System Safety Performance Metrics for Skeletal Structures Avinash M. Nafday1 Abstract: A system safety performance metric is a tool to quantify the degree of safety of a structural system for a given performance objective. Classically, safety of skeletal structure as a complete system, designed by memberbased structural codes, was evaluated by checking for system stability. Reliability oriented research has focused on ultimate strength and probability of failure as system safety performance objectives, whereas earthquake engineering evaluations have considered displacement as the performance objective. This paper explores and clarifies the connection between system safety and structural redundancy concepts, provides alternative interpretation of structural system redundancy, and proposes two nondimensional system safety performance metrics for skeletal structures with stability as the performance objective. The proposed structural system safety performance metrics conveniently range from 0 to 1, and use algebraic properties of stiffness matrix to 1 measure the minimum distance of stiffness matrix from a set of singular matrices; and 2 quantify the degree of linear dependency of column vectors of the stiffness matrix, respectively. Also, approaches for identifying critical structural member and quantifying failure path importance are addressed. DOI: 10.1001/ASCE003394452000134:3499 CE Database subject headings: Structural safety; Redundancy; Structural stability; Progressive failure. Introduction The investigation report for the collapses of the World Trade Center towers was released by the National Institute of Standards and Technology NIST in October 2000 NIST 2000 and includes recommendations for improving building safety through code changes and consideration of progressive collapse. At a followthrough meeting, it was pointed out Sunder 2000 that unlike other industries and applications, structural engineers do not have an objective metric for measuring the safety performance of the structure as a complete system. A structural system level safety performance metric is a tool to quantify the degree of safety of a structural system or compare the safety of one structural system relative to another system for a given performance objective. As the current codebased skeletal structural design is member oriented, system level safety can be considered in terms of “system redundancy” for performance objective of overall structural system stability Nafday 2000. Previously, a flawed metric called degree of redundancy, defined in terms of the degree of structural system indeterminacy, was used even though it only provides a necessary but not sufficient condition for statically indeterminate structure and does not guarantee structural system stability. Webster’s Dictionary defines the word “redundant” as something that is superfluous, extra, in excess, or more than enough. The prevalent redundancy definition for skeletal structures implies excess of structural members over that required for a statically determinate analysis. An implicit belief that the extra members also lead to a better system safety performance in terms of more stable structure, higher load carrying capacity or reliability has been shown to be erroneous. Gorman 1984 has shown examples where reliability of structural systems was found to decrease with increase in the degree of indeterminacy. Sebastian 2000 has discussed some interesting results about how a statically indeterminate truss can also be a mechanism and higher degree of indeterminacy itself can enhance collapse likelihood for trusses and frames. A review of past failure case investigations shows a plethora of examples that structures often fail due to lack of “true” redundancy in seemingly highly indeterminate structures e.g., see the discussion in MurthaSmith 1988, and damage investigation reports from the Oklahoma City incident and the Northridge earthquake or due to failure of a critical component e.g., Kansas City Hyatt Regency Walkways Collapse. The degree of indeterminacy is not, in general, related to any measure of system safety performance, especially when indeterminacy is due to weak members. It also ignores contributions of topological and geometrical positioning of members, their importance or criticality in alternative load paths, material behavior and applied loading on the structures to system safety performance. In linear algebraic terms, degree of indeterminacy is basically the degree of underdetermination for the linear system of equations for solving the structural problem and it is important for the civil engineering profession to discard using indeterminacy and redundancy in a synonymous manner. The current memberbased design in structural codes does not explicitly consider system safety performance during structural design and the level of safety to be built in new designs is usually provided on the basis of intuition and past experience. Simple, commonly accepted, practically usable, and objective numerical metrics for structural system safety will provide a systematic way of addressing system safety at the design stage for various performance objectives. This paper explores, interprets, and clarifies the connection between system safety and redundancy concepts and structural 1CSLC, Marine Facilities Division, 200 Oceangate, Long Beach, CA 90000. Note. Associate Editor: Shahram Sarkani. Discussion open until August 1, 2000. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on August 29, 2000; approved on December 14, 2000. This paper is part of the Journal of Structural Engineering, Vol. 134, No. 3, March 1, 2000. ©ASCE, ISSN 00339445/2000/3499–500/$25.00. JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2000 / 499 Downloaded 17 Jul 2000 to 216.47.136.160. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org redundancy is interpreted as “additional” or “extra” safety in appropriate performance objective units provided by consideration of the structure as a “system,” over and beyond safety provided by memberbased structural design code. Two simple, practically usable, and nondimensional system safety performance metrics for skeletal structures are proposed, with overall stability as performance objective. The proposed system safety performance metrics use algebraic properties of structural stiffness matrix and are based on 1 measuring the minimum distance of the stiffness matrix from a set of singular matrices; and 2 quantifying the degree of linear dependency of column vectors of the stiffness matrix, respectively. Also, procedures for identifying critical structural members and numerical measures for quantifying importance of failure paths are addressed. System Performance Objectives Structural system performance objectives depend on the goal of evaluation. These objectives may have varied units of measurement and initial reference markers origin for measurement. If system efficiency is the goal, a statically determinate structural system would be the ideal Sunder 2000. Structural weight may be a performance objective in aerospace applications. Other structural performance objectives such as stress, mass, buckling load factor, deformation, interstory drift, or natural frequency are also often used. For example, monitoring and measurement of system response characteristics natural frequencies and mode shapes of offshore platforms are used for inservice inspection of member damage. PEER’s performance based earthquake engineering methodology Porter 2000 uses system level performance objectives of repair costs, casualties, loss of use duration dollars, deaths, and downtime. System safety is the goal here and many alternative performance objectives are available, depending on the application. Even though degree of redundancy is discredited as a system safety metric since it is not a system property, intuitively, system redundancy is an ideal system safety performance metric, provided it is defined with rigor for the desired performance objective. This is important since the term redundancy seems to have varied and conflicting interpretations in structural engineering literature and is often defined in an ambiguous manner. For example, redundancy has variously been defined in terms of the capability of structure to carry additional load after one or more of its members have failed, availability of ordinarily not required capacity, conditional probability of failure given first member failure, ratio of system failure probability to any member failure probability, etc. The ambiguous nature of the underlined phrases makes it difficult to compute redundancy in a scientifically rigorous manner and reach consistent results during structural analysis by different engineers. Even nonambiguous definitions may not be adequate. For example, structures with multiple load paths are said to be redundant as these structures generally remain safe even with failure of some members and redundancy is defined in terms of number of alternative load paths. Sebastian 2000 has discussed problems with this definition. Moreover, this is not a usable definition as extra load paths do not guarantee safety or higher collapse loads since it is possible for multiple members in a path to fail immediately after the failure of the first member, depending on postfailure member behavior brittle or ductile. Also, there is an inherent conflict between increased reliability due to longer failure paths and decreasing reliability due to larger number of failure paths. In a classical sense, system redundancy for skeletal structures was only a geometric/topological configuration based stability oriented concept but reliability studies Frangopol 1987 have interpreted system redundancy in terms of strength or probability of failure. Damage or fault tolerance measures such as reserve strength or residual strength are often used. Reserve strength implicitly assumes that failure occurs when the ultimate load is reached, though in practice the failure can be from other causes like fatigue, gross deformation, etc. Residual strength in a structure once a critical member has failed