Sổ tay kết cấu thép - Section 10

COLD-FORMED STEEL DESIGN

10.1SECTION 10

COLD-FORMED STEEL DESIGN

Cold-formed members for most application are designed in accordance with the Specificationfor the Design of Cold-Formed Steel Structural Members, American Iron and Steel Institute,Washington, DC Generally referred to as the AISI Specification, it applies to members cold-

formed to shape from carbon or low-alloy steel sheet, strip, plate, or bar, not more than in thick, used for load carrying purposes in buildings With appropriate allowances, it canbe used for other applications as well The vast majority of applications are in a thicknessrange from about 0.014 to 0.25 in.

1-The design information presented in this section is based on the AISI Specification andits Commentary, including revisions being processed The design equations are written in

dimensionless form, except as noted, so that any consistent system of units can be used Asynopsis of key design provisions is given in this section, but reference should be made tothe complete specification and commentary for a more complete understanding.

The AISI Specification lists all of the sheet and strip materials included in Table 1.6 (Art.

1.4) as applicable steels, as well several of the plate steels included in Table 1 (A36, A242,A588, and A572) A283 and A529 plate steels are also included, as well as A500 structuraltubing (Table 1.7) Other steels can be used for structural members if they meet the ductilityrequirements The basic requirement is a ratio of tensile strength to yield stress not less than1.08 and a total elongation of at least 10% in 2 in If these requirements cannot be met,alternative criteria related to local elongation may be applicable In addition, certain steelsthat do not meet the criteria, such as Grade 80 of A653 or Grade E of A611, can be used

10.2 SECTION TEN

FIGURE 10.1 Typical cold-formed steel members.

for multiple-web configurations (roofing, siding, decking, etc.) provided the yield stress istaken as 75% of the specified minimum (or 60 ksi or 414 MPa, if less) and the tensile stressis taken as 75% of the specified minimum (or 62 ksi or 428 MPa if less) Some exceptionsapply Suitability can also be established by structural tests.

As the name suggests, the cross section of a cold-formed member is achieved by a bendingoperation at room temperature, rather than the hot rolling process used for the heavier struc-tural steel shapes The dominant cold forming process is known as roll-forming In thisprocess, a coil of steel is fed through a series of rolls, each of which bends the sheetprogressively until the final shape is reached at the last roll stand The number of roll standsmay vary from 6 to 20, depending upon the complexity of the shape Because the steel isfed in coil form, with successive coils weld-spliced as needed, the process can achieve speedsup to about 300 ft / min and is well suited for quantity production Small quantities may beproduced on a press-brake, particularly if the shape is simple, such as an angle or channelcross section In its simplest form, a press brake consists of a male die which presses thesteel sheet into a matching female die.

In general, the cold-forming operation is beneficial in that it increases the yield strengthof the material in the region of the bend The flat material between bends may also showan increase due to squeezing or stretching during roll forming This increase in strength isattributable to cold working and strain aging effects as discussed in Art 1.10 The strengthincrease, which may be small for sections with few bends, can be conservatively neglected.

Alternatively, subject to certain limitations, the AISI Specification includes provisions forusing a section-average design yield stress that includes the strength increase from cold-

forming Either full section tension tests, full section stub column tests, or an analyticalmethod can be employed Important parameters include the tensile-strength-to-yield-stress

TABLE 10.1 Safety Factors and Resistance Factors Adopted by the AISI Specification

Flexural members(a) Bending strength

Sections with stiffened or partially stiffened compression flanges1.670.95Sections with unstiffened compression flanges1.670.90

Beams having one flange through-fastened to deck or sheathing (C- or Z-sections)1.670.90Beams having one flange fastened to a standing seam roof system1.670.90(b) Web design

Shear strength controlled by yielding (Condition a, Art 10.12.4)1.501.00Shear strength controlled by buckling (Condition b or c, Art 10.12.4)1.670.90Web crippling of single unreinforced webs1.850.75

Web crippling of two nested Z-sections1.800.85Stiffeners

(a) Groove welds

(b) Arc spot welds

Shear, minimum edge distance2.500.60 / 0.70

Connected part, longitudinal loading

Weld length / sheet thickness⬍252.500.60Weld length / sheet thicknessⱖ252.500.55Connected part, transverse loading2.500.60

