Mechanical performance of AH joints and influence on the stability behaviour of single-layer cylindrical shells

Thin–Walled Structures 146 (2020) 106459

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Mechanical performance of AH joints and influence on the stability behaviour of single-layer cylindrical shells Qinghua Han a, b, c, Chenxi Wang c, Ying Xu a, b, c, *, Xiaoning Zhang c, Yiming Liu c a

Key Laboratory of Earthquake Engineering Simulation and Seismic Resilience of China Earthquake Administration, Tianjin University, Tianjin, 300350, China Key Laboratory of Coast Civil Structure Safety of China Ministry of Education, Tianjin University, Tianjin, 300350, China c School of Civil Engineering, Tianjin University, Tianjin, 300350, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Assemble hub joint Mechanical performance Eccentric load Combined spring model Stability bearing capacity Single-layer cylindrical shell

Previous studies on the mechanical performance of AH (assembled hub) joints mainly focused on the axial ca­ pacity and the bending capacity of the joint system. However, the joint system in latticed shells is subjected to combined axial forces and bending moments in practice. In this paper, the mechanical performance of joints under eccentric load is investigated experimentally and numerically. The eccentric bearing capacity, as well as the failure mechanism of the novel joint system, are derived and compared with the axial loading and pure bending conditions. The correlation curve with 95% guaranteed rate is determined by non-linear regression analysis. On this basis, a combined non-linear spring model is established to simulate the mechanical perfor­ mance of joints in single-layer cylindrical shells for stability analysis. The results indicate that for orthogonal type cylindrical latticed shells with diagonals or crossed cables, the stability reduction factor can reach a range of up to 0.8 to 0.96. However, for cylindrical latticed shells with lamella grids or three-way grids, the stability reduction factor is less than 0.62. This situation can be improved by strengthening the joint system or reducing the intersection angles of members. When the intersection angle decreases from 90� to 60� , the stability reduction factors of lamella shells will increase by 34%–47%. This means that cylindrical shells with AH joints have sufficient stability to meet engineering requirements by rational design.

1. Introduction The design, development and application of assembled joints has been one of the active research topics in the field of spatial structures. Lopez et al. carried out an experimental study on the flexural perfor­ mances of semi-rigid joints in single-layer latticed shells and further established the analysis model based on the component method [1,2]. Nooshin et al. studied the influence of axial compression on the bending moment-rotation curves of MERO joints [3,4]. The results indicated that axial compression leads to an increase in initial bending stiffness and a decrease in the ultimate bending capacity. Fan et al. investigated the mechanical performance of Socket joints under combined axial forces and bending moments and further proposed several new types of assembled joints, include the Bolt-column joints, Gear-bolt joints, and BCP joints, etc. [5–8]. Feng et al. conducted an experimental study on the mechanical performance of the Ring-sleeve joint system [9]. The failure mode of joints under a combined shear, moment and axial load

was obtained. Han et al. proposed a novel assembled hub joint [10,11]. The practical formulas of axial and flexural bearing capacities were derived. All of the above research indicated that the mechanical per­ formance of assembled joints under complex loading conditions is distinctly different from single loading cases. Meanwhile, how to consider the mechanical properties of joints in overall structural analysis is another issue of concern for scholars. Considerable research is focused on the influence of joints on the sta­ bility behaviour of latticed shells. El-Sheikh et al. and Nooshin et al. pointed out that the stability bearing capacities of single-layer latticed shells with bolted joints are mainly influenced by bending stiffness of joints [12,13]. This influence can be reduced by increasing the size of �pez et al. proposed the formula for calculating the bolts and sleeves. Lo stability bearing capacities of single-layer latticed domes considering joint rigidity, which was verified by a model test [14,15]. Kato et al. found that the influence of the joint rigidity on the stability behaviour of shells is more significant than the initial imperfections [16]. Guo et al.

