Shape and force control of prestressed cable-strut structures based on nonlinear force method

Abstract

Cable-strut structures are flexible structures which are very sensitive to their shape variation. Addition of actuators in selected members of cable-strut structures provides an efficient way to control the structural shape, thus enhances the structural performance. Although a large number of research studies have been performed to achieve shape control via linear force method, it has been proved that nonlinear force method is more accurate due to the existence of nonlinearity characters in cable-strut structures. The aim of this article is proposing an approach based on the nonlinear force method to change the shape of cable-strut structures under external load with respect to different control targets. The formulas and calculation procedure of structure’s nonlinear response to external load and length variation of members are established based on nonlinear force method. Then, solving strategy for determining the minimum required number of actuators and testing the feasibility for arrangement plan of actuators is provided for different control targets. Two numerical cases are demonstrated using the proposed approach to obtain the structural behavior before and after the control process. The effectiveness of the algorithm can be verified through numerical results.

Keywords

force control Moore–Penrose pseudoinverse nonlinear force method prestressed cable-strut structures shape control singular value decomposition

Introduction

Unlike traditional structures, cable-strut structures including cable truss, cable dome, and tensegrity are flexible structures composed of tensioned cables and compressed struts. Structural geometry and prestress have an essential role in providing stiffness for the structures to maintain the structural stability and resist external load. Resulting from lightweight and high efficiency of its material, cable-strut structures are implemented in numerous long-span roof structures. Shape of cable-strut structures has a close relationship with prestress distribution of members and further to the stiffness of the whole structure (Wang, 1998).

Structures having the ability to change their shape by actuators are defined as adaptive structures whose performance is controlled by a control system (Wada and Das, 1991). These structures have the ability to optimize their shape and improve the capability to adapt to changing environments. Applying actuators on active members to alter the length of members has been proved to be an effective way to adjust structural shape and improve the adaptability of the structure to external environment (Shea et al., 2002). A computational control framework was proposed by Adam and Smith (2007, 2008) and a five-module active tensegrity structure experiment was carried out to prove the possibility of structural control by altering the length of members. Korkmaz (2011) gives an overview of feasible applications of active structures and investigates the active control performance for cable members of a tensegrity bridge (Korkmaz et al., 2012).

Nowadays, the displacement-based finite element method (FEM) is commonly used in analysis of various structures (Gasparini and Gautam, 2002; Kahla and Kebiche, 2000). For assemblies with infinitesimal mechanisms, the stiffness matrix when using FEM may not have full rank. Then, the displacement-based method becomes invalid (Xu and Luo, 2009). Although this problem can be dealt with numerically, the force method (FM) gives a more satisfactory solution based on the advantage that the FM does not need to integrate the stiffness matrix. With the clear concept of mechanics and the advantages to determine the modes of inextensional mechanisms and states of the self-stress, the FM is often employed in indeterminate structures (Pellegrino and Calladine, 1986; Pellegrino et al., 1992; Pellegrino and Van Heerden, 1990), such as tension structures and cable-strut structures. The FM is used in analysis of prestressed mechanisms and a saddle-shaped cable net experiment has been carried out to validate this approach (Pellegrino, 1990). To derive the formulae of member forces and nodal displacements for a given structure, singular value decomposition (SVD) method is adopted in order to analyze the equilibrium equations.

The linear FM has been proved the possibility for applying shape control in a plane truss (You, 1997). Altering the length of some members is helpful to fix manufacturing errors of members and correct the location of selected nodes. In order to extend the scope of application of FM, Luo and Lu (2006) introduced geometrically nonlinear analysis procedure into traditional FM to provide a more satisfactory solution. The correctness of the algorithm is evidenced by a pin-connected bar assembly and three-cable assembly numerical examples. Xu and Luo (2009) applied this method in prestressed cable structures to control specified nodal displacements without external load.

The purpose of this article is to apply the nonlinear force method (NFM) in shape and force control of cable-strut structures with external load. The layout of the article is as follows. In section “NFM for prestressed cable-strut structures,” internal force and nodal displacement increment equations of variation length of active members by NFM are obtained for the loading state before and after the control process. In section “Controllability of the structure and controlling amount for active members,” the mathematical models for shape control, force control, and synchronous control of shape and force are established. Then, the minimum number of actuators and controlling amount of specified target is determined using the properties of the Moore–Penrose pseudoinverse. Furthermore, a tension truss structure and a Geiger cable dome are performed in section “Numerical examples” to validate the proposed scheme and the results of structural response are compared with the results from nonlinear FEM. Finally, conclusions and suggestions for future work are provided in section “Conclusion.”

