stability degradation and redundancy in damaged structures benjamin w. schafer puneet bajpai...
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Stability Degradation and Redundancy in Damaged Structures
Benjamin W. Schafer
Puneet Bajpai
Department of Civil Engineering
Johns Hopkins University
Acknowledgments
• The research for this paper was partially sponsored by a grant from the National Science Foundation (NSF-DMII-0228246).
There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know.
Donald RumsfeldFebruary 12, 2002
Building design philosophy
service loads(always present)
live loads(transient)
environmentalhazards
wind/seismic(transient)
Pf?service loads
(always present)
live loads(transient)
environmentalhazards
wind/seismic(transient)
Pf?
traditional design for environmental hazards
service loads(always present)Pf?
unforeseendamage
service loads(always present)Pf?
unforeseendamage
augmented design forunforeseen hazards
Overview• Performance based design
Extending to unknowns: design for unforeseen events
• Example 1Stability degradation of a 2 story 2 bay planar moment frame (Ziemian et al. 1992) under increasing damage
• Example 2Stability degradation of a 3 story 4 bay planar moment frame (SAC Seattle 3) under increasing damage
Impact of redundant systems (bracing) on stability degradation and Pf
• Conclusions
PBD and PEER framework equation
|)(||||| IMdIMEDPdGEDPDMdGDMDVGDV
)IM(d|IMEDPdG|EDPDVGPf
IM = Hazard intensity measure
spectral acceleration, spectral velocity, duration, …
components lost,volume damaged,% strain energy released, … driftEigenvalues of Ktan after loss
inter-story drift,max base shear,plastic connection rotation,…
EDP = Engineering demand parameter
condition assessment,necessary repairs, …
DM = Damage measure
failure (life-safety), $ loss,downtime, …
DV = Decision variable
probability of failure (Pf),
mean annual prob. of $ loss, 50% replacement cost, …
v(DV) = PDV
IM: Intensity MeasureInclusion of unforeseen hazards through damage• Type of damage
– discrete member removal* – brittle!– strain energy, material volume lost, ..– member weakening
• Extent/correlation of damage– connected members – single event– concentric damage, biased damage, distributed
• Likelihood of damage– categorical definitions (IM: n=1, n/ntot= 10%)
– probabilistic definitions (IM: N(,2))
Damage- Insertion
* member removal forces real topology change, new load paths are examined, new kinematic mechanisms are considered, …
EDP: Engineering Demand Parameter• Potential engineering demand parameters include
– inter-story drift, inelastic buckling load, others…
• Primary focus is on stability EDP, or buckling load: cr – single scalar metric– avoiding disproportionate response means avoiding stability
loss for portions of the structure, and – calculation is computationally cheap, requires no iteration and
has significant potential for efficiencies.
• Computation of cr involves:
intact: (Ke - crKg(P)) = 0
damaged: (Ker - crKg
r(Pr)r = 0
Example 1: Ziemian Frame
• Planar frame with leaning columns. Contains interesting stability behavior that is difficult to capture in conventional design.
• Thoroughly studied for advanced analysis ideas in steel design (Ziemian et al. 1992).
• Also examined for reliabiity implications of advanced analysis methods(Buonopane et al. 2003).
B1:W27x84 B2:W36x170
B3:W21x44 B4:W27x102
C1:
W8x
15
C2:
W14
x132
C3:
W14
x120
20’ 48’
20’
15’
C4:
W8x
13
C5:
W14
x120
C6:
W14
x109
4.9 k/ft
10.5 k/ft
B1:W27x84 B2:W36x170
B3:W21x44 B4:W27x102
C1:
W8x
15
C2:
W14
x132
C3:
W14
x120
20’ 48’
20’
15’
C4:
W8x
13
C5:
W14
x120
C6:
W14
x109
4.9 k/ft
10.5 k/ft
(Ziemian et al. 1992)
Analysis of Ziemian Frame• IM = Member removal
– single member removal: m1 = ndamaged/ntotal = 1/10
– multi-member removal: m1 = 1/10 to 9/10– strain energy of removed members
• EDP = Buckling load (cr)– load conservative or non-load conservative?– exact or approximate Kg?– first buckling load, or tracked buckling mode?
