safety assessment of a steel frame using lrfd and sbra methods

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Safety Assessment of a Steel Frame Using LRFDand SBRA Methods

Leo Václavek1; I.-Hong Chen, Ph.D., M.ASCE2; and Pavel Marek, F.ASCE3

Abstract: Powerful computers on each designer’s desk, improvements in simulation technique, and advances in the reliability assessmenttheory make it possible to consider a transition from prescriptive concepts, such as partial factor design �PFD�, to fully probabilisticconcepts applicable in design practice, such as the feasible simulation-based reliability assessment �SBRA� method. Using a pilotexample, the main differences between the PFD assessment procedure and SBRA procedure are demonstrated and described. Attention isfocused on the substance of individual steps starting with the assignment �including a representation of loading, mechanical, andgeometrical properties of the structure and characteristics of imperfections� and concluding the comparison with safety limits expressedvis-à-vis PFD by a ratio of demand and capacity �to be less than 1.0� and regarding SBRA by comparing the probability of failure Pf anddesign �target� probability Pd, i.e., Pf � Pd. The potential, efficiency, and advantages of the proposed SBRA method are elucidated.

DOI: 10.1061/�ASCE�SC.1943-5576.0000028

CE Database subject headings: Structural reliability; Load and resistance factor design; Monte Carlo method; Probability; Frames;Steel.

Author keywords: Structural reliability; Load and resistance factor design; Representation of variables; Monte Carlo method; Proba-bilistic reliability assessment; Steel frame.

Introduction

Advances in computer technology make possible a transitionfrom structural reliability assessment concepts developed in theprecomputer era to fully probabilistic concepts equivalent to thepotential of modern computer technology. Such a transition re-quires a reengineering of the entire reliability assessment proce-dure starting from the new representation of loading �e.g., by loadduration curves; see Marek et al. �1995�� to the reliability ex-pressed by comparing the obtained probability of failure Pf andthe target probability Pd specified in the codes by the investor orby others having jurisdiction.

The deterministic approach to the structural reliability assess-ment, such as allowable stress design �ASD� has been graduallyreplaced in the past three decades by a semiprobabilistic �or “pre-scriptive”� approach, called the LRFD in the United States, andthe partial factors design in European eurocodes. The reliability ofstructures is now considered to be a function of defined criticalfailure modes. The concept of a limit-state surface separating thereliability domain into “safe” and “unsafe” domains has been gen-

1Associate Professor, Faculty of Mechanical Engineering, VŠB-Technical Univ. of Ostrava, 17 Iistopadu 15, 708 33 Ostrava-Poruba,Czech Republic �corresponding author�. E-mail: [email protected]

2Professional Engineer, Civil and Structural Engineers, LandtechConsultants, 3845 Beacon Ave., Suite D, Fremont, CA 94538. E-mail:[email protected]

3Professor and Senior Researcher, Institute of Theoretical and AppliedMechanics, Academy of Sciences of the Czech Republic, Prosecká 76,190 00 Praha, Czech Republic. E-mail: [email protected]

Note. This manuscript was submitted on September 17, 2008; ap-proved on April 14, 2009; published online on April 16, 2009. Discussionperiod open until July 1, 2010; separate discussions must be submitted forindividual papers. This paper is part of the Practice Periodical on Struc-tural Design and Construction, Vol. 15, No. 1, February 1, 2010.
©ASCE, ISSN 1084-0680/2010/1-63–72/$25.00.
PRACTICE PERIODICAL ON STRUCTUR

Pract. Period. Struct. Des. Co

erally accepted and is increasingly used in structural reliabilitytheory, design applications, and in the codes �e.g., EuropeanCommittee for Standardization �CEN� �2005��. Please note thatthe AISC �2005� standards and specifications list both LRFD andASD side by side in every section. To change the ASD designformat to the same failure criteria as LRFD, it rephrases “allow-able stress” to “allowable strength” and it assigns new load con-version factors which match old ASD results. In other words, thepresented format for ASD is different but in essence it leads tosimilar reliability.

Current specifications for structural steel design, such as AISCand eurocodes, are primarily based on the partial factors conceptusing the “design point” approach, reliability index �, load andresistance factors, and different failure criteria as the main tools inevaluating reliability. The reliability check is not explicitly de-fined nor fully explained in the specifications. It may be arguedthat the target reliability is set to predefined values and the com-plete reliability assessment scheme is somewhat hidden in whatmight be called a “black box.” The designer’s creative work isthus limited to interpretation of regulations and instructions con-tained in the codes to meet the predefined reliability.

With modern computer technology, however, methods basedon sampling and simulations are very effective. The simulationtechnique is a convenient and very powerful tool for the analysisof loads, load effect combinations, resistance, safety, durability,and serviceability with regard to single as well as multicomponentvariables. Taking into consideration the potential of current per-sonal computers, a pilot reliability assessment scheme has beendeveloped based on a representation of individual variables bybounded histograms and calculating the probability of failure Pf

�see, e.g., the simulation-based reliability assessment, or, SBRAmethod, documented in a textbook by Marek et al. �1995��. Usingthe SBRA method, all input quantities are expressed by nonpara-metric distributions �histograms�. Some of these histograms are
obtained by evaluating sets of long-term recordings �e.g., wind
AL DESIGN AND CONSTRUCTION © ASCE / FEBRUARY 2010 / 63

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velocity and direction leads to two-component histograms� whileother histograms are determined from large sets of test data �e.g.,yield stress�. Such an approach superbly represents the physicalsubstance of individual variables. To make the introduction of thesimulation procedure easily accessible to the designer, a directversion of the Monte Carlo-simulation technique can be applied.

This paper compares and delineates structural safety assess-ment procedures/criteria of the AISC-LRFD code with the proba-bilistic SBRA method. �Probabilistic in this context is defined as“of, based on, or affected by probability or randomness.”� Eachcomponent of the design criteria and each major step of the de-sign procedure are presented. The assignment selected is commonand identical for both approaches, however, substantial differ-ences must be considered in each step of the assessment proce-dure. Some of the sections are supplemented with appendices tomore closely specify the given problems. The advantages as wellas disadvantages of LRFD and of SBRA are specified.

