By Przemyslaw Litewka
Phenomena happening in the course of a touch of 2 our bodies are encountered in lifestyle. in fact virtually all types of movement is expounded to frictional touch among a relocating physique and a flooring. in addition, modeling of easy and extra advanced methods as nailing, slicing, vacuum urgent, stream of machines and their parts, rolling or, ultimately, a numerical simulation of vehicle crash checks, calls for taking touch into consideration. hence, its research has been a subject matter of many study efforts for a very long time now. notwithstanding, it's author’s opinion that there are really few efforts concerning touch among structural components, like beams, plates or shells. the aim of this paintings is to fill this hole. It matters the beam-to-beam touch as a selected case of the 3D solids touch. A numerical formula of frictional touch for beams with shapes of cross-section is derived. additional, a few powerful tools for modeling of delicate curves representing beam axes are provided. part of the ebook can also be dedicated to learn a few facets of thermo-electro-mechanical coupling in touch of thermal and electrical conductors. Analyses in each bankruptcy are illustrated with numerical examples displaying the functionality of derived touch finite elements.
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Additional info for Finite Element Analysis of Beam-to-Beam Contact
To solve this strongly non-linear problem the incremental-iterative Newton-Raphson method can be applied. Its use requires calculation of the linearisation of the weak form. The linearisation of the additional contact terms leads to the following formulae: ΔδΠ εN = ε N Δg N δg N + ε N g N Δδg N , ΔδΠ λN = δλ N Δg N + Δλ N δg N + λ N Δδg N . 23) provide the residual vector and the tangent stiffness matrix, respectively, for the contact finite element. 6). 24) Δx sn = x sn,s Δξ sn + Δu sn . After some basic operations the necessary kinematic variables for the normal contact can be expressed as (Wriggers and Zavarise, 1997) Δg N = (Δu mn − Δu sn ) o n , δg N = (δu mn − δu sn ) o n , Δδg N = (Δδu mn − Δδu sn ) o n + (δu mn,m Δξ mn − δu sn,s Δξ sn )o n + + (Δu mn,mδξ mn − Δu sn,sδξ sn )o n + + (x mn,mm Δξ mnδξ mn − x sn,ss Δξ snδξ sn )o n + + 1 (δu mn + x mn,mδξ mn − δu sn − x sn,sδξ sn ) gN × (1 − n ⊗ n )(Δu mn + x mn,m Δξ mn − Δu sn − x sn,s Δξ sn ).
53) ( ) − e p s ε Tm g Tmn (F ) t 2 T − x mn − x mpf e p s ε Ts gTsn (F ) t 2 T x sn − x spf p s ε Tm FT t (1 − t m ⊗ t m ) S m − (F ) t 2 (F T t ) o t m t mT S m T ⎫ Δu ⎪⎡ M ⎤ FT (1 − t s ⊗ t s ) S s ⎬⎢ ⎥. 54) ⎡Δu M ⎤ Δp s = Ps ⎢ ⎥. ⎢⎣ Δu S ⎥⎦ Now all the required variables are discretised. 31) are used. 3. In this way the residual vector and the tangent stiffness matrix related to friction for the beam-to-beam contact finite element can be derived. 6 the total vectors and matrices for frictional contact can be found.
58) and the auxiliary vectors are R 2 = S mT t m , R 3 = S sT t s . 61) while for the approach with one resultant friction force – pε = με N g N pm s m R 2 , R Tm pε R Ts = με N g N p s s s R 3 . 65) x sn,ss Fs Fs + Z s + Z s T T − t s T x sn,s (a sm R m + a ss R s ) + Ws . In Eq. 4) is used. 68) for the friction forces considered as the components of one resultant friction force. 45). In the Lagrange multipliers metod, after introduction of friction, the vector of unknowns has three more elements – one Lagrange multiplier for the normal contact – λN, and two multipliers for friction – λTm and λTs.