NFM for prestressed cable-strut structures

The classic FM consists of three equations: equilibrium equations, compatibility equations, and constitutive equations. They can be written as

At = f

(1)

Bd = e

(2)

e = e_{0} + Ft

(3)

where $A$ , $B$ , and $F$ are equilibrium matrix, compatibility matrix, and flexibility matrix, respectively. The vectors $f$ , $d$ , $e$ , and $e_{0}$ denote, respectively, nodal loads, nodal displacements, elongations in members, and initial elongations in members. Equilibrium matrix and compatibility matrix have been proved by Calladine (1978) to have the relationship that $A^{T} = B$ . Theoretically, characteristics of structures can be derived from these three equations and the equations are established in the initial configuration.

Due to the nonlinear characteristics of the structure, the equilibrium matrix $A$ is a function of $d$ . Then, equation (1) can be written in incremental form as follows in step k

A^{k} (d) δ t^{k} = δ f^{k}

(4)

Similarly, compatibility equation and constitutive equation can be written as

B^{k} (d) δ d^{k} = δ e^{k}

(5)

δ e^{k} = F^{k} (d) δ t^{k}

(6)

To solve the nonlinear problem, an iterative approach similar to Newton–Raphson method which is frequently used in the nonlinear FEM is introduced. Equilibrium matrix $A$ is updated in every step.

Apply SVD of $A$

A = [\begin{matrix} U_{r} & U_{q} \end{matrix}] [\begin{matrix} S_{r} & 0 \\ 0 & 0 \end{matrix}] {[\begin{matrix} V_{r} & V_{s} \end{matrix}]}^{T}

(7)

The solution of equilibrium equation (4) can be written as the combination of the particular solution and the general solution

δ t = (V_{r} S_{r}^{- 1} U_{r}^{T}) δ f + V_{s} α

(8)

in which $α$ is a coefficient vector of the general solution and can be determined according to the principle of virtual work.

Stage I: structure analysis before control process

The relationship between final equilibrium matrix $A$ after iteration and internal force $t$ should satisfy equation (1) under the action of $f$ . In the iteration process, the coefficient vector $α$ becomes

α = - {(V_{s}^{T} F V_{s})}^{- 1} (V_{s}^{T} F V_{r} S_{r}^{- 1} U_{r}^{T} δ f)

(9)

where $δ f$ is the nodal redundant load in every step. Then, $δ t$ can be received using equation (8). For displacement increment $δ d$ , in general

δ d = B^{+} δ e + [I - B^{+} B] β

(10)

where $β$ is a vector that deals with inextensional displacements and it exists for kinematically indeterminate assemblies. While, for kinematically determinate structures and kinematically indeterminate structures that have proper prestress to acquire stiffness to restrict inextensional displacements, $β$ equals to zero. In the following, structures belong to these two cases are taken into consideration. Thus, equation (10) can be written in a simplified form

δ d = B^{+} δ e

(11)

The iteration process analogous to Newton–Raphson method is utilized to solve the nonlinear equilibrium equation, which is shown in Figure 1. Finally, the response of the structure will converge at the specified criterion.

Figure 1.

Iteration process of NFM in stage I.

For the calculation procedure of nonlinear structural analysis in stage I, Luo and Lu (2006) can be referred to for details.

Stage II: structure analysis after control process

The goal of calculation in stage II is to achieve structural response under external load $f$ and length variation in active members $e_{s}$ . Equation (8) illustrates that internal force variation in members is the linear combination of states of self-stress under constant nodal force in stage II, which means $δ t_{s} = V_{s} α$ . The coefficient vector $α$ of stage II can be rewritten as

α = - {(V_{s}^{T} F V_{s})}^{- 1} V_{s}^{T} δ e_{s}

(12)

And then, the internal force increment is given by

δ t_{s} = - V_{s} {(V_{s}^{T} F V_{s})}^{- 1} V_{s}^{T} δ e_{s}

(13)

The total elongation in members contains active length variation $δ e_{s}$ and length variation caused by the deformation of members. Final elongation of members can be expressed as

