Effect of joint stiffness and size on stability of three-way single-layer cylindrical reticular shell

Abstract

The main aim of the present article is to study the effect of joint stiffness and joint size on load-carrying capacity of single-layer cylindrical reticular shell. One normalized joint bending stiffness index κ_b and three proposed normalized indexes, that is, normalized joint axial stiffness κ_a, normalized joint shear stiffness κ_s, and normalized joint torsional stiffness κ_t, are used to evaluate the stiffness of joint. Through a large number of numerical computations, the main conclusions are summarized as follows: κ_b has a significant effect on limit load of reticular shell, and this effect has a close relationship to rise-to-span ratio of reticular shell. If κ_b is larger than 30, the joint can be treated as rigid joint. The relationship between the logarithm of κ_b and limit load of reticular shell can be expressed by the logistic formulation. Overall rigidity and load-carrying capacity of reticular shell are greatly influenced by joint axial stiffness. If κ_a is larger than 30, the effect of joint axial stiffness on load-carrying capacity of reticular shell is no longer obvious. Otherwise, the load-carrying capacity will be markedly reduced. The relation between the logarithm of κ_a and limit load of reticular shell can be fitted by the Dose–response formulation. The load-carrying capacity of reticular shell is also influenced by joint torsional stiffness and joint shear stiffness to some extent. The relation between the logarithm of κ_s and limit load can be fitted by the Asymptotic formulation. The effect of joint size on overall rigidity and limit load of reticular shell is evident and cannot be neglected. The limit load gradually decreases with the decrease in joint size, and there is an approximate linear relationship between limit load and joint size.

Keywords

axial stiffness bending stiffness joint size semi-rigid joint shear stiffness single-layer cylindrical reticular shell torsional stiffness

Introduction

Single-layer cylindrical reticular shell is one of the most efficient structures due to the characteristics of light self-weight, strong spanning ability, high quality manufacturing, and so on. Hence, it is widely used in stadium, exhibition hall, coal storage shed, terminal building, and so forth. In the design and analysis, stability of reticular shell is the key issue. Stability of this kind of structure is affected by many factors, for instance, overall shape, grid form and size, load pattern, joint stiffness, various imperfections, and so on.

The mechanical performance of single-layer cylindrical reticular shell is obviously influenced by joint stiffness, especially joint bending¹ stiffness. However, in the traditional design and analysis, the stiffness of joint is usually treated as infinite, namely, rigid joint, which can facilitate the analysis and design indeed. However, the actual overall rigidity and load-carrying capacity of reticular shell may be different, and a large or a small calculation error definitely exists under this hypothetical² condition.

Actually, stiffness of joint in single-layer cylindrical reticular shell is finite, not infinite. Based on this issue, lots of effort has been paid on the mechanical behavior of joint stiffness. Many types of joint systems are adopted in practical single-layer cylindrical reticular shell,³ such as welded ball joint, weld ribbed-ring joint, bolted disk joint, bolt ribbed-cylinder joint, screw ball joint, and so on. The bending stiffness of each connection system is obviously different.⁴ Normalized bending stiffness κ_b is larger than 30 for welded ball joint, and κ_b is larger than 20 and less than 50 for welded ribbed-ring joint. For bolted disk joint, κ_b is larger than 10 and less than 30. For bolted ribbed-cylinder joint, κ_b is larger than 10 and less than 30. For bolted cylinder joint, κ_b is larger than 1.0 and less than 10. κ_b is usually larger than 0.1 and less than 1.0 for screw joint system.

In the past years, lots of efforts have been paid on the investigation of effect of joint bending stiffness on mechanical performance of space structures. Taniguchi et al.⁵ carried out experimental research on mechanical performance of space trusses, and developed the regression curve of bending moment–rotation relationship of joint system. Saka and Heki⁶ researched the influence of joint stiffness on the load-carrying capacity of space truss using a mechanical model with rigid end parts and springs. Ma and colleagues^7,8 developed a finite element (FE) model to simulate semi-rigid joint in space structures, and research the moment–rotation relationship of joint to verify the accuracy of the developed model. Chenaghlou et al.⁹ proposed an exponential model to predict of moment–rotation behavior of semi-rigid joint in space latticed structure. Ding et al.¹⁰ developed a multi-step FE model for welded hollow spherical joint. Lopez et al.^11,12 carried out experimental research studies on the rotational stiffness of joints in single-layer latticed structures to establish geometrical parameters for a semi-rigid joint, and presented how to improve the stiffness of the joint. Ma et al.¹³ developed a new semi-rigid joint system, which is referred to as the bolt-column (BC) joint, and conducted a series of tests considering different parameters. Furthermore, the bending stiffness, moment resistance, rotational capacity, and failure mode of the joint were evaluated by a three-dimensional (3D) FE model of the joint. Ma et al.¹⁴ developed a new joint, named the bolt-column-plate (BCP) joint. Combined with theoretical investigation, the mechanical performance of the joint under both static and quasi-static loads was researched. The numerical results show that the bending capacity significantly increases with increasing eccentric distance of the axial forces, while the initial bending stiffness is almost the same for joints subject to pure bending. Wang et al.¹⁵ investigated the influence of the connections’ behavior and imperfection on stability of the cable-stiffened lattice shell. The experiment and numerical results show that the limit load of the lattice shells with bolted connections decreases slowly than the decrease in joint stiffness. Zhu et al.¹⁶ researched the in-plane stability of members in single-layer reticulated latticed shells with aluminum alloy gusset joints by experimental and theoretical analysis, and the effective length factors of members in reticulated shell with the joints are obtained.

