Load Factor Design Applied to Truss Members in Design of Greater New Orleans Bridge No

Home , Factor of safety

Record 903 22 Transportation Research Load Factor Design Applied to Truss Members in Design of Greater New Orleans Bridge No. 2

JOHN M. KULICKI

theory' loadingt The application of load fastor design principles to the design of truss br¡dgos uncertainties in strength, di.mensions. The ¡s ¡llustrated. The recommendations presented were developed during pre- analysis, ¡naterial properties and l¡minary des¡gn of Greater New Orleans Bridge No. 2 and were appl¡ed dur¡ng factor 5/3 ís incorporated to allolv for overloads. final design. Significant sav¡ngs in construct¡on Gost ¡esulted' A specification format version of these recommendat¡ons is currently before the American The feaEure that most distinguishes LFD frorn Association of State H¡ghways and Transportation Officials Subcomm¡ttee on service load design is the use of different nulti- Bridges and Structures for possible adoption as a "guide specification"' pliers on the tlead antl live loadings. Structural nenbers designed by LFD will have a more uniforn The following general description of the loail factor capacity for live loacl (in terrns of nultiPles of design (LFD) method as it applies to beam and girder live loads) than the same members designecl by the bridges of moderate span is taken frorn the Highway service load nethod. The same is true of structures Structures Design Handbook of u.S. Steel CorPoration of varÍous span lengths. (r): Section 1.2.22 of the Anerican Association of State Highvtay and Transportation Official"s (AÀSH1[O) Members designed bY the Load Factor method are bridge specifications (2) states that "when long proportioned for nuttiples of the desígn loads' span structures are being designed by load factor They are required to neet certain criteria for clesign, the rmultipliers'should be increased if in thràe theoretical load levels: t) Maxirnu¡n Design the engineer's judgnent, anticipated loads, service Load, 2) Overload, and 3) Service Load' The conditions or ¡naterials of construcEion are cliffer- Èhe Maxírnun Design Load and Overload requirements are ent than anticipateil by the specification." In basecl on multiples of the service loads with case of long-span structures' for nost elenenÈs of certain other coefficients necessary to ensure the structure the ratio of clead load to total loail the requireil capabilities of the structure. Ser- is greater than it is in rnotlerate-length structures. vice loads are defined as the same loads as used Furthermore, the current AASHTO specifications do in working stress design. not fully treat the evaluation of truss menber The Maxinun Design Loatl criteriâ ensures Èhe capacity. Therefore' design criteria that deal with structurers capability of withstanding a few proposeil load factors and ¡nethoils of computing mem- passages of exceptionally heavy vehicles (sinul- ber caPacities are requireil before truss ilesign by Laneousty in nore than one lane), in tirnes of LFD can procee¿l. extreme ernergêncy' that rnaY induce significant permanent deformations without failure' SETECTION OF I,OAD FACTORS The overload criteria ensures control of per- ¡nanent defornations in a ne¡nber' caused by oc- The fornula for Group I "nultipliers'r, or "load casional overweight vehicles equal to 5/3 the factors", given above for beniling problerns {¡naxi- design live and imPact loads (simultaneously in rnum design loact = 1.3 [D + 5/3lL + I)l] is shown ¡nore than one lane), that htould be objèctionablê as curve A in Figure 1, which rel-ates factor of to riding quality of the structure. safety for bending and tension nernbers to the Per- The Service Load criteria ensures that the centage of totat loaal--either dead loa¿l (upper live load deflection and fatigue life (for as- scale) or live toad plus impact (lower scale). The suned fatigue loading) of a menber are controlled conventional factor of safety against first yiel-d in within acceptable linits. the service load nethod is 1.82' and this is shown Monents' shears and other forces are deter- as curve B. It has not been unco¡nrnon in long-sPan mined by assuning elastic behavior of the sfruc- bridge design to allow 10 percent overstress in trrre, except for a continuous bean of conpact members that carry nostly dead load. This corre- section where negative nonents over supports' sponds to a factor of safety of 1.65. The transi- deternined by elastic analysis, may be reducecl by tion to 10 percent allov¡able oversEress often used a maximum of 10t. This reduction' howevêr' must by Modjeski and Masters occurs vrhen the dead load is bê accompanietl by an increase in the rnaximum nore than 75 percent of lhe total l-oad. This is positive mornent equal to the average clecrease of shown as curve C. The GrouP I load factors proposed the negative monents in thê sPan. here lrere developed by starting with a line that The monents' shears or forces to be sustained would inlercept (a) the point corresponding to a by a stress-carryinq steel nember âre cornputed factor of safety of 1.65 at 75 percent dea¿l loacl and fron the fotlowing forrnulas for the three loading (b) the poinc at $thich the AASHTO service load and levels. For Group I toailing: LFD mêthods have the same factor of safeLy--i.e", 40 percent dead load. with sone rounding off, the Service Load: D+(t+I) proposed load factors result in