δ e_{s}^{'} = δ e_{s} + F δ t_{s} = [I - F V_{s} {(V_{s}^{T} F V_{s})}^{- 1} V_{s}^{T}] δ e_{s}

(14)

Substitute equation (14) into equation (11) gives

δ d_{s} = B^{+} [I - F V_{s} {(V_{s}^{T} F V_{s})}^{- 1} V_{s}^{T}] δ e_{s}

(15)

For the sake of preventing cable relaxation and fulfilling stress constraints for members, $δ e_{s}$ can be decomposed into a number of small amount length variations and check internal force for all members at every end of step. After each control step, structural configuration needs to be updated to eliminate nodal redundant load. The nonlinear control procedure is summarized as follows:

The initial geometrical configuration of stage II is the equilibrium configuration after applying external load. Initialize $t^{0} = t_{f}$ , $d^{0} = d_{f}$ , $e^{0} = e_{f}$ , and $k = 0$ ( $t_{f}$ , $d_{f}$ , and $e_{f}$ are internal force, nodal displacement, and member elongation under external load $f$ ). Number of sub steps is n and each controlling amount in step k is $δ e_{s}^{k} = δ e_{s} / n$ .

Calculate $A^{k}$ and carry out SVD of $A^{k}$ ; obtain flexibility matrix $F^{k}$ .

Get force increment $δ t_{s}^{k}$ from equation (13).

Get elongation increment $δ e_{s}^{' k}$ from equation (14).

Get displacement increment $δ d_{s}^{k}$ from equation (15).

Calculate the internal force, member elongation, and nodal displacement in the next step

t^{k + 1} = t^{k} + δ t_{s}^{k}, e^{k + 1} = e^{k} + δ e_{s}^{' k}, d^{k + 1} = d^{k} + δ d_{s}^{k}

Obtain nodal redundant load $δ f^{k} = f - A^{k} t^{k}$ and use Newton–Raphson method to find the equilibrium configuration.

Set $k = k + 1$ after finding the equilibrium configuration of the structure.

Loop step (2) to (8) until $k = n$ .

Controllability of the structure and controlling amount for active members

The stiffness of structure increases through changing the structure’s shape and internal force to resist external force and finally achieve an equilibrium state. However, considering the fact that using a large number of actuators is neither cost effective nor maintenance friendly, the number, position, and length variation of actuators need to be determined.

Rewriting equations (15) and (13) in the following expression. These two equations give the relationship for shape and force control

δ d_{s} = S_{d} δ e_{s}

(16)

δ t_{s} = S_{t} δ e_{s}

(17)

where $S_{d} = B^{+} [I - F V_{s} (V_{s}^{T} F V_{s})^{- 1} V_{s}^{T}]$ and $S_{t} = - V_{s} (V_{s}^{T} F V_{s})^{- 1} V_{s}^{T}$ . Let the target and current displacement vector and internal force vector are $d^{*}$ , $t^{*}$ and $d$ , $t$ . The amount need to be controlled can be written as

δ d_{s} = d^{*} - d

(18)

δ t_{s} = t^{*} - t

(19)

In most cases, it is not necessary to control every free node but to control some selected nodes in the structure. For cable-strut structure with membrane materials, displacements of the upper nodes have the priority to be concerned in order to keep the slope of the roof in order to prevent folds or tear of membranes.

Shape control

The shape of structural configuration is an important section for cable-strut structures in the structural design. The mathematical model of structural shape control is given as

\begin{matrix} Find δ e_{s} = {δ e_{s}^{1}, δ e_{s}^{2}, δ e_{s}^{3}, \dots, δ e_{s}^{n_{a}}} \\ s . t . d = d^{*}, At = f \end{matrix}

(20)

To get the relationship between member elongation quantity of actuators and displacement control quantity, divide $δ e_{s}$ into two parts: elongation quantity of active members $δ e_{s}^{c} (n_{a} \times 1)$ and members without actuators $δ e_{s}^{u} ((n - n_{a}) \times 1) = 0$ . Similarly, $d_{s}^{c} (m_{c} \times 1)$ includes $m_{c}$ degrees of freedom need to be controlled in the displacement vector, and $d_{s}^{u} ((m - m_{c}) \times 1)$ includes $m - m_{c}$ degrees of freedom which do not need to be controlled. Then, equation (16) can be rearranged into