However, the previous studies mainly focus on the effect of joint bending stiffness and joint size on mechanical performance and stability of space structures, and most of them were restricted to single-layer reticulated dome. Furthermore, the relationship between joint stiffness/size and limit load of reticular shell has not been clarified well. Hence, the main aim of the article is to systematically research the effect of joint stiffness (i.e. joint bending stiffness, joint axial stiffness, joint torsional stiffness, and joint shear stiffness) and joint size on load-carrying capacity of single-layer cylindrical reticular shell with three kinds of grid numbers and four kinds of rise-to-span ratios. One normalized joint bending stiffness index κ_b introduced and three normalized indexes κ_a, κ_s, and κ_s proposed by the present authors are used to evaluate the stiffness of joint, and more than 1500 numerical simulations are carried out to investigate the effect of joint stiffness and joint size on load-carrying capacity of reticular shell.

Semi-rigidly jointed model of single-layer cylindrical reticular shell

For the sake of simplicity, all members in cylindrical single-layer reticular shell are supposed to be rigidly jointed in traditional numerical analysis. Under the circumstances, all kinds of joint stiffness (bending stiffness, axial stiffness, shear stiffness, and torsional stiffness) are considered as infinite, and joint size is neglected. In reality, the body of joint belonging to semi-rigid joint can deform under member force and external load, and its stiffness is a finite value. To figure out the effect of joint stiffness and joint size on load-carrying capacity of single-layer reticular shell, a semi-rigidly jointed model is put forward in this study, shown in Figure 1(a). Each member in reticular shell is simulated by several beam elements, and n is the number of elements and n is taken as 10 in the present study. In Figure 1(a), l is the distance between the centers of two joints $i^{'}$ and $j^{'}$ . Two rigid beams are appended at both two ends of each member to consider the joint size, and their lengths equal to $2 λ l$ , that is to say, diameter of joint is $2 λ l$ . The actual length of member is $(1 - 2 λ) l$ , where $λ$ is a factor (far less than 0.5) depending on actual joint size. Usually, $2 λ$ is in the range of $[0.03, 0.14]$ .

Figure 1.

Mechanical model considering joint size and joint semi-rigid stiffness: (a) semi-rigidly jointed model and (b) illustrative diagram of different joint stiffness.

Due to the fact that wall thickness and stiffness of joint is much larger than the ones of member, material constitutive model of joint is assumed as linear in this mechanical model. Spring element (namely, Combin14 in ANSYS package) connecting ordinary beam and rigid beam, shown in Figure 1(a), is used to simulate the semi-rigidity of joint. In ANSYS package, the ordinary beam and rigid beam are all simulated by Beam188 element, which is suitable for analyzing slender to moderately stubby/thick beam structures. Shear deformation effects are also included. This element is based on Timoshenko beam theory. The elasticity modulus of ordinary beam is taken as actual value according to steel material used, and the elasticity modulus of rigid beam is amplified by 1000 times to realize the infinity of joint stiffness.

Illustrative diagram of a joint linking to six members is shown in Figure 1(b). Semi-rigid joint is simulated by the spring element Combin14. There are two coincident nodes at the junction of ordinary beam and rigid beam. One belongs to ordinary beam, and the other one belongs to rigid beam. The two nodes have the same coordinates, that is to say, they coincide and the length of spring element is zero. There are totally six springs between the two nodes for each junction of ordinary beam and rigid beam. Three of them are used to simulate translational stiffness (actually, one axial stiffness $K_{a x}$ , and two shear stiffness $K_{s y}$ and $K_{s z}$ , shown in Figure 1(b)), and other three springs are utilized to simulate rotational stiffness (in fact, two bending stiffness $K_{b y}$ and $K_{b z}$ , and one torsional stiffness $K_{t x}$ ). For the connection shown in Figure 1(b), there are totally 36 springs in this connection system. Semi-rigid connection/joint can be effectively realized by this method.

Load-carrying capacity of reticular shell with semi-rigid joint

Model description

Three-way single-layer cylindrical reticular shell is taken as analytical model with span S = 30 m and longitudinal size L = 36 m in this study, shown in Figure 2. To comprehensively investigate the effect of joint stiffness and joint size on load-carrying capacity of single-layer cylindrical reticular shell, four kinds of rise-to-span ratios (f/S = 1/3, 1/4, 1/5, and 1/6, where f and S are rise and span of reticular shell, respectively) and three kinds of gird numbers 12 × 12, 10 × 10, and 8 × 8 are considered in the present study. Different types of member length are also depicted in Figure 2 and Table 1, and most of members in the single-layer cylindrical reticular shell belong to Length 3 and Length 4. Range of member length and member slenderness ratio is tabulated in Table 1.

Figure 2.

Geometric size, boundary condition, and type of member of single-layer cylindrical reticular shell: (a) grid number 12 × 12, (b) grid number 10 × 10, and (c) grid number 8 × 8.

Table 1.

Range of member length and member slenderness ratio of the three-way single-layer reticulated shell.

Grid number	f/S	1/3	1/4	1/5	1/6
12 × 12	Range of member length	[1.500, 3.516]	[1.500, 3.260]	[1.500, 3.139]	[1.500, 3.071]
12 × 12	Range of slenderness ratio	[27.45, 64.35]	[27.45, 59.67]	[27.45, 57.44]	[27.45, 56.21]
10 × 10	Range of member length	[1.800, 4.217]	[1.800, 3.911]	[1.800, 3.765]	[1.800, 3.685]
10 × 10	Range of slenderness ratio	[32.94, 77.17]	[32.94, 71.58]	[32.94, 68.91]	[32.94, 67.44]
8 × 8	Range of member length	[2.250, 5.265]	[2.250, 3.911]	[2.250, 4.705]	[2.250, 4.605]
8 × 8	Range of slenderness ratio	[41.18, 96.36]	[41.18, 89.42]	[41.18, 86.10]	[41.18, 84.27]

BEAM188 is used to simulate members of single-layer cylindrical reticular shell. For the convenience of analysis, all members have the same cross section $Φ 159 \times 4.5$ , and the external diameter and thickness of the member are 159 and 4.5 mm, respectively. All nodes at the bottom of reticular shell in the longitudinal edge are hinged and all nodes in the lateral edge along span direction are only supported in the vertical direction, shown in Figure 2. External load is uniformly exerted in the vertical direction. The material of steel pipe is Q235 with yield stress 235 MPa and elasticity modulus 2.1 × 10⁵ MPa, and strain hardening rate is taken as 0.02. In the numerical stability analysis, both geometrical nonlinearity and material nonlinearity are all considered, and the arc-length method is used to gain limit load of reticular shell. Due to limited space, initial imperfections are not considered in the present study, and research object is the perfect reticulated shell with semi-rigid joint.