Maximum load 1.5 + 413 (L + I)l < capacity (1) Overload: D+5/3 (L+I) design = ÍD

Maximum Design Load: I.30 [D+5/3 (L+I) I The corresponcling overload provision is

where: D=DeådLoad Overload = D + 413 (L + I) < 80 percent of fìrst yield capacity Q) L = Live Loacl I = IÍìPact Load comparison of the maxinum design load ancl overload provisions shovts that the overload provision does The factor 1.30 is inclutleil to conpensate for not control. Transportation Research Record 903 23

Figure 1.. Load multipl¡er to first yield versus relative proportions of dead This load and live load, interaction equation contains two sirnplífy_ ing assumptions. The first is that the shape of the interaction eguation is a straight joining 2.5 2.5 Iine the PoínÈs (P = Pv, M = O) and (p O, M M^). This 'to = = AASHTO SERVICE LOAD is known be a conservative assumption lår 2.t7 I wide-flange shapes bent about their MODIFIEO SERVICE LOAO major axis anrl o 2.O rectangular shapes. AlI shapes under consideration gJ 2.O can bè considered in B this range. The second assump_ t .a2 1.82 Fg tion ís that the plastic shape factor for thê net øz \\ section Section gô ---?----e r .65 lZ, 1.7.15), the gross effective L2 ¡ .5 section, and the o6U r.5 net section with atl holes renoved F t is the same. These are reasonable assu¡nptions, co PROPOSEO ¡.3 _u, especially since the nonent portion of the interac- a'Þ tion curve is usually less than 5 percent of the AASHTO .olo FACTOR a2 t.o t-o tot,a1. ãts o Compress ion ) Two interaction equations are used, basically âs o.5 o.s discussed in Section f.7.69(B) (1) of the AÀSHTO specifications (2): GROUP I LOAD¡NG (r,/0.8s Ase n",;+(uc7u,{r - tp(Ás"XFe)l}) < r.0 (s) OEAD o"h 500/o tooo/" (P/0.8s As" + (M/Mo) (L+¡) too"/. 500h Fy) < 1.0 (6)

