290_15

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Welded

Built-Up and

Rolled

Heat-Treated

T-l

Steel

Columns

A514 STEEL

BEAM COLUMNS

by

C

K

Yu and L Tall

This work

has been carried

out

as part

of an investigation

sponsored

by

the United States

Steel Corporation.

Technical

Guidance

was

provided by Task

Group of the Column

Research

Council.

Fri tz

Engineering Laboratory

Department

of Civil

Engineering

Lehigh University

Bethlehem

Pennsylvania

October 968

Fritz

Laboratory

Report No

290.15

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TABLE OF CONTENTS

ABSTRACT

INTRODUCTION

Stress-Strain Relationship

Residual

Stress

Assumptions

2 MOMENT CURVATURE THRUST RELATIONSHIP

PAGE

2

3

4

Basic

Concepts

5

Strain Reversal Effect 7

E ffects of

Residual

Stresses

and Mechanical

Properties

9

3

LOAD DEFLECTION RELATIONSHIP

2

Numerical

Procedure

3

Unloading Effect

E ffects of

Residual

Stresses and Mechanical

Properties 7

Interaction

Curves

for

A5 4

Steel Beam Columns 18

4

EXPERIMENTAL INVESTIGATION

Test Procedure

Test

Resul ts In-Plane

Behavior

Test Results Local Buckl ing

5 SUMM RY N

CONCLUSIONS

6

ACKNOWLEDGEMENTS

7

NOMENCLATURE N DEFINITIONS

8.

FIGURES

9.

REFERENCES

19

19

21

22

28

31

32.

35

60

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ABSTRACT

ultimate s trength load d e f ~ m a t i o n behavior and local

buckling

phenomenon of Welded and Rolled A514 Steel beam columns

were investigated

analytically and

experimentally. The

beam columns

were

subjected

to axial thrust and to moments applied a t the two

ends to cause bending about the

strong

axis of the members I t was

assumed

that

the

beam columns are permitted to

deflect

only in the

plane of the

applied

load.

Because

of the n on li ne ar it y o f the

s t ress s t ra in

curve

and

the par t icu lar

p atter ns o f r es id ua l s t ress

the behavior

of

an

A514

s tee l

beam column

could differ from that

of a mild

steel

beam column of rolled shape. Numerical

resul ts

were

obtained by means

of

a computer

and are presented

in the forms

of

both interact ion

curves and moment rotation c u r v e s ~ Strain

reversal

effect and the unloading effect of

reversed

curvatures are included

in the computation . . F ull scale

te s t s have

been conducted

to provide

a check on the theoret ical analysis . I t was shown that

extrapolation

procedures from ultimate strength

s o ~ u t i o n s

for A

steel

provide

an

approximate

but

conservative

estimate

of the strength of A514

s tee l

shapes.

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INTRODUCTION

The term beam column denotes a member

is subject

simultanec

lsly to ax ial thrust and bend:·.ng. The

bending

moment

in

the

member may be

caused

by externally applied end moments

eccentr ici ty

of

longitudinal

forces, in i t i a l out-of-straightness

of

axially loaded columns,

o r t ransve rs e forces

in a dd itio n

to

axia l

forces and end moments. The types of beam-columns which

a re s ub je ct

to

constant

axial

force and varying end moments

are

investigated

in

th is study The beam columns s tudied a re assumed to be la teral ly

supported,

that i s they f a i l

in

the

bending

plane without twisting.

The

determination

of

the

ult imate s tr en gth o f a beam column

i s a problem

in

which inelast ic action

must

be considered. Extensive

research

has been carr ied out

in the

study of

the

behavior of la teral ly

supported wide-flange shapes under combined moment and axia l force,

including the

effect

of

residual stresses. 1,2,3,4,5 The methods

and

solutions

previously

developed a re app li cabl e

only

to

materials

which

have an elast ic-perfect ly-plast ic s t ress s t rain relat ionship

and

are

res t r ic ted

to

residual stress

p atte rn s o f

cooling

af te r

rol l ing

of A 6 s tee l

shapes.

Both

rolled

and welded

A5 4 s tee l shapes are considered

here. The ultimate strength, the load-deformation behavior

and

the

local

buckling

phenomenon

of

the

beam-columns

are

investigated.

The discussion includes s t rain reversal and

unloading

effects.

1

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-2

The theoretical analysis is compared with fu l l scale experiments,

and the comparison indicates a good correlat ion.

The

purpose

of th is report

is to

investigate th e s tre ng th

of both

welded

and rolled beam-columns made of A514 s tee l

and

to

present a sol ut ion to

the

overal l load-deformation character is t ics

of non-linear

material

including

the

consideration

of

strain

reversal

and

unloading.

Stress-Strain

Relationship.

The s t ress-s t rain curve for A514 s teel can be described by the

f

1

·

h . 6

o oWlng

t r e e

equatlons :

when

1

a

1.0 0.005

1.517) 0.3647

L

-

1.5

-

.0 -

E

y

y

0.3276

£

1.517)5

-

y

when

0.8

0-

1.0

2)

a

y

and

a

1.0

0.005

1.517)

0=-

=

-

v

y

when

1.0

3)

:f

Y

in

which a is s t ress

j

is

the

yield stress

determined

by

the 0.2

y

7)

l-

is

s t rain

and

is the yield strain

=

rJ::/O

f f se t

method ,

y

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-3

Figure shows

the complete s t ress-s t rain curve for

A5 4

s tee l .

y

comparing

th is to

that

for mild

s teel , i t is seen that A5 4

s tee l has

a

lower proportional

l imit

stress and tha t s t rain hardening

occurs

immediately

af te r the

ending

of

the

t ransi t ion range,

continues

unt i l the tensi le strength is reached, and then s tar ts

to unload. This representative s t ress-s t rain curve for

A5 4

s tee l

was

determined by

averaging

the results

of 58 standard

STM tension

specimen

tes ts . The

average

values

of

yield s t ress ,

modulus of

elast ici ty and strain hardening

modulus are

2 Ksi,

28,900

Ksi and

144 Ksi, respectively.

Residual

Stress.

Figure

2 shows

the ide aliz ed p atte rn s o f residual stress

distr ibut ion

in heat -t re at ed ro ll ed shapes and welded H-shapes

with

flame-cut

plates,

a l l of

A5 4

s tee l .

These

idealized

patterns

are

approximation of the results

obtained

from an extensive investigation

h

d

8,9,10

on t e reSl ual stresses

ln

A5 4

s tee l

shapes an plates • For

11 d

h

t ·

10 . d·

h h d

ro

e s

apes, prevlous

~ v s

19atlon

ln

lcates

t

a t

t e

magnltu

e

and pattern of residual

stresses

essent ial ly

are

independent of the

yield stress of the

s tee l

i f

s tee l

is not heat- treated

af te r roll ing.

Heat-treatment

apparently reduces

the

residual s t ress

magnitudes

as,

for example, in rolled

A5 4 s teel

sh apes. Furthermore, because of the

high yield stress of the s teel , the residual s t ress magnitude

in

rolled

A5 4 s teel ,

i f

compared ona

nondimensionalized basis with respect

to i t s yield s t ress , is

much

smaller

than

that for s t ruc tura l

carbon

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-4

s teels Thus, the effect

of

residual s t ress could

be

less

for rol led

A l

s teel

beam-columns

than

for

those

0;

A s teel For

welded

H-shapes, the flame-cutting of

the

component plates creates tensi le

residual

stress

at

the

cut edges;

th is

pattern

of

residual

s t ress

distr ibution

is

completely

dif ferent from that

in

rolled shapes.

Therefore,

a separate and dif ferent ana ly sis f or

both

rolled

and

welded shapes

of

A l s tee l is

needed.

Assumptions.

