Research Article | Open Access

# Modelling and Structural Design for Parallel Umbrella-Shaped Cable-Strut Structures Based on Stationary Potential Energy Principles

**Academic Editor:**Jose A. Lozano-Galant

#### Abstract

A method for the modelling and structural design of a parallel umbrella-shaped cable-strut structure (PUSC) is presented. First, simplified calculation models of a PUSC are built. Next, based on the principle of stationary potential energy, the relationships among the cable sectional areas, prestress forces, vector height, sag height, overall displacement, and local deformation are proposed. Then, the static responses of the PUSC under vertical loads and wind loads are put forward. Finally, a calculation model of a 100 m-span PUSC is developed and optimized to verify the feasibility of the proposed method. The results show that when the combinations of the loading, variation ranges of the vector height and sag height, and material properties of the components are given, the sectional areas of the cables, dimensions of the inner strut, and prestress forces of these components can be obtained. A greater external load requires a corresponding increase in vector height and sag height to increase the overall stiffness, leading to larger sectional dimensions of the components and a greater prestress of the entire structure. Therefore, the total weight of the cables and inner struts are determined. Moreover, because the weight of the cables decreases and the weight of the inner struts increases as the vector height and sag height increase, the total weight of the cables and struts decreases sharply during the initial stage, decreases gradually during the second stage, and increases slowly during the last stage after reaching the minimum value. For the optimal design of the calculation model, using the vector height and sag height as design variables provides an adequate geometric stiffness and a suitable prestress for the PUSC to fulfill the requirements of all the loading combinations.

#### 1. Introduction

In the last two decades, cable-strut structures, such as cable-dome structures and cable-truss structures, have been widely used in various large-span projects [1] and aviation antennas [2]. A cable-strut structure is a type of prestressed structure that includes a large number of flexible cables and struts. The major advantages of cable-strut structures are as follows: (1) large structural rigidity with the introduction of prestress forces in cables and struts, (2) low weight of the entire structure with high-strength metal materials used as the cables, (3) slender struts with compressive forces added in the axial direction, and (4) simple node configurations with adjacent components connected by hinged joints. As society develops, more complex cable-strut structures will be used to cover wider spans with relatively lower project costs. For these applications, the parallel umbrella-shaped cable-strut structure (PUSC) is a well-suited approach. A PUSC is a double-layer cable-strut roof system, and each substructure of a PUSC is a triangular cable net with one central compressive strut. Due to their long spans, high clearances, low construction costs, and elegant shapes, PUSCs have been widely used in various large-span projects, such as the AstroMesh antennas [3] and the roof of the Sony Centre in Berlin’s Potsdamer Square [4].

Unlike traditional steel structures [5], the axial pretension forces in the cables and axial precompression forces in the struts provide PUSCs the rigidity and ability to resist external loads [6]. Therefore, the structural design of PUSCs is related to both the structural shape and the initial prestress distribution and requires in-depth study.

Currently, most research on structural design is focused on rigid structures, and many theories and methods have been proposed based on energy principles, e.g., a combination of the total complementary energy minimization theory and a modified sequential quadratic programming (SQP) algorithm discussed by Ohkubo* et al*. [7], a metaheuristic algorithm developed by Toklu* et al*. [8], and a particle swarm optimization (PSO) algorithm presented by Temür* et al*. [9]. However, research on the structural design of large-span steel structures is lacking, with most of the current studies predominantly limited to trusses [10, 11], as discussed by Kawamura et al. [12].

Numerous studies have been conducted on the form-finding and force-finding of cable-strut structures, such as the force-density method presented by Schek [13], the dynamic relaxation method proposed by Motro* et al*. [14], the nonlinear force method (NFM) presented by Luo [15], and the nonlinear dynamic finite element method (NDFEM) proposed by Luo et al. [1]. More recently, Zhou et al. [16] proposed the modified double singular value decomposition method to acquire feasible prestress states of cable-strut structures without changing its predefined shape.

Most of these works involve form-finding and force-finding and are barely related to the structural design of PUSCs. These studies are mainly based on given prestress distributions or given configuration modes of entire structures with predetermined support node locations. If all of these parameters are not given, the structural design will be more intricate, yet such a situation is likely, especially in the initial design stage. However, few studies have focused on design problems under these conditions, and these studies have predominantly focused on antennas [17].