PERCÊNÎ OF TOTAL LOAD where

Agu = gross effective areai It is felt that the load fåctor relâtion proposed 'cr - critical load [2, Section I.j.69(A)] with here is more appropriate for those truss menbers a suitable effective length factor (K), that carry high percentages of dead load__i.e., more c= equivalent moment factor taken as 0.g5 or than 75 percent. Simitarlyr the improbability of 1.00, as appropriate; live load positioned on the structure so as to naxi_ Mu= ¡naximum bending strenqth, reduced for nize menber loads supports the somewhaÈ lower total lateral buckling as inclicateil in the next capacity required by the proposêd metho¿l for me¡nbers sect ion i that carry high percentages of live load. re - (0.85) (,r2) (s,)/(Kr/r\2 ín plane of The proposed load factors for groups other than bending; Group I have been selected to yielcl essentially the %= (Fy) (f ) (Sq.); anrl sa{ìe resul,ts as service J-oad design, as given below sgu = section modulus at end, reduced for access (case IfA is specifically intended for laúeral truss holes, if any. mernbers) ! Computation of Bending Strength Basic Factor of Safety,/ Group Group Overstress Factor Load Factor Box Members II I.82/I.2s = 1.46 I.46 (D + vr) IIA Typical box-shaped truss ¡nembers have such high 1. 6ol¡t Iateral-torsional TII I 82/I.2s = I.46 1.46 [D + (L + r) stiffness that the reduction in bending strength arising from lack of + 0.3w + wL + LFI port lateral sup_ xI I.82/I.60 is minimal. The bending capacity can be com_ = 1.14 1.14 (D + Hw) puted as follows: COMPUTATION OF MEMBER CAPACITY Mu = F, sg" {r - o.oo+r [Fy ss" L \arG/ù EAJÇ] ] gl Tension where The capacity of tension menbers is evaluâtêd by Sgu = gross effective sectíon modulus about using the tvro interaction equations shown below: bending axis, L - Iength of nember, + [P/(FyXAn)] [M/(snXFvXÐ] < 1.0 (3) s/t = length of a side divided by its thickness, A = area enclosed within [P(F"\An)] + [M/(sñ)(F")(Ð] < 1.0 (4) center lines of plates of box nembers, and Iy = monent of inertia about the nonbendíng where axis ("vertical axis"). P= factored axial load; H-Shaped Members Bent parallel Fy= yield point; Àbout Axis to Flange Aa, = (2, net area Section 1.7.15); H-shaped sections bent about their najor axis (the M= factored dead load moment; axis parallel to the flanges) are very susceptible sn= net section modulus (2, Section 1.7.15); to lateral torsional buckling. The elastic critical f= plastic shape factor computed on the basis stress at which buckling is imminent is of gross effêctive properties (t = EAv/ sq); o". (1/S*") = V[(rz n4 c q¡1xr¡2 | + IG4 h2 rjï)14 (KL)4] (s) 4,, st.atíca1 nornent of gross effectíve areasi Þq gross section modulus; where F; tensile strength of st.eel; An= net area, aLl holes re¡novedi and Sg" = gross effective section nodulus about "n net section modulus, aII holes renoved. najor axis; Transportation Research Record 903 24 provisions in moment of inertia; The service load r¡'idth-thickness Iv = minor-axis be \'triilên âs ë = shear tnodulusi Lhe AASI1TO spccificotiono cen approxi- .t = St. venant torsional constant' (16) mately t btt/3, b/t = NSL/V0.55 oc./1.25 K = effective length factor for colu¡nn buckì'lng âbout weak axisi and or h = dePÈh of web plate PIus flange thickness' (t7) o., = (1.25 NsL'?/0.ssxt/b)'? Fv, then Mu = If ocr < I/2 _õcr .PS.t of 0.85 if > r/2 Fr,'then Mu = Fy sg. fl For LFD with a maxinu¡n conpressive stress (Frl4o".)' "; I . õct, tTha"a*pr"ssion for o", above can also be used x 0.85)/0.s51 (t/b)'? Ns; (18) for modifietl H-shaPeat rneñbers conposed of two chan- 0.85 o". = N1p2 (t þ)2 = 10.2s (as flanges) anal a v¡eb plate' nels Therefore, H-Shaped llenbers Bent About Axis Parallel to I'ileb (le) NLF =v[(O.ss x 1.25)/0.ss] N5¡ about their minor axis ¿lo not H-shaPe¿t ne¡nbers bent (20) exni¡l.t laterat-Èorsional buckling, and thelr full b/i = N¡pffi". may be used' Therefore' in this ptastic capacity given in the second caset The resulting values are column of Table I along with K-values reconnended by (e) Steel Construction' the M" = 1.5 Fy Sge the Anerican Institute of resulting coeffícient, a recommenclecl coefficient Ratios for Plates (Nr.n), anil a naxivnum b/t ratio' The recommendecl b,/i' coefficients (Nl¡.) vrere selected to agree, the AASHTo buckling stress for plates can be where possible, with The coefficients in Critical elastic for solid rib arches' written as Ioad factor provisions The existing AASHTo provisions for stiffened provisions for o",=Kn2 Elt2(L-pr(bh)2 (10) plates containeat in the load factor 'cornposite box girders (adjusteil for 85- percent of million psi and u = 0'3 and ru*i*ut stress) olr Preferably, the loâd factor Substituting E = 29 are appl"icable to solving for b/E provisions for solid rib arches Yields ètiftened plates in truss ne¡nbers' (11) b/t=(sl2o\aKy\6; Fatigue Design defined by this equation AAsHTo shifts the curve as in conventional to account for the observed behavior of plates' Fatigue design proceeds exactly which lndicates Lhàt residual strc66e6 an¿l out-of- service loail dlesign. flatness reduce the strength of plates of inÈernedi- ate slenderness below that which would be in¿licâted Connection Desiqn by sinple elåsÈic stability analysis' This shift is by multiplying the equätion above by The loa¿l factor atlowable stresses are taken from "ã.o.pii"n"a0.6, which results in Sections 1.7.?f(A) and L.7.72(C) of the AÀSHTO specifications except as nodified below' The overÌoad blt=Qo12flnÆ- (12) provision, Section L.7.72(Cl, will control the design ot fricEion joints. The corrêsponding clesign proposeil load For. the case of a simply suPporte¿l plate' the ¡nini- caóacities deterninecl by using the num value of K is 4.0. This value of K and the factors are as follows: of a factor of safety that results in a introduction + + R/31 $rorking stress of 0.55 o"r/I.25:IíeLð Group I: I.s tD + 4/3lL I)l = I.5 tl lF.,l Im] [a] bft=arll3lJw (13) Group II: t.¿6 tD + wl = 1.46 Fv tml [a] GToUP IIA: I.60w = 1.60 Fv [n] fal + + 0.3W +lrlL + LFI = For mai¡r plates of truss ncmbera, the AASHTî speci- Group fTT: 1.46 tD + L I fications use I.46 Fv [m] [a] Group XI ! I.14 tD + Iil{l = 1.14 Fv tml tal (14) btt = 4o}OA/oq The value of F' is obtained from Tables t'7'4ICl For LFD, the ¡naximttm cnmpressive stress is 0'85 and I.7.41C2 in the AASITTO sPecifications, m is the o.rr which leads to nu¡nber of bolts, a is the area per bolt' and R is the ratio of live load and impact force to total (1s) on btr = s66ol\/;; force. A procedure could also be developed based altowing friction bolts to slip into bearing at The èxact values of K, to be used for plate con- factoreal loa¿ls. other conilitions of support' The ilesign of wel¿ls is based on AASHTo Section ponents of ¡nembers for design allow- are funcÈions of the ¿legree of suPport, which will l.7.lL(2). No nodtification of state¿l vary fron nember to nember' Actually, Èhe plate ables is envisionecl at this ti¡ne. nenbêr is a characterístic sirångttt of a fabricated Pins of the whole cross section, not of ân indiviilual Evebar piate. The existing coêfficients for b/t ratios for pins not by The proposed aLlowable bearíng stress on lruss ¡ne¡nbers are the Product of theory tempered Th.is val-ue is nonideal charac- subject to rotation is 1.35 Fv' experience ancl allowances for many Brirlge Design Code (Ð, plates in me¡nbers. Therefore, the pro- basád on the ontario Highway teiistics of a value of I.50, in which 0 = 0'9 fôr cedure described belovr has been used in developing which uses b/t reguirements. steê1. Transportation Research Record 903 25

