Calfem - A finite element toolbox to MATLAB, version 3.3

eam2tsTwo dimensional Timoshenko beam elementThe evaluation of the section forces is based on the solutions of the basic equationsEA d2 ūd¯x 2 + q¯x = 0EI d3 θd¯x 3 − q ȳ = 0EI d4¯vd¯x 4 − q ȳ = 0(The equations are valid if qȳ is not more than a linear function of ¯x). From theseequations, the displacements along the beam element are obtained as the sum of thehomogeneous and the particular solutionswhereandu =⎡⎢⎣ū(¯x)¯v(¯x)θ(¯x)⎤⎥⎦ = u h + u pu h = ¯N C −1 G a e u p =¯N =⎡⎢⎣⎡⎢⎣ū p (¯x)¯v p (¯x)θ p (¯x)1 ¯x 0 0 0 00 0 1 ¯x ¯x 2 ¯x 30 0 0 1 2x 3(x 2 + 2α)⎤⎤⎥⎦ =⎡⎢⎣q¯x L¯x2EA (1 − ¯x L )qȳL 2¯x 224EI (1 − ¯x L )2 +q ȳL¯x(1 − ¯x 2GA k s L )⎥⎦ α = EIGA k sqȳL 2¯x12EI (1 − 2¯x L )(1 − ¯x L )⎤⎥⎦⎡C =⎢⎣1 0 0 0 0 00 0 1 0 0 00 0 0 1 0 6α1 L 0 0 0 00 0 1 L L 2 L 30 0 0 1 2L 3(L 2 + 2α)⎤⎥⎦⎡a e =⎢⎣⎤u 1u 2⎥. ⎦u 6The transformation matrix G and nodal displacements a e are described in beam2e.Note that the transpose of a e is stored in ed.Finally the section forces are obtained fromN = EA dūd¯xV = GA k s ( d¯vd¯x − θ)dθM = EId¯xELEMENT 5.6 – 10