Bending strength AISC 9th EditionAllowable Stress in BendingThe allowable bending stress depends on the following criteria: the geometric shape of the cross-section; the axis of bending; the compactness of the section; and a length parameter.I-SectionsFor I-sections the length parameter is taken as the laterally unbraced length, l22, which is compared to a critical length, lc. The critical length is defined as Af is the area of compression flange.Major Axis of BendingIf l22 is less than lc, the major allowable bending stress for Compact and Noncompact sections is taken depending on whether the section is welded orrolled and whether fy is less than or equal to 65 ksi or greater than 65 ksi.

For Compact sections:Fb33 = 0.66 Fy if fy 65 ksi, (ASD F1-1)Fb33 = 0.60 Fy if fy > 65 ksi. (ASD F1-5)

For Noncompact sections:If the unbraced length l22 is greater than lc, then for both Compact and Noncompact I-sections the allowable bending stress depends on the l22 /rT ratio.

and Fb33 is taken not to be less than that given by the following formula:where,rT is the radius of gyration of a section comprising the compression flange and 1/3 the compression web taken about an axis in the plane of the web,Ma and Mb are the end moments of any unbraced segment of the member and Ma is numerically less than Mb; Ma / Mb being positive for double curvature bending and negative for single curvature bending. Also, if any moment within the segment is greater than Mb, Cb is taken as 1.0. Also, Cb is taken as 1.0 for cantilevers and frames braced against joint translation (ASD F1.3).The program defaults Cb to 1.0 if the unbraced length, l22, of the member isredefined by the user (i.e., it is not equal to the length of the member). Theuser can overwrite the value of Cb for any member by specifying it.The allowable bending stress for Slender sections bent about their major axisis determined in the same way as for a Noncompact section. Then the followingadditional considerations are taken into account.If the web is slender, the previously computed allowable bending stress is reducedas follows:F'b33 = RPGReFb33,

In the above expressions, Re is taken as 1, because currently the programdeals with only non-hybrid girders.If the flange is slender, the previously computed allowable bending stress istaken to be limited, as follows.F'b33 Qs (0.6 Fy), where (ASD A-B5.2a, A-B5.2d)Qs is defined earlier.

Fb33=Allowable bending stress assuming the section is non-compact, andF'b33=Allowable bending stress after considering web slendernessMinor Axis of BendingThe minor direction allowable bending stress Fb22 is taken as follows:For Compact sections:Fb22 = 0.75 Fy if fy 65 ksi, (ASD F2-1)

For Noncompact and Slender sections:

Channel SectionsFor Channel sections, the length parameter is taken as the laterally unbracedlength, l22, which is compared to a critical length, lc. The critical length is defined as

Major Axis of BendingIf l22 is less than lc, the major allowable bending stress for Compact and Noncompactsections is taken depending on whether the section is welded orrolled and whether fy is greater than 65 ksi or not.For Compact sections:Fb33 = 0.66 Fy if fy 65 ksi, (ASD F1-1)Fb33 = 0.60 Fy if fy > 65 ksi. (ASD F1-5)For Noncompact sections:

If the unbraced length l22 is greater than lc, then for both Compact and Noncompact Channel sections the allowable bending stress is taken as follows:

The allowable bending stress for Slender sections bent about their major axisis determined in the same way as for a Noncompact section. Then the followingadditional considerations are taken into account.If the web is slender, the previously computed allowable bending stress is reducedas follows:F'b33= ReRPGFb33 (ASD G2-1)If the flange is slender, the previously computed allowable bending stress istaken to be limited as follows:F'b33 = Qs (0.60 Fy) (ASD A-B5.2a, A-B5.2d)The definitions for rT, Cb, Af, Aw, Re, RPG, Qs, Fb33, and F'b33 are given earlier.Minor Axis of BendingThe minor direction allowable bending stress Fb22 is taken as follows:Fb22 = 0.60 Fy (ASD F2-2)T Sections and Double AnglesFor T sections and Double angles, the allowable bending stress for both majorand minor axes bending is taken as,Fb = 0.60 FyBox Sections and Rectangular TubesFor all Box sections and Rectangular tubes, the length parameter is taken as the laterally unbraced length, l22, measured compared to a critical length, lc.

