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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).

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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

  • 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

Building design philosophy
  • 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

IM = Hazard intensity measure

EDP = Engineering demand parameter

DM = Damage measure

DV = Decision variable

v(DV) = PDV

spectral acceleration, spectral velocity, duration, …

inter-story drift,

max base shear,

plastic connection rotation,…

condition assessment,

necessary repairs, …

failure (life-safety), $ loss,

downtime, …

probability of failure (Pf),

mean annual prob. of $ loss, 50% replacement cost, …

components lost,

volume damaged,

% strain energy released, …


Eigenvalues of Ktan after loss

PBD and PEER framework equation
im intensity measure

Damage- Insertion

IM: Intensity Measure

Inclusion 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(m,s2))

* member removal forces real topology change, new load paths are examined, new kinematic mechanisms are considered, …

edp engineering demand parameter
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: lcr
    • 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 lcr involves:

intact: (Ke - lcrKg(P))f = 0

damaged: (Ker - lcrKgr(Pr)fr = 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).

Example 1: Ziemian Frame

(Ziemian et al. 1992)

analysis of ziemian frame
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 (lcr)
    • load conservative or non-load conservative?
    • exact or approximate Kg?
    • first buckling load, or tracked buckling mode?
  • DV = Probability of failure (Pf)
    • Pf = P(lcr<1)
    • Pf = P(lcr =0)
    • Pf = P that a kinematic mechanism has formed
single member removal



Load conservative?

Single member removal

lcr-intact= 3.14

Solution? exact approximate

(Ker - lcrKgr(Pr)fr = 0 (Ker - lcrKgr(P)fr = 0

l cr1 f 1 pairs


Pf = P(lcr<1) = 1/10


lcr1 , f1 pairs
mode tracking
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
Multi-member removal (Stability Degradation)

damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10

fragility p l cr 1
Fragility: P(lcr < 1)

lcr<1 = Failure

damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10

progressive collapse p c







Pd = probability that lcr=0 at state nd

Progressive Collapse, Pc

Pc|(nd = n4) =


Pc|(nd = n4) = 40 %

Pd is cheap to calculate only requires

the condition number of Ker!

im n d vs se
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

Example 2: SAC/Seattle 3 Story*

*This model modified from the paper, member sizes are Seattle 3.

decision making and p f
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%)

with braces: Pf = 27%

no braces: Pf = 44%

Pf $  decision

IM1 = N(2,2)

IM2 = N(10,2)

  • 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.
why member removal
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:

Dlcr: Change in the buckling load as members are removed from the frame

DPf*: Change in the Pf as the mean yield strength is varied in the frame (Buonopane et al. 2003)