The

assumptions

made in the

theoret ical analysis

are

as

follows:

1

the members are

perfectly

st ra ight

2

the

ef fec t

of

shear

is insignif icant and can be neglected.

3

the

thrust

is applied

f i r s t and

then

kept con sta nt a s

the

end

moments

in crea se or decrease.

4

the members

are bent with

respect

to strong

axis

and weak

axis

buckling and

la te ra l tors ional

buckling is effectively

prevented.

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2.

MOM NT

CURV TURE THRUST REL TIONSHIP

Basic ConL2pts.

A prerequisi te

to performing ultimate

strength

analyses

of

beam columns

is

a knowledge

of the r e la t ionsh ip ex is ting between

the bending moment and the

axial

force acting on the cross-section, and

the result ing

curvature.

The

basic equations are

~

d

A

and

~

y • d

M

J.

A

s

shown

in Figure 3,

y

is

t he d is tance

of

a

f ini te

element

area

d from the bending axis and

~ i s

the stress in . th is element. P

4

5

is

t he app lied

thrust

and M the internal moment. The stress

at

J.

each

element i s

a function of

stra in, £ and

t he re fo re t he

stress-

strain relat ionship

must

be defined

f i r s t .

Generally, the

monotonic

stress-stra in

relat ionship

can

be

descr ibed well

by

the

data obtained

from a

t n ~ o n

specimen t e s t ,

and recoraed

or r ep re sent ed

by

a

mathematical equation a s

=f c

-5-

6

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-6

However,

i f the s t ress s t ra in relat ionships are history-dependent ,

Eq. 6

is

invalid

i f

the

strain

reverses. In this

study, the

incremental

s t ress s t ra in

relat ionship

is

shown

in

Figure

4,

and

defined as

c

=

f

l

for

=

E :

£.,.

E.

2 ·f

-

.

for

- f

[

£; . 7

-

2

c

=

f

f [ 1

for

£

< E.; .

in

which

and

are the

largest

compressive

stress

and

strain

to

which th e mate ria l of

any

element has been

subjected. The sign

convention used

here

is

plus for compression, and

minus

for

tension.

The

t o t a l

s t ra in

a t

any

point in

a

loaded

beam-column

i s

composed

of

a

residual

s t ra in

, a constant s t ra in

over the

r

entire

cross-section

due to

the

presence

of

thrus t

, and

c

the

s t ra in

due

to

curvature,

[ ¢

That

is

£

=

+ +

£ -

r c

V

8

Here

9

where ¢

is

the

curvature

a t the

section

under consideration.

When

the

s t ress s t ra in

relat ionship

is

known,

i t is obvious

that

i f

P

is

specif ied,

and

by

assuming

a

value

for

the curvature

¢,

the

corresponding M can be determined by

sat isfying both

eqs.

4 and

5.

J.

I f

the thrus t

is

applied

f i r s t on the

member

and held

constant

through

the whole loading process,

a

moment-curvature

relat ionship

can be

established.

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-7

The numerical procedure for the

determination

of the

11

moment-thrust-curvature curve is a t r ia l-and er ror process

For a given residual s tr es s d i st ri bu ti on, [ i s known; and

for

the

r

given curvature

is known.

assuming an £ value

for

the

c

whole cross-sect ion, the to ta l

stra in and

therefore the stress

a t

each e lement area

is

determined.

The summation

of to ta l

internal

forces must be

equal

to

the

given

P, otherwise

E

c

must

be revised

unt i l Eq. 4 i s sat is f ied .

Then, the

corresponding

can be

J.

evaluated

by means of Eq. 5. i nc reas ing the value of and

repeating

the

calculation, a complete

moment-curvature

relat ionship

can be determined for a sRecified

thrus t

P.

In this

study, th e s t ress s t ra in relat ionship

of

the

material and residual s tr e ss d is tr ibu ti on is programmed

in

subroutine

subprogram

forms.

Both the materi al p roper ti es and

the

s t ra in reversal

effect

are included.

Strain

Reversal Effect.

Strain

reversal is defined here as

the unloading

s t ress s t ra in

relat ionship

which

i s

dif ferent from the monotonic s t ress s t ra in

relat ionship,

as

shown in Figure 4.

Usually,

i t i s

convenient

to assume tha t s t ra in reversal will be

defined

by

the

monotonic curve .

In Figure 5a,

the

progress of applied s t ra in on a

section

for

a given constant

thrus t

is shown.

There

are basically two modes by

which the strain reversal

can

influence

the

M-0-P curves. Firs t

as shown

in

Figure

5b,

i f

the

[

p

£

l ine runs

acro ss th e s t ra in

rc

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-8

reversal zone where [ is

the strain

a t the

proportional

l imit

p

and

[

is

the compressive residual strain

a t

a

point) ,

or the

. rc c

in i t i a l

applied

strain due

to

a

given constant

thrus t ,

is larger

than

th e sma lle st

value of E

-

£

the

s t rain reversal will

p rc

affect

the result ing

M 0 P curves. The region affected by

strain

reversal is shown as

the shaded area in

Figure 5b. The

second mode

is when

the

curvature is very large, then the tensi le strain near

the

convex

side can

be larger than tha t a t

the

proportional: l imit

as

shown

in

Figure

5c.

Then

a t

further

l oad ing , r eve rsed

tensile

strain

i nf luences the

M 0 P curves.

A

combination

of these two modes

of strain

reversal

is

also

possible,

i f

the

applied thrust is high and

the curvature is

large.

However i t was found that only a t

extremely

large

curvature,

will t en si le s tr ai n

reversal

be effective.

From th is observation, i f the thrust ra t io P is the

y y

axia l

force

corresponding

to

yield

stress

level)

is

less

than

«

-

r

l

p

rc

then

the

reversed s t rain does not affect the results since

i t

is

s t i l l within the elas t ic range, except when the curvature rat io 0 0

pc

is very

large. However when the

applied thrust ra t io

is

larger

than

cJ

- r

10 pronounced differences could

occur i f

the

. p

rc

y .

strain-reversal ef fec t is neglected.

To

demonstrate the effect of

strain reversal , a set

of

curves is presented

in

Figure 6. The section

is a welde2 A514 s teel H-shape built-up :rom flame-cut plates. The

M 0 P curves were piotted

for

varying from

0.5 to 0.9.

I t is

y

c le ar th at for less than 0.7

proport ional l imi t·

a e is

0.8

Y P Y

and maximum

compressive

residual -stress rr

10

= 0.1), the case

in

which

rc

y

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-9

strain reversal is considered y l ~ r s u l t s in curves which are

identical

with

the

corresponding

one

in

which

the

s t ress s t rain

relat ionship

is assumed

to follow the

mOTlotonic s t ress s t rain curve

only

•. However, for

equal to 0.8 and 0.9, signif icant

.differences

are shown for the two cases. Therefore, th e in flu ence

of strain reversal is pronounced and should be taken into

account

i f

th e se ctio n exh ib its a combination of c o m p r e ~ s i v e residual stresses

and thrust which

cause

yielding

immediately

af ter

thrust

is

applied.

Effects of Residual Stresses and

Mechanical

Properties.

In

addit ion

to t he conside ra ti on of the effect of s t rain

reversal ,

the

pattern of distr ibut ion and

magnitude

o f r es idua l

stress

also

change

the shape

of

the

curve. Figure

7 presents

three

types of residual stress distributions which represent

the

idealized

residual

stresses

in A

rolled

low-carbon

s tee l

section,

B rol led

heat- treated

A5

s tee l section

and C

welded

buil t-up

A5

s teel shapes with

flame-cut

plates. I f the mechanical

properties are assumed

to

be elast ic-perfect ly plast ic the

curves

for the three

types

of residual stress distr ibut ion are curves 1 ,

2 and

4

in Figure 7.

I t

is noticed that there

are

signif icant

differences

among them

in

the

elas t ic p las t ic

range.