10.4 SECTION TEN

TABLE 10.1 Safety Factors and Resistance Factors Adopted by the AISI Specification (Continued )

factor,␾(e) Flare groove welds

(b) Tension strength on net section

With washers, double shear connection2.000.65With washers, single shear connection2.220.55Without washers, double or single shear2.220.65

(e) Tensile strength of bolts2.00 / 2.250.75

* Fuis tensile strength and Fsyis yield stress.

ratio of the virgin steel and the radius-to-thickness ratio of the bends The forming operationmay also induce residual stresses in the member but these effects are accounted for in theequations for member design.

The nominal loads for design should be according to the applicable code or specificationunder which the structure is designed or as dictated by the conditions involved In the absenceof a code or specification, the nominal loads should be those stipulated in the American

Society of Civil Engineers Standard, Minimum Design Loads for Buildings and Other tures, ASCE 7 The following loads are used for the primary load combinations in the AISISpecification:

Struc-D ⫽ Dead load, which consists of the weight of the member itself, the weight of allmaterials of construction incorporated into the building which are supported by the mem-ber, including built-in partitions; and the weight of permanent equipment

E⫽Earthquake load

L⫽Live loads due to intended use and occupancy, including loads due to movable objectsand movable partitions and loads temporarily supported by the structure during mainte-

nance (L includes any permissible load reductions If resistance to impact loads is taken

into account in the design, such effects should be included with the live load.)

Lr⫽Roof live load

The AISI Specification is structured such that nominal strength equations are given for various

types of structural members such as beams and columns For allowable stress design (ASD),the nominal strength is divided by a safety factor and compared to the required strengthbased on nominal loads For Load and Resistance Factor Design (LRFD), the nominalstrength is multiplied by a resistance factor and compared to the required strength based onfactored loads These procedures and pertinent load combinations to consider are set forthin the specification as follows.

ASD Strength Requirements.A design satisfies the requirements of the AISI Specification

when the allowable design strength of each structural component equals or exceeds therequired strength, determined on the basis of the nominal loads, for all applicable loadcombinations This is expressed as

where R⫽ required strength

Rn⫽ nominal strength (specified in Chapters B through E of the Specification)

⍀ ⫽safety factor (see Table 10.1)

Rn/⍀ ⫽allowable design strength

ASD Load Combinations In the absence of an applicable code or specification or if the

applicable code or specification does not include ASD load combinations, the structure andits components should be designed so that allowable design strengths equal or exceed theeffects of the nominal loads for each of the following load combinations:

in-10.6 SECTION TEN

Composite Construction under ASD For the composite construction of floors and roofs

using cold-formed deck, the combined effects of the weight of the deck, the weight of thewet concrete, and construction loads (such as equipment, workmen, formwork) must beconsidered.

LRFD Strength Requirements.A design satisfies the requirements of the AISI Specification

when the design strength of each structural component equals or exceeds the requiredstrength determined on the basis of the nominal loads, multiplied by the appropriate loadfactors, for all applicable load combinations This is expressed as

where Ru⫽required strength

Rn⫽nominal strength (specified in chapters B through E of the Specification)

␾ ⫽resistance factor (see Table 10.1)␾Rn⫽design strength

LRFD Load Factors and Load Combinations In the absence of an applicable code or

specification, or if the applicable code or specification does not include LRFD load nations and load factors, the structure and its components should be designed so that designstrengths equal or exceed the effects of the factored nominal loads for each of the followingcombinations:

Several exceptions apply:

1 The load factor for E in combinations (5) and (6) should equal 1.0 when the seismic load

model specified by the applicable code or specification is limit state based.

2 The load factor for L in combinations (3), (4), and (5) should equal 1.0 for garages, areas

occupied as places of public assembly, and all areas where the live load is greater than100 psf.

3 For wind load on individual purlins, girts, wall panels and roof decks, multiply the load

factor for W by 0.9.

4 The load factor for Lrin combination (3) should equal 1.4 in lieu of 1.6 when the rooflive load is due to the presence of workmen and materials during repair operations.

