Aluminum Structural Member Design

Transcription

1 Aluminum Structural Member Design Randy Kissell, P.E. TGB Partnership 1 Outline 1 Introduction 2 Aluminum alloys and tempers 3 Aluminum material properties 4 Aluminum structural design overview 5 Axial tension 6 Axial compression 7 Flexure Designing Aluminum Structural Members 2 1

2 1. Introduction Why learn aluminum structural design? To design aluminum structures To better understand steel structural design Designing Aluminum Structures 3 Examples of Aluminum Structures Curtain walls and storefronts Roofing and canopies Space frames Tanks and vessels (corrosive & cryogenic) Portable structures (scaffolding, ladders) Highway products (signs, light poles, bridge rail) Designing Aluminum Structures 4 2

3 Science Land Egg (164 wide) courtesy of Temcor Designing Aluminum Structures 5 Science Land Egg courtesy of Temcor Designing Aluminum Structures 6 3

4 The Aluminum Association Founded in 1933, 50+ members are the major US producers AA writes most standards on aluminum; has worldwide influence Contact: Wilson Blvd, Suite 600 Arlington, VA ; pubs Designing Aluminum Structures Aluminum Design Manual Designing Aluminum Structures 8 4

5 Aluminum Design Manual (ADM) Now issued every 5 years Latest edition is ADM 2010 Previous editions in 2005, st edition (1994) was compilation of several AA pubs previously issued separately; most importantly, the Specification for Aluminum Structures (SAS) Designing Aluminum Structures 9 Aluminum Design Manual Contents The Aluminum Design Manual (ADM) provides aluminum structural design tools Part I Specification for Aluminum Structures Part II Commentary Part III Design Guide Part IV Material Properties Part V Section Properties Part VI Design Aids Part VII Illustrative Examples Designing Aluminum Structures 10 5

6 Specification for Aluminum Structures (SAS) The Specification for Aluminum Structures is Part I of the Aluminum Design Manual SAS is also called the Aluminum Specification Adopted in BOCA, UBC, SBC, and IBC It s the source of all aluminum structural design requirements in the US Designing Aluminum Structures 11 Specification for Aluminum Structures (SAS) 2010 edition is a major rewrite It s a unified Specification, with both: Allowable Strength Design For buildings and bridges Load and Resistance Factor Design For buildings only Load factors from ASCE 7 (= 1.2D + 1.6L ) Designing Aluminum Structures 12 6

7 2010 Specification for Aluminum Structures (SAS) A. General Provisions B. Design Requirements C. Design for Stability D. Design of Members for Tension E. Design of Members for Compression F. Design of Members for Flexure G. Design of Members for Shear H. Design of Members for Combined Forces and Torsion J. Design of Connections Designing Aluminum Structures Specification for Aluminum Structures (SAS) L. Design for Serviceability M. Fabrication and Erection Appendices 1. Testing 3. Design for Fatigue 4. Design for Fire Conditions 5. Evaluation of Existing Structures 6. Design of Braces for Columns and Beams New in 2010 Designing Aluminum Structures 14 7

8 2. Aluminum Alloys and Tempers Wrought alloy designation system Aluminum temper designation system Material specifications Designing Aluminum Structures 15 Aluminum Isn t Just One Thing Like other metals, aluminum comes in many alloys Alloy = material with metallic properties, composed of 2 or more elements, of which at least one is a metal Different aluminum alloys can have very different properties Alloying elements are usually < 5% Designing Aluminum Structures 16 8

9 Alloys Alloys of iron are called steel Carbon < 0.5% or it s cast iron C, Cr, Ni, Mb, Cu, Ti used Alloys of aluminum are called aluminum alloys Si, Fe, Cu, Mn, Mg, Cr, Ni, Zn, Ti used Designing Aluminum Structures 17 Designating Aluminum vs. Steel Steel is identified by ASTM no. and grade Example: ASTM A 709 Grade 50 Steel ASTM specs are for one alloy and a few products (for example, shapes and plate) Aluminum is identified by AA no., temper, and product Example: 6061-T6 sheet Aluminum ASTM specs are for many alloys and one product (e.g., ASTM B 209) Designing Aluminum Structures 18 9

