Report - PEER - University of California, Berkeley

f yd =f yk /γ s , of the concrete and steel strengths and the value ε yd =f yd /Ε s of ε y =f y /E s androunding up a coefficient, the following condition for DC M or H beams is derived:0.0018 fcdρ1 ,max = ρ2+(6)ε yd µ φ f ydwhere ρ 2 is the compression reinforcement ratio and both ρ 1 and ρ 2 are normalized tothe width b of the compression zone. With this rounding, the safety factor of 1.35 onµ φ from Eq.(2) mentioned in (iii) above increases to 1.6 if the values γ c =1.5, γ s =1.15recommended in Eurocode 2 for the persistent and transient design situation areadopted, and to 1.4 if the values γ c =1, γ s =1 are used instead, as recommended inEurocode 2 for the accidental design situation (1.6/1.4≈γ s =1.15). This “theoretical”safety factor can be compared to the ratio of: (a) the real value of (ρ 1 -ρ 2 ) in beamscyclically tested to flexural failure, to (b) the value obtained from Eqs. (6) and (2) forthe value of µ θ =µ δ at beam ultimate deflection. The median value of the ratio in 52beam tests is 0.725 for γ c =1, γ s =1, or 0.825 for γ c =1.5, γ s =1.15. Being less than 1.0,these values suggest that Eq.(6) is unsafe. Nonetheless, if the value of µ θ =µ δ isdetermined as the ratio of beam ultimate drift not to the experimental drift at yielding,but to the value M y L s /3(0.5EI) that corresponds to the effective elastic stiffness of0.5EI assumed in Eurocode 8, the median ratio in the 52 tests becomes 2.5 for γ c =1,γ s =1, or 2.85 for γ c =1.5, γ s =1.15, above the “theoretical” safety factors of 1.4 or 1.6.2.2.3 Confining Reinforcement at the Base of DC M or H Columns or WallsSimilar is the derivation of the confining reinforcement required by Eurocode 8 at thebase of columns or walls (in the boundary elements). Setting: φ u =ε cu,c /ξ cu h and φ y =λε y /h with λ = 1.85 for columns and λ = 1.45 for walls, as derived from test results,using again Eq.(5) but this time for the confined core after spalling of the concretecover (δ 1 ≈0) and with all variables in both sides of Eq.(5) normalized to h instead ofd, using Eq.(3) for the ultimate strain of confined concrete, rounding-up coefficientsand using the design strengths of materials, we obtain the Eurocode 8 requirement forconfining reinforcement of symmetrically reinforced (ω 1 =ω 2 ) columns or walls:bcαω wd = 30µφεyd ( ν d + ω ν d ) − 0. 035(7)bowhere b c is the width of the compression flange and b o the corresponding width of theconfined core. In columns Eurocode 8 neglects ω vd in the parenthesis, compared to ν d .With the rounding-up, the safety factor of 1.65 in columns or 1.1 in walls on the valueof µ φ from Eq.(2) — as mentioned in (iii) above — becomes nearly 2.65 for columnsor 2.25 for walls. This “theoretical” safety factor can be compared to the ratio of: (a)the value of αω wd +0.035 required from Eqs. (7) and (2) in columns or walls cyclicallytested to flexural failure for the value of µ θ at member ultimate deflection, to (b) thereal value of αω wd +0.035 (proportional to the available value of µ φ according toEq.(7)) in the test. The median value of the ratio in 626 cyclic tests of columns withnon-zero ν d is 0.88 for γ c =1, γ s =1, or 0.92 for γ c =1.5, γ s =1.15. The correspondingmedian values in 49 cyclic wall tests is 0.93 for γ c =1, γ s =1, or 0.96 for γ c =1.5,7