Report - PEER - University of California, Berkeley

i. The well-known relation µ θ - µ φ that employs the plastic hinge length, L pl ;ii. An empirical relation for L pl , fitted to hundreds of cyclic test results on memberswith flexure-controlled failure, for ultimate curvatures computed assuming: (a) asteel ultimate strain, ε su , equal to the minimum values of 2.5% and 5% given inEurocode 2 for steel Classes A or B and to ε su = 6% for steel Class C, and (b) theultimate strain of confined concrete given by the Eurocode 2 relation:ε cu,c = 0.0035 + 0.1αω w (3)where ω w =ρ w f yw /f c is the volumetric mechanical ratio of confining steel withrespect to the confined concrete core and α the confinement effectiveness ratio:⎛2⎛ s ⎞⎛ ⎞ ⎞= ⎜ ⎟⎜ −h s⎟⎜ −h⎟−∑biα 1 1 1(4)⎝ 2bo⎠⎝2ho⎠⎝6hobo⎠In Eq. (4) b o and h o are the dimensions of the confined core to the hoop centerlineand b i the spacing of laterally restrained longitudinal bars on the perimeter;iii. Rounding-up the values of L pl resulting from (ii) above for the range of memberparameters common in buildings into a single one: L pl = 0.185L s , where L s is theshear span at the member end. Then Eq.(2), with µ θ replacing µ δ , gives a safetyfactor on µ φ for given µ θ , which is on average equal to 1.65 for columns, 1.35 forbeams or 1.1 for walls, within the range of possible values of q for DC M and Hbuildings and for the usual range of L pl for the 3 types of concrete members.Once a beam-sway plastic mechanism is ensured, the demand value of µ θ at thosemember ends where plastic hinges may form (at beam ends and the base of columnsand walls) is about equal to the global displacement ductility factor, µ δ . Hence Eq.(2).Members are detailed to provide the value of µ φ =φ u /φ y from Eq.(2). This isachieved on the basis of the definition of φ u as φ u =ε cu /ξ cu d, with ξ cu computed as:( 1 − δ1)( ν + ω1− ω2) + ( 1 + δ1)ωvξcu= (5)⎛ ε c⎞( 1 − δ1) ⎜1− + 2ωv3ε⎟⎝ cu ⎠where ω 1 , ω 2 , ω v are mechanical ratios of tension and compression reinforcement andof the (web) vertical bars between them, ν =N/bdf c is the axial load ratio, δ 1 =d 1 /d thedistance of the tension or compression reinforcement from the corresponding extremefibers, normalized to d, ε c =0.002 the strain of concrete at f c and ε cu its ultimate strain.2.2.2 Maximum Tension Reinforcement Ratio at the Ends of DC M or H BeamsTaking φ y =1.5f y /E s d, as derived from beams tests at yielding, ω v =0, ν=0 andε cu =0.0035 at the unconfined extreme compression fibers, the upper limit of the beamtension reinforcement ratio, ρ 1 , is derived. Using the design values, f cd =f ck /γ c ,6