strength of cold-formed steel box columns

8/17/2019 Strength of Cold-Formed Steel Box Columns

1/17

Missouri University of Science and Technology

Scholars' Mine

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Strength of cold-formed steel box columnsN. E. Shanmugam

S. P. Chiew

S. L. Lee

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2/17

ABSTRACT

Ninth

International

Specialty Conference on Cold-Formed Steel Structures

St. Louis Missouri U.S.A. November

8-9,1988

STRENGTH OF COLD FORMED

STEEL

OOX

COLUMNS

N.E.

Shanmugam

1

M.ASCE S.P.

Chiew

2

S.L. Lee

3

F.ASCE

Cold-formed

s t ee l box

columns have two obvious modes of

fa i lure ;

they

can

reach the

ul t imate capac i ty

e i the r by

overal l column

buckl ing

or l oca l buckling. This paper i s

concerned with

a

numerical

method to

pr e d i c t the ul t imate load-car ry ing capaci ty

of

cold-formed s t ee l box

columns

subj

ected

to

axial

force

and

unequal

end

moments. The method

accounts for the ef f ec t of

loca l

p la te buckling and in i t i a l

imperfec t ions

upon

the u l t imate s t reng th of columns.

Moment-curvature- thrust r e l a t ionsh ips are

developed by

using

piecewise

l i nea r

s t r e s s - s t r a in curves; they are incorporated in to the column

-analysis

in which

the

d i f fe ren t i a l equation

of bending

i s numerically

in tegra ted .

Use

of a

su i tab le fa i lu re c r i t e r ion and

numerical

procedure

makes t poss ib le to

obtain column

curves .

For design purposes,

column

curves from which ul t imate s t reng th

of

column under ax ia l

or

eccentr ic

loading condi t ions

can

be

eas i ly obtained

are presented.

INTRODUCTION

Thin-wal led s t i f fened compression

elements

are commonly

encountered

in

cold-formed

s t ee l box columns. When such columns are

subj

ected to

compressive

loading, wi th the onse t

of buckling

the

growth

of out

of plane

def lec t ions in

the pla te elements r esu l t s in changes in

the s t re ss

pa t te rn

which

in

tu rn reduces

the pla te

s t i f fn es s . This

reduction causes the f a i lu re of

the

column a t a

load

l ess

than

i t s

c l a s s i ca l

Euler

buckling load.

The

design

of

cold-formed s t ee l box

columns, therefore ,

requires

the

cons idera t ion of

local

p la te

buckling,

overa l l column

buckl ing and

the

i n t e r a c t i on

between the local

and

overa l l buckling. A

close-form eva lua t ion

of

ul t imate s t rength of such

columns i s

well

nigh

impossible

and

the

use of numerical

procedure,

therefore , becomes necessary .

Numerous inves t igators

6,8,11,17)

have studied the ef f ec t

of

loca l

buckl ing on

the s t r eng th of

pla te elements

subjected to

compressi

ve loading.

Rigorous

methods

using

la rge def lect ion

theory

coupled with f i n i t e dif fe rence

methods

(12,14)

and

e las to -p las t i c f i n i t e

1 . Senior

Lecture r , Department

of

Ci

v i I

Engineering,

National

Univers i ty

of

Singapore,

Kent Ridge,

Singapore

0511.

2. St ruc tura l Engineer, T.H. Chuah and Associa tes (Pte) Ltd, 190

Middle

Road,

Fortune

Centre, 11-04,

Singapore

0718.

3.

Professor and Head, Department

of Civi l

Engineering, National

Univers i ty of Singapore, Kent Ridge, Singapore 0511.

129


3/17

130

element formulat ions (5 ,7) have been proposed for

the

ul t imate s t rength

ana lys is

of

p la t e elements .

For

design

off ice

use, however, simple

methods using the

ef f ec t ive

width

concept

were presented by Von

Karman

(16)

and

Winter

(18) .

Ilimy

of

the design

spec i f i ca t ions

1,2 ,4)

for

cold-formed s t ee l s t ruc tu ra l elements have accepted the

ef fec t ive

width

pr inc ip le

and it has been proved

to

yie ld sa t i s f ac tory

r e su l t s .

