Steel Box Girder Bridges Design Guides & Methods

Transcription

1 Steel Box Girder Bridges Design Guides & Methods CONRAD P. HEINS IN MEMORIAM CONRAD P. HEINS September, 97 December 4, 98 During the past decade, there has been extensive use of steel box girders for straight and curved highway and transit structures. ' 4 To meet the need for use of such structural elements, design criteria had to be established. Therefore, the purpose of this paper is to present information relative to the design criteria in addition to information on preliminary plate sizes, design aids, and computer-aided design of steel box girder bridges. INTRODUCTION Box girders have become a prominent element in the construction of major river crossings, highway interchanges, and transit systems. These types of structural elements are particularly attractive because of their high torsional stiffness, which is required when the bridge is curved. With the advent of these bridges, appropriate design specifications ' ' design guides 5 ' 6 ' 7 computer solutions 8 ' 9 are required. Here is a summation of this information: DESIGN SPECIFICATIONS There are at present a set of standard specifications, which pertain to straight box girders for highway bridges. Guide specifications are also being used for curved box structures, but to date have not been incorporated into the standard code. Further research has also been conducted, which has resulted in a tentative strength or load factor design code for curved bridges. All three of these codes ' ' have been studied and the appropriate criteria for the design of each element of a box (i.e., top flange, bottom flange, web) categorized according to working stress method or strength method as given in Tables and. The working stress criteria has recently been incorporated into a design oriented computer program. 9 In addition to these basic specifications, ' ' a new code 4 has been proposed for consideration, but has yet to be adopted. DESIGN GUIDES Flange Areas In the design of any complex structure in which the section changes and the forces are not readily computed, it is useful to have data or empirical equations to select plate geometry, which can then be incorporated in a computer program 9 to automate the bridge design. Such information has been developed 5 ' 6 ' 7 and has resulted in the following: i) Single-span bridge ii) A B = \?>d\\-j\ Two-span bridge A + B=- (0.005L - 0.L + ) k A- B = \ M A ^ Conrad P. Heins was Professor', Institute for Physical Science and Technology and Civil Engineering Department, University of Maryland, College Park, Maryland. This paper is the 98 T R. Higgins Lectureship Award winner. A\ = 0AAA + B A- T = \MA%^ ty FOURTH QUARTER / 98

2 Item Table. Working Stress Design Requirements Straight Curved Compression Flange (positive moment) 'H i b 4400 t = y/t y and F b = 0.55 F y r //\ i lr/ 4ir E PBPW and p w = p w \ or p w, ; + U) w Pw\ =". - ( ^ - (W 75 l/b [ (0. -//fl) ] Pw : +0.6F if (+) use smaller p w \ or p W fb J w / \ (-)usepa,i Jb Compression Flange (negative moment) b ^ 640 < = \f~fy Jb tk 0.55 F y 640 6,00 - ^ 60 or i=r t y/fy f b * 0.S5F y -,00 b - J =r < - ^ 60 y/fy t 0.4F y f b ^ 57.6 X 0 6 tv TT/,00-6/^ sin \ 760 V^jj t ^ VTy x F b = 0.55 F y ^ - X = + % ,00 = < - ^ _ or 60 /, 0.5 U l f 7r/,00-6v / F/^ sin l I,00-640X if 6,00 Fb is smaller of the following: F b = ) -0 6 A ft = X ma Jv -X 0 6 [bf ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

3 Table. {continued) 070 y/k \/T y fb ^ 0.55/*; Straight 070 y/k W 6650 y/~k = < - s 60 or -=ry/fy t yjfy f b * 0.55/; - 0.4/*; 6650 y/k w?± < - * y/k-- y/t y y t sin y/k 070^ \Z~Fy *i ^=0.55^y^9^) X\ = \(n>\) X x = m*'» ^K ^4 Curved (/,/W ) / K> =- (n+ l) 5.4 fb 4.4 X \0 6 K \wl Stiffener requirement with longitudinal Stiffener 0 = 0.0 K^n 4 for n > 0.5AT forrz = 070^* w 66b0y/K _ V^V / \/Fy F b = 6650 y/k X ~ y/f y 0.6/*; + 0.4/*; { sin -=. *-= y y! 6650 V ^AT - 070y/KX^ Where A X = Ht 6650^^ w < s 60 /*"/, is smaller value of F h = 4.4/T - A0 6 ^ = 4.4AT - x 0 6 /, ^ i4.4(/:,) l-l -X 0 6 FOURTH QUARTER / 98