Considering these aspects, this paper addresses a structural design method that can be applied to PUSCs without determining the initial prestress distributions and even without knowing the sectional areas of the cables. The simplified models and basic assumptions are presented first, followed by the calculation models in different states. Subsequently, the relations among the forces of the cables and struts, the sectional areas of the cables, the specifications of the struts, the vector height, and the sag height are derived. Using the minimum total weight of all the cables and struts as the design objective and the values of vector height and sag height as the design variables, the structural design, including the initial structural shape, the initial prestress, and the parameters of the structural components, of a PUSC can be obtained. Finally, the roof of the Shijiazhuang International Exhibition Centre is used as an example to verify the applicability of the structural design method. This method is capable of producing a PUSC with a minimal total weight and reasonable mechanical performance.

#### 2. Description of the PUSC

##### 2.1. Simplified Models and Basic Assumptions

A PUSC is a cable-strut structure that comprises* n *substructures (where* n* is a positive integer), and each substructure consists of two ridge cables, two diagonal cables, and one inner strut. The joints between adjacent components are hinged. The simplified calculation model of a PUSC is shown in Figure 1(a), and the simplified model of a substructure is shown in Figure 1(b).

**(a)**

**(b)**

##### 2.2. Calculation Models in Different States

According to the loads added on the structure, a PUSC has three predominant states: the initial geometric state, the initial prestress state, and the loading equilibrium state.

(1) The initial geometric state is an equilibrium state without building prestress or adding external loads. This state corresponds to the natural form of a PUSC under its self-weight and is primarily controlled by its initial geometric shape.

(2) The initial prestress state is an equilibrium state that considers the effect of prestress only. The initial prestress state of a PUSC is the basis for deriving the structural forces and deformations under complex external loads and is primarily controlled by the initial geometric shape and the distribution mode of the prestress. The simplified calculation model for the initial prestress state of a PUSC is shown in Figure 2.

(3) The loading equilibrium state is an equilibrium state that considers the effect of both the prestress and external loads. This state is primarily controlled by the initial geometric shape, the prestress, and the external loads. The external structural loads include vertical loads (e.g., dead load, live load, and snow load) and wind loads (e.g., suction wind load and pressure wind load), where the vertical loads are directed towards the ground, the suction wind load is directed upward and perpendicular to the ridge cables, and the pressure wind load is directed downward and perpendicular to the ridge cables. In this case, to determine the structural response of a PUSC in the loading equilibrium state, the mechanical theories of PUSCs under vertical loads and wind loads must be determined first.

Because the substructures of a PUSC are mutually parallel, a load (i.e., a vertical load or a wind load) added to the ridge cables of the PUSC is uniformly distributed. Details of the deformations of a PUSC under different loads are shown in Figure 3.

**(a)**

**(b)**

**(c)**

#### 3. Structural Performance of PUSCs Based on the Stationary Potential Energy Principle

##### 3.1. Stationary Potential Energy Principle

Different displacements may occur under different loading conditions for a cable-strut structure. The total potential energy of the structure can be expressed as

For the linear elastic cable elements, only axial deformation is considered. In this case,* U* is expressed as

When adding an external load, the potential energy of the entire structure is

Considering (2), (3), (4), and (5), the expression of the total potential energy of the structure is

Applying the principle of stationary potential energy [18], (7) can be obtained to describe the static equilibrium equation of the structure.

##### 3.2. Structural Performance of PUSCs in the Initial Prestress State

According to Figure 2, the configuration equations for the ridge cables and the diagonal cables in the initial prestress state are defined as follows:

Assuming that* H*_{r1} is known,* T*_{r1} and* T*_{d1} are

Hence,* Π* isand* H*_{d1} is

##### 3.3. Structural Performance of PUSCs in the Vertical Loading Condition

Under a vertical load, the ridge cables of parallel PUSCs are directed vertically downwards. When adding the uniform vertical line load, , the deformations of the ridge cables are caused by and , and the deformations of the diagonal cables are caused by only. In this case,* T*_{r2,v} is

Because a cable follows a parabolic shape under a uniform line load [19], can be expressed as

The deformation of the ridge cables is

The deformations caused by and can be expressed as follows:where the elongation of the ridge cables caused by isThe elongation of the ridge cables caused by isand the total elongation of a ridge cable is

The deformation of the diagonal cables can be expressed as

The elongation of the diagonal cables is caused by only and can be expressed as

Then, the potential energy of the total structure can be expressed as

Combining (6), (7), (21), and (24), the total energy of the entire structure is

According to (7), the partial derivative of (25) is

Assuming that = 0, = 0, and = 0, the left side of the equal sign in (26) can be expressed as

After performing these computations, (27) is equal to 0. In this case, (26) is correct, i.e., (26) meets the principle of stationary potential energy.