Composite Construction under LRFD For the composite construction of floors and roofs

using cold-formed deck, the following additional load combination applies:

1.2DS⫹1.6CW⫹1.4C (10.3)

where DS⫽weight of steel deck

CW⫽weight of wet concrete

C⫽construction load (including equipment, workmen, and form work but excludingwet concrete

10.5SECTION PROPERTY CALCULATIONS

Because of the flexibility of the manufacturing method and the variety of shapes that can bemanufactured, properties of cold-formed sections often must be calculated for a particularconfiguration of interest rather than relying on tables of standard values However, properties

of representative or typical sections are listed in the Cold-Formed Steel Design Manual,American Iron and Steel Institute, 1996, Washington, DC (AISI Manual ).

Because the cross section of a cold-formed section is generally of a single thickness of

steel, computation of section properties may be simplified by using the linear method With

this method, the material is considered concentrated along the centerline of the steel sheet

and area elements are replaced by straight or curved line elements Section properties arecalculated for the assembly of line elements and then multiplied by the thickness, t Thus,the cross section area is given by A⫽L⫻t, where L is the total length of all line elements;the moment of inertia of the section is given by I ⫽ I⬘ ⫻ t, where I⬘ is the moment ofinertia determined for the line elements; and the section modulus is calculated by dividing

I by the distance from the neutral axis to the extreme fiber, not to the centerline of the

extreme element As subsequently discussed, it is sometimes necessary to use a reduced or

effective width rather than the full width of an element.

Most sections can be divided into straight lines and circular arcs The moments of inertiaand centroid location of such elements are defined by equations from fundamental theory aspresented in Table 10.2.

The design of cold-formed steel differs from heavier construction in that elements of

mem-bers typically have large width-to-thickness (w / t) ratios and are thus subject to local

buck-ling Figure 10.2 illustrates local buckling in beams and columns Flat elements in pression that have both edges parallel to the direction of stress stiffened by a web, flange,

com-lip or stiffener are referred to as stiffened elements Examples in Fig 10.2 include the topflange of the channel and the flanges of the I-cross section column.

To account for the effect of local buckling in design, the concept of effective width isemployed for elements in compression The background for this concept can be explainedas follows.

Unlike a column, a plate does not usually attain its maximum load carrying capacity atthe buckling load, but usually shows significant post buckling strength This behavior isillustrated in Fig 10.3, where longitudinal and transverse bars represent a plate that is simplysupported along all edges As the uniformly distributed end load is gradually increased, thelongitudinal bars are equally stressed and reach their buckling load simultaneously However,as the longitudinal bars buckle, the transverse bars develop tension in restraining the lateraldeflection of the longitudinal bars Thus, the longitudinal bars do not collapse when theyreach their buckling load but are able to carry additional load because of the transverserestraint The longitudinal bars nearest the center can deflect more than the bars near theedge, and therefore, the edge bars carry higher loads after buckling than do the center bars.The post buckling behavior of a simply supported plate is similar to that of the gridmodel However, the ability of a plate to resist shear strains that develop during buckling

also contributes to its post buckling strength Although the grid shown in Fig 10.3a buckled

into only one longitudinal half-wave, a longer plate may buckle into several waves as

illus-trated in Figs 10.2 and 10.3b For long plates, the half-wave length approaches the widthb.

After a simply supported plate buckles, the compressive stress will vary from a maximumnear the supported edges to a minimum at the mid-width of the plate as shown by line 1 of

10.8 SECTION TEN

TABLE 10.2 Moment of Inertia for Line Elements

Source:Adapted from Cold-Formed Steel Design Manual, American Iron and Steel Institute, 1996,

Washington, DC.

FIGURE 10.2 Local buckling of compression elements (a) In beams; (b) incolumns (Source: Commentary on the Specification for the Design of Cold-

Formed Steel Structural Members, American Iron and Steel Institute, Washington,DC, 1996, with permission.)

Fig 10.3c As the load is increased the edge stresses will increase, but the stress in the

mid-width of the plate may decrease slightly The maximum load is reached and collapse isinitiated when the edge stress reaches the yield stress—a condition indicated by line 2 of

Fig 10.3c.

