Transcript Stability Degradation and Redundancy in Damaged Structures

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 Rumsfeld
February 12, 2002
Building design philosophy
traditional design for
environmental hazards
augmented design for
unforeseen hazards
Overview
• Performance based design
Extending to unknowns: design for unforeseen events
• Example 1
Stability degradation of a 2 story 2 bay planar moment frame
(Ziemian et al. 1992) under increasing damage
• Example 2
Stability 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
 DV    G DV | DM | dG DM | EDP dG EDP | IM | d ( IM ) |
IM = Hazard
intensity measure
spectral acceleration, spectral
velocity, duration, …
components lost,
volume damaged,
% strain energy released, …
EDP = Engineering
demand parameter
inter-story drift,
max base shear,
plastic connection rotation,…
drift
Eigenvalues of Ktan after
loss
DM = Damage
measure
condition assessment,
necessary repairs, …
DV = Decision
variable
failure (life-safety), $ loss,
downtime, …
v(DV) = PDV
probability of failure (Pf),
mean annual prob. of $ loss,
50% replacement cost, …
Pf   G DV EDP | dG EDP IM | d (IM )
IM: Intensity Measure
Inclusion of unforeseen hazards through damage
• Type of damage
Damage– discrete member removal* – brittle!
– strain energy, material volume lost, ..
– member weakening
Insertion
• 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(m,s2))
* 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))f = 0
damaged: (Ker - crKgr(Pr)fr = 0
Example 1: Ziemian Frame
10.5 k/ft
15’
B2:W36x170
C2:W14x132
20’
C6:W14x109
B4:W27x102
C3:W14x120
C4:W8x13
B1:W27x84
C1:W8x15
• 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).
B3:W21x44
C5:W14x120
4.9 k/ft
20’
48’
(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 removal
Load conservative?
C3
B2
C2
C1
B1
C6
B4
C5
C4
B3
cr-intact= 3.14
Solution?
no
yes
load conservative
no
yes

solution
exact
approx.
member
cr1
cr
cr3
%
C1
3.09
3.09
3.13
C2
0.84
0.84
1.50
C3
1.44
1.44
1.19
C4
3.17
3.17
3.14
C5
2.20
2.20
3.14
C6
3.14
3.14
1.05
B1
3.89
1.42
2.04
B2
3.07
1.90
1.92
B3
3.50
1.76
2.59
B4
3.98
3.14
3.13
*est. Pf for intact structure under a unit increase in mfy Buonop
exact
(Ker - crKgr(Pr)fr = 0
approximate
(Ker - crKgr(P)fr = 0
cr1 , f1 pairs
Pf = P(cr<1) = 1/10
BEAM REMOVAL
COLUMN REMOVAL
Mode tracking
Eigenvectors of the intact structure fi form an
eigenbasis, matrix Fi. We examined the eigenvectors for
the damaged structure fjr in the Fi basis, via:
(fjr)F = (Fir)-1(fjr)
The entries in (fjr)F 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)
cr<1 = Failure
damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10
Fragility
Progressive Collapse, Pc
Pc nd  ni   Pi
k
 Pd
d i  1
Pd = probability that cr=0 at state nd
Pc|(nd = n4) =
P4
P5
P6
P7
P8
P9
FAIL
Pc|(nd = n4) = 40 %
Pd is cheap to calculate only requires
the condition number of Ker!
Fragility
IM = strain energy removal?
SE = ½dTKd
IM = nd vs SE
Distribution of SEintact=SEdamaged?
Example 2: SAC/Seattle 3 Story*
12 k/ft
W18X40
3 stories @ 13’
12 k/ft
W24X84
12 k/ft
W24X76
*This model modified from the paper, member sizes are Seattle 3.
W14X176
W14X159
• Planar moment frame
with member selection
consistent with current
lateral design standards.
• Considered here, with and
without additional braces
60
0X
W1
4 bays @ 30’
Intact buckling mode shapes (fi)
Computational effort
m ≈ an4
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) = N(7%,7%)
with braces:
Pf = 0.2%
no braces:
Pf = 0.7%
IM2 = N(10,2) = N(37%,7%)
IM1 = N(2,2)
IM2 = N(10,2)
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…
no
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:
loadyes
conservative
no
yes
cr intact = 3.14
cr intact = 3.14
solution
exact
approx.
exact
approx.
cr: Change
in the buckling
cr1cr
cr f*
cr3
%cr
%Pf*
cr1
cr member
cr3
%
%P
load
as
members
are
C1 3.13
3.09
3.09
3.13
-2
-20
3.09
3.09
-2
-20
removed -73
from the -1
frame
C2 1.50
0.84
0.84
1.50
0.84
0.84
-73
-1
C3 1.19
1.44
1.44
1.19
-54
-11
1.44
1.44
-54
-11
C4 3.14
3.17
3.17
3.14
1
-1
3.17
3.17
1
-1
P
*:
Change
in
the
C5 3.14
2.20
2.20
3.14f
-30
4 Pf as the
2.20
2.20
-30
4
C6 1.05
3.14
3.14
1.05
14 is varied
3.14
3.14
0
14
mean yield0 strength
B1 2.04
3.89
1.42
2.04
-55
-22
3.89
1.42
-55
-22
in the frame
(Buonopane
et
B2 1.92
3.07
1.90
1.92
-39
-26
3.07
1.90
-39
-26
al. 2003) -44
B3 2.59
3.50
1.76
2.59
-3
3.50
1.76
-44
-3
B4 3.13
3.98
3.14
3.13
0
11
3.98
3.14
0
11
*est.
Pf forinintact
structure under
a unit increase in mfy Buonopane (2003)
ure under a unit
increase
mfy Buonopane
(2003)