CHAPTER 6 RESISTANCE FACTOR FOR THE DESIGN OF COMPOSITE SLABS

Transcription

1 CHAPTER 6 RESISTANCE FACTOR FOR THE DESIGN OF COMPOSITE SLABS 6.1. Geeral Probability-based desig criteria i the for of load ad resistace factor desig (LRFD) are ow applied for ost costructio aterials. The desig requireets have to isure satisfactory perforace of structures. The ai advatage of the approach is the ability to achieve a uifor level of reliability for structural ebers, or to ipose a certai level of reliability (higher or lower) of soe certai parts of the structures. This gives a strog ratioale to the load ad resistace factors as copared to the desig safety factors of the allowable stress desig. Additioally, a uified desig strategy as to settig up coo load cobiatios ad load factors ca be obtaied. I this part of the study, resistace factors, φ, for the flexural desig of coposite slabs were evaluated based o test data of 39 full scale coposite slab species. The tests were perfored at the Structures ad Materials Laboratory of Virgiia Polytechic Istitute ad State Uiversity, Blacksburg, Virgiia. The φ factors evaluated correspod to the SDI-M ethod ad direct ethod described i Sectio Review of Probabilistic Cocepts of Load ad Resistace Factor Desig Discussios o the probabilistic cocepts of the LRFD approach are give i detail by ay sources (Corell 1969, Lid 1971, Ag ad Corell 1974, Galabos ad Ravidra 1977, Ravidra ad Galabos 1978, Elligwood et al. 1980, Load ad 1986, Hsiao et al. 1990, Chapter 6. Resistace Factor 71

2 Geschwider et al. 1994, Coetary o 1996, Barker ad Puckett 1997). I priciple the followig iequality applies: φ R γ i Q i i (6-1) i which, R = oial resistace, Q i = load effect, φ = resistace factor ad γ i = load factor. The probability of failure ca be expressed by: p f = 1 Φβ ( ) (6-2) where Φ is the stadard oral probability fuctio, ad β is the reliability idex Reliability Idex The reliability idex, β, used i Eq. (6-2), i a log-oral forat, ca be expressed by: β = λ ζ R λ Q R l Q 2 2 R + ζq VR VQ (6-3) where λ, ζ ad V, respectively, deote the log-oral ea, log-oral stadard deviatio ad the coefficiet of variatio. Subscript R ad Q deote the resistace ad the load effect, respectively. R ad Q are the eas of resistace ad load, respectively. Itroduce a liearizatio give by: ( ) 2 2 R Q Q V + V = α V R + V (6-4) the Eq. (6-3) ca further be approxiated as: Chapter 6. Resistace Factor 72

3 β = α l Q R ( VR + VQ) (6-5) where accordig to Lid (1971), for / 3 V / V 3, α = 0.75 gives a good approxiatio 1 Q R with ±6% axiu error. Equatio (6-5) fors the basis equatio for the AISC ad AISI load ad resistace factor desig specificatio for structural steel ad cold-fored steel. Fro Eq. (6-5), the cetral safety factor ca be expressed as: θ = R Q = e αβ ( VR+ VQ) (6-6) By iiizig the error of this cetral safety factor, Galabos ad Ravidra (1977) suggested a value of α = 0.55 which was later adopted i AISC LRFD. The reliability idex, β, ca be deteried fro Eq. (6-5). As a illustratio, the followig table shows soe β values ad the correspodig probability of failure, p f. Table 6-1. β vs. p f β p f x x x x 10-2 AISC-LRFD uses the followig β values: β = 3.0, for ebers, uder DL + LL or Sow β = 4.5, for coectios, uder DL + LL or Sow β = 2.5, for ebers, uder DL + LL + Wid β = 1.75, for ebers, uder DL + Earthquake Chapter 6. Resistace Factor 73

4 whereas the AISI-LRFD uses the followig β values: β = 2.5, for ebers β = 3.5, for coectios Galabos et al. (1982) give β values for various structural ebers uder coditios of ratio of basic specific live load to oral value of dead load equal to 1, 2 ad 5. Rages of β values were also give by Elligwood et al. (1980). These values of β rage betwee for reiforced cocrete ebers ad for steel ebers AISC LRFD Approach for the Resistace Factor Usig the cetral safety factor give i Eq. (6-6), the followig iequality ca be writte: R θ Q (6-7) which leads to: R αβvr e Q e αβv Q (6-8) or, φ R γ Q (6-9) i which, R ad Q are oial values of resistace ad load, φ αβ = R V e R R (6-10) γ αβ = Q e Q V Q (6-11) Further, the ea resistace, R, ca be expressed i ters of the oial resistace ad statistical paraeters that represet the variability of aterial stregth ad stiffess, M, Chapter 6. Resistace Factor 74

