# A Simplified Calculation Method of Length Adjustment of Datum Strand for the Main Cable with Small Sag

Authors: Jun He; Chuanxi Li; Hongjun Ke; Yang Liu; Yuping Zhang; Chuangwen Dong; Hongli Li; Zhe Zhang
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

In order to overcome the complicated iterative process of the cable length adjustment based on catenary theory and large error of length adjustment for cable with a small sag based on the parabola theory, this paper firstly develops a direct and simply calculation method based on parabola theory, considering the influence of elastic elongation on the cable unstressed length, which can apply for datum strand of the main cable and catwalk bearing rope with a small sag to improve the construction accuracy of the datum strand of suspension bridges. Then, the applicability of the proposed cable length adjustment formula under different conditions of the sag-span ratio is analyzed and compared with other calculation methods based on the theory of catenary or parabola. Finally, numerical examples are presented and discussed to illustrate the accuracy and efficiency of the proposed analytical method.


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## Body

## 1. Introduction

Cable-supported structures such as suspension bridges have been recognized as the most appealing structures due to their aesthetic appearance as well as the structural advantages of the cables [1–6]. It is well known that cables cannot behave as structural members until large tensioning forces are induced. Therefore, in order to design a cable-supported structure economically and efficiently, it is extremely important to determine the optimized initial cable tensions or unstrained lengths.Generally, designers cannot determine the initial shape arbitrarily when cable structures are considered. The initial shape is determined to satisfy the equilibrium condition between dead loads and internal member forces including cable tensions in the preliminary design stage, because cable members display strongly geometric nonlinear behaviors and the configuration of a cable system cannot be defined in the stress-free state. The process determining the initial state of cable structures is referred to as “shape finding,” “form finding,” or “initial shape or initial configuration’’ [7–13].Cables in cable-supported structures present highly nonlinear behavior, so there have been various studies of the cables. Cable analysis can be separated into two categories: parabolic approach and catenary approach.The catenary approach aims at obtaining the exact solutions of cable behavior. This approach was originally presented by O’Brien and Francis [14]. Irvine et al. subsequently derived the flexibility matrix of a two-dimensional inclined cable [15, 16]. For the analysis of suspension bridges, three-dimensional catenary cable elements were later developed by several researchers [11, 17]. In particular, there are two catenary-type analytical elements available which can be used to model the cables with a large sag in suspension bridges: (1) inextensible catenary elements: the cable elements adopted are infinitely stiff in the axial direction and cannot experience any increment of the length; (2) elastic catenary elements: an elastic catenary curve is defined as the curve formed by a perfectly elastic cable, which obeys Hooke’s law and has negligible bending resistance when suspended from its ends and subjected to gravity.In contrast, the parabola approach provides an approximate solution. To account for a cable’s sag effect, Ernst proposed the equivalent modulus of elasticity for a parabolic cable [18]. The simplicity of Ernst’s formula has made it widely used not only in the research field but also for practical designs of cable-supported structures such as suspension bridges. Later, Ren et al. [19] proposed a two-dimensional horizontal parabolic cable element that includes the vertical stiffness as well as the horizontal stiffness determined by Ernst’s formula.Generally, the main cable should be erected before the installation of the main girder for earth-anchored suspension bridges; the configuration of the main cable under construction stage is very important. In order to reach the initial configuration of the main cable, the configuration of the datum strand should be controlled precisely. The key to precise control is how to calculate the cable length adjustment and what is the relationship between cable length adjustment ΔS and sag adjustment Δf [20].Based on the abovementioned cable analysis approaches, the cable length adjustment can be calculated using a parabolic or catenary approach. The calculation of cable length adjustment based on catenary theory can provide exact results, but a complicated iterative method must be used [20]. The calculation formula of cable length adjustment based on quasi-catenary theory (adopt inextensible catenary elements) can represent explicitly by the ratio (c) of applied distribution load to the horizontal component of cable force, but the solution of c also need to use a complicated iterative method. The cable length adjustment based on parabola theory is a direct method, but the error of the adjustment amount is large when the sag is small (generally, the sag-span ratio is less than 1/30), since the effect of elastic elongation on the unstressed cable length of the cable strand is not considered.In order to overcome the complicated iterative process of the cable length adjustment based on catenary theory and large error of length adjustment for cable with small sag based on the parabola theory, we aim to find a simple and direct calculation method having both high search efficiency and accuracy. This paper starts from the basic principle of parabolic theory, considering the influence of elastic elongation on the unstressed cable length, and establishes a simplified and direct calculation method which can apply for datum strand of the main cable and catwalk bearing rope with a small sag. Then, the applicability of cable length adjustment formula based on parabolic theory is analyzed under different conditions of sag-span ratios. Finally, numerical examples are presented and discussed to illustrate the accuracy and efficiency of the proposed analytical method.


