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Welded Continuous Frames and Their Components
END MOMENT - END ROTATION CHARACTERISTICS FOR BEAM -COLUMNS
by
Theodore V. Galambosand
Maxwell G. Lay
This work has been carried out as part of an investigationsponsored jointly by the Welding Research Council and theDepartment of the Navy with funds furnished by the fo11pwing:
American Institute of Steel Construction.American Iron and Steel InstituteInstitute of Research, Lehigh UniversityColumn Research Council (Advisory)Office of Naval Research (Contract Nonr 610 (03»Bureau of ShipsBureau of Yards and Docks
Reproduction of this report in whole or in part ispermitted for any purpose of the United StatesGovernment.
Fritz Engineering LaboratoryDepartment of Civil Engineering
Lehigh UniversityBethlehem, .Pa.
May 1962
Fritz Engineering Laboratory Report No. 205A.35
205A.35
S YN 0 PSI S
i
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This report discusses the end moment - end rotation behavior of
wide flange steel beam-columns. It is shown that the theoretical moment
rotation curves agree well with those obtained from tests on full size
columns~ provided that failure occurs by excessive deflection in the
plane of bending. As the rotation capacity of beam-columns is of con
siderable interest, comparisons are made between the theoretical and
experimental values of this parameter.
Local and lateral-torsional buckling may also contribute to the
failure behavior of a beam-column and so approximate methods are pre
sented.which allow prediction of both the occurrence and effect of these
phenomena. The amended rotation capacities are also compared with ex
perimental results.
Curves are presented which enable rotation capacity to be pre
dicted for any combination of slenderness ratio~ applied load and ratio
of end moments. Interaction curves for local buckling and lateral-tor
sional buckling are also presented .
205A.35
TAB LEO F CON. TEN T S
ii
1. INTRODUCTION 1
1.1 APPLICATION OF DEFORMATION CHARACTERISTICS 1
1.2 PREVIOUS WORK 2
1.3 PURPOSE OF THIS REPORT 2
3.
LOAD-DEFORMATION BEHAVIOR OF BEAM-COLUMNS
EFFECT OF LOCAL AND LATERAL TORSIONAL BUCKLING
3.1 LOCAL BUCKLING
a. Flange Buckling
b. Web Buckling
c. Effect on Rotation Capacity
3.2 LATERAL TORSIONAL BUCKLING EFFECT
a. Unbraced Columns
b. Braced Columns
3.3 SUMMARY
4
6
6
7
8
10
11
11
13
14
4. EXPERIMENTAL SOURCES 16
5. COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL. RESULTS 18
..
5.1 GENERAL
5.2 LOCAL. BUCKLING
5.3 LATERAL TORSIONAL BUCKLING
5.4 BRACED COLUMNS
18
18
19
20
...
205A.35 iii
TAB L.E 0 F CON TEN T S (continued)
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205A.35-1
1. I N T ROD U C T ION
Column research in the past has usually been directed towards the
strength properties of columns and beam-columns~l) However for many
plastic design and stability investigations it is also necessary to know
the deformation characteristics of the members, particularly in the ulti
mate load. region.
For beam-columns these"characteristics may be defined by three
parameters at each joint: the end moment (M), the end rotation (8) and
the axial force (P). This report will consider the theoretical and ex-
perimental relationships between these parameters for beam-columns of
rolled steel wide-flange sections which are bent about their major axis.
1.1 APPLICATION OF DEFORMATION CHARACTERISTICS
Plastic design methods assume that a structure becomes a mechanism
at its ultimate load due to the formation of a sufficient number of plastic
hinges within the structure. All but the last hinge to form must undergo
a certain amount of inelastic rotation and it is necessary to ensure that
the member is capable of sustaining this rotation. Methods of calculating
the rotation requirement have been discussed elsewhere~2) However there
is little information on the rotation capacity which a beam-column can
provide to meet this requirement.
Frames may also be analyzed by methods which check the stability
of the structure at each load increment. The stability check. requires a
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205A.35
structure to return to its original configuration after a virtual dis-
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turbance has been applied. As these procedures are based on deformation
analyses(3) (4) , they require a knowledge of the load deformation history
of the beam-columns for both the analysis and the stability check. Such
information is contained in the P-M-9 relationships. (5)
1.2 PREVIOUS WORK
For the elastic range it is possible to derive direct analytical
(6)relationships between P, M and 9 . In the inelastic range the discon-
tinuities associated with yielding make such derivations impossible and
therefore inelastic P-M-9 curves are obtained by numerical and graphical
procedures. The moment-rotation curves so derived include the effect of
residual stresses and are also applicable in the unloading range. A
collection of such M-P-9 curves is presented in Ref. 7.
1.3 PURPOSE OF THIS REPORT
There are two aspects of the present theory which require further
discussion:
(1) The moment-rotation curves only apply to in-plane bending
and do not consider the possibility of either lateral
torsional or local buckling~7)
(2) The moment-rotation curves have not yet been experimentally
verified although they are-based on moment-curvature rela-
tionships which are the outcome of .extensive experimental
work~8)
205A.35 -3
,
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This report provides additional information on these two aspects.
Theoretical limits are given for .local and lateral-torsional buckling .
Experimental results are abstracted from previous column tests and are
compared with these limits and with the moment-rotation curves .
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205A.35
2. LOA D - D E FOR MAT ION B E H A V 10 R
OF BEAM-COLUMNS
Column deflection curves form the basis for obtaining load
deflection curves forbeam~columns~5) These column deflection curves:...
give the shape that an axially loaded column would take under a given
initial slope, 8. Figure 1 ~hows a set of such curves for constanto
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"
axial load. This particular family of curves is for strong axis bending
of as-rolled wide-flange members .and includes the effect of residual
stresses. However the curves do not include the effect of local or
lateral-torsional buckling.
Figure 2 presents a family of moment-rotation curves for constant
axial load. They are derived.from curves such as Fig. 1 by finding those
column deflection curve segments which satisfy the given conditions of
(5)axial load, slenderness ratio and end-moment ratio. The slopes at the
ends of each segment are measured and this combination of moment and
rotation gives a point on the moment-rotation curve. The curves are for
various values of slenderness ratios. It is also possible to draw 8o
contours which connect column segments derived from the same column de-
flection curve (dashed curves in Fig. 2.).
The resulting M-8 curves (Fig. 2) show how both moment capacity
and rotation capacity decrease with increasing slenderness ratio.
Figure 3 illustrates the effect of various axial load values. As the
axial load increases both the moment and the rotation capacity of the beam
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205A.35 -5
columns decrease. The influence of end-restraints can be seen from Fig. 4.
