Elastic Analysis of Bridge Cable Wires Under Static Loading

S. B. Singh, Professor, and Piyush Sood, Undergraduate Research Assistant,
Department of Civil Engineering, Birla Institute of Technology and Science, Pilani


This article deals with comparison between the responses of CFRP and Steel cable wires when subjected to their self-weight and external vertical load considering the elasticity of the cables. The responses of the two wire types, each from the CFRP manufacturers, CFCC (Tokyo Rope Mfg. Co., Ltd.) and NACC Strand (Nippon Steel Chemical Co., Ltd.) and a steel wire, have been compared keeping in mind their elongation. The externally applied load is considered as equally spaced point forces to make the analysis realistic, whereas the self-weight is considered as the uniformly applied load along the cable length. Furthermore, the results of the extensible and the inextensible analysis of the cables have been compared. It is also shown how the theory can be used in the form-finding of the elongated cables under vertical static load and reducing the area of the cables. Numerical methods using MATLAB have been employed to find the various solutions.

Introduction
Long span bridge systems, such as suspension bridges, have been an indispensable part of the history of civil engineering. A strong cable system is an integral part of these megastructures and it is crucial that it can take the dead and the imposed loads along with its own self-weight. With advances in construction with composites, fibre reinforced materials have provided an alternative for the high strength steel (HS Steel) cable systems that are currently being used. In the past decades, these steel cables have been deteriorating due to corrosion and fatigue [1, 2].

Fibre reinforced wires are made of fibres embedded in a cohesive bulk resin. The relative volume of these two constituents determines the mechanical and volumetric properties of the composite [1]. The introduction of fibre-reinforced polymers (FRP) for cables was proposed in 1982 by Meier et al. [2]. Due to their excellent elastic properties and no yielding point on the longitudinal stress-strain diagram under short-term loading and ambient temperature, FRP is suited for the structures bearing heavy load [3]. The high strength along with the low self-weight, resistance to corrosion and chemical attacks provide favourable benefits. The high tensile properties are only valid in the longitudinal direction though. Due to poor inter-laminar shear properties, it becomes difficult to anchor the FRP wire bundles, but a solution was provided by Meier et al. [4, 5].

For the cable purpose, carbon fibre reinforced polymer (CFRP) wires have been found to be more advantageous than glass fibre reinforced polymer (GFRP) and aramid fibre reinforced polymer (AFRP) due to higher modulus of elasticity [6]. The structural response of a CFRP wire was derived on the basis of simplified mechanics of material by Mallick [7]. A major setback with CFRP wires is their higher cost with respect to the steel wires [8, 9]. Due to their several favourable properties, CFRP cables have found applications in some of the mega projects [10].

In this article, the responses of two CFCC wires manufactured by Tokyo Rope Mfg. Co., Ltd., two NACC wires by Nippon Steel Chemical Co., Ltd. [11] and a steel wire have been compared (Table 1). The analysis performed is based on the work by Irvine and Sinclair [12]. In the high load-bearing structures such as suspension bridges, there is a considerable increase in the length of the cable due to the elasticity of the cables as the internal stresses are significant. This also results in the increase in the dip of the cable and reduction in the cable tension. An analytical approach has been established to find the shape of an elastic cable. The elastic parabola approach, followed here, is only applicable when the sag-to-span ratio is 1:8 or less, as in Irvine’s theory [12].

Methods Used
In contrast to the classical solution [12], the equations used here are not in terms of the non-dimensional parameters. Furthermore, instead of choosing a fixed origin, as given in the classical solution, the location of the origin can be arbitrarily chosen to make use of the presented equations. The following assumptions have been made on the basis of classical theories:

1. The flexural rigidity of the cables has been ignored, which implies that they are considered to be perfectly elastic.

2. The bending moment and torsional moment is assumed to be absent.

3. The cable material is assumed to be linearly elastic homogeneously in the axial direction.

4. The basic principle of conservation of mass has been assumed, that is, the mass of the unstrained length of the cable would remain the same even after the elongation of the cable due to the elasticity.

In this article, an extensible cable refers to the cable in which elongation due to elasticity is considered, whereas an inextensible cable refers to the cable in which no elongation is considered. The design dip refers to the dip of the inextensible cable, whereas the actual dip refers to the final dip of the cable after elongation due to elasticity.

Figure 1 shows an inextensible cable section in x-z plane (with an arbitrary origin). The horizontal span of the cable is h and the vertical dip of the cable is d. The self-weight wC of the cable acts along the arc-length s of the cable and the external load wL is assumed to remain constant throughout the horizontal span h of the cable. Y and Z are the two endpoints of the cable with the coordinates (xY,zY) and (xZ,zZ), respectively. Point B is marked as the lowest point on the cable element with the coordinates (xB,zB). The inextensible cable analysis is an important part of this approach because the length of the cable and the tension in the cable obtained by this analysis are used as the initial values to obtain the solutions of the equations for the extensible analysis.