{\begin{matrix} δ d_{s}^{c} \\ δ d_{s}^{u} \end{matrix}} = [\begin{matrix} S_{d}^{(m_{c} \times n_{a})} & S_{d}^{(m_{c} \times (n - n_{a}))} \\ S_{d}^{((m - m_{c}) \times n_{a})} & S_{d}^{((m - m_{c}) \times (n - n_{a}))} \end{matrix}] {\begin{matrix} δ e_{s}^{c} \\ 0 \end{matrix}}

(21)

Furthermore, incremental relationship between controlling amount for actuators and the displacement to be controlled have a simpler expression

δ d_{s}^{c} = S_{d}^{(m_{c} \times n_{a})} δ e_{s}^{c}

(22)

After obtaining equation (22), the next step is to decide whether a possible controlling scheme $δ e_{s}^{c}$ can satisfy the target $δ d_{s}^{c}$ . The verification process depends on the analysis of vector $S_{d}^{(m_{c} \times n_{a})}$ . However, generally $S_{d}^{(m_{c} \times n_{a})}$ is a rectangular matrix that Cramer’s rule becomes invalid.

The rank of the matrix can be used to identify the existence of solutions for equation (22). Assume that the rank of $S_{d}^{(m_{c} \times n_{a})}$ is $r_{d}$ and the rank of augmented matrix $[S_{d}^{(m_{c} \times n_{a})} | δ d_{s}^{c}]$ is $r_{d}^{'}$ , the only way for equation (22) to have solutions is to ensure that $r_{d} = r_{d}^{'} \leq n_{a}$ . The rank of the coefficient matrix cannot be found before the structure configuration and location of actuators control scheme are determined. However, for most cases, $S_{d}^{(m_{c} \times n_{a})}$ is a full rank matrix or close to a full rank matrix. In order to achieve a feasible solution, the principle of actuator placement should make that $n_{a} \geq m_{c}$ , which means the minimum required number of actuators is not less than the number of degrees of freedom to be controlled.

Another effective way to find the existence of solutions is using the property of the Moore–Penrose pseudoinverse. Write $S_{d}^{(m_{c} \times n_{a})}$ as $S_{d}$ , the sufficient and necessary conditions for the existence of solutions of equation (22) are

S_{d} S_{d}^{+} δ d_{s}^{c} = δ d_{s}^{c}

(23)

If equation (23) is satisfied, then the general solution of equation (22) can be expressed as

δ e_{s}^{c} = S_{d}^{+} δ d_{s}^{c} + (I - S_{d}^{+} S_{d}) y

(24)

All the possible solutions to equation (24) can satisfy equation (22) and control objectives can be set to find an optimum solution. For simplicity, the minimum norm solution is selected corresponding to $y = 0$

δ e_{s}^{c} = S_{d}^{+} δ d_{s}^{c}

(25)

After the control of the incremental form is achieved, the ultimate control vector can be obtained.

The process in shape control can be briefly described as follows:

Initialize $t^{0} = t_{f}$ , $d^{0} = d_{f}$ , $e^{0} = e_{f}$ , $k = 0$ , and controlling amount $e_{s}^{c (0)} = 0$ . Calculate control target $δ d_{s}^{c (0)}$ .

Calculate $A^{k}$ and carry out SVD of $A^{k}$ ; obtain $S_{d}^{k}$ .

Continue calculation to check whether equation (23) holds. If equation (23) is satisfied, proceed to next step. Otherwise, exit the loop. If equation (23) does not hold, which means the existing layout scheme of actuators is unable to meet the control requirements, a new scheme needs to be selected.

Determine $δ e_{s}^{c (k)}$ from equation (25).

Follow the procedures in section “Stage II: structure analysis after control process” from steps (2) to (7) to find the equilibrium configuration. Calculate internal force, member elongation, nodal displacement in next step, and controlling amount for the equilibrium configuration

t^{k + 1} = t^{k} + δ t_{s}^{k}, e^{k + 1} = e^{k} + δ e_{s}^{' k}, d^{k + 1} = d^{k} + δ d_{s}^{k}, e_{s}^{c (k + 1)} = e_{s}^{c (k)} + δ e_{s}^{c (k)}

Finish iteration if $δ d_{s}^{c (k + 1)}$ satisfies the convergence condition. Otherwise, set $k = k + 1$ and loop steps (2) to (5).