Assignment

An unbraced planar steel frame �shown in Fig. 1� consists of twocantilevered flexible fixed columns �1, 2�, two leaning columns�3, 4� that could be considered simple pin ended, and three cross-

Table 1. Loading Data

Loading

Symbol UnitMaximum

value

F1

DLa kN �kips� 400 �89.92�

LLLb kN �kips� 200 �44.96�

SLLc kN �kips� 100 �22.48�

F2

DL kN �kips� 600 �134.9�

LLL kN �kips� 400 �89.92�

SLL kN �kips� 200 �44.96�

F3

DL kN �kips� 400 �89.92�

LLL kN �kips� 200 �44.96�

SLL kN �kips� 200 �44.96�

F4

DL kN �kips� 400 �89.92�

LLL kN �kips� 200 �44.96�

SLL kN �kips� 200 �44.96�

W WLd kN �kips� �20 ��4.50�

EQ ELe �0.035� �F1+F2+F3+F4��Ti TDf Ranges from �21°C to +40°CaDead load.bLong-lasting load.cShort-lasting load.dWind load.eEarthquake load.f

α, �T2 α, �T3α, �T1 F4

l 4=

6m

3

F3

l 3=

5.4

m

1

E1I1

F1

M1k1l 1=

6m

2

E2I2

F2

M2k2l 2=

7.6

m

d1 = 10 m d3 = 10 md2 = 10 m

EQ ,W

4

Fig. 1. Unbraced frame with leaning columns

Temperature difference.

64 / PRACTICE PERIODICAL ON STRUCTURAL DESIGN AND CONSTRUC


bars. Columns and crossbars are connected by means of pinjoints. It is assumed that the displacements of all components outof the frame plane are disallowed.

The frame is loaded in its plane: �1� by vertical forces F1, F2,F3, F4, and horizontal forces wind �W� and earthquake �EQ� and�2� by forced deformation of the crossbars caused by temperaturevariation �T1, �T2, and �T3 �with respect to the temperatureduring erection�. The thermal coefficient of expansion of thecrossbars is �=12�10−6 K−1. Each of vertical forces Fi is thesum of mutually uncorrelated dead load, long-lasting load �LLL�,and short-lasting load �SLL�. �The maximum values of each loadand range of temperature differences are indicated in Table 1.�All forces are mutually uncorrelated, with the exception of theEQ loading which is correlated with vertical forces Fi, so thatEQ= �0.035� �F1+F2+F3+F4� at the instant of an earthquake.The forces W and EQ may act to the left or to the right indepen-dently of each other.

The cantilevered columns 1 and 2 are W sections W 16�57and W 16�77, respectively. Cross section characteristics�A—area of cross section, I—moment of inertia, Sel—elastic sec-tion modulus, Spl—plastic section modulus� of these profiles,yield stress fy, and modulus of elasticity E are seen in Table 2.

The effect of residual stresses is implicitly considered inLRFD column curves. The SBRA method shows the effects ofresidual stress included in the equivalent geometrical imperfec-tions. Small initial curvatures with amplitudes f1 and f2 of fixedcolumns 1 and 2 and unavoidable eccentricities e1 and e2 offorces F1 and F2 are shown in Fig. 2. Imperfections a3 and a4

represent the initial deviations of upper ends of columns andthe points of application of forces F3 and F4 from a verticalline through the lower end of leaning columns 3 and 4 �see alsoFig. 2�. The range of the forenamed geometrical imperfections isgiven in Table 3.

The linear M −� relation Mi��i�=ki��i, i=1,2 between themoment Mi �at the fixed end of the ith column�, and the relativerotation �i at the fixed end are intended to describe the support

Table 2. Material and Geometrical Properties

Column Profile Symbol, unit Nominal values

1 and allothermembers

W 16�57 fy N /mm2 �ksi� 344 �50�

E N /mm2 �ksi� 210,000 �30,458�

A m2 �in.2� 1.08�10−2 �16.8�

I m4 �in.4� 3.15�10−4 �758�

Sel m3 �in.3� 1.51�10−3 �92.2�

Spl m3 �in.3� 1.73�10−3 �105�

2 W 16�77 fy N /mm2 �ksi� 344 �50�

E N /mm2 �ksi� 210,000 �30,458�

A m2 �in.2� 1.46�10−2 �22.6�

I m4 �in.4� 4.62�10−4 �1,110�

Sel m3 �in.3� 2.20�10−3 �134�

Spl m3 �in.3� 2.46�10−3 �150�

31

E1I1

k1

2

E2I2

k2

d1 d3d2

a3 a4

l4

f1 e1 e2f2

4

l1 l3l2

Fig. 2. Initial undeformed shape of the frame

TION © ASCE / FEBRUARY 2010

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flexibility conditions of cantilevered columns 1 and 2. �The nu-merical values of elastic stiffness k1 and k2 are stated in Table 3.�

In the following, the main components of the safety assess-ment procedure are specified according to LRFD and SBRAmethods. It should be mentioned that a practical approach to theunbraced frames with leaning columns is proposed in Geschwind-ner �1994�.

Loading and Load Combination

Significant differences between the representation of loads ac-cording to both LRFD and SBRA methods can be observed. WithLRFD, each load is expressed by its nominal value and by itscorresponding load factor. The load combinations are determinedusing a set of rules specified in the codes AISC �2005� and ASCE�2005�. A completely different representation of loads has beenintroduced in the SBRA method �Marek et al. 1995�. Each load isexpressed by a load duration curve �and/or corresponding histo-gram� and the load effect combinations are calculated using theMonte Carlo sampling technique. Such an approach is equivalentto the potential of the available computer technology �for moredetails see numerous examples described in a textbook �Mareket al. 2003� and in Appendixes I and II�.