* Corresponding author. Key Laboratory of Earthquake Engineering Simulation and Seismic Resilience of China Earthquake Administration, Tianjin University, Tianjin, 300350, China. E-mail address: [email protected] (Y. Xu). https://doi.org/10.1016/j.tws.2019.106459 Received 5 June 2019; Received in revised form 18 September 2019; Accepted 10 October 2019 Available online 23 October 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Assembled hub joints.

[17,18] proposed the theoretic calculation formulas of stability bearing capacities of single-layer latticed domes with Temcor joints. Liu et al. obtained the correction coefficient of stability of aluminium latticed shells [19]. The influence of joint rigidity and the skin effect were considered in their analysis. Based on the above studies, many researchers have proposed the simplified numerical models of joints to simulate the stiffness charac­ teristics in integral structure analysis. Fan et al. proposed the 3D beam element-spring model to simulate the mechanical behaviour of semirigid joints and further carried out a parametric analysis on the stabil­ ity of single-layer latticed shells with different joint rigidity [20,21]. The joints used in lattice shells are classified into three categories according to their analysis: rigid, semi-rigid and pinned [22]. Chen et al. proposed the double element method to evaluate the influence of joint stiffness on the buckling capacity of suspend-dome structures [23]. The above models have the advantages of simplicity, convenience and fast calcu­ lation speed. However, most of them are unable to consider the reduc­ tion of bearing capacity and the degradation of joint stiffness in the elasto-plastic phase. Therefore, Han et al. established a nonlinear spring model of welded hollow spherical joints based on the theoretical solutions of plastic bearing capacity and critical deformations to eval­ uate the elasto-plastic stability of single-layer latticed shells [24]. The bending capacity and the axial capacity of AH joints have been

previously studied thoroughly. However, the real bearing status of AH joints in latticed shells is more complicated. In this paper, the eccentric bearing capacity of AH joints was studied experimentally and numeri­ cally. Non-linear regression analysis was carried out to determine the correlation curve based on the parametric analysis results. Moreover, the combined nonlinear spring model of AH joints was proposed to simulate the mechanical performance of AH joints under complex loading conditions. The stability analysis of cylindrical latticed shells with AH joints was carried out considering the influence of joint rigidity. Some effective methods were proposed to improve the stability of Table 1 Material properties of the components. Components

Steel grade

E (GPa)

fy (MPa)

fu (MPa)

εu

Et (%)

Hub Bolt Sleeve Closure plate Rectangular tube

IC22 35CrMo IC22 Q235B Q235B

205.96 213.04 207.07 205.35 206.21

308.48 911.97 279.64 297.84 322.93

479.65 1211.31 425.02 453.52 476.06

0.087 0.051 0.182 0.153 0.192

24.8 9.3 27.4 27.2 29.1

Note: E is the elastic modulus, fy and fu are the yield strength and the ultimate strength, respectively; εu is the ultimate strain; Et is the elongation after fracture. 2

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Fig. 2. Setup of the eccentric loading test.

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In the eccentric compression test, a hinge connection and fixed connection were installed at the top and bottom of the specimens, respectively. Lateral supports were arranged on both sides to prevent instability of the specimens. The contact surface of the supports and the loading frame was smoothed to minimize the influence of friction. First, a preloading of 2 kN was applied after the installation of specimen. Then, the test was conducted by force control in the elastic stage of joints. Subsequently, when the joints exhibited nonlinearity, the loading mode changed to the deformation control. The minimum sus­ tained time for each loading level was 5 min [11]. In the eccentric loading tests, the load data were collected by a force sensor on the test machine. Four laser displacement sensors (D1 to D4) were installed on each specimen to obtain the axial deformation and rotation of joints, as shown in Fig. 3. Assuming that the displacement indications at each measuring point are d1, d2, d3 and d4, respectively, the axial deformation and rotation of joints can be expressed as:

Fig. 3. Deformation of joint under eccentric load.

latticed shells with AH joints.

ðd1 þ d2 þ d3 þ d4 Þ 4 � �� ðd1 d3 Þ ðd2 d4 Þ θ ¼ arctan þ arctan 2 l l δ¼

2. Mechanical behaviour of AH joints under eccentric load 2.1. Test programme

(1) (2)

where l is the vertical distance between D1 (D2) and D3 (D4).

