P-Delta and Minimum Base Shear
H. Krawinkler, 2/22/07
Min. Base Shear • V = (0.8ZNvI/R)W for drift limitation? • Is it for ground motion uncertainty or “modeling” uncertainty or both? • Let’s say explicitly what it is for! • Should be the same “enforcement” as for code-conforming structures • If for modeling uncertainties and alternative design, then it belongs to Level 3 – if for collapse safety? H. Krawinkler, 2/22/07
Collapse Safety + Drift ! P-Delta P-Delta is controlled by – P (large in lower stories) – Delta - but inelastic ! – Collapse mechanism – Length of post-yield “plateau” – Effective post yield stiffness – Deterioration – Frame problem very diff. from wall problem H. Krawinkler, 2/22/07
Pushover Deflection Profiles, without and with P-delta -- N =18, T = 3.6 -- Frame
H. Krawinkler, 2/22/07
Global Pushover Curve, without and with P-Delta -- N =18, T = 3.6 -- Frame
H. Krawinkler, 2/22/07
Global Pushover Curve, LA-20, without and with PROOF DRIFT ANGLE vs. NORMALIZED BASE SHEAR Pushover (NEHRP '94 k=2 pattern): LA 20-Story, Pre-Northridge, M1, M 1-NPD
0.14 Model M1 Model M1-NPD ASD Base Shear (UBC '94)
0.12
) W / V ( 0.1 r a e h 0.08 S Vy e s a B 0.06 d e z i l a 0.04 m V r o ASD N0.02 0 0
0.01
0.02
0.03
Roof Drift Angle
H. Krawinkler, 2/22/07
0.04
0.05
Sensitivity to Strain Hardening, Pushover, LA-20 ROOF DRIFT ANGLE vs. NORMALIZED BASE SHEAR Pushover: LA 20-Story, Pre-Northridge, Model M2,
= 0%, 3%, 5%, 10%
0.20 Strain-Hardening = 0% Strain-Hardening = 3% Strain-Hardening = 5% Strain-Hardening = 10%
) W / 0.15 V ( r a e h S e s 0.10 a B d e z i l a 0.05 m r o N 0.00 0.00
0.01
0.02
0.03
0.04
0.05
0.06
Roof Drift Angle
H. Krawinkler, 2/22/07
0.07
0.08
0.09
0.10
Elastic and Inelastic Stability Coefficient N =18, T = 3.6 -- Frame GLOBAL PUSHOVER CURVES N=18, T 1=3.6, BH, Peak Oriented Model, LMSR-N, s=0.03,
c/ y =Inf,
c=N.A,
s,c,k,a
=Inf,
=5%,
=0
1.2 s,0 K 0=0.04K 0
y
1
V / V , 0.8 h t g n e r 0.6 t S d e z 0.4 i l a m r o 0.2 N
i=0.37
e=0.09
0
Pushover Curve with PPushover Curve w/o P-
Effects Effects
0 0
0.5
H. Krawinkler, 2/22/07
1 1.5 2 Normalized Roof Displacement,
2.5 r
/
yr
3
Effect of P-Delta on Median Collapse Capacity (Deteriorating Frame Systems) EFFECT OF P-
ON MEDIAN [S a,c (T1)/g]/
N=Var, T 1=Var, BH, Peak Oriented Model, LMSR-N, =5%,
s=0.03,
c/ y =4,
c=-0.10,
s,c,k,a
=Inf,
=0
20
15 / ] g / )
1
T ( c
Ref. Frames w/o P-D, T1=0.1N
10
Ref. Frames w/o P-D, T1=0.2N
, a
Ref. Frames with P-D, T1=0.1N
S [
Ref. Frames with P-D, T1=0.2N
5
0 0
H. Krawinkler, 2/22/07
1
2 Period (sec)
3
4
Effect of P-Delta on Median Collapse Capacity (as function of base shear yield coefficient) Frames versus Walls (8-story) P-Delta Effect on N = 8, T 1 = 1.2,
c (MRF)
P-Delta Effect on
= var., Stiff.&Str. = Shear, SCB = 2.4-1.2,
p = 0.03,
pc / p =
5.0,
= 0.05
N = 8, T
= 20, M c /M y = 1.1
= 0.8,
p = 0.02,
3
c (SW)
= var., Str. = var.,
pc / p = 1.0,
= 0.05
= 20, M c /M y = 1.1
6 Without P-Delta
Without P-Delta
2.5
5 With P-Delta
With P-Delta
2 ) g / a S (
1
4 ) g / a S (
1.5
c
3
c
1
2
0.5
1
0
0 0
0.05
0.1
0.15
0.2
Base shear coefficient
H. Krawinkler, 2/22/07
0.25
0.3
0.35
0
0.1
0.2
0.3
Base shear coefficient
0.4
0.5
0.6
So, what’s the point? • P-Delta, which is amplified by deterioration, causes collapse (not the only source) • P-Delta effect is very sensitive and not straight forward to predict • We should safeguard against prediction errors • But min. base shear does not look like the right vehicle to do so • In codes: establish a limit on P!/(Vyh)???? H. Krawinkler, 2/22/07
