Reliability- B ased Structural Design

Dibyendu Mukherjee 0 Vmderbilt University 0 Nashville Smkaran Mahadevan a Vaaderbilt University a Nashville

Key words: Re!iaLiiicy. Sensicivity, Limir; State, Target Reiiabiiities, Structures. Design. System Reliabllicy, F d u r e Modes, Saiesy Factor, System Factors

This paper will present the use of state-of-the-art re- liability techniques to develop efficient structural design guidelines for civil engineering structures in a manner tha t includes overall structural system effects. Probabilistic de- sign is done by explicitly accounting for the uncertainties in the different variables and their influence on structural performance. This approach has been used to develop a set of load and resistance factors instead of just a single safety factor to address all the uncertainties. However. the reli- abilities of the individual components, when placed in the overail configuration, may not remain the same as when the components are considered separately. Therefore, this paper proposes that the reliability of the structure be CY- plicitly satisfied at the component level as well as a t the system level during design. Two important considerations will be addressed in this regard: (i) the influence of sys- tem effects on the element reliability, and (ii) formulation of component design to ensure overall system reliability. As a result, system factors are developed for inclusion in structural design.

1. INTRODUCTION

The goal of a design engineer is to assure the safety and performance of a system. Many decisions that are required during the process of planning and design are made under conditions of uncertainty or without complete information. Therefore, the assurance of performance may be realisti- cally stated only in terms of probability of success or fail- ure. In the past, however, structural design has been done deterministically, conservative nominal values for struc- tural resistance, and loading conditions, with experience- based design margins. This approach performs satisfac- torily when there is extensive past experience about the structure's performance under different conditions. How- ever, i L does not provide a quantitative estimate of the level of safecy assured and the degree of design margin sensitivity to any of ihe parameters. The performance re- quirements and cost reduction goals in the design of new systems require such information and are therefore increas- ingly challenging the deterministic approach. A reliability assessment methodology based on a mathematical consid- eration of the uncertainties in the basic parameters is able

t o provide a solution to the above problem, leading to a rational redesign process that improves the reliability to required levels without undue conservatism.

The basic design procedure involves satisfying a criterion of the form:Factored Resistance 2 Effect of factored Loads. In the common case where the total load effect is a linear combination of individual loads, we need to satisfy

n

i = l

In this formula the left side represents the resistance capac- ity of the structural element under consideration, and the right side denotes the forces which the element is expected to support during its intended life (load effects). The term Rn is a nominal resistance corresponding to a faiiure state and 6 is the resistance factor, which is less than unity and which reflects the degree of uncertainty associated with the determination of the resistance. The term rQ is the prod- uct of the load Q (i.e., the force on the member or the element in the form of bending moment, shear force? axial force etc.) and a load factor y, generally larger than unity, which accounts for the degree of uncertainty inherenc in the determination of the forces Q. In general $R,, may repre- sent a number of failure states (e.g., yielding and fracture in a tension member) for each element, and XY=l=ly;&; reflects the largest of several load combinations. The Probabilistic design methodoiogy is already being applied successfully to steei and concrete building scructures, offshore structures, and bridge structures.

The key to successful implementation of probablistic de- sign is to explicitly account for (i) the uncertaincies in dif- ferent variables, and (ii) their influence on structural per- formance.

This approach can be used to develop a sec of load and resistance factors instead of just a single safety factor to ad- dress all the uncertainties. The probabilistic design factors should be developed in such a manner that the practicing engineer needs to be familiar with only the situations for the application of these factors and not the mathematical detaiis of the procedure that deveioped them.

