Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 178 Structural elements Figure . Local and global coordinate systems structure since the local reference frames differ necessarily along the horizontal beam and along the legs. It is precisely this transformation which produces the coupling terms between bending and axial terms which arise in the K matrix. Let us consider the beam element of Figure which is assumed to lie in the Oxz plane. The local frame can be transformed into the global frame by the product of the translation IO and the plane rotation a. Neither the forces nor the displacements are modified by a translation which thus can be disregarded. Obviously the same is not true as far as the rotation is concerned. The rotation is described by the matrix R which relates the displacements by q R q . The transformation rule of the node labelled J is Xj Yj Zj ỶxJ ỶyJ ỶzJ cos a 0 sin a 0 0 0 0 1 0 0 0 0 sin a 0 cos a 0 0 0 0 0 0 cos a 0 sin a 0 0 0 0 1 0 0 0 0 sin a 0 cos a YJ zj K J ty J rzj_ More generally the rotation matrix takes the form R 0 . 0 R J Rx x mx nx with R y my ny _ zz mz nz _ Straight beam models Hamilton s principle 179 where Ĩ mịi Uịi are the director cosines of the local axes as expressed in the global system. Since energy is invariant through a transformation of axes the element matrices and force vectors are found to comply with the following conditions q qJ K n qi q J q q j K n q J K n R t K n R qi J M n qi qJ _ q qrj M n qi J M n R t M W R q qJ Qfl q qJ io I Q fl R T QI J J J J Once the element matrices and vectors are suitably transformed into the global system of coordinates the finite element model can be assembled according to the same rules as described in subsection . Colour Plate 3 shows the elastic response of a portal frame to a concentrated force as computed by using the finite element software CASTEM 2000 CAS 92 . The finite element model uses two nodes beam elements. The material is steel and the beams have uniform rectangular cross-sections
