1

4-11 & 12 Compression Member

2

2

lEICPcr

π=

( )

gyrationofradiusthe

ratiosslenderneswhere2

2

2

2

2

=

=

=

==

kAkI

klklEC

AlEIC

APcr ππ

PIN-ENDED COLUMNS

( )2

2

2

2

22

where/

AkIklE

AlEAr

APcr

cr

=

===ππ

σ

x

y

P

y

P P

x

M

0

0

2

2

2

2

=+∴

=+

=

PydxydEI

PyM

MdxydEI

Define EIPk =2

kxCkxCy

ykdxyd

cossin

0

21

22

2

+=

=+

•  Boundary Conditions ( )( ) 0sin0

000

1

2

=→=

=→=

klClyCy

sin kl = 0 Buckling equation

2

2)2(

2

2)1(

2

22)(

4

3,2,1

lEIPP

lEIPP

nlEInP

crcr

crcr

ncr

π

π

π

==

==

==

0or 1 == Cnkl π

Columns Fixed-Free Ends P

y

2

2)1(

4lEIPcr

π=

2

2)2(

49lEIPcr

π=

( )

EIPk

kykdxyd

yPMdxydEI

=

=+

−==

2

222

2

2

2

where

δ

δ

δ++=

+=

kxCkxCyyy PH

cossin:Solution

21

Boundary Conditions: ( ) ( ) δ=== )(and00,00 ly

dxdyy

δ

y

P

M

δ++= kxCkxCy cossin 21

C2=-δ and C1=0. ( )Kxy cos1−=δ

Solution: δ=0 or cos Kl=0

5,3,12

== nnKl π2

22

4lEInPcr

π=

4-13 Intermediate-Length Columns

Parabolic (J.B. Johnson) formula

CES

b

Sakl

kl

klba

AP

y

y

cr

12

where

when

2

1

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

⎟⎠

⎞⎜⎝

⎛≤⎟

⎠

⎞⎜⎝

⎛−=

π

Point T is chosen to be

2yS

( )2/12

1

2

2

2

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛

===

y

ycr

SEC

kl

SklEC

AlEIC

AP

π

ππ

2

4-14 Columns with Eccentric Loading

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛=+−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛==⎟

⎠

⎞⎜⎝

⎛=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

==

==

−=+

EAP

kl

kec

AP

IcM

AP

EIPlPeePM

lEIPelxy

xEIPx

EIP

EIPley

ylxyxsCB

EIPey

EIP

dxyd

c 2sec1

2sec

12

sec2

1cossin2

tan

000:..

2max

max

2

2

σ

δ

δ

( ) ( ) ]2sec[1 2 AEPklkecS

AP yc

+=

When σc reaches Syc,

4-14 Columns with Eccentric Loading

2kecEccentricity ratio:

( ) ( ) ]2sec[1 2 AEPklkecS

AP yc

+=

When σc reaches Sy,

Secant column formula

For the steel with Syc=40ksi

4-15 Struts or Short Compression Member ⎟⎠

⎞⎜⎝

⎛ +=+=+= 21kec

AP

IAPecA

AP

IMc

AP

cσ

equation above theuse Otherwise,

fomulacolumnsecantheuse,If

282.0

2

2/1

2

⎟⎠

⎞⎜⎝

⎛>

⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛

kl

kl

PAE

kl

Compared to Secant Column Formula

For less than 1% of e

How short the member to use the equation above?

4-16 Elastic Stability Localized buckling where the max. bending occurs

Due to the compression at the bottom fiber

Recommend FEM analysis

3

4-17 Shock and Impact

Vibration

4-18 Suddenly Applied Loading

33 lEIyFk ==

( )

mk

Wkg

kWhtBtAy

ghghggty

ghthy

gty

hyWhykygW

hyWygW

==

+++=

===

==

=

>+−−=

≤=

ω

ωω

Define

`sin`cos:)2(toSolution

22

2,tillvalidisThis2

:)1(toSolution

)2(

)1(

1

1

2

4-18 Suddenly Applied Loading

[ ]

2/1

2/1

max

2/12

2/12

1

21

21

`cos2

2

sin2cos

`sin2`cos;;0

2and0`:ConditionInitial

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛++==

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛++=−=

++−⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛=

==−

+++−===

===−=

WhkWWkF

Whk

kW

kWhy

kWht

kWh

kWy

kWh

kWC

CkWhandCkWLetkWht

kWht

kWyvBA

ghyhyttt

δ

δ

φω

φφ

ωωω