Yield interaction surface of an enhanced LeTourneau 116C chord

Marine Structures 14 (2001) 379}396

Yield interaction surface of an enhancedLeTourneau 116C chord

Peter Marshall, David Lewis*, Doug StockLewis Engineering Group, 11530 Galbreath Drive, Houston, TX 77066, USA

Abstract

Limitations of, and improvements to, the beam-column design procedures of SNAMEBulletin 5-5 are described. Results from 25 inelastic FEM analyses are used to de"ne thebase-case failure surface for a LeTourneau tear-drop chord section of non-zero length. Applica-tion to two design examples is demonstrated. Supplementary studies were used to de"nea knocked-down stress}strain curve, and to show that using base case results is conservative formore complex situations. � 2001 Elsevier Science Ltd. All rights reserved.

Keywords: O!shore; Jack-up; MODU; Beam-columns; Failure surface; Box sections

1. Introduction

The IADC Jack-up Committee was organised after several existing MODUs wererejected on the basis of SNAME Technical and Research Bulletin 5-5 [1], whereasthese same units had been evaluated and accepted under prior standards, using thesame environmental and site conditions. The committee has funded several projects toidentify the sources of unexpected conservatism and improve Bulletin 5-5, includingthis project on the capacity of LeTourneau-type chords, to deal with conservatism inthe beam-column formulation for singly symmetric sections. Based on preliminaryresults from this study [2], corrections and enabling language have been made for thesecond edition of SNAME 5-5.

The beam-column interaction equations given in AISC-LRFD and SNAMEB8.1.4.1 are a reasonable lower bound "t to the plastic interaction surface, for doublysymmetric column sections. For signi"cant axial load ('20% of capacity), the generic

* Corresponding author. Tel.: #1-281-397-9037; fax: #1-281-440-9645.

0951-8339/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 9 5 1 - 8 3 3 9 ( 0 0 ) 0 0 0 5 5 - 1

Fig. 1. `Type 4a chord interation curves for P/P�"0.5.

interaction equation is

�P

��

) P��#

8

9��M

��

) M��

�#�

M��

��

) M��

��

��.

Normally, the designer refers axial loads to the elastic neutral axes of a member forease of correlation with global "nite element analyses. However, with di!erent yieldstrengths present in the cross-section, maximum axial load is applied at a di!erentlocation; this is the so-called centre of squash. Moments from frame analysis must beadjusted to the new reference, being particularly careful to get the signs right.

Fig. 1 shows a cut through the interaction envelop for axial load equal to half of thesquash load. In the prescribed formula, o!-axis bending is handled with the exponent`�a. Linear addition of the two bending stresses (exponent of 1.0) applies to sectionsthat fail as soon as they yield at the corners. Quadratic interaction (exponent of 2.0)would apply for circular sections. All other doubly symmetric sections, such as Type4 chord, are presumed to fall between these limits.

This formulation misses important aspects of jack-up leg chords having an asym-metric `tear-dropa shape. In elastic stress-based design, negative M

�helps M

�, as it

relieves axial compression on the back plate. Several large classes of jack-up rigs havebeen successfully designed to exploit this behaviour. The fully plastic section envelopealso shows o!-axis capacity exceeding that on the adjoining axis for this quadrant.The inelastic behaviour of realistic beam columns falls between the elastic and plasticsection envelopes, re#ecting important inelastic P-delta e!ects.

SNAME Bulletin 5-5 essentially prescribes linear addition of the two bendinge!ects, with only a little relief at the bulge, when the exponent is derived from theirFig. 8.4. The basic form of the mandated interaction equation is incapable of givingextra o!-axis capacity, even with an exponent of in"nity. It also denies obvious

380 P. Marshall et al. / Marine Structures 14 (2001) 379}396

asymmetry in the M�

behaviour when axial load is present. The shaded area showsunnecessary and inconsistent conservatism in the simpli"ed LRFD equations.

If the stakes are large enough, we should be willing to do a rigorous analysis,avoiding the conservatism that accompanies approximation. The reliability formatbehind LRFD design codes is easily adapted to more complex failure algorithms, anddoes not require closed-form equations.