Generally

speaking, the

M-0-P curve for

the rolled

structural-carbon s tee l

section, which has the

largest

compressive residual stress ra t io

r

l r among

the three,

exhibits a smoothe r knee whereas the

rc

y

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-10

rol led Leat- treated A5l4

s teel

shapes, for which

the

compressive

residual

stress

ra t io is

th e sma lle st

and thus residual s tr es s e ff ec t

the

least show a

sharper knee.

Aside

from

the

effect

of residual stresses, the mechanical

p rope rt ie s a lso play an

important.role

in the

curve; in

Figure 7, curves 2 and 3 are the curves for sections

with

ident ical residual stress distr ibut ion but different mechanical

properties; one is of

elast ic

perfect ly-plast ic type and

the

other

. is representative of A5l4

steel .

For material with a non-linear

type of

stress s tra in

curve,

such

as that of A5l4 stee l the M-0-P

curve is

lower in the knee

portion

than that for which

an

elast ic

perfect ly-plast ic stress s tra in

curve is

assumed. However for

curvature greater than

that

a t the

end of. the

knee, curve 3 is

above

curve 2 , due to the

strain-hardening

property of the A5l4

steel .

Curves 4 and 5 are

also

presented

in

Figure 7

for

welding-type

residual

stresses

and a similar

behavior

is observed.

For

most pract ical beam-columns,

the

internal moments

for

a

la rg e por tion of the

member

are within the knee range of the

curve during the

loading

process.

Therefore,

the shape of the knee

has

a pronounced influence on th e

load-deformation

relat ionship

and

th e u ltlmate

s tr en gth o f

the

beam-columns. This leads

to

the

emphasis

on

the basic assumptions of the residual stress

distr ibut ion

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290.15

as well as of the shape of the

s t ress s tr in

curve

and of the

st rain y=versal phenomenon in

the

when

thrust i s applied f i r s t

and yielding occurs before the

application

of moment he assumption

th t th ru st i s a pp li ed b ef or e the moment

approximates

the actual

behavior

of

multi story

frames

in

which most of the axia l forces

in

the

columns

are

due to the dead

load and

moments

to the l ive

load.

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3. LOAD-DEFLECTION RELATIONSHIP

In the general practice

for

the design o f p lanar

structures

i t

i s

often

suff icient to know the ultimate

stre ng th o f

a beam-

column. However,

in

pl st ic design, e spec ia ll y for mu lt i- st ory

bui ldings, i t

i s necessary to determine the

maximum

moment

of

a

(12)

jo int

of

a subassemblage • Therefore,not only the

ultimate

moment capacity but

also

the

complete

load-deformation curve

of

each individual

beam-column

must

be known. The

most

pract ical and

useful

way

of presenting t he load-de fl ec tion

r el at ion sh ip o f

a

beam-column

i s the

end moment vs.

end

rotation curve.

There

a re genera lly two types of numerical i nt eg ra ti on for

the determination of load vs. deformation curves of a beam

column.

O

h

h

d

. N k .

. d (13)

ne 0

t e two met

0

s lS ewmar s

numerlca lntegratlon proce ure

•

The

merit

of Newmark s method i s that i t can be

applied

to any kind

of

end

conditions.and the

interative

process

converges

reasonably

f st However, Newmark s numerical integration diverges

i f

the

assumed end moment

is larger

than

t he u ltima te

load, and

the

descending branch of the M-G curve becomes very

diff icul t

to obtain.

The other numerical

method

i s the so-called stepwise integrat ion

. (14)

procedure. This method

has been used extensively in the

development

of

column

deflection

curves

(CDC s).

However,

during

the c o n s · ~ r u c t i o n o f these CDC s, i t wa assumed tha t reversed

internal moments would

s t i l l

follow the monotonically

increased

curve.

Therefore,

end moment vs. end rotation M-G)

curves

obtained from these CDC s do not in clud e the unloading

e f f e c ~ ·

-12-

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-1 3

f th is unloading e f f e c t

is to be considered,

then

a t

each

i n t e g r a t i o n

s t a t i o n

o f

th e

beam-coluID

th e pr esent

moment

must

be

compared with i t s h i s t o r y t o determine

th e

corresponding

curvature. Consequently, i f

a

s e r i e s

o f CDC s are

to

b e d ev elo ped

in th e same manner,

th e lo c atio n o f

th e s eg me nt w hi ch corresponds

t o a p a r t i c u l a r

beam-column)

o f a

D

must be known beforehand

.

so

t h a t th e

h i s t o r y

o f CDC s

ca n

be.made i d e n t i c a l to that

o f the

beam-column in

question.

This

is impossible

fo r

most cases

except

fo r a few p a r t i c u l a r

e n d- lo ad in g c o nd i ti on s·

as shown

in

Figure 8;

1) equal en d moments sin g le

cur vat ur e) ,

where th e mid-height o f

th e beam-column

is always a t th e

peaks o f

th e CDC s, o r

2)

equal

en d moments double

curvature)

an d

3) one end

pinned zero en d

moment

where

f o r case 2)

th e mid-height

an d fo r

case

3) th e zero

moment

en d

are always

a t on e

en d o f th e CDC s.

Therefore,

i nt eg ra ti on o f CDC s f o r case

1)

can always be in i t ia ted a t th e

quar t er p o i n t s , where th e slopes are

zero,

an d fo r cases

2)

and 3 ) ,

a t the

zero def l ect i on p o i n t ,

an d then th e loading e f f e c t can be

considered. Of course, this negates th e

advantage o f

using CDC s,

t h a t

i s

t h a t they

are

assumed

t o

be

h i s t o r y independent an d

hence

may

be used fo r beam-columns o f an y en d conditions an d l engt h.

Numerical Procedure.

The unloading behavi?r o f

beam-columns

can

be included

in

th e

M G curve,

i f th e following

numerical in teg ratio n

procedure

7/25/2019 290_15


290.15

-14

is

employed.

1.

Subdivide

the length of the

member

which is

under

a

constant

thrus t into n in tegrat ion

stat ions as

shown

in Figure

9a.

The distance between any two adjacent

stat ions on the

deflected

member is

\ =

L/

n-l»

approximately equal to the arc length

within

the

segment

•

2.

Assume

tha t

the

segment

in

each sub

length

is

a

circular arc.

3. Assume

an

end rotation and

an

end moment a t stat ion 1.

4.

Determine the

curvature a t

stat ion 1

from

the

curve. I f present M

l

is less than the previous

maximum

M

l

, the unloading

M-P-0

curve

is

to

apply .

5.

As shown in Figure 9b, deflection

at

stat ion 2,

n

th

moment

t

st tion

?\

), and the slope a t

/2

•

.

A

ta t ion

2,

G =G

2 1

6.

7. Determine

O

2

from

the curve,

and continue the

i nt eg ra ti on in the same

manner as from

step

4 to

6 .

That i s ,

v. = 1\.

J.

Sin

G

i

_

l

- 1/2 0

i

-

l

1\

v.

1

J .-

G

; :

G 1

1\

J.

J .-

M -

M

\

i - I

M

M

i

_

l

+ PV

l

-

1 n

2

Cos

G.

1-1/2

i - I

J.

L

J .-

2

7/25/2019 290_15


290.15 -15

I f

the assumed

M

l

and G

l

are cor rec t ,

then a t

the

nth

sta t ion

v should be zero, or equal to a g iven value i f

sidesway of the

beam-column is allowed. Otherwise M

l

must be decreased or increased i f v i s sma ll er th an or

larger than the

given

end deflection,

and

steps 3 to

7 repeated

unt i l

v is within a cer tain allowable error .

9. Increase G

l

and

increase or

decrease

M

l

a cer tain amount

and

rep eat the whole

process from

step 1 to step 8

unt i l the

complete

M as needed is obtained.