## 2. The Complete Solution of Unstressed Cable Length Based on Parabolic Theory

### 2.1. Basic Equations

Following the theory of Irvine [15] and on the basis of the assumption of a parabolic cable, the self-weight is distributed uniformly along the horizontal direction, and the ratio of the sag at the midpoint to the horizontal length is kept relatively small, that is, 1/8 or less. Furthermore, the cable’s cross section, elastic modulus, and density are considered to be constant along its length, and the cable is under a small strain. The geometry of an inclined parabolic cable considered in this study is shown in Figure 1; a uniform cable is suspended between two rigid supports with a horizontal distance of l, a vertical height difference of h, a sag of f at midspan, and subjected to a uniform distributed load q along the horizontal length of the cable; the value of q can be calculated as follows:(1)q≈γ⋅A⋅h2+l2l=γAcosθ,where E is the elastic modulus of the suspension cable; A and γ are the cross-sectional area and the density of the suspension cable, respectively; and θ is the horizontal angle of the connecting line between two ends of the suspension cable.Figure 1
A parabolic cable segment.Based on a parabolic configuration, the cable ordinate with respect to thex-axis, shown in Figure 1, is expressed as follows:(2)y=4fl2xl−x+hlx.The sagf is calculated from ql2/8H, in which H is the horizontal component of the cable force, because q⋅l represents the total weight, and its value does not vary as the shape changes and it can be replaced by the self-weight ww=γ⋅A per unit unstressed length multiplied by the unstressed length S0. Thus, the sag f can be expressed as follows:(3)f=ql28H=wS0l8H.The total lengthS and its elastic elongation ΔSq of a parabolic cable (stressed cable length) can be presented as equations (4) and (5), along with equations (2) and (3):(4)S=∫0l1+dydx2dx=l216fD11+D12−D21+D22+lnD1+1+D12D2+1+D22,(5)ΔSq=∫0lε⋅ds=∫0lHEA1+dydx2dx=HEA∫0l1+dydx2dx=HEAl+16f23l+h2l=γl28E⋅f⋅cosθl+16f23l+h2l,where D1=h+4f/landD2=h−4f/l.Therefore, the unstressed length S0 can be determined as follows:(6)S0=S−ΔSq,=l216fD11+D12−D21+D22+lnD1+1+D12D2+1+D22−γl28E⋅f⋅cosθl+16f23l+h2l.


### 2.2. Relationship between Cable Length Adjustment and Sag Variation

We consider the unstressed lengthS0 in equation (6) as a function of cable sag f; differentiating equation (6) with respect to f, we get(7)dS0df=l216fD2D4−D1D3+lnD6−lnD5+D7+γl8E⋅f2⋅cosθh2+l2−2γl3E⋅cosθ,where D3=1+D12, D4=1+D22, D5=D1+D3, D6=D2+D4, and D7=D3+D41+1/D3D4+D12/D3+D22/D4⋅4f/l.


## 2.1. Basic Equations

Following the theory of Irvine [15] and on the basis of the assumption of a parabolic cable, the self-weight is distributed uniformly along the horizontal direction, and the ratio of the sag at the midpoint to the horizontal length is kept relatively small, that is, 1/8 or less. Furthermore, the cable’s cross section, elastic modulus, and density are considered to be constant along its length, and the cable is under a small strain. The geometry of an inclined parabolic cable considered in this study is shown in Figure 1; a uniform cable is suspended between two rigid supports with a horizontal distance of l, a vertical height difference of h, a sag of f at midspan, and subjected to a uniform distributed load q along the horizontal length of the cable; the value of q can be calculated as follows:(1)q≈γ⋅A⋅h2+l2l=γAcosθ,where E is the elastic modulus of the suspension cable; A and γ are the cross-sectional area and the density of the suspension cable, respectively; and θ is the horizontal angle of the connecting line between two ends of the suspension cable.Figure 1
A parabolic cable segment.Based on a parabolic configuration, the cable ordinate with respect to thex-axis, shown in Figure 1, is expressed as follows:(2)y=4fl2xl−x+hlx.The sagf is calculated from ql2/8H, in which H is the horizontal component of the cable force, because q⋅l represents the total weight, and its value does not vary as the shape changes and it can be replaced by the self-weight ww=γ⋅A per unit unstressed length multiplied by the unstressed length S0. Thus, the sag f can be expressed as follows:(3)f=ql28H=wS0l8H.The total lengthS and its elastic elongation ΔSq of a parabolic cable (stressed cable length) can be presented as equations (4) and (5), along with equations (2) and (3):(4)S=∫0l1+dydx2dx=l216fD11+D12−D21+D22+lnD1+1+D12D2+1+D22,(5)ΔSq=∫0lε⋅ds=∫0lHEA1+dydx2dx=HEA∫0l1+dydx2dx=HEAl+16f23l+h2l=γl28E⋅f⋅cosθl+16f23l+h2l,where D1=h+4f/landD2=h−4f/l.Therefore, the unstressed length S0 can be determined as follows:(6)S0=S−ΔSq,=l216fD11+D12−D21+D22+lnD1+1+D12D2+1+D22−γl28E⋅f⋅cosθl+16f23l+h2l.


## 2.2. Relationship between Cable Length Adjustment and Sag Variation

We consider the unstressed lengthS0 in equation (6) as a function of cable sag f; differentiating equation (6) with respect to f, we get(7)dS0df=l216fD2D4−D1D3+lnD6−lnD5+D7+γl8E⋅f2⋅cosθh2+l2−2γl3E⋅cosθ,where D3=1+D12, D4=1+D22, D5=D1+D3, D6=D2+D4, and D7=D3+D41+1/D3D4+D12/D3+D22/D4⋅4f/l.


## 3. Simplified Calculation of Cable Length Adjustment Based on Parabola Theory

### 3.1. Traditional Simplified Cable Length Adjustment Formula without considering the Effect of Elastic Elongation

The total lengthS of parabolic cable in equation (4) can be expanded as series of (f/l) in the following equation, and only the first two items are adopted [21]:(8)S=l1cosθ+8cos3θ3⋅fl2.Equation (8) is called the traditional simplified cable length formula without considering the effect of elastic elongation.Differentiating equation (8) with respect to f, we obtain(9)dSdf=16cos3θ3⋅fl.Equation (9) is the traditional simplified cable length adjustment formula without considering the effect of elastic elongation, only the first two items of the expansion series are considered, and the value of f/l is required small enough (less than 1/8) to ignore the high-order terms. The literature [22] pointed out that the error is relatively large when calculating the cable length adjustment of the side span (the sag is small) using equation (9).


