Lecture -10 Beams

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CE-409: Lecture 10Prof. Dr Akhtar Naeem Khan

*Design of Steel Beams The development of a plastic stress distribution over the cross-section can be hindered by two different length effects: Lateral Torsional buckling of the unsupported length of the beam/member before the cross-section develops the plastic moment Mp. Local buckling of the individual plates (flanges and webs) of the cross-section before they develop the compressive yield stress Fy.


*Lateral Torsional Buckling A simply supported beam can be subjected to gravity transverse loading.Due to this loading the beam will deflect downward and its upper part will be placed in compression and hence will act as compression member.


*Lateral Torsional Buckling Beams are generally proportioned such that moment of inertia about the major principal axis is considerably larger than that of minor axis.This is done to make Economical Beams.


*Lateral Torsional Buckling

As result they are weak in resistance to Torsion and Bending about the Minor axis.If its Y-axis is not braced perpendicularly, it will buckle laterally at much smaller load than would otherwise have been required to produce a vertical failure.










*Lateral Torsional Buckling A laterally unsupported compression flange will behave like a column and tend to buckle out of plane between points of lateral support. However because the compression flange is part of a beam x-section with a tension zone that keeps the opposite flange in line, the x-section twists when it moves laterally. This behavior is referred to as lateral torsion buckling. Simply it is a sidewise buckling of beam accompanied by twist.


*Lateral Torsional Buckling Embedment of top flange in concrete slab provides lateral support to beam, except when the beam is cantilever.Lateral bracing will be adequate (both for strength & stiffness) if each lateral brace is designed for 2% of compressive force in the flange of beam it braces.( this thumb rule is based on lab test results).


*Lateral Torsional Buckling Consider a doubly symmetric prismatic beamBoth ends simply supported w.r.t x & y axis but Held against rotation about z-axis.It is subjected to a uniform bending moment Mx


*Lateral Torsional Buckling Moment at which Lateral Torsional buckling begins is given by:Mn = Mcr =


*Lateral Torsional Buckling The critical moments for beams with end moments and beams with transverse loads acting through shear center can be given byWhere Cb is a coefficient which depends on variation in moments along the span and K is an effective length coefficient depending on restraint at supports. Values of Cb and K are given in table 5-1


*Lateral Torsional Buckling Values of Cb developed by curve fitting to data from numerical analysis of LTB of simple beams acted only by end-moments is given by:Another equation obtained by working on numerical test data of beam-column behaviour is

Cb=1.75 + 1.05(M1/M2) + 0.3(M1/M2)2 2.3

Cb= 1/ [0.6 0.4(M1/M2) ] 2.3

Where M1 is smaller of two end moments.M1/M2 is +ve for reverse curvature.


*Lateral Torsional Buckling Accurate equation for Cb, if moment diagram within the un braced length is not a straight line


*Inelastic LTB If stress is proportional to strain, Mx,cr for elastic LTB is valid as given.But for critical stress, Fcr exceeding Fy, Mx,cr is given by


*Inelastic LTB The equation can be solved in a simplified manner by using an equivalent radius of gyration which is obtained by equating the critical bending stress to the tangent modulus critical stress for columns


*Local Buckling of Beam Elements Concept of Compact, Non-Compact, And Slender Elements and Sections.


*Local Buckling of Beam Elements For establishing width-thickness ratio limits for elements of compression members, the LRFD classification divides members into three distinct classification as follows.CompactNon-compactSlender


*Local Buckling Compact ElementsIf the slenderness ratio (b/t) of the plate element is less than lp, then the element is compact. It will locally buckle much after reaching Fy


*Local Buckling Non-compact ElementsIf the slenderness ratio (b/t) of the plate element is less than lr but greater than lp, then it is non-compact. It will locally buckle immediately after reaching Fy


*Local Buckling Slender ElementsIf the slenderness ratio (b/t) of the plate element is greater than lr then it is slender. It will locally buckle in the elastic range before reaching Fy


*Local Buckling Compact SectionsA section that can develop fully plastic moment Mp before local buckling of any of its compression element occurs.Non-compact SectionsA section that can develop a moment equal to or greater than My, but less than Mp, before loca buckling of any of its element occurs.Slender sectionsIf any one plate element is slender, the section is slender.


*Important NoteThus, slender sections cannot develop Mp due to elastic local buckling. Non-compact sections can develop My but not Mp before local buckling occurs. Only compact sections can develop the plastic moment Mp.

All rolled wide-flange shapes are compact with the following exceptions, which are non-compact.W40x174, W14x99, W14x90, W12x65, W10x12, W8x10, W6x15 (made from A992) Local Buckling


*Local Buckling contd;If the beam x-section is to develop the yield moment My, the compression flange must be able to reach yield stress and the web/webs, must be able to develop corresponding bending stresses.Local Buckling of the flange and/ or web can prevent these limits from being attained.More restrictive limits must be observed if a beam x-section is to attain the fully plastic moment Mp.


*Local BucklingFor uniformly compressed laterally simply supported on one unloaded edge and free on the other, the critical stress is

Plated used in structural members are long enough to warrant neglecting the second term, so


*Local Buckling Following limits of late slenderness (b/t) which preclude premature local buckling of compression flange of beams are available.Projecting ElementFlange of BoxSince these limits are not well defined, they differ somewhat from one specifications to another refer table 5-3


*Local BucklingLimiting values of beam flange and web slenderness

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Lecture -10 Beams