Bridges for Service Life Beyond 100 Years: Service Limit State Design (2014)

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310 The derivation of the resistance prediction equation for a prestressed concrete girder subjected to flexural loading is shown in this appendix. Figure D.1 displays the stress dis- tribution diagram for a typical prestressed concrete bridge girder at various stages of loading. In this study, decom- pression is considered as the stress state producing a zero stress at the extreme bottom fibers of the prestressed girder. Using axial force equilibrium, we have Equation D.1: A f A f f b c c h c b b c h f c h h c b b c h h f f c b c c h b b c h h b b s s f f f f w f f f f f w ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = − − − − − − − − − − = − − − − − − −       2 2 2 2 (D.1) ps ps ct 0 0 ct 1 1 ct ct 0 2 2 0 1 2 where As = area of nonprestressing steel; Aps = area of prestressing steel in tension zone; b = prestressed beam top flange width; b0 = effective deck width transformed to the beam material; bw = web thickness; c = depth of neutral axis from the from extreme com- pression fiber; fct = calculated stress in concrete at the top fiber; fps = calculated stress in prestressing steel; fs = calculated stress in nonprestressing steel; hf = deck thickness; and hf1 = top flange thickness. The stress in the prestressing steel can be calculated as follows (Equation D.2): (D.2)ps ps ps ps se ce ps ctf E E E E f d c cc p( )= ε = ε + ε + −   By rearranging Equation D.2, the stress in the concrete at the top fiber can be calculated as follows (Equation D.3): (D.3)ct ps ps se ce ps f f E E c E d c c p( ) ( ) = − ε + ε  − From strain compatibility, we have Equation D.4, (D.4)ctf E E E f d c cs s s s c s( )= ε = − where c = depth of neutral axis from the from extreme com- pression fiber; dp = distance from extreme compression fiber to centroid of prestressing steel; ds = distance from extreme compression fiber to centroid of nonprestressing steel; Ec = modulus of elasticity of concrete; Eps = modulus of elasticity of prestressing steel; Es = modulus of elasticity of nonprestressing steel; fct = stress in the concrete at the top of the beam after losses at service load; ece = strain in concrete at the level of prestressing steel after losses at dead load state; eps = strain in prestressing steel after losses at service load; es = strain in nonprestressing steel after losses at service load; and ese = strain in prestressing steel after losses at dead load state. Substitute Equation D.3 and Equation D.4 into Equa- tion D.1 to obtain Equation D.5: 2 2 (D.5) ps ps ps ps se ce ps 0 2 2 0 1 2 A f f E E E d c b c c h b b c h h b b A E d c E c p f f f w s s s c i( ) ( ) ( ) ( ) ( ) ( ) ( ) = − ε + ε  − − − −   − − − − − −   A p p e n d i x d Derivation of the Resistance Prediction Equation of Prestressed Concrete Bridge Girders

311 Equation D.5 can be simplified and rewritten as a quadratic equation with unknown c, neutral axis depth, as follows in Equa- tion D.6: i 2 2 1 2 1 2 0 (D.6) 2 ps ps ps ps ps se ce 0 1 1 ps ps ps ps ps se ce 0 2 1 2 c b A f E f E E b h bh b h b h A E E c b A f E f E E b b h b b h h A E d E w c f f w f w f s s c w c f w f f s s s c ( ) ( ) ( ) ( ) ( ) + − ε + ε  + + − − +       − − ε + ε  − − + − + +       = The moment resistance can be expressed as follows (Equa- tion D.7): 1 6 2 2 3 2 2 2 3 (D.7) ps ps ct 0 2 2 0 ct 1 2 ct 1 M A f d A f d f b c c h b b c f c h c h h b b c f c h h n p s s s f f f f w f f ( ) ( ) ( ) ( ) = + − + − − +    + − − − + +    where Mn = nominal moment resistance. fs is calculated using Equation D.3, and fct is calculated using Equation D.4. The depth of neutral axis from the compression face, c, can be computed from Equation D.6. Also, assuming a linear elastic relationship in the behavior of the prestressing steel, we have Equation D.8, (D.8)se se ps f E ε = Then Equation D.9 follows, 1 (D.9)ce ps se 0 2 0 A f E A e I M e E Ic c D c ε = +     − where Ac = area of concrete at the cross section considered; e0 = eccentricity of the prestressing force with respect to the centroid of the section; Ec = modulus of elasticity of concrete; Eps = modulus of elasticity of prestressing steel; fse = effective stress in prestressing steel after losses; I = moment of inertia; and MD = dead load moment. Considering an uncracked section under service loads and plane section remains plane, the linear strain distribution diagram is as follows in Figure D.2: From Figure D.2, the relationship between top and bottom strain is as follows (Equations D.10 and D.11): i (D.10)ps d c h c p c − − ε = ε (D.11)cb ps pt d c h c f E E f p c i − − = ∆ Figure D.1. Stress distribution diagrams for a typical prestressed concrete bridge girder at various stages of loading. Tension Compression Compression Compression Compression Compression Tension (a) Initial State (b) Dead Load State Compression (c) Decompression @ Bottom of Girder (d) Decompression @ Level of Prestressing (e) Service Load State b hf2 hf1 hf b0 bw b1 h ds dp c Figure D.2. Strain distribution at service loads. Compression Tension dp h c

312 where ec = strain in concrete at bottom fiber; Ec = elastic modulus of concrete; Dfpt = change in prestressing tendons stress between decompression and the stress in concrete at the bot- tom of the girder reaching fct assuming uncracked section; and fcb = concrete allowable tensile stress at the bottom of the girder. According to the current AASHTO LRFD Bridge Design Specifications (2012), 0.19cbf fc= ′ or 0.0948cbf fc= ′, depend- ing on the exposure conditions. Then fps for the uncracked section should be as follows (Equation D.12): f f f M= ∆ + ( ) (D.12)ps pt ps Dec For the cracked section, the fps can be calculated by Equation D.13: (D.13)ps ps ps Decf f f M= ∆ + ( ) where fps(MDec) = the stress in prestressing steel at decompression; Dfps = the increase in the prestressing steel stress beyond the decompression state for cracked members; and MDec = the decompression moment. Dfps in Equation D.13 can be calculated based on the equa- tion of maximum crack width at the bottom of prestressed concrete girder. In this study, Equation D.14, developed by Nawy and Huang (1977), was used: 5.85 10 0 (D.14)max 5 psw A ft ( )= × βΣ ∆− where S0 = sum of reinforcing element circumferences; At = area of concrete in tension; and b = ratio of distance from neutral axis of beam to concrete outside tension face to distance from neutral axis to steel reinforcement centroid. By rearranging Equation D.14, Dfps can be calculated using Equation D.15: 0 5.85 10 (D.15)ps max 5f w At i i i ∆ = β Σ × − Dfpt varies according to the maximum allowable tensile stress at the bottom of the concrete girder. Moreover, fps(MDec) can be calculated using Equation D.16: f f E M M IE A E E A e I M D c c c [ ] = + − + +         ( ) 1 1 (D.16)ps se ps Dec ps ps 0 2Dec The decompression moment at the level of prestressing strands, MDecp, can be calculated using Equation D.17: M f M e E I A E E A e I IE e I A e I A e E A E E A e I IE D c c c c c c c = − + +               +     − + +         1 1 1 1 (D.17)Decp se 0 ps ps ps 0 2 0 0 2 ps 0 ps ps ps 0 2 The decompression moment at the bottom fiber of the concrete girder, MDecb, can be calculated using Equation D.18: 1 1 1 1 (D.18) Decb se 0 ps ps ps 0 2 0 ps 0 ps ps ps 0 2 M f M e E I A E E A e I IE y I A e y I A e E A E E A e I IE D c c c b c b c c c = − + +               +     − + +         where Ac = area of concrete at the cross section considered; Aps = area of prestressing steel in tension zone; e0 = eccentricity of the prestressing force with respect to the centroid of the section at supports; yb = distance from centroidal axis to extreme bottom fiber; Ec = modulus of elasticity of concrete; Eps = modulus of elasticity of prestressing steel; fse = effective stress in prestressing steel after losses; I = moment of inertia; MD = dead load moment; MDecb = decompression moment at the bottom of the girder; and MDecp = decompression moment at the level of the pre- stressing strands. References AASHTO LRFD Bridge Design Specifications, 6th ed. 2012. American Association of State Highway and Transportation Officials, Washing- ton, D.C. Nawy, E., and P. Huang. 1977. Crack and Deflection Control of Pre- tensioned Prestressed Beams. Journal, Precast/Prestressed Concrete Institute, Vol. 23, No. 3, pp. 30–43.