### 3.2. Improved Cable Length Formula considering the Effect of Elastic Elongation

Similarly, expanding the first item of the right side in equation (6) as series of (f/l) and considering the effect of elastic elongation, adopting the first two items, we get(10)S0=l1cosθ+8cos3θ3⋅fl2−γl28E⋅f⋅cosθl+16f23l+h2l.Equation (10) is called the improved cable length formula considering the effect of elastic elongation.Differentiating equation (10) with respect to f, we obtain(11)dS0df=16cos3θ3⋅fl+γl38E⋅f2⋅cosθ1+tan2θ−2γl3E⋅cosθ.Equation (11) is the simplified cable length adjustment formula considering the effect of elastic elongation; we call equation (11) as simplified cable length adjustment formula I. The latter two items at the right side of equation (11) consider the effect of elastic elongation.


### 3.3. Further Improved Cable Length Formula considering the Effect of Elastic Elongation

Generally, the densityγ and elastic modulus E of parallel steel wires (or steel stands) are 80 kN/m3 and 2.0 × 105 MPa, respectively; substituting them into equation (11), we get(12)dS0df=16cos3θ3⋅fl+5×10−8l3f2⋅cosθ1+tan2θ−2.6667×10−7lcosθ,where the unit of the span and height is meter.Let cable length adjustment tolerance (Δ(ΔS)) be less than 1 mm. For the midspan of a suspension bridge, if we set Δf = 200 mm, cosθ ≈1, l ≤ 3000 m, the effect of the third item in the right side of equation (12) on the cable length adjustment is ΔS3=2.6667×10−7l/cosθΔf≤0.16mm, which is much smaller than (Δ(ΔS)); therefore, the third item can be ignored in this condition. For the side span, if we set Δf = 100 mm, l ≤ 1000 m, and cos θ ≥ 1/3 (generally, suspension bridges satisfied these conditions), then the effect of the third item in the right side of equation (12) on the cable length adjustment is ΔS3=2.6667×10−7l/cosθΔf≤0.08mm, which is also much smaller than (Δ(ΔS)), thus, the third item can be ignored.Therefore, for stay cables in a cable-stayed bridge or main cable in both middle and side span of a suspension bridge, the third item in the right side of equation (12) or equation (11) can be ignored.Thus, equation (11) can be further simplified as(13)dS0df=16cos3θ3⋅fl+γl38E⋅f2⋅cosθ1+tan2θ.Equation (13) is the further simplified cable length adjustment formula considering the effect of elastic elongation; we call equation (13) as simplified cable length adjustment formula II.From equation (13), when the first item in the right side plays a leading role, it can be found that the smaller the f/l, the faster the variation for the sag at midspan. However, when the first item in the right side can be ignored, the second item plays a leading role and the sag at midspan changes more slowly with the cable length if the value of f/l is small enough.Similarly, equation (12) is further simplified to the following equation for a suspension cable which is made of parallel steel wires (or steel stands):(14)dS0df=16cos3θ3⋅fl+5×10−8l3f2⋅cosθ1+tan2θ.


### 3.4. Calculation of Cable Length Adjustment

The cable length adjustment amount (ΔS) is expressed as follows if the sag difference (Δf) is known:(15)ΔS=dS0¯dfΔforΔS=dS¯dfΔf,where dS0¯/df (or dS¯/df) is the average value of dS0/df (or dS/df) when f is changed from f0 to f0+Δf (i.e.f=f0∼f0+Δf), in which f0 is the sag before adjustment and f0+Δf is the target sag adjustment. Since the average value is not easily obtained, it usually is replaced by the value of dS0/df at f=f0 [19, 21]. If sag difference (Δf) is relatively large, the value of dS0/df changes obviously when the sag changes from f0 to f0 + Δf; thus, the value of dS0¯/df is taken as the value of dS0/df at the midpoint of the interval f0,f0+Δf, i.e., at f=f0+Δf/2 [19].


## 3.1. Traditional Simplified Cable Length Adjustment Formula without considering the Effect of Elastic Elongation

The total lengthS of parabolic cable in equation (4) can be expanded as series of (f/l) in the following equation, and only the first two items are adopted [21]:(8)S=l1cosθ+8cos3θ3⋅fl2.Equation (8) is called the traditional simplified cable length formula without considering the effect of elastic elongation.Differentiating equation (8) with respect to f, we obtain(9)dSdf=16cos3θ3⋅fl.Equation (9) is the traditional simplified cable length adjustment formula without considering the effect of elastic elongation, only the first two items of the expansion series are considered, and the value of f/l is required small enough (less than 1/8) to ignore the high-order terms. The literature [22] pointed out that the error is relatively large when calculating the cable length adjustment of the side span (the sag is small) using equation (9).


## 3.2. Improved Cable Length Formula considering the Effect of Elastic Elongation

Similarly, expanding the first item of the right side in equation (6) as series of (f/l) and considering the effect of elastic elongation, adopting the first two items, we get(10)S0=l1cosθ+8cos3θ3⋅fl2−γl28E⋅f⋅cosθl+16f23l+h2l.Equation (10) is called the improved cable length formula considering the effect of elastic elongation.Differentiating equation (10) with respect to f, we obtain(11)dS0df=16cos3θ3⋅fl+γl38E⋅f2⋅cosθ1+tan2θ−2γl3E⋅cosθ.Equation (11) is the simplified cable length adjustment formula considering the effect of elastic elongation; we call equation (11) as simplified cable length adjustment formula I. The latter two items at the right side of equation (11) consider the effect of elastic elongation.