depends on whether the failure of critical member causes small or significant loss of strength. The evaluation of residual strength requires extensive calculations due to the need for calculating collapse load of damaged and intact structures. Bertero and Bertero 1999 indicate that strength reduction oriented performance objectives are incomplete descriptors for brittleness of the system in earthquake engineering, because the associated displacement is not considered. As system safety performance quantification depends on the purpose of application, it may be relevant to consider multiple performance objectives such as stability, strength, displacement, probability of failure, etc., and appropriate system safety metrics need to be defined for these performance objectives to achieve system redundancy for each of these objectives. Structural redundancy can be interpreted as additional or extra safety in appropriate performance objective units from consideration of the structure as a system, over and beyond that achieved by member based design code. Thus, redundancy is basically a system level property of structure and system safety performance metrics also provide measures of structural redundancy for the performance objectives considered. Redundancy is meant to provide additional safety in units of performance objective and system redundancy measures are no different than global structural system safety performance metrics distance from initial reference value to limiting value in units of specified performance objective stability, strength, displacement, etc. for memberbased code design. It is axiomatic that the system safety performance metric be nondecreasing function of the performance objective, be defined in a scientifically rigorous manner, produce consistent results, and preferably be nondimensional for ease of comparison. System safety performance measurement requires selection of initial and limiting reference markers for measurement. For example, initial reference for reserve strength is in terms of working loads, residual strength is defined with reference to predefined damage state of structure. Limiting reference markers limit state may be in terms of collapse of the system, failure of prespecified number of components, etc. To summarize, definition of system safety metric requires selection of performance objective, unit for objective measurement, initial and limiting reference markers, and whether deterministic or probabilistic measure is desired. Computational requirements for probabilistic calculations are enormous and there are no universally accepted standard probabilistic analytical techniques, unlike deterministic structural analysis methods. Another difficulty with probability based definition is that performance can be improved by reducing uncertainty for a given mean. The emphasis here is on measures of system safety for deterministic skeletal structures under deterministic loads. 500 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2000 Downloaded 17 Jul 2000 to 216.47.136.160. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org System Safety Performance Metrics System safety performance metric is a structural system property and ideally must account for all relevant parameters— configuration, member sizes, material properties, connection types, location, and applied loads. These parameters are captured in the structural stiffness matrix, K, and the real nn matrix K will be considered to be a point in Rn2, where each entry of K forms a coordinate in Rn2. For a skeletal structural system, set of K matrices form a region in this space, divided into safe and unsafe regions. The set of singular K matrices represents various failure states of the system. The properties of skeletal structure’s K matrix can be used to define system safety performance metrics or redundancy measures for the case of system stability as objective, correcting the earlier flawed definition of degree of redundancy. Note that the eigenvalue analysis for the stability performance objective generates elastic buckling load multiplier as safety metric which may correspond to only partial failure of the structure. Minimum Distance from Structural Stiffness Matrix to Set of Singular Matrices The traditional concept of system safety for skeletal structures is closely associated with system stability and a stable structure is one which is in a state of static equilibrium. As loads are increased during analysis by matrix based displacement method, an indication of stability is provided by the stiffness matrix K, since if the structure is unstable, this matrix is singular and cannot be inverted. As a singular stiffness matrix represents an unstable structure; for system safety, it is desirable to have the stiffness matrix K for the skeletal structure “farther away” in appropriate distance measure from the set of noninvertible singular matrices representing various unstable states for that structure. Demmel 1987 has shown that the closer a given matrix K is to the set of noninvertible matrices, the