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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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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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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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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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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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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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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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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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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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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
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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
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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.
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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
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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 .
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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-
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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
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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 .
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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
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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
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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
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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
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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
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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 .
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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-
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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.
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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.
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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
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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
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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
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IGUR S
5
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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
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290 15
olled
Welded
y
37
Tension
ompression
Fig
Idealized Average Residual
Stress
Distribution in A5l Steel Shapes
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290 15
38
b
~
1
~ b
N
W
d
X
r W
t
y
Fig
Arrangement of Finite Area Elements
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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
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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
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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
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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
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I
l
M
p
P
M
M
p
M
p
p
se
C
se
se 3
Fig
Loading
Conditions
for Beam Columns
0
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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
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29 15
1 01
1.0
b MOMENT
CURVATURE
CURVE
WITH UNLOADING
t p
Fig
10 om nt Curvature Curve With
Unloading
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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-
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290.15
-61
10.
Odar E. Nishino F. and Tall L.
RESIDUAL
STRESSES
IN
ROLLED
HEAT TREATED T-l
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