Specifications Prestressed Concrete

Transcription

1 AASHTO LRFD Bridge Design Speciications Prestressed Concrete RICHARD A. MILLER, PhD, PE, FPCI PROFESSOR UNIVERSITY OF CINCINNATI AASHTO-LRFD Speciication, 4th Edition General This module covers prestressed concrete superstructure elements. Segmental boxes are NOT covered. Topics which are related to reinorced concrete only are covered in another module. Concrete structures are covered in Chapter 5. Chapter 5 uses a uniied approach reinorced concrete and prestressed concrete are covered in the same chapter. Loads and load combinations related to concrete are covered in Chapter 3. Analysis o concrete structures is covered in Chapter 4. Do Not Duplicate Prestressed Concrete: Slide #2

2 General LRFD equations are in KSI units! Example Modulus o Rupture: r = ksi = 0.530ksi = psi = 530 psi r LRFD STD. SPEC. In most cases, the equations are simply the old Standard Speciications equations converted to ksi units psi 1000 psi / ksi 7.5 = ksi 7.5 5ksi = 1000 = ksi Do Not Duplicate Prestressed Concrete: Slide #3 5.4 Material Properties Materials must meet AASHTO LRFD Bridge Construction Speciications. Unless speciied otherwise, all provisions apply or strengths up to 10 ksi (Art ). Some provisions allow up to 15 ksi. There is an eort to extend all provisions to 18 ksi. I a provision does not allow higher strength, use a maximum o 10 ksi in the calculations. Decks must have a minimum strength o 4 ksi. Do Not Duplicate Prestressed Concrete: Slide #4

3 5.4 Material Properties A current problem with the LRFD Speciications is that some provisions allow strengths up to 18 ksi, but many are limited to 15 ksi or the deault o 10 ksi. So what do you do i you are using a high strength concrete and a speciic provision does not allow that strength? Use the highest strength allowed by that provision. For example, assume a 15 ksi strength is speciied but a particular provision has not been veriied or that strength. For that particular provision, you must use a concrete strength o 10 ksi or your calculations (you may still use 15 ksi concrete in the structure, you just cannot take advantage o the additional strength or that particular provision). However, i other provisions allow the use o 15 ksi concrete, you can use 15 ksi or those provisions. Do Not Duplicate Prestressed Concrete: Slide #5 5.4 Material Properties Shrinkage and Creep For calculation o creep and shrinkage, the engineer may use: Articles and CEB-FIP Model Code ACI 209 For prestressed concrete the loss equations include creep and shrinkage. The main use o these provisions or prestressed concrete is or calculating restraint moments or continuous or live load bridges. These are veriied to 15 ksi. The creep equations do not work or strengths over 15 ksi. Do Not Duplicate Prestressed Concrete: Slide #6

4 5.4 Material Properties Shrinkage and Creep Creep Coeicient ( Art ) : ψ k k k k ( t, t ) = vs hc td i 1. 9k vsk hck k tdt i V = S = H 5 = 1+ ' ci t = 61 4 ci ' + t H = Relative Humidity t = time rom irst loading to time being considered t i = time o irst loading V/S = volume to surace ci = concrete strength at time o prestress transer or time o irst load (RC). I unknown, assume = 0.8 c. Std. Spec did not have a creep coeicient. Previous versions o LRFD use a dierent equation. It is similar to the ACI equation using t 0.6 /(10+ t 0.6 ). Do Not Duplicate Prestressed Concrete: Slide #7 5.4 Material Properties Shrinkage and Creep Shrinkage ( Art ) : 3 ( 0.48x ) ε sh = k vsk hsk k td 10 k k k k vs hs td V = S = H 5 = 1+ ' ci t = 61 4 ci ' + t H = Relative Humidity t = time rom end o cure to time being considered V/S = volume to surace ci = concrete strength at time o prestress transer or time o irst load (RC). I unknown, assume = 0.8 c. Std. Spec. set shrinkage = Previous editions o LRFD used an ACI type equation with a term o t/(35+t). Do Not Duplicate Prestressed Concrete: Slide #8

5 5.4 Material Properties Modulus o Rupture There are now 3 deined Moduli o Rupture or normal weight concrete: For Arts (crack control) and (I e ): 0.24 c ksi (= 7.5 c in psi units) For Art (minimum reinorcement): 0.37 c ksi (= 11.5 c in psi units) For Art (shear) (this is new in 2007): 0.20 c ksi (= 6 c in psi units) Note that the value or Article (shear) ONLY applies to the new, simpliied method. Do Not Duplicate Prestressed Concrete: Slide #9 5.4 Material Properties Modulus o Elasticity & Poisson s Ratio E = 33,000K w ' 1.5 c 1 c c µ = ( ) (5425) ( ) Where: K 1 = Aggregate actor. Taken as 1.0 unless determined by testing or as approved by a jurisdiction. w = concrete unit weight in kc c = concrete strength ksi E is basically the old Standard Speciications equation converted to ksi units and with an aggregate correction actor added. µ is unchanged rom Standard Speciications. Do Not Duplicate Prestressed Concrete: Slide #10

6 3.4 - Loads and Load Factors 3.4.1: Load Factors and Load Combinations For prestressed girders, the ollowing service load combinations are most common: Service I: Used or compression and transverse tension in prestressed concrete. Service III: Used or longitudinal tension in prestressed concrete girders. Service IV: Used or tension in prestressed columns, or crack control. Strength I: Basic load combination. Fatigue : Fatigue o reinorcement does NOT need to be checked or ully prestressed components designed using Service III (Art ) Strength II-V and Extreme Event I and II are checked as warranted. Service II is or steel and never applies to prestressed concrete. Do Not Duplicate Prestressed Concrete: Slide # Loads and Load Factors 3.4.1: Load Factors and Load Combinations Load Combination DC DD DW EH EV ES EL Table Load Combinations and Load Factors LL IM CE BR PL LS WA WS WL FR Use One o These at a Time TU CR SH TG SE EQ IC CT CV STRENGTH I (unless noted) γ p /1.20 γ TG γ SE STRENGTH II γ p /1.20 γ TG γ SE STRENGTH III γ p /1.20 γ TG γ SE STRENGTH IV γ p / STRENGTH V γ p /1.20 γ TG γ SE Do Not Duplicate Prestressed Concrete: Slide #12