Force control

If the internal force of some members is chosen to be the control target, the mathematical model of structural force control is given as

\begin{matrix} Find δ e_{s} = {δ e_{s}^{1}, δ e_{s}^{2}, δ e_{s}^{3}, \dots, δ e_{s}^{n_{a}}} \\ s . t . t = t^{*}, At = f \end{matrix}

(26)

Similar to the procedure to obtain equation (21), rearrange equation (17) into

{\begin{matrix} δ t_{s}^{c} \\ δ t_{s}^{u} \end{matrix}} = [\begin{matrix} S_{t}^{(n_{c} \times n_{a})} & S_{t}^{(n_{c} \times (n - n_{a}))} \\ S_{t}^{((n - c_{c}) \times n_{a})} & S_{t}^{((n - n_{c}) \times (n - n_{a}))} \end{matrix}] {\begin{matrix} δ e_{s}^{c} \\ 0 \end{matrix}}

(27)

where $t_{s}^{c} (n_{c} \times 1)$ includes $n_{c}$ degrees of freedom need to be controlled in the internal force vector, and $t_{s}^{u} ((n - n_{c}) \times 1)$ includes $n - n_{c}$ degrees of freedom which do not need to be controlled.

Furthermore, the simpler expression is

δ t_{s}^{c} = S_{t}^{(n_{c} \times n_{a})} δ e_{s}^{c}

(28)

The way to check the existence of solutions and find the solutions for equation (28) is similar to obtaining equations (23) to (25) in section “Shape control.” The procedure which is applied in the force control program can refer to that in shape control. Replace $δ d_{s}^{c}$ by $δ t_{s}^{c}$ and $S_{d}$ by $S_{t}$ in the control process in section “Shape control” to accomplish force control.

Synchronous control

The adjustment procedure to control specified nodal displacements may be accompanied by the decreasing member force in some structural members. On the other hand, improving internal force in bars of force control is also probable to cause large displacements in a number of nodes. Therefore, it is necessary to consider whether the nodal displacement and the internal force of the members can meet the specified requirements at the same time. The mathematical model is illustrated as

\begin{matrix} Find δ e_{s} = {δ e_{s}^{1}, δ e_{s}^{2}, δ e_{s}^{3}, \dots, δ e_{s}^{n_{a}}} \\ s . t . d = d^{*}, t = t^{*}, At = f \end{matrix}

(29)

So, both equations (22) and (28) should be fulfilled in synchronous control. The combined equation is

{\begin{matrix} δ d_{s}^{c} \\ δ t_{s}^{c} \end{matrix}} = [\begin{matrix} S_{d}^{(m_{c} \times n_{a})} \\ S_{t}^{(n_{c} \times n_{a})} \end{matrix}] δ e_{s}^{c}

(30)

The reduced equation of equation (30) has the form that

δ x_{s}^{c} = S^{((m_{c} + n_{c}) \times n_{a})} δ e_{s}^{c}

(31)

Write $S^{((m_{c} + n_{c}) \times n_{a})}$ as $S$ , and the equation to check the existence of solution is as equation (32)

S S^{+} δ x_{s}^{c} = δ x_{s}^{c}

(32)

The required controlling amount can be expressed as

δ e_{s}^{c} = S^{+} δ x_{s}^{c}

(33)

Calculation flow of synchronous control is similar to that in section “Shape control.” Replace $δ d_{s}^{c}$ by $δ x_{s}^{c}$ and $S_{d}$ by $S$ in the calculation procedure to fulfill synchronous control.

Numerical examples

Example 1

As showed in Figure 2, a space cable-strut structure consists of eight cables and one strut. The diameter, Young’s modulus, and ultimate stress of cable are 8 mm, 185 GPa, and 1850 MPa, respectively. Circular hollow section $ϕ 50 \times 4$ with a Young’s modulus of 206 GPa is used for the strut. A total of three states of self-stress and zero inextensional mechanisms can be found using the SVD method. Initial prestress vector is identified based on the equilibrium of each free node. Prestress vector of upper cable, lower cable, and strut is [29.01, 29.01, −28.14] kN. The value of downward external load on the upper node is 20 kN.

Figure 2.

Space cable-strut structure: (a) front view and (b) axonometric view.

Shape control

The displacement in z direction of the upper node under external load is −9.02 mm. The control target is to ensure that the upper node back to the initial position (0, 0, 500) mm. The number of degrees of freedom needs to be controlled is 3, including x, y, and z directions.