A

simple

ana ly t i ca l

method

was presented

recent ly

by

the authors

(10,15) for

predic t ing

the s t r eng th of thin-walled

welded s tee l

box

columns

subjec ted to

ax ia l

load and end moments.

The

method accounts

for

loca l buckl ing of

component pla tes , welding r es idua l

s t r esses and

i n i t i a l column imper fec t ions .

The

method i s extended for

the

ana lys is of cold-formed s t ee l box columns in

the

present s tudy .

Homent-curvature- thrust r e l a t ionsh ips are developed by

using

piecewise

l inea r

s t r e s s - s t r a i n curves .

They

are incorporated in to the column

ana lys i s in which the

d i f fe ren t i a l equat ion

of bending i s numerical ly

i n t eg ra t ed .

Use of

a

su i tab le f a i lu re

c r i t e r i on and numerical procedure

makes it

poss ib le

to

obta in the ul t imate s t r eng th . Resul ts are

presented in the form

of

column

curves

which can be used

read i ly

by

des igners .

THEORY

An exact ana lys is for u l t imate

s t rength of cold-formed

s tee l box

columns i s

complica ted

because

of

the local

buckl ing of

component p la t es

and the non- l inear i ty of the

s t r e s s - s t r a i n

curves.

However, the

so lu t ion

can

be

great ly

s imp l i f i ed

by

adopting approximate

l inear i sed

s t r e s s - s t r a in curves .

Box columns

can be

t r ea ted as

an assemblage

of

long p la t es supported along the longi tud ina l edges

as

shown in Figure 1

Local

buckl ing of

the

component

pla tes ,

which are under

ax ia l

compression, i s

allowed for

by applying

an appropr ia te

load-shor ten ing

curve, while

those under t ens ion

are t r ea t ed by assuming an

e las t i c -pe r fec t ly

p l a s t i c

s t r e s s - s t r a i n curve.

Simpl i f ied piecewise

l i ne a r

s t r e s s - s t r a i n curves (Figure

2b) based

on

approximat ions

of s t r e s s - s t r a i n curves by Moxham

Cr is f ie ld ,

Harding

e t a l . and L i t t l e (3)

were

proposed by Shanmugam e t al .

(15) .

These

curves represent

unwelded

p la t es having s lenderness

r a t ios

equal

to

80,

55,

40 and

30

or

l e s s , and

i n i t i a l

imperfec t ion

of

b 1

000,

b

being the width of the

p la t e in a d i rec t ion normal to

the

compressive

loadings .

These

curves

have been used

to account

for the local buckl ing

of the

component p la t es

of

box columns and

the

moment-curvatures- thrust

H - ~ - P ) re la t ionsh ips

for column cross - sec t ions were

developed

as

explained

in the fol lowing sec t ions .

MOMENT CURVATURE THRUST

RELATIONSHIPS

t becomes imperat ive

to

formulate the moment-curvature- thrust

r e l a t ionsh ips for ind iv idua l

cross-sect ions

in order

to

determine the

equi l ibr ium curves def in ing

the

domain of s t ab le equi l ibr ium of moment,

t h r u s t and column l eng th . The M ~ P r e l a t ionsh ip i s

computed

numerical ly using

the

method

s imi l a r to tha t adopted by Nishino e t

a l .

13).

The

fol lowing assumptions are made in the

ana lys i s

which

follows:


4/17

131

( i )

the

mater ia l i s homogeneous and i so t rop ic in both

the

e l a s t i c

and

p l a s t i c

s t a t e s ,

( i i ) e l a s t i c - pe r f e c t l y p l a s t i c


r e la t ion

Figure

2a)

i s

assumed

for f l ange p la t e under

t ens ion

and

the

local buckl ing of the f lange p la t e

under compression

i s allowed by

applying an

appropr ia te


curve

from

the

se t

of

curves given in Figure 2b.