Load-carrying capacity of rigidly jointed cylindrical reticular shell

To verify the effect of joint stiffness and joint size on load-carrying capacity of three-way single-layer cylindrical reticular shell, limit loads of rigidly jointed reticular shells with different grid numbers (12 × 12, 10 × 10, and 8 × 8) and rise-to-span ratios (f/S = 1/3, 1/4, 1/5, and 1/6) are first determined in this section. Limit loads of all rigidly jointed reticular shells listed in Table 2 are obtained by FE method considering both material and geometrical nonlinearity. Table 2 shows that limit load of reticular shell gradually reduces with the decrease in rise-to-span ratio and grid number. Figure 3 shows scatter plot of internal force of rigidly jointed reticular shells in limit state (grid number 12 × 12 and rise-to-span ratio 1/3).

Table 2.

Limit loads of rigidly jointed single-layer cylindrical reticular shells (unit: kN/m²).

f/S	12 × 12	10 × 10	8 × 8
1/3	8.28079	7.48777	6.13217
1/4	7.65328	6.66253	5.74638
1/5	6.25144	5.44373	4.64433
1/6	5.20927	4.53667	3.84676

Figure 3.

Scatter plot of internal force (grid number 12 × 12, f/S = 1/3): (a) axial force, (b) bending moment MY, (c) bending moment MZ, (d) shear force SFZ, and (e) shear force SFY.

Effect of joint bending stiffness on load-carrying capacity of reticular shell

Definition of normalized joint bending stiffness

The index for the bending rigidity of joint can be defined as normalized bending stiffness¹⁷

κ_{b} = K_{b} \frac{(1 - 2 λ) l}{E I}

(1)

where $K_{b} = K_{b y} = K_{b z}$ is the joint bending stiffness, $N \cdot m / rad$ ; and $K_{b y}$ and $K_{b z}$ are joint bending stiffness about y and z axes shown in Figure 1(b). $(1 - 2 λ) l$ is the length of member shown in Figure 1(a). E and I are elasticity modulus and moment of inertia, respectively.

In this section, emphasis is mainly paid on effect of joint bending stiffness on load-carrying capacity of reticular shell, so $K_{b}$ are taken as finite values and the other three kinds of joint stiffness (i.e. axial stiffness $K_{a x}$ , torsional stiffness $K_{t x}$ , and shear stiffness $K_{s}$ , shown in Figure 1(b)) are taken as infinite values. There are many types of joint systems used in practical space frame structures, such as welded ball joint, weld ribbed-ring joint, bolted disk joint, bolt ribbed-cylinder joint, bolted cylinder joint, screw ball joint, and so forth. Joint bending stiffness of each type of joint system is apparently different.⁴ κ_b is usually greater than 30 for welded ball joint, and greater than 20 and less than 50 for welded ribbed-ring joint. κ_b is greater than 10 and less than 30 for bolted disk joint. κ_b is greater than 10 and less than 30 for bolted ribbed-cylinder joint. κ_b is greater than 1.0 and less than 10 for bolted cylinder joint. κ_b is greater than 0.1 and less than 1.0 for screw joint. To research the effect of joint bending stiffness on load-carrying capacity of reticular shell, 11 κ_b are considered here, that is, κ_b equals to 0.1, 0.3, 0.5, 1, 2, 3, 4, 5, 10, 30, and 100, respectively. The different values of κ_b represent different joint system.

Stability analysis of reticular shell with different κ_b

Load-carrying capacity of reticular shell is obviously affected by joint bending stiffness.^5,6 To clarify the influence of joint bending stiffness on limit load of the reticular shell, 11 different magnitudes of joint bending stiffness mentioned in the previous section are considered in this section. In the following numerical analyses, limit load of reticular shell is determined considering both geometrical and material nonlinearity.

The single-layer cylindrical reticular shell consists of four different types of length of member, shown in Figure 2, so it is impossible to define normalized bending stiffness κ_b as a unique value. However, diagonal member in the reticular shell, denoted by black solid line (Length 4) shown in Figure 2, is the main member to support external load and self-weight of reticular shell. Hence, diagonal member is used to define $κ_{b}$ , and λ is taken as 0.03. Values of EI/((1 − 2λ)l) for all cases are all listed in Table 3. κ_b is taken as 0.1, 0.3, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 10, 30, and 100, respectively. The joint with κ_b = 100 can be treated as rigid joint.

Table 3.

Value of EI/((1 − 2λ)l) (unit: N·m/rad).

f/S	12 × 12	10 × 10	8 × 8
1/3	414,459	345,577	276,754
1/4	446,937	372,570	298,245
1/5	464,297	387,006	309,732
1/6	474,471	395,461	316,458

All limit loads and their reductions of reticular shells with 11 different magnitudes of joint bending stiffness are shown in Figure 4. Figure 4(a) shows that the limit load of reticular shell (grid number 12 × 12) gradually decreases with the decrease in κ_b. Compared with limit loads 8.43348, 7.83769, 6.4126, and 5.34018 kN/m² of the four reticular shells (f/S = 1/3, 1/4, 1/5, and 1/6) with κ_b = 100 shown in Figure 4(a), limit load reductions of reticular shell with κ_b = 30 are 0.86%, 1.83%, 1.68%, and 2.06%, respectively, shown in Figure 4(b), and limit load reductions of reticular shells with κ_b = 10 are 3.07%, 4.59%, 6.51%, and 7.66%, respectively. Hence, if κ_b is larger than 30, the effect of joint bending stiffness on load-carrying capacity of reticular shell is not obvious, and it can be considered as rigid joint. If κ_b is less than 30, the load-carrying capacity of reticular shell will obviously reduce. For instance, limit load reductions are 5.97%, 10.45%, 13.70%, and 15.35%, respectively, for the four reticular shells with κ_b = 5. Furthermore, the effect of joint bending stiffness on limit load of reticular shell has a close relationship to rise-to-span ratio, and limit load reduction increases with the decrease in rise-to-span ratio of reticular shell.

