Tie Rod-Equivalent Non-Linear Constitutive Law for Uniformly Loaded Cables

Cables are typically used in engineering applications as tensile members. Relevant examples are the main cables of suspension bridges, the stays of cable-stayed bridges, the load-bearing and stabilizing cables of tensile structures, the anchor cables of floating mooring structures, the guy-ropes for ship masts, towers, and wind turbines, the copper cables of electrical power lines. Since cables are characterized by non-linear behavior, analysis of cable structures often requires advanced techniques, like non-linear FEM, able to consider geometric non-linearity. Nevertheless, a traditional simplified approach consists in replacing the cable with an equivalent tie rod, characterized by a suitable non-linear constitutive law. Currently used equivalent constitutive laws have been derived by Dischinger, Ernst and Irvine. Since the equivalence is restricted to taut cables, characterized by small sag to chord ratios, these traditional formulae are not appropriate for uniformly loaded sagging cables: the main cables of suspension bridges are a particularly emblematic case. Despite some recent attempts to find more refined solutions, the problem is still open, since closed form solutions of general validity are not available. In the paper, general analytical formulae of the non-linear constitutive law of the equivalent tie rod are proposed, distinguishing two relevant cases, according as the length of the cable can vary or not. The expressions, derived by applying the general form of the theorem of virtual work, can be applied independently on the material, on the sag to chord ratio, on the load intensity and on the stress level, so allowing the replacement of the whole cable with a single equivalent tie rod. The expressions are critically discussed referring to a wide parametric study also in comparison with the existing formulae, stressing the influence of the most relevant parameters.

A Novel Shape Finding Method for the Main Cable of Suspension Bridge Using Nonlinear Finite Element Approach

The determination of the final cable shape under the self-weight of the suspension bridge enables its safe construction and operation. Most existing studies solve the cable shape segment-by-segment in the Lagrangian coordinate system. This paper develops a novel shape finding method for the main cable of suspension bridge using nonlinear finite element approach with Eulerian description. The governing differential equations for a three-dimensional spatial main cable is developed before a one-dimensional linear shape function is introduced to solve the cable shape utilizing the Newton iteration method. The proposed method can be readily reduced to solve the two-dimensional parallel cable shape. Two iteration layers are required for the proposed method. The shape finding process has no need for the information of the cable material or cross section using the present technique. The commonly used segmental catenary method is compared with the present method using three cases study, i.e., a 1666-m-main-span earth-anchored suspension bridge with 2D parallel and 3D spatial main cables as well as a 300-m-main-span self-anchored suspension bridge with 3D spatial main cables. Numerical studies and iteration results show that the proposed shape finding technique is sufficiently accurate and operationally convenient to achieve the target configuration of the main cable.

STABILITY ANALYSIS OF THE GRAVITY ANCHORAGE OF A SUSPENSION BRIDGE BASED ON LARGE-SCALE FIELD TESTS

The stability of gravity anchorages is critical to the safe operation of suspension bridges. The purpose of this paper is to study the stability of a gravity anchorage under the tension of main cables. Large-scale load tests and direct shear tests under the nature state and the saturated state were carried out in the adit at the bottom of the gravity anchorage, because the bottom surface of the gravity anchorage is below the design water level of a reservoir to be built. Representative rock samples in the anchorage area were selected for indoor tests. The basic parameters of the rock mass and the characteristic value of subsoil bearing capacity were obtained based on the field and indoor tests results. The static theory was used to calculate the anti-sliding stability coefficient, anti- overturning stability coefficient and contact stress of the gravity anchorage. The finite element software was used to calculate the horizontal displacement, vertical settlement and contact stress of the gravity anchorage. The calculation results met the relevant requirements, and it can be considered that the gravity anchorage was stable under the tension of main cables. It would provide a reference for the stability study of the gravity anchorage of suspension bridges.

Probabilistic Assessment of the Safety of Main Cables for Long-Span Suspension Bridges considering Corrosion Effects

This paper presents a reliability analysis to assess the safety of corroded main cables of a long-span suspension bridge. A multiscale probability model was established for the resistance of the main cables considering the length effect and the Daniels effect. Corrosion effects were considered in the wire scale by relating the test results from accelerated corrosion tests to the corrosion stages and in the cable scale by adopting a corrosion stage distribution of the main cable section in NCHRP Report 534. The load effects of temperature, wind load, and traffic load were obtained by solving a finite element model with inputs from in-service monitoring data. The so-obtained reliability index of the main cables reduces significantly after operation for over 50 years and falls below the design target value due to corrosion effects on the mechanical properties of the steel wire. Multiple measures should be taken to delay the corrosion effects and ensure the safety of the main cables in the design service life.

Parametric analysis of the dynamic characteristics of a long-span three-tower self-anchored suspension bridge with a composite girder

Three-tower self-anchored suspension bridge (TSSB) is more and more favored because of its beautiful structure and strong adaptability to terrain and geological conditions. However, there are few engineering practices and related researches on super long-span three-tower self-anchored suspension bridges. A three-dimensional finite element model for the Fenghuang Yellow River Bridge, with the world’s longest span of its kind under construction, is established using the ANSYS finite element program, and the structural dynamic characteristics of the super long-span TSSB are studied and compared with those of several bridges of the same type or with similar spans. In addition, the influence of the key design parameters such as the stiffening girder stiffness, tower stiffness, main cable and suspender stiffness, central buckle, and longitudinal constraint system on the dynamic characteristics of the structure is analyzed. The results show that the first mode of the TSSB is longitudinal floating, the lower-order modes are dominated by vertical bending modes, while the higher-order modes are primarily vibration modes of the main cables, and the torsional modes exhibit strong coupling with the lateral sway of the towers and main cables. The frequency of the first antisymmetric vertical bending mode of the TSSB has an inversely proportional relationship with the main span length. Compared with a double-tower ground-anchored suspension bridge and cable-stayed bridge with similar spans, the TSSB has the lowest frequency for the first antisymmetric vertical bending mode and the highest frequency for the first symmetric vertical bending mode, with a more pronounced coupling with the towers and main cables in the torsional modes. Analysis of the structural parameters shows that the frequencies of the longitudinal floating mode, first antisymmetric vertical bending mode, first symmetric vertical bending mode, and first torsional mode are most sensitive to the longitudinal bending stiffness of the side tower, central buckle, vertical bending stiffness of the stiffening girder, and torsional stiffness of the stiffening girder, respectively. The research findings and relevant conclusions can provide basic data for response analysis of long-span TSSBs under dynamic loads and offer an engineering reference for the design of similar bridges around the world.