Theoretical analysis and experimental study on sensitivity of element-length error in cable-strut tensile structures

Abstract

In a real cable-strut tensile structure, the element-length errors are inevitable. To understand their effects on the bearing capacity of a cable-strut tensile structure, the element-length error sensitivity analysis was investigated in this study. First, mathematical model of the element-length error was proposed based on stochastic theory. By combining the balance equation, geometric equation, and physical equation, the fundamental equation between the pre-stress deviation and element-length error was derived. After that, pre-stress deviation statistics characteristic was achieved with the help of statistical theory and the element-length error sensitivity analysis method was formulated. Then, a cable-strut tensile structure model with a diameter of 5.0 m was designed and fabricated to validate the proposed method. The element-length was set adjustable in order to simulate the element-length errors. Making use of the measured internal forces induced by element-length errors, the error sensitivity of each kind of element was achieved. In addition, a finite element model was also established with the commercial software ANSYS. The element-length errors were simulated by the changes of element-length due to temperature variations. The results of the three models coincided with each other satisfactorily, verifying the effectiveness of the proposed mathematical model. It was found that different elements had different error sensitivities. The error sensitivity of the hoop cables was most prominent, the ridge cables and diagonal cables the second, and the struts the third.

Keywords

cable-strut tensile structure element-length error error sensitivity analysis model experiment statistics characteristic

Introduction

The cable-strut tensile structure is a sort of flexible structure composed of cables in tension and struts in compression. Because of the high strength as well as the adjustable stiffness distribution of the tension cables, the cable-strut tensile structure possesses excellent bearing capacity. As the stiffness of tension cables is provided with the help of pre-stress, the exact pre-stress distribution is the premise of the superior bearing performance of a cable-strut tensile structure. In the last two decades, a lot of work on self-stress mode analysis (Pellegrino, 1993; Pellegrino and Calladine, 1986) and optimal pre-stress design (Chen and Dong, 2013; Yuan et al., 2007) have been reported. Most of them were carried out in ideal conditions which ignore the construction errors. However, due to the complicated working conditions, construction errors are inevitable in a cable-strut tensile structure. As a result, the real parameters of a cable-strut tensile structure may deviate from the ideal ones, such as the pre-stress (Lin and Wang, 2007; Qin et al., 2007). Previous studies have shown that the bearing capacity of a cable-strut tensile structure is very sensitive to the pre-stress deviation (Gao et al., 2005; Peng and Wu, 2004). It therefore necessitates the evaluation of the effects of construction errors and the elimination or reduction of their negative effects.

Construction errors mainly include element-length error, installation deviation error, pin hole machining error, node or anchorage size error, and temperature deviation error. In the last two decades, a lot of error sensitivity analysis studies had been reported (Guo et al., 2009; Sheng et al., 2008; Wang et al., 2012; Zhang, 2008; Zong et al., 2012). The methods proposed by them are mainly divided into two types. The first type method is established based on the probability theory, which models the errors with a group of stochastic errors conforming to a certain distribution. The second type is a kind of orthogonal design method, which simulates the errors with some groups of deviation levels. The error sensitivity analysis can then be conducted by evaluating their effects on the distribution of initial pre-stress as well as the structure static and dynamic behaviors. So far, the construction errors which conform to not only a certain distribution but also the mathematical statistics law have not been studied.

In this study, the element-length error of a cable-strut tensile structure, which is an important construction error that affects the bearing performance, was investigated by approaches of a mathematical model, a scale-down model, and a finite element model (FEM). First, the mathematical model of element-length error was proposed based on stochastic theory. By combining the balance equation, geometric equation, and physical equation, the fundamental equation between the pre-stress deviation and element-length error was derived. After that, pre-stress deviation statistics characteristic was achieved with the help of statistical theory and the element-length error sensitivity analysis method was formulated. Then, a cable-strut tensile structure model with a diameter of 5.0 m was designed and fabricated to validate the proposed method. The element length was set adjustable in order to simulate the element-length errors. Making use of the measured internal forces induced by element-length errors, the error sensitivity of each kind of element was achieved. In addition, an FEM was also established with the commercial software ANSYS. The element-length errors were simulated by the changes of element length due to temperature variations. The results of the three models coincided with each other satisfactorily, verifying the effectiveness of the proposed mathematical model. It was found that the error sensitivity of the hoop cables was most prominent, the ridge cables and diagonal cables the second, and the struts the third.

