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)