The numerical in tegrat ion procedure

suggested above

i s

essent ial ly

the

same

as

that

used in the development of

CDC s The

po int o f d iffe ren ce

is

the fac t tha t the in tegrat ion

is

carried

out

on

th e def le ct ed

shape of

the

member for fixed stat ions. Thus

the

history of every

stat ion

can

be

recorded,

and

the unloading ef fec t

can

be

taken into

account.

Unloading Effect .

The M curve

for

the

equal

end moments single curvature

case i s considered here. For th is pa rt icu la r s i tua ti 0n, the

slope

a t

the

mid-height

point is always zero

and

the

internal

moment

a t

th is

point

always

increases

during

the

whole

loading

history.

Numericc l integration can be

simplif ie j

by start ing a t the mid-heigLt

15

point

and

working toward the end with only one

half

of

the member

The example

given

here i s

for

the

M curves of

A514

s tee l

residual

stresses

of both the

rolled and welded type are considered.

7/25/2019 290_15


290.15 -16

The actual

moment-curvature

relat ionship

for

an

A

s tee l

16)

beam under reversed loading has been presented by Popov .

t

was

observed

that

when the moment

is

re ve rse d, th e

in i t ia l unloading

portion

for a moment curvature

hysteresis

loop i s

approximately

. 16)

Ilnear . In the present study, elas t ic unloading of moment is

postulated. The relat ionship is

therefore represented

by

the

fol lowing equa tions

see

Figure 10).

for

=

=

M.,P)

<

=

M. - M.)/EI

Where and M.

are the

largest curvature and internal moment,

respectively,

to

which

the

cqlumn has

been subject

to any stat ion

The M G curves for A514 s tee l

beams-columns

with slenderness

rat ions ranging from

20 to

40 are presented in Figures 11

and 12.

t is apparent ~ h a t

there

is l i t t l e

difference

between

the

M G

curves including the unloading

effect and

excluding i t The reason

for

th is

are that th e portio ns of the

member that do

unload are the

less highly loaded

regions,

for example when

L/r =

20,

the

moments

a t

the unloading

region

are around 0 .8

M ,which

is

approximately

pc

on

the s ta r t

of

the

knee of

the

curve

where the elas t ic

unloading

effect

i s

not

pronounced, and

also

most

of

the

deformation

of the

~ o l u m n

continues

to

come from Lhe regions under monotonic

loading. From Figures

10

and

11,

i t

can also

be

seen

that

the unloadin

7/25/2019 290_15


7/25/2019 290_15


290.15

Interaction Curves for A514 Steel Beam Columns

-18

The interaction curves between PIP

and

M M for equal end

y

moment conditions

symmetrical

bending)

are

shown

in Figure

14 for

A514 s t e e l beam-columns

with

slenderness

r a t i o s

equal

to

and 60

Beam columns of rolled

heat- treated

shapes show higher

ultimate

s t r e n g t ~ than those

of

welded

buil t -up

shapes.

This can be understood

as the consequence of the

smaller effect

of

residual

stresses

on

the

M P 0 curves for rolled shapes than

that

for

welded

shapes.

The r e s u l t s obtained for A514 s t e e l beam-columns from

direct

interaction may be compared with the

so lu t ions extrapola ting

from

the r e s u l t s obtained previously for A36 s t e e l

beam-columns.

12 )

For

beam-columns made of s t e e l other than A ~ the

slenderness

r a t i o

12)

must be adjusted according to the followlng formula:

+ )

x

equivalent

J

The extrapola tion solu t ion

wil l

be exact i f the residual s t r e s s

to yieid s t r a i n r a t i o and the

residual

s t r e s s patterns are the

same

as

those

of rolled A36 s t e e l s h p e s ~ and i f the

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

curve

i s

elast ic-perfect ly

p l a s t i c . For A514 s t e e l beam-columns these

two conditions cannot be s t i s f i e ~ and t h u s ~

yield

only an

approximate

solution. The interaction curve e t e ~ m i n e from

t h i s extrapolation

procedure

i s presented

in

Figure 14 for

the

case

L/r

=

20 . I t

i s

shown

tha t t he ext rapo la ti on sol ut ion

i s

lower than the

corresponding

Ilexact solution .

7/25/2019 290_15


4

EXPERIMENTAL INVESTIGATION

An

experimental invest igat ion

of

the behavior

of

beam-columns

made

of

A514 high

strength

o n s t r u t i ~ n l

al loy s tee l has been

carried out.

The program consisted

of t es t s of

two

ful l-scale

beam-

columns, one a rolled 8WF4 shape

and

the other a

welded

IlH71 shape

(flame-cut plates . The members were tested

in an as-delivered

condition;

no

attempt

was made

to eliminate

rol l ing

or welding

residual

stresses by

annealing.

The magnitude and

distr ibut ion

of the residual

stresses were determined by the method

of

sectioning,,(17), and

i t

was

.

(8

9

found

tha t

they were

close

to the

resul t s of prev10us

measurements

and

hence the

idealized residual stress

dis t r ibut ions

as shown

in

Figure 2 were

used

for

the

theoret ical

predict ions.

The beam-columns

were

tested under equal end moment (single curvature) conditions.

Test

Procedure.

The

procedure

for

test ing

beam-columns

has

been

described

. d . 1 1 4,

18)

d f I h

ln e ta l prev10us y an only a

r1e

out 1ne 1S glven ere.

The general

set-up of the

beam-column specimen is sho\ Tn

in

Figure15a. The horizontal moment arms are

r ig idly

welded to the

end

of

the

column.

The sizes

of the

beams are comparatively larger

than

that

of

the column so

tha t

the beam sections remain in the

elas t ic

range

during the

whole

loading

process.

Pinned-end f ixtures

were

ut i l ized

to

ensure

that there

arp

no

end

moments

other

than

those

imposed by the moment arms,applied

a t

the column ends. In

Figure

15a

i t

can be seen tha t the axia l force

in

the column is

made up of the

direct

f or ce app li ed by the tes t ing machine, P and

-19-

7/25/2019 290_15


In the

early

290.15

-20

the

jack force,

F.

s imul at e t he si tuat ion

e xistin g in the lower

stories of a

multi-story

,frame and

to be in

accord

with the

assumptions for the

theoret ical analysis ,

the

te s t s were

performed

with

the

axial

load held constant.

Thus t

each increment

of

load

or deformation, the

direct

force, P, was adjusted so

that

the to t l

force in

the

column

remained

t 0.55 P where P is the yield load

of the column.

The

direct

axial force, P, was f i r s t applied ·to the

column;

the beam-to-column

joints were rotated

by

applying

the

jac k fo rce to

the

ends of

the

moment

arms.

The column was therefore forced

into

a symmetrical curvature mode

of

deformation. In order to preclude

any deformation out of the

plane

perpendicular

to th e stro ng

axis ,

the

column was braced t

the third

points

by

two sets of l t e r l

braces. The

l te r l

braces used

were designed

for t he l abor ato ry

19

t s t n ~

of large

structures

permitted

to sway

stages of loading, that i s

in

the

el s t ic

range,

approximately

equal

increments

of moment were applied to the column. In

the

inel s t ic range, comparatively larger deformations

occur for the

same amount

of

moment

increment,

therefore, end

rota t ions

instead

of moment

are

used as a

basis for loading in. order

to

obtain

a

complete

load-deformation

curve

with approximate ly evenly

distr ibuted

tes t points.

At

each increment

of

load or

end ro ta t ion ,

the

end

rotations

were

measured by level bars see

Figure

l5b . The mid-height deflectio

in the

bending

plane

as well as out -o f-pl ane, o f

the

column was

also

measured

by

mechanical

dial

gages. SR 4 gages

were

mounted t

the

7/25/2019 290_15


290.15

-21

•

beam and column junctions

as well as a t

several

o ther loca tions

along

the

column, as showm

in

Figure

15b,

to

determine

s t r a i n

distr ibution

in

the

column

or

to

serve

as

a means

for

checking

moments.