LRFD Loading and Load Combination

The nominal values of individual loads are obtained by dividingthe maxima �listed in Table 1� by the corresponding load factor.These nominal loads are factored and combined according to dif-ferent load combos per AISC �2005�. The most critical loadcombo will control the final frame design �see Appendix I formore details�.

Simulation-Based Reliability Assessment Loadingand Load Combination

Using a generator of pseudorandom numbers, in each MonteCarlo-simulation step a new set of loads is created and applied inthe multiple repeated analysis of the frame. Load duration curvesand corresponding bounded histograms of loads are described in

Table 3. Geometrical Imperfections and Support Conditions of FixedColumns

Column

Geometrical imperfections

Symbol, unit Range

1 f1 mm �in.� �20 ��0.787�

e1 mm �in.� �30 ��1.181�

2 f2 mm �in.� �25 ��0.984�

e2 mm �in.� �38 ��1.496�

3 a3 mm �in.� �27 ��1.063�

4 a4 mm �in.� �30 ��1.181�

Column

Fixed �elastic� support conditions

Symbol, unit Nominal value

1 k1 kNm/rad 8�104

2 k2 kNm/rad 1.5�105

detail in Appendix II.



Material Properties

For identical material used in any frame design, different aspectsof the material are adopted in the evaluation procedure. In LRFD,the yield stress of the selected steel grade is expressed by a ”dis-crete minimum” value while with the SBRA method the yieldstress is represented by abounded histogram.

LRFD Material Properties

The A992 Grade 50 steel is the most commonly available steel fora W-shape in the United States. Thus, A992 is used in this study.

Simulation-Based Reliability Assessment MaterialProperties

A bounded histogram of steel A992 yield stress �see Fig. 3� isused in the study. The histogram can be obtained using a statisti-cal analysis of test results �for more details see Marek et al.2003�.

Local and Global Imperfections

In the safety assessment of a steel frame, it is necessary to con-sider local and global geometrical imperfections. According toLRFD, the effect of imperfections is traditionally included in theassessment formulas and adjustment of column curves while withthe SBRA method the individual imperfections are expressed byvariables related to the shape of individual members of the frameand by variables defining the imperfections of the global geom-etry of the frame. It is worth noting that in the 2005 LRFD,any second-order analysis method that considers both P−� andP− effects may be used.

LRFD Local and Global Imperfections

The global imperfections can be expressed by deterministic val-ues of the out-of-vertical position of columns. In LRFD, thesecond-order P− and P−� effects are traditionally accountedthrough amplification factors B1 and B2 in member maximummoment calculation �see LRFD specification or Appendix III fordetailed explanations�.

Simulation-Based Reliability Assessment Localand Global Imperfections

The SBRA method has adopted from European standards an ideaof equivalent geometrical local and global imperfections. Theseare not actual construction tolerances but, because they are in-tended to represent the effect of a number of factors, are likely tobe larger than such tolerances �details regarding the variable local

Range from345 to 450 MPa

(50 to 65 ksi)

Steel A992yield stress

Fig. 3. Substitute for A992 yield stress histogram

and global imperfections are explained in Appendix IV�.


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Resistance and Reference Values

The frame analysis must correspond to the “rules of the game” ofsafety assessment. In LRFD, the resistances are combined withdifferent resistance factors �R�, which are associated with differ-ent failure mechanisms. The majority of failure mechanisms arebased on the design yield stress of the applied steel grade and onthe formation of the plastic mechanism. For compact steel sec-tions in bending, which are fully braced against lateral torsionalbuckling, the resistance factors times the internal forces corre-sponding to the formation of the fully plastic mechanism definesthe resistance applied in the safety check criteria. In the SBRAmethod, the reference value can be expressed by the onset ofyielding the steel frame or on a tolerable total or permanent plas-tic deformation. The frame analysis must follow these rules of thegame.

LRFD Resistance and Reference Values

In LRFD, the resistances for beam and column members are ofnominal axial capacity and nominal moment capacity times thecorresponding reduction factor of 0.90 and 0.90. In otherwords, only 90 and 90% of the nominal axial and bending capac-ity are used during the frame design �refer to Appendix V for adetailed description of pertinent nominal resistance factors�.

Simulation-Based Reliability Assessment Resistanceand Reference Values

In the SBRA method, the resistance for beam and column mem-bers is derived from the histogram of yield stress. With regard tothe actual scatter of yield stress, the elastic bending capacity canbe higher in many simulation steps than the full plastic capacity�for details see Appendix VI�.

Frame Analysis

Two totally different approaches to the frame analysis are appliedin the LRFD and in the SBRA methods. In LRFD just a single setof input data are considered in the analysis of the frame. WithSBRA, on the other hand, the analysis is repeated many times�e.g., 107 times� while in each simulation step the pseudorandomnumber generator is used for random sampling of all input vari-ables �loads, imperfections, material and geometrical properties,and others�, i.e., the frame is recalculated many times and theresponse of the structure �expressed, e.g., by stresses or strains� isstored and made ready for the final calculation of the probabilityof failure.

LRFD Frame Analysis

In the 2005 AISC, a direct second-order analysis may be used.However, LRFD usually requires a first-order analysis of bothsway and nonsway occasions. The critical load combo for theassignment is 1.2D+1.0E+L. �The first-order analysis results areshown in Fig. 4.� These results are then modified �B1, B2, and Kfactors� to obtain the final second-order demands.

Simulation-Based Reliability Assessment FrameAnalysis

Using the SBRA method, the “strength stability concept” is ap-
plied. The analysis is carried out according to the second-order


theory. Initial imperfections �see Fig. 2� are taken into account inthe computational model used, such that P− and P−� effectsare by themselves included. No individual stability check of col-umns is necessary; resistance is related to the onset of yielding.�A deformed shape of the frame after loading is shown in Fig. 5.�

Safety Assessment

In the LRFD method, demand D divided by capacity C must beless than 1.0, i.e., D /C�1.0, where D expresses the combinationof the assigned load effects. With the SBRA method and referencevalue defined by the onset of yielding, the calculated probabilityof failure Pf is compared with the target probability Pd containedin the codes �ČSN 1998� or required by the investor or deter-mined by the authorities having jurisdiction.

