A spherical lattice shell composed of six-bar tetrahedral units: Configuration,structural behavior,and prefabricated construction

Abstract

A novel spherical lattice shell composed of six-bar tetrahedral units whose horizontal projections are quadrilaterals is proposed. The lattice shell provides an advantage in industrialized production and prefabricated construction by comprising a plurality of similarly shaped modular units. Kinematic analysis for the lattice shell, based on which several strengthening structures and joint rigidity arrangements are proposed, is carried out. The structural behavior of a basic structure of the lattice shell, whose chord and web members are designed as beam and bar elements, respectively, is studied systematically. Construction details of the nodal joints and six-bar tetrahedral modular units are suggested, followed by a discussion about the construction scheme. A tool called A-shaped installing frame is designed to realize the prefabricated construction without scaffold. A test model of 10-m span was built to verify the feasibility of the nodal joint construction as well as the prefabricated construction scheme.

Keywords

kinematic analysis prefabricated construction six-bar tetrahedral modular unit spherical lattice shell stability behavior static behavior structural configuration

Introduction

Lattice shell is one of the most widely used space structures for its high adaptive capacity and excellent structural behavior. These features have encouraged the architects and structural engineers in the past several decades to develop numerous lattice shell structural systems, which can be applied to buildings with different spans, functions, plane shapes, surface forms, and supporting conditions (Chilton, 1999; Dong et al., 2000, 2012; Makowski, 1984, 1986).

Although impressive flexibility has been demonstrated in practical applications, some conventions still need to be followed in the design of lattice shells, for example, it is a mandatory requirement in China’s Technical Specification for Space Frame Structures (JGJ 7-2010, 2010) that rigid connections between members shall be adopted in single-layer lattice shells. However, studies on some single-layer lattice shells with folded forms (Dong and Xing, 2011; Dong and Zheng, 2012) showed that these structures are able to provide sufficient load-carrying capability when some or all nodal joints are designed as hinged connections, resulting in simpler and lighter nodal joint systems and a substantial reduction in the cost of manufacture and construction. This has pointed out a way of developing new structural systems which is based on the innovation of structural form and/or topology.

On the other hand, industrialized production and prefabricated construction have always been the key issues in the economization of building structures. Space grid structures constructed of modular units, such as Space Deck System (Bolton and Desmond, 1960) and CUBIC Space Frame System (Kubik and Kubik, 1991), have made successful realizations of the thoughts by permitting a maximum amount of ground-level construction and reducing the workload on site. In China, the modernization of building industry has been a hot issue in recent years, and many researchers, design institutes, and construction companies are devoted to facilitating the process. However, developing feasible structural systems remains the most essential work for the researchers and engineers.

This article aims to provide a prefabricated lattice shell structural system that serves the building industrialization. A lattice shell composed of six-bar tetrahedral modular units, whose horizontal projections are quadrilaterals, is proposed. The lattice shell provides an advantage in industrialized production and prefabricated construction by comprising a plurality of similarly shaped modular units. The structural configuration of the lattice shell is introduced in detail, followed by a kinematic analysis. The structural behavior of the lattice shell, including the static performance as well as the linear and nonlinear stability behaviors, is investigated. Construction details of the nodal joints and six-bar tetrahedral modular units, which make it convenient for the industrialized production and prefabricated construction of the structural system, are suggested. An A-shaped installing frame with simple construction is developed as an essential tool for the prefabricated construction of the lattice shell where no scaffolds are employed, and the detailed installation procedure is given subsequently. Finally, a test model of 10-m span was built, in order to verify the feasibility of the construction of the nodal joint and the modular unit as well as the proposed construction scheme.

Structural configuration

Figure 1 illustrates a six-bar tetrahedral unit that comprises one upper chord member, one lower chord member, and four web members and has a quadrilateral horizontal projection. It can be readily proved that the unit is kinematically determinate if its four nodal joints are hinged. The said unit differs from the conventional triangular pyramid unit shown in Figure 2, which consists of three chord members and three web members and usually has a triangular horizontal projection.

Figure 1.

Six-bar tetrahedral unit.

