ENCE 710
Design of Steel Structures
V. Lateral-Torsional Buckling of
Beams
C. C. Fu, Ph.D., P.E.
Civil and Environmental Engineering Department
University of Maryland
Introduction
Following subjects are covered:
 Lateral Torsional Buckling (LTB)
 Flange Local Buckling (FLB)
 Web Local Buckling (WLB)
 Shear strength
 Lateral Bracing Design
Reading:
 Chapters 9 of Salmon & Johnson
 AISC LRFD Specification Chapters B (Design Requirements)
and F (Design of Members for Flexure)
2
Introduction
A beam can fail by reaching the plastic moment and
becoming fully plastic (see last section) or fail
prematurely by:
1.
LTB, either elastically or inelastically
2.
FLB, either elastically or inelastically
3.
WLB, either elastically or inelastically
If the maximum bending stress is less than the
proportional limit when buckling occurs, the failure
is elastic. Else it is inelastic.
For bending bMn(b = 0.9)
3
Design of Members for Flexure
(about Major Axis)
4
Lateral Torsional Buckling (LTB)






Compact Members (AISC F2)
Failure Mode
Plastic LTB (Yielding)
Inelastic LTB
Elastic LTB
Moment Gradient Factor Cb
5
Lateral Torsional Buckling (cont.)

Failure Mode
A beam can buckle in a
lateral-torsional mode
when the bending
moment exceeds the
critical moment.
6
Lateral Torsional Buckling (cont.)

Nominal Flexural Strength Mn



Lb  L p
Lp  Lb  Lr
plastic when
inelastic when
elastic when
and M n  M p
and M p  M n  M r
and
Mn  Mr
Lb  Lr
Mn
C b  1.0
Mp
plastic
inelastic
elastic
Mr
Lb
Lp
Lr
7
Lateral Torsional Buckling (cont.)
I-Beam
in a
Buckled
Position
8
Lateral Torsional Buckling (cont.)

Elastic LTB
 coupled differential equations for rotation and lateral
translation
(8.5.10)
d
d 3
M z  GJ
where
Mz
z

G
J
E
Cw
=
=
=
=
=
=
=
dz
 ECw
dz 3
moment at location z along member axis
axis along member length
angle of twist
shear modulus
torsional constant (AISC Table 1-1 for torsional prop.)
modulus of elasticity
warping constant (AISC Table 1-1 for warping)
9
Lateral Torsional Buckling (cont.)

Plastic LTB (Yielding)

Flexural Strength
M n  M p  Fy Z
(AISC F2-1)
where Z= plastic section modulus & Fy= section yield stress
 Limits


Lateral bracing limit
Lb  Lp  1.76ry
Flange and Web width/thickness limit
E
Fy
(AISC F2-5)
(AISC Table B4.1)
(Note: Lpd in Salmon & Johnson Eq. (9.6.2) is removed from AISC 13th Ed.)
10
Lateral Torsional Buckling (cont.)

Inelastic LTB

Lp  Lb  Lr
Flexure Strength (straight line interpolation)

 Lb  Lp 
  M p
M n  Cb  M p  M p  M r  
L L 
p 

 r

(9.6.4)
or

 Lb  Lp 
  M p
M n  Cb  M p  M p  0.7 Fy S x  
L L 
p 

 r

(AISC F2-2)
11
Lateral Torsional Buckling (cont.)

Elastic LTB

Lb  Lr
Flexure Strength
(AISC F2-3)
M n  Fcr S x  M p
Fcr 
Cb 2 E
 Lb 
 
 rts 
2
Jc  Lb 
 
1  0.078
S x ho  rts 
2
(AISC F2-4)
(The square root term may be conservatively taken equal to 1.0)
(c in AISC F2-8a,b for doubly symmetric I-shape, and channel, respectively)

Limit
E
Lr  1.95rts
0.7 Fy
rts 
2
I y Cw
Sx
 0.7 Fy S x ho 
Jc
1  1  6.76

S x ho
Jc 
 E
2
(AISC F2-6)
(AISC F2-7)
12
Lateral Torsional Buckling (cont.)

Moment Gradient Factor Cb


The moment gradient factor Cb accounts for the variation
of moment along the beam length between bracing
points. Its value is highest, Cb=1, when the moment
diagram is uniform between adjacent bracing points.
When the moment diagram is not uniform
(9.6.3)
12.5M max
Cb 
(AISC F1-1)
2.5M max  3M A  4 M B  3M C
where
Mmax= absolute value of maximum moment in unbraced length
MA, MB, MC= absolute moment values at one-quarter, one-half, and
three-quarter points of unbraced length
13
Cb for a Simple Span Bridge
14
Nominal Moment Strength Mu as
affected by Cb
15
Flange Local Buckling (FLB)





Compact Web and Noncompact/Slender
Flanges (AISC F3)
Failure Mode
Noncompact Flange
Slender Flange
Nominal Flexural strength, Mn = Min (F2, F3)
16
Flange Local Buckling (cont.)

Failure Mode
The compression flange
of a beam can buckle
locally when the
bending stress in the
flange exceeds the
critical stress.
17
Flange Local Buckling (cont.)

Nominal Flexural Strength Mn



bf / 2t f   p
plastic when
inelastic when
elastic when
and M n  M p
and M p  M n  M r
Mn  Mr
and
 p  bf / 2t f  r
bf / 2t f  r
Mn
Mp
compact
noncompact
slender
Mr
λp
λr
λ
bf
tf
18
Flange Local Buckling (cont.)

Noncompact Flange (straight line
interpolation)

Flexure Strength
    pf
M n  M p  M p  0.7 Fy S x  
 
pf
 rf




(AISC F3-1)
19
Flange Local Buckling (cont.)