• DV = Probability of failure (Pf)– Pf = P(cr<1)– Pf = P(cr =0)– Pf = P that a kinematic mechanism has formed
Single member removalLoad conservative? no yes
Solution? exact approximate(Ke
r - crKgr(Pr)r = 0 (Ke
r - crKgr(P)r = 0
cr-intact= 3.14
load conservative no yes solution exact approx.
member cr1
cr
cr3 %
cr %Pf*C1 3.09 3.09 3.13 -2 -20C2 0.84 0.84 1.50 -73 -1C3 1.44 1.44 1.19 -54 -11C4 3.17 3.17 3.14 1 -1C5 2.20 2.20 3.14 -30 4C6 3.14 3.14 1.05 0 14B1 3.89 1.42 2.04 -55 -22B2 3.07 1.90 1.92 -39 -26B3 3.50 1.76 2.59 -44 -3B4 3.98 3.14 3.13 0 11
*est. Pf for intact structure under a unit increase in fy Buonopane (2003)
cr intact = 3.14
C1
C2
C3
C4
C5
C6
B1 B2
B3 B4
C1
C2
C3
C4
C5
C6
B1 B2
B3 B4
BEAM REMOVAL
cr1 , 1 pairs COLUMN REMOVAL
Pf = P(cr<1) = 1/10
Mode trackingEigenvectors of the intact structure i form an eigenbasis,
matrix i. We examined the eigenvectors for the damaged
structure jr in the i basis, via:
(jr) = (i
r)-1(jr)
The entries in (jr) provide the magnitudes of the modal
contributions based on the intact modes.
Multi-member removal (Stability Degradation)
damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10
Fragility: P(cr < 1)
damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10
cr<1 = Failure
Fragility
P4 P5 P6 P7 P8 P9 FAIL
k
iddiid PPnnP
1c
Pd = probability that cr=0 at state nd
Pc|(nd = n4) =
Progressive Collapse, Pc
Pc|(nd = n4) = 40 %
Pd is cheap to calculate only requiresthe condition number of Ke
r!
Fragility
IM = strain energy removal?SE = ½dTKd
IM = nd vs SE
Distribution of SEintact=SEdamaged?
Example 2: SAC/Seattle 3 Story*
• Planar moment frame with member selection consistent with current lateral design standards.
• Considered here, with and without additional braces
*This model modified from the paper, member sizes are Seattle 3.
3 stories @ 13’
12 k/ft
12 k/ft
12 k/ft
W24X76
W14X
159
W14X
176
W24X84
W18X40
4 bays @ 30’
W10X60
3 stories @ 13’
12 k/ft
12 k/ft
12 k/ft
W24X76
W14X
159
W14X
176
W24X84
W18X40
4 bays @ 30’
W10X60
Intact buckling mode shapes (i)
Computational effort
m ≈ n4
Computational effort and sampling
Stability degradation
Fragility and impact of redundancy
Fragility and mode tracking
i.e.,
Decision-making and Pf
IM1 = N(2,2) IM2 = N(10,2)
IM1 ~ N(2,2) = N(7%,7%)
with braces: Pf = 0.2%
no braces: Pf = 0.7%
IM2 = N(10,2) = N(37%,7%)
with braces: Pf = 27%
no braces: Pf = 44%
Pf $ decision
Conclusions• Building design based on load cases only goes so far.• Extension of PBD to unforeseen events is possible.• Degradation in stability of a building under random
connected member removal uniquely explores building sensitivity and provides a quantitative tool.
• For progressive collapse even cheaper (but coarser) stability measures may be available via condition of Ke.
• Computational challenges in sampling and mode tracking remain, but are not insurmountable.
• Significant work remains in (1) integrating such a tool into design and (2) demonstrating its effectiveness in decision-making, but the concept has promise.
Can we transform unknown unknowns into known unknowns? Maybe a bit…
Why member removal?• Member removal forces the topology to change – this explores
new load paths and helps to reveal kinematic mechanisms that may exist. Standard member sensitivity analysis does not explore the same space, consider:
load conservative no yes solution exact approx.
member cr1
cr
cr3 %
cr %Pf*C1 3.09 3.09 3.13 -2 -20C2 0.84 0.84 1.50 -73 -1C3 1.44 1.44 1.19 -54 -11C4 3.17 3.17 3.14 1 -1C5 2.20 2.20 3.14 -30 4C6 3.14 3.14 1.05 0 14B1 3.89 1.42 2.04 -55 -22B2 3.07 1.90 1.92 -39 -26B3 3.50 1.76 2.59 -44 -3B4 3.98 3.14 3.13 0 11
*est. Pf for intact structure under a unit increase in fy Buonopane (2003)
load conservative no yes solution exact approx.
member cr1
cr
cr3 %
cr %Pf*C1 3.09 3.09 3.13 -2 -20C2 0.84 0.84 1.50 -73 -1C3 1.44 1.44 1.19 -54 -11C4 3.17 3.17 3.14 1 -1C5 2.20 2.20 3.14 -30 4C6 3.14 3.14 1.05 0 14B1 3.89 1.42 2.04 -55 -22B2 3.07 1.90 1.92 -39 -26B3 3.50 1.76 2.59 -44 -3B4 3.98 3.14 3.13 0 11
*est. Pf for intact structure under a unit increase in fy Buonopane (2003)
cr intact = 3.14
cr: Change in the buckling load as members are removed from the frame
Pf*: Change in the Pf as the mean yield strength is varied in the frame (Buonopane et al. 2003)
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