Figure 2.

Conventional triangular pyramid unit.

The spherical lattice shell composed of the six-bar tetrahedral modular units is demonstrated in Figure 3(a), with one modular unit highlighted. In practical application, the open top is generally enclosed with inner-ring members (Figure 3(b)) due to construction requirements. The numbers of modular units distributed in radial and circumferential directions are p and q, respectively (Dong et al., 2014), leading to a total number of pq. The structural configuration of the spherical lattice shell can be designated as T_pq for simplicity.

Figure 3.

Spherical lattice shells composed of six-bar tetrahedral modular units: (a) basic structure without inner-ring members and (b) basic structure with inner-ring members.

Although working like a double-layer space grid, the lattice shell has a rather simple configuration compared with lattice shells comprising triangular or rectangular pyramid units, while the latter need a series of additional upper (or lower) chord members to form complete structures (Dong et al., 2006; JGJ 7-2010, 2010). The modular units forming the lattice shell can be grouped into p specifications, each of which contains q similarly shaped modular units, thereby permitting the implementation of mass production. The modular units form a series of concentric circular rings in the lattice shell, and each ring corresponds to a specification of modular units. If the inner-ring members are employed, q individual members should be added. These are all the components needed to form the lattice shell. Industrialized production and prefabricated construction can be carried out for its simple and regular construction.

It can be found from Figure 3 that the shell is two-way folded, that is, circumferentially folded if viewed from outside and radially folded from inside. To be specific, if the upper chord members are regarded as ridge lines while the radial connecting lines of lower chord nodal joints as valley lines, the lattice shell is circumferentially folded when viewed from outside; on the other hand, if the lower chord members and the circumferential connecting lines of the upper chord nodal joints are treated as ridge and valley lines, respectively, the lattice shell can be viewed as radially folded when observed from inside. Thus, the form of the building is enriched enormously and the structural performance and architectural aesthetics are able to complement each other.

Kinematic analysis of the spherical lattice shell

If all the nodal joints of the lattice shell are hinged, the basic structures in Figure 3 will turn into bar assemblies and kinematic analyses can then be carried out using Maxwell’s rule (Maxwell, 1864) as well as the equilibrium matrix theory (Pellegrino, 1993; Pellegrino and Calladine, 1986).

The number of six-bar tetrahedral modular units in the basic structure shown in Figure 3(a) is pq, and each modular unit contains six bars, just as indicated by its name. Therefore, the total number of bars is C = 6pq. On the other hand, the numbers of nodal joints and constraints at the periphery are J = (2p+1)q and C ₀ = 3q, respectively, thus

D = 3 J - C - C_{0} = 3 (2 p + 1) q - 6 pq - 3 q \equiv 0

(1)

So this structure is statically and kinematically determinate according to Maxwell’s rule.

Nevertheless, the structure is proved to be kinematically indeterminate if the singular value decomposition (SVD) of its equilibrium matrix (Pellegrino, 1993) is executed. Regarded to be able to offer an accurate criterion for the movability of bar assemblies, the equilibrium matrix theory reveals that the basic structure in Figure 3(a) has p mechanisms when q is even, each of which corresponds to the movement of a specific ring of modular units, as illustrated in Figure 4.

Figure 4.

Mechanisms of the basic structure of T_pq where p = 5 and q = 16: (a) first order, (b) second order, (c) third order, (d) fourth order, and (e) fifth order.

Theoretically, there are two methods available for the elimination of the mechanisms. The first one is to add extra bars between nodal joints with relative movements, so as to establish some additional restraints within the structure. It can be proved that the minimum number of bars required for the basic structure in Figure 3(a) to be kinematically determinate is p, and each ring of modular units should be assigned with one bar at least. Basic structures strengthened by extra bars are referred to as strengthening structures herein. Two types of strengthening structures can be formed by this method, as shown in Figure 5. It should be noted that the inner-ring members in Figure 3(b) can also be deemed as the said extra bars but can only eliminate one mechanism which corresponds to the movement of the modular units belonging to the innermost ring (Figure 4(a)), whereas the other p − 1 mechanisms remain, that is, p − 1 extra bars are still demanded for the basic structure with inner-ring members to become kinematically determinate. The extra bars are generally arranged symmetrically in practical applications, for the sake of satisfaction of architectural and structural design requirements. Two examples are provided in Figure 6.

