Lehigh University
Lehigh Preserve
Fritz Laboratory Reports
Civil and Environmental Engineering
1954
Residual stress and the compressive strength of
steel . Welding Journal, 33 (12), p. 589-s,
(December 1954), Reprint No. 96 (54-3)
A. W. Huber
L. S. Beedle
Follow this and additional works at: http://preserve.lehigh.edu/engr-civil-environmental-fritz-labreports
Recommended Citation
Huber, A. W. and Beedle, L. S., "Residual stress and the compressive strength of steel . Welding Journal, 33 (12), p. 589-s, (December
1954), Reprint No. 96 (54-3)" (1954). Fritz Laboratory Reports. Paper 1510.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1510
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[email protected].
Final Report on the Pilot Investigation
' ',
RESIDUAL STRESS AND THE COMPRESSIVE
STRENGTH OF STEEL
by
A. W. Huber and L. S. Beedle
,I
This work has been carried out
as a part of an investigation sponsored
jointly by the Column Research Council,
the Pennsylvania Department of Highways
and the Bureau of Public Roads
Fritz Engineering Laboratory
of Civil Engineering and Mechanics
Lehigh University
Bethlehem, Penna.
De~artment
'
..
1 December 1953
Fritz L2boratory Report No. 220A.9
TAB L E
•
oF
CON TEN T S
Page
I.
II.
SYNOPSIS
3.
4.
5.
2.
3.
4.
5.
6.
3.
4.
5.
V.
2.
3.
4.
Test Program .
Coupon Tests· .
Residual Strain Measurements
Cross-Section Tests.
Column Tests
Coupons.
Residual Stresses
Cross-Section Specimens
Columns.
DISCUSSION
1.
2.
3.
4.
5.
6.
7.
VII.
•
TEST RESULTS .
1.
VI.
.
Rectangular Shape-Known Residual Stress •
WF Shape of Two Rectangles-Known Residual Str'ess .
WF Shapes - llExact" Solution for Known
•.
Residual Stress.
Cross-Section Analysis - Rectangular .
Specimen ~
. . .
Cross -Section Analysis - WF Shape (Approx.) .
Cross-Section Analysis - WF Shape ("Exact!l)
DESCRIPTION OF TESTS
1.
2.
2
Nature of Residual Stress.
Influence of Residual Stress.
Approach to the .Problem
Compressive Strength of Steel
'rest Program .
THEORETICAL ANALYSIS
1.
IV.
1
INTRODUCTION .
1.
2.
III.
.
8
9
11
11
12
12
14
15
16
19
19
20
22
23
25
28
28
29
31
33
34
Compressive Strength
Magnitude and Distribution of Residuai Stress .
Variation of Residual Stresses. . . .
Influence of Residual Stress Distribution
Upon Column Strength. . . .
Influence of Residual Stress-Correlation.
Between Theory and Tests. . . . .
Correlation with Current Design Specifications.
Recomnendations for Future Research
CONCLUSIONS
3
4
5
•
34
36
38
39
40
44
48
49
TAB L E
.
VIII .
IX.
REFERENCES
· 53
X.
APPENDIXES
· 55
3.
4.
j
Page
· 52
2.
',.
CON TEN T S
Continued
ACKNOWLEDGEMENT .
1.
.
oF
Nomenciature
.
Rectangular Specimen. Containing Residual.
Stress .
•
WF Specimen Containing Residual Stress.
Sample Calculation
· 55
· 57
· 59
· 65
-1
220A~9
I.
SYNOPSIS
•
The scope of the investigation upon which this report
is based was to determine whether residual stresses set up
during cooling appreciably reduce the strength of compression
members of structural steel and, if s-o, to find methods for
predicting this reduction with satisfactory engineering accuracy.
The report shows that the strength of axially loaded
columns of 8 WF 31 shape cannot be determined from results of
small coupon tests but that due consideration must be given to
residual stresses.
The theories described give adequate explana-
tion of experimental behavior.
of steel is also reported •
.
,
Data on the compressive properties
-2
220A.g
.,
..
II. I N T ROD U C T ION
For years the strength of metal columns has been computed on the basis of the stress-strain curve.
Such curves have
been determined from coupons cut from the cross-sections of
rolled shapes.
In cutting these coupons, residual stresses
(sometimes called "locked Upll or "initial l ! stresses) have been
reJ.eased so that the measured stress-stJ:'ain curves have necessarily omitted any possible influence of residual stress.
The strength of axialJ.y loaded columns free from residual stresses is given by the tan~ent modulustheory(IS). To
apply the theory it is only necessary to have information 6n the
compressive stress-strain relation in order to obtain the com-
.
plate licolumn curve" of stress plotted against slenderness. ratio .
As numerous tests have shown*, this procedure permits the prediction of column strength for annealed material.
Differences
between theory and experiment can be attributed to such causes
as initial curvature and eccentricity and both can be kept small
in carefully conducted laboratory tests.
When small coupons are tested
~.n
compression the re-
sulting stress-strain diagram approaches an idealized curve
consistj.ng of. two straight lines, one at a slope equal to the
elastic modulus E, the other
ho~izontal
(average stress in the plastic range).
at the yield stress level
With such a curve it
would seem that the tangent modulus theory was of little value
in the case of structural steel.
A consideration of residual
stresses, however, shows its applicability.
*
3~e
p. 20 of Reference 6.
. -3
220A.9
1. The Nature of Residual stress
Residual stresses are those that are left,in a member
after it has been formed into a finished product.
steel, residual stress
. a.
~ay
In structural
be due to several causes:
Uneven cooling of shapes immediately after hot
rolling.
b.
Fabrication operations such as cold-bending
("gaging"), punching,
shearing~
cambering and
weJ.ding.
Only the first type (cooling residual stress) is considered in
this paper.
The mechanism for the formation of cooling residual
stresses has already been described (4, 16,1~8).
The flange tips
(which cool the most rapidly) are left in compression while the
flange Qenter is left in tension.
As a general rule, the part
of the member cooling most slowly will be left in residual
Fig. 1
tension~
indicates a typical shape of residual stress pattern.
It was observed in tests at Lehigh that yielding in
the flanges of WF sections started at lower loads than theoretically predicted; therefore measurements of residual stresses
were made
(11). These measurements and later experimental
work
,
on beams of WF shapes (18) confirmed beyond doubt the presence of
residual stress.
..
,
Magnitudes of compressive residual stress up
to 20,000 psi have been observed in as-delivered rolled steel
bpame.
(12)
-4
220A,g
2.
.•.
...
Influence of Residual stress
From the nature of residual
stres~es
it can be seen
that their presence has a definite influence upon the over-all
stress-strain relation of structural steel.
By partial yielding
of a cross-section under an applied uniform stress below the
"nominal" yield point, increased strains result in the remaining
elastic parts of the cross-section, thus making the stressstrain relationship non-linear.
Assuming the "applicability of
the tangent-modulus concept there will be a consequent influence
of residual stress upon column strength.
The influence of residual stresses on bending members
and on
•
colt~ns
has been briefly described in Reference 9
and
more extensively in Reference 18 .. In brief, referring to Fig.
2,. if a compressive residual stresss of magnitude O'rc were
present at the flange tips, then yielding would commence when
O'p = O'y - O'rc
The over-all stress-strain relationship for a column would
thereafter be non-linear: but With sufficient straining the
average stress would reach yield-point magnitude.
Thus, the influence of residual stress in columnS is to
make the stress-strain relationship non-linear above a residualproportional limit, O'p.
Assuming the applicability of the
tangent-modulus concept) there will be a consequent influonce ort
column strength.
This is illustrated in Fig. 3,
the right-hand
sketch being the familiar column curve.
Why aren I t these stresses Wiped out
yields?
They are wiped
out~
mencing at the stress, 'O'p'
whe~
the membe .."
but it is a gradual process, comAs stress is increased, more and more
-5
220A.9
of this stress is relaxed by plastic deformation until, at Point
.'
'.
3, Fig. 2;
the member is completely plastic and the stress is
at yield-point level, cry'
Prior to this study a few isolated tests on as-delivered
columns also indicated that the strength
W&8
lov,er than predicted
on the basis of small coupons cut from the cross-section and
this reduction, could not be attributed entirely to such causes
as initial curvature and eccentricity.
This discussion shows that the following is necessary:
a.
Develop methods for predicting the strength
of columns containing resiqual stress.
b.
Examine experimental techniques for the proper
correlation with theory and for routine laboratory use.
c•
With this in hand, examine current specifications
and formulas to determine whether present design
rules require modification either for safety or
economy.
These considerations led to the program here reported.
3. Approach to the Problem
As reported by Salmon (14) in 1921, as early as 1888,
it was recognized that ('previous strains" were of direct in...,
fluence upon the strength of columns.
Although other sources of
llinitial stresses" were mentioned, stresses set up du,e to cooling
aftGr rolling were not discussed by Salmon.
He notes that
!:qonsidere suggested using test pieces that had undergone exactly
the same processes as the member itself"-- and this was one apprca8h used in the investigation reported in this paper.
220A·9
-6
Recognition of the role of residual stresses is a con-
.'
tribution of, the Column Research Council (8)
•
was established in an effort to remove the confusion and lack
which organization
of harmony in column formulas in many specifications now in use.
Bruce Johnston was active in efforts to initiate studies of
residual stress influence.
In 1946 a Research Committee of
Column Research Council stated the following as one of their
recommendations for further research:
"Rolled sections, sections built up by welding, and
sections fabricated by bolting or riveting generally
have mateJ:'ial residual stresses, both compression and
tension, in the member.
,.
In addition, the member may
have residual moments and shears incident to relative
deformations in the fabrication of the structure of
which the member is a part. The effect of these residual stresses on the strength of compression members
is subject to question".
Research carried out at
th~
University of Illinois on
the influence of welding residual stresses, was reported upon
in 1935(17). Cooling stresses were not studied in that investigation which involved the study of bui,lt-up members.
More re-
cently studies have been conducted on model members with artifi-
.
clally-induced residual stresses.
Test results were reported*
which show a substantial reduction in strength
residual stresses.
*
attributab~.e
to
Recently osgood(l3) has discussed the
S., Cherry, P. Y. Chow, and W.J. Austin, "Experimental Studies
of Model Columns ll , Structural Research Series No. 34, University
of Illinois, July 1952.
-7
220A.9
general problem of residual stresses in columns.
11,
At Lehigh
the problem has been investigated for steel columns(18). Based
on those investigations this paper outlines two approaches for
•
a solution of the problem and its application to WF structural
steel columns.
The agreement between the two methods is very
close and in addition is corroborated for 8 WF 31 sections by
column tests.
The first method is an analytical solution requiring
(1) a knowledge of resid~aJ_ stress distribution (measured or
assumed) and (2) the material properties (modulus of elasticity,
yield stress level).
As shown by the Lehigh study(l8) the
strength of the column can be expressed as a function of the
moment of inertia of the unyielded portion of the cross-section
(elastic moment of inertia, Ie) and the slenderness ratio.
The second method involves an analytical solution
and requ:i,res only one measurement:
the average stress-strain
relation determined from a short member of sufficient length to
retain the original magnitude of residual stresses.
This is
called the "cross..:.section" test since it involves testing the
complete cross-section of the specimen.
Column strength can be
expressed as a function of the tangent modulus obtained from
the stress-strain curve of the "cross-section specimen" and the
...
slenderness ratio with due allowance for bending about the
'
strong or weak axis.
This second method does not require the
special measurement of residual stresses. '
-8
220A.9
4. Compressive Strength of Steel
Up to this point attention has been directed to the
influence of residual stresses.
Of considerable additional
importance is the basic strength of steel in compression, or
more particularly the yiela stress level, a y (Fig. 3).
A few
careful studies have been made of the strength of steel in compression and in one, (7)
comparison was made with results of
tension tests.
Compression testing is considerably more difficult
to accomplish than tension testing, and correlation between the
two is essential if tension test data is to continue to be the
basis for colwm1 strength formulas.
In the mill tension test, the specimen is invariably
selected from the web.
Since this. material ordinarily tests
at higher strength than flange material and since more than
75%
of the load capacity of columns is 'in the flanges, then there is
a question as to whether or not the tension test is representative of column strength.
Careful machining is required of compression test
coupons and usually a number of specimens are required from one
cross-section before a good average may be made.
Thus the
"cross-section" test previously mentioned offers possibilities
of an economical substitute for a large number of coupon tests .
•
,
Answers to the follOWing questions are then desired:
a.
vfuat is the strength of full cross-sections
of the various rolled shapes (v&, angle, channel)
and how do their properties compare with those
given by coupon tests?
-9
b.
How do the compressive properties so determined
compare with the mill-type tension tests of
specimens cut from the web?
•
Can a correlation
be established that will be useful in arriving
at reliable compressive strengths of structural
shapes?
The answers to these questions will be sought in
future investigations.
E~ploratory
relationships nlay be seen
in the data presented herein.
5. Test Program
The fo:llowing general program was considered necessary
. to accomplish the purpose of the study:
•
1.
Tests of as-delivered, axially loaded, pin-ended
columns in a range of slenderness ratios, (residual stresses to be measured on pieces immediately adjacent to the columns).
Columns were to
be free to bend about either one axis or the other.
2.
Tests of annealed columns.
3.
Tests of tension and compression coupons cut
adjacent to the other test pieces.
4.
"Cross-section" tests.
5.
Meaurement of residual stress distribution,
magnitudes and variation.
The study was limited to ASTM grade A-7 steel rolled to 8 WF 31
shape, one of the most frequently used shapes for building
columns.
-10
220A.9
It is the purpose of this paper to describe theoreti-
cally the influence of
res~dual
stress, present test results
in substantiation of the theory, give data on compressive pro-
•
perties of steel, and tocorrp.late the observations
speCifications and design formulas.
w~th
current
-11
220A.9
III.
THE 0 RE TIC A L
A N A L Y SIS
The ideas presented in earlier investigations (13,18)
,~
..
are extended here for practical application to WF sections .
The subsequent derivations are made under the following assrunptions:
1.
The stress-strain behavior of fibers of the
material is characterized by two straight lines
of slopes E and zero (ideal stress-strain diagram for steel).
2.
Plane sections remain plane after deformation.
3.
The residual stress distributj.on is symmetrical
with respect to both axes of the
~W
sections;
the residual stresses are constant along the
column.
The derivations are divided into two parts: (1) those
based on measured or assumed residual stress patterns and (2)
those based on the lIcross-section" test.
For completeness some
of the material presented elsewhere is repeated here.
1.
Becta0gula~
Shape - Known Residual Stress
c.
When a steel column is strained above the residualproportional limit, portions of the cross-section yield (POint 1
of Fig. 2).
')
If
th~
yielded parts are perfectly plastic, the
bending stiffness of those parts reduces to zero.
The theoreti-
cal buckling strength will then be equivalent to that of a new
column whose moment of inertia, Ie' is the moment of inertj.a of
the remaining elastic cross-section(18), or
pc·,
."..?-1:;'
I
_. ,'-,_~ ..~I?
,L'
220A·9
-12
or
PerlA =
•
•
----.;;.-
The action of a rectangular specimen is described in Appendix 2
and it is shown there how the column curve may be obtained for
particular residual stress distributions.
