Keywords: warping function, secondary torsional moment deformation effect, secondary torsion constant, stiffness matrix, boundary element method.

One of the problems often encountered in engineering practice is the analysis of space frames or grid systems subjected to transverse, bending or twisting loading. However, accurate analysis of three-dimensional beam structures is difficult to achieve for three reasons. First, commercial programs in general (with very few exceptions) consider six degrees of freedom at each node of a member of a space frame, ignoring in this way the torsional warping effects due to the corresponding restraint at the ends of the member. Several researchers tried to overcome this inaccuracy by developing a 14x14 member stiffness matrix including additional warping degrees of freedom at the ends of the member. Secondly, the aforementioned commercial programs are unable to compute accurately the shear correction factors and the torsion constant of the members' cross section, thus often ignoring shear deformations arising from shear forces. Though these deformations are quite small in most civil engineering applications, they may be dominant in short span beams. Finally, commercial beam finite element packages completely ignore shear deformations due to secondary torsional moments.

In this paper a boundary element method is developed for the construction of an extended 14x14 stiffness matrix and the corresponding nodal load vector of a member of arbitrary doubly symmetric constant cross-section taking into account both torsional warping and shear deformation effects arising from shear forces and secondary torsional moments. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors employing a strain energy approach. Eight boundary value problems with respect to the variable along the beam total angle of twist, to the primary and secondary torsional warping functions, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure boundary element method approach, where only boundary discretization is used. The evaluation of the shear deformation coefficients is accomplished from the aforementioned warping and stress functions.

Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the method developed. The main conclusions that can be drawn from this investigation are:

a)

In bars of closed shaped cross sections the secondary torsional moment deformation effect has an important influence and should be considered in the analysis.

b)

The validity of the results obtained from the proposed model compared with those obtained from a three-dimensional finite element solution using shell or solid finite elements is remarkable.

c)

The discrepancy of the deflections and the internal stress resultants arising from the ignorance of the warping degrees of freedom at the ends of a member necessitates the utilization of the 14x14 member stiffness matrix.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £130 +P&P)