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The in-plane effective length of members in aluminum alloy reticulated shell with gusset joints
Thin-Walled Structures 123 (2018) 483–491
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Full length article
The in-plane effective length of members in aluminum alloy reticulated shell with gusset joints
T
⁎
Shaojun Zhu, Xiaonong Guo , Xian Liu, Shuyu Gao College of Civil Engineering, Tongji University, Shanghai 200092, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Aluminum alloy gusset joint Semi-rigid joint Effective length In-plane stability
This paper presents an analytical research on the in-plane stability of members in single-layer reticulated latticed shells with aluminum alloy gusset (AAG) joints. The influence of two kinds of actual boundary conditions are considered, namely, the stiffness of the AAG joint (k1) and restraint of adjacent members (k2). In order to explore the influence of k1, an experimental program including 5 member specimens with AAG joints is introduced, and finite element (FE) models are established and verified. Based on the test and FE results, the effective length of the specimens are obtained. Subsequently, theoretical analysis is conducted to investigate the effect of k2, with accounting for the stiffness reduction caused by axial compressive force. Finally, the effective length factors of members in reticulated shell with AAG joints are derived and tabulated. Numerical examples are presented to illustrate the necessity of considering the influence of k1 and k2, and verify the good accuracy of the given effective length factors.
1. Introduction Stability is essential throughout the design process of a reticulated shell, where the members mainly subject compressive loads. The stability of reticulated shells includes two levels, i.e. global stability and local stability. A shell often fails with global instability, presented as large out-of-plane deflection of joints and members. Generally speaking, the global stability of a shell controls its ultimate bearing capacity. On the other hand, in-plane buckling of a single member, which is a typical type of local instability, cannot be ignored due to the further collapse it may cause [1]. Once local instability occurs, the buckling member deactivates, leading to redistribution of internal force and even the domino effect of global instability. It is fairly wasteful to let the structure fail by local instability, since the bearing capacity of most members is not utilized. As a consequence, in-plane buckling of shell members should be considered in the design of a reticulated shell. A practical method is the effective length method, which is widely used in determining the buckling load of members. By introducing the effective length factor (hereinafter, the K factor), second-order analysis is avoided. For regular frames, the K factors of the compressive members were proposed by Kavanagh [2] and tabulated in the AISC standard [3]. As for special frames, the research findings were also abundant. The K factors of tapered members in steel gabled frames were investigated by Saffari [4], based on extended slope-deflection equations. Kishi [5,6] considered the non-linearity of the semi-rigid joint
⁎
stiffness, and proposed the K factors of columns in braced and unbraced frames, by means of physical equations as well. Girgin [7] emphasized that for irregular frames, the application of K factors in design codes may lead to rather erroneous results, because the derivation of the K factors only considered local stiffness distributions. A simplified procedure was developed, utilizing a simple quotient based on the results of a fictitious lateral load analysis which was available for both regular and irregular frame members. Hellesland [8] highlighted that under special analysis applications, negative restraints may be involved. Approximate formulas for K factors were derived of compressive members with both positive and negative end constraints. From Hellesland's research results, it can be concluded that section of the member, joint stiffness and end constraint must be taken into consideration in order to determine the K factor of a compressive member. In China, the K factors of members in common spatial structures are given in JGJ7-2010 [9]. However, for single-layer shells with gusset joints, the specified K value is not given. For single-layer aluminum alloy shells, the in-plane and out-of-plane K factors of members are recommended to be 0.9 and 1.6 respectively, by GB50429-2007 [10], but the joint types are not specified. Aluminum alloy gusset (AAG) joint is widely applied in single-layer aluminum alloy reticulated shells, and researches revealed that it is a typical semi-rigid joint system [11]. Thereby, the aforementioned values are unavailable. The out-of-plane buckling of the member is linked to global instability involving most members, and numerous researchers have
Corresponding author. E-mail addresses: [email protected] (S. Zhu), [email protected] (X. Guo), [email protected] (X. Liu), [email protected] (S. Gao).
https://doi.org/10.1016/j.tws.2017.10.033 Received 21 August 2017; Received in revised form 9 October 2017; Accepted 17 October 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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studied the global instability of shells with semi-rigid joints [12–15]. Therefore, this paper is mainly focused on the in-plane effective length factor of the member in reticulated shells with AAG joints. Due to the fact that the section of the shell members generally remain the same along the length, the K factor is related to the AAG joint stiffness (denoted as k1), and end constraint (denoted as k2). Firstly, in order to explore the influence of k1, an experimental program is introduced. Finite element (FE) models are established to simulate the mechanical performance of the specimens, and the effective length of the specimens are obtained. Subsequently, theoretical analysis is carried out to obtain the influence of k2. Finally, the formula of effective length factor is proposed, and validated with a FE reticulated shell model.
Table 1 Information of specimens. Specimen
Plate thickness (mm)
Beam section
Total length (mm)
Net length (mm)
1mA
5
× 50 × 4
1000
760
1mB
5
× 50 × 4
1000
760
1mC
5
× 50 × 4
1000
760
1.2mA
5
× 50 × 4
1200
960
1.2mB
5
I100 ×5 I100 ×5 I100 ×5 I100 ×5 I100 ×5
× 50 × 4
1200
960
2. Test program uncovered by the gusset plate, see Fig. 1. Three specimens had the total length of 1.0 m, and two specimens had the total length of 1.2 m. For convenience, the specimens were labelled with their total length. For example, specimen “1.2mB” denoted the second specimen with the length of 1.2 m. The detailed information of the specimens is tabulated in Table 1.
2.1. Specimens Tests on 2 series (5 in total) of AAG shell members were conducted under axial compressive load. The overall configuration of the specimens is shown in Fig. 1. The joint zone was designed according to the research results of Ref. [16]. The diameter of the gusset plate was 240 mm, and the thickness was 5 mm. The member was connected to the gusset plate by 8 stainless steel M6 bolts hand tightened in 6.5 mm drilled holes. All of the members had the section dimension of I100 × 50 × 4 × 5, denoting that the height of the section is 100 mm, the width of the flange is 50 mm, the thickness of the web and the flange is 4 mm and 5 mm respectively, as plotted in Fig. 2. The total length of the specimen l was defined as the distance between the centers of the gusset plates, and the net length ln was defined as the length of the member
2.2. Materials Aluminum alloy 6061-T4 [17] was adopted to be the material of the gusset plates and the extruded members. The material of the bolts was austenitic stainless steel A2-70 [18]. Tensile tests were conducted to obtain the mechanical properties of the material, according to the Chinese mechanical testing standard [19]. The tensile specimens were cut directly from the gusset plates and the members, as shown in Fig. 3. Tensile tests on the stainless bolts were performed in Ref. [11]. All of the test results are given in Table 2. 2.3. Test arrangement The elevation view of the test device is shown in Fig. 4. Both of the ends of the specimens were connected to the steel clamp using several stainless steel bolts, see Fig. 5(a). In order to simulate the fixed-end boundary condition, a steel sleeve (shown in Fig. 5(b)) was covered on the end of the hydraulic jack. To monitor the response of the specimens, three types of measuring points (demonstrated in Fig. 4) were arranged: (1) 4 strain gauges SG1–SG4 were placed on the edges of the flange, at the mid-section of the specimen. The axial load and eccentricity could be calculated using the readings of the strain gauges.
Fig. 1. Overall configuration of AAG shell member specimens.
Fig. 3. Tensile specimens. (a) Specimens cut from the plate. (b) Specimens cut from the beam member.
Fig. 2. Configurations of the members.
