Redundancy theory and evaluation index of cable

Abstract

Redundancy theory of cable–strut system is developed for evaluation of structural design. This redundancy theory extends system-level redundancy to member-level redundancy and takes pre-force into account by introducing the second-order deformation of cable and strut. The linear and nonlinear redundancy matrices are derived through potential energy equations in the form of linear and nonlinear deformations, respectively. The characteristics of structural geometry, topology, and material property are included in both linear and nonlinear redundancy matrices. Nonlinear redundancy matrix is also related to pre-force. Two sets of redundancy-based indexes are proposed for evaluation of cable–strut system. The system-level index set is used to evaluate structural redundancy and the strengthening effect of the pre-force on the structural redundancy. The member-level index set is used to evaluate the distribution of redundancy and structural uniformity. Finally, three groups of 3-bar prisms and two groups of cable domes are designed and analyzed to verify the effectiveness of the proposed redundancy-based indexes. The results show that the two sets of redundancy-based indexes are able to reflect pre-force level and evenness of distribution of strain energy and thus can be used as structural evaluation indexes.

Keywords

cable–strut system member-level redundancy redundancy theory structural evaluation index system-level redundancy

Introduction

Cable–strut system (Wang, 1998) is a self-equilibrium pin-jointed system comprising compressive struts and tensile cables. This kind of lightweight system is widely used in large-span structures (Cadoni and Micheletti, 2012; Cai et al., 2016; Zhang et al., 2015) because of high structural performance, esthetics, and cost-effectiveness. The need for redundancy in cable–strut system should be recognized, and the system deserves much attention during its structural design. Studies related to redundancy analysis of various structures, including bridge structures (Frangopol and Nakib, 1989; Ghosn et al., 2010; Zhu and Frangopol, 2012, 2015), ship structures (Blagojević and Žiha, 2012; Frangopol et al., 2016), timber structures (Kirkegaard, 2012; Kirkegaard et al., 2010), and frame and truss structures (Kanno and Ben-Haim, 2011; Kohta et al., 2013; Schafer and Bajpai, 2005; Ye and Zhu, 2016; Zhang et al., 2016), have been investigated in last decades. However, few studies have been conducted to quantify the redundancy of cable–strut systems.

Structural redundancy (Frangopol et al., 1992) depends on many factors, such as the topology, geometry, material properties, intensity of external loads, initial pre-force in members, influence of secondary systems, and correlation among random variables. All the above factors make it difficult to quantify redundancy for evaluation of structural systems. During the last decades, probabilistic (Blagojević and Žiha, 2012; Frangopol and Curley, 1987; Frangopol and Nakib, 1989; Ghosn et al., 2010) and deterministic redundancy measures (Ghosn and Moses, 1998; Moses, 1990; Pandey and Barai, 1997; Schafer and Bajpai, 2005) are proposed to evaluate the structural design. Time-dependent redundancy (Biondini and Frangopol, 2017; Frangopol et al., 2016; Okasha and Frangopol, 2010a, 2010b) is also presented and measured recently. The simplest deterministic measure of redundancy is the degree of static indeterminacy. Ströbel (1995) established the fundamental redundancy theory for structural systems and extended the static indeterminacy into member-level with respect to its mechanical properties. Yuan et al. (2016) introduced the concept of system-level redundancy of structures composed of beam–column members and extended it to member-level. Chen et al. developed redundancy theory for flexible systems, and investigated physical meanings and engineering applications of system-level redundancy (Zhou et al., 2017, 2015; Zhu et al., 2013).

The focus in this work is on the development of redundancy evaluation method of cable–strut system considering system-level and member-level redundancy. The redundancy proposed in this article depends on the factors including structural topology, geometry, material properties, and initial pre-force. Other factors such as external force and correlation among random variables are not considered. A linear redundancy theory is first proposed to quantify the redundancy of cable–strut system using both system-level and member-level static indeterminacy. A nonlinear redundancy theory is then established on the basis of linear redundancy theory by introducing the effect of pre-force on structural redundancy. Two sets of redundancy-based indexes are proposed for evaluation of cable–strut system. The system-level indexes are used to evaluate the structural redundancy and the strengthening effect of the pre-force on the structural redundancy, while the member-level indexes are used to evaluate the uniformity of redundancy distribution across the structure. Finally, three groups of 3-bar prisms and two groups of cable domes are designed and analyzed to verify the effectiveness of the proposed evaluation indexes of cable–strut system.

Redundancy theory and index of cable–strut system

In this section, linear redundancy theory of cable–strut system (LRTCS) is developed based on linear potential energy equations, and nonlinear redundancy theory of cable–strut system (NRTCS) is developed based on nonlinear potential energy equations. Two sets of redundancy-based indexes are then proposed, respectively, based on linear and nonlinear redundancy theories as redundancy measures of cable–strut system. Both linear and nonlinear redundancy-based indexes (LR and NR, respectively) can reflect the effect of geometry, topology, and material property on structural redundancy and its distribution, while the nonlinear set can additionally demonstrate the effect of pre-force. Both linear and nonlinear sets contain two indexes. One is system-level index related to structural redundancy and the other is member-level index related to the distribution of member redundancy.

