rise/span ratio (fH/L); the joint model (stiffness); the initial geometric imperfection; and the load distribution.

Beforehand, a typical single-layered reticulated timber dome with K6 BASS joints was selected as an example to study the features of the created models. The span of the dome is 40 m, and the rise-to-span ratio is 1/3. The dome includes 6 nested rings from the edge to the center. The dead loads were transferred as the equivalent nodal mass with the MASS21 elements. Besides, while calling the BEAM189 element, the joints of the presented domes were assumed to be pinned, semi-rigid (stiffness provided by the tests and/or estimations), and rigid. The geometric imperfections were fixed at 1/1000. Dead loads and live loads were distributed on the dome, assuming that all the nodes on each of the rings carried the same amount of load, whose magnitude is shown in Fig. 20.

The first three buckling modes and their frequencies are illustrated in Fig. 21. In addition, a factor is also provided with each of the modes. The system reaches its stability capacity when the initial loads (shown in Fig. 20) were multiplied with the factors. It is shown in the figure that the second and third modes of the dome usually occur right after the global bucking from the first mode. Most of these modes are hybrid modes including local torsional motions

-0.02 0 0.02 0.04 0.060

10

20

30

40

50

θ (rad)

Mu (k

N-m

)

Specimen 1 (J4)Specimen 1 designJ4 tested meanSpecimen 2Specimen 2 designSpecimen 3Specimen 3 design

Fig. 18 Estimation of the moment–rotation curves of the three exam-ples of BASS joints

Fig. 19 Performance of J8 BASS joints along the weak axis


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and global vertical motions. Shown from the above stud-ies, the BASS joints provide not only the rotational stiffness along the main axis, but also the rotational stiffness along the weak axis. With such characteristics, the torsional per-formance of the timber domes with BASS joints is not as important as the horizontal performance.

Comparing the results from the case with pinned joints and those from the case with semi-rigid joints, it can be con-cluded that the rotational joint stiffness ought to be con-sidered during the design. The pinned assumption for the joints is not recommended as it leads to a soft truss system whose buckling capacity is very limited. Furthermore, with

the provided BASS joints, the dominating modes of the dome were approaching those of the cases with rigid joints. Moreover, to better understand the horizontal behavior of the dome with BASS joints, the internal forces of the system are provided in Fig. 22.

In Fig. 22, the buckling capacity versus deformation curves for two selected nodes with large deformations was also provided. In this study, the ultimate buckling capacity was used as the performance index to evaluate the global stability of the domes. The capacity is defined as the buck-ling force of the force–displacement curves that could be tracked using the arc-length method for the node that has the maximum vertical displacement.

4.2 Span and rise‑to‑span ratio

Two spans (i.e., L = 30 m and 50 m) and three rise-to-span ratios (i.e., fH/L = 1/3, 1/4, and 1/5) were selected to study their influences on the stability of the timber domes. The geometric imperfections were fixed at 1/1000. Figure 23 shows the ultimate buckling capacities of the timber domes whose BASS joints were modeled with the rigid and the semi-rigid assumptions. For the semi-rigid joint properties, the tested moment–rotation behaviors were directly applied to the model as all the joints were the same as the tested joints.

The first observation from the Fig. 23 is that the domes with larger spans have lower ultimate buckling capaci-ties. Besides, the increase of the rise-to-span ratios leads to the improvement of the buckling resistance. It can also be concluded that the consideration of semi-rigidity of the BASS joints is important while evaluating the stability of the timber domes. The rigid joint assumption leads to an

Fig. 20 Nodal loads for stability analysis (from the center node to the other nodes of the 6 nested rings)

Fig. 21 First three modes of a typical single-layered reticu-lated timber dome

PinnedJoints

factor1=0.57 factor2=0.69 factor3=0.69

Semi-rigidJoints


RigidJoints



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overestimation of the buckling capacities. It is recommended that the rise-to-span ratio is designed in the range between 1/4 and 1/3.

4.3 Joint models and rotational stiffness

Shown from the above analysis, the proposed bilinear joint model is a simplified mathematical model that can reason-ably represent the rotational behavior of the proposed joints. Specifically, the elastic joint stiffness is determined by the timber material, the bolts, and the design of the joint. In this section, the joint models of the timber domes were assigned with different elastic stiffness to explore their influences on the structural stability.

Figure 24 shows the ultimate buckling capacity of the timber domes with different joint stiffness of the BASS joints. The joint stiffness was sampled at 0.1 kN m/rad (close to a pinned joint), 205 kN m/rad, 266 kN m/rad (close to the test results), 843 kN m/rad, and over 1000 kN m/rad (close to rigid joint). The geometric imperfections were fixed at 1/1000. It is noted from the figure that the ultimate buckling capacities of the domes increased with the increase of the joint stiffness. The pinned joints lead to extremely low buck-ling capacities for both the 30-m and 40-m domes.