2. WF Shape Consisting of Two Rectangles - Known Residual Stress
If the web is neglected and if it is assumed that a
\{F
shape consists of two equivalent rectangles, then, referring
x
to Fig. a,
0"
-.........:
=
Ie
y
I
in which k =
,
neglects the
own axes.
k
1\)'
(Flexure about x-x axis)y <-. ,
.
~~
. _._.__L___ _
= k 3 (Flexure about y-y axis)
FIG. a
xo •. The above equation is an apprOXimation that
__
b/2
moment of inertia of the rectangles about their
Equatj.ons for the critical slenderness ratio are the
same as in AppendiX 2; the weak axis of the rectangle is the
strong (x-x) axis of the WF shape.
3. WF Shapes - "Exact II Solution for Known Residual Stress
The previous solutions have all assumed that the web
of a WF shape contains no residual stress. Numerous measurements
have been made indicating that this is not the case.
A solution based on
coup~~
test results and a known
residual stress pattern which takes into account the web residuals and is applicable to WF shapes may be obtained using the
approaches previously mentioned.
220A.9
LE
\:-.+---= .I
,---J,
¥I:I\
I
E
lFx
I.
bpI
-13
Ip
\
i
\
I
\
I
I
I
I
I
I..,,)
u
z .' /
1/
Li'kJ
I
p
FIG. b
Considering the column j.n Fig. b the strain
E
will
increase uniformly until bending starts (corresponding to the
tangent modulus load of the Engesser-Shanley theory).
'y
rrhen the
following equilibrium equation foJ." the infinitessimal bent
position about the y-axis results:
Pu + . \ E(x) (bE) (x)dA = 0
A-"
2
From the relationship 6E
d u then is obtained
= dz2
x
2
Pu + d u
dz 2
E(x) x 2 dA -- 0
\
) A
which corresponds to the Euler differential equation
f01~
E =
constant.
The integral in Eq. 2 is readily evaluated for the
"ideal" stress-strain curve of structural steel.
JAE(x) x 2 dA
=
E
\ x 2 dA
Ae
= EI e
Ie is the moment of inertia of the elastic part of the crosssection.
The critical stress, i.8. the stress at which the
column may start to bend, can therefore be written in the form
-14
220A.9
C1 cr
= -rr 2
EIe/I
(1)
(L/r)2
For the solution of Eq. 2 the relation between
be known.
C1 cr
and Ie must
This may be don8 by obtaining an expression for ocr in
terms of the yield condition (xo)-and by obtaining an expression
for-Ie in terms of the geometry at the yield condition.
been done in Appendix 3.
(a)
This has
Two cases have been considered:
Yielding does not commence in the web until
the flanges are fully plastic.
(b)
The web starts to yield before the flanges
have completely yielded.
If the tensile stress at any point in the web is less than that
in the flange, then Case (b) is applicable.
In Figs. 43- and 46 typical stress-strain curves have
been drawn and the applicable equations are tabulated there.
A sample solution has been carried out in Appendix
4.
In the previous three sections it was assumed that the
yield stress level and the residual stress distribution were
known.
Using the same original assumptions the problem can be
approached in a different manner, based on application of tangent
modulus theory to a "cross-section" test.
4. "Cross-Section" Analysis - Rectangular Specimen
,"'
Suppose a rectangular specimen containing residual
stress and short enough to prevent failure as a column were
loaded in compression.
Et
do
Now,
= dE = Ae/A
(18)
. E
-15
220A·9
In a rectangular element bent about its weak axis,
IeII = Ae/A.
For flexure about the strong (:y"-y) axis,
IelI
= {Ae /A)3
Et/E
=
Since
Ae/A = k
then Eq. 1 becomes, for the two flexure axes:
(J
cr
(4)
lrhe values for Et are determined from the test of the full
".
rec t angu 1 ar cross-sec"tlon.
S·lnce _Et
1
= .(,
E
+'1"
L-.l.1S
"1
"d
approacn
s 10tH
give the same result as that in Section 1, previously.
5. Cross-Section Analysis -
VW Shapes (~pprox.)
A demonstration that tangent-modulus theory is
ap~
plicable to the cross-section test of a WF shape may be seen
from the following which is based on a suggestion made by B. G.
Johnston.
It is assumed that the residual stress distribution
is such that the flange is fUlly plastic before the web yields.
Using
Et
E
Ae
A
=
in which
A
= Aw +
Ae
AF
Aw + k AF
=
and where k = xo/b/2 as before, then
EtA
=
E (Aw + k AF)
and solving for k as a function of Et,
k = AEt
AW~
AF
AFE
-"\
,
-
(5)
-16
220A.9
Nm'l, approximately, neglecting the moment of i.nertia of the web
about its own axis,
Iq,y.
= 1<:3
Iy
Then, for the weak axis
(6)
where k is computed by Eq. 5.
For flexure about the strong axis, neglecting the
moment of inertia of the web
Ie,x . k
Ix
Then, from Eq. 1,
7T 2Ek
ocr = (1,/1')2
v111ere
k is given by Eq.
5.
The slenderness ratio, L/r, for o·ritical stress at a
given Et may then be computed.
k in Eq. 5 may be expressed approxJ.matel;y as k
Eq. 7 then becomes_ 7T2Et
ocr - (L/r) 2
::!::
Et/E.
(8)
6. Cross-Section Test - "Exact" Solution, WF Shape
Assume a WF-section subjected to compression, the
length being sufficiently small to prevent failure as a column.
If no residual stresses were present the stress-strain curve
would be linear up to the yield stress level.
i"
Due to residual
stresses, the curve would deviate from the slope E at a stress
crp = cry - arc (proportional limit for the cross-section test).
Above the proportional limit (see Appendix 3) the strain is
given by
220A.9
-17
1
(9)
E = E (cry - crrxo )
for yielding in the flanges only.
After the flanges have become
completely plastic similar e:::cpressions for strains at sUbsequent
stages may be obtained and are tabulated in Figs. 43 and 46. The
average stress is given by Eq. 10 (see Appendix 3)
Ae
4t f'b/2
°1 = 0y - A °rx o - AJ
°rxdx
. X
o
Using
Eqs.
10 and
9
the average stress-strain
curve of the cross-section can be computed.
of the curve is given by
do
do dxo
Et = dE = dxo €d
..
(10)
The tangent modulus
.
By differentiating Eq. 10
'
do
dx o
From Eq. 9,
dE
dx o
Then
=!
A
(4tx o + Wd l )
1
E
(11)
Eq. 11 expresses the tangent modulus Et in terms of x o '
derivation is for Yo
= 0,
rl'his
which means that Eq. 11 is valid as
long as the web remains elastic.
The relationship between the moment of inertia Ie and
....
the yield condition as defined by Xo
wa~
obtained in Appendix 3
Since both Eqs. 31 and 11 are in terms of Xo '(Yo = 0)
it is possible to express Ie in terms of Et . Solving Eq. 11
(Eq.31).
for
Xo
and substituting it into Eq. 31, the following equations
are obtained relating Ie to Et after neglecting some small terms,
220A.9
E
lex
Ix
-18
2
= _AE_t_-...:::3~A_wE_
E ley =E
Iy
AF + Aw
3
~Et
(12)
_
A~
3
,
\AFE
AF
'It will be recognized that the second of Eqs. 12 may be" expressed
as the quantity Ek 3
(see Eq. 5).
proximately equal to Ek.
The first of Eq. 12 is ap-
Eq. 12 can be used until either the
web starts to yield or the flanges have completely yielded. For
·the latter case the limiting Et is obtained from Eq. 11 by
setting
= O.
Xo
Therefore it is seen that if Et is determined from a
'.
'.
"cross-section" stress-strain curve, Eq. 1 may be solved with
the aid of Eq. 12.
The various solutions derived in Sections 5 and 6 are
compared in Fig. 4 considering flexure about the strong axis.
(The various experiments for flexure about the weak axis are
identical.)
pared.
Eqs.7, 8, and Eq. 1 using expression 12 are com-
The "exact ll solution is used in the later portion of
this paper;
it involves only a slight amount of extra work
beyond the approximation of Eq. 7.
In some applications, direct
use of the tangent modulus (Eq. 8) may be sufficiently precise.
This completes the presentation of theoretical methods
for computing the influence of residual stress on column
strength.
A description of test methods used to obtain a corre-
lation with theory follows in the next chapter.
220A.9
IV.
DES C RIP T ION
o
F
-19
'rESTS
1. Test Program
...
,
The tests performed In.thls investigation are shown
•
in Table 1.
Note that the same test number applies to one series
of tests conslsting generally of coupon tests, residual stress
measurement, cross-section test, and column tests.
and column tests shown by test number "205" and
of another program.*
material.
rt
Some coupon
205A" were part
The specimen designation identifies the
For example in the designation IB2.l, "I" is the
heat, "B" is the rolling or ingot, "2" is the posi.tion
{middJ.e
third) in the rolling and "1" is a specimen cut out of this
piece.
Fig. 5 j.llustrates this designative procedure and shows
where all of the material in the test program was located.
The type and number of tests is arranged according to
material condition in Table 2.
The tests covered as-delivered
and annealed material, two heats, two rollings in one heat and
several locations in two rollings.
Only one shape, the 8 WF 31, was used in all tests.
The as-delivered material used in the colu.'1ln tests, cross-section
tests, residual strain measurements and coupon tests came from
one ingot rolled in two parts (designated as rollings A and B
in Fig~
5). The member used for the investigation of the
variation of residual stresses along its length came from a
different heat.
The two annealed columns and one annealed
cross-section specimen were heat treated at the same time.
second cross-section specimen was annealed separately.
* "Welded Continuous Frames and Their Components", sponsored
by the Welding Research Council and the Office of Naval
Research.
A
-20
220A.9
In summary, the following tests were performed as
,
.
..
part of this propram and will be des'cribed in detail later .
a.
Tension and compression tests of small coupons
cut from positions next to the cross-section
specimens.
b.
Six sets of residual strain measurements performed on pieces next to the cross-section
specimens.
c.
Six "cross-section tests" of which two were
annealed.
d.
Three axial column tests where bending. was
allowed in the weak direction.
columns were annealed specimens.
Two of these
The results
of four additional axially-loaded column tests,
•
(three ,"strong-axis I', one "weak-axis") are
also included.
e.
Study of the variation of residual stresses
along a 9 ft. member and the change in residual
stress distribution within the member as it was
shortened by successively cutting pieces from
each end.
2. Coupon Tests
..
The dimensions of the tension coupons were selected
according to ASTMStandards:
full thickness of the material
(flanges or web) and 1 1/2 in. width over an 8 in. gage length.
The dimensions of the compression coupons followed
the recommendations of Research Committee A of the Column
220A.9
~21
Research Council*:
.
full thickness of the material, a coupon
length smaller than 4 1/2 times the thickness and greater than
twice the width plus gage
•
length~
In addition the gage length
should be chosen greater than the width but smaller than twice
the width of the coupon.
The dimensions of the coupons cut from
the web \'Tere approximately 1.35" x 0.29" x 0.40 11 -0.50 11 ** and
1.90 11 x 0.43 11 X 0.55 11 -0.75 11 for those cut from the flanges.
Some coupon test results included in Table I were already available when this program was started.
Their dimensions were about
the same as those given above except for the width which was
1.12".
All coupons were tested in a 60,000# hydraulic testing
machine.
strain
The strains were measured in most cases
gages~
wj.. th
mechanical
in a few cases they were measured with bonded
electrical resistance strain gages (Baldwin SR-4).
An 8-in. Moore extensometer was used in tension tests,
while a pair of Huggenberger gages of 1/2" gage length were
used in compression tests.
Some of the compression coupons
were tested into the strain-hardening range by an averaging
compre·ssometcr replacing the Huggenberger gages after passing
of the yield point.
on a coupon.
Fig. 6 shows the compressometer in place
On the opposite s,ides of the vertical cross-bar
SR-4 gages are mounted so that strains may be measured up to
15% with an apprOXimate sensitivity of 0.0002 in./in.
The coupons were numbered according to their location
in the cross-section as shown in Table 3.
*
"Notes on Compression T,:=sting of Metals and the ColumnStrength Curve ll , 1951.
**
Length x thickness x width
-22
220A.9
Some of the compression coupons were provided with
..
bearing blocks and a spherical bearing on top (Fig. 1 ), others
were tested in a sUbprcss (Fig.6').
All coupons were carefully
aligned before the test to about 1/3 of the estimated yield
load.
The alignment was considered satisfactory when the dif-
ference of strain readings on opposite sides of the specimens
was from 2 to 3 times the sensitivity of the gage at 1/3 of the
yield load.
Where necessary thin aluminum foils were used as
shims.
The testing speed in the elastic range was approximately 1 micro-in./in./sec. for all coupons.
In the plastic
range the valve opening set in the elasti.c range was kept con-
'.
stant.
Only coupons in series T-O were tested beyond
hardening.
strain-,
All other tests were dlscontinued at a strain of
from two to four times the yield-point strain.
3. Residual Strain Measurements
The sectioning method was adopted for the measurement
of residual' strains because of its simplicity. (11)
Strains
were measured over a 10-in. gage length by a 1/10,000 Whittemore
Fig. 8
strain gage ona series of previously laid out holes.
shows a typical lay-out for the sectioning process.
It was
cutomary on this pilot program to take readings on both sides
of the web and of each fla.nge and to make the spacing of lines
I
'.
of gage holes equal to 1/2 - in.
The 11 in. section to be cut
out of the beam was at least two feet distant from the ends.
Care was taken to select sections free from "yield lines" produced in the cold straightening
o~eration at~the
mill.
A stan-
dard 10-in. mild steel bar was attached to the specimen to
220A.9
-23
observe changes in temperature during readings.
The average
error in these measurements corresponds to a stress of approximately +600 psi.
Following an initial set of readings on
drilled and reamed holes serving as gage points, a second set
of readings was taken when the II-in. section was sawed out of
the beam as a unit.
The final strains were obtained after the
II-in. section had been sawed up into strips of 1/2 in. Width,
each strip containing a pair of gage holes.
4. Cross-Section Tests:
The length of the first of ·the I'cross -section II specimens was 20 in.
26 in.
The other five had a length between 24 in. and
Fig. 22 shows that this length is sufficient to retain
most of the original magnitUde of residual stresses in the
center part.
•
Further comment on this point will be made later.
The specimens were tested in compression in an 800,000lb. screw-type testing machine as shown in Fig. 9.
To obtain
a uniform distribution of applied stress, bearing plates and a
spherical bearing block were used between the specimen and the
head of the testing machine.
milled flat.
The ends of' the specimens were
Thin copper.sheets were inserted between the
specimen and the bearing plates on the first test as it was
hoped that this arrangement would give a more uniform stress
distribution.
However the yielding of those sheets influenced
unfavorably head rotation measurements and since no improvement
could be observed,
sh~s
were eliminated in later tests.
The strains were measured by SR-4 gages of the A-II
type (I-in. gage length) except for one test where A-9 type
gages (6-in. gage length) were applied.
In addition, the average
-24
220A·9
strain over a IG-in. gage length at the flange centers was
measured on three specimens by a frame-dial gage combination
illustrated in Fig.
•
four flange corners
9~
The shortening of the specimens at the
was measured between top and bottom bear-
ing plates with the aid of four dial gages and thin wire
ex-
tensions; also they indicated any possible rotation of the head.
