Design of Concrete Structures Using High-Strength Steel Reinforcement (2011)

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68 A, B & C Ramberg-Osgood parameters Ab area of reinforcing bar (in2) Ac area of core of spirally reinforced compression member measured to the out- side diameter of spiral (in2) Acv area of concrete at shear interface (in2) Ag gross area of section (in2) Aps area of prestressing steel (in2) As area of tension reinforcement (in2) As′ area of compression reinforcement (in2) Asp area of spiral reinforcement (in2) As,min minimum area of longitudinal steel (in2) As,max maximum area of longitudinal steel (in2) Asb area of longitudinal steel to produce a balanced failure (in2) Atr area of each stirrup or tie crossing the potential plane of splitting adjacent to the reinforcement being developed, spliced, or anchored (in2) Av area of transverse reinforcement within distance s (in2) Avf area of shear-friction reinforcement (in2) ADT average daily traffic ADTT average daily truck traffic a depth of equivalent rectangular stress block (in.); length of beam shear span (in.) b width of compression face of a member (in.) bc diameter of core of spirally reinforced compression member measured to the outside of the spiral (in.) bv width of web (in.) COV coefficient of variation c distance from the extreme compression fiber to the neutral axis (in.); required concrete cover over the reinforcing steel (in.); cohesion factor (psi) cb smaller of: distance from center of bar to nearest concrete edge or one-half the center-center spacing of bars being developed (in.) cc concrete clear cover (in.) DL dead load (AASHTO LRFD §3.6.1) d distance from compression face to centroid of tension reinforcement (in.) d′ distance from compression face to centroid of compression reinforcement (in.) Notation

69 db diameter of reinforcing bar (in.) dc minimum concrete cover measured to center of reinforcing bar (in.) dsp diameter of spiral reinforcement (in.) dt distance from compression face to extreme tension reinforcement (in.) dv effective shear depth (in.) Ec modulus of elasticity of concrete (ksi) Ecalc experimentally determined secant modulus of elasticity of reinforcing steel (ksi) Ep modulus of elasticity of prestressing tendons (ksi) Es modulus of elasticity of reinforcing bars (ksi) f c′ specified compressive strength of concrete for use in design (ksi) f cc′ confined concrete strength (ksi) f co′ unconfined concrete strength (ksi) fct average splitting tensile strength of lightweight aggregate concrete (ksi) ff permissible fatigue stress range (ksi) fmax maximum stress in fatigue cycle (ksi) fmin minimum stress in fatigue cycle (ksi) fpo “locked-in” stress in prestressing steel (ksi) fpu specified tensile strength of prestressing steel (ksi) fr permissible reinforcing steel stress range (ksi) fs tensile stress in tension reinforcement (ksi); reinforcing steel stress at service load, often taken as 0.60fy (ksi) f s′ tensile stress in compression reinforcement (ksi) fscr reinforcing steel stress at cracking load (ksi) fsp tensile stress in spiral reinforcement (ksi) fss tensile stress in steel reinforcement at the service limit state (ksi) fu ultimate (tensile) strength of reinforcing steel (ksi) fy yield strength of reinforcing steel (ksi) fyh specified yield strength of transverse reinforcement (ksi) fyt yield strength of the stirrup reinforcement (ksi) H height of a column (in.) h overall thickness or depth of a member (in.) Icr moment of inertia of fully cracked concrete section (in4) Ie effective moment of inertia of cracked concrete section (in4) Ig moment of inertia of gross concrete section (in4) K1 fraction of concrete strength available to resist interface shear (psi) K2 limiting interface shear resistance (psi) Ktr transverse reinforcement factor L length of simple span beam (in.) LLtruck vehicular live load (AASHTO LRFD §3.6.1) LLlane lane live load (AASHTO LRFD §3.6.1) LLfatigue fatigue live load (AASHTO LRFD §3.6.1) ld development length (in.) ldb basic development length for straight reinforcement to which modification fac- tors are applied to determine ld (in.) ldh development length of standard hook in tension as measured from critical sec- tion to outside end of hook (in.)