The critical length is defined as

where Ma and Mb have the same definition as noted earlier in the formula for Cb. If l22 is specified by the user, lc is taken as 1,200b/Fy in the programMajor Axis of BendingIf l22 is less than lc, the allowable bending stress in the major direction of bending is taken as:Fb33 = 0.66 Fy (for Compact sections) (ASD F3-1)Fb33 = 0.60 Fy (for Noncompact sections) (ASD F3-3)If l22 exceeds lc, the allowable bending stress in the major direction of bendingfor both Compact and Noncompact sections is taken as:Fb33 = 0.60 Fy (ASD F3-3)The major direction allowable bending stress for Slender sections is determinedin the same way as for a Noncompact section. Then the following additionalconsideration is taken into account. If the web is slender, the previouslycomputed allowable bending stress is reduced as follows:F'b33 = ReRPGFb33 (ASD G2-1)The definitions for Re, RPG, Fb33 and F'b33 are given earlier.If the flange is slender, no additional consideration is needed in computingallowable bending stress. However, effective section dimensions are calculatedand the section modulus is modified according to its slenderness.Minor Axis of BendingIf l22 is less than lc, the allowable bending stress in the minor direction ofbending is taken as:Fb22 = 0.66 Fy (for Compact sections) (ASD F3-1)Fb22 = 0.60 Fy (for Noncompact and Slender sections) (ASD F3-3)If l22 exceeds lc, the allowable bending stress in the minor direction of bendingis taken, irrespective of compactness, as:Fb22 = 0.60 Fy (ASD F3-3)Pipe SectionsFor Pipe sections, the allowable bending stress for both major and minor axesof bending is taken as

Fb = 0.66 Fy (for Compact sections), and (ASD F3-1)Fb = 0.60 Fy (for Noncompact and Slender sections). (ASD F3-3)Round BarsThe allowable stress for both the major and minor axis of bending of roundbars is taken as,Fb= 0.75 Fy. (ASD F2-1)Rectangular and Square BarsThe allowable stress for both the major and minor axis of bending of solidsquare bars is taken as,Fb= 0.75 Fy. (ASD F2-1)For solid rectangular bars bent about their major axes, the allowable stress isgiven byFb= 0.60 Fy, andthe allowable stress for minor axis bending of rectangular bars is taken asFb= 0.75 Fy. (ASD F2-1)Single-Angle SectionsThe allowable flexural stresses for Single-angles are calculated based on their principal axes of bending (ASD SAM 5.3).Major Axis of BendingThe allowable stress for major axis bending is the minimum considering thelimit state of lateral-torsional buckling and local buckling (ASD SAM 5.1).The allowable major bending stress for Single-angles for the limit state of lateral-torsional buckling is given as follows (ASD SAM 5.1.3):where, Fob is the elastic lateral-torsional buckling stress as calculated below.The elastic lateral-torsional buckling stress, Fob, for equal-leg angles is taken as

and for unequal-leg angles, Fob is calculated aswhere,t = min(tw, tf),l = max(l22,l33),Imin = minor principal moment of inertia,Imax = major principal moment of inertia,Smajor = major section modulus for compression at the tip of one leg,rmin = radius of gyration for minor principal axis,

z = coordinate along the major principal axis,w = coordinate along the minor principal axis, andzo = coordinate of the shear center along the major principal axis withrespect to the centroid.w is a special section property for angles. It is positive for short leg in compression,negative for long leg in compression, and zero for equal-leg angles(ASD SAM 5.3.2). However, for conservative design in the program, it is always taken as negative for unequal-leg angles.

In the previous expressions, Cb is calculated in the same way as is done for Isections, with the exception that the upper limit of Cb is taken here as 1.5 insteadof 2.3.

The allowable major bending stress for Single-angles for the limit state of local buckling is given as follows (ASD SAM 5.1.1):where,t = thickness of the leg under consideration,b = length of the leg under consideration, andQ = slenderness reduction factor for local buckling.(ASD A-B5-2, SAM 4)In calculating the allowable bending stress for Single-angles for the limit stateof local buckling, the allowable stresses are calculated considering the factthat either of the two tips can be under compression. The minimum allowablestress is considered.Minor Axis of BendingThe allowable minor bending stress for Single-angles is given as follows (ASD SAM 5.1.1, 5.3.1b, 5.3.2b):In calculating the allowable bending stress for Single-angles, it is assumedthat the sign of the moment is such that both the tips are under compression.The minimum allowable stress is considered.General SectionsFor General sections, the allowable bending stress for both major and minoraxes bending is taken as,Fb = 0.60 Fy.