The SR-4 gages and dial gages
~t
the flange edges served for
alignment and in addition supplied yielding and local buckling
information.
The alignment of the specimens was made at loads not
exceeding one-half of the expected residual-proportional limit
(the load corresponding to yield stress minus the maximum compressive residual stress).
I
The alignment was considered satis-·
~
factory if the maximum deviation of one corner gage from the
I
average of the four corner gages was less than
maximum alignment load.
5%
at the
A small amount of non-uniformity in
distribution of applied stress is not considered critibal as
long as the gages which are used to represent
t~e
average
stress~
strain curve are close to the average strain.
A qualitative picture of the yielding process was
obtained by the flaking of the mill scale made clearly visable
by whitewashing the specimens with hydrated lime
(F~g.
10).
According to the assumption of linear strain increase
•
over the total section, it is necessary to measure strains
representing the average strain of the cross-section at those
locations expected to remain elastic for the greatest period
of time.
For the 8 WF 31 shape, these locations are in the web
and at the flange centers which are usually found to contain
220A.9
-25
tensile residual stresses.
Theoretically, strains measured in
the plastic portions should give the same strains as in the
~lastic
..
portions; however, the mechanism by which yielding
occurs in thin planes (yield lines) makes the measured strains
depend on the gage length and the results tend to be erratic.
The loads were applied inappropriate increments which
were determined from the load vs. strain curve plotted during
the test.
After passing of the proportional limit, readings were
taken after the maintained load and the strains had but negligable changes within a period of 15 minutes, the time required
to take all readings.
The criterion used was a change of less
than 1/2% in the load and less than
during the period.
i
~
5%
of the measured strains
After the maximum load was passed a small
increment of strain was applied by movement of the head of the
testing machine and the readings were taken after both load
and strains had stabilized.
5. Column Tests
The columns were tested in the same 800,000# testing
machine as the cross-section specimens .. The axial load was
applied through knife edges.
A detailed description of the
testing frame and the testing set-up is contained in Reference
5.
Strains were measured by SR-4 gages of A-II type (I-in.
gage length) at locations close to both 'ends and at the center
of the columns.
The deflections were measured with dial gages
located at the quarter points and at the center of the column.
Deflections were also measured perpendicular to the plane of
bending at the ends and at the center.
-26
220A·9
The alignment was checked by the SR-4 and the deflection gages.
A very good alignment is essential tO,approximate
as closely as possible the conditions assumed in theory as
eccentricity (and also initial curvature) may reduce appreciably
the expected maximum loads.
Accidental eccentricities were
eliminated by relative movement of the column bases with respect
to the base fixtures.
of wedges.
Uneven bearing was remedied by movement
Both operations could be made while a small load
was maintained on the column.
The effect of initial curvature
can be reduced somewhat by application of a slight eccentricity;
however, it can never be completely removed.
reported herein all the
For the tests
effects described above were small
compared with the reductions attributed to residual stress except for the column test 205A T-15 (Fig. 27) which had a relatively large amount of curvature.
The alignment was made at
loads not ,exceeding about 1/3 of the estimated maximum load.
,The deflection and strains at the center and near the ends were
measured for the accuracy of the alignment.
In most cases it
was possible to obtain a deflection at the center of less than
0.1 in. at maximum load.
The load was applied in appropriate increments determined from a load-deflection graph plotted during the test.
Again, near the maximum load a criterion had to be used to avoid
appreciable changes of load, strains or deflections while
readings were taken.
Readings were taken if the change in load
was less than 1/2% and the change in deflection at the center
of the column was less than 5% of the measured deflection.
The
-27
220A.9
tests
we~e
continued beyond the maximum load until the load
had dropped to about 2/3 of the maximum value.
In weak axis
tests this drop occurred qUite suddenly without a possibility
of measuring intermediate values.
Whitewashing of the as-
delivered columns gave a qualitative picture of the development
of yield lines.
Although
thecross~sectionswere usually
free
from cold-bend yield lines, their presence could not be avoided
in the columns.
In all cases, the center part of the member
was substantially free from cold-bend yield lines.
examples are recorded in Fig. 11.
I
T
Typical
-28
220A.9
V. T EST
RES U L T S
For the tests performed as shown in Table 1 and 10-
•
cated in the various rollings as shown in Fig. 5, the following
is a brief presentation of results for each type of test.
These
results are discussed and evaluated in Chapter VI.
1. Coupons
The results of all available coupon data are listed
in Table 3.
Values of E, cr p ' cruy ' and cry are shown and corre$ponding tensile and compressive data are listed side by side.
When a coupon showed the Bauschinger effect, the results were
not used in determining average
for crp . The weighted
In computing average
val~es
averages are summarized in Table 4.
properties, the values obtained from individual coupons at a
·f
particular cross-section were weighted in proportion to the
flange and web areas.
~tress-strain
The terms used in connection with coupon
curves are defined graphically in Fig. 12.
The modulus of elasticity, E, was determined from a
large-scale plot of the stress-strain curve and was taken as
the slope of the average line through test points up to about
the proportional limit.
The maximum deviation of data points
from this line was between 20 and40 microinch/inch.
The pro-
portional limit, crp ' was defined arbitrarily as the stress at
"
which the average curve crossed the lower boundary of the
scatter band (Fig.l2).
The yield stress level cry was determined
from the average stress in the plastic range.
-29
220A.9
Compression coupons tested into the strain-hardening
range are shown in Fig. 13.
Typical stress-strain curves for
individual compression coupons are
•
Fig.15
~hown
in Figs. 14,16, and 17 .
gives the stress-strain curves for a set of tension
coupons.
The average compressive stress-strain curve to be
used for later comparison with cross-scction and column tests
was determined from the above compression tests.
The technique
used is based on the assumption that plane sections
~emain
plane
after deformation, and, thus, the axial strains are uniform
across the section for each load increment.
After plotting the
individual stress-strain curves to a large scale, the stress
values are read from the various curves for predetermined strain
values.
Frequent strain intervals are selected in the region
of the yield point. Individual stress readings of flange material
are averaged for each of the selected strains and the same is
done for the web material.
Finally, for each of the stratn
values, a weighted average of stresses is determined in proportion to the relative flange and web areas. The average stressstrain curve is then plotted directly from this data.
2. Residual Stresses
The residual stresses were computed from the measured
strains using an elastic modulus of 30 x 106. psi .
The distribu-
tion of these stresses in three as-delivered specimens is shown
in Fig. 18.
The results from T4 are not shown but were qUite
similar to those obtained in TO.
represent two different rollings.
These particular distributions
In Fig. 19 are shown the
220A.9
-30
results of residual stress measurements at three different locations in a single piece of material.
In both Figs.1B and 19 the
distributions are shown separately for the flanges and the web.
Except for a few measurements on T6 (Fig. 19) the web isih residual tension (a maximum value-of +16,000 psi was measured); the
flanges are in compression at the edges (maximum va'1uc
_. -17,500
,psi) and change to tension at the flange centers (maximum value
==
+11,000 psi).
The magnitude and distribution of residual stresses
in two annealed specimens is shown in Fig. 20
and indicates
that practically all the residuals were eliminated.
The results of the study on the variation of residual
stresses along a member are shown in Fig. 21.
A member 9..,.ft.
J.ong received a layout of holes as shown in the upper slcetches.
At three locations a complete sectioning was made (76 pairs of
holes) and the resulting stresses are shown in Fig. 19.
At the
other locations a partial sectioning was made involving 32 pairs
of holes and at the circled stations (Fig.21).
The change of
the residual stresses along the beam at the web center, the
flange center and at the flange edges is shown in the lower
portion of Fig. 21.
The region containing cold-bend yield
lines is to be noted.
Other measurements on
th~
beam also served to study
the changes of residual stress at a section for different
-;
.
-
lengths of cross-section.
As the sectioniI!-g process was per-
formed at various stations along the member shown in Fig. 22
the member finally consisted only of the piece shown as 6-7
with a length of 3l-in.
At this stage, no change in readings
had been observed over the gage length A-A.
The remaining
member (6-7) was then further reduced in length (25; 19, 14, and
220A,9
-31
11 inches) and finally cut into strips.
The residual stress
at this location is plotted vs .. the length of "cross-section"
I
•
..
in Fig. 22 for the same locations as 'before, that is, at the
web center, flange centers and flange edges.
This figure shows
that the residual stress distribution in the gage length A-A
did not change appreciably until the total length was reduced
to 19 in.
Since all of the cross-section specimens were longer
than 19 in., they had sufficient length to preserve almost completely the original residual stress pattern in their center
portions.
Of course this was important if the cross-section
test was to reflect correctly the influence of residual stresses.
3. Cross-section Specimens
The results of measurements of E, crp ' and cry determined from the cross-section tests are summarized in Table 5
and compared with corresponding weighted compression coupon resuIts.
The average stress-strain curves for
typical
cross-section specimens are presented in Figs. 23 to 25.*(Fig.
25 is for the annealed material.)
The average curve determined
from compression coupons is also shown on each figure.
In most cases strains were measured at several locations and
these results are also presented in Fig.23 to 25.
Theoretically
these latter curves should be identical according to the assump't
tion of uniform strain.
The stress vs. tangent modulus curves needed to computethe column curves are shown in Fig. 26.
(They also show
the difference between measurements on the same specimen at
different locations.)
In Figs. 23 and 24 the strain E5 is the theoretical strain
which the cross-section should reach the yield stress level.
See Fig. 46.
*at
220A.9
-32
Column curves based on the
,
stress~strain
curves of
coupons and of cross-sections and using the relationship in
Chapter III, Section 6 of the report are shown in Figs. 27, 28,
•
•
29,
and 30. (Column test results are shown in each case for
comparison.)
Fig. 27 (T-O) compares results from web SR-4'gages
and flange SR-4 gages.
Fig. 28 (T4) also contains column curves
from residual stress measurements as will be described below.
Column curves for the two sets of annealed material are shown
in Fig. 29.
In order to compare all the cross-section test .results;
Fig.30
has been prepared.
It presents the data on a non-
dimensional basis to eliminate the effect of differences in the
material properties.
The values of E and cry used in the figure
were taken from the corresponding stress-strain relations.
cry
was the maximum average stress carried by the cross-sections; .
it was also applied to the column test results.
curves~cr
y
For the coupon
was taken as the weighted average of the yield stress
level.
By the method of Chapter III, Section 3 of this report
column curves from residual stress measurements are plotted
in
Fig. 28 in addition to the curves based on cross-section tests.
Straight lines were used to approximate the actual residual
stress pattern. The magnitude of cry was taken as 93% of the" value
determined from coupon tests.
Fig. 31
corresponds with Fig. 30
and presents all the results on a non-dimensional basis.
2201L9
-33
4. Columns
The maximwn average stresses carried by the columns
~.
have been plotted on Figs.27 and 28 for as-delivered mat~rial
and in Fig. 29 for the annealed material.
pare
Fig. 30 and 31
com-
all of the column tests to the various theoretical curves
on a non-dimensional basis.
Load vs. center deflection is plotted for a number of
tests in Fig. 32.
Theoretically, a concentrically 10aded, ideal
column should not deflect until the tangent-modulus load is
reached.
start.
In tests, slight deflections are observed from the
Table 6 gives·the approximate values of. eccentricity and
center deflection if the observed deflections were attributed
to (a) initial curvature or (b) initial eccentricity only.
The
initial curvature was assumed to be a sine curve with amplitude
"a".
The table also shows the corresponding vaJ.ues of ec/r 2 .
A comparison with a .study recently published (10) shows that
except for 205A T-15 the reductions due to eccentricity were
small when compared with the theoretical reductions due 'GO residual stresses.
All columns were within AISC rolling tolerances
of 1/8 11 per 10 ft. length of member. (2)
-34
220A.9
VI.
DIS C U S S ION
1. Compressive Strength
The compressive strength of steel as measured by the
yield stress level is not as great as that obtained from a milltype tension test; likewise, although compression and tension
coupons give results that are almost identical (on the
ave~age),
the compressive strength of full cross-sections is less than
that obtained by averaging the results of small coupons tested
in compression.
The comparison between the results of tension and
compression coupons shown in Table 4, supports the usual assumption of identical behavior of steel in tension and compression.
The elimination of compression testing of coupons (in the case
•
of rolled structural steel shapes) is thus considered as warranted, particularly in view of larger variation in properties
due to other causes.
There was considerable variation in coupon properties
(both compression and tension) particularly between web and
flange material (TabJ.e
3).
On the average, the yield stress
level was 10% higher for web material than for flange material.
Since web material is used for the mill tests, it can be expected that, due to this cause alone, average compressive
strength will be from 5 to 10%
mill.
les~
than that reported by the
The difference in strength has been attributed to the
more rapid cooling rate of the web material as compared to the
thicker flange material.(9)
-35
220A·9
Difference in strength of as much as 10%
wa~
also
(~able
observed between different locations in the same flange.
3). However, when weighted averages are compared for different
locations in the same ingot, the agreement is much better
(2-l/2% in the compression tests).
Some coupons exhibited a
Bauschinger effect (coupons #5 in Fig.13, #5 in Fig.
and #8 in Fig. 15, and #1 in Fig. 16).
lL~,
#4
In the case of the
latter, "co ld-bend ll yield lines produced during straightening
of beams in the
mil~
were positively identified on the coupon.
It is the previous tensile strain that accounts for this
behavior.
The difference in strength between full cross-sections
and the weighted averages obtained from coupons is evident in
Table 5.
The maximum stress carried by the cross-sections
varies from 90 to 96% of the yield stress level of the compression coupons.
This reduction .also appeared in the case of
annealed tests and has also been observed in other tests.
The
difference in strength may possibly be due to size of specimen,
shape of specimen or both.
At any rate, because the entire
member is used, test of a full cross-section is a more rational
indication of strength than is the average obtained from a
large number of coupon tests.
The cross-section is also less
expensive for laboratory use than are coupons, the cost ratio
being about 2 to 3 for cross-section cost vs. cost of 4 compressioncoupons.
Incidentally, the agreement between modulus of elasticity, E, as obtained from compression coupons and· from crosssections· was excellent.
(Table 5).
Four of the coupon sets
-36
220A.9
agree within one-half of one percent with the corresponding
cross-sections and the maximwn difference in E is 3%.
The
average value ~as 29.6 x 10 6 psi.
F1.g.33 surn.rnarizes diagrammatically the influence of
several factors on the strength of steel'in comparison with the
value reported by the mill.
The mill report is usually, though
not always, based on the upper yield point.· (In
Table
4
the upper yield point in tension was 5% higher than the lower
yield stress level.)
The differen6e in rate of strain could
account for a further reduction of 10%;. difference in strength
between web and flange accounts for from 5 to 10% reduction;
finally, the full cross-section is from 5 to 10% weaker than
that indicated by
•
cou~on
tests.
Thus a full cross-section'
tested in the laboratory in compression may 'be expected to show
,a lower yield stress level of from 20 to 40% less than the
"yield point ll reported·by the mill.
Futul""e work is aimed at
specifying more precisely the strength in compression as compared With the mill results.
,The effect of annealing is, of course, to reduce the
compressive strength somewhat (Table 3).· Even in the annealed
state the web was still stronger than the flange.
2. MagnitUde and Distribution of Residual Stress
Fj.gs .18 and 19 indicate that compressive residual
stresses at the flange tips amounted to as much as -17,500 psi
and in the web reached values of -5,600 psi.