Figure 4.

Limit load and its reduction of reticular shell with different joint bending stiffness (2λ = 0.06): (a) limit load of reticular shell with different κ_b (12 × 12); (b) limit load reduction of reticular shell with different κ_b (12 × 12); (c) limit load of reticular shell with different κ_b (10 × 10); (d) limit load reduction of reticular shell with different κ_b (10 × 10); (e) limit load of reticular shell with different κ_b (8 × 8); and (f) limit load reduction of reticular shell with different κ_b (8 × 8).

Figure 4(c) shows that the limit loads of reticular shells (grid number 10 × 10) gradually decrease with the decrease in κ_b. Compared with limit loads 7.656, 6.934, 5.676, and 4.750 kN/m² of the four reticular shells (f/S = 1/3, 1/4, 1/5, and 1/6) with κ_b = 100 shown in Figure 4(c), limit load reductions of reticular shell with κ_b = 30 are 1.09%, 2.13%, 1.70%, and 2.07%, respectively, shown in Figure 4(d), and limit load reductions of reticular shell with κ_b = 10 are 4.18%, 4.23%, 6.61%, and 7.71%, respectively. Similarly, if κ_b is larger than 30, the effect of joint bending stiffness on load-carrying capacity of reticular shell is not obvious, and it can be considered as rigid joint. If κ_b is less than 30, the joint bending stiffness obviously reduces the load-carrying capacity of reticular shell. Figure 4(e) shows that the limit loads of reticular shells (grid number 8 × 8) also gradually decrease with the decrease in κ_b. Similarly, if κ_b is larger than 30, it can be considered as rigid joint.

Based on the variation of curves in Figure 4(a), (c), and (e), authors of the present article found that the relationship between the logarithm of κ_b and limit load of reticular shell can be expressed by the following logistic formulation

P_{c r} = \frac{a}{1 + e^{- k (\log κ_{b} - c)}}

(2)

where P_cr is the limit load of reticular shell, κ_b is normalized bending stiffness. a, c, and k are constant parameters, and all of them are tabulated in Table 4. Figure 4(a), (c), and (e) shows that the computational results accord well with the logistic formulation, which also can be judged by the values of adjusted coefficient of determination (Adj. R-Square) tabulated in Table 4.

Table 4.

Values of a, c, and k for all cases.

Parameter	12 × 12				10 × 10				8 × 8
Parameter	1/3	1/4	1/5	1/6	1/3	1/4	1/5	1/6	1/3	1/4	1/5	1/6
a	8.5064	8.0431	6.6000	5.5298	7.6761	6.9867	5.7341	4.8133	6.2518	5.9473	4.8873	4.0942
c	−0.2785	−0.0787	−0.0300	0.0020	−0.1664	−0.058	−0.0117	0.0189	−0.1410	−0.0415	0.0139	0.0560
k	2.7666	2.6139	2.3880	2.2836	2.8516	2.7177	2.4684	2.3257	2.7564	2.8989	2.4994	2.4053
Adj. R-Square	0.9839	0.9985	0.9996	0.9998	0.9949	0.9980	0.9996	0.9997	0.9920	0.9962	0.9994	0.9997

Failure mechanism of reticular shells with different κ_b is discussed here. Typical load and nodal vertical displacement curves of reticular shells with rise-to-span ratio 1/4 are shown in Figure 5, from which it can be seen that reticular shells (grid numbers 12 × 12, 10 × 10, and 8 × 8) with different κ_b exhibit discrepant mechanical behavior, and limit load gradually decreases with the decrease in κ_b. The typical failure modes after peak loads of the reticular shells with different κ_b are shown in Figure 6, from which it can be concluded that failure mode of reticular shell changes if κ_b is smaller than 1–3. Local instability occurs when κ_b is larger than 2 or 3, while overall instability happens when κ_b is smaller than 1 or 2. Hence, initial stiffness of reticular shell under vertically uniformly distributed loads considered is different when κ_b changes, and initial stiffness of reticular shell with smaller κ_b (e.g. smaller than 1 or 2) is slightly larger than the one of reticular shell with larger κ_b.

Figure 5.

Load–vertical displacement curves of reticular shells with different κ_b (f/S = 1/4, 2λ = 0.06): (a) 12 × 12, (b) 10 × 10, and (c) 8 × 8.

Figure 6.

Failure mode of reticular shell with different magnitudes of κ_b (f/S = 1/4, 2λ = 0.06): (a) 12 × 12, (b) 10 × 10, and (c) 8 × 8.

Effect of joint axial stiffness on load-carrying capacity of reticular shell

Definition of normalized joint axial stiffness

The index for joint axial stiffness of joint proposed by authors of the article can be defined as normalized axial stiffness

κ_{a} = K_{a x} \frac{(1 - 2 λ) l}{E A}

(3)

where $K_{a x}$ is the axial stiffness of joint, $N / m$ , shown in Figure 1(b). $(1 - 2 λ) l$ is the actual length of member. E and A are elasticity modulus and cross-sectional area of member, respectively.

In this section, the main attention is paid on effect of joint axial stiffness and combination of joint axial stiffness and bending stiffness on load-carrying capacity of reticular shell, so $K_{a x}$ and $K_{b}$ are taken as finite values and the other two kinds of joint stiffness (i.e. torsional stiffness $K_{t x}$ and shear stiffness $K_{s}$ ) are taken as infinite ones. Eleven normalized axial stiffness κ_a are taken into account here, that is, κ_a equals to 0.1, 0.3, 1.0, 1.2, 1.5, 2.0, 3.0, 5.0, 10, 30, and 100, respectively.