Selection of the number of actuators should comply with the following four principles:

The $n_{a}$ (number of actuators) is not less the degrees of freedom need to be controlled, which means $n_{a} \geq 3$ .

For cable-strut structure, actuators have a simple form and are easy to manufacture when placed on cables. So, the actuators are preferred to be located on cables.

During the control process, asymmetric deformation of the structure will lead to a negative effect on the performance of the structure. So, placing the actuators on the same type of cables helps to keep the structural symmetry and the strut to be straight. Then, the possible plans are the structure with four actuators or eight actuators.

Less number of actuators is preferred as more actuators lead to more economic costs and energy consumptions. If four actuators are sufficient for achieving the target, then the plan to use four actuators is better than the plan to use eight actuators.

Finally, four actuators are selected to arrange in the upper cables. Then, 4.56 mm from nonlinear calculation can be received to revise the upper node to the initial position. The internal force of members during stage I and stage II can be expressed in Table 1. In order to verify the proposed approach, substitute the controlling amount into ANSYS for nonlinear analysis. The results of structural response using NFM are compared with the results from nonlinear FEM based on ANSYS. The error of the internal force of members between these two methods is less than 0.2 kN.

Table 1.

Internal force for shape control.

Location of members	Initial state (kN)	Stage I (kN)		Stage II (kN)
Location of members	Initial state (kN)	NFM	FEM	NFM	FEM
Upper cable	29.01	19.136	19.228	8.234	8.408
Lower cable	29.01	38.775	38.864	28.850	29.023
Strut	−28.14	−38.249	−38.337	−27.988	−28.157

NFM: nonlinear force method; FEM: finite element method.

The requirement of shape control can be achieved by arranging the actuators on the upper four cables; however, decreasing of internal force happens in all members. Cable relaxation may occur without control of internal force.

Force control

The elongation or shortening of certain member bars will cause the stress redistribution of the whole structure. That is, the elongation of members will decrease the internal force of all members, while the shortening of members will tend to increase their inner force. It is observed that the upper cables may slack during the shape control procedure. So, the control target in force control is to maintain the internal force of upper cables to 29.01 kN. To compare the effectiveness of location of actuators, both cases of arrangement actuators, that is, on upper cables and lower cables, are considered.

Actuators on upper cables

Controlling amount of nonlinear calculation is −4.03 mm. Negative values indicate that the actuator is shortened. The internal force and node position are presented in Tables 2 and 3.

Table 2.

Internal force for force control when the actuators are on upper cables.

Location of members	Initial state (kN)	Stage I (kN)	Stage II (kN)
Upper cable	29.01	19.14	29.01
Lower cable	29.01	38.78	47.24
Strut	−28.14	−38.25	−47.26

Table 3.

Node position for force control when the actuators are on upper cables.

Upper node location	Initial state (mm)	Stage I (mm)	Stage II (mm)
x	0	0	0
y	0	0	0
z	500	490.98	483.27

From Tables 2 and 3, control target can be achieved through the nonlinear algorithm. It should be noticed that to keep the force level in upper cables, node position in stage II continues to decrease if the actuators are placed on upper cables.

Actuators on lower cables

Shortening actuator on upper cables gives the structure a tendency to continue the downward displacement. Instead, shortening actuator on lower cables increases the prestress level and gives the structure a tendency to rise which is verified in the following.

Four actuators are placed on lower cables and −4.52 mm is applied for each actuator. Tables 4 and 5 illustrate a better performance of the structure when compared to plan A.

Table 4.

Internal force for force control when the actuators are on lower cables.

Location of members	Initial state (kN)	Stage I (kN)	Control state (kN)
Upper cable	29.01	19.14	29.01
Lower cable	29.01	38.78	49.66
Strut	−28.14	−38.25	−48.15

Table 5.

Node position for force control when the actuators are on lower cables.

Upper node location	Initial state (mm)	Stage I (mm)	Stage II (mm)
x	0	0	0
y	0	0	0
z	500	490.98	500.16

After control procedure of plan B, the results show that internal forces of upper cables are the same with those in initial state. Moreover, the upper node location increases and becomes close to initial position.

