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Chapter

# Flexural members

DOI link for Flexural members

Flexural members book

# Flexural members

DOI link for Flexural members

Flexural members book

## ABSTRACT

Acp = area enclosed by outside perimeter of concrete cross section, in.2 (mm2)

Af = area of FRP reinforcement in2. (mm2) Afb = area of flexural reinforcement producing balanced failure, in.2

(mm2) Af,bar = area of one FRP bar, in.2 (mm2) Af,i = area of reinforcement in i-th layer, in.2 (mm2) Af,min = minimum area of FRP reinforcement needed to prevent failure

of flexural members upon cracking, in.2 (mm2) Af,sh = area of shrinkage and temperature FRP reinforcement per

linear foot, in.2 (mm2) Afv = amount of FRP shear reinforcement within spacing s, in.2 (mm2) Afv,min = minimum amount of FRP shear reinforcement within spacing

s, in.2 (mm2) Aoh = area enclosed by centerline of the outermost closed transverse

torsional reinforcement, in.2 (mm2) As = area of tension steel reinforcement, in.2 (mm2) At = area of one leg of a closed stirrup resisting torsion within spacing

s, in.2 (mm2) Avf = area of shear friction reinforcement perpendicular to the plane of

shear a = depth of equivalent rectangular stress block, in. (mm) b = width of rectangular cross section, in. (mm) bo = perimeter of critical section for slabs and footings, in. (mm) beff = effective width of the slab, in. (mm) bw = width of the web, in. (mm) C = Compressive force, lb (N) CE = environmental reduction factor for various fiber types and

exposure conditions

c = distance from extreme compression fiber to the neutral axis, in. (mm)

c = spacing or cover dimension cb = distance from extreme compression fiber to neutral axis at bal-

anced strain condition, in. (mm) ccr = cracked neutral axis depth, in. (mm) D = diameter of circular cross section, in. (mm) d = distance from extreme compression fiber to centroid of tension

reinforcement, in. (mm) db = diameter of reinforcing bar, in. (mm) dc = thickness of concrete cover measured from extreme tension

fiber to center of bar or wire location closest thereto, in. (mm) di = distance from centroid of ith layer of longitudinal reinforcement

to geometric centroid of cross section, in. (mm) e = ratio of εfu over εcu df = effective depth of the FRP reinforcement, in. (mm) Ec = modulus of elasticity of concrete, psi (MPa) Ef = design or guaranteed modulus of elasticity of FRP defined as

mean modulus of sample of test specimens (Ef = Ef,ave), psi (MPa) Ef,ave = average modulus of elasticity of FRP, psi (MPa) Es = modulus of elasticity of steel, psi (MPa) f ′c = specified compressive strength of concrete, psi (MPa) fr = modulus of rupture of concrete, psi (MPa) ff = stress in FRP reinforcement in tension, psi (MPa) ffb = strength of bent portion of FRP bar, psi (MPa) ffd = design tensile strength, psi (MPa) ffe = bar stress that can be developed for embedment length le, psi (MPa) ffr = required bar stress, psi (MPa) ff,s = stress level induced in FRP by sustained loads, psi (MPa) ffu = design tensile strength of FRP, considering reductions for ser-

vice environment (ffu = CEf *fu), psi (MPa) f*fu = guaranteed tensile strength of FRP bar, defined as mean tensile

strength of sample of test specimens minus three times standard deviation (f *fu = ffu,ave – 3σ), psi (MPa)

ffv = tensile strength of FRP for shear design, taken as smallest of design tensile strength ffu, strength of bent portion of FRP stirrups ffb, or stress corresponding to 0.004Ef, psi (MPa)

fs = allowable stress in steel reinforcement, psi (MPa) fu,ave = mean tensile strength of sample of test specimens, psi (MPa) fvf = shear friction stress in reinforcement fy = specified yield stress of nonprestressed steel reinforcement, psi

(MPa) h = overall height of rectangular member, in. (mm) I = moment of inertia, in.4 (mm4)

Icr = moment of inertia of transformed cracked section, in.4 (mm4) Ie = effective moment of inertia, in.4 (mm4) Ig = gross moment of inertia, in.4 (mm4) K1 = parameter accounting for boundary conditions k = ratio of depth of neutral axis to reinforcement depth kb = bond-dependent coefficient km = neutral axis depth to reinforcement depth ratio at midspan l = span length of member, ft (m) la = additional embedment length at support or at point of inflec-

tion, in. (mm) lbhf = basic development length of FRP standard hook in tension,

in. (mm) ld = development length, in. (mm) le = embedded length of reinforcing bar, in. (mm) fct = tensile strength of concrete, psi (MPa) fft = tensile strength of transverse reinforcement, psi (MPa) kcreep-R = creep rupture factor ld,fi,t,T>Tcr = embedment length of a bar with a temperature exceeding 122°F