Tensile residual
stresses were as much as + 16,000 psi in the web and + 11,000
psi at the flange centers.
These are average stresses' calcu-
-37
220A.9
lated from measurements on both sides of the webs and outstanding
flanges .. The difference between readings on such pairs of holes
was in most cases of the same order of magnitude as the error
of the measurements.
However, a trend of slightly higher
stresses seems to be indicated on the outside flange surfaces
as compared with the inside surfaces.
There were several typical patterns (distributions)
of residual stress in flange and web (Fig.18,19).
The distri-
bution of flange stresses was in some cases linear and in others
parabolic, hyperbolic, or ffS-shaped(l.
The distribution of
stresses in flange and web are interrelated since the condition
of equilibrium requires that the resultant web force balance
the resultant in the two flanges.
For example, in Fig. i.8, Tl
shows a relatively high resultant compressive force in the
flanges; as a consequence, the tensile force in the web is
relatively high.
Although not so marked, a similar correlation
is seen between TO and T2.
In Fig. 19, the
flang~
is so nearly
in equilibrium that there is little resultant force in the web.
The influence of the different distributions of residuals will
be examined in Scction 4 below.
The annealed material
(Fig~
20) had residual stresses
of such small magnitude that the measuring method employed was
not sufficiently sensitive to determine them precisely.
Fig. 20
Thus
cannot be considered to represent an actual stress
distribution since the recorded values are.of the same magnitude
as the possible errors.
However, the purpose of these measure-
ments was only to verify that annealing had eliminated the
residuals.
220A.9
-38
3. Variation of Residual
Stresse~
The magnitude of the stresses at flange tips show
relatively small variation.
Stresses at flange centers show
larger variation (Fig. 18) and those in the web show the greatest
of all.
Considering first the material from one ingot (Fig. 18)
there is some variation in distribution of stress in the flanges
of different rollings and a much bigger variation in the webs.
This latter variation has only a small influence because column
strength is primarily determined by the residual stresses in
the flanges.
The material from the second h ea t ha d mu c h
higher tensile stresses at the flange centers (Fig. 19).
Except
for magnitude 'of compressive stress at flange edges, the variations between two different heats were rather large (Compare
Fig.18 and 19).
Measurements at different locations on the same
beam indicate good uniformity (Fig. 21,19) except for the end
which contained yield lines from cold bending.
Considering the
variation along the member, the full magnitude of residual
stress is reached after a short distance of approximately 1 ft.
from the ends (Fig. 21).
This is in agreement with the results
presented in Fig. 22 where a minimum length of about 3 times the
section depth or flange width is seen to be sufficient to preserve the residual stress pattern at the center of the specimen.
In stmmary, the assumption made earlier that the
distribution of cooling residual stresses is uniform along a
column seems to be reasonable.
It is further borne out by column
tests, where yielding is usually oberved to occur rather uniformlyalong the member.
-39
220A·9
4. Influence of Residual Stress Distribution upon Column Strength
Three of the possible distributions of residual stress
are considered as shown in the top diagram of Fig. 34.
The web
stresses were assumed to be constant and of a value to satisfy
equilibrium.
The analysis was carried out until the flanges
had yielded completely or until the web had l"eached the yield
point
~s
was the case for the parabolic distribution (curve III).
Column curves were then drawn for each case and for flexure
about either the strong or the weak axis as indicated.
The greatest reduction was caused by the hyperbolic distribution
(curve I).
The result obtained from Tl, which had a residual
stress distribution appioaching Type I is in confirmation of the
prediction of Fig. 34.
The cross-section column curve plotted
in Fig. 30 is the lowest for this test.
In Fig.35
an S-shaped pattern has been used as the
basis for computing a column curve.
Comparison is shown with
the curve obtained from a linear distribution of residual
stresses.
A fair average of these cases is supplied by the
linear pattern of residual stress.
In general a large area of
compressive residual stresses in the flanges results in a lower
column curve.
A limiting case would be compressive residual
stresses throughout the flanges and tension in the web.
The
column curve would become horizontal at a stress cry - crrc as
shown in Fig. 35.
Recent model tests* confirmed this behavior.
* S. Cherry, P. Y. Chow, and W. J. Austin, ftExperimental Studies
of Model Columns ", Structural Research Serie.s No. 34, University of Illinois, July 1952.
-40
220A.9
5. Influence of Residual Stress - Correlation between
Theory and Tests
Of first importance in considering the influence of
residual stress is the reduction in strength of columns below
the strength predicted on the basis of average compression
coupon curves.
Fig. 27 serves as illustration; by no means
could "coupon" curves serve as a rational basis for predicting
as-delivered column strength.
The reduction amounts to as much
as 35%.
Secondly, the influence of residual stress is different
depending on the flexure axis.
As predicted by theory, the
influence of residual stress is more pronounced for columns bent
about the weak axis.
Q
The two methods for predicting theoretically the
influence of residual stresses were treated in Chapter I I I of
this
pap~r
and have the same fundamental basis.
a relationship between average applied stress,
mom~nt
of inertia, Ie,is desired.
In both cases
0,
and elastic
To find this relation certain
material properties are required and these must be furnished by
tests.
These methods will now be discussed and correlated with
column tests.
(a)
Method based on residual stress distribution:
The method requires the measurement of residual stresses
by sectioning a piece of the material or by some other process.
Numerous coupons must be tested and a factor (90-95%) applied to
the yield stress level to arrive at an equivalent basic compressive strength.
The residual stress pattern must then be approxi-
mated by an appropriate analytical expression and the resulting
-41
220A.9
column curve computed according to Section 3 of Chapter III of
,
the report.
Errors can be made in the measurement of residual
stresses (~ 600 psi.) and in the measurement of E and cry in the
coupons.
Care must be taken that residuals are measured in a
piece free of cold-bending yield lines.
It is impractical to go
to too much refinement in approximating analytically the residual
. stress distribution.
Figs. 34 and 35
show further, that a
reasonable approximation to the pattern produces a satisfactory
column curve.
In Fig. 28 test results are compared with typical
curves computed from residual stress patterns.
is satisfactory.
The agreement
The same comparison for all tests, the results
being plotted non-dimensionally, is shovvn in Fig. 31.
I
~
~
In all
these calculations a factor of 93% was applied to cry (c) obtained
from coupons to give the basic compressive strength.
(b)
Method based on cross-section test:
Correlation between actual
col~~n
tests and the results
of the analysis of "cross -sections" containing residual stresses
is shown in Figs. 27,28, 29, and 30.
The agreement between
theory and test is as good as is the "residual stress" method,
and the scatter in results obtained from different cross-sections
is not unreasonable.
The results of both the "cross-section"
and the "residual stress" analyses are shown in Fig. 28
with test results.
In this respect there is little
choosing one over the other.
together
basis for
-42
220A.9
The "cross-section" method for determing the influence
of residual stress on column strength requires a length of
material sufficiently long to retain most of the residual stress
intensity at the center.
This may be accomplished if the length
is of the order of three times the depth of member or width of
flange.
Testing of the full cross-section provides, directly,
values of E, cry' and 0rc'
By determining the values of Et at
various stress levels and using the expressions in Section 6 of
Chapter III, column curves are obtained.
There are considerable advantages of time and money
in the cross-section scheme as compared with the residual stress
method.
'J>
Of course, errors may be made in determining E and cry'
Care must be taken in the conduct of the test, particularly in
the alignment.
Theoretically, strain measured at different
locations should be identical.
However, variation is often
observed and this leads to differences in the stress vs. tangentmodulus relationship (Fig. 26).
Considering the worst discrep-
ancy between different sets of gages (as observed in T-O) two
se~arate
column curves have been drawn in Fig. 27.
In the one
case SR-4 I s were mounted on the web and in the other on the
flanges.
Even for this extreme case, the difference is small
compared with the reduction from the average coupon curve.
As-
suming that a testing machine of sufficient capacity is avail-'
able these tests' suggest that the cross-section method is preferable to the method that requires the measurement of residual
stress.
-43
220A·9
The "cross-section" method allows the determination of
,
the column curve until the flanges have completely yielded (the
•
limiting value of Et is found in Chapter III). The corresponding stress is so close to the yield stress level that the curve
may simply be extended to Oy without serious error.
Fair agreement was observed between the predicted
= 0y - °rc) and the loo.d at which flaking of
first yielding (Op
whitewash occurred.
'Thus the cross-section test permits a rough
estimation of arc' the compressive stress at the flange edges.
Incidentally, the residual stress magnitude and distribution in the flanges may be reconstructed from the measured
stress-strain curve of a cross-section specimen.
is as follows:
The procedure
the average strain in terms of the residual
stress at a point was given by Eq.9'as
- drx )
0
The tangent modulus was given by Eq.ll as
€
Et
=
E (0
A
=
Y
+ wd )
E (4tx
A.
0
l
(11)
Since Oy, E, Et , and € are determined from the test, 0rx may
o
be determined from Eq" 9 for discreet values of E and the corresponding Xo may be obtained from Eq.ll. A comparison between
computed values of residual stress with actual measurements is
shown in Fig. 36:
the agreement is quite good.
The variation
of the calculated stresses is due in part to the experimental
error involved in obtaining the tangent modulus, Et .
Column test 205A.T15 (Fig. 27) showed a reduction
below the expected value; this was due to a relatively large
initial curvature present in the member.
-44
220A.9
The results of annealed tests, shown in Fig. 29, exhibit
the difference between coupon yield stress level and cross-
.
section strength that was also seen in the as-delivered material.
No column tests were made for T-3 material.
the full yield load.
T-5b almost carried
The longer column, T-5a, showed some re-
duction which is probably due to
b~th
out of straightness and
eccentricities, their influence on the deflection curve being
seen in Fig. 32.
The theoretical column curves based on cross-
section tests shows clearly that residual stresses have largely
been eliminated by annealing.
This series of annealed tests
gives conclusive evidence of the presence and influence of residual stresses.
..
The summary curve of Fig. 30 also contains
of the study of annealed material.
the results
Comparison of the weak axis
tests (solid circles are for annealed material, open circles asdelivered), show the reduction in strength due to residual
stress which is ip confirmation of the theory.
6. Correlation with Current Design Specifications
'In Fig. 37 the average cross-section curve is plotted.
It will be noted that the curve for buckling in the strong
direction is
app~oximately
parabolic in shape, while the weak
axis curve shows approximately linear behavior.
The two curves
could be expressed by the following simple relationships:
~
Vy-;-y
= '"y _ arc
Tr
v
1
220A·9
-45
where arc is the residual stress at the flange tips.
Using the
average material properties measured in the tests (Oy = 37,000
psi, E = 29.6 x 10 6 psi and arc = -14,000 psi) Eq.13 may be
(14)
These curves are also shown in Fig.37
together with appropriate
test results.
To compare with these,curves given by three specifications (AISC (2), AREA (3), and AASHO (1)) are also shown in
Fig. 37.
Assuming that Eq. 14 (plotted in Fig. 37 ) to be a
correct representation of true column strength, the implied
factor of safety inherent in these three specifications is shown
in the lower portion in
considerably.
200) to 2.4.
F1g~
36. The factor of safety varies
For the AREA-AASHO it varies from 1.6 (at L/r
=
For the AISCformula it varies from 1.90 to 2.18.
Because two theoretical curves have been used, there are two
sets of "safety-factor ll .curves in Fig.
38,
one for flexure
about tpe strong axis, the other for weak axis bending.
Bleich(6)
suggests the use of a constant factor of safety, albeit, one of
greater magnitude than used for tension members.
This seems
fully justified if the factor is applied to a column curve that
describes actual strength fairly precisely.
Using a constant
factor of safety equal to the minimum exhibited by the AISC
formula (1.90) a possible working formula would be
0x-x
= -19,500
+ 0.58
0y_y = -19,500 + 65.2
{L/r)2
t/r
(L/r < 110)
-46
220A·9
For L/r> 110, the Euler hyperbola would be used as the column
curve for both flexure axes and
cr -
1r 2E
1.9 (L!r)2
,
L/r
~
( 16)
110
The curves of Eqs. 15 and 16 are plotted in Fig. 37.
One conclusion to be gained from Fig.37 is that for
most efficient utilization of material, two design curves should
be used:
one for the strong axis and one for the weak axis.
The following comment on a similar point was made by Jonathan
Jones, Chairman of the Practical Applications Committee of
Column Research Council:
"Past conditions have permitted us to be somewhat
prodigal of materials, but that picture is changing.'
Materials are obviously going to grow more and more
expensive, and their employment without waste is becoming of greater importance. In the design of compression members we have to a large extent coasted
along, using rules and formulas derived from simple
experiments made, and deductions drawn, long ago. We
have established the rules either for average cases,
meaning that some structural elements are not as well
protected as they should be, or we have established
them for the 'worst case', thus wasting material in
cases less severe."
A question might logically be raised as to why the
lI
spec ification"
in Eq.13.
yield point of 33,000
psiwa~
not used for cry
Actually, this is a minimum yield point stress, a
value that must always be exceeded in a mill-type tension test.
The value used for cry in Eq.13 (37,000 psi) was found to be the
average basic compressive strength of material rolled to ASTM
,
IW
A-7 Specification.
A statistical mean eventually $hould be
established for the basic compressive strength and design formulas based upon it.
The factor of safety is intended to cover,
-47
220A.9
among other things, "underrun in dimensions and physical pro-
'.
perties", and on this basis, would ,cover those rare cases in
which the material just met the specifications.
A second question relates to eccentricities.
In the
past it was common to explain the reduction in column strength
in" the region up to L/r = 100 as due to accidental eccentrj.cities
and initial curvature.
(In fact, this was done earlier ih the
paper for the annealed members).
This study shows, however,
that reductions in strength due to residual stresses are very
large in themselves and it is not rational to explain them on
the basis of eccentricities and crookedness.
Reductions due to residual stresses and eccentricities
•
are summarized in a paper resulting from a project currently
underway. (10)
It is clear from figures contained in that study*
that column strength could be described by selecting an appropriate value of eccentricity sufficient to cover the influence
of residual' stresses.
However, both eccentricity andr0sidual
stress must be considered in arriving at a completely rational
column formula intended to include known eccentricity as a
factor.
It must be admitted, of course, that the suggested
"rational" formulas shown on Fig. 37 offer no startling departure
from present practice insofar as the AISC formula is concerned.
Test results have figured promi0ently in arriving at column
.'
formulas for specifications.
Such tests as were made on as-
delivered columns and thus automatically included the influence
of residual stresses; hence
col~~n
* See Figs.37 ,38 of Ref. 10.
formulas would be expected to
-48
220A.9
be in at least "fair agreement with the findings of this investigation.
•
The following is quoted from an earlier Lehigh report. (18)
flIt is probable that column design formulas have
included empirically the reduction in carrying capacity
due to residual stress, although the percentage reduction due to this factor has not gene~ally been appreciated. This indicates once more that experience and
prior satisfactory performance is now, as it has always
been, the important factor in structural design."
Fig.37
indicates that some savings may be realized
in column design by using formulas 15 especially if the first of
these is used for flexure about the strong axis.
This is con-
sldered as secondary, however, to their rational explanation of
column behavior.
7. Recommendations for Future Research
Before it is possible to make final design recommenda-
:.
tions certain additional work is necessary.
