Load and Resistance Factor Design
THEODORE V. GALAMBOS
Load and Resistance Factor Design, abbreviated as LRFD, ample, beam theory is more accurate than column theory is a scheme of designing steel structures and structural (e.g., in Ref. 1,0 = 0.85 for beams and 0 = 0.75 for col components which is different from the traditionally used umns), or that the uncertainties of the dead load are smaller allowable stress format, as can be seen by comparing the than those of the live load (e.g., in Ref. 2, yD = 1.2 and 7^ following two inequalities: = 1.6). LRFD thus has the potential of providing more consistency, simply because it uses more than one factor. Rn/F.S. > ±Qm (1) The purpose of this paper is to describe the development 1 of an LRFD specification for steel structures. 4>Rn > t yiQni (2) SIMPLIFIED PROBABILISTIC MODEL 1 The strength R of a structural member and the load effect The first of these inequalities represents the allowable stress Q are both random parameters, since their actual values case, while the second represents the LRFD design crite cannot be determined with certainty (Fig. 1). The strength rion. The left side in each case is the design strength, and of a structure, often referred to also as its resistance, is de the right is the required strength* The term R defines the n fined in a popular sense as the maximum force that it can normal strength as given by an equation in a specification, sustain before it fails. Since failure is a term tfrat is associ and Q i is the load effect (i.e., a computed stress or a force n ated with collapse, it is more useful, in the context of such as bending moment, shear force axial force, etc.) de structural behavior, to define strength as the force under termined by structural analysis for the loads acting on the which a clearly defined limit state is attained. Such limit structure (e.g., live load, dead load, wind load, etc.). The states are, for example, the plastic mechanism, the plastic term F.S. represents the Factor of Safety, 0 is termed the moment, the overall or component buckling load, fracture, resistance factor, and the 7 's are the load factors associated ? fatigue, deflection, vibration, etc. Not all of these limit states with each load effect Q . z cause "collapse" in the popular sense, and so it is appro The coefficients F.S. > 1.0, 0 < 1.0, and 7, > 1.0 all priate to define strength as "the limit state which deter serve the same purpose; they account for the uncertainties mines the boundary of structural usefulness." inherent in the determination of the nominal strength and Structural behavior is thus satisfactory if Q < R, while the load effects due to natural variation in the loads, the on the contrary, Q > R is unacceptable. Since Q and R are material properties, the accuracy of the theory, the precision random, it is theoretically not possible to state with certainty of the analysis, etc. The fundamental difference between LRFD and the allowable stress design method is, then, that the latter employs one factor (i.e., the Factor of Safety), while the Ptobabi I ity former uses one factor with the resistance and one factor Density each for the different load effect types. LRFD, by employing more factors, recognizes the fact that, for ex-
Theodore V. Galambos is the Harold D. folley Professor of Civil Engineering at Washington University in St. Louis. * The terms in italics in this paragraph are the adopted terms used in Refs. 1 and 2. Fig. 1. Probabilistic description of Q and R
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ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION where R and Q are the mean values and_F# and VQ are the coefficients of variation (VR = OR/R, VQ = (TQ/Q, Mean where a denotes the standard deviation; see Ref. 6) of R and Q, respectively. It can be seen in Fig. 2 that the mag nitude of /3 effectively positions the coordinates with respect to the distribution curve. When (3 is larger, the probability of exceeding the limit state is smaller, i.e., the "reliability" is increased; the converse is true when /3 decreases (see Fig. 3). The reliability index (3 can thus serve as a comparative
