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 ).

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 P f Conclusions

PBD and PEER framework equation IM = Hazard intensity measure spectral acceleration, spectral velocity, duration, … components lost, volume damaged, % strain energy released, … drift Eigenvalues of K tan 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 (P f ), mean annual prob. of $ loss, 50% replacement cost, … v(DV) = P DV

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/n tot = 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: (K e - cr K g (P))  = 0 damaged:(K e r - cr K g r (P r )  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). (Ziemian et al. 1992)

Analysis of Ziemian Frame IM = Member removal –single member removal: m 1 = n damaged /n total = 1/10 –multi-member removal: m 1 = 1/10 to 9/10 –strain energy of removed members EDP = Buckling load ( cr ) –load conservative or non-load conservative? –exact or approximate K g ? –first buckling load, or tracked buckling mode? DV = Probability of failure (P f ) –P f = P( cr <1) –P f = P( cr =0) –P f = P that a kinematic mechanism has formed

Single member removal Load conservative? noyes Solution? exactapproximate (K e r - cr K g r (P r )  r = 0 (K e r - cr K g r (P)  r = 0 cr-intact = 3.14

BEAM REMOVAL cr1,  1 pairs COLUMN REMOVAL P f = P( cr <1) = 1/10

Mode tracking Eigenvectors of the intact structure  i form an eigenbasis, matrix  i. We examined the eigenvectors for the damaged structure  j r in the  i basis, via: (  j r )  = (  i r ) -1 (  j r ) The entries in (  j r )  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

P4P4 P5P5 P6P6 P7P7 P8P8 P9P9 FAIL P d = probability that cr =0 at state n d P c |(n d = n 4 ) = Progressive Collapse, P c P c |(n d = n 4 ) = 40 % P d is cheap to calculate only requires the condition number of K e r !

Fragility

IM = strain energy removal? SE = ½d T Kd

IM = n d vs SE Distribution of SE intact =SE damaged ?

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.

Intact buckling mode shapes (  i )

Computational effort m ≈  n 4

Computational effort and sampling

Stability degradation

Fragility and impact of redundancy

Fragility and mode tracking i.e.,

Decision-making and P f IM 1 = N(2,2)IM 2 = N(10,2) IM 1 ~ N(2,2) = N(7%,7%) with braces: P f = 0.2% no braces:P f = 0.7% IM 2 = N(10,2) = N(37%,7%) with braces: P f = 27% no braces:P f = 44% P f  $  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 K e. 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:  cr : Change in the buckling load as members are removed from the frame  P f *: Change in the P f as the mean yield strength is varied in the frame (Buonopane et al. 2003)