larger is its condition number K with respect to the operation of matrix inversion and condition number is also exactly inversely proportional to the distance to the hypersurface of the nearest singular matrix in the set. This original result for Euclidean norm is due to Eckart and Young 1939 and was subsequently developed further in Kahan 1966 and Demmel 1987. Thus, for skeletal structural system, the shortest distance from the structure’s stiffness matrix K to the set of singular matrices is equal to the reciprocal of the condition number K of matrix K for the operation of inversion. This shortest distance, S, is also the system safety performance metric for the performance objective of stability and is given as S = 1/K 1 Condition number with respect to operation of inversion measures the sensitivity of problem solution to small changes in the input. The singular matrices are illposed i.e., their condition number is infinite with respect to the problem of inversion. Thus, as a structural matrix gets closer to the set of illposed ones, its condition number approaches infinity. Illposed problems typically form lower dimensional hypersurface due to linear dependencies, as discussed in the next section in space of Rm2, where mn. The condition number of the stiffness matrix K of a linear system is a nonnegative number defined in terms of the matrix norm. Many different norms may be used, and though the condition numbers will vary for each norm, the general rule that large condition numbers indicate sensitivity will hold true. A commonly used definition for relative condition number K for any arbitrary norm is K = K K−1 2 For Euclidean matrix norm, Turing’s Ncondition number is defined as K = n−1K K−1 3 with Euclidean matrix norm defined as K=ijkij21/2. Therefore, system safety performance metric for the performance objective of stability is given as S = n/K K−1 4 This value indicates how stable with respect to overall collapse a given skeletal structure is for the loading conditions under consideration. Turing’s Ncondition number has an average value of n and value of 1 for orthogonal matrix. As the minimum value of condition number is 1 and maximum is infinity, the reciprocal of condition number, i.e, system safety performance metric of the skeletal structure for the performance objective of overall stability, S, will conveniently range between 0 and 1 higher value indicating more stable system, 0S 1 5 The system safety metric, S, is applicable to both intact and damaged skeletal structures, using appropriate stiffness matrices for evaluation. It is hypothesized that this definition may potentially be applicable in the context of a continuum. Example 1: For the planar truss given in Fig. 1, the 55 stiffness matrix K has been derived in Gutkowski 1990 and is given in the following the common factor EA/L has been taken out: 1.000 0 − 1.000 0 0 0 1.00 0 0 0 − 1.000 0 2.000 0 − 0.640 0 0 0 1.000 0.480 0 0 − 0.640 0.480 1.640 The inverse of matrix K, given by matrix K−1 is as follows: 2.410 0 1.410 − 0.300 0.640 0 1.000 0 0 0 1.410 0 1.410 − 0.300 0.640 − 0.300 0 − 0.300 1.230 − 0.480 0.640 0 0.640 − 0.480 1.000 Fig. 1. Planar truss from Gutkowski 1990, with permission JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2000 / 500 Downloaded 17 Jul 2000 to 216.47.136.160. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org The Euclidean norms for K and K−1 are: K = 3.600 K−1 = 4.214 From Eq. 4, structural system safety performance metric for the objective of system stability is S = n/K K−1 = 5/3.600/4.214 = 0.33 Comparing this to average value of 1/ n=0.45, the system safety performance metric is below average. Even though the linkage between a condition number and system safety performance is an attractive theoretical result, this requires computation of matrix inverse. Therefore, next an alternative definition for structural system safety performance metric is proposed. Degree of Linear Dependency in Stiffness Matrix In structural stiffness matrix, near singularity occurs when some of the rows or columns vectors for the matrix are almost linearly dependent, resulting in hyperplanes defined by the nearly dependent equations to be approximately parallel or column vectors to be almost collinear. As the degree of linear dependency increases, stiffness matrix tends toward singularity, indicating unstable structural system. Thus, structural system instability is linked to degree of linear dependency of rows or column vectors of the stiffness matrix. The degree of linear dependency can be measured by the determinant of K, denoted by K, which gives the volume of the geometrical parallelepiped defined by the column vectors of the stiffness matrix. The more the system is linearly dependent, causing structure to be unstable and stiffness matrix to be close to singular, the closer are the column vectors and resulting in smaller volume for the geometrical parallelepiped. Unstable configurations with singular K will lead to determinant i.e., volume of stiffness matrix K to be zero. As the determinant