In (R/Q) measure of reliability between various design methods, types of members, and types of loading, and it has generally been preferred to use /3, rather than the probability of ex ceeding the limit state, in developing the new LRFD Fig. 2. Definition of the Reliability Index specifications (Refs. 1,4,7,8, for example). Typical values of /3 encountered range from 2 to 6, each increase of one unit for any structure that Q< R. Even for the most carefully corresponding very roughly to one order of magnitude of designed and constructed structure there is a small but finite decrease in the probability of exceeding a limit state. chance that Q> R, i.e., that the limit state can be exceeded. A satisfactory structural design specification is one which EXAMPLE OF THE DETERMINATION OF THE RELIABILITY INDEX minimizes this chance to an acceptably low level. It is possible, by using a simplified probabilistic ap The use of Eq. (3) will be illustrated next. This is not a proach, to quantify the statistical parameters which de design office exercise (more will be said about that later), scribe the probability of exceeding a limit state. Figure 2 but a scheme whereby actual designs were "calibrated" is an identical representation of Fig. 1, using as the abscissa prior to the development of the resistance and load factors the ratio In (R/Q). When In (R/Q) < 1, the limit state has which are to be used in the design office. been exceeded, and the shaded area in Fig. 2 is the proba A compact two-span continuous beam will be used for bility of this event. Since the probabilistic distributions of illustration. R and Q are not known very precisely, a method has been The mean strength and the coefficient of variation of a devised which operates only with the mean and the stan compact beam is equal to:5 dard deviation of the random parameters.3 This method is called the First-Order Second-Moment probabilistic R=Rn (PMF) (4) 4 2 2 2 analysis. More refined methods, which also take into ac VR = (VR + VM + VF )V2 (5) count the distribution, are also available and have been used (Ref. 4) in the actual development of the load factors pro The term Rn is the nominal strength posed in Ref. 2. Rn=Mp= Zx Fy (6) According to the first-order probabilistic method, one 5 where Mp is the nominal plastic moment based on the can determine a reliability index /3: handbook value of the plastic section modulus Zx and the In (R/Q) specified yield stress Fy. 2 2 (3) VvR + vQ The coefficients P, MyF (representing abbreviations for Professional, Material, Fabrication) are mean values of the following random parameter ratios: P = test/prediction M = actual static yield stress/specified yield stress
F = actual Zx /handbook Zx All available data were examined and, based on the in terpretation of these data, it was decided that the following statistical values appropriately represent the total popu lation of compact steel beams: (R/Q) P = 1.06, Vp = 0.07 (41 indeterminate beam tests, _ Ref. 9) M = 1.05, VM = 0.10 (Ref. 10)
Fig. 3. Comparative description of the Reliability Index F= 1.00, VF = 0.05 (Ref. 5)
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THIRD QUARTER / 1981 Design according to the 1978 AISC Specification, 1 I i 1 1 i 1 1 I 1 1 1 i 1 1 | rmtrr ffffl'" Parti
—t 4- 1 wD = 57.5X32 = 1840 plf 2 Fig. 4. Two-span beam example Live load reduction for AT = 896 ft and D/L = 57.5/50: 0.49
wL = 50(1 - 0.49)(32) = 809 plf 2 Obviously there is a certain element of judgment involved Mmax = wL /8 = 3115 kip-in. (at center support) in these values, but they represent the best that a number 0.9X3115 _. , of experienced people could come up with on the basis of Srea = = 118 in.3 the available information. Substitution of these values into q 0.66 X 36 3 3 Eqs. (4) and (5) gives R = \AlZxFy and VR =0.13. Use W21x62, Sx = 127 in. ; Zx = 144 in. 11 Plastic analysis of the beam in Fig. 4 gives: Thus R = 1.17 X 144 X 36 = 6065 kip-in. 2 OP + wL)L With this value of R, and with V = 0.13, Q = 2333 M = R P 11.66 (J) kip-in. and VQ = 0.11, the reliability index [Eq. (3)] is: (3 = 5.6. The right side of Eq. (7) is the load effect Q. It consists of the dead and the live load effect, and it can be written as: Design according to the 1978 AISC Specification, Part 2 Q = Q.D + QL (8) wD = 1.7 X 1840 = 3128 plf The mean and the standard deviation are then:6 wL = 1.7X809 = 1387 plf 2 Q=QB + QL (9) __ (4.515 X28 )(12) . 3 Z =1 lm VQ = (VD2QD2+VL2QL2)V2/(QD + QL) (10) ^~ 11.66X36 ° - 3 A study of the load effect statistics (Ref. 2) gives the Use Wl8x50, Z^. = 101 in. following values: The resulting (3 is 3.5. Design according to the proposed AISC LRFD Qj> = 1.05QnZ) , ^=0.10 (11) criteria (Ref. 1) QL = QUL, V =0.25 (12) L The design criterion is Eq. (2). where the subscript n relates to the nominal code-specified