Even though one of
the most frequently employed column shapes was used in the test
program,
applicability of the theory should be verified for
other shapes and sizes of rolled sections.
Secondly, a statis-
tical collection of data on yield stress level, cry' and residual
stress magnitude and distribution for various shapes is required.
In this regard, of practical importance is the relation between
the basic compressive yield stress level and the mill test yield
point determined at maximum permitted strain rate.
A third
important problem is the influence on column strength of cold;
straightening.
These and related problems are being studied in
a continuing program at Lehigh.
220A.9
-49
VII.
CON C L U S ION S
The following are some of the conclusions resulting
•
from the Pilot Program on the.influence of residual stress on
column strength and are limiteci to those cases in which the
residual stress distribution in
v&
shapes is approximately sym-
metrical.
1.
The presence of residual stresses results in a con-
siderable lowering of column strength below values predicted on
the basis of coupon tests.
co~pressive
the basic
2.
..
It may amount to as much as 30% below
yield-point at L/r = 90.
(Fig. 27,30)
The distribution of residual stresses in the flanges
has more influence on the strength of the column than do residual
stresses in the web.
The more of the flange that is in compres-
sion the greater is the reduction (Fig. 34).
In most cases an
assumed linear distribution will result in a satisfactory column
curve.
Compressive residual stresses up to 17,500 psi were
measured at the flange tips. (Fig. 18).
3.
influence.
The direction of possible bending is of significant
Columns of the same slenderness ratio carry less
load when allowed to bend in the weak direction than when allowed
to bend in the strong direction. (Fig. 30)
4.
The reduction of column strength.due to unavoidable
small eccentricities and crookedness is small compared with the
large r.eductions caused by residual stresses.
5.
The variation of residual stresses along a member
free from cold-bend yield lines is small
(Fig. 21).
The
variations of residual stresses within material from one ingot
220A.9
-50
are relatively small (Figs. 18 3 19)3 but large variations may
exist between material from different heats.
6.
The magnitude and distribution of residual stresses
present in a long beam is preserved at the center part of a piece
cut out of the beam if the length is about three times the depth
of the section.
7.
The yield stress level of a full cross-section is as
much as 10% lower than the value obtained from compression
coupons (Fig. 333 Table 5).
material.
It is possibly due
This also applies to annealed
to
an influence of size and
shape of specimen.
8.
•
Factors such as upper yield point, strain rate, and
web strength vs. flange strength when added to the reduction due
to size or shape effect may result in a full cross-section teste1
in the laboratory in compression showing a yield stress level of
from 20 to 40% less than the "yield point" reported by the mill.
(Fig. 33). Further research should establish this percentage
more precisely.
9.
Modulus of Elasticity, E, and yield stress level, cry,
are obtained in satisfactory manner from the test of a full
cross-section. (Table
5). Residual stress magnitude and distri-
bution may also be determined from the cross-section test.
Further 3 the column curve computed from the cross-section stressstrain relation by application of tangent modulus theory is in
good agreement with the solution based on residual stress measurements (Fig. 28) and with test results (Fig. 30).
This test is
therefore an effective laboratory tool for obtaining basic
compressive properties of steel and for predicting column strength.
220A.9
-51
10.
EXisting column specifications (1,2,3) exhibit a large
variation of reserve strength for slenderness ratios less than
..
120 (1<090 to 2.47) when compared with the strength of as-delivered
axially-loaded columns.
11.
Rather simple working. formulas have been developed
(Eqs. 15,16) which are based on the results of the tests and
analyses.
While these do not depart markedly from existing for-
mulas (Fig. 37), they offer increased economy for the shorter
columns and constitute a rational explanation of column strength.
As more statistical data is collected on. the properties of rolled
steel shapes (E, ~y' arc) and in particular on the relationship
..
II
•
..
~,
between mill tension tests and the basic yield stress, a definite
formula for recommended practice maybe put forward.
220A.g
-52
VIII.
A C K NOW LED GEM E N T
This work has been carried out at Fritz Engineering
..
Laboratory of which Professor William J. Eney is Director .
The authors wish to express their appreciation to
the sponsors of this research program, namely, the Column Research Council, the Pennsylvania Department of Higrnqays, and
the Bureau of Public Roads.
Acknowledgement is also expressed
for the suggestions received from members of Research Committee
A (Column Research Council),particularly those of its chairman,
Bruce G. Johnston.
The interest shown by members of the Lehigh Project
Subcommittee (Welding Research Council ),of which T. R. Higgins
.lIt
is chairman, is also sincerely appreciated.
This group spon-
sored the program of column tests from which some data was used
in this report and acknowledgement is due to Robert L. Ketter,
Research Instructor, who is conducting that program.
Acknowledgement is also due to Kenneth R. Harpel,
foreman, and the staff of machinists and technicians at the
Fritz Engineering Laboratory .
..
220A.9
-53
IX.
REF ERE N C E S
to
(1)
The American Association of State Highway Officials
"Standard Specifications for Highway Bridges",. ·1950.
(2)
American Institute of Steel Construction (A.I.S.C.)
"Steel Construction Manual", 1951.
(3)
American Railway Engineering Association
ttSpecifications for Steel Railway Bridges" 1949.
(4)
American Welding Society "Welding. Handbook" 3rd
Ed., 1950.
(5)
L. S. Beedle, J. A. Ready and B. G. Johnston" "Test
of Columns under Combined Thrust and Moment", Proceedings SESA Volume VIII, No. 1~ 1950.
(6)
F~ Bleich, "The Buckling Strength of MetalStructures", McGraw-Hill Book Co., 1st Ed. (1952).
B. G. Johnston and F. Opila, "Compression and
Tension Tests of Structural Alloy~" Proc. ASTM, Vol.41,
1941.
•
(8)
Bruce G. Johnston and the Project Supervisors "A
Survey of Progress 1944 - 1951", Column Research Council
of Engineering Foundation, Bulletin No.1, January 1952.
(9)
B. G. Johnston, C. H. Yang". and L. S. Beedle,"An
Evaluation of Plastic Analysis as Applied to Structural
Design", The \'lelding Journal Research Supplement, Vol.
32, No.5, pp. 224-s to 239-s, May, 1953.
(10)
R. L. Ketter, E. L. Kaminsky, and L. S. Beedle,
"Plastic Deformation of Wide-Flange Beam-Columns"
Proceedings Separate # 330, ASCE, Vol. 79, 1953.
(11)
W. W. Luxion and B. G. Johnston" IlP1astic Behavior
of Wide Flange Beams", The Welding Journal Research
Supp1ement,Vol.27,No.11,pp.538-s to 554-s, November" 1948,
(12)
T. Mathar, "Determination of Inherent Stresses by
Measuring Deformations of Drilled Holes ll , NACA" T.M.
702, March, i933.
(13)
W. R. Osgood, "The Effect of Residual' Stress on
Column Strength"" Proceedings of the First National
Congress of Applied Mechanics. June, 1951
(14)
E. H. Salmon, "Columns", Oxford Technical Publications (1921) '.
220A.g
-54
(15)
F. R.. Shanley, "Applied Column TheOI'y", Transaction
of the American Society of Civil Engineers,Vol.l~5J1950.
(16)
R. G. Trenting, H. B. Wishart, J. J. Lynch, and
D. G. Richards, "Residual Stress Measurements", A.S.M.
33rd National Metal Congress, October 15 - 19, 1951.
(17)
W. M. Wilson and R. L. Bl"'own, liThe Effect of Residual Longitudinal Stresses upon the Load-Carrying
Capacity of Steel Columns ", University of Illinois
Bulletin No. 280, November 1935.
. (18)
C. H. Yang, L. S. Beedle and B. G. Johnston, "Residual Stress and the Yield Strength of Steel Beams",
The Welding Journal Research Supplement, Vol. 31, No.4,
pp. 205-s to 229-s, April, 1952.
226A.9
-55 .
X.
A P PEN D I XES
Appendix 1: Nomenclature
A
=
cross-sectional area
Ae
-
area of the unyielded part of the cross-section
AF
=
=
area of both flanges of WF shape
approximated.' web area
=
flange width·
Aw =
wdJ..
b
depth of section
d
d,.L = d-t
=
depth of WF section between center lines,of
flanges
E
=
Youngrs modulus of elasticity
E+·v
=
tangent modulus
I
=
moment of inertia
Ie
=
moment of inertia of the unyielded (elastic)
part of the. cross-section
L
=
total length of a pin-ended column
L/r
=
slenderness ratio
load on a column
P
=
critical load on a column.:; in the case of axial
load alone it is Pe for buckling in the elastic
range.
=
Euler buckling load for a pin-ended column
=
tangent modulus .load, the load at which bending
of a perfectly straight column may commence.
=
axial load corresponding to yield point stress
across entire section.
I>
=
radius of gyration in the plane of bending.
t
;:::
flange thickness
w
=
web thickness
~.
distance from the center of the elastic flange
area to the beginning of the yielded area
=
distance from the center of the plastic web area
to the beginning of the elastic area
-56
220A.9
.,
E
=
unit strain
O'p
proportional limit stress
O'cr
=
=
=
O'y
=
yield stress level; average stress in the plastic
range
O'y(c)
=
coupon yield' stress level
O'uy
=
upper yield point stress
O'rx
.-
residual stress in the flanges
O'ry
=
residual stress in the web
p
0' = -
..
:
.'
A
applied average stress on a column
critical stress
220A.9
-57
Appendix 2
Rectangular Specimen Containing Residual Stress
(Referred to in Chapter III,8ection I)
Fig. 39 illustrates the action of a rectangular speci-
men under increasing load and shows the strain and the stress
distributions.
At p-= 0 no load is applied and residual strains
only are present in the specimen, Line (b).
The distributions
of Lines, (c), (d L (e), (f), and (g) are self -evident.
When
the specimen is strained by loads Pl to P4 a uniform strain
(Line c) must be added to the initial residual strain (Line b)
to give the total strain (Line d).
At P2 the total strains have
just reached the yield point at the edges. Stresses are still
proportional to strains.
When the load reaches P , however, strains are greater
3
than the elastic limit and the stress ~ diagram in Line e results.
The superimposed stress, shown by Line f, is determined simply
by substracting the residual stress (Line b) from the total
stress distribution (Line e).
P is thus equal to the area
3
cross-hatched in Line f times the thickness.
Finally at P4 the cross-section is completely yielded.
Sketch h of Fig. 39 shows a resultant stress-strain diagram for
the whole specimen With certain critical points
noted.*The
column curve (including the influence of the assumed residual
.
'
stress pattern) may be determined from the various yield and
For illustration a 10-lnch wide specimen of cry = 40 ksi ami
-crrc::t: crro = 20 ksi was used. The dotted curve is for. a para·bolic distribution (-crrc = 20 ksi, crro = 10 ksi).
*
-58
220A·9
If k
Xo
b/2 '
Ie
I
(flexure about x-x axis)
x
•
Ie
I
y
= k3
(flexure about y-y axis)
Per is obtained from the stress condition (cross-hatched area of
Line (f) of Fig. 39).
For a linear distribution of residual
stress,*
O'er = Pcr/A = O'y
-.~
2
(O'rc-O'ro) (1 + k )
-
O'ro
For a -parabolic distribution of residual stress, +
3
O'er = Pcr/A = O'y - 1:. (arc-O'ro) (1 + 2K ) - O'ro
3
".
The critical L/r is then determined by solution ofEq.
L/r
I'x
=
L/r
Iy
= 7T"
7T"
-'1
1.
....._-)
Ek/ 0'cr
1-Ek 3/0' cr
I
Sketch (i) Fig. 39 is the column curve that corresponds
to the straight-line pattern.
As is eVident, a rather slight
deviation from linearity of the stress-strain curve (Fig. 39-h)
produces a pronounced influence on column strength .
.
~
* If the flange is in equilibrium, - arc
+ If the flange is in equilibrium, - arc
=+
=
a ro
+ 20'ro
220A.9
-59
Appendix 3
In this Appendix a solution is obtained for the in-
fluence of residual stress on WF columns,
the solution being
based on a known residual stress distribution and known yield
stress level, Oy.
A relationGh~p b~tween
00r and the yield
by writing the
co~dition
equil~Lbrium
rhrection of the colunm.
the average critical stress
(described by x o ) may be obtained
condition for forces in the axial
Consider the residual stress distri-
bution. as illustrated j.n Fig. 40 and 41. * It is assumed that
sufficient external load has been applied to cause yielding to
occur at the areas cross-hatched.
.'
The meaning of the various
symbols is indicated on the sketch.
The applied load at the
ends of the column causes a uniform stress distribution to be
superposed on the existing residual stress pattern (Fig. 41).
The edges of the flanges will reach the yield stress level when
the applied uniform stress attains the value,
°p. = Ap
- 0 Y - 0 rc
-.
With further increase of the load a part of the flanges will
yield, and depending on relative magnitudes of residual stresses
in the flange and web, part of the web may also yield before the
flanges have completely yielded.
ThUS, two possibilities must
be considered:
(I)
The web stays elastic until the flanges have
completely yielded;
The assumed
*equilibrium
is
r dA
AJor
residual stress distribution must be such that
satisfied:
= 4t Jb/2 0rx dx + 2.w Jdl/2
.
0rydy = 0
0 0 .
-60
220A.9
(II) The
w~b
starts to yield before'the flanges
have completely yielded.
CASE I: I1Web Elastic"
•
First consider that only the flange edges are yielded
(the load is slightly greater than crp.A).
A uniform strain,
is added to the eXisting residual strains (see Fig. 42).
€,
The
resultant stress distribution is obtained as shown in detail C
by the same process as described in Appendix 2.
The total force
applied to the member is ,equal to the sum of the cross-hatched
area in web and flange, which corresponds to the applied or
superimposed stress.
The applied force in tne flange is made up of a plastic
part, A P p and an elastic part 6Pe' In the plastic part crx =
cry - crrx ' The stress in the elastic part corresponds to € and
is obtained at x = Xo as being cry - cr",x
... 0 .
In the web the
stresses are all elastic and equal, also, to cry - crrxo .
There-
fore
or
The integral corresponds to the area a - b - c -d, in Fig. 42.
Dividing P through by A, and using the relationship,
•
Ae
= 4txo +
Fig. 42.)
.
01
wdl
(the unhatched portion of cross-section in
J
b
P
4t
/2
=A
= -A
Xo ( (jy
-
Ae
crrx )dx + -'
A (0y - crrxo )
(18)
-61
220A.9
Simplifying the integral and rearranging terms the following
expression is obtained for the applied stress just after first
yielding has occurred.*
•
_ 4t
P
A
<11 = A = <1y - Ae
The applied strain,
El
A
E,
Sb/2
(10)
Xo
is given by
1
(9)
= E (<1y - <1rxo )
The corresponding point on a typical stress-strain curve is
shown in Fig. 43.
Eqs.
9 and 10 also apply for a rectangular
cross-section if the corresponding Ae is taken.
For a fully plastic flange and elastic web
It
and, from Eq. 10, with.A e = Aw
_ 4t
<12 = <1y - Aw <1 ro
A
A
Xo =
0
rb/2 <1rxdX
Jo
This point is plotted as Point 2 on the stress-strain curve of
Fig. 43.
As the load is increased further (<1rw~<1rt) the increase in load is carried by the web.
