[american society of civil engineers structures congress 2011 - las vegas, nevada, united states...

Performance Indicators for Structures and Infrastructures

Dan M. Frangopol1 and Duygu Saydam2

1 Professor and the Fazlur R. Khan Endowed Chair of Structural Engineering and Architecture, Department of Civil and Environmental Engineering, ATLSS Engineering Research Center, Lehigh University, 117 ATLSS Dr., Bethlehem, PA 18015-4729, USA, PH (610) 758-6103; FAX (610) 768-4115; email: [email protected] (Corresponding author) 2 Graduate Research Assistant, Department of Civil and Environmental Eng., ATLSS Engineering Research Center, Lehigh University, 117 ATLSS Dr., Bethlehem, PA 18015-4729, USA, PH (610) 758-6270; FAX (610) 738-5553; email: [email protected] ABSTRACT

Design, assessment and prediction of the performance of structures and infrastructures based on component analysis have been the conventional methodologies enforced by the specifications. Nevertheless, system-based performance indicators can provide more efficiency and dependability in performance-based design, assessment, and prediction of structures and infrastructures. The objective of this paper is to provide a general overview of performance indicators applicable to civil structures and infrastructures. Definitions and formulations of selected performance indicators are presented. Information on what level (section, component, system, system of systems) these indicators are useful is provided. Interaction between life-cycle cost and lifetime performance is presented. INTRODUCTION Civil infrastructural systems are subjected to deterioration in strength and performance due to the aggressive environmental conditions (e.g., corrosion) and aging of the materials they are composed of. Therefore, the safety of these systems is highly affected by this deterioration. In order to avoid the consequences of structural failures, maintenance programs are carried out by the responsible authorities. It is necessary to predict the life-cycle performance of these systems accurately to establish a rational maintenance program. However, difficulties arise in prediction of life-cycle performance because of the complexity and high uncertainty in loading and deterioration processes. Consequently, it is essential to use proper indicators to evaluate the structural performance in a quantitative manner.

So far, many researchers focused on quantifying structural performance with deterministic and probabilistic indicators such as safety factor and reliability index, respectively. National design codes take uncertainty into account by including specific factors in the computation of structural resistance and load. However, the prediction of structural performance under deterioration may require the use of several performance indicators. For example, system reliability index is an adequate

1215Structures Congress 2011 © ASCE 2011

Structures Congress 2011

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measure for quantifying the safety of a structure with respect to ultimate limit states, but system redundancy index is required to evaluate the availability of warning before final failure. Moreover, performance indicators related to damage tolerance of structures, such as vulnerability and robustness are essential to consider for structures under deterioration and local damage together with the indicators related to ultimate limit states. To provide a desired safety level of structures, the values of performance indicators under consideration shouldn’t down-cross required threshold levels. On the other hand, life-cycle cost of structures is another measure which decision makers have to balance with the performance indicators. Nevertheless, it is evident that evaluating structural performance requires considering multiple indicators simultaneously. The purpose of this paper is to review several structural performance indicators used to evaluate the performance of civil structural and infrastructural systems. The formulation of these indices and several associated references are presented. Information on what level (section, component, system, system of systems) these indicators are useful is provided. Background of balancing cost and performance is also briefly indicated. LIMIT STATES IN PERFORMANCE QUANTIFICATION Performance evaluation of cross-sections, members, parts of structures, and overall structures is based on limit states defining the failure domain under specific loading conditions. The limit states defining the failure modes of components are included in design codes. For instance, AASHTO LRFD Bridge Design Specifications (2007), regardless of type and analysis used, requires that ∑ ≤ niii RQ φγη (1) is satisfied for all specified load effects and combinations, where Qi is the load effect, Rn is the nominal strength, γi is load factor, iφ is resistance factor and ηi is load modifier. Eq. (1) applies to all strength limit states, service limit states and fatigue and fracture limit states. The factors multiplying the load effects and nominal strength exist to ensure a predefined safety level of the component. However, if the purpose is to evaluate the performance of an existing structure or design with respect to different target performance levels, the equation defining limit states must be in the pure form. A general representation of limit states which is used in reliability analysis in terms of a performance function g(X) can be expressed as

0),...,,()( 21 == nXXXgg X (2) where X = (X1, X2, …, Xn) is a vector of random variables of the system, and the performance function g(X) determines the state of the system as [g(X) > 0] = “Safe state”, and [g(X) < 0] = “Failure state”.