Design of steel beam (ASD, Allowable Stress design) Design code: AISC Allowable Stress Design 9th edition, 1989. Design requirements 1. Maximum bending stress, fb must not exceed allowable stress, Fb. 2. Deflection should not exceed allowable limit. 3. Maximum shear stress, fv shall not exceed allowable shear stress.

Design procedure: 1. Calculate design load. 2. Calculate design moment, M and bending stress, fb. 3. Select a trial beam size and calculate allowable bending stress, Fb (see below) 4. Calculate deflection and check with allowable deflection ratio. 5. Calculate design shear and shear stress, fv. 6. Calculate allowable shear stress, Fv.

Determine bending stress and shear stress Bending stress shall be determined as

fb= M/S

where M is design moment, S is section modulus.

Determine if section is compact, non-compact or slenderClassification of sectionfor a section to qualify as compact, its flanges must be continuously connected to the web or webs and the width-thickness ratios of its compression elements must not exceed the applicable limiting width-thickness ratios fromTable B5.1.

Steel sections that do not qualify as compact are classified as noncompact if the width-thickness ratios of the compression elements do not exceed the values shown for noncompact in Table B5.1.

If the width-thickness ratios of any compression element exceed the latter applicable value, the section is classified as a slender element section.

LIMITING SLENDERNESS RATIOSFor members whose design is based on compressive force, the slendemess ratio Kl/r preferably should not exceed 200. If this limit is exceeded, the allowable stress shall not exceed the value obtained from Equation (E2-2).

Determine Design WidthFor unstiffened elements which are supported along only one edge, parallel to the direction of the compression force, the width shall be taken as follows: a. For flanges of I-shaped members and tees, the width b is half the full nominal width.b. For legs of angles and flanges of channels and zees, the width b is the full nominal dimension.c. For plates, the width b is the distance from the free edge to the first row of fasteners or line of welds.d. For stems of tees, d is taken as the full nominal depth.

For stiffened elements, i.e., supported along two edges parallel to the direction of the compression force, the width shall be taken as follows:a. For webs of rolled, built-up or formed sections, h is the clear distance between flanges.b. For webs of rolled, built-up or formed sections, d is the full nominal depth.c. For flange or diaphragm plates in built-up sections, the width b is the distance between adjacent lines of fasteners or lines of welds.d. For flanges of rectangular hollow structural sections, the width b is the clear distance between webs less the inside corner radius on each side.If the corner radius is not known, the flat width may be taken as the total section width minus three times the thickness.For tapered flanges of rolled sections, the thickness is the nominal value halfway between the free edge and the corresponding face of the web.

Design of steel beam with W, I shape or Channel

Slender section design only for compression elements

Maximum width-thickness for compression flange for W, I and Channel section a. Compact section: bf/t 65/Fy. b. Non-compact section: bf/t 95/Fy. Determine UNSUPPORTED LENGTH Simply supported beams 1. For simply supported beam, the top flange is in compression. If the beam is directly attached to roof deck or floor slab, the compression flange is fully supported. The unsupported length Lb is 0. 2. When the beam supporting joists or other beams, and its flange is directly attached to the supported joists or other beams, the unsupported length is the spacing of the joists or other beams.

Cantilever beams: 1. For cantilever beam, the compression flange is at the bottom of the beam. If the bottom flange is unbraced, the unsupported length is the length of the cantilever beam. 2. If bracing is provided at the bottom flange, the unsupported length is the spacing between bracings.

Continuous beams: 1. For the positive moment portion of the beam, the compression flange is at the top of the beam. The unsupported length is determined as a simply supported beam. 2. For the negative moment portion of the beam, the compression flange is at the bottom of the beam. The unsupported length is determined as a cantilever beam.

Determine allowable bending stress Fb (ASD) W, I shape and channel hot-roll section bending about its major axis or shear center 1. Compact section: allowable bending stress,

Fb = 0.66 Fy if Lb Lc

where Fy is yield strength of steel members. Lb is laterally unsupported length of the compression flanges, Lc is the smaller of 76 bf/Fy or Lc = 20,000/[(d/Af)Fy] bf is the width of the flange,Af is area of the flange.