4 Table. (continued) Straight Use same formula, but use K] instead ofat Curved A'i=a\. (n + \) H + 0.(n + l) I, > St u ^Af J 7/) J It > 0.0(n + l) a/ - E a a: spacing of transverse stiffener f s : maximum longitudinal bending stress Af Area of flange including longitudinal stiffener - < 50 Same 5.65 X 0 7 F v f < < -I Jv (d/ty d,000. -< _ < 70 t VT h d 0 < l.sd c ( 0.87( - C) ^ [ Vl+WoA0 J.X0 8 [l+(^o) ] <.0 Fy{dA) do = stiffener spacing If do/r < 0.0 use straight girder criteria Ifdo/R >0.0 d,000 _ t < l F b d 0 < \.Sd U" F v U9_ 0, j + 4(^ 0.87( - C) C + VI + (da/d) \ < 70 c =.XlQ8 l+(^o), iq Fy(d/t) Stiffener Criteria 0.9 J = 5 J - 0 > 5.0 Stiffener Criteria I > d 0 t* J / = do X =.0 for < 0.78 d X>5.0 X =.0 + [do Z 4 ;0.78 < <.0 d d Z = 0.95 Rt b 600 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 4

5 Table. Strength Design Requirements Straight b 00 t " \/T y fb = F bs p B PiL Curved PB + i(, + ±) (-00 6 \ 66/ U = R F bs = i7(l - X ) U\ [Fy_ 7T \6j 00 b 4400 if < - < =z VTy t y/fy fb < F b y Fby = F bs p B P PB Pu/ = [o.-0.-^ P B Fy/F bs - and p w = p w \ or p w, ; I I +-- Rb l-^(l--^ /A \ 756, // (0. -l/r) P : + 0.6(/ // 6 ) b ^ p= then F cr = ' VTy 640 6,00 < : < - Fy t VFy then ^cr = 0.59 F y \\ sin -,00-- C - / 760 b,00 (+) use smaller p^i or p w fb Jw.. (-) use p wx Jb F y b R x f v^ -L and - ^ i= V f V^; then F b - F y A #i 6 /? -7= < ~ ^ =z or 60 V^ / y/fy thenf 6 =6. X 0 6 W- - also if flk 6. X 0% -j thenf cr = 05 X 0 6 (;/6) V ' = VFy F b = FyA A= V 5 FOURTH QUARTER / 98

6 Table. {continued) Item Compression Flange (negative moment) With Stiffener t.j JUL. b Compression Flange (negative moment) With Stiffener 070VT t VT < 070VT w 6650y^ y/fy C = ' Straight 0.59 F v sin 6650VT - - y/fy y / 6650VT VTy 580VT F cr = 6.X\0 6 K(t/w) Is ^ 4>t w Stiffener Criteria 4> = 0.07/s: n 4 forn =,,4,5] 0.5AT forr = J and b',600 I'' < VTy 6': depth of stiffener f: I slate thickness of stiffener Curved 070v^ /?! = i + A '"VM' ra + 4 I R VT ViM 4 - o W <A - - 4>!+4 tef(?j =«F A: = 4 AT, = 5.4 / < *= and - < = Vl t VTy thenf b =F y A Ri w R =z < - < == or 60 VTy t VTy /?-" uvty\ then F b = F y < A { - sin -I Ro-Ri R <w < 60 F 6 = 6. X0 6 AT - - F F also if -~ < j v < L and - ^* * " VTy F b =F y A \/ V -flk 6. X 0 6 AT, - \wj R\,R and A are given under compression flange without stiffener section and <K< (Vw* ) / A, = ^ < 5.4 (n+) /, > (/)t W : (f) = 0.07 AT n 4 for n > 0 = 0.5 K*n = 6,600 t ' VTy Stiffener Criteria ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 6

7 Table. (continued) Item Straight Curved Without Stiffener < 50 V u <.05 X low/d or d -< 50 d With Transverse Stiffener V u < 0.58 Fydt d 6,500 </o^l.5</.5 P V u < V u d -< 50 d 0 <\.5d <0.5SF y dt or 0.87( - C) K<F C + Vl+(<V</) J : V < 0.58 F y dtc : F p = 0.58 F y dt C= 8,000 + (d/d 0 ) -0.<.0 C= {8,000 (*/</) V Wo) - 0. <.0 With Transverse and Longitudinal Stiffener m d 7,000 - < t VT y d 0 < \.5d and longitudinal stiffener is d/5 for compression flange. Shear requirements in accordance with transversely stiffened web criteria. 6,500 y/ty,- 8! +M (f d 7,000 < - < -.9 t yffy! " d 0 <.5 d and longitudinal stiffener is J/5 for compression flange shear requirements in accordance with transversely stiffened web criteria. Transverse Stiffener Criteria Longitudinal Stiffener Criteria b/t :, 600 WTy b = projected width of stiffener and the gross area is,4 > [0ASBdt(\ - C)(V/V U ) - \St ]Y :.6 =.0 for stiffener pairs B =.8 for single angles B =.4 for single plates C = 8,000 (*/</) A / ^ ^ ^ ^ and F u = as given previously Y = ratio of web plate to stiffener plate yield strengths / > d Q tv J =.5(d/d 0 ) - > 0.5 V_ 600 t' ~ y/ty I = dt*.4 ^-0. Same as straight except / = [.5(d/d 0 ) -}X< 0.5 X =.0 when (d</d) < 0.78 and ido/d ] A d 0 X = \ + { Z 4 when 0.78 < < d " 0.95Jp Z = /ft Same criteria as straight,000. S t = Section modulers of transverse stiffener S s Section modulers of longitudinal stiffener 7

8 iii) Three-span bridge A% = {U - 7) 6.4* n A B=TZ ( L i - 5) 5k A T = n (Li - 00).6 AB = (0.964L -.65Ll kn X0' - _ 70) A~r = 0.95Ar - 0.0U(Ar) \ - 5.4/6 Ai = Wke->-*«> : k = N BF y d w R X 600? Exterior section Support Interior section F y = yield point of material at specified section (ksi) L, L\ 9 L = span length (ft) WR = roadway width (ft) NB = number of boxes d = girder depth (inches) n = L^/L\, L\ = exterior span, L = interior span A%, A j = total top flange area (in. ) in positive or negative moment region A~B, AB = total bottom flange area (in. ) in positive or negative moment region Box Girder Geometry To select the final cross-sectional dimensions of a box girder bridge, along its length, many designs are required. To facilitate such designs, a study 6 was conducted to optimize the cross sections of single, twoand three-span straight box girder bridges. The specific geometry associated with these bridges are:. Parametric details Span length single-span: L = 50 ft, 00 ft, 50 ft two-span: LI = 50 ft, 00 ft, 50 ft L = N.L\ y N=.0,.,.4,.6 three-span: LI = 50 ft, 00 ft, 50 ft L = JV.L,AT=.0,.,.4,.6 L equals end span for two span or L equals center span for three span symmetrical bridge. depth: d/l < V5 Top flange: b/t <. (positive moment region) Bottom flange width: 80 in., 00 in., 0 in. Bottom flange stiffener: ST 7.5 X 5. (negative moment region) Concrete slab 8.5 in. Steel type ^46, F y = 6 ksi N = 9,tf = 7,/c = 4ksi Unit weight: steel 490 pcf, concrete 50 pcf General parameters parapit: 00 lbs./ft wearing surface: 5 lbs./ft miscellaneous concrete: lbs./ft miscellaneous steel: %. Procedure The determination of the correct plate geometry the various bridges, involved the following pre dure: Fix span length L Select web depth d = L/5 Select bottom flange width W = 80 in. Select web thickness Select top flange width b < t Determine dead-load moments Determine location of cross sectional chanj using data given in Tables and 4 and Fig. Revise sections and computed dead-load, live- forces and stress. Revise per specifications. Set bottom flange width W = 00 in., repeat.. Results The procedure outlined above was followed for design of 8 bridges. The results of these designs single, two span and three span bridges are tabuh in Tables 5, 6, and 7. Bracing Requirements The required cross diaphn bracing area, 0 as shown in Figs. and, can be de mined from the following; Sb t*,. 9N - (in. ) *6 ^ 750 d (d+b) s = Diaphragm spacing (in.) b = Width of box (in.), at bottom flange d = Depth of box (in.) * = n/ J = weighted section thickness (in (d -r b) A = Total cross sectional plate area (in. ) ai diaphragm location Ab = Required area of cross diaphragm bra( (in.) The bracing spacing requirement is given by the lowing: / R \V s - XL h^r ^r\ - 0 in - \00L / ~ L = Span length (ft) R = Radius of girder (ft) Top lateral bracing is utilized in stiffening the box du shipment and erection. Such bracing can also provide ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 8

9 Table Span No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. At/Ab ** / -80,/ eral stiffness to create a pseudo closed box and thus minimize the warping stresses. The required area for such bracing, as shown in Fig. 4, is given by; A b i ^ 0.06 (in. ) AM = Required area of lateral bracing (in. ) Natural Frequency The designer is often required to evaluate the vertical natural frequency/, especially if the structure is subjected to train loadings. Such evaluation, for curved structures, has been determined and has resulted in the following equations: /= IT k L EL + El w GK T L ' fc R M / (cps) Table 4. Location of Section Changes for Negative Moment L<49 Length (ft) 49 < L < 8 8<L<5 No. of Cros. sect. 4 5 Negative Region Moment Xi 0.09L 0.08L 0.065L x 0.9L 0.7L 0.6L x 0.8L 0.5L x 4 0.0L 9 FOURTH QUARTER / 98

10 i Dead Load Moment Diagram Fig.. Example of location of section changes : EI X = bending stiffness (kip-in. ) EI W = warping stiffness (kip-in. 4 ) GKT - torsional stiffness (kip-in. ) R = radius (in.) M = mass (w/g) (kip-sec /in.) L = exterior span length (in.) k = Bn + Cn + D (for simple spans, k = ) and B, C y D are constants defined as: Two span Three span and n = L- mttt - ior /L ext erior.0 < n <.7. B C D As approximations for the torsional properties, the following expressions may be used; _ t'{b'd'y (d+b) t'b' d / (\ -b'/d') Lw ~ 4 ( + b'/d'} >\ b' average width of box d' = average depth of box t f = average plate thickness Ultimate Strength The ultimate strength determination of a curved box girder requires consideration of the interaction between the bending moment and torque. A com- ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 0

11 Table 5. Single-Span Section Dimensions Span (ft) No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB J FOURTH QUARTER / 98

12 Table 6. Two-Span Section Dimensions SPANS (ft) ** No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

13 Table 6. Two-Span Section Dimensions SPANS (ft) No. cross sect. Depth i Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. 4 AT/AB FOURTH QUARTER / 98

14 Table 6. Two-Span Section Dimensions SPANS (ft) No. cross sect. J Depth J Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

15 Table 6. Two-Span Section Dimensions SPANS (ft) No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB j **L-L. prehensive laboratory study, in which composite and noncomposite negative and positive sections were tested, has resulted in the following interaction equation: : Mp = plastic bending strength M = design bending moment Tp = plastic torsional strength T = design torsional moment Subsequent examination of typical box girders and their moment capacities, as controlled by the current AASHTO specifications and as given in Table, has also permitted development of a series of design charts 7 which permit rapid evaluation of these moments. Computerized Design The general response of single or continuous curved box girder bridges can be predicted by the solution of a series of coupled differential equations, when written in difference form as given in Fig. 5. These equations have been subsequently incorporated into a computer program, 9 which automates the design/ analysis of prismatic or nonprismatic straight or curved box girders as governed by the AASHTO criteria. ' The box girder may be either composite or noncomposite construction and can have integral transverse diaphragms spaced along the box and contain top lateral bracing. The 5 FOURTH QUARTER / 98

16 Table 7. Three-Span Box Dimensions SPANS (ft) No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. ATMB ** J ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

17 Table 7. Three-Span Box Dimensions SPANS (ft) No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB FOURTH QUARTER / 98

18 Table 7. Three-Span Box Dimensions SPANS (ft) No. cross sect, j Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION

19 Table 7. Three-Span Box Dimensions SPANS! (ft) No. cross sect. Depth Top Flange Width Bottom Flange Width Bottom Stiffener A Ix no. AT/AB **L\-L-L. basic configuration of a typical box and the type of cross diaphragms is shown in Figs., and 4. The computer output contains influence line ordinates, stresses on top and bottom flanges at locations along the span due to dead load, superimposed dead load, and live load plus impact. The stress resultants include the effects of bending, warping and distortion, utilizing the automatically computed section properties. Stress envelopes are given for fatigue design. Specifications (AASHTO) are used to establish allowable stresses, web and flange stiffening requirements and shear connector spacing, as given in Tables and. Resulting girder deflections and rotations, due to sequential concrete placements, can also be determined for specified length of pours. Composite/noncomposite actions may be assured after the concrete hardens. The entire output sequence is as follows: Basic Data Job description Girder geometry Structural details Concrete properties Loading properties Section details: span length, plate sizes, section properties, stiffener and bracing details, dead loads Pouring sequence geometry 9 FOURTH QUARTER / 98

20 tsw Wsw Wc -Ntc so Clear roadway width CROSS DIAPHRAGM (A b ) Inside overhang Fig.. Structural details Stresses Dead-load normal stress Superimposed dead-load normal stress Live-load normal stress (positive and negative moment) Forces Moment envelope Deflection envelope Shear envelope Vertical reaction envelope Torsion reaction envelope Torsion envelope Bimoment envelope Normal stress envelope Stress range envelope d/t y b/t requirements, web stress Theoretical web stiffener requirement Total stresses Shear connector spacing requirements Fatigue criteria Pouring sequence deflections Natural frequency Pouring sequence rotations Right Flange Fig.. Cross section Left Right Transverse Stiff Bottom Flange Stiff. Ef^pERINeaQlJBNi?!/; AMERICAN INSTITUTE OF STEEL CONSTRUCTION 40

21 Diaphragm spacing I i ^T^ < < <t X / X X te Afr = Bracing member area. Fig. 4. Bracing types (top Nf Ab lateral) Field-Test Comparison to Theoretical Results The static and dynamic response of a full scale bridge structure, when subjected to a known truck loading, was examined during the Fall The bridge consisted of twin steel boxes (4.5 ft X 8.8 ft) in composite action with a 9V -in. concrete slab. That part of the bridge under test was a three-span continuous with span lengths 00 ft, 0 ft, and 0 ft and centerline radius of,7 ft. The bridge was designed as a two-lane structure. The deformations and strains throughout the structure were measured during the application of the test vehicle. The resulting static load data were then examined and the results compared to the data obtained by the previously described analytical technique. 8 ' 9 In summary, the resulting induced stresses, at various locations along the structure, are described in Table 8. The sections are located as follows: Section A 0.4 (exterior span) Section B l.oli (first interior support) Section C O.5L (midspan of interior span) Examination of the data given in Table 8 indicates reasonable correlation between theory and experiment and the importance of the top lateral bracing during the dead-load response. The resulting girder deflections at Section A and Section C are given in Table 9. The data shown in this table show comparisons between theory and tests, indicating reasonable correlation especially for live load effects. The discrepancy in the dead-load results is due to the sequential placement of the concrete and time dependent composite actions. leivv R ^/EIwV h (EI x + CK t ) _ T 6 EIw + h (EI w + CK t ) <*) h (EI x + CK t ) R EI W : -m z h -EI W 4EI W + h CK t -[ 6EI W + h CKt + h 4 EI x " R J 4EI W + h GKt -EIw n- n- n+ n+ EIw R + EI X ) + h CK t R 6K t \R / R -(^ +EI X ) -q y h EI W ' V + h (EI> c + CK t ) R, R t r 6 /EIw\h (EI x + CKn! t/eiwv h (EI x + 6K t ) EIw R Fig. 5. Curved girder finite difference 4 equation FOURTH QUARTER / 98

22 Cross Sections A B C Table 8. Prototype Bridge Test-Stresses Loading DL LL + I Total DL LL + I Total DL LL + I Total Test (ksi) With bracing Theory (ksi) , Without bracing Table 9. Prototype Bridge Test-Vertical Deflections Cross Section A C Loading DL LL + I Total DL LL + I Total Test (in.) Theory With bracing Without bracing In general, results indicate the curved girder finite difference theory provides an excellent technique for box girder design. CONCLUSIONS This paper presents the results of various research which has permitted a better understanding of steel box girder bridges and the development of design criteria. Through use of these design data, more efficient and rapid design of such structures can be achieved, and a better service to the public provided. REFERENCES. Standard Specifications for Highway Bridges 'th Edition, AASHTO, Washington, D.C, Guide Specifications for Horizontally Curved Highway Bridges AASHTO, Washington, DC, Galambos, T. V. Tentative Load Factor Design Criteria for Curved Steel Bridges Rept.No. 50, Washington Univ., St. Louis, Mo., May Wolchuck, R., and R. Mayrbaurl Proposed Design Specifications for Steel Box Girder Bridges FHWA Rept. TS-80-05Jan Hems, C. P. Box Girder Bridge Design AISC Engineering Journal, Vol. 5, No. 4, Dec Heins, C. P., and L. J. Hua Proportioning Box Girder Bridges ASCE St. Dw. journal, Vol 06, No. ST, Nov Heins, C. P., and D. H Hall Designers Guide to Steel Box Girder Bridges Bethlehem Steel Corp., Bethlehem, Pa., Heins, C. P., and J. C. Olenick Curved Box Beam Bridges Analysis Computers and Structures Journal, Vol. 6, pp. 65-7, Pergamon Press, London, Heins, C P., and F. H. Sheu Computer Analysis of Steel Box Girder Bridges Civil Engineering Dept. Report, Univ. of Maryland, June Heins, C. P., and J. C. Olenick Diaphragms for Curved Box Beam Bridges ASCE St. Dw. Journal, Vol. 0, No. ST0, October Heins, C. P., and A. Sahin Natural Frequency of Curved Box Girder Bridges ASCE St. Dw. Journal, Vol. 05, No. ST, Dec Heins, C. P., and R. Humphrey Bending and Torsion Interaction of Box Girders ASCE St. Div. Journal, Vol. 05, No. ST5, May Heins, C P., et al. Curved Steel Box Girder Bridges: A Survey ASCE St. Dw. Journal, Vol. 04, No. ST, Nov Heins, C P.,et al. Curved Steel Box Girder Bridges: State of Art ASCE St. Dw. Journal, Vol. 04, No. ST, Nov Heins, C P., and W. H. Lee Curved Box Girder Field Test ASCE St. Dw. Journal, Vol. 07, No. ST, Feb Yoo, C H., J. Buchanan, C P. Heins, and W. L. Armstrong Loading Response of a Continuous Box Girder Bridge Proceedings ASCE Specialty Conference on Metal Bridges, St. Louis, Mo., Nov Heins, C P., and J. Y. Shyu Moment Capacity of Box Girders Institute for Physical Science and Technology Report, University of Maryland, June 98. ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION 4