Forcing the member towards single curvature deformation reduces both the
moment and rotation capacities.
It can be seen that at least some of the M-8 curves have an in-
elastic rotation range at ultimate moment, In order to measure rotation
capacity for plastic design purposes and also to give a measure of relative
curve sizes a simple definition will be proposed. The rotation capacity,
Rc ' of a beam column is defined as the ratio of two rotations:
( 1)
...
•
where 8p and 8E
are defined in Fig. 5. Both are measured at a moment
level 5% below the maximum moment. 8E is the rotation at this level during
loading and 8 p is the rotation at the same level during unloading. The
choice of 0.95 as the measuring level for R is to some extent arbitrary.c
However, examination of the experimental curves showed that at this level
rapid unloading was either in process or was immdnent .
..
205A.35
3. E F F E C T 0 FLO CAL AND L ATE R A L
TOR S ION ALB U C K LIN G
-6
•
It has been shown that a suitable collection of basic M-9 curves
are available(7) and it has been noted that these curves neglect the effect
of local and lateral-torsional buckling, These phenomena will cause devi-
ations from the available theoretical M-9 curves. If this deviation occurs
before the peak of the curve, both .moment and ro·tation capacity will be re-
duced, If it occurs after the peak, rotation capacity mayor may not be
affected. Lateral-torsional buckling may be prevented by a suitable bracing
system. However local buckling is a function of the geometry of the cross
section and will start when a certain critical strain has beenreached,(9)
3.1 LOCAL BUCKLING
Curves have been published(9) which give the critical strain for
" web or ,flange buckling as a function of the plate width-thickness ratio ....
For a given axial load and a given cross section it is possible to compute,
from equilibrium conditions, the moment Ms corresponding to the relevant
critical strain. The column deflection curves are then searched and any
curves in which the maximum moment is less than Ms are considered safe
from local buckling. In this way a critical 90 , 90c ' may be found for
each value of axial load and cross section (Fig.6). This critical 90 is
marked on M-9 curves such as Fig. 7 and represents the 90 contour at which
local buckling will occur. M-9 curves are not used beyond this contour.
205A.35 -7
This simple searching technique applies only if the maximum column
•
moment does not occur at the end of the column, that is if the maximum
column moment is also the maximum column deflection curve moment. This is
the case for a beam-column such as AA' in Fig. 8. However many columns are
similar to BB I, and have their peak moment at a joint, that is at the end
of the member. The critical e for such columns will be greater thano
that found previously and it will be a function of slenderness ratio and
end restraint in addition to the normal variables. Thus an amended curve
for local buckling may be shown on Fig. 7 (dashed line) and this will be
more liberal than the previous cut-off contour. It is obtained by a some-
what more involved search of the column deflection curves. The boundary
between the two cases occurs when the node of the column deflection curve
is at the end of the column.
a. Flange Buckling
Curves which show the critical strains at which local
buckling is assumed to occur for various bit ratios are re-
produced in Fig. 9 from Ref. 9. The solid line shows the curve
selected as governing these investigations. The maximum bit of
9.25 is slightly more liberal than that recommended for design
(l~in the Commentary on Plastic Design in Steel (bit = 8.50)
but seems more in accord with later experimental results,
especially as it is now possible to specify the precise loading
conditions.
It can be seen that there are some members (bit> 9.25)
which cannot be assumed safe against local buckling and others
(b/t~ 7.25) which are not prone to local (flange) buckling.
...
..
205A.35 -8
The following results apply to the large group of sections
falling in between, and, they give the loading at which local flange
buckling might be expected .
The values of critical 80 and axial load are determined
graphically from the column deflection curves, as previously
described. When 80 is plotted as a function of P it is found
that the section type causes little variation in the curves for
constant bit. In addition the variation between the curves for
the extreme cases of bit = 7.25 to bit = 9.25 form a small band
whose width is generally less than the presently available 80
increments. Fig. 10 shows the analytical results for the case of
equal end moments (solid lines) and also for the case where the
column is pinned at one joint and where the maximum column moment
occurs at the other joint (dashed lines).
From such curves it is now possible to predict the point
of occurrence of local buckling once the bit ratio is known.
b. Web Buckling
Calculation of the critical 80 v~lues when web bucklingI
governs follows a procedure similar to that used when flange
buckling is critical. Fig. 11 (taken from Ref. 9) gives values
of the critical'web b~ckling strains. The solid lines give th~
relationship between P/~ cry and df/w for various values of these
critical strains as measured py the maximum strain in the com-
pression flange. The dashed line shows the limit recommended in
the Plastic Design~ommentary(lO) for a section with d/df = 1.05
.,
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205A,35 -9
and A/Aw = 2.0. If the axial load and the section properties
are known the critical strain may be read directly from Fig. 11.
The column deflection curves are searched to find the
curves in which the critical conditions are first attained. The
resulting values of eoc are shown. in Fig. 12, although for clarity
the cases where the maximum moment occurs at the end of the column
have been omitted. It has also been assumed that d = 1.05 df .
It was found that eoc is not sensitive to changes in section,
provided that d/w is constant. Furthermore the curves for the
extreme cases of d/w = 36 and 53 themselves form a narrow band.
When Fig. 10 (flange buckling) and Fig. 12 (web buckling)
are compared it is seen that the values of eoc are almost identical
for the two cases. This occurrence greatly simplified local
buckling criteria as one set of curves may be used for both
cases. Fig. 13 illustrates this final result of the local buckling
analyses.
The bands enclosed by the curves A,B and C in, Fig. 13
represent the extremes of variation with section properties.
Curve A is for b/ t = 9.25 or d/w = 53, Curve B for b/t. = 8.25 or
d/w = 36, and Curve C for bit = 7.25. The dotted lines apply to
the p = 0 cases where the maximum moment occurs at the end of
the column, whereas A, Band C apply directly to the f = + 1.0 case.
For r = - LO the ~ = 0 curves may be used provided the slenderness
ratio of the column is taken as half its actual value.
205A.35 -10
For most of Fig. 13 the width of the band is less than the
P presently available(7) (1962) 80
increments. The close coincidence
between the various critical 80
curves arises from the relatively
large increases in strain which occur between column deflection
curve increments in this region. If the curves are investigated
for a hinge formation criterion a similar critical 80
range is
indicated. In addition some of the results presented later in
dicate that Fig. 13 may be a critical range, even for sections
whose bit and dlw values appear safe. This would arise from the
very large strain increases occurring in the critical region
rather than from' any limitations on the basic local buckling
•
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criteria.