Figure

16

shows the photographs taken a t the beginning

and end

of the

t e s t

The

occurrence

of local buckling of the compressed

flange

was

determined by measuring the out-of-plane

deformations

of the

flanges

a t

f iv e l ocat ions

in the

v i c i n i t y o f mid-height of the beam column

with an

inside

micrometer.

Test Resul ts ( In -P lane Behavior).

The

resul t s

of the

t e s t s

can

best

be

presented

in the

form

of end moment vs.

end-rotation

curves as shown

in

Figures 17 and 18.

In Figure 17

the

M 9 curve for

the

8WF4

A514 s t e e l beam column i s

shown.

Figure

18

contains

the

M 9

curve for the llH71

welded

A514

s t e e l beam-column. The moments indicated

by open

po in ts r ep resen t

the

t o t a l applied

moment determined from the hydraulic

jack lo ad.

The

length

of

the

moment arm i s

the

distance from

the

centerl ine of

the

column

to the

center

of

the

rod

to which the hyd raul ic ja ck i s

connected. The

end

moments

are

also checked by th e rea ding of the

dynamometer which i s inser ted

in

series with

the

jack and by four

sets of

SR 4

s t r a i n

gages which

were

affixed

to the

loading beam

near

i t s junction

with

the

column.

The

difference

between the moment

readings by

these

three

means are shown in

F igu re 19.

I t i s apparent

tha t

they

are rather consistent .

7/25/2019 290_15


290.15

-22

The length

used to

compute slenderness rat ios

of

the columns

were the

distances

between the

points of

intersection

of the

centerl ines

of the

column

and loading beams. For both beam-columns,

the

slenderness ra t io

L/r , is 40 . Comparison of the experimental

resul t s

with

the theore t ica l reveals

that

the

test ing

p oin ts are

above

the theore t ica l ly obtained M G curve

Figures 7

and 18).

This

discrepancy i s due

in par t to

the fact tha t

the actual

slenderness

rat io

has

been

reduced somewhat

by the insta l la t ion of jo int

s t i f feners and to the fact that

the actual

s t ress s t rain relat ionship

determined

from

tension coupon

tes ts shows a

sl ight ly

higher

proportional l imi t

than that

of the average typica l

s t ress s t rain

curve as

shown

in

Figure

1, on

which the

theoret ical

analysis

was

based. The

t es t s

are compared

also to the

theory

in

a plot

of

M M

pc

vs.

L/r as

shown

in

Figure

20. The

difference between

theory and

x

t es t s

is

approximately

of the

theoret ical

p redi ct ion for

both

rolled and

welded

buil t -up shapes.

From Figure

20, i t can also be

observed tha t

th e d iffe re nce o f u ltim ate

s tr en gth fo r rolled and

welded shapes vanished for

low

slenderness ra t ios . This i s apparently

because of

the

fact

tha t the in ternal

moments

in the

greater portion

of

the

member are within the

st ra in hardening

region a t

ultimate

load,

and hence

the

residual

stress

ef fec t

becomes insignif icant .

Test

Results Local

Buckling).

The local buckling deformation9

of the

flanges

of the

beam

columns were measured during the

process

of test ing. I t

was

observed

tha t

hardly any web buckling occurred,

and tha t the

b it ra tio

of the

7/25/2019 290_15


290.15

-23

flange

provided

sufficient

st i f fness

to ensure

the

development

of

rotation capac ity o f the beam-column. That is the local buckling

of the flange

occurred

af te r

the

moment

capacity

had dropped

five

percent below i t s m x mum value

ult imate

strength .

I t is di f f icu l t in general to analyze the buckling

of

plates

in

which residual

stresses exis t

The

local buckling problem of

columns or beam-columns with residual

stresses

becomes more

i nvolved because not only the

compressive

r es idual s tr e ss es could

induce

yielding before the

local

buckling

st ress

i s

reached,

but

also because of

the

u nc ertain ty o f the

coeff icient

of res t ra in t

h

1 15 I h H

a t t e Junctlon

component

pa t e s

n t e case

-sectlons,

i t

was found that

the

assumption that

the

flange i s divided into

two canti levers which have fu l l deflection res t ra in t but zero

20

rotation res t rain t

is

close to rea l i ty although conservatlve

•

In

th is

study,

this

same

res t ra in t

condition

was

assumed

for the

flange

of the

beam-columns. I t i s not in te nded to perform an

elaborate theore t ica l analysis with

respect

to the local buckling

behavior of columns

or

beam-columns here. Experimental resul t s

are compared

to

the theoret ical analyses ava ila ble f or the purpose

of

determining

the

m x mum

allowable

b it

rat io for th e fla ng e, such

that local

buckling does not occur

before

the

attainment

o f yie ld

st ress

and for ·the prediction

of the

l Jcal buckling

point

for a

beam-column. Reference

21

p re se nts th e

theoret ical and

experimental

investigations

of the pla te

buckling under different

edge

conditions

and

residual stress patterns. Figure 21 shows a plot

of

7/25/2019 290_15


290.15 -24

curves for plates simply supported and

o y

b

t

s,

cr

cr

y

free

a t unloaded

edges, and simply supported on both loading

edges.

Here, band t denote the width

and

the thickness of the plate ,

respectively. Even though the stress strain law, elas t ic perfectly-

plast ic and

residual s t ress

patterns used in

Ref.

21

are

dif ferent

from those

used in

th is study, the solutions

may

provide a

reasonable

theoret ical

p redi ct ion for the buckling

of

A514 s teel plates in the

determination

of

the cr i t ica l t ra t io . Figure 21 shows that

h ~ h pattern

of

~

values when

1.0 are nearly the

same,

approximatel

=

residual stress var ies , the

maximum

0

cr

y

therefore assumed

that

the

maximum

allowable tt is

even

equal to 0.45.

=

t max

rat io for

a y ie lded f lange would be

0.45

x

k

y

For A514 s teel with E

=

9 x 10

3

ksi 0

=

110

ksi .

y

max

=

14.5

where band t are the width

and

thickness

of

the f lange , respec tively .

The

t rat io of

both

sections tested are

equal

to 14.35

which is

sl ight ly

less

than

the

maximum

t for a y ie lded f lange.

In

order

to

have

some

experimental

verif icat ion

of the

approximate

cr i t ica l

t rat io

determined

and, in

addition, to

furnish

some data

relat ing mechanical properties , stub columns cut from

the

same

pieces

which

also provided the beam column specimens

were

tested

under

uniform

axia l

load.

The

cr i t ica l local buckling strains

were determined

experimentally by the so-called top of the knee method,,(22).

Figure

7/25/2019 290_15


290.15 -2 5

22 shows the load vs. out -o f- pl ane

deflection

curves

of

the flange

plates a t

several

measured

points. Then, the cr i t i ca l local buckling

strain

were

determined

as

shown

in

Figures

23

and

24

as

the

solid

points. Fo r both cases,

the

f lange buckl ing stress or cr i t i ca l

s t ress ~ is approximately equal to the

yield

stress

of

the

section.

Therefore, i t seems reasonable to put

t

=

14.5

as the m x mum

allowable

width-thickness rat io (or cr i t i ca l

width-thickness

ra t io

i f

the flange i s to

be

designed to withstand the yield load, P

Y

without local

buckling.

Theoret ical ly; i f

the

cr i t i ca l s t ress

is equal to the

yield

s t ress the

cr i t i ca l

s t rain

s equal

to the sum

of

the

strain

cr

a t

the

yield point

and the

m x mum tensi le residual s t ra in . That

i s

=

1.52

r t

=

0;006

for

rolled

heat- treated

shapes,

and

=

3.04

=

0.0115

for

welded built-up shapes.