Potential energy equation based on deformation

Potential energy equations are the basis of the redundancy theory developed in this article. To make the redundancy theory applicable in cable–strut system, the potential energy equations are derived in the form of first- and second-order deformations. It should be noted that the material discussed here is assumed to be linear elastic, and the cross-sectional area of each member keeps constant. The initial installation error $\tilde{D}$ is also taken into consideration in the potential energy formula since it is inevitable in practical cases. Therefore, the nonlinear structural potential energy $Π$ based on nonlinear deformation can be obtained in equation (1)

\begin{matrix} Π = \\ \underset{strain energy}{\underset{︸}{(V_{L}^{T} + V_{NL}^{T}) (N_{σ} + C \tilde{D}) + \frac{1}{2} (V_{L}^{T} + V_{NL}^{T}) C (V_{L} + V_{NL})}} \\ - \underset{internal work}{\underset{︸}{S^{T} (L + V_{L} + V_{NL} - \tilde{L})}} - \underset{external work}{\underset{︸}{P^{T} Δ X}} \end{matrix}

(1)

where $V_{L}$ is the linear deformation vector, $V_{NL}$ is the second-order nonlinear deformation vector, $N_{σ}$ is the pre-force vector, $C$ is the diagonal matrix of member stiffness, $\tilde{D}$ is the initial installation error vector, $S$ is the internal force vector, $L$ is the initial member length vector, $\tilde{L}$ is the member length vector after deformation, $P$ is the external load vector, and $Δ X$ is the nodal displacement vector. Linear deformation vector $V_{L}$ and the second-order nonlinear deformation vector $V_{NL}$ are associated with nodal displacement vector $Δ X$ by linear compatibility matrix $B_{L}$ and nonlinear compatibility matrix $B_{NL}$ (Qian and Yang, 2003), respectively, as expressed in equations (2) and (3)

V_{L} = B_{L} Δ X, \frac{\partial V_{L}}{\partial Δ X} = B_{L}

(2)

V_{NL} = \frac{1}{2} B_{NL} Δ X, \frac{\partial V_{NL}}{\partial Δ X} = B_{NL}

(3)

where $B_{L}$ is only related to structural topology and independent of $Δ X$ , and $B_{NL}$ is a function of the nodal displacement vector $Δ X$ .

Based on the small displacement assumption, equation (1) is simplified by neglecting the higher order term in $(1 / 2) (V_{L}^{T} + V_{NL}^{T}) C (V_{L} + V_{NL})$ and then expressed as equation (4)

\begin{matrix} Π = \underset{strain energy}{\underset{︸}{(V_{L}^{T} + V_{NL}^{T}) (N_{σ} + C \tilde{D}) + \frac{1}{2} V_{L}^{T} C V_{L}}} \\ - \underset{internal work}{\underset{︸}{S^{T} (L + V_{L} + V_{NL} - \tilde{L})}} - \underset{external work}{\underset{︸}{P^{T} Δ X}} \end{matrix}

(4)

As a subcase of nonlinear potential energy, the formula of linear potential energy $\overset{⌢}{Π}$ could be obtained if only the first-order linear deformation $V_{L}$ is considered as shown in equation (5)

\begin{matrix} \overset{⌢}{Π} = \underset{strain energy}{\underset{︸}{V_{L}^{T} (N_{σ} + C \tilde{D}) + \frac{1}{2} V_{L}^{T} C V_{L}}} \\ - \underset{internal work}{\underset{︸}{S^{T} (L + V_{L} - \tilde{L})}} - \underset{external work}{\underset{︸}{P^{T} Δ X}} \end{matrix}

(5)

The linear potential energy expressed in equation (5) is used to develop the linear redundancy theory, meanwhile the nonlinear potential energy expressed in equation (4) could be used to develop the nonlinear redundancy theory.

Linear redundancy theory

According to equation (5), the linear potential energy $\overset{⌢}{Π}$ is a function of linear deformation vector $V_{L}$ , internal force vector $S$ , and nodal displacement vector $Δ X$ . Then, the equilibrium, constitutive, and compatibility equations are obtained by taking the derivative of potential energy $\overset{⌢}{Π}$ with respect to $Δ X$ , $V_{L}$ , and $S$ , respectively, as shown in equations (6) to (8)

\frac{\partial \overset{⌢}{Π}}{\partial Δ X} = {(\frac{\partial \tilde{L}}{\partial Δ X})}^{T} S - P = 0

(6)

\frac{\partial \overset{⌢}{Π}}{\partial V_{L}} = N_{σ} + C \tilde{D} + C V_{L} - S = 0

(7)

\frac{\partial \overset{⌢}{Π}}{\partial S} = - (L + V_{L} - \tilde{L}) = 0

(8)

According to equation (8), the derivation of member length vector with respect to nodal displacement $\partial \tilde{L} / \partial Δ X$ is equal to linear compatibility matrix $B_{L}$ , as shown in equation (9)

\frac{\partial \tilde{L}}{\partial Δ X} = \frac{\partial (L + V_{L})}{\partial Δ X} = \frac{\partial V_{L}}{\partial Δ X} = B_{L}

(9)

By substituting equations (7) and (9) into equation (6), the equilibrium equation can be rewritten as

B_{L}^{T} C B_{L} Δ X + B_{L}^{T} C \tilde{D} + B_{L}^{T} N_{σ} - P = 0

(10)

where $B_{L}^{T} C B_{L}$ is elastic stiffness matrix $K_{L}$ .

For kinematic indeterminate structure (Qian and Yang, 2003), the elastic stiffness matrix $K_{L}$ is rank deficiency, which means that it is not invertible. Thus, the nodal displacement vector $Δ X$ could be obtained by solving equation (11)

Δ X = K_{L}^{+} (P - B_{L}^{T} C \tilde{D} - B_{L}^{T} N_{σ}) + (I_{n_{n}} - K_{L}^{+} K_{L}) ξ

(11)

where $K_{L}^{+}$ is the Moore–Penrose inverse (Ben-Israel and Greville, 2003) of the elastic stiffness matrix; $I_{n_{n}}$ is $n_{n} \times n_{n}$ identity matrix, $n_{n}$ is the number of node freedom; and $ξ$ is an arbitrary vector. The dot product of linear compatibility matrix $B_{L}^{T}$ and pre-force vector $N_{σ}$ is $0$ since the structure is designed to be self-equilibrium at initial state, and the term $(I_{n_{n}} - K_{L}^{+} K_{L}) ξ$ shall be neglected when the cable–strut system is stiffened by pre-stress. By substituting equation (11) into equation (7), the internal force $S$ can be rewritten as equation (12)

S = S_{e} + S_{0} = C B_{L} K_{L}^{+} P + R_{L} C \tilde{D} + N_{σ}

(12)

where $S_{e} (= C B_{L} K_{L}^{+} P)$ is internal force caused by external force; $S_{0} (= R_{L} C \tilde{D} + N_{σ})$ is internal force caused by initial pre-force and initial installation error; $R_{L} (= I_{n_{m}} - C B_{L} K_{L}^{+} B_{L}^{T})$ is the linear matrix of member force redistribution; and $I_{n_{m}}$ is $n_{m} \times n_{m}$ identity matrix, $n_{m}$ is the number of members. To explore the properties of the structure without considering the external loads, the following emphasis is put on the linear force redistribution matrix $R_{L}$ . It is easy to figure out that $R_{L}$ is an idempotent matrix (Horn and Johnson, 1990; Shi and Wei, 1996)

\begin{matrix} R_{L} R_{L} = (I_{n_{m}} - C B_{L} K_{L}^{+} B_{L}^{T}) (I_{n_{m}} - C B_{L} K_{L}^{+} B_{L}^{T}) \\ = I_{n_{m}} - C B_{L} K_{L}^{+} B_{L}^{T} = R_{L} \end{matrix}

(13)

Thus, the trace of the matrix $R_{L}$ is equal to its rank

trace (R_{L}) = rank (R_{L}) = rank (I_{n_{m}}) - rank (C B_{L} K_{L}^{+} B_{L}^{T})

(14)

where $rank (C B_{L} K_{L}^{+} B_{L}^{T})$ is calculated using equation (15)

rank (C B_{L} K_{L}^{+} B_{L}^{T}) = rank (K_{L} K_{L}^{+}) = n_{r}

(15)

where $n_{r}$ is the rank of the linear compatibility matrix $B_{L}$ . Then, the trace of the matrix $R_{L}$ can be calculated by substituting equation (15) into equation (14) as presented in equation (16)

trace (R_{L}) = rank (R_{L}) = n_{m} - n_{r} = s

(16)

where $s$ is the static indeterminacy of the structure.

According to the matrix theory (Horn and Johnson, 1990; Shi and Wei, 1996), the summation of diagonal entries of matrix $R_{L}$ is equal to the static indeterminacy s of the structure. Define the ith diagonal entry $r_{l_{i}}$ of $R_{L}$ as linear redundancy of the ith member, and define the trace of $R_{L}$ as LR of the structure, that is

r_{l_{i}} = {(R_{L})}_{ii}

(17)

LR = trace (R_{L}) = s

(18)

Therefore, the matrix $R_{L}$ not only indicates linear structural redundancy (static indeterminacy) but also can be used to distribute linear structural redundancy to each member.