Mathematical model of element-length error

Element-length error mainly comes from the original length calculation error and machining error. The original length calculation error is the deviation between the approximate curve equation and the real shape. In addition, temperature or humidity changes, material properties changes, and other factors may also contribute to element-length errors. In this study, it is assumed that the errors are independent of each other and the probability of positive or negative errors is the same. According to the Lindberg–Levy central limit theorem, the total length error obeys the normal distribution N∼(µ,σ²), where µ is the average error and $σ^{2}$ is the error variance. For a random error variable obeying normal distribution, the proportion of error falling in $[μ - 3 σ, μ + 3 σ]$ is 99.74%. Assume the element-length error falls in the range of [a, b], then

μ = \frac{a + b}{2} σ = \frac{1}{6} (b - a)

(1)

where a and b are the upper and lower limits of the length error, respectively. As per the technical specification for cable structure (JGJ257-2012), the manufacture errors of cable-strut tensile structure are shown in Tables 1 and 2. Making use of equation (1), the average value and standard deviation of the cable and strut length deviation distribution difference can be obtained, as shown in Tables 3 and 4.

Table 1.

Allowable error of cable length.

Cable length, L (m)	The allowable deviation, $Δ L$ (mm)
≤50	±15
50 < L ≤ 100	±20
>100	L/5000

Table 2.

Allowable error of strut length.

Strut length, L (m)	The allowable deviation, $Δ L$ (mm)
≤5	±5
5 < L ≤ 10	±10
>10	±15

Table 3.

Cable-length error characteristic.

Cable length, L (m)	The average, $μ (mm)$	Standard deviation, $σ (mm)$
≤50	0	5
50 < L ≤ 100	0	6.67
>100	0	L/15,000

Table 4.

Strut-length error characteristic.

Strut length, L (m)	The average, $μ (mm)$	Standard deviation, $σ (mm)$
≤5	0	1.67
5 < L ≤ 10	0	3.33
>10	0	5

The relationship between element-length error and pre-stress deviation

To analyze the effects of element-length error on the pre-stress deviation, it is desirable to establish the relationship between the element-length error and the pre-stress deviation. In a cable-strut tensile structure, set b as the number of elements, n as the number of nodes, c as the number of support constraints, and $e_{0} (b \times 1)$ as the element-length error vector. The equilibrium equation (equation (2)), physical equation (equation (3)), and geometric equation (equation (4)) can be established as

At = P

(2)

t = M (e - e_{0})

(3)

Bd = e

(4)

where t is the internal force vector (b × 1); P is the nodal load vector ((3n − c) × 1); d is the node displacement vector ((3n − c) × 1); e is the member expansion (b × 1); e ₀ is the initial member expansion (b × 1); M is the element stiffness matrix (b × b); $M_{ii} = E_{i} A_{i} / l_{i}$ , E_i, A_i, and l_i are the elastic modulus, section area, and length of element i; A is the equilibrium matrix $((3 n - c) \times b)$ ; B is the coordinate matrix $(b \times (3 n - c))$ ; and $B = A^{T}$ . Combining equations (2), (3), and 4 leads to

t = M (A^{T} {(A M A^{T})}^{- 1} P + M (A^{T} {(A M A^{T})}^{- 1} A M - I) e_{0} = t_{p} + t_{e}

(5)

d = {(A M A^{T})}^{- 1} (P + A M e_{0})

(6)

Equation (5) reflects the relationship among the initial defect length $e_{0}$ , load P , and axial force t . Equation (6) reflects the relationship among the initial defect length $e_{0}$ , load P , and node displacement d . In above equations, $I$ is the unit matrix, $t_{p}$ is the additional axial force caused by load P , and $t_{e}$ is the additional axial force caused by the initial defect length $e_{0}$ . So when the load P is constant, the pre-stress deviations and shape deviations of the structure are caused by the original length variation only. If the member length changes by $δ e_{0}$ , the corresponding pre-stress and shape changes are

δ t = M (A^{T} (AM A^{T})^{- 1} AM - I) δ e_{0} = S_{t} δ e_{0}

(7)

δ d = (AM A^{T})^{- 1} AM δ e_{0} = S_{d} δ e_{0}

(8)

where $S_{t} = M (A^{T} (AM A^{T})^{- 1} AM - I)$ is the stress sensitivity matrix $(b \times b)$ ; $S_{d} = (AM A^{T})^{- 1} AM$ is the shape sensitivity matrix $((3 n - c) \times b)$ . The parameter $(S_{t})_{ij}$ reflects the magnification between the $j th$ element original length change and the ith element pre-stress deviation. The parameter $(S_{d})_{ij}$ reflects the magnification between the $j th$ element original length change and the ith nodal displacement deviation.