Stability analysis of reticular shell with different joint axial stiffness

Mechanical performance and load-carrying capacity of reticular shell are influenced by joint axial stiffness. To figure out the effect of joint axial stiffness on load-carrying capacity of reticular shell, 11 different magnitudes of joint axial stiffness and four kinds of joint bending stiffness are taken into consideration here.

Similar to section “Stability analysis of reticular shell with different κ_b,” the diagonal member is used to define normalized axial stiffness $κ_{a}$ , and λ is taken as 0.03. Values of EA/((1 − 2λ)l) are all listed in Table 5. κ_a is taken as 0.1, 0.3, 1.0, 1.2, 1.5, 2.0, 3.0, 5.0, 10, 30, and 100, respectively. The joint with κ_a = 100 can be treated as rigid joint.

Table 5.

Value of EA/((1 − 2λ)l) (unit: N/m).

f/S	12 × 12	10 × 10	8 × 8
1/3	138,786,044	115,720,315	92,674,274
1/4	149,661,959	124,759,116	99,870,618
1/5	155,474,861	129,593,353	103,717,103
1/6	158,881,822	132,424,515	105,969,521

All limit loads and their reductions of reticular shells with 11 magnitudes of joint axial stiffness are shown in Figure 7. Figure 7(a) shows that the limit load of reticular shell (κ_b = 100) first gradually decreases then suddenly drops with the decrease in κ_a. Compared with limit loads of all reticular shells with κ_a = 100 shown in Figure 7(a), limit load reductions of reticular shells with κ_a = 30 are between 0.45% and 2.53%, shown in Figure 7(b), and limit load reductions of reticular shell with κ_a = 10 are between 1.84% and 6.69%. Hence, if κ_a is larger than 30, the effect of joint axial stiffness on load-carrying capacity of reticular shell is not obvious, and it can be considered as rigid joint. If κ_a is less than 30, the load-carrying capacity of reticular shell will obviously reduce.

Figure 7.

Limit load and its reduction of reticular shell with different magnitudes of κ_a (2λ = 0.06): (a) limit load of reticular shell with different κ_a (κ_b = 100); (b) limit load reduction of reticular shell with different κ_a (κ_b = 100); (c) limit load of reticular shell with different κ_a (κ_b = 30); (d) limit load reduction of reticular shell with different κ_a (κ_b = 30); (e) limit load of reticular shell with different κ_a (κ_b = 10); and (f) limit load reduction of reticular shell with different κ_a (κ_b = 10).

Figure 7(c) further shows that the limit loads of reticular shells (κ_b = 30) first gradually decrease then abruptly reduce with the decrease in κ_a. Compared with limit loads of all reticular shells with κ_a = 100 shown in Figure 7(c), limit load reductions of reticular shells with κ_a = 30 are between 0.23% and 2.59%, shown in Figure 7(d), and limit load reductions of reticular shells with κ_a = 10 are between 0.90% and 9.16%. Hence, if κ_a is larger than 30, it can be considered as rigid joint. Otherwise, the load-carrying capacity of reticular shell will obviously reduce. For instance, limit load reduction is up to 3.15%–15.85% for the reticular shells with κ_a = 5. Figure 7(e) also shows that the limit loads of reticular shells (κ_b = 10) first gradually decrease then abruptly reduce with the decrease in κ_a. Similarly, if κ_b is larger than 30, it can be considered as rigid joint.

From Figure 7 it also can be seen that limit load reduction of reticular shell with different magnitudes of κ_a is apparently affected by rise-to-span ratio f/S. For reticular shell with f/S = 1/3, limit load reduction of reticular shell with larger κ_a (e.g. larger than 2) reduces slowly compared with other rise-to-span ratios, while limit load reduction of reticular shell with small κ_a (e.g. smaller than 2) reduces markedly compared with other rise-to-span ratios.

Based on the variation of curves in Figure 7(a), (c), and (e), authors of the present article found that the relationship between the logarithm of κ_a and limit load of reticular shell can be fitted the following Dose–response formulation

P_{c r} = A 1 + \frac{A 2 - A 1}{1 + 10^{(\log x 0 - \log κ_{a}) p}}

(4)

where P_cr is the limit load of reticular shell, κ_a is normalized axial stiffness. A1, A2, p, and logx0 are constant parameters, and only some of them are listed in Table 6 due to the limit of length of this article. Figure 7(a), (c), and (e) shows that the computational results accord well with the Dose–response formulation, which also can be judged by the values of adjusted coefficient of determination (Adj. R-Square) shown in Table 6.

Table 6.

Values of A1, A2, logx0, and p (κ_b = 100).

Parameter	12 × 12				10 × 10				8 × 8
Parameter	1/3	1/4	1/5	1/6	1/3	1/4	1/5	1/6	1/3	1/4	1/5	1/6
A1	2.99234	2.86210	2.94859	2.76924	2.55642	2.46633	2.55756	2.40739	2.21210	2.10825	2.15199	2.04430
A2	8.50046	7.96053	6.53331	5.47548	7.68522	7.00465	5.68701	4.77212	6.27260	5.99665	4.86159	4.05956
logx0	−0.1052	0.06611	0.12033	0.11253	−0.09096	0.07387	0.11448	0.11376	−0.12876	0.08375	0.11075	0.11454
p	1.41813	0.94591	0.94084	0.90934	1.31271	0.92541	0.93547	0.90250	1.45437	0.93807	0.91481	0.90429
Adj. R-Square	0.99780	0.99941	0.99984	0.99985	0.99809	0.99963	0.99991	0.99985	0.99773	0.99951	0.99985	0.99974

Failure mechanism of reticular shells with different κ_a is discussed here. Representative load and nodal vertical displacement curves of reticular shells are shown in Figure 8, which shows that overall rigidity and load-carrying capacity of reticular shell are remarkably influenced by joint axial stiffness. With the decrease in κ_a, the overall rigidity of reticular shell gradually decreases because the slope of load-displacement curve gradually becomes flat in the elastic range.

