Beam-Columns

Members Under Combined Forces

Most beams and columns are subjected to some degree of both bending and axial load

e.g. Statically Indeterminate Structures

P1

P2

C

E

A

D

F

B

Interaction Formula

REQUIRED CAPACITY

Pr Pc

Mrx Mcx

Mry Mcy

2.0 0.19

8

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

2.0 0.12

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

2.0 0.19

8

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

2.0 0.12

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

Axial Capacity Pc

877.0

44.0or

71.4 658.0

otherwiseF

QFF

QF

E

r

KLifQF

F

e

ye

yy

F

QF

cr

ey

gcrn AFP

Axial Capacity Pc

Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional)

Fe:

Theory of Elastic Stability (Timoshenko & Gere 1961)

Flexural Buckling Torsional Buckling2-axis of symmetry

Flexural Torsional Buckling1 axis of symmetry

Flexural Torsional BucklingNo axis of symmetry

22

/ rKL

EFe

AISC EqtnE4-4

AISC EqtnE4-5

AISC EqtnE4-6

Axial Capacity Pc

LRFD

ncc PP

strength ecompressiv design ncP

0.90 ncompressiofor factor resistance c

Axial Capacity Pc

ASD

c

nc

PP

strength ecompressiv allowable cnP

1.67 ncompressiofor factor safety c

Moment Capacities

2.0 0.19

8

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

2.0 0.12

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

Moment Capacity Mcx or Mcy

2

2

2

078.01

ts

b

oxtsb

bcr r

L

hS

Jc

rL

ECF

rbp

brpxcr

ppr

pbrppb

pbp

n LLL

LLMSF

MLL

LLMMMC

LLM

M

for

for

for

xyr SFM 7.0

REMEMBER TO CHECK FOR NON-COMPACT SHAPES

Moment Capacity Mcx or Mcy

rp

rpxcr

ppr

prpp

pp

n

MSF

MMMM

M

M

for

for

for

REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

Moment Capacity Mcx or Mcy

nbc MM b

nc

MM

LRFD ASD

90.0b 67.1b

Axial Demand

2.0 0.19

8

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

2.0 0.12

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

Axial Demand Pr

ur PP

LRFD ASD

ar PP factored service

Demand

2.0 0.19

8

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

2.0 0.12

c

r

cy

ry

cx

rx

c

r

P

Pfor

M

M

M

M

P

P

Second Order Effects & Moment Amplification

W

P P

M

y

ymax @ x=L/2 =

Mmax @ x=L/2 = PwL2/8 + P

additional moment causes additional deflection

Second Order Effects & Moment Amplification

Consider

Mmax = P

additional moment causes additional deflection

P

H H

P

Design Codes

AISC Permits

Second Order Analysis

or

Moment Amplification MethodCompute moments from 1st order analysis

Multiply by amplification factor

Braced vs. Unbraced Frames

ltntr MBMBM 21

ASDfor

for LRFD

strengthmoment required

a

u

r

M

M

M

Eq. C2-1a

Braced vs. Unbraced Frames

ltntr MBMBM 21 Eq. C2-1a

Mnt = Maximum 1st order moment assuming no sidesway occurs

Mlt = Maximum 1st order moment caused by sidesway

B1 = Amplification factor for moments in member with no sidesway

B2 = Amplification factor for moments in member resulting from sidesway

Braced Frames

2-C2 EquationAISC 11

1

1

er

m

PaP

CB

Pr = required axial compressive strength

= Pu for LRFD

= Pa for ASD

Pr has a contribution from the P effect and is given by

ltntr PBPP 2

Braced Frames

2-C2 EquationAISC 11

1

1

er

m

PaP

CB

a = 1 for LRFD

= 1.6 for ASD

21

2

1LK

EIPe

Braced Frames

Cm coefficient accounts for the shape of the moment diagram

Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

4-C2 AISC 4.06.02

1

M

MCm

M1: Absolute smallest End Moment

M2: Absolute largest End Moment

Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

2-C2 Commentary AISC 11

e

rm P

aPC

C2.1-C Table Commentary AISC

1-2

2

LM

EI

o

o

COSERVATIVELY Cm= 1

Unbraced Frames

ltntr MBMBM 21 Eq. C2-1a

Mnt = Maximum 1st order moment assuming no sidesway occurs

Mlt = Maximum 1st order moment caused by sidesway

B1 = Amplification factor for moments in member with no sidesway

B2 = Amplification factor for moments in member resulting from sidesway

Unbraced Frames

Unbraced Frames

Unbraced Frames

1

1

1

2

2

e

nt

P

PaB a = 1.00 for LRFD

= 1.60 for ASD

ntP = sum of required load capacities for all columns in the story under consideration

2eP = sum of the Euler loads for all columns in the story under consideration

Unbraced Frames

22

2

2LK

EIPe

Hme

HLRP

2

Used when shape is knowne.g. check of adequacy

Used when shape is NOT knowne.g. design of members

Unbraced Frames

22

2

2LK

EIPe

I = Moment of inertia about axis of bending

Hme

HLRP

2

K2 = Unbraced length factor corresponding to the unbraced condition

L = Story Height

Rm = 0.85 for unbraced frames

H = drift of story under consideration

H = sum of all horizontal forces causing H

Homework

• 6.2-1

• 6.2-2

• 6.5-2

• 6.5-6

• 6.6-1