From Fig. 13 cut~off lines may be drawn on the M-8 curves,
as has already been indicated in Fig. 7. When the column de
flection curve node falls within the beam-column length these
cut-offs correspond to a 80
contour.
c. Effect on Rotation Capacity
The effect.~f local buckling on rotation capacity, as de
fined in Fig~ 5, is shown in Fig. 14. The assumption that un
loading occ4rs at local buckling may sometimes be conservative as
local buckling does not always cause immediate unloading. If the
local buckling contour falls beyond the defined rotation capacity
range there is no effect at all '(Fig. 14, lower curve). If it
falls within the range (upper curve in Fig. 14) it is assumed
that the load falls off with a characteristic identical to the
205A.35
original curve, AB.
-11
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•
Rotation capacity values obtained in this manner are
shown in Fig. 19. They will be discussed later in this report.
3.2 LATERAL TORSIONAL BUCKLING EFFECT
a. Unbraced Columns
At present no general solution is available for the
determination of M-8-P curves for lateral-torsional buckling.
A first movement solution has been given for the inelastic case
with a zero moment gradient(ll) (~ = + 1.00). The solution for
an 8 WF 31 section using this method is shown in Fig. 15 as the
family of solid curves. Results for three other sections. are
also shown.
The first movement solution only gives the moment at which
the equilibrium bifurcates. It cannot provide information on the
rotation capacity. However, consideration of experimental results
has shown that a good estimate of rotation capacity can be ex-
pressed by:
R ' c(2)
"
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r
where R' = rotation capacity as amended by lateral-ctorsional buckling
ML = critical moment for lateral-torsionalbuckling ,~
~ = in-plane maximum moment
R' = in-plane rotation capacityc
205A.35 -12
To apply the theoretical solution(ll) for ML to other loading
cases,Massonnet's equation(12) for equivalent moments can be
..used. That is,
~
~ = ~j 0.3 + 0.4( + 0.3132 (3)
I
where ME is the equivalent moment.
This allows unequal end moments to be converted~o an
equivalent pair of equal and opposite end moments. If ML) ME
then lateral-torsional buckling is not critical for the particular
beam-column under consideration.
The analytical method(ll) predicts moments rather than
rotations. Hence the results can not be presented in the manner
adopted for modifying the M-8 curves for local buckling. The
rotation value results as read from M-8 curves are also ill-
conditioned for moment values near the maximum. It is an in-
herent advantage of column deflection methods that they predict
rotations and thus lead automatically to well-condi tioned, unique
solutions.
It has been shown(ll) that a significant section constant
in lateral-torsional buckling is the ratio DT where
(inch units) (4)
*Work on the same topic at Cambridge University produceda similar cross sectional constant "T" w:here T = G x AK d2/I2 ,
TG = modulus of rigidity (Ref.l3).
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205A.35
where KT St. Venant's torsional coefficient
A = cross sectional area
d = section depth
-13
Results are available (see Fig. 15) for four sections (27 WF 94,
8 WF 31, 14 WF 142, 14 WF 246) whose DT
values are 219, 925, 1580
and 3712 respectively. If DT
>1000 there are loading conditions
for the section which preclude lateral-torsional buckling.
Figure 16 gives values of ML/~ to be used in the rotation
capacity equation. As ~ is a function of L/ry rather than L/rx
it is not possible to present the results more generally.
In Figure 19, later in this report, these ratios and
Eq. (2) are used to obtain the rotation capacity values for an
8 WF 31 section (DT = 925).
b. Braced Columns
The above principles for finding rotation capacity may
be applied to the segments between lateral braces for intermit-
tently braced columns (Fig. 17). The moments at A and E are
given and hence the moments at the brace points B, C and D can
be found directly from the revelant column deflection curve.
The curve used depends on the rotation capacity required and can
be selected by considering charts such as Fig. 2, with due regard
for local buckling (Fig. 13).
As the moment at the brace points depends on the par-
ticular column deflection curve used the results of analysis of
..
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205A,35 -14
a braced column can be presented more in the manner used for
local buckling. However bracing dimensions are too variable to
present general results and so the method will be illustrated by
an example. Fig, 17 shows a braced column - it is actually Test
A8 to be discussed later - and it will be checked to see whether
it is capable of delivering the full rotation capacity with the
bracing used. Local buckling occurs within the plastic plateau
(Fig. 2) and so the critical 80
for P/Py = 0.3 will be 0,06
(Fig. 13). From this column deflection curve the bending moment
diagram shown in Fig, 17 is constructed (the moments between
braces are represented by straight lines), Next the ratio of
moments givesf3 and substitution in Eq, (3) gives ME/MM for each
segment, ~ is known for each segment so ME is found. From Fig,
15 ML is obtained for each segment. As ME/ML is greater than 1,0
for segments AB and BC the column is unstable and thus the bracing
is not sufficient to allow attainment of the available rotation
capacity.
Checking for eo = 0.05 shows that the bracing is adequate
for this degree of loading.
3~3 SUMMARY
Figure 18 shows the in-plane rotation capacities and also the
zones of influence of lateral-torsional and local buckling. The rotation
capacity values given are taken directly from the original M-e curves(7)
and do not include the effects of local or lateral-torsional buckling,
•
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205A.35
Figure 19 gives the rotation capacity results as ammended by
local and lateral-torsional buckling. The values were obtained by ap-
plying the modifications discussed beforehand to the in-plane rotation
capacities in Fig. 18. The modifications apply strictly to the 8 WF 31
section. For other shapes the appropriate modification factors may be
-15
obtained from Figs. 13 and 16. However, for most cases the 8 WF 31 will
give conservative but useable results.
It is seen that local buckling will precipitate failure in the
f3 = 0 case, if the axial loads are low. Otherwise lateral-torsional
buckling is usually critical.
a. Limits at Zero Load
The limits of Fig. 19 as axial load approaches zero
can be found by considering available beam test results. For
the f3 = + 1.0 case at zero load the curvature distribution
will be uniform (Fig. 22a). Thus the critical local buckling
strain is reached simultaneously-along th: beam. A simple
analysis shows that the expected Rc value is 11.9, if 1atera1
( 15)torsional buckling does not occur. - Beam tests indicate this
to be .so if L/ry is less than about 40. If L/ry :> 40 1atera1
torsional buckling will seriously curtail the rotation capacity.
As rx/ry can have values of the
in Fig. 19 «(3 = 1.0) will tend
magnitude of three, the curves
to qefinite 1imit~, less than~ ~.
or equal to 12. Their precise determination requires further
inves tiga tion.
For thef3 = 0 case the situation is somewhat different.
There the curvature distribution is not uniform but exhibits a
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205A,35 -16
pronounced peak which will be many times the averagecurva-
ture (Fig 22b), Visually, or by conjugate beam theory, it
can be seen that the rotation capacity as affected by local
buckling will decrease asf' decreases from 1.0 to O. Analysis
gives a value of Rc = 2 for bit = 9.25 and Rc = 4 for bit = 7.25,
at zero axial load. Tests(16) on beams with moment gradients
indicate that this tendency exists. However, there is also
an indication that the local buckling theory is conservative
and that strains in excess of the critical may occur without
local buckling. Fig. 19 gives then a lower bound to these
results and tends to a limit of Rc = 2.0 at zero axial ~oad.
~
The appa~ent peak in the Rc curves for~= 0 arises
from the geometry of the Rc definition,~nd from the more favor
able curvature distributions which res~lt from the addition of
the moments due to axial load (Fig. 22c).
;
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205A,35 ... 17
4. E X PER I MEN TAL SOU R C E S
This section presents experimental results in order to check the
•
theoretical processes discussed earlier in the report. Many tests have
been done on centrally loaded columns, but relatively few on beam-columns.
Of the reported beam-column tests only a limited number have been oriented
towards finding moment-rotation characteristics and of those tests the
rotations were often such that the recording instruments did not function
throughout the entire loading range,
This report is confined to a discussion of the "T" and "A" series.,.tests performed at Lehigh University betweeh 1948 and 1959, These tests
are reported in Ref, 14, and Table I of this report gives the test de-
tails relevant to this investigation, The M-9 curves obtained are shown
in the Appendix,
Of the 42 tests in the two series ("Til Series - unbraced;
"A" Series - braced) only 20 were completely suitable for use in this
investigation, Some of the remaining tests were discontinued before un-
loading occurred (4), some were pin~ended (5), in some the rotation
readings were unreliable (5), and in two tests the moments were applied
before the axial load. There were six tests which were discontinued
before unloading but which gave some useful rotation information and these
are also recorded together. with the twenty fully applicable test~.
"
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205A,35
Rotation capacity was measured by applying the definition of
Fig, 5 to the experimental M-8 curves, The results of this process
are shown in Table II for all tests, and Figs, 20 and 21 show fifteen
of the results graphically, Fig. 20 is for the unbraced tests with
r=s = + 1.0 and P/Py = 0,12. It can be seen that good agreement is ob
tained between the test results and the predicted curve.
Fig. 21 is for the braced ~~'I series tests, In this case the
correlation between theory and experiment is not as consistent as it
was for thef3 = 0 case, The variations between theory and experiment
will be discussed in the next section, however, Figs, 20 and 21 show
that a reasonable estimate of rotation capacity can be expected from
the preceding theory,
-18
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205A.35
5. C OM PAR I S O.N BET WEE N E X PER I M E NT A L
AND THE 0 RET I C A LR E S U L T S
5.1 GENERAL
-19
Table II records the predicted and experimental rotation capacity
results and also the difference between these two quantitites.The pre-
dicted rotation capacity includes the effects of lateral-torsional
buckling and local buckling where these are applicable, For the tests
carried to completion this difference (theory-experiment) had an average
value of - 0,06 and a range from - 1.10 to + 0.70. The majority of the
results were conservative and the most serious overestimate was + 0,70
for test A4 (See Fig. AS), This overestimate arose partly from the
experimental maximum moment being M/M = 0.60 as compared with thep
theoretical value of M/Mp = 0,55, Hence the rotation capacity was measured
at M/Mp = 0.57 which was still 3-1/2% above the theoretical moment. The
discrepancy can thus be traced to the definition of Rc which was chosen
in order to account for the more critical case of the predicted moment
being greater than the actual moment,
5,2 LOCAL BUCKLING
Table II indicates those tests for which an 8 WF 31 analysis
predicted a local buckling failure mode, Failure by local buckling,was
observed in six of the ten tests. The four remaining tests were all
4 WF 13 sections with high slenderness ratios, but it is noted that three
205A.35 -20
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4 WF 13 sections with lower slenderness ratios were amongst the six which
failed by local buckling. The 4 WF 13 is a section for which bit = 6.04
and from Fig. 9 local buckling is not expected to occur under any circum-
stances, however it is also a section which is relatively strong torsionally
and in short members local buckling is likely when the strains become
large and are not relieved by lateral-torsional buckling. In thi~; case
the limits of the Haaijer analysis(9) are exceeded and it is desirable to
base all calculations on the 8 WF 31 section although this will sometimes
'lead to conservative results.
In tests A9 and Tl3 local puckling was observed when theoretically
predicted but unloading did not occur until late~ in the test. In test
A9 the end rotation at unloading was twice itsva~ue when local buckling
was first observed and this would account for the large underestimate of
rotation capacity for this test. The inability of local buckling to always
cause immediate unloading will lead to conservative results, on the other
hand it is not considered desirable at this stage to utilize members in
which some local buckling has occurred.
From the limited samples available it would appear that the theory
predicts both the point of occurrence and the effect of local buckling,, ..
but is likely to be conservative,in both estimate~o
5.3 LATERAL-TORSIONAL BUCKLING
Ten completed tests failed by lateral-torsional buckling (Table II)
a~di~ nine of these this failure mode had been predicted. The method
also appears to give a good estimate of rotation capacity in all(~ ranges.
205A.35
5.4 BRACED COLUMNS
-21
The out-of-plane slenderness ratio L between brace points inr y
the "A" Series tests was about 40 (Table I). Lateral movements were ob-
s~rved in a number of tests, but lateral-torsional buckling was only con-
sidered as the failure mode in test A8. ~he three tests on the torsionally
strong 4 WF 13 (AS, A6 and A7) had no predicted out-of-plane buckling and
their results agreed well with those predicted by the basic M-8 curves
(Table II). Of the six remaining tests three (A2, A3 and A4) were 8 WF 31
tests and some reduction in rotation capacity was expected and~:observed..