Figure 8.

Load–vertical displacement curves of reticular shells with different κ_a (f/S = 1/5, 2λ = 0.06): (a) 12 × 12, (b) 10 × 10, and (c) 8 × 8.

Effect of joint torsional stiffness on load-carrying capacity of reticular shell

Definition of normalized joint torsional stiffness

The index for joint torsional stiffness of joint developed by the present authors can be defined as normalized torsional stiffness

κ_{t} = K_{t x} \frac{(1 - 2 λ) l}{G I_{t}}

(5)

where $K_{t x}$ is the torsional stiffness of joint, $N \cdot m / rad$ , shown in Figure 1(b). $G I_{t}$ is the torsional stiffness of member. G and I_t are shear modulus and torsional moment of inertia, respectively.

In this section, the main attention is exerted on the effect of joint torsional stiffness and coupling effect of joint torsional and bending stiffness on load-carrying capacity of reticular shell, so $K_{t x}$ and $K_{b}$ are taken as finite values and the other two kinds of joint stiffness (i.e. axial stiffness $K_{a x}$ and shear stiffness $K_{s}$ ) are taken as infinite ones. Seven normalized torsional stiffness κ_t are taken into account, that is, κ_t equals to 0.0001, 0.001, 0.01, 0.1, 1, 10, and 100, respectively.

Stability analysis of reticular shell with different joint torsional stiffness

To figure out the influence of joint torsional stiffness on load-carrying capacity of single-layer cylindrical reticular shell, seven different magnitudes of joint torsional stiffness are taken into consideration. Similarly, the diagonal member is used to define normalized torsional stiffness $κ_{t}$ , and λ is taken as 0.03. Values of GI_t/((1 − 2λ)l) are all presented in Table 7. κ_t is taken as 0.0001, 0.001, 0.01, 0.1, 1, 10, and 100, respectively. The joint with κ_t = 100 can be treated as rigid joint.

Table 7.

Value of GI_t/((1 − 2λ)l) (unit: N·m/rad).

f/S	12 × 12	10 × 10	8 × 8
1/3	312,742	260,765	208,833
1/4	337,250	281,133	225,049
1/5	350,348	292,027	233,717
1/6	358,026	298,407	238,793

All limit loads and their reductions of reticular shells with seven different magnitudes of joint torsional stiffness are shown in Figure 9. Figure 9(a) shows that the limit load of reticular shell (κ_b = 100) first gradually decreases then smoothly reduces with the decrease in κ_t. Compared with limit loads of all reticular shells with κ_t = 100 shown in Figure 9(a), limit load reductions of reticular shells with κ_t = 10 are between 0.31% and 2.31%, shown in Figure 9(b), and limit load reductions of reticular shells with κ_t = 1 are between 3.19% and 7.67%. Hence, if κ_t is larger than 10, it can be considered as rigid joint. Otherwise, the load-carrying capacity of reticular shell will obviously decrease, for instance, limit load reduction is between 3.58% and 11.53% for the shells with κ_t = 0.1.

Figure 9.

Limit load and its reduction of reticular shells with different magnitudes of κ_t: (a) limit load of reticular shell with different κ_t (κ_b = 100); (b) limit load reduction of reticular shell with different κ_t (κ_b = 100); (c) limit load of reticular shell with different κ_t (κ_b = 30); (d) limit load reduction of reticular shell with different κ_t (κ_b = 30); (e) limit load of reticular shell with different κ_t (κ_b = 10); and (f) limit load reduction of reticular shell with different κ_t (κ_b = 10).

Figure 9(c) also shows that the limit loads of reticular shells (κ_b = 30) first gradually decrease then smoothly reduce with the decrease in κ_t. Compared with limit loads of all reticular shells with κ_t = 100 shown in Figure 9(c), limit load reductions of reticular shells with κ_t = 10 are between 0.50% and 1.97%, shown in Figure 9(d), and limit load reductions of reticular shells with κ_t = 1 are between 2.33% and 7.96%. Hence, if κ_t is larger than 10, the effect of joint torsional stiffness on load-carrying capacity of reticular shell is not obvious, and it can be considered as rigid joint.

Figure 9(e) also shows that the limit loads of reticular shells (κ_b = 10) first gradually decrease then smoothly reduce with the decrease in κ_t. Similarly, if κ_t is larger than 10, it can be considered as rigid joint.

Representative load and nodal vertical displacement curves of reticular shells are shown in Figure 10, from which it can be seen that load-carrying capacity and post-buckling behavior of reticular shell is slightly influenced by joint torsional stiffness.

Figure 10.

Load–vertical displacement curves of reticular shells with different κ_t (8 × 8, f/S = 1/5, κ_b = 100).

Effect of joint shear stiffness on load-carrying capacity of reticular shell

Definition of normalized joint shear stiffness

The index for joint shear stiffness of joint proposed by authors of this article can be defined as normalized shear stiffness

κ_{s} = K_{s} \frac{(1 - 2 λ) l}{G A}

(6)

where $K_{s} = K_{s y} = K_{s z}$ is shear stiffness of joint, $N / m$ ; and $K_{s y}$ and $K_{s z}$ are joint shear stiffness about y and z axes, respectively, shown in Figure 1(b). G and A are shear modulus and cross-sectional area of member, respectively.

In this section, emphasis is mainly paid on the effect of joint shear stiffness and coupling effect of joint shear stiffness and bending stiffness on load-carrying capacity of reticular shell, so $K_{s}$ and $K_{b}$ are taken as finite values and the other two kinds of joint stiffness (i.e. axial stiffness $K_{a x}$ and torsional stiffness $K_{t x}$ ) are taken as infinite values. Twelve normalized shear stiffness κ_s are taken into account, that is, κ_s equals to 0.001, 0.002, 0.003, 0.004, 0.005, 0.01, 0.02, 0.03, 0.1, 1, 10, and 100, respectively.