7 3.4 - Loads and Load Factors 3.4.1: Load Factors and Load Combinations Load Combination Table Load Combinations and Load Factors (cont.) DC DD LL Use One o These at a Time DW IM EH CE EV BR TU ES PL CR EL LS WA WS WL FR SH TG SE EQ IC CT CV EXTREME EVENT I γ p γeq EXTREME EVENT II γ p FATIGUE LL, IM, & CE ONLY Do Not Duplicate Prestressed Concrete: Slide # Loads and Load Factors 3.4.1: Load Factors and Load Combinations Load Combination Table Load Combinations and Load Factors (cont.) DC DD DW LL IM Use One o These at a Time EH CE EV BR TU ES PL CR EL LS WA WS WL FR SH TG SE EQ IC CT CV SERVICE I /1.20 γ TG γ SE SERVICE II / SERVICE III /1.20 γ TG γ SE SERVICE IV / Do Not Duplicate Prestressed Concrete: Slide #14

8 3.4 - Loads and Load Factors 3.4.1: Load Factors and Load Combinations Service III applies only to LONGITUDINAL TENSION in prestressed girders. The modiier to (LL+IM) is 0.8. The modiier is < 1 because it was ound that the tensile capacity o prestressed girders is underestimated. This is largely because the loss o prestressing orce is usually overestimated and a lower bound is used or the tensile strength (modulus o rupture). Do Not Duplicate Prestressed Concrete: Slide #15 AASHTO-LRFD Distribution Factors or Precast/Prestressed Concrete Elements AASHTO-LRFD Speciication, 4th Edition

9 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear The simpliied distribution actors may be used i: Width o the slab is constant Number o beams, N b > 4 Beams are parallel and o similar stiness Roadway overhang d e < 3 t Central angle < Article Cross section conorms to AASHTO Table Note: Multiple presence actors are NOT used with simpliied distribution actors. Do Not Duplicate Prestressed Concrete: Slide #17 This is part o Table showing common precast/ prestressed concrete bridge types. The letter below the diagram correlates to a set o distribution actors. Do Not Duplicate Prestressed Concrete: Slide #18

10 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Beam and Slab Bridges would be a Type k bridge. Moment distribution actors - LRFD Table b-1: Two or more lanes loaded: d DFM = (S/9.5) 0.6 (S/L) 0.2 (K g /12.0Lt s3 ) 0.1 One lane loaded: DFM= 0.06+( S/14 ) 0.4 ( S/L ) 0.3 (K g /12.0Lt s3 ) 0.1 Do Not Duplicate Prestressed Concrete: Slide #19 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear S = girder spacing (t) 3.5 < S < 16.0 L = span length (t) 20 < L < 240 t s = slab thickness (in) 45< 4.5 t s < 12.0 N b = Number o Beams N b > 4 K g = n(i g + A g e g2 ) (in 4 ) 10,000 < K g < 7,000,000 n = E c,beam /E c,slab I g = gross moment o inertia, non composite girder (in 4 ) A g = gross area, non composite girder (in 2 ) e g = distance between centers o gravity o the non composite beam and slab. (in) I N b = 3, use the lesser o the equations above with N b = 3 and the lever rule. Do Not Duplicate Prestressed Concrete: Slide #20

11 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Beam and Slab Type k bridge Shear Distribution Factors - LRFD Table a-1: Two or more lanes loaded: DFV = ( S/12 ) - ( S/35 ) 2 One lane loaded: DFV = ( S/25 ) Do Not Duplicate Prestressed Concrete: Slide #21 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear 3.5 < S < 16.0 t. 20 < L < 240 t. 4.5 < t s < 12.0 in. N b > 4 I N b = 3; use the lever rule. Do Not Duplicate Prestressed Concrete: Slide #22

12 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Beam and Slab Bridge Type k Exterior Moment Two or more lanes loaded: gext = egint de e = One lane loaded use the Lever Rule LRFD Table d-1 g = DFM d e = distance rom edge o the traic railing to the exterior web o the exterior beam. The term d e is positive when the railing is outboard (shown) and negative when the railing is inboard < d e < 5.5 t. Do Not Duplicate Prestressed Concrete: Slide #23 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Beam and Slab Bridge Type k Exterior Shear Two or more lanes loaded: gext = egint de e = One lane loaded use the Lever Rule LRFD Table b-1 g = DFV -1.0 < d e < 5.5 t. Do Not Duplicate Prestressed Concrete: Slide #24

13 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Beam and Slab Bridge Type k Longitudinal Beams on Skewed Supports Any number o lanes loaded; multiply DFM by: (LRFD Table c-1) c ( tanθ ) 1 K c1 = Lts 0.25 S L θ = Angle o skew; 30 o < θ < 60 o ; i θ<30 o, c 1 = 0; i θ>60 o then θ=60 o L = Span, 20 < L < 240 t S = Beam Spacing, 3.5 < S < 16 t N b > Do Not Duplicate Prestressed Concrete: Slide #25 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Beam and Slab Bridge Type k Longitudinal Beams on Skewed Supports Correlation Factor or Load Distribution Factor or Support Shear at Obtuse Corner - (LRFD Table c-1) Lt + K g tanθ θ = Angle o skew; 0 o < θ < 60 o ; L = Span, 20 < L < 240 t S = Beam Spacing, 3.5 < S < 16 t N b > 4 3 s 0.3 Do Not Duplicate Prestressed Concrete: Slide #26