In the three tests on the torsionally weak 8 B 13 (A8, A9 and A10) , large
rotation capacity reductions occurred as predicted. Local buckling
•occurred in test A9, this phenomenon being independent of the bracing
spacing, It would appear that for torsionally weak sections the
bracing requirements for attainment of full rotation capacity could be-
."come excessive,
5.5 MOMENT-ROTATION CURVES
The accuracy of prediction of rotation capacity is also a measure
of the accuracy of the prediction of the general M-8 curve shape. In
the "A" Series braced tests (Table II) AS, A6 and A7 had no predicted or
observed lateral or local buckling effects, hence they should best fit
the basic M-8 curves derived by OjalvoP). Fig, A6 in the Appendix~
shows this to be the case and these tests are strong confirmation of the"
Ojalvo theory. Tests AS and A6 show particularly good correlation. In
addition these are 4 WF 13 tests whereas the column deflection curves
were derived for the 8 WF 31. For tests A2, A3 and A4 the correlation
..
205A.35 .... 22
is still good although the O)alvo curves appear conservative. As ex-
pected the results for the tests (A8, A9 and A10) on the torsionally
weak 8 B 13 do not show good agreement, due to out-of-plane action.
However for the "T" Series tests there are some marked. deviations
between the test results and portions of the in-plane M-8curves (Figs.
A.l - A.4). These arise mainly from (1) differences in maximum moment,
'.and (2) differences in rotation capacity and indicate that the basic
,i:l '
curves should only be used where in-plane, local buckling free, behavior',,'
is assured.
5.6 ROTATION CAPACITY VALUES
It has .been shown that for local buckling the results based on
the 8 WF 31 may be used for all sections. For lateral-torsional buckling
the same applie,s if the section has a similar DT value (925) or if it is
stronger and greater refinements are not required. In these cases
Fig. 19 gives Rc values directly. Otherwise Eq. 2 is evaluated and used
in conjunction with Fig. 18. (Note that L/ry may also vary.)
Attention is drawn to the magnitude of both the theoretical and
experimental Rc values. For the ~w¢pty.completed tests in Table II the
average experimental,value was only 1.62. Theoretically, it can be seen
from Fig. 19 that only columns with low slenderness ratio and axial loads
can be expected to provide Rc :> 3.0.
•
205A.35
There are many columns which do not fall into these categories .
Highly loaded columns in tall buildings or slender columns in low
buildings must be analyzed carefully if they are required to provide
some rotation capacity during the course of their structural behavior.
Finally it is noted that rotation capacities would not normally
be required to an accuracy of greater than 0.25.
~23
•
205A.35 -24
6. CON C L U S ION S
This report has presented an analysis of the presently available
'0
moment-rotation curves for beam-columns. Certain modifications have been
proposed and the basic theory and the modifications have been checked
against experimental results. The following conclusions may be stated:
a. The methods formulated for the prediction of local and
lateral-torsional buckling and their effect on rotation capacity
give results which appear to be reasonably consistent when ap-
plied to test results.
b. As the work in this report is largely based on the
( 5)theory presented by Ojalvo ,possibly the most interesting
outcome of the experimental analyses is that they offer con-
firmation of Ojalvo's theory and do not disclose any incon-
sistencies. When the assumptions of in-plane behavior are
fulfilled the agreement is excellent. When local and lateral-
torsional buckling occur there appears justification for using
the approximate modifications proposed in this report to predict
rotation capacities.
c. A method has been presented which gives an interaction
curve between axial load and column deflection curve for local
buckling. It has been shown that one curve provides a suffiently
accurate prediction of flange and web buckling and that the be-
..
..
•
205A.35 ...·25
havior of all sections may be reduced to a few simple cases •
In this form the criteria may be applied to any column loading
case and presents a limit on the usable length of the moment
rotation curve. The estimates of rotation capacity so obtained
are sometimes conservative as local buckling is frequently not
catastrophic.
d. A less complete solution is presented for lateral-
torsional buckling, and more refined methods of analysis are
required for the final solution of this problem.
e. Most columns appear to have a.definite rotation capacity
although in many cases it is not very great. Because of this it
seems necessary .to check the rotation requirements of hinges
forming in the columns of plastically designed frames. If the
requirement exceeds the capacity predicted by this report it
would seem advisable to either use a stronger member or to carry
out the design by·one of the stability methods now available.
f. For the spacing of lateral bracing of beam-columns it
has been shown that the column deflection curve may be used to
obtain the actual in-plane moments at a brace. Hence design
checks may proceed more logically. The bracing requirements
presently used(.lO) appeared adequate although care must be taken
if the member is torsionally weak, as' it is unlikely that the
full rotation requirement will be attained.
•
-0
.'
205A.35 -26
'7. ACKNOWLEDGMENTS
This study is part of a general investigation
''Welded Continuous Frames and Their Components" currently
being carried out at Fritz Engineering Laboratory of the
Civil Engineering Department of Lehigh University under
the general direction of Lynn S. Beedle. The investigation
is sponsored jointly by the Welding Re~earch Council, and
the Department of the Navy, with funds furnished by the
American Institute of Steel Construction, the American Iron
and Steel Institute, Lehigh University Institute of Research,
the Bureau of Ships, and the Bureau of Yards and Docks. The
Column Research Council acts in an advisory capacity.
The authors express their thanks to all those whose
experimental and theoretical work has been drawn upon and used
in this report, and to Mrs. Dorothy Fielding who did the typing
and Mr. Richard Sopko for his assistance with the drawings.
•
-~
205A035 -27
80 NOM E N C L A T U R-E
2b Flange width of section
d Depth of Section
df Distance between flange centroids
r x Strong acis radius of gyration
r y Weak axis radius of gyration
t Flange thickness
w Web thickness
A Cross-sectional area
Aw Web area
DT Lateral-torsional buckling coefficient, Eq. (4)
L/r Slenderness ratio
M End moment
ME Equivalent moment, Eqo (3)
ML Lateral-torsional buckling moment
MM Maximum moment
MS
Local buckling critical moment
~ Moment at first yield
P Axial Load
P = A (JY Y
Rotation capacity
R'c
Rotation capacity modified for lateral-torsional buckling
Ratio of enp moments on a column, the larger moment inthe denominator and the moments measured in oppositedirections.