Synchronous control

If both control targets in shape control and force control are taken into consideration at the same time, the total degrees of freedom increase to 7. Eight actuators are placed on all cables considering the symmetry of the structure. Controlling amounts −0.04 and −4.36 mm are obtained for actuators on upper cables and lower cables, respectively. Internal force and upper node location for synchronous control are summarized in Tables 6 and 7, respectively.

Table 6.

Internal force for synchronous control.

Location of members	Initial state (kN)	Stage I (kN)	Stage II (kN)
Upper cable	29.01	19.14	29.01
Lower cable	29.01	38.78	48.32
Strut	−28.14	−38.25	−47.67

Table 7.

Node position for synchronous control.

Upper node location	Initial state (mm)	Stage I (mm)	Stage II (mm)
x	0	0	0
y	0	0	0
z	500	490.98	500

From the above calculation, shape control and force control target including 7 degrees of freedom can be attained by placing eight actuators on the cables.

Example 2

Cable dome was first proposed by Geiger et al. (1986) who were inspired by Fuller’s idea of tensegrity. Benefit from its lightweight character, it is widely used in large span structures, for example, Georgia Dome in Atlantic, Redbird Arena in Bloomington-Normal, Amagi Dome in Japan, and Taoyuan Arena in Taiwan.

A Geiger cable dome is shown in Figure 3 which is implemented to perform synchronous control analysis in the following. Initial prestress in cables and struts is obtained from the approach of integral feasible prestress (Yuan and Dong, 2002) through the equilibrium matrix theory. The cable achieves a Young’s modulus of 185 GPa and an ultimate stress of 1850 MPa. For the strut, Young’s modulus is 206 GPa and the allowable stress is 310 MPa. The cross-section areas of cables and struts are 100 and 500 mm², respectively.

Figure 3.

Space cable-strut structure: (a) section plan and (b) axonometric view.

Initial prestress is shown in Table 8.

Table 8.

Initial prestress of Geiger cable dome.

Member type	1	2	3	4	5	6	7
Value of stress (N)	12,500	26,282.5	12,500	26,282.5	24,807	−9302.5	−8682.5

The primary reason for the failure of cable dome is due to large deflections of the structure or cable relaxation of inner upper cables. Hence, the upper central node displacement and inner upper cable force are selected as control target. Total number of degrees of freedom needed to control is 9. Considering the principle that the number of actuators should be not less than the degrees of freedom, twelve actuators are used in the control procedure. Since actuators on lower cables tend to get a better performance for the structure, all actuators are arranged on the lower radial cables. Controlling amounts of 0.23 mm for member type 3 and −6.64 mm for member type 4 are gotten from nonlinear calculation. Analysis results of the characteristic response of cable dome are given in Tables 9 and 10.

Table 9.

Internal force for synchronous control in cable dome.

Member type	Initial state (N)	Stage I (N)	Stage II (N)
1	12,500	4585	12,500
2	26,282.5	16,506	32,909
3	12,500	11,119	18,759
4	26,282.5	33,831	50,923
5	24,807	31,916	48,112
6	−9302.5	−8373	−14,163
7	−8682.5	−11,250	−16,959

Table 10.

Node position for synchronous control in cable dome.

Central upper node location	Initial state (mm)	Stage I (mm)	Stage II (mm)
x	0	0	0
y	0	0	0
z	950	942.07	950

Synchronous control can be attained through arranging actuators on lower radial cables. The utility of actuator contraction is to improve the stiffness of the whole structure and give the structure an upward moving trend. Shape and internal force goal can be reached at the same time via the synchronous control approach.

Conclusion

A new control approach based on NFM for the analysis of cable-strut structures is proposed in this article. The benefit of the approach is that controlling amount can be attained through the calculation process. In addition, whether the proposed layout of actuator scheme is available for different control targets can be identified during the calculation.

The practicability of this method applied in the loading and control analysis of cable-strut structures has been verified by demonstrating two numerical examples. Numerical results indicate that the proposed calculation procedure is able to reach the intended goal for shape, force, and synchronous control.

The control technique introduced in this article can be applied to increase the performance of cable-strut structures with manufacturing errors, load modifications, and damage occurrence. Further research to find the best location of actuators still needs to be taken into consideration for the better applicability of this technique.

Footnotes

Appendix 1 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 Zhejiang Provincial Natural Science Foundation of China (LZ14E080001), the National Key Technology R&D Program (2012BAJ07B03) and the National Natural Science Foundation of China (Grant Nos 51278461 and 51578492).

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