Stability analysis of reticular shell with different joint shear stiffness

Load-carrying capacity of reticular shell may be influenced by joint shear stiffness to a certain degree. To clarify the effect of joint shear stiffness on load-carrying capacity of reticular shell, 12 magnitudes of joint shear stiffness are taken into consideration here. Similarly, the diagonal member is used to define normalized shear stiffness $κ_{s}$ , and λ is taken as 0.03. Values of GA/((1 − 2λ)l) are all shown in Table 8. κ_s is taken as 0.001, 0.002, 0.003, 0.004, 0.005, 0.01, 0.02, 0.03, 0.1, 1, 10, and 100. The joint with κ_s = 100 is treated as rigid joint.

Table 8.

Value of GA/((1 − 2λ)l) (unit: N/m).

f/S	12 × 12	10 × 10	8 × 8
1/3	52,362,500	43,660,046	34,965,019
1/4	56,465,868	47,070,289	37,680,123
1/5	58,659,014	48,894,196	39,131,361
1/6	59,944,424	49,962,363	39,981,175

All limit loads and their reductions of reticular shells with 12 different magnitudes of joint shear stiffness are shown in Figure 11. Figure 11(a) shows that the limit loads of reticular shells (κ_b = 100) first smoothly reduce then abruptly drop with the decrease in κ_s. Compared with limit loads of all reticular shells with κ_s = 100 shown in Figure 11(a), limit load reductions of reticular shells with κ_s = 1 are between 0.04% and 0.21%, shown in Figure 11(b), and limit load reductions of reticular shells with κ_s = 0.1 are between 0.41% and 2.24%. Then, limit load of reticular shell sharply decreases if κ_s is smaller than 0.1. Hence, if κ_s is larger than 0.1, the effect of joint shear stiffness on load-carrying capacity of reticular shell is not obvious, and it can be considered as rigid joint. Otherwise, if κ_s is less than 0.1, the joint shear stiffness obviously reduces the load-carrying capacity of single-layer cylindrical reticular shell.

Figure 11.

Limit load and its reduction of reticular shell with different magnitudes of κ_s: (a) limit load of reticular shell with different κ_s (κ_b = 100); (b) limit load reduction of reticular shell with different κ_s (κ_b = 100); (c) limit load of reticular shell with different κ_s (κ_b = 30); (d) limit load reduction of reticular shell with different κ_s (κ_b = 30); (e) limit load of reticular shell with different κ_s (κ_b = 10); and (f) limit load reduction of reticular shell with different κ_s (κ_b = 10).

The same conclusion can be drawn for reticular shell with κ_b = 30 and 10. Hence, the effect of joint shear stiffness on load-carrying capacity of shell is not obvious and can be neglected unless κ_s is less than 0.1.

Based on the variation of curves in Figure 11(a), (c), and (e), authors of the present article found that the relationship between the logarithm of κ_s and limit load of reticular shell can be fitted the following Asymptotic formulation

P_{c r} = a - b \cdot c^{\log κ_{s}}

(7)

where P_cr is the limit load of reticular shell, κ_s is normalized shear stiffness. a, b, and c are constant parameters, and some of them are listed in Table 9 due to the limit of length of this article. Figure 11(a), (c), and (e) shows that the computational results accord well with the Asymptotic formulation, which also can be judged by the values of adjusted coefficient of determination listed in Table 9.

Table 9.

Values of a, b, and c (κ_b = 100).

Parameter	12 × 12				10 × 10				8 × 8
Parameter	1/3	1/4	1/5	1/6	1/3	1/4	1/5	1/6	1/3	1/4	1/5	1/6
a	8.60026	8.01626	6.5174	5.43295	7.73379	7.00748	5.67695	4.73644	6.37947	5.98546	4.85373	4.03429
b	0.08117	0.04885	0.01648	0.00587	0.05120	0.03994	0.01502	0.0058	0.02165	0.03949	0.01626	0.00605
c	0.25450	0.22493	0.17323	0.13133	0.22422	0.21838	0.17527	0.13659	0.17947	0.22798	0.18853	0.14556
Adj. R-Square	0.99534	0.99895	0.99926	0.99670	0.99480	0.99574	0.99957	0.99743	0.98256	0.99473	0.99958	0.99834

A typical failure mode of shells with κ_s = 0.01 is shown in Figure 12, and six load-displacement curves of reticular shells with κ_s = 100, 1, 0.1, 0.01, 0.003, and 0.001 are depicted in Figure 13, from which it can be seen that reticular shells with different κ_s exhibit different load-carrying capacity and mechanical performance. If value of κ_s is too small, large shear deformation at conjunction of joint and member will occur, shown in Figure 12, which leads to the obvious reduction of load-carrying capacity of reticular shell.

Figure 12.

Failure modes of reticular shells with weak joint shear stiffness (κ_b = 100, κ_s = 0.01, 2λ = 0.06).

Figure 13.

Load–vertical displacement curves of reticular shells with different κ_s (8 × 8, f/S = 1/5, κ_b = 100).

Effect of joint size on load-carrying capacity of reticular shell

In this section, main attention is paid on the effect of joint size and combination of joint size and bending stiffness on load-carrying capacity of reticular shell, so 2λ and $K_{b}$ are taken as different values and the other three kinds of joint stiffness (i.e. $K_{a x}$ , $K_{t x}$ , and $K_{s}$ ) are taken as infinite values. Twelve joint sizes are taken into account here, that is, 2λ equals to 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.11, 0.12, 0.13, and 0.14, respectively. Load-carrying capacity of reticular shell is affected by joint size. To verify the influence of joint size on load-carrying capacity of the reticular shell, 12 joint sizes are taken into consideration here. Similarly, the diagonal member with length 5.234 m is used to define the size of joint.

