Curved Steel I-girder Bridge LFD Guide Specifications (with 2003 Edition) C. C. Fu, Ph.D., P.E. The BEST Center University of Maryland October 2003

Transcription

1 Curved Steel I-girder Bridge LFD Guide Specifications (with 2003 Edition) C. C. Fu, Ph.D., P.E. The BEST Center University of Maryland October 2003

2 Guide Specifications ( ) 2.3 LOADS 2.4 LOAD COMBINATIONS AND LOAD FACTORS 2.5 DESIGN THEORY 2.6 FATIGUE 2.7 EXPANSION AND CONTRACTION 2.8 BEARINGS 2.9 DIAPHRAGMS, CROSS FRAMES, AND LATERAL BRACING 2.12 NONCOMPOSITE GIRDER DESIGN

3 ** 2003 New Guide Specifications Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Section 8 Section 9 Section 11 Section 12 Section 13 Limit States Loads Structural Analysis Flanges with One Web Webs Shear Connectors Bearings I Girders Splices and Connections Deflections Constructibility

4 ** 2003 New Guide Specifications Section STRENGTH LIMIT STATE (same as 2002; LFD only) 2.3 FATIGUE LIMIT STATE (use modified AASHTO LRFD Art with load factor of 0.75 instead; Uncracked section is used as per Art ) 2.4 SERVICEABILITY LIMIT STATE (same as 2002; Deflection and Concrete Crack Control) 2.5 CONSTRUCTIBILITY LIMIT STATE (This limit state considers deflection, strength of steel and concrete and stability during critical stage of construction)

5 ** 2003 New Guide Specifications Section 9 Deflection - Max. Preferable Span-to-Depth Ratio Las 50 = 25 d F - Live Load Deflections D LL+I < L/800 D LL+I+sidewalk < L/1000 y ** Effective for the full width and full length of the concrete deck

6 2.3 LOADS! Uplift must be investigated, without reduction for multiple lane loading and during concrete placement AASHTO 3.17 Check D + 2 (L + I) AASHTO and 1.5 (D + L + I) Anchor bolts! Impact same as current AASHTO ( I = 50 / (L + 125) 0.3 AASHTO Eq. 3.1 )** ** Revised in 2003.

7 ** 2003 New Guide Specifications Section 3! Impact Art I-Girders. Load Effect Girder bending moment, torsion and deflections Reactions, shear, cross frame and diaphragm actions Art Box Girders. Load Effect Girder bending moment, torsion and deflections Reactions, shear, cross frame and diaphragm actions Impact Factor Vehicle Lane Impact Factor Vehicle Lane Art Fatigue - 15%

8 2.3 LOADS (cont.)! Centrifugal Force same as current AASHTO ( C = S 2 D = 6.68 S 2 / R AASHTO Eq. 3.2 ) ( S = the design speed in miles/hr )! Thermal Forces do not have to be considered if temperature movements are allowed to occur (more in Article 1.6 & 1.7)! Superelevation redistribution of wheel load

9 2.4 LOAD COMBINATIONS AND LOAD FACTORS (A) Construction Loads (Construction Staging) 1.3 (D p + C) (B) Service Loads (Stress range values and live load deflections) D + L r (C) where L r L I CF = L + I + CF = Basic live load = Impact loads = Centrifugal force Overloads D + 5/3 L r

10 2.4 LOAD COMBINATIONS AND LOAD FACTORS (cont.) For all loadings less than H20 Group IA: 1.3 [D L r ] (Single lane) Group II: 1.3 [D + W + F + SF + B + S + T] (EQ may replace W and ICE may replace SF) Group III: 1.3 [D + L r + 0.3W + WL + F + LF] (D) Maximum Design Loads Group I: 1.3 [D + 5/3 L r ] For uplifting reaction: 1.3 [0.9D + 5/3 L r ]

11 2.5 DESIGN THEORY! General Analysis should be of entire structure However, primary bending moment curvature effects may be neglected if the following conditions exist: No. of Girders Limiting Central Angle 2 3 or 4 5 or more 1 Span 2 o 3 o 4 o 2 or more Spans 3 o 4 o 5 o

12 2.5 DESIGN THEORY! Torsion Diaphragms must be provided and designed as primary members! Nonuniform Torsion otherwise known as warping or lateral flange bending, this effect always must be taken into account! Composite Design may be used, but shear connectors must be designed using procedures that will be discussed at a later point

13 ** 2003 New Guide Specifications Section 5 The effects of curvature may be ignored if! Girders are concentric,! Bearing lines are not skewed more then 10 degrees from radial, and! The arc span divided by the girder radius is less then 0.06 radians where the arc span, L as, shall be taken as follows: For simple spans: L as = arc length of the girder, For end of spans of continuous members: L as = 0.9 times the arc length of the girder, For interior spans of continuous members: L as = 0.8 times the arc length of the girder.