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LEVELS OF PERFORMANCE QUANTIFICATION In regard to the purpose and type of analysis, structural performance can be quantified at cross-section level, member (component) level, overall structure (system) level, and group of structures (system of systems) level. The strength of a component under different loading conditions can be expressed in terms of the capacity of its most critical section when stability problems are not accounted for. Under consideration of stability problems, the performance is quantified at component level. In most of the current building design codes, strength requirements are based on component strength. Although this kind of approach may ensure an adequate level of safety of components, it lacks the information about the interaction between the components and overall performance of the whole structure. However, performance at system level is of concern in performance-based design. Structural reliability theory offers a rational framework for quantification of system performance by also including the uncertainties both in the resistance and the load effects. Overall system performance for a limit state can be evaluated by assuming that the system limit state is series or parallel or series-parallel combination of component limit states (Ditlevsen and Bjerager, 1986; Hendawi and Frangopol, 1994). The domain Ω, representing the violation of system limit state can be expressed in terms of component limit states as (Ang and Tang 1984):

(a) for series system ( ){ }Un

kk Xg

1

0=

<≡Ω

(b) for parallel system ( ){ }In

kk Xg

1

0=

<≡Ω

(c) for series-parallel system ( ){ }UIn

k

c

jjk

n

Xg1 1

, 0= =

<≡Ω

where cn is the number of components in the n-th cut set. The performance of a system which consists of a number of subsystems depends not only on the performance of the subsystems but also on the interaction of these subsystems. An example of a system of systems is a highway bridge network, where each bridge is a system itself and interacts with the other bridges for the performance of whole network by means of traffic flow. DEFINITIONS OF COMMON PERFORMANCE INDICATORS Safety factor. The simplest deterministic interpretation of the safety factor, when there is a linear relationship between the load and the stress produced by the load, is

all

uSFσσ

= (3)

where SF represents the safety factor, σu is the maximum usable stress which can be either the yield stress or the ultimate stress, and σall is the allowable stress which is usually defined in structural design codes.



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The most common use of safety factor is design at section and component levels. Allowable stress design is based on the concept that the maximum stress in a cross-section or a component shall not exceed a certain allowable stress under normal service conditions (AISC, 2005). The limiting stress, which can be yield stress or stress at instability or fracture, is divided by a safety factor to provide the allowable stress. The safety factor is used to provide a design margin over the theoretical design capacity to allow for consideration of uncertainty due to any components of the design process including calculations, material strengths, and manufacture quality. Load and resistance factors. Load and resistance factor design is based on the ultimate strength of critical member sections or the load carrying capacity of members (Ellingwood et al., 1980). In Load and Resistance Factor Design (LRFD), the resistances R and the load effects Q are usually considered as statistically independent random variables. If the resistance R is greater than the load effect Q, a margin of safety exists. On the other hand, since resistance and load effect are random variables, there is a probability that resistance is smaller than load effect. This probability is related to the overlap area of the frequency distributions of the resistance and load effect and their dispersions (AISC, 2005). Figure 1 illustrates the probability density functions (PDFs) of the resistance and load effect. In this figure E(Q) and E(R) represent the mean values of load effect and resistance, respectively.

Figure 1 – Probability density functions of resistance and load effect

Load and resistance factors are used for design at section and component levels. LRFD suggests the use of resistance factor and partial load factors to account for the uncertainty in the resistance and the load effect. The partial load factor approach was originally developed during 1960s for reinforced concrete structures. It gives the opportunity for live and wind loads to have greater partial load factors than the dead load due to the fact that live loads and wind loads have greater uncertainty. The condition of safety with respect to the occurrence of failure mode i including the reduced resistance and factored loads using resistance and partial load factors can be written as

fQ(q)

fR(r)

E(Q) E(R)

Prob

abili

ty D

ensit

y Fu

nctio

n

Resistance and Load Effect

r, q



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...++≥ LiLiDiDiii QQR γγφ (4) where R is the member resistance, iφ is the resistance factor, QD and QL are the dead and live load effects, and γD and γL are the partial load factors associated with the dead and live load, respectively. Probability of failure. The stochastic nature of the structural resistance and the load effects can be described by their probability density functions. The probability of failure of any section, component or system is defined as the probability of occurrence of the event that resistance is smaller than the load effects and can be evaluated by solving the following convolution integral:

∫∞

=≤=0

)()()0( dssfsFgPP QRf (5)

where g is the performance function, R is the resistance in a certain failure mode, Q is the load effect in the same failure mode, FR is the cumulative distribution function of R, and fQ is the probability density function of the load effect Q.

Probability of failure is the basis for most probabilistic performance indicators. It is used at all levels (section, component, system, system of systems). In many cases, it is impossible or very demanding to evaluate Pf by analytical methods. Therefore, numerical methods such as Monte Carlo Simulations are used. Reliability index. The reliability of a structure can be expressed in terms of either probability of failure or its corresponding reliability index. As a measure of reliability, reliability index can be defined as the shortest distance from the origin to the limit state surface in the standard normal space. For normally distributed independent variables reliability index can be calculated as

)()()()(

22 QRQERE

σσβ

+

−= (6)

where β indicates the reliability index, E(R) and E(Q) are the mean values of the resistance and load effect, and σ(R) and σ(Q) are the standard deviations of the resistance and load effect, respectively. For more complex problems, first and second order reliability methods (FORM and SORM) which approximately provide the reliability index by searching the most probable point on the failure surface (gj=0), are the most common methods to compute reliability index. The probability of failure and reliability index are approximately related to each other as follows

)(1 βΦ−=fP (7) where Φ(.) indicates the standard normal distribution function.



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Cumulative time probability of failure. The probability of failure within a certain period of time is called cumulative time probability of failure. The basic approach considers only one random variable, the time at which the component or system fails. Cumulative time probability of failure up to time tf can be calculated as (Frangopol and Okasha, 2008)

∫=≤=ft

ff duuftTPtF0

)()()( (8)

where F(tf) is the area under the probability density function f(u) of the time to failure to the left of tf. Cumulative time probability of failure is a time-dependent performance indicator. Survivor function. Survivor function is the probability that a component or system survives until time t. It is equal to the reliability function, which is the probability that a component or system is still functioning at time t. Survivor function is the complement of the cumulative time probability of failure and can be expressed as

∫∞

=>=−=ft

duuftTPtFtS )()()(1)( (9)

where F(tf) is the area under the probability density function f(u) of the time to failure to the left of tf. Survivor function is a time-dependent performance indicator. Hazard function. Hazard function is a measure of risk associated with an item at time t. It is also known as hazard rate or failure rate. Hazard function also can be defined as the conditional probability of failure in a time interval t+dt, given that a component has survived until time t (Ramakumar, 1993). Hazard function is the ratio of probability density function to the survivor function and can be expressed as

)()()(

tStfth = (10)

where h(t) indicates the hazard function, f(t) and S(t) represent the probability density function and survivor function, respectively. Hazard function is a time-dependent performance indicator. Redundancy. There are several definitions and indicators for structural redundancy in the literature. A measure of redundancy, in the context of availability of warning before system failure, was proposed by Frangopol and Curley (1987) as

)(

)()(1

sysf

sysfdmgf

PPP

RI−

= (11)



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where Pf(dmg) is the probability of damage occurrence to the system, and Pf(sys) is the probability of system failure. A measure of redundancy, as the availability of alternative load path after sudden local damage, was proposed by Frangopol and Curley (1987) as

damagedintact

intactRIββ

β−

=2 (12)

where βintact is the reliability index of the intact system and βdamaged is the reliability index of the damaged system. Redundancy is a system performance measure. However, it is also applicable at the section and component levels as a measure of warning with respect to failure. AASHTO LRFD Bridge Design Specifications (2007) considers redundancy in bridge structures based on the study of Frangopol and Nakib (1991). The load modifier ηi in Eq. (1), which accounts for redundancy level, is based on the redundancy definition in Frangopol and Nakib (1991). Application of redundancy concept to deteriorating structures can be found in Okasha and Frangopol (2009, 2010a). Vulnerability and damage tolerance. In structural engineering, vulnerability is one of the key measures used to capture the essential feature of damage tolerant structures. A probabilistic measure of vulnerability was proposed by Lind (1995), defined as the ratio of the failure probability of the damaged system to the failure probability of the undamaged system

),(),(

0 QrPQrP

V d= (13)

where rd indicates a particular damaged state, r0 indicates a pristine system state, Q is the prospective loading, P(rd, Q) represents the probability of failure of the system in the damaged state, P(r0, Q) represents the probability of failure of the system in the pristine state, and V refers to vulnerability of the system in state rd for prospective loading Q. The value of vulnerability is 1.0 if the probabilities of failure of the damaged and intact systems are the same. Lind (1995) also defined the damage tolerance of a structure as the reciprocal of vulnerability. Vulnerability and damage tolerance are system performance indicators. Risk. The most common formulation of risk in engineering is multiplication of probability of occurrence by the consequences of an event. Direct risk is the one associated with the damage occurrence itself while indirect risk is associated with the system failure as a result of the damage. Direct and indirect risks are formulated as (Baker et al., 2006)

∫ ∫=x y EEDDirDir dydxxfxyfCR )()|(| (14)