14 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Lever Rule: Assume a hinge develops over each interior girder and solve or the reaction in the exterior girder as a raction o the truck load k 36k This is or one lane loaded. Multiple Presence Factors apply 1.2 is the MPF MH 1.2Pe RS = 0 1.2Pe 1.2e R= DF = S S In the diagram, P/2 are the wheel loads; P is the resultant orce. All three loads are NOT applied at the same time. 8 t Note that truck cannot be closer than 2 rom the barrier ( ) Do Not Duplicate Loads & Analysis: Slide #27 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Minimum Exterior DFM: (Rigid Body Rotation o Bridge Section) DF Ext, Min N = N L b + b L N L - Number o loaded lanes under consideration N b e - Number o beams or girders - Eccentricity it o design truck or load rom CG o pattern o girders (t.) x - Distance rom CG o pattern o girders to each girder (t.) X Ext - Distance rom CG o pattern o girders to exterior girder (t.) X Ext N N x 2 e (C d-1) Do Not Duplicate Loads & Analysis: Slide #28

15 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Adjacent Box Girders Adjacent box girders with shear keys and a cast-in-place overlay are Type sections. Adjacent box girders with shear keys, but no cast-inplace deck, are Type g sections. Type g sections may or may not be laterally post-tensioned. Lack o lateral post-tensioning causes a reduction o the distribution actor. Do Not Duplicate Prestressed Concrete: Slide #29 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Interior Box Girders The ollowing distribution actors may be used or a Type (composite deck) or a Type g (non-composite) bridge IF the girders are suiciently connected together meaning they achieve transverse lexural continuity. This can be done with lateral post-tensioning o at least 250 psi (Commentary ; paragraph 12). The Commentary urther states that bridges without a structural t overlay and which h use untensioned transverse rods should NOT be considered as suicient to achieve transverse lexural continuity, unless demonstrated by testing or experience (Commentary , paragraph 14). Do Not Duplicate Prestressed Concrete: Slide #30

16 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Interior Box Girders Type (composite deck) or g with lateral PT - LRFD Table b-1 Moment: Two lanes loaded DFM = k ( b/305 ) 0.6 ( b/12.0l ) 0.2 ( I/J ) 0.06 One lane loaded DFM = k(b/33.3l) 0.5 (I/J) 0.25 Do Not Duplicate Prestressed Concrete: Slide #31 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Interior Box Girders k =25(N 2.5 N b ) > N b = number o beams 5 < N b < 20 b = width o beam, in 35< b < 60 in L = span o beam, t 20< L < 120 t I = moment o inertia o beam, in 4 J = St. Venant torsional constant, in 4 For preliminary design, ( I/J ) 0.06 = 1.0 Do Not Duplicate Prestressed Concrete: Slide #32

17 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Interior Box Girders Distribution Factors or Shear - LRFD Table a-1 Two Lanes Loaded: DFV = (b/156) 0.4 (b/12l) 0.1 (I/J) 0.05 (b/48) One Lane Loaded: DFV = (b/130l) 0.15 (I/J) < N b < 20 35< b < 60 in 20< L < 120 t 25,000 < J < 610,000 in 4 40,000 < I < 610,000 in 4 These are used or both composite and non-composite; even i the girders are NOT suiciently connected. Do Not Duplicate Prestressed Concrete: Slide #33 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Type g box with NO lateral PT DFV (distribution actor or shear) does not change. It is the same or Type g structures with and without lateral PT. DFM is dierent. For Type g structures without lateral PT, the old Standard Speciications equations are used. NOTE: The Standard Speciications equations were based on wheel loads and the LRFD equations are based on axle loads; so the equations changed by a actor o 2. Do Not Duplicate Prestressed Concrete: Slide #34

18 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Distribution Factor or Moment - LRFD Table b-1 DFM = S/D S = width o precast beam (t) D = ( N L )+1.4N L (1-0.2C) 2 when C < 5 D = ( N L ) when C > 5 Where: N L = number o traic lanes C = K(W/L) < K Do Not Duplicate Prestressed Concrete: Slide #35 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear C = K(W/L) < K Where: ( 1+ µ ) I K = J For Preliminary Design Beam Type K Nonvoided d rectangular beams Rectangular beams with circular voids: 0.8 Box section beams 1.0 Channel beams 2.2 T-beam 2.0 Double T-beam 2.0 W = overall width o bridge measured perpendicular to the longitudinal beam (t) L = span (t) µ = Poisson s ratio = 0.2 or concrete ( ) Do Not Duplicate Prestressed Concrete: Slide #36

19 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear J 2 4 A S t Where: A = Area enclosed by the centerline o the webs and langes. S = length o a web or lange centerline. t = thickness o the corresponding web or lange. Do Not Duplicate Prestressed Concrete: Slide #37 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear The bending moment or exterior beams is determined by multiplying the distribution actor or interior beams by a actor, e, which accounts or the distribution o load to the exterior girder. Note that this applies to type g even i there is no lateral post-tensioning. Lack o lateral posttensioning is accounted or in the DVM. Minimum exterior distribution actor based on rigid body rotation does not apply to adjacent box girders. Do Not Duplicate Prestressed Concrete: Slide #38

20 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Exterior Box Girders Multiplier or Moment Types and g -LRFDTable46222d d Two or more lanes loaded: g ext = eg interior Where: e = ( d e / 25 ) > 1 d e=distance rom edge o the traic railing to the exterior web o the exterior beam. The term d e is positive when the railing is outboard (shown) and negative when the railing is inboard. d e < 2.0 UNIT IS FEET! g= DFM Do Not Duplicate Prestressed Concrete: Slide #39 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Exterior Box Girder Multiplier or Moment Types and g -LRFDTable46222d d One lane loaded: g ext = eg interior e = ( d e / 30 ) > 1 d e < 2.0 t. e e accounts or the distribution o load to the exterior girder Do Not Duplicate Prestressed Concrete: Slide #40