The post buckling strength of a plate element can be considered by assuming that afterbuckling, the total load is carried by strips adjacent to the supported edges which are at auniform stress equal to the actual maximum edge stress These strips are indicated by the

dashed lines in Fig 10.3c The total width of the strips, which represents the effective widthof the element b, is defined so that the product of b and the maximum edge stress equals

the actual stresses integrated over the entire width The effective width decreases as theapplied stress increases At maximum load, the stress on the effective width is the yieldstress.

Thus, an element with a small enough w / t will be able to reach the yield point and will

be fully effective Elements with larger ratios will have an effective width that is less thanthe full width, and that reduced width will be used in section property calculations.

The behavior of elements with other edge-support conditions is generally similar to thatdiscussed above However, an element supported along only one edge will develop only oneeffective strip.

Equations for calculating effective widths of elements are given in subsequent articles

based on the AISI Specification These equations are based on theoretical elastic buckling

theory but modified to reflect the results of extensive physical testing.

10.10 SECTION TEN

FIGURE 10.3 Effective width concept (a) Buckling of grid model; (b) buckling ofplate; (c) stress distributions.

10.7MAXIMUM WIDTH-TO-THICKNESS RATIOS

The AISI Specification gives certain maximum width-to-thickness ratios that must be adhered

For flange elements, such as in flexural members or columns, the maximum flat

width-to-thickness ratio, w / t, disregarding any intermediate stiffeners, is as follows:

Stiffened compression element having one longitudinal edge connected to a web or flangeelement, the other stiffened by

(a) a simple lip, 60

(b) other stiffener with IS⬍Ia, 90

(c) other stiffener with ISⱖIa, 90

Stiffened compression element with both longitudinal edges connected to other stiffenedelements, 500

Unstiffened compression element, 60

In the above, ISis the moment of inertia of the stiffener about its centroidal axis, parallel to

the element to be stiffened, and Iais the moment of inertia of a stiffener adequate for theelement to behave as a stiffened element Note that, although greater ratios are permitted,

stiffened compression elements with w / t⬎250, and unstiffened compression elements with

w / t⬎30 are likely to develop noticeable deformations at full design strength, but ability todevelop required strength will be unaffected.

For web elements of flexural members, the maximum web depth-to-thickness ratio, h / t,

disregarding any intermediate stiffeners, is as follows:

Unreinforced webs, 200

Webs with qualified transverse stiffeners that include (a) bearing stiffeners only, 260(b) bearing and intermediate stiffeners, 300

The effective width for load capacity determination depends on a slenderness factor␭definedas

1.052 w ƒ

␭ ⫽ 兹k 冉 冊t 冪E (10.4)

where k ⫽plate buckling coefficient (4.0 for stiffened elements supported by a web alongeach longitudinal edge; values for other conditions are given subsequently)ƒ⫽maximum compressive stress (with no safety factor applied)

E⫽Modulus of elasticity (29,500 ksi or 203 000 MPa)

10.12 SECTION TEN

FIGURE 10.4 Illustration of uniformly compressed stiffened element (a) Actual element; (b) stress oneffective element (Source:Specification for the Design of Cold-Formed Steel Structural Members, Amer-ican Iron and Steel Institute, Washington, DC, 1996, with permission.)

For flexural members, when initial yielding is in compression, ƒ⫽Fy, where Fyis the yieldstress; when the initial yielding is in tension, ƒ⫽the compressive stress determined on thebasis of effective section For compression members, ƒ⫽column buckling stress.

The effective width is as follows:

when␭ ⱕ0.673, b⫽w (10.5)when␭ ⬎0.673, b⫽␳w (10.6)where the reduction factor␳is defined as

␳ ⫽(1⫺0.22 /␭) /␭ (10.7)Figure 10.4 shows the location of the effective width on the cross section, with one-halflocated adjacent to each edge.

Effective widths determined in this manner, based on maximum stresses (no safety factor)define the cross section used to calculate section properties for strength determination How-ever, at service load levels, the effective widths will be greater because the stresses aresmaller, and another set of section properties should be calculated Therefore, to calculateeffective width for deflection determination, use the above equations but in Eq 10.4, sub-stitute the compressive stress at design loads, ƒd.

