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Com © Springer-Verlag Berlin Heidelberg 2010 40 3 Friction in Beam-to-Beam Contact In the Lagrange multipliers method the friction forces are defined in the following way FTm = λTm , FTs = λTs . 4) Therefore, the introduction of friction in this method requires two additional unknowns – the Lagrange multipliers, λTm and λTs, for each pair of contacting beam edges or axes. It is worth to note, that the friction forces FTm and FTs do not represent a force pair of action-reaction type. They are rather two independent quantities related to two independently considered displacements along two contacting beams.

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.

22) Usually, the location of contact points is determined using directly the local coordinates. But in the case of contact between beams with rectangular cross-sections where the local co-ordinates are related to the beam axis and the contact points lie on the edges, it is more convenient to use the position vectors. Such an approach was suggested in the paper by Agelet de Saracibar (1997). Besides, independently of the cross-section shape, such a formulation allows for an easier treatment of the case, when the contact point moves from one finite element to another.

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