Dynamic analysis of 3-D beam elements including warping and shear deformation effects

Abstract In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.

[1]  P. Waldron.  Elastic analysis of curved thin-walled girders including the effects of warping restraint , 1985 .

[2]  Yeong-bin Yang,et al.  A procedure for analysing space frames with partial warping restraint , 1984 .

[3]  Robert J. Reilly.  Stiffness Analysis of Grids Including Warping , 1972 .

[4]  Justín Murín,et al.  3D-beam element with continuous variation of the cross-sectional area , 2002 .

[5]  J. R. Hutchinson.  Shear coefficients for Timoshenko beam theory , 2001 .

[6]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[7]  Walter D. Pilkey,et al.  BEAM STIFFNESS MATRIX BASED ON THE ELASTICITY EQUATIONS , 1997 .

[8]  E. Sapountzakis,et al.  A BEM solution to transverse shear loading of beams , 2005 .

[9]  Walter D. Pilkey,et al.  On the shear deformation coefficient in beam theory , 1994 .

[10]  John T. Katsikadelis,et al.  Boundary Elements: Theory and Applications , 2002 .

[11]  John T. Katsikadelis,et al.  Non-linear dynamic analysis of beams with variable stiffness , 2004 .

[12]  V. G. Mokos,et al.  Warping shear stresses in nonuniform torsion by BEM , 2003 .

[13]  Richard H. Gallagher,et al.  Finite element analysis of torsional and torsional–flexural stability problems , 1970 .

[14]  R. Clough,et al.  Dynamics Of Structures , 1975 .

[15]  Evangelos J. Sapountzakis.  Solution of non-uniform torsion of bars by an integral equation method , 2000 .

[16]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[17]  C. Bach.  Elastizität und Festigkeit: Die für die Technik wichtigsten Sätze und deren erfahrungsmäßige Grundlage , 1889 .

[18]  Dr.-Ing. Dieter Stojek.  Zur Schubverformung im Biegebalken , 1964 .

[19]  N. G. Stephen.  Timoshenko’s Shear Coefficient From a Beam Subjected to Gravity Loading , 1980 .

[20]  Frank E. Weisgerber,et al.  Torsion constant for matrix analysis of structures including warping effect , 1996 .

[21]  Peter Waldron,et al.  Stiffness Analysis of Thin‐Walled Girders , 1986 .

[22]  A. N. Bercin,et al.  Finite element modelling of the coupled bending and torsional free vibration of uniform beams with an arbitrary cross-section , 1997 .

[23]  Aleksandar Prokić.  On triply coupled vibrations of thin-walled beams with arbitrary cross-section , 2005 .