Elements with stress gradients include webs subjected to compression from bending aloneor from a combination of bending and uniform compression For load capacity determination,

the effective widths b1 and b2 illustrated in Fig 10.5 must be determined First, calculatethe ratio of stresses

␺ ⫽ƒ / ƒ2 1 (10.8)where ƒ1and ƒ2are the stresses as shown, calculated on the basis of effective section, withno safety factor applied In this case ƒ1is compression and treated as⫹, while ƒ2can beeither tension (⫺) or compression (⫹) Next, calculate the effective width, be, as if theelement was in uniform compression (Art 10.8.1) using ƒ1for ƒ and with k determined as

FIGURE 10.5 Illustration of stiffened element with stress gradient (a) Actual element; (b) stress on fective element varying from compression to tension; (c) stress on effective element with non-uniform com-pression (Source:Specification for the Design of Cold-Formed Steel Structural Members, American Ironand Steel Institute, Washington, DC, 1996, with permission.)

10.14 SECTION TEN

FIGURE 10.6 Illustration of uniformly compressed unstiffened element (a) Actual element; (b) stresson effective element (Source: Specification for the Design of Cold-Formed Steel Structural Members,

American Iron and Steel Institute, Washington, DC, 1996, with permission.)

The effective widths for uniformly compressed unstiffened elements are calculated in the

same manner as for stiffened elements (Art 10.8.1), except that k in Eq 10.4 is taken as

0.43 Figure 10.6 illustrates the location of the effective width on the cross section.

The effective width for unstiffened elements (including edge stiffeners) with a stress gradientis calculated in the same manner as for uniformly loaded stiffened elements (Art 10.9.1)

except that (1) k in Eq 10.4 is taken as 0.43, and (2) the stress ƒ3is taken as the maximumcompressive stress in the element Figure 10.7 shows the location of ƒ3 and the effectivewidth for an edge stiffener consisting of an inclined lip (Such lips are more structurallyefficient when bent at 90⬚, but inclined lips allow nesting of certain sections.)

ELEMENTS WITH EDGE STIFFENER

A commonly encountered condition is a flange with one edge stiffened by a web, the otherby an edge stiffener (Fig 10.7) To determine its effective width for load capacity determi-nation, one of three cases must be considered The case selection depends on the relation

between the flange flat width-to-thickness ratio, w / t, and the parameter S defined as

For each case an equation will be given for determining Ia, the moment of inertia required

for a stiffener adequate so that the flange element behaves as a stiffened element, ISis themoment of inertia of the full section of the stiffener about its centroidal axis, parallel to the

element to be stiffened A⬘Sis the effective area of a stiffener of any shape, calculated bymethods previously discussed The reduced area of the stiffener to be used in section property

calculations is termed ASand its relation to A⬘S is given for each case Note that for edgestiffeners, the rounded corner between the stiffener and the flange is not considered as partof the stiffener in calculations The following additional definitions for a simple lip stiffener

illustrated in Fig 10.7 apply The effective width dS⬘ is that of the stiffener calculated cording to Arts 10.9.1 and 10.9.2 The reduced effective width to be used in section property

ac-FIGURE 10.7 Illustration of element with edge stiffener (a) Actual element; (b) stress on effective element andstiffener (Source:Specification for the Design of Cold-Formed Steel Structural Members, American Iron and SteelInstitute, Washington, DC 1996, with permission.)

calculations is termed dSand its relation to dS⬘is given for each case For the inclined stiffener

of flat depth d at an angle␪as shown in Fig 10.7,

IS⫽(d t sin ␪) / 12 (10.13)

A⬘ ⫽SdS⬘t (10.14)

Limit d / t to 14.Case I: w / tⱕS / 3

For this condition, the flange element is fully effective without an edge stiffener so b⫽

buckling coefficient kaand other terms are determined as follows:

For a simple lip stiffener with 140⬚ ⱖ␪ ⱖ40⬚ and D / wⱕ0.8 (see Fig 10.7),

ka⫽5.25⫺5(D / w)ⱕ4.0 (10.17)

dS⫽C d2 S⬘ (10.18)For other stiffeners,

The nominal tensile strength, Tn, of an axial loaded tension member is the smallest of threelimit states: (1) yielding in the gross section, Eq 10.22; (2) fracture in the net section awayfrom the connections, Eq 10.23; and (3) fracture in the net section at connections (Art.10.18.2)

Tn⫽A Fgy (10.22)

Tn⫽A Fnu (10.23)

where Agis the gross cross section area, Anis the net cross section area, Fy is the design

yield stress and Fuis the tensile strength.