5 fabricatio, F, ad the ucertaities ivolved i the assuptios of the egieerig desig equatio, P (Ravidra ad Galabos 1978): ( ) R = R M F P (6-12) where M, F ad P are the eas of M, F, ad P, respectively. Accordigly, the coefficiet of variatio of the resistace ca be approxiated by usig: R M F P V ( V ) + ( V ) + ( V ) (6-13) i which, V M, V F ad VP are, respectively, the coefficiets of variatio of M, F ad P. Here, Eq. (6-13) assues idepedet relatios aog M, F ad P variables. By usig Eq. (6-12), the resistace factor give by Eq. (6-10) ca be odified to: φ ( ) = M FP e αβv R (6-14) AISI LRFD Approach for the Resistace Factor The AISI specificatio for cold-fored steel structures follows a differet approach i deteriig the resistace factor, φ. The approach is based o the research by Hsiao et al. (1990). Istead of usig Eq. (6-10), it starts by expressig the effective resistace i ters of the oial loads ad load factors ultiplied by a deteriistic coefficiet, c, that relates the load itesities to the load effect ad is give by: D φ R = c ( γ D D + γ L ) = γ D γ L + L L c L (6-15) where γ D ad γ L are the dead ad live load factors, ad D ad L are the oial values of the dead ad live load. Siilarly, the ea of the load effect ca be expressed as: Chapter 6. Resistace Factor 75

6 Q = 1.05 D L + 1 c L (6-16) Notice that i the last equatio, D = 105. D ad L = L were used (based o load statistic by Elligwood et al. 1980). Fro Eqs. (6-15) ad (6-16), oe obtais: R Q = ψ φ R R (6-17) with, D ψ = γ + γ / 1.05 D D L L L + 1 (6-18) By cobiig Eqs. (6-3), (6-12) ad (6-17), a expressio of the resistace factor ca be obtaied: φ ψ( F P ) = M e -β V 2 R V Q + 2 (6-19) Usig this equatio, deteriatio of the α coefficiet ca be avoided. However, the coefficiet of variatio of the load has to be kow Statistical Data Evaluatio of the resistace factor, φ, as give by Eq. (6-14) or (6-19) requires statistical values of the paraeters ivolved. These data are available fro the lab tests coducted o the coposite slab species previously etioed. However, larger sets of database are preferred to give ore represetative values of eas, stadard deviatios, ad coefficiets of variatio of the afore-etioed paraeters. Therefore, statistical values fro other sources that were based o larger sets of database were used. These values are the statistical values of the cocrete copressive stregth, f c ', which was based o the study by Chapter 6. Resistace Factor 76

7 MacGregor (1997), ad steel deck yield stress, f y, which was based o the study o cold-fored steel ebers by Hsiao et al. (1990). For these two paraeters, the data obtaied fro the lab tests fro the coposite slab species were used as a copariso oly. Data obtaied fro the lab tests, which are ot available elsewhere fro larger sets of database, were used for the deteriatio of the resistace factor. These data are the statistical data of deck thickess, t, axiu ad iiu shear bod stregth at the iterface of steel deck - cocrete, f s,ax ad f s,i, respectively Material Factor, M The aterial factor, M, represets the variability of the stregth ad stiffess of the aterial. I this case, M is affected by the variability of f c ', f y, f s,ax ad f s,i data of these paraeters are listed i Table Statistical Table 6-2. Statistical data of f c ', f y, f s,ax ad f s,i µ σ V fc' (MacGregor, 1997) 3940 psi 615 psi fc' (test) 3867 psi 878 psi fy (Hsiao et al. 1990) fy fy fy (test) fy fy fs,ax fs,ax fs,ax fs,i fs,i fs,i Note: µ = ea, σ = stadard of deviatio, V = coefficiet of variatio Assuig that those paraeters are statistically idepedet, coefficiets of variatio of the aterial factor ca be approxiated by: V = V + V = MSDI, f c ' f y (6-20) for the SDI-M ethod while for the Direct ethod: V = V + V + V + V = (6-21) M, Direct f c ' fy fs, ax fs, i Chapter 6. Resistace Factor 77

8 for the SDI-M or direct desig procedure, respectively. The ea values of M for the SDI-M ad direct desig procedures ca be evaluated fro: fc ', y, f M c' SDI, = = fc ' f y fc' M, Direct = f f c', c ' f f f y, y µ µ f fy y = ( f ) ( f ) s,ax f s,ax s,i f s,i (6-22) µ µ f µ µ f f f fc ' fy fs,ax M, Direct = c ' y s,ax' fs,i s,i = (6-23) Fabricatio Factor, F The fabricatio factor, F, represets the variability of the aufacturig process. I this case, the variability of the steel deck thickess, t, is cosidered. The statistical data for this steel deck thickess are listed i Table 6-3. These data were based o the easureet coducted o the steel decks that were used for the coposite slab specie tests. Table 6-3. Statistical data of t µ t σt Vt t t Based o the above statistical values, V F ad F ca be coputed as follow: VF = Vt = (6-24) F = t t = µ, t t = (6-25) Professioal Factor, P The professioal factor, P, takes ito accout the ucertaities of the desig equatio. This professioal factor is defied as (Ravidra ad Galabos 1978, Geschwider et al. 1994): Chapter 6. Resistace Factor 78