Thus, when a constant external vertical load acts on a weightless cable, the cable assumes the shape of a parabola represented by Eqn (6), where x and z can represent the x-coordinate and the z-coordinate of the cable element, respectively. By substituting the coordinates of the endpoint of the cable section(xY,zY) (or (xZ,zZ)) and the value of the external uniform vertical load wL in Eqn (6), we find out the horizontal component of the tension force H. Recognising the vertical component of the cable tension V=H tan q, an expression for the cable tension can be written as

where x* is the variable of integration.

If we substitute xZ for x in Eqn (8), we can calculate the unstrained length L0 of the cable. It should be noted that in the inextensible analysis, the cable length L0 and the shape of the cable only depend on the horizontal span and the vertical depth of the cable. The change in the external weight does not alter these properties of the cable.

The load to the main cable of a suspension bridge is transferred through a number of hanger cables in real life situation. To make the things more realistic, we will assume that the point forces act at the various points where the hanger cables are attached to the main cable. Let there be NH hangers attached to the main cable which equals to the number of point forces acting on the main cable. Figure 3 shows the cable element acted upon by various equal point forces F1, F2, F3,….., FNH and self-weight wC. It should be noted that L0 represents the initial inextensible length of the cable whereas Lf is the length of the strained cable. The final shape of the cable is assumed to be a function of the unstrained arc-lengths, that is x=fx (s), z=fz (s) and p=fp (s). A point j on the cable will have the arc-length coordinate pj=fp (sj), and Cartesian coordinates (xj,zj) =(fx (sj),fy (sj)) in the strained configuration where 0 = s1 <.. ..<sj <sj+1 <.. ..<sNH <L0. Although the functions fx (s), fz (s) and fp (s) are continuous functions of arc-length s, but the derivatives of these functions will be discontinuous at all the points where the point loads have been applied, owing to the zero flexural rigidity of the cable. This doesn’t comply with the real-life situation, but if the jumps between the derivatives are small, then this theory can be applied to the suspension bridge cables safely. Furthermore, there is no point force associated with the arc-length coordinate s=L0, but it turns out to be convenient if we define sNH+1=L0; however, there is no point force associated with this node.

where sN is the axial stress in the cable element, A0 is the cross-sectional area of the cable element, which is derived considering the maximum tension, and E is the Young’s modulus of elasticity for the cable element. The relative equivalent modulus of elasticity EC, as given by Ernst [13], is used in the supporting example.

For the strained cable element, the equations for the horizontal and the vertical equilibrium can be written as,
Using the chain rule for Eqns (16) and (18), we can eliminate dp, so that x can be expressed as a function of s.

After solving the above differential equation and eliminating the constants of integration by making the use of first end-point coordinate and the points where the point forces are acting, the following recurrence relation is formulated. To generalise the equation in order to handle the case of pure gravitational load, as well as the interval s0 =s <s1 if NH = 1, two additional point node forces F-1=F0 =0 are created, which are assumed to act at the arc-length co-ordinates s-1=s0 =0.
To use Eqns (20) and (22), we need to have the values for H and VY.By inserting x = xZ and s = L0 in Eqn (20) and z = zZ and s = L0 in Eqn (22), we get a set of non-linear equations which can be solved numerically for H and VY. The initial approximates of H and VY can be obtained by plotting these equations in a 3-D plane along with the plane f (H,V) = 0.

For the length of the stretched cable, we start with the following expression.
As dx / ds and dz / ds are discontinuous functions at the points of application of the forces, we will have to find the length of each segment separately and sum it up.

Example
For the supporting example, a cable with a horizontal span h = 1400 m and a design dip d =115 m is analysed. The co-ordinates of the end points of this cable are assumed to:

(xY , zY) = (0,0); (xZ , zZ) = (1400, 0)

Therefore, the coordinates of the lowest point of this cable will be

(xB , zB) = (700, - 115)

A uniform vertical load of wL=110 kN/m is transferred to the cable by NH = 80 hanger cables.

In the derivation of the form finding equation for an inextensible cable, it was assumed that the cable is weightless. But to get a more accurate result, the weight of the cable is assumed to act as a constant external vertical force (distributed along the span of the bridge as opposed to the actual weight which is distributed along the cable arc length) while cables themselves are assumed to be weightless; that is, an external uniform force of wL* = wL + wC will be used for this inextensible analysis procedure. Being symmetric in shape, the maximum cable tension Tmax, inextensible will occur at the end points of the cable. By substituting the values for xY, zY and wL* in Eqn (6), the horizontal component of Tmax, inextensible is found. A factor of safety (FOS) of 2 has been implied to calculate the number of wires in a cable. The number of wires, the total area and the unit-weight of the cable, thus obtained, have been listed in Table 2. The initial unstrained length L0 of the cable, calculated using Eqn (8), is 1,424.79 m.