Figure 5.

Strengthening of basic structure by adding extra bars: (a) extra bars between upper chord nodal joints and (b) extra bars between lower chord nodal joints and inner-ring nodal joints.

Figure 6.

Strengthening structures for practical application: (a) upper chord strengthening and (b) lower chord strengthening.

From another point of view, the mechanisms of the basic structures can also be eliminated by restraining relative rotations between adjacent members, that is, setting some members as beam elements. Figure 7 provides some options for the joint rigidity arrangement of the basic structure with inner-ring members (Figure 3(b)), accounting for symmetry and feasibility. A fully rigid lattice shell, all of whose members shall be designed as beams, is demonstrated in Figure 7(a), while in Figure 7(b) the ends of the web members are pinned to the rigid connections between the chord and inner-ring members, making the nodal joints hybrids of rigid and hinged parts. The structure in Figure 7(c) only leaves the upper chord and inner-ring members rigidly connected, and the ends of lower chord and web members are all pinned to the nodal joints. The fully hinged lattice shell in Figure 7(d) consists of bars and is the same as the original basic structure in Figure 3(b). This method is particularly efficient in cases that do not expect extra bars to be employed and has an advantage in industrialized production and prefabricated construction due to the availability of more unified construction details of nodal joints and modular units for the entire structure. In the remaining sections of this article, the structure shown in Figure 7(b) will be taken as an example to discuss the structural behavior, construction details of nodal joints and modular units, as well as realization of building industrialization of the lattice shell.

Figure 7.

Joint rigidity arrangements for the basic structure with inner-ring members: (a) all nodal joints are rigid (fully rigid); (b) inner-ring, upper, and lower chord members are rigidly connected, and web members are pinned to the nodal joints; (c) inner-ring and upper chord members are rigidly connected, and lower chord and web members are pinned to the nodal joints; and (d) all nodal joints are hinged (fully hinged, kinematically indeterminate).

Static performance of the spherical lattice shell

In this section, the distributions of the internal forces and nodal displacements of the structure in Figure 7(b), to which uniform load is applied, will be investigated through an example. The calculation is carried out using ANSYS, and in the finite element model, the inner-ring, upper (radial), and lower (circumferential) chord members are modeled with BEAM188 (all in red solid lines in Figure 7(b)) and the web members are modeled with LINK180 (in black dashed lines in Figure 7(b)). Hinged supports are employed at the structure’s periphery.

In this example, the values of 5 and 12 are separately assigned to parameters p and q. The structure spans a distance of 50 m while the ratios of rise, thickness, and diameter of the open top to the span are set to be 1/4, 1/30, and 1/6, respectively. Circular steel tubes of specifications Φ203 × 14, Φ180 × 5, and Φ114 × 5, the cross-sectional areas of which are 83.13, 27.49, and 17.12 cm², are assigned to the upper chord (including the inner-ring) members, lower chord members, and web members, respectively. The steel consumption is 28.3 kg/m². Uniform load of 2.0 kN/m² that covers the entire area of the lattice shell (including the open top area) is applied to the structure. The elasticity modulus of steel is 2.06 × 10¹¹ N/m². Figure 8 shows the geometric dimensions of the structure and the numbering of the nodal joints and internal forces.

Figure 8.

Geometric dimensions of the structure and numbering of nodal joints and internal forces: (a) plan view and (b) profile view.

A static calculation is made based on the foregoing conditions. The internal forces of the members, support reactions, as well as the axial and flexural stresses are presented in Table 1. The flexural stress is calculated by the larger moment on the two ends of a member. Table 2 illustrates the vertical (w), horizontal (Δ), and total (u) nodal displacements. The positive direction for the vertical displacements is straight up and the direction from the center of the structure to the support is taken as positive for the horizontal displacements.

Table 1.

Internal forces, support reactions, and axial/flexural stresses of the lattice shell.

Member number	Axial force (kN)	Axial stress (MPa)	Flexural stress (MPa) (% of axial stress)	Member number	Axial force (kN)	Axial stress (MPa)
N ₁	−132.2	−15.90	0.74 (4.65%)	N _1,2	11.3	6.58
N ₃	−167.7	−20.18	0.91 (4.51%)	N _2,3	−7.4	−4.35
N ₅	−209.0	−25.14	3.82 (15.19%)	N _3,4	16.1	9.43
N ₇	−252.4	−30.36	4.98 (16.40%)	N _4,5	−8.5	−4.97
N ₉	−276.7	−33.29	4.98 (14.96%)	N _5,6	19.4	11.36
H ₁	−294.4	−35.42	0.42 (1.19%)	N _6,7	−5.6	−3.25
H ₂	−70.4	−25.62	0.00 (0%)	N _7,8	24.2	14.15
H ₄	−75.3	−27.40	0.00 (0%)	N _8,9	8.5	4.96
H ₆	−56.4	−20.53	0.00 (0%)	N _9,10	17.2	10.07
H ₈	−11.7	−4.24	0.00 (0%)	N _10,11	0.6	0.38
H ₁₀	−13.5	−4.92	0.00 (0%)	R ₁₁	182.6	–
				Z ₁₁	216.5	–

Table 2.

Vertical, horizontal, and total nodal displacements of the lattice shell.

Joint number	Displacements (mm)			Joint number	Displacements (mm)
Joint number	Vertical (w)	Horizontal (Δ)	Total (u)	Joint number	Vertical (w)	Horizontal (Δ)	Total (u)
1	−5.93	−0.72	5.97	2	−6.21	−0.78	6.26
3	−5.52	−1.02	5.61	4	−6.25	−1.43	6.42
5	−5.37	−1.49	5.57	6	−6.04	−1.49	6.22
7	−3.96	−1.36	4.18	8	−4.40	−0.39	4.41
9	−0.82	0.28	0.87	10	−1.44	−0.53	1.54

It can be found from the above results that

The upper chord members are all in compression, and the absolute values of the internal forces increase outwardly from the center to the periphery of the structure. The maximum force in the upper chord members appears in the members directly connected to the supports. The lower chord members are also in compression but vary without an obvious regularity. The maximum force in the lower chord members is approximately 27% of that in the upper chord members. The forces in the web members vary in a tension-and-compression rule near the open top, with relatively small values. The maximum force in the web members does not exceed 10% of that in the upper chord members. The inner-ring members (H ₁) bear the maximum force in the structure.

The maximum absolute values of vertical, horizontal, and total displacements are observed at nodal joints 4, 5, and 4, respectively. The vertical and total displacements of upper chord nodal joints decrease from the center to the periphery regularly. The horizontal displacement of joint 9, the nodal joint in the vicinity of the support, is outward from the center of the lattice shell, which is opposite to the other joints. The maximum total displacement, 6.42 mm, is approximately 1/8000 of the span, indicating the considerably high stiffness of the structure.

The flexural stresses in upper chord and inner-ring members are small compared with their axial stresses, since they are merely about 15% of the latter. There are no flexural stresses in the lower chord members, that is, the members carry only axial forces and no moments, although they are rigidly connected at both ends.

Stability behavior of the spherical lattice shell

The stability of the spherical lattice shell is evaluated based on the example in section “Static performance of the spherical lattice shell.” The calculations involve the eigenvalue buckling analysis and nonlinear analysis considering geometrical and material nonlinearities accounting for initial imperfections.

The first five types of eigenvalue buckling modes and their corresponding eigenvalues, of which the first eight orders emerge in pairs, are listed in Table 3. The vertical deformation and circumferential deformation are coupled in Mode I and a saddle-shaped deformation is generated as a consequence. Circumferential squeezing between the six-bar tetrahedral modular units is discovered in the other four types of modes and the buckling of chord members is involved from Mode III to Mode V. The minimum eigenvalue, which corresponds to the first buckling mode, is 4.526. Eigenvalues of the other four modes are not far from each other, as the ratios to the first eigenvalue increase from 1.8 to 2.2.