70 lhb basic development length of standard hook in tension (in.) lp length of plastic hinge at the base of a column (in.) M moment applied at the section (kip-in) Ma moment at which Ie is calculated (kip-in) Mcr moment to cause cracking in concrete section (kip-in) Mn nominal flexural resistance (kip-in) Mu factored moment at the section (kip-in) m variable exponent in Branson Equation N number of cycles of fatigue loading n modular ratio = Es/Ec or Ep/Ec P applied axial load (kips) P′ axial load capacity after spalling of concrete (kip) Pc permanent net compressive force across interface (kips) Pn nominal axial resistance of a section (kip) Po axial load capacity before spalling of concrete (kip) r/h ratio of base radius to height of rolled-on transverse deformations S fatigue stress range (ksi) s spacing of transverse shear reinforcing steel (in.); spacing of longitudinal rein- forcing steel (in.) smax maximum permitted spacing of transverse reinforcement (in.) tb distance from the extreme tension fiber to the center of the closest bar (in.) V applied shear (kip) Vc nominal shear resistance provided by tensile stresses in the concrete (kip) Vcr applied shear to cause cracking (kip) Vn nominal shear resistance of the section considered (kip) Vni shear friction capacity (kip) Vs shear resistance provided by shear reinforcement (kip) Vu ultimate shear capacity (kip) vu shear stress on the concrete (ksi) w crack width (in.); self weight of a beam (kip/in) wc unit weight of concrete (pcf); limiting crack width (in.) wmax maximum crack width at the extreme tension fiber (in.) w/c ratio of water to cementitious materials y neutral axis depth (in.) α angle of inclination of transverse reinforcement to longitudinal axis (°) β factor to account for amplification of strain calculated at the bar level to that at the surface due to strain gradient; factor relating effect of longitudinal strain on the shear capacity of concrete, as indicated by the ability of diagonally cracked concrete to transmit tension β1 factor relating depth of equivalent rectangular compressive stress block to neu- tral axis depth γ load factor γd, γe crack control exposure condition factor Δ shear displacement parallel to shear friction interface (in.) Δcr shear displacement parallel to shear friction interface at cracking load (in.) Δu shear displacement parallel to shear friction interface at ultimate load (in.)

71 δp inelastic portion of lateral deflection of column (in.) δy elastic portion of lateral deflection of column (in.) ε strain (in./in.) εcc peak strain of confined concrete (in./in.) εco peak strain of unconfined concrete (in./in.) εcu ultimate compressive strain of concrete (in./in.) εrupture reinforcing steel strain at rupture (in./in.) εs strain in tension steel (in./in.); reinforcing steel strain at service load (in./in.) εs′ strain in compression steel (in./in.) εscr reinforcing steel strain at cracking load (in./in.) εsh concrete shrinkage strain (in./in.) εsp strain in spiral reinforcement (in./in.) εsu reinforcing steel strain at ultimate load (in./in.) εt net tensile strain in extreme tension steel at nominal resistance (in./in.) εu reinforcing steel strain at ultimate strength fu (in./in.) εy reinforcing steel strain at yield strength fy (in./in.) θ angle of inclination of diagonal compressive stresses (°) λ factor reflecting the reduced mechanical properties of lightweight concrete µ friction factor µφ curvature ductility ν Poisson’s Ratio ρ reinforcement ratio = As/bd ρ′ reinforcement ratio = As′/bd ρb balanced reinforcement ratio ρg reinforcement ratio = As/Ag ρs ratio of spiral reinforcement to total volume of column core σsu tensile stress in interface steel reinforcement at ultimate shear load (ksi) τ shear capacity of reinforcing steel (ksi) τcr concrete shear stress at cracking (ksi) τu concrete shear stress at ultimate capacity (ksi) ϕ material resistance factor ϕu curvature at ultimate strength ϕy curvature at yield strength Ψc factor modifying ACI development length equation based on concrete strength Ψe factor modifying ACI development length equation based on reinforcement coating Ψs factor modifying ACI development length equation based on reinforcement size Ψt factor modifying ACI development length equation based on reinforcement location