## 3.3. Further Improved Cable Length Formula considering the Effect of Elastic Elongation

Generally, the densityγ and elastic modulus E of parallel steel wires (or steel stands) are 80 kN/m3 and 2.0 × 105 MPa, respectively; substituting them into equation (11), we get(12)dS0df=16cos3θ3⋅fl+5×10−8l3f2⋅cosθ1+tan2θ−2.6667×10−7lcosθ,where the unit of the span and height is meter.Let cable length adjustment tolerance (Δ(ΔS)) be less than 1 mm. For the midspan of a suspension bridge, if we set Δf = 200 mm, cosθ ≈1, l ≤ 3000 m, the effect of the third item in the right side of equation (12) on the cable length adjustment is ΔS3=2.6667×10−7l/cosθΔf≤0.16mm, which is much smaller than (Δ(ΔS)); therefore, the third item can be ignored in this condition. For the side span, if we set Δf = 100 mm, l ≤ 1000 m, and cos θ ≥ 1/3 (generally, suspension bridges satisfied these conditions), then the effect of the third item in the right side of equation (12) on the cable length adjustment is ΔS3=2.6667×10−7l/cosθΔf≤0.08mm, which is also much smaller than (Δ(ΔS)), thus, the third item can be ignored.Therefore, for stay cables in a cable-stayed bridge or main cable in both middle and side span of a suspension bridge, the third item in the right side of equation (12) or equation (11) can be ignored.Thus, equation (11) can be further simplified as(13)dS0df=16cos3θ3⋅fl+γl38E⋅f2⋅cosθ1+tan2θ.Equation (13) is the further simplified cable length adjustment formula considering the effect of elastic elongation; we call equation (13) as simplified cable length adjustment formula II.From equation (13), when the first item in the right side plays a leading role, it can be found that the smaller the f/l, the faster the variation for the sag at midspan. However, when the first item in the right side can be ignored, the second item plays a leading role and the sag at midspan changes more slowly with the cable length if the value of f/l is small enough.Similarly, equation (12) is further simplified to the following equation for a suspension cable which is made of parallel steel wires (or steel stands):(14)dS0df=16cos3θ3⋅fl+5×10−8l3f2⋅cosθ1+tan2θ.


## 3.4. Calculation of Cable Length Adjustment

The cable length adjustment amount (ΔS) is expressed as follows if the sag difference (Δf) is known:(15)ΔS=dS0¯dfΔforΔS=dS¯dfΔf,where dS0¯/df (or dS¯/df) is the average value of dS0/df (or dS/df) when f is changed from f0 to f0+Δf (i.e.f=f0∼f0+Δf), in which f0 is the sag before adjustment and f0+Δf is the target sag adjustment. Since the average value is not easily obtained, it usually is replaced by the value of dS0/df at f=f0 [19, 21]. If sag difference (Δf) is relatively large, the value of dS0/df changes obviously when the sag changes from f0 to f0 + Δf; thus, the value of dS0¯/df is taken as the value of dS0/df at the midpoint of the interval f0,f0+Δf, i.e., at f=f0+Δf/2 [19].


## 4. Applicability of Traditional Simplified Cable Length Adjustment Formula for Small Sag-Span Ratio

### 4.1. Cable in the Middle Span

LetΔf=200mm and cosθ=1 and in equation (14) 5×10−8l3/f2⋅cosθ1+tan2θ×0.200≤ΔΔS, then(16)fl≥fl=l108ΔΔS.The unit of l in equation (16) is meter.Equation (16) shows the applicable range of traditional simplified cable length adjustment for midspan. When the ratio of sag to span in midspan is larger than the critical value f/l, the traditional simplified equation for cable length adjustment meets the accuracy requirements. And f/l is proportional to l and inversely proportional to ΔΔS.As long as the requirement ofΔΔS is not too strict, the length adjustment of datum strand cable for most suspension bridges can meet the requirement of equation (16), so the traditional simplified cable length adjustment equation can be adopted in midspan. Take the datum strand cable at midspan in Huangpu Suspension Bridge as an example, the horizontal distance l=1105.662m, the sag at midspan f=99.221m, and let ΔΔS=0.002m; substituting them into equation (16), we get f/l=99.221/1105.622=0.08974≥l/108ΔΔS=0.0743. Thus, equation (16) can be satisfied.


### 4.2. Cable in the Side Span (or the Main Cable of the Suspension Bridge with a Single Tower)

LetΔf=100mm and in equation (14), 5×10−8l3/f2⋅cosθ1+tan2θ×0.100≤ΔΔS, then(17)fl≥fl=5⋅l109cos3θΔΔS.The unit of l in equation (17) is meter. Equation (17) shows the applicable range of traditional simplified cable length adjustment for side span. When the ratio of sag to span in the side span is larger than the critical value f/l, the traditional simplified equation for cable length adjustment meets the accuracy requirements. And f/l is proportional tol and inversely proportional to ΔΔS and cos3θ.When the value ofΔΔS adopts an acceptable value satisfying engineering accuracy, the length adjustment of datum strand cable at side span can be calculated by equation (13) and the calculated value by the second item in equation (13) will be smaller or larger than the value of ΔΔS; therefore, the length adjustment of datum strand cable at side span should be calculated considering the influence of elastic elongation.From equations (16) and (17), it can be found that the sag-span ratio f/l should not be too small when traditional simplified cable adjustment formulas are used without considering the influence of elastic elongation.