Following presentation of the foregoing results at the Jack-Up Conference in 1997,several changes were made to Bulletin 5-5, as follows:

� B8.1.4.1 would get enabling language for alternative more advanced methods in lieuof the interaction equations given, not just for the determination of `�a.

� B8.1.4.7 would liberalise the determination of `�a.� BC8.1.4.1 would acknowledge that favourable interaction between M

�and

M�

cannot be reproduced by the prescribed `designa equation, and that recourse tothe actual plastic interaction surface is advisable.

� BC8.1.4.7 `alternativelya would expand description of how such direct recourse isto be carried out, and mention the desirability of considering elasto-plastic bucklingwhen kL/r'0, as follows:

&&At present, Figs. C8.1.8}C8.1.11 cover only fully plastic section consider-ations, and their use for a beam-column member is based on the assumptionthat the member being evaluated is su$ciently short/compact that elasto-plastic stability (buckling at large strains) is not a consideration. Violating thisassumption may lead to errors on the unsafe side. Updated informationcovering elasto-plastic stability may be generated in the future, and shouldpreferentially be used for member evaluations''.

This paper describes application of this new enabling language, by developingand using the beam-column plastic failure interaction surface of an EnhancedLeTourneau-Type 116C Chord.

2. Finite element studies

The detailed "nite element analyses reported here were performed at DigitalStructures Inc., Berkeley, California, and described at length in a separate report (inpreparation).

2.1. FEM software, mesh, and solution strategy

Analytical modelling was performed using the 20-node quadratic brick elementwith reduced integration (C3D20R) available in the ABAQUS program. These arehigher-order isoparametric elements, and can achieve good results with a relativelycoarse mesh. Calibration with a box column used 10, 20, and 30 layers for a symmetrichalf-column, with 12}16 elements per layer. The main study used 19 layers with 49brick elements per layer. Additional eight-node elements model the "llet welds. It isclear from the results achieved that the solution strategy was able to capture the

P. Marshall et al. / Marine Structures 14 (2001) 379}396 381

beam-column P-delta e!ect, together with non-linear stress}strain behaviour.The models were loaded by gradually increasing axial load until failure (i.e. nofurther increase in load could be sustained). The solution was stable upto thepoint of maximum load, well into the non-linear range, but the post-bucklingunloading was not captured. For the base case, k"1 and Cm"1, pinnedend columns with eccentric axial loads at the ends are used, resulting in proportionalloading to failure.

2.2. Dexnition of stress}strain curve to reproduce the nominal column curve

Two types of steel are used in the LeTourneau-type 116 chords. The rack has 85 ksiSMYS (speci"ed minimum yield strength, assumed to be at 0.2% o!set) and 97 ksiUTS (ultimate tensile strength, corresponding strain not measured but assumed to be12%). The rest of the section is lower strength, 70 ksi SMYS and 80 ksi UTS.Examination of mill test data from the manufacturer con"rmed that these steels hadrounded stress}strain curves and a low UTS/YS ratio.

Three types of stress}strain curves were generated for each steel. The idealisedelasto-plastic is comparable to "nite element studies in which no attempt is made toreproduce practical column behaviour, as in#uenced by residual stresses, imperfec-tions, and other factors. In lieu of actually measuring residual stresses and imperfec-tions, the tangent modulus approach was adopted to reproduce the SNAME columncurve, following the precedents of Refs. [3}5]. This results in a stress}strain curvehaving the following characteristics: Initial modulus of about 0.88E, to reproduce theknockdown in elastic buckling. Continuously reducing tangent modulus above 44%of yield, e.g. 0.88E at 0.44F

�, 0.66E at 0.66F

�, and 0.44E at 0.88F

�. Passes through

yield at 0.2% strain. Strain hardens to UTS at about 12% strain. Its de"nition isfurther described in Fig. 2. A "nal type of stress}strain curve with bilinear strain-hardening, steeper initially (which should be bene"cial) and derived from mill test datawas used in two of the "nite element analyses. Stock's detailed report also discussessome di!erences between the uniaxial stress}strain curve from a tension test, and thethree-dimensional #ow rule used in the "nite elements. Detailed discussions of theseareas are outside the scope of this paper.

