Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Civil and Environmental Engineering


Eric M. Lui


Delta girder, finite element analysis, lateral-torsional buckling, residual stresses, shear capacity, torsional properties

Subject Categories



In steel structures, I-sections are commonly used for beams and columns. These cross-sections usually lack lateral rigidity and torsional stiffness. An effective method to improve their lateral rigidity and overall flexural resistance is to weld two inclined rectangular plates to the compression flange and the compression portion of the web of hot-rolled or welded I-section to form what is known as a Delta girder. This mixed cross-section, i.e., cross-section composed of an open profile attached to a closed profile, can provide enhanced torsional stiffness and hence noticeably higher lateral-torsional buckling (LTB) capacity for the beam. While Delta girders can be used for any beams, their main applications are the design of crane runway and bridge beams and strengthening of existing beams.

The main objectives of this dissertation are to study the static behavior of these girders and to provide a set of design equations for their nominal flexural and shear capacities. The research includes deriving closed-form equations for the cross-section properties of Delta girders. These equations are then verified against solutions obtained numerically. Using these cross-section properties, the theoretical lateral-torsional buckling capacity of Delta girders are determined and compared against results obtained from a finite element (FE) analysis. The results show that the theoretical LTB equation derived for general monosymmetric sections can be applied to these Delta girders. Additionally, it is shown that a simplified expression for the coefficient of monosymmetry βx derived for I-sections can be used in the computation of the elastic LTB capacity of Delta girders. A parametric study is then performed based on elastic LTB capacity to demonstrate the effectiveness of Delta girders in achieving a favorable capacity-to-weight ratio when compared to standard I-section members.

A refined three-dimensional (3D) nonlinear inelastic FE models are then developed to examine the capacity of simply-supported Delta girders under uniform bending and pure shear. The models take into considerations the effects of initial geometrical imperfections and residual stresses on the behavior of Delta girders. The FE model and the modeling techniques used are verified against the experimental result of a test beam that failed by inelastic LTB. The analysis covers a comprehensive range of Delta girder dimensions based on the dimensions of standard hot-rolled European H- and I-sections. A sensitivity study on the effects of using reduced imperfections magnitudes shows up to 18.2% increase in the LTB capacity of the girder.

Based on the FE LTB simulation results, it is shown that the buckling curve in the AISC (2016a) specifications overestimates the buckling capacity of Delta girders by an average of 9% and a maximum value of 21%. On the other hand, buckling curves “a” and “b” for rolled sections or equivalent welded sections case in the EuroCode 3 (2005) for Delta girders with d⁄bc ≤2 and d⁄bc >2, respectively, provide an average difference of only 2% and a maximum difference of 7% in comparison to the FE results. Hence, these two curves are recommended for the LTB design of Class 1 (compact) Delta girders. Additionally, design recommendations are provided for selecting the proper delta stiffeners dimensions based on the cross-section geometries of the corresponding I-sections.

Shear capacity equations for Class 1 (compact) Delta girders are proposed based on FE simulation results. The equations provide the option of selecting a conservative value that ignores strain hardening in the cross-section or a value that allows for some strain hardening to occur. In comparison to I-sections, the Delta girders analyzed in this study show an increase in shear capacity in the range of 41% to 89%. Furthermore, it is shown that in contrast to I-sections, yielding is a gradual process in Delta girders due to the presence of a non-uniform elastic shear stress distribution in the cross-section.


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