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IMPROVED DESIGN ASSESSMENT OF LTB OF I-SECTION MEMBERS VIA MODERN COMPUTATIONAL METHODS Improved Design Assessment of LTB of I Section Members Donald W. White (with credits to Dr.

1 IMPROVED DESIGN ASSESSMENT OF LTB OF I-SECTION MEMBERS VIA MODERN COMPUTATIONAL METHODS Improved Design Assessment of LTB of I Section Members Donald W. White (with credits to Dr. Woo Yong Jeong & Mr. Oguzhan Toğay) School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta, GA USA The STEEL Conference 2 D. White, Georgia Tech 1

2 IMPETUS FOR THIS WORK Efficient, more rigorous assessment of Web tapered members with Multiple taper Steps in the cross section (CS) geometry Doubly & singly symmetric CS geometry Ordinary frame members including Impact of general lateral & torsional bracing Benefits of end restraint & member continuity across braced points Influence of general moment gradient and other load & displacement boundary condition effects 3 IMPETUS FOR THIS WORK Lack of sufficient rigor of Direct Analysis (DM) type approaches for assessment of 3D member limit states & stability bracing requirements Lack of sufficient computational efficiency of advanced (plastic zone) analysis methods Difficulty of correlation between advanced analysis results and Specification resistances Desire for improvement upon traditional K & C b factor approximations Desire for improvement upon traditional strength interaction eqs. 4 D. White, Georgia Tech 2

3 INTRODUCTION Column inelastic effective length factors have been used extensively in the past to achieve improved accuracy and economy in the design of steel frames Using a buckling analysis with inelastic stiffness reduction factors,, the following effects can be captured quite rigorously & efficiently for columns, beams & beam columns: Loss of member rigidity due to the spread of plasticity Various end restraints Various bracing constraints and other load & displacement boundary conditions Continuity across braced points 5 AISC a FOR COLUMNS W/ NONSLENDER ELEMENTS P 09. ( ) P 09. ( ) P P P c n e a e e a e cpn P c y apy Pe. P 4 e cp n 0658 for Py 9 cp Pn Pe y cpn P c y aPy Pn 0.877aPy cp n Pn ln ln cp y cp cp n y ln 0.877a ln0.658 cp y cp P P u P u n for u a ln cp y cp cp y y P u for cp y a 1 6 D. White, Georgia Tech 3

4 ANALYSIS STEPS Build a model of the structure Apply the desired LFRD factored loads. These loads produce the internal axial forces P u Reduce EI x, EI y, EC w and GJ by SRF = 0.9 x 0.877x a Solve for the inelastic buckling load Vary the applied loads by the scale factor Calculate τ a at the current load level Iterate until the assumed in the calculation of a is the same as that determined from the buckling analysis The resulting P u is a rigorous calculation of c P n accounting for all member continuity, bracing and/or end restraint effects 7 EXAMPLE COLUMN INELASTIC BUCKLING ANALYSIS SABRE2 (using the inelastic reduction factor a ) gives c P n = 1153 kip This result matches with a traditional iterative calculation (Yura 1971) using an inelastic K = 0.861, based on a = D. White, Georgia Tech 4

5 COLUMN STIFFNESS REDUCTION FACTORS (SRFs) Net SRF P u P u a ln cp y cp y P / P u y P u P u b 4 1 P y P y 9 BEAM ltb MODEL, COMPACT & NONCOMPACT WEB MEMBERS FL Mu bmn bme 0.9ltbMe For m where m F M yc b yc ltb 1 F F 4 2 M Y X m ltb 2 M 2Fyc X m 2Y E L b max. LTB For where: yc b yc m 1 R pc L L r p Lp Fyc 1 Y m F rt rt r t E 1.95 L 1 RpcF yc M X S h J 2 xc o R M max. LTB pc yc 10 D. White, Georgia Tech 5

6 COLUMN VS. BEAM FACTORS FOR A W21X Beam LTB Tau Factor Column Tau Factor Pu P, c ye b M M u max 11 REQUIREMENTS FOR INELASTIC LTB ANALYSIS The software must rigorously include EC w in addition to EI x, EI y & GJ in the context of doubly symmetric I section members For singly symmetric members, the behavior associated with the monosymmetry factor, β x, also must be included The 0.9 x ltb factor should be applied equally to the member elastic stiffness contributions EI y, EC w and GJ for the execution of the buckling analysis Required number of elements: At least 4 elements per unbraced length are required to capture the behavior for frame elements based on cubic Hermitian interpolation of the transverse displacements and twists along the element length 12 D. White, Georgia Tech 6

7 GENERAL PURPOSE THIN WALLED OPEN SECTION FRAME ELEMENT Implemented in SABRE2 (available at white.ce.gatech.edu/sabre) 13 ELASTIC LTB BENCHMARK 14 D. White, Georgia Tech 7