Yielding starts in the
web at a stress <13 (see Fig. 44) given by
<13
=
<1 y - Aw <1rw - 4t Jb/2<1 dx
A
A
0
rx
(20)
Then
The point is plotted in Fig. 43. Eq. 20 follows from 19 by , .
wdl
adding
(O"ro - O"rw) which is the increase in average stress.
A
*
The sign convention is the usual one: compressive stresses
negative, tensile stresses positive. Consistent with thiS
convention, negative values are to be substituted for <1y '
220A.9
-62
If the load is further increased, strain and stress
configurations are as shown in Fig. 44.
given by
.
4t (b/2
•
cr4 ::::
The average stress is
AJ o
2~ ryo. .
w
(cry-crrx)dX + A}o (cry-crry)dy + ;(d l -2YO ) (cry-O"ryo)
(21) ,
This may be simplified to
A crry _ 4t
cr4 :::: cr - ~
A
y
0
A
1
b/ 2
(22)
0
Finally, when the complete section has yielded, the following
equation is obtained from (22) by setting Yo
cr :::: cr _ 4t (b/2 cr dx - 2w \dl/2 cr dy
5
rx
AJ0
ry
Y
A Jo
Then
(23)
.According to Eq. 17 the sum of two integrals in Eq. 23 must be
zero, and
(pOint 5 in Fig. 43)
case crrw > crrt then Eq. 22 becomes
Ae
4t b / 2 cr., d x - 2w Jd l / 2
__,cr
- -rx
A 0
A ryo
A
yo
J
(24)
The equations are summarized on Fig. 43.
CASE II: IIWeb Plastic II
Case II differs from Case I in that the web starts to
yield before the flange becomes completely plastic •
Referring to Fig. 45, the web will reach the yield
..
stress when
a-b = c-d
•
or
cry - crrx o = crrw
from which the condition for yield to commence in the web is
220A.9
-63
and the average stress, from Eq. 10 for this condition is
P
Ae
4t Jb/2
°rxdx
°a = A = 0y - Jr °rw - A Xo
With further increase in load the applied stress will
be as pictured by the solid lines Fig. 45.
must have the
E
. same value for the flanges and the web, and therefore
or
(26 )
°rxo :: °ry0
Eq. 26 is important because it gives the relationship between
Xo and Yo'
The average stress for both web and flange partially
yielded is then,
4t
0b = A
+
2w
A
5'
b/2 .
4txo
(Oy-Orx)dx + - ( 0y-Orx ) +
x
A
o
JYO
0
0
w
(Oy-Ory )dy + A
which may be simplified to
_
Ae
0b - 0y - - 0rx
A
0
(28)
In the above expression
Ae
= 4xot +
wd l - 2yo
Eventually as load is increased, the flanges will be
completely yielded (x o
condition that
'.
= 0),
'or
Oro :: °ryo = °rx o
Yo may be determined from the
-64
220A.9
for Xo : 0, from Eq. 28, ror com~lete yiel~,ing of the flanges
YO
Ae
4t fb/2
d ( 30 )
cr c = cry - A Oro - A0
O'rxdX - A
0
O'ry Y
'2Wj
which is the same as Eq. 22 except aro is substituted for a ry •
o
For further yielding, Eq. 22 for 0"4 applies ,until
finally the yield stress level is reached.
A typical curve is plotted in Fig. 46 and the equations
are also summarized there.
There have now been obtained relationships between cr
and x o .
To solve the problem posed by Eq. 1 in the text, a
relationship must be found between Ie and the yield condition
as described by the dimensions
Xo
and Yo'
This relationship
is approximately as follows:
(31) .
for the strong and weak axes respectively.
Ie have been expressed in terms
to solve Eq. 1 for L/r.
Appendix
'.
4.
Xo
Since both cr and
and Yo' it is now possible
A sample calculation is presented in
-65
~20A.9
Appendix 4
Sample Calculation
b
~-~----.,-~.~~~
I
t
I
J
Xo!
I
-'
::-1
:
!
Consider 'he distribution
shown in Fig.
c .
The flange
stresses are
2
=b
O"rx
(O"rc - O"ro) x,+ O"ro
and the constant web stress follows
from Eq. 17 (Appendix 3)
=-
O"rw
FIG. c
(33)
bdt (O"rc + O"ro)
w1
The applied average stress for a partially plastic flange is
.,
:t:
obtained from Eq. 10 after substituting for O"rx
In terms of
Xo
using Eq. 32.
as the variable,
= ~y
0"
o
- O"ro - O"rc-O"ro (4t x o 2 + 2wdl
A
b
b
Xo
+ bt)
(34)
The question arises as to whether the web yields before the
flanges are fully plastic.
This may be determined from
the
condition that at yield of the web,
O"rx o
= O"rw
This was discussed in Appendix 3.
If O"ro is less than O"rw as
determined from Eq. 33 then the flange
is fUlly plastic before
•
the web yields.
Otherwise
condition that O"rx
•
o
= O"rw'
Xo
is determined from Eq. 25 for the
which gives,
b 2t
x
o
= -----:.;.---------2wdl
O"ro - O"rc
(35)
220A,9
-66
For illustration consider a numerical
an 8WF31 shape
Then from Eq. 32
°rx
=
... 4x +
Ore
=
-
°ro
=
+ 5 ksi
0y
=
-
E
= 30~000
5~
examp1e~
11 ksi
39.8 ks1
ksi
a = + 9.4 ksi
from Eq. 33
and from Eq. 34 the average stress at a given
o 1s
X
.
2
a = - 38.8 + 0.964x o + 0.377xo
<0rw
Since oro
~
38.8
ksi~
(36)
the flanges· will be completely plastic before the
web reaches the yield point.
is
using
(The average stress at this point
1evel~
only 21/2% lower than the yield stress
indicating that the web is of little influence for this case.)
I ex and I ey are determined from Eq. 31 for the condition that Yo =
o.
may be solved for critical L/r by the use of Eqs. 36and31. It
is advantageous to compute the column curves in tabular
assuming successive values for
Eq.-
X
o
Ix~1 x~
II
a
(36)
I ex
(31)
Xo
from b/2 to 0:
l!1r
Iey
(31)
.
form~
2
EI ea/I
-
,
{L/r lx
(1)
~L/r
ly
(1)
-=--=
I
I
The column curve of the above example is plotted in
Fig. 34, curve II.
Table 1: TEST PROGRAM
Shape 8WF31
Test No.
•
Type of Test
Heat
Material
Condition
~eCimen
Type of
No.of
Residual Specimens
Stress
Deaig..:.
I
~tiqn
Compression Coupons
Res icll:al 5t):'2s s
CrosL '-be,-;':~i(,n
I
I
I
as-delivered I IB2.1
I 1-82.2
132.2
205*
205A T-18
205A T-15
Tension ,Coupons
Column
Column
I
I
I
132.1
1:32.4
IB2.5
T-l
T-l
T-l
Tension Coupons
Residual Stress
Cross-Section
I
I
I
Compression Coupons
Tension Coupons
Column
I
I·
I
T-2
T-2
T-2
Compression .Coupons
Residual Stress
Cross-Section
I
I
I
205
205
205A
_. T-25
Compression Coupons
Tension Coupons
Column·
I
I
I
T-3
T-3
T-3
Compression Coupons
Residual Stress
Cross-Section
I
I
I
annealed
T-4
T-4
T-4
T-4
Compression Coupons
Residual Stress
Cross-Section
Column
I
I
I
I
as-delivered
Tension Coupons
Column
Column
I
I
I
T-O
T-O
T-O
205
205
205A T-ll
as-delivered
IA2. 2
cooling
1
"
i
205
205A T-18
,205A T-15
I
T-5
T-5
T-5
T-5a
T-5b
Ii
T-6
I
I
i I
4
-I
'
COOlin;;- ~81
IA2.2
IA2.2·
1
IA2.1
IA2.1
IA2.3
3
4
1
1-.
as-delivered
4
l'
IAl.l
IAl.1
IAl.l
cooling
I
I; iI
1
3
4
1
I
IAl.3
IAl.3
IAl.l
I
i
I
I
Compression Coupons
Residual Stress
Cross-Section
. Column
_~~lumn
I Variation
of Residual Stress
II
II
II
II
II
II
~
I
IB2a.2
IB2a.2
IB2a.2
N~gligibJe
i
t
cooling
IB2.3
IB2.3
IB2.3
IB2.3
annealed
I
a s-delivered
4
1
1
1
II
IB2.1 I
IB2.4
IB2.5
4
1
1
I
I
I
I
i
I
I
1
i
I
cooling
t
I
IIB3a.4 Negligible, 4
IIE3a .L~
1
1
IIB3a.4
1
IIB3a.3
1
IIB3a.2
IIB3.5
I
1
4
l
I
I
,
-I
..
I
I
I
1
i
8
i
* Designation "205" and "205A ", indicates test performed as part· of
another program sponsored by WRC and ONR.
TRBLE I
I
1
,
.'
-_. :
Table 2: ARRANgEMENT' OF TESTS ACCORDINQ TO MATERIAL
SPECIMEN
·1 NO. OF COUPONS RESIDUAL STRESS
DESIGNATION !TENS. l COMPR.
MEASUREMENTS
CROSS~SECT!ON
TESTS
.
COLUMN TESTS
TEST NO. IL/~..
'1'2
T2
205A-T25~82
Tl .
Tl
205A-Tll
-
..
As-delivered Material
.,
,
.~
IAI
4
IA2
12
-
11
3
56
..' - - -
IB2
4
TO,T4
TO,T4
8
T4'
158
. 205A-T15 i 42
205A-Tl~ ~~~.
i
!
.
-
IIB3
I
-
I
I
!
measurements
along 9'beam
I-
I --
.
I:
J~n',ealed
-
-
T-6, eight
Material
,
IB2a
-
4
T3
T3
IIB3a
-
4
T5
T5
- ~I
T5a
T5b
I! 8'')),.
c,... ,J
.h.l
I
* The number 2'05A des'1gnates those column tests. conducted under a .
separate program sponsored by the Welding Research Council.
TRBLE 2
Table 3: 8WF31 lvIl\'rERUL PROP~RTIES
(~ll values in ki~s Per squ~re in.)
~
S
DEL I V ERE D
Compression Coupons
iviate-
rial
E
O'p
a u.y la y(c)
1
2
3
6
7
F- 8
l\.ve.- 1*
29,250
30,750
29,750
29:800
27,500
28,250
29,220
17.0
36.0
34.0
32.0
31.0
34.0
30.6
37.4
38.5
37.3
37.7
37.7
37.7
36.8
37.4
37.8
37.3
37.7
37.7
37.4
W- 4
w- 5
!l.ve.- 1
29,000
25,500
27,250
36.0
17.0
26.5
43.1
42.3
42.7
43.1
42.3
42.7
FFFFF-
ewc- ~/~
F6
!l.ve.- 2* 29,000 ·29.6
FFFil.ve.-
1
7
8
1
32,800
30,500
30,500
31,270
39.0
38.8
40.0
38.9
40.0
39.6
39.5
38.3
39.7
39.2
Willi.
F7
F8
Tension Coupons
iviate-
ria.l
U1.3
F2
Spaci-
men
No.
111.1.1
( T-2)
Fl
Speci. men
No.
K
°'p
0'
uy
0'
y( c)
F- 2 30,300
F- 3 30,200
F- 6 30,200
.ll.ve.- 1 30,230
39.8
45.6
43.0
·42.8
37.9
39.2
40.3
39.1
W... 5 30,200
. 44.8
43.3
.Q.ve. - 2 30,220
43.3
40.2
IH.3
i
U.2.1
F- 1
F- 7
29,750
30,400
F- 8 30,200
!l.ve.- 1 30,.120
38.8
39.7
41.0
39.8
39.2
39.6
40.5
39.8
FFF!l.ve.-
3
6
7
1
30,200
29,400
29,800
29,800
41.1
40.9
40.7
40.9'
w-
5
30,500
47.5
l\.ve.- 2
30,010
42.5
U.2.1
'.
Table 3: Con' d
l S
DELIVERED
Compression Coupons
Mate- Specinial
men
No.
* Ave.-l
* b e.• -2
E
op
Tension Coupons
OUy
oy( c; .
Mate- Snecirial . men
E
O"p
No.
0"
uy
0" y(
c)
Average value
Weighted ~verage in proportion to flange and web areas.
TRBLE 3
Table 4: SUMMARY OF COUPON TEST RESULTS:
Compression Coupons (as-delivered)
(Average Values in ksi)
IAl
Flange
Web
29,900 (9'*
28,T50 (2) .
-
.,
I
Ave.-2**
t
:
29~580! (11)
Flange
30,120 (3)
IB2
Flange
Web
28,940 (6)
30,000 l2)
............................... ,
..................... ,
Total
._ _.
-
29.6(~)
-
IIA2
38.4 (8~*.
42.7 ~2~
30.6 (6 )*
26.5 ( 2 ).
38.0 (9)'
42.7 (2)
,
;
..
:
39.4 ~16)
39·2 (11)
39.8 (3)
39.8 (3)
30.4 (6)
30.0 (2)
39.6 (6)
43.6 (2)
39.6
43.3
30.3 (8)
40.6 (8)
40.5 (8)
,
.
~§~
\~}
:
Ave.-2
"
,
29,200 (8)
;
,
Ave. -2
,
29,580 (22)
29.9 (16)
,
,
.
.
40.0
(21rl- 39.8
'
_-:_--':'--L'
__-(22)'
::~.:....
Tension Coupons (as-delivered)
(Average Values in ksi)
IAl
Flange
Web
..............................................................................,
30,230 (3)
30,200 (1)
4442 . 88. (3 )
1
()
.
~
:
·.· ·
Ave.-2
IA2
Flange
Web
30,010 (9)
29,270 (3)
Ave.-2
29,820 (12)
l
I
30,210 (4)
39.1 (3)
43~3 (1;
43.3 (4)
40.1 (4)
32.0 (6)
27.7 (2)
39.1 (9)
42.6 (2)
37.4 (6)
35.7 (2)
30.9 (8)
3g.9 (11)
37.0 (8)
.
.
....................................................................................................................................................................................................................................................................................................................
........................................................................ ~
IB2
............. ,
,
.
Flange
Web
30,090 (3)
30,200 (1)
Ave.-2
30,120 (4)
43.5 (3)
46.6 (1)
_
40.5 (3<
44.2 (1;
.44.2 (4)41.4 (4) !
...........................................................................................................................................................................................................................................................................................:
Total
Ave.-2
I
I'
.
29,970 (20)
30.9 (8)
I---·-----l.-....::..::~_.:.--=--L--_·
--l...
41.6 (19)
..l
I
38.. 9 !).6)~i
~______
t
I
Mill Report Tension Test (as-delivered)
Web
-
r
I.
I
I
*N'J.rn.ber of specimens.
**i.tlelghted average in proportion of flange and web areas.
--,!
!
I
43.3
.
-_._-_._".,,
TRR'uL ~
L.
~4.:. .
