【正文】 de?nition of an original “hybrid” ?nite element has the general behaviour of a ring except for the axial direction where its stiffness is related to the local stiffness that governs the behaviour of the bolted assembly. Meshing the ring with several elements allows taking into account nonlinear stiffness distribution in the assembly and in particular the effect of the load application height. The evolving contact area is modelled by contact springs and usingan iterative solving technique.Fig. 1. Slewing bearings.3. Modelling and assumptionsThe speci?c loading on these systems is an offcentre axial load (a normal crane loading type) resulting in a large overturning moment. This will build up an internal load on the bearings grooves (as shown in Fig. 1). At this stage of development, the intensity of the external load is not important. It is suf?cient to apply the same loading on both models: the 2D numerical model and the 3D ?nite elements model used to tune the ?rst one up.To build the numerical model, several simpli?cations were made:? for modelling purposes, we consider only the most loaded bolt and the associated sector。? the outer ring only is modelled. Thus the external forces are replaced by the rolling elements load asan equivalent load which increase the working load on bolts。? the loading as well the formulation of the speci?c elements are considered axisymmetric。? the mounting is considered extremely rigid.Fig. 2 presents the principle underlying the new modelling. On the lefthand side is the sketch of the numerical model and on the righthand side is the equivalent ?nite elements model.Fig. 2. Modelling principle.As Fig. 2 shows, the bearing ring model consists of three types of elements:a. The plate elements based on the circularplate model as described byVadean [11] and developed from Roark’s analytical formulas [12]. They are axisymmetric elements with two DOFs/node (y translation and z rotation) and their role is? to represent the ring bending according to the axial direction OY。? to characterize displacements and particularly the boundary separation of the ring from its mounting. Coupled to springs elements they are able to model the variable contact zone between the ring and the mounting according to the preload installed and the external load applied. Consequently to different contact status, the stiffness matrix will be adjusted and a nonlinear loading of the threaded element is produced.b. The socalled hybrid elements which make it possible to take into account the part pression stiffness, as well as the speci?c bending stiffness of a tube along radial direction OX. The three DOFs per node enable the structure to be loaded with a force system equivalent to the external load Fe.c. The spring elements that model the contact with the mounting. They characterize the elastic behaviour of the interface and the unilateral contact. Springs stiffness will be a parameter of the model tuning.The bolt has the formulation of an equivalent beam as described later in this paper.. Determining the axial stiffness of the bearing sectorIn order to calculate the axial stiffness of the bearing sector we have used the improvement made by MASSOL [13] to the formulation of Rasmussen [14] for an elementary cylindrical assembly (Fig. 3).The calculation of the equivalent section, noted Ap, makes it possible to determine the Kp stiffness of the parts. The relations used are （1）Fig. 3. Dimensions of an elementary assembly.Fig. 4. Dimensions for the axial stiffness sector calculation.with the following dimensionless quantities: In the case of our bearings, the sections are not circular. Overall dimensions X and Y are considered as indicated in Fig. 4.If the diameterDp=3*Da (3)is not inscribed in the sector section, the following expression is used:Dp=(x+y)/2 (4)The total axial stiffness of the sector is calculated with Eqs. (1) and (2) considering the length lp equal to the height of the bearing ring. Axial stiffness Kp as well as the equivalent section Ap is thus obtained considering the whole angular sector.. Hybrid elementsAs shown in Fig. 2, the bearing ring is meshed using three hybrid elements in relation with the three main parts: one element is assigned for the part between the upper surface of the ring and the load application origin on the raceway。 a second element is assigned for the reduced section determined by the bearing raceway and a third for the lower part between the raceway and the mounting. The intermediate nodes face the rolling elements contact points, so that the external force is applied to one of them.r – average radius of elementt – radial thickness of elementL height of elementu, v, θ – DOFs in local CSFig. 5. Parameters of tube (cylindrical shell) element.Fig. 6. Cylindrical shell element matrix.. Hybrid element stiffness matrixDue to the relatively lowradial thickness pared to the outer diameter and to the height, the behaviour of the outer ring is similar to a tube which is loaded on its inner surface. The degrees of freedom of an elementary tube as well as the principal parameters are showed in Fig. 5. Its representation is based on a general formulation of a cylindrical shell [15].For our bearing, it is important to take the speci?c bending of a cylindrical shell into account, as well as the radial displacements, produced by a radial force (or load ponent). The general stiffness matrix of such element according to Rockey [15] is shown in Fig. 6, where all terms kij are expressed using ther, t , L parameters (Fig. 5), E—Young’s modulus and —Poisson number.The stiffness matrix of the hybrid element is based on the fo