Table 1. Determination of service load width-thickness rat¡os. Type of Plate Coefficient K-Vatuea Coefficient Nlr, Max b/t Ratio Main s560 4 s 660 Cover 5700 45 6950 5 OJJS Perforation 67 50 50 Basic plate 8 340 6.97 7477 8000 Edge of perforation 55 2260 0.7 2370 2200 nlt6b âRecommended bMain by AISC. membersTsecondary members.

Figwe 2, lnteraction cürves for p¡ns subiected to shear and moment, shear. The progression of yield for each of these points was fron the top and botton = o (DRUCKER) tora,ârd t.he middle rþ-("r/ in order of layer position. The r.o ' von Mises yiêld criterion was used in these conputations because Ít is incorporated into the existint AASHTo provisions load factor t*'(ü)' = l.O for girder design. The calculations vrere repeated with the Trescä yield criterion and little difference h'as observed. The von Mises yietd cri_ terion is o2 ¡ 3r2 = ou2i the Tresca cri_ u /v\3 terion is o2 + 4r2 x =" = o.,2. -t'.1%/ 4. points narked # ,.r" obtained for a círcular + shåpe by using the same conputer progran ( PROPOSÊD ) for ratios è of shear and moment, which caused the progression of o.5 plasticity procee¿l = + to either fron the rniddle layer = out Eo both edges in order of layer position or to start at the middle and then proceed to total plas_ + tification in an order that did not bear any rela_ ( HODGE tion ) to the order of the layer position. This i¡n_ plies a discontinuous strain field, a phenonenon that can exist in plastic flow. These points are regarded as informative but less reliable than the points rnarXed $ because not al_l implications of the discontinuoús strain field on o (i.e., the type of bound o o.5 t.o upper or lower) have been evaluatáá. 5. The radial_ line in Figure 2 represents the division between ra!ios of shear and noment for which first yield occurs in shear or bending. Above o APPROXIMATE LOWER EOUND SOLUIION -CIRCLE the radial line, first yield results from bending, o APPROXIMATE LOWER BOUND rå! SOLUT¡ON.SOUARE below it, first yield results from shear. Y MORE EXACI SOLUTION. CIRCLE 6. Plasticity theory NOOITIO¡¡ÂL indicates that the failure + SOLUT¡ONS FOR A qIRCLE criterion must be convex. Therefore, the the. shape of -i1[eractíon curve from the lowest point ¡nark-e

Table 2. Comparâtive chord desiqn$.