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Table 2 Tensile test results. Specimen
Elastic modulus (MPa)
Nominal yield strength (MPa)
Ultimate strength (MPa)
Plate Beam Bolts
69,088 65,364 206,000
184.40 177.40 470.00
213.58 206.80 725.00
Fig. 6. Typical failure mode.
the position of the specimen according to the readings of strain gauges SG1–SG4 until the specimen is axially loaded. (2) Loading: Apply axial load and collect data continuously. At the initial stages, use force control loading scheme. An increment of 10% of the estimated ultimate load is applied at each stage, and last for at least 5 min. In order to ensure the safety of the experiment and prevent sudden buckling, when the axial load reached 80% of the ultimate load, switch to the displacement control loading scheme. When the load dropped to 60% of the ultimate load, the test program was ended.
Fig. 4. Test device and the arrangements of the measuring points.
2.5. Test results The typical failure mode of the specimen is shown in Fig. 6, i.e. global buckling about the minor axis. The ultimate load and the failure modes of the specimens are summarized in Table 3, the load-deflection and load-strain curves are shown in Figs. 7 and 8, respectively. It is observed in Figs. 7 and 8 that twist occurred at the initial parts of the curves. This was because when the load overcame the friction, slippage occurred between the gusset and flange. Subsequently, the bolt contacted the hole wall, and the stiffness increased. When the ultimate load was reached, the deflection grew rapidly. It can be seen in Table 3 and Fig. 7 that specimen “1mC” presented higher stiffness and bearing capacity, because the diameter of the bolt hole and the initial imperfection were relatively smaller. Besides, specimen “1.2mA” presented super rigidity owing to perfect alignment.
Fig. 5. Details of the test device. (a) Steel clamp. (b) Steel sleeve on the jack.
Consequently, the strain gauges were used to align the specimens, and validate the axial load of the jack throughout the loading process. (2) 2 vertical linear variable differential transducers (LVDTs) LD1–LD2 were placed along the length direction of the specimen. The axial relative displacement between the two ends of the member could be obtained, and the eccentricity could be monitored, based on the readings of these LVDTs. (3) 3 LVDTs were placed at the mid-section of the specimens, 1 along the major axis of the section (D2), and 2 along the minor axis of the section (D1 and D3). They were used to measure the horizontal displacement of the web and the flange, and control the loading process when non-linearity occurs.
3. FE simulation In order to explore the detailed performance of the specimens, the analysis was developed using FE analysis. FE models were established in ABAQUS [20] to simulate the behavior of the specimens. Table 3 Test results.
2.4. Test procedure The experimental process included 2 procedures: (1) Trial loading: After the installation of a specimen, a preload which is 30% of the estimated ultimate load is applied. The aim of trial loading is to make sure that the instruments are working properly, and eliminate the gaps between the components. Meanwhile, adjust 485
Specimen
Pu (kN)
PFE (kN)
Failure mode
Relative error
1mA 1mB 1mC 1.2mA 1.2mB
129.29 132.53 145.71 118.28 118.91
127.93 127.93 147.13 116.49 118.88
Global Global Global Global Global
1.05% 3.47% 0.97% 1.51% 0.03%
buckling buckling buckling buckling buckling
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on ultimate loads and load-displacement curves. The ultimate loads of the FE models and the relative errors are tabulated in Table 3, and the comparison on the load-displacement curves are shown in Fig. 10. It can be observed that the maximum relative error of the ultimate load is 3.47%, and the minimum is 0.03%, and the load-displacement curves agree well. Therefore, the FE models could accurately reveal the behavior of the specimens under axial compressive load. 4. Analysis of in-plane effective length As mentioned before, when the AAG shell member is subjected to inplane buckling, it is restrained by the semi-rigidity of the AAG joint [24] and the rotation stiffness provided by adjacent members, denoting k1 and k2, respectively. The calculating diagram of the isolated member in AAG shell is demonstrated in Fig. 11, where R is the radius of the gusset plate, ln = l − 2R. Whereafter, the influence of k1 and k2 will be analyzed.
Fig. 7. Load-displacement curves of the specimens.
4.1. Influence of k1 – stiffness of AAG joint In order to study the influence of k1, the adjacent members are assumed to be perfectly rigid (k2 = ∞), which is coincident with the boundary conditions of the experiment. According to its physical meaning, the effective length l0 is equal to the distance between the two inflection points of the buckling member, as illustrated in Fig. 12. At the inflection points of the member, only axial and shear forces exist, and there is no bending moment. Hence, the location of the inflection points can be determined easily from the FE results. The Mises stress and displacement contour plots of specimen 1.2mB are shown in Fig. 13. It can be observed that the deformation curve of the specimen is close to that of a both-end-fixed member. The inflection points are located where the stress reached a minimum value, which are easily determined in Fig. 13(b). The distance of the inflection points was measured to be 510 mm, thus the effective length l0 of this member is 510 mm. The effective lengths of the specimens measured from the inflection points are listed in Table 4, along with the total length l and the net length ln. In order to obtain the K factors, the ratios of l0 to l and ln are also given in Table 4. According to Table 4, the following conclusions can be drawn: (1) The gusset plates of AAG joint can strengthen the end boundaries to a certain extent, enabling the member to have a shorter effective length (0.393l–0.421l) than that of fixed-end member (0.5l). (2) It is more rational to relate l0 to ln, where the influence of R is eliminated, and the K factors presented smaller fluctuations (0.517–0.526). Therefore, k1 enables the member to have almost the same effective length as that of a fixed-end member with the net length of ln.
Fig. 8. Load-strain curves of specimen 1.2mA.
3.1. FE model In the FE models (shown in Fig. 9), the geometric dimensions were the same as those of the specimens, as described in the experimental program. The bolt gap was taken into consideration, which denoted that the diameter of the bolt hole was larger than that of the bolt. The steel clamps were simulated as plates with enormous elastic modulus. Fixed boundary conditions were applied at both ends of the model, and axial displacement load was applied at one end. The reduced integration elements C3D8R were adopted to mesh all the components. To simulate the important interfaces between two parts, contact elements were used, and the friction coefficient for all contact surfaces was 0.3 [21]. The mechanical properties were taken from the tensile test results (Table 2). The strain-stress relationship of stainless steel was the bilinear model. For aluminum alloy, the Ramberg-Osgood model [22] and Steinhardt suggestion [23] were considered.
4.2. Influence of k2 – restraint of adjacent members 3.2. Validation of the FE model Pan [25] developed an improved method to obtain the effective length of the members in reticulated shell. It conducted buckle analysis on a whole shell model, with the compressive load applying only to the target member. The advantage of this method is that it can accurately simulate the end boundaries of the target member, and obtain the buckling modes and critical loads with a smaller computation cost. However, the shell majorly bears loads through compressive internal forces in the members, so actually it is inadequate to ignore the stiffness reduction of the adjacent members caused by compression. Therefore, it is speculated that the stiffness of k2 is contributed by the rotation stiffness of the adjacent members, and it is also influenced by their internal compressive forces. To calculate the value of k2, a simplified local zone is taken out (shown in Fig. 14), where the farther ends of the members are conservatively regarded as hinged. When there is no internal forces, the rotation stiffness of k2 at node i of the target member can be given as:
The validation of the FE model was performed through comparison
Fig. 9. FE model of the specimens.
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Fig. 10. Comparison on the load-displacement curves. (a) 1mA. (b) 1mB. (c) 1mC. (d) 1.2mA. (e) 1.2mB.
k1j
k 2j
j R
k2i i
ln
k1i
k1j
inflection points
k1i
l
R
j
l0
i Fig. 12. The effective length obtained from the inflection points.
Fig. 11. Calculating diagram of the AAG shell member.
k2i =
∑ k2m m
k2m = (1)
3Em Im lm2 (lm − 2R)(lm2 − lm R + R2)
(2)
in which Em and lm are the elastic modulus and total length of member m, respectively; Im is moment of inertia about the minor axis of the uniform I-shaped section of member m; R is the radius of the gusset
where m is the number of the adjacent member (m = 1–5). The value of k2m can be obtained using structural mechanics: 487
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Fig. 13. Contour plots of specimen 1.2mB. (a) x-direction displacement. (b) Mises stress.