All limit loads and their reductions of reticular shells with 12 different joint sizes are shown in Figure 14, from which it can be seen that the limit load gradually decreases with the decrease in joint size, and there is an approximate linear relationship between limit load and joint size. For reticular shell with κ_b = 100, limit load reductions of shell with 2λ = 0.03 are between 5.17% and 9.38% compared with ones with 2λ = 0.14, shown in Figure 14(b). Hence, the effect of joint size on load-carrying capacity of reticular shell is obvious.

Figure 14.

Limit load and its reduction of reticular shell with different joint sizes: (a) limit load of shell with different joint sizes (κ_b = 100); (b) limit load reduction of shell with different joint sizes (κ_b = 100); (c) limit load of shell with different joint size (κ_b = 30); (d) limit load reduction of shell with different joint size (κ_b = 30); (e) limit load of shell with different joint size (κ_b = 10); and (f) limit load reduction of shell with different joint size (κ_b = 10).

The similar conclusion can be obtained for reticular shell with κ_b = 30 and 10. Hence, the effect of joint size on load-carrying capacity of reticular shell is obvious and cannot be neglected.

Six load-displacement curves of reticular shells with 2λ = 0.04, 0.06, 0.08, 0.10, 0.12, and 0.14 are depicted in Figure 15, which shows that overall rigidity and load-carrying capacity of shells are apparently affected by joint size.

Figure 15.

Load–vertical displacement curves of reticular shells with joint sizes (8 × 8, f/S = 1/5, κ_b = 100).

Conclusion

In this article, the effect of joint stiffness and joint size on load-carrying capacity of single-layer cylindrical reticular shell is systematically investigated. Meanwhile, one normalized joint bending stiffness index κ_b introduced and three indexes, that is, normalized joint axial stiffness κ_a, normalized joint torsional stiffness κ_t, and normalized joint shear stiffness κ_s proposed by the present authors are used to evaluate the stiffness of joint. Through a large number of numerical computations, the main conclusions obtained are summarized as follows:

(1) Normalized joint bending stiffness κ_b has an evident effect on limit load of reticular shell. Compared with four reticular shells (f/S = 1/3, 1/4, 1/5, and 1/6) with κ_b = 100, limit load reductions of reticular shell with κ_b = 30 are 0.86%, 1.83%, 1.68%, and 2.06%, respectively, and limit load reductions of reticular shell with κ_b = 10 are 3.07%, 4.59%, 6.51%, and 7.66%, respectively. Hence, if κ_b is larger than 30, the connection of reticular shell can be treated as rigid joint. Otherwise, the joint with weak bending stiffness obviously reduces the load-carrying capacity of reticular shell. In addition, the effect of joint bending stiffness on limit load of reticular shell has a close relationship to rise-to-span ratio of reticular shell, and limit load reduction increases with the decrease in rise-to-span ratio. The relationship between the logarithm of κ_b and limit load of reticular shell can be expressed by the logistic formulation. Failure mode of reticular shell is affected by the magnitude of κ_b.

(2) Overall rigidity and load-carrying capacity of reticular shell are remarkably influenced by joint axial stiffness. With the decrease in κ_a, the overall rigidity and limit load of reticular shell gradually decreases. Compared with limit loads of all reticular shells with κ_a = 100, limit load reductions of reticular shells with κ_a = 30 are between 0.45% and 2.53%, and limit load reductions of reticular shell with κ_a = 10 are between 1.84% and 6.69%. Hence, If κ_a is larger than 30, the effect of joint axial stiffness on load-carrying capacity of reticular shell is not obvious, and it can be considered as rigid joint. If κ_a is less than 30, the load-carrying capacity of reticular shell will be markedly reduced. The relationship between the logarithm of κ_a and limit load of reticular shell can be expressed by the Dose–response formulation.

(3) The load-carrying capacity of reticular shell is also influenced by joint torsional stiffness. The limit load of reticular shell first obviously decreases then smoothly reduces with the decrease in κ_t. If κ_t is larger than 10, the connection of reticular shell can be considered as rigid joint. If κ_t is less than 10, the joint torsional stiffness obviously reduces the load-carrying capacity of reticular shell.

(4) The effect of joint shear stiffness on load-carrying capacity of shell is not obvious and can be neglected unless κ_s is less than 0.1. Limit load of reticular shell first smoothly reduces then abruptly drops with the decrease in κ_s. The relationship between the logarithm of κ_s and limit load of reticular shell can be expressed by the Asymptotic formulation. Failure mode of reticular shell is affected by κ_s. If value of κ_s is too small, large shear deformation at conjunction of joint and member occurs, which further leads to the obvious reduction of load-carrying capacity of reticular shell.

(5) The effect of joint size on overall rigidity and load-carrying capacity of reticular shell is obvious and cannot be neglected. The limit load gradually decreases with the decrease in joint size, and there is an approximate linear relationship between limit load and joint size.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The supports of the National Natural Science Foundation of China (No. 51408490), the China Scholarship Council (No. 201706305044), are gratefully appreciated.

ORCID iD

Hui-jun Li

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Effect of joint stiffness and size on stability of three-way single-layer cylindrical reticular shell

Abstract

Keywords

Introduction

Semi-rigidly jointed model of single-layer cylindrical reticular shell

Load-carrying capacity of reticular shell with semi-rigid joint

Model description

Load-carrying capacity of rigidly jointed cylindrical reticular shell

Effect of joint bending stiffness on load-carrying capacity of reticular shell

Definition of normalized joint bending stiffness

Stability analysis of reticular shell with different κb

Effect of joint axial stiffness on load-carrying capacity of reticular shell

Definition of normalized joint axial stiffness

Stability analysis of reticular shell with different joint axial stiffness

Effect of joint torsional stiffness on load-carrying capacity of reticular shell

Definition of normalized joint torsional stiffness

Stability analysis of reticular shell with different joint torsional stiffness

Effect of joint shear stiffness on load-carrying capacity of reticular shell

Definition of normalized joint shear stiffness

Stability analysis of reticular shell with different joint shear stiffness

Effect of joint size on load-carrying capacity of reticular shell

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Stability analysis of reticular shell with different κ_b