The AH joint system consists of a hollow hub, high-strength bolts, sleeves, closure plates and rectangular structural members. A total of six groups of specimens with varying loading directions and eccentricities were designed to investigate the mechanical performance of AH joints under eccentric loads. Specimens ET-1 to ET-3 correspond to the eccentric tension cases. The eccentricities were taken as 20, 50 and 120 mm, respectively. Similarly, specimens EC-1 to EC-3 correspond to the eccentric compression case, and the eccentricities are the same as the tension cases. The geometric parameters of the specimens are shown in Fig. 1. The material properties of the components were obtained from the tensile tests on coupons, as shown in Table 1. The setup of the eccentric loading test is shown in Fig. 2. A horizontal hydraulic jack with a loading capacity of 300 t was used in the eccentric tension test, and a vertical hydraulic jack with a loading capacity of 500 t was used in the compression test. In the eccentric tension test, the specimens were connected to the test machine with pin joints. The vertical distance between the centre of the pin joints and the centre of the hub is the eccentricity e. Triangle stiff­ eners were arranged at the intersection of the end plate and the rect­ angular tube to avoid local tearing damage of the tube.

2.2. FE model In this study, the FE models of the AH joints under eccentric load were established in ANSYS, as illustrated in Fig. 4. All the components were modelled by Solid element 185. The sleeves were simplified as hollow cylinders, which have the same cross-sectional area as the real components. The thread connections were simulated by bond con­ straints [15]. The contact pairs (Target surfaces and Contact surfaces) were generated between the other contacting components. The friction coefficient at the contact interface was set as 0.4 [25]. The mesh around the contact interfaces was densified for accuracy. The boundary conditions employed in the FEA were consistent with those in the test. According to symmetry, constraints in the Y direction were introduced at the symmetric plane and constraints in the X direc­ tion were introduced at the loading plane. A concentrated load parallel to the Y-axis with the eccentricity of e was applied to the end of the rectangular tube. The tri-linear material model was adopted in the nu­ merical simulation, and the material properties were obtained from the

Fig. 4. FE model. 4

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Fig. 5. Failure mode of joints under eccentric tensile load.

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tensile tests [10].

tensile stiffness and the elastic bending stiffness were less than 7% and 1%, respectively. The test results and the simulation results of the initial stiffness and the plastic capacities are compared in Table 2. The errors of the bearing capacities are less than 6%, and the errors of the initial stiffness are less than 13%. It was verified that the FE model of the AH joint has high accuracy and validity.

2.3. Mechanical performance of AH joints under eccentric tension The failure modes of ET-1, ET-2 and ET-3 in the test and the Von Mises stress nephograms obtained from FEA are compared in Fig. 5. In the eccentric tension tests, the eccentricity decreased with the tensile deformation. For ET-1, the eccentricity gradually approached zero at the later stage of loading. Therefore, the failure mode was very close to that in the axial tension test [11]. The hub had uniform plastic deformation after the test. Punching failure of the hub happened on the side of eccentric loading. By contrast, ET-2 has slight bending deformation under the eccentric load. The joint failed due to the fracture of the hub at the side of eccentric loading. Similarly, ET-3 had notable non-uniform deformation in the loading process, and the failure of the joint was also caused by fracture of the hub. The axial load-deformation curves and the bending moment-rotation curves of ET-1, ET-2, and ET-3 are shown in Fig. 6. Han et al. pointed out that the plastic capacity of AH joints is more applicable to guide the structural design [10]. Therefore, the plastic capacities of joints NTP and MP were extracted by the graphing method [26], as shown in Fig. 6. The results indicate that the tensile capacity of AH joints decreased with the eccentricity. The tensile capacity of ET-3 was only approximately 57% of that of ET-1. Comparatively, the bending capacity and the ultimate rotation angle increased with the eccentricity. The bending capacity and the ultimate rotation angle of ET-3 were approximately 1.6 and 3.1 times those of ET-1, respectively. During the plastic phase, the tensile stiffness of joints increased with the plastic deformation, and the bearing mechanism of the hubs grad­ ually changed from bending to tension. Since fracture of the hub happened suddenly to ET-2 and ET3, the load-deformation curves decreased sharply at the end of loading. It is also noted that the elastic stiffness of the AH joints was mini­ mally affected by the eccentricity. The variation ratios of the elastic

