ENCE 710 Design of Steel Structures

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


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


I-Beam in a Buckled Position

9


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


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


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


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


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




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


Noncompact Flange (straight line interpolation) Flexure Strength

(AISC F3-1)

pfrf

pfxyppn SFMMM

7.0

20


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




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


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


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


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


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

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ENCE 710 Design of Steel Structures