ENCE 710 Design of Steel
Structures
V. Lateral-Torsional Buckling of Beams
C. C. Fu, Ph.D., P.E.Civil and Environmental Engineering
DepartmentUniversity of Maryland
2
IntroductionFollowing subjects are covered: Lateral Torsional Buckling (LTB) Flange Local Buckling (FLB) Web Local Buckling (WLB) Shear strength Lateral Bracing DesignReading: Chapters 9 of Salmon & Johnson AISC LRFD Specification Chapters B (Design
Requirements) and F (Design of Members for Flexure)
3
IntroductionA beam can fail by reaching the plastic moment and
becoming fully plastic (see last section) or fail prematurely by:
1. LTB, either elastically or inelastically2. FLB, either elastically or inelastically3. WLB, either elastically or inelasticallyIf 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)
4
Design of Members for Flexure(about Major Axis)
5
Lateral Torsional Buckling (LTB)
Compact Members (AISC F2) Failure Mode Plastic LTB (Yielding) Inelastic LTB Elastic LTB Moment Gradient Factor Cb
6
Lateral Torsional Buckling (cont.)
Failure ModeA beam can buckle in a lateral-torsional mode when the bending moment exceeds the critical moment.
7
Lateral Torsional Buckling (cont.)
Nominal Flexural Strength Mn
plastic when and inelastic when and elastic when and
pb LL pn MM
rbp LLL rnp MMM
rb LL rn MM
inelastic plastic elastic
1.0Cb
nM
rM
pM
pL bL
rL
8
Lateral Torsional Buckling (cont.)
I-Beam in a Buckled Position
9
Lateral Torsional Buckling (cont.)
Elastic LTB coupled differential equations for rotation and lateral
translation (8.5.10)
whereMz = moment at location z along member axis z = axis along member length = angle of twistG = shear modulus J = torsional constant (AISC Table 1-1 for torsional
prop.)E = modulus of elasticity Cw = warping constant (AISC Table 1-1 for warping)
3
3
dz
dEC
dz
dGJM wz
10
Lateral Torsional Buckling (cont.)
Plastic LTB (Yielding) Flexural Strength (AISC F2-1)
where Z= plastic section modulus & Fy= section yield stress
Limits Lateral bracing limit
(AISC F2-5)
Flange and Web width/thickness limit (AISC Table B4.1)
(Note: Lpd in Salmon & Johnson Eq. (9.6.2) is removed from AISC 13th Ed.)
ZFMM ypn
yypb F
ErLL 76.1
11
Lateral Torsional Buckling (cont.)
Inelastic LTB Flexure Strength (straight line interpolation)
(9.6.4)
or
(AISC F2-2)
ppr
pbrppbn M
LL
LLMMMCM
ppr
pbxyppbn M
LL
LLSFMMCM
7.0
rbp LLL
12
Lateral Torsional Buckling (cont.)
Elastic LTB Flexure Strength
(AISC F2-3)
(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 (AISC F2-6)
(AISC F2-7)
pxcrn MSFM
27.0
76.6117.0
95.1
Jc
hS
E
F
hS
Jc
F
ErL oxy
oxytsr
x
wy
ts S
CIr 2
2
2
2
078.01
ts
b
ox
ts
b
bcr r
L
hS
Jc
rL
ECF
rb LL
13
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)(AISC F1-1)
whereMmax= 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
CBAb MMMM
MC
3435.2
5.12
max
max
14
Cb for a Simple Span Bridge
15
Nominal Moment Strength Mu as affected by Cb
16
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)
17
Flange Local Buckling (cont.)
Failure ModeThe compression flange of a beam can buckle locally when the bending stress in the flange exceeds the critical stress.
18
Flange Local Buckling (cont.)
Nominal Flexural Strength Mn
plastic when and inelastic when and elastic when and
pn MM
rnp MMM
rn MM
pff tb 2/
rffp tb 2/
rff tb 2/
noncompact compact slender
nM
rM
pM
pλ f
f
t
bλ
rλ
19
Flange Local Buckling (cont.)
Noncompact Flange (straight line interpolation) Flexure Strength
(AISC F3-1)
pfrf
pfxyppn SFMMM
7.0
20
Flange Local Buckling (cont.)
Slender Flange Flexure Strength
(AISC F3-2)
(kc shall not be less than 0.35 and not greater than 0.76)
Limit(AISC Table B4.1)
2
9.0
xc
n
SEkM
w
cth
k/
4
21
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)
22
Web Local Buckling (cont.)
Failure ModeThe web of a beam can also buckle locally when the bending stress in the web exceeds the critical stress.
23
Web Local Buckling (cont.)
Nominal Flexural Strength Mn
plastic when and inelastic when and elastic when and
pn MM
rnp MMM
rn MM
p
rp
r
noncompact compact slender
nM
rM
pM
pλ wt
hλ
wtλh
rλ
24
Web Local Buckling (cont.) Compression Flange Yielding
Flexural Strength
(AISC F4-1)
where Rpc= web plasticification factor (AISC F4-9a, b) & Fy= section yield stress
Limits (AISC Tables B4.1)
xcypcycpcn SFRMRM
ytpb F
ErLL 1.1
25
Web Local Buckling (cont.)
LTB (Inelastic) Flexure Strength
(AISC F4-12)
where FL= a stress determined by AISC F4-6a, b
ppfrf
pfxcLycpcycpcbn MSFMRMRCM
rbp LLL
26
Web Local Buckling (cont.) LTB (Elastic)
Flexure Strength(AISC F4-3)
(AISC F4-5)
Limit (AISC Table B4.1)
(AISC F4-8)
ycpcxccrn MRSFM
2
2
2
078.01
t
b
ox
t
b
bcr r
L
hS
J
rL
ECF
2
76.61195.1
J
hS
E
F
hS
J
F
ErL oxL
oxLtr
rb LL
27
Web Local Buckling (cont.) Compression FLB (Noncompact Flange)
Flexure Strength
(AISC F4-12)
Compression FLB (Slender Flange) Flexure Strength
(AISC F4-13)
(kc shall not be less than 0.35 and not greater than 0.76)
ppfrf
pfxcLycpcycpcn MSFMRMRM
2
9.0
xc
n
SEkM
w
cth
k/
4
28
Web Local Buckling (cont.)
Tension Flange Yielding Flexure Strength
(AISC F4-14)
Rpt = web plastification factor to the tension flange yielding limit
xtyptytptn SFRMRM
xcxt SS
(a) hc/tw ≤ pw Rpt=Mp/Myt (AISC F4-15a)
(b) hc/tw > pw (AISC F4-15b)
yt
p
pwrw
pw
yt
p
yt
ppt M
M
M
M
M
MR
1
29
Shear Strength Failure Mode Shear-Buckling Coefficient Elastic Shear Strength Inelastic Shear Strength Plastic Shear StrengthFor shear vVn(v = 0.9 except certain
rolled I-beam h/tw≤2.24√E/Fy, v = 1.0)
Vn=0.6FyAwCv (AISC G2-1)
30
Shear Strength (cont.)
Failure ModeThe web of a beam or plate girder buckles when the web shear stress exceeds the critical stress.
31
Shear Strength (cont.)
Nominal Shear Strength Vn (v = 0.9) plastic when and inelastic when and elastic when and
r
y
r y 8.0
p
cr
inelastic plastic elastic
nV
rV
pV
pλ λ
rλ
r
32
Shear Strength (cont.)
AISC G2 Nominal Shear Strength Vn
(a) For (AISC G2-3)
(a) For (AISC G2-4)
(a) For (AISC G2-5)
yw
v
w F
Ek
t
h10.1
y
v
wy
v
F
Ek
t
h
F
Ek37.110.1
w
y
v
v
th
FEk
C10.1
wyw
v
t
h
F
Ek37.1
yw
vv
Fth
EkC 2
51.1
0.1vC
33
Lateral Bracing Design
34
Lateral Bracing DesignAISC 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
35
Lateral Bracing Design (cont.)
AISC Provisions – LRFD Stability Bracing Design for Beams
1. Relative bracing
Φ=0.75
2. Nodal bracing
0
008.0Re
h
CMPquired drrb
0
41Re
hL
CMquired
b
drrb
0
02.0Re
h
CMPquired drrb
0
101Re
hL
CMquired
b
drbr
36
Lateral Bracing Design (cont.)
AISC Provisions – LRFD Stability Bracing Design for Columns
1. Relative bracing
Φ=0.75
2. Nodal bracing
rrb PPquired 004.0Re
b
rrb L
Pquired
21Re
rrb PPquired 01.0Re
b
rbr L
Pquired
81Re