Table 4 Effective lengths of the specimens. Specimen
l0 (mm)
l (mm)
ln (mm)
l0/l
l0/ln
1mA 1mB 1mC 1.2mA 1.2mB
396 396 393 505 505
1000 1000 1000 1200 1200
760 760 760 960 960
0.396 0.396 0.393 0.421 0.421
0.521 0.521 0.517 0.526 0.526
3 4
k23
k25 5
k22
k24
i
2 Fig. 15. Analysis on the reduction factor. (a) Calculating diagram. (b) FE model.
k21
applicability of Eq. (3) is discussed by means of FE analysis. The calculating diagram of the FE analysis is shown in Fig. 15(a). Two kinds of load conditions were considered: (1) N = 0, M = 1; (2) N = pNE, M = 1 (0 < p < 1). The actual reduction factors are calculated as:
1 Target member
ηm =
Fig. 14. Components of k2.
Nm NEm
(3)
where Nm is the actual compressive load of member m, and NEm is the Euler load of member m. For member m which is hinged at both ends, its Euler load NEm can be calculated as:
NEm =
π 2Em Im lm 2
(5)
where k2m(i) indicates the rotation stiffness of k2m under load condition (i). The FE models were established using ANSYS [26]. The components were simulated by element BEAM188, and the elastic modulus of the joint zones were considered 10 times of the beam member [27], as demonstrated in Fig. 15(b). The FE results are presented in Fig. 16. It can be seen that the FE results are all above the line of Eq. (3), indicating that the reduction factor given by Eq. (3) is the lower limit. Owing to its simplicity, Eq. (3) is considered to be the reduction factor in the following derivation process. However, the aforementioned theoretical and numerical analysis was only conducted to validate the applicability of Eq. (3). In an practical structure, the restraint at the joint zone can be very strong, thus the member m should be assumed to be “hinged at the farther end” and “fixed at the near end”, to calculate NEm in Eq. (3). Therefore, in an actual structure, the reduction factor ηm should be calculated as:
plate. When the compressive force exists in member m, a reduction occurs on k2m due to the second-order effect. For the convenience of analysis, suppose the deformation of the member under bending moment has a shape of the sine curve. Denote the reduction factor as ηm, then its expression can be derived by solving the equilibrium differential equation:
ηm = 1 −
k2m (2) k2m (1)
(4)
As the actual deformation curve may not be exactly a sine curve, the 488
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Fig. 18. Comparison on the load-displacement curves. Fig. 16. p-η distribution.
Fig. 17. Calculating diagram of the member under compression. Fig. 19. Overall configuration of the shell models.
ηm = 1 −
Nm =1− NEm
Nm π 2Em Im (0.7lm)2
4.3. In-plane effective length factor of the AAG shell member (6) Based on the aforementioned analysis, the calculating diagram of a shell member under compression can be described as Fig. 17. The influence of k1 has been considered by introducing the net length of the member. Denote ω2 = N/EI, then the bending equilibrium differential equation can be solved as:
It is worth noting that Eq. (4) can also be applied, which leads to the result inclining to the safer side, while Eq. (6) is of higher accuracy and more economical. As a result, it is proposed that considering the effect of stiffness reduction, k2 can be calculated as:
k2i =
∑ ηm k2m
2 i ⎞ ⎡ i ⎛ i + i ⎞ ωl cos ωl ⎤ + i ω4l 4 sin ωl ω2l n2 ⎢ ⎜⎛ + ⎟ sin ωl n − ⎜ ⎟ n n⎥ n n k k k k k 2j ⎠ 2j ⎠ 2i k2j ⎝ 2i ⎣ ⎝ 2i ⎦ + ωl n (2 − 2 cos ωl n − ωl n sin ωl n ) = 0
(7)
m
where ηm is obtained by Eq. (6).
(8)
Table 5 K factors of AAG shell members. i/k2j i/k2i
0
0.05
0.1
0.2
0.3
0.4
0.5
1
2
3
4
5
≥ 10
0 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 ≥ 10
0.500 0.524 0.546 0.578 0.599 0.615 0.626 0.656 0.675 0.682 0.686 0.689 0.694
0.524 0.549 0.570 0.603 0.626 0.642 0.654 0.685 0.706 0.714 0.718 0.721 0.726
0.546 0.570 0.592 0.625 0.648 0.665 0.677 0.710 0.732 0.741 0.746 0.748 0.754
0.578 0.603 0.625 0.660 0.684 0.701 0.715 0.750 0.773 0.783 0.788 0.791 0.797
0.599 0.626 0.648 0.684 0.709 0.727 0.741 0.777 0.803 0.812 0.818 0.821 0.828
0.615 0.642 0.665 0.701 0.727 0.746 0.760 0.798 0.824 0.834 0.840 0.843 0.850
0.626 0.654 0.677 0.715 0.741 0.760 0.774 0.813 0.840 0.851 0.856 0.860 0.867
0.656 0.685 0.710 0.750 0.777 0.798 0.813 0.855 0.885 0.896 0.902 0.906 0.914
0.675 0.706 0.732 0.773 0.803 0.824 0.840 0.885 0.916 0.928 0.934 0.939 0.947
0.682 0.714 0.741 0.783 0.812 0.834 0.851 0.896 0.928 0.940 0.947 0.951 0.960
0.686 0.718 0.746 0.788 0.818 0.840 0.856 0.902 0.934 0.947 0.954 0.958 0.967
0.689 0.721 0.748 0.791 0.821 0.843 0.860 0.906 0.939 0.951 0.958 0.963 0.971
0.694 0.726 0.754 0.797 0.828 0.850 0.867 0.914 0.947 0.960 0.967 0.971 0.981
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Fig. 20. Buckling modes. (a) Loading condition (1). (b) Loading condition (2).
Table 6 Critical loads and K factors of L1. B/m
l/mm
ln/mm
Pcr(1)/kN
Pcr(2)/kN
μ(2)
i/k2①
i/k2③
μEq
RE
8.3 10 12 14 16 18 20 22
1200 1446 1735 2024 2313 2602 2891 3180
960 1206 1495 1784 2073 2362 2651 2940
249.40 156.43 100.54 69.87 51.33 39.28 31.03 25.13
141.31 92.52 61.81 44.21 33.19 25.83 20.68 16.92
0.707 0.696 0.686 0.680 0.675 0.672 0.669 0.667
0.744 0.559 0.474 0.432 0.398 0.376 0.361 0.349
0.260 0.240 0.228 0.221 0.216 0.213 0.210 0.208
0.748 0.730 0.719 0.711 0.705 0.700 0.697 0.694
5.80% 4.89% 4.81% 4.56% 4.44% 4.17% 4.19% 4.05%
presented. In addition, the buckling modes and critical loads are compared under various loading conditions to verify the necessity of introducing the stiffness reduction factor.
where i is the linear rigidity of the member, i = EI/ln. Denote the K factor as μ, which has the following relationship with ω:
ωl n =
π μ
(9)
5.1. FE model
Substitute Eq. (9) into Eq. (8):
π2 ⎡⎛ i i ⎞ π i i ⎞π π⎤ i2 π 4 π + − ⎜⎛ + cos ⎥ + sin ⎜ ⎟ sin ⎟ 4 μ2 ⎢ ⎝ k2i k2j ⎠ μ k k μ μ k k μ μ 2j ⎠ 2i 2j ⎝ 2i ⎣ ⎦ π π π π + ⎛⎜2 − 2 cos − sin ⎞⎟ = 0 μ⎝ μ μ μ⎠
The FE model of a shell with AAG joints was established in ANSYS [26]. The model was validated against the experimental results in Ref. [27], as shown in Fig. 18. The FE curve agreed well with the test loaddisplacement curve, indicating that the FE model can reveal the behavior of the shell accurately. The overall configuration of the models are shown in Fig. 19. The basic model is a Kiewitt-6 shell with 4 rings, and the rise-to-span ratio is 1/4. The section of the beam members are I100 × 50 × 4 × 5, and the radius of the gusset plate is 120 mm. The spans of the shell are chosen to be 8.3, 10, 12, 14, 16, 18, 20 and 22 m. The support condition is hinged support. With L1 (noted in Fig. 18) being the target member, two kinds of loading conditions are considered: (1) Apply compressive load only to L1, as recommended by Ref. [25]; (2) Apply vertical load on the nodes (①–④, as illustrated in Fig. 19), to simulate actual load distribution.