The

por t ion of

the

web

p la t e under compression i s t r ea t ed in a s imi lar

manner

to tha t

of the compression f l ange by

applying the


curve with the

assumption t ha t

b i s equal to the depth of

the

compression

zone

( i i i ) the ioca l buckl ing of the p la t e

due to

shear i s

ignored,

( iv)

v )

(vi)

( v i i )

the s t r a i n

d i s t r i bu t i on

i s

l i ne a r

across the depth

of

the cross - sec t ion

Figure 3) ,

the res idua l s t r esses in each component p la t e are

assumed

to

be i n se l f - equ i l ib r ium and

dis t r ibu ted

in the form shown in

Figure

4,

the

e f f e c t

of s t r a i n

r ever sa l

i s

negl ig ib le ,

and

the de f lec t ions

are

smal l so t ha t curvature can be

expressed by

the

second der iva t ives of def l ec t ions .

Consider the

cross - sec t ion

as shown in Figure 3(a) .

The neutra l

axis i s located a t a dis tance R from the concave

extreme

f ib re of the

cross sec t ion . For purpose of obta in ing the M P

re la t ionsh ip ,

the

cross sect ion

i s

d i sc re t i sed i n to n elements . The s t r a i n a t any

element

i can

be expressed

in the

non-dimensional

form

in

which

e:

e

c

1 )

t o t a l s t r a in a t element i ,

pos i t ive

i f in tension

s t r a i n

a t the centro id of the sec t ion

j> curvature non-dimens ional i sed by

the curvature

a t

i n i t i a l

yie ld ing

for bending, 2e:

y

/d

Yi

dis tance

of

the

cen t re

of element

i

from

the

cent ro idal

axis

e: . res idual s t r a in a t element i

~

e

y

yie ld s t r a i n

d

depth

of cross sec t ion

The corresponding s t r e s s

cr

i s obta ined by making use of

the

appropr ia te s t r e s s - s t r a in

curves

~ ven in Figure

2b,

cr being posi t ive

i f in

tens ion .


5/17

132

The ax ia l force and bending moment carr ied

by

the c ro ss - se c t io n

can

be

eas i ly obta ined by

using the equi l ib r ium

equat ions

n

p

E

2 a)

n

m

E

Z

i ~ 1

2 b)

where

p

m

n

t o t a l number

of

elements

A

re

of

cros s s ec t i on

Z p l as t i c modulus

of sec t ion

M .

area of element

i

P

y

squash load

My

p l as t i c moment yz

The

moment

developed about

the

cen t ro ida l

axis

of

a

cross-sec t ion

can be

determined numerica l ly

fo r

any

given value of

ax ia l

t h rus t

and

curva ture 4> The

computations

involve the determinat ion of the correc t

pos i

t i o n

of the n e u t ra l ax i s . Success i ve values of R can be

i n t e rpo la t ed and the correc t value

i s

determined such t ha t the

ne t

compress ive fo rce from the reSUlt ing s t r a i n dis t r ibu t ion

defined

by

R

and

> obtained

from

Eq.

2 a)

toge ther with Eq.

matched the given ax ia l

t h rus t . The r e s u l t i n g moment

can be

c a lc u la te d

from

Eq. 2 b)

and

the

procedure repeated

fo r

other values of 4>. Typica l M-4>-P

r e l a t i onsh ips

obtained

are given in Figure

5 . The

v a r i a t i o n

of

m M My

wi

th

respec t

to

> ~ ~ fo r p P P

0 .3

i s

plo t t ed

in Figure 5 for

pla te

s l e n d e r n e s ~

i t 40, and 80.

ULTIMATE

STRENGTH

OF COLU 1NS

Box

columns sub jec ted to e c c e n t r i c

load can

be t rea ted

by

cons ider ing the

can t i l eve r column under

the

ac t ion

of a x ia l force

P,

t r a n sv e rse

shear force Q

and

bending moment

M

a t

the f ree

end

as

shown

in Figure 6 . In i t i a l imperfec t ion i s assumed to be represented

by

i n i t i a l curva ture which

i s

constan t

throughout

the length of the

column.