Next, we move on to the extensible analysis. In this case, the cable is acted upon by its self-weight wC and a number of points of forces. The hanger cables are equally spaced and therefore, the space between any two hanger cables is ?x=1400/80=17.5 m. Each hanger point force is equal to Fj = wL . ?x = 1925 kN. For each hanger position on an unstrained cable, the arc-length is found out by making use of the Eqn (8) at

The x and z co-ordinates of the strained configuration of the cable are given by Eqns (20) and (22), respectively. To use these equations, we need to determine the values of H and VY. As the cable, in the example, is symmetric, the vertical reaction VY (= VZ) at the end points is calculated considering the vertical equilibrium (only valid for symmetric cables, otherwise solutions will have to found out by plotting the planes as described earlier) as

If z = zZ and s = L0 are inserted in Eqn (22) along with the other constants, it turns out that VY is the solution for this equation which is independent of H. Further, the horizontal component H of the tension is found out by implying Eqn (22), which gives a non-linear equation for H. To easily obtain the solution for this equation, the horizontal component of inextensible cable tension is used as an initial approximate. The slope of the strained cable (steel cable) is plotted against the span of the bridge in Figure 4. Finally, the cable length is calculated using Eqn (23). The results from the inextensible and extensible analysis have been shown in Table 4.

Results and Discussions
Practicality of Extensible Cable Analysis
Figure 4 depicts the slope of the cable along the span of the bridge. As the jumps are very small, the theory can be used in practical situations. Elasticity of the cable can’t be ignored while designing the heavy load carrying cable systems due to the significant elongation and the resulting increase in the dip of the cable. Elongations due to the other factors like temperature changes, slipping of cable over the pulley, etc. have not been considered in the paper. The sag of the cables considering elasticity, along with the inextensible cable, has been plotted against the span of the bridge in Figure 5.

Also, extensible analysis results show a considerable reduction in the maximum tension in the cable with respect to inextensible analysis, as shown in Table 4. The variation of percentage change in the maximum tension between inextensible and extensible analysis with change in actual dip of the cable is shown in Figure 6. So, if cables are designed keeping their extensibility in mind, the area of the cables can be reduced by a good extent. As the bridge girders are hanged one at a time to the main rope, it allows enough time for the cable to elongate before the next girder is hanged, and hence the maximum inextensible tension is never reached.

CFRP vs. Steel Cable
As observed from Table 2, for the same external load, the unit weight of every CFRP cable is considerably less than that of the steel cable. Due to the lower self-weight, the total area of the cable and the maximum cable tension (Table 3) are significantly reduced as well. Table 3 shows the results obtained from the extensible and inextensible analysis of the cables. Due to the higher stiffness of the steel cable, its elongation and actual dip are the least. This difference is compensated by the much superior strength of the CFRP cables though. If various CFRP cables are compared, it turns out that CFCC, the thinnest wires manufactured by Tokyo Rope, is the most effective. The 5 mm CFCC wire cable can handle the same external load with the least total area and the self-weight. The cost of the CFRP wires still remains a problem, but they are expected to fall with the development of material technology and competitive market.

Form Finding of the Cable
Due to substantial elongation of the cable length, the lowest point on the centreline of the cable goes well beyond the prescribed design value when elasticity is considered. It was found that the actual dip varies linearly with the design dip (Figure 7). If the final intended dip (actual dip) is known, for example, 125 m, we can find the initial dip (design dip), 117 m for steel wire type, which should be provided so that the intended final cable shape is obtained. As the inextensible approach is followed to do the commencing analysis, the initial length L0, horizontal reaction H and tension Tmax, inextensible can be found out using parabolic equations.

Conclusions
Based on the results, the following conclusions may be drawn;
1. The classical theory presented by Irvine and Sinclair can be safely applied for the analysis of the suspension bridge cables under the static load and self-weight, keeping in mind that the sag to span ratio should be 1:8 or less. An extensible cable analysed under the action of point load gives a more accurate and realistic results. The elasticity of such cables cannot be ignored.
2. Considering the maximum extensible tension for designing the cable, instead of the maximum inextensible tension, can help to reduce the area of the cable. There is a significant change in the percentage change in maximum tension for the two kinds of analysis when the dips are shallow.
3. Due to the superior tensile properties and lower self-weight, a CFRP cable has proved to be a valuable alternative to the steel cables. The cost of CFRP wires still remains a problem, but it is expected to fall with the development of material technology and a competitive market. The 5 mm CFCC wire manufactures by Tokyo Rope Mfg. Co., Ltd. is best suited for the use in bridges due to the minimum cable diameter and self-weight.

References
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