National Academies Press: OpenBook

LRFD Minimum Flexural Reinforcement Requirements (2019)

Chapter: List of Notations

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Page 100
Suggested Citation:"List of Notations." National Academies of Sciences, Engineering, and Medicine. 2019. LRFD Minimum Flexural Reinforcement Requirements. Washington, DC: The National Academies Press. doi: 10.17226/25527.
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Page 100
Page 101
Suggested Citation:"List of Notations." National Academies of Sciences, Engineering, and Medicine. 2019. LRFD Minimum Flexural Reinforcement Requirements. Washington, DC: The National Academies Press. doi: 10.17226/25527.
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Page 101
Page 102
Suggested Citation:"List of Notations." National Academies of Sciences, Engineering, and Medicine. 2019. LRFD Minimum Flexural Reinforcement Requirements. Washington, DC: The National Academies Press. doi: 10.17226/25527.
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Page 102

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100 a = distance from loading frame to concentrated crack A = area of concrete section Act = area of cross section between flexural tension face and center of gravity of gross section Afl = ratio of direct tension strength and flexural cracking strength Aps = area of prestressing strand Aps,min. = minimum area of prestressed steel As = area of non-prestressed steel As,min. = minimum area of reinforcement b = beam width bt = mean width of tension zone; for T-beam with flange in compression, only width of web is taken into account bw = beam web width or beam effective width c = neutral axis depth Cc = compression force in concrete d = beam effective depth ds = effective depth of beam to centroid of reinforcement e = eccentricity Ec = Young’s modulus of concrete Eps = Young’s modulus of prestressing strand Es = Young’s modulus for mild steel reinforcement f c′ = concrete compressive strength f ′ci = concrete compressive strength at time of prestressing fcm = average or mean compressive strength fcpe = compressive stress in concrete due to effective prestress forces at extreme fiber fctm = mean value of axial tensile strength of concrete fctm(t) = mean value of axial tensile strength of concrete at time being considered fcr = anticipated flexural cracking strength of full-depth member fpe = effective stress in prestressing strand fpi = initial stress in prestressing strand fps = stress in prestressing strand at ultimate flexure fpu = ultimate stress in prestressing strand (just before rupture) fpy = yielding stress in prestressing strand fr = modulus of rupture of concrete fse = effective stress in prestressing steel after losses fsp = concrete splitting strength (from split cylinder test) List of Notations

101 fsu = stress in prestressed steel at failure fu = ultimate stress in mild reinforcement (just before rupture) fy = stress in mild reinforcement at specified yield strain F = applied load GF = fracture energy h = height I = moment of inertia j = internal moment arm ratio jd = distance between centroid of compression zone and tensile force due to post-tensioning KIC = fracture toughness of concrete lpt = length of unbonded prestressed strand L = span length Mcr = cracking moment mcr = theoretical cracking moment (fry/I) Mcre = experimental cracking moment Mcr = proposed cracking moment Mdnc = total unfactored dead load moment acting on section Mn = nominal flexural strength Mo = maximum moment capacity, or nominal moment at overstrength Mo/Mcr = overstrength moment ratio Mr = factored moment resistance Mu = factored ultimate moment n = modular ratio Np = brittleness number Pf = force due to prestressing S = section modulus Sc = section modulus for the extreme fiber of the composite section Snc = section modulus for the extreme fiber of the monolithic or noncomposite section Tcr = tensile force Tpt = tensile force from prestressing strand Wcr = crack width yb = centroid location from extreme bottom fiber yc = distance from centroid of compression zone to bottom of girder ypt = distance from center of prestressed strand to bottom of girder yt = centroid location from extreme top fiber Y-hinge = distance from centroid of compression zone to bottom of girder α = strength modification factor β1 = factor relating depth of compressive stress block to depth of neutral axis γ1 = flexural cracking variability factor γ2 = prestress variability factor γ3 = ratio of yield to ultimate reinforcement for nonprestressed steel Δo = displacement at failure Δcr = displacement at first cracking Δo/Δcr = displacement ductility beyond cracking limit state ecl = compressive controlled strain limit in extreme tension steel ept = incremental prestressed strand strain es = strain in prestressed strand esu = ultimate strain in prestressed strand et = net tensile strain in extreme tension steel at nominal resistance

102 etl = tension-controlled strain limit in extreme tension steel etm = maximum net tensile strain eu = ultimate strain q1,q2 = rotation of girder on each side of concentrated crack µΔ = displacement ductility r = reinforcement ratio (As/bd) rL = longitudinal reinforcement ratio rmin. = minimum reinforcement ratio φ = resistance factor

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TRB’s National Cooperative Highway Research Program (NCHRP) Research Report 906 includes proposed revisions to the AASHTO LRFD Bridge Design Specifications minimum flexural reinforcement provisions for load and resistance factor design (LRFD) with detailed design examples illustrating the application of the proposed revisions.

According to the AASHTO LRFD Bridge Design Specifications, minimum reinforcement provisions are intended to reduce the probability of brittle failure by providing flexural capacity greater than the cracking moment. There was a concern with the current American Association of State Highway and Transportation Officials (AASHTO) LRFD minimum flexural reinforcement requirements, especially when applied to pretensioned or post-tensioned concrete flexural members.

A number of deliverables for the project are provided in three appendices to the contractor’s final report that are available online. They include the following:

Appendix A: Test Girder Drawings,

Appendix B: Design Examples, and

Appendix C: Parametric Study Results.

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