Dead Load Without 10 Perænt Overstres With l0 Percent Overstr€ss Percentage Service Load Type of Total Service Load Membe¡ Service Load Area/Load of Service Load Service Load Area/Load Factor Ar€a Design Area Factor Area Chord No Typeu Steel Design LFD Area Design Area NA 68 287.85 326.61 1.13 NA Top NU9.NU7 T 42t.99 1.05 T ^572A572 76 402.60 464.19 l.l5 NU7-NU5 525.78 I .05 NU5.NU4 T 81 498.60 578.35 1.16 5 35 .28 1.05 T A572 81 5 l2.10 588.81 1.15 NU4-NU2 ^572 l.l7 445.87 LO7 T A572 86 418.35 490.45 CIJ21U4 437.86 1.08 T A572 86 404.85 481.ó5 1.19 cu4.cu5 272.55 l.l0 CU5{U7 T 4572 86 247.98 299.81 L2l 1.15 352.36 1.05 C A588 85 335.06 386.95 SU3-SU5 Ll4 429.41 1.04 su5-su7 C A588 85 4t2.O6 469.88 I .16 528.36 1.06 AU2.AU4 T A572 82 499.35 581.19 1.t7 5 17.60 t.07 AU4-AU5 T A572 82 485.8 5 5 69.35 t .t7 398.53 1.06 AU5.AU7 T 11572 79 37 6.23 438.39 11 357.06b 1.03 NA NA Bottom NL8-NL6 C A514 346.59b '18 436.94 l.l0 398.69 1.00 NL6-NL4 C A514 398.22 47 5 .66 1.03 A5 14 84 462.75 523.69 l.l3 NL4.NL2 C l.l3 475.66 1.03 NL2.NLO C As 14 84 462.7 5 523.69 497.81 l.l4 449.84 1.03 NLO{L2 C 4514 86 436.94 497.8t. l4 449.84 1.03 cL2-CL4 C A514 86 436.94 l 35 7.06D 1.07 354.66b 1.06 cL41L6 C A514 86 3?3.69b 1.18 270.41 1.07 SL4-SL6 T A572 85 251.85 297.45 324.58 1 18 295.07 |.07 SL6-SL7 T 85 27 5 .85 523.69 l.l3 475.66 1.03 ALO.ALz C ^572A5 14 84 462.7 5 1.1 3 47 5 .66 1.03 ALz-AL4 c 4514 84 462.7 5 523.69 l.l3 382.8 1 t.o2 AL4-AL6 c A5 14 80 37 6.38 424.03 1.03 33 3.69b 1.00 AL6-AL8 c A514 76 ß3.69b 344J9b

Nolei T = tenslon and c = comp¡esion. alf is given' tension member, net a¡ea is given; lf compresslo¡ membe¡, Sfoss Ùea bMembe¡ desigt limited by b/t tequirements.

criterion results in a range from shoutd be evaluateal by using the following equations: percent overstress 1.00 to 1.10 and an average of l'05' Me = (D3/o Qr) Gy) REFERENCES

= (1TD2 (FyYVi Q2) vp 14) 1. Highway Slructures Design Handbook' U'S' SteeL Q3) Corp. Pittsburgh' PA, n.d. (M/Mp)+(v/vp)3< o.9s 2. Standard' Specifications for Highway Briilges, I2th ed. AASHTo, washington' DC, ].977 (interi¡ns through 1982). 3. Ontario Highway Bridge Design Code' Ontario Ministry of Transportation and conmunications' CONCLUSIONS Downsview,1979. Shear on the Plastic members an¿l 4. D.c. Drucker. The Effect of Table 2 cornpares designs Ôf 25 chord Appl-ied llechanics' possible vrith strength ¿le- Benclinq of Beams. Journal of illustrates the savíngs of Mechanical Engineers' Dec' sign. The ¡nenbers shown were generålly controllêd American society by strength requirenents rather than fatique or 1956. In the 5. P.G. Ho¿lge' Jr. Interaction Curves for Shear and nìnimurn plate sizes; exceptions are noted' of Applied both design nethods $¡ould result Bencling of Ptastic Bea¡ns. Journal latter t$ro cases, American Society of ÞlechänicaI Engi- in the same design. Comparison of thê results in Mechanics, which the l0 percent overstress service load cri- neers, sept.1"957. terion was noÈ invoked shows that the ratio of ser- tFD area ranges fron I'03 vice-loacl-design area to Bridges' to 1.21 and averages 1.14. Inclusion of the I0 Publicøtion of this paper sponsored by C.ammíttee on Steel