(10)
where the values of k2i and k2j are computed by Eq. (7). The K factors μ solved by Eq. (10) under various stiffness ratio (i/k2) are tabulated in Table 5. It is worth noting that the effective length l0 should be calculated as:
l 0 = μl n
(11)
The K factors in Table 5 ranged from 0.500 to 0.981. Therefore, it is also conservative to take the net length of the member as its effective length, that is:
l0 = ln
(12)
5.2. Buckling modes and critical loads
The value given by Eq. (12) is close to but more accurate than the recommended effective length in Ref. [10] (0.9l).
The first-order buckling modes of the shells under the 2 loading conditions are plotted in Fig. 20. It is observed that the main difference is how severely member L1 buckled. For loading condition (1), no compressive forces existed in the adjacent members (L2–L11), enabling them to fully restrain member L1. As for loading condition (2), the vertical load led to compressive interior forces in members L2–L11.
5. Validation of the proposed K factors To illustrate the application of the proposed K factors to practical engineering, one numerical example of a whole AAG shell model is 490
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by National Natural Science Foundation of China under Grant no. 51478335.
With the end restraint being weakened, L1 buckled more severely than being fully restrained. Table 6 presents the critical loads of the shells, where B is the span of the shell, and Pcr(i) denotes the critical load of member L1 under loading condition (i). Pcr(1) is at least 50% larger than Pcr(2), indicating that ignoring the stiffness reduction factor may overestimate the buckling capacity of the members. Hence, it is necessary that Eq. (3) is introduced in the derivation process.
References [1] K. Abedi, G.A.R. Parke, Progressive collapse of single-layer braced domes, Int. J. Space Struct. 11 (3) (1996) 291–306. [2] T.C. Kavanagh, Effective length of framed columns, J. Struct. Div. 86 (1960) 1–22. [3] AISC, Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, Illinois, America, 1986. [4] H. Saffari, R. Rahgozar, R. Jahanshahi, An efficient method for computation of effective length factor of columns in a steel gabled frame with tapered members, J. Constr. Steel Res. 64 (4) (2008) 400–406. [5] N. Kishi, W.F. Chen, Y. Goto, et al., Effective length factor of columns in flexibly jointed and braced frames, J. Constr. Steel Res. 47 (1–2) (1998) 93–118. [6] N. Kishi, W.F. Chen, Y. Goto, Effective length factor of columns in semi-rigid and unbraced frames, J. Struct. Eng. 123 (3) (1998) 313–320. [7] K. Girgin, G. Ozmen, E. Orakdogen, Buckling lengths of irregular frame columns, J. Constr. Steel Res. 62 (6) (2006) 605–613. [8] J. Hellesland, Mechanics and effective lengths of columns with positive and negative end restraints, Eng. Struct. 29 (12) (2007) 3464–3474. [9] Ministry of Housing and Urban-Rural Construction of the People’s Republic of China, JGJ7-2010 Technical Specification for Space Frame Structures, China Architecture and Building Press, Beijing, 2010 (in Chinese). [10] Ministry of Construction of the People's Republic of China, GB50429-2007 Code for Design of Aluminum Structures, China Planning Press, Beijing, 2008 (in Chinese). [11] X. Guo, Z. Xiong, Y. Luo, et al., Experimental investigation on the semi-rigid behaviour of aluminium alloy gusset joints, Thin-Walled Struct. 87 (2015) 30–40. [12] X. Wang, S. Dong, Tangent stiffness matrix of members of latticed shell considering joint's stiffness, Eng. Mech. (1999) (In Chinese). [13] X. Guo, Z. Shen, Analysis of stability of single-layer spherical reticulated shells with semi-rigid joints, Build. Sci. Res. Sichuan (2004) (In Chinese). [14] M.A. Dabaon, M.H. El-Boghdadi, O.F. Kharoob, Experimental and numerical model for space steel and composite semi-rigid joints, J. Constr. Steel Res. 65 (8–9) (2009) 1864–1875. [15] A. López, I. Puente, M.A. Serna, Numerical model and experimental tests on singlelayer latticed domes with semi-rigid joints, Comput. Struct. 85 (7–8) (2007) 360–374. [16] X. Guo, Z. Xiong, Y. Luo, et al., Block tearing and local buckling of aluminum alloy gusset joint plates, KSCE J. Civ. Eng. 20 (2) (2016) 820–831. [17] GB 50429-2007, Code for Design of Aluminum Structures, (2007) (In Chinese). [18] GB/T 3098.6-2000, Mechanical Properties of Fasteners: Boles, Screws and Studs Made of Stainless-steel, (2001) (In Chinese). [19] GB/T 228-2002, Metallic Materials-tensile Testing at Ambient Temperature, (2002) (In Chinese). [20] ABAQUS 6. 14-1, Dassault Systèmes, Providence, RI, USA, 2014. [21] EN1999-1-1:2007 Eurocode 9: Design of Aluminum Structures-General Rules and Rules for Buildings. [22] W. Ramberg, W.R. Osgood, Description of stress-strain curves by three parameters, NACA Rep. 656 (1939). [23] O. SteinHardt, Aluminum constructions in civil engineering, Aluminium 47 (1971) 31–39 (254-61). [24] X. Guo, Z. Xiong, Y. Luo, et al., Application of the component method to aluminum alloy gusset joints, Adv. Struct. Eng. 18 (11) (2015) 1931–1946. [25] S. Pan, Y. Li, Calculation method of effective length of members in single-layer spherical lattice shells, Spat. Struct. 22 (2) (2016) 3–8 (in Chinese). [26] ANSYS Multiphysics 16.0, Ansys Inc., Canonsburg, Pennsylvania, USA, 2015. [27] Z. Xiong, X. Guo, Y. Luo, et al., Experimental and numerical studies on single-layer reticulated shells with aluminium alloy gusset joints, Thin-Walled Struct. 118 (2017) 124–136.
5.3. Comparison on K factors The K factors of loading condition (2) are listed in Table 6. They are determined by the critical loads, as:
μ(2) =
π 2EI Pcr(2) l n2
(13)
The stiffness ratios i/k2i are given in Table 6, where k2i indicates the rotation stiffness devoted by the adjacent members at node i. Thus the theoretical K factors μEq can be obtained from Table 5. The relative errors (RE) of the theoretical value μEq and the FE result are also tabulated. It is found that the theoretical values are all larger than the FE results, with an average RE of 4.61%. As a result, the proposed K factors can accurately estimate the in-plane buckling capacity of the member, and are on the safe side. 6. Conclusion In this paper, the in-plane stability of members in single-layer reticulated latticed shells with AAG joints was investigated by means of experimental and theoretical analysis. The main conclusions can be summarized as follows: (1) An experimental program on fixed-end AAG shell members were finished. FE models were established and verified. The test and FE results revealed that the existence of AAG joint zone (denoted as rotation spring k1) relates the effective length of the member with its net length. (2) The stiffness of end restraint (denoted as rotation spring k2) was obtained through theoretical derivation, considering the reduction on rotation stiffness caused by compression. (3) The K factors of AAG shell members calculated by Eq. (10) were listed in the form of a table for engineering service. (4) The proposed K factors were applied to a numerical example. It was seen that the proposed values are accurate and conservative. Acknowledgement The authors gratefully acknowledge the financial support provided
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Full length article
The in-plane effective length of members in aluminum alloy reticulated shell with gusset joints
T
⁎
Shaojun Zhu, Xiaonong Guo , Xian Liu, Shuyu Gao College of Civil Engineering, Tongji University, Shanghai 200092, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Aluminum alloy gusset joint Semi-rigid joint Effective length In-plane stability
This paper presents an analytical research on the in-plane stability of members in single-layer reticulated latticed shells with aluminum alloy gusset (AAG) joints. The influence of two kinds of actual boundary conditions are considered, namely, the stiffness of the AAG joint (k1) and restraint of adjacent members (k2). In order to explore the influence of k1, an experimental program including 5 member specimens with AAG joints is introduced, and finite element (FE) models are established and verified. Based on the test and FE results, the effective length of the specimens are obtained. Subsequently, theoretical analysis is conducted to investigate the effect of k2, with accounting for the stiffness reduction caused by axial compressive force. Finally, the effective length factors of members in reticulated shell with AAG joints are derived and tabulated. Numerical examples are presented to illustrate the necessity of considering the influence of k1 and k2, and verify the good accuracy of the given effective length factors.
1. Introduction Stability is essential throughout the design process of a reticulated shell, where the members mainly subject compressive loads. The stability of reticulated shells includes two levels, i.e. global stability and local stability. A shell often fails with global instability, presented as large out-of-plane deflection of joints and members. Generally speaking, the global stability of a shell controls its ultimate bearing capacity. On the other hand, in-plane buckling of a single member, which is a typical type of local instability, cannot be ignored due to the further collapse it may cause [1]. Once local instability occurs, the buckling member deactivates, leading to redistribution of internal force and even the domino effect of global instability. It is fairly wasteful to let the structure fail by local instability, since the bearing capacity of most members is not utilized. As a consequence, in-plane buckling of shell members should be considered in the design of a reticulated shell. A practical method is the effective length method, which is widely used in determining the buckling load of members. By introducing the effective length factor (hereinafter, the K factor), second-order analysis is avoided. For regular frames, the K factors of the compressive members were proposed by Kavanagh [2] and tabulated in the AISC standard [3]. As for special frames, the research findings were also abundant. The K factors of tapered members in steel gabled frames were investigated by Saffari [4], based on extended slope-deflection equations. Kishi [5,6] considered the non-linearity of the semi-rigid joint
⁎
stiffness, and proposed the K factors of columns in braced and unbraced frames, by means of physical equations as well. Girgin [7] emphasized that for irregular frames, the application of K factors in design codes may lead to rather erroneous results, because the derivation of the K factors only considered local stiffness distributions. A simplified procedure was developed, utilizing a simple quotient based on the results of a fictitious lateral load analysis which was available for both regular and irregular frame members. Hellesland [8] highlighted that under special analysis applications, negative restraints may be involved. Approximate formulas for K factors were derived of compressive members with both positive and negative end constraints. From Hellesland's research results, it can be concluded that section of the member, joint stiffness and end constraint must be taken into consideration in order to determine the K factor of a compressive member. In China, the K factors of members in common spatial structures are given in JGJ7-2010 [9]. However, for single-layer shells with gusset joints, the specified K value is not given. For single-layer aluminum alloy shells, the in-plane and out-of-plane K factors of members are recommended to be 0.9 and 1.6 respectively, by GB50429-2007 [10], but the joint types are not specified. Aluminum alloy gusset (AAG) joint is widely applied in single-layer aluminum alloy reticulated shells, and researches revealed that it is a typical semi-rigid joint system [11]. Thereby, the aforementioned values are unavailable. The out-of-plane buckling of the member is linked to global instability involving most members, and numerous researchers have
Corresponding author. E-mail addresses: [email protected] (S. Zhu), [email protected] (X. Guo), [email protected] (X. Liu), [email protected] (S. Gao).
https://doi.org/10.1016/j.tws.2017.10.033 Received 21 August 2017; Received in revised form 9 October 2017; Accepted 17 October 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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studied the global instability of shells with semi-rigid joints [12–15]. Therefore, this paper is mainly focused on the in-plane effective length factor of the member in reticulated shells with AAG joints. Due to the fact that the section of the shell members generally remain the same along the length, the K factor is related to the AAG joint stiffness (denoted as k1), and end constraint (denoted as k2). Firstly, in order to explore the influence of k1, an experimental program is introduced. Finite element (FE) models are established to simulate the mechanical performance of the specimens, and the effective length of the specimens are obtained. Subsequently, theoretical analysis is carried out to obtain the influence of k2. Finally, the formula of effective length factor is proposed, and validated with a FE reticulated shell model.
Table 1 Information of specimens. Specimen
Plate thickness (mm)
Beam section
Total length (mm)
Net length (mm)
1mA
5
× 50 × 4
1000
760
1mB
5
× 50 × 4
1000
760
1mC
5
× 50 × 4
1000
760
1.2mA
5
× 50 × 4
1200
960
1.2mB
5
I100 ×5 I100 ×5 I100 ×5 I100 ×5 I100 ×5
× 50 × 4
1200
960
2. Test program uncovered by the gusset plate, see Fig. 1. Three specimens had the total length of 1.0 m, and two specimens had the total length of 1.2 m. For convenience, the specimens were labelled with their total length. For example, specimen “1.2mB” denoted the second specimen with the length of 1.2 m. The detailed information of the specimens is tabulated in Table 1.
2.1. Specimens Tests on 2 series (5 in total) of AAG shell members were conducted under axial compressive load. The overall configuration of the specimens is shown in Fig. 1. The joint zone was designed according to the research results of Ref. [16]. The diameter of the gusset plate was 240 mm, and the thickness was 5 mm. The member was connected to the gusset plate by 8 stainless steel M6 bolts hand tightened in 6.5 mm drilled holes. All of the members had the section dimension of I100 × 50 × 4 × 5, denoting that the height of the section is 100 mm, the width of the flange is 50 mm, the thickness of the web and the flange is 4 mm and 5 mm respectively, as plotted in Fig. 2. The total length of the specimen l was defined as the distance between the centers of the gusset plates, and the net length ln was defined as the length of the member
2.2. Materials Aluminum alloy 6061-T4 [17] was adopted to be the material of the gusset plates and the extruded members. The material of the bolts was austenitic stainless steel A2-70 [18]. Tensile tests were conducted to obtain the mechanical properties of the material, according to the Chinese mechanical testing standard [19]. The tensile specimens were cut directly from the gusset plates and the members, as shown in Fig. 3. Tensile tests on the stainless bolts were performed in Ref. [11]. All of the test results are given in Table 2. 2.3. Test arrangement The elevation view of the test device is shown in Fig. 4. Both of the ends of the specimens were connected to the steel clamp using several stainless steel bolts, see Fig. 5(a). In order to simulate the fixed-end boundary condition, a steel sleeve (shown in Fig. 5(b)) was covered on the end of the hydraulic jack. To monitor the response of the specimens, three types of measuring points (demonstrated in Fig. 4) were arranged: (1) 4 strain gauges SG1–SG4 were placed on the edges of the flange, at the mid-section of the specimen. The axial load and eccentricity could be calculated using the readings of the strain gauges.
Fig. 1. Overall configuration of AAG shell member specimens.
Fig. 3. Tensile specimens. (a) Specimens cut from the plate. (b) Specimens cut from the beam member.
Fig. 2. Configurations of the members.
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Table 2 Tensile test results. Specimen
Elastic modulus (MPa)
Nominal yield strength (MPa)
Ultimate strength (MPa)
Plate Beam Bolts
69,088 65,364 206,000
184.40 177.40 470.00
213.58 206.80 725.00
Fig. 6. Typical failure mode.
the position of the specimen according to the readings of strain gauges SG1–SG4 until the specimen is axially loaded. (2) Loading: Apply axial load and collect data continuously. At the initial stages, use force control loading scheme. An increment of 10% of the estimated ultimate load is applied at each stage, and last for at least 5 min. In order to ensure the safety of the experiment and prevent sudden buckling, when the axial load reached 80% of the ultimate load, switch to the displacement control loading scheme. When the load dropped to 60% of the ultimate load, the test program was ended.
Fig. 4. Test device and the arrangements of the measuring points.
2.5. Test results The typical failure mode of the specimen is shown in Fig. 6, i.e. global buckling about the minor axis. The ultimate load and the failure modes of the specimens are summarized in Table 3, the load-deflection and load-strain curves are shown in Figs. 7 and 8, respectively. It is observed in Figs. 7 and 8 that twist occurred at the initial parts of the curves. This was because when the load overcame the friction, slippage occurred between the gusset and flange. Subsequently, the bolt contacted the hole wall, and the stiffness increased. When the ultimate load was reached, the deflection grew rapidly. It can be seen in Table 3 and Fig. 7 that specimen “1mC” presented higher stiffness and bearing capacity, because the diameter of the bolt hole and the initial imperfection were relatively smaller. Besides, specimen “1.2mA” presented super rigidity owing to perfect alignment.
Fig. 5. Details of the test device. (a) Steel clamp. (b) Steel sleeve on the jack.
Consequently, the strain gauges were used to align the specimens, and validate the axial load of the jack throughout the loading process. (2) 2 vertical linear variable differential transducers (LVDTs) LD1–LD2 were placed along the length direction of the specimen. The axial relative displacement between the two ends of the member could be obtained, and the eccentricity could be monitored, based on the readings of these LVDTs. (3) 3 LVDTs were placed at the mid-section of the specimens, 1 along the major axis of the section (D2), and 2 along the minor axis of the section (D1 and D3). They were used to measure the horizontal displacement of the web and the flange, and control the loading process when non-linearity occurs.
3. FE simulation In order to explore the detailed performance of the specimens, the analysis was developed using FE analysis. FE models were established in ABAQUS [20] to simulate the behavior of the specimens. Table 3 Test results.
2.4. Test procedure The experimental process included 2 procedures: (1) Trial loading: After the installation of a specimen, a preload which is 30% of the estimated ultimate load is applied. The aim of trial loading is to make sure that the instruments are working properly, and eliminate the gaps between the components. Meanwhile, adjust 485
Specimen
Pu (kN)
PFE (kN)
Failure mode
Relative error
1mA 1mB 1mC 1.2mA 1.2mB
129.29 132.53 145.71 118.28 118.91
127.93 127.93 147.13 116.49 118.88
Global Global Global Global Global
1.05% 3.47% 0.97% 1.51% 0.03%
buckling buckling buckling buckling buckling
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on ultimate loads and load-displacement curves. The ultimate loads of the FE models and the relative errors are tabulated in Table 3, and the comparison on the load-displacement curves are shown in Fig. 10. It can be observed that the maximum relative error of the ultimate load is 3.47%, and the minimum is 0.03%, and the load-displacement curves agree well. Therefore, the FE models could accurately reveal the behavior of the specimens under axial compressive load. 4. Analysis of in-plane effective length As mentioned before, when the AAG shell member is subjected to inplane buckling, it is restrained by the semi-rigidity of the AAG joint [24] and the rotation stiffness provided by adjacent members, denoting k1 and k2, respectively. The calculating diagram of the isolated member in AAG shell is demonstrated in Fig. 11, where R is the radius of the gusset plate, ln = l − 2R. Whereafter, the influence of k1 and k2 will be analyzed.
Fig. 7. Load-displacement curves of the specimens.
4.1. Influence of k1 – stiffness of AAG joint In order to study the influence of k1, the adjacent members are assumed to be perfectly rigid (k2 = ∞), which is coincident with the boundary conditions of the experiment. According to its physical meaning, the effective length l0 is equal to the distance between the two inflection points of the buckling member, as illustrated in Fig. 12. At the inflection points of the member, only axial and shear forces exist, and there is no bending moment. Hence, the location of the inflection points can be determined easily from the FE results. The Mises stress and displacement contour plots of specimen 1.2mB are shown in Fig. 13. It can be observed that the deformation curve of the specimen is close to that of a both-end-fixed member. The inflection points are located where the stress reached a minimum value, which are easily determined in Fig. 13(b). The distance of the inflection points was measured to be 510 mm, thus the effective length l0 of this member is 510 mm. The effective lengths of the specimens measured from the inflection points are listed in Table 4, along with the total length l and the net length ln. In order to obtain the K factors, the ratios of l0 to l and ln are also given in Table 4. According to Table 4, the following conclusions can be drawn: (1) The gusset plates of AAG joint can strengthen the end boundaries to a certain extent, enabling the member to have a shorter effective length (0.393l–0.421l) than that of fixed-end member (0.5l). (2) It is more rational to relate l0 to ln, where the influence of R is eliminated, and the K factors presented smaller fluctuations (0.517–0.526). Therefore, k1 enables the member to have almost the same effective length as that of a fixed-end member with the net length of ln.
Fig. 8. Load-strain curves of specimen 1.2mA.
3.1. FE model In the FE models (shown in Fig. 9), the geometric dimensions were the same as those of the specimens, as described in the experimental program. The bolt gap was taken into consideration, which denoted that the diameter of the bolt hole was larger than that of the bolt. The steel clamps were simulated as plates with enormous elastic modulus. Fixed boundary conditions were applied at both ends of the model, and axial displacement load was applied at one end. The reduced integration elements C3D8R were adopted to mesh all the components. To simulate the important interfaces between two parts, contact elements were used, and the friction coefficient for all contact surfaces was 0.3 [21]. The mechanical properties were taken from the tensile test results (Table 2). The strain-stress relationship of stainless steel was the bilinear model. For aluminum alloy, the Ramberg-Osgood model [22] and Steinhardt suggestion [23] were considered.
4.2. Influence of k2 – restraint of adjacent members 3.2. Validation of the FE model Pan [25] developed an improved method to obtain the effective length of the members in reticulated shell. It conducted buckle analysis on a whole shell model, with the compressive load applying only to the target member. The advantage of this method is that it can accurately simulate the end boundaries of the target member, and obtain the buckling modes and critical loads with a smaller computation cost. However, the shell majorly bears loads through compressive internal forces in the members, so actually it is inadequate to ignore the stiffness reduction of the adjacent members caused by compression. Therefore, it is speculated that the stiffness of k2 is contributed by the rotation stiffness of the adjacent members, and it is also influenced by their internal compressive forces. To calculate the value of k2, a simplified local zone is taken out (shown in Fig. 14), where the farther ends of the members are conservatively regarded as hinged. When there is no internal forces, the rotation stiffness of k2 at node i of the target member can be given as:
The validation of the FE model was performed through comparison
Fig. 9. FE model of the specimens.
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Fig. 10. Comparison on the load-displacement curves. (a) 1mA. (b) 1mB. (c) 1mC. (d) 1.2mA. (e) 1.2mB.
k1j
k 2j
j R
k2i i
ln
k1i
k1j
inflection points
k1i
l
R
j
l0
i Fig. 12. The effective length obtained from the inflection points.
Fig. 11. Calculating diagram of the AAG shell member.
k2i =
∑ k2m m
k2m = (1)
3Em Im lm2 (lm − 2R)(lm2 − lm R + R2)
(2)
in which Em and lm are the elastic modulus and total length of member m, respectively; Im is moment of inertia about the minor axis of the uniform I-shaped section of member m; R is the radius of the gusset
where m is the number of the adjacent member (m = 1–5). The value of k2m can be obtained using structural mechanics: 487
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Fig. 13. Contour plots of specimen 1.2mB. (a) x-direction displacement. (b) Mises stress.
Table 4 Effective lengths of the specimens. Specimen
l0 (mm)
l (mm)
ln (mm)
l0/l
l0/ln
1mA 1mB 1mC 1.2mA 1.2mB
396 396 393 505 505
1000 1000 1000 1200 1200
760 760 760 960 960
0.396 0.396 0.393 0.421 0.421
0.521 0.521 0.517 0.526 0.526
3 4
k23
k25 5
k22
k24
i
2 Fig. 15. Analysis on the reduction factor. (a) Calculating diagram. (b) FE model.
k21
applicability of Eq. (3) is discussed by means of FE analysis. The calculating diagram of the FE analysis is shown in Fig. 15(a). Two kinds of load conditions were considered: (1) N = 0, M = 1; (2) N = pNE, M = 1 (0 < p < 1). The actual reduction factors are calculated as:
1 Target member
ηm =
Fig. 14. Components of k2.
Nm NEm
(3)
where Nm is the actual compressive load of member m, and NEm is the Euler load of member m. For member m which is hinged at both ends, its Euler load NEm can be calculated as:
NEm =
π 2Em Im lm 2
(5)
where k2m(i) indicates the rotation stiffness of k2m under load condition (i). The FE models were established using ANSYS [26]. The components were simulated by element BEAM188, and the elastic modulus of the joint zones were considered 10 times of the beam member [27], as demonstrated in Fig. 15(b). The FE results are presented in Fig. 16. It can be seen that the FE results are all above the line of Eq. (3), indicating that the reduction factor given by Eq. (3) is the lower limit. Owing to its simplicity, Eq. (3) is considered to be the reduction factor in the following derivation process. However, the aforementioned theoretical and numerical analysis was only conducted to validate the applicability of Eq. (3). In an practical structure, the restraint at the joint zone can be very strong, thus the member m should be assumed to be “hinged at the farther end” and “fixed at the near end”, to calculate NEm in Eq. (3). Therefore, in an actual structure, the reduction factor ηm should be calculated as:
plate. When the compressive force exists in member m, a reduction occurs on k2m due to the second-order effect. For the convenience of analysis, suppose the deformation of the member under bending moment has a shape of the sine curve. Denote the reduction factor as ηm, then its expression can be derived by solving the equilibrium differential equation:
ηm = 1 −
k2m (2) k2m (1)
(4)
As the actual deformation curve may not be exactly a sine curve, the 488
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Fig. 18. Comparison on the load-displacement curves. Fig. 16. p-η distribution.
Fig. 17. Calculating diagram of the member under compression. Fig. 19. Overall configuration of the shell models.
ηm = 1 −
Nm =1− NEm
Nm π 2Em Im (0.7lm)2
4.3. In-plane effective length factor of the AAG shell member (6) Based on the aforementioned analysis, the calculating diagram of a shell member under compression can be described as Fig. 17. The influence of k1 has been considered by introducing the net length of the member. Denote ω2 = N/EI, then the bending equilibrium differential equation can be solved as:
It is worth noting that Eq. (4) can also be applied, which leads to the result inclining to the safer side, while Eq. (6) is of higher accuracy and more economical. As a result, it is proposed that considering the effect of stiffness reduction, k2 can be calculated as:
k2i =
∑ ηm k2m
2 i ⎞ ⎡ i ⎛ i + i ⎞ ωl cos ωl ⎤ + i ω4l 4 sin ωl ω2l n2 ⎢ ⎜⎛ + ⎟ sin ωl n − ⎜ ⎟ n n⎥ n n k k k k k 2j ⎠ 2j ⎠ 2i k2j ⎝ 2i ⎣ ⎝ 2i ⎦ + ωl n (2 − 2 cos ωl n − ωl n sin ωl n ) = 0
(7)
m
where ηm is obtained by Eq. (6).
(8)
Table 5 K factors of AAG shell members. i/k2j i/k2i
0
0.05
0.1
0.2
0.3
0.4
0.5
1
2
3
4
5
≥ 10
0 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 ≥ 10
0.500 0.524 0.546 0.578 0.599 0.615 0.626 0.656 0.675 0.682 0.686 0.689 0.694
0.524 0.549 0.570 0.603 0.626 0.642 0.654 0.685 0.706 0.714 0.718 0.721 0.726
0.546 0.570 0.592 0.625 0.648 0.665 0.677 0.710 0.732 0.741 0.746 0.748 0.754
0.578 0.603 0.625 0.660 0.684 0.701 0.715 0.750 0.773 0.783 0.788 0.791 0.797
0.599 0.626 0.648 0.684 0.709 0.727 0.741 0.777 0.803 0.812 0.818 0.821 0.828
0.615 0.642 0.665 0.701 0.727 0.746 0.760 0.798 0.824 0.834 0.840 0.843 0.850
0.626 0.654 0.677 0.715 0.741 0.760 0.774 0.813 0.840 0.851 0.856 0.860 0.867
0.656 0.685 0.710 0.750 0.777 0.798 0.813 0.855 0.885 0.896 0.902 0.906 0.914
0.675 0.706 0.732 0.773 0.803 0.824 0.840 0.885 0.916 0.928 0.934 0.939 0.947
0.682 0.714 0.741 0.783 0.812 0.834 0.851 0.896 0.928 0.940 0.947 0.951 0.960
0.686 0.718 0.746 0.788 0.818 0.840 0.856 0.902 0.934 0.947 0.954 0.958 0.967
0.689 0.721 0.748 0.791 0.821 0.843 0.860 0.906 0.939 0.951 0.958 0.963 0.971
0.694 0.726 0.754 0.797 0.828 0.850 0.867 0.914 0.947 0.960 0.967 0.971 0.981
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Fig. 20. Buckling modes. (a) Loading condition (1). (b) Loading condition (2).
Table 6 Critical loads and K factors of L1. B/m
l/mm
ln/mm
Pcr(1)/kN
Pcr(2)/kN
μ(2)
i/k2①
i/k2③
μEq
RE
8.3 10 12 14 16 18 20 22
1200 1446 1735 2024 2313 2602 2891 3180
960 1206 1495 1784 2073 2362 2651 2940
249.40 156.43 100.54 69.87 51.33 39.28 31.03 25.13
141.31 92.52 61.81 44.21 33.19 25.83 20.68 16.92
0.707 0.696 0.686 0.680 0.675 0.672 0.669 0.667
0.744 0.559 0.474 0.432 0.398 0.376 0.361 0.349
0.260 0.240 0.228 0.221 0.216 0.213 0.210 0.208
0.748 0.730 0.719 0.711 0.705 0.700 0.697 0.694
5.80% 4.89% 4.81% 4.56% 4.44% 4.17% 4.19% 4.05%
presented. In addition, the buckling modes and critical loads are compared under various loading conditions to verify the necessity of introducing the stiffness reduction factor.
where i is the linear rigidity of the member, i = EI/ln. Denote the K factor as μ, which has the following relationship with ω:
ωl n =
π μ
(9)
5.1. FE model
Substitute Eq. (9) into Eq. (8):
π2 ⎡⎛ i i ⎞ π i i ⎞π π⎤ i2 π 4 π + − ⎜⎛ + cos ⎥ + sin ⎜ ⎟ sin ⎟ 4 μ2 ⎢ ⎝ k2i k2j ⎠ μ k k μ μ k k μ μ 2j ⎠ 2i 2j ⎝ 2i ⎣ ⎦ π π π π + ⎛⎜2 − 2 cos − sin ⎞⎟ = 0 μ⎝ μ μ μ⎠
The FE model of a shell with AAG joints was established in ANSYS [26]. The model was validated against the experimental results in Ref. [27], as shown in Fig. 18. The FE curve agreed well with the test loaddisplacement curve, indicating that the FE model can reveal the behavior of the shell accurately. The overall configuration of the models are shown in Fig. 19. The basic model is a Kiewitt-6 shell with 4 rings, and the rise-to-span ratio is 1/4. The section of the beam members are I100 × 50 × 4 × 5, and the radius of the gusset plate is 120 mm. The spans of the shell are chosen to be 8.3, 10, 12, 14, 16, 18, 20 and 22 m. The support condition is hinged support. With L1 (noted in Fig. 18) being the target member, two kinds of loading conditions are considered: (1) Apply compressive load only to L1, as recommended by Ref. [25]; (2) Apply vertical load on the nodes (①–④, as illustrated in Fig. 19), to simulate actual load distribution.