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ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION Research on the LRFD method for steel building struc LEVEL I. tures, intended as a companion to the AISC Specification, Used in Design Office started in 1969 at Washington University, and by 1978 a With 0's and y's Prescribed draft of such a specification was published.16 These ten tative LRFD criteria had also been tested in several design offices.17 It became obvious early in the LRFD research that the f major difficulty, once the method and the format were LEVE1 II. agreed upon, was the treatment of the loads and the deci First-Order Second-Moment Used to Develop Level I. Codes sions on the magnitudes of the load factors. There was a Probabilistic Analysis clear need for a broader stance on this matter than could
Load and Probabilistic Resistance Judgment Statistics Model
LEVEL III. 1 Full Probabilistic Analysis - Research Analysis for Special Structures
Past Experience; Design Office Current Fig. 5. Hierarchies of probabilistic methods Specifications Studies
Designers were always aware of the uncertainties in herent in the design process, and this explains the evolution of the "factor of safety." Experience with aircraft design Specification during World War II indicated that the probabilistic nature Logic of the design parameters could somehow be quantified. The basic notions of this quantitative probabilistic approach r were formulated in the 1950's by a series of papers au thored by Freudenthal, and the premises advanced in these LRFD Criteria papers still hold true today.13 At the same time the idea of using multiple factors was Fig. 6. Flowchart of LRFD specification suggested in England by Pugsley,14 and a specification using them was formulated in the Soviet Union. Simulta neously, discussions within the American Concrete Institute eventually resulted in the familiar ACI ultimate design Load Statistics *• | *• Resistance Statistics method, with its 0-factors and its load factors. By the beginning of the 1960's, then, there was available a comprehensive probabilistic theory, and there were a Probabilistic Analysis number of design codes in use which used multiple factors arrived at by intuitive means. The 1960's and early 1970's B's for Current Specifications brought about first the simple, and then an increasingly more sophisticated, First-Order Second-Moment method, whereby multiple design factors could be generated from Selection of Target g s statistical data and using probabilistic theory. There were many contributors to this development, e.g., J. R. Benja min, G. A. Cornell, N. C. Lind, A. Ang, E. Basler, N. Probabilistic Analysis Ditlevsen, to mention only a few, and the first design specification for steel structures based on this approach was Search for Optimum 8 issued in 1974 in Canada. At present (1981) the various Y ' s and to's approaches are ordered into hierarchies, as shown in Fig. 5. Many countries and regions of the world are presently either adopting or considering the adoption of LRFD type codes. Load Factors Resistance Factors Recommended in to be Determined by In the U.S. the development of LRFD for steel structures ANSI A58.1 Materials Specifications also has its origins in plastic design for buildings,11 and in the formulation of Load Factor Design for steel bridges.15 Fig. 7. Flowchart of load and resistance factor development