The stub column tes ts show

tha t

the

cr i t i ca l strains are close

to

these

values, (Figure

4.23 and 4.24)

and hence these two

cr i t i ca l

strains are

taken

as th e standa rd

for

determining

the local

buckling of beam-columns

i f t is less than

14.5 .

The

plate

buckling problems in the strain hardening range

h b d

· d b H . . (23) d d d b L

(20)

f

ave een

s tu

1e y aa1Jer a n

exten

e y ay or

application

to

beam,

beam cQlumn and column

members.

Lay

stated

Local buckling wil l not be cr i t i ca l

unt i l

a

cr i t ica l

region

is

strain-hardened , and aclopted

an aspect ra t io Lib,

of 1.2 as the

7/25/2019 290_15


290.15

-26

minimum cri terion for local

buckling

where is the half wave-length

of

the

local buckle and b i s

the width

of the

flange.

In this

study

the

same

Lib

1.2

is used for

the

determination of local

buckling point

on

the

theoret ical

M

curve.

t i s

taken as

the

local buckling moment when the minimum

applied

s t ra in in a

length

of

1.2b

i s equal

to

the

cr i t ica l

s t ra in For

beam-columns

with

thin component plates i t i s generally suff ic ient for the.prediction

of local

buckling

to assume that

the

s t ra in

applied

to

the

flange

is uniform across the thickness

and

width i f the

beam-columns are

bent

about the

strong axis . For beam-columns with equal end

moments

which cause

symmetrical bending the

local buckling

point

can

be

determined

theoret ical ly as an end rotat ion a t which

the

s tra in a t th e lo ca ti on

0.6b

from the mid-height

reaches

the

cr i t ica l

s t ra in The

experimental

local buckling

point

was determined again

by

the top of the knee

method.

In Figures17

and 18 local buckling

points

are shown as cross

marks.

t is

observed

that there is good

correlat ion between the local

buckling

points

determined theoret ical ly

and experimentally. Also i t

i s

i nt er es tin g t o

notice

that

for

welded buil t-up shapes

the

occurrence of loca l

buckling

i s a t a.

comparatively

la rger

end rotat ion than tha t

for

rol led

heat- treated

shapes. Apparently

this

is because the higher tens i le residual

s t resses in

the

welded

shape

which

in cre as e th e

value of

the

cr i t ica l

s t ra in

necessary

to cause

to ta l

y ie ld in g o f th e fla ng e. This

indicates

tha t welding residual stresses

can

actually

in cre as e th e

rotat ion

7/25/2019 290_15


290.15

7

c ap ac ity o f

the

beam column i f the

termination of rot t ion

capaci i s

taken as the local

bucklL.g point .

Furthermore

the

in i t i t ion of loc l buckling

does

not seem

to

reduce

to

s tr en gth o f

beam c olumns dramatical ly. The M

curves · s t i l l follow

t he i r

or ig in l

path for

some distance unt i l

p r o n o u n ~

out of plane

deflect ions

of

the

flanges are observed.

f further

study on the

post loc l

buckling behavior confirms

th is

in the future the

use

of beam columns may be extended

beyond

the

local

buckling

point .

7/25/2019 290_15


5 SUMM RY

N ON LUS ONS

The strength

of

rolled and welded H-shaped beam-columns

made

of

1 514

steel

has

been

investigat

cd

both

theoret ical ly

and

experimentally.

The

effect

of the residual

stresses

due to

cooling

af ter

welding or ro l l ing,

and

the

effect

of

strain

reversal and

of the

mechanical

properties

of

the

s tee l

are

included

in the

determination

of

m o ~ e n t c u r v t u r e t h r u s t

relationships. The load-deformation behavior of beam-columns

was studied by carrying

out

numerical

integration

on fixed

stat ions

of

th e def le cted shapes

of

the

beam-columns; thus,

the

unloading

effects

due to reversed curva tures

can be

included. A computer

program was developed

to

perform

the computation.

Based

on

this

study, th e following

conclusions

may be made:

The mechanical p rope rtie s o f

the

material , the pattern

and magnitude of residual s t resses , and the

strain

reversal

effect ,

a l l are

important

in the f inal shape

of M-P-0 curves, which,

in

turn ,

are

the

sole basis

for

the

determination

of

the load-deformation characterist ics

of beam-columns.

2. For beam-columns, the

effect

of

stra in

re ve rsa l th at

is consideration of the unloading

stress s tra in

relationship

is

more

pronounced for non-linear

materials

than for l inear materials

i f

o ther condi tions, tha t

is

residual

stresses

and

thrust

are

ident ica l .

-28-

7/25/2019 290_15


290.15

29

Consideration of the unloading effect

generally

is not pronounced immediately af te r

t he u lt im a te

load.

Signi f ican t d i fferences between

t he i nc lu si on

and

the

exclusion

of the unloading

effect

can be shown

only

a t

large

rotations of

the descending

part of

the moment-

rotation

curve.

Thus

for simplici ty

the unloading

s t ress s t ra in relat ionship may be assumed

ident ical

to tha t for loading.

4 Two fu l l scale beam column t e s t s one a

rol led

8WF40

shape and

the

other a welded I lH71 shape

were

conducted.

A comparison between

the

theoret ical curves and

the

corresponding experimental M G curves

showed

tha t

the

theory

can

predict not only

the

u lt imate s tr eng th

but

also the complete

history of

a beam column

with

good

accuracy.

5 .

Comparing the

direct integration

solutions of th is

study

to

the extrapolation so lu t ions

obtained from

previous i nv es ti ga ti on s i n A36

steel

shapes i t

i s

shown

that for A514

s tee l shapes

both

rol led

and welded

buil t up

the

direct

integration

solutions

provide

a

higher

ultimate strength.

hence extrapolation

procedure

may provide an approximate but conse rvat ive

estimate of

the strength of A5 4

s tee l

shapes.

7/25/2019 290_15


290.15 -30

he

local

buckling

point of

beam columns was determined

both experimenta lly and theoret ical ly. t was

found

that th e regio na l cri ter ion

provides a

sufficient

basis fo r

the prediction

of

flange

local buckling

of

beam columns bent with respect to

the

strong axis and

that the stub

column

tes ts supply vi t l data for

the

determination of cr i t ic l

str in

when the compression

flanges buckle local ly.

7/25/2019 290_15


CKNOWLEDGEMENTS

Tpis investigation was o n u t e ~ t the Fritz

Engineering

Laboratory Department

of

Civil

Engineering

Lehigh

University

Bethlehem Pennsylvania.

The

United States

Steel

Corporation sponsored

the

study

and

appreciation s

due

to Charles

G

Schilling

of

that

corporation

who

provided much information

and gave

many

valuable

comments olumn

Research

Council Task Group 1 under

the

chairmanship of John Gilligan

provided

valuable

guidance.