After obtaining linear member redundancy, the structural uniformity considering geometry, topology, and material property can be evaluated using linear member redundancy distribution across the structure. The quotient of standard deviation of the linear member redundancy over their mean value is defined as an index for evaluation of structural uniformity and named as linear redundancy-based distribution index $(LRD)$ . Equation (19) shows the definition of LRD

LRD = \frac{\sqrt{\frac{1}{n_{m}} \sum_{i = 1}^{n_{m}} (r_{l_{i}} - {\bar{r}}_{l})}}{{\bar{r}}_{l}}

(19)

where ${\bar{r}}_{l}$ is the mean value of the linear member-redundancy.

Since the linear redundancy matrix $R_{L}$ contains the information of geometry, topology, and material property, LR indicates the linear redundancy property in system level, and LRD can comprehensively reflect linear redundancy distribution in member level.

Nonlinear redundancy theory

Similarly, the equilibrium, constitutive, and compatibility equations based on nonlinear deformations can be obtained by taking the derivative of nonlinear potential energy $Π$ with respect to $Δ X$ , $V_{L}$ , and $S$ , respectively. The resulting constitutive equation is the same with equation (7), while the equilibrium and compatibility equations are changed into equations (20) and (21), respectively

\frac{\partial Π}{\partial Δ X} = {(\frac{\partial \tilde{L}}{\partial Δ X})}^{T} S - B_{NL}^{T} S + B_{NL}^{T} (N_{σ} + C \tilde{D}) - P = 0

(20)

\frac{\partial Π}{\partial S} = - (L + V_{L} + V_{NL} - \tilde{L}) = 0

(21)

where the second-order deformation term $V_{NL}$ represents the characteristics of large deformation. According to equation (21), the derivation of member length vector with respect to nodal displacement $\partial \tilde{L} / \partial Δ X$ is equal to summation of linear and nonlinear compatibility matrix $B_{L} + B_{NL}$ , as shown in equation (22)

\begin{matrix} \frac{\partial \tilde{L}}{\partial Δ X} = \frac{\partial (L + V_{L} + V_{NL})}{\partial Δ X} = \frac{\partial V_{L}}{\partial Δ X} + \frac{\partial V_{NL}}{\partial Δ X} \\ = B_{L} + B_{NL} \end{matrix}

(22)

By substituting equations (7) and (22) into equation (20), the equilibrium equation can be rewritten as

K_{L} Δ X + B_{L}^{T} C \tilde{D} + B_{NL}^{T} (N_{σ} + C \tilde{D}) - P = 0

(23)

In equation (23), the dot product of nonlinear compatibility matrix $B_{NL}^{T}$ and pre-force vector $N_{σ}$ can be written as a function of nodal displacement vector $Δ X$ as shown in equation (24)

B_{NL}^{T} N_{σ} = K_{S} Δ X

(24)

where $K_{S}$ is the geometric stiffness matrix (Němec et al., 2016; Qian and Yang, 2003; Wriggers, 2008), which contains the information of initial pre-stress. Detailed proof of equation (24) is shown in Appendix 2. By substituting equation (24) into equation (23), it is obtained that

(K_{L} + K_{S}) Δ X + (B_{NL}^{T} + B_{L}^{T}) C \tilde{D} - P = 0

(25)

For a stable cable–strut structure, the stiffness matrix $(K_{L} + K_{S})$ is full rank and has inverse matrix. The nodal displacement vector can then be expressed as

Δ X = {(K_{L} + K_{S})}^{- 1} [P - (B_{NL}^{T} + B_{L}^{T}) C \tilde{D}]

(26)

Thus, the expression of internal force $S$ can be obtained by substituting equation (26) into equation (7)

S = S_{e} + S_{0} = C B_{L} {(K_{L} + K_{S})}^{- 1} P + R_{NL} C \tilde{D} + N_{σ}

(27)

where $R_{NL} (= I_{n_{m}} - C B_{L} (K_{L} + K_{S})^{- 1} (B_{L}^{T} + B_{NL}^{T}))$ is the nonlinear matrix of force redistribution. With small displacement of cable–strut system, matrix $B_{NL}^{T}$ is much smaller than matrix $B_{L}^{T}$ . In this condition, terms of $B_{L}^{T} + B_{NL}^{T}$ can be simplified into $B_{L}^{T}$ by neglecting $B_{NL}^{T}$ . Thus, the simplified nonlinear redundancy matrix $R_{NL}$ can be expressed as equation (28) and used to evaluate the nonlinear redundancy properties of cable–strut systems in their initial configuration

R_{NL} = I - C B_{L} {(K_{L} + K_{S})}^{- 1} B_{L}^{T}

(28)

Similarly, define the ith diagonal entry $r_{n l_{i}}$ of the matrix $R_{NL}$ as nonlinear redundancy of the ith member and define the trace of $R_{NL}$ as NR of the structure, that is

r_{n l_{i}} = {(R_{NL})}_{ii}

(29)

NR = trace (R_{NL})

(30)

Therefore, the matrix $R_{NL}$ can indicate nonlinear structural redundancy and be used to distribute nonlinear structural redundancy to each member. Herein, the nonlinear structural redundancy considers the effect of pre-force.