2. SIGiVIF ICA lVCE 0 F R EL IA B IL I T Y- BA SED DESIGiV

0149-144>(195/$4.00 01995 IEEE 1995 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY' Symposium 207

Design factors on load and resistance variables corre- sponding to an acceptable failure probability have been derived by researchers to be used in the structural de- sign process. These load and resistance factors have to be unique in the sense tha t they should apply to a wide range of design conditions and material properties. This objective is achieved by a judicial combination of g o u p - ing and optimization. T h e principal benefits of such an approach may be itemized as follows (Ref. 2):

More consistent reliability is attained for different de- sign situations because the different variabilities of the strength properties and loads are considered explicitly and independently;

The desired reliability level can be chosen to reflect the consequences of failure;

I t simplifies the design process by encouraging the same design philosophy and procedures to be adopted for all the different materials that make up the entire structure;

It is a tool for exercising judgement in non-routine situations; and,

It provides a tool for updating standards in a rational manner.

This approach has resulted in the development of the MSC Load and Resistance Factor Design Code (1986) (Ref. 9) used extensively for structural steel design. The same methodology has also been developed for the design of highway bridge superstructures, and offshore structures.

3. R EL Id4 BIL IT Y- BASED DESIGLV iMETHODOLOGY

The first step in reliability-based design is the estima- tion of failure probability for each performance criterion. A mathematical model which relates the resistance and load variables is formulated. This is referred to as the per- f o m a n c e function. Let this function be expressed as g ( X ) , where X is the vector of input random variables related to the resistances or the loads. Failure occurs when g ( X ) < 0. The probability of failure PJ = P ( g ( X ) < 0) is computed a s (Ref. 1)

Pf = l(x)<o fx(zW (2)

where fx(z) is the probability distribution of x. Ln gen- eral, the computation of the above integral is very difficult. In addition, the limit s ta te functions are usually nonlinear and some of the random variables have non-normal distri- butions. An analytical approximation of the failure prob- ability is available. All the variables (,U) are converted to equivaient standard normal Variables (.U'). A linear ap- proximation is constructed for the limit s ta te at the point of minimum distance on the limit s ta te from the origin.

This point represents t h e most probable limit state com- bination of the random variables, and is referred to as the Most Probable Point ( M P P ) as shown in Fig. 1. The fail- ure probability is then estimated as

(3) where ,B is the distance from the origin to MPP, and is the cumulative distribution function of a standard nor- mal variable. The gradient vector of the limit state at the M P P provides the sensitivity factor of the limit state with respect to each random variable. These sensitivity factors are subsequently used to derive the load and resistance factors. Joint. pdf

! I I

. . MPP

,

I' x; Figure 1: Failure Probability Estimation

The basic variables (XI, X 2 , ...., Xn) are transformed to the standard normal space X as,

(4)

where px, and ox, are the mean and s tandard deviations of the i-th variable X i respectively. In terms of the reduced variables, X,!, the limit s ta te equation would be,

I t can be shown tha t the point on the failure surface with minimum distance to the origin is the most probable failure point and h i s minimum distance may be used as a measure of reliability. Thus i t can be shown tha t ,

where the derivatives (%),, are evaluated at

(z:, z;, ...., zk). Thus, the most probable failure point on the failure surface (in scalar form) becomes,

, I

208 1995 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium

in which,

(*)‘

4- = (8)

are the direction cosines along the axes c ; . Thus the re- quired partial safety factors are obtained as

19)

In general, the detrrmi.natioti of 2: requires an iterative solution. For this purpose. the following algorithm may be used (Ref. 1 ) :

1. Assume a:: and obtiain

2 . Evaluate (-/$)- and (r;

3 . Obtain x; = px, - cr;oax, (11)

4. Rrpeat Steps( 1) through ( 3 ) until convergence is ob- tained.

The required design factors are then obtained with Eq. 9.

In t,he above discussion. the variables are assiimed to be normal and uncorrelacrd. For non-normal variables: fix, and g,yt should be replaced hy t.he equivalent, normal p:;

anti r;t (Ref. 1) . If t)he variablrs are corraiasecl. thr X variables are transformed to an uncorrelatect Y variable space and ail the above c,aiculations are performed in the Y-space. Thus, the partial safety f x t o r s include the sta- tistical information about the design variables: and their individual influences on structurai performance.

