ENCE 455ENCE 455 Design of Steel Structuresg

lV. Beam-Columns

C. C. Fu, Ph.D., P.E.Civil and Environmental Engineering DepartmentCivil and Environmental Engineering Department

University of Maryland

Beam-ColumnFollowing subjects are covered:

Interaction formulasMoment amplificationMoment amplificationStability: Braced vs. unbraced framesDesign of beam-columngTrusses with top-chord loads

Reading:Ch 6 f S iChapters 6 of SeguiAISC Steel Manual Specification Chapters C (Stability Analysis and Design) and Appendix 1 (Inelastic Analysis and

2

y g ) pp ( yDesign), 6 (Stability Bracing for Columns and Beams) and 7 (Direct Analysis Method)

Interaction FormulasDoubly and Singly Symmetric Members in Flexure and Compression

(H1 1 )P 018 ⎟⎞

⎜⎛ uyMMP

(H1-1a)

(H1-1b)

2.0≥nc

u

PPForφ

0.198

≤⎟⎟⎠

⎜⎜⎝

++nyb

uy

nxb

ux

nc

u

MMM

PP

φφφ

20<uPFor 01≤⎟⎟⎞

⎜⎜⎛

++ uyuxu MMP

Unsymmetric and Other Members in Flexure and C i

2.0<ncP

Forφ

0.12

≤⎟⎟⎠

⎜⎜⎝

++nybnxbnc MMP φφφ

Compression(H2-1)

0.1≤++ bzbwa

Ff

Ff

Ff

3

bzbwa FFF

(Segui Example 6.1)

Moment Amplification

)]/(1/[)( 00max ePPMyyPM −=+=

)/(11

ePPAmpFactor

−=

ltntr MBMBM 21 +=

4

B1, Mnt: Amplification factor & maximum moment for braced frameB2, Mlt: Amplification factor & maximum moment for unbraced frame

Braced Frames

11

1

1 ≥−

=

e

r

m

PP

CB α1e

5Single-curvature bending Reverse-curvature bending

Evaluation of Cm

M )(4.06.02

1M

MCm −=

)/PP(1 e1rαΨ+=mC

AISC Eq. C2-4

AISC Eq. C-C2-2

6Case of no transverse loads(Segui Examples 6.3 & 6.5)

11

12 ≥

ΣΣ−

=nt

PPB α

Unbraced Frames1

2Σ eP

AISC Eq. C2-3

Unbraced frames

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Superposition of braced & unbraced framesUnbraced frames

(Segui Example 6.6)

AISC Chapter C – Stability AnalysisAISC Chapter C Stability Analysis and Design

2nd-order flexural strength Mr

(C2-1a)MBMBM += ( )2nd-order axial strength Pr

(C2-1b)

ltntr MBMBM 21 +=

PBPP += (C2 1b)where

(C2-2)

ltntr PBPP 2+=

11 ≥= m

PCB α (C2 2)

(C2-3)

11

1−

e

rP

Pα

11≥=B

8

(C2-3)11

2

2 ≥

ΣΣ−

=

e

ntP

PB α

AISC Chapter C – Stability AnalysisAISC Chapter C Stability Analysis and Design (cont.)

B1 is an amplifier to account for second order effects caused by displacement between brace points (P-δ)B2 is an amplifier to account for second order effects caused by displacements of braced points (P-Δ)If B 1 05 i i i hIf B1≤1.05, it is conservative thatMr=B2(Mnt+Mlt)

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AISC Chapter C – Stability AnalysisAISC Chapter C Stability Analysis and Design (cont.)

Cm is a coefficient assuming no lateral translation of frame (no transverse loading)

M (C2-4)Pe1 is the elastic critical buckling resistance with zero sidesway

)(4.06.02

1M

MCm −=

2EIΠsidesway(C2-5)

ΣPe2 is the elastic critical buckling resistance for the ( )21

2

1 LKEIPe

Π=

e2 gstory

For moment frames (C2-6a)( )22

2

2 LKEIPe

ΠΣ=Σ

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For all types (C2-6b) H

MeHLRPΔΣ

=Σ 2

C : coefficient assuming no lateralCm: coefficient assuming no lateral translation of the frame

For beam-columns not subject to transverse loading – Equation (C2-5)For beam-columns subject to transverse loading: analysis or conservatively as 1

11

C : coefficient assuming no lateralCm: coefficient assuming no lateral translation of the frame (cont.)

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Summation ofSummation of forces acting on all columns in one story of ya multistory building framebuilding frame

13 14

15

Primary plus secondary bendingPrimary plus secondary bending moment by end moments

Braced frame -(Segui Example 6.8)

Unbraced frame -

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(Segui Example 6.10)