Table 5: CROSS-SECTION TEST RESULTS AND COMPARISON
WITH COMPRESSION COUPONS
Cross-Section Tests*:
r
Condition
Material
)-
1:82
IA2
:LAl
!132
I
I Test
-'---I
As-delivered I T··O
As·-de1ivered I r~·'-l
ft3-deliv-=:red T-·2
-de1i vered T-4
-------l!:.:- - - -
I
I
-
t~~:ra.ge _I
IB2a
1133a
I!
i
crp
vy -C5rc
First
Yield
Line
29,ltoO
30,100
29,600
29,340
25.6
28.0
25.2
27.0
23.41
24.70
25 ?~
-' (
22.82 I
I
29.65
28.26
24 . 89
29.16
29,610
26.4
29,760
29,600
26.0
31.0
I
.
cry
I
l
I
31
37.
36.20
37.77
36. '72
-----37.00
..
IT-5
T-3
Annealed
Al!nealed
E
-
_-------
~ §r-:~6 __
Compression Coupons*:
--M;t~riall
r•
I
Test I
I
IB2
IA2
IAl
IB2
----
Condition
As-delivered
As-delivered
As-delivered
As-delivered
T-O
T-l
T-2
T-4
Average
IB2a
IIB3a
IT-3
T-5
Annealed
Annealed
S3
cry! Oy(c)
.Oy( c)
.922
.909
.963
.907
I
I
39.8
.925
36.0
33.9
.927
.926
1
29,200 I 30.3
30,120
29,580 29.6
29,200 30.3
40.5
39.8
39.2
40.5
29,580
29.9
30,840 35.6
29,650 I 33.3
I
-
I
I
Table 6: EQUIVALENT ECCENTRICITIES OR
INITIAL CURVATURES
ec/r 2
a
!
(inches)
(kips)
L
(inches)
bG.(inches)
05A T-15
05A T-ll
200
200
146
194
0.018
0.002
0.097
0.005
.032
.002
0.119
0.007
. strong
axis
,?05A T-25
T-4
-T""5a
T-5b
200
164
116
164
116
0.025
0.016
0.045
0.012
0.020
0.038
0.037
0.027
.020
.038
0"'-7
--
0.026
0.048
0.046
t aJ~:i.8
Test
P
~OO
200
200
*A11 values in ksi.
e
(inches)
.
o
(1;")t7
. ,_. ;
I
0.036
weak
i
TRBLES 5G6
I
I
.L
------~
II
i
I
I
1TF: i
(+ )
t
11
T
//1----··
v
FlANGE
STFESSES
(+-)
/
I
I
I I
.'
(+)
I
II
:i!'
~l
WEB
STRESSES
~
t
~
(-)
FIG. 1: TYPICAL RESIDUAL STRESSES IN WF SHAPE
............................................................................................................................................................................;
(j
y
•
STRESS
.
,
-
1
i~-.:JTi!
!
rn
-;:11
STRUN .. ----,~
FIG. 2: INFLUENCE OF RESIDUAL STRESS ON
STRESS-S~RAIN
CtffiVE
('
'
)
I l.f L.
•
r
--\
"""
/ /0,,',-,
(
'\
cry
crp
IDEM,
COUPON
--~-
cr _ 11"2 Et \
--
t -
(KL/r)·~
'.
__
~-------
€
- - - - -
/',
... ~
E
KL/r
Member Free Of Residual Stress (Coupon)
Member Containing Residual Stress
FIG. 3: INFLUENCE OF RESIDUAL STRESS ON THE "COLUMN CURVE"
·'
..
I
Oross-section Test T-Q
I
40_
U.r
r.n
1%1
~
[f.)
10
80
t/r
120
160
•
FIG. 4: COMPARJ;SON BETWEEN "EXACT" AND APPROXIMATE SOLUTIONS
FIG. 4
8W31
INGOT
DU.GR~
HEAT I
.,/~
'\.'
\
)) \., Ingot
I.
Bottom
I ngo t
Hot saw cut
Top
'---' · - - - - - r - - - - , - - - - - - + I - - - - . , . . - - - - - - . - 1B_3_· __
...
I Bl._ _;....__
I_B2_ _ L..-_ _
~,~-._.:-J--I_U--.!.--I-,~_2---'---I-A3--4:,
,
.I
\
~, \
I
I
i
,!
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i
180'
.-r--.
First rolling
INGOT DH.GRiW
Ingot
Bot tom
[
!.
.
IUIL..
24'
l_. II.1\3.L-=_.
24'
I
f-----!
HEAT II
r Hot saw cut
----,-'_
--,
..
180'
-------.-Second rolling
I
i_--::'''::'''-1
I
I
,-_._--
Fi rst roll ing
---1---
Ingot
Top
.J._2-I!33_J
::..- - _.._
56' .. _-_.!:
._--_.. _,
Second roll ing
FIG. 5a: LOCATION OF TEST SPECIMENS
I
FIG. 50
llL1. 3
lU.l
205*
Ii!!)
2C54.· T-25*
T-2
-,f ~
E,_,t.:...-I
I
•
col
i
11 L._ ._ _J
12'-10 11 t
; . c - - - -.. '---i
16'-0 11
:
I1.....0:-----------I
14\.2.1
r:82.1
U.2.2
I:82.4·
IIB3a.3
T-5a
IIIm/
U.2.3
I:82.2
IB2.3.
I:82a.2
IIB3a.2
T-5b
IIB3.5
T-6
IIB3a.4
T-5
>}~~~::~~;:>:~_~i)~-~:~~;~~~~::~~:~~ ~0»~~~\~:.~~:~~~
r
!~~M~~
_
~--~~o~.--------l-.--J~.t!~~~-O·I;-L-~~=Q·I~l.....L'.-~-Q.~L- o~i
!---,,_.
. . _ - - - - - - - _ . - _.. _- ...._ - - - - - . ._---~
........•.. ,
t
c
I:82.5
.
,
tension coupons
c-s···cross-section specimen
compression coupons r····residual stress specimen
specimen
annealed material
col·~column
!~\Z%.~1
... Tests made und.er a separate program sponsored by WHO & ONR
IFIG. 5b: CUTTING DIAGRAMS
I
FIG. 5b
FIG. 6
AVERAGING COMPRESSOMETER IN PLACE ON A COUPON
FIG. 7
HUGGENBURGER EXTENSOMETER IN PLACE ON A COUPON
6,7
.
Section A
gage holes for Whittemore
strain gage
\
",'
\
\
..
/-I
/-I
..
\
\
cuts
\
\
\
\
\
/-I
Q
\
\
\
\
\\
\
\
\
\
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.
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\
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o\\~.
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,
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1 "-~i 1...J-·
1;
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2
SECTION A
I
\
\
0 \
\
\
\
\
\
\
\
\,
\
\
\
\
\
\
\
\
,
\
\
\
\
\
\
\
--------_\!
\-
(')
/
\
\
r
i·
i..
1:
'-'.#
1 ILJ. :::)
•
FIG .. 9
CROSS - SECTION SPECIMEN
T - 1
,
FIG. 10
CROSS - SECTION SPECIMEN T-l AFTER TEST
(PERMANENT SET 0.0022 in./in.)
10
-1
I
I
,
~
I
I
0
II
C\l
r-l
"
I
, !i
r-l
!
!,
i'
I
--~ _.
•
South Flange
FIG. lla: COLD-BEND YIELD LINES ON COLUMN 205A T-25
.. ,
4·
North J!'lange
,
,
-,I
,
~
.. I
-j
I
~TT
IIIII
J
I
-o
~\
-
ex:>
<h.....
!
i.
North Flange'
.
r
"
N /
---,
South Flange
FIG. llb: COLD-BEND YIELD LINES ON COLUMN T-4
FIG.
,II
.
•
l
\ STRESS~STRUN CURVE._1
1--
.
FIG. 12: GRAPHICAL DEFINITION OF TERMS
•
1132.1 (T.;.O)
........ -.
,/
50 '~... --. - - • ---*
,'-
.-'
.-
-.-_...
...
•
I
I
30 i2
~
'3
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I
I
20
i
·
i
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10
I
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p.02
in/in I
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7
: 8
!
I
I
.
I
i
I
FIG. 13: COMPRESSION COUPON STRESS -·STRAIN CURVES" IB2.1 (T-O)
FIG~S
12& 13
r-:
lli.s-IlellieredT-·---1
I
I
•
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,
,-
I
i
i! - . -
r :'
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.
40
I
. 1B2.3 ( T-4) -
1B2.1 (T-O)
I
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l2..:.?_~~ in! in
FIG. 14: COMPRESSION COUPON STRESS-STRAIN CURVES
•.
..
· U.2. 2 (T-l)
......
:
•
I
30
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40 I-
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FIG. 15: TENSION COUPON STRESS-STRAIN
CUR\~S
FiG.Sf 4[~ 5
f
:------.:.-==-----·1
;
1
! as-Delivered. I
.I
IU.l (T-2)
40
r
I.•.•••.• .-
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1
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_
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[:4
.
r;t.; 5
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FIG. 16: COMPHESSION COUPON STRESS -STR1\IN CURVES
40
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I
30
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r
.
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IV
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FIG. 17: ANNEALED COMPRESSION COUPON STRESS-STRAIN CURVES
FIG:S 16 G/7
'.
t----- Compression
Compressio_~~~
't,
Stress KSI
WEB DISTRIBUTION
!
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I(
+ 5
l:l
0
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1
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Q
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--<D----(D-- T-1
0
T-2
o
FIG. 18:
F lr~
"...,-
.,
/8
..
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1 "i
I
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i
l~
f2]
kR1
I
9' - 0"
j-.f
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2)
--i
FLl!.NGE DISTRIBUTION
-5
,
-10 . . . 15
I --t
Ccmpressi.on
8\J\F31
.
WEB DI8r~.'.p.I:JtJTl Ol'if
'1/
+5
I
II
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(I
II
o
o
- - ~~;-~~;
_ _ __
----
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-(9-2
...
-(1)-3
RESIDU'~-;';RESS3~-(~SI)"--'1
-----I'---:'-'----:------i
!'
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FIG. 19: DISTRIBUTION OF RESIDUAL STRESS AT THREE LOCATIONS
IN A SINGLE BEAM (T-6;
...
Compression
Compression
...
/
-!.O
-5
!
Sh'eS2 KSI
•
l'
FIG. 20: RESIDUAL STRESS IN ANNEALED MATERIAL
I('
'In
F,lJ. L!.j
.~
'.'-"',
2
\J. /~ 3 ,'r
I'
68) 10 12 14
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4
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5-2~' .~) 11 a~;i,!.5
I
I
,
II
I
~~311
c: Complete sectioning
P: Partial sectioning
(circles on location
sketch) ,
I
I
I
If'
I
.r
LOCATION OF GAGE ~OLE.~i
............................................................................................................................ ,
o
1
,
3
p
p
4
6
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c
p
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p
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o
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CUTT-ING LA Y(;~-;-'I
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,
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20 lSI
-.-I
lZl
lZl
III
d
~ ~ 10 KS1
lZl
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t:5
A
H
lZl
d
~ ~-10
KS1
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~-20
o
'
KSI
FIG. 21: DISTRIBUTION OF RESIDUAL STRESSES ALONG BEAM
?I
......
f
9' - 0"
f.e-----.----.--.-.- --.--...-.-. --------.. .----.----.--..----.
. ---.--.--....--......---'---"-.----.-'-' . -....--.-_. ---1
..
i '
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10 in. gage length
:
1
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)
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0
.,
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STRESS
KSI
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10
~
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rn -10
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i
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/
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Stress
Flange Centers
.__... Q.
._--'--
Stres s
Web Centers
--·0-- .- .- .- .-.-
~3"
141119" 25" 31"
.~
._ ..- - .-.
-_.. ~verage
---?-CROSS-SECTION LENGTH
. '
"-0"
'O---O---O-
.
.r-"
.Q.verage Stresii
-L-.. _olJ,.a!.);ge .. :Eldge B
~
0
0
FIG. 22: RESIDUAL STRESS AT SECTION A-A FOR DIFFERENT
liCROSS-SECTION II LENGTHS
~'-IG.
??
j._------_._---_._---_.
-------_._----- ..
i'·
I
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I
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t Il
I\.verage
4,verage
Compression
Coupons
Web
i
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Flange
SR-4' s
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//
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J
.. - i,-- -
_ _ _L
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4 gages
--l.-
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._
STR4.IN
FIG. 23: CROSS-SECTION STRESS-STRAIN CURVES (T-G)
FIG. 23.
.
-'-------r
!
'
,berage
Compression
Coupons
,~verage
Web
SR-4' s
4.verage
Flange
SR-4's
I
I
10 in. Gage
I
I
I
40
l-
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1
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T-4
,~s-De1 iva red
lOll Gage
I
-
2 gages
I
/
\.£1__Jgage
1
SR-4(11l)
2 gages
...---l.l_-1
" 1 gage
.'_.
J
II
STRUN
.
FIG. 24: CROSS-SECTION STRESS-STRAIN CURVES (T-4)
'f
FIG. 24
+
Average
Compression
Coupons
Flange
SR-4
.tverage
Web SR-4's
i
i
/
I
30
lI
;
..
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1
/
6
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o
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r
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10 in. Gaga
10"
G~GE
\/
1:ge iljl-I~ S::(1")
l!
I
1 gage
1
-- ~7\ l-;age
i
,,001 in/in'
:-·~---·--·---·-··-'·i
STRUN
FIG. 25: ANNEALED CROSS-SECTION STRESS-STRAIN CURVES (T-3)
FIG. 25
,~verage
/'------.
'~
40
I-=~_-__- . , -....-./~:~----tl
Compression Coupons
--_.. _'' ' /"
,- '-- . . _.. . . _(,
///~'---_.-//
.- - __ --t__
""\ --7\"'\ . '~'>-'
I
~
.
30
t:>
-Web SR-4's
1
"I
I
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in.~\
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\ II
\
,
i
I
SR-4 s
-10 in. 'Gage
,
L-l1eb SR-4's
(J
20
IT-o"l
I T-zi
1_ _ :
10
.l--
--J '--
'-
.J,
--J
I'-"'---"--"-'-'-'---'~
30,000 KSI
i
FIG. 26: STRESS-TANGENT MODULUS RELATION OF CROSS-SECTION TESTS
FIG, 26
T-O
,~s-Deli vered
40
30
1-4
tI)
.~
CI)
tI)
.
~
tI)
20
'.
10-
----£ross-section Test-Strong ~xis
- --Cross-section Test-Weak ,~xiS
a
o
40
Strong~xis
Column Tests
Weak ~xis Column Tests
80
120
160
tlr
FIG. 27: COLUMN CURVES AND COLUMN TEST RESULTS
FIG. 27
r
T4
As-Doli vered
cr (c)
/
40
-
y
~~- - - - -~:-7
~-'~~O
=-- - /
.
. . ,....
" ... ,
. . ......
0 ""'-'...
'''"'I
-, ....J............
30 -
........
Y
c·- ..L'
... ..
'~\/-
/'
I"......"
Coupons·
\.
.
,
1- -<.- 9
. . ":::" .......
Average
,-- Compression
'v
'/-
,/,\\
\
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...........
10 in .
Gage
Residual
Stress
Pattern
\
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20
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8
to
l!l
N
I
- - Cross-section Test - Stro
axis
- - -Cross-section Tost - Weak axis
10
--'Residual Stress Pattern-Strong axis
-
-Residual Stress Pattern-Vleak axis
o Strong .~xis Column Tests
_u
I
o I Weak
40
~xis C lumn Tests
80
120
160
Llr
FIG. 28: COLUMN CURVES AND COLUMN TEST RESULTS
FIG. 28
. T3, T51
.4.nnea1ed I
,
40
-crY(C~T-I>.5
-T-1S
-
-----
_c.__cr!. __
-'-----(
\
-=-:-~ :0_ ~~.Gage)
.
