Large sag with a bending stiffness catenary is a subject that draws attention in the realm of fatigue analysis, estimation of suspension cable sag for bridge cable hoisting, and ocean engineering of the employment of mooring systems. However, the bending stiffness is the cause of boundary layers at the anchorage of cables, thereby finding a solution of the differential equation can be extremely difficult. Previous studies have tackled this problem with the perturbation method; yet, due to the complexity of the matching process and solution finding, the method might not be an ideal solution for engineering applications. Moreover, the finite difference method and the finite element method in numerical analysis can often be ineffective because of inappropriate parameter configuration and the drastic variation of functions in the boundary layers. Therefore, this study proposed a novel bending moment expression of a large sag catenary. The expression was derived from the sag identified using bending moment equations, and a solution was found by applying the WKB method (Wentzel-Kramers-Brillouin method) to overcome the complex problem of boundary layers. Consequently, a simple solution of various mechanical properties in a cable with bending stiffness and large sag could be obtained.

Slim tension members have been comprehensibly applied to civil and ocean engineering to create cables of cable-stayed and suspension bridges that satisfy traffic demands and mooring systems that satisfy the demands of mining deep-sea petrol and natural gas. Such structural problems could be understood as general cable problems where the mechanical behavior of cables varies significantly with the magnitude of tension. In the study of cables of bridges, researchers often regard the ratio of midpoint sag to span as an appropriate standard for simplifying the parabola theory. When the ratio of midpoint sag to span is 1/8, the horizontal force happens to be the total cable weight. When the ratio of midpoint sag to span is smaller than 1/8 (horizontal force greater than the total cable weight), the resulting parabola is appropriate and correct. However, when the ratio of midpoint sag to span is greater than 1/8 (horizontal force smaller than the total cable weight), a greater error can be found in the parabola [

The problem, however, of incorporating the bending stiffness effect in cables results from the complex differential equations. Bending stiffness is the cause of boundary layers in anchorages and rapid variations of bending moments occur near the differential regions of the cable anchorage. Burgess [

In dealing with the boundary layer problems, the perturbation method consists of two main techniques [

To correct the weaknesses of a conventional perturbation method, this study proposed a novel catenary bending moment equation. The core idea originated from the comparison between two catenary models, which consisted of anchorages with the same horizontal distance and vertical elevation. The self-weight and horizontal force between the two models were also identical. The only difference is that one model contains bending stiffness but the other does not. The bending moment solution of the model with a bending stiffness catenary can be expressed by that of the model without a bending stiffness catenary. By winding around the difficulty of a conventional perturbation method, which requires a fourth order differential equation for finding the solution, the proposed equation could directly identify cable sag with the help of bending moment equations.

To satisfy the demands of different cable forces and to verify the correctness of the proposed bending moment equation, two respective second order differential equations of parabola and catenary incorporating bending stiffness were established and discussed. The parabola was expressed by a small sag linear moment-curvature relation, and the solution of the parabola was used to verify the catenary result. The catenary was expressed by a large sag nonlinear moment-curvature relation; moreover, the WKB method (Wentzel-Kramers-Brillouin method is named after physicists Wentzel, Kramers, and Brillouin, who all developed it in 1926.) in the perturbation method [

Figure

Free-body of static equilibrium of catenaries without bending stiffness.

Consider

As shown in Figure

Free-body of static equilibrium of a suspended cable with its self-weight distributed uniformly without bending stiffness.

The boundary condition is set to the hinges at the two ends and the solution can be expressed by a parabola. Consider

The two hinges are at the same elevation level (

Equation (

As shown in Figure

Free-body of static equilibrium of catenaries with bending stiffness.

The horizontal tensile force

The equilibrium of cable segments yields

Compilation yields

The moment-curvature relation of a large-sag cable is

Due to the difficulty in finding a solution to (

Dimensionless parameter

Boundary conditions of hinges (

The two hinges are at the same elevation level (

Dimensionless bending moment equation of hinges on a horizontal cable is

Bending moments applied with compression on the upper part of the cable are regarded as positive. The substitution of the fixed-end boundary condition shows that the slope of the hinge pivot is not equal to zero but to the slope of the cable (

For a horizontal parabola (

The dimensionless bending moment equation for the fixed-end on a horizontal cable is

Equations (

To overcome the difficulty of finding a solution using (

The comparison model between number 1 and number 2 catenaries.

Based on (

The mechanical relation existing between number 1 and number 2 catenaries was independent and indirect. The two catenaries had two definitely different shapes,

Mathematically, (

By integrating (

Another integration could obtain the bending moment function,

Equation (

Constants of integration

The dimensionless bending moment equation is

Equation (

This study used the parabola to verify (

The correctness of (

The moment-curvature relation of the large sag catenary is

Given that (

The large horizontal tensile force and small bending stiffness make

First, the homogeneous solution of (

The WKB approximation method developed by Wentzel, Kramers, and Brillouin [

In the classical Sturm-Liouville equation [

Equation (

The result of

The final result is

The total sag is

For a horizontal catenary (

A solution could be found by substituting (

The substitution of (

The substitution of (

Boundary conditions of fixed-ends (

Substitute (

Substitute (

Substitute (

Figure

Comparisons between the total sag of parabolas and catenaries.

Fundamental mechanics show that the difference between (

First the influence of bending moment sag,

Define the ratio of span midpoint bending moment sag to hinged catenaries corresponding to division of (

Under four different

The ratio of span midpoint bending moment sag to hinged catenaries (

Define the ratio of span midpoint bending moment sag to fixed-end catenaries corresponding to division of (

Figure

The ratio of span midpoint bending moment sag to fixed-end catenaries (

Define the ratio of span midpoint bending moment sag of parabolas to catenaries corresponding to division of (

The ratio differences between bending stiffness and solutions of parabolas and catenaries were compared using (

The ratio differences between bending stiffness and solutions of parabolas and catenaries.

Figure

The comparison between the parabola and catenary fixed-end bending moments (

Define the ratio of fixed-end bending moments of parabolas to catenaries corresponding to division of (

Figure

The ratios of influences of bending stiffness on fixed-end bending moments.

Figure

The distribution of parabola and catenary bending moments.

This study proposed a novel large sag catenary bending moment expression that allows finding the sag directly from the bending moment equation and finding the WKB catenary solution using the WKB method. The matching of cable and beam segments is no longer necessary, thereby overcoming the complexity of boundary layers. Moreover, with only a single equation, the proposed method could simultaneously describe the drastic function variations inside and outside the boundary layer. The method provides a simple calculation of cables with bending stiffness and large sag and fulfills the engineering needs for tensile cable fatigue stress analysis and estimation of suspension cable sag for bridge hoisting.

Statistical analysis revealed a significantly different influence on the behavior of cables with bending stiffness given that the boundary condition was hinges or fixed-ends. Bending stiffness was regarded as effective when

The authors declare that there is no conflict of interests regarding the publication of this paper.