## 1 Introduction  
As typical connection members between arch ribs and bridge decks, hangers of arch bridges are required to be robust enough to sustain dead and live loads. According to the magnitude of the flexural rigidity, the hangers of arch bridges are divided into flexible and rigid hangers [1, 2, 3]. As typical flexible hangers, steel wires and steel rebars are extensively applied in deck-type arch bridges to coordinate the diverse mechanical properties between ribs and decks. However, flexible hangers are unable to provide essential rigidity and stability to adapt to the requirements of small deformation and vibration. Recently, rigid hangers have been more commonly applied in half-through arch bridges, especially in high-speed railway arch bridges, to reduce the dynamic responses of railway bridges under cyclic loads, such as winds, earthquakes and running trains [4, 5, 6, 7, 8, 9, 10]. Initially, concrete or prestressed concrete was used to produce rigid hangers. However, concretes frequently suffer from cracking, assembly and replacement problems. Recently, profile steels, such as rectangular and H-shaped steels, have been used extensively to manufacture rigid hangers due to their high installation and maintenance efficiency [11; 12]. As replacements for traditional cables, rigid steel hangers have been considered specimens with infinite fatigue lives for very small vibration and stress ranges, and the wind- or train-induced fatigue lives for cable-stayed arch bridges have been within the design life requirements [13; 14; 15]. Compared to flexible hangers, rigid hangers sustain more complicated stress under transportation loads and environmental actions due to their considerable cross-sectional stiffness. Among all hangers of whole arch bridges, rigid hangers near the roots of arch ribs have to suffer from higher stress with obvious multiaxial properties due to their short lengths [16; 17]. Under the action of dynamic live loads, the shortest rigid hangers are subjected to repeated actions of tensile, shear and bending stresses, inducing a higher fatigue-cracking risk. However, the most likely positions of fatigue cracking for the shortest rigid hangers are located at the connections between the hangers and bridge-deck beams [18]. The reason for this risk is the more serious and complicated multiple-axle stress field due to the special geometric profile and short lengths of the connected hangers. To execute fatigue assessments with significant multiple-axle properties, various multiple-axle fatigue methods have evolved and been applied based on different assumptions or theories. Three typical methods have been established to predict the fatigue life by transforming the complex stress or strain state into a simple equivalent stress or strain. These three typical methods are the static-strength method, the equivalent energy method and the critical plane method [19; 20]. Among them, the critical plane method has been extensively applied in actual projects because of its specific definition of fatigue damage, and the simplification of the critical stress state. According to their different assumptions for failure rules, fatigue damage functions were built by using tensile or shear-related critical planes, where fatigue-failure is dependent on the evolution or accumulation of tensile or shear stress amplitudes. In this paper, a multiaxial fatigue evaluation method was first proposed based on the critical-damage plane methodology by applying the shearing stress in the critical-damage plane as the fatigue parameter, where the critical plane is assumed to be vertical to the maximum nominal stress. Then, fatigue assessment can be achieved by using the Basquin equation to convert this multiaxial fatigue problem into a fatigue problem of pure torsion.