The

equi l ib r ium

condi t ion

and the curva ture-d i sp lacement

r e l a t i o n are given, respec t ive ly in the non-dimens ional form

as

Ar

pw

+

qx)

3

mf

Z

d

2

w

(4) + 4>i)

2r

4)

dx

2

d


6/17

133

in

which

m

f Mf/My

q

Q/P ; g

Y Y

x

x;; f ;

w

W/r

and

~ i

~ i / ~ y

The

def lected

shape

of

a

column

for

given values

of

mf

p

and

q

ana

prescn.oea values

of

~ .

can oe

ootained

oy in tegra t ing t:q. 4 J.n

view

of

Eq. 3

and the m ~ m e n t - c u r v a t u r e - t h r u s t re la t ionship

for a

par t i cu la r

cross - sec t ion developed ea r l i e r . In order

to s impli fy the

in tegra t ion

of Eq. 4 a numerical

procedure

was

adopted. The method is

explained in re fe rence 15

and

hence it i s

not

repeated

here in .

Wi

th

the

help

of the

numerical

in tegra t ion technique the re la t ion

between

m

and

x can

be obtained

for a se t

of

p, q

and and various

assumed

values of

m

f

•

The m-x re la t ionships thus obtained: are plot ted

as shown in

Figure

7

and

they

are refer red

to

as equil ibrium

curves.

Applying Horne s (9) s t ab i l i t y cr i te r ion the envelope of the equil ibrium

curves can

be constructed. The envelope

i s the boundary of the

s table

equilibrium domain

of

the cant i lever column

of cer ta in

length

subjected

to

the

combined act ion

of

end

moment, axia l

th rus t

and shear .

The

var ia t ion

of

the

moment capaci ty along the length of the

column

for a

par t icular

value

of ax ia l thrus t and shear i s gi

ven by

the envelopes in

Figure 7.

t i s

more useful

to presen t the ul t imate

s t rength

of

box columns

in the

form

of

column

curves. The equations developed for can t i l eve r

column can be extended

to

the trea tment of

simply

supported box columns

subjected

to

unequal end moments with

-1

K 1 as shown

in

Figure 8.

The

simply

supported

column

may

be

t rea ted

as

two can t i l eve r

columns

of

lengths

X

and

x2

with f ixed ends a t poin t

O.

The par t of the

colum to

the r ight of point

0

corresponds

to

the cant i lever column of Figure

6.

With

Q

= M 1

-

K)/L whi l s t the

par t to l e f t

of

0

corresponds to

a

column

under a t ransverse

load Q opposite in sense to

tha t shown

in

Figure 6

and hence care should be exercised in using the appropriate envelopes.

A

typica l

envelope

constructed for the whole length of

a

column

with par t i cu la r

values

of

and

q

and various

magnitudes of p i s

shown

in

Figure

9. The

curves t..re

plot ted

on the m-A

plane in which

A

is

normalised s lenderness r a t i o given

by

5)

The envelopes on the l e f t correspond

to

the cant i lever column of

length

x1 and on

the

r igh t

to

the

cant i lever column

of

length

x2 For given

values of p and q, a t the l imi t of s t ab i l i t y the poin t (m, A

1

) and

(Kffi , A

2

)

must

be on the envelopes corresponding to p

and q. The

c r i

t i c a l

values of A,

and

x

2

are

not

known a p r io r i

however,

they mus t

sa t i s fy

the fol lowing

equat ions:

6)


7/17

134

z

m 1

- K)

q

Ar ITA

7)

A

value

of A

i s

f i r s t

assumed along with the

prescr ibed

values

of K , m

and p.

q i s determined from Eq. 7

and the

values of 1.1

and 1.2 and

hence A are

obtained

from the envelopes corresponding

to the

se t of

value

of p

and

q.

This

process i s

repeated

unt i l

the assumed value

of

A

matches

the

computed one. A faml.ly of column curves p-A

curves)

for

various values of

m and

K can be constructed. The

s teps

involved

in

the const ruct ion of column curves are i l l u s t r a t ed in Figure 10.