The derivation of load and resistance factors requires a target, value of the reliability index. T h e reliability index varies deprnding on the stmicture and loading type. There- fore. by repratedly det,ermining p for a large number of existing sr,ructures with varying loading conditions whose members have brrn designed acc,ording to the AISC LRFD specifications. repwsentat-ive values of / j may be seiectrd to reflect thr reliability of satisfactory current designs.

4.1 A I,S(? LK FD ,Spfrificntion.i; .4 (iirecr connrq i i enc r of t ,hr a l iove t,echriiqiie is t h e .AIS( ‘

Load and Resistance FactcJr Design ( :ode ~ised extensiveiy by engineers in the U.S. as a guidr to structural steel ti?-

sign. This code enabies t,hr designer t,o selecr, partial ioad

anti resistance factors for a range of loads and load combi- nat,ions. Since loads bring about a negative effect on th? rrliability of a structure. the partial load factors are 11~11-

ally greater than unity (except, for counter-acting ioatis), while che resistances which contribute to the reliability arr provided factors that are less than or equal to unity.

In general. there are two types of limit states (Ref. 2): ( i ) ultimate limit states under which the structure or com- ponent is considered to have failed in its capacity to carry load; and (ii) serviceability limit states under which the function of the striictiire is impaired. T h r LRFD method basically deals with the ultimate limit states as these are of particular concern in standards and spec.ifications which are intended to protect the structiire fro 111 cat as t ro p hi c dam age.

4.1.1 LRFD Load Factom

below (Ref. 9): The LRFD load factors and load combinations are shown

e 1.4D, *1.2D, + 1.6L, 01.2D, + l.6.Sn -I- (0.5L, OT O.XW,j 0l.2D7, 4- l.:3w,2 + O.nL, 0l .2Dn + 1.5& + (O.SL, 07’ O.%.S’,) *o.9Dn - ( l.:lwn 07‘ 1.5zn)

Here D, is the nominal dead load, L , is the nominal live load. W, is the nominal wind load, 5; is the nominai snow load, E,, is the nominal earthquake load. The above fac- tors have also been deveioped based on varying levels of reliability. The target reliability indices chosen for gravity !oads ( D , L , and S) is 3.0, that for wind !oads is 2.5. and ft3r earthquake loads it is 1.73. The target .+levels were chosen to be similar to the safety levrls obsrrvrd in rxist- ing design procedures that wrre found to be aclequate for different, cases.

4.1.2 L H F D Rcsis tanc~ Facto,rs The foilowing is a list of resist,ance factors for some types

of members and connections provitled in the LRFD sprci- fication (Ref. 9):

o Trrision Mrmbers: Yield Limit State, c j = 0.913 Fracture Limit State. Q = 0.75

e (:ompression Members: fi, = 0.85

e Flexural Membrrs: cp = 0.85

0 Wrids: a = 0.90, 0 80. 0 75 irkpending on thr type of connect 10 ns)

4.2 High, irjuy H7‘2dg1~ ,S,upCrstru(’turr.s ( i h o s n anti Moses ( l W 3 ) have used probahilisr,ic tipsign

format to dev~ lop faccors chat, m a y be used i,y t,hr engineer during the design of bridge nierril>ers. T h e research is based o n non-linear analysis of highway hridge syst.Pms followed

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by the application of reliability techniques. The factors that have been developed may be used to improve member capacities and provide a safer design.

5 . .YYhYTEiVf EFFECT.? T h e individual components of a struc,ture interact with

one another to form a single coherent system. Current design specifications do not explicitly address this system effect and deal only with the design of individuai com- ponencs. The reliability of each component is assured by designing each limit state to have a low probability of vio- lation. However, the reliabilities of the individual compo- nents, when placed in the overall configuration, may not remain the same as when the components are considered separately. Therefore, the reliability of the structure has to be satisfied at the component level as well as at the sys- tem level. There are two important considerations ac this point: ( i ) how such system effects influence the element reliability. and (ii) how to perform component design to ensure overall system reliability.

