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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. 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

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  1. Stability Degradation and Redundancy in Damaged Structures Benjamin W. Schafer Puneet Bajpai Department of Civil Engineering Johns Hopkins University

  2. Acknowledgments • The research for this paper was partially sponsored by a grant from the National Science Foundation (NSF-DMII-0228246).

  3. 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

  4. traditional design for environmental hazards augmented design for unforeseen hazards Building design philosophy

  5. 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

  6. 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, … drift Eigenvalues of Ktan after loss PBD and PEER framework equation

  7. 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, …

  8. 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

  9. 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)

  10. 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

  11. no yes Load conservative? Single member removal lcr-intact= 3.14 Solution? exact approximate (Ker - lcrKgr(Pr)fr = 0 (Ker - lcrKgr(P)fr = 0

  12. COLUMN REMOVAL Pf = P(lcr<1) = 1/10 BEAM REMOVAL lcr1 , f1 pairs

  13. 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.

  14. Multi-member removal (Stability Degradation) damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10

  15. Fragility: P(lcr < 1) lcr<1 = Failure damage states: 10 – 17 – 32 – 56 – 85 – 102 – 84 – 41 – 10

  16. Fragility

  17. P4 P5 P6 P7 P8 P9 Pd = probability that lcr=0 at state nd Progressive Collapse, Pc Pc|(nd = n4) = FAIL Pc|(nd = n4) = 40 % Pd is cheap to calculate only requires the condition number of Ker!

  18. Fragility

  19. IM = strain energy removal? SE = ½dTKd

  20. IM = nd vs SE Distribution of SEintact=SEdamaged?

  21. 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.

  22. Intact buckling mode shapes (fi)

  23. Computational effort m ≈ an4

  24. Computational effort and sampling

  25. Stability degradation

  26. Fragility and impact of redundancy

  27. i.e., Fragility and mode tracking

  28. 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)

  29. 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.

  30. Can we transform unknown unknowns into known unknowns? Maybe a bit…

  31. 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)

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