205A.35
• Em
cry9
90
90c
:Jarc
0y
•
•
Maximum strain in compression flange
Yield strain
End rotation
Column deflection curve slope at zero deflection
Critical column deflection curve for local buckling
Abscissae of M-9 curve defined in Fig. 5
Residual stress parameter
Yield stress
... 28
·' .TABLE I
TEST DATA (FROM REF .14 )
!l-Force Ratio Bracing Spacing
Test L/rx Section Value P/Py . (Measured From Maximum Moment End)
T-7 111 4 WF 13 - 0.56 0.26T-12 55 8 WF 31 + 1,00 0.12T-16 41 8 WF 31 + 1.00 0.12T-17
CIl.j.J 56 4 WF 13 - 0.50 0.12
T-19CIl
8 WF 31Q) 28 + 1.00 0012E-l
T-20"C
56 4 WF 13 + 1 0 00 0,12 NoT-21 Q) 56 4 ToJF 13 - 0,55 0,47 Bracing
(J
T-23 til 83 4tIF 13 0 0.111-1
T-26 .0 84 4WF 13 + 1.00 0.12TM 31 § 112 4 WF 13 0 0.12T-32 112 4 WF 13 + 1.00 0.12
A-2 55 8 WF 31 0 0.65 71" .·..·12.1" (one brace)A-3 55 8 WF 31 0 0.32 67.5" - 124.5" (one brace)A-4 CIl 55 8 WF 31 0 0.49 67" - 125" (one brace)
.j.J
A-5 CIl 110 4 WF 13 0 0.33 36'1 37" - 119 " (two braces)Q)
A-6 E-l 112 4WF 13 0 0.50 36" 37" - 119" (two braces)A-7 "C 112 4 WF 13 0 0,16 36" 37" - 119" (two braces)
Q)
A-8 (J 52 8 B 13 0 0,30 30" . 34" 36" - 67.5" (three braces)til
A-9 1-1 52 8 B 13 0 0.12 30 " 30" 38" - 70" (three braces)l:l:l
A-10 52 8 B 13 0 0.60 30" 30" 38" - 70" (three braces)
T-4 "C 55 ti tJF 31 - 0.50 0.12Q)
T-9 ::l 111 4 WF 13 - 0.56 0.10t:: CIl
T-13 .~ .j.J 55 8 WF 31 0 0.12 No.j.J CIl
T-24 t:: Q) 83 4 WF 13 - 0.52 0.12 BracingoE-l
T-29 (J 84 4 WF 13 - 1.00 0.13CIl
T-30 ~ 112 4 WF 13 - 1.00 0,120
Section COilstants:
·Dor (in) rx/ry
4WF 13 2360 1. 73 cN
8WF 31 925 1. 73 \0
8 B 13 364 2.75
•
•
zo...(,)IJJ...J~IJJo
p
.IP is constant I
COLUMN LENGTH
Fig. 1 COLUMN DEFLECTION CURVES
"
p
-31
.. .. •
IWN
0.130.120.11
p .'P =0.3.
yO'"RC =O. 3 O"yoy =33 ksiStrong Axis Bending
0.10
I/
I/
I
0.09
II
II
I
0.07 0.08
ROTATION
IIIII,I I, MI~~/.~~jZ1:;:::::::Ie==k~
II
! -I-III,
I
,I,,
II .III
0.05 0.06
8. END
0.040.02' 0.03
II,,
O.o
0.1
0.7
tr:lZ? 0.5~ Mi~M~ . y 0.4~t:::l·1
~~t-3Ho·zC":l
,~ 02<: " •tr:ltJ)
•
1.0
MM O.5y
P=0.4Py
-33
P =0.12 P.
P=0.20P.
13= 0L/r=60
-I-P=0.3Py '
MP ,p_,0-0----0-+
• 0.048
0.08 0.12
1.0
Fig. 3 INFLUENCE OF AXIAL LOAD ON M-~ CURVES
13 =+1.0
0.04
8
{3=0
0.08
13 = -1.0
P= 0.3ACYStrong AxisL/r= 60
'-:I-, ,13M M
P+-0--0---o-+p
0.12
Fig. 4 INFLUENCE OF END RESTRAINT, ON M-e CURVES
-34
Fig. 5
MAXIMUM MOMENT MMMM t--.L-------------===-=---__
ENDMOMENT
M
DEFLECTION
C.RITICAL 80 CONTOUR
p COLUMN LENGTHFig. 6 CRITICAL COLUMN .DEFLECTION CURVE·
p,
,
F[f,
0.8
MM OA- ~ ,y
o 0.048, END ROTATION
L/r=40
---I-,~AMENDEDI CUT-OFF·
/I ,
II
I
M-8CURVES
-35
Fig. 7 CRITICAL 9 CONTOURo .
DEFLECTION =~
BSI. AAI
•
1,!\
p - -p
t~L/rx MAXIMUM MOMENTS
Fig. 8 LOCATION OF. COLUMN SEGMENTS
-36
40
30
20
"P=oP=O.OI
EQ.. _
~= Coefficient of Restraint
10
Ef ..0---------·'o 5 10
bIt.15 20 25
Fig. 9 FLAN.GE LOCAL BUCKLING
-- --..
~=o~, .----~ =7.~5.-----~~.....~. -- --
........ - --., --
~=9.25--~
Solid Lines:~ =·+1.0
0.6
0.2
P/~ 0.4
•
. 0.02 0.04. 0.06CRITICAL 80
0.08 0.10
Fig. 10 CRITICAL 90 ,VALUES FOR FLANGE LOCALBUCKLING
. -,37
~.
PAway
0.75
0.50
0.25
o30 40
AA =2.0w
1;-=1.05
50 60
ALLOWABLE WEB DIMENSIONS
•
0.60..>-
Q:0
Ii~ 0.4
0«~~w0
...J.
~ 0.2X«•
.- ',.'
'.0.02 0.04 0.06
CRITICAL 80
0.08 0.10
Fig. 12 CRITICAL 80 VALUESFQR WEB LOCAL BUCKLING
ABC
-38
~ =9.25curveA<d
w= 53
~ =8.25curveB<d .
-=36w
CurveC-~=7.25
0.60
050
0.40
030
0.20
0.10
~= +1.0
Values For~=O
t~~
........." .......... ""-.........; """ L/r=30~ ......................~~ .......................
~........... ..................... " L/r=40 "
""~"" ........... ,~ ........... ~/r=50 .......................
.......... "" .................. L/r=60 .............................. ..
.................................
•
0.04 0.05 0.06
80
0.07 0.08 0.09
Fig. 13 CRITICAL COLUMN DEFLECTION. CURVE FOR
FLANGE AND WEB LOCAL BUCKLING
I.
..
M-My
. .' . .
ROTATION CAPICITY AMEN EDFOR LOCAL BUCKbJ.NG
•. '~~:',;...:~, ~_.~.;,; .:.".•:..;.~ ", 0;. ".
-...:"?:......\:~: .. "'-"•..,.':,-' '.,
IN-PLANE ROTATI NCAPACITIES
\i80e
LOCALBUCKLINGCONTOUR
END ROTATION, 9
. Fig. 14:: EF_F~CT OF LOCAL BUCKLING ON ROTATION CAPACI TY
-39
-40
•••••• DT:: 3712
-. -e_- DT:: 1580
---- DT:: 925
219
300
•..··......
