【正文】 元素的高度 u， v，θ 局部自由度 圖 5 管（圓柱表面）元素的參數(shù) 圖 6 圓柱表面單元矩陣 混合單元剛度矩陣 由于相比于外徑和高度相對較低的徑向厚度，在外圈的運動方式是和其內(nèi)表面裝載的管子相似的。該管的基本自由度以及主要參數(shù)顯示如圖 5。其代表性是以圓柱表面的基本公式[15]為基礎(chǔ)。 對于我們軸承，重要的是要考慮到一個具體的圓柱表面彎曲，以及由一個徑向力（或壓力）造成的徑向位移。這種根據(jù)羅基 [15]如圖 6所示的元素的剛度矩陣很普遍。在圖 6中，所有的表達方式 kij都使用 R， T， L參數(shù)（圖 5）， E電子楊氏模量和 泊松數(shù)字來表達。 混合元素的剛度矩陣是以圓柱表面元素公式為基礎(chǔ)的。為了準確的在軸向剛度建模，與 39 圓柱表面單元矩陣的拉伸力的表達方式對應的行和列已被等效剛度的梁的公式所取代。因為輕微的影響力，所以連接表達方式設(shè)置為零，正如數(shù)值試驗表明。在圖 6（給予部分坐標）采用的坐標轉(zhuǎn)換程序根據(jù)整個模型的坐標系統(tǒng)和編號矩陣圖控制的原因提出來的。 在全球 CS的管狀物元素矩陣的拉伸自由度是（行和列）六方面。該矩陣轉(zhuǎn)換得到的最后形式是如圖 7 介紹的雜交元矩陣。 在新的條件下截面積 Ap 和以前提出的橫截面計算使用改進的 RAS 穆森公式 [14]是相等的。 此外，為了考慮負載點的應用高度，總軸向剛度（或相反的靈活性）必須用不同的元素在非均勻模式下來分配，正如在 。 圖 7 管狀物元素矩陣轉(zhuǎn)化成雜交元矩陣 上部靈活的 Sp1 銻螺栓的靈活性 下部靈活的 Sp2 40 圖 8 螺栓裝配示意圖 考慮外在負載應用程序的起源 正 如 Guillot[9]和最近以來的張 [10]所示的外負載應用這個地方 ,對螺栓裝配行為 ,計算拉力的標準及帶有螺紋部件的彎曲瞬間補充度有極其重大影響。對于一個軸向載荷 ,螺栓的裝配可以按照圖 8所示來代表。 圖 9 實際區(qū)域的壓縮 41 圖 10 自適應的靈活性 眾所周知 ,和初始狀態(tài)的預加負荷 Q比較 ,外力導致螺栓受力增大。 螺栓所受總力 Fb為 Fb=Q+Sp2*Fe/(Sp+Sb) (5) 全部零件的靈活性 Sp=Sp1+Sp2 (6) 是什么讓零件的靈活性不均勻分布的厚度的計算復雜化了，事實上 ,在壓縮條件下的頭螺栓的模樣 ,取決于裝配的水平 ,看起來就像一個體積接近于被切去頂端的形狀的圓錐 (圖9)。 符合標準的實際情況是通過合理的算法來計算一個壓縮零件的靈活性。 零件可由兩個或多個分區(qū)分開?？紤]一個兩部分組裝零件隔斷案例 (圖 10),這個方法 如下 : 通過改良的拉斯穆森的 [14]計算橫截面面積 Ap,然后全部零件的靈活性。 Ap?Sp=Lp/ApEp (7) 42 Finite Elements in Analysis and Design 42 (2021) 298–313 Bolted joints for very large bearings— numerical model development Aurelian Vadean ?, Dimitri Leray , Jean Guillot aDepartment of Mechanical Engineering, Ecole Polytechnique de Montreal, . Box 6079, Station CentreVille,Montreal, Qu233。bec, Canada H3C 3A7 bLaboratoire de Genie Mecanique de Toulouse COSAM, INSA Toulouse, 135 Avenue de Rangueil,Toulouse Cedex 4, 31077, France Abstract The conventional theory of bolted joints does not take the plexity of external loads into account, neither its related joint stiffness nor the contact nonlinearities. This article deals with a 2D numerical model allowing fast and precise calculation of the fastening bolts for very large diameter bearings subjected to an overturning moment. The originality of the modelling lies in the use of a particular ?nite element that behaves like a ring, except in the axial direction. Its axial stiffness is the local stiffness that governs the behaviour of the bolted assembly. The model was tuned upon 3D ?nite elements simulations and provides excellent results for several types of bearings. Keywords: Bolted joints。 Numerical model。 Slewing bearings。 Finite elements analysis 1. Introduction Providing fast and accurate results is one of the challenges of practical engineering mainly during the early stages of the design process. The manufacturers of different mechanical systems involving bolted joints need suitable calculation models that allow integrated solutions. Numerous models approximate the parts and bolt stiffness using cones,spheres, equivalent cylinders or other analytical models [1–4]. According to the conventional theory, which was originally developed for loads that are centred or slightly offcentre, the stiffness of the member is constant. However, ?nite elements simulations as well as experimental results showstrong nonlinearities due to the changing contact area [5–7]with the external load. Stiffness nonlinearities were studied by Grosse [8] and Guillot [9] and they propose a nonlinear model but only for platelike con?gurations. Another weakness of the conventional theory lies in the way the socalled load factor is calculated. The load factor tries to measure the amount of the external force which is transmitted to the bolt. The location where external forces are applied on themember governs the load factor and the way themember stiffness is distributed. Zhang [10] developed a new analyticmodel of bolted joints and takes into consideration the stiffness reduction associated with the residual force on the assembly, pression deformation caused by external force and dimensions changing due 43 to member rotation. This model has its limitation and is not applicable to bolted assemblies when the members are of different geometry or when the external forces are not symmetric about the member interface. The speci?cmodel we are proposing for large bearings can serve as base for a genericmodel of circular ?anges which can take into account the different nonlinearities as well as different con?gurations or geometries by appropriate stiffness distribution. 2. The slewing bearings The model this article proposes is suitable for speci?c bolted joints for large diameter bearings. These large bearings (up to 13m(43 ft)) also called ―slewing bearings‖, are used for cranes, radar dishes, tunnelboring machines, etc. The two bearing rings are clamped to the main frame by preloaded high strength bolts. One or both rings are provided with gear teeth to enable the swing drive to operate. The connection is like a thick and narrow cylindrical ?ange on a very rigid frame fastened with a large number of bolts. Another particularity of the system is the important and variable overturning moment. The bearings are subjected to radial and axial loads of same importance. The three types of bearings under study, ball bearings, crossedroller bearings and threerow roller bearings, are presented in Fig. 1. Due to the plexity of structure and the particularity of bolted joint loading, neither traditional models nor nonlinear models are appropriate. Thus we have developed a new model that takes into