The statistical features of the pre-stress deviation

As mentioned earlier, this study assumes that errors are independent of each other and the probability of positive or negative errors is the same. According to the Lindberg–Levy central limit theorem, the total error obeys the normal distribution N∼(µ_ei, σ_ei), where $μ_{ei}$ and $σ_{ei}$ are the average value and standard deviation for the ith element-length error, respectively. As the length error of each element distributes independently, the distribution of initial pre-stress deviation of each element also obeys the normal distribution N∼(µ_ti, σ_ti), where $μ_{ti}$ and $σ_{ti}$ are the average value and standard deviation of the ith element internal force, respectively, that is

μ_{ti} = \sum_{j = 1}^{b} S_{tij} μ_{ej}

(9)

σ_{ti}^{2} = \sum_{j = 1}^{b} S_{tij}^{2} σ_{ej}^{2}

(10)

Making use of equations (9) and (10), the pre-stress deviation caused by the element-length error, that is, the element-length error sensitivity, can be evaluated.

Experimental study

Model design

To validate the proposed error sensitivity analysis method, a cable-strut tensile structure model with a diameter of 5.0 m was fabricated in laboratory, as shown in Figure 1. The model is composed of 12 pieces of cable-strut units and 24 pieces of supporting elements, which form a self-balance system. The supporting platform consists of ring beams and columns, as shown in Figure 1(b). The cable-strut unit, as shown in Figure 2, consists of tension cables and compression struts. The tension cables, including hoop cables, ridge cables, and diagonal cables, are made of high-strength steel wire with a diameter of 7 mm. The length of cables can be slightly adjusted by the cable sleeve. The compression strut shown in Figure 3 comprises strut 1 and strut 2 with a section parameter of Φ15 × 3. To adjust the length of strut, the strut also has sleeves. The model has two kinds of nodes. The first kind is used for connecting cables and struts, as shown in Figure 4. For simplicity, it is denoted as node A hereafter. The second kind is used for the connections of outer cables and the supporting platform, as shown in Figure 5. Likewise, it is referred to as node B. One end of the screw is connected with cable and the other end is connected with screw cap. Thus, the length of cable can be adjusted by tightening or loosening the screw cap.

Figure 1.

A cable-strut tensile structure model: (a) the whole model and (b) element of each symmetrical unit.

Figure 2.

Basic element.

Figure 3.

Struts.

Figure 4.

Partial node A.

Figure 5.

Node B.

Due to inevitable construction errors, six pieces of cable-strut units shown in Figure 6 were measured in the experiment although the tensile structure is symmetrical. All kinds of members except for the hoop cables were measured in each of the six units, implying each kind of member has six measurement points. For hoop cables, the measurement positions were located in units 1, 4, 7, and 10 only. Electrical resistances foil strain gauges with 5-mm gauge length were employed to measure the member internal force and a data logger was used to collect the readings of the strain gauges. In order to eliminate the errors induced by the temperature changes and eccentric force, two strain gauges were mounted on opposite surfaces of each measurement point. They were then connected with a temperature compensation strain gauge to form a half bridge circuit. In addition, the nodal displacement was measured with dial gauge.

Figure 6.

Measurement position.

In order to improve the measurement precision, eliminate the errors induced by the section dimension and material differences, and verify the validity of the resistance foil strain gauges, the relationships between the internal force and the strain must be obtained in advance, with the help of the least square method, that is

N = k ε

(11)

where N and ε represent the internal force and the strain obtained by strain gauges, respectively; k is the coefficient to be decided, which was obtained with the help of the least square method, that is

k = \frac{\sum_{i} N_{i} ε_{i}}{\sum_{i} ε_{i}^{2}}

(12)

As an illustration, Figure 7 shows the strain–force curves of partial cables in unit 4. The coefficient k is summarized in Table 5. It is seen that the force–strain relation is almost linear and the coefficients of different elements usually differ from each other.