(50°C) lthf = length of tail beyond hook in FRP bar, in. (mm) M = maximum positive moment lb-in.(N-mm) m = non-dimensional moment parameter Ma = maximum moment in member at stage deflection is computed,

lb-in. (N-mm) MC, MT = contributions to nominal moment capacity for circular section,

lb-in. (N-mm) Mcr = cracking moment, lb-in. (N-mm) Mn = nominal moment capacity, lb-in. (N-mm) Mnb = nominal moment strength corresponding to balanced failure,

lb-in. (N-mm) Ms = moment due to sustained load, lb-in. (N-mm) Mu = factored moment at section, lb-in. (N-mm) nf = ratio of modulus of elasticity of FRP bars to modulus of elasticity

of concrete pcp = outside perimeter of concrete cross section, lb-in. (N-mm) rb = internal radius of bend in FRP reinforcement, in. (mm) s = stirrup spacing or pitch of continuous spirals, and longitudinal

FRP bar spacing, in. (mm) T = temperature, oC T1, T2 = tensile force corresponding to A1 and A2, lb (N) Tf,i = tensile force in ith layer, lb (N) T, Tmax = tensile force and maximum tensile force, lb (N) Tn = nominal torsional moment strength, lb-in. (N-mm) Tg = glass transition temperature, °F (°C)

Tu = factored torsional moment at section, lb-in. (N-mm) t = time, minutes tslab = slab thickness, in. (mm) u = average bond stress acting on the surface of FRP bar, psi (MPa) Vc = nominal shear strength provided by concrete, lb (N) Vf = shear resistance provided by FRP stirrups, lb (N) Vn = nominal shear strength at section, lb (N) Vs = shear resistance provided by steel stirrups, lb (N) Vu = factored shear force at section, lb (N) w = maximum crack width, in. (mm) wc = maximum crack width at the tension face of a flexural member,

in., (mm) x, xb = distance of N.A. from compression edge and at balanced condi-

tion, in. (mm) yt = distance from centroidal axis of gross section, neglecting rein-

forcement, to tension face, in. (mm) α = angle of inclination of stirrups or spirals α = top bar modification factor α = ratio of x over d α1 = ratio of average stress of equivalent rectangular stress block

to f ′c αL = longitudinal coefficient of thermal expansion, 1/°F (1/°C) αT = transverse coefficient of thermal expansion, 1/°F (1/°C) β = ratio of distance from neutral axis to extreme tension fiber to

distance from neutral axis to center of tensile reinforcement β1 = factor relating depth of equivalent stress block to neutral axis

depth β2 = factor representing the influence of the load duration and

repetition βd = reduction coefficient used in calculating deflection γ = ratio of d over h or of d over D for columns Δ(cp+sh) = additional deflection due to creep and shrinkage under sus-

tained loads, in. (mm) (Δi)sus = immediate deflection due to sustained loads, in. (mm) (Δ/l)max = limiting deflection-span ratio εc = strain in concrete εcu = ultimate strain in concrete εf = strain in FRP reinforcement εf i = reinforcement strain in ith layer εo = maximum strain of unconfined concrete corresponding to fc εfu = design rupture strain of FRP reinforcement ε*fu = guaranteed rupture strain of FRP reinforcement defined as the

mean tensile strain at failure of sample of test specimens minus three times standard deviation (ε*fu = εu,ave – 3σ)

εm = reinforcement tensile strain at midspan εu,ave = mean tensile strain at rupture of sample of test specimens εv = shear friction strain in reinforcement εy = design yield strain η = ratio of distance from extreme compression fiber to centroid

of tension reinforcement (d) to overall height of flexural member (h)

λ = multiplier for additional long-term deflection μ = coefficient of subgrade friction for calculation of shrinkage and

temperature reinforcement ξ = time-dependent factor for sustained load ρ′ = ratio of steel compression reinforcement, ρ′ = As′/bd ρb = FRP reinforcement ratio producing balanced strain conditions ρf = average deterioration factors for modulus of elasticity at a spe-

cific temperature T in oC ρf = FRP reinforcement ratio ρf′ = ratio of FRP compression reinforcement ρfv = ratio of FRP shear reinforcement ρf,ts = reinforcement ratio for temperature and shrinkage FRP

reinforcement ρmin = minimum reinforcement ratio for steel σ = standard deviation σc = compressive stress, psi (MPa) χ = curvature ϕ = strength reduction factor ωf = tension reinforcement index ωf b = tension reinforcement index corresponding to balanced failure

4.1 INTRODUCTION

In this chapter, the design of slabs and beams is discussed. Following the conventional design procedure, this chapter first investigates the structural analysis of flexural members and elaborates on the parameters that define the input and determine the output of such analysis methods. Next, flexural design with fiber-reinforced polymer (FRP) bars is discussed and it is demonstrated how their mechanical behavior can divert the design process from methods that are well established for steel bars. Flexural design is completed by detailing and explaining the serviceability provisions of FRP reinforced concrete (FRP RC) flexural members that, when compared to steel RC, play a more prominent role in the overall design process. Shear design of flexural members with or without FRP transverse reinforcement concludes this chapter. Chapters 6-8 and 10 propose design examples that clarify the topics covered in this chapter.