31

7/25/2019 290_15


7/25/2019 290_15


290.15

w

~ ~

z

£c

cr

£p

E

rc

£r t

Est

y

£¢

c i

33

Displacement

in

the x ~ y ~

and

z d i r e t i o n s ~

respect ively

Coordinate x e s ~ coordinates of the point with

respect to

x ~ y ~

and z axes

Strain

Strain

due

to ax ia l load

ri t ical

s t ra in

Strain a t proportional l imit

Residual

s t ra in

Maximum

compressive residual

s t ra in

Maximum

t ens i le residual

s t ra in

Strain a t s ta r t of

s t ra in

hardening

Total s t ra in

Yield s t ra in I

y

Strain

due

to curvature

Largest

s t ra in

any element area experienced

End

rotat ion

of a member

Slenderness

f u n t i o n ~ distance between two adjacent

in tegrat ion s ta t ions

Summation

Curvature

Curvature a t

M

p

Curvature a t M

pc

7/25/2019 290_15


290.15

cr

rc

O rt

4

Stress

ri t ical

stress

Stress

a t

proportional

l imit

Residual stress

aximum compressive residual s t ress

aximum tens i le residual stress

Yield s t ress

determined by 0 2

offset

method

for

non-linear

s t ress s t ra in

relat ionship

Largest

stress any element

area

experienced

Rolled

wide-flange

shape

Welded

H-shape

7/25/2019 290_15


IGUR S

5

7/25/2019 290_15


290 15

36

_ = = =

500

Transition

Range

Strain Hardening Range

lastic

Range

ProportionaI Limit

up Uy=o a

o

1 : - - - - - _ - - _ 1 - - - - - - - L - - - L _ . L . - - - - - - - - . l - - _ . . L - - - - - 2 : - - - - - - _ - - - - - - _ L . . . - - - - - - _ - - : 3 ~ - - -

STR IN RATIO EE

U

y

l

VER €E TYPI l

STR SS

STR IN

GIoIRV

0 5

1 0

li

u

y

STR SS

R TIO

Fig

Typical Stress Strain Curve for A5

Steel

7/25/2019 290_15


290 15

olled

Welded

y

37

Tension

ompression

Fig

Idealized Average Residual

Stress

Distribution in A5l Steel Shapes

7/25/2019 290_15


290 15

38

b

~

1

~ b

N

W

d

X

r W

t

y

Fig

Arrangement of Finite Area Elements

7/25/2019 290_15


290 15

39

STR IN

-

- - -

Unloading

tress-

Strain Curve

Monotonic tress-

Strain Curve

Compression

Stress

Monotonic Stress tr in

Curve

TENSION

STRESS

~

Fig

Unloading Stress Strain urve

7/25/2019 290_15


Bending xis

Strain Reversal Zone

HI

I

-40

Tension

Compression

d ~ t

pplied Strain

Due

to

Thrust

Only,€cI

290.15

0 STR IN DIAGRAM

Strain

Line

1---1-- : : ; , ._ +--- - - - - - - - -4 tPreSSion

nsion

E ective Strain Reversal

Area

on

the Section

I

I

STRAIN

COMPRESSIVE

STRESS

b COMPRESSIVE STRAIN REVERS L

-:. .=:;:-===-------- ---i

Compression

1----+-==--:: -==-------1

t

nsion

Effective Strain Reversal

Area on

the

Section

TENSILE STRESS

O STR IN

c TENSILE STRAIN REVERSAL

Fig. 5

Typical Strain Diagram

for

a Beam-Column

7/25/2019 290_15


1

0.9

0.8

0.7

1L

Mpc

0.5

0.4

0.3

0.2

1

i

~ l

--

...

/

./

/ J

Trc

=

1 T

y

Trt

= 3CT

y

1 < 4

...........

_ :

= 9 ,

~ [urc=

T

y

y

1 I - 1 - J - - ~ - - - -

8

111-ifL--. . - ----- 0.7

Assumed Residual

I H e l r - - - ~ - - - - - - - - - 0.6 Stress Pattern

w.fJr--I---r------- 5

Strain Reversal

Included

Strain Reversal

Neglected

o

1.0

2.0

3.0

.

4.0

c/>/A

pc

5.0

6.0

7.0

8.0

Fig.

6

Moment Curvature Thrust Relationship

I

1=

I

7/25/2019 290_15


Curve No

Residual Stress

Stress Strain

Curve

I 4 5

A B B C C

E E

E Elastic Perfectly Plastic

E Curve

P A514 Steel

E Curve

1 0

M

Mpc

. . . .

: =0 55

y

Residual Stress Distributions

I

u

u

1

rt

y urc=O lu

y

C

8.0

:

=

0 55

y

7.0

.0

0

B

4.0

4> 1-

pc

A

3.0

.0

0

Fig

Comparison of

Moment Curvature Thrust

Curves

7/25/2019 290_15


I

l

M

p

P

M

M

p

M

p

p

se

C

se

se 3

Fig

Loading

Conditions

for Beam Columns

0

7/25/2019 290_15


290.15

..J

- . : : 5 _ - t - - - ~ n

I

44

M

L

M

R

L

0.

P ~ ~ ~ X

= 1

a

Radius: ~

Fig.

b

Numerical Procedure

for

Calculating Load Deflection

Relationship

7/25/2019 290_15


29 15

1 01

1.0

b MOMENT

CURVATURE

CURVE

WITH UNLOADING

t p

Fig

10 om nt Curvature Curve With

Unloading

7/25/2019 290_15


290.15

o a

0 7

0.6

M

0.5

Mpc

END

MOMENT

0.4

0.3

0.2

1

o

0 01

Unloading ffe t Considered

Unloading

ffe t

Neglected

20

aYF40

Braced

P=0.55

P

y

A514 Steel

0.02

0.03 0.04 0.05

END ROTATION

8

RADIAN

0.06

46

Fig. 11

End Moment vs. End

Rotation

Curves

7/25/2019 290_15


290 15

0 8

0.7

0.6

. L.

M

0 5

pc

END

MOMENT

0 4

0 3

0 2

1

o

0.01

Unloading Effect Considered

Unloading

Effect

Neglected

=

20

. r

Welded

Built UP·

II

H

Braced

P=0 55P

y

A514

Steel

0 02 0 03 0 04

0 05

END

ROTATION 8

RADIAN

47

Fig 12 End Moment

vs

End Rotation Curves

7/25/2019 290_15


0.5

0.4

M

M

pc

0.3

0.2

1

o

Elastic Perfectly Plastic ] - Curve

A514 Steel ] -

E Curve

P =0.55 P

y

:

1 0.02

END ROTATION

RADIANS

p

0.03

Welded Shapes

With

Flame-Cut Plates

0.04

1 0

il l

o

l

Fig 3

Comparison of

End Moment

and

End Rotation Curves

I

1=

ro

7/25/2019 290_15


tv

o

Rolled Heat Treated 8 I

4

}

Welded IIH71

With Flame Cut

plates· A514 Steel

Extrapolation Solution From

A3 6

Steel Shapes

p

M

1

8

6

y

4

M

P

2

o

2

4 6

M/Mp

8

1

Fig 4 Interact ion Curves for A514 Steel eam Columns

I

7/25/2019 290_15


290.15

-50

Compressive Force

P

WeldedConnectlon

7

\

\

r

F

I

lD

I

I

I

=<t

I

F

0 GENERAL LAYOUT

Location of

Bracing

Dynamometer

Crossheod

I j

n ~ h ~

ne

Third t [ T ~

Point

rt

1

- -

t

Posi tion

j

H

At Posit ion ®

Hydraulic

Jock

Location of Bracing

I

H

At Position @

Location

of SR4

Gages

Fig. 5

b INSTRUMENTATION

Detail

of

the Beam Column Specimen

7/25/2019 290_15


N

i ll

o

a

Beg inn ing o f Tes t

b

End o f T es t

Fig 6 eam Column

in

Testing Machine

l

7/25/2019 290_15


0 5

Rolled

Heat - Treated

A514 Steel

tv

O

o

I

l

0 02 0 03 0 04

END ROTATION 8 RADIANS

0 4

M

M

pc

0 3

END

MOMENT

0 2

1

o

1

P

M

8VF

Braced

=0 55

b = 40

r

x

Experiment

0 05

Fig Load Deformation Relationship and Test Results

l

tv

7/25/2019 290_15


Welded H 71 With Flame Cut

Plates A 514 teel

Braced

P = 0.55 P

y

= 4

x

0.5

4

M

0.3

M

pc

END

MOMENT

0.2

1

o

0.01

P

M

8

0.02 0.03

END

ROTATION.

