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Publication Number: FHWA-HRT-05-083
Date: August 2007

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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.

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The following basic cable properties are required:

T0, T1= Lower and upper cable tensions, respectively (kN, kips)
H= Horizontal component of the cable tension (kN, kips)
m= Cable mass per unit length (kg/m, kips/ft)
A= Total cross-sectional area of cable strands (mm2, inch2)
E= Cable modulus of elasticity (MPa, ksi)
σUTS= Cable yield stress–ultimate tensile stress (MPa, ksi)
h= Cable rise (m, ft)
L= Cable run (m, ft)
s= Cable sag (m, ft)

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

The equation reads lowercase W is equal to lowercase M times lowercase G.
(88)

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

The equation reads L superscript star is equal to the square root of lowercase H squared plus L squared.
(89)

cable chord length (m, ft); and

The equation reads theta is equal to arc tangent open parentheses lowercase H over L close parentheses.
(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:

The equation reads lowercase Y prime is equal to differential in lowercase Y over differential in lowercase X is equal to the hyperbolic sine of open parentheses lowercase W over H times lowercase X plus K subscript 1 close parentheses.
(91)

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

The equation reads lowercase Y is equal to H over lowercase W times the hyperbolic cosine of open parentheses lowercase W over H times lowercase X plus K subscript 1 close parentheses plus K subscript 2.
(92)

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

x = 0 , y = 0

x = L, y = h

Thus as shown in equations 93 and 94:
The equation reads K subscript 1 is equal to arc hyperbolic sine of open parentheses lowercase W times lowercase H all over 2 times H times the hyperbolic sine of the sum of lowercase W times L over 2 times H close parentheses minus the sum of lowercase W times L over 2 times H.
(93)

The equation reads K subscript 2 is equal to negative H over lowercase W times the hyperbolic cosine of open parentheses K subscript 1 close parentheses.
(94)

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

The equation reads C is equal to H over lowercase W open parentheses hyperbolic sine of  open parentheses lowercase W times L over H plus K subscript 1 close parentheses minus hyperbolic sine of open parentheses K subscript 1 close parentheses, close parentheses.
(95)

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

Lower slope (rad):

The equation reads theta subscript 0 is equal to arc tangent of open parentheses hyperbolic sine of open parentheses K subscript 1 close parentheses, close parentheses.
(96)

Upper slope (rad):

The equation reads theta subscript 1 is equal to arc tangent open parentheses hyperbolic sine of open parentheses hyperbolic sine of open parentheses lowercase W times L over H plus K subscript 1 close parentheses, close parentheses.
(97)

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

Lower end reaction (kN, kips):

The equation reads V subscript 0 is equal to H times open parentheses tangent open parentheses theta subscript 0 close parentheses, close parentheses.
(98)

Upper end reaction (kN, kips):

The equation reads V subscript 1 is equal to H times open parentheses tangent open parentheses theta subscript 1 close parentheses, close parentheses.
(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:

The equation reads V subscript 1 minus V subscript 0 is equal to lowercase W times C.
(100)

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

Lower tension (kN, kips):

The equation reads T subscript 0 is equal to H times open parentheses secant open parentheses theta subscript 0 close parentheses, close parentheses.
(101)

Upper tension (kN, kips):

The equation reads T subscript 1 is equal to H times open parentheses secant open parentheses theta subscript 1 close parentheses, close parentheses.
(102)

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

The equation reads sigma subscript axial is equal to T subscript 1 over A.
(103)

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

The equation reads Ratio subscript U-T-S is equal to sigma subscript axial over sigma subscript U-T-S.
(104)

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

The equation reads delta is equal to H times L over A times E that sum times open bracket lowercase W times lowercase H squared over 2 times H times L all times the hyperbolic cotangent of open parentheses lowercase W times L over 2 times H plus 1 over 2 plus H over 2 times lowercase W times L times the hyperbolic sine of open parentheses lowercase W times L over H close parentheses, close parentheses, close bracket.
(105)

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

The equation reads USL subscript 0 is equal to C minus delta.
(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:

The equation reads lowercase R is equal to lowercase W times L divided by 2 times H is less than or equal to 0.5.
(107)

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

The equation reads the hyperbolic sine of open parentheses lowercase X close parentheses is equal to lowercase X plus lowercase X superscript 3 over 6 plus lowercase X superscript 5 over 120 plus et cetera, et cetera, et cetera.
(108)

The equation reads the hyperbolic cosine of open parentheses lowercase X close parentheses is equal to 1 plus lowercase X superscript 2 over 2 plus lowercase X superscript 4 over 24 plus lowercase X superscript 6 over 720 plus et cetera, et cetera, et cetera.
(109)

The equation reads the arc hyperbolic cotangent of open parentheses lowercase X close parentheses is equal to 1 over lowercase X plus lowercase X over 3 minus lowercase X superscript 3 over 45 plus 2 times lowercase X superscript 5 over 945 minus et cetera, et cetera, et cetera.
(110)

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

The equation reads C is equal to open parentheses L superscript star close parentheses, open bracket 1 plus the cosine of theta squared over 6 times lowercase R squared plus open parentheses the cosine of theta superscript 2 over 45 minus the cosine of theta superscript 4 over 72 close parentheses times lowercase R superscript 4 plus et cetera, et cetera, et cetera close parentheses.
(111)