LRFD Safety Assessment

Following LRFD design procedure, members are considered ad-equate when demands are less than the capacity, i.e., D /C�1.0.With D /C closer to 1, the design is more economical in terms ofmember size. �Refer to Table 4 for a summary of the D /C ratiofor different frame members and Appendix VII for details ondemand/capacity calculations.�

(a) Analysis without lateral translation

!!"#$!%& !''#$(%& !""#")!&

!''#$(%&

!!"#$ !''#'

!""#"

!''#'

(b) Analysis with lateral translation

!*#!('&

!($#$

+#$!*%#(

,#"

Fig. 4. LRFD first-order analysis results without and with LT

31

E1I1

k1

2

E2I2

k2

d1 d3d2

a3 a4

l4

f1 e1 e2f2

4

l1 l3l2

δ1 δ3 δ2 δ4

F1 F3 F2 F4�T1 �T2 �T3

M2

M1

EQ, W

Fig. 5. Deformed shape of the frame after loading


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Simulation-Based Reliability AssessmentSafety Assessment

Taking into account the SBRA safety criterion, the calculatedprobability of failure Pf must be less than the target probabilityPd. The target probability Pd recommended by Code ČSN 731401�ČSN 1998� is Pd=5�10−4, 7�10−5, and 8�10−6 for reduced,common, and enhanced reliability levels, respectively. Calculatedprobabilities of failure Pf are in Table 5 �for more details seeAppendix VIII�.

Summary and Conclusions

The focus of this study is a hypothetical confrontation of twostructural reliability assessment methods using a steel framesafety assessment as a common point of reference, i.e., the pre-scriptive partial factors method specified in contemporary U.S.standards, i.e., LRFD, and the probabilistic SBRA method. Thefocus has been on the qualitative differences between the twoapproaches. It is not a simple matter to compare prescriptive de-sign procedures given in LRFD with the probabilistic SBRA ap-proach. Wide-ranging rules of the game are to be considered inthe comparison, i.e., a completely different representation of loadsand analysis of the load combination, the definition of referencevalues, the introduction of local and global imperfections, and thesubstance of reliability criteria including the definition of thesafety limit.

In the LRFD approach, the demand/capacity ratio of eachstructural element is checked individually to ensure the adequacyof the whole structural system. Traditionally, second-order effectsincluding member imperfection and P-delta effects are either im-plicitly or explicitly included in the member demand/capacitycheck. The LRFD approach is an improvement over the ASD byintroducing different load and resistance factors to produce amore rational and economic design. Either ASD or LRFD hasbeen the foundation for steel building design in the United States.Both approaches are based on demand/capacity check of the in-dividual member. Either is very accurate for a simple element andcomponent; however, it may be tedious and produce an uneco-nomic design for a complicated structural system. To produce aneconomic and time saving design even for a complicated struc-tural system, with the continuing advancement and innovation ofcomputer technology, different types of advanced analysis havebecome feasible such as “second-order refined plastic hingeanalysis,” which is accepted as the direct second-order analysis inAISC �2005� �Chen and Chen 1999; Kim and Chen 1996; Liew

Table 4. Summary for LRFD Design

MemberDemand/capacity

check

Column 1 0.971

Column 2 0.781

Table 5. Safety Assessment by Course of SBRA

MemberCalculated probability

of failure

Column 1 3.1�10−6

Column 2 Less than 10−6



et al. 1993� or alternatively, the probabilistic SBRA method�Marek et al. 1995; Pustka et al. 2006�.

In the SBRA method all input variables including actions areexpressed by bounded histograms �Marek et al. 2003�. By usingthe Monte Carlo technique, each simulation step generates a newcombination of random variables �considering variable actionsand their combination�, imperfections, material properties, andothers. This procedure is applicable for linear as well as for non-linear structural analysis. In this way, the reliability check of aframe structure is significantly simplified. The stability problemsare solved using the “strength and second-order theory stabilityapproach,” without determining buckling lengths and bucklingfactors. The reference level regarding safety is defined by theonset of yielding or to the tolerable elastoplastic deformations.The same methodology can be applied to studies of differentsecond-order theory problems considering elastic response ofstructure. The main limitation of the probabilistic SBRA approachis prospective hard- and software. The dramatic improvements incomputer technology, especially parallel processing, is very en-couraging and promising—if not exciting.

Failure probability Pf versus target probability Pd is a goodway to quantify a design. This assignment shows Column 1 iswell designed and Column 2 is overdesigned, according to LRFD.The D /C ratios correlate well with the probability of failure Pf incomparison with SBRA. In real practice, both LRFD and SBRAdesign procedures should yield safe frame designs. The LRFDapproach produces a design that meets predefined failure reliabil-ity while the SBRA approach discloses the failure reliability ex-plicitly. With this additional information, designers, in the future,can easily adopt new complex approaches �such as the perfor-mance based design, see Galambos �2006��, which are, in point offact, the current trends of future codes.

Concluding Comment

We live in a world defined by variables. The reliability of struc-tures depends on numerous variables as well. The advances incomputer technology facilitate the application of a probabilisticapproach so as to investigate the interaction of these variables andto express the reliability by comparing the calculated probabilityof failure and the target probability.

Numerous examples described not only in SBRA papers but intextbooks as well are drawing increasing attention to the qualita-tive difference between the current approaches applied in codes�deterministic and prescriptive� and to significant advances insteel construction resulting from the application of the probabilis-tic method using simulation techniques and powerful computers.The computer era necessitates a transition from the deterministicto the probabilistic approach which leads to a nontraditional reli-ability assessment of structures and without a doubt to more effi-cient design and to an improved understanding of the actualperformance of structures.