10 Wrought Alloy Designation System Number Main Alloy Strength Corrosion 1xxx > 99% Al Fair Excellent 2xxx Cu High Fair 3xxx Mn Fair Good 4xxx Si Good Good 5xxx Mg Good Good 6xxx Mg Si Good Good 7xxx Zn High Fair 8xxx others Designing Aluminum Structures 19 Wrought Alloy Key 1 st digit denotes main alloying element 3 rd and 4 th digits are sequentially assigned 2 nd digit denotes a variation Example: 2319 is AlCu alloy (2xxx), variation on composition is identical to 2219 except slightly more Ti (grain refiner to improve weld strength); both have 6.3% Cu Designing Aluminum Structures 20 10

11 1xxx Alloys (pure Al) Common uses: Electrical conductors Corrosive environments Examples 1060 (99.60% aluminum) 1100 (99.00% aluminum) Pro: corrosion resistant, good conductors Con: Not very strong Designing Aluminum Structures 21 2xxx Alloys (Al-Cu) Common Uses Aircraft parts, skins Fasteners Example 2024 Pro: Strong Con: Not very corrosion resistant; hard to weld Designing Aluminum Structures 22 11

12 3xxx Alloys (Al-Mn) Common Uses Roofing and siding Gutters and downspouts Examples 3003, 3004, 3105 Pro: Formable, good corrosion resistance Con: Not that strong Designing Aluminum Structures 23 Common Uses 4xxx Alloys (Al-Si) Welding and brazing filler metal Example 4043 Pro: Flows well Con: Lower ductility Designing Aluminum Structures 24 12

13 5xxx Alloys (Al-Mg) Common Uses Marine applications Welded plate structures Examples 5052, 5083 Pro: Strong, even when welded Con: Hard to extrude; those with 3%+ Mg can have corrosion resistance issues Designing Aluminum Structures 25 6xxx Alloys (Al-Mg 2 Si) Common Uses Structural shapes Pipe Examples 6061, 6063 Pro: Good combination of strength and corrosion resistance, very extrudable Con: Lose considerable strength when welded Designing Aluminum Structures 26 13

14 Common Uses 7xxx Alloys (Al-Zn) Aircraft parts Two classes With copper (example: 7075) Without copper (example: 7005) Pro: Very strong (7178-T6 F tu = 84 ksi) Con: Not very corrosion resistant; hard to weld Designing Aluminum Structures 27 How Alloys are Strengthened Alloying elements (Mg is good example) Treatment: Strain hardening (cold working) Heat treatment Heat treatable: 2xxx, 6xxx, 7xxx Non-heat treatable: 1xxx, 3xxx, 5xxx Designing Aluminum Structures 28 14

15 Annealed Condition Before tempering, alloys start in the annealed condition (-O suffix) Annealed condition is weakest but most ductile Tempering increases strength, but decreases ductility Most alloys are annealed by heating to 650 o F (melting point is about 1100 o F) Designing Aluminum Structures 29 Strain Hardening Mechanical deformation at ambient temps For sheet and plate, deformation is by rolling to reduce the thickness Some non-heat treatable alloys undergo a stabilization heat treatment Purpose: to prevent age softening Only used for some Al-Mg (5xxx) alloys Designing Aluminum Structures 30 15

16 Strain Hardened Tempers Temper F tu (ksi) Description 5052-O 25 Annealed 5052-H32 31 ¼ hard 5052-H34 34 ½ hard 5052-H36 37 ¾ hard 5052-H38 39 Full hard Designing Aluminum Structures 31 Effect of Strain Hardening 5052 Designing Aluminum Structures 32 16

17 Heat Treating 1) Annealed material is solution heat treated 6061-O is heated to 990 o F, then quenched Resulting temper is 6061-T4 2) Solution heat treated material is precipitation heat treated (artificially aged) 6061-T4 is heated to 350 o F and held for 8 hrs Resulting temper is 6061-T6 Designing Aluminum Structures 33 Heat Treatment Tempers T1 through T4: naturally aged (6005-T1) T5 through T9: artificially aged (6063-T6) Artificial aging makes the stress-strain curve flatter after yield, which affects the inelastic buckling strength Designing Aluminum Structures 34 17