2.3. Calibration and demonstration results covering the foregoing

The foregoing knocked-down stress}strain curve mimics the e!ects of residualstresses and geometric imperfections, to reproduce the SNAME column curve in thelambda range of 0.25}1.5. Lambda is the non-dimensional slenderness (unity whereEuler crosses yield). Fig. 3 shows the results of the calibration study. The axiallyloaded column results are for a homogeneous box section of a moderate yield strengthsteel (70 ksi SMYS with strain hardening to 105 ksi at 3% strain). The FEM resultsclosely follow the target SNAME column curve, and lie between SSRC curve 1 (whichincludes 100-ksi Q&T welded box sections) and SSRC curve 2 (which includes weldedA-36 box columns). The higher results at low lambda show the bene"cial e!ect ofstrain hardening, also observed in prior research data. The slightly lower results at


Fig. 2. Knock-down material stress}strain curve.

Fig. 3. Axial loaded columns.


Table 1Calculated plastic capacities with (1) classical mechanics and "nite element method, (2) elasto-plastic andmodi"ed non-linear stress}strain behaviour, (3) pure moment cases with zero applied axial load

Method�� Major-axismoment

Major-axismoment�

453 biaxialmoment

Minor-axismoment

1203 biaxialmoment�

1353 Biaxialmoment

cam-ep� M�

100,500 67,300 48,069 67,536M

�42,900 64,638 53,721 43,437

Resultant M

100,500 79,810 64,638 72,087 80,566fea-ep M

�102,173 108,407 70,252 68,431

M�

44,781 64,569 43,617Resultant M

102,173 100,407 83,311 65,469 81,150

fea-nl M�

106,220 72,673 51,600M

�46,320 66,938 56,910

Resultant M

106,220 86,180 66,938 76,820 86,180Ratiosfea-ep/cam-ep 1.017 * 1.044 1.013 * 1.007fea-nl/fea-ep 1.040 * 1.034 l.022 * 1.062fea-nl/cam-ep 1.057 * 1.080 1.036 1.066 1.070

�Moment orientation in degrees is de"ned with respect to Dyer's interaction diagram (Fig. 5).�cam * classical applied mechanics, fea * "nite elementanalysis, ep * least-plastic without P-delta.�This M

�is determined from a chord length equal to two times the basic length.

�With pure axial load, P�*12,460 kips for the cam-ep method.

�Biaxial moment, theoretical value is based on Dyer's interaction diagram (Fig. 5).

high lambda may be the result of using L/3000 eccentricity in addition to the knockeddown stress}strain curve.

2.4. Pure axial and bending loads

Results for pure axial loading of the LeTourneau-type 116C tear-drop section arealso shown in Fig. 3. These have the more realistic bilinear strain hardening describedearlier. At the low lambda's of interest, these also lie above the SNAME curve, whenload is applied at the squash centre; thus we see little or no detrimental e!ect frominitial elastic curvature. However, when load is applied at the elastic centre, earlyyielding in the back plate produces signi"cantly lower capacity. Therefore, the squashcentre has been adopted as the reference point for subsequent studies of theserelatively short beam-columns.

Pure bending results are given in Table 1. Finite element results using the idealelasto-plastic stress}strain curve are 1.7}4.4% higher than the corresponding resultsfrom classical applied mechanics, suggesting some arti"cial sti!ness due to theelement formulation, mesh size, or #ow rule. Much of this error, due to end conditionsdisappears when a longer chord section is used. The initially softened `realisticastress}strain curves with strain hardening show a further 2.3}4.0% increase over "niteelement results using the ideal elasto-plastic stress}strain curve. Since this is due tostrain hardening, it represents usable extra capacity. The bottom line is 4.6}8.0%more bending capacity than classical applied mechanics.


Fig. 4. Deformed shape base case.

2.5. Base case interaction surface for one kL/r

Type 116 chord was modelled at its actual length, 134 in (3.4 m) node to node.A good clean reference base case is with proportional loading, k"1, and Cm"1, i.e.single curvature eccentrically loaded columns. Doing six directional slices through theinteraction surface, each at three load levels, makes 25 inelastic FEM analysis cases,including the pure axial and pure moment cases. Closer spacing of the slices was usedin the favourable quadrant (compression on the rack) to de"ne more detail near thepeak of the elastic interaction surface.