After obtaining member redundancy, the structural uniformity considering geometry, topology, material property, and pre-force can be evaluated using member redundancy distribution across the structure. The quotient of standard deviation of the nonlinear member redundancy over their mean value is defined as an index for evaluation of structural uniformity and named as nonlinear redundancy-based distribution index $(NRD)$ . Equation (31) shows the definition of NRD

NRD = \frac{\sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(r_{n l_{i}} - {\bar{r}}_{nl})}^{2}}}{{\bar{r}}_{nl}}

(31)

where ${\bar{r}}_{nl}$ is the mean value of the nonlinear member-redundancy. Compared with LR and LRD, NR and NRD can reflect the properties of the structure more effectively, because they also take the pre-force into account.

Redundancy analysis

The proposed LR is equal to the static indeterminacy s so that it can represent the linear structural redundancy, which is determined by geometry, topology, and material property. The NR introduces the effect of pre-force on the structural redundancy on the basis of LR. Thus, for the cable–strut system, the effect of pre-force on the structural redundancy can be evaluated using difference between NR and LR, that is, NR − LR. Similarly, the LRD can reflect the distribution of member redundancy resulting from geometry, topology, and material property, while the NRD introduces the influence of pre-force. For flexible system, like cable–strut system that needs pre-force to establish the structure, NRD is a better index to evaluate the distribution of member redundancy.

In this section, three groups of 3-bar prisms and two groups of cable domes are designed and analyzed to verify the effectiveness of NR − LR and NRD on representing the effect of pre-force on the structural redundancy and the distribution of member redundancy, respectively. A new concept named structural uniformity is defined to demonstrate the evenness of distribution of member strain energy and could be calculated by commercial FEM software ANSYS. The NRD value is then compared with structural uniformity value to show that the redundancy-based indexes can be used in structural design as evaluation indexes.

3-bar minimal regular prism

A 3-bar minimal regular prism (Skelton and de Oliveira, 2009) is the simplest self-balanced three-dimensional (3D) tensegrity unit, as shown in Figure 1. It is composed of three bars (struts) and nine strings (cables), which could be divided into four member groups. The member groups include vertical bars, top strings, bottom strings, and vertical strings. According to Skelton and de Oliveira (2009), the twist angle $α$ between the top and bottom triangles is set to be $30^{°}$ to keep the prism stable. The static indeterminacy of the prism is 1, which means the prism has only one self-stress mode. The geometry of the structure is determined by the radius $r$ of top and bottom triangles and the thickness $Δ$ of prism. In order to verify the effectiveness of the proposed redundancy-based indexes for cable–strut system design, three cases of 3-bar prisms are analyzed in this section. In Case 1, four prisms are designed with the same geometry, same member sectional area, and different pre-force level. In Case 2, nine prisms are designed with the same geometry, same pre-force, and different member sectional area. In Case 3, nine prisms are designed with the same member sectional area, same pre-force in vertical bars, and different geometry. In these three cases, the tensile modulus of bars is $2.1 \times 1 0^{5} MPa$ , the compressive strength of bars is 345 MPa, the tensile modulus of strings is $1.7 \times 1 0^{5} MPa$ , and the tensile strength of strings is 1600 MPa.

Figure 1.

3-bar prism: (a) top and (b) perspective views.

LR and NR of 3-bar prism

In Case 1, four 3-bar prisms with the same geometry, same member sectional area, and different pre-force level is discussed. The radius of these prisms is $r = 0.2 m$ , and the thickness of the prism is $Δ = 0.4 m$ . Table 1 lists the sectional area of each member group. Table 2 shows the initial pre-force in each of the four prisms. The pre-force level of prism #1 is deemed the benchmark level, and the pre-force level of prisms #2, #3, and #4 is 50%, 150%, and 200% of the benchmark level, respectively.

Table 1.

Sectional area of all the members in Case 1.

Member group	Vertical bar	Top string	Bottom string	Vertical string
Sectional area, A (unit: $\times 10^{- 6} m^{2}$ )	61.24	2.50	2.50	2.50

Table 2.

Initial pre-forces in four prisms in Case 1.

Prism no.	Initial pre-force, $N_{σ}$ (unit: N)
	Vertical bar	Top string	Bottom string	Vertical string
#1	−1000.0	359.6	359.6	743.0
#2	−500.0	179.8	179.8	371.5
#3	−1500.0	539.4	539.4	1114.5
#4	−2000.0	719.2	719.2	1486.0

Table 3 shows the value of LR and NR calculated using LRTCS and NRTCS, respectively, and the resulting value of NR − LR. As presented in Table 3, all the four 3-bar prisms have the same value of LR, which is equal to the value of static indeterminacy s of these prisms. Different initial pre-force level results in different value of NR. By comparing the NR value of prisms with their pre-force level, it is shown that the higher the pre-force level, the greater the NR value. The value of NR − LR is obtained by subtracting LR from NR. If the NR − LR value of prism #1 is taken as benchmark $NR - LR$ value, it can be seen that the NR − LR value of prisms #2, #3, and #4 is 50%, 150%, and 200% of the benchmark so that the NR − LR value is proportional to the pre-force level in 3-bar prism. Therefore, LR value can represent the structural redundancy provided by geometry, topology, and material, and NR − LR value can accurately reflect the extra structural redundancy provided by pre-force. Both LR and NR − LR can be used as evaluation index for structural design of cable–strut system.