21 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Exterior Box Girders Multiplier or Shear Types and g - LRFD Table b 23b-1 Two or more lanes loaded: g ext = eg 48 1 b int 48 b 0.5 b de e = d e < < b < 60 in g = DFV Do Not Duplicate Prestressed Concrete: Slide #41 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Multiplier or Shear Types and g - LRFD Table b-1 One lane loaded: g ext = eg interior e = ( d e / 20 ) > 1 d e < 2.0 t. Do Not Duplicate Prestressed Concrete: Slide #42

22 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Skewed Box Girders Multiplier or Moment - LRFD Table c-1 22c ( tan θ) < 1.0 θ = skew angle I θ > 60 0 use θ = 60 0 This is optional. Do Not Duplicate Prestressed Concrete: Slide #43 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear When the skew angle o a bridge is small, say, less than 20 o, it is oten considered sae to ignore the angle o skew and to analyze the bridge as a zero-skew bridge whose span is equal to the skew span. This approach is generally conservative or moments in the beams, and slightly unsae (<5%) or slab-on-girder decks or longitudinal shears. The LRFD Speciications Table e-1 lists reduction multipliers or moments in longitudinal beams. The previous slide illustrates the multiplier or spread box beams, adjacent box beams with concrete overlays or transverse posttensioning and double tees in multi-beam decks or Types (b), (c), () and (g). Do Not Duplicate Prestressed Concrete: Slide #44

23 Distribution Factors or Precast/Prestressed Concrete Elements Distribution Factor Method or Moment and Shear Correlation Factor or Load Distribution Factor or Support Shear at Obtuse Corner Types and g - (LRFD Table c-1) This is mandatory. 12.0L d tanθ 0 o < θ < 60 o 20 < L < 240 t. 17 < d < 60 in d is depth o the girder 35 < b < 60 in b is width o the lange 5 < N b < 20 Do Not Duplicate Prestressed Concrete: Slide #45 AASHTO-LRFD Flexure and Axial Loads AASHTO-LRFD Speciication, 4th Edition

24 Flexure and Axial Loads Deinitions o various d terms or Do Not Duplicate Prestressed Concrete: Slide #47 Flexure and Axial Loads AASHTO LRFD now uses the same terminology as ACI This is a uniied method or prestressed and reinorced concrete members. Article deines 3 states: Tension Controlled Compression Controlled Transition In all cases, extreme iber compressive strain = (Article ). Values above are allowed or conined cores. Do Not Duplicate Prestressed Concrete: Slide #48

25 Flexure and Axial Loads Assumptions or Strength and Extreme Event Limit States Deinition o Section Types Extreme tensile steel strain when the extreme concrete compressive strain = ε t > ε t < y / E s (may use = 0.002) Type o section Tension controlled Compression controlled > ε t > y / E s Transition For all prestressing or Grade 60 non-prestressed steel, ε t may be assumed = in place o y /E s or compression controlled. The ACI 318 code, upon which this provision is based, requires lexural members (that is, members with a superimposed axial load o < 0.1 c A g ) to have ε s > AASHTO does not impose this requirement. Do Not Duplicate Prestressed Concrete: Slide #49 Flexure and Axial Loads Assumptions or Strength and Extreme Event Limit States Deinition o strain conditions or determining tension or compression control. Note that tensile strain in the steel closest to the tensile ace is used. Balanced condition is when ε t = ε y. For Grade 60 steel and all prestressing steel, ε y may be taken as Note that or prestressing steel, ε t is the tensile strain which occurs in the steel ater the pre-compression in the concrete is lost. Do Not Duplicate Prestressed Concrete: Slide #50

26 Flexure and Axial Loads Assumptions or Strength and Extreme Event Limit States For a prestressed beam, it is important to understand the deinition o ε t. d t Begin by considering the strain condition o the beam at the point where the only loads are the prestressing orce and the beam sel weight. In this condition, the top o the beam is usually in tension (due to the prestressing). There is a net tensile strain in the prestressing steel o ε p1. This is the initial pull minus any strain lost due to prestress losses. At the level o the steel, there is a compressive strain the concrete, ε c. Do Not Duplicate Prestressed Concrete: Slide #51 Flexure and Axial Loads Assumptions or Strength and Extreme Event Limit States d t As load is applied, the strain proile changes, the bottom decompresses and eventually reaches a point where the CONCRETE strain at the level o the steel is 0. This is called decompression. I there were no losses (except or elastic shortening), the strain in the steel, ε p2 at this point would be the initial pull. The actual strain in the steel, with losses, can be calculated by mechanics. Do Not Duplicate Prestressed Concrete: Slide #52

27 Flexure and Axial Loads Assumptions or Strength and Extreme Event Limit States d t This is the condition at M n. The compressive strain in the concrete is The total strain in the prestressing steel is the sum o the strain in the steel at decompression, ε p2, and the strain developed between decompression and the ultimate state, ε t. The speciications only regulate the strain developed between decompression and the ultimate state, ε t. The additional strain in the prestressing steel, ε p2 is not part o the speciication. Do Not Duplicate Prestressed Concrete: Slide #53 Flexure and Axial Loads Resistance Factors Φ = 0.9 tension controlled reinorced concrete members 1.0 tension controlled prestressed concrete members 0.75 compression controlled members with spirals or ties (except or members in Seismic Zones 3 & 4) 0.90 shear and torsion 0.70 shear and torsion lightweight concrete For transition members, use a linear interpolation o the Φ actor based on the extreme tensile steel strain. Do Not Duplicate Prestressed Concrete: Slide #54