A deformed plot of the "nite element model of the symmetric half-column is shownin Fig. 4. The plane of symmetry is at the origin, and loads are applied via a 2-in-thicksti! element layer at the far end. The lateral de#ections are substantially larger thanwould be calculated for the elastic P-delta e!ect, increasing the original eccentricitiesand moments by 22%. The de#ections shown corresponds to about 2% uniformbending strain about each axis, just prior to ultimate collapse.

Target loading eccentricities were selected to match Dyer's plastic section interac-tion surface (without column buckling), as shown in Fig. 5 for the LeTourneau-type116 chord [6]. This surface is non-dimensionalised to unity at the following limitingcapacities:

plastic moment about the X-axis 100,500 kip-in,plastic moment about the>-axis 64,000 kip-in,axial load at the squash centre 12,460 kip.


Fig. 5. Biaxial moment vectors on interactive curves (after dyer) and corresponding directions on chordsections.

Cuts through the interaction surface were made at normalised angles of 0, 45, 90,120, 135, and 1803 (03 represents positive bending about the strong X-axis, rack intension). The normalised 1203 direction is very close to the peak of the favourableinteraction quadrant. `Directiona is the moment vector as plotted on the interactionsurface (with respect to the squash centre) normalised to plastic moment capacitiesabout the principal axes. Actual loading directions vary, as the two principal axeshave substantially di!erent bending capacities. As computed from the direction ofload eccentricities, the actual angles are, respectively, 0, 32.5, 90, 132.2, 147.5, and 1803.De#ections and orientation of the neutral axis go o! in yet other directions, emphasis-ing the weak axis. All results are referenced to the original normalised directions. Foreach of these six directions, pure bending (0% axial) and axial load eccentricitiestargeted for 30, 70 and 90% of the squash load were selected, as shown in Table 2.

Numerical results for the eccentric axial load cases are tabulated in Table 3. Failureis de"ned as the last numerically stable load increment, using small load increments inthe face of rapidly increasing de#ections. The failure load for these relatively stocky


Table 2Finite element analyses load matrix

Direction (deg) Target load (kip) Speci"ed eccentricity (in)

ey Ex

000 3738 !24.08 0.008722 !4.03 0.00

11,214 !0.99 0.00045 3738 !14.00 8.92

8722 !2.65 1.6911,214 !0.63 0.40

090 3738 0.00 14.908722 0.00 3.01

11,214 0.00 0.69120 3738 13.96 15.40

8722 3.63 4.0011,214 0.90 0.99

135 3738 19.00 12.108722 4.40 2.80

11,214 1.25 0.80180 3738 24.08 0.00

8722 3.16 0.0011,214 1.78 0.00

Table 3Beam-column interaction capacity (lambda"0.327)�

Target (P

) vs achieved ultimate loads (P��

)

Axial yield ratio (P

/P�)

0.3 0.7 0.9P

3738 8722 11,214

fea-nl achieved P��

load:

Direction note (1) P��

P��

P��

(deg) (kip) (kip) (kip)

000 3591 8375 11,263045 3372 8022 10,581090 3364 7883 10,537120 3310 7800 10,581135 3414 8238 10,781180 3567 7993 10,306

�Note: (1) `Directiona is the load vector as plotted on the interation curveswith respect to squash center; normalized to plastic moment capacitiesabout the principal axes.


Fig. 6. Thrust-moment interaction at 03.

beam-columns falls short of the classical applied mechanics failure envelope for thiscross-section by as much as 11%. The biggest shortfall is in the favourable quadrantthat we are seeking to exploit. This is despite the strain-hardening boost in purebending capacity, and is due to the inelastic local P-delta e!ect. Neglecting this canlead to errors on the unsafe side and possibly larger than what the partial safety factoron resistance accommodates. Most of the runs are for constant strain hardening, andthese are the results, which are carried forward. Two runs with more favourablebilinear strain hardening show either little di!erence or a modest (3%) improvement.

Consistent data from Tables 1}3 were used in a spreadsheet to generate theinteraction plots shown in Figs. 6}11. These are slices through the failure surface, atthe normalised loading directions. Axial load is that at failure, not some target whichwas never reached. Moment is the resultant, failure axial load (Table 3) times totalpreselected eccentricity (Table 2), applied at the end of the member, referenced to thesquash centre. Thus, local P-delta e!ects are not included in the stated moments,though of course they are very important in determining the inelastic failure load. Forthe pure moment cases (FEA-NL from Table 1), the de"nitions are obvious. Protocolsfor using these results directly in a design (or reassessment) situation are discussed insubsequent sections.

2.6. Parametric variations on the ewects of k and Cm

In anticipated applications, the e!ective length factor k will be incorporated in thedetermination of pure axial capacity P

�, using the appropriate column curve, and

the e!ect of global P-delta e!ects will be re#ected in the loads applied to the member.




The proper handling of the local P-delta e!ect (due to de#ections within the member)and the proper use of Cm are the subject of ongoing special investigations. The abilityof this strategy to represent the behaviour of cases more complex than the base casewill be tested by full elasto-plastic modelling of several such cases (e.g. see Fig. 12).




Although SNAME anticipates that these re"nements would be incorporated into thecalculation of applied moment M

�, results to date show that using the base case

interaction with unampli"ed applied moments is always on the safe side. While theremay be scope for using the more favourable plastic section envelope for lower k, and



for discounting secondary bending moments with Cm, the analyses performed to datedo not provide an unequivocal basis for implementing these re"nements. Finally,given that we have gone to all the trouble of such rigorous inelastic beam-columnanalyses, we should be able to use whatever capacity is demonstrated by the base casefailure envelope, without any further arbitrary conservatism.

3. Protocols for application

The anticipated protocol is to use the base case beam-column failure surface insteadof Dyer's section plastic envelope, as described in the revised SNAME B C8.1.4.7`alternativelya. This requires that the beam-column failure surface be generated foreach chord design of interest (about 25 inelastic FEM cases each).The design protocol is as follows: Using the applied loads on the member from an

elastic global analysis, accounting for global P-delta e!ects and foundation non-linearities, but not local P-delta e!ect or Cm, compare these directly with the failureenvelope from the base case inelastic beam column FEM analysis, i.e. k"1 andCm"1. Both loads and the failure envelope are modi"ed by their respective partialsafety factors. It is necessary to convert moments from their elastic centre reference inthe global analysis, to the squash centre reference used in the failure envelope.

Fig. 13 shows the complete failure envelope for the LeTourneau-type 116 C chordanalysed in this study. This is plotted in terms of actual, not normalised moments,preferably with equal scales in X, Y, and other radial directions. Since the inelastic


Fig. 12. Additional special cases. (a) Simple special models (non-proportional loading). (b) Complex specialmodels (proportional loading).

FEM analyses did not match the target 30, 70, and 90% of axial capacity, the contourmoments at these axial loads must be interpolated from the directional slices of Figs.6}11. The slices and interaction surface are also reduced to incorporate SNAME'sprescribed resistance factors of F"0.85 for axial load and �"0.90 for bendingmoment.

Factored moments and axial load from two actual design cases are also plotted inthe "gure. A ray from the origin indicates the actual direction of these loads. They arein the quadrant with the favourable bulge that Bulletin 5-5 short-changed.

One example is from Lewis' earlier comparison with SNAME calculation proced-ure. Fig. 14 shows a factored slice through the actual loading direction, interpolated


Fig. 13. Factored beam column failure surface.

from the full interaction surface (Fig. 13). The resultant moment and axial load arenow plotted on the slice, and a proportional loading line from the origin is used toscale the interaction ratio (ratio of the loading point distance to the interaction surfacedistance). Using the proposed new protocol, the IR of 0.87 represents better than a 7%improvement over the old SNAME generic interaction equation.

Using the failure envelope for the zero-length cross-section (as given in the SNAMEcommentary), rather than the one for a realistic beam-column, would of courseimprove the IR, but be technically incorrect and on the unsafe side as it ignores theinelastic P-delta e!ects.

The second case is from Doug Stock's global rig analysis for IADC. Repeatingthe approach described in the Lewis Example, its actual direction slice is plotted inFig. 15. Correcting for squash centre vs. elastic centre changes the indicated IR by 5%.

4. Summary, conclusions, and recommendations

The generic LRFD interaction equations have serious limitations when applied tosingly symmetric sections, like the LeTourneau tear-drop chord. These limitationshave been partially addressed in the second edition of SNAME Bulletin 5-5.

To implement new language regarding direct recourse to the beam-column failuresurface for non-zero length, 24 non-linear FEM analyses were made for the type 116Cchord results for the base case of k"1 and Cm"1 were "rst plotted as failureenvelopes for directional slices at 0, 45, 90, 120, 135, and 1803. After scaling by the


Fig. 14. Slice through failure surface*Lewis case.

LRFD resistance factors, and interpolation, a contour plot of the complete failuresurface is de"ned, as in Fig. 13.

For design application, the factored design loads and moments are compared to thefailure surface* "rst on the contour plot, then on a new directional slice through thedesign load case. The IR is de"ned as the ratio of the resultant moment and loadvector, to the corresponding vector at the failure surface.

Supplementary studies derived a knocked-down stress}strain curve to mimic thee!ect of residual stresses and geometric imperfections, and demonstrated that usingthe base case is always on the safe side for more complex cases involving k and Cm lessthan unity. Continuing these studies to further validate and re"ne the method areneeded.

Recommendations for future work are listed below:

1. Increased usability of the current jackup #eet justi"es more rigorous analysis thatavoids conservative approximations. The inelastic FEM protocol presented here is


Fig. 15. Slice through failure surface*Stock case.

a step in this direction. Still less approximation would be involved if the criticalinelastic members were modelled as a substructure in the global rig analysis.

2. Initial analysis results of a half-length chord case has implied much of the knock-down relative to the full plastic envelope has been recovered. If additional supple-mentary runs show apparent bene"ts of lower `ka are retained for all loadingdirections in unambiguous and practical cases, then the protocol should be modi-"ed to account for these bene"cial e!ects. Incorporate results of other special case(see Fig. 12) to modify results for the classical pinned}pinned base case.

3. In the absence of completion of the protocol recommended in this study Bulletin5-5, users might consider using Dyer's plastic section envelope. However, com-puted moments must account for inelastic P-delta e!ects in their analyses. Thee!ects of inelastic P-delta, at ultimate failure, may be estimated based on de#ectionof the member assuming bending strain of 10 times yield about each loaded axis ofthe member.


4. LeTourneau is performing a full-scale test of one of their Type 4 chord con"gura-tions. This test includes the improved stress}strain data for the two materials usedin the fabrication of the chord investigated in our study. LeTourneau will alsodevelop realistic residual stress data for the material and welding procedure used inthe construction of the chords. The data from these tests should be incorporated ina reanalysis of a limited number of the 24 basic case FEA analyses. If the resultsdemonstrate signi"cant changes to the conclusions of the original study, a com-plete reanalysis of all base cases should be performed.

5. Investigate and justify alternative column curves that better describes the behav-iour of various jackup chord con"gurations. Use bilinear strain-hardening behav-iour in the 24 FEA cases to demonstrate the bene"t in the desired quadrant.

6. Increase the partial factor on resistance for bending and compression in recogni-tion of the comprehensive FEA analyses using the protocol in this paper, whichincludes elastic and inelastic P-delta e!ects. Evaluate the issue of safety factors asapplied to the jackup assessment.

Acknowledgements

Support from the IADC Jack-up Committee is gratefully acknowledged.

References

[1] SNAME. Recommended practice for site-speci"c assessment of mobile jack-up units. SNAME Bulletin5.5, 1st ed., 1994.

[2] Lewis DR. IADC Jack-up Committee Contribution to SNAME Bulletin 5-5. Presented at OTCTopical Luncheon, 1998.

[3] Marshall PW. An overview of recent work on cyclic inelastic behavior and system reliability, paneldiscussion on stability of o!shore structures. Proceedings of SSRC, 1982.

[4] Sherman DR. Ultimate capacity of tubular members. Report CE-15, Shell Oil Co, 1975 (also ASCEpaper 14627, STJ June 1979).

[5] Sherman DR. Interpretive discussion of beam-column test data. University of Wisconsin-MilwaukeeReport to Shell Oil Co (with addendum), 1980.

[6] Dyer A. Plastic strength interaction equations for jack-up chords. MSc thesis, University of She$eld,1992.


Documents

Yield interaction surface of an enhanced LeTourneau 116C chord