14 ** 2003 New Guide Specifications Section 4 Simplified formula for the lateral bending moment in I- girder flanges due to curvature. Where: 6Ml M lat = 5RD 2 Eq. (4-1) M lat = lateral flange bending moment (k-ft) M = vertical bending moment (k-ft) l = unbraced length (ft) R = Girder radius (ft) D = web depth (in)

15 ** 2003 New Guide Specifications Section 4 Methods (Art. 4.3)! Approximate Methods: V-Load method for I-girder bridges M/R Method for Box-girder bridges! Refined Methods: Finite Strip method. Finite Element method. (including gird model considering torsional warping)

16 Equation & Solution for Concentrated Torsion M DESCUS Alternate Solution for Torsion and Warping Differential Equations M EC ω 1 = 2 a φ φ Z Z φ = A + B cosh + C sinh + a a MZ GJ Equation & Solution for Uniform Torsion m Equation & Solution for Linear Varying Torsion m 1 a 2 φ = φ φ = A + BZ 1 φ φ = 2 a φ = A + BZ m EC ω + C cosh mz LEC ω + C cosh Z a Z a + + Dsinh Dsinh Z a Z a 2 mz 2GJ 3 mz 6GJL

17 DESCUS Alternate Solution for Torsion and Warping Differential Equations Fixed-Fixed End Solution for Uniform and Linear Varying Torsion m Reference: Torsion Analysis published by Bethlehem Steel

18 DESCUS Alternate Solution for Torsion and Warping Differential Equations Fixed-Hinged End Solution for Uniform and Linear Varying Torsion m

19 DESCUS Alternate Solution for Torsion and Warping Differential Equations Stresses induced by Torsion & Warping Formula 1 τ = Gt φ t Formula 2 τ ω s = ES s φ ω t Formula 3 σ ω EW φ s = ns

20 DESCUS Alternate Solution for Torsion and Warping Differential Equations Stresses induced by Bending Formula 4 σ b = M b I y Formula 5 τ b = VQ It

21 2.5 DESIGN THEORY (cont.) Overload ** (f b + f w ) overload < 0.8 F y noncomposite < 0.95 F y composite ** Modified in 2003 New Specifications Art. 9.5 Permanent Deflections as (f b ) overload < 0.95 F y Continuously braced flanges & partially braced tension flanges (f b ) overload < F cr1 = F bs ρ w ρ s < 0.95 F y Partially braced compression flange (f b,web comp. ) overload < F cr 0.9Ek = 2 D t w F y

22 2.6 FATIGUE Same as Straight Bridge except for the contribution of torsional stress AASHTO Standard Spec. Allowable Fatigue Stress Range (Redundant or nonredundant; stress cycles & stress category) Stress cycles Stress category AASHTO LRFD (** 2003 New Guide Specs.) Infinite fatigue life (ADTT single-lane $ 2000 Trucks/day F n = ( F) threshold Finite fatigue life (ADTT single-lane < 2000) F n = F = ( A/N ) 1/3

23 ** 2003 New Guide Specifications Section 4 " Fatigue Load Design truck only with constant 30' between 32-kip axles. (LRFD Art with a load factor of 0.75) ADTT single-lane = p x ADTT where p = 1.0 for one-lane bridge = 0.85 for two-lane bridge = 0.80 for three-lane or more bridge Class of Highway ADTT/ADT Rural Interstate 0.2 Urban Interstate 0.15 Other Rural 0.15 Other Urban 0.10