18 Tempers Summarized -H is for strain hardened tempers 1xxx, 3xxx, 5xxx alloys Higher 2 nd digit: stronger, less ductile -T is for heat treated tempers 2xxx, 6xxx, 7xxx alloys T4 = solution heat treated T5 and greater = precipitation heat treated Designing Aluminum Structures 35 ASTM Wrought Aluminum Specifications B 209 Sheet and Plate B 210 Drawn Seamless Tubes B 211 Bar, Rod, and Wire B 221 Extruded Bars, Rods, Wire, Profiles and Tubes B 241 Seamless Pipe and Seamless Extruded Tube B 247 Die Forgings, Hand Forgings, Rolled Ring Forgings Designing Aluminum Structures 36 18

19 ASTM Wrought Aluminum Specifications B 308 Standard Structural Profiles B 316 Rivet and Cold Heading Wire and Rod B 429 Extruded Structural Pipe and Tube B 928 High Magnesium Aluminum Alloy Sheet and Plate for Marine and Similar Service (has corrosion resistance reqs) There are others not included in SAS, including some used structurally Designing Aluminum Structures Aluminum Material Properties Strengths Modulus of Elasticity, Poisson s Ratio Ductility Effect of Welding on Properties Designing Aluminum Structures 38 19

20 Aluminum s Stress-Strain Curve Stress F y After yield, plastic behavior Linear region at low strains slope is E Strain Designing Aluminum Structures 39 Yield Strength Aluminum doesn t exhibit a definite yield point like mild carbon steel stress-strain curve is more like high strength steel s 0.2% offset is used to define yield Gage length for yield is 2 in. (50 mm) Artificially aged tempers have different shape stress-strain curve vs. nonartificially aged tempers Designing Aluminum Structures 40 20

21 Types of Strengths Type of Stress Yield Ultimate Tension F ty F tu Compression F cy Shear 0.6F ty F su Designing Aluminum Structures 41 Some Aluminum Alloy Strengths Alloy-temper, product F ty (ksi) F tu (ksi) 5052-H32 sheet & plate H116 plate < 1.5 thick 6061-T6 extrusions T5 extrusions < 0.5 thick Designing Aluminum Structures 42 21

22 Modulus of Elasticity, Poisson s Ratio Modulus of Elasticity (Young s Modulus) E Measures stiffness and buckling strength Compressive E = 1.02(Tensile E) Varies by alloy; E c = 10,100 to 10,900 ksi for SAS alloys Compares to 29,000 ksi for steel Poisson s ratio ν Average value = 0.33 Shear Modulus G = 3800 ksi = E/[2(1+ ν)] Designing Aluminum Structures 43 Ductility Ductility : the ability of a material to withstand plastic strain before rupture Fracture Toughness: Aluminum doesn t have a transition temperature like steel Elongation (e) Notch-Yield Ratio = (F tu of standard notched specimens)/f ty If notch-yield ratio > 1, that s good Designing Aluminum Structures 44 22

23 Effect of Welding Other than alloying, strength comes from strain hardening or artificial aging Heating (like welding) erases these effects So welding reduces strengths: For -H tempers, to annealed (-O) For -T tempers, to solution heat treated(-t4) Reduction is least for 5xxx alloys Some 2xxx, 7xxx aren t weldable Designing Aluminum Structures 45 Welded Strengths Welded strengths are in SAS Table A.3.5 Notation: add w to subscript F tuw = welded F tu AWS D1.2 Table 3.2 gives same F tuw as SAS Table A.3.5 To qualify groove weld procedures, F tuw must be achieved Designing Aluminum Structures 46 23

24 Effect of Welding on Strength Designing Aluminum Structures Aluminum Structural Design Overview Limit states Strength limit state design: Allowable Strength Design (ASD) Load and Resistance Factor Design (LRFD) Analysis Designing Aluminum Structures 48 24

25 Limit States A structural engineer considers limit states Static strength available strength > required strength Serviceability (deflection, vibration, etc.) Fatigue Designing Aluminum Structures 49 What a Structural Engineer Does Analysis: determine forces, moments in the structure (required strength) Use the same methods for all materials But beware: since aluminum is more flexible than steel, 2 nd order effects may be more significant Design: proportion the aluminum structure to safely resist the loads (provide available strength) Designing Aluminum Structures 50 25

26 ASD vs. LRFD Allowable Strength Design (ASD): (strength)/(safety factor) > load effect allowable strength > load effect Load & Resistance Factor Design (LRFD): (strength)(resistance factor) > (load factor)(load effect) design strength > (load factor)(load effect) The difference is load factors Designing Aluminum Structures 51 Safety/Resistance Factors for Aluminum Building Structures Limit State Safety Factor Ω Resistance Factor Yield Ω = 1.65 = 0.90 Rupture Ω = 1.95 = 0.75 Fastener rupture 1.2Ω = 2.34 = 0.65 Designing Aluminum Structures 52 26

27 Aluminum Safety Factors Stress type yielding buckling or rupture Axial tension Bending tension Axial compression Bending compression Shear Designing Aluminum Structures 53 SAS Section C.2: Analysis Must Account for: Axial, flexural, and shear deformations Second-order effects (P- and P-δ) Geometric imperfections (use construction and fabrication tolerances) Effect of inelasticity on flexural stiffness (use τ b I in place of I ) Uncertainty in stiffness and strength (use 0.8E in place of E, i.e ksi) Designing Aluminum Structures 54 27

28 Loads for Analysis For LRFD, use load factors prescribed for LRFD and use the resulting forces and moments produced by the analysis to determine the required strengths P r For ASD, use 1.6 times the ASD loads, then divide the resulting forces and moments produced by the analysis by 1.6 to determine the required strengths P r Designing Aluminum Structures 55 Loads for Analysis Design method Load Combination Analysis Results Required Strength Available Strength LRFD ASD 1.2D + 1.6L 1.6(D + L) P LRFD P ASD P LRFD P ASD /1.6 R n e.g ksi R n /Ω e.g ksi Designing Aluminum Structures 56 28

29 Second-Order Analysis Many structural analysis applications include 2 nd order effects Most applications address P-Δ (effect of loads acting on the displaced location of joints) Some don t include P-δ (effect of loads acting on the deflected shape of member between joints) To check, compare results to benchmark examples given in 2005 AISC Spec commentary Section 7.3 Designing Aluminum Structures 57 P-Δ and P-δ Effects P-Δ Δ P P-δ P δ P P Designing Aluminum Structures 58 29

30 Geometric Imperfections Tolerance on out-of-plumbness specified by the designer This is the structure analyzed Designing Aluminum Structures 59 Effect of Inelasticity on Flexural Stiffness Factor τ b on flexural stiffness for inelasticity is a function of how highly stressed the structure is τ b ranges from: 1.0 for P r <0.5P y /α = 0.31P y for ASD 0 for P r = P y /α = 0.62P y for ASD In between, τ b = (4α P r /P y )(1 α P r /P y ) α = 1.0 for LRFD, 1.6 for ASD Designing Aluminum Structures 60 30

31 5. Axial Tension SAS Chapter D covers axial tension Tensile limit state is reached at: Rupture on the net section (Ω = 1.95) Yield on the gross section (Ω = 1.65) Same criteria as in AISC for steel It s assumed that the net section exists only over a short portion of the member length, so yielding there won t cause much elongation Designing Aluminum Structures 61 Net and Gross Sections Rupture on the net section Yield on the gross section Designing Aluminum Structures 62 31

32 Allowable Tension Stress Example 6061-T6 Extrusions: F ty = 35 ksi, F tu = 38 ksi Allowable stress on the gross section: F /Ω = F ty /Ω = 35/1.65 = 21.2 ksi Allowable stress on the net section: F /Ω = F tu /(Ω k t ) = 38/[(1.95)(1.0)] = 19.5 ksi Net section always governs Designing Aluminum Structures 63 Allowable Tensile Stress Example 6063-T5 Extrusions: F ty = 16 ksi, F tu = 22 ksi Allowable stress on the gross section: F /Ω = F ty / Ω = 16/1.65 = 9.7 ksi Allowable stress on the net section: F /Ω = F tu /(Ω k t ) = 22/[1.95)(1.0)] = 11.3 ksi Gross or net section could govern Designing Aluminum Structures 64 32

33 Tension Coefficient k t k t is a notch sensitivity factor For alloys in SAS, k t > 1 only for : 2014-T6, 6005-T5, and 6105-T5, k t = T6 and 6070-T6, k t = 1.1 Designing Aluminum Structures 65 LRFD Tension Example 6061-T6 Extrusions: F ty = 35 ksi, F tu = 38 ksi LRFD design stress on the gross section: F = y F ty = 0.90(35) = 31.5 ksi LRFD design stress on the net section: F = u F tu /k t = 0.75(38)/(1.0) = 28.5 ksi So just like ASD, net section governs. Designing Aluminum Structures 66 33

34 Shear Lag in a Channel flanges are not connected and not fully stressed at member end Designing Aluminum Structures 67 Effective Area in Tension (A e ) SAS Section D.3.2 If all parts of x-section aren t connected to joint, full net area isn t effective in tension Example: Channel bolted through its web only (not flanges) SAS addresses angles, channels, tees, zees, and I beams Designing Aluminum Structures 68 34

35 Effective Net Area A e Effective net area = where A n = net area x = eccentricity in x direction y = eccentricity in y direction L = length of connection in load direction A e > A n of connected elements Designing Aluminum Structures 69 Effective Net Area Example For a tee bolted through its flange only: Other examples are in ADM Part II D.3.2 When only a single row of fasteners is used, L = 0 and A e = A n of connected elements only Designing Aluminum Structures 70 y neutral axis of tee 35

36 6. Axial Compression Column = axial compression member SAS Chapter E addresses columns Column strength is the least of: Member buckling strength Local buckling strength Interaction between member buckling and local buckling strengths Designing Aluminum Structures 71 Member Strength Member s compressive limit states are Yielding (squashing) Inelastic buckling Elastic buckling Designing Aluminum Structures 72 36

37 Yielding, Inelastic Buckling, and Elastic Buckling Designing Aluminum Structures 73 Elastic Buckling Elastic buckling stress = F e = E / 2 E is the only material property that elastic buckling strength depends on = kl/r = largest slenderness ratio for buckling about any axis All other things equal, F e for aluminum is 1/3 F e for steel since E a = E s /3 Designing Aluminum Structures 74 37

38 Inelastic Buckling Inelastic buckling strength = 0.85[B c - D c (kl/r)] B c (y intercept) and D c (slope) are buckling constants that depend on F cy and E Calculate them by SAS equations in: Table B.4.1 for O, H, T1 thru T4 tempers Table B.4.2 for T5 thru T9 tempers B c and D c are tabulated in ADM Part VI Table 1-1 (unwelded) and 1-2 (welded) Designing Aluminum Structures 75 Member Buckling 0.85 factor accounts for member out-ofstraightness k = 1 for all members (see Section C.3) Allowable member buckling strengths really haven t changed from 2005 SAS: Designing Aluminum Structures 76 38

39 Yielding Yield strength is simply F cy Yielding depends only on material strength Designing Aluminum Structures 77 Buckling Strength vs. Slenderness Stress Yielding F c = F cy Inelastic Buckling F c = 0.85(B c D c kl/r) Elastic Buckling F c = 0.85π 2 E/(kL/r) 2 Slenderness ratio kl/r Designing Aluminum Structures 78 39

40 Slenderness Limits S 1, S 2 S 1 is the slenderness for which yield strength = inelastic buckling strength S 2 is the slenderness for which inelastic buckling strength= elastic buckling strength Slenderness ratios (kl/r) are not limited by S 1 and S 2 ; S 1 and S 2 are just the limits of applicability of compressive strength equations Designing Aluminum Structures T6 Column Strength ADM Part VI, Table 2-19 gives allowable stresses based on SAS rules Inelastic buckling allowable is always < F cy /Ω = 21.2 ksi, so S 1 = 0 For kl/r < 66, F c /Ω = (kL/r) For kl/r > 66, F c /Ω = 51,350/(kL/r) 2 S 2 = C c = 66 Designing Aluminum Structures 80 40

41 Slenderness Limits Demonstrated For kl/r = S 2 = 66: Inelastic buckling allowable stress is (66) = 11.9 ksi Elastic buckling allowable stress is 51,350/(66) 2 = 11.8 ksi 11.9 ksi Difference is only due to round off in allowable stress expressions Designing Aluminum Structures 81 Column Example What s the allowable member buckling compressive stress for a column given: 6061-T6 Pinned-end support conditions Length = 95 Shape is AA Std I 6 x 4.03 r x = 2.53, r y = 0.95 No bracing Designing Aluminum Structures 82 41

42 Column Example Answer Column will buckle about minor axis since slenderness ratio kl/r is larger there: kl/r = (1.0)(95 )/0.95 = 100 Since kl/r = 100 > S 2 = 66 (buckling is elastic), so F c /Ω = 51,350/(100) 2 = 5.1 ksi We need to check local buckling, too Designing Aluminum Structures 83 Flexural & Torsional Column Buckling Deflected shape Undeflected shape Flexural Buckling (lateral movement) Torsional Buckling (twisting) Designing Aluminum Structures 84 42

43 Types of Column Member Buckling Flexural (lateral movement) Torsional (twisting about longitudinal axis) Torsional-flexural (combined effect) Designing Aluminum Structures 85 Local Buckling Local buckling is buckling of an element of a shape (i.e., a flange or web) Buckle length width of element If local buckling strength of all elements > yield strength, shape is compact, and local buckling isn t a concern Since aluminum shapes vary widely (extrusions, cold-formed shapes), we can t assume aluminum shapes are compact Designing Aluminum Structures 86 43

44 Local Buckling buckled shape Web Buckling Flange Buckling Designing Aluminum Structures 87 Elements of Shapes are Called: Element Flange or web Component Plate Designing Aluminum Structures 88 44

45 Dividing a Shape Into Elements Designing Aluminum Structures 89 Elements of Shapes Cross sections can be subdivided into two types of elements: Flat elements (slenderness = b/t ) Curved elements (slenderness = R b /t ) Longitudinal edges of elements can be: Free Connected to another element Stiffened with a small element Designing Aluminum Structures 90 45

46 Element Support Conditions supported edge stiffened edge free edge Designing Aluminum Structures 91 Elements in Uniform Compression Addressed by the SAS B Flat element supported on one edge (flange of an I beam or channel) B Flat element supported on both edges (web of I beam or channel) B Flat element supported on one edge, other edge with stiffener B Flat element supported on both edges, with an intermediate stiffener B Curved element supported on both edges Designing Aluminum Structures 92 46

47 Local Buckling Strengths Yielding F c = F cy Inelastic buckling F c = B p D p (kb/t) Elastic buckling F c = 2 E /(kb/t) 2 Postbuckling F c = k 2 (B p E) 1/2 / (kb/t) k = edge support factor k = 5.0 for elements supported on 1 edge k = 1.6 for elements supported on both edges k 2 = postbuckling factor 2 (Table B.4.3) Designing Aluminum Structures 93 Postbuckling Strength Only elements of shapes have postbuckling strength members do not Postbuckling strength is not recognized by SAS for all types of elements Postbuckling strength is only recognized for elements so slender that they buckle elastically Designing Aluminum Structures 94 47

48 Element Strength vs. Slenderness Designing Aluminum Structures T6 Column Flange Strength Yielding F c /Ω = F cy /Ω S 1 = 6.7 F c /Ω = 35/1.65 = 21.2 ksi Inelastic buckling F c /Ω = [B p D p (5.0b/t)]/ Ω S 2 = 12 F c /Ω = b/t Elastic buckling F c /Ω = [ 2 E /(5.0b/t) 2 ]/ Ω S 2 = 10.5 F c /Ω = 2417 /(b/t) 2 Postbuckling F c /Ω = [k 2 (B p E) 1/2 /(5.0b/t)]/ Ω F c /Ω = 186 /(b/t) Designing Aluminum Structures 96 48

