Direct Strength Design for Cold-Formed Steel Members with Perforations. Progress Report 5 C. Moen and B.W. Schafer AISI-COS Meeting February 2008

Transcription

1 Direct Strength Design for Cold-Formed Steel Members with Perforations Progress Report 5 C. Moen and B.W. Schafer AISI-COS Meeting February 28

2 outline Objective Summary of past progress Simplified methods for predicting elastic buckling Local buckling Element-based approximation Finite strip approximation Distortional Buckling Finite strip approximation Global Buckling Beams Columns Conclusions Future Work

3 objective Development of a general design method for cold-formed steel members with perforations. Direct Strength Method Extensions P n = f (P y, P cre, P crd, P crl )? How do the DSM curves need to change? Gross or net, or some combination? Explicitly model hole(s)? Accuracy? Efficiency? Identification? Just these modes?

4 progress report 1 highlight elastic buckling modes in a column P crl =.42P y,g Distortional modes unique to a column with a hole L P crl =.42P y,g P crd1 =.52P y,g L L+DH LH New insight we consider these modes as local buckling now: L (plate) and DH2 LH LH (unstiffened strip) P crd2 =.54P y,g D D+L P crd =1.15P y,g P crd3 =1.16P y,g

5 progress report 2 highlight influence of multiple holes on elastic buckling h hole /h=.66 h hole /h=.44 h hole /h=.19 f cr,holes /f cr,no holes h hole /h= New insight:.8 Decrease in f cr when hole spacing becomes small S>2L.75 hole sufficient to avoid largest.2 2interactions, 3 4 but 5 we also must limit S>1.5h S/L hole.2 S L hole h h hole S/L hole S > 5L hole or holes interact with one another

6 progress report 3 highlight impact of hole on beam distortional buckling Holes decrease critical elastic distortional buckling moment, modeling required M crd,hole /M crd,no hole (now we have.2 some conservative.4.6 finite strip.8 approx...) h hole /H

7 progress report 3 highlight column tests Column axial load (kips) Slotted hole influences post-peak load path and NH H -2 decreases.2 column.4 ductility Column axial displacement (inches)

8 progress report 4 highlight nonlinear FEA and initial stress/strain state * bending the corners... DETAIL A H H H coiling, uncoiling, and flattening Results sensitive to solution controls, new method for prediction of residual stresses, initial state of member important, collapse modeling improved, but still ongoing in this last stage of research.

9 DSM Option progress report 4 highlight preliminary DSM for columns with holes Description Test-to-Predicted Ratio by Failure Mode Local Distortional Global Yielding average STDEV average STDEV average STDEV average STDEV 4 cap P nl and P nd P n /P ne Option 4 - cap P n l, P nd DSM Local Prediction Local Controlled Test DSM DSM Local Yield Prediction Control Local Yield Controlled Controlled Test Test DSM Yield Control Yield Controlled Test DSM for Local buckling shown here λ =(P l ne /P cr l ).5

10 outline Objective Summary of past progress Simplified methods for predicting elastic buckling Local buckling Element-based approximation Finite strip approximation Distortional Buckling Finite strip approximation Global Buckling Beams Columns Conclusions Future Work

11 elastic buckling prediction Shell FE eigenbuckling analysis completely general not widely available can be difficult to interpret Finite strip modifications for buckling analysis global, distortional, local must be modeled separately subject to some limitations which are still being determined, though appears promising Element-based methods useful for simple approximation in DSM also potentially useful for traditional k effective width methods simplified methods

12 element-based local buckling prediction - treat each element separately -find f cr for each element - find minimum f cr for the section P = crl f crl, min A g this is the classic plate buckling solution method, but the wrinkle is that now we must also consider plates (elements) with holes.

13 buckling of stiffened element with holes f = crl min( f l, f lh ) Plate buckling (L) Unstiffened strip (LH) v

14 buckling of stiffened element with holes f = crl min( f l, f lh ) Plate buckling (L) Unstiffened strip (LH) f l k l = k l = 4 2 π E ( ν ) t h 2 f lh, net k l h Yu and Schafer (27), accounts for length of strip = k lh 2 π E 2 12( 1 ) t ν hstrip.2 = Lhole hstrip.6 2 Converts net section to gross section stress f h = flh, net h lh 1 hole

15 ABAQUS study of stiff. elem. with holes SS 145 simulations: hole shape: 9% slotted, 5% circular, 5% square SS SS SS Simply Supported h hole /h S/L hole S/h h/t Min Max SS L hole h hole h S

16 accuracy of elastic buckling prediction P crl-abaqus / P crl-predicted Plate buckling controls Unstiffened strip controls Buckling mode transitions from unstiffened strip to plate buckling between holes at h/h hole =.6 For elements with S/L hole >2, S/h> h hole /h

17 accuracy of elastic buckling prediction P crl-abaqus / P crl-predicted Plate buckling controls Unstiffened strip controls Buckling mode transitions from unstiffened strip to plate buckling between holes at h/h hole = h hole /h

18 accuracy of elastic buckling prediction P crl-abaqus / P crl-predicted Plate buckling controls Unstiffened strip controls Buckling mode transitions from unstiffened strip to plate buckling between holes at h/h hole = h hole /h ON TO LOCAL BUCKLING BY FSM (SKIP THE REST OF ELEMENT METHODS)

19 development of geometric limits P crl-abaqus / P crl-predicted Plate buckling controls Unstiffened strip controls S/h S/h Plate buckling controls Unstiffened strip controls S/L hole S/L hole Prediction accuracy decreases for small S/h and S/L hole

20 development of geometric limits P crl-abaqus / P crl-predicted Plate buckling controls Unstiffened strip controls Remove data for S/h S/h Plate buckling controls Unstiffened strip controls S/L hole S/L hole

21 development of geometric limits P crl-abaqus / P crl-predicted Plate buckling controls Unstiffened strip controls Remove data for S/h 1.5 and S/L hole S/h Plate buckling controls Unstiffened strip controls Remove data for S/L hole 2 and S/h S/L hole

22 impact of hole location (offset) L hole h hole δ h S h As h strip or δ h increase the unstiffened strip model becomes quite conservative. unstiffened strip ABAQUS elastic buckling mode shape

23 accuracy of method with offset holes 3 P P l, /P crl-abaqus cr / l P, crl-predicted δ hole /L hole.15 Plate buckling controls Unstiffened strip controls 43 ABAQUS models, regularly spaced slotted holes, plate width and hole offset varied δ hole /h δ hole /h if δ h >.15L hole or h strip >.4h expect overly conservative predictions

24 unstiffened element with holes SS SS Simply Supported SS FREE 9% circular holes, 91% slotted holes SS h hole /h S/L hole S/h h/t Min Max simply supported L hole h strip h hole h S free

25 unstiffened element with holes f crl = min( f l, f lh ) Plate buckling (L) Unstiffened strip (LH) f l = k 2 π E 12 1 ( 2 ν ) t h 2 since h strip <h this mode never controls mathematically, nor is the mode shape observed in ABAQUS. k Lhole = hstrip Empirical factor derived from ABAQUS results

26 development of correction Physical limit, prevents local buckling of strip between free edge and hole P crl-abaqus / P crl-predicted L hole /h strip 1 Prediction (k=.43) Prediction (with α) leads to.. Lhole k = hstrip L hole /h strip

27 problematic mode for longer holes Holes do not influence the buckled shape of unstiffened plates until the hole width becomes large relative to plate width (L hole /h strip >1). Buckling of the strip at the free edge of the plate changes the shape of the local buckling mode.

28 prediction accuracy for offset holes in unstiffened elements P crl-abaqus P l, / /P P cr crl-predicted l, Shift tow ards SS edge Shift tow ard free edge δ hole /h δ hole /h

29 Outline Objective Summary of past progress Simplified methods for predicting elastic buckling Local buckling Element-based approximation Finite strip approximation Distortional Buckling Finite strip approximation Global Buckling Beams Columns Conclusions Future Work

30 Elastic buckling prediction Shell FE eigenbuckling analysis completely general not widely available can be difficult to interpret Finite strip modifications for buckling analysis global, distortional, local must be modeled separately subject to some limitations which are still being determined, though appears promising Element-based methods useful for simple approximation in DSM also potentially useful for traditional k effective width methods

31 FSM local buckling prediction P crl = min( P l, P lh ) h hole local buckling (gross cross section) local buckling (net cross section) Pinned corners in CUFSM isolate local buckling of cross section (origins of this idea go back to Sputo and Tovar, and others)

32 FSM local buckling prediction P crl = min( P l, P lh ) h hole local buckling (gross cross section) Pinned corners in CUFSM isolate local buckling of cross section half-wavelength = L hole

33 FSM local buckling prediction P crl = min( P l, P lh ) λ l =L hole =4 in. h hole examination at the half-wavelength the hole creates is the practical extension of the unstiffened strip approach, but now flange/web/lip interaction is included! Pinned corners in CUFSM isolate local buckling of cross section half-wavelength = L hole

34 web hole comparison with ABAQUS.5.45 λ l =L hole =4 in. ABAQUS CUFSM Approx. Method λ l =L hole.4 P crl P cr l / /P P y,g y-gross P lh P l S with increasing web hole h hole /h h hole /h

35 flange hole comparison with ABAQUS.5.45 ABAQUS CUFSM Approx. Method λ l =L hole P crl P l / /P P y,g y-gross P lh 362S with increasing flange hole b hole /b b hole /b

36 outline Objective Summary of past progress Simplified methods for predicting elastic buckling Local buckling Element-based approximation Finite strip approximation Distortional Buckling Finite strip approximation Global Buckling Beams Columns Conclusions Future Work

37 FSM distortional buckling this approach remains under development, but appears generally promising h hole t hole t hole = λd L λ d hole t reduced thickness at hole location... t λ d λ d L

38 impact of reduced thickness on FSM 1 9 without hole with hole λ d (determined at local minimum of no hole curve) P cr, kips P crd (includes influence of hole) half-wavelength, in.

39 web hole comparison with ABAQUS (distortional buckling) CUFSM Approx. Method ABAQUS Pure D ABAQUS DH P P crd /P crl / P y,g y-gross S h hole /h h hole /h

40 outline Objective Summary of past progress Simplified methods for predicting elastic buckling Local buckling Element-based approximation Finite strip approximation Distortional Buckling Finite strip approximation Global Buckling Beams Columns Conclusions Future Work

41 global buckling approximations cre, y cre, x cre, φ cre, y 2 cre, x = 2 ro ro ( )( )( ) ( ) o o P P P P P P P P ( P P) weighted properties approach P x P y e.g., I= I g LNH + L I net L H it can be shown that for flexural buckling of a members with holes symmetric about the midlength this is exact weighted thickness approach h hole t hole = L L L H t t hole t easy to implement in CUFSM (CUTWP) or other software

42 example of global modes weak axis flexural P cre =6.96 kips flexural-torsional P cre =1.64 kips Global column buckling modes (SSMA 12S162-68, h hole /h=.5)

43 weak axis flexural buckling 1.5 P cre-abaqus P cre,abaqus /P / cre,prediction P cre-predicted 1.5 weighted thickness weighted properties net section net section conservative weighted t has wrong trend for large holes h hole /h h hole /h

44 weak axis flexural buckling 1.5 P cre-abaqus P cre,abaqus /P / cre,prediction P cre-predicted 1.5 weighted thickness weighted properties net section classical global equations even without holes are slightly different than ABAQUS due to plate deformations in the web of the 12S at these lengths (note CUFSM and ABAQUS agree) net section conservative weighted t has wrong trend for large holes h hole /h h hole /h

45 flexural-torsional buckling 1.5 P cre-abaqus P cre,abaqus / /P P cre,prediction cre-predicted 1.5 weighted thickness weighted properties net section h hole /h h hole /h torsion approximations require further study, particularly w.r.t. C w

46 global buckling of beam with holes M cre GJ + π L 2 = EI y ECw 2 Lateral-torsional buckling (SSMA 12S162-68, h hole /h=.5)

47 lateral-torsional buckling prediction 1.5 weighted thickness weighted properties net section M cre,abaqus /M cre,prediction 1.5 torsion approximations require further study, particularly w.r.t. C w - but agreement is much better for weak-axis torsional coupling of LTB in a beam, than strong-axis torsional coupling of FTB in a column h hole /h

48 Outline Objective Summary of past progress Simplified methods for predicting elastic buckling Local buckling Element-based approximation Finite strip approximation Distortional Buckling Finite strip approximation Global Buckling Beams Columns Conclusions Future Work

49 Conclusions (on this progress report) Local Buckling We now have viable prediction methods for stiffened and unstiffened elements with holes (uniaxial compression) FSM can be used to improve element-based predictions if due care is taken in modifying a traditional FSM model Distortional Buckling FSM method with weighted thickness over distortional half-wave produces conservative predictions in limited stud and appears to be a promising/convenient approach. Additional study underway. Global Buckling Weighted cross-section properties (or even weighted thickness properties) work well for capturing flexural buckling Torsional mode predictions become unconservative as hole size increases, actual J and C w are not captured with weighted methods C w is of particular concern and under current study

50 Final project phase Completion of simplified methods for elastic buckling prediction Extending database of member strengths for cross-sections with holes continuing nonlinear FEA collapse analysis for columns and beams in specific geometric ranges where data is scarce and where DSM prediction equations need further study Specification and commentary provisions for DSM of members with holes Recommendations for main specification provisions for members with holes