24 ** 2003 New Guide Specifications Section 4 Fatigue Limit State ( ()f) # ()F) n (LRFD Eq ) Infinite Fatigue Life (ADTT single-lane $ 2000 trucks/day) High traffic volume F n = ()F) threshold = 24 ksi for Category A 16 B 12 BN 10 C 12 CN 7 D 4.5 E 2.6 EN

25 ** 2003 New Guide Specifications Section 4 Fatigue Limit State (cont.) Finite Fatigue Life (ADTT single-lane < 2000 trucks/day) ( 365)( 75)( )( ADTT) single lane N = n Category A ()F) TH (ksi) Fn = F = A N n ( F ) TH A 2.5 H R > 40! R < 40! B 1.2 H Simple-span BN 6.1 H C 4.4 H Continuous CN 4.4 H (1) Interior D 2.2 H Support E 1.1 H (2) Elsewhere EN 3.9 H

26 2.7 EXPANSION AND CONTRACTION Must allow thermal movements to take place in directions radiating from fixed supports 2.8 BEARINGS! Must permit horizontal movements in directions radiating from fixed supports! Must allow angular rotation in a tangential vertical plane! Must hold down in cases of uplift

27 2.9 DIAPHRAGMS, CROSS FRAMES, AND LATERAL BRACING Same as current AASHTO except as follows:! Diaphragms/Crossframes shall be provided at each support and at intervals between Diaphragms/Crossframes shall extend in a single plane across full width of bridge! Diaphragms/Crossframes need not be located along skew at interior supports! Diaphragms/Crossframes should be full depth design as main structural element! Diaphragms/Crossframes/Lateral Bracing should be framed to transfer forces to flanges and webs, as necessary

28 2.9 DIAPHRAGMS, CROSS FRAMES, AND LATERAL BRACING (cont.) Suggested Diaphragm/Crossframe spacing Centerline Radius of Bridge (feet) Below to to Over (B) Lateral Bracing Max. stress of the bottom lateral wind bracing Suggested Max. Spacing (feet) ƒ b = ƒ d (DF) br ƒ d = Max. Stress in Crossframes from grid analysis ( DF) ( 016. L 10) 43 / ( SG) = SD bl L L R

29 ** 2003 New Guide Specifications Section 9 Intermediate cross frames should be spaced as nearly uniform as practical to ensure that the flange strength equations are appropriate. Equation (C9-1) may be used as a guide for preliminary framing. l = 5 36 r σ Rb f Eq. (C9-1) Where: l r σ R b f = cross frame spacing (ft) = desired bending stress ratio, = girder radius (ft) = flange width (in) f l f b

30 ** 2003 New Guide Specifications Section 5 fl 0. 5F y Eq. (5-1) where: F y = specified minimum yield stress (ksi) When f b is greater than or equal to the smaller of 0.33F y or 17 ksi, then: f l f b 0.5 Eq. (5-2) The unbraced arc length, l, in feet between brace points at cross frames or diaphragms shall satisfy the following: l 25b and R /10 where: b = minimum flange width in the panel (in) R = minimum girder radius within the panel (ft) Eq. (5-3)

31 2.12 NONCOMPOSITE GIRDER DESIGN (B) Maximum Normal Flange Stress! Compact Sections Compression Flange Requirement: (** New: 18) Allowables: where F bs = F y (1-3 λ 2 ) b 3200 t F F y = 1 F ρ ρ cr bs B w Eq. (5-4) where λ = 1 12l π b F y E ** Revised in 2003 ** 2003 New Guide Specs use the same equation, except l/b f in all the equations to be consistent with the units used

32 2.12 NONCOMPOSITE GIRDER DESIGN (cont.) ρ b = 1 + b l 2l l 1 f b 6 f R ** Revised in 2003 ρ w = l R + f f w b l l Rbf F bs ρ b F y ** Revised in 2003 ρ ρ 10. B w Tension Flange f b < F bu = F y ** Superseded

33 ** 2003 New Guide Specifications Section 5 " Compact Compression (Art ) F cr = smaller F F cr1 cr 2 = = F F bs y ρ b ρ w f l 3 (Eq. 5-4) (Eq. 5-5) F cr for Partially Braced Tension Flange is the same as F cr for compact compression. F cr for Continuously Braced Tension Flange is F y