If is small relative to* L* and assuming that /*L*≈0, (26) is

By substituting (15) into (28), is

Assuming that all of the cables reach their design strength after loading and that the values of and are both given, the following equations can be obtained:

This results in the following equations:

In this case, the sectional areas of the ridge cables and diagonal cables in the vertical loading condition, and , can be denoted by using (15), (29), and (33).where

##### 3.4. Structural Performance of PUSCs in the Suction Wind Loading Condition

In contrast to the vertical loads, the ridge cables move upwards perpendicular to their lengths under suction wind loads. The maximum local deformation of the ridge cables is

The elongation of the ridge cables is

Hence, the potential energy of the total structure is

Combining (6), (7), (38), and (39), is

Assuming that all of the cables reach their design strength after loading and that the values of* f*_{sw} and are both given, the following equations can be obtained:

In this case, the sectional areas of the ridge cables and diagonal cables in the suction wind loading condition can be denoted by using (37), (40), and (43).where

##### 3.5. Structural Performance of PUSCs in the Pressure Wind Loading Condition

Under a pressure wind load, the ridge cables move downwards perpendicular to their lengths with a maximum local deformation as follows:

The elongation of the ridge cables is

Afterwards, the vertical displacement of the inner strut can be expressed using the same derivation process as that used for the suction wind loading condition.

Assuming that all of the cables reach their design strength after loading, the following equations are obtained:

In this case, the sectional areas of ridge cables and diagonal cables in the pressure wind loading condition can be denoted by using (47), (49), and (52).where

##### 3.6. Structural Performance of PUSCs in the Condition Considering Both a Suction Wind Load and Vertical Loads

Considering that the effect of vertical loads may be either greater or lesser than that of a suction wind load, two cases are listed as follows.

*(1) Case #1: The Effect of the Vertical Loads Is Greater Than That of the Suction Wind Load*. The vertical displacement of the inner strut,* w*_{vs}, and the local deformations of the ridge cables,* f*_{vs}, can be denoted as follows:

Assuming that all of the cables reach their design strength after loading, the following equations can be obtained:

In this case, the sectional areas of the ridge cables and diagonal cables in the condition considering both a suction wind load and vertical loads, and , can be denoted by using (56), (57), and (60).where

*(2) Case #2: The effect of the Suction Wind Load Is Greater Than That of the Vertical Loads*. The vertical displacement of the inner strut, , and the local deformations of the ridge cables,* f*_{vs}, can be denoted as follows:

Assuming that all of the cables reach their design strength after loading, the following equations can be obtained:

In this case, the sectional areas of the ridge cables and diagonal cables in the condition considering both a suction wind load and vertical loads, and , can be denoted by using (64), (65), and (68).where

##### 3.7. Structural Performance of PUSCs in the Condition Considering Both a Pressure Wind Load and Vertical Loads

Adding the effect of both the pressure wind load and the vertical loads, the vertical displacement of the inner strut, , and the local deformations of the ridge cables,* f*_{vp}, can be denoted as follows:

Assuming that all of the cables reach their design strength after loading, the following equations can be obtained:

In this case, the sectional areas of the ridge cables and diagonal cables in the condition considering both a pressure wind load and vertical loads, and , can be denoted by using (72), (73), and (76).where

##### 3.8. Design of Other Structural Parameters

Generally, the forms of the external load added on a PUSC can be divided into three types, as follows: Type (a): only vertical loads, as discussed in Section 3.3; Type (b): vertical loads with a suction wind load, as discussed in Section 3.6; Type (c): vertical loads with a pressure wind load, as discussed in Section 3.7.

After the form of the external load is determined, the values of* H*_{r1}, , and are defined as follows:

The forces of the cables and inner struts in the initial prestress state can be calculated as

The compression force of the inner struts in the loading condition is

The relevant provision in [20] indicates that, for the inner struts that bear axial compressive forces, the buckling load should follow (84). where* φ *decreases with an increase in the slenderness ratio of the inner struts.

After obtaining the sectional areas of all the components, the weight of all the cables, the weight of the inner struts, and the total weight of a PUSC can be calculated as follows: