Vertical Incremental Dynamic Analysis for Assessing Progressive Collapse Resistance and Failure Modes of Structures

Transcription

1 The 4th International Workshop on Reliable Engineering Computing (REC 2010) Vertical Incremental Dynamic Analysis for Assessing Progressive Collapse Resistance and Failure Modes of Structures Dagang Lu, Ph.D. Prof. School of Civil Engineering Harbin Institute of Technology Harbin , China Singapore, March 4, 2010

2 Overview of the Presentation Background and Motivation Vertical Nonlinear Dynamic Analysis for Progressive Collapse Vertical Incremental Dynamic Analysis for Progressive Collapse Case Study: RC Framed Structures Conclusions and Future Works

3 Background and Motivation REC2010 is focused on robust design in the context of hazards, risk and uncertainty. Then, what is robustness in nature?

4 Background and Motivation Structural Robustness the ability of a structure to withstand local damage that may arise by accidental actions without global failure that is disproportionate to the triggering cause. the tolerance to damage from extreme loads or accidental loads. A damage tolerant (i.e. robust) system would be forgiving, fail-safe and foolproof.

5 Background and Motivation Progressive Collapse A collapse that is triggered by localized damage that cannot be contained and leads to a chain reaction of failures resulting in a partial or total structural collapse, where the final failure is disproportionate to the local damage from the initiating event.

6 Background and Motivation Progressive Collapse Three Historic Events in Progressive Collapse

7 Background and Motivation Global Overturning Collapse Collapse of a high-rise apartment in Shanghai, June 27, 2009

8 Background and Motivation Progressive Collapse under EQ The Great Wenchuan Earthquake, May 12, 2008

9 Background and Motivation Lateral Incremental Collapse The Great Wenchuan Earthquake, May 12, 2008

10 Background and Motivation International Guidelines of Preventing Progressive Collapse Joint Committee of Structural Safety (JCSS), Probabilistic Model Code, Interagency Security Committee (2001), ISC Security Criteria for New Federal Office Buildings and Major Modernization Projects. GSA 2003, Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects. DoD 2005, Unified Facilities Criteria (UFC), Design of Buildings to Resist Progressive Collapse.

11 Background and Motivation Chinese Guidelines of Preventing Progressive Collapse Chinese National Standard (1984). Unified Standard for Reliability-Based Design of Building Structures (GBJ68-84) Chinese National Standard (2002). Code for Design of Concrete Structures (GB ) Chinese National Standard (2003). Code for Design of Steel Structures (GB ) Chinese National Standard (2008). Unified Standard for Reliability-Based Design of Engineering Structures (GB )

12 Background and Motivation Progressive Collapse Analysis Based on Alternative Load Path (ALP) Approach Linear-elastic Static Procedure (LSP) Linear-elastic Dynamic Procedure (LDP) Nonlinear Static Procedure (NSP) Nonlinear Dynamic Procedure (NDP)

13 Background and Motivation Motivation of This Research How to perform nonlinear dynamic progressive analysis based on alternative load path still relies heavily on engineering judgment. The problem of column removal and its effects is very complex. Few researchers have considered the effects of the duration and the instant of element removing on damaged structures. How to evaluate the ultimate bridging-over capacity and impact resistant ability of the areas upon and below the removed elements still poses a difficult problem that has been explored by few researchers.

14 Overview of the Presentation Background and Motivation Vertical Nonlinear Dynamic Analysis for Progressive Collapse Vertical Incremental Dynamic Analysis for Progressive Collapse Case Study: RC Framed Structures Conclusions and Future Works

15 Vertical Nonlinear Dynamic Analysis Used as a baseline step to determine the capacity of a structural system when one load carrying member is suddenly lost. The loads that cause progressive collapse of a structure: Primary load: the action that causes the structural element to fail, such as blast pressure due to explosion and vehicular impact; Secondary loads: result from internal static and dynamic loads, are due to sudden changes in the load path through the structure geometry caused by abrupt loss of one or many load carrying members;

16 Vertical Nonlinear Dynamic Analysis In order to simulate the phenomenon that one load carrying member is abruptly removed: The member forces should be suddenly removed after a certain time had elapsed while the gravity load remained unchanged; The stiffness matrix of the system needs to be suddenly changed; However, This may cause difficulty in the analytical modeling process!

17 Vertical Nonlinear Dynamic Analysis Modeling of sudden removal of a load carrying member: All member forces are obtained first from the structural model subjected to the applied load; Then the structure is re-modeled without a column with its member forces (P, V, and M) applied to the structure as lumped forces to maintain equilibrium position; The progressive collapse analysis starts from the moment that the structure is already deformed by the applied load, which reflects the loading situation quite realistically.

18 Vertical Nonlinear Dynamic Analysis Loading scheme in vertical nonlinear dynamic analysis Load D+0.25L Load p 0 P design t 0 t 1 Time t 0 t 1 Time Vertical load Unbalanced load A B C D The effects of the duration of element removing are considered by applying the unbalanced force to the remaining structure in sub-steps; The failed column is removed within the time interval (t 0, t 1 ).