As with all of the member design provisions, these nominal strengths must be dividedby a safety factor,⍀, for ASD (Art 10.4.1) or multiplied by a resistance factor,␾, forLRFD (Art 10.4.2) See Table 10.1 for⍀and␾values for the appropriate member orconnection category.

In the design of flexural members consideration must be given to bending strength, shearstrength, and web crippling, as well as combinations thereof, as discussed in subsequentarticles Bending strength must consider both yielding and lateral stability In some appli-cations, deflections are also an important consideration.

10.12.1Nominal Strength Based on Initiation of Yielding

For a fully braced member, the nominal strength, Mn, is the effective yield moment basedon section strength:

Mn⫽S Fey (10.24)

where Seis the elastic section modulus of the effective section calculated with the extreme

fiber at the design yield stress, Fy The stress in the extreme fiber can be compression ortension depending upon which is farthest from the neutral axis of the effective section Ifthe extreme fiber stress is compression, the effective width (Art 10.8–10.10) and the effective

section can be calculated directly based on the stress Fyin that compression element ever, if the extreme fiber stress is tension, the stress in the compression element depends onthe effective section and, therefore, a trial and error solution is required (Art 10.22).

For this condition, the nominal strength, Mn, of laterally unbraced segments of singly-,

dou-bly-, and point-symmetric sections is given by Eq 10.25 These provisions apply to I-, Z-,C-, and other singly-symmetric sections, but not to multiple-web decks, U- and box sections.

Also, beams with one flange fastened to deck, sheathing, or standing seam roof systems aretreated separately The nominal strength is

10.18 SECTION TEN

For singly-symmetric sections, x-axis is the axis of symmetry oriented suchthat the shear center has a negative x-coordinate For point-symmetric sec-tions, use 0.5 Me.

Alternatively, Mecan be calculated using the equation for doubly-symmetric

I-sections or point-symmetric sections given in (b).

Me⫽CsA␴ex[ j⫹Cs兹j2⫹r (o2␴t/␴ex)] / CT Ffor bending (10.30)about the centroidal axis perpendicular to the symmetry axis for singly-symmetric sections only

Cs⫽ ⫹1 for moment causing compression on the shear center side of the centroid

Cs⫽ ⫺1 for moment causing tension on the shear center side of the centroid␴ex⫽(K L / r )x␲2xEx2 (10.31)␴ey⫽(K L / r )y␲2yEy2 (10.32)

where Mmax⫽absolute value of maximum moment in the unbraced segment

MA⫽absolute value of moment at quarter point of unbraced segment

MB⫽absolute value of moment at centerline of unbraced segment

MC ⫽absolute value of moment at three-quarter point of unbraced segment Cbispermitted to be conservatively taken as unity for all cases.

For cantilevers or overhangs where the free end is unbraced, Cb⫽1.0 Formembers subject to combined compressive axial load and bending moment

(Art 10.15), Cb⫽ 1.0.

E⫽Modulus of elasticity

CT F⫽0.6⫺0.4 (M1/ M2) (10.35)where

M1 is the smaller and M2 the larger bending moment at the ends of the

unbraced length in the plane of bending, and where M1/ M2, the ratio of end

moments, is positive when M1and M2have the same sign (reverse curvaturebending) and negative when they are of opposite sign (single curvature bend-ing) When the bending moment at any point within an unbraced length islarger than that at both ends of this length, and for members subject to com-

bined compressive axial load and bending moment (Art 10.15), CT F⫽1.0.

ro⫽Polar radius of gyration of the cross section about the shear center

ro⫽兹rx2⫹ry2⫹xo2 (10.36)

rx, ry⫽Radii of gyration of the cross section about the centroidal principal axes