Figure 7.

Strain–force curves of typical cables in unit 4.

Table 5.

Coefficient k of cables in unit 4.

Loading order	Weight (kg)	RC1	RC2	DC1	DC2	Strut 1	Strut 2	THC	LHC
0	0	0	0	0	0	0	0	0	0
1	20	3	3	3	3	7	6	4	2
2	40	6	6	6	5	12	12	8	5
3	60	10	10	10	8	18	17	10	7
4	80	13	13	14	10	24	23	14	10
5	100	16	17	16	13	29	29	17	13
6	120	20	20	20	15	35	34	20	15
Coefficient k		60.01	59.01	59.17	76.81	33.26	34.16	57.24	78.13

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Numerical analysis

The element-length error sensitivity of the model structure was also studied by finite element analysis with the commercial software ANSYS. The change in element length was simulated by temperature changes. The temperature change ΔT was described as

Δ T = \frac{Δ L}{α L}

(13)

where $Δ L = 3$ is the element-length change, which was set as 3 mm in this article; $α = 1.2 \times 10^{- 5}$ is the linear expansion coefficient; and $L$ is the designed element length.

Model research

The ridge cable 1 (denoted as RC1), diagonal cable 1 (denoted as DC1), strut 1, strut 2, top hoop cable (denoted as THC), and lower hoop cable (denoted as LHC) in unit 4 were all elongated 3 mm, respectively. The variations of internal forces of the elements in unit 4, unit 3, and unit 2 were shown in Tables 6 to 11. Positive value indicated an increase in the element force and vice versa. It can be seen that the length change of any element can cause the variation of forces of the whole structure, which indicated the tensile structure has an excellent entire balance performance. The length changes of THC and LHC both induced some 27% variation in internal forces. The length changes of ridge cable 1 and diagonal cable 1 both caused about 15% change in internal forces. The length changes of strut 1 and strut 2 both generated about 3% variation in internal forces. Therefore, the error sensitivity of the hoop cables is the most prominent; the sensitivity of ridge cables and diagonal cables the second and that of struts the third. It is also found that the element force changes in unit 4 were almost similar to the element force changes in unit 2 and unit 3. For example, they were all about 15% when the length of the ridge cable 1 and diagonal cable 1 were elongated 3 mm, which again indicates the tensile structure has an excellent entire balance performance. In addition, the measured values and computed results of the error sensitivity analysis were generally consistent with each other. The small differences between them were attributed to the errors such as element-length error, geometric error in boundary platform, and adjustment length error.

Table 6.

The element internal force changes caused by the diagonal cable 1 elongation 3 mm in unit 4 (kN).

Unit no.	Element		DC1	DC2	RC1	RC2	Strut 1	Strut 2	THC	LHC
Unit no.	Pre-stress		1.75	1.66	1.60	1.51	−0.31	−0.31	2.86	3.14
4	Numerical ANSYS values	Axial force	1.48	1.40	1.34	1.27	−0.26	−0.26	2.41	2.68
		Deviation %	−15.50	−15.67	−16.05	−16.00	−16.18	−15.66	−15.74	−14.52
	Measured values	Axial force	1.65	1.28	1.50	1.42	−0.31	−0.31	2.21	2.40
		Deviation %	−5.92	−23.04	−6.00	−5.90	0.00	0.00	−22.89	−23.44
	Measured error %		11.35	−8.74	11.98	12.02	19.30	18.56	−8.49	−10.43
3	Numerical ANSYS values	Axial force	1.51	1.43	1.34	1.27	−0.26	−0.26	2.41	2.68
		Deviation %	−13.93	−14.02	−15.68	−15.79	−14.72	−14.72	−15.74	−14.52
	Measured values	Axial force	1.65	1.28	1.29	1.42	−0.31	−0.31
		Deviation %	−5.77	−23.07	−19.16	−6.39	0.00	0.00
	The measured error %		9.55	−10.48	−4.08	11.23	17.30	17.35
2	Numerical ANSYS values	Axial force	1.50	1.42	1.34	1.27	−0.26	−0.26	2.41	2.68
		Deviation %	−14.56	−14.66	−15.74	−15.83	−15.05	−15.06	−15.73	−14.57
	Measured values	Axial force	1.41	1.56	1.50	1.34	−0.31	−0.31
		Deviation %	−19.51	−5.94	−5.77	−11.54	0.00	0.00
	The measured error %		−5.77	10.23	11.84	5.12	17.73	17.76

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Table 7.