8 ELASTIC LTB BENCHMARK Typ.14 dof prismatic element (10 elem) stepped using avg. depth in ea. elem. Typ.14 dof prismatic element (10 elem) stepped using smallest depth in ea. elem. 15 BASIC BEAM COLUMN EXAMPLE USING SABRE2 From AISC/MBMA Design Guide 25 Point Brace, i = k/in 11.3 kips 90 in 144 in 1800 kip-in Dimensions: At left end: b ft = 6 in t ft = in b fc = 6 in t fc = in h = 12 in t w = in At right end: b ft t ft = 6 in = in b fc = 6 in t fc = in h = 24 in t w = in F y = 55 ksi Simply-supported end conditions SABRE2 available at: white.ce.gatech.edu/sabre 16 D. White, Georgia Tech 8

9 SRF DIAGRAM 17 SABRE2 VS DG25 vs DG25 Solution A DG25 Solution C 18 D. White, Georgia Tech 9

BEAM ltb MODEL RESULTS (W21X44 BEAMS) 1.2 1.0 Uniform Moment M n /M p 0.8 0.6 0.4 Moment Gradient (C b = 1.75) AISC Specification Chapter F 0.2 0.0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.

10 BEAM ltb MODEL RESULTS (W21X44 BEAMS) Uniform Moment M n /M p Moment Gradient (C b = 1.75) AISC Specification Chapter F Moment Gradient, Inelastic Buckling Analysis (SABRE2) L b /L p 19 BEAM ltb MODEL RESULTS (W21X44 BEAMS) Uniform Moment M n /M p Moment Gradient (C b = 1.75) AISC Specification Chapter F Moment Gradient, Inelastic Buckling Analysis (SABRE2) L b /L p 20 D. White, Georgia Tech 10

11 W21X44 BEAM LTB RESISTANCE VS. POINT BRACING STIFFNESS (Flexurally & torsionally simply-supported end conditions, Uniform Moment, One intermediate point brace at compression flange, L br1 = 8.1 ft, L br2 = 4.9 ft ) Inelastic Buckling Analysis AISC 2016 Commentary Provisions including L q 21 W21X44 BEAM LTB RESISTANCE VS. POINT BRACING STIFFNESS (Flexurally & torsionally simply-supported end conditions, Uniform Moment, One intermediate point brace at compression flange, L br1 = 8.1 ft, L br2 = 4.9 ft ) Inelastic Buckling Analysis AISC 2016 Commentary Provisions including L q 22 D. White, Georgia Tech 11

One intermediate point brace at compression flange, L br1 = 8.1 ft, L br2 = 4.

ASSESSMENT Mp 1.40, My D 160, tw 2Dc 109, tw 2Dcp 32, tw Dc 3.413, Dcp D 1.

12 W21X44 BEAM LTB RESISTANCE VS. POINT BRACING STIFFNESS (Flexurally & torsionally simply-supported end conditions, Uniform Moment, One intermediate point brace at compression flange, L br1 = 8.1 ft, L br2 = 4.9 ft ) Inelastic Buckling Analysis AISC 2016 Commentary Provisions including L q 23 TRANSFER GIRDER ASSESSMENT Mp 1.40, My D 160, tw 2Dc 109, tw 2Dcp 32, tw Dc 3.413, Dcp D Dc Lateral brace (TYP) P 0.5P x x x x x x x x 3 at L b = 45 ft = 135 ft Critical Middle Unbraced Length: C b = 1.10 K = Girder Factored Load Capacity: P max = 361 kip from manual calcs. 24 D. White, Georgia Tech 12

13 INELASTIC BUCKLING MODE P max = 376 kip 25 MOMENT & SRF DIAGRAMS MOMENT SRF 26 D. White, Georgia Tech 13

CROSS SECTION UNITY CHECK 27 BEAM COLUMN SRF Calculate the UC value with respect to the cross section strength from Eqs. H1 1 UC = P u / c P ye + 8/9 M u / b M max for P u / c P ye >0.

14 CROSS SECTION UNITY CHECK 27 BEAM COLUMN SRF Calculate the UC value with respect to the cross section strength from Eqs. H1 1 UC = P u / c P ye + 8/9 M u / b M max for P u / c P ye >0.2 UC = P u /2 c P ye + M u / b M max for P u / c P ye < 0.2 Use the UC value in the a & ltb eqs. instead of P u / c P ye & M u / b M max Pu / cpye Determine the angle atan Mu / bmmax Calculate the net SRF applied to EC w, EI y & GJ as A SRF 0.9 x x e 1 0.9R o o 90 A 90 a b ltb g 28 D. White, Georgia Tech 14

15 BEAM COLUMN ltb MODEL W21X44 RESULTS Simply supported members, moment gradient loading c P n / c P y L = 7.5 ft L = 10 ft L = 15 ft Fully Effective Cross Section Plastic Strength b c M n // c b M p 29 ROOF GIRDER EXAMPLE (ADAPTED FROM AISC 2002) 2 in (G' = 1 kip/in) 30 D. White, Georgia Tech 15

16 ROOF GIRDER EXAMPLE = b M n = 230 kip ft c P n = 18.4 kips 31 SRF, GRAVITY LOAD CASE AT BUCKLING LOAD Net SRF Position along girder length (ft) 32 D. White, Georgia Tech 16

1.2 (Dead + Collateral + Self-Weight) + 1.