5.1 Systcm-A ffwted Compo~ten t Re,liability The word component may refer to a bolt or weld a t a

joinc, or a member in the structure. In order t,o design these components, the AISC LRFD specification (Ref. 9) provides load and resistance factors that correspond to a target reliability index. As mentioned earlier, the reliabil- ity index of each component analyzed individually may be different from the reliability index of the component ob- tained when considering the effect of the entire structure. The latter is defined to be system-affected element reliabil- ity. Such system effects on individual component reliabii- ity have so far not been accounted for explicitly ir, design procedures. Current research by the authors is attempt- ing to develop a method to account for the system effects (both sGructura1 and probabilistic) on individual compo- nent desiqn. The effects of the system on individual el- ements (system-affected element reliability) is accounted for by providing revised sectional properties corresponding to the target reliability IeveIs after a non-linear reiiabilicy analysis is performed.

5.2 O w r a l l S‘ystem Reliability The concept of overall system reliability involvps multi-

ple failure modes. The failure of the diifPrent componencs themselves o r their combinations may constitiice a partic- ular failure mode of the system. T h e overall system faiiure probability is computed as the probability of the union nf failure events of the system in the different modes. Failure event of chr system in a mode is given hy the intersection of the failure events of a number of componrnts. Therefore. the overall system reliability problem involve3 the compu- tation of the probability of the union of the intersection of the component failure events. However, the problem becomes more complicated when the possibiiic:; of mul- tiple compoiient failure modes is inclutirtl. which means that each of the individual members niay themselves fail

Load Load Load

Figure 2 : Motleiing of structural member failure

in more than one failure mode. Furthermore, an individuai element may undergo a brittle failure as in shear failure, semi-brittle failure as in buckling failure, or i t may undergo a ductile failure a s in bending or tensile failure. Brittle fail- ures like shear or buckling result in immediate failure and the member loses its capacity to resist the applied shear or axial load. Semi-brittle failures like buckling result in the partial loss of the initial failure strength capacity and the member continues to function a t a reduced capacity. Ductile failures like bending allow the member to maintain its load carrying capacity even after failure (first yieid) . These three failure types may be modeled as shown in Fig. 2.

4 branch and bound selection procedure (Ref. 17) is used to identify the dominant failure modes, in order to reduce the amount of computation. Overall system relia- bility is estimated using only these significant failure modes of the system.

t5.$ ,Yystcm Factors The effects of the system on the individual component

design will have to be eventually incorporated in the de- sign process. The structural engineer is oniy interested in designing the individual elements that constitute the sys- tem. T h e design should be such as to withstand the dam- ages in the system once it is operational. The reliability of the individual elements against failure is currently en- sured by the application of load and resistance factors. It is proposed that similar factors be developed to assure the overall system reliability. These factors are to be used in conjunction with the existing LRFD factors. This way, the designer can use existing design methods. but with an ad- ditional factor, known as the system factor. The load and resistance factor design formac will be followed in general to d ~ v e l o p these system factors. It is desirable that the ex- isting factors that account only for component reliability be left unaltered because of the engineers’ famiiiarity with them.

6. .-iPPLICATIO!V EXAMPLE

X portai frame is designed as per the AIS(’ LRFD d ~ - sign specification considering only the bending limit state for the sake of illustration XI1 the loads are treated ab

random variables and the rpsistance vector consists of the plastic moment capacities of the sections used (.U,,, , MP2,

210 1995 PROCEEDINGS Annuai RELIABILITY and MAINTAINABILITY Symposium

+ 4-1 Seaions when System-

.. Load Comb. j Mem. Z L R F D 1 Z,,,,,, 1 0; i 106.0 106.0 1.00

~ :I 23.7 40.7 0.58 11.3 40.7 0.28

85.7 0.73

81.7 1 81.7 1 1.00 0.9D-1.3W i 1 2

1 1 0.9D;l.:;W 1 3 1 62.7

LL 69.0 106.0 0.65

1 2 3 4

methodology is applicable to a wide range of structural systems under different ioading conditions for each mem- ber. The calibration process is simiiar to that done un- der the earlier LRFD study (Ref. 2 ) , with the addicional consideration of system effects. The analysis in section 6 provides a separate system factor under different loading conditions for each member. This is not practical for de-