Acknowledgments

Support for this project has been provided by the Grant Agency ofthe Czech Republic �Project Nos. 103/07/0557 and 103/05/H036�,by the Institute of Theoretical and Applied Mechanics �ITAM�,Academy of Sciences of the Czech Republic, and by VŠB Tech-nical University of Ostrava, Czech Republic. Dr. I.-Hong Chen
and both cowriters would like to express deepest gratitude to

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Professor Wai-Fah Chen, dean, University of Hawaii. Without thisprofessor’s involvement, this international cooperation could nothave come about.

Appendix I. LRFD Load Combinations

The appropriate critical load combination shall be used to deter-mine the required strength of the structure and its elements.Please note that listing combinations are from ASCE 7-05. Thesecombinations are to be used in 2005 LRFD, unless the local gov-erning code states otherwise.1. 1.4�D+F�;2. 1.2�D+F+T�+1.6�L+H�+0.5�Lr or S or R�;3. 1.2D+1.6�Lr or S or R�+ �L or 0.8W�;4. 1.2D+1.6W+L+0.5�Lr or S or R�;5. 1.2D+1.0E+L+0.2S;6. 0.9D+1.6W+1.6H; and7. 0.9D+1.0E+1.6H.D=dead load �the weight of the structural elements and perma-nent features on the structure�; E=earthquake load �dependingon the applicable code�; F=load due to fluids with well-definedpressures and maximum height; H=load due to lateral earthpressure, ground water pressure, or pressure of bulk materials;L=live load �occupancy and moveable equipment�; Lr=rooflive load; R=rain load; S=snow load; T=self-straining force; andW=wind load. The loadings listed in Table 1 are maximum val-ues. To convert maximum to nominal loadings, loadings are di-vided by the maximum load factor and substituted into differentload combinations as nominal loads. For example, both LLL andSLL are live loads �L�, the maximum load factor for live load�L� is 1.6; therefore nominal L= �LLL /1.6+SLL /1.6�. Loadcombo �Item 5� would be equal to 1.2�DL /1.4�+1.0�EL /1.0�+ �LLL /1.6+SLL /1.6�, where DL, EL, LLL, and SLL are load-ings listed in Table 1.

Appendix II. Simulation-Based ReliabilityAssessment Method and Loadings

In structural design, loading is one of the most significant vari-ables affecting reliability. Practically all types of loads arerandomly variable quantities represented by a correspondingprobability density function. Considering some selected loadings�e.g., LLL or SLL�, the probability density function correspond-ing to these loads do not correspond to any parametrical distribu-tion. This problem in the SBRA method is very easily resolved bythe introduction of so-called “load duration curves” and corre-sponding bounded histograms. The load duration curves can bederived on the basis of measured load data or an engineer’s esti-mate. �Procedures for self-made load duration curves andbounded histograms, including a histogram database of commontypes of loads, can be found in a book by Marek et al. �2003�.�Load duration curves and corresponding bounded histogramsused in the frame analysis in this study �e.g., dead load, LLL,SLL, wind load, earthquake load, and temperature difference� areshown in Fig. 6. �The illustrated histograms were taken from thedatabase of the book by Marek et al. �2003�.�

In modeling loads, it is important to understand the depen-dence connecting different loads and load types. The SBRAmethod makes it possible to consider various conditional relationsbetween individual loads �independent loads, loads depending on
the existence of other loads, loads that cannot appear on a struc-


ture at the same time, mutually correlated loads, etc.�. In theframe analysis proposed in this study, the loads are considered asmutually independent except EQ �earthquake load�, which is cor-related with vertical forces so that EQ= �0.035� �F1+F2+F3

+F4� at the instant of an earthquake.One of the most important problems affecting the reliability

analysis of frame structure, according to the second-order theory,is the combination of loads and other random variables. It is wellknown that the principles of superposition and proportionality arenot valid from the standpoint of the nonlinear analysis. The com-binations of load effects corresponding to the particular load casescannot be used to find the most critical response of the structure.It is necessary to carry out the combinations of all random vari-ables �not only of particular actions� already on the level of inputdata and the analysis must be separately done for all consideredcombinations. Such an analysis is fully compatible with the prin-ciples of the SBRA method, where in each simulation step a newset of input random variables is generated. This is a new gener-ated set of random variables after it enters the transformation

0.82

0.98

0.90

0 0.2 0.80.4 0.6 1 Histogram

Load Duration Curve

Dead load0.82

0.98

0.900.86

0.94 0.94

0.86

1

0

0.4

0.8

0.2

0.6

Histogram

Long lastingload

Load Duration Curve

0 0.2 0.80.4 0.6 1

1

0

0.4

0.8

0.2

0.6

Histogram

Load Duration Curve1

0

0.4

0.8

0.2

0.6

0 0.2 0.80.4 0.6 1

1

0

0.4

0.8

0.2

0.6

Short lastingload

Histogram0 0.2 0.80.4 0.6 1

1

0.5

0-0.5

-1

1

0.5

0-0.5

-1

Load Duration Curve

Wind load

Histogram

Load Duration Curve

0 0.2 0.80.4 0.6 1

1

0.5

0-0.5

-1

1

0.5

0-0.5

-1Earthquake load

Histogram0 0.2 0.80.4 0.6 1

3.9

2.4

0.9-0.6

-2.1

3.9

2.4

0.9-0.6

-2.1

Load Duration Curve

Temperature difference

Fig. 6. Dimensionless bounded histograms representing variousloads �right side� and corresponding load duration curves �left side�

model �for details, see Marek et al. 1995�. The result of the simu-


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lation process is a set of searched load effects �stresses in crosssections, bending moments and internal forces, and deformations�corresponding to individual combinations of input variables. Suchobtained results may be used for the probabilistic analysis andfollowing the reliability evaluation of the structure and its com-ponents.

Appendix III. LRFD Second-Order Effects

In 2005, LRFD analysis methods are expanded to any second-order analysis method that considers both P−� and P− effects,which may be used. In previous LRFD versions, the second-ordereffects, “member imperfection,” and “P-delta effects” are usuallyincluded as follows.