RADIAN)

4

Experiment

0.05

tv

l O

o

I

tTl

Fig 8 Load Deformation Relationship and Test Results

tTl

W

7/25/2019 290_15


290 15

54

\

\

\

5 0

1 5

M·

l 1

~ ~ ~ ~ : a t d ~ ~ ~ -

d

105 01

M

s

1 o - - < f - - - - - O - : : ~ _ + _ _ - - - : ; . , . _ _ k ~ ~ = - = - -

d

95 01

o 0 0

I

~ 0 03

EN ROT TION

0 04

Mj Hydraulic Jack Reading

d ynamometer Reading

M

s

Strain age Reading

Fig 19 Error the Moment Readings

7/25/2019 290_15


N

l D

o

0 8

Rolled Heat Treated

M

max

Wide

Flange Shapes

pc

0 6

M

L x

Welded Built Up

Shapes With

0 4

Flame

ut

Plates

M

: =

0 55

0 2

P

y

A5 4 Steel

o

20

L

r

x

40

60

Fig

omparison

etweenTest

Results and

Theoretical

Solutions

I

U

U

7/25/2019 290_15


290.15

-56

1 5

.0

.5

1 rtl=IO rcl·

1. 0

. - - ~ - - - - - - - ~

o

J

J cr 0.5

y

0 With Res idua l Stress of Cooling Pattern

1.5

Elastic L imit

0.5

I .0

. . . . . . .

r

5

y

~ r u =

2t

1

E

b) With Residual Stress of Welding Pattern

Fig.

21

Plate Buckling Curves Plates Simply

Supported and Free

Unloading Edges

7/25/2019 290_15


tv

c

a

.

J1

2

a

7 1

4

11

:

7

1

4

ali

®

2

Location

1 0

2 0

Load - Deflection Curves

L oca l B uckli ng Tests on

on II H 71 Shope

o

0.5

1 0

1.5

DEFLECTION IN.)

Fig. 22

Load-Local Deformation

Curves

I

J1

J

7/25/2019 290_15


290 15

15

58

cr

123

ksi

rLO O uckling oint

Rolled Heat Treated

8VF4

A514 Steel

Stub

Column

Test

29 W 1

LO

KIPS

1

5

o

5

STRAIN 10

3

i ~ l i n

10

Fig 3 A Stub Column Test

7/25/2019 290_15


290.15

3.0

59

II H

Welded Built Up Shape

A514 Steel

Stub Column Test

290 I H

LOAD

10

KIPS

2.0

1

o

S T ~ N

10 in.lin.

10

Local

Buckling

Point

F ig . 24

Stub

Column Test

7/25/2019 290_15


9. REFERENCES

1.

Galembos T.

V.

and

Ketter

R.

L

OLUMNS UNDER

COMBINED

BENDING

ND

THRUST

Transactions ASCE Vol. 126 Part 1 1961 p. 1.

2. Ketter

R. L.

FURTHER STUDIES

ON

THE STRENGTH OF BEAM COLUMNS

Journal of the Structural Division ASCE Vol. 87.

No. ST6 Proc. Paper 2910 August ·1961 p. 135.

3.

Ojalvo

M. and Fukumoto Y.

NOMOGR PHS FOR THE SOLUTION OF BEAM COLUMN PROBLEMS

Bulletin No.

78

Welding

Research Council

1962.

4. Van Kuren T. C. and Galambos T. V.

BEAM COLUMN

EXPERIMENTS

Journal of the Structural

Division

ASCE. Vol. 90 No. ST2

Proc.

Paper 3876

April 1964 p. 233.

5.

Lu L. W. and Kamalvand

H.

ULTIMATE

STRENGTH OF

LATERALLY LO DED

COLUMNS

Journal

of

the Structural Division

ASCE

Vol. 94

ST6.

Proc.

Paper 6009 June 1968.

6.

Yu C. K.

INELASTIC

OLUMNS

WITH

RESIDUAL

STRESSES

Ph.D Dissertation Lehigh University Bethlehem a ~

April 1968.

University Microfilms Ann

Arbor

Michigan.

7 American Society for Testing and

Materials

STM STANDARDS

Part

3

A370-6l7 1961.

8. Odar E. Nishino

F.

and Tall L.

RESIDUAL

STRESSES IN T-l

CONSTRUCTIONAL

ALLOY

STEEL

PLATES WR Bull. No. 121 April 1967.

9.

Odar

E.

Nishino

F.

and

Tall

L.

RESIDUAL STRESSES

IN

WELDED BUILT UP T-l SHAPES

WR Bull. No. 121 April 1967.

-60-

7/25/2019 290_15


290.15

-61

10.

Odar E. Nishino F. and Tall L.

RESIDUAL

STRESSES

IN

ROLLED

HEAT TREATED T-l

SHAPES WRC

Bull. No.

121 April 1967.

11.

Parikh

B.

P.

ELASTIC PLASTIC ANALYSIS

ND

DESIGN

OF UNBRACED

MULTI STORY STEEL FRAMES Ph.D

Dissertation

Lehigh University Bethlehem

Pa.

1966 University

Microfilms Ann Arbor

Michigan.

12.

Driscoll

G.

C. J r et al

•

. PLASTIC DESIGN OF

MULTI STORY FRAMES

Notes on Plastic Design of

Multi-Story

Frames

Lehigh

University

Bethlehem

Pa.

1965.

13. Godden, W G.

NUMERICAL

ANALYSIS ON BE M ND COLUMN

STRUCTURES

Prentice-Hall

Inc.

Englewood Cliffs New Jersey

1965.

14.

Ojalvo

M

RESTRAINED COLUMNS

Proc. ASCE 86 EM5 , October

1960.

15 .

Tall

L. Editor-in-Chief

STRUCTURAL

STEEL DESIGN

Ronald

Press

New

York

1964

16.

Popov, E.

P.

and Franklin H.

A.

STEEL

BEAM TO COLUMN CONNECTIONS

SUBJECTED

TO

CYCLICALLY

REVERSED

LOADING

Prel iminary Reports

1965 Annual Meeting

Structural

Engineer

Association

of Ca li fo rn ia

February

1966.

17. Huber,

A. N.

and Beedle L. S.

RESIDUAL STRESS

ND

THE COMPRESSIVE

STRENGTH

OF

STEEL

Welding Journal Vol. 33 December,

1954.

18 . Lay,

M. G. Aglietti R. A.

and Galambos, T.

V.

TESTING

TECHNIQUES

FOR

RESTRAINED

BEAM COLUMNS

Fritz Laboratory Report 30. 270.7 October 1963 .

7/25/2019 290_15


290 15

62

19. Yarimci ·

E.

Yura J. A.

and Lu L

W

TECHNIQUES

FOR TESTING

STRUCTURES PERMITTED TO

SWAY

Experimental Mechanics

August

1967.

20

Lay

M.

THE STATIC

LOAD DEFORMATION

BEHAVIOR

OF

PLANAR

STEEL STRUCTURES

Ph D

Dissertation

Lehigh

University Bethlehem

Pa. 1964

University

Microfilms Ann

Arbor Michigan.

21 Nishino F

BUCKLING STRENGTH OF COLUMNS

ND THEIR

COMPONENT

PLATES

Ph D

Dissertation

Lehigh

University

Bethlehem

Pa.

1964 University

Microfilms

Ann Arbor Michigan.

22 Hu P

C.

Lundquist

E

E.

and

Batdorf

S

B.

EFFECT OF

SM LL DEVIATIONS

FROM FLATNESS ON

EFFECTIVE

WIDTH ND

BUCKLING OF

PLATES IN

COMPRESSION

NACA TN

1124

1946.

23

Haaijer

G.

PLATE BUCKLING

IN THE

STRAIN HARDENING RANGE

ASCE

Journal

Mechanics

Division

Vol. 83

No.

EM2 April

1967 •

Documents

290_15