Beam M e c l " S

Sway Mechanisms M m 9 10

Figure 3: The portal frame with its collapse mechanisms anti limit state functions

M p 3 ) . All random variables are considered to be normal. T h e yield stress fy is taken to be a constant (36.0 ksi). T h e individual members are designed based on the LRFD design equation, OR 2 (0.9D It 1.3W) and their section moduli are denoted as ZLRFD. The optimized sections obtained from overall system reliability considerations are compared to those obtained from the system-affec,ted ele- ment reliability method ancl the higher of the two is chosen as Zsllstzm to ensure safety from all aspects. The ratio of Zsystom and ZLRFD gives the system factors (&) which are shown in Table 1. For each load case. a set of sys- tem factors are obcained which will result. in a new set of members for the portal.

L gg = M p l + a-zl,? t 3MP, - T P ~ - H P I (20)

910 = M p l f 4 M p 2 f iWp3 - 7 Pz - HPI (21) L 2

Table 2 shows the original design equation modified by the system factors. I t is apparent that by using the mod- ified design equations, the sections obtained will be heav- ier. The heavier section is given priority of choice over the lighter one. Thus, a new set of members are obtained for each load case. Since the wind load effect on the struc- ture is reversible, the design of the portal frame in real life would be: symmetrical. Hence. the same sections will be provided for both the columns. In both Table 1 and 2. ;\iI refers to member.

L. Comb. I M 1 Mod. Dsgn. Eqn. 1 0.9D-1.3W 1 3 1 0.52 R > 0.9D-1.3W I

0.25R 3 0.9D+1.3W

0.38R > 0.9D-i-l.3W

Table 2 : Modified Design Equations.

The design produced by the proposed method is cali- brated with respect to a n accepcable target reliability index and current design standards. This is essential to obtain

a parcicuiar group are compared to a preassigned target system reliability index value. .An optimization procPdurP is used to minimize the deviation of the system 3 value from the targec system /) value. The value of the resis-

Thr porcai wich ics various S ~ S C ~ K I failure mechanisms are shown in Fig. 3 . The expressions for t he ten mrciia- nisms shown in Fig. 2 are given below.

L 2

tance factor for which this minimization is achieved is the system factor for tha t member. Thus, a unique system ( 1 2 ) 91 = .VIpl i, 2 M p 2 + - -F'Z

1995 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium 21 1

factor for each member under a particular load combina- tion is attained. These system factors can then be easily

[IO] Verderainie, V., Aerostructural Safety Factor Criteria us- ing Deterministic Reliability, AIAA Space Program and

incorporated in the design proc,ess. Technologies (:onference, Huntsville, Alabama. March 24- 27. Paper no. 92-1583, 1992.

[11] Xiao, Q., Structural Reliability Under Cumulative Dam- 8. CONCL LTSION

This paper presents a n overview of the use of proba- bilitic methods in the design of structural systems. State- of-the-art reliability and advanced probabilistic. techniques are currently being used to refine the design process. Cur- rent research is extending these methods to incorporate the overall system effects. T h e probabilistic techniques are quite general and may be applied to a wide range of structural systems. The techniques are being applied to building, highway, offshore, mechanical and aerospace structures at present. The same methods may be extended to other engineering systems using the terms ‘capacity’ and ‘demand’ instead of ‘resistance’ and ‘load’ respectively.

[11 Ang, A.H-S., and Tang, W.K., Probability Concepts in Engineering Planning and Design - Volume 11, John Wiley and Sons, New York, 1984.

[2 ] Ellingwood, B.. Gdarnbos, T.V., MacGregor, J.G., and Cornell, C:. A., Development of a Probability Based Load and Resistance Criterion for American National Standard A58, NBS Special Publication 577, U.S. DOC/NBS, June 1980.

age, Ph.D Dissertation, Vanderbilt University, Nashville, Tennessee, 1993.

RIO G R A P HIES

Dibyendu Mukherjee, M.S. Vanderbilt University Dept. of Civil & Environmental Engg. P.O.Box 1625. Station B Nashville, TN 37235 Phone: (615) 322-5164 E-mail: dipu074(~vuse.vanderbilt.edu

Dibyendu Mukherjee is a Graduate student a t Vanderbilt University. He is working towards his Ph.D in the field of Prob- abilistic Analysis and Design of Frame Structures and System Effects on individual component design. His B.S. is from Ja- davpur Univeristy, India in 1986. From 1986 to 1988 he was a Design Engineer at Paharpur Cooling Towers, India. He has a Master’s Degree from The University of Tennessee, Knoxville in 1990. He has worked in various companies in the U.S. since then. He was an Assistant Engineer at the City Engineer’s of- fice at OakRidge Municipality, OakRidge, T N and worked as a Structural Engineer at Allen & Hoshall Inc., Knoxville, TN during the summer of 1992. He is student member of ITE and XSCE. He has a publication in the ITE Journal.

[3] Galanibos, T.V., Reliability of Structural Steel Systems, Structural Engineering Report No. 88-06, Department of ( : i d and Mineral Engineering Institute of Technology, {Jniversity of Minnesota, August 1988.

Sankaran Mahadevm, Ph.D Vanderbilt UTniversity Dept. of (:ivil & Environrnentd Engg. P.O.Box 6077, Station B

[4] Ghosn, M., and Moses, F., Redundancy in Highway Bridge Superstructures. Prepared for NCHRP, TRB. NRC, de- partment of C i d Engineering, The City College of The City University of New York, New York. New York, September 1993.

[5] Madsen, 8. O., Krenk, S. , Lind., N. C.: Methods of Struc- tural Safety, Prentice-Hall, Inc., Englewood Cliffs, N J! 1986.

[6] Mahadevan, S., and Haldar, A., Efficient Algorithm for Stochastic Structural Optimization. .Journal of Structural Engineering, AS(:E, Vol. 115, No. 7, July 1989. pp. 1579- 1597.

[7] Mahadevan, S., Stochastic Finite Element-Based Struc- tural Reliability Analysis and Optimization, Ph.D Thesis, Georgia Institute of Technology, Atlanta. Georgia, 1988.

[8] Mahadevan, S., and Haldar, A., Stochastic FEM-Based validation of LRFD, Journal of Structural Engineering, ASCE, Vol. 117, No. 5, May 1991, pp. 1393-1413.

[9] Manual of Steel (:onstruction: Load and Resistance Factor Design, American Institute of Steel (.:onstruction (AISC:), First Edition, 1986, New York, N.Y.

Nashville, TN ,37235 Phone: (615) 322-3040 E-mail: mahasl (avuse.vanderbilt.edu

Sankaran Mahadevan is an Associate Professor of Civil and Environmental Engineering at Vanderbilt University. He ob- tained his B.S. from Indian Institute of Technology, Kanpur, India: M.S. from Rensselaer Polytechnic Institute; and Ph.D. from Georgia Tech, all of them in Civil (Structural) Engineer- ing. His research efforts are primarily in the area of reliability and risk assessment, applied to structural engineering. His spe- cific contributions to this field are under the topics of probabilis- tic finite element analysis, reliabdity-based design optimization, systems reliability, and time-variant reliability. He has applied these concepts to the reliability estimation and design of build- ing and hydraulic structures, and gas turbine engine structures. He has authored over forty publications in the area of structural reliability. In 1992, he received the ASME award for Best Paper at the Structures. Dynamics and Materials (SDM) Conference. In 1991 and 1992, he received Outstanding Teacher awards at Vanderbilt University.

212 1995 PROCEEDINGS Annual RELIABILITY and MAINTAINABILITY Symposium