The element internal force changes caused by the ridge cable 1 elongation 3 mm in unit 4 (kN).

Unit no.	Element		DC1	DC2	RC1	RC2	Strut 1	Strut 2	THC	LHC
Unit no.	Pre-stress		1.75	1.66	1.60	1.51	−0.31	−0.31	2.86	3.14
4	Numerical ANSYS values	Axial force	1.49	1.42	1.33	1.25	−0.26	−0.26	2.40	2.69
	Numerical ANSYS values	Deviation %	−14.64	−14.58	−16.91	−17.10	−16.18	−15.67	−15.95	−14.35
	Measured values	Axial force	1.65	1.28	1.40	1.42	−0.31	−0.31	2.70	2.89
		Deviation %	−5.92	−23.04	−12.00	−5.90	0.00	0.00	−5.72	−7.81
	The measured error %		10.21	−9.91	5.91	13.51	19.30	18.58	12.16	7.63
3	Numerical ANSYS values	Axial force	1.50	1.42	1.35	1.28	−0.26	−0.26	2.40	2.69
	Numerical ANSYS values	Deviation %	−14.30	−14.39	−15.37	−15.47	−14.74	−14.74	−15.95	−14.35
	Measured values	Axial force	1.65	1.37	1.39	1.32	−0.31	−0.31
		Deviation %	−5.77	−17.30	−12.78	−12.78	0.00	0.00
	The measured error %		10.01	−3.35	3.13	3.24	17.33	17.38
2	Numerical ANSYS values	Axial force	1.50	1.42	1.34	1.27	−0.26	−0.26	2.40	2.69
	Numerical ANSYS values	Deviation %	−14.35	−14.43	−16.00	−16.10	−15.07	−15.09	−16.01	−14.34
	Measured values	Axial force	1.41	1.76	1.50	1.25	−0.31	−0.31
		Deviation %	−19.51	5.94	−5.77	−17.30	0.00	0.00
	The measured error %		−6.01	23.81	12.20	−1.41	17.77	17.79

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Table 8.

The element internal force changes caused by the strut 1 elongation 3 mm in unit 4 (kN).

Unit no.	Element		DC1	DC2	RC1	RC2	Strut 1	Strut 2	THC	LHC
Unit no.	Pre-stress		1.75	1.66	1.60	1.51	−0.31	−0.31	2.86	3.14
4	Numerical ANSYS values	Axial force	1.80	1.70	1.65	1.56	−0.33	−0.32	2.95	3.23
		Deviation %	3.00	2.71	3.31	2.99	6.47	1.49	3.04	2.76
	Measured values	Axial force	1.75	1.79	1.60	1.42	−0.31	−0.32	2.86	3.14
		Deviation %	0.00	7.68	0.00	−5.90	0.00	3.42	0.00	0.00
	The measured error %		−2.91	4.84	−3.21	−8.63	−6.08	1.90	−2.95	−2.68
3	Numerical ANSYS values	Axial force	1.80	1.70	1.64	1.55	−0.32	−0.32	2.95	3.23
		Deviation %	2.69	2.57	2.95	2.82	2.91	2.87	3.04	2.76
	Measured values	Axial force	1.75	1.66	1.60	1.51	−0.31	−0.31
		Deviation %	0.00	0.00	0.00	0.00	0.00	0.00
	The measured error %		−2.56	−2.46	−2.81	−2.69	−2.79	−2.72
2	Numerical ANSYS values	Axial force	1.80	1.70	1.64	1.56	−0.32	−0.32	2.95	3.23
		Deviation %	2.75	2.64	3.03	2.92	2.99	2.95	3.04	2.76
	Measured values	Axial force	1.75	1.76	1.60	1.51	−0.32	−0.31
		Deviation %	0.00	5.94	0.00	0.00	3.33	0.00
	The measured error %		−2.67	3.23	−2.93	−2.81	0.35	−2.85

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Table 9.

The element internal force changes caused by the strut 2 elongation 3 mm in unit 4 (kN).