Member Imperfection

In the LRFD design, column curves were established by closelyfitting structural stability research council Curve 2. They implic-itly account for the second-order effects, residual stresses, and aninitial out of straightness of 1/1,500. �The column strength curvesfor the design strength Pn of columns are presented in Chapter Eof the LRFD specification, AISC �2005��

Pn = AgFy�0.877

�c2 � for �c 1.5

Pn = AgFy�0.658�c2� for �c � 1.5

where Ag=gross area of member; Fy =specified yield stress; and�c=slenderness parameter, which is expressed as

�c =KL

� · r�Fy

E

where K=effective length factor; L=length of member; r=radius of gyration; and E=modulus of elasticity. KL /r should betaken as the larger of the effective slenderness ratios for strong-and weak-axis buckling. For the leaning column, one of LRFD’sallowed methods is the further adjustment of the K factor to K2

for Pn calculation only and is used in the calculations of thispaper

K2 =��2EI

L2Pr � Pr

�2EI

�Kn2L�2 ��5

8Kn2

where Kn2=K value determined directly from the alignment chart.

P-Delta Effect

In LRFD, the second-order P− and P−� effects are accountedfor by amplification B1 and B2 factors in member maximum mo-ment calculation. The member maximum moment Mr can be de-cided by the following approximate second-order analysisprocedure, according to LRFD �in Chapter C�

Mr = B1Mnt + B2Mlt

where Mnt=required flexural strength in the member assumingthere is no lateral translation �NT� of the frame; Mlt= requiredflexural strength in the member as a result of lateral translation�LT� of the frame only; and B1= P− moment amplification fac-
tor, which can be expressed as


B1 =Cm

1 −Pr

Pel

where Cm=equivalent moment factor, generally expressed asCm=0.6−0.4�M1 /M2� �when not subjected to the transverse

loading between support�where M1 /M2=ratio of the smaller to larger end moment in a

nonsway situation, positive when the member is bent in reversecurvature, and negative when bent in single curvature; Cm=1.0 ordetermined by analysis �when subjected to transverse loading be-tween the support�; Pr=required second-order axial strengthusing LRFD load combinations; Pe1=�2EI / �KL�2, where K is aneffective length factor in the plane of bending based on the as-sumption that side sway is prevented; and B2= P−� moment am-plification factor, which can be expressed as

B2 =1

1 − �Pnt� �oh

�HL�

or

B2 =1

1 −�Pnt

�Pe2

where �Pnt= total vertical load supported by the story usingLRFD load combinations, including gravity column loads; �oh

=lateral interstory deflection; �H=summation of all story hori-zontal forces producing �oh; and L=story height

�Pe2 = ��2EI/�KL�2�

where K=effective length factor not including the leaning columneffect in the plane of bending, assuming that side sway is free.

Appendix IV. Simulation-Based ReliabilityAssessment Method and Imperfections

The probabilistic analysis of steel frames does not only enter thecombinations of particular actions but also the combinations of allrandom variables, i.e., including imperfections, cross-sectionalcharacteristics, yield stress, and others. The current standards�AISC 1993; CEN 2002� give formulas only for combining par-ticular actions. If we want to use a strength stability concept �i.e.,the assessment according to the second-order theory includingP−� and P− effects without determining buckling lengths andbuckling factors�, it is necessary to consider the structure with allglobal as well as local imperfections. It does not suffice to com-bine only particular actions but what must also be taken intoaccount is the combination of randomly variable imperfections,cross-sectional characteristics, and other randomly variable quan-tities. This approach can be carried out very easily and transpar-ently using the probabilistic SBRA method.

The analytical transformation model based on the second-order theory was developed by the writers for structures of thetype whose particular arrangement and shape is shown in Figs. 1,2, and 5, i.e., unbraced frames �see Václavek 2006�. The modelstands for unbraced frames with an arbitrary number �at least 1�of fixed columns and arbitrary number of leaning columns, con-nected at the top by crossbars. The model makes it possible to
explicitly calculate stress and deformation response of the struc-

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ture exposed to loads, i.e., flexible fixed columns with initialcrookedness and leaning columns �see vertical forces Fi and hori-zontal force H�.

The analytical transformation model of the unbraced frameshown in Figs. 1, 2, and 5, is expressed by Eqs. �1�–�8� �Václaveket al. 2005�. In addition, the model simultaneously facilitates cal-culating explicitly stresses and deformations of the structure ex-posed to the vertical forces F1, F2, F3, and F4, to the horizontalforce H=W+EQ, and to the forced deformation induced by tem-perature differences �Ti. Horizontal displacements 1 and 2 �seeFig. 5� are

1 =

W + EQ + i=3

4

Fi

ai

li+

i=1

2FiVi

Gili+ D

i=1

2Fi

Gili−

i=3

4Fi

li

�1�

2 = 1 + i=1

2

di�i�Ti �2�

The following symbols were used in Eqs. �1� and �2�:

D =F3

l3d1�1�T1 −

F2

G2l2i=1

2

di�i�Ti +F4

l4i=1

3

di�i�Ti �3�

Gj =tan�� jlj�

Pj� jlj− 1 �4�

Pj = 1 −Fjlj

kj

tan�� jlj�� jlj

�5�

Vj = ej� 1

Pj cos�� jlj�− 1� + f j Pj�1 − �2� jlj

��2��−1

− 1��6�

� j =� Fj

EjIj, j = 1,2 �7�

Bending moments M1 and M2 at the fixed sections of cantileveredcolumns 1 and 2 are �j=1,2�

Mj = Fj��1 +1

Gj� j −

Vj

Gj+ ej + f j� j = 1,2 �8�

In Eqs. �1�–�3� i=summation suffix and j in Eqs. �4�–�8�=freesuffix; Fi , Fj =vertical forces; Ej =modulus of elasticity; Ij

=moment of inertia; li , lj =lengths of columns; di=length ofcrossbar; ej , f j , ai=geometrical imperfections; j =horizontaldisplacement; kj =elastic stiffness �fixed support condition�;�i=thermal coefficient of expansion; and �Ti=temperature dif-ference. Small initial crookedness of fixed columns 1 and 2 �seeFig. 2� are supposed to be in the shape of a quarter wave of cosinefunction.