## 4.1. Cable in the Middle Span

LetΔf=200mm and cosθ=1 and in equation (14) 5×10−8l3/f2⋅cosθ1+tan2θ×0.200≤ΔΔS, then(16)fl≥fl=l108ΔΔS.The unit of l in equation (16) is meter.Equation (16) shows the applicable range of traditional simplified cable length adjustment for midspan. When the ratio of sag to span in midspan is larger than the critical value f/l, the traditional simplified equation for cable length adjustment meets the accuracy requirements. And f/l is proportional to l and inversely proportional to ΔΔS.As long as the requirement ofΔΔS is not too strict, the length adjustment of datum strand cable for most suspension bridges can meet the requirement of equation (16), so the traditional simplified cable length adjustment equation can be adopted in midspan. Take the datum strand cable at midspan in Huangpu Suspension Bridge as an example, the horizontal distance l=1105.662m, the sag at midspan f=99.221m, and let ΔΔS=0.002m; substituting them into equation (16), we get f/l=99.221/1105.622=0.08974≥l/108ΔΔS=0.0743. Thus, equation (16) can be satisfied.


## 4.2. Cable in the Side Span (or the Main Cable of the Suspension Bridge with a Single Tower)

LetΔf=100mm and in equation (14), 5×10−8l3/f2⋅cosθ1+tan2θ×0.100≤ΔΔS, then(17)fl≥fl=5⋅l109cos3θΔΔS.The unit of l in equation (17) is meter. Equation (17) shows the applicable range of traditional simplified cable length adjustment for side span. When the ratio of sag to span in the side span is larger than the critical value f/l, the traditional simplified equation for cable length adjustment meets the accuracy requirements. And f/l is proportional tol and inversely proportional to ΔΔS and cos3θ.When the value ofΔΔS adopts an acceptable value satisfying engineering accuracy, the length adjustment of datum strand cable at side span can be calculated by equation (13) and the calculated value by the second item in equation (13) will be smaller or larger than the value of ΔΔS; therefore, the length adjustment of datum strand cable at side span should be calculated considering the influence of elastic elongation.From equations (16) and (17), it can be found that the sag-span ratio f/l should not be too small when traditional simplified cable adjustment formulas are used without considering the influence of elastic elongation.


## 5. Numerical Examples

### 5.1. Example 1

A flexible cable in References [23, 24] was adopted as an example; the geometrical and material parameters are as follows: l = 210.925 m; h = 110.485 m; E = 2.0 × 105 MPa; f0 = 1.082276 m; A = 0.011 m2; γ = 72.5 kN/m3; and Δf = 67.357 mm. The unstressed cable length S0 and cable length adjustment ΔS will be solved. The calculation results by a perfect solution of parabola theory, improved simplified formula, traditional simplified formula, quasi-catenary theory, and the theory of catenary are shown in Table 1.Table 1
Comparison of the calculation results of the unstressed length and cable length adjustment.
Item
Theory of catenary
Quasi-catenary theory
Perfect solution of parabola theory
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
Unstressed cable length (m)
237.56707
237.56706
237.57255
238.12012
237.55485
/
Cable length error (m)
0
−0.00001
0.00548
0.55305
−0.01222
/
Cable length adjustment (mm)
33.60
35.95
34.4279
13.21
34.4278
33.9906
Relative error of adjustment
0%
7.0%
2.5%
−60.7%
2.5%
1.2%It can be found that for the suspension cable withf/l = 1/194.5, h/l = 0.524, and l = 210.925 m, unstressed cable length calculation using perfect solution of parabola theory has enough accuracy, as well as the improved simplified formula (but with the increase of sag-span ratio, the calculation error for unstressed cable length gradually increases since improved simplified method only adopts the first two items in the expansion series); cable length adjustment amount calculation using perfect solution of parabola theory, improved simplified formulas I and II, has sufficient accuracy; the absolute error does not exceed 1 mm, while the relative error does not exceed 2.5%; but the calculation absolute error is large using traditional simplified method, especially the error of cable length adjustment is too large to be accepted.


### 5.2. Example 2

The calculation error of length adjustment for datum strand in the middle span using traditional and improved simplified formula was compared in this example.The known conditions for a datum strand in the middle span of the Guangzhou Huangpu suspension bridge [20] (Figure 2) are as follows: l = 1105.622 m; h ≈ 0; E = 2.02 × 105 MPa; and γ = 78.495 kN/m2. Comparison of length adjustment results of datum strand in the middle span with different sag to span ratio under the condition that the sag f was reduced 20 cm (Δf = −20 cm) as shown in Table 2.Figure 2
Main cable system of Huangpu Suspension Bridge (unit: m).Table 2
Comparison of length adjustment results for datum strand in midspan with different sag-span ratios.
Sag to span ratio
Sagf0 (m)
Exact solution ΔS (mm)
Perfect solution of parabola theory
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
1/3
368.541
263.2710
252.8554
−4.0
355.5655
35.1
355.6049
35.1
355.6622
35.1
1/5
221.124
187.5098
182.6118
−2.6
213.3426
13.8
213.5538
13.9
213.6111
13.9
1/8
138.203
126.8748
125.1264
−1.4
133.3432
5.1
133.9733
5.6
134.0305
5.6
1/11.14
99.221
94.3145
93.5452
−0.8
95.7347
1.5
97.0108
2.9
97.0681
2.9
1/15
73.708
72.3814
72.0170
−0.5
71.1206
−1.7
73.4794
1.5
73.5367
1.6
1/20
55.281
57.1265
56.9534
−0.3
53.3429
−6.6
57.5804
0.8
57.6377
0.9
1/50
22.112
47.6561
48.0732
0.9
21.3426
−55.2
48.1141
1.0
48.1714
1.1
1/70
15.7946
66.457
67.2319
1.2
15.3346
−76.9
67.2471
1.2
67.3044
1.3