34 2.12 NONCOMPOSITE GIRDER DESIGN (cont.) (B) Maximum Normal Flange Stress (cont.)! Non-Compact Sections Compression Flange Requirement: Allowables: 3200 b 4400 < F t y Fy ** Superseded f b # F by = F bs ρ w ρ s (f b + f w ) # F y where Tension Flange F bs = F y (1-3 λ 2 ) (Same definition for ρ b and ρ w ) (f b + f w ) # F by = F y ** Superseded

35 ** 2003 New Guide Specifications Section 5 " Non-Compact Compression (Art ) b t f f E 1.02 ( f + ) b f l 23 Eq. (5-7) F cr = F smaller F cr cr 2 1 = Fbsρbρ = F y w f l Eq. (5-8) Eq. (5-9)

36 ** 2003 New Guide Specifications Section 5 ρb = 1+ ρ ρ w1 w2 1 l 12l R 75b f 1 = f l 12l 1 1 f b 75b f 12l b f.95 + l , R = f l fb 0 2 When f l f b 0 ρ w = min(ρ w1, ρ w2 ); f l f b < 0, ρ w = ρ w1 ** Revised in 2003 ** Revised in 2003 ** Revised in 2003

37 ** 2003 New Guide Specifications Section 5 " Partially Braced Tension Flanges (Art. 5.3) F cr = smaller F F cr1 cr 2 = = F F bs y ρ b ρ w f l 3 Eq. (5-10) Eq. (5-11) " Tension Flanges and Continuously Braced Flanges (Art. 5.4) F cr = F y F cr = smaller F F cr1 cr 2 = = F F bs y ρ b ρ w f l 3

38 ** 2003 New Guide Specifications Section 6 " Unstiffened Web (Art. 6.2) for R 700 feet for R > 700 feet D t w 100 Eq. (6-1) t D w ( R 700) Eq. (6-2) Where: D = distance along the web between flanges (in) t w = web thickness (in) R = minimum girder radius in the panel (ft)

39 ** 2003 New Guide Specifications Section 6 " Web Bending Stresses (Art , & 6.4.1) where: F cr 0.9Ek = 2 D t w k = bend-duckling coefficient = 7.2 (D/D c ) 2 for unstiffened webs = 9.0 (D/D c ) 2 for transversely stiffened webs D 2 ds = k = 5.17( ) for 0.4 d for longitudinally s Dc = D 2 ds k = 11.64( ) for < 0.4 stiffened webs D d D c s Eq. (6-3) D c = depth of web in compression (in) d = distance between long. stiffener and comp. flange (in) c F y

40 2.12 NONCOMPOSITE GIRDER DESIGN (cont.) (C) Web Design! Webs Without Stiffeners V u = C V p where V p = 0.58 F y D t for D t w < 6000, F y K for 6000, K D 7500, F t F y w y K C = 1.0 C 6000, = D t w K F y

41 2.12 NONCOMPOSITE GIRDER DESIGN (cont.) for D t w > 7500, F y K C = K D t w 2 F y where buckling coefficient K = 5 ** 2003 New Guide Specs use the same equation, except E is included in all the equations to be consistent with LRFD format.

42 ** 2003 New Guide Specifications Section 6 " Transversely Stiffened Webs D/t w < 150 For R 700 feet d o = D Eq. (6-7a) For R > 700 feet [ ( R 700) ] D D d o = 3 Eq. (6-7b) where: R = minimum girder radius in the panel (ft)

43 2.12 NONCOMPOSITE GIRDER DESIGN (cont.)! Transversely Stiffened Girders V u = C V p (Calculations for C and V p are the same as for webs without stiffeners except 5! Requirements for Transverse Stiffeners I # d o t 3 J where K = 5 + J do D 2 2 D = X do

44 ** 2003 New Guide Specifications Section 6 " Transverse Web Stiffeners Width-to-Thickness Ratio Requirement b t s s 0.48 E F y Eq. (6-13) Where: t s = stiffener thickness (in) b s = stiffener width (in) F y = specified minimum yield stress of stiffener (ksi)

45 ** 2003 New Guide Specifications Section 6 Moment of Inertia Requirement I = d t 3 ts o w Eq. (6-14) where: J = 2 X 0.5 Eq. (6-15) d / D J X = 1.0 for a 0.78 Eq. (6-16) X = a Z 1,775 Eq. (6-17) a = aspect ratio d o = D 2 Z = 0.079do 10 Eq. (6-18) Rtw d o = actual distance between transverse stiffeners (in) R = minimum girder radius in the panel (ft)