The SBRA method applies to any of particular imperfections,distribution �using bounded histograms�, corresponding to themeasured or estimated values. The frame analysis given in thispaper takes into account bounded normal distributions of imper-fections. Nominal values of E ,A , I ,Sel in Table 2 and k1 ,k2 inTable 3 are thought of as mean values of bounded normal distri-
butions �for examples see Fig. 7�. Variability of geometrical im-


perfections f1, f2, e1, e2, a3, and a4 �as in Table 3; see also Fig. 2�is expressed using bounded normal distribution with a mean valuezero �an example is shown in Fig. 7�.

Appendix V. LRFD Resistance Factors

In LRFD, the resistance factors for the different types of struc-tural members are as follows:

c = 0.90 �resistance factor for compression�

b = 0.90 �resistance factor for flexure�

t = 0.90 �resistance factor for tensile yielding�

t = 0.75 �resistance factor for tensile rupture�

sf = 0.75 �resistance factor for shear rupture�

v = 1.00 �resistance factor for shear�

Please note the listing resistance factors are based on the2005 LRFD. There is a slight difference in both load and resis-tance factors from previous versions. This is anticipated and ex-pected because in LRFD, the load factor and resistance factor arecalibrated against each other for failure probability.

Appendix VI. Simulation-Based ReliabilityAssessment Method and Reference Values

The resistance of frame structure and its elements is derived fromthe yield stress of steel fy taking into account the strength stabilityconcept used together with the SBRA method. The results of a

209000179000 194000 224000 239000

Modulusof elasticity E

[MPa]

Moment of inertia

W 16x57Mean value316×10-6[m4]

291 303 315 327 339

56000 68000 80000 92000 104000

Elasticstiffness k1

[kNm/rad]

-27 -1-14 12 25

Equivalent geometricalimperfection a3

[mm]

Fig. 7. Bounded histograms, some of the imperfections—input ran-dom variables

long-standing research and measurement of yield stress of bar


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profiles can be used for the probabilistic assessment using theSBRA method. �Fig. 3 illustrates the design histogram of the yieldstress of steel A992.�

An appropriate establishment of reference value is one of themost important problems affecting the safety assessment of framestructures. The reference value is generally a boundary value ex-ceeding the defined reliability assessment. The onset of yieldingin a checked cross section is very often considered as the refer-ence value; the safety is so related to exceeding elastic resistanceRel �see Fig. 8�. The tolerable plastic or total elastoplastic defor-mations �deflections of beams, rotations of joints� can be alsocomplemented as appropriate reference values. The exceedance ofsuch deformations may limit the proper function of the frame andthereby the structure is not reliable �for instance, some structuralelements must be reconstructed or replaced�. An increased workdifficulty of elastoplastic analysis prevents more extensive utili-zation of elastoplastic reference values Rel,pl. The utilization offull plastic resistance of the structure and its components Rpl

is not suitable for probabilistic reliability assessment for theaforementioned reasons concerning tolerable plastic deformations�except for some accidental design situations, such as fire orearthquake�.

In the presented example of frame analysis using the SBRAmethod, only the elastic resistance Rel was complemented as thereference value. But it is necessary to realize that the elastic bend-ing carrying capacity �due to the actual scatter of yield stress� canbe higher in many simulation steps than the full plastic capacitycorresponding to the values of yield stress fy at the initial sectionof the histogram of yield stress �see Fig. 3 and elastic sectionmodulus Sel and plastic section modulus Spl in Table 2�.

Appendix VII. LRFD Demand/Capacity Calculations

The procedure for LRFD demand/capacity checks of differentstructural members is the same. Therefore, the calculations forColumn 1 and Column 2 are presented as ## \ ## for solutions of

Fig. 8. Relationship between loading and deformation

the same equation �“NA” is “not applicable”�.



Given

Ag = 16.8 \ 22.6 in.2

Iy,col = 758 \ 1,110 in.4

Sy = 92.2 \ 134 in.3

Zy = 105 \ 150 in.3

Lcol = 19.685 \ 24.934 ft

Fy = 50 \ 50 ksi

E = 30,458 \ 30,458 ksi

Iy,beam1 = 758 in.4

Iy,beam2 = 758 in.4

Lbeam1 = 32.808 ft

Lbeam2 = 32.808 ft

From Analysis

Pnt=119.2\199.9 kip �NT�Plt=0.0\0.0 kip �LT�Pr= Pnt+ Plt=119.2\199.9 kip �lateral translation+NT�Other leaning columns’ �Pnt=266.6\266.6 kip �NT�Other leaning columns’ �Plt=0.0\0.0 kip �LT�Other leaning columns’ �Pr=�Pnt+�Plt=266.6\266.6 kip�LT+NT�Mnt=0.0\0.0 k-ft �NT of the frame�M1=0.0\0.0 k-ft �smaller of end moment when NT�M2=0.0\0.0 k-ft �larger of end moment when NT�Mlt=162.2\147.6 k-ft �with LT of the frame�

Calculation—Axial Load

r = Sqrt�Iy,col/A� = 6.72 \ 7.00 in.