### 5.3. Example 3

The calculation error of length adjustment for datum strand in the side span using traditional and improved simplified formula was compared in this example.The known conditions for a datum strand in the side span of the Guangzhou Humen suspension bridge [22] are as follows: l = 298 m; h = 96.798; E = 2.0 × 105 MPa; and γ = 78.358 kN/m2. Comparison of the calculation results of length adjustment for datum strand in the side span with different sag to span ratio under the condition that the sag f was reduced 8.7 cm (Δf = −8.7 cm) as shown in Table 3.Table 3
Comparison of length adjustment results for datum strand in side span with different sag-span ratios.
Sag to span ratio
Sagf0 (m)
Exact solution ΔS (mm)
Perfect solution of parabola theory ΔS (mm)
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
1/8
37.25
48.2564
48.1414
49.8407
3.5
49.9282
3.7
49.9354
3.7
1/10
29.8
/
39.0885
39.8610
2.0
40.0019
2.3
40.0090
2.4
1/20
14.9
/
20.3755
19.9017
−2.3
20.4884
0.6
20.4955
0.6
1/35.98
8.2830
12.9359
12.9419
11.0379
−14.7
12.9611
0.1
12.9682
0.2
1/50
5.9600
/
11.6553
7.9261
−32.0
11.6624
0.1
11.6696
0.1
1/70
4.2571
/
13.0155
5.6450
−56.6
13.0180
0.0
13.0251
0.1
1/100
2.9800
19.0276
19.1201
3.9343
−79.4
19.1209
0.0
19.1280
0.0From Tables2 and 3, the following can be found:(1)
The applicable range of sag to span ratio for four solutions based on parabola theory is different. There is no lower limit of sag to span ratio for the methods of the perfect solution of parabola theory, improved simplified formulas I and II, but the lower limit for traditional simplified formula is about 1/30. The upper limit for the perfect solution of parabola theory is the highest for all four solutions; the upper limit for other three simplified formula is about 1/8, since those three methods ignore the effect of high-order items of expansion series. The calculation error of length adjustment for datum strand by the methods of perfect solution of parabola theory, improved simplified formulas I and II, is less than 4% for the suspension cable whose sag-span ratio is less than 1/8.(2)
From the calculation results by the methods of the perfect solution of parabola theory, the improved simplified formula I and II, the finding “the smaller the sag-span ratio, the faster the sag variation in midspan with the change of cable length” is right only in a certain range of sag-span ratio.


## 5.1. Example 1

A flexible cable in References [23, 24] was adopted as an example; the geometrical and material parameters are as follows: l = 210.925 m; h = 110.485 m; E = 2.0 × 105 MPa; f0 = 1.082276 m; A = 0.011 m2; γ = 72.5 kN/m3; and Δf = 67.357 mm. The unstressed cable length S0 and cable length adjustment ΔS will be solved. The calculation results by a perfect solution of parabola theory, improved simplified formula, traditional simplified formula, quasi-catenary theory, and the theory of catenary are shown in Table 1.Table 1
Comparison of the calculation results of the unstressed length and cable length adjustment.
Item
Theory of catenary
Quasi-catenary theory
Perfect solution of parabola theory
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
Unstressed cable length (m)
237.56707
237.56706
237.57255
238.12012
237.55485
/
Cable length error (m)
0
−0.00001
0.00548
0.55305
−0.01222
/
Cable length adjustment (mm)
33.60
35.95
34.4279
13.21
34.4278
33.9906
Relative error of adjustment
0%
7.0%
2.5%
−60.7%
2.5%
1.2%It can be found that for the suspension cable withf/l = 1/194.5, h/l = 0.524, and l = 210.925 m, unstressed cable length calculation using perfect solution of parabola theory has enough accuracy, as well as the improved simplified formula (but with the increase of sag-span ratio, the calculation error for unstressed cable length gradually increases since improved simplified method only adopts the first two items in the expansion series); cable length adjustment amount calculation using perfect solution of parabola theory, improved simplified formulas I and II, has sufficient accuracy; the absolute error does not exceed 1 mm, while the relative error does not exceed 2.5%; but the calculation absolute error is large using traditional simplified method, especially the error of cable length adjustment is too large to be accepted.


## 5.2. Example 2

The calculation error of length adjustment for datum strand in the middle span using traditional and improved simplified formula was compared in this example.The known conditions for a datum strand in the middle span of the Guangzhou Huangpu suspension bridge [20] (Figure 2) are as follows: l = 1105.622 m; h ≈ 0; E = 2.02 × 105 MPa; and γ = 78.495 kN/m2. Comparison of length adjustment results of datum strand in the middle span with different sag to span ratio under the condition that the sag f was reduced 20 cm (Δf = −20 cm) as shown in Table 2.Figure 2
Main cable system of Huangpu Suspension Bridge (unit: m).Table 2
Comparison of length adjustment results for datum strand in midspan with different sag-span ratios.
Sag to span ratio
Sagf0 (m)
Exact solution ΔS (mm)
Perfect solution of parabola theory
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
1/3
368.541
263.2710
252.8554
−4.0
355.5655
35.1
355.6049
35.1
355.6622
35.1
1/5
221.124
187.5098
182.6118
−2.6
213.3426
13.8
213.5538
13.9
213.6111
13.9
1/8
138.203
126.8748
125.1264
−1.4
133.3432
5.1
133.9733
5.6
134.0305
5.6
1/11.14
99.221
94.3145
93.5452
−0.8
95.7347
1.5
97.0108
2.9
97.0681
2.9
1/15
73.708
72.3814
72.0170
−0.5
71.1206
−1.7
73.4794
1.5
73.5367
1.6
1/20
55.281
57.1265
56.9534
−0.3
53.3429
−6.6
57.5804
0.8
57.6377
0.9
1/50
22.112
47.6561
48.0732
0.9
21.3426
−55.2
48.1141
1.0
48.1714
1.1
1/70
15.7946
66.457
67.2319
1.2
15.3346
−76.9
67.2471
1.2
67.3044
1.3


