Capacity of Singly Symmetric Beam Column Jack-up Chord
Authors: Lewis David R. , Stock Doug , Marshall Peter 
Published: October 2008

Abstract
Limitations of, and improvements to, the beam-column design procedures of SNAME Bulletin 5-5 are described. Results from 25 inelastic FEM analyses are used to define the basecase failure surface for a LeTourneau tear-drop chord section of full bay length. A further 16 cases were run for k=0.5 and reverse curvature for establishing a more realistic upper bound on capacity. Supplementary studies were used to validate a knocked-down stress-strain curve, and to show that using base case results is conservative for more complex situations.

Introduction
The IADC Jack-up Committee has funded several projects to improve SNAME Technical and Research Bulletin 5-5A (ref. 1), including this project on the capacity of LeTourneau type chords which, among other things, addresses the conservatism in the beam-column formulation for singly symmetric sections. Based on preliminary results from this study (ref. 2), corrections and enabling language have been included in the second edition of SNAME 5-5.
The beam-column interaction equations in AISC-LRFD and SNAME ¶8.1.4.1 give a reasonable lower bound fit to the plastic interaction surface, for doubly symmetric column sections. For significant axial load (&gt;20% of capacity), the generic interaction is shown in Eq. (1).
(Mathematical equation available in full paper)
Normally the designer refers axial loads to the elastic neutral axes of a member for ease of correlation with global finite element analyses. However, with different yield strengths present in the cross section, maximum axial capacity occurs when the load is applied at a different location; this is the so-called center of squash. Moments from frame analysis must be adjusted to the new reference, being particularly careful to get the signs right.
Figure 1 shows a cut through the interaction envelop for axial load equal to half of the squash load (axial compressive force causing full yielding of the section). In the prescribed formula, off-axis bending is handled with the exponent "?". Linear addition of the two bending stresses (exponent of 1.0) applies to sections that fail as soon as they yield at the corners. Quadratic interaction (exponent of 2.0) would apply for circular sections. All other sections, doubly and singly symmetric (e.g. Type 4 Chords), are presumed to fall between these limits.
This formulation misses important aspects of jack-up leg chords having a singly symmetric "tear-drop" shape. In elastic stress-based design, negative Mx helps My, as it relieves axial compression on the back plate. Several large classes of jackup rigs have been successfully designed to exploit this behavior. The fully plastic section envelope also shows offaxis capacity exceeding that on the adjoining axis for this quadrant. The inelastic behavior of realistic beam columns falls between the elastic and plastic section envelopes, reflecting important inelastic P-delta effects.
SNAME Bulletin 5-5A essentially prescribes linear addition of the two bending effects, with only a little relief at the bulge, when the exponent is derived from their Figure 8.4. The basic form of the mandated interaction equation is incapable of giving extra off-axis capacity, even with an exponent of infinity.