**1. Modeling of tapered elements**

Stability calculation of tapered members is always a difficult problem despite its popularity in steel hall construction. ConSteel had very efficient modeling and analysis tools providing an easy way for the structural design of such models. For the stability analysis the segmented uniform beam element method was used where a member with I or H cross section and with variable web depth is divided into

*n*segments and the depth of each segment is taken equal to the real depth measured at the middle of the segment. The lengths of the segments were taken equal, except at both ends where additional shorter segments are added in order the better approximate the real depth of the elements to be modeled. Each segment was modeled with a traditional 7 DOF beam finite element of uniform cross section. Such model captures correctly the in-plane displacements, but cannot consider accurately the additional torsion coming from the axial stresses due to warping in the flanges which are not parallel with the reference line in case of tapered elements. Therefore this approximation may cause slight inaccuracy in calculating buckling modes involving torsional displacements like flexural-torsional buckling of columns or lateral-torsional buckling of beams especially in such cases where the beam flanges are largely unrestrained.Modelling web tapered elements with a sequence of segments of constant depth2. New analysis model for tapered members in ConSteel 11 SP1 |

In order to improve further the accuracy of the stability analysis of structural models including tapered members in ConSteel 11 SP1 a new tapered finite element has been introduced. A basics of this unique finite element have been published just recently by more researchers however ConSteel as a pioneer, is the only commercial software which implemented it into the buckling analysis. The mentioned problems arising from the non-parallel flanges can be fixed by considering appropriate additional terms in the element stiffness matrix. The final stiffness matrix can be written as a sum of original stiffness matrix and the additional terms:

Where

**is for the original stiffness matrix with uniform cross section and***K*_{S}**contains the additional terms valid for double an monosymmetric I and H cross sections.***K*_{T}
The additional terms in

**use the following special cross section parameters:***K*_{T}
Where I

_{flzT }and I_{flzB }are the intertias of the flange related to*z*axis, for upper (*) and bottom (*_{T}*) flanges, respectively*_{B}*,**a*and a_{T }*is the distance between the centerline of upper and lower flange and the line parallel with the reference axis of the element and passing through the shear center of the middle cross section, as seen on the picture below for double symmetric I and H cross section.*_{B}Definition of a_{T }and a_{B }in case of double symmetric I and H sections |

Additionally d

_{aT}/d_{x }and d_{aB}/d_{x }mean the angle between the upper and lower flanges and the line parallel with the reference line of the element and passing through the shear center of the middle cross section. As an approximation these can be expressed as:
where

*a*_{flT }and*a*_{flB }are the angles between the flanges and the element reference line, ẟ_{shear}is the angle between the lines passing through the centers of gravity and shear centers of the extreme cross sections of the elements.Definition of a_{flT }and a_{flB} in case of double symmetric I and H sections |

**3. Comparison of results**

**This part shows some validation examples for the accuracy of the implemented new finite element compared to published numerical results and analysis by shell elements. The examples show the very high accuracy of this element even in the most challenging buckling cases where the segmented uniform beam element method yields some extent of inaccuracy.**

**3.1 Tapered cantilever**

Tapered cantilever with welded I section. The initial height is 600 mm and the end height is the 300 mm. The length of the cantilever is variable. The critical force is calculated by the ConSteel software using tapered csBeam7 tap model. The ConSteel results are compared to the reference Ansys shell models [B. Asgarian, M. Soltani, Lateral –Torsional Buckling of Non-Prismatic Thin—Walled Beams with Non-Symmetric Cross Section. Procedia Engineering 14 (2011) 1653-1644] and other numerical solutions [Andrade A, Camotim D. Lateral-torsional buckling of singly symmetric tapered beams: Theory and applications. Journal of Engineering Mechanics 2005; 131(6):586–97. ]. Additionally results obtained with SABRE2 (1) software have also been provided.

Fig. Tapered cantilever with welded I section (equal flanges) |

Fig. tapered csBeam7 model |

**Summary of results:**

**3.2 Yang and Yau & Andrade and Camotim simply-supported web-tapered member**

ConSteel tapered beam (7tap) results are compared with results publshed in (2)

**Pcr forces [kip] in function of tapering factor α**

**References**

1: SABRE2 software Dr Donald w. White

© 2017 Georgia Institute of Technology http://www.white.ce.gatech.edu/node/24

2: Improved Design Assessment of LTB of I-Section Members via Modern Computational Methods, 2016 NASCC D. White, Georgia Tech