Table 3.

Stability analysis results of the spherical lattice shell.

Mode number	Eigenvalue buckling		Nonlinear stability coefficient
Mode number	Buckling mode	Stability coefficient	Nonlinear stability coefficient
I (Orders 1 and 2)		4.526	2.292
II (Orders 3 and 4)		8.548	3.133
III (Orders 5 and 6)		9.211	3.450
IV (Orders 7 and 8)		9.451	3.166
V (Order 9)		9.517	3.392

An elastic–perfectly plastic constitutive model is employed in the nonlinear stability analysis. The von Mises yield criterion is followed, with the yield strength set to 215 MPa. Geometric and material nonlinearities are both taken into account and eigenmode-affined initial imperfections are introduced. The five eigenvalue buckling modes in Table 3 are introduced as imperfections separately, in order to compare the influences of different imperfection patterns on the stability performance of the lattice shell. The amplitudes of the imperfections are set to be 1/300 of the span. It should be clarified that the member flexure in the buckling modes is excluded when the imperfections are being introduced, that is, only the nodal displacements in the buckling modes are introduced into the structure (Fan et al., 2012; Tian et al., 2012).

The last column in Table 3 provides the nonlinear stability coefficients. The minimum stability coefficient, 2.292, is obtained with the imperfection pattern analogous to Mode I and is about half of the corresponding eigenvalue buckling load factor. Stability coefficients ranging between 1.3 and 1.5 times the minimum one are obtained when introducing the other four patterns of imperfections.

Construction details of nodal joints and modular units

The spherical lattice shell shown in Figure 7(b) has demonstrated satisfying static and stability behaviors in the foregoing discussions, which endows it with bright prospects for practical engineering. As introduced in the end of section “Kinematic analysis of the spherical lattice shell,” the configuration of the structure makes a clear requirement that the chord and inner-ring members should have rigidly connected ends and the web members should be pinned to the nodal joints. Besides, the nodal joint needs to be composed of two separate parts that belong to two adjacent modular units, so as to facilitate the prefabrication of modular units. And it should be convenient to secure the two parts together for ease of construction of the structural system.

Construction details satisfying the above requirement for the nodal joint are shown in Figure 9. Only one part of the nodal joint is given due to symmetry. The shown part is constructed of a flange, a half hollow sphere, and a pair of ear plates. Coupling between flanges of the two separate parts of a nodal joint forms the rigid connection between two adjacent chord members. A half hollow sphere with diameter smaller than that of the flange is welded onto the latter and an end of the chord member. A pair of ear plates which the web members are pinned to are welded on the half hollow sphere, providing a simple but effective way to form the hinged connections.

Figure 9.

Construction details of the nodal joint.

A six-bar tetrahedral modular unit can be assembled by joining the ends of four web members and those of one upper chord member and one lower chord member together, as shown in Figure 10. Modular units should be prefabricated at ground level such as in the factory and installed on site. Similarly shaped modular units can be stacked up (Figure 11) during transportation and construction in order to reduce the space requirement. With the nodal joint construction described above, adjacent modular units can be joined to each other by simply securing the flanges together with fasteners. A 10-m span test model of the spherical lattice shell was built to verify the feasibility of the construction details of the nodal joints and the modular units, as presented in Figure 12.

Figure 10.

Six-bar tetrahedral modular unit.

Figure 11.

Stacking of similarly shaped modular units.

Figure 12.

Ten-meter span test model of the spherical lattice shell.

A-shaped installing frame and prefabricated construction scheme

As stated in section “Structural configuration,” a series of concentric circular rings are formed by several groups of similarly shaped modular units by the configuration of the lattice shell. Based on this characteristic, the principle followed during construction is defined as that the structure is constructed with the modular units and extends upwardly and inwardly ring by ring from its periphery while no scaffolds are employed, which will permit mass production of the modular units as well as easy and economical construction of the structure. To realize this idea, a simple but essential tool referred to hereinafter as A-shaped installing frame is developed, which is shown in Figure 13(a). This tool can provide a temporary support for a newly installed modular unit before the latter is completely secured to the existing part of the structure and will be used repeatedly in the construction procedure.