28 Flexure and Axial Loads Resistance Factors Phi Factor Prestressed: Strain = Phi = 0.92 Prestressed Reinorced Compression Tension Controlled Transition Controlled Extreme Steel Strain dt 0.75 φ = ( ) c dt 0.75 φ = c ( ) Do Not Duplicate Prestressed Concrete: Slide #55 Prestressed Members Reinorced Members Flexure and Axial Loads Resistance Factors Eect o New Resistance Factors It is allowable to design lexural members with extreme iber steel strains < This is done by increasing the area o steel. However, in general, the Φ actor is reduced at a slightly lower rate than moment resistance is gained. There is a slight increase in M n but it is minimal. Thus, there is little eect on the allowable moment by increasing the amount o steel above that required to bring the extreme iber steel strains to Do Not Duplicate Prestressed Concrete: Slide #56

29 Flexure and Axial Loads Resistance Factors For tension controlled partially prestressed members: φ = PPR 010PPR A ps py PPR = A ps py + A s y (554213) ( ) ( ) PPR = Partial prestressing ratio A ps = Area o prestressing steel py = Yield strength o the prestressing steel A s = Area o mild steel y = Yield strength o the mild steel Do Not Duplicate Prestressed Concrete: Slide #57 Flexure and Axial Loads The stress block remains the same as Standard Speciications. Analysis o reinorced concrete RECTANGULAR beams is the same as Standard Speciications. HOWEVER, there are some dierences with prestressed concrete. Do Not Duplicate Prestressed Concrete: Slide #58

30 AASHTO-LRFD Prestressed Beams with Bonded Tendons AASHTO-LRFD Speciication, 4th Edition Prestressed Beams with Bonded Tendons Flexural Members The value o ps can be ound rom (i pe > 0.5 pu ): c ) = ( ) 104 (573112) ps pu 1 py k k d = ( ) p pu Then: 0.85 'b a = A c a =βc 1 ps ps c 0.85 c 'bβ 1c = Aps pu 1 k d p Aps pu c = pu 0.85 c' β 1b + kaps d p Stress in the steel, ps, can also be ound rom strain compatibility analysis. Do Not Duplicate Prestressed Concrete: Slide #60

31 Prestressed Beams with Bonded Tendons Flexural Members c = c A ps pu 0.85 ' β b + ka 1 ps d pu p c = depth o neutral axis b = width o compression block A ps = area o TENSILE prestressing steel d p = depth to centroid o tensile prestressing steel pu = tensile strength o prestressing steel py = yield strength o prestressing steel β 1 = stress block actor same as Std. Spec. Do Not Duplicate Prestressed Concrete: Slide #61 Prestressed Beams with Bonded Tendons Flexural Members I there is mild (nonprestressed) tensile steel, A s and mild compression steel A s both with a yield stress o y, the equation or c becomes:.85 ' ' ' 1 Aps pu + As y As ' y ' c = pu 0.85 c' β 1b+ kaps d c bβ1c+ As y = As y + Aps pu k d p ( ) The engineer must do an analysis to see i the compression steel yields. I the compression steel does not yield, the actual stress is substituted or y into equation p c Do Not Duplicate Prestressed Concrete: Slide #62

32 Prestressed Beams with Bonded Tendons Flexural Members Sometimes, things change or the better!!!! Std. Spec And LRFD 2005 Interim Editions 1 through h 3 o LRFD In Editions 1-3 o the LRFD Speciications, the β actor was applied to the lange as well as to the web. This made no sense. It was changed with the 2005 Interim back to the old deinition. Now it is the same deinition as ACI 318 and Std. Spec. Do Not Duplicate Prestressed Concrete: Slide #63 Prestressed Beams with Bonded Tendons Flexural Members The T beam equation returns to normal: a a Mn = Aps ps dp + As y ds 2 2 a a h As' y' ds' c' ( b bw) h ( ) Again the engineer must do an analysis to see i the compression steel yields. I the compression steel does not yield, the actual stress is substituted or y into equation Do Not Duplicate Prestressed Concrete: Slide #64

33 Prestressed Beams with Bonded Tendons Flexural Members The LRFD Speciications give only this equation: a a a a h Mn= Apsps dp + Asy ds A' s ' s d' s ' c( b bw) h I the section is NOT a T beam, b = b w and: a a a Mn = Apsps dp + Asy ds A' s ' s d' s I there is no compression steel: a a Mn= Aps ps dp + As y ds 2 2 I there is no non-prestressed tensile steel: a Mn = Apsps dp 2 Do Not Duplicate Prestressed Concrete: Slide #65 Prestressed Beams with Bonded Tendons Flexural Members For prestressed T- Beams: c = pu c β1bw + kaps d p ( ) A + A A ' ' 0.85 ' b b h 0.85 ' ps pu s y s y c w ( ) b w = web width b = lange width h = lange thickness Do Not Duplicate Prestressed Concrete: Slide #66

34 AASHTO-LRFD Prestressed Beams with Unbonded Tendons AASHTO-LRFD Speciication, 4th Edition. Prestressed Beams with Unbonded Tendons Flexural Members The stress in the prestressing steel can be ound rom: l d p c < ps = pe le 2l i l = e 2+ Ns e = eective tendon length l i = length o tendon between anchorages N s = Number o support hinges crossed by the tendon between anchorages or discretely bonded points. pe = Eective stress in the steel ater losses. py ( ) ( ) Do Not Duplicate Prestressed Concrete: Slide #68

35 Prestressed Beams with Unbonded Tendons Flexural Members For rectangular beams: c = For T-beams: c = A + A A ' ps ps s y s y 0.85 ' β b c 1 ' ( ) A + A A b b h ' ' 0.85 '( ) ps ps s y s y c w 0.85 ' β b c 1 w ( ) Do Not Duplicate Prestressed Concrete: Slide #69 Prestressed Beams with Unbonded Tendons Flexural Members For unbonded tendons, the equations or c require the value o ps, but the equation or ps requires the value o c. The two equations can be solved simultaneously in a closed orm, but most people will not do this. Thus, inding ps becomes an iterative procedure. The Commentary (C ) gives an equation or a irst estimate o ps (in ksi): ps = + 15 pe (C ) Do Not Duplicate Prestressed Concrete: Slide #70