Unit no.	Element		DC1	DC2	RC1	RC2	Strut 1	Strut 2	THC	LHC
Unit no.	Pre-stress		1.75	1.66	1.60	1.51	−0.31	−0.31	2.86	3.14
4	Numerical ANSYS values	Axial force	1.80	1.71	1.65	1.56	−0.32	−0.34	2.95	3.23
		Deviation %	2.91	2.87	3.19	3.13	1.54	8.58	3.07	2.79
	Measured values	Axial force	1.75	1.66	1.60	1.51	−0.30	−0.30	2.86	3.14
		Deviation %	0.00	0.00	0.00	0.00	−3.33	−3.42	0.00	0.00
	The measured error %		−2.83	−2.79	−3.09	−3.04	−4.80	−11.05	−2.97	−2.71
3	Numerical ANSYS values	Axial force	1.80	1.70	1.64	1.55	−0.32	−0.31	2.95	3.23
		Deviation %	2.72	2.60	2.98	2.85	2.94	0.99	3.07	2.79
	Measured values	Axial force	1.75	1.66	1.60	1.51	−0.31	−0.31
		Deviation %	0.00	0.00	0.00	0.00	0.00	0.00
	The measured error %		−2.59	−2.48	−2.84	−2.71	−2.82	−0.90
2	Numerical ANSYS values	Axial force	1.80	1.70	1.64	1.56	−0.32	−0.32	2.95	3.23
		Deviation %	2.78	2.67	3.06	2.94	3.02	2.98	3.07	2.79
	Measured values	Axial force	1.75	1.66	1.60	1.51	−0.31	−0.31
		Deviation %	0.00	0.00	0.00	0.00	0.00	0.00
	The measured error %		−2.70	−2.59	−2.96	−2.83	−2.91	−2.87

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Table 10.

The element internal force changes caused by the top hoop cable elongation 3 mm in unit 4 (kN).

Unit no.	Element		DC1	DC2	RC1	RC2	Strut 1	Strut 2	THC	LHC
Unit no.	Pre-stress		1.75	1.66	1.60	1.51	−0.31	−0.31	2.86	3.14
4	Numerical ANSYS values	Axial force	1.30	1.23	1.14	1.08	−0.23	−0.23	2.03	2.33
	Numerical ANSYS values	Deviation %	−25.80	−25.84	−28.76	−28.83	−27.17	−26.69	−28.89	−25.82
	Measured values	Axial force	1.44	1.28	1.21	1.24	−0.27	−0.27	2.21	2.16
		Deviation %	−17.75	−23.04	−24.00	−17.70	−13.30	−13.66	−22.89	−31.25
	The measured error %		10.85	3.77	6.68	15.63	19.05	17.76	8.43	−7.32
3	Numerical ANSYS values	Axial force	1.30	1.23	1.14	1.08	−0.23	−0.23	2.03	2.33
	Numerical ANSYS values	Deviation %	−25.80	−25.84	−28.76	−28.83	−27.17	−26.69	−28.89	−25.82
	Measured values	Axial force	1.45	1.18	1.19	1.03	−0.26	−0.27
		Deviation %	−17.30	−28.84	−25.55	−31.94	−14.83	−14.38
	The measured error %		11.52	−3.99	4.56	−4.32	16.99	16.87
2	Numerical ANSYS values	Axial force	1.30	1.23	1.14	1.08	−0.23	−0.23	2.04	2.33
	Numerical ANSYS values	Deviation %	−25.81	−25.85	−28.83	−28.91	−27.24	−27.23	−28.84	−25.80
	Measured values	Axial force	1.41	1.27	1.04	1.16	−0.27	−0.27
		Deviation %	−19.51	−23.76	−34.61	−23.07	−13.33	−14.21
	The measured error %		8.51	2.84	−8.11	8.23	19.14	17.91

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Table 11.

The element internal force changes caused by the lower hoop cable elongation 3 mm in unit 4 (kN).