The ultimate buckling capacities could improve signifi-cantly when some semi-rigidity (i.e., 205 kN m/rad) was considered. In addition, increasing the joint rotational stiff-ness from 266 to 843 kN m/rad only marginally increases

Fig. 22 Internal forces of the single-layered reticulated timber dome with BASS joints, a moment diagram; b axial diagram; c out-of-plane shear diagram; d in-plane shear diagram


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Fig. 23 The impact of rise-to-span ratio on the ultimate buckling capacity, a span = 30 m; b span = 40 m

Fig. 24 The impact of joint stiffness on the ultimate buckling capacity


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the ultimate buckling capacity. If the semi-rigidity could be considered in the model, there is no need to significantly enhance the joint for higher buckling capacities.

4.4 Geometric imperfections

The geometric imperfections of civil structures are unavoid-able due to the manufacturing and assembling errors. To consider these geometric imperfections, consistent mode imperfection method was commonly used while analyzing the stability of the spatial structures. The geometric imper-fections can be determined by the least eigenvalue buckling mode and the maximum amplitude of the imperfections. In this study, a few amplitudes of imperfection were selected as: 0, L/3000, L/1500, L/1000, L/500, and L/300.

The influence of geometric imperfections on the ulti-mate buckling capacity of the reticulated timber domes with proposed semi-rigid joints is shown in Fig. 25. It could be noticed from the figure that the buckling capacities were smaller given larger initial geometric imperfections. In addi-tion, the consideration of joint semi-rigidity also lowers the stability capacities of the domes. Such effect is slightly more significant for the domes with smaller spans than those with larger spans.

It is also noticed that when the amplitude of geometric imperfection is larger than L/1000, the domes lose most of their ultimate buckling capacities. Therefore, the geomet-ric imperfections shall be considered while evaluating the stability of the reticulated timber domes with BASS joints. Besides, the manufacturing and assembling procedures shall be controlled to keep the maximum amplitude of the imper-fections within L/1000. It is recommended to minimize the

imperfections as global stability of the domes are sensitive to them.

4.5 Load distribution

The live loads and the dead loads on the reticulated domes are usually distributed loads. The full-span dead load (pDL) of the single-layered reticulated timber domes contains not only the loads from structural components, but also the loads from the roof, the lightening, the facilities, etc. In this study, pDL is first fixed at 0.96 kN/m2. In addition, a half-span live load, pLL, was considered and its magni-tude was set as 0%, 25%, 50%, and 100% of the dead load. The loads were applied on a 40-m span reticulated timber dome. The geometric imperfections were fixed at L/1000.

Then, the influence of load distribution on the ultimate buckling capacity is shown in Fig. 26. It could be seen that the general trend of the buckling capacities is decreasing while increasing the magnitude of the half-span live load. Besides, the largest deformation might happen at differ-ent locations. As the loads were changing, the buckling modes might also change, especially with the rigid joint assumption.

Generally concluded from the figure, larger rise-to-span ratios lead to more stable reticulated domes. Besides, smaller live loads are more desired while considering the stability of the structure. Furthermore, comparing the results of the 30 m and 40 m domes, the 30 m solution has got larger buckling resisting capacities. Most importantly, the semi-rigid joint assumption, which is more realistic compared with the rigid joint assumption, leads to much smaller buckling resistance.

Fig. 25 The impact of geometric imperfections on the ultimate buckling capacity, a span = 30 m; b span = 40 m


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5 Conclusions

There are finite studies currently available on the joint and structural behaviors of timber domes. This paper investi-gates a novel bolted connection (i.e., the BASS joint) that is available for small-span and mid-span reticulated timber domes. The BASS joint uses the steel bolts to connect the timber members with the angled slotted-in steel plates. Focusing on the proposed joint, full-scale experimental tests were carefully designed to investigate the material properties as well as the semi-rigid mechanical properties while considering axial loads. Finite element simulations of the proposed connections were also provided, and their accuracies were calibrated with the experimental tests. Then, the semi-rigidity of the proposed connections is explored, and the design formula is provided for differ-ent joint angles and heights. Finally, based on the tested and simulated results, the mid-span reticulated timber domes were hypothetically designed, and the stability of the domes was systematically investigated. The following conclusions could be drawn from the stability analysis:

(1) The rotational stiffness of the joint ought to be consid-ered during the design. The semi-rigid joint behavior is crucial regarding to the overall stability of the dome;

(2) Smaller spans, larger rise-to-span ratios, less geometric imperfections, and smaller loads lead to more stable domes with BASS joints.

Future work regarding to the similar joint designs could include: (1) A more accurate model of hand computation. Specifically, the definition of center of rotation, the magni-tude of dowel forces at the tension and compression sides, the coefficients considering the angle of the joints and the steel plate, and the ultimate rotation at failure could be fur-ther studied. (2) Along with the static stability analysis per-formed in this study, the dynamic stability problems and the progressive collapse problems could also be further investigated.

Acknowledgements The authors gratefully acknowledge National Nat-ural Science Foundation of China (Grant Nos. 51878476, 51778460, and 51708418) and National Key R&D Program of China (Grant No. 2017YFC0703507) and for supporting this research.

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