36 AASHTO-LRFD Components with Both Bonded and Unbonded Tendons AASHTO-LRFD Speciication, 4th Edition Components with Both Bonded and Unbonded Tendons Flexural Members Article allows two methods: Article a Detailed Analysis In this method, a detailed, strain compatibility is used. Article b Simpliied Analysis Shown on the ollowing slide A psb = area o bonded tendons A psu = area o unbonded d tendons Do Not Duplicate Prestressed Concrete: Slide #72

37 Components with Both Bonded and Unbonded Tendons Flexural Members Simpliied Analysis - The stress in the UNBONDED tendons may be conservatively taken as the eective stress ater losses: pe. p For T-beams: c = A psb pu + A For rectangular beams: psu c = A + A 0.85 ' + pe + As y As ' y ' 0.85 c ' pu 0.85 c ' β1bw + kaps d psb pu psu pe pu c β1b kaps d p p ( b b ) w h Do Not Duplicate Prestressed Concrete: Slide #73 AASHTO-LRFD Moment Capacity AASHTO-LRFD Speciication, 4 th Edition

38 Moment Capacity Flexural Resistance For T-beams (where a>h ): a a Mn = Apsps dp + As y ds 2 2 a a h ( ) ' ' ' + A 0.85 ' s y ds c b bw h ( ) For rectangular beams, b=b w; thus equation becomes: a a a M n = Aps ps d p + As y ds As ' y ' ds ' Do Not Duplicate Prestressed Concrete: Slide #75 Moment Capacity Flexural Resistance In the preceding equations: d p d s d s y y = distance rom the extreme compression iber to the prestressing steel. = distance rom the extreme compression iber to the non-prestressed tensile steel. = distance rom the extreme compression iber to the non-prestressed compression steel. = yield strength o the non-prestressed tensile steel. = yield strength o the non-prestressed compression steel. Do Not Duplicate Prestressed Concrete: Slide #76

39 Moment Capacity Limits or Reinorcement Minimum reinorcement (Article ): It is the smaller o: φm n > 1.2 M cr same as in Std. Spec. φm n > 1.33M u LRFD added Do Not Duplicate Prestressed Concrete: Slide #77 Moment Capacity Limits or Reinorcement For the minimum reinorcement requirement, the cracking moment M cr is ound rom: S c M = S ( + ) M 1 S cr c r cpe dnc c r Snc ( ) S c = composite section modulus r = modulus o rupture = 0.37 c (ksi units) cpe = compressive stress in the concrete due to eective prestressing orce, at the extreme tensile iber or applied loads. M dnc = Unactored dead load moment on the non-composite or monolithic section. S nc = Non-composite section modulus. Do Not Duplicate Prestressed Concrete: Slide #78

40 Moment Capacity Limits or Reinorcement Maximum reinorcement provision was dropped with 2005 Interim No longer needed with new deinitions o tension controlled, compression controlled and transition. LRFD previously used a c/d ratio. This can still be used: c 3 d t 8 c 3 d t 5 3 c 3 > > 5 d 8 t Tension Controlled ε t > Compression Controlled ε t <0.002 Transition Do Not Duplicate Prestressed Concrete: Slide #79 Moment Capacity Limits or Reinorcement Maximum reinorcement is now controlled by ε t. To determine ε t, calculate c. Then, using similar triangles: ε t = d t c c Do Not Duplicate Prestressed Concrete: Slide #80

41 Moment Capacity Limits or Reinorcement Maximum Reinorcement This is more restrictive that Std. Speciication or previous editions o LRFD. For reinorced sections, 0.75ρ bal was used. This was a strain o in the steel. For prestressed, Std. Spec. c/d e ratio was limited to This corresponded to a strain o c d Tension Controlled ε t > t c d e e 0.42 c 0.45 d Previous Editions ε t > Std. Speciications, RC. Do Not Duplicate Prestressed Concrete: Slide #81 Moment Capacity Control o Cracking by Distribution o Reinorcement γ e d ( ) c s s 700 s 2 β d βs = ( h d ) s = spacing o reinorcement closest to the tension ace. γ e = exposure actor; 1 or Class 1 and 0.75 or Class 2 ODOT uses 0.75 or decks, 1 or everything else d c = cover to extreme tension iber s = Steel service limit state h = overall thickness or depth Does not apply to slabs designed using the empirical method (ODOT does not allow empirical design). It applies to all other concrete components where the service tensile stress exceeds 0.8 r = 0.8(0.24) c = 0.20 c c c Do Not Duplicate Prestressed Concrete: Slide #82

42 Moment Capacity Moment Redistribution ODOT does not permit moment redistribution Do Not Duplicate Prestressed Concrete: Slide #83 Moment Capacity Delection and Camber Prestressed members are usually designed as uncracked at service loads. Instantaneous delections and cambers are then calculated using the gross moment o inertia, I g. I the delection is calculated using I g, long term delection can be ound by multiplying the instantaneous delection by 4. For prestressed members, the Commentary (C ) allows the multipliers given in the PCI Design Handbook to be used or long term camber/delection values. Do Not Duplicate Prestressed Concrete: Slide #84

43 Prestressing and Partial Prestressing Stress Limitations or Prestressing Tendons Table Stress Limits or Prestressing Tendons py = yield stress o prestressing steel Tendon Type pu = ultimate strength o prestressing steel Condition Stress-Relieved Strand and Plain High- Strength Bars Pretensioning Low Relaxation Strand Deormed High-Strength Bars Immediately prior to transer ( pbt ) 0.70 pu 0.75 pu At service limit state ater all losses ( pe ) 0.80 py 0.80 py 0.80 py Post-Tensioning Prior to seating short-term term pbt may be allowed py py py At anchorages and couplers immediately ater anchor set Elsewhere along length o member away rom anchorages and couplers immediately ater anchor set 0.70 pu 0.70 pu 0.70 pu 0.70 pu 0.74 pu 0.70 pu At service limit state ater losses ( pe ) 0.80 py 0.80 py 0.80 py Do Not Duplicate Prestressed Concrete: Slide #85 Prestressing and Partial Prestressing Stress Limits or Concrete Table Temporary Tensile Stress Limits in Prestressed Concrete Beore Losses, Fully Prestressed Components. (Partial) Bridge Type Location Stress Limit Other than Segmentally Constructed Bridges In precompressed tensile zone without bonded reinorcement In areas other than the precompressed tensile zone and without bonded reinorcement In areas with bonded reinorcement (reinorcing bars or prestressing steel) suicient to resist the tensile orce in the concrete computed assuming an uncracked section, where reinorcement is proportioned using a stress o 0.5 y, not to exceed 30 ksi. For handling stresses in prestressed piles Compression Limit at Transer N/A ci <0.2(ksi) 0.24 ci (ksi) ci (ksi) 0.6 ci (ksi) Do Not Duplicate Prestressed Concrete: Slide #86

44 Debonding and Harping I the tensile stresses at the end o girder are above 0.24 ci, then the stress must be reduced either by debonding the strand or harping the strand. I debonding is used, no more than 25% o the total number o strands may be debonded and not more than 40% in any single row may be debonded. (Art ) Do Not Duplicate Prestressed Concrete: Slide #87 Prestressing and Partial Prestressing Stress Limits or Concrete Table Compressive Stress Limits in prestressed Concrete at Service Limit State Ater Losses, Fully Prestressed Components. Location Stress Limit In other than segmentally constructed bridges due to the sum o eective prestress and permanent loads 0.45 c (ksi) In segmentally constructed bridges due to the sum o eective prestress and permanent loads 0.45 c (ksi) In other than segmentally constructed bridges due to live load and one-hal the sum o eective prestress and permanent loads 0.40 c (ksi) Due to the sum o eective prestress, permanent loads, 060φ 0.60φ w c (ksi) and transient loads and during shipping and handling Do Not Duplicate Prestressed Concrete: Slide #88

45 Prestressing and Partial Prestressing Stress Limits or Concrete Table Tensile Stress Limits in Prestressed Concrete at Service Limit State Ater Losses, Fully Prestressed Components. (Partial) Bid Bridge Type Location Stress Limitit Other than Segmentally Constructed Bridges Tension in the Precompressed Tensile Zone Bridges, Assuming Uncracked Sections For components with bonded prestressing tendons or reinorcement that are subjected to not worse than moderate corrosion conditions For components with bonded prestressing tendons or reinorcement that are subjected to severe corrosive conditions For components with unbonded d prestressing tendons 0.19 c (ksi) c (ksi) No Tension Again, these are Std. Spec. limits in ksi units. 0.19(1000) 0.5 = 6 Do Not Duplicate Prestressed Concrete: Slide #89 AASHTO-LRFD Loss o Prestressing Force AASHTO-LRFD Speciication, 4 th Edition

46 5.9 Prestressing and Partial Prestressing Loss o Prestress Loss o prestressing orce was changed with the 3 rd Edition. Like creep and shrinkage, the changes are based on the results NCHRP Report 496 Prestressed Losses in Pretensioned High Strength Concrete Bridge Girders These provisions are applicable up to 15 ksi concrete Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress The basic equations: Pretensioned Members: = + ( ) pt pes plt Post-tensioned Members: = (5951-2) ( ) pt pf pa pes plt Do Not Duplicate Prestressed Concrete: Slide #92

47 5.9 Prestressing and Partial Prestressing Loss o Prestress pt = Total loss o prestressing orce (ksi). pf = Loss due to riction (ksi). pa = Loss due to anchorage set (ksi). pes = Loss due to elastic shortening (ksi). plt = Loss due to long term shrinkage and creep o the concrete and relaxation o the steel (ksi). pa is usually given by the manuacturer. Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress Friction losses: Loss due to riction between an internal tendon and a duct wall: ( kx + µα ) pj ( e ) = 1 ( b-1) pf Loss due to riction between an external tendon and a single deviator pipe: pf = µ ( α ) pj ( 1 e ) ( b-2) Do Not Duplicate Prestressed Concrete: Slide #94

48 5.9 Prestressing and Partial Prestressing Loss o Prestress pj = initial jacking stress in the tendon (ksi). x = length o tendon rom the jacking point to the point being considered (t). K = wobble riction coeicient (per t. o tendon) µ = riction coeicient. α = sum o the absolute value o angular change o prestressing steel path rom jacking end (or nearest jacking end i jacked rom both ends) to point under consideration. (radian) Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress Table b-1 Friction Coeicients or Post-Tensioning Tendons. Steel Duct K µ Wire or Strand Rigid or Semi rigid galvanized metal sheathing Polyethylene Rigid steel deviator bar or external tendons HS Bar Galvanized metal sheathing Values or K and µ should be ound rom experimental data. I such data is absent, values rom the table above may be used. Do Not Duplicate Prestressed Concrete: Slide #96

49 5.9 Prestressing and Partial Prestressing Loss o Prestress Elastic Shortening, pretensioned members: pes = E E p ct cgp ( a-1) E ct = modulus o elasticity o the concrete at transer or at time o load Elastic Shortening, Post-tensioned Members: N 1 Ep pes = ( b-1) cgp 2N E ci Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress cgp = concrete stresses at the center o gravity o the prestressing tendons due to prestressing orce immediately ater transer (pretensioning) or immediately ater jacking (post-tensioning) and the sel-weight o the member at the sections o maximum moment (ksi). In pretensioned members, at transer, cgp may be calculated by assuming the stress in the prestressing tendon ater release = 0.9 pi ; where pi is the initial prestressing stress (jacking stress) in the tendons. E p = Elastic Modulus o the prestressing strand (ksi). E ci = Elastic Modulus o the concrete at the time o transer or time o load application (ksi). N = number o identical strands. Do Not Duplicate Prestressed Concrete: Slide #98

50 5.9 Prestressing and Partial Prestressing Loss o Prestress Long Term Losses For standard, precast, pretensioned members subject to normal loading and environmental conditions: pi Aps plt = 10 γhγ st + 12γhγ st + pr ( ) Ag γ h = H ( ) 5 γ st = ( ) 1 + ci Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress pi = prestressing steel stress immediately PRIOR to transer. H = Average annual relative humidity in percent (e.g.70 not 0.7) pr = 2.5 ksi or LoLax 10 ksi or stress relieved γ h = humidity actor γ st = strength actor Do Not Duplicate Prestressed Concrete: Slide #100

51 5.9 Prestressing and Partial Prestressing Loss o Prestress To use the plt equation, the ollowing criteria must be met: Members are pretensioned Normal weight concrete is used Members are moist or steam cured Prestressing is by bar or strand with normal and low relaxation properties Average exposure conditions and temperatures. Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress This table can be used to estimate time dependent losses in prestressed members which do not have composite slabs and are stressed ater attaining a compressive strength o at least 3.5 ksi. Type o Beam Section Rectangular Box Girder Single T, Double T, Hollow core and Voided Slab Table Time-Dependent Losses in ksi. Level Upper Bound Average Upper Bound Average Upper Bound Average PPR is the partial prestressing ratio. For wires and Strands with pu = 235,250 or 270 ksi For Bars with pu = 145 or 160 ksi PPR PPR PPR PPR ' c PPR 6.0 ' c PPR ' c PPR 6.0 Do Not Duplicate Prestressed Concrete: Slide #102

52 5.9 Prestressing and Partial Prestressing Loss o Prestress Lump Sum Losses: For lightweight concrete, the stresses in the table are increased 5 ksi. For low relaxation strand, the values in the table are reduced by: 4 ksi or box girders 6 ksi or rectangular beams and solid slabs 8 ksi or single T s, double T s, hollow core and voided slabs. Do Not Duplicate Prestressed Concrete: Slide # Prestressing and Partial Prestressing Loss o Prestress For post-tensioned members, the Reined Method or estimation o time dependent losses must be used. However, this method is based on NCHRP 496, but requires a large amount o calculation. Since longitudinal post-tensioning is not common in Ohio, the method is not presented here. However, it can be ound in Article 5954otheLRFDSpeciications Speciications. Do Not Duplicate Prestressed Concrete: Slide #104

53 AASHTO-LRFD Bond/Development Length AASHTO-LRFD Speciication, 4 th Edition 5.11 Bond and Development Length Transer Length For ully bonded d strands, the transer length rom the end o the girder is assumed to be 60d b, where d b is the bar or strand diameter. Do Not Duplicate Prestressed Concrete: Slide #106

54 5.11 Bond and Development Length Development Length Development length or ully bonded strand is given by: 2 =κ d 3 l ( ) d ps pe b Do Not Duplicate Prestressed Concrete: Slide # Bond and Development Length Development Length Where: l d = development length ps = steel stress at strength limit state pe = eective prestressing stress ater all losses d b = strand diameter κ =1.0 or pretensioned panels, piles and other pretensioned members with a depth < 24 inches. = or pretensioned members with a depth > 24 inches = 2.0 or debonded strand Do Not Duplicate Prestressed Concrete: Slide #108

55 5.11 Bond and Development Length Development Length In previous editions o the LRFD Speciications, bond stress was assumed linear e.g, i the bonded length was only ½ the development length, it was assumed that the strand could only develop 0.5 ps. This assumption is still true or TRANSFER LENGTH; e.g at ½ the transer length it is assumed only 0.5 pe is developed. However, stress in the steel beyond the transer length, but less than the development length, can now be calculated by a bilinear ormula. Do Not Duplicate Prestressed Concrete: Slide # Bond and Development Length Development Length l px 60db = + l 60 d ( ) px pe ps pe d b ( ) Where: px = stress at x rom the end o the girder pe = eective stress in the steel ater all losses ps = stress in the steel at the strength limit state l px = length were the stress is being calculated l d = development length d b = strand diameter Do Not Duplicate Prestressed Concrete: Slide #110

56 5.11 Bond and Development Length Development Length Within the transer length (which is 60d b ): px px pe = l ( ) 60d b Do Not Duplicate Prestressed Concrete: Slide # Bond and Development Length Development Length Do Not Duplicate Prestressed Concrete: Slide #112

57 AASHTO-LRFD Shear AASHTO-LRFD Speciication, 4 th Edition Shear and Torsion 5.6 Design Considerations Important things about the shear section This section has the provisions o the LRFD Speciications, through the 2007 changes. This section concentrates the provisions as they apply to prestressed concrete; both pretensioned and posttensioned. Segmental box girder bridges and spliced girders are NOT covered. Reinorced concrete is covered in another section. Do Not Duplicate Prestressed Concrete: Slide #2

58 5.8 - Shear and Torsion Strut-and-Tie Model Strut and Tie Model Strut and tie can be used or analysis o anchorage zones and support regions. It is also useul or deep ootings, pile caps and sections where the depth is more than ½ the span. This model is covered in Article Strut and tie will not be discussed as part o this module. It will be covered in another presentation. Do Not Duplicate Prestressed Concrete: Slide # Shear and Torsion General Requirements V r = φ V ( ) n V u V r V n = nominal shear resistance given in Article (kip) φ = 0.9 normal weight concrete φ = 0.7 lightweight concrete V u = Factored shear at the cross section being considered. I there is signiicant torsion present, this term is modiied or torsion. Do Not Duplicate Prestressed Concrete: Slide #4