17 CLEAR SPAN FRAME EXAMPLE SYM 33 BUCKLING MODE & CONTROLLING LIMIT STATE INFORMATION 1.2 (Dead + Collateral + Self-Weight) Uniform Snow 34 D. White, Georgia Tech 17

18 AXIAL FORCES AT STRENGTH LIMIT 35 MOMENTS AT STRENGTH LIMIT 36 D. White, Georgia Tech 18

19 SRF VALUES AT STRENGTH LIMIT 37 b M max VALUES AT STRENGTH LIMIT 38 D. White, Georgia Tech 19

20 c P ye VALUES AT STRENGTH LIMIT 39 CROSS SECTION UNITY CHECKS AT STRENGTH LIMIT 40 D. White, Georgia Tech 20

21 ADVANTAGES OF BUCKLING ANALYSIS APPROACH More general and more rigorous handling of all types of bracing, end restraint & continuity effects Substantially cleaner, more streamlined & less error prone member strength calculations Consistent bracing stiffness & member strength assessments More accurate capture of Moment gradient and other load & displacement b.c. effects Tapered & stepped member geometry effects via a continuous representation of the corresponding SRF values along the member lengths 41 COMPLEMENTARY RESEARCH Trahair, N.S. and Hancock, G.J. (2004). Steel member strength by inelastic lateral buckling, Journal of Structural Engineering, ASCE, 130(1), Trahair, N.S. (2009). Buckling analysis design of steel frames, Journal of Constructional Steel Research, 65(7), Trahair, N.S. (2010). Steel cantilever strength by inelastic lateral buckling, Journal of Constructional Steel Research, 66(8-9), pp Kucukler, M., Gardner, L., Macorini, L. (2014). A stiffness reduction method for the in-plane design of structural steel elements, Engineering Structures, 73, Kucukler, M., Gardner, L., Macorini, L. (2015a). Lateral torsional buckling assessment of steel beams through a stiffness reduction method, Journal of Constructional Steel Research, 109, Kucukler, M., Gardner, L., and Macorini, L. (2015b). Flexural-torsional buckling assessment of steel beam-columns through a stiffness reduction method. Engineering Structures, 101, Kucukler, M., Gardner, L., and Macorini, L. (2015c). In-plane design of steel frames through a stiffness reduction method. Journal of Constructional Steel Research. in press. Gardner, L. (2015). Design of Steel Structures to Eurocode 3 and Alternative Approaches. Proceedings, International Symposium on Advances in Steel and Composite Structures, Hong Kong, D. White, Georgia Tech 21

DESIGN METHOD REQUIREMENTS Traditional Methods Effective Length Method (ELM),, INELASTIC,, } BUCKLING ANALYSIS 2 nd Order Elastic Analysis Joint out of alignment (gravity only load cases) Advanced

22 DESIGN METHOD REQUIREMENTS Traditional Methods Effective Length Method (ELM),, INELASTIC,, } BUCKLING ANALYSIS 2 nd Order Elastic Analysis Joint out of alignment (gravity only load cases) Advanced Methods AISC (2016) App. 1.2 (Elastic Analysis) INELASTIC,, } BUCKLING ANALYSIS 2 nd Order Elastic Analysis Stiffness reductions 0.8 & 0.8 Joint out of alignment Member out of straightness Direct Analysis Method (DM),, INELASTIC,, } BUCKLING ANALYSIS 2 nd Order Elastic Analysis Stiffness reductions 0.8 & 0.8 Joint out of alignment AISC (2016) App. 1.3 (Inelastic Analysis) 2 nd Order Inelastic Analysis 0.9 & 0.9 Spread of yielding including residual stress effects Joint out of alignment Member out of straightness 43 For TWOS members, For TWOS members, THANKS FOR YOUR ATTENTION! I will be happy to address any questions SABRE2 available at: white.ce.gatech.edu/sabre 44 D. White, Georgia Tech 22

MODULE C: COMPRESSION MEMBERS

MODULE C: COMPRESSION MEMBERS This module of CIE 428 covers the following subjects Column theory Column design per AISC Effective length Torsional and flexural-torsional buckling Built-up members READING: