LRFD Minimum Flexural Reinforcement Requirements (2019)

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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