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THIRD QUARTER / 1981 be taken by one single materials-oriented specification the data for hot-rolled steel structural members is given in group, and consequently a study was initiated under the Tables 2, 3, and 4. umbrella of the ANSI A58 Load Factor Subcommittee at On the basis of the available load and resistance statistics the National Bureau of Standards during 1979. The aim and on their (1979) current structural design specifications, of this research was to arrive at load factors which would the values of the reliability index (3 inherent in all types of be commonly applicable to all structural materials: hot- materials used in building construction were determined, rolled and cold-formed steel, aluminum, reinforced and much in the same way as the example illustrated above for prestressed concrete, timber, and masonry. The recom mendations of this research were published in 1980.4 The Table 2. Material Property Statistics load factors 7 incorporated into the draft of the AISC 1 LRFD Specification are the same factors which have been Mean 2 recommended in ANSI A58.1-81. Property Value M VM The schematic process of arriving at the LRFD Speci fication is flow-charted in Figs. 6 and 7. These charts show, Static yield stress, flanges 1.05 Fy 1.05 0.10 in a very simplistic way, the ingredients of arriving at the Static yield stress, webs, plates 1.10 Fy 1.10 0.11 new criteria. One should note that judgment and experience Modulus of elasticity E 1.00 0.06 play a crucial role in the process. It is by no means a one Static yield stress in shear 1.11 Fy/y/3 1.11 0.10 way procedure which is based solely on probabilistic methods. Such methods provide a set of tools which permit Poisson's ratio 0.3 1.00 0.03 the decision makers to make rational rather than intuitive Tensile strength \A0Fu 1.10 0.11 judgments. Tensile strength of weld, ou 1.05 FEXX 1.05 0.04
Shear strength of weld 0.84 au 0.84 0.10 THE ANSIA58.1 LOAD FACTORS Tensile strength of H.S.S. bolt, A325, au \.20FU 1.20 0.07
The recommended common load factors, as developed for Tensile strength of H.S.S. bolt, A490, ANSI A58 responsible for the various load types, e.g., live Fu = Specified tensile strength load, wind load, etc. Resistance data were obtained from FEXX = Specified tensile strength of weld metal the literature on the various materials involved. The data for steel structures were taken from a series of papers and Table 3. Modeling Statistics research reports generated in the research at Washington University, where similar information was also collected Type Member Model P Vp on cold-formed and aluminum structures. A sampling of Tension member AgFy or A F 1.00 0 Tension member AaFv or Aebu 1.00 Compactt W-beamW-beam: Uniform momenmoment Mp 1.02 0.06 Table 1. Load Statistics Continuous Mechanism 1.06 0.07 Mean Coeff. Distribution W-beam,, limit state LTBLTB: Load Type Value of Var. Type Elastic SxFcrSxFcr 1.03 0.09 Inelastic Fig. 10 1.06 Fig. 10 1.06 0.09 Dead 1.05 Dn 0.10 Normal Beam-column Interaction Eq 1.02 0.10 Interaction Eq. Max. Lifetime Live Ln 0.25 Type I Plate girdergirder: Flexure Mu 1.03 0.05 A.P.T. Live 0.24 L0 0.8-0.4 Gamma Mu Shear Vu 1.03 0.11 Max. Lifetime Wind 0.78 Wn 0.37 Type I CompacComnactt compositcomnosite beabeam Mu 0.99 0.08 Max. Lifetime Snow 0.82 Sn 0.26 Type II Mu Max. Annual Snow 0.20 Sn 0.73 Lognormal Notations: A, = Gross area Notations: Ae = Effective net area A.PT. = Arbitrary-Point-in-Time Mp = Plastic moment Lifetime = 50 yr. = Elastic section modulus sx Dn, Wn,Sn,L0 = Code-specified load intensities (ANSI-A58.1) Fcr = Elastic critical stress Ln =L0(0.25 + 15/V^) LTB = Lateral-torsional buckling Ai — Influence area (2 AT for beams, 4 AT for columns) Mu = Ultimate moment capacity AT — Tributary area = Ultimate shear capacity vu 78 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION Table 4. Resistance Statistics Table 5. Recommended Load Factors \ADn Type Member R VR \.2Dn+ 1.6L„ \.2D + \.6S + (0.5L or 0.8 W^) Tension member, yield 1.05 0.11 n n n 1.2Z) + \.3W + (0.5LJ Tension member, fracture 1.10 0.11 n n \.2D + \.5E + (0.5L or 0.2S„) Compact beam, uniform moment 1.07 0.13 n n n 0.9D -(\.3W or\.5E ) Compact beam, continuous 1.11 0.13 n n n Elastic beam, LTB 1.03 0.12 Notations: Inelastic beam, LTB 1.11 0.14 D = Dead load Beam-column 1.07 0.15 L = Live load due to occupancy Plate girder, flexure 1.08 0.12 W = Wind load (50 yr mean recurrence interval