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 1 W 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)