G ⫽Shear modulus (11,000 ksi or 78 000 MPa)

Kx, Ky, Kt⫽Effective length factors for bending about the x- and y-axes, and for twistingLx, Ly, Lt⫽Unbraced length of compression member for bending about the x- and y-axes,

and for twisting

xo⫽Distance from the shear center to the centroid along the principal x-axis, taken

as negative

J ⫽St Venant torsion constant of the cross section

C ⫽Torsional warping constant of the cross section

TABLE 10.3 R values for Simple Spans

(See Eq 10.40)

Section depth, d, in Profile R*dⱕ 6.5C or Z0.706.5⬍ d ⱕ 8.5 C or Z0.658.5⬍ d ⱕ 11.5 Z0.508.5⬍ d ⱕ 11.5 C0.40

* For simple spans, multiply R by the correctionfactor r to account for the effects of compressed

insulation between the sheeting and the member:

For uncompressed batt insulation of thickness ti, the

L⫽ Unbraced length of the member

Iyc⫽ Moment of inertia of the compression portion of a section about the gravityaxis of the entire section parallel to the web, using the full unreduced sectionOther terms are defined in (a).

Deck or Sheathing

If the tension flange of a beam is screwed to deck or sheathing and the compression flangeis unbraced, such as a roof purlin or wall girt subjected to wind suction, the bending strengthlies between that for a fully braced member and an unbraced member This is due to therotational restraint provided by the spaced connections Therefore, based on numerous tests,

the AISI Specification gives the nominal strength in terms of a reduction factor R applied to

the nominal strength for the fully braced condition (Art 10.12.1):

10.20 SECTION TEN

panel characteristics, span length (33 ft maximum), fasteners, and fastener spacing (12 inmaximum).

Standing Seam Roof System

If the flange of a supporting beam is fastened to a standing seam roof system, the bendingstrength generally lies between that for a fully braced member and an unbraced member, butmay equal that for a fully braced member The strength depends on the details of the system,as well as whether the loading is gravity or uplift, and cannot be readily calculated There-

fore, the AISI Specification allows the nominal strength to be calculated by Eq 10.40, butwith the reduction factor R determined by representative tests of the system Test specimensand procedures are detailed in the ‘‘Base Test Method’’ given in the AISI Manual Alterna-

tively, the rules for discrete point bracing (Art 10.12.2) can be used.

The AISI specification gives three equations for nominal shear strength of beam webs for

three categories or conditions of increasing web slenderness Condition (a) is based on ing, condition (b) is based on inelastic buckling, and condition (c) is based on elastic buck-ling.

yield-(a) For h / tⱕ0.96兹Ek / Fvy

Vn⫽0.60F hty (10.41)(b) For 0.96兹Ek / Fvy⬍h / tⱕ1.415兹Ek /Fvy

h ⫽Depth of the flat portion of the web measured along the plane of the web

kv⫽Shear buckling coefficient determined as follows:

1 For unreinforced webs, kv⫽5.34

2 For beam webs with transverse stiffeners satisfying AISI requirements

when a / hⱕ1.0

kv⫽4.00⫹ 2 (10.44)

(a / h)when a / h⬎1.0

kv⫽5.34⫹ 2 (10.45)

(a / h)

where a⫽the shear panel length for unreinforced web element

⫽the clear distance between transverse stiffeners for reinforced web elements.

For a web consisting of two or more sheets, each sheet is considered as a separate elementcarrying its share of the shear force.

Combinations of bending and shear may be critical at locations such as near interior supports

of continuous beams To guard against this condition, the AISI Specification provides

tradi-tional interaction equations, which depend on whether the beam web is unreinforced ortransversely stiffened Although similar in concept, for clarity, separate equations are givenfor ASD and LRFD Symbols have common definitions except as noted.