49 Column Local Buckling - Flange Shape is AA Std I 6 x 4.03: flange slenderness = b/t b/t = ( )/2/0.29 = 6.6 S 1 = 6.7 > 6.6 so 4 F c /Ω = 21.2 ksi R 0.29 Designing Aluminum Structures 97 Column Local Buckling - Web Shape is AA Std I 6 x 4.03: web slenderness = b/t b/t = [6 2(0.29 )]/0.19 = 28.5 S 1 = 7.6 < 28.5 < 33 = S 2, so F c /Ω = b/t = F c /Ω = (28.5) = R Designing Aluminum Structures 98 49

50 Weighted Average Allowable Compressive Stress of I 6 x 4.03 F cf /Ω = 21.2 ksi F cw /Ω = 19.0 ksi A f = 2(4 )(0.29 ) = 2.32 in 2 A w = (6 2(0.29 ))(0.19 ) = 1.03 in 2 F ca /Ω = 21.2(2.32) (1.03) ( ) = 20.5 ksi 0.19 R Designing Aluminum Structures 99 The local buckling mode is shown to the right. Note, that there is no translation at the folds, only rotation. The load factor is 0.10, so elastic critical local buckling (P crl ) occurs at 0.10P y in this member. Designing Aluminum Structures

51 Local Buckling/Member Buckling Interaction If the elastic local buckling stress < member buckling stress, member buckling stress must be reduced (SAS Section E.5) Reduced member buckling stress is F rc : F rc = (F c ) 1/3 (F e ) 2/3 where F c = elastic member buckling stress F e = elastic local buckling stress This only governs if elements are very slender and postbuckling strength is used Designing Aluminum Structures 101 Local/Member Buckling Interaction Example Flange elastic buckling stress F ef Web elastic buckling stress F ew Designing Aluminum Structures

52 Local/Member Buckling Interaction Example Member elastic buckling stress F c Since F e = 47.9 ksi > F c = 8.5 ksi, the member buckling strength need not be reduced for interaction between local and member buckling Designing Aluminum Structures 103 Column Design Summary Column strength is the least of: Member buckling strength Local buckling strength Interaction between member and local buckling strengths Designing Aluminum Structures

53 7. Flexure Beam = flexural member Beam strength is limited by: Rupture (Ω = 1.95) Yielding (Ω = 1.65) Buckling (local or member) (Ω = 1.65) Buckling can be: Flexural stress buckling Shear stress buckling M M Designing Aluminum Structures 105 Yielding and Rupture in Beams F y F y F u neutral axis M y M p M u F y F y F u cross section yielding onset full yielding rupture Designing Aluminum Structures

54 Shape Factors Sec. No. Element M p /M y M u /M y F Uniform stress F.6.1 Curved element F Flexural stress Designing Aluminum Structures 107 Yielding and Rupture Example 6061-T6 extruded round tube For rupture, F b /Ω = 1.24 F tu /(k t Ω) for 6061-T6, F b /Ω = 1.24(38)/[(1.0)(1.95)] F b /Ω = 24 ksi For yielding, F b /Ω = 1.17 F ty / Ω for 6061-T6, F b /Ω = 1.17(35)/(1.65) F b /Ω = 25 ksi So F b /Ω = 24 ksi (the lesser of the two) Designing Aluminum Structures

55 Major Axis Bending Load Top flange (in compression) buckles laterally Undeflected shape Deflected shape at ultimate load Bottom flange (in tension) stays In place Lateral-torsional buckling Designing Aluminum Structures 109 Weak Axis Bending Load Undeflected shape Deflected shape at ultimate load Lateral buckling usually does not govern Designing Aluminum Structures