Slender Flange

Flexure Strength
Mn 
0.9 Ekc S x
kc 
4
h / tw
2
(AISC F3-2)
(kc shall not be less than 0.35 and not greater than 0.76)

Limit (AISC Table B4.1)
20
Web Local Buckling (WLB)






Compact or Noncompact Webs (AISC F4)
Failure Mode
Compact Web (Yielding)
Noncompact Web
Slender Web
Nominal Flexural Strength, Mn=min (compression
flange yielding, LTB, compression FLB, tension
flange yielding)
21
Web Local Buckling (cont.)

Failure Mode
The web of a beam can
also buckle locally
when the bending
stress in the web
exceeds the critical
stress.
22
Web Local Buckling (cont.)

Nominal Flexural Strength Mn



plastic when
inelastic when
elastic when
and M n  M p
and M p  M n  M r
and
Mn  Mr
  p
 p    r
  r
Mn
Mp
compact
noncompact
slender
Mr
λp
λr
h
tw
h
λ
λ
23
Web Local Buckling (cont.)

Compression Flange Yielding

Flexural Strength
M n  R pc M yc  R pc Fy S xc
(AISC F4-1)
where Rpc= web plasticification factor (AISC F4-9a, b) &
Fy= section yield stress

Limits (AISC Tables B4.1)
E
Lb  Lp  1.1rt
Fy
24
Web Local Buckling (cont.)

LTB (Inelastic)

Lp  Lb  Lr
Flexure Strength

    pf
M n  Cb  R pc M yc  R pc M yc  FL S xc  
 
pf

 rf

  M p


(AISC F4-12)
where FL= a stress determined by AISC F4-6a, b
25
Web Local Buckling (cont.)

LTB (Elastic)

Flexure Strength
Lb  Lr
(AISC F4-3)
M n  Fcr S xc  R pc M yc
C E
J  Lb 
 
Fcr  b 2 1  0.078
S x ho  rt 
 Lb 
 
 rt 
2

Limit
(AISC F4-5)
2
(AISC Table B4.1)
E
Lr  1.95rt
FL
2
J
F S h 
1  1  6.76 L x o  (AISC F4-8)
S x ho
E J 
26
Web Local Buckling (cont.)

Compression FLB (Noncompact Flange)

Flexure Strength

    pf
M n   R pc M yc  R pc M yc  FL S xc  
 
pf

 rf


  M p


(AISC F4-12)
Compression FLB (Slender Flange)

Flexure Strength
Mn 
0.9 Ekc S x
kc 
4
h / tw
2
(AISC F4-13)
(kc shall not be less than 0.35 and not greater than 0.76)
27
Web Local Buckling (cont.)

Tension Flange Yielding S xt  S xc

Flexure Strength
M n  R pt M yt  R pt Fy S xt
(AISC F4-14)
Rpt = web plastification factor to the tension flange yielding limit
(a) hc/tw ≤ pw
(b) hc/tw > pw
Rpt=Mp/Myt
(AISC F4-15a)
(AISC F4-15b)
Mp  Mp
     pw  M p
 
R pt  

 1 




 M yt  M yt
  rw   pw  M yt
28
Shear Strength
Failure Mode
 Shear-Buckling Coefficient
 Elastic Shear Strength
 Inelastic Shear Strength
 Plastic Shear Strength
For shear vVn(v = 0.9 except certain rolled Ibeam h/tw≤2.24√E/Fy, v = 1.0)
Vn=0.6FyAwCv
(AISC G2-1)

29
Shear Strength (cont.)

Failure Mode
The web of a beam or
plate girder buckles
when the web shear
stress exceeds the
critical stress.
30
Shear Strength (cont.)

Nominal Shear Strength Vn (v = 0.9)



plastic when
inelastic when
elastic when
and
and
and
  p
  r
  r
  y
  0.8 y
  cr
  r
Vn
Vp
plastic
inelastic
elastic
Vr
λp
λr
λ
31
Shear Strength (cont.)

AISC G2 Nominal Shear Strength Vn
(a)
For
(a)
(a)
For
For
k E
h
 1.10 v
tw
Fyw
1.10
kv E
k E
h

 1.37 v
Fy
tw
Fy
1.37
kv E
h

Fyw t w
Cv  1.0
(AISC G2-3)

kv E 
1
.
10

Fy 
 (AISC
Cv  
h


tw




 1.51Ek 
v
Cv  

2
 h  F 
  tw  y 
G2-4)
(AISC G2-5)
32
Lateral Bracing Design
33
Lateral Bracing Design
AISC Provisions – Stability
Bracing Design for Beams


1. For stiffness βreqd,
βreqd = 2 βideal
2. For nominal strength Fbr,
(a) Fbr = βideal (2Δ0);
(b) Fbr = βideal (0.004Lb)
Where βideal = Pcr/Lb
34
Lateral Bracing Design (cont.)
AISC Provisions – LRFD Stability Bracing Design for Beams
1. Relative bracing
Re quired
Φ=0.75
Re quired
2. Nodal bracing
Re quired
Re quired
Prb 
0.008M r Cd
h0
1  4M C 
r d

 rb  
  Lb h0 
Prb 
0.02 M r Cd
h0
1  10M r Cd
br  
  Lb h0



35
Lateral Bracing Design (cont.)
AISC Provisions – LRFD Stability Bracing Design for Columns
1. Relative bracing
Re quired
Prb  0.004 Pr
Re quired
 rb   r 
  Lb 
Φ=0.75
2. Nodal bracing
Re quired
Re quired
1  2P 
Prb  0.01Pr
1  8Pr
br  
  Lb



36