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∫ ∫ ==x y EEDIndirIndir dydxxfxyfyDFPCR )()|()|( | (15)

where CDir and CIndir are the direct and indirect consequences, and fZ(z) is used to denote the probability density function of a random variable Z. E, D and F represent the hazard occurrence, damage occurrence, and system failure, respectively. These integrals can be computed with numerical integration or Monte Carlo Simulation. Risk is applicable at component and system levels as well as system of systems level. Robustness. Robustness is one of the key measures in the field of progressive collapse and damage tolerant structures. Although robustness is recognized as a desirable property in structures and systems, there is not a widely accepted theory on robust structures. Maes et al. (2006) defined robustness of a system as

si

s

i PP

R 0min= (16)

where Ps0 is the system failure probability of the undamaged system, and Psi is the system failure probability assuming one impaired member i. Baker et al. (2008) state a robust system to be one where indirect risks do not contribute significantly to the total system risk. An index of robustness is proposed as

IndDir

DirRob RR

RI

+= (17)

where IRob is the index of robustness, and RDir and RInd are the direct and indirect risks, respectively. This index varies between 0 and 1.0 with larger values representing a larger robustness. Additional robustness indicators are indicated in Biondini and Frangopol (2010), Biondini et al. (2008), Ghosn and Frangopol (2007) , and Ghosn et al. (2010). Robustness is a system performance indicator. Resilience. Bruneau et al. (2003) defined resilience as the ability of the system to reduce the chances of a shock, to absorb a shock if it occurs (abrupt reduction of performance), and to recover quickly after a shock (re-establish normal performance). According to their definition, a resilient system exhibits (a) reduced failure probabilities, (b) reduced consequences from failures in terms of lives lost, damage, and negative economic and social consequences, and (c) reduced time to recovery. The analytical definition of resilience is based on the concept of functionality, also called “serviceability” or “quality of infrastructure” (Bruneau et al. 2003; Cimellaro et al. 2010). Then, resilience is defined as the integral of the functionality in time as

∫=ht

t

dttQR0

)( (18)

where t0 is the time at which the extreme event occurs, and th is the investigated time horizon. Resilience is a system performance indicator. Resilience is also applied to



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system of systems such as highway bridge networks. Application of resilience concept to highway bridge networks can be found in Bocchini and Frangopol (2011), Frangopol and Bocchini (2011). Life-cycle cost. One of the most important measures in evaluation of structural performance is life-cycle cost. The proper allocation of resources can be achieved by minimizing the total cost while keeping structural safety at a desired level. The expected total cost during the lifetime of a structure can be expressed as (Frangopol et al., 1997)

FREPINSPMTET CCCCCC ++++= (19) where CT is the initial cost, CPM is the expected cost of routine maintenance cost, CINS is the expected cost of inspections, CREP is the expected cost of repair, and CF is expected failure cost. BALANCING COST AND PERFORMANCE Life-cycle cost and performance level of a structure are two conflicting criteria that should be balanced. Significant amount of research has been done in the area of optimum planning for life-cycle management of civil structures and infrastructures (Chang and Shinozuka, 1996; Ang and De Leon, 1997; Frangopol et al. 1997, 2001; Ang et al. 1998; Estes and Frangopol, 1999; Frangopol and Kong, 2001; Okasha and Frangopol, 2010b). The interaction between the components of life-cycle cost and initial performance level (i.e., reliability index) is qualitatively illustrated in Figure 2. Higher performance levels require higher initial cost. However, if the initial performance level is high, failure cost, preventive maintenance cost and repair cost are lower. CONCLUSIONS In this paper, common performance indicators for civil structures and infrastructures are presented. Some indicators are used to define performance at cross-section and component levels while others are used to quantify performance of systems and groups of systems.

The matter of which indicators should be considered depends on the priorities and objectives of the decision makers. In most design processes, the main purpose is minimizing the life-cycle cost while maintaining safety at an acceptable level. If time-dependent analysis is required, time-dependent indices such as cumulative probability of failure, survivor function, hazard function or cumulative hazard function may be essential. Furthermore, structures are subjected to damage and deterioration during their lifetime. In this case, not only basic safety measures such as reliability index but also the measures related to damage tolerance of structures such as vulnerability, redundancy, and robustness may be beneficial.



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Figure 2 – Interaction between costs and initial performance level

The performance indicators for civil structures and infrastructures are not limited to those mentioned in this paper. More effort should be placed on determination of performance indicators and their use in risk communication and optimal decision making under uncertainty. ACKNOWLEDGMENTS The support from (a) the National Science Foundation through grants CMS-0638728 and CMS-0639428, (b) the Commonwealth of Pennsylvania, Department of Community and Economic Development, through the Pennsylvania Infrastructure Technology Alliance (PITA), (c) the U.S. Federal Highway Administration Cooperative Agreement Award DTFH61-07-H-00040, and (d) the U.S. Office of Naval Research Contract Number N00014-08-1-0188 is gratefully acknowledged. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organizations. REFERENCES

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Initial CostFailure Cost

Maintenance and Repair Cost

Preventive Maintenance Cost

Initial Performance Level

Cos

ts



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