Table 3.

LR and NR of prisms in Case 1.

Prism no.		#1	#2	#3	#4
$r_{l_{i}}$	Vertical bar	0.0182	0.0182	0.0182	0.0182
	Top string	0.0444	0.0444	0.0444	0.0444
	Bottom string	0.0444	0.0444	0.0444	0.0444
	Vertical string	0.2262	0.2262	0.2262	0.2262
LR		1.0000	1.0000	1.0000	1.0000
$r_{n l_{i}}$	Vertical bar	0.0182	0.0183	0.0183	0.0183
	Top string	0.0452	0.0467	0.0467	0.0475
	Bottom string	0.0452	0.0467	0.0467	0.0475
	Vertical string	0.2276	0.2302	0.2302	0.2315
NR		1.0173	1.0087	1.0259	1.0345
NR − LR		0.0173	0.0087	0.0259	0.0345

LR: linear redundancy-based indexes; NR: nonlinear redundancy-based indexes. Bold values represent the LR, NR, NR-LR values of prism #1-#4 respectively, they are the values used to show the relationship between redundancy and pre-force level.

LRD and NRD of 3-bar prism

In Case 2, nine 3-bar prisms with the same geometry, same pre-force, and different member sectional area are designed. The mass of prisms is also controlled to be the same. The geometry parameter and pre-force of prisms in this case are the same with prism #1 in Case 1. Table 4 lists the member sectional area in prisms in Case 2. As shown in Table 4, nine prisms are divided into three sets. The sectional area of bars remains the same in each set, while bar becomes much slender from Set #1 to Set #3. The sectional area of strings within prism #1, #4, and #7 are set to be the same, respectively. The sectional area of cables within prism #2, #5, and #8 are set proportional to its pre-force distribution, respectively. The sectional area ratio between vertical to bottom string within prism #3, #6, and #9 is set larger than corresponding pre-force ratio, respectively, while the sectional area of top string is set to be the same with bottom string in these three prisms. The nine prisms in Case 2 are loaded under central symmetric external forces. These forces point to the geometric center of prism as shown in Figure 2.

Table 4.

Member sectional area in prisms in Case 2.

Prism no.	Sectional area, A (unit: $\times 10^{- 6} m^{2}$ )
	Set #1			Set #2			Set #3
	#1	#2	#3	#4	#5	#6	#7	#8	#9
Vertical bar	61.24	61.24	61.24	58.18	58.18	58.18	55.12	55.12	55.12
Top string	2.50	1.79	1.40	4.04	2.89	2.26	5.58	3.99	3.13
Bottom string	2.50	1.79	1.40	4.04	2.89	2.26	5.58	3.99	3.13
Vertical string	2.50	3.69	4.34	4.04	5.97	7.02	5.58	8.24	9.69

Figure 2.

The central symmetric forces for the prisms.

Figure 3 shows LRD and NRD values of prisms in Case 2 using LRTCS and NRTCS, respectively. As shown in Figure 5, LRD and NRD of prisms #2, #5, and #8 rank the second in each set and are larger than those of prisms #3, #6, and #9, respectively. It indicates that the most even distribution of member redundancy is not achieved when members are sized proportional to their pre-force. The value of NRD is close to LRD in each prism.

Figure 3.

(a) LRD and (b) NRD values of prisms in Case 2.

A new concept named structural uniformity (SU) is introduced herein to demonstrate the evenness of distribution of member strain energy under specified loads. Equation (32) shows the definition of SU

SU = \frac{\sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(Π_{s_{i}} - {\bar{Π}}_{s_{i}})}^{2}}}{{\bar{Π}}_{s_{i}}}

(32)

where $Π_{s_{i}}$ is the strain energy in ith member and ${\bar{Π}}_{s_{i}}$ is the mean value of member strain energy. The numerator $\sqrt{1 / n \sum_{i = 1}^{n} {(Π_{s_{i}} - {\bar{Π}}_{s_{i}})}^{2}}$ is the standard deviation of member strain energy over the structure. Thus, SU can reflect the evenness of distribution of member strain energy and can be calculated by ANSYS. Figure 4 shows SU value in prisms in Case 2. As shown in Figure 4, SU value of prisms in Case 2 demonstrates the same trend with LRD and NRD.

Figure 4.

SU value of prisms in Case 2.