map) Plate girder, shear 1.14 0.16 £ = Snow load (50 yr mean recurrence interval map) Compact composite beam 1.04 0.14 E — Earthquake load Notation: LTB = Lateral-torsional buckling RESISTANCE FACTORS the two-span beam of Fig. 4, but using a more advanced 4 The loads and the load factors in the LRFD design crite method. rion [right side of Eq. (2)] are now set by ANSI, and the The resulting /3's were then examined, and on the basis task of the materials specification groups is then to develop of this across-materials survey, it was determined that, on (j)Rn values which are consistent with the target reliabilities the average, structural design in the U.S. inherently had inherent in the load factors. Reference 4 gives the proba the following reliability index values: bility-based method, as well as charts and a computer Gravity loads only: /3 = 3.0 program to facilitate this task. The resistance factors are Gravity plus wind loads: /3 = 2.5 to be developed for the load combination \.2Dn + 1.6Ln and for a target reliability index of (3 = 3.0. While there was considerable fluctuation, these values The following simple example, based on the simplified 4 turned up most frequently, and the decision was made by approach of Eq. (3), will illustrate the method. For a simply the ANSI A58 subgroup working on the problem to base supported beam of compact shape, Ref. 9 provides the the new load factors on these values of /? as target relia following statistical data: bilities. Accordingly a new cycle of probabilistic analyses was made (see Fig. 7), and the load factors listed in Table ?= 1.02, FP = 0.06, M = 1.05, 8 were finally developed; these were recommended for VM =0.1, F= 1.00, ^ = 0.05 adoption in the new ANSI A58.1 standard.2 Therefore, using Eqs. (4) and_(5), R = 1.07i?„ and VR = The load combinations are of the following general 0.13. The load effect data is: Q = l.05Dn + Ln , where Ln form: = Lo(0.25 + 15/V2^r) [Eqs^) and (13)], and VQ is given by Eq. (10). Noting that D = 1.05Z>„, VD=0A.L 1.2Dn + 7I-QI-n+ £ yQjn 05) = Ln, VL = 0.25 (Table 1), and letting where Qzn is the dominant transient load effect, with yt X = 0.25 + 15A/2l7 (16) based on the load taking on its maximum lifetime (50 yr) then value, and Qjn are the other transient load effects, with the 7/s being determined for the arbitrary-point-in-time Q = Dn (1.05 + L0X/Dn) (17) values of the loads. For example, the combination 2 2 VQ = [0.011025 + 0.0625 (L0X/Dn) }^ /Q (18) 1.2Z)n + \.(>Sn + 0AWn For the design according to the AISC 1978 Specification, stipulates the maximum snow load in the 50 yr lifetime of Part 2: the structure, while the wind load is based on the maximum Rn = \.lDn(\+L0X'/Dn) (19) annual value. The load factors and load combinations in Table 5 are where presently (1981) being balloted for adoption in the ANSI X' = 1 - 0.0008 A > 0.4 or 1 - 0.23(1 + D /L ) load standard.2 They are meant to apply to building T n 0 (20) structures made from any of the traditional building ma terials. whichever is larger. THIRD QUARTER / 1981 For the design according to the AISC LRFD Specifi Table 6. Tentatively Recommended Resistance Factors cation, Type Member 0* Rn=Dn(\.2 + \.6L0X/Dn)/(t> (21) Tension member, limit state: yield 0.90 Substituting R/RnyRn, Q, VR and VQ into Eq. (3), the Tension member, limit state: fracture 0.70 variation of /3 with the tributary area and the live-to-dead Columns, rolled W sections 0.75 load ratio L0/Dn can be determined, as seen in Figs. 8 and Columns, all other type sections 0.70 9 for a value of 0 = 0.85. It is seen that the LRFD design Beams, all types and all limit states 0.85 gives a nearly constant reliability of approximately /3 = 3.0 Fillet welds 0.75 over the whole range of parameters. H.S.S. bolts, tension 0.75 H.S.S. bolts, shear 0.65 By performing similar analyses on other types of struc tural members and components, 0-factors were derived, * These are not necessarily the values which will be finally adopted. and some of these are given in Table 6. It should be noted that these 0-factors provide roughly /3 = 3.0 for members, and /3 = 4.0 for connectors. This is consistent with the traditional practice of providing higher factors of safety for the latter. The ^-factors listed in Table 6 are not necessarily the values which will finally appear in the AISC LRFD Specification. They are at present (May 1981) still under O discussion. The point to be made here, however, is that, whatever final value they are assigned, the method pre sented here will permit an evaluation of the consequences on the reliability index /3. O 1978 AISC Specification LIMIT STATE: Lateral-Torsional Buckling O LRFD Specification, 0 = 0.85 Mr=SK(f\j-l°) I I I I I I I I I I 400 800 1200 1600 2000 Tributary Area, ft Xp=3oo//iT Fig. 8. Reliability Index for simply supported compact wide- > flange beam, live-to-dead load ratio, L0/Dn = 1.0 LIMIT STATE: Flange Local ^KMr ^v. 2looo O^ \-.^(^ V-i^|Vfy^ --o LIMIT STATE: Web Local Buckling O 1978 AISC Specification O LRFD Specification, 0 - 0.85 -L L /D _L rh'f Fig. 9. Reliability Index for simply supporte d compact wide- Fig. 10. Nominal strength of wide-flange beams under flange beam, tributary area = 400ft2 uniform moment 80 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION LRFD-type specifications which are now (1981) appearing throughout the world. These specifications employ several resistance factors and load factors to account for the various types of uncertainties which underlie design. The reliability is to be interpreted as being "notional," i.e., it is a com parative concept. It should not be confused with actual structural failures, which are the result of errors and k omissions. Only the natural statistical variation of the pa rameters is included, and, as in other traditional specifi cations, human errors must be guarded against by other SUMMARY AND CONCLUSIONS REFERENCES An attempt has been made here to summarize briefly the 1. American Institute of Steel Construction Tentatiye Spec existing development of the AISC LRFD Specification. ification for Load and Resistance Factor Design, Fabrication Emphasis was placed on the numerous sources on which and Erection of Structural Steel for Buildings Fourth Draft, this document stands. It is one of the several new Feb. 8, 1981. 81 THIRD QUARTER / 1981 2. American National Standards Institute Building Code 9. Yura, J. A., T. V. Galambos, and M. K. Ravindra The Requirements for Minimum Design Loads in Building and Bending Resistance of Steel Beams Journal of the Struc Other Structures ANSI A58.1-81 draft. tural Division, ASCE, Vol. 104, No. ST9, Sept. 1978. 3. Cornell, C A. A Probability-Based Structural Code 10. Galambos T. V. and M. K. Ravindra Properties of Steel Journal, American Concrete Institute, Vol. 66, No. 12, Dec. for Use in LRFD Journal of the Structural Division, 1969. ASCE, Vol. 104, No. ST9, Sept. 1978. 4. Ellingwood, B., T. V. Galambos, J. G. Mac Gregor, and C 11. Beedle, L. S. Plastic Design of Steel Frames John Wiley A. Cornell Development of a Probability-Based Load and Sons, 1958. Criterion for American National Standard A58 National 12. Ang, A.H.-S. Structural Safety—A Literature Rev Bureau ojStandards Special Publication 577, June 1980. iew Journal of the Structural Division, ASCE, Vol. 98, 5. Ravindra, M. K. and T. V. Galambos Load and Resistance ST4: April 1972. Factor Design for Steel Journal of the Structural Division, 13. Freudenthal, A. M. Safety, Reliability and Structural ASCE, Vol. 104, No. ST9, Sept. 1978. Design Journal of the Structural Division, ASCE, Vol. 87, 6. Benjamin, J. R. andC. A. Cornell Probability, Statistics, ST3, March 1961. and Decision for Civil Engineers McGraw-Hill Book Co., 14. Puglsey, A. The Safety of Structures E. Arnold Pub 1970 lishers, London, 1966. 7. The Nordic Committee on Building Regulations Recom 15. Vincent, G. S. Tentative Criteria for Load Factor Design mendations for Loading and Safety Regulations for Struc of Steel Highway Bridges AISI Bulletin No. 15, 1969. tural Design NKB Report No. 36, Nov. 1978. 16. Galambos, T. V. Proposed Criteria for LRFD of Steel 8. Canadian Standards Association Steel Structures for Building Structures AISI Bulletin No. 27, Jan. 1978. Buildings—Limit States Design CSA Standard S16.1- 17. Wiesner, K. B. LRFD Design Office Study Engineering 1974. Journal, AISC, First Quarter, 1978. 82 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION