structural steel design and construction

9 Roger L. BrockenbroughPresidentR. L. Brockenbrough & Associates, Inc.Pittsburgh, PennsylvaniaSTRUCTURAL STEELDESIGN ANDCONSTRUCTION

The many desirable characteristics ofstructural steels has led to their wide-spread use in a large variety of appli-cations. Structural steels are available in

many product forms and offer an inherently highstrength. They have a very high modulus ofelasticity, so deformations under load are verysmall. Structural steels also possess high ductility.They have a linear or nearly linear stress-strainrelationship up to relatively large stresses, and themodulus of elasticity is the same in tension andcompression. Hence, structural steels behaviorunder working loads can be accurately predictedby elastic theory. Structural steels are made undercontrolled conditions, so purchasers are assured ofuniformly high quality.

Standardization of sections has facilitateddesign and kept down the cost of structural steels.For tables of properties of these sections, seeManual of Steel Construction, American Instituteof Steel Construction, One East Wacker Dr.,Chicago, IL 60601-2001 www.aisc.org.

This section provides general information onstructural-steel design and construction. Any useof this material for a specic application shouldbe based on a determination of its suitabilityfor the application by professionally qualiedpersonnel.

9.1 Properties of StructuralSteels

The term structural steels includes a large number ofsteels that, because of their economy, strength,ductility, and other properties, are suitable for load-carrying members in a wide variety of fabricatedstructures. Steel plates and shapes intended for useinbridges, buildings, transportationequipment, con-struction equipment, and similar applications aregenerally ordered to a specic specication ofASTMand furnished in Structural Quality according tothe requirements (tolerances, frequency of testing,and so on) of ASTM A6. Plate steels for pressurevessels are furnished in Pressure Vessel Qualityaccording to the requirements of ASTM A20.

Each structural steel is produced to speciedminimummechanical properties as required by thespecic ASTM designation under which it isordered. Generally, the structural steels includesteels with yield points ranging from about 30 to100 ksi. The various strength levels are obtained byvarying the chemical composition and by heattreatment. Other factors that may affect mechanicalproperties include product thickness, nishingtemperature, rate of cooling, and residual elements.

The following denitions aid in understandingthe properties of steel.

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Source: Standard Handbook for Civil Engineers

Yield point Fy is that unit stress, ksi, at whichthe stress-strain curve exhibits a well-dened in-crease in strain without an increase in stress. Manydesign rules are based on yield point.

Tensile strength, or ultimate strength, is thelargest unit stress, ksi, the material can achieve in atensile test.

Modulus of elasticity E is the slope of thestress-strain curve in the elastic range, computedby dividing the unit stress, ksi, by the unit strain,in/in. For all structural steels, it is usually taken as29,000 ksi for design calculations.

Ductility is the ability of the material to under-go large inelastic deformations without fracture. Itis generally measured by the percent elongation fora specied gage length (usually 2 or 8 in). Struc-tural steel has considerable ductility, which isrecognized in many design rules.

Weldability is the ability of steel to be weldedwithout changing its basic mechanical properties.However, the welding materials, procedures, andtechniques employed must be in accordance withthe approved methods for each steel. Generally,weldability decreases with increase in carbon andmanganese.

Notch toughness is an index of the propensityfor brittle failure as measured by the impact energy

necessary to fracture a notched specimen, such as aCharpy V-notch specimen.

Toughness reects the ability of a smoothspecimen to absorb energy as characterized by thearea under a stress-strain curve.

Corrosion resistance has no specic index.However, relative corrosion-resistance ratings arebased on the slopes of curves of corrosion loss(reduction in thickness) vs. time. The reference ofcomparison is usually the corrosion resistance ofcarbon steel without copper. Some high-strengthstructural steels are alloyed with copper andother elements to produce high resistance toatmospheric deterioration. These steels develop atight oxide that inhibits further atmosphericcorrosion. Figure 9.1 compares the rate of re-duction of thickness of typical proprietary cor-rosion-resistant steels with that of ordinarystructural steel. For standard methods of esti-mating the atmospheric corrosion resistance oflow-alloy steels, see ASTM Guide G101, AmericanSociety of Testing and Materials, 100 Barr HarborDrive West Conshchoken, PA, 19428-2959, www.astm.org.

(R. L. Brockenbrough and B. G. Johnston, USSSteel Design Manual, R. L. Brockenbrough &Associates, Inc., Pittsburgh, PA 15243.)

Fig. 9.1 Curves show corrosion rates for steels in an industrial atmosphere.

9.2 n Section Nine



STRUCTURAL STEEL DESIGN AND CONSTRUCTION

9.2 Summary of AvailableStructural Steels

The specied mechanical properties of typicalstructural steels are presented in Table 9.1. Thesesteels may be considered in four general categories,depending on chemical composition and heattreatment, as indicated below. The tensile proper-ties for structural shapes are related to the sizegroupings indicated in Table 9.2.

Carbon steels are those steels for which (1) themaximum content specied for any of the follow-ing elements does not exceed the percentagesnoted: manganese1.65%, silicon0.60%, andcopper0.60%, and (2) no minimum content isspecied for the elements added to obtain a desiredalloying effect.

The rst carbon steel listed in Table 9.1A36is a weldable steel available as plates, bars, andstructural shapes. The last steel listed in the table.A992, which is available only for W shapes (rolledwide ange shapes), was introduced in 1998 andhas rapidly become the preferred steel for build-ing construction. It is unique in that the steel hasa maximum ratio specied for yield to tensilestrength, which is 0.85. The specication alsoincludes a maximum carbon equivalent of 0.47percent to enhance weldability. A minimum aver-age Charpy V-notch toughness of 20 ft-lb at 70 8Fcan be specied as a supplementary requirement.The other carbon steels listed in Table 9.1 areavailable only as plates. Although each steel isavailable in three or more strength levels, only onestrength level is listed in the table for A283 andA285 plates.

A283 plates are furnished as structural-qualitysteel in four strength levelsdesignated as GradesA, B, C, and Dhaving specied minimum yieldpoints of 24, 27, 30, and 33 ksi. This plate steel is ofstructural quality and has been used primarily foroil- and water-storage vessels. A573 steel, which isavailable in three strength levels, is a structural-quality steel intended for service at atmospherictemperatures at which improved notch toughnessis important. The other plate steelsA285, A515,and A516are all furnished in pressure-vesselquality only and are intended for welded construc-tion in more critical applications, such as pressurevessels. A516 is furnished in four strength levelsdesignated as Grades 55, 60, 65, and 70 (denotingtheir tensile strength)having specied minimum

yield points of 30, 32, 35, and 38 ksi. A515 hassimilar grades except there is no Grade 55. A515steel is for intermediate and higher temperatureservice, whereas A516 is for moderate and lowertemperature service.

Carbon steel pipe used for structural purposesis usually A53 Grade B with a specied minimumyield point of 35 ksi. Structural carbon-steel hot-formed tubing, round and rectangular, is furnish-ed to the requirements of A501 with a yield point of36 ksi. Cold-formed tubing is also available inseveral grades with a yield point from 33 to50 ksi.

High-strength, low-alloy steels have speciedminimum yield points above about 40 ksi in thehot-rolled condition and achieve their strength bysmall alloying additions rather than through heattreatment. A588 steel, available in plates, shapes,and bars, provides a yield point of 50 ksi in platethicknesses through 4 in and in all structuralshapes and is the predominant steel used instructural applications in which durability isimportant. Its resistance to atmospheric corrosionis about four times that of carbon steel. A242 steelalso provides enhanced atmospheric-corrosionresistance. Because of this superior atmospheric-corrosion resistance, A588 and A242 steels providea longer paint life than other structural steels. Inaddition, if suitable precautions are taken, thesesteels can be used in the bare, uncoated conditionin many applications in which the members areexposed to the atmosphere because a tight oxide isformed that substantially reduces further cor-rosion. Bolted joints in bare steel require specialconsiderations as discussed in Art. 9.36.

A572 high-strength, low-alloy steel is usedextensively to reduce weight and cost. It is pro-duced in several grades that provide a yield pointof 42 to 65 ksi. Its corrosion resistance is the same asthat of carbon steel.

Heat-Treated Carbon and High-Strength, Low-Alloy Steels n This group iscomprised of carbon and high-strength, low-alloysteels that have been heat-treated to obtain moredesirable mechanical properties.

A633, Grades A through E, are weldable platesteels furnished in the normalized condition toprovide an excellent combination of strength (42 to60 ksi minimum yield point) and toughness (upto 15 ft-lb at 2 75 8F).

Structural Steel Design and Construction n 9.3




Table 9.1 Specied Mechanical Properties of Steel*

ASTM Designation Plate Thickness, in

ANSI/ASTM Groupor Weight/ft forStructural Shapes

Yield Pointor Yield

Strength, ksiTensile

Strength, ksi

Carbon Steels

A36 To 8, incl To 426 lb/ft, incl. 36 5880Not applicable Over 426 lb/ft 36 58over 8 Not applicable 32 5880

A283, Grade C None specied Not applicable 30 5570A285, Grade C To 2, incl Not applicable 30 5575A516, Grade 55 To 12, incl Not applicable 30 5575A516, Grade 60 To 8, incl Not applicable 32 6080A516, Grade 65 To 8, incl Not applicable 35 6585A516, Grade 70 To 8, incl Not applicable 38 7090A573, Grade 58 To 112, incl Not applicable 32 5871A573, Grade 65 To 112, incl Not applicable 35 6577A573, Grade 70 To 112, incl Not applicable 42 7090A992 Not Applicable All w shapes 5065 65

High-Strength, Low-Alloy Steels

A242 To 34, incl Groups 1 and 2 50 70Over 34 to 1

12, incl Group 3 46 67Over 112 to 4, incl Groups 4 and 5 42 63

A588 To 4, incl Groups 15 50 70Over 4 to 5, incl 46 67Over 5 to 8, incl 42 63

A572, Grade 42 To 6, incl Groups 15 42 60A572, Grade 50 To 4, incl Groups 15 50 65A572, Grade 60 To 114, incl Groups 1 and 3 60 75A572, Grade 65 To 114, incl Groups 1 and 3 65 80

Heat-Treated Carbon and High-Strength, Low-Alloy Steels

A633, Grade C and D To 212, incl Not applicable 50 7090Over 212 to 4, incl 46 6585

A633, Grade E To 4, incl 60 80100Over 4 to 6, incl 55 7595

A678, Grade C To 34, incl Not applicable 75 95115Over 34 to 1

12, incl 70 90110Over 112 to 2, incl 65 85105

A852 To 4, incl Not applicable 70 90110A913, Grade 50 Not applicable Groups 15 50 65A913, Grade 60 Not applicable Groups 15 60 75A913, Grade 65 Not applicable Groups 15 65 80A913, Grade 70 Not applicable Groups 15 70 90

(Table continued )

9.4 n Section Nine




A678, Grades A through D, are weldable platesteels furnished in the quenched and temperedcondition to provide a minimum yield point of 50to 75 ksi.

A852 is a quenched and tempered, weathering,plate steel with corrosion resistance similar to thatof A588 steel. It has been used for bridges andconstruction equipment.

A913 is a high-strength low-alloy steel for struc-tural shapes, produced by the quenching and self-tempering process, and intended for buildings,bridges, and other structures. Four grades providea minimum yield point of 50 to 70 ksi. Maximumcarbon equivalents range from 0.38 to 0.45 percent,and the minimum average Charpy V-notch tough-ness is 40 ft-lb at 70 8F.

Heat-Treated, Constructional-AlloySteels n Heat-treated steels that contain alloyingelements and are suitable for structural appli-cations are called heat-treated, constructional-alloysteels. A514 (Grades A through Q) covers quen-ched and tempered alloy-steel plates with a mini-mum yield strength of 90 or 100 ksi.

Bridge Steels n Steels for application inbridges are covered by A709, which includes steelin several of the categoriesmentioned above. Underthis specication, Grades 36, 50, 70, and 100 aresteels with yield strengths of 36, 50, 70, and 100 ksi,respectively. The grade designation is followed bythe letter W, indicating whether ordinary or highatmospheric-corrosion resistance is required. An

Table 9.1 (Continued)

ASTM Designation Plate Thickness, in

ANSI/ASTM Groupor Weight/ft forStructural Shapes

Yield Pointor Yield

Strength, ksiTensile

Strength, ksi

Heat-Treated Constructional Alloy Steel

A514 To 212, incl Not applicable 100 110130Over 212 to 6, incl 90 100130

* Mechanical properties listed are speciedminimumvalues exceptwhere a specied range of values (minimum tomaximum) is given.The following properties are approximate values for all the structural steels: modulus of elasticity29,000 ksi; shear modulus

11,000 ksi; Poissons ratio0.30; yield stress in shear0.57 times yield stress in tension; ultimate strength in shear 23 to34 times tensile

strength; coefcient of thermal expansion6.5 1026 in/in/8F for temperature range 250 to 150 8F.

Table 9.2 Wide-Flange Size Groupings for Tensile-Property Classication

Group 1 Group 2 Group 3 Group 4 Group 5

W24 55, 62 W40 149, 268 W40 277328 W40 362655 W36 920W21 4457 W36 135210 W36 230300 W36 328798 W14 605873W18 3571 W33 118152 W33 201291 W33 318619W16 2657 W30 99211 W30 235261 W30 292581W14 2253 W27 84178 W27 194258 W27 281539W12 1458 W24 68162 W24 176229 W24 250492W10 1245 W21 62147 W21 166223 W21 248402W8 1048 W18 76143 W18 158192 W18 211311W6 925 W16 67100 W14 145211 W14 233550W5 16, 19 W14 61132 W12 120190 W12 210336W4 13 W12 65106

W10 49112W8 58, 67





additional letter, T or F, indicates that CharpyV-notch impact tests must be conducted on thesteel. The T designation indicates the material is tobe used in a nonfracture-critical application asdened by the American Association of StateHighway and Transportation Ofcials (AASHTO).The F indicates use in a fracture-critical application.A trailing numeral, 1, 2, or 3, indicates the testingzone, which relates to the lowest ambient tempera-ture expected at the bridge site. See Table 9.3. Asindicated by the rst footnote in the table, theservice temperature for each zone is considerably

less than the Charpy V-notch impact-test tempera-ture. This accounts for the fact that the dynamicloading rate in the impact test is severer than that towhich the structure is subjected. The toughnessrequirements depend on fracture criticality, grade,thickness, and method of connection. Additionally,A709-HPS70W, designated as a High PerformanceSteel (HPS), is also available for highway bridgeconstruction. This is a weathering plate steel, de-signated HPS because it possesses superior welda-bility and notch toughness as compared to conven-tional steels of similar strength.

Table 9.3 Charpy V-Notch Toughness for A709 Bridge Steels*

Grade

MaxThickness,in, Inclusive

Joining/FasteningMethod

Min AvgEnergy,ft-lb

Test Temp, 8F

Zone1

Zone2

Zone3

Non-Fracture-Critical Members

36T 4 Mech/Weld 15 70 40 10

50T, 50WT 2 Mech/Weld 15 70 40 102 to 4 Mechanical 152 to 4 Welded 20

70WT 212 Mech/Weld 20 50 20 210212 to 4 Mechanical 20212 to 4 Welded 25

100T, 100WT 212 Mech/Weld 25 30 0 230212 to 4 Mechanical 25212 to 4 Welded 35

Fracture-Critical Members

36F 4 Mech/Weld 25 70 40 10

50F, 50WF 2 Mech/Weld 25 70 40 102 to 4 Mechanical 25 2102 to 4 Welded 30 210

70WF 212 Mech/Weld 30 50 20 210212 to 4 Mechanical 30 210212 to 4 Welded 35 210

100F, 100WF 212 Mech/Weld 35 30 0 230212 to 4 Mechanical 35 230212 to 4 Welded 45 NA

* Minimum service temperatures: Zone 1, 0 8F; Zone 2, ,0 to 230 8F; Zone 3, ,230 to 260 8F. If yield strength exceeds 65 ksi, reduce test temperature by 15 8F for each 10 ksi above 65 ksi. If yield strength exceeds 85 ksi, reduce test temperature by 15 8F for each 10 ksi above 85 ksi.

9.6 n Section Nine




Lamellar Tearing n The information onstrength and ductility presented generally pertainsto loadings applied in the planar direction (longi-tudinal or transverse orientation) of the steel plateor shape. Note that elongation and area-reductionvalues may well be signicantly lower in thethrough-thickness direction than in the planardirection. This inherent directionality is of smallconsequence in many applications, but it doesbecome important in the design and fabricationof structures containing massive members withhighly restrained welded joints.

With the increasing trend toward heavy welded-plate construction, there has been a broaderrecognition of occurrences of lamellar tearing insome highly restrained joints of welded structures,especially those in which thick plates and heavystructural shapes are used. The restraint inducedby some joint designs in resisting weld-depositshrinkage can impose tensile strain high enough tocause separation or tearing on planes parallel tothe rolled surface of the structural member beingjoined.

The incidence of this phenomenon can bereduced or eliminated through use of techniquesbased on greater understanding by designers, de-tailers, and fabricators of the (1) inherentdirectionality of constructional forms of steel, (2)high restraint developed in certain types ofconnections, and (3) need to adopt appropriateweld details and welding procedures with properweld metal for through-thickness connections.Furthermore, steels can be specied to be pro-duced by special practices or processes to enhancethrough-thickness ductility and thus assist inreducing the incidence of lamellar tearing.

However, unless precautions are taken in bothdesign and fabrication, lamellar tearing may stilloccur in thick plates and heavy shapes of suchsteels at restrained through-thickness connections.Some guidelines for minimizing potential pro-blems have been developed by the AmericanInstitute of Steel Construction (AISC). (See TheDesign, Fabrication, and Erection of HighlyRestrained Connections to Minimize LamellarTearing, AISC Engineering Journal, vol. 10, no. 3,1973, www.aisc.org.)

Welded Splices in Heavy Sections n

Shrinkage during solidication of large weldscauses strains in adjacent restrained material thatcan exceed the yield-point strain. In thick material,

triaxial stresses may develop because thereis restraint in the thickness direction as well asthe planar directions. Such conditions inhibit theability of the steel to act in a ductile mannerand increase the possibility of brittle fracture.Therefore, for building construction, AISCimposes special requirements when splicing eitherGroup 4 or Group 5 rolled shapes, or shapes builtup by welding plates more than 2 in thick, ifthe cross section is subject to primary tensilestresses due to axial tension or exure. Includedare notch toughness requirements, the removalof weld tabs and backing bars (ground smooth),generous-sized weld access holes, preheatingfor thermal cutting, and grinding and inspectingcut edges. Even when the section is usedas a primary compression member, the sameprecautions must be taken for sizing theweld access holes, preheating, grinding, andinspection. See the AISC Specication for furtherdetails.

Cracking n An occasional problem known ask-area cracking has been identied. Wide angesections are typically straightened as part of themill production process. Often a rotary straight-ening process is used, although some heaviermembers may be straightened in a gag press.Some reports in recent years have indicated a po-tential for crack initiation at or near connections inthe k area of wide ange sections that have beenrotary straightened. The k area is the regionextending from approximately the midpoint of theweb-to-ange llet, into the web for a distanceapproximately 1 to 1-12 in. beyond the point oftangency. Apparently, in some cases, this limitedregion had a reduced notch toughness due tocold working and strain hardening. Most of theincidents reported occurred at highly restrainedjoints with welds in the k area. However, thenumber of examples reported has been limitedand these have occurred during construction orlaboratory tests, with no evidence of difcultieswith steel members in service. Research hasconrmed the need to avoid welding in the karea. AISC issued the following recommendationsconcerning fabrication and design practices forrolled wide ange shapes:

. Welds should be stopped short of the k area fortransverse stiffeners (continuity plates).





. For continuity plates, llet welds and/or partialjoint penetration welds, proportioned to trans-fer the calculated stresses to the column web,should be considered instead of complete jountpenetration welds. Weld volume should beminimized.

. Residual stresses in highly restrained joints maybe decreased by increased preheat and properweld sequencing.

. Magnetic particle or dye penetrant inspectionshould be considered for weld areas in or nearthe k area of highly restrained connectionsafter the nal welding has completely cooled.

. When possible, eliminate the need for columnweb doubler plates by increasing column size.

Good fabrication and quality control practices,such as inspection for cracks, gouges, etc., at ame-cut access holes or copes, should continue to befollowed and any defects repaired and groundsmooth. All structural wide ange members fornormal service use in building construction shouldcontinue to be designed per AISC Specicationsand the material furnished per ASTM standards.

(AISC Advisory Statement, Modern Steel Con-struction, February 1997.)

Fasteners n Steels for structural bolts arecovered by A307, A325, and A490 Specications.A307 covers carbon-steel bolts for general appli-cations, such as low-stress connections andsecondary members. Specication A325 includestwo type of quenched and tempered high-strengthbolts for structural steel joints: Type 1medium-carbon, carbon-boron, or medium-carbon alloysteel, and Type 3weathering steel with atmos-pheric corrosion resistance similar to that of A588steel. A previous Type 2 was withdrawn in 1991.

Specication A490 includes three types ofquenched and tempered high-strength steel boltsfor structural-steel joints: Type 1bolts made ofalloy steel; Type 2bolts made from low-carbonmartensite steel, and Type 3bolts having atmos-pheric-corrosion resistance and weathering charac-teristics comparable to that of A588, A242, andA709 (W) steels. Type 3 bolts should be speciedwhen atmospheric-corrosion resistance is required.Hot-dip galvanized A490 bolts should not be used.

Bolts having diameters greater than 112 in areavailable under Specication A449.

Rivets for structural fabrication were includedunder Specication A502 but this designation hasbeen discontinued.

9.3 Structural-Steel Shapes

Most structural steel used in building constructionis fabricated from rolled shapes. In bridges, greateruse is made of plates since girders spanning overabout 90 ft are usually built-up sections.

Many different rolled shapes are available:W shapes (wide-ange shapes), M shapes (mis-cellaneous shapes), S shapes (standard I sections),angles, channels, and bars. The Manual of SteelConstruction, American Institute of Steel Con-struction, lists properties of these shapes.

Wide-ange shapes range from a W4 13 (4 indeep weighing 13 lb/lin ft) to a W36 920 (36 indeep weighing 920 lb/lin ft). Jumbo columnsections range up to W14 873.

In general, wide-ange shapes are the mostefcient beam section. They have a high proportionof the cross-sectional area in the anges and thus ahigh ratio of section modulus to weight. The 14-inW series includes shapes proportioned for use ascolumn sections; the relatively thick web results ina large area-to-depth ratio.

Since the ange and web of a wide-ange beamdo not have the same thickness, their yield pointsmay differ slightly. In accordance with design rulesfor structural steel based on yield point, it istherefore necessary to establish a design yieldpoint for each section. In practice, all beams rolledfrom A36 steel (Art. 9.2) are considered to have ayield point of 36 ksi. Wide-ange shapes, plates,and bars rolled from higher-strength steels arerequired to have the minimum yield and tensilestrength shown in Table 9.1.

Square, rectangular, and round structural tubu-lar members are available with a variety of yieldstrengths. Suitable for columns because of theirsymmetry, these members are particularly useful inlow buildings and where they are exposed forarchitectural effect.

Connection Material n Connections arenormally made with A36 steel. If, however,higher-strength steels are used, the structural sizegroupings for angles and bars are:

Group 1: Thicknesses of 12 in or less

9.8 n Section Nine




Group 2: Thicknesses exceeding 12 in but notmore than 34 in

Group 3: Thicknesses exceeding 34 in

Structural tees fall into the same group as thewide-ange or standard sections from which theyare cut. (A WT7 13, for example, designates atee formed by cutting in half a W 14 26 andtherefore is considered a Group 1 shape, as is a W14 26.)

9.4 Selecting StructuralSteels

The following guidelines aid in choosing betweenthe various structural steels. When possible, a moredetailed study that includes fabrication anderection cost estimates is advisable.

A basic index for cost analysis is the cost-strength ratio, p/Fy, which is thematerial cost, centsper pound, divided by the yield point, ksi. Fortension members, the relative material cost of twomembers, C2/C1, is directly proportional to thecost-strength ratios; that is,

C2C1

p2=Fy2p1=Fy1

(9:1a)

For bending members, the relationship dependson the ratio of the web area to the ange area andthe web depth-to-thickness ratios. For fabricatedgirders of optimum proportions (half the totalcross-sectional area is the web area),

C2C1

p2p1

Fy1

Fy2

1=2(9:1b)

For hot-rolled beams,

C2C1

p2p1

Fy1

Fy2

2=3(9:1c)

For compression members, the relation depends onthe allowable buckling stress Fc, which is a functionof the yield point directly; that is,

C2C1

Fc1=p1Fc2=p2

(9:1d)

Thus, for short columns, the relationship appro-aches that for tension members. Table 9.4 givesratios of Fc that can be used, along with typicalmaterial prices p from producing mills, to calculaterelative member costs.

Higher strength steels are often used forcolumns in buildings, particularly for the loweroors when the slenderness ratios is less than 100.When bending is dominant, higher strength steelsare economical where sufcient lateral bracing ispresent. However, if deection limits control, thereis no advantage over A36 steel.

On a piece-for-piece basis, there is substantiallyno difference in the cost of fabricating and erectingthe different grades. Higher-strength steels, how-ever, may afford an opportunity to reduce thenumber of members, thus reducing both fabrica-tion and erection costs.

9.5 Tolerances for StructuralShapes

ASTM Specication A6 lists mill tolerances forrolled-steel plates, shapes, sheet piles, and bars.Included are tolerances for rolling, cutting, section

Table 9.4 Ratio of Allowable Stress in Columns of High-Strength Steel to That of A36 Steel

SpeciedYield Strength

Fy , ksi

Slenderness Ratio Kl/r

5 15 25 35 45 55 65 75 85 95 105 115

65 1.80 1.78 1.75 1.72 1.67 1.62 1.55 1.46 1.35 1.22 1.10 1.0360 1.66 1.65 1.63 1.60 1.56 1.52 1.47 1.40 1.32 1.21 1.10 1.0355 1.52 1.51 1.50 1.48 1.45 1.42 1.38 1.33 1.27 1.20 1.10 1.0350 1.39 1.38 1.37 1.35 1.34 1.32 1.29 1.26 1.22 1.17 1.10 1.0345 1.25 1.24 1.24 1.23 1.22 1.21 1.19 1.17 1.15 1.12 1.08 1.0342 1.17 1.16 1.16 1.15 1.15 1 14 1.13 1.12 1.10 1.08 1.06 1.03





area, and weight, ends out of square, camber, andsweep. The Manual of Steel Construction con-tains tables for applying these tolerances.

The AISC Code of Standard Practice gives fab-rication and erection tolerances for structural steel forbuildings. Figures 9.2 and 9.3 show permissibletolerances for column erection for a multistorybuilding. In these diagrams, a working point for acolumn is the actual center of the member at eachend of a shipping piece. Theworking line is a straightline between the members working points.

Both mill and fabrication tolerances should beconsidered in designing and detailing structuralsteel. A column section, for instance, may have anactual depth greater or less than the nominal depth.An accumulation of dimensional variations, there-fore, would cause serious trouble in erection of abuilding with many bays. Provision should bemade to avoid such a possibility.

Tolerances for fabrication and erection ofbridge girders are usually specied by highwaydepartments.

Fig. 9.2 Tolerances permitted for exterior columns for plumbness normal to the building line.(a) Envelope within which all working points must fall. (b) For individual column sections lying within theenvelope shown in (a), maximum out-of-plumb of an individual shipping piece, as dened by a straightline between working points, is 1/500 and the maximum out-of-straightness between braced points isL/1000, where L is the distance between braced points. (c) Tolerance for the location of a working point at acolumn base. The plumb line through that point is not necessarily the precise plan location, inasmuch asthe 2000 AISC Code of Standard Practice deals only with plumbness tolerance and does not includeinaccuracies in location of established column lines, foundations, and anchor bolts beyond the erectorscontrol.

9.10 n Section Nine




9.6 Structural-Steel DesignSpecications

The design of practically all structural steel forbuildings in the United States is based on twospecications of the American Institute of SteelConstruction. AISC has long maintained a tradi-tional allowable-stress design (ASD) specication,including a comprehensive revised specicationissued in 1989, Specication for Structural Steelfor BuildingsAllowable Stress Design and PlasticDesign. AISC also publishes an LRFD specica-tion, Load and Resistance Factor Design Speci-cation for Structural Steel for Buildings. Otherimportant design specications published by AISCinclude Seismic Provisions for Structural SteelBuildings, Specication for the Design of SteelHollow Structural Sections, Specication for theDesign, Fabrication and Erection of Steel SafetyRelated Structures for Nuclear Facilities, andSpecication for Load and Resistance FactorDesign of Single-Angle members.

Design rules for bridges are given in StandardSpecications for Highway Bridges, (AmericanAssociation of State Highway and TransportationOfcials, N. Capitol St, Suite 249 N.W., Washing-ton, DC 20001, www.ashto.org). They are some-what more conservative than the AISC Specica-tions. AASHTO gives both an allowable-stressmethod and a load-factor method. However, themost recent developments in bridge design are

reected in the AASHTO publication. LRFDBridge Design Specications.

Other important specications for the design ofsteel structures include the following:

The design of structural members cold-formedfrom steel not more than 1 in thick follows the rulesof AISI Specication for the Design of Cold-Formed Steel Structural Members (American Ironand Steel Institute, 1101 17th St., N.W., Washington,DC 20036-4700, www.aisc.org. See Sec. 10).

Codes applicable to welding steel for bridges,buildings, and tubular members are offered byAWS (American Welding Society, 550 N.W. LeJoneRoad, Miami, FL 33126).

Rules for the design, fabrication, and erection ofsteel railway bridges are developed by AREMA(American Railway Engineering and Maintenance-of-Way Association, 8201 Corporate Drive, Suite1125, Landover, Md., 20785-2230). See Sec. 17.

Specications covering design, manufacture,and use of open-web steel joists are availablefrom SJI (Steel Joist Institute, www.steeljoist).See Sec. 10.

9.7 Structural-Steel DesignMethods

Structural steel for buildings may be designedby either the allowable-stress design (ASD) orload-and-resistance-factor design (LRFD) method

Fig. 9.3 Tolerance in plan permitted for exterior columns at any splice level. Circles indicate columnworking points. At any splice level, the horizontal envelope dened by E lies within the distances Ta andTt from the established column line (Fig. 9.2a). Also, the envelope Emay be offset from the correspondingenvelope at the adjacent splice levels, above and below, by a distance not more than L/500, where L is thecolumn length. Maximum E is 112 in for buildings up to 300 ft long. E may be increased by

12 in for eachadditional 100 ft of length but not to more than 3 in.





(Art. 9.6). The ASD Specication of the AmericanInstitute of Steel Construction follows the usualmethod of specifying allowable stresses thatrepresent a failure stress (yield stress, bucklingstress, etc.) divided by a safety factor. In the AISC-LRFD Specication, both the applied loads and thecalculated strength or resistance of members aremultiplied by factors. The load factors reectuncertainties inherent in load determination andthe likelihood of various load combinations. Theresistance factors reect variations in determiningstrength of members such as uncertainty in theoryand variations in material properties and dimen-sions. The factors are based on probabilistic deter-minations, with the intent of providing a morerational approach and a design with a more uni-form reliability. In general, the LRFD method canbe expected to yield some savings in materialrequirements but may require more design time.

Factors to be applied to service loads for variousloading combinations are given in Art. 15.5. Rulesfor plastic design are included in both specica-tions. This method may be applied for steels withyield points of 65 ksi or less used in braced andunbraced planar frames and simple and continu-ous beams. It is based on the ability of structuralsteel to deform plastically when strained past theyield point, thereby developing plastic hinges andredistributing loads (Art. 6.65). The hinges are notanticipated to form at service loads but at thehigher factored loads.

Steel bridge structures may be designed byASD, LFD, or LRFD methods in accordance withthe specications of the American Association ofState Highway and Transportation Ofcials(AASHTO). With the load-factor design (LFD)method, only the loads are factored, but with theload-and-resistance-factor (LRFD) method, factorsare applied to both loads and resistances. For loadfactors for highway bridges, see Art. 17.3. Railroadbridges are generally designed by the ASDmethod.

9.8 Dimensional Limitationson Steel Members

Design specications, such as the AmericanInstitute of Steel Construction Specication forStructural Steel BuildingsAllowable StressDesign and Plastic Design and Load andResistance Factor Design for Structural SteelBuildings and the American Association of State

Highway and Transportation Ofcials StandardSpecications for Highway Bridges and LRFDBridge Design Specications set limits, maximumand minimum, on the dimensions and geometryof structural-steel members and their parts. Thelimitations generally depend on the types andmagnitudes of stress imposed on the members andmay be different for allowable-stress design (ASD)and load-and-resistance-factor design (LRFD).

These specications require that the structure asa whole and every element subject to compressionbe constructed to be stable under all possiblecombinations of loads. The effects of loads on allparts of the structure when members or theircomponents deform under loads or environmentalconditions should be taken into account in designand erection.

(T. V. Galambos, Guide to Stability DesignCriteria for Metal Structures, 5th ed., JohnWiley &Sons, Inc., New York.)

Vibration Considerations n In large openareas of buildings, where there are few partitions orother sources of damping, transient vibrationscaused by pedestrian trafc may become annoying.Beams and slender members supporting such areasshould be designed with due regard for stiffnessand damping. Special attention to vibration controlshould be given in design of bridges because oftheir exposure to wind, signicant temperaturechanges, and variable, repeated, impact and dyna-mic loads. Some of the restrictions on member di-mensions in standard building and bridge designspecications are intended to limit amplitudes ofvibrations to acceptable levels.

Minimum Thickness n Floor plates inbuildings may have a nominal thickness as smallas 18 in. Generally, minimum thickness availablefor structural-steel bars 6 in or less wide is 0.203 inand for bars 6 to 8 in wide, 0.230 in. Minimumthickness for plates 8 to 48 in wide is 0.230 in andfor plates over 48 in wide, 0.180 in.

The AASHTO Specication requires that, exceptfor webs of certain rolled shapes, closed ribs inorthotropic-plate decks, llers, and railings, struc-tural-steel elements be at least 516 in thick. Webthickness of rolled beams may be as small as0.23 in. Thickness of closed ribs in orthotropic-platedecks should be at least 316 in. No minimum isestablished for llers. The American Railway

9.12 n Section Nine




Engineering and Maintenance-of-Way AssociationManual for Railway Engineering requires thatbridge steel, except for llers, be at least 0.335 inthick. Gusset plates connecting chords and webmembers of trusses should be at least 12 in thick. Inany case, where the steel will be exposed to asubstantial corrosive environment, the minimumthicknesses should be increased or the metalshould be protected.

Maximum Slenderness Ratios n TheAISC Specications require that the slendernessratio, the ratio of effective length to radius ofgyration of the cross section, should not exceed200 for members subjected to compression inbuildings. For steel highway bridges the AASHTOSpecication limits slenderness ratios for com-pression members to a maximum of 120 for mainmembers and 140 for secondary members andbracing. The AREMA Manual lists the followingmaximum values for slenderness ratios for com-pression members in bridges: 100 for main mem-bers, 120 for wind and sway bracing, 140 for singlelacing, and 200 for double lacing.

For members in tension, the AISC Specicationslimit slenderness ratio to a maximum of 300 inbuildings. For tension members other than rods,eyebars, cables, and plates, AASHTO species forbridges a maximum ratio of unbraced length toradius of gyration of 200 for main tension mem-bers, 240 for bracing, and 140 for main-memberssubject to stress reversal. The AREMA Manuallimits the ratio for tension members to 200 forbridges.

Compact Sections n The AISC andAASHTO specications classify structural-steelsections as compact, noncompact, slender, or hy-brid. Slender members have elements that exceedthe limits on width-thickness ratios for compactand noncompact sections and are designedwith formulas that depend on the differencebetween actual width-thickness ratios and themax-imum ratios permitted for noncompact sections.Hybrid beams or girders have anges made ofsteel with yield strength different from that for thewebs.

For a specic cross-sectional area, a compactsection generally is permitted to carry heavierloads than a noncompact one of similar shape.Under loads stressing the steel into the plastic

range, compact sections should be capable offorming plastic hinges with a capacity for inelasticrotation at least three times the elastic rotationcorresponding to the plastic moment. To qualify ascompact, a section must have anges continuouslyconnected to the webs, and thickness of its ele-ments subject to compressionmust be large enoughto prevent local buckling while developing a fullyplastic stress distribution.

Tables 9.5 and 9.6 present, respectively, maxi-mum width-thickness ratios for structural-steelcompression elements in buildings and highwaybridges. See also Arts. 9.12 and 9.13.

9.9 Allowable Tension in Steel

For buildings, AISC species a basic allowable unittensile stress, ksi, Ft 0.60Fy, on the gross crosssection area, where Fy is the yield strength of thesteel, ksi (Table 9.7). Ft is subjected to the furtherlimitation that it should not exceed on the net crosssection area, one-half the specied minimumtensile strength Fu of the material. On the netsection through pinholes in eyebars, pin-connectedplates, or built-up members, Ft 0.45Fy.

For bridges, AASHTO species allowabletensile stresses as the smaller of 0.55Fy on the grosssection, or 0.50Fu on the net section (0.46Fy for100 ksi yield strength steels), where Fu tensilestrength (Table 9.7). In determining gross area, areaof holes for bolts and rivets must be deducted ifover 15 percent of the gross area. Also, open holeslarger than 114 in, such as perforations, must bededucted.

Table 9.7 and subsequent tables apply to twostrength levels, Fy 36 ksi and Fy 50 ksi, theones generally used for construction.

The net section for a tension member with achain of holes extending across a part in anydiagonal or zigzag line is dened in the AISCSpecication as follows: The net width of the partshall be obtained by deducting from the grosswidth the sum of the diameters of all the holes inthe chain and adding, for each gage space in thechain, the quantity s2/4g, where s longitudinalspacing (pitch), in, of any two consecutive holesand g transverse spacing (gage), in, of the sametwo holes. The critical net section of the partis obtained from the chain that gives the least netwidth.





Table 9.5 Maximum Width-Thickness Ratios b/t a for Compression Elements for Buildingsb

Description ofElement

ASD and LRFDc ASDc LRFDc

Compactlp Noncompactd Noncompactlr

Projecting ange element ofI-shaped rolled beams andchannels in exure

65=Fy

p95=

Fy

p141=

FL

p g

Projecting ange element ofI-shaped hybrid or weldedbeams in exure

65=Fy

p95=

Fyt=kc

pe 162=

FL=kcc

ph

Projecting ange element ofI-shaped sections in purecompression, plates projectingfrom compression elements;outstanding legs of pairs of anglesin continuous contact; anges ofchannels in pure compression

Not specied 95=Fy

p95=

Fy

p

Flanges of square and rectangularbox and hollow structural sectionsof uniform thickness subject tobending or compression; angecover plates and diaphragm platesbetween lines of fasteners or welds

190=Fy

puniform comp.

160=Fy

pplastic anal.

238=Fy

p238=

Fy

p

Unsupported width of cover platesperforated with a succession ofaccess holes

Not specied 317=Fy

p317=

Fy

p

Legs of single-angle struts; legs ofdouble-angle struts with separators;unstiffened elements; i.e., supportedalong one edge

Not specied 76=Fy

p76=

Fy

p

Stems of tees Not specied 127=Fy

p127=

Fy

pAll other uniformly compressedstiffened elements; i.e., supportedalong two edges

Not specied 253=Fy

p253=

Fy

p

Webs in exural compressiona 640=Fy

p760=

Fy

p970=

Fy

pD/t for circular hollow sections f

In axial compression for ASD 3,300/FyIn exure for ASD 3,300/Fy

(Table continued )

9.14 n Section Nine




For splice and gusset plates and other connec-tion ttings, the design area for the net sectiontaken through a hole should not exceed 85% of thegross area. When the load is transmitted throughsome but not all of the cross-sectional elementsfor example, only through the anges of a Wshapean effective net area should be used (75 to90% of the calculated net area).

LRFD for Tension in Buildings n The limitstates for yielding of the gross section and fracture inthe net section should be investigated. For yielding,the design tensile strength Pu, ksi, is given by

Pu 0:90FyAg (9:2)where Fy specied minimum yield stress, ksi

Ag gross area of tension member, in2For fracture,

Pu 0:75FuAe (9:3)where Fu specied minimum tensile strength,

ksi

Ae effective net area, in2In determining Ae for members without holes,when the tension load is transmitted by fasteners orwelds through some but not all of the cross-sectional elements of the member, a reductionfactor U is applied to account for shear lag. Thefactor ranges from 0.75 to 1.00.

9.10 Allowable Shear in Steel

The AASHTO Standard Specication for High-way Bridges (Art. 9.6) species an allowableshear stress of 0.33Fy , where Fy is the speciedminimum yield stress of the web. Also see Art.9.10.2. For buildings, the AISC Specication forASD (Art. 9.6.) relates the allowable shear stress inexural members to the depth-thickness ratio,h/tw, where tw is the web thickness and h is theclear distance between anges or between adja-cent lines of fasteners for built-up sections. Indesign of girders, other than hybrid girders, largershears may be allowed when intermediate stiffen-ers are used. The stiffeners permit tension-eldaction; that is, a strip of web acting as a tensiondiagonal resisted by the transverse stiffenersacting as struts, thus enabling the web to carrygreater shear.

9.10.1 ASD for Shear in Buildings

The AISC Specication for ASD species the fol-lowing allowable shear stresses Fv, ksi:

Fn 0:40Fy h=tw 380=Fy

q(9:4)

Fv CnFy=2:89 0:40Fy h=tw . 380=Fy

q(9:5)

Table 9.5 (Continued)


ASD and LRFDc ASDc LRFDc

Compactlp Noncompactd Noncompactlr

In axial compression for LRFD Not specied 3,190/FyIn exure for LRFD 2,030/Fy 8,990/FyIn plastic design for LRFD 1,300/Fyab width of element or projection (half the nominal width of rolled beams and tees; full width of angle legs and Z and channel

anges). For webs in exural compression, b should be taken as h, the clear distance between anges (less llets for rolled shapes) ordistance between adjacent lines of fasteners; t should be taken as tw , web thickness.

bAs required in AISC Specications for ASD and LRFD. These specications also set specic limitations on plate-girder components.cFy speciedminimumyieldstressof thesteel,ksi,but forhybridbeams,useFyt , theyieldstrength,ksi, ofanges;Fb allowablebending

stress, ksi, in the absence of axial force; Fr compressive residual stress in ange, ksi (10 ksi for rolled shapes, 16.5 ksi for welded shapes).dElements with width-thickness ratios that exceed the noncompact limits should be designed as slender sections.ekc 4.05/(h/t)0.46 for h/t. 70; otherwise kc 1.fD outside diameter; t section thickness.gFL smaller of (Fyf 2 Fr) or Fyw, ksi; Fyf yield strength, ksi, of anges and Fyw yield strength, ksi, of web.hkcc 4/(h/t)0.46 and 0.35 kcc 0.763.





where Cn 45,000kn/Fy(h/tw)2 for Cn , 0.8

36,000kn=Fy(h=tw)

2q

for Cn . 0.8

kn 4.00 5.34/(a/h)2 for a/h , 1.0 5.34 4.00/(a/h)2 for a/h . 1.0

a clear distance between transverse stif-feners

The allowable shear stress with tension-eld actionis

Fn Fy

289Cn 1 Cn

1:151 (a=h)2

p" #

0:40Fy (9:6)

Cn 1

Table 9.6 Maximum Width-Thickness Ratios b/ta for Compression Elements for Highway Bridgesb

Load-and-Resistance-Factor Designc

Description of Element Compact Noncompact d

Flange projection of rolled orfabricated I-shaped beams

65=Fy

p235

1

fc

2Dctw

qvuut

h

Webs in exural compressionwithout longitudinal stiffeners

640=Fy

p2Dctw

1150fc

pAllowable-Stress Designe


(Compression Members)fa , 0.44 Fy

fa 0.44 FyFy 36 ksi Fy 50 ksi

Plates supported on one side andoutstanding legs of angles

In main members 51/fa

p 12 12 11In bracing and other secondarymembers

51=fa

p 16 12 11Plates supported on two edges orwebs of box shapes f

126/fa

p 45 32 27Solid cover plates supported on twoedges or solid websg

158/fa

p 50 40 34Perforated cover plates supportedon two edges for box shapes

190/fa

p 55 48 41ab width of element or projection; t thickness. The point of support is the inner line of fasteners or llet welds connecting a plate

to the main segment or the root of the ange of rolled shapes. In LRFD, for webs of compact sections, b d, the beam depth, and fornoncompact sections, b D, the unsupported distance between ange components.

bAs required in AASHTO Standard Specication for Highway Bridges. The specications also provide special limitations onplate-girder elements.

cFy specied minimum yield stress, ksi, of the steel.dElements with width-thickness ratios that exceed the noncompact limits should be designed as slender elements.efa computed axial compression stress, ksi.fFor box shapes consisting of main plates, rolled sections, or component segments with cover plates.gFor webs connecting main members or segments for H or box shapes.hDc depth of web in compression, in; fc stress in compression ange, ksi, due to factored loads; tw web thickness, in.

9.16 n Section Nine




When the shear in the web exceeds Fn , stiffenersare required. See also Art. 9.13.

The area used to compute shear stress in a rolledbeam is dened as the product of the web thicknessand the overall beam depth. The webs of all rolledstructural shapes are of such thickness that shear isseldom the criterion for design.

At beam-end connections where the top angeis coped, and in similar situations in which fail-ure might occur by shear along a plane through thefasteners or by a combination of shear along aplane through the fasteners and tension along aperpendicular plane, AISC employs the blockshear concept. The load is assumed to be resistedby a shear stress of 0.30Fu along a plane throughthe net shear area and a tensile stress of 0.50Fu onthe net tension area, where Fu is the minimumspecied tensile strength of the steel.

Within the boundaries of a rigid connection oftwo or more members with webs lying in a com-mon plane, shear stresses in the webs generally arehigh. The Commentary on the AISC Specicationfor buildings states that such webs should bereinforced when the calculated shear stresses, suchas those along plane AA in Fig. 9.4, exceed Fv; thatis, when SF is larger than dctwFv, where dc is thedepth and tw is the web thickness of the memberresisting SF. The shear may be calculated from

SF M10:95d1

M20:95d2

Vs (9:7)

where Vs shear on the sectionM1 M1L M1GM1L moment due to the gravity load on the

leeward side of the connection

M1G moment due to the lateral load on theleeward side of the connection

M2 M2L 2M2GM2L moment due to the lateral load on

the windward side of the connection

M2G moment due to the gravity load on thewindward side of the connection

9.10.2 ASD for Shear in Bridges

Based on the AASHTO Specication for HighwayBridges, transverse stiffeners are required whereh=tw exceeds 150 and must not exceed a spacing, a,of 3h, where h is the clear unsupported distancebetween ange components, tw is the web thick-ness, and all dimensions are in inches. Wheretransverse stiffeners are required, the allowableshear stress, ksi, may be computed from

Fn Fy

3C 0:87(1 C)

1 (a=h)2p

" #(9:8)

where C 1.0 when htw,

190k

pFy

pC 190

k

p

(h=tw)Fy

p when 190k

pFy

p htw 237

k

pFy

pC 45,000

k

p

(h=tw)2Fy

p when htw.

237k

pFy

pSee also Art. 9.13.

Table 9.7 Allowable Tensile Stresses in Steel forBuildings and Bridges, ksi

Buildings Bridges

YieldStrength

OnGrossSection

OnNet

Section*

OnGrossSection

OnNet

Section*

36 22.0 29.0 20.0 29.050 30.0 32.5 27.5 32.5

* Based on A36 and A572 Grade 50 steels with Fu 58 ksi and65 ksi, respectively.

Fig. 9.4 Rigid connection of steel members withwebs in a common plane.





9.10.3 LRFD for Shear in Buildings

Based on the AISC Specications for LRFD forbuildings, the shear capacity Vu, kips, of exuralmembers with unstiffened webs may be computedfrom the following:

Vu 0:54FywAw when h=tw 4171=Fyw

q(9:9)

Vu 0:54FywAw417

1=Fyw

ph=tw

!

when 4171=Fyw

q, h=tw 523

1=Fyw

q(9:10)

Vu Aw 131,000(h=tw)

2

when 5231=Fyw

q, h=tw 260

(9:11)

where Fyw specied minimum yield stress ofweb, ksi

Aw web area, in2 dtwStiffeners are required when the shear ex-

ceeds Vu (Art. 9.13). In unstiffened girders, h/twmay not exceed 260. For shear capacity withtension-eld action, see the AISC Specication forLRFD.

9.10.4 LFD Shear Strength Designfor Bridges

Based on the AASHTO Specications for load-factor design, the shear capacity, kips, may be com-puted from:

Vu 0:58FyhtwC (9:12a)for exural members with unstiffened webs withh/tw , 150 or for girders with stiffened webs buta/h exceeding 3 or 67,600(h/tw)

2.

C 1:0 when ktw, b

bh=tw

when b htw 1:25b

45,000kFy(h=tw)

2when

h

tw. 1:25b

where b 190 k=Fypk 5 for unstiffened websk 5 b5=(a=h)2c for stiffened webs

For girders with transverse stiffeners and a/h lessthan 3 and 67,600(h/tw)

2, the shear capacity isgiven by

Vu 0:58Fydtw C 1 C1:15

1 (a=h)2

p" #

(9:12b)

Stiffeners are required when the shear exceeds Vu(Art. 9.13).

9.11 Allowable Compressionin Steel

The allowable compressive load or unit stress for acolumn is a function of its slenderness ratio. Theslenderness ratio is dened as Kl/r, where K effective-length factor, which depends on restraintsat top and bottom of the column; l length ofcolumn between supports, in; and r radius ofgyration of the column section, in. For com-bined compression and bending, see Art. 9.17.For maximum permissible slenderness ratios, seeArt. 9.8. Columns may be designed by allowable-stress design (ASD) or load-and-resistance-factordesign (LRFD).

9.11.1 ASD for Building Columns

The AISC Specication for ASD for buildings(Art. 9.7) provides two formulas for computingallowable compressive stress Fa, ksi, for mainmembers. The formula to use depends on therelationship of the largest effective slendernessratio Kl/r of the cross section of any unbracedsegment to a factor Cc dened by Eq. (9.13a).See Table 9.8a.

Cc 2p 2E

Fy

s 756:6

Fyp (9:13a)

where E modulus of elasticity of steel 29,000 ksi

Fy yield stress of steel, ksi

9.18 n Section Nine




When Kl/r is less than Cc,

Fa [1 (Kl=r)2=2C2c ]Fy

F.S.(9:13b)

where F.S. safety factor 5=3 3(Kl=r)=8Cc (Kl=r 3)=8C3c(See Table 9.8b).

When Kl/r exceeds Cc,

Fa 12p2E

23(Kl=r)2 150,000

(Kl=r)2(9:13c)

(See Table 9.8c.)The effective-length factor K, equal to the ratio

of effective-column length to actual unbracedlength, may be greater or less than 1.0. TheoreticalK values for six idealized conditions, in which jointrotation and translation are either fully realized ornonexistent, are tabulated in Fig. 9.5.

An alternative and more precise method ofcalculating K for an unbraced column uses anomograph given in the Commentary on theAISC Specication for ASD. This method requirescalculation of end-restraint factors for the topand bottom of the column, to permit K to be deter-mined from the chart.

9.11.2 ASD for Bridge Columns

In the AASHTO bridge-design Specications, al-lowable stresses in concentrically loaded columnsare determined from Eq. (9.14a) or (9.14b). WhenKl/r is less than Cc,

Fa Fy

2:121 (Kl=r)

2

2C2c

(9:14a)

When Kl/r is equal to or greater than Cc,

Fa p2E

2:12(Kl=r2) 135,000

(Kl=r)2(9:14b)

See Table 9.9.

9.11.3 LRFD for Building Columns

For axially loaded members with b/t , lr given inTable 9.5, the maximum load Pu, ksi, may becomputed from

Pu 0:85AgFy (9:15)

where Ag gross cross-sectional area of themember

Fcr (0:658l2c )Fy for l 1.5

Fcr 0:877l2c

Fy for l . 1.5

Table 9.8b Allowable Stresses Fa, ksi, in SteelBuilding Columns for Kl/r 120

Kl/rYield Strength of Steel Fy, ksi

36 50

10 21.16 29.2620 20.60 28.3030 19.94 27.1540 19.19 25.8350 18.35 24.3560 17.43 22.7270 16.43 20.9480 15.36 19.0190 14.20 16.94100 12.98 14.71110 11.67 12.34*120 10.28 10.37*

* From Eq. (9.13c) because Kl/r . Cc.

Table 9.8c Allowable Stresses, ksi, in SteelBuilding Columns for Kl=r . 120

Kl=r Fa

130 8.84140 7.62150 6.64160 5.83170 5.17180 4.61190 4.14200 3.73

Table 9.8a Values of Cc

Fy Cc

36 126.150 107.0





l Klrp

Fy

E

rThe AISC Specication for LRFD also presents

formulas for designing members with slenderelements.

9.11.4 LFD for Bridge Columns

Compression members designed by load-factordesign should have a maximum strength, kips,

Pu 0:85AsFcr (9:16)

where As gross effective area of column crosssection, in2.

For KLc=r 2p 2E=Fy

p,

Fcr Fy 1Fy

4p 2E

KLcr

2" #(9:17a)

For KLc=r .2p 2E=Fy

p,

Fcr p2E

(KLc=r)2 286,220

(KLc=r)2

(9:17b)

where Fcr buckling stress, ksiFy yield strength of the steel, ksiK effective-length factor in plane of

buckling

Lc length of member between supports, inr radius of gyration in plane of buck-

ling, in

E modulus of elasticity of the steel, ksi

Fig. 9.5 Values of effective-length factor K for columns.

Table 9.9 Column Formulas for Bridge Design

YieldStrength,

ksi

Allowable Stress, ksi

Cc Kl=r , Cc Kl=r Cc

36 126.1 16.98 2 0.00053 (Kl/r)2

50 107.0 23.58 2 0.00103 (Kl/r)2

90 79.8 42.45 2 0.00333 (Kl/r)2 135,000/(Kl/r)2

100 75.7 47.17 2 0.00412 (Kl/r)2

9.20 n Section Nine




Equations (9.17a) and (9.17b) can be simpliedby introducing a Q factor:

Q KLcr

2 Fy2p 2E

(9:18)

Then, Eqs. (9.17a) and (9.17b) can be rewritten asfollows:For Q , 1.0:

Fcr 1Q2

Fy (9:19a)

For Q . 1.0:

Fcr Fy

2Q(9:19b)

9.12 Allowable Stresses andLoads in Bending

In allowable-stress design (ASD), bending stressesmay be computed by elastic theory. The allowablestress in the compression ange usually governs theload-carrying capacity of steel beams and girders.

(T. V. Galambos, Guide to Design Criteria forMetal Compression Members, 5th ed., John Wiley& Sons, Inc., New York.)

9.12.1 ASD for Building Beams

The maximum ber stress in bending for laterallysupported beams and girders is Fb 0.66Fy if theyare compact (Art. 9.8), except for hybrid girdersand members with yield points exceeding 65 ksi.Fb 0.60Fy for noncompact sections. Fy is theminimum specied yield strength of the steel, ksi.Table 9.10 lists values of Fb for two grades of steel.

Because continuous steel beams have consider-able reserve strength beyond the yield point, aredistribution of moments may be assumed whencompact sections are continuous over supports

or rigidly framed to columns. In that case, negativegravity-load moments over the supports may bereduced 10%. If this is done, the maximum positivemoment in each span should be increased by 10%of the average negative moments at the span ends.

The allowable extreme-ber stress of 0.60Fyapplies to laterally supported, unsymmetricalmembers, except channels, and to noncompact-box sections. Compression on outer surfaces ofchannels bent about their major axis should notexceed 0.60Fy or the value given by Eq. (9.22).

The allowable stress of 0.66Fy for compactmembers should be reduced to 0.60Fy when thecompression ange is unsupported for a length, in,exceeding the smaller of

lmax 76:0bf

Fyp (9:20a)

lmax 20,000Fyd=Af

(9:20b)

where bf width of compression ange, ind beam depth, inAf area of compression ange, in2

The allowable stress should be reduced even morewhen l/rT exceeds certain limits, where l is theunbraced length, in, of the compression ange andrT is the radius of gyration, in, of a portion of thebeam consisting of the compression ange andone-third of the part of the web in compression.

For102,000Cb=Fy

p l=rT 510,000Cb=Fyp , useFb 2

3 Fy(l=rT)

2

1,530,000Cb

" #Fy (9:21a)

For l=rT .510,000Cb=Fy

p, use

Fb 170,000Cb(l=rT)

2(9:21b)

where Cb modier for moment gradient[Eq. (9.23)].

When, however, the compression ange is solidand nearly rectangular in cross section and its areais not less than that of the tension ange, theallowable stress may be taken as

Fb 12,000Cbld=Af

(9:22)

Table 9.10 Allowable Bending Stresses inBraced Beams for Buildings, ksi

Yield Strength,ksi

Compact(0.66Fy)

Noncompact(0.60Fy)

36 24 2250 33 30





When Eq. (9.22) applies (except for channels), Fbshould be taken as the larger of the valuescomputed from Eqs. (9.22) and (9.21a) or (9.21b)but not more than 0.60Fy .

The moment-gradient factor Cb in Eqs. (9.20) to(9.22) may be computed from

Cb 1:75 1:05M1M2

0:3 M1M2

2 2:3 (9:23)

where M1 smaller beam end momentM2 larger beam end moment

The algebraic sign ofM1/M2 is positive for double-curvature bending and negative for single-curvature bending. When the bending moment atany point within an unbraced length is larger thanthat at both ends, the value of Cb should be taken asunity. For braced frames, Cb should be taken asunity for computation of Fbx and Fbywith Eq. (9.65).

Equations (9.21a) and (9.21b) can be simpliedby introduction of a new term:

Q (l=rT)2Fy

510,000Cb(9:24)

Now, for 0.2 Q 1,

Fb (2Q)Fy

3(9:25)

For Q . 1,

Fb Fy

3Q(9:26)

As for the preceding equations, when Eq. (9.22)applies (except for channels), Fb should be taken asthe largest of the values given by Eqs. (9.22) and(9.25) or (9.26), but not more than 0.60Fy.

9.12.2 ASD for Bridge Beams

AASHTO (Art. 9.6) gives the allowable unit (tensile)stress in bending as Fb 0.55Fy (Table 9.11). Thesame stress is permitted for compression when the

compression ange is supported laterally for its fulllength by embedment in concrete or by othermeans.

When the compression ange is partly sup-ported or unsupported in a bridge, the allowablebending stress, ksi, is

Fb (5 107Cb=Sxc)(Iyc=L)

0:772J=Iyc 9:87(d=L)2

q 0:55Fy

(9:27)

where L length, in, of unsupported ange be-tween connections of lateral supports,including knee braces

Sxc section modulus, in3, with respect to thecompression ange

Iyc moment of inertia, in4, of the com-pression ange about the vertical axis inthe plane of the web

J 13 (bct3c btt3t Dt3w)bc width, in, of compression angebt width, in, of tension angetc thickness, in, of compression angett thickness, in, of tension angetw thickness, in, of webD depth, in, of webd depth, in, of exural member

In general, the moment-gradient factor Cb may becomputed from Eq. (9.23). It should be taken asunity, however, for unbraced cantilevers andmembers in which the moment within a signicantportion of the unbraced length is equal to or greaterthan the larger of the segment endmoments. If coverplates are used, the allowable static stress at thepoint of cutoff should be computed from Eq. (9.27).

The allowable compressive stress for bridgebeams may be roughly estimated from the ex-pressions given in Table 9.12, which are based on aformula used prior to 1992.

Table 9.12 Allowable Compressive Stress inFlanges of Bridge Beams, ksi

Fy Max l=b Fb

36 36 20 2 0.0075 (l/b)2

50 30 27 2 0.0144 (l/b)2

Table 9.11 Allowable Bending Stress in BracedBridge Beams, ksi

Fy Fb

36 2050 27

9.22 n Section Nine




9.12.3 LRFD for Building Beams

The AISC Specication for LRFD (Art. 9.6) permitsuse of elastic analysis as described previously forASD. Thus, negative moments produced by gravityloading may be reduced 10% for compact beams,if the positive moments are increased by 10% ofthe average negative moments. The reduction isnot permitted for hybrid beams, members ofA514 steel, or moments produced by loading oncantilevers.

For more accurate plastic design of multistoryframes, plastic hinges are assumed to form atpoints of maximum bending moment. Girders aredesigned as three-hingedmechanisms. The columnsare designed for girder plastic moments distribu-ted to the attached columns plus the moments dueto girder shears at the column faces. Additionalconsideration should be given to moment-end ro-tation characteristics of the column above and thecolumn below each joint.

For a compact section bent about the major axis,however, the unbraced length Lb of the com-pression ange where plastic hinges may form atfailure may not exceed Lpd given by Eqs. (9.28) and(9.29). For beams bent about the minor axis andsquare and circular beams, Lb is not restricted forplastic analysis.

For I-shaped beams, symmetric about both themajor and the minor axis or symmetric about theminor axis but with the compression ange largerthan the tension ange, including hybrid girders,loaded in the plane of the web,

Lpd 3480 2200(M1=M2)Fyc

ry (9:28)

where Fyc minimum yield stress of compressionange, ksi

M1 smaller of the moments, in-kips, atthe ends of the unbraced length ofbeam

M2 larger of the moments in-kips, at theends of the unbraced length of beam

ry radius of gyration, in, about minor axisThe plastic momentMp equals FyZ for homogenoussections, whereZ plastic modulus, in3 (Art. 6.65),and for hybrid girders, it may be computed fromthe fully plastic distribution. M1/M2 is positive forbeams with reverse curvature.

For solid rectangular bars and symmetric boxbeams,

Lpd 4930 2900(M1=M2)Fy

ry 2900ry

Fy(9:29)

The exural design strength is limited to0.90Mp or 0.90Mn, whichever is less. Mn is deter-mined by the limit state of lateral-torsional buck-ling and should be calculated for the region of thelast hinge to form and for regions not adjacent to aplastic hinge. The Specication gives formulas forMn that depend on the geometry of the section andthe bracing provided for the compression ange.

For compact sections bent about the major axis,for example, Mn depends on the following un-braced lengths:

Lb the distance, in, between points bracedagainst lateral displacement of the com-pression ange or between points bracedto prevent twist

Lp limiting laterally unbraced length, in, forfull plastic bending capacity

300ry=Fyf

pfor I shapes and channels,

Lb Lr 3750(ry=Mp)=

JA

pfor solid rectangular

bars and box beams, Lp LrFyf ange yield stress, ksiJ torsional constant, in4 (see AISC Manual

of Steel Construction on LRFD)

A cross-sectional area, in2Lr limiting laterally unbraced length, in, for

inelastic lateral buckling

For doubly symmetric I-shaped beams andchannels

Lr ryX1

FL

1

1 X2F2L

qr(9:30)

where FL smaller of Fyf 2 Fr or FywFyf specied minimum yield stress of

ange, ksi

Fyw specied minimum yield stress ofweb, ksi

Fr compressive residual stress in ange 10 ksi for rolled shapes, 16.5 ksi forwelded sections





X1 (p=Sx)EGJA=2

p

X2 (4Cw/Iy)(Sx/GJ)2E elastic modulus of the steelG shear modulus of elasticitySx section modulus about major axis, in3

(with respect to the compressionange if that ange is larger than thetension ange)

Cw warping constant, in6 (see AISC Man-ualLRFD)

Iy moment of inertia about minor axis,in4

For the aforementioned shapes, the limiting buck-ling moment Mr, ksi, may be computed from,

Mr FLSx (9:31)

For doubly symmetric shapes and channelswith Lb Lr , bent about the major axis

Mn Cb Mp (Mp Mr)Lb LpLr Lp

Mp (9:32)

where Cb 12:5Mmax2:5Mmax 3MA 4MB 3MC

Mmax absolute value of maximum momentin the unbraced segment, kip-in

MA absolute value of moment at quarterpoint of the unbraced segment, kip-in

MB absolute value of moment at centerlineof the unbraced segment, kip-in

MC absolute value of moment at three-quarter point of the unbraced segment,kip-in

Also, Cb is permitted to be conservatively taken as1.0 for all cases.

(See T. V. Galambos, Guide to Stability DesignCriteria for Metal Structures, 5th ed., JohnWiley &Sons, Inc., New York, for use of larger values of Cb.)

For solid rectangular bars and box section bentabout the major axis,

Lr 58,000ry

Mr

JA

p(9:33)

and the limiting buckling moment is given by

Mr FySx (9:34)

For doubly symmetric shapes and channelswith Lb . Lr , bent about the major axis,

Mn Mcr CbMr (9:35)

where Mcr critical elastic moment, kip-in.For shapes to which Eq. (9.30) applies,

Mcr Cb pLb

EIyGJ IyCw pE

Lb

2s(9:36a)

For solid rectangular bars and symmetric boxsections,

Mcr 57,000CbJA

pLb=ry

(9:36b)

For determination of the exural strength ofnoncompact plate girders and other shapes notcovered by the preceding requirements, see theAISC Manual on LRFD.

9.12.4 LFD for Bridge Beams

For load-factor design of symmetrical beams, thereare three general types of members to consider:compact, braced noncompact, and unbraced sec-tions. The maximum strength of each (moment,in-kips) depends on member dimensions andunbraced length as well as on applied shear andaxial load (Table 9.13).

The maximum strengths given by the formulasin Table 9.13 apply only when the maximum axialstress does not exceed 0.15FyA, where A is the areaof the member. Symbols used in Table 9.13 aredened as follows:

Dc depth of web in compressionFy steel yield strength, ksiZ plastic sectionmodulus, in3 (SeeArt. 6.65.)S section modulus, in3b0 width of projection of ange, ind depth of section, inh unsupported distance between anges, in

M1 smaller moment, in-kips, at ends of un-braced length of member

Mu FyZM1/Mu is positive for single-curvature bending.

9.24 n Section Nine




9.13 Plate Girders

Flexural members built up of plates that formhorizontal anges at top and bottom and joined tovertical or near vertical webs are called plate girders.They differ from beams primarily in that their webdepth-to-thickness ratio is larger, forexample, exceeds760=

Fb

pin buildings, where Fb is the allowable

bending stress, ksi, in the compression ange.The webs generally are braced by perpendicular

plates called stiffeners, to control local bucklingor withstand excessive web shear. Plate girdersare most often used to carry heavy loads or for longspans for which rolled shapes are not economical.

9.13.1 Allowable-Stress Design

In computation of stresses in plate girders, themoment of inertia I, in4, of the gross cross sectiongenerally is used. Bending stress fb due to bendingmoment M is computed from fb Mc/I, where c isthe distance, in, from the neutral axis to the extremeber. For determination of stresses in bolted orriveted girders for bridges, no deduction need bemade for rivet or bolt holes unless the reduction inange area, calculated as indicated in Art. 9.9,exceeds 15%; then the excess should be deducted.For girders for buildings, no deduction need bemade provided that

0:5FuAfn 0:6FyAfg (9:37a)

where Fy is the yield stress, ksi; Fu is the tensilestrength, ksi; Afg is the gross ange area, in

2; andAfn is the net ange area, in

2, calculated asindicated in Art. 9.9. If this condition is not met,member exural properties must be based on aneffective tension ange area, Afe, given by

Afe 5FuAfn

6Fy(9:37b)

In welded-plate girders, each ange shouldconsist of a single plate. It may, however, comprisea series of shorter plates of different thicknessjoined end to end by full-penetration groovewelds. Flange thickness may be increased ordecreased at a slope of not more than 1 in 2.5 attransition points. In bridges, the ratio of com-pression-ange width to thickness should notexceed 24 or 103=

fb

p, where fb computed maxi-

mum bending stress, ksi.The web depth-to-thickness ratio is dened as

h/t, where h is the clear distance between anges,in, and t is the web thickness, in. Several designrules for plate girders depend on this ratio.

9.13.2 Load-and-Resistance-FactorDesign

The AISC and AASHTO specications (Art. 9.6)provide rules for LRFD for plate girders. These arenot given in the following.

Table 9.13 Design Criteria for Symmetrical Flexural Sections for Load-Factor Design of Bridges

Type of Section MaximumBendingStrength

Mu, in-kips

FlangeMinimumThicknesstf , in**

WebMinimumThicknesstw, in**

MaximumUnbraced

Length Lb, in

Compact* FyZ (b0 Fyp )=65:0 (d Fyp )=608 ([3600 2200(M1=Mu)]ry)=Fy

Braced noncompact* FyS (b0 Fyp )=69:6 (Dc Fyp )=487 (20,000Af )=(Fyd)

Unbraced See AASHTO Specication

* Straight-line interpolation between compact and braced noncompact moments may be used for intermediate criteria, except thattw d

Fy

p=608 should be maintained.

** For compact sections, when both b0=tf and d=tw exceed 75% of the limits for these ratios, the following interaction equation applies:

d

tw 9:35 b

0

tf 1064

Fyfp

where Fyf is the yield strength of the ange, ksi; tw is the web thickness, in; and tf ange thickness, in.





9.13.3 Plate Girders in Buildings

For greatest resistance to bending, as much of aplate girder cross section as practicable should beconcentrated in the anges, at the greatest distancefrom the neutral axis. This might require, however,a web so thin that the girder would fail by webbuckling before it reached its bending capacity. Topreclude this, the AISC Specication (Art. 9.6)limits h/t. (See also Art. 9.8).

For an unstiffened web, this ratio should notexceed

h

t 14,000

Fy(Fy 16:5)p (9:38)

where Fy yield strength of compression ange,ksi.

Larger values of h/t may be used, however, ifthe web is stiffened at appropriate intervals.

For this purpose, vertical plates may be weldedto it. These transverse stiffeners are not required,though, when h/t is less than the value computedfrom Eq. (9.38) or given in Table 9.14.

With transverse stiffeners spaced not more than1.5 times the girder depth apart, the web clear-depth-to-thickness ratio may be as large as

h

t 2000

Fyp (9:39)

(See Table 9.14.) If, however, the web depth-to-thickness ratio h/t exceeds 760=

Fb

pwhere Fb, ksi, is

the allowable bending stress in the compressionange thatwould ordinarily apply, this stress shouldbe reduced to F0b, given by Eqs. (9.40) and (9.41).

F0b RPGReFb (9:40)

RPG 1 0:0005AwAf

h

t 760

Fbp

1:0

(9:41a)

Re 12 (Aw=Af )(3a a3)

12 2(Aw=Af )

1:0 (9:41b)

where Aw web area, in2Af area of compression ange, in2a 0.6Fyw/Fb 1.0

Fyw minimum specied yield stress, ksi, ofweb steel

In a hybrid girder, where the ange steel has ahigher yield strength than the web, Eq. (9.41b)protects against excessive yielding of the lower-strength web in the vicinity of the higher-strengthanges. For nonhybrid girders, Re 1.0.

Stiffeners on Building Girders n The shearand allowable shear stress may determine requiredweb area and stiffener spacing. Equations (9.5) and(9.6) give the allowable web shear Fn , ksi, for anypanel of a building girder between transversestiffeners.

The average shear stress fn , ksi, in a panel of aplate girder (web between successive stiffeners) isdened as the largest shear, kips, in the paneldivided by the web cross-sectional area, in2. As fnapproaches Fn given by Eq. (9.6), combined shearand tension become important. In that case, thetensile stress in the web due to bending in its planeshould not exceed 0.6Fy or (0.825 2 0.375fn/Fn)Fy ,where Fn is given by Eq. (9.6).

The spacing between stiffeners at end panels, atpanels containing large holes, and at panels ad-jacent to panels containing large holes, should besuch that fn does not exceed the value given by Eq.(9.5).

Intermediate stiffeners, when required, shouldbe spaced so that a/h is less than 3 and less than[260/(h/t)]2, where a is the clear distance, in,between stiffeners. Such stiffeners are not requiredwhen h/t is less than 260 and fn is less than Fncomputed from Eq. (9.5).

An innite combination of web thicknessesand stiffener spacings is possible with a particulargirder. Figure 9.6, developed for A36 steel, facil-itates the trial-and-error process of selecting asuitable combination. Similar charts can be deve-loped for other steels.

The required area of intermediate stiffeners isdetermined by

Ast 1 Cn2

a

h (a=h)

21 (a=h)2

p" #

YDht (9:42)

Table 9.14 Critical h/t for Plate Girders inBuildings

Fy, ksi 14,000=Fy(Fy 16:5)

p2000=

Fy

p36 322 33350 243 283

9.26 n Section Nine




where Ast gross stiffener area, in2 (total area, if inpairs)

Y ratio of yield point of web steel toyield point of stiffener steel

D 1.0 for stiffeners in pairs 1.8 for single-angle stiffeners 2.4 for single-plate stiffeners

If the computed web-shear stress fn is less than Fncomputed from Eq. (9.6),Astmay be reduced by theratio fn/Fn .

The moment of inertia of a stiffener or pair ofstiffeners should be at least (h/50)4.

The stiffener-to-web connection should bedesigned for a shear, kips/lin in of single stiffener,or pair of stiffeners, of at least

fns hFy

340

3s(9:43)

This shear may also be reduced by the ratio fn/Fn .Spacing of fasteners connecting stiffeners to the

girder web should not exceed 12 in c to c. If in-termittent llet welds are used, the clear distancebetween welds should not exceed 10 in or 16 timesthe web thickness.

Bearing stiffeners are required on webs whereends of plate girders do not frame into columns orother girders. They may also be needed under

concentrated loads and at reaction points. Bearingstiffeners should be designed as columns, assistedby a strip of web. The width of this strip may betaken as 25t at interior stiffeners and 12t at the endof the web. Effective length for l/r (slendernessratio) should be 0.75 of the stiffener length. See Art.9.18 for prevention of web crippling.

Butt-welded splices should be complete-pen-etration groove welds and should develop the fullstrength of the smaller spliced section. Other typesof splices in cross sections of plate girders shoulddevelop the strength required by the stresses at thepoint of splice but not less than 50% of the effectivestrength of the material spliced.

Flange connections may be made with rivets,high-strength bolts, or welds connecting ange toweb, or cover plate to ange. They should beproportioned to resist the total horizontal shearfrom bending. The longitudinal spacing of thefasteners, in, may be determined from

P Rq

(9:44)

where R allowable force, kips, on rivets, bolts, orwelds that serve length p

q horizontal shear, kips/inFor a rivet or bolt, R AnFn , where An is the

cross-sectional area, in2, of the fastener and Fn

Fig. 9.6 Chart for determining spacing of girder stiffeners of A36 steel.




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