56 Lateral-Torsional Buckling (LTB) Section Shape Slenderness F.2 Open shapes F.3 Closed shapes F.4 Rectangular bars F.5 Single Ls Designing Aluminum Structures 111 Slenderness Ratio for Beams Slenderness ratio depends on unbraced beam length L b L b = length between bracing points or between a brace point and the free end of a cantilever beam. Braces: restrain the compression flange against lateral movement, or restrain the cross section against twisting Appendix 6 addresses brace design Designing Aluminum Structures

57 6061-T6 I-Beam Strength Inelastic buckling F c /Ω = [B c D c (1.2L b /r y )]/Ω S 2 = 79 F c /Ω = (L b /r y ) Elastic buckling F c /Ω = [ 2 E /(1.2L b /r y ) 2 ]/ Ω F c /Ω = 87,000 /(L b /r y ) 2 Designing Aluminum Structures 113 LTB Example What s the allowable LTB stress for a beam given: 6061-T6 Length = 86 Shape is AA Standard I 12 x 14.3 r y = 1.71 No bracing Designing Aluminum Structures

58 LTB Example Answer Slenderness ratio is L b /r y = 86 /1.71 = 50.3 Since L b /r y = 50.3 < 79 = S 2, F c /Ω = (L b /r y ) F c /Ω = (50.3) = 17.7 ksi Remember to check local buckling, too Designing Aluminum Structures 115 Moment Variation Over Beam Length (C b ) If moment varies along the beam, account for this using C b. For doubly sym sections: C b = 12.5M max (2.5M max + 3M A + 4M B + 3M C ) where M A = moment at ¼ point M B = moment at midpoint M C = moment at ¾ point Designing Aluminum Structures

59 Moment Gradient Factor C b C b = 1.0 (min) M M C b = 2.3 M M C b = 1.14 w Designing Aluminum Structures 117 Moment Gradient Factor C b Replace unbraced length L b with L b /C b for tubes and L b / C b for single web beams So L b can be reduced to as little as 0.43L b = L b /2.3 Designing Aluminum Structures

60 Effective r y (= r ye ) F.2.1 allows using r y for r ye It s easy to determine, but conservative When L b / r y > 50, it s worth determining r ye using F.2.2. It s more work, but more accurate Designing Aluminum Structures 119 Transverse Load Location Load acts toward shear center (smallest bending strength) Load applied at neutral axis Load acts away from shear center (greatest bending strength) Designing Aluminum Structures

61 r ye for Shapes Symmetric About the Bending Axis Load applied toward shear center Load applied at shear center, or no load Load applied away from shear center Designing Aluminum Structures 121 I 12 x 14.3 Beam, 200 long SAS Apply r ye L b /r ye F b /Ω Section Load (in.) (ksi) F F above s.c F at s.c F at s.c F below s.c Designing Aluminum Structures

62 Local Buckling of Beam Elements Beam elements in uniform compression (flanges) are treated like column elements in uniform compression (see B.5.4) Designing Aluminum Structures 123 How Can Beam Elements Be In Uniform Compression? To be precise, they can t But the SAS idealizes elements parallel to the neutral axis as being in uniform compression Example: the flange of an I-beam or channel - there is some variation in the compressive stress across the flange thickness, but it s neglected Designing Aluminum Structures

63 Beam Flange Stress neutral axis Compression Tension cross section bending stress Designing Aluminum Structures 125 Beam Elements in SAS Elements in Flexure (Webs) B Flat element - both edges supported (web of I beam or channel) B Flat element - compression edge free, tension edge supported B Flat element with a longitudinal stiffener both edges supported (see B for stiffener requirements Designing Aluminum Structures

64 Compression Bending Strength Compression bending strength is the lesser of: Member buckling strength Local buckling strength Designing Aluminum Structures 127 Weighted Average Bending Strength (SAS F.8.3) compression side I f c cf c cw I w c tf c tw I f tension side Designing Aluminum Structures

65 Thank You Please contact me with questions Farmview Rd, Hillsborough, NC For a more extensive aluminum seminar: go to click on View a complete list of ASCE courses, scroll down alphabetically to Aluminum Structural Design (or call ) Designing Aluminum Structures