In Case 3, other nine 3-bar prisms with the same member sectional area, same pre-force in vertical bar, and different geometry are analyzed. The member sectional area, length, and pre-force of vertical bar are set the same as the prism #1 in Case 1. Table 5 shows geometric parameters of these nine prisms. The ratio of thickness $Δ$ and diameter $2 r$ of prisms varies from 0.6 to 1.4. The pre-force distribution within prism changes with its geometry as shown in Table 6. The prisms in Case 3 are also loaded by the same central symmetric forces with Case 2. Figure 5 shows value of LRD, NRD, and SU of prisms in Case 3. As shown in Figure 5, these index value increases with increase of $Δ / (2 r)$ , which indicates that smaller $Δ / (2 r)$ will produce more even distribution of mechanical property and member redundancy across the prism. In addition, values of LRD, NRD, and SU demonstrate the same sequence of prism as $Δ / (2 r)$ varies from 0.6 to 1.4. Because 3-bar prism cannot be constructed without pre-force, NRD value is more valuable than LRD in this kind of structure.

Table 5.

Geometric parameters of prisms in Case 3.

Prism no.	#1	#2	#3	#4	#5	#6	#7	#8	#9
$Δ / Δ (2 r) (2 r)$	0.6	0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4
$r (m)$	0.2445	0.2331	0.2217	0.2106	0.2000	0.1900	0.1805	0.1717	0.1635
$Δ (m)$	0.2934	0.3263	0.3547	0.3791	0.4000	0.4179	0.4332	0.4464	0.4578

Table 6.

Pre-forces of prisms in Case 3.

Prism no.	Initial pre-force, $N_{σ}$ (N)
	#1	#2	#3	#4	#5	#6	#7	#8	#9
Vertical bar	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000
Top string	439.7	419.1	398.7	378.7	359.6	341.6	324.6	308.7	294.0
Bottom string	439.7	419.1	398.7	378.7	359.6	341.6	324.6	308.7	294.0
Vertical string	574.7	625.6	670.4	709.3	743.0	771.9	796.9	818.4	837.0

Figure 5.

(a) LRD, (b) NRD, and (c) SU values of prisms in Case 3.

Based on the results of Case 2 and Case 3, it can be concluded that redundancy-based index NRD can effectively reflect the distribution of mechanical properties in cable–strut system. The more uniform the distribution of mechanical property, the smaller the NRD value. Thus, NRD can be used together with LR and NR − LR as evaluation indexes for structural design.

Geiger dome

Two cases of Geiger dome with span of 50 m are analyzed in this section. Each dome comprised 272 members, which can be divided into 13 groups of cables and 4 groups of struts. The top and isometric views of Geiger dome are shown in Figure 6, and the number of member groups is shown in Figure 7. The nodes in the outermost circle are constrained nodes, while the rest are free nodes. In Case 4, three domes are designed with the same geometry, same member sectional area, but different pre-force level (Figure 8). Domes are designed with the same member sectional area, same pre-force in central vertical bars, but For Case 5, Geometric parameters of dome #1 is the same with dome in Case 4 (Figure 8), and geometric parameters of dome #2 and #3 are shown in Figure 9. In these two cases, the material properties are the same with prisms.

Figure 6.

(a) Top view and (b) isometric view of Geiger dome.

Figure 7.

Numbers of member groups in Geiger dome.

Figure 8.

Geometric parameters of domes in Case 4 (unit: m).

Figure 9.

Geometric parameters of (a) dome #2 and (b) dome #3 in Case 5 (unit: m).

LR and NR of Geiger dome

In Case 4, three domes with the same geometry, same member sectional area, but different pre-force level are discussed. Figure 8 shows the geometry parameters of these domes. Table 7 lists the sectional area of each member group of these domes, and Table 8 lists the pre-force of each member group of dome #1. The pre-force level of dome #1 is deemed the benchmark level, and the pre-force level of prisms #2 and #3 is 200% and 300% of the benchmark level, respectively.

Table 7.

Member sectional area of domes in Case 4.

Member group	1	2	3	4	5	6	7	8	9
Sectional area, A (unit: $\times 10^{- 6} m^{2}$ )	177.0	177.0	177.0	177.0	98.3	98.3	177.0	177.0	177.0
Member group	10	11	12	13	14	15	16	17
Sectional area, A (unit: $\times 10^{- 6} m^{2}$ )	177.0	177.0	177.0	236.0	236.0	236.0	1260	1260

Table 8.

Member pre-force of dome #1 in Case 4.

Member group	1	2	3	4	5	6	7	8	9
Initial pre-force, $N_{σ}$ (unit: kN)	22.6	29.6	44.9	68.6	6.87	15.2	23.7	38.8	57.9
Member group	10	11	12	13	14	15	16	17
Initial pre-force, $N_{σ}$ (unit: kN)	17.4	37.6	56.4	87.2	−1.13	−3.91	−8.80	−18.7

Table 9 shows the values of LR and NR of Geiger domes calculated using LRTCS and NRTCS, respectively, and the resulting value of NR − LR. The results are similar to those of prisms. As presented in Table 9, all the three domes have the same value of LR, which is equal to the value of static indeterminacy s of these domes. Different initial pre-force level results in different value of NR. The NR − LR value is proportional to the pre-force level in Geiger dome.

Table 9.

LR and NR of domes in Case 4.

Dome no.	#1	#2	#3
LR	1.000	1.000	1.000
NR	4.776	8.552	12.328
$NR - LR$	3.776	7.552	11.328

LR: linear redundancy-based indexes; NR: nonlinear redundancy-based indexes.

NRD of Geiger dome

In Case 5, three Geiger domes with the same member sectional area, same pre-force in central vertical bar, but different geometry are analyzed. Take the dome #1 in Case 4 as the benchmark dome in this case as well. The geometry of rest two domes can be obtained by keeping the positions of bottom and outermost nodes unchanged and rising the positions of top nodes, as shown in Figure 9. The pre-force distribution within dome changes with its geometry as shown in Table 10. The domes in Case 5 are loaded by vertical forces as shown in Figure 10.

Table 10.

Member pre-force of dome #2 and dome #3 in Case 5.

Dome #2
Member group	1	2	3	4	5	6	7	8	9
Initial pre-force $N_{σ}$ (unit: kN)	22.6	27.3	38.1	56.3	4.65	10.9	17.2	40.2	57.9
Member group	10	11	12	13	14	15	16	17
Initial pre-force $N_{σ}$ (unit: kN)	11.6	22.4	39.7	90.4	−1.13	−3.61	−7.48	−19.4
Dome #3
Member group	1	2	3	4	5	6	7	8	9
Initial pre-force $N_{σ}$ (unit: kN)	22.6	26.2	34.6	50.4	3.57	8.70	13.8	42.9	57.8
Member group	10	11	12	13	14	15	16	17
Initial pre-force $N_{σ}$ (unit: kN)	8.68	20.5	30.7	96.3	−1.13	−3.46	−6.79	−20.7

Figure 10.

The vertical forces for the domes.

Figure 11 shows value of NRD and SU of domes in Case 5. The sequences of NRD and SU are also the same, indicating that NRD could reflect the uniformity of strain energy in this case. The redundancy matrix could reflect the force redistribution when the disturbance to each member is the same or very close. Since it is hard to find the external forces which introduce the similar disturbance to each member of real structures in practical engineering projects, the sequence of NRD might not match the sequence of SU in some real cases.

Figure 11.

(a) NRD and (b) SU values of domes in Case 5.

Conclusion

Redundancy theory of cable–strut system is developed as a new perspective in this article for evaluation of structural design. The redundancy proposed in this article depends on the factors including structural topology, geometry, material properties, and initial pre-force. Other factors such as external force and correlation among random variables are not considered. Linear potential energy equations are derived based on linear deformation and used to develop LRTCS. Nonlinear potential energy equations are derived based on nonlinear deformation and used to develop NRTCS. Both LRTCS and NRTCS extend system-level redundancy to member-level redundancy. Two sets of redundancy-based indexes are then proposed, respectively, based on linear and nonlinear redundancy theories for evaluation of cable–strut system. Three groups of 3-bar prisms and two groups of Geiger domes are designed and analyzed to verify the effectiveness of these proposed indexes. The results of analytical cases show the following:

The LR based on LRTCS can represent the system-level structural redundancy provided by geometry, topology, and material property;

The difference between NR and LR, namely, NR − LR, can accurately reflect the extra system-level structural redundancy provided by pre-force in Case 1 and Case 4;

The LRD, NRD, and SU show the same sequence of prism when the member sectional area varies in Case 2;

The LRD, NRD, and SU increase and demonstrate the same sequence of prism as $Δ / (2 r)$ varies from 0.6 to 1.4 in Case 3;

The NRD and SU increase and demonstrate the same sequence of dome as the top nodes rise in Case 5;

The NRD is more valuable than LRD for cable–strut system, and the more uniform the distribution of mechanical property, the smaller the NRD value.

Therefore, redundancy-based indexes LR, NR − LR, and NRD can be used as evaluation indexes for structural design of cable–strut system.

Footnotes

Appendix 1

Appendix 2 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 National Natural Science Foundation of China (Grant Nos 51578492 and 51878600) and Natural Science Foundation of Zhejiang Province (Grant No. LZ14E080001).

ORCID iDs

Xingfei Yuan

Shuo Ma

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Redundancy theory and evaluation index of cable–strut system

Abstract

Keywords

Introduction

Redundancy theory and index of cable–strut system

Potential energy equation based on deformation

Linear redundancy theory

Nonlinear redundancy theory

Redundancy analysis

3-bar minimal regular prism

LR and NR of 3-bar prism

LRD and NRD of 3-bar prism

Geiger dome

LR and NR of Geiger dome

NRD of Geiger dome

Conclusion

Footnotes

Appendix 1

Appendix 2

Declaration of Conflicting Interests

Funding

ORCID iDs

References