Lecture Notes #5Lateral buckling effect

1

Professor Guowei MaOffice: 160

Tel: 61-8-6488-3102Email: ma@civil.uwa.edu.au

Laterally Unrestrained Beams

Lateral deflection

Twist

Lateral torsional buckling of beam

Buckled point at midspan

Dead weight load applied vertically

Buckled position

Unloaded position

Clamped at root

Response of a slendercantilever beam to vertical loading - LTB

• When designing a steel beam, it is usual to think first of the need to provide adequate strength and stiffness against vertical bending.

• Thus, a typical cross section of a beam normally has a stiffness in the vertical plane that is much greater than that in the horizontal plane.

• When a slender structural element is loaded in its stiff plane (axially in the case of the strut), there exists a tendency for it to fail by buckling in a more flexible plane (by deflecting sideways in the case of a beam).

• It is illustrate the response of a slender cantilever beam to a vertical end load. The phenomenon is termed lateral-torsional buckling (LTB).

• LTB involves both lateral deflection and twisting about a vertical axis through the web.

B

BP

z

yx

P

uSection BB

Strut buckling

B

B

z

yx

M

M

Beam (lateral torsional)buckling

Section BBEIx > EIyEIx > GJ

u

Figure 35 Similarity between strut buckling and beam buckling

Lateral Torsional Stability• Lateral-torsional instability influences

the design of laterally unrestrained beams in much the same way that flexural buckling influences the design of columns.

• Thus, the bending strength in the presence of lateral torsional instability (Buckling Resistance Moment) would be a function of:• beam’s (lateral) slenderness, • end/support conditions, • shape of cross section, • bending moment profile along the

span, etc.

Dead weight load applied vertically

Buckled position

Unloaded position

Clamped at root

Influencing Factors:

(1) Unbraced span lengthIt is the distance between points at which lateral deflection is prevented.

(2) Shape of cross-sectionThe web and the tension flange are more important for relatively shallow sections (eg. UCs) than for deep slender sections (eg. UBs).

(3) Distribution (profile) of Moments along the spanWhen moment is nonuniform, the force in the compression flange will no longer

be constant. The members are expected to be more stable than similar members under a more high uniform pattern of moment.

(4) End restraintEnd Restraints influence the buckling shape of the beam (Figure 36).

(5) Presence of lateral RestraintsIf a beam is laterally continuous as illustrated in Figure 37, buckling involves the whole span with the more stable segments restraining the critical segment. A safe result may be obtained by basing design on the strength of the weakest segment.

Free to rotate about horizontal axis only

Bending moment diagrams

Buckled shapes (plan view) and effective lengthsLE LE

Free to rotate about both axes normal to longitudinal axis

Free to rotate about vertical axis only

Effect of end restraint in plan or elevation on LTB

LE

A

B

CD Beam loaded by crossing

beams which provide lateral support to points B & C

Plan view of buckled shapeAB C

D

Note: Laterally, beam AD is a continuous beams with intermediate supports at B and C.Vertically, beam AD is a simply supported beam with point loads at B and C.

AB C

D Elevation view of buckled shape

Section Restraint

9

• Full restraint (F)— prevents the lateral displacement of the critical flange of 

the cross‐section and prevents twisting of the section.

• Partial restraint (P)— prevents the critical flange of the cross‐section from 

displacing laterally and partly prevents the section from twisting.

• Lateral restraint (L)— prevents lateral displacement of the critical flange 

without preventing the twist of the section.

• Continuous lateral restraint (C)—a critical flange restraint provided 

continuously by a concrete slab, chequer plate or timber floor with the 

requirement that the segment ends are fully or partly restrained . 

• Lateral rotation restraint (LR)— prevents rotation of the critical flange about 

the section’s minor axis.

• Full lateral restraint—a beam or beam segment with F, P or L restraints to the 

critical flange 

10

• Critical flange—the flange that would displace laterally and rotate further if 

the restraints were removed. This is the compression flange of a simple beam 

and tension flange of a cantilever.

• Critical section—the cross‐section that governs the beam design with the 

largest ratio of M* to Ms.

• Segment—a portion of a beam between fully (F) or partially (P) restrained 

cross‐sections. Restraint combinations (left and right) can be FF, PP or FP.

• Segment length, l—length of the beam between restraints type F, P or L. For a 

beam having FF or PP end restraints and no mid‐span restraints, the segment 

length is equal to the beam span. 

11

12

13

14

15

Member capacity

16

sssmb MMM

• Segments fully or partially restrained at both ends

αm = a moment modification factor

α s = a slenderness reduction factor

Ms = the nominal section moment capacity determined in 

accordance with Clause 5.2 for the gross section

Moment Amplification Factor αm

(i) 1.0 

(ii) a value obtained from Table 5.6.1

(iii)

(iv) a value obtained from an elastic buckling analysis

17

5.2

7.12*

42*

32*

2

*

MMM

M mm

Mm* = maximum design bending moment in the segmentM2* , M4* = design bending moments at the quarter points of the segmentM3* = design bending moment at the midpoint of the segment;

18

Slenderness Reduction Factor αs

19

oa

s

oa

ss M

MMM

36.02

(A) Moa = Mo, where Mo is the reference buckling moment; or(B) the value determined from an elastic buckling analysis in 

accordance with Clause 5.6.4.

2

2

2

2

e

w

e

yo L

EIGJ

L

EIM

E, G  = the elastic moduli (see Clause 1.4)Iy, J, and Iw = section constants (see Clause 1.4)le  = the effective length determined in accordance with Clause 5.6.3.

Member Capacity

20

• Segments unrestrained at one end

ob

s

ob

ss M

MMM

36.02

sssb MMM

Effective Length Le

21

LkkkL rlte

kt = a twist restraint factor given in Table 5.6.3(1)kl = a load height factor given in Table 5.6.3(2)kr = a lateral rotation restraint factor given in Table 5.6.3(3)

22

23