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• Appendix G. Introduction to Mechanics of Inclined Cables

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Appendix G. Introduction to Mechanics of Inclined Cables

STAY CABLE STATIC PROPERTIES

This section provides mathematical descriptions of the various properties of a typical stay cable. Exact mathematical functions are given and then power series approximations (for relatively tight cables) are provided for ease of computation and to allow checking of exact equations.

A typical stay cable is shown in figure 147:

Figure 147. Drawing. Incline stay cable properties.

The following basic cable properties are required:

The following properties can then be calculated, as shown in equations 88 through 90:

(88)

cable weight per unit length (kN/m, kips/ft) (g = acceleration due to gravity);

(89)

cable chord length (m, ft); and

(90)

angle of chord to horizontal.

Catenary Equations (Exact Solutions)

Coordinates of cable nodes are defined using the parameters described above and the catenary equations for this cable.⁽⁵⁷⁾

From the free body diagram for any point on the cable, the slope is shown in equation 91:

(91)

Integrating this expression yields the equation for the cable coordinates (equation 92):

(92)

The constants K1 and K2 can be obtained from evaluating the boundary conditions:

x = 0 , y = 0

x = L, y = h

(93)

(94)

The length of the cable, C (m, ft), is then shown in equation 95:

(95)

The slopes at the ends of the cables are then shown in equations 96 and 97:

Lower slope (rad):

(96)

Upper slope (rad):

(97)

Following from these angles, the cable end vertical reactions are shown in equations 98 and 99:

Lower end reaction (kN, kips):

(98)

Upper end reaction (kN, kips):

(99)

By way of verification, it may be checked that the difference in the cable end vertical reactions should equal the mass of the cable as shown in equation 100:

(100)

The cable end tensions are shown in equations 101 and 102:

Lower tension (kN, kips):

(101)

Upper tension (kN, kips):

(102)

The maximum cable axial stress (MPa, ksi) is shown in equation 103:

(103)

The maximum ratio of the ultimate tensile stress utilized is then shown in equation 104:

(104)

The elastic elongation of the cable is (m, ft) (equation 105):

(105)

Finally, the unstressed length of the cable (m, ft) is defined as shown in equation 106:

(106)

Power Series Approximations to Exact Solutions

It may be more convenient to express the exact hyperbolic relationships in terms of power series expansions of the hyperbolic functions. This is particularly effective for tight cables, defined as those for which have the relationship defined in equation 107:

(107)

The relationships in equations 108 through 110 are used and terms up to and including r⁴ are retained:

(108)

(109)

(110)

Then, the cable length C is defined as shown in equation 111:

(111)

The lower end slope is defined as shown in equation 112:

(112)

The upper end slope is defined as shown in equation 113:

(113)

The cable elongation is defined as shown in equation 114:

(114)

STAY CABLE DYNAMIC PROPERTIES

Dynamic properties of stay cables may be examined in two ways:

• The “taut string” simplified approximation (figure 148A).
• The theoretically correct dynamic properties for an inclined cable (figure 148B).

Figure 148. Drawing. Definition diagram for a horizontal cable (taut string), compared to the definition diagram for an inclined cable.

Taut String

For the taut string, it is assumed the sag of the cable is small compared to its length, the cable is perfectly elastic, and the cable is inextensible. The accuracy of the taut string approximation is usually adequate for practical purposes.

The natural frequencies, ω_i, of stay cables can be approximated (often accurately enough) using the expressions for a taut string (figure 148A) as shown in equation 115: ⁽⁵⁸⁾

(115)

which yields the natural circular frequencies in radians/s.

The more practical form (equation 116):

(116)

yields natural frequencies in Hz.

Inclined Cable

In the case of an inclined cable (figure 148B), it is convenient to deal with transformed coordinates x^* measured along the cable chord from point A) and z^* (perpendicular distance from the chord to the profile), rather than with coordinates x and z. After performing the necessary transformations we obtain equation 117:⁽⁵⁹⁾

(117)

so that (equation 118)

(118)

where (equations 119 through 121):

(119)

(120)

(121)

Solving the equations of motion using the above transformed axes, the natural frequencies of the out-of-plane modes are given by equation 122:

(122)

and those of the antisymmetric in-plane modes are shown in equation 123:

(123)

Frequencies of symmetric in-plane modes are given by the following transcendental equation (equation 124:

(124)

For all of the above equations:

(125)

(126)

in which (equation 127):

(127)

is a quantity usually only slightly greater than the length of the cable chord itself. Substituting into the transcendental equation yields the equation 128, the roots of which are the natural frequencies in question:

(128)

Equations shown are of general form and applicability. As special cases, they contain results for a horizontal cable (θ = 0°) and a vertical cable (θ = 90°).

Formulation Including Sag

More complicated expressions exist for cables with larger sag ratios such as the main cables of suspension bridges. For such cables, natural frequencies can be obtained by solving the following frequency equations (equations 129 and 130):⁽⁵⁹⁾

For antisymmetric in-plane modes:

(129)

For symmetric in-plane modes:

(130)

where:

(131)

(132)

(133)

is the span of the cable, while l is the length of the catenary hanging between the supports, and s is the cable sag. A, B, C, and D are parameters which depend on the value of Ψ₀ .

Natural frequencies of cable stays with larger sag can be estimated using equation 134:⁽⁶⁰⁾

(134)

where:

For asymmetric modes (equation 135):

(135)

For symmetric modes (equation 136):

(136)

Full-Scale Cable Properties

For many stay systems the term √T /m is more or less constant as the cables are usually designed to a maximum working stress of 0.45 X GUTS (guaranteed ultimate tensile stress). The term, √T /m , has been calculated for the stay cables of three bridges as listed in table 18 and shown in figure 149. The first two modes of vibration as a function of cable unstressed length (USL) are shown in figure 150.

Figure 149. Graph. Cable √T/ m versus cable unstressed length: Summary of Alex Fraser, Maysville, and Owensboro bridges.

Figure 150. Graph. Cable frequency versus cable unstressed length: Summary of Alex Fraser, Maysville, and Owensboro bridges.

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