RESULTS AND

DISCUSSION

The

proposed

method

was appl ied

to the

analysis of

the behaviour

of

box columns

having

d i f fe ren t p la te

slenderness

ra t ios . The

parameters were

so chosen

tha t

they show

the

influence of

end

moment,

the

moment

ra t io and

column

s lenderness .

Column

curves which were

generated from

the

resul t s are

presented

in Figures 11-13. In each of

these f igures curves are given for moment ra t ios equal

to

1.0,

0.0

and

-1

.0 ,

respect ively .

K equal

to one

represents

the case of

a column

under

equal end

moments.

Resul t s

are presented for end moment M/My)

equal

to

0.0,

0.1 and 0.3

and

pla te s lenderness

ra t io of

30, 40, 55

and

80.

All these

curves

were

obtained

assuming

i n i t i a l

curvature , .,

equal to

0.1

and f ixed

value of

res idual s t ress

level shown

in

Figure 4

The

re su l t s for K =

0.0

and

-1 .0 c lea r ly show

the

e f fec t

of

unequal

end

moments upon the s t r eng th

of axia l ly

loaded

box

columns.

For a l l

values of

m the column

s t rength

increases

with

the

decrease in

K

values . For example consider column curves

in

Figures 11

a) and

13 a)

in which K

1.0 and

-1.0

r espec t ive ly . For

A 0.6

and

m 0.3 the

s t rength

of

column

for

K equal

to

1.0 i s 0.56 and the corresponding

value for

K equal to 1.0 is 0.72

an increase of about

29

percent .

I t

i s

a l so

observed t ha t

the

columns under s ing le curvature bending

K =

0.0,

1.0) are normally weaker than those under double curvature

bending K = -1 .0 . This e f fec t

i s

found

to

be more signi f icant in the

case of

intermediate column

range. Simi lar observat ions can

be made

for

a l l values of K in the case

of

columns with other slenderness values

a l so .

t

i s

obvious from the

f igures t ha t

the column

s t rength drops

s ign i f i can t ly

as the pla te s lenderness

increases.

For

axia l ly loaded

shor t

columns th i s

reduction i s

approximately

20 percent when

the

s lenderness

ra t io of the component

pla te i s

increased from

30

to

55.

The

f igures

also show

tha t the column s t rength

i s independent

of A in

the

lower range of

column

s lenderness , th i s range

increasing

with

pla te

s lenderness . At low b i t r a t i o

the

in terac t ive buckling fa i lu re occurs

for the

whole

range of A, whereas for

la rger

b i t ra t ios

the

fa i lure i s

mainly

due to local

buckling

upto

a ce r t a in value of A.

The

in terac t ive

buckl ing becomes predominant t h e rea f t e r .

CONCLUSIONS

A simple ana ly t ica l

method to

the analysis

of

cold-formed s t ee l

box

columns

pinned

a t

t he i r ends

and

subjected

to

axia l

load and unequal


8/17

135

end

moments has been presented. The method i s

capable of predic t ing

the

load-carrying

capaci ty

of these columns which experience

loca l

buckling

of

component p la tes and may have i n i t i a l column imperfect ions. Resul ts

presented in

the form of column curves

c lear ly show the adverse ef fec t

of .

the

local buckl ing

of

component

pla tes .

The ult imate ca.pacity

of

these columns i s reduced

s ign i f i can t ly

for l arger b i t

rat ios

of

component pla tes . The columns subjected to double curvature bending

ha'le

been found to be stronger than those

under s ingle

curvature

bending.

The

column curves provide the means for

est imat ing

the

ult imate

s t rength of

cold-formed s tee l

box

columns subjected

to

loads

with unequal eccen t r ic i ty a t the

ends.

They should

be

usefu l design

tools and in termediate values can

be

obtained by i n t e rpo la t ion .

APPENDIX I - REFERENCES

1 .

AISI,

Specif ica t ion

for the

design

of

cold-formed s t ee l

s t ruc tu ra l members , American I ron and Steel Ins t i tu t e ,

Washington

D C

1983,

46

pp.

2.