.....
250
----DT =
-1-{3=+ID
....." ..... ..'.\.
\\
\.\\ .\ :
•\ :.\ :\ :'· .
. \ :\ l\l\i.
,\\\ :\ :.\ .\\ .\ :\ .
..........................
100 150· 200
SLENDERNESS RATIO,L/ry
Fig. 15 INTERACTION CURVES FOR LATERAL-TORSIONAL
50
'.\.
\.\
"
"
"
".,""- '.,
'.
\\
\ ,,",,
\ '. ,\ ,.\ ". .,, .... ,
" .. '\ •• P
\ .•'P."-. .y\., ..
. ':e.\...
~<
\\
",",,. ,' ..\ ~
\ .". "\ \..". ,\ \.
\. , .." . : P, , . -=0.3
" \. Py" .." .\. ." .~.\.: ,'~.,: , .
\ :£'=0.5',..py "\. ,.... \
\,,,t,
o
0.2
.0.1
0.3
0.5
0.7
0.6
0.8
MMp
0.4
..
BUCKLING
-41
L= 50ry1.0 •.••.•••••.••••.••.••••• e ....
0.60.50.40.3
f.Py
·I·1 r• III .U'io· ...I 0<·
0.20.1
.'., --".---' ....' ;"'t-~__.. ....
\; ---" \:" .." ' .... ':..... ........... .... ............. i ....._-
~ . -.....r:, '~~ I'00 i ',<"'"'. , ~~ I \ 0
• \11"I J-
\,,,\\\\\\
,,,.,,. ,(,, .~
\ 0,q'J-
\\\\\
0.4
0.2 L--__.....I...__....L......&.-.......__L--_---L-.LL. ...L- I.-
o
0.3
0.5
0.8
0.6
0.7
.... ...l -.... ~ :~~?ry LIFIOOry
09 .'lo.. ." •••• L=50ry •••I .. ' • •• •., _. _._._._....... e.. --.-- .. .'. ..'. ..- -......,.. , ....._._ : e. • •... ......-._.-._. , ..... .........~ .'... ., ..
~ : L 200 ,. .'.•• : = ry'\ ,
L= 50ry
•
..
•
Fig. 16 REDUCTION FACTOR (MdMM) TO ALLOW FOR
LATERAL-TORSIONAL BUCKLING
-42
75.6
35.3
,I
19.8
9.25
PIACTy=0.30(1~ DI~NS
A SM=0.866My3d'
BIoo--'---.--..L......f-+-+----L..-+-+-+-----1--+-t-+--, 64.25"
TestA8
C~...-___Ir___-----II"-+-_+_----t-----+__+_---~t--+---100.5" 31.0 118.1
.b...=197ryL=168"
BRACINGPOINTS
D~04-_____jr___-----L.._+_-------I.__+_----&_._+--
8813
Et----------L....--------------L--P/Auy=0.30
C0.622,<PR
1T
0.610' <PbAL
0.0 0.548 0.44 0.24 0.54
0.710 0.857- 0.62 0.53 .605
BENDINGMOMENT'DIAGRA
o
M=0.866My
80=0.06 0.918 0.9600.8660.832
.....--10.795My
0.7800.891 0.795 0.710
E
B
8BI3
'..
•
Fig 0 17 , CALCULATIONS, FOR A BRACED COLUMN
-43
L/r=60
L/r=4b
L/r=50
_ - 0-,-..-".
,,/ 'LOCAL BUCKLINGZONE.
L/r=80/L/r=120 L/r= 100
0.2
113 =+ 1.0 I
4.0 6.0 . 8.0Rc ,ROTATION CAPACITY•
0.6
2.0
L/r=80L/r=60
t--+---l~~r-' L/r =40
L/r=30~-~../ L/r=20
LOCAL./4-BUCKLING
·10.0
2.0 4.0 6.0 8.0ROTATION CAPACITY
10.0
Fig. 18 IN-PLANE ROTATION CAPACITIES ( = 0, = + 1.0)
-44
•
02
,, I
II
., 1.0 2.0 3.0 ~ 4.0ROTATION CAPACITY, Rc
5.0
I~=+1.010.6
~,
•1.0 2.0 3.0 4.0
ROTATION CAPACITY, Rc
5.0
Fig. 19 AMENDED ROTATION CAPACITIES <(S= 0, ~ = + LO)
u
a:: 3.0>-....C3~~.02.0
zo~6a:: 1.0
THEORETICALCURVE
{3=+1.0PP =0.12y
CompletedTests
o Test Results
-45
•
20 . 40 60 80 '100 120
SLENDERNESS RATIOFig. 20 RES ULTS FOR UNBRACED TES TS (~= + 10 0)
•
'.
4.0
ua:: 3.0>-'....a~~02.0zo'~
~oa:: 1.0 4YFt3 '2 BRACE;S
8BI3
3 BRACES
~VF31
I-BRACE
0.1 0.2 0.3 0.4 0.5 0.6
P/PyFig. 21 RESULTS FOR BRACED TESTS (~= 0)
•
¢CRIT ,---------------., ¢ C~IT
CURVATURE DIAGRAM
-46
CURVATURE DIAGRAM
( b)
•
¢CRIT
CURVATURE DIAGRAM
M
P~~=========~---p(C)
.. Fig. 22 CURVATURE DISTRIBUTIONS FOR VARIOUS
LO~ING CASES
••
.•
•
205A.35
APPENDIX
. EXPERIMENTAL AND THEORETICAL
MOMENT-ROTATION CURVES
(Also see Table I)
-47
-48
T32 TEST
~ T32 THEORY~
:-..
Theoretical Curves Are .For In- Plane Behavior
T 7 ;L/rx=1I1 ;~=-0.56 P/~=0.26
TI2 ; L/rx=55;~=+ 1.00 P/~=O.l2
T32;L/rx=1I2 ;~=+ 1.00 P/Py=O.l2
--........." ........./ "'"
/ "/ "
I "I ,I "-
/ "T12 THEORY
IIIIIIIIIIII
.IIII
IIIIIII
0.8
0.1
O.
0.5
0.7
0.2
0.3
·0.4
1!lM ..
p•
•..
0.02 0.04 0.06
8, END ROTATION
0.08 0.10
Fig. Al TEST RESULTS (T SERIES)
'..