46 ! Longitudinally Stiffened Girders Not required if One stiffener required (D/5 from the compression flange) if Two stiffeners required (D/5 from both flanges) if! Requirements for Longitudinal Stiffeners 2.12 NONCOMPOSITE GIRDER DESIGN (cont.) D t F d R d R y o o ,. D t F d R d R y o o ,.. D t F y ,

47 ** 2003 New Guide Specifications Section 6 " Longitudinal Web Stiffeners Moment of inertia Requirement I ls Dt 3 w ( 2 2.4a 0.13)β Eq. (6-19) where: Z β = +1 when the longitudinal stiffener is on the side 6 of the web away from the center of curvature β = Z +1 when the longitudinal stiffener is on the side 12 of the web toward the center of curvature

48 ** 2003 New Guide Specifications Section 6 " When single transverse stiffeners are used, they preferably shall be attached to both flanges. When pairs of transverse stiffeners are used, they shall be fitted tightly to both flanges. " Transverse stiffeners used as connection plates shall be attached to the flanges by welding or bolting with adequate strength to transfer horizontal force in the cross members to the flanges. " Two longitudinal stiffeners may be used on web section where stress reversal occurs. " Longitudinal and transverse stiffeners preferably shall be attached on opposite sides of the web. " Bearing stiffeners shall be placed in pairs, bolted or welded to the girder web or diaphragm

49 CURVED COMPOSITE I-GIRDERS 2.13 GENERAL 2.14 EFFECTIVE FLANGE WIDTH 2.15 NONCOMPOSITE DEAD LOAD STRESSES 2.16 COMPOSITE SECTION STRESSES 2.17 SHEAR CONNECTORS

50 2.13 GENERAL! Girders shall be proportioned for ordinary bending and for nonuniform torsion (also called warping or lateral flange bending)! Usual moment of inertia method shall be used for bending! Any rational method shall be used for nonuniform torsion! For fatigue, perform stress analysis including bending and nonuniform torsion

51 2.14 EFFECTIVE FLANGE WIDTH AASHTO Standard Spec. (Art ;**2003 Art ) not to exceed: ¼ of the span length of the girder the distance center to center of girders 12 x the least thickness of the slab AASHTO LRFD (Art ) For interior beams, the least of: ¼ of the effective span length 12 x the average thickness of the slab + greater of (web thickness, ½ the width of the top flange of the girder) the average spacing of adjacent beams

52 2.15 NONCOMPOSITE DEAD LOAD STRESSES! (f b ) DL1 # F b determined from Art (B)! (f b + f w ) DL1 # F y determined from Art (B)! b t f DLI where f DL1 = (f b + f w ) DL1 ** Revised in 2003, same equations for noncomposite and composite

53 2.16 COMPOSITE SECTION STRESSES! Compact within Concrete f b # F y! Noncompact, compression flange braced by cross-frames or diaphragms f b # F by f b + f w # F y braced by concrete f b + (f w ) DL # F y ** Revised in 2003, same equations for noncomposite and composite

54 2.17 SHEAR CONNECTORS Same as current AASHTO for straight girders, except as modified by Article 1.19 (A)Fatigue (B) Ultimate Strength ** Covered in 2003 Section 7 Shear Connectors

55 ** 2003 New Guide Specifications Section 7 - Strength N P = φ S sc u Eq. (7-1) where: φ sc =0.85 S u = strength of one shear connector according to AASHTO Article (kip) P = force in slab at point of maximum positive live load moment given by Equation (7-2) (kip) P = P p F p Eq. (7-2)

56 ** 2003 New Guide Specifications Section 7 where: P p F p = longitudinal force in the slab at point of maximum positive live load moment computed as the smaller of P 1p or P 2p from Equations (7-3) and (7-4) (kip) = radial force in the slab at point of maximum positive live load moment computed from Equation (7-5) (kip) P = 1 p where: A s = area of steel girder (in 2 ) A s F y Eq. (7-3) P 2 p = f c b d t d Eq. (7-4) where: f c = specified 28-day compressive stress of concrete (ksi) b d = effective concrete with as specified in Article (in) t d = average concrete thickness (in)