Austral ian

Standard Rules for

the use of

cold-formed

s tee l

in

s t ructures ,

AS

1538-1974.

3.

Bradfield,

C.D. and

Chladmy,

E. (1979), A review

of

the

e la s t i c -p la s t i c analys is of s t ee l p la tes loaded in in -p lane

compression , Report

CUED/D

Struct /TR 77,

Dept. of

Engrs.

Cambridge Univ. , Cambridge,

U.K.

4 . Bri t i sh

Standard Struc tura l

use

of

steelwork in building

Part

4

Code

of

Pract ice

for

design

of f loors with prof i led sheet ing,

BS

5950: Par t 4:

1982.

5 . Crisf ie ld , M.A. (1975), Ful l - range analys is

of

s tee l plates

and

s t i f fened

panels

under uni-axia l compression, Proc. Ins tn . Civ.

Engrs . ,

Par t

2, 59, 595-625.

6 . Dwight, J .B. and Moxham K.E. (1969), ' 'Welded s t ee l pla tes in

compression , The Structural

Engineer, 47(2) ,

49-66.

7. Frieze,

P.A.,

Dowling, P.J . and Hobbs, R.E. (1975),

Steel

box

girder

bridges parametric s tudy of

p la t es

in

compression ,

CESLIC

Report

BG39 Engrg. Struc tures Lab. , Civ. Engrg. Dept . ,

Imperial

CoI l . ,

Londong, U.K.

8. Harding,

J .E . , Hobbs,

R.E.

and Neal, B.G.

(1977),

The

e l a s to -p l a s t i c

analysis of

imperfect

square

p la t es under

in-plane

loading , Proc. Ins tn . Civ. Engrs . , Par t 2, 63, 137-158.

9.

Horne, M.R. (1956), The

e l as t i c -p l as t i c

theory

of

compression

members , J l .

Mech. Phys. Solids,

Vol.

4, 104-120.

10.

Lee,

S.L. ,

Shanmugam

N.E.

and Chiew,

S.P.

(1988),

Thin-walled

box columns under arb i t ra ry

end

loads ,

J l .

Struct . Engrs . , Am.

Soc. Civ. Engrs . , 114, in press .

11. Li

t t l e , G.H. (1974), Rapid

analys is of p la te

collapse

by l ive

energy minimisat ion ,

In t . J .

Mech.

Sci . ,

19, 725-244.

12. Moxham K.E. (1971),

Theoret ical determinat ion

of

the s t rength

of

welded s t e e l

p la t es

in

compression , Report No. CVED/C

StructjTR2 •

13. Nishino, F. , Kanchanalai ,

T.

and Lee, S.L. (1972), ·Ult imate

s t rength

of wide f lange and box columns ,

Proc.

Colloq. on

Centra l ly Compressed Stru ts , Par is , France, IABSE 254-265.

14.

Rhodes, J . and Harvey,

(1971),

Pla tes

in

uniaxia l

compression

with

various

support condi t ions a t

the

unloaded boundaries · , . ..


9/17

136

J . Mech.

Sci .

13,

787-802.

15.

Shanmugam N.E.,

Chiew, S.P.

and Lee, S.L. (1987),

Strength of

thin-walled

square

s tee l box

columns ,

. : J : . . : I : : . . : . ~ . : : : S . . : : t : . : r . . : : u ~ c : . . : t : . . : . ~ . : : : E : : . n : ; ; g . r : . . ; g L : . : . J , ~ . : ; A : : : m : . . : . . . .

Soc. C i v ~ Engrs. ,

113(4),

818-831.

16.

Von Karman

T.,

Sechler, E.E. and Donnel,

L.H.L.

(1932), The

strength

of

thin pla tes in compression ,

..=T..=r..=a:.;n;;;:s;..;.:..-.:.;Am=._S;:.o=c.:..--=M,::e ,c ,h;:..:...

17.

Engrs. , 54, No.2 .

WalKer A.C. 1 ~ o ~ , rne

post-oucKll.ng

oehaviour

supported square

panels ,

Aeronautical Quarterly,

203-222.

of

silllfily

20,

August,

18.

Winter, G., (1947),

Strength

of thin

steel compression

flanges ,

Trans. Am.

Soc.

Civ. Engrs. ,

112, 527-54.

APPENDIX NOT TION

The following symbols are used

in

this

paper:

A

area

of

column

cross-sect ion ,

b width

of

a compressed pla te element

d

depth of

column

cross-sect ion ,

L length of

simply-supported

column,

M moment a t

any

sect ion along

the

column,

M

f

fixed end moment

of

cant i lever

column,

My plas t ic

moment and equals to cryz

n to ta l number

of

elements

in

cross-sect ion,

P

axial

force,

Py

squash load

and

equals

to

cryA

Q t ransverse shear force,

r radius of gyration,

W t ransverse

deflection,

X distance along

the

length of cant i lever column,

Xl

length

of l e f t

cant i lever

column,

X

2

length

of

r ight cant i lever column,

Yi distance

of

the centre of element i from the centroidal axis,


10/17

37

z plas t ic sect ion modulus

i

area of

element i ,

E normal

s t ra in ,

EC

s t ra in

a t

the

centroidal

axis

of

section

Ei to ta l s t ra in

a t

element i ,

Eri residual s t ra in a t element i ,

Ey s t ra in

a t

yie ld

point

K moment

ra t io ,

normal s t ress ,

rc compressive residual s t ress ,

y

yield s t ress ,

curvature

caused by

bending

i n i t i a l curvature

curvature

a t i n i t i a l yielding

for

pure bending moment and

equals to 2E

y

/d ,

t ~

t ~ i / ~ y

nondimensionalised

slenderness

ra t io

of simply-supported

column

A X 1 / n r ) ~ ,

A2 X 2 / n r ) ~ .


11/17

I

I

I

J

I

M

P

R

O

C

M

P

R

O

.

L

T

=

C

Z

-

U

·

1

C

M

P

O

~

T

O

P

L

T

3

T

N

O

I

J

0

N

O

F

G

1

L

n

o

C

m

p

P

a

e

n

a

B

C

u

m

n

1

5

b

b

b

1

5

b

1

0

b

b

0

5

0

5

f

T

=

1

f

o

a

b

;

i

1

0

1

5

E

=

E

/

E

y

a

T

o

2

0

2

5

C

0

5

+

0

4

E

+

0

0

(

b

/

-

E

)

:

b

3

b

/

=

4

0

I

I

i

=

5

I

b

=

a

D

0

0

C

.

_

L

_

L

L

_

7

0

0

0

1

0

1

5

2

0

2

5

E

:

E

/

E

y

b

C

m

p

e

o

F

G

2

S

m

p

e

P

e

w

i

s

L

n

S

e

S

a

n

C

u

v

C

;

>

0


12/17

c

d

~

p

a

p

a

e

2

p

a

J

I

b

I

l

a

c

o

o

I

b

o

u

n

s

o

s

h

w

n

l

o

n

I

1

1

·

~

~

~

1

&

~ I

_

~

X

i

s

I

c

s

a

n

d

s

b

o

F

G

3

T

n

W

a

e

B

C

u

m

t

e

o

w

d

t

~

2

r

.

~

d

b

n

4

T

T

l

>

F

9

1

J

>

l

·

F

G

4

R

d

S

t

e

D

i

s

b

o

0

9

0

8

0

7

0

5

I

0

4

E

0

3

0

2

D

p

P

=

O

b

=

i

5

b

=

o

V

,

i

i

i

i

,

D

0

2

D

0

4

0

5

0

6

0

7

0

8

0

9

,

2

,

1

4

5

6

1

J

4

1

I

4

1

y

F

G

5

T

c

C

v

v

D

o

m

C

o

d

A

s

M

,

t

M

p

~

-

-

~

-

~

.

X

F

G

6

C

a

e

C

u

m

w

h

I

n

a

C

v

u

e

c

<

D


13/17

p

P

=

3

b

3

q

=

0

1

C

c

=

0

0

E

-

j

E

0

-

0

2

e

0

4

-

0

6

-

0

8

L

w

E

o

1

I

I

o

0

2

0

4

0

6

0

8

1

1

2

1

4

1

6

1

8

2

2

2

2

4

2

2

B

3

3

2

X

=

X

/

E

Y

L

M

I

I

M

P

.

P

1

I

X

l

X

2

I

(

a

S

n

e

-

C

v

u

e

B

e

n

<

1

0

M

t

M

P

;

»

~

,

t

i

(

b

I

D

e

-

C

v

u

e

B

e

n

<

-

1

0

I

F

G

7

E

b

u

m

C

u

v

a

E

o

1 0

0

6

0

4

,

E

0

E

0

0

e

0

2

-

0

4

-

0

6

-

O

·

-

1

-

1

2

2

L

C

e

R

g

C

e

b

3

q

=

0

1

=

C

c

=

o

0

1

6

1

O

B

0

4

0

0

4

0

8

1

2

1

6

A

=

X

/

n

f

,

F

G

9

E

o

o

E

b

u

m

C

u

v

F

G

8

S

m

p

y

u

e

C

u

m

n

~


14/17

q

p

>

>

l

a

i

=

C

c

=

c

a

m

m

l

}

g

v

t

t

e

q

L

m

I

e

A

1

>

C

T

;

C

c

a

m

m

i

e

e

g

v

>

0

3

>

0

m

p

p

l

e

-

R

g

C

e

>

I

b

q

=

i

C

r

c

c

a

m

m

l

}

V

e

=

e

>

I

d

e

c

u

m

n

c

v

c

b

c

u

e

F

G

1

C

o

u

o

o

C

u

m

n

C

v

I

-

HI

-


15/17

'

2

_

0

·

8

Q

.

.

Q

.

0

·

6

I

0

.

O

·

4

O

·

2

0

0

0

2

0

·

4

0

6

'

2

_

o

·

a

Q

.

.

Q

.

0

·

6

I

0

0

.

4

0

·

2

0

°

0

2

0

·

4

0

6

0

·

8

,

2

'

4

,

6

'

8

A

=

(

L

V

E

Y

l

a

0

·

8

,

'

2

'

4

,

6

,

8

A

=

(

L

m

l

E

I

e

'

2

-

-

-

-

-

-

r

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

.

-

-

-

-

-

.

I

0

.

0

·

4

0

·

2

1

I

:

=

_

8

Q

.

0

·

6

I

e

0

·

4

0

·

2

o

·

2

0

·

4

0

·

6

0

8

,

'

2

'

4

'

6

'

8

A

=

(

L

m

l

v

£

Y

b

r

I

I

f

-

i

=

a

-

I

e

=

1

0

_

-

.

-

M

=

D

f

.

-

0

1

-

-

-

D

·

I

-

J

o

0

2

0

4

0

6

0

·

8

,

'

2

1

4

'

6

,

8

A

=

L

T

E

y

(

d

F

G

1

1

-

C

u

m

n

C

v

(

K

1

0

.

.


16/17

I I

~

c : ; : ~

..:.

u;o

......

./

J

I II

>----::::: .l

7A

7

V

/

VJ/

'-..,-

t l

-

...

--

\::

0

-

..

...J

II

c.

I I

J

..,t

co CD -z

c:::t

C ~ d I ~

=

d

1

1

Il

?

t----

L n c::"

' ' ' '

j

I ,

I--

V ~

:

i

L/

V

/

J

"''. ..,

I

~ ' .

~

--

\::

-

. ...J

J

I

0 -

I I

t<

f.O

I

C.

6

'"

CD CD -z

=

=

c:: Q

d

=d

o

o

N


17/17

I

I

I ~ ~

t - -

/IA

o J

_

II I

- II

:;; u

v J

I

?

l ~ ' I i

l i l l ~

v

.,

-of

c::

c::

=

c::

Ad

d

=d

.. .

..,

c::o

c::

c::

c::

Ad d

=d

00

.......

1=-

..........

6-

I I

,,?r

strength of cold-formed steel box columns

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