9 P = test predictio (6-25) The predictio is the resistace of the slab as predicted by the desig equatio based o the easured (actual) values of its paraeters. Based o the lab tests perfored o the aforeetioed full-scale coposite slab species, the followig statistical data is obtaied: Table 6-4. Statistical data of P µ p = P σp Vp SDI Direct Load Statistic Iforatio regardig statistical data of the load i ters of the coefficiet of variatio, V Q, is eeded for the AISI approach as show i Eq. (6-19). For this reaso, statistical data of dead ad live loads were take fro a special publicatio of the Natioal Bureau of Stadards (Elligwood et al. 1980). These data are suarized i Table 6-5. D ad L deote the oial dead ad live loads. Table 6-5. Statistical data of dead ad live loads µ σ V D 1.05 D D 0.10 L 1.00 L L 0.25 For the cobiatio of the dead ad live loads give by: Q = γ D D + γ L L (6-26) the ea ad stadard of deviatio of this cobiatio ca be expressed by: µ γ µ γ µ Q = + D D L L (6-27) Chapter 6. Resistace Factor 79

10 σq = Var(Q) = γ D σ D + γ L σ L (6-28) assuig that the distributio of D ad L are statistically idepedet. I Eq. (6-28), Var(Q) deotes the variace of Q. By substitutig values fro Table 6-5 ito Eq. (6-27) ad Eq. (6-28), the coefficiet of variatio of Q ca be obtaied as: V Q 2 D 2 D = γd γ L / 1.05 γd + γ L L 2 L (6-29) 6.4. The Resistace Factor I this study, the AISI-LRFD approach is adopted. The AISC-LRFD approach is used to give a copariso. Cosiderig the fact that coposite slabs are geerally used i steel fraed structures, a β (reliability idex) value greater tha 3.0 is ot cosidered ecessary (β=3.0 for steel ebers). Therefore, β=3.0 is chose as the target reliability idex (AISI uses β=2.5 as the basic case). The fial result of φ factors, however, is rouded to the closest 0.05 ad hece, the actual β values used will ot be exactly 3.0. A iiu liit of β=2.5 is used. A load cobiatio with γ D = 1.2 ad γ L = 1.6 as give i the SDI Coposite Deck Desig Hadbook (Heagler et al. 1997) is used as the basic load case. The cobiatio usig γ D = 1.4 ad γ L = 1.0 is ot cosidered because the ratio of dead to live load is typically < 3.0 for coposite slabs. A rage of dead to live load ratios betwee 0.5 (short to oral spa slabs with relatively heavy live load, approxiately 100 psf) ad 1.5 (log spa slabs up to 20 ft with relatively light live load, approxiately 50 psf) is cosidered. Based o the statistical data preseted i sectio 6.3 ad equatios give i sectio 6.2, φ factors for several values of D/L (0.5, 1.0 ad 1.5) were coputed ad the results are listed i Table 6-6 ad Table 6-7 for the SDI-M ad direct desig procedures, respectively. Agai, these results are based o the AISI-LRFD approach preseted i sectio Chapter 6. Resistace Factor 80

11 Table 6-6. Calculated φ factors for SDI-M ethod (AISI-LRFD Approach) β D/L Table 6-7. Calculated φ factors for direct ethod (AISI-LRFD Approach) β D/L Based o the results i Tables 6-6 ad 6-7, φ = 0.90 is chose for the SDI-M ethod ad φ = 0.85 is selected for the direct ethod. For copariso, φ factors coputed by usig the AISC-LRFD approach are listed i Table 6-8 for the SDI-M ethod ad Table 6-9 for the direct ethod for several cobiatios of α ad β values. This later approach is ot iflueced by the ratio of the dead to live load (D/L). As show i these tables, the choice of α betwee 0.65 to 0.75 show relatively close results to the AISI approach. Table 6-8. Calculated φ factors for SDI-M ethod (AISC-LRFD Approach) α β Table 6-9. Calculated φ factors for direct ethod (AISC-LRFD Approach) α β Chapter 6. Resistace Factor 81

12 6.5. Cocludig Rearks Resistace factors for the flexural desig of coposite slabs based o the SDI-M ad direct ethods have bee preseted. The AISI LRFD approach for evaluatig the resistace factor was adopted. By this approach, the deteriatio of the α coefficiet is ot ecessary. A target reliability idex β=3.0 ad iiu liit of β=2.5 were used. This choice was based o the target reliability β=3.0 for steel ebers (AISC-LRFD), ad the lower boud β=2.5 used i AISI-LRFD for the basic load case. The resultig resistace factors are φ=0.90 for the SDI-M ethod ad φ=0.85 for the direct ethod. These φ values were based o a rage of dead to live load ratios betwee 0.5 ad 1.5, which is represetative of typical coposite slab desigs. Chapter 6. Resistace Factor 82