ASD Method.For beams with unreinforced webs, the required flexural strength, M, andrequired shear strength, V, must satisfy the following:

⍀bM ⍀vV

0.6冉 冊 冉 冊Mnxo ⫹ Vn ⱕ1.3 (10.47)where⍀b⫽factor of safety for bending (Table 10.1)

⍀v⫽factor of safety for shear (Table 10.1)

Mn⫽nominal flexural strength when bending alone exists

Mnxo⫽nominal flexural strength about the centroidal x-axis determined in accordance

with AISI, excluding lateral buckling

Vn⫽nominal shear force when shear alone exists

LRFD Method.For beams with unreinforced webs, the required flexural strength, Mu, and

required shear strength, Vu, must satisfy the following:

冉 冊 冉 冊␾bMnxo ␾vVn

For beams with transverse web stiffeners the required flexural strength, Mu, and the

required shear strength, Vu, shall not exceed ␾bMn and ␾vVn, respectively When

Mu/ (␾bMnxo)⬎0.5 and Vu/ (␾vVn)⬎0.7, then Muand Vumust satisfy the following tion equation:

0.6冉 冊 冉 冊␾bMnxo ⫹ ␾vVn ⱕ1.3 (10.49)where␾b⫽resistance factor for bending (Table 10.1)

␾v⫽resistance factor for shear (Table 10.1)

Mn⫽nominal flexural strength when bending alone exists

At points of concentrated loads or reactions, the webs of cold-formed members are

suscep-tible to web crippling If the web depth-to-thickness ratio, h / t, is greater than 200, stiffeners

10.22 SECTION TEN

TABLE 10.4 Equation Numbers for Nominal Strength of Webs at Concentrated Load or Reaction*

Load spacing

Single web shapesStiffened or

partially stiffenedflanges

Double web shapes(back-to-back C’s

or similar)All types of flangesLoad on either flange or opposing loads,

bearing edges spaced⬎1.5h Beam endsBeam interiors

Eq 10.50a†Eq 10.50d

Eq 10.50bEq 10.50d

Eq 10.50cEq 10.50e

Opposing loads, bearing edges spaced

ⱕ1.5h Beam endsBeam interiors

Eq 10.50fEq 10.50h

Eq 10.50ƒ

Eq 10.50h

Eq 10.50gEq 10.50i

* When Fyⱖ66.5 ksi (459 MPa), the value of kC3should be taken as 1.34 in Eq 10.50a, 10.50b, and 10.50f

** Beam ends condition applies when distance from edge of bearing to end of beam is⬍1.5h; otherwise, beam interiors condition

Table 10.4 indicates the equation to be used for the various conditions The nominal

strength to resist a concentrated load or reaction, per web, Pn(kips or N ), is calculated from

the following equations:

Pn⫽t kC C C C [331349␪ ⫺0.61(h / t)][1⫹0.01(N / t)](10.50a)

Pn⫽t kC C C C [2173 4 9 ␪ ⫺0.28(h / t)][1⫹0.01(N / t)](10.50b)When N / t⬎60, the factor [1⫹0.01(N / t)] may be increased to [0.71⫹0.015(N / t)]

Pn⫽t F C (10.0y 6 ⫹1.25兹N / t)(10.50c)

Pn⫽t kC C C C [538129␪ ⫺0.74(h / t)][1⫹0.007(N / t)](10.50d )When N / t⬎60, the factor [1⫹0.007(N / t)] may be increased to [0.75⫹0.011(N / t)]

C1⫽1.22⫺0.22k(10.51a)C2⫽1.06⫺0.06R / tⱕ1.0 (10.51b)

h / t 1

⫽冋1.10⫺665册k, when h / t⬎66.5 (10.51i)h / t 1

C8⫽冋0.98⫺865册k(10.51j )C9⫽1.0 for U.S customary units, kips and in

⫽6.9 for metric units, N and mm

C␪⫽ 0.7⫹0.3(␪/ 90)2 (10.51k)Fy⫽ design yield stress of the web, ksi (MPa)

h⫽ depth of the flat portion of the web measuredalong the plane of the web, in (mm)

loads distributed over unequal bearing lengths,

the smaller value of N appliesR⫽ inside bend radius

␪ ⫽ angle between the plane of the web and theplane of the bearing surfaceⱖ45⬚, but notmore than 90⬚

For beams with unreinforced flat webs, combinations of bending and web crippling near

concentrated loads or reactions must satisfy interaction equations given in the AISI