2.4. Mechanical performance of AH joints under eccentric compression The failure modes of EC-1, EC-2 and EC-3 and the Von Mises stress nephograms in FEA are compared in Fig. 7. The initial eccentricities were 20 mm, 50 mm, and 120 mm, respectively. The eccentricity kept growing with the axial deformation in the eccentric compression test. The test results indicate that the hubs have obvious plastic deformation in the loading process, and the failures of EC-1, EC-2, and EC-3 were all caused by elastoplastic instability of the hubs. The non-uniform defor­ mation of the hub was positively related to the initial eccentricity. The axial load-deformation curves and the bending moment-rotation curves of EC-1, EC-2, and EC-3 are shown in Fig. 8. The plastic capacities of joints NCP and MP were extracted by the graphing method [26]. The results indicated that the eccentric loading led to a significant reduction of the compressive bearing capacity. The compressive capacity of ET-3 was only approximately 42% of that of ET-1. All the load-deformation curves have obvious extreme points and then dropped steadily in the later stage of loading. It was further demonstrated that the failure of joints is caused by instability of the hubs. The moment-rotation curves of EC-1, EC-2, and EC-3 had an enduring rise in the whole loading process. The plastic bending capacity under eccentric compression was positively related to the initial ec­ centricity. The plastic bending capacity of ET-3 was approximately 1.5 times that of ET-1. Similar to the eccentric tension case, the variation of eccentricity had little influence on the initial stiffness of AH joints. The test results and the simulation results are compared in Table 3.

Fig. 6. Mechanical performance of AH joints under eccentric tension.

Table 2 Initial stiffness and plastic capacities of AH joints in the eccentric tension test. No.

KTE (kN/mm) Test

FEA

Test

FEA

Test

FEA

Test

FEA

ET-1 ET-2 ET-3

287.8 288.6 268.5

296.4 308.8 303.4

244.3 208.9 140.1

257.8 212.4 143.4

1202.9 1250.1 1215.1

1178.8 1325.1 1290.4

7.9 9.8 12.6

8.3 10.4 13.4

NTP (kN)

MP (kN⋅m)

KM E (kN⋅m/rad)

6

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Fig. 7. Failure mode of joints under eccentric compressive load.

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Fig. 8. Mechanical performance of AH joints under eccentric compression. Table 3 Initial stiffness and plastic capacities of AH joints in the eccentric compression test. No.

KCE (kN/mm)

EC-1 EC-2 EC-3

MP (kN⋅m)

KM E (kN⋅m/rad)

NCP (kN)

Test

FEA

Test

FEA

Test

FEA

Test

FEA

349.3 342.4 322.3

380.8 366.4 335.2

257.8 185.9 109.2

247.7 197.2 112.0

1273.5 1250.1 1320.2

1388.1 1325.1 1402.1

9.8 12.2 14.3

10.1 12.6 15.2

Table 4 Joint specifications in the parametric analysis (mm). No. FE-1 FE-2 FE-3 FE-4 FE-5 FE-6 FE-7

Bolt M27 M30 M30 M27 M30 M30 M30

Member

Closure plate

Sleeve

b

h

Hub Grade

D

H

S

t

tp

Ds

Ls

100 100 100 100 100 100 100

200 200 200 200 200 200 200

IC22 IC22 IC22 IC22 IC22 IC22 IC22

200 200 200 200 200 200 200

200 200 200 100 300 300 300

100 100 100 60 90 150 210

20 25 30 20 20 20 20

27 30 33 27 30 30 30

52 55 55 52 55 55 55

45 50 50 45 50 50 50

Note: b and h are the width and height of the structural member; D, H, and t are the outer diameter, height and thickness of the hub, S is the pitch of bolts; tp is the thickness of the closure plate; Ds and Ls are the outer diameter and length of the sleeves.

The errors of the bearing capacities are less than 6%, and the errors of the initial stiffness are less than 9%. 2.5. Parametric analysis of the eccentric bearing capacities of AH joints Parametric analysis of the eccentric bearing capacities was carried out to obtain the correlation curve between the axial capacities and the bending capacities of AH joints. Seven FE models with different geo­ metric parameters were established in ANSYS, as listed in Table 4. The ratios of e/S were taken as 0.05, 0.1, 0.2, 0.5, 0.8, 1.0, 1.2, 3.0 and 10.0. Both the eccentric tension case and the eccentric compression case were considered in the parametric analysis. The parametric analysis results are shown in Fig. 9. A power model of non-linear regression was adopted to fit the correlation curve of AH joints under eccentric load. The cor­ relation formula with 95% guaranteed rate can be expressed as: � �1:4 � �1:4 N M þ ¼1 (3) N0 M0 where N0 and M0 are the design capacities of AH joints under axial load and bending moment, respectively. Referring to Eurocode 3 [27], the safety factor of the design capacities is suggested to be 1.25.

Fig. 9. Correlation curve of AH joints. 8

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Fig. 10. Combined nonlinear spring model.

3. Stability analysis of single-layer cylindrical shells with AH joints

the elastoplastic phase and the plastic phase, as shown in Fig. 11. The tensile stiffness of the nonlinear springs increased with axial deforma­ tion in the plastic phase. Conversely, the compressive stiffness contin­ uously decreased throughout the loading process and finally declined to zero when the ultimate bearing capacity was reached. The length of the rigid bar is defined as L, which satisfies:

3.1. Establishment of the combined nonlinear spring model The mechanical performance of joints may have a significant influ­ ence on the stability behaviour of single-layer latticed domes. Therefore, a refined numerical model was established to simulate the stiffness and bearing capacities of AH joints under complex loads, which consisted of a certain number of nonlinear spring combinations and rigid bars, as shown in Fig. 10. Each connection of the hub and the rectangular tube was simulated by three nonlinear springs, one spring A and two springs B. The out-of-plane bending moment was carried by spring B, and the axial force was carried by all three springs. The load-deformation curve of the nonlinear springs was divided into three phases: the elastic phase,

L¼

δE ϕE

(4)

where δE and ϕE are the elastic critical deformation and the elastic rotational angle of joints, respectively. The axial capacity and the bending capacity of the AH joints are represented by Njoint and Mjoint , and the axial stiffness and the bending stiffness are represented by KNjoint and KM joint , respectively. According to

Fig. 11. Load-deformation curve of nonlinear springs. 9

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the conditions of compatible displacement and static force equilibrium, the initial stiffness and the elastic bearing capacities of the nonlinear springs can be expressed as: KspringB ¼

KM joint 2L2

(5)

KspringA ¼ K Njoint NE; ​ springB ¼

2⋅KspringB

ME; joint 2L

NE; ​ springA ¼ NE; joint

(6) (7)

2⋅NE; ​ springB

(8)

Similarly, the plastic bearing capacities of the nonlinear springs are expressed as follows: NP;springB ¼

MP;joint 2L

NP; springA ¼ NP;

joint

(9) 2⋅NP; springB

(10)

In the plastic phase, the expressions of the ultimate tensile capacities and the ultimate compressive capacities are derived separately: N CU; ​ springB ¼

MU; joint 2L

N CU; ​ springA ¼ N CU;joint N TU; ​ springB ¼

(11) 2⋅N CU; ​ springB

T MU; joint N U;joint ⋅ 2L N CU;joint

N TU; ​ springA ¼ N TU;joint

2⋅N TU; ​ springB

(12) (13) (14)

The critical deformations of the nonlinear springs were consistent with those of the AH joint system under axial loads, which are repre­ sented by δE , δP , and δU , respectively. Since the initial stiffness, bearing capacities and the critical deformations of AH joints have already been derived in the previous studies, the feature points of the nonlinear springs were able to be determined based on Eq. (4) to Eq. (14). 3.2. Verification of the combined nonlinear spring model The eccentric loading tests in Section 2 were simulated by the combined spring model and the solid element model. The loaddeformation curves and the moment-rotation curves are compared in Fig. 12. The results showed that the accuracy of the combined spring model is close to that of the solid element model. However, the computing time is only approximately 1/200 to 1/100 of that of the solid element model. The error of the initial axial stiffness between the test results and the spring model results ranged from 5.3% to 13.2%, the error of the initial bending stiffness ranged from 7.3% to 14.4%, and the error of the plastic bearing capacities ranged from 3.1% to 13.6%. 3.3. Definition of the stability reduction factor In this study, the stability reduction factor ϕK was introduced to evaluate the influence of joint behaviour on the stability capacities of single-layer cylindrical latticed shells [24], and it is described as follows: ϕK ¼

λ’U λU

(15)

where λU is the ultimate load factor of the shell with ideal rigid joints obtained by dual nonlinear stability analysis, and λ’U is the ultimate load factor of the shell with semi-rigid joints. The rotational stiffness ratio β was adopted to describe the joint ri­ gidity, which is the ratio of the joint rigidity to that of the connecting

Fig. 12. Comparison of the test results and the FE results.

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Fig. 13. The first-order buckling modes of eigenvalue analysis.

member [24]. Since the value of β varies with the specifications of joints and members, the uniform expression of the rotational stiffness ratio is proposed: n X

β¼

βi i¼1

Ii Imax

considered in the stability analysis. The cylindrical latticed shells with a span of 30 m and height of 10 m were established in ANSYS. The fixed hinge supports were introduced at two longitudinal sides. The dead load was taken as 1.0 kN/m2, and the live load was taken as 0.5 kN/m2. The live load is applied on the half-span of the cylindrical shells since the half-span distribution of the live load is more disadvantageous for cy­ lindrical shells supported on two longitudinal sides. An initial imper­ fection of 1/300 span was introduced in the nonlinear full-range analysis according to the consistent imperfect buckling analysis method. The first-order buckling modes of eigenvalue analysis are shown in Fig. 13. The deformation amplification factor is 5. Two grid forms including the lamella grids and the three-way grids were considered. Meanwhile, three specifications of AH joints with different joint rigidities were

(16)

where βi is the rotational stiffness ratio that corresponds to the ith connection, n is the number of connections, Ii is the moment of inertia of member i, and Imax is the maximum moment of inertia of all the members. 3.4. Stability analysis of single-layer cylindrical shells In this study, the effects of joint rigidity and grid forms were

Table 6 Specification of members in cylindrical latticed shells.

Table 5 Specification of joints in cylindrical latticed shells (mm). No.

Bolt

AH-1 AH-2 AH-3

M48 M48 M45

Hub

Closure plate

Sleeve

D

H

S

t

tp

Ds

Ls

300 300 300

300 300 300

200 200 200

74 62 52

50 50 45

80 80 75

80 80 75

No.

h � b � tm (mm)

I (mm4)

A (mm2)

Position

TM1 TM2

300 � 200 � 12

1.41 � 108

1.14 � 104

200 � 100 � 8

2.30 � 107

4.54 � 103

Diagonals/Lateral members Longitudinal members

Note: I and A are the moment of inertia and the section area of structural members. 11

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Fig. 14. Load factor-displacement curves for single-layer cylindrical shells.