-49
T26,TEST
Theoretical Curves AreFor 1n - Plane Behavior
T16; L/rx=41;~=+1.0 P/Py=O.l2T17; Llrx=56;~=-0.50 P/Py=O.l2T26; L/rx=84;~=11.00 P/Py=0.12
~,,,,"
\ \ T26,THEORY\
\\\
--- ------------------"..... TI7,TEST T17, THEORY
/ _.::..:---I ~....." .......
I / ....... ,I
/ ,I "I I "
I I "I "I I '}16, THEORY
I II II II II I
IIII
0.1
0.6
0.9
0.5
0.7
0.8
0.4
MMp•
'.
•, 0.02 0.04 0.06 '0;08
8, END ROTATION
0.10
•
Fig. A2TESTRESULTS (TSERIES)
"",.... .----.. ........ -- ----------0.9 ,.. -""'-Y.'/ T31 --" --/ / .........THEORY. / / ..........
I / ....................I .......I I
0.8 I II T31I TESTI,
0.7 ,I
. l Tl90.6 I TEST
IM IMp I
I0.5 t -----I /' --• I / T21 ---.
l / THEORYI I
0.4 III1
0.3 Theoretical Curves AreFor In- Plan.e Behavior
T19; L/rx=28; ~=+ 1.0 P/Py=O.l2
0.2T21; L/rx=56 ;~= -0.55P/Py=0.47T3t· L/r =112' ~ == 0 PIP. =012, x, y •
•
-50
•
"0.02 0.04 0.068, END ROTATION
0.08 0.10
.Fig. A3 TEST RESULTS (T SERIES)
-51
•;., .
•
0.02
Fig. A4
0.04 0.06 0.08
8 I END ROTATION .
TEST RESULTS (T SER~ES)
0.10
• • .. ..
0.090.080.07
LocalBuckling
Theory -----Experiment 0--0----0
0.06
-I-8'*"31
Strong Axis BendingL/rx=55p=o
. TEST A-3
P/Py =0.326
--- ------------- --- ....... -.... --.....' .... ,
......
""'\\
\\\\\\\\\
0.03 0.04 0.05
8 t END ROTATION
0.02
-,........
"\
\
TEST A-2 \P/Py =0.647.. \
0.01o
0.1
0.2
0.3
0.7
0.8
MMp
0.4
Fig. AS TEST RESULTS (A SERIES) IVIN·
- .53
0.7
- --- - - - - Theory _::1:_0---0----0 Experiment :.z.l:
0.6
4VFI3Strong Axis Bending
~=O
0.070.06
TESTA7
P/Py=O.l6L/rx=1I2
----------
0.050.03 0.04
9, END ROTATION
TEST A6
P/Py =0.502
L/rx=112
0.02
TEST A5
P/Py=0.33
L/rx=1I1
"...--,,..."" ,... \\\\\\
0.01
0.3
0.1
0.4
0.2
0.5
MoMp
,.
,
••
Fig. A6TEST RESULTS (A SERIES)
-54
TEST 'A-9P/~=01l20
"
-----.....,'",\
\\\\\\\
TEST A-8P/Py =0.300
Local Buckling,a:::.Jb-o:=:-o:-:-~::o=..:~~~ - - - - - - - -
- -- - ----Theoryo 0 0 Experiment
1.0
0.7
0.9
0.8
0.5
II
0.4II
~I-·I ~
III 8BI3
0.3 I Strong AxisI
TEST A-IO BendingI
P/Py =0.600 . L/r. =52 ~=Ox ,
0.2
0.1
• o 0.02 004 006 008
8, END ROTATION
0.10
Fig. A7 ,TEST RESULTS (A SERIES)'
205A.35
REF ERE N.C E S
1. Austin, W. J.STRENGTH AND DESIGN OF METAL BEAM COLUMNS,ASCE Proceedings, 87 (ST-4), April 1961
2. Driscoll, G. C., Jr.ROTATION CAPACITY REQUIREMENTS FOR BEAMS AND PORTAL FRAMES,Ph,D. Dissertation, Lehigh University, 1958
3. Ojalvo, M, and Lu, L.ANALYSIS OF FRAMES LOADED IN. THE PLASTIC RANGE,ASCE Proceedings, 87 (EM-4), August 1961
-55
4. Ojalvo, eM. and Levi, V.COLUMN,DESIGN IN CONTINUOUS STRUCTURES,Fritz Laboratory Report No, 278.4, July 1961
, 5. Ojalvo, M.RESTRAINED COLUMNS,ASCE Proceedings, 86 (EM-5), October 1960
6. Bleich, F.BUCKLING STRENGTH OF METAL STRUCTURES,McGraw-Hill, 1952
7. Ojalvo, M. and Fukumoto, Y.NOMOGRAPHS FOR THE SOLUTION OF BEAM-COLUMN PROBLEMS,Fritz Laboratory Report No, 278.5, July 1961
8. Ketter, R., Kaminsky, E. and Beedle, L. S.PLASTIC DEFORMATION OF WIDE-FLANGE BEAM-COLUMNS,ASCE Transactions, Vol. 120, 1955
"9. Haaijer, G. and Thurlimann, B,ON INELASTIC BUCKLING IN STEEL,ASCE Proceedings, 84 (EM-2), April 1958
10. ASCECOMMENTARY ON PLASTIC DESIGN IN STEEL,ASCE Manual No. 41
11. Galambos, T, V.INELASTIC LATERAL-TORSIONAL BUCKLING OF ECCENTRICALLYLOADED WF COLUMNS,Ph.D. Dissertation, Lehigh University, 1959
REF. ERE N C·E S (continued)
12. CRCGUIDE TO DESIGN CRITERIA FOR METAL COMPRESSION MEMBERS~
. CRC, 1960
13. Horne, M. R.THE STANCHION PROBLEM IN CONTlNUOUS STRUCTURES DESIGNEDBY THE PLASTIC THEORY,B. W. R. A. Report FE 1/42
14. Van Kuren, C. and Galambos, T. V.BEAM-COLUMN EXPERIMENTS,Fritz Laboratory Report No. 205A.30, July 1961
15. Lee, G. C. and Galambos, T. V.POST-BUCKLING STRENGTH OF WIDE-FLANGE BEAMS,Proc. ASCE, Vol. 88 (EM1) , February 1962, p. 59
-56
.,
16, Kusuda, T" Sarubbi, R. and Thur1imann, B.THE SPACING OF LATERAL BRACING IN PLASTIC DESIGN,Fritz Laboratory Report No. 205E.11
DATE DUE
\.x.\\\ \iH u1~1VfRSl/fJDEC 9 1984
lL...i8RA~/. ~~
·f
\,
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