Figure 13.

A-shaped installing frame: (a) working principle and (b) details of connection between the main frame and the temporary strengthening bar.

The components that form the A-shaped installing frame include a main frame, a front cable, two backstay cables, a position adjuster, a pair of connectors, and several temporary strengthening bars. First, the temporary strengthening bars should be connected at both ends to adjacent innermost nodal joints of the existing structure. The main frame is then connected to the temporary strengthening bars via the horizontal axles on its feet and the connectors, as shown in Figure 13(b). The front cable is connected at one end to the position adjuster situated at the top of the main frame and at the other end to the newly installed modular unit, while the two backstay cables anchor the A-shaped installing frame itself and the newly installed modular unit to the existing structure. The position adjuster is able to go up and down so as to adjust the newly installed modular unit to its designed position.

The A-shaped installing frame can lie on the existing structure in standby state when no modular units are being installed and rotate about the horizontal axles on its feet to stand up when switching to the working state. In the working state, the direction of the A-shaped installing frame can be perpendicular either to the horizontal plane (Figure 14(a)) or to the tangent plane of the spherical surface of the lattice shell (Figure 14(b)). The latter will form an angle Φ between the frame and the horizontal plane, but can permit a consistent front cable length throughout the construction procedure.

Figure 14.

Options for the directions of the A-shaped installing frame in working state: (a) perpendicular to horizontal plane and (b) perpendicular to the tangent plane of the spherical surface of the lattice shell.

The detailed procedure of installing one ring of modular units is outlined in Figure 15. Figure 16 shows the working state of the A-shaped installing frame during the construction of the 10-m span test model.

Figure 15.

Procedure of installing one ring of modular units.

Figure 16.

Construction of the lattice shell using the A-shaped installing frames.

Summary

A novel spherical lattice shell composed of six-bar tetrahedral modular units whose horizontal projections are quadrilaterals is studied systematically. The lattice shell combines the advantages of single-layer and double-layer lattice shells with a rather simple structural configuration and provides an additional advantage in industrialized production and prefabricated construction by comprising a plurality of similarly shaped modular units. Some conclusions can be drawn based on the foregoing discussions:

The basic structure of the lattice shell with hinged nodal joints is kinematically indeterminate. Adding a certain amount of extra bars onto the basic structure and adopting rigid connections between some members are both effective ways to eliminate the mechanisms. The latter is usually more favorable in practical applications because it preserves the features of the structural configuration to a greater extent and is more advantageous in building industrialization.

For a spherical lattice shell under uniform load, internal forces of large values are observed mainly in the upper chord and inner-ring members, while the maximum internal forces in the lower chord and web members do not exceed 25% of that in the former. The axial stress is dominant in the upper chord and inner-ring members while the maximum flexural stress is about 15% of the axial stress in the same member, and there are no flexural stresses in the lower chord members.

The spherical lattice shell demonstrates good stability behavior. The minimum nonlinear buckling load accounting for geometric and material nonlinearities and initial imperfections is about 50% of the corresponding eigenvalue buckling load.

For a 50-m span spherical lattice shell designed under uniform load of 2.0 kN/m², the steel consumption is only 28.3 kg/m² without taking into account the weight of nodal joints indicating the significant economy of the structural system.

The construction details for both the nodal joint and the six-bar tetrahedral modular unit are illustrated. The chord members are secured to each other by flanges, while the web members are pinned to the ear plates on the nodal joints.

A simple but essential tool called A-shaped installing frame is developed for the prefabricated construction of the lattice shell. The detailed procedure for installing one ring of modular units is given.

A 10-m span test model for the spherical lattice shell has been built, and the feasibility of the construction details for the nodal joints and modular units as well as the practicability of prefabricated construction scheme were verified. No welding was needed on site during construction, permitting easy and fast construction and providing benefit for quality assurance.

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: This work was supported by the Natural Science Foundation of China (51478420).

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