l = distance between cross sections braced against twist or lateral displacement of the compression flange, in. For cantilevers braced against twist only at the support, l may conservatively be taken as the actual length. rT = radius of gyration of a section comprising the compression flange plus lh of the compression web area, taken about an axis in the plane of the web, in. Af = area of the compression flange, in. Cb = 1.75 + 1.05 (M1/M2)+ 0.3 (M1/M2)^2 but not more than 2.3* where the larger bending moment at the ends of the unbraced length, taken about the strong axis of the member, and where M1/M2, the ratio of end moments, is positive when M1 have the same sign (reverse curvature bending) and negative when they are of opposite signs (single curvature bending). When the bending moment at any point within an unbraced length is larger than thatat both ends of this length, the value of Cb shall be taken as unity. When computing Fbx to be used in Equation (Hl-1), Cb may be computed by the equation given above for frames subject to joint translation, and it shall be taken as unity for frames braced against joint translation. Cb may conservatively be taken as unity for cantilever beams.**

*It is conservative to take Cb as unity. For values smaller than 2.3, see Table 6 in the Numerical Values Section. **For the use of larger Cb values, see Galambos (1988).2. Non-compact section: allowable bending stress

Fb = 0.60 Fy if Lb 76 bf/Fy

3. Compact or non-compact section,

Fb is the larger of the following:

When THEN [ASD F1-6]

When THEN [ASD F1-7]

For any value of l/rt [ASD F1-8] USED FOR BENDING OF CHANNELS

Note: for channels bent, allowable stress is determined from [F1-8]. Where rT (inch) is radius of gyration of a section comprising the compression flange plus 1/3 of the compression web area, taken about an axis in the plan of web. (note: rT is available in AISC steel table for most W and I section) . Equations (Fl-6) and (Fl-7) may be refined to include both St. Venant and warping torsion by substituting a derived value for rT. The equivalent radius of gyration, rTequiv, can be obtained by equating the appropriate expression giving the critical elastic bending stress for the compression flange of a beam with that of an axially loaded column (Galambos, 1988).

wherely = minor axis moment of inertia of Sx = major axis section modulusAf is area of flange, Cb = 1.75+1.05(M1/M2)+0.3(M1/M2) but not more than 2.3. Cb must be taken as unity when computing Fbx for use in Equation (Hl-1) for frames braced against joint translation (Galambos, 1988).M1 and M2 are the smaller and the larger applied moments M1/M2 is positive if M1 and M2 have the same sign. W and I shape hot-roll section bending about its minor axis, 1. Compact section: allowable bending stress,

Fb = 0.75 Fy

2. Non-compact section: allowable bending stress

Fb = 0.60 Fy

BENDING Y-AXIS OF I- AND SOLID ROUND & SOLID RECTANGULAR & SQUARE BAR

ALLOWABLE STRESS: BENDING OF BOX MEMBERS, RECTANGULAR TUBES AND CIRCULAR TUBESProvision for compact circular members is given in Table B5.1

Supplement No. 3 (1974) to the 1969 Specification added Equation (F3-2), an unsupported length criteria for compact tubular members with rectangular cross sections. The equation recognizes the effect of moment gradient, and tests have shown it to be conservative (Sherman, 1976).

Box-type members are torsionally very stiff (Galambos, 1988). The critical flexural stress due to lateral-torsional buckling, for the compression flange of a box-type beam loaded in the plane of its minor axis so as to bend about its major axis, can be obtained using Equation (E2-1) with an equivalent slenderness ratio, by the expression

1. Members with Compact SectionsCompact section:- For members bent about their strong or weak axes - members with compact sections as defined in Sect. B5 and - flanges continuously connected to the webs, the allowable stress is

Fb = 0.66 Fy (F3-1)

Box-shaped memberif box-shaped section, To be classified as a compact section, a box-shaped member shall have, in addition to the requirements in Sect. B5:- a depth not greater than 6 times the width, 6bd- a flange thickness not greater than 2 times the web thickness and tf2tw- a laterally unsupported length Lb less than or equal toLc = (1,950 + 1,200 M1/M2)b/Fy (F3-2) (Lb Lc)

except that it need not be less than 1,200 (b/Fy),

where M1 is the smaller bending moment at the ends of the unbraced lengthM2 the larger bending moment at the ends of the unbraced length, taken about the strong axis of the member, and M1/M2, the ratio of end moment:- positive when M1and M2 have the same sign (reverse curvature bending)- negative when M1and M2 are of opposite signs (single curvature bending).

2. Members With Noncompact SectionsFor box-type and tubular flexural members that meet the noncompact section requirements of Sect. B5, the allowable stress is Fb = 0.60 Fy (F3-3)

- Lateral bracing is not required for a box section whose depth is less than 6 times its width. if(6bd)- Lateral-support requirements for box sections of larger depth to-width ratios must be determined by special analysis.Summary

BENDING OF SLENDER SECTIONS