Gtop = ��Ic/Lc�/��Ib/Lb� = 10.0 \ 10.00

Gbottom = 1.0 \ 1.0

K �from nonsway chart�=0.85\0.85Nonsway �c= �KL� / �r��Sqrt�Fy /E�=0.386\0.468K �from sway chart�=2.1\2.1Sway �c= �KL� / �r��Sqrt�Fy /E�=0.952\1.156Pe2=AgFy /�c

2=926.0\845.6 kip �from sway �c��Pr=585.7\585.7 kip �summation of columns 1 and 2 andleaning columns��Pe2=1 ,771.1\1 ,771.1 kip �summation of columns 1 and 2�Adjusted leaning column’s Kadj=Sqrt��2EI�Pr / �L2Pr�Pe2��=3.366\2.483Adjusted leaning column’s sway �c= �KadjL� / �r��Sqrt�Fy /E�=1.527\1.367Fcr= �0.877 /�c

2�Fy, if �c 1.5, otherwise �0.658�c2�Fy

=18.82\22.87 ksi �from adjusted sway �c�cPn=0.90FcrAg=284.51\465.07 kip �from adjusted sway �c�
Pr /cPn=0.419\0.430 �from sway �c�

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Calculation—Moment

Pe1=AgFy /�c2=5,652.0\5 ,158.7 kip �from nonsway �c�

Cm = 0.6 − 0.4�M1/M2� = 0.60 \ 0.60

B1=Cm / �1− Pr / Pe1�=1.000\1.000 �greater than 1�Pe2=AgFy /�c

2=926.0\845.6 kip �from sway �c��Pnt=585.7\585.7 kip �summation of columns 1, 2, andleaning columns��Pe2=1,771.1\1,771.1 kip �summation of columns 1 and 2�

B2 = 1/�1 − �Pnt/�Pe2� = 1.494 \ 1.494

Mr = B1Mnt + B2Mlt = 242.3 \ 220.5 k-ft

Assume compact section and no lateral torsional buckling

bMn = 0.9Mp = 0.9ZyFy = 4,680 \ 6,705 k-in. = 390.0 \ 558.8 k-ft

Interaction „Demand/Capacity…

If Pr /cPn�0.2, Pr / �2cPn�+Mr /bMn otherwise, Pr /�cPn�+ �8 /9��Mr /bMn�=0.971\0.781�OK\OK�.

Appendix VIII. Simulation-Based ReliabilityAssessment Method and Safety Criterion

The safety assessment using the SBRA method is expressed bycomparing the calculated probability of failure Pf �using theMonte Carlo-simulation technique� with the target probability Pd

given, for instance, in Czech code ČSN731401 �ČSN 1998�. Thevalue of probability of failure Pf determines the probability thatthe resistance of the structure or their elements �expressed byintroduced reference value RV� will be exceeded by the calcu-lated LE. This relation is written formally by the expression

P��RV − LE� � 0� = Pf � Pd

Considering the RV, defined in the solved example by theonset of yielding, i.e., the probability of failure of the individualcolumn equals the probability of exceeding the yield stress fy inthe most loaded fibers of the investigated cross section of thecolumn �disregarding the effect of residual stresses�, the safetycondition is expressed by the formula

Pf = P��fy − �� 0� � Pd

where Pd=target probability for safety assessment and �=maximum value of stress in the most loaded fibers of thechecked cross section. Assuming an elastic response of the struc-ture to the loading, the maximum stress � in the outer fibers of thesection can be determined by the equation



� = abs� M

Sel� + abs�N

A�

where M�kNm� and N�kN�=values of bending moment and nor-mal force, respectively, at the checked section while Sel �m3� andA �m2� are the section modulus and area of cross section, respec-tively. Note that the values M and N are different if a transforma-tion model based either on the first-order or on the second-ordertheory is applied.

References

AISC. �1993�. Load and resistance factor design specification, 2nd Ed.,AISC, Chicago.

AISC. �2005�. Steel construction manual, 13th Ed., AISC, Chicago.ASCE. �2005�. Minimum design loads for buildings and other structures,

ASCE standard, revision of ASCE 7-02, ASCE, Reston, Va.Chen, I. H., and Chen, W. F. �1999�. “Practical advanced analysis for

seismic frame design.” Adv. Struct. Eng., 2�4�, 237–263.ČSN. �1998�. “Design of steel structures, appendix A.” ČNI 731401, Pra-

gue, Czech Republic �in Czech�.European Committee for Standardization �CEN�. �2002�. “Basis of struc-

tural design.” Eurocode EN 1990, Brussels.European Committee for Standardization �CEN�. �2005�. “Design of steel

structures. Part 1.1: General rules and rules for buildings.” Eurocode3, Brussels.

Galambos, T. V. �2006�. “Structural design codes: The bridge betweenresearch and practice.” Proc., IABSE Symp.—Report Responding toTomorrow’s Challenges in Structural Engineering, Budapest, Hun-gary, 2–10.

Geschwindner, L. F. �1994�. “A practical approach to the ‘leaning’ col-umn.” Eng. J., 31�4�, 141–149.

Kim, S. E., and Chen, W. F. �1996�. “Practical advanced analysis forbraced steel frame design.” J. Struct. Eng., 122�11�, 1266–1274.

Liew, J. Y. R., White, D. W., and Chen, W. F. �1993�. “Second-orderrefined plastic hinge analysis for frame design: Part I.” J. Struct. Eng.,119�11�, 3196–3216.

Marek, P., Brozzetti, J., Guštar, M., and Tikalsky, P. �2003�. Probabilisticassessment of structures using Monte Carlo simulation. Background,exercises, and software, 2nd Ed., Academy of Sciences of the CzechRepublic, Prague, Czech Republic.

Marek, P., Guštar, M., and Anagnos, T. �1995�. Simulation-based reliabil-ity assessment for structural engineers, CRC, Boca Raton, Fla.

Pustka, D., Křivý, V., Václavek, L., and Marek, P. �2006�. “Selectedstructural stability problems using the SBRA method.” Proc., IABSESymp. 2006—Responding to Tomorrow’s Challenges in Structural En-gineering, Budapest, Hungary.

Václavek, L. �2006�. Analytical computational models, second ordertheory and probabilistic reliability assessment of structures, VŠBTechnical University of Ostrava, Ostrava, Czech Republic �in Czech�.

Václavek, L., Marek, P., and Konečný, P. �2005�. “SBRA approach tothe stability assessment of structural components and systems.” The2005 Joint ASCE/ASME/SES Conf. on Mechanics and Materials(McMat2005) Proc. �CD-ROM�, G. Z. Voyadjis and R. J. Dorgan,eds., Louisiana State Univ., Baton Rouge, La.


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safety assessment of a steel frame using lrfd and sbra methods

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