19 Vertical Nonlinear Dynamic Analysis The main procedure of vertical nonlinear dynamic analysis Firstly, apply the gravity load; Secondly, remove the column from the unloaded structure; Finally, use nonlinear dynamic analysis. Control criteria The yielding state of the structure; The ultimate state of the structure; The collapse state of the structure.

20 Overview of the Presentation Background and Motivation Vertical Nonlinear Dynamic Analysis for Progressive Collapse Vertical Incremental Dynamic Analysis for Progressive Collapse Case Study: RC Framed Structures Conclusions and Future Works

21 Vertical Incremental Dynamic Analysis Progressive collapse resistance defined as the ultimate downward loading capacity of the column-removed building the ultimate bridging-over capacity and impact resistant ability of the areas upon and below the removed elements

22 Vertical Incremental Dynamic Analysis Lateral IDA Vamvatsikos, D. and Cornell, C. A. Incremental dynamic analysis. Earthquake Engineering and Structural Dynamics, 31(3): , (a) Single-record IDA curve (b) Multiple-record IDA curves

23 Vertical Incremental Dynamic Analysis Vertical IDA A series of vertical nonlinear dynamic analyses under different dynamic secondary loadings are conducted; A step function multiplied by D L is used to simulate the dynamic loading applied to the column-removed building; The magnitude of the step function is increased gradually till extremely large deflection occurs at the column-removed point; The P-Delta effect and large displacement are considered in the nonlinear dynamic analysis; The peak displacement response of each time history is collected to construct the load-displacement envelopes for the incremental dynamic analysis.

24 Vertical Incremental Dynamic Analysis Flowchart of Vertical IDA Initial value Initial value:0.01 No Search backward for formula Initial incremental formula Convergence or not Yes Search forward for formula Initia incremental formula: f = k 1 f + + k k Search backward for formula: f = 1 f + LS ( f + fls )/3 Search forward for formula: f = k 1 f + k ( flf + fk)/3 k amplitude modulation for the kth time f k amplitude modulation factor for this time k k Satisfied End Precision requirements Convergence or not Not satisfied f + amplitude modulation factor for the (k+1)th time k f LF f LS 1 Last failure factor Last successful factor

25 Vertical Incremental Dynamic Analysis Two element-removing scenarios in Vertical IDA Case (a) Case (b) Case (a) is regarding to the remaining structure above the removed element; Case (b) is concerned with the remaining structure below the removed element.

26 Vertical Incremental Dynamic Analysis Two element-removing scenarios in Vertical IDA Case (a) Case (b) For case (a), we need to remove the lost column firstly, then apply the load α(d+0.25l) to the floor which is below the lost column. For case (b), we need apply the load α(d+0.25l) to the floor which is above the left column in the second floor, then remove this column, and finally apply the dynamic analysis to the damaged structure.

27 Overview of the Presentation Background and Motivation Vertical Nonlinear Dynamic Analysis for Progressive Collapse Vertical Incremental Dynamic Analysis for Progressive Collapse Case Study: RC Framed Structures Conclusions and Future Works

28 Case Study: The RC Framed Structure Design and modeling of the case study structure D C B A (a) Plan (b) Elevation (c) Details

29 Case Study: The RC Framed Structure Design and modeling of the case study structure Table I. Layout of vertical load Uniformly distributed load (kn/m) Concentrated load (kn) Location Dead load Live load Dead load Edge node Live load Internal node Dead load Live load Head floor Middle floor Ground floor Table II. Structural vibration periods direction The first mode Before the gravity load is applied (s) The second mode The third mode The first mode After the gravity load is applied (s) The second mode The third mode X Y

30 Case Study: The RC Framed Structure Simulation Platform

31 Case Study: The RC Framed Structure Material properties used in the simulation Unconfined Confined Confined Table III. Parameters of concrete material properties (C35) column beam Peak stress Peak strain Ultimate stress Ultimate strain Concrete is modeled by Concrete01, confinement is specified implicitly by using the confined stress-strain relationships proposed by Mander, Priestley and Park; Reinforcing steel is modeled by Steel02 with 1% strain hardening.

32 Vertical NDA: Motivation To evaluate the progressive collapse potential of the model structure, which was designed according to the conventional design code without considering progressive collapse.

33 Vertical NDA: Criterion Yielding state of the structure the yielding of the rebars in tension Ultimate state of the structure the strain of core concrete at the edge reaches its ultimate strain Collapse state of the structure the dynamic instability of the structure or the divergence of the program.

34 Vertical NDA: Results The overall response w.r.t the development of plastic hinges θ max θ y M θ / max θ y M u M y θ y θu θ c θ Node vertical displacement (mm) υ max Time (s) It is found that the plastic hinges occur at the ends of the beams, and the plastic deformations are rather small, there isn t any plastic hinge which occurs in columns, so the design structure will not collapse.

35 Vertical NDA: Results The local reactions w.r.t internal forces of the adjacent members 2.2 The ratio of the internal force is calculated by R0 γ = R t R 0 R t The axial force variations where: is the force of the column before the failure of column B1; is the force of the column after the failure of column B1. 2 C1 1.8 N 0 /N t A D1 A1 C1 D Time (s)

36 Vertical NDA: Results The local reactions w.r.t internal forces of the adjacent members The shear force variations A1 0 D1 Q 0 /Q t C Time (s) The bending moment variations 40 A1 A1 C1 D1 20 M 0 /M t 0-20 D1 C Time (s) A1 C1 D1

37 Vertical NDA: Results The local reactions w.r.t internal forces of the adjacent members A1 2 C1 10 A1 20 N 0 /N t A1 Q 0 /Q t 0-10 D1 C1 M 0 /M t 0-20 D1 C1 1.2 D Time (s) Time (s) Time (s) Column D1 plays as a support which absorbs the unbalanced force of the structure. The main response of the damaged structure is the vertical vibration in the damaged bays after column B1 is removed. The damaged structure also vibrate in the horizontal direction, this is mainly due to the geometric asymmetry of the structure itself after column B1 is removed.

38 Vertical NDA: Results The internal force of the beams which are originally supported by the lost column 10 x 104 BC5 8 BC4 Axis force ( N ) BC1 BC3 BC5 BC4 BC2 BC3 BC2 BC Time ( s )

39 Vertical NDA: Results The internal force of the beams which are originally supported by the lost column 5 x 104 AB5 BC5 0 Shear force (N) AB5-1 AB1-1 BC2-1 BC1-1 BC5-1 AB1 BC2 BC Time (s)

40 Vertical NDA: Results The internal force of the beams which are originally supported by the lost column 2 x AB5 BC5 Moment ( N mm ) BC2-1 BC1-1 BC5-1 AB1-2 AB5-2 AB2-2 AB2 AB1 BC2 BC Time (s)

41 Vertical NDA: Results The internal force of the columns in column line B N 0 /N t M 0 /M t B5 B4 B3 B Time (s) (a) before the failure of Column B Time (s) (b) after the failure of Column B1 The internal forces of the remaining columns in column line B is unloaded to zero quickly after column B1 is removed, this means that the columns in column line B have lost their ability to resist the applied load.

42 Vertical IDA: Two Scenarios Case (a) Case (b) the remaining structure above the removed element; the remaining structure below the removed element.

43 Vertical IDA: Results Vertical IDA for the substructure above the removed column θ max θ max α max = α max = Vertical IDA Pushdown analysis Plastic rotation of the structure in the instable state obtained by pushdown analysis and vertical IDA

44 Vertical IDA: Results Vertical IDA for the substructure above the removed column IDA analysis Pushdwn analysis Load factor α Vertical displacement( mm ) Comparison of IDA curve and pushdown curve It is found that, if one floor of the structure has collapsed, then the structure will collapse when a column below the collapsed floor is removed, and the damage range is between the collapsed floor and the floor which is supported by the removed column

45 Vertical NDA: Results Vertical IDA for the substructure below the removed column The failure modes of substructure below the removed column obtained by (a) pushdown analysis and (b) vertical IDA (a) (b) It is found that the structure has the ability to resist the dynamic impact caused by the collapse of upper structures as long as the critical member below the lost column does not fail.

46 Overview of the Presentation Background and Motivation Vertical Nonlinear Dynamic Analysis for Progressive Collapse Vertical Incremental Dynamic Analysis for Progressive Collapse Case Study: RC Framed Structures Conclusions and Future Works

47 Conclusions The structural damage only happens in the damaged bays, where plastic hinges occur at the ends of the beams, and the plastic deformations are rather small, so the design structure will not collapse. The two adjacent beams supported by those columns become one large-span beam, the motion of each layer is consistent with each other. If one floor of the structure has collapsed, then the structure will collapse when a column below the collapsed floor is removed, and the damage range is between the collapsed floor and the floor which is supported by the removed column. The structure has the ability to resist the dynamic impact caused by the collapse of upper structures as long as the critical member below the lost column does not fail.

48 Future Works How to consider the randomness in element removing and uncertainties in structural systems? Random Vertical IDA! X.H. Yu, D.G. Lu, P.Y. Song, et al. Stochastic incremental dynamic analysis considering random system properties. The 14th World Congress on Earthquake Engineering (WCEE14), paper No. S

49 Acknowledgements The support of the following projects is greatly appreciated: Major Research Program of National Natural Science Foundation (grant No ) ; National Natural Science Foundation (grant No ) ; National Natural Science Foundation (grant No ) ; Earthquake Engineering Research Special Foundation (grant No ).

50 Dagang School of Civil Engineering, HIT Thank you for your attention! Questions or comments?