(10)
where the values of k2i and k2j are computed by Eq. (7). The K factors μ solved by Eq. (10) under various stiffness ratio (i/k2) are tabulated in Table 5. It is worth noting that the effective length l0 should be calculated as:
l 0 = μl n
(11)
The K factors in Table 5 ranged from 0.500 to 0.981. Therefore, it is also conservative to take the net length of the member as its effective length, that is:
l0 = ln
(12)
5.2. Buckling modes and critical loads
The value given by Eq. (12) is close to but more accurate than the recommended effective length in Ref. [10] (0.9l).
The first-order buckling modes of the shells under the 2 loading conditions are plotted in Fig. 20. It is observed that the main difference is how severely member L1 buckled. For loading condition (1), no compressive forces existed in the adjacent members (L2–L11), enabling them to fully restrain member L1. As for loading condition (2), the vertical load led to compressive interior forces in members L2–L11.
5. Validation of the proposed K factors To illustrate the application of the proposed K factors to practical engineering, one numerical example of a whole AAG shell model is 490
Thin-Walled Structures 123 (2018) 483–491
S. Zhu et al.
by National Natural Science Foundation of China under Grant no. 51478335.
With the end restraint being weakened, L1 buckled more severely than being fully restrained. Table 6 presents the critical loads of the shells, where B is the span of the shell, and Pcr(i) denotes the critical load of member L1 under loading condition (i). Pcr(1) is at least 50% larger than Pcr(2), indicating that ignoring the stiffness reduction factor may overestimate the buckling capacity of the members. Hence, it is necessary that Eq. (3) is introduced in the derivation process.
References [1] K. Abedi, G.A.R. Parke, Progressive collapse of single-layer braced domes, Int. J. Space Struct. 11 (3) (1996) 291–306. [2] T.C. Kavanagh, Effective length of framed columns, J. Struct. Div. 86 (1960) 1–22. [3] AISC, Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, Illinois, America, 1986. [4] H. Saffari, R. Rahgozar, R. Jahanshahi, An efficient method for computation of effective length factor of columns in a steel gabled frame with tapered members, J. Constr. Steel Res. 64 (4) (2008) 400–406. [5] N. Kishi, W.F. Chen, Y. Goto, et al., Effective length factor of columns in flexibly jointed and braced frames, J. Constr. Steel Res. 47 (1–2) (1998) 93–118. [6] N. Kishi, W.F. Chen, Y. Goto, Effective length factor of columns in semi-rigid and unbraced frames, J. Struct. Eng. 123 (3) (1998) 313–320. [7] K. Girgin, G. Ozmen, E. Orakdogen, Buckling lengths of irregular frame columns, J. Constr. Steel Res. 62 (6) (2006) 605–613. [8] J. Hellesland, Mechanics and effective lengths of columns with positive and negative end restraints, Eng. Struct. 29 (12) (2007) 3464–3474. [9] Ministry of Housing and Urban-Rural Construction of the People’s Republic of China, JGJ7-2010 Technical Specification for Space Frame Structures, China Architecture and Building Press, Beijing, 2010 (in Chinese). [10] Ministry of Construction of the People's Republic of China, GB50429-2007 Code for Design of Aluminum Structures, China Planning Press, Beijing, 2008 (in Chinese). [11] X. Guo, Z. Xiong, Y. Luo, et al., Experimental investigation on the semi-rigid behaviour of aluminium alloy gusset joints, Thin-Walled Struct. 87 (2015) 30–40. [12] X. Wang, S. Dong, Tangent stiffness matrix of members of latticed shell considering joint's stiffness, Eng. Mech. (1999) (In Chinese). [13] X. Guo, Z. Shen, Analysis of stability of single-layer spherical reticulated shells with semi-rigid joints, Build. Sci. Res. Sichuan (2004) (In Chinese). [14] M.A. Dabaon, M.H. El-Boghdadi, O.F. Kharoob, Experimental and numerical model for space steel and composite semi-rigid joints, J. Constr. Steel Res. 65 (8–9) (2009) 1864–1875. [15] A. López, I. Puente, M.A. Serna, Numerical model and experimental tests on singlelayer latticed domes with semi-rigid joints, Comput. Struct. 85 (7–8) (2007) 360–374. [16] X. Guo, Z. Xiong, Y. Luo, et al., Block tearing and local buckling of aluminum alloy gusset joint plates, KSCE J. Civ. Eng. 20 (2) (2016) 820–831. [17] GB 50429-2007, Code for Design of Aluminum Structures, (2007) (In Chinese). [18] GB/T 3098.6-2000, Mechanical Properties of Fasteners: Boles, Screws and Studs Made of Stainless-steel, (2001) (In Chinese). [19] GB/T 228-2002, Metallic Materials-tensile Testing at Ambient Temperature, (2002) (In Chinese). [20] ABAQUS 6. 14-1, Dassault Systèmes, Providence, RI, USA, 2014. [21] EN1999-1-1:2007 Eurocode 9: Design of Aluminum Structures-General Rules and Rules for Buildings. [22] W. Ramberg, W.R. Osgood, Description of stress-strain curves by three parameters, NACA Rep. 656 (1939). [23] O. SteinHardt, Aluminum constructions in civil engineering, Aluminium 47 (1971) 31–39 (254-61). [24] X. Guo, Z. Xiong, Y. Luo, et al., Application of the component method to aluminum alloy gusset joints, Adv. Struct. Eng. 18 (11) (2015) 1931–1946. [25] S. Pan, Y. Li, Calculation method of effective length of members in single-layer spherical lattice shells, Spat. Struct. 22 (2) (2016) 3–8 (in Chinese). [26] ANSYS Multiphysics 16.0, Ansys Inc., Canonsburg, Pennsylvania, USA, 2015. [27] Z. Xiong, X. Guo, Y. Luo, et al., Experimental and numerical studies on single-layer reticulated shells with aluminium alloy gusset joints, Thin-Walled Struct. 118 (2017) 124–136.
5.3. Comparison on K factors The K factors of loading condition (2) are listed in Table 6. They are determined by the critical loads, as:
μ(2) =
π 2EI Pcr(2) l n2
(13)
The stiffness ratios i/k2i are given in Table 6, where k2i indicates the rotation stiffness devoted by the adjacent members at node i. Thus the theoretical K factors μEq can be obtained from Table 5. The relative errors (RE) of the theoretical value μEq and the FE result are also tabulated. It is found that the theoretical values are all larger than the FE results, with an average RE of 4.61%. As a result, the proposed K factors can accurately estimate the in-plane buckling capacity of the member, and are on the safe side. 6. Conclusion In this paper, the in-plane stability of members in single-layer reticulated latticed shells with AAG joints was investigated by means of experimental and theoretical analysis. The main conclusions can be summarized as follows: (1) An experimental program on fixed-end AAG shell members were finished. FE models were established and verified. The test and FE results revealed that the existence of AAG joint zone (denoted as rotation spring k1) relates the effective length of the member with its net length. (2) The stiffness of end restraint (denoted as rotation spring k2) was obtained through theoretical derivation, considering the reduction on rotation stiffness caused by compression. (3) The K factors of AAG shell members calculated by Eq. (10) were listed in the form of a table for engineering service. (4) The proposed K factors were applied to a numerical example. It was seen that the proposed values are accurate and conservative. Acknowledgement The authors gratefully acknowledge the financial support provided
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