adopted for comparison in the stability analysis. For AH-1 and AH-2, the plastic capacities of joints were no less than 1.2 and 1.0 times the fullsection plastic capacities of members. Considering the fabrication error, corrosion and other factors, the plastic capacity of AH-3 is less than the full-section plastic capacity of the members as a reference group. The specifications of joints and members are listed in Table 5 and Table 6. The uniform rotational stiffness ratio of AH-1, AH-2 and AH-3 equalled 7.5, 6.0 and 4.0, respectively. Fig. 14 shows the load factor-vertex displacement curves of cylin­ drical shells with different joint rigidities. The stability bearing capac­ ities of cylindrical shells with different grid forms are approximately

equivalent when the joints are considered to have ideal rigidity. For latticed shells with lamella grids, the stability reduction factors were in the range of 0.48–0.63. For latticed shells with three-way grids, the stability reduction factors were in the range of 0.50–0.66. The results showed that the joint rigidity has a great influence on the stability bearing capacities of cylindrical shells with lamella grids or three-way grids. It is also demonstrated that the arrangement of the longitudinal members has a minor effect on improving the stability of cylindrical shells with lamella grids. Fig. 15 and Fig. 16 show the Mises stress nephograms of cylindrical shells when reaching the ultimate bearing capacities. The results show

Fig. 15. Mises stress nephograms of cylindrical shells with lamella grids. 12

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Fig. 16. Mises stress nephograms of cylindrical shells with three-way grids.

Fig. 17. The stability analysis results of cylindrical shells with different member intersection angles. 13

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Fig. 18. Mises stress nephograms of cylindrical shells with different member intersection angles.

that the number of joints entering the elasto-plastic stage is negatively related to the joint rigidity. For cylindrical shells with AH-1 or AH-2, a considerable number of joints enter the elasto-plastic stage. For cylin­ drical shells with AH-3, a subset of joints enters the elasto-plastic stage or even the plastic stage. By contrast, the member stresses in cylindrical shells with ideal rigid joints are much higher than those in semi-rigid jointed shells. The results showed that the reduction of stability was mainly caused by a stiffness decline of the joint system instead of member failure. The distributions of stress were basically identical for the two grid forms.

in the last numerical example. The intersection angles of diagonals θ were taken as 90� , 75� and 60� , respectively. The stability analysis re­ sults of cylindrical shells with different member intersection angles are shown in Fig. 17. When the intersection angle decreases from 90� to 60� , the load factor increased by 66%–83%, and the stability reduction factor increased by 34%–47%. The stability of cylindrical shells with lamella grids was improved effectively by reducing the intersection angles of members. The stability reduction factor ϕK can reach up to 0.85 when the intersection angle equals to 60� . The distribution of Mises stress was altered after reducing the intersection angles of diagonals. The value of member stress and the number of elasto-plastic joints both increased, as shown in Fig. 18. After the intersection angle was reduced, more sufficient development of plasticity could be achieved when reaching the ultimate bearing ca­ pacity of the shell.

4. Effective methods to improve the stability of single-layer cylindrical shells Based on the above analysis, the insufficient joint stiffness may cause a considerable decrease in the stability bearing capacities of cylindrical latticed shells. In addition, there is no significant difference in stress distribution between the shells with lamella grids and the shells with three-way grids. In this section, effective methods to improve the sta­ bility of cylindrical shells were attempted by altering the intersection angles of members. The specifications of joints and members were consistent with those

5. Statistical analysis of the stability reduction factors The stability reduction factor is very useful in guiding the practical design of latticed shells with novel assembled joints. It can be used to modify the safety factor in non-linear stability analysis when consid­ ering the effects of joint rigidity. In this section, the parametric analysis 14

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Fig. 19. The variation of ϕK for single-layer cylindrical shells with AH joints.

In all study cases, fixed hinge supports were introduced at two lon­ gitudinal sides. The dead load was taken as 1.0 kN/m2, and the live load was taken as 0.5 kN/m2. The safety factors of shells in dual nonlinear stability analysis were all in the range of 2.3–3.0 under the ideal rigid joint assumption. The length of members was in the range of 3 m–5 m. The slenderness ratios of members was determined according to the requirements of Chinese standard JGJ7-2010 [28]. The variation of the stability reduction factors with the rotational stiffness of joints is shown in Fig. 19. It is obvious that the stability reduction factor ϕK is positively related to the rotational stiffness ratio β. In most cases, ϕK increases with the structural span and decreases with the rise-span ratio. For orthogonal type cylindrical shells with diagonals or crossed cables, the stability reduction factor can reach up to 0.8 to 0.96 when the bending capacity of joints is greater than that of the members (AH-1 or AH-2). The reduction of stability was less than 20% when considering the joint rigidity. However, for the orthogonal type cylindrical shells with AH-3, the stability reduction factor was only in a range of approximately 0.64–0.77. The significant reduction of stability was caused by the failure of joints prior to the failure of structural members. The stability of orthogonal type cylindrical shells was able to

Table 7 Parametric analysis results of ϕK . Grid forms

AH-1

AH-2

AH-3

Orthogonal type with crossed cables Orthogonal type with unidirectional diagonals Orthogonal type with crossed diagonals Lamella type Three-way girds

0.96–0.95 0.93–0.90

0.86–0.84 0.85–0.81

0.69–0.65 0.71–0.67

0.92–0.90 0.61–0.58 0.62–0.59

0.88–0.84 0.54–0.50 0.54–0.51

0.77–0.71 0.47–0.41 0.48–0.42

on the stability reduction factors was carried out to obtain the reference value of ϕK in project design. The effects of the grid form, span, rise-span ratio and joint rigidity were considered in the analysis. Two structural spans were considered in the analysis: 30 m and 60 m. The rise-span ratios were taken as 1/5, 1/4, 1/3 and 1/2. AH-1, AH-2 and AH-3 were adopted to consider the vari­ ation of joint rigidity. Five available grid forms were involved: orthog­ onal type with crossed cables, orthogonal type with unidirectional diagonals, orthogonal type with crossed diagonals, lamella type (θ ¼ 90� ) and three-way grids. 15

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Thin-Walled Structures 146 (2020) 106459

be improved by increasing the number of diagonals or arranging crossed cables in quadrilateral grids. The stabilities of cylindrical shells with lamella grids or three-way grids were relatively poor compared with the orthogonal types. The stability reduction factors were less than 0.62, which meant that the reduction in stability is greater than 38%. This situation can be improved by strengthening the joint system or reducing the intersection angles of members. The parametric analysis results of the stability reduction factor ϕK are summarized in Table 7. The recommended values of ϕK for engi­ neering applications are suggested to be 0.8 for orthogonal types with diagonals or crossed cables, and 0.5 for lamella types and three-way girds.

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6. Conclusions In this paper, an experimental study and FE analysis are carried out to investigate the mechanical performance of AH joints under eccentric loads. The correlation curve between the axial capacity and the bending capacity was proposed to guide the practical design of the novel joint system. Thereafter, a combined nonlinear spring model was introduced into the whole structural model to evaluate the influence of joint rigidity on the stability of cylindrical latticed shells. The main conclusions are as follows: 1. Under eccentric loads, the bending capacity of AH joints is positively related to the initial eccentricity, while the tensile or compressive capacity is negatively related to the eccentricity. However, the variation of eccentricity has little influence on the initial stiffness of AH joints. 2. The failure modes of AH joints under eccentric tension are similar to the axial tension cases, including hub fracture and punching failure around the bolt holes. By contrast, joint failure is caused by insta­ bility of the hubs under eccentric compression. 3. The combined nonlinear spring model of AH joints is proposed and further verified by the test results, which can simulate the mechan­ ical performance of AH joints under complex loading conditions during the whole loading process. 4. For orthogonal type cylindrical latticed shells with diagonals or crossed cables, the stability reduction factor can reach up to 0.8 to 0.96 and is suggested to be 0.8 in engineering applications. The stability of orthogonal type cylindrical latticed shells can be improved by increasing the number of diagonals or arranging crossed cables in quadrilateral grids. 5. The stabilities of latticed shells with lamella grids or three-way grids are relatively poor compared with the shells with orthogonal grids. This situation can be improved by strengthening the joint system or reducing the intersection angles of members. The stability reduction factor is suggested to be 0.5 in engineering applications. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments This research was financially supported by the National Natural Science Foundation of China (No. 51525803 & 51978458) and the Scientific and Technological Development Plans of Tianjin Construction System (No.2014–02).

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