Unit no.	Element		DC1	DC2	RC1	RC2	Strut 1	Strut 2	THC	LHC
Unit no.	Pre-stress		1.75	1.66	1.60	1.51	−0.31	−0.31	2.86	3.14
4	Numerical ANSYS values	Axial force	1.29	1.23	1.14	1.08	−0.23	−0.22	2.05	2.32
	Numerical ANSYS values	Deviation %	−26.07	−26.11	−28.42	−28.48	−27.13	−27.64	−28.43	−26.22
	Measured values	Axial force	1.44	1.28	1.21	1.24	−0.27	−0.27	2.21	2.40
		Deviation %	−17.75	−23.04	−24.00	−17.70	−13.30	−13.66	−22.89	−23.44
	The measured error %		11.25	4.16	6.17	15.07	18.97	19.32	7.73	3.77
3	Numerical ANSYS values	Axial force	1.29	1.23	1.14	1.08	−0.23	−0.22	2.05	2.32
	Numerical ANSYS values	Deviation %	−26.06	−26.12	−28.43	−28.48	−27.13	−27.64	−28.43	−26.22
	Measured values	Axial force	1.45	1.18	1.19	1.03	−0.26	−0.27
		Deviation %	−17.30	−28.84	−25.55	−31.94	−14.83	−14.38
	The measured error %		11.91	−3.63	4.08	−4.79	16.92	18.42
2	Numerical ANSYS values	Axial force	1.29	1.22	1.14	1.08	−0.23	−0.23	2.05	2.32
	Numerical ANSYS values	Deviation %	−26.17	−26.23	−28.39	−28.45	−27.19	−27.18	−28.40	−26.17
	Measured values	Axial force	1.41	1.17	1.04	1.16	−0.27	−0.28
		Deviation %	−19.51	−29.70	−34.61	−23.07	−13.33	−10.66
	The measured error %		9.04	−4.69	−8.67	7.54	19.05	22.72

THC: top hoop cable; LHC: lower hoop cable; RC: rigid cable; DC: diagonal cable.

Element-length error sensitivity analysis of the model structure based on the proposed method

The mathematical models of the element-length error were first established based on equation (1) and Tables 3 and 4. Then, the sensitivity matrix $S_{t}$ was achieved based on equations (7), (9), and (10). The statistical characteristics of the pre-stress deviation were shown in Table 12. As it is seen, the error standard deviation ratios of the hoop cable $σ_{HC}$ , ridge cable (diagonal cable) $σ_{RC (DC)}$ , and strut $σ_{strut}$ were, respectively, 8.43, 4.45, and 0.87, which were almost consistent with the numerical analysis results as well as the experimental results. Therefore, different kinds of elements had different element-length error sensitivities. The error sensitivity of the hoop cables was most significant as it induced largest error standard deviation ratios, the ridge cables and diagonal cables the second, and the struts the third, which were also consistent with the numerical analysis results and experimental results. Furthermore, the error standard deviation ratios induced by ridge cable 1 and diagonal cable 1 were both 4.70, indicating they had the same length error sensitivity. In the same way, strut 1 and strut 2, ridge cable 2 and diagonal cable 2, THC, and LHC had the same length error sensitivity, respectively. It again validated the consistence between the mathematical analysis results, FEM analysis results, and experiment results. Therefore, the proposed error sensitivity analysis method has sufficient accuracy.

Table 12.

Statistical features of pre-stress deviation.

Element	Statistical features
Element	$μ_{ti}$	$σ_{ti}$
Diagonal cable 1	0	4.70
Diagonal cable 2	0	4.45
Ridge cable 1	0	4.70
Ridge cable 2	0	4.45
Strut 1	0	0.87
Strut 2	0	0.87
Top hoop cable	0	8.43
Lower hoop cable	0	8.43

Conclusion

This study carried out an investigation into the element-length error sensitivity of a cable-strut tensile structure. A total of three models, including a mathematical model, a scale-down model, and an FEM, were established to carry out the element-length error sensitivity analysis. The following findings were obtained: (1) the results of the three models coincided with each other satisfactorily, verifying the effectiveness of the proposed mathematical model; (2) different elements had different error sensitivities. In this study, the error sensitivity of the hoop cables was most prominent, the ridge cables and diagonal cables the second, and the struts the third; and (3) the study results of this article can not only help to find out those elements with prominent error sensitivity, whose lengths should be controlled accurately, but also provide a basis for further study, such as the controlling and optimization theories of structural pre-stress distribution. So, it is a very important and meaningful work of this article both to real project applications and theory studies.

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 National Natural Science Foundation of China (Grant Nos 51578422 and 51308080), the Natural Science Foundation of Zhejiang Province (Grant No. LY14E080019), the Science and Technology Project of Zhejiang Province (Grant No. 2014C33013), the Science and Technology Project of Ministry of Housing and Urban-Rural Development of China (2014-K3-005), and Qinglan Project of Jiangsu Province.

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