## 5.3. Example 3

The calculation error of length adjustment for datum strand in the side span using traditional and improved simplified formula was compared in this example.The known conditions for a datum strand in the side span of the Guangzhou Humen suspension bridge [22] are as follows: l = 298 m; h = 96.798; E = 2.0 × 105 MPa; and γ = 78.358 kN/m2. Comparison of the calculation results of length adjustment for datum strand in the side span with different sag to span ratio under the condition that the sag f was reduced 8.7 cm (Δf = −8.7 cm) as shown in Table 3.Table 3
Comparison of length adjustment results for datum strand in side span with different sag-span ratios.
Sag to span ratio
Sagf0 (m)
Exact solution ΔS (mm)
Perfect solution of parabola theory ΔS (mm)
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
1/8
37.25
48.2564
48.1414
49.8407
3.5
49.9282
3.7
49.9354
3.7
1/10
29.8
/
39.0885
39.8610
2.0
40.0019
2.3
40.0090
2.4
1/20
14.9
/
20.3755
19.9017
−2.3
20.4884
0.6
20.4955
0.6
1/35.98
8.2830
12.9359
12.9419
11.0379
−14.7
12.9611
0.1
12.9682
0.2
1/50
5.9600
/
11.6553
7.9261
−32.0
11.6624
0.1
11.6696
0.1
1/70
4.2571
/
13.0155
5.6450
−56.6
13.0180
0.0
13.0251
0.1
1/100
2.9800
19.0276
19.1201
3.9343
−79.4
19.1209
0.0
19.1280
0.0From Tables2 and 3, the following can be found:(1)
The applicable range of sag to span ratio for four solutions based on parabola theory is different. There is no lower limit of sag to span ratio for the methods of the perfect solution of parabola theory, improved simplified formulas I and II, but the lower limit for traditional simplified formula is about 1/30. The upper limit for the perfect solution of parabola theory is the highest for all four solutions; the upper limit for other three simplified formula is about 1/8, since those three methods ignore the effect of high-order items of expansion series. The calculation error of length adjustment for datum strand by the methods of perfect solution of parabola theory, improved simplified formulas I and II, is less than 4% for the suspension cable whose sag-span ratio is less than 1/8.(2)
From the calculation results by the methods of the perfect solution of parabola theory, the improved simplified formula I and II, the finding “the smaller the sag-span ratio, the faster the sag variation in midspan with the change of cable length” is right only in a certain range of sag-span ratio.


## 6. Engineering Application

### 6.1. Bridge Overview

Lishui Bridge on the expressway from the city of Zhangjiajie to Huayuan is a suspension bridge with a single span, two towers (no hangers at two side spans), and steel truss girders [24]. The deck system uses a composite section with steel-stiffened girder and concrete slab. The main cable is arranged with dimensions (200 + 856 + 190) m; the ratio of sag to span is 1/10 at the main span; the bridge uses 69 pairs of hangers; the standard spacing of these hangers is 12 m; and the distance from the end hanger to the tower is 20 m. The bridge arrangement is shown in Figure 3.Figure 3
Bridge layout of Lishui Bridge on expressway from Zhangjiajie to Huayuan (unit: m).


### 6.2. Cable Length Adjustment

The sag of the datum strand and catwalk bearing rope at both main span and side span were corrected according to the cable length adjustment calculated using the modified simplified formula II to get the target sag. The elevation error of the erection of datum strand and catwalk bearing rope is less than 1 cm, much higher than the required accuracy in construction specifications.Table4 lists target geometry parameters of the datum strand and catwalk bearing rope at the main span and side span in Zhangjiajie direction. In addition, the main mechanical parameters of catwalk bearing rope and datum strand are listed in the following: the density γ of catwalk bearing rope and datum strand are 123.9 kN/m3 and 77 kN/m3, respectively, and the elastic modulus of catwalk bearing rope and datum strand are 1.21 × 105 MPa and 1.96 × 105 MPa, respectively.Table 4
The target geometry parameters of the datum strand and catwalk bearing rope under erection stage.
Item
Target unstressed length (m)
Position
Mileage (m)
Elevation (m)
l (m)
h (m)
f (m)
Catwalk bearing rope at main span
863.989
Anchorage point at Zhangjiajie side
2545.782
592.072
Middle span
2970.000
522.551
848.406
7.601
73.322
Anchorage point at Huayuan side
3394.188
599.673
Catwalk bearing rope at side span
213.937
Anchorage point at girder end
2347.093
494.047
Middle point at side span in Zhangjiajie
2442.523
541.093
190.859
97.991
1.950
Anchorage point at tower end
2537.952
592.038
Datum strand at main span
869.797
Tangent point at tower end in Zhangjiajie
2543.967
593.374
Middle span
2969.920
520.267
851.937
7.614
76.914
Tangent point at tower end in Huayuan
3395.904
600.988
Datum strand at side span
218.338
Tangent point at girder end
2343.368
493.770
Middle span
2440.621
538.664
194.505
98.986
4.599
Tangent point at tower internal end
2537.873
592.755Table5 shows cable length adjustment process of the datum strand and catwalk bearing rope at the main span and the side span in Zhangjiajie direction from the beginning to the target state using different methods. The calculated results indicate that the cable length adjustment can quickly reach the goal with high accuracy using the improved simplified formula II.Table 5
The length adjustment of the datum strand and catwalk bearing rope.
Item
Target sag to span ratio
Target sagf0 (m)
Sag before adjustedf0 (m)
Exact solution
Perfect solution of parabola theory
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
ΔS (mm)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
Catwalk bearing rope at main span
1/11.57
73.322
73.508
86.11
85.43
−0.79
85.72
−0.45
88.32
2.56
88.43
2.69
Catwalk bearing rope at side span
1/97.87
1.950
2.136
68.89
68.94
0.07
7.14
−89.64
68.94
0.07
68.96
0.11
Datum strand at main span
1/11.08
76.914
77.147
109.91
109.22
−0.63
112.18
2.07
113.32
3.11
113.37
3.15
Datum strand at side span
1/42.29
4.599
4.821
25.17
25.16
−0.02
19.82
−21.27
25.16
−0.03
25.18
0.02