57 ** 2003 New Guide Specifications Section 7 where: L p = arc length between an end of the girder and an adjacent point of maximum positive live load moment (ft) R = minimum girder radius over the length, L p (ft) " Between M LL+ and M LL- - where: P T F T F p = P P L R P = + Eq. (7-5) Eq. (7-6) = longitudinal force in the concrete slab at point of greatest negative live load moment computed from Equation (7-7) (kip) = radial force in the concrete slab at point of greatest negative live load moment computed from Equation (7-10) (kip) p 2 2 P T F T

58 ** 2003 New Guide Specifications Section 7 where: P n P = P + = the smaller of P 1n or P 2n (kip) P 2n T P = 1n p A = s f F c P y b d n t d Eq. (7-7) Eq. (7-8) Eq. (7-9) L F = n Eq. (7-10) T PT R where: L n = arc length between a point of maximum positive live load moment and an adjacent point of greatest negative live load moment (ft) R = minimum girder radius over the length, L n (ft)

59 ** 2003 New Guide Specifications Section 7 - Fatigue 2 ( V ) ( ) 2 fat F V sr = + where: V fat = longitudinal fatigue shear range/unit length (k/in) F fat = radial fatigue shear range/unit length (k/in) F fat = wr where: A bot = area of bottom flange (in 2 ) σ flg = range of fatigue stress in the bottom flange (ksi) l = distance between brace points (ft) R = minimum girder radius within the panel (ft) w = effective length of deck (in) A bot σ flgl fat Eq. (7-11) Eq. (7-12)

60 where: ** 2003 New Guide Specifications Section 7 Eq. (7-13) F CR = net range of cross frame force at the top flange (kip) where: F Eq. (7-14) n = number of shear connectors in a cross section fat Z r = shear fatigue strength of an individual shear connector determined as specified in AASHTO LRFD Article (kip) V sr = range of horizontal shear for fatigue from Equation (7-11) (kip) = F CR w nz p = V sr r

61 CURVED HYBRID GIRDERS 2.18 GENERAL 2.19 ALLOWABLE STRESSES 2.20 PLATE THICKNESS REQUIREMENTS ** not covered in 2003 new Guide Specifications

62 2.18 GENERAL! Pertains to hybrid I-girders having a vertical axis of symmetry through the middle-plane of the web plate

63 2.19 ALLOWABLE STRESSES (A) Bending-Noncomposite Girders! Allowable flange stress Equal to allowable stress from Article 2.12 multiplied by reduction factor R = 1 2 ( 1 α ) ( 3 + α ) 6+ BΨ( 3 Ψ) BΨ Ψ Ψ where F y = yield strength of compression flange B = Ψ= web area tension flange area C of tension flange depth of section

64 2.19 ALLOWABLE STRESSES (Cont.) ( / ) w α = α 1 + f f b t α = web yield strength tension flange yield strength f w / f b t = absolute value of (lateral flange bending stress / bending stress) for tension flange If f / f 1 α, then R = 1 w b t α

65 2.19 ALLOWABLE STRESSES (Cont.) (B) Bending Composite Girders! Allowable flange stress positive moment region Same as for noncomposite girder! Allowable flange stress negative moment region Same as for noncomposite girder, except α is defined as follows: T when f f w b c 2Ψ 1 = 1 Ψ, α α T when 2 Ψ 1 f w Ψ < < 1 Ψ f 1 b c α ( Ψ) 1

66 2.19 ALLOWABLE STRESSES (Cont.) = + α α 1 1 f f w b c Ψ Ψ where is the same as f f f f w b c w b t except that it is for the compression flange ( ) if, then R = 1 f f w b Ψ Ψ α 1 1

67 2.19 ALLOWABLE STRESSES (Cont.)! Shear (same as homogeneous section in Art. 2.12(C)! Fatigue V u = C V p Same as current AASHTO for straight hybrid girders

68 2.20 PLATE THICKNESS REQUIREMENTS! Web plate thickness requirements Same as for non-hybrid curved girders, except that f b in the formulas is replaced by f b / R! Flange plate thickness requirements Same as for non-hybrid curved girders

69 THE END Questions??