;--..JI1-5
SR-
I
s\(
30 -
\
20·-
10,
- - Cross-section Test
--•
40
~verage
Compression Conpons
~ealed Weak axis Column Tests
80
120
160
Llr
FIG. 29: COLUMN CURVES AND COLUMN TEST RESULTS
ANNEALED MATERIAL
,FIG. 29
/,-'--~s-dolivered
Compression Coupons
-iUlnealed Cross-section (T 5)
Cros s-so cti n (T 3)
Comnression Coupons
(T3, T5)
0.8
0.6
..
'
.'
I
r-I
lC
r-I
,~
~
<Xl
~
.CJy
CJ
lC
@
~
to
0
N
.0
lC
r-I
r-I
ro
lC
lC
"2
E-i
e!.
~ ~
'<;jl
<J!
to
L()
@
@
I
~
.~
~
0.4
"-
Tests-Strong~s
Cross-section
Cross-seC'jjion Tests-Weak ~xis "'-."
----.r;' -- T-O (Web SR-4! sy
- - . - T-l (10 ia. G'aga)
_ .•• ---- T-2 (Web SR-4 IS)
-.-_.... T-4 (10 in. Gage)
Strong .~xis Column Tests
0.2
0
0 Weak
•
0.2
•
0.4
0.6
Column Tests
.~ealed Weak ,~xis Column Tests
1
I
0.8
!:-1
A.= Tr
.~xis
I
I
1.0
1.2
1.4
1.6
1.8
cry I L
r
E
FIG. 30: SUMMARY OF CROSS-SECTION AND COLUMN TEST RESULTS
FIG. 30
\
\
\
0.6 -
0.4
f-
0.2
I-
~,4.nnealed Weak ·~xis Column 'Tests
•
I
I
0.2
0.4
, I
0.6
!
'
I I
0.8
j
.",
}'\-
I
I
J
I
1.0
1.2
1.4
1.6
~1 ~-i
le8
L
r
FIG. 31: COLUMN CURVES FROM LINEARIZED RESIDUAL STRESSES
FIG 31
'.'
LOll)
I
I
,-
I
i 0.10"
...
COLiJJvTN DEFLECTION CURVES
'.
.----------'--------------:===~---:-~
L~nnea.ledj
LC;l1)
lOoK
COLUMN DEFLECTION CURVES
FIG.32
,.
/\... j
_
-U.Y~P.
rLab07-:a'~ory .~~s~ (web
I
.
/
L
COl.1.j))!",)
.
,__ .1 _, 9, -=. 10%
I
(U-p:?er Y.J? vs Lower Y.F.)
5 - 10% (St~ein rate)
5 - 10% (web vs flange)
I
t
Mill Type tension tes'G (webmater:1.al)
'7-'
L
..
--5-=-10% (coupon va cross-section)
Cross-section test (compression)
Laboratory test (flange coupon)
•
STRUN __...
FIG. 33: INFLUENCE OF SEVERAL VARIABLES ON THE
YIELD STRESS LEVEL, cry
FIG.33
"Of
°rx'
I
l--..., :.-:-=---cl-- x
lcrrw~constant
.
•
r=:"
8
l1\KSI
j
I
i
I
I
I
in.
'fiF 31
.(;.) I~ 4~
'...._.
..
'V
_---_ .._-_._.
__ __ _._._.__
..._...
..
._.-._._..•.
_-------
'T
!
l
---"1
Ii
I
".\
<J:rc
j
i
.1.
,
30
•
= III KSI
o
20 -
10
----Weak axis
40
80
120
160
L/r
FIG. 34: INFLUENCE OF RESIDUAL STRESS DISTRIBUTION UPON
COLUMN STRENGTH
i
o rx
0rx
=8
Cos (if xl 4) -3
--f
~lO
!
!11
-5
1I
o
20
160
L/r
FIG. 35: INFLUENCE OF RESIDUAL STRESS DISTRIBUTION
UPON COLUMN STRENGTH
KSI
_.
Compression
,
Compression
--4--
J-'
I~/
~/
/'
\
I
\
•
==*1:
T-O
I~=F====-~"::--="
!
10 Ksi
~
!...
10 Ksi
!
~
!..
\\
//
'\.\\
\."-
T-l
l
f\..
I
.t'.
~
'J
~
i
,:~
,//
1/
h
10
,-_.,
10 Ksj.
Ksi
~-""'i
,
, ,"/'
y-
-~\
/~
",
\
T-2
,'..'
7
~
---
Measured Residual Stresses
Computed from cross-aection Tests
FIG·36; ,jor·~PJl)1ISON OF MEASURED RESIDUAL STRESSES 1VITH STRESSES
CQr.tPUrED FROM CROSS-SECTION TESTS
40 1- - - - - - - - - - - - - - - - - - - - -1
"--T'
i
i
I
;
!(Jrc
30
•
I1
I
(J
-_
,I,
.. ~--.,
10
o
40
80
160
120
200
L/r
FIG. 37: COMPARISON WITH SPECIFICATION FORMULAS
"/::",~:;",
,
,
-,
,
-,
.. -,
-,
,_
-,
-
-,
,
-.- , .. " ..
,_
.
-
,
~
2.5f-
"' ,,-----.. -
-" '- ""'-,
_
".-....
-
2.0
-__
"~4.RE4.-4.4.SHO
.... ,
-
,"':""
-
_:0.. - ; : : - -
--.
Strong .4.xis
4/; ___
",;
r- '- '- ,,-:,:::.:usc-
-.l'..
- ---r
__
!
t-
......... _
t-- -
--'- -
_
".
""'"
--:::-.........
-~~
__ ...
------_1--:-/
---
-...J....,c'- _.- -
Weak iLxis->'
" .... _
-
,/'
______"
-" -'- -- -
"-
""- "-
'-
-
",'
./
/
--.
-- --
l
I
I
I
1. 5~-----l...----~-----L.----..L-.-----!
o
40
80
120160
2"0
L/r
FiG. 38: THE FACTOR OF SAFETY AS IMPLIED BY
SPECIFICATION FORMULAS COMPARED WITH EQ. 15
FIG:S 37 &38
1'=0
1'1
1'2
:
u
I
D
(a)
(')
;
I
t
~
:,
( b) Res idua1
Strain
(c);\.dded
Strain
!
1
l!
I
I'
t ,
t t
t
~.~,
•. '
l//
'I
~
1 ._--
.r.Y
(d) Total
Tey1/
StrainJl.
(a)Total
.
S t r e ss
Distr.
/"\
'\J
/"'I' , .
, I~ I1
/j
I: I II I
U~~!I
.V·' . ~~
.... _.... _.
1,!!~
1. _.
.
"
.. ---_._--
'r
.'
-"--'-,11,
".'. .!
If"
t-·······
r
p 4 1Il
o
-"
I
l"J
),.
,
J I
. '
( g)Yield
Condi t:1on ' - - - ]
P3f.~ ._.._
.
. :'1
(t) SUperimposed. 0y
StresSAS '.. - - - - - - -
J
~-u.J
~
j
~
•
1'4
1'3
,
1
1
7----:0.....------
_.._. /.~ /:
t
I,,p
l' 2/4. ' ... _ ...__."
1'1 / ~ '._.... ...
(h)
l _ ._ _ ._.
0.001
E
0.002
.50
L/r
1Q 0
150
FIG. 3g
p
(-)
't
Ir- --I (+)+_~
j
c
ITTIIl[
~
(+)
f--.I
Y
i' II,j]n
_
-
'j-
_·w
__
.d._
'd.
'
-,:,' , /
(J
(_)
ry
.l.. Yo
-- .----- " ':~~:~:
---r-'..
"'T
(Jrw
p
WEB
DI STRIBUTI ON
I
FIG. 40: RESIDUAL STRESS IN
WF SHAPE
FIG. 41: PARTIALLY YIELDED
CROSS-SECTION - NOMENCLATURE
€y
...•,
-----···1
I
_..cry
_
._--_...._...
_._._-- ._
_-_
-
~.===\
.- . j..
--....
.
._--.' .
.. _.... .. - . " ' - " .
' - " " . _ ' ..
....
-.... ~,
_~.-
WEB
6
. I
'··::~~~~~··=· -.
_-_,- __ '.
._-., .. ,
"
.
...
,
I
€y
! -l:-
I
{,
i
---'_.
I
cry;I
.-.l.._
.
I
,€
...__!-
Li i
;~
__
, ..
-, .
-_.. .. _ -. , ..
.-. __ .. _... ...
.__ .
.~_.
~
l..~___
__.J
l
.__-{:a \
'--)
STR~IN
DISTRIBUTICN
t
,/"-",
--_.~)
!
""
~'-"".'-
_ • • • • __ • . •
i : .! ;
) /!:1:"':
.!
. ,
_.¥w' _
,-_
_~~
I
!
~:::~~ .':::.:.'
'-'.-_ ....
••
_ - -_.
. ....
).
--~,,_.
/
L-(Oy -
c"-d
.U'PLIED STRESS
orx o )
FIG. 42: STRAIN AND STRESS DISTRIBUTIONS FOR WF SHAPE -FLANGE PARTIALLY YIELDED
FIG. 42
,."
,
linear
-\-----------,.-c.-..-..----- 5
cry
cr4
'"
/'
-~4
-------
---
cr3
cr2
-0{
FLANGE
---------1.
..-' 2
/
cr1 ---------y'l
b
f
crp
til
til
~
/ ' . - ...:
(1)"
til
I
A
E:1
H
$
1".
I
Flange
EiasticPlastic
/
(4)
(2 )
~'
..
T
!:1~~.:.;jDITIillH
-l
I~
I I
Web Starts
To Yield
Flange
Plastic
.
.
,U'PLIED
STRA.IN
...................................................................................................................................................
,
ITLImILr.lwllillJ
Web
ElasticPlastic
Fully
Plastic
E
.
STRESSES
i
STRAINS
r
(
)
crp=cry _ o r c .
; E p - 1:. cry-crrc
Ae
4t ( ' b / 2 1
E
.
''\="y - A "rxo -p: ~ X "rxdx
(lO) . "1 = ~ ("y-"rx )
o
o
......................................................
cr2 =cry
j
i111
crro
- -4·t
Aw cr
rw
A
-4tA
- -Aw
A
cr3=cry
-
cr4 =cry
-AeA crry 0
crS=cry
'
A
-
" b/2
~o
crrx dx .
.· .... ·..........·......·..·....:.. . ·(19 )
crrxdx
....................................( 20)
,b/2
~o
.-b/2
4t
A \- 0 crrxdx
-2w
A
(Yo
\
crry dY'''-(22 )
_) 0
................................... (23)
::::
-1
E2
=
E3
= -
E
(cry-crro )
1 (cry-cr )
rw
E
1
E4 = -E (cry-crrYo )
ES
=
-E1
FIG. 43: IDEALIZED AVERAGE STRESS-STRAIN CURVE FOR
WF SHAPE CONTAINING RESIDUAL STRESS - CASE I
(cry-crrt)
FIG. 4-]
.
~\'~I\I\I'!II!III\ Iii
r:
iI
--_.- --++ -- --_._Ij
ii
l\iTli;:r[Tji\ji\\ifi!~I!ii\\!i I' 'IIi 1111iTi~
t.-\-.-· .·~·
i--'
I.-- .
/---_._.
t-·_·-.·
/
I. . <',(",.
I I "
j
"!.
]/1
;
/,
I
I
:
•
;
. ,
I1 .:
-!.
;
!:
•
'I
I. Ii',
l:
'!
iii'
;I
;. I,
I
I
I' )'. ;.
I~':
,:.
l
/i
--- _.._.•.•........
- . - - - - - - -.. "1
\
II.
~I_:~~;~~~-~::~V~=\
,-.--_ -_
{~ _.. =~:~~=::?::~l:
I
_.. _---._ ..--
..
1-
.
I'
.r'.-l~
I
('\
~
(2\
--<:..'../
'G 1,.3"
....
1, -'\V
, I,.
!
1
~
I
. i ' ' ) ' ' ' l i·i ! I
/I ' i i -..J 'I : :
'
;,,,,
'
;
I ''-i I I
! Vi) ,/,\~,I \" I
: /l : ;
, ',i :' I..
"
. /'
... _.-._.-....-... i
..
_ -
I :'1 ' . I
I
i/
L--==-:~;
_--_.. __ ..• ..
;
iii
:
\
I, I·.t". I~ :i
i· I ~ J !'j
I . , I I , ; ,...1
ii,
I I I I' ....
~
!
-+, I'-1-1I -!/-J" +-, ,h . --+-.,
i ,I
.
I"I - -
I,
:~)
-~~i
,\,/
' - ....
STRUN
DISTRIBUTION
.U'PLIED
STRESS
FIG. 44: STRESS AND STRAIN DISTRIBUTIONS FOR WF SHAPE
WEB PARTIALLY YIELDED
~
FIG.4'+I;
i
!
O'rt
I
......
,
i ; ; i
j
1
I
!
/
i
',-'·-·-jllf----'-'--'I
----tcr .~~~! I
Iill
,1 '
,
i+' _--Ho.!
'.
I
.-1--'; - _ . _ -
L.:..J..L_
O'ryo
__ ~.
'~~~~i-I
I
I , " .
I--:::-€-·~I
I
O'ro
.'
O'rx~
RESIDUU.
STRESS
I :
.""
\ arc
•
/
/
/
'" " "-
----;~)
STRQ.IN
DISTRIBUTION
I
/-~
------.f, C )
'_/
.U'PLIED
STRESS
FIG. 45: STRESS AND STRAIN DISTRIBUTION FOR WF SHAPE -FLANGE AND WEB PARTIALLY YIELDED
FIG.45
cry 1 - - - - - - - - - - ---:-=~--------·----~T
cre '.
_.
.
.__~c
crb
___-----.--.-~~
b
cra
cr1 ..._ ~ -
-$--'--
;fa
----1
/
cr .
p
L-
..............................~;;~.~~.~~
/
Flange
Web Just Fle.nge & Web
Elasticat Yield
ElasticP.::l.::a.=..s..::..:ti:...:c:...-_--'-r-~o.=..;i n;.:..t..:--_----::F-=·l.::::a.=..s..:.:ti:...:c'--
Flange
Plastic
;\,PFLIED STRUN E
~;~.~.~~~
. . . . ."
· . · . ·.· · ·
=(Jy_P~e (Jrxo 4X J:~2
.
r·..· ·.·.· ·. ·.·.·.i
.
~ Ep = E (cry-crrc )
crp=cry-crrc
(Jl
Flange &
Web
..=.",Plas tic
(JrxdXm(lO)
............. :
(25)
i
E
1
= - ((jy-crrxo )
1
E
I Ea = -E1
1
Eb
YO
crrxdx -
J
o·
crry d vv .....(29)
=E
EC -
-E1
(cry-crrw )
(cr -cr
)
Y rx o
((jy-eYro )
1
E
= E (cry-crrt )
5
FIG. 46: IDEALIZED AVERAGE STRESS-STRAIN CURVE FOR WF SHAPE
CONTAINING RESIDUAL STRESS-CASE II
FIG.4·0