## 6.1. Bridge Overview

Lishui Bridge on the expressway from the city of Zhangjiajie to Huayuan is a suspension bridge with a single span, two towers (no hangers at two side spans), and steel truss girders [24]. The deck system uses a composite section with steel-stiffened girder and concrete slab. The main cable is arranged with dimensions (200 + 856 + 190) m; the ratio of sag to span is 1/10 at the main span; the bridge uses 69 pairs of hangers; the standard spacing of these hangers is 12 m; and the distance from the end hanger to the tower is 20 m. The bridge arrangement is shown in Figure 3.Figure 3
Bridge layout of Lishui Bridge on expressway from Zhangjiajie to Huayuan (unit: m).


## 6.2. Cable Length Adjustment

The sag of the datum strand and catwalk bearing rope at both main span and side span were corrected according to the cable length adjustment calculated using the modified simplified formula II to get the target sag. The elevation error of the erection of datum strand and catwalk bearing rope is less than 1 cm, much higher than the required accuracy in construction specifications.Table4 lists target geometry parameters of the datum strand and catwalk bearing rope at the main span and side span in Zhangjiajie direction. In addition, the main mechanical parameters of catwalk bearing rope and datum strand are listed in the following: the density γ of catwalk bearing rope and datum strand are 123.9 kN/m3 and 77 kN/m3, respectively, and the elastic modulus of catwalk bearing rope and datum strand are 1.21 × 105 MPa and 1.96 × 105 MPa, respectively.Table 4
The target geometry parameters of the datum strand and catwalk bearing rope under erection stage.
Item
Target unstressed length (m)
Position
Mileage (m)
Elevation (m)
l (m)
h (m)
f (m)
Catwalk bearing rope at main span
863.989
Anchorage point at Zhangjiajie side
2545.782
592.072
Middle span
2970.000
522.551
848.406
7.601
73.322
Anchorage point at Huayuan side
3394.188
599.673
Catwalk bearing rope at side span
213.937
Anchorage point at girder end
2347.093
494.047
Middle point at side span in Zhangjiajie
2442.523
541.093
190.859
97.991
1.950
Anchorage point at tower end
2537.952
592.038
Datum strand at main span
869.797
Tangent point at tower end in Zhangjiajie
2543.967
593.374
Middle span
2969.920
520.267
851.937
7.614
76.914
Tangent point at tower end in Huayuan
3395.904
600.988
Datum strand at side span
218.338
Tangent point at girder end
2343.368
493.770
Middle span
2440.621
538.664
194.505
98.986
4.599
Tangent point at tower internal end
2537.873
592.755Table5 shows cable length adjustment process of the datum strand and catwalk bearing rope at the main span and the side span in Zhangjiajie direction from the beginning to the target state using different methods. The calculated results indicate that the cable length adjustment can quickly reach the goal with high accuracy using the improved simplified formula II.Table 5
The length adjustment of the datum strand and catwalk bearing rope.
Item
Target sag to span ratio
Target sagf0 (m)
Sag before adjustedf0 (m)
Exact solution
Perfect solution of parabola theory
Traditional simplified formula
The improved simplified formula I
The improved simplified formula II
ΔS (mm)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
ΔS (mm)
Error (%)
Catwalk bearing rope at main span
1/11.57
73.322
73.508
86.11
85.43
−0.79
85.72
−0.45
88.32
2.56
88.43
2.69
Catwalk bearing rope at side span
1/97.87
1.950
2.136
68.89
68.94
0.07
7.14
−89.64
68.94
0.07
68.96
0.11
Datum strand at main span
1/11.08
76.914
77.147
109.91
109.22
−0.63
112.18
2.07
113.32
3.11
113.37
3.15
Datum strand at side span
1/42.29
4.599
4.821
25.17
25.16
−0.02
19.82
−21.27
25.16
−0.03
25.18
0.02


## 7. Conclusions

(1)
The improved simplified formulas considering the influence of elastic elongation on the cable unstressed length was derived for a suspension cable with small sag to develop parabola theory of cable analysis.
(2)
In comparison with the catenary theory (exact method) and quasi-catenary theory, three proposed cable length adjustment formulas considering the effect of elastic elongation are direct calculation methods which do not need iteration and programming. Especially for improved simplified formula II, the calculation work is further reduced.
(3)
The applicability and accuracy of three proposed cable length adjustment formulas at both middle and side span for suspension bridges were verified through numerical examples and practical engineering projects.
(4)
The sag variation rate with the change of cable length depends on sag-span ratio, when the sag-span ratio is greater than a certain value (such as 1/60); the smaller the sag-span ratio, the faster the sag variation at the middle span with the change of cable length; however, when the sag-span ratio is less than a certain value (such as 1/60), the smaller the sag-span ratio, the slower the sag variation at middle span with the change of cable length.



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*Source: 6075893-2019-01-22.xml*