of stiffness matrix K can be arbitrarily changed by multiplying with a constant, it is necessary that the K matrix be normalized by dividing the ith row by j=1 n kij 2 1/2 Then, the determinant of normalized stiffness matrix KN, denoted by KN, can serve as a measure of system stability. In general, this determinant increases with increasing independence of vectors constituting the matrix KN, reaching maximum value of ±1 for the set of linearly independent vectors. Thus, if the determinant of the normalized matrix KN is small compared to ±1, the system is close to singular, i.e., with low structural system safety for performance objective of system stability. Therefore, another absolute value system safety metric, S, in the range of 0 to 1 can be defined for the stability performance objective as S = KN 6 0S 1 7 where KN=normalized stiffness matrix for the intact structure. The determinant is zero, if and only if, at least two vectors of a matrix are linearly dependent. The zero determinant value does not tell if more than two vectors are linearly dependent, only rank of matrix can provide this information about the degenerate case. Note that orthogonal stiffness matrix is the ideal to strive for since besides easily available matrix inverse, its condition number is 1 also determinant is 1. The determinant may be obtained from the product of singular values from singular value decomposition. The product of eigenvalues also provides the determinant and for positive definite matrix K, all eigenvalues are positive. In preliminary designs for large systems, it may not be necessary to calculate the exact determinant, as a simple upper bound is provided by Hadamard’s Inequality. If CI represents the Euclidean norm of Column I of K, this inequality states that S C1*C2*¯Cn 8 Example 2: Again using the planar truss example from Gutkowski 1990 given in Fig. 1, the 55 stiffness matrix K is given in Example 1. The normalized stiffness matrix KN given in the following is obtained by dividing each row by j=1 n kij 2 1/2 0.700 0 − 0.700 0 0 0 1.000 0 0 0 − 0.430 0 0.860 0 − 0.275 0 0 0 0.900 0.433 0 0 − 0.351 0.263 0.899 The determinant of KN is calculated to be 0.15, i.e., S = KN = 0.15 Stiffness matrices are a treasure trove of important information on key characteristics of the structure and their algebraic properties may be exploited to define system safety metrics for other performance objectives or infer important structural properties. For example, an indicator for low system safety is provided when the matrix K has widely different right and left Moore Penrose pseudoinverses during singular value decomposition, and any distance measure between left and right inverses can be chosen as a safety metric. Also, when matrix K is represented as ACAT with square matrix C representing material properties, the nonpositivedefiniteness of C is an indicator of brittle material. Importance Measures for Critical Member and Failure Paths System safety performance metrics discussed earlier considered overall stability of the system. Stability is basically concerned with changes to external parameters causing perturbation in the structural system. Robustness, on the other hand, deals not with external perturbation but the ability of the system to withstand changes in system topology. It is well known that many structures survive despite local failures due to availability of alternative load paths “failsafe” structures whereas some highly indeterminate structures can fail immediately, due to failure of a single critical member “weakestlink structures”. Survival after removal of members defines robustness of the structure and can be quantified using the importance measure for the removed member, indicated by contribution of the member to system safety. Let KN * be normalized stiffness matrix after removal of members in the system. As discussed, structural system safety is a function of the degree of independence of the vectors constituting the stiffness matrix and determinant of the normalized stiffness matrix KN determines the overall degree of dependency for the system. Then, the importance measure “I” for the removed member or failure path sequence of removed members is defined as 500 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2000 Downloaded 17 Jul 2000 to 216.47.136.160. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org I = KN/KN * 9 Note that KN is volume of the geometrical shape which is spanned by the vectors of matrix KN for “intact condition” and KN * is similar volume under “damaged condition.” The higher the importance measure, the more critical the member or failure path for survival of the structure and these measures range from 1 to infinity. The importance measure of infinity indicates that removal of members results in structural failure, with the stiffness matrix determinant of zero. Alternatively, as the stiffness matrix K can be represented as ACAT, with square matrix C representing material properties, the rectangular matrix A capturing geometric properties may be useful to define robustness properties or configuration redundancy of the structure. Configuration or topological redundancy is a function of how members are arranged to form the system analogous to system shape factor and some configurations may be inherently better at providing higher system margin for a given performance objective. For example, in earthquakes, some configurations are better at dissipating energy than others and for the performance objective of energy dissipation, optimal configuration can be found. It is important to remember that the original redundancy definition in structural analysis was basically a configuration oriented concept and empirically determined redundancy factors in National Earthquake Hazards Reduction Program NEHRP and International Building Code IBC are also configuration based. It is hypothesized that the reciprocal of the condition number of matrix A may possibly be considered as a system shape factor. Example 3: For the planar truss given in Fig. 2a, the 44 stiffness matrix K for displacements u2 ,v2 and u3 ,v3 at Joints 2 and 3 has been derived in Au and Christianson 1993 and is given as follows: 3 −1 −2 0 − 1 3 0 0 − 2 0 3 1 0 0 1 3 The corresponding normalized stiffness matrix KN is 0.800 − 0.267 − 0.534 0 − 0.316 0.949 0 0 − 0.534 0 0.800 0.267 0 0 0.316 0.949 And the determinant KN is 0.20. Now, to evaluate the importance of Member 5, let us remove Member 5 as shown in Fig. 2b. The stiffness matrix K* after removal of Member 5 is given as Au and Christianson 1993: 2 0 −2 0 0 2 0 0 −2 0 3 1 0 0 1 3 The normalized stiffness matrix after removal of Member 5, KN * , is 0.700 0 − 0.700 0 0 1 0 0 − 0.534 0 0.800 0.267 0 0 0.316 0.949 and KN * is 0.119. Therefore, the importance measure for Member 5 is given by I=KN / KN * =0.200/0.119=1.68 and the volume spanned by vectors of stiffness matrix reduces by 40.5% due to removal of Member 5. The progressive collapse has attracted increasing attention and computation of importance measures can be useful in identification of member to member failure propagation by removal of members in turn and assessing the ability of the structure to maintain integrity. Conclusions The current memberbased design using structural codes does not explicitly consider system safety performance of skeletal structures. The importance of system safety performance metrics to quantify the degree of safety of skeletal structural system for performance objectives such as stability, ultimate strength, probability of failure, displacement, etc., is discussed. The connection between system safety and redundancy concepts is explored and structural redundancy is interpreted as additional or extra safety in appropriate performance objective units provided by consideration of the structure as a system over and beyond safety provided by memberbased design code. Two simple, practically usable, and nondimensional system safety performance metrics for skeletal structures are proposed, with stability as performance objective. Both these metrics range from 0 to 1 and are based on using algebraic properties of stiffness matrices to 1 measure the minimum distance of structural Fig. 2. Planar truss from Au and Christiano 1993, with permission from PrenticeHall JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2000 / 500 Downloaded 17 Jul 2000 to 216.47.136.160. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org stiffness matrix from a set of singular matrices; and 2 quantify the degree of linear dependency of column vectors of stiffness matrix, respectively. Also, approaches for identifying critical structural member and quantifying failure path importance are addressed. Notation The following symbols are used in this paper: CI Column I of matrix K; I importance measure of structural member or failure path; K structural system stiffness matrix for intact structure; KN normalized structural system stiffness matrix for intact structure; K* structural system stiffness matrix for damaged structure; KN * normalized stiffness system matrix for a damaged structure; n number of rows or columns in stiffness matrix K; u2 ,u3 horizontal joint displacements at Nodes 2 and 3, respectively; v2 ,v3 vertical joint displacements at Nodes 2 and 3, respectively; S determinant of normalized structural stiffness matrix KN or volume of the hypersurface formed by column vectors of KN; S shortest distance from structural system stiffness matrix K to set of singular matrices; and K condition number of K. References Au, T., and Christianson, P. 1993. Fundamentals of structural analysis, PrenticeHall, Englewood Cliffs, N.J. Bertero, R. D., and Bertero, V. 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S. 2000. “Opening remarks at the meeting to discuss IBC code change proposal for progressive/disproportionate collapse.” Meeting of NIBS Building Code Experts and Structural Engineering Organizations, NIST, Gaithensburg, Md. 500 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2000 Downloaded 17 Jul 2000 to 216.47.136.160. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org