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

The equation reads theta subscript 0 is equal to the arc tangent of open bracket lowercase H over L minus lowercase R times lowercase S over L plus lowercase H times lowercase R squared over 3 times L minus lowercase S times the cosine of theta squared over 6 times L times lowercase R superscript 3 minus lowercase H times lowercase R superscript 4 over 45 times L et cetera, et cetera, et cetera close bracket.
(112)

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

The equation reads theta subscript 1 is equal to the arc tangent of open bracket lowercase H over L plus lowercase R times lowercase S over L plus lowercase H times lowercase R squared over 3 times L plus lowercase S times the cosine of theta squared over 6 times L times lowercase R superscript 3 minus lowercase H times lowercase R superscript 4 over 45 times L et cetera, et cetera, et cetera close bracket.
(113)

The cable elongation is defined as shown in equation 114:

The equation reads delta is equal to H times L times secant theta squared over A times E times open bracket 1 plus lowercase R superscript 3 over 3 minus lowercase R superscript 4 over 45 times open parentheses 1 minus 4 times L squared over open parentheses L superscript star close parentheses squared close parentheses etc, etc, etc close bracket.
(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.

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

The equation reads omega subscript lowercase I is equal to lowercase I times pi over L times the square root of T over lowercase M.
(115)

which yields the natural circular frequencies in radians/s.

The more practical form (equation 116):

The equation reads lowercase F subscript lowercase I is equal to lowercase I over 2 times L times the square root of T over lowercase M.
(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)

The equation reads lowercase X superscript star is equal to lowercase x times secant theta plus lowercase Z times sine theta, and lowercase Z superscript star is equal to lowercase Z times cosine theta, and L superscript star is equal to L times secant theta, and H superscript star is equal to H times secant theta.
(117)

so that (equation 118)

The equation reads lowercase Z bar is equal to 1 over 2 times lowercase X bar times open parentheses 1 minus lowercase X bar close parentheses, open bracket 1 minus varepsilon bar over 3 times open parentheses 1 minus 2 times lowercase X bar close parentheses, close bracket.
(118)

where (equations 119 through 121):

The equation reads lowercase Z bar is equal to lowercase Z star over the quotient of lowercase MG times L star squared times the cosine of theta over H star.
(119)

The equation reads lowercase X bar is equal to lowercase X star over L star.
(120)

The equation reads varepsilon bar is equal to lowercase M-G times open parentheses L star close parentheses times the sine of theta all over H star.
(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:

The equation reads omega bar subscript lowercase N is equal to lowercase N times pi, where lowercase N is equal to 1 comma 2 comma 3 comma, et cetera, et cetera, et cetera.
(122)

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

The equation reads omega bar subscript lowercase N is equal to 2 times lowercase N times pi, where lowercase N is equal to comma 2 comma 3 comma, et cetera, et cetera, et cetera.
(123)

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

The equation reads the tangent of omega bar subscript lowercase N over 2 is equal to omega bar subscript lowercase N over 2 minus 4 over open parentheses lambda star close parentheses squared times open parentheses omega bar subscript lowercase N over 2 close parentheses superscript 3.
(124)

For all of the above equations:

The equation reads omega bar subscript lowercase N is equal to open parentheses omega subscript lowercase N star close parentheses times open parentheses L star close parentheses all over the square root of H star over lowercase M.
(125)
The equation reads open parentheses lambda star close parentheses squared is equal to open parentheses lowercase MG times open parentheses L star close parentheses times the cosine of theta over H star close parentheses squared times open parentheses L star over H times open parentheses L subscript lowercase E star close parentheses over E-A, close parentheses.
(126)

in which (equation 127):

The equation reads L subscript lowercase E star is equal to open parentheses L star close parentheses, open bracket 1 plus 1 over 8 times open parentheses lowercase M-G times open parentheses L star close parentheses times the cosine of theta over H star close parentheses squared, close bracket.
(127)

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

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(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:

The equation reads tangent of open parentheses omega bar times A plus the quantity B over omega bar close parentheses is equal to a negative C divided by omega bar.
(129)

For symmetric in-plane modes:

The equation reads tangent of open parentheses omega bar times A plus the quantity B over omega bar close parentheses is equal to omega bar divided by D.
(130)

where:

The equation reads omega bar squared is equal to omega squared times lowercase L over the sum of 2 times lowercase G times the tangent of psi subscript 0.
(131)

The equation reads the tangent of psi subscript 0 is equal to 4 times lowercase S over lowercase L divided by the sum of 1 minus open parentheses 2 times lowercase S over lowercase L close parentheses squared.
(132)

The equation reads L is equal to lowercase L times open parentheses arc sine of lowercase H close parentheses times open parentheses tangent of psi subscript 0 close parentheses all divided by tangent of psi subscript 0.
(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)

The equation reads lowercase F subscript lowercase N is equal to lambda over pi times the square root of lowercase G over 8 times lowercase S.
(134)

where:

For asymmetric modes (equation 135):

The equation reads lambda is equal to lowercase N times pi where lowercase N is equal to comma 2 comma 3 comma, et cetera, et cetera, et cetera.
(135)

For symmetric modes (equation 136):

The equation reads lambda minus tangent of lambda all divided by lambda superscript 3 is equal to GAMMA and GAMMA is equal to lowercase M times L all divided by the quantity 128 times E times A open parentheses lowercase S divided by L close parentheses to the power 3 times cosine to the power 6 of theta.
(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.

Table 18. Stay cable property comparison.
BridgeCable TypeAverage √T /m
Alex Fraser Helical twist parallel wire231
Maysville Grouted 7 wire parallel strand188
Owensboro Grouted 7 wire parallel strand201

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

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

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