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Fritz Engineering Laboratory Report No. 33181
Space Frames with Biaxial Loading in Columns
. by
Wai F. ChenSakda Santathadaporn
FRITZ EN(~INEER'NG
LABORATORY LIBRARY
RECENT DEVELOPMENTS IN THE STUDYOF COLUMN BEHAVIOR
UNDER BIAXIAL LOADING
Space Frames with Biaxial Loading in Columns
RECENT DEVELOPMENTS IN THE STUDY OF
COLUMN BEHAVIOR UNDER BIAXIAL LOADING
by
Wai F. Chen·
Sakda Santathadaporn
This work has been carried out as a part ofan investigation sponsored jointly by the Welding Research Council and the Department of theNavy with funds furnished by the following:
American Iron and Steel. InstituteNaval Ship Systems Command
Naval Facilities Engineering Command
Reproduction of this report in whole or inpart is permitted for any purpose of the UnitedStates Government.
Fritz Engineering LaboratoryDepartment of Civil Engineering
Lehigh UniversityBethlehem, Pennsylvania
April 1968
Fritz Engineering Laboratory Report No. 331.1
331.1
ABSTRACT
i
Columns in a multi-story building framework are usually designed
to resist bending moments acting in the plane of the frame. An actual
column, however, is frequently subjected to bending moments acting in
two perpendicular directions in addition to an axial load. Except in
Great Britain, no such considerations are now included in column design.
It is the purpose of this paper to survey the present stand of the
problem of columns under biaxial loading. The survey deals with two
equally important aspects of progress: New solutions emerging from the
analytical study and experimental results reported from laboratory
tests, with emphasis placed on the fundamental concepts advanced by
various investigators.
Three typical solutions ranging from an almost exact to a rather
crude approximation have been selected for a rather detailed discussion.
Comparison of the analytical work with test results is reviewed.
Directions of further study are indicated.
331.1
ABSTRACT
TABLE OF CONTENTS
ii
page
i
1. INTRODUCTION 1
2. HISTORICAL DEVELOPMENTS 4
3. BASIC CONCEPTS 12
4. BIAXIALLY LOADED COLUMNS OF H SECTION 16
5 . APPROXIMATE SOLUTION OF THE BIAXIALLY LOADED COLUMN PROBLEM 19
6 . THE INTERACTION ~URFACE RELATING P, M AND M 22x y
7 . TESTS ON BIAXIALLY LOADED COLUMNS 25
8. FUTURE DEVELOPMENTS OF THEORY OF BIAXIALLY LOADED COLUMNS 28
9. CONCLUSION 30
10 . ACKNOWLEDGMENTS 31
11. FIGURES 32
12 . NOTATION 46
13. REFERENCES 47
331.1 -1
1. INTRODUCTION
A three-dimensional space structure is often tr~ated as a
collection of two-dimensional planar structures; that is, structures
with all their members lying in a single plane and with all the
loads applied in the same plane. This procedure is equivalent to
setting a number of secondary interaction bending moments and tor
ques equal to zero. The columns in a planar frame are therefore
designed to resist bending moments acting in the plane of the frame.
While this idealization has resulted in satisfactory designs
in the past, it does not necessarily represent the true loading
condition existing in a space structure and may not give th~ op
timum design. In an actual building framework, the columns are
frequently subjected to bending moments acting in two perpendicular
directions in addition to an axial load, (commonly called "biaxial
loading Jl). The biaxial moments may resul-t from the space actio.n
of the entire framing system (Fig. la), or from an axial load
biaxial1y located with respect to the principal axes of the column
cross section (Fig. Ib).
Although the loading conditions shown in Fig. la Bnd Fig. Ib
are statically equivalent to each other and often are considered
identical in terms of stress resultants, the inelastic behavior of
these two columns may be quite different, depending on the detailed
loading program acting at the ends of the column in Fig. 1a.
331.1
Plastic action is load path dependent and usually requires
step-by-step calculations that follow the history of loading. If
P, M and M increase proportionally (or sometimes termed radialx y
loading), the inelastic behavior of the column in Fig. 1a is equiva-
lent to that of Fig. lb. The term radial comes from the plot of
1*the loading path in a generalized stress space as shown dia-
grammatically in Fig. 2 [see path (i)]. Different loading paths
for Fig. 1a are also shown in the figure. In the path O-B-A as
marked by (ii) and (iii), the column is first loaded axially to
point B; and then the axial load P is held constant while the column
is loaded to failure by two end moments M and M which increasex y
proportional in magnitude from zero. In the path O-B-C-A as ffi8rked
by (ii), (iv) and (v), the column is first loaded axially up to
-2
point B and then bent by M to C while keeping P constant and finallyy
bent by M to failure while keeping P and M constant. Loading pathx y
O-D-A can be interpreted in 8 similar manner and was found especially
convenient for the experimental investigations of an elastically
restrained column which is also restrained against sway. (See
Fig. 15 and will be discussed in Section 7)~
There is only one solution reported for biaxially loaded columns
. 2with a loading program other than the proportional loading. Most
solutions are limited to isolated columns with the eccentric loading
shown in Fig. lb. The discussion in this paper will also be con-
cerned with this particular case.
* Superscript is used to denote reference number.
331~1
The purpose of this paper is to survey the present status of
elastic-plastic behavior of columns under biaxial loading. A com
plete account of recent developments in this area must necessarily
deal with two equally important aspects: New solutions emerging
from analytical studies of columns under biaxial loading and ex
perimental results reported from laboratory testing. In this
paper, emphasis is placed on the fundamental concepts emerging
from discussions of this problem by various investigators.
-3
331.1 -4
2. HISTORICAL DEVELOPM:ENTS
Solutions that describe the elastic behavior of columns with
various end conditions are the most highly developed aspect of
column research. These theories and important solutions are given
3 to 13in several texts as well as papers. Here the names of
Timoshenko and Gere, Bleich, Vlasov, Goodier and Johnston, among
others, can be mentioned, but their work is also comparatively
14recent. The contribution made by Wagner in 1929 for the torsional
buckling of thin-walled open sections to the currently established
solutions for torsion and flexure buckling gives an indication of
the rapid progress. in this area.
Analytical studies on elastic columns with a thin-walled open
section loaded biaxially with re"spect to the principal axes of the
column cross section are very extensive. Goodier,9,lO,11 following
the related work on flexural-torsional buckling of Wagner,14 Pret-
15 16schner and Kappus, extended the governing differential equations
to include columns under biaxial bending with identical loading
conditions at each end. Goodier's equations are simplified 'by the
assumption that the twisting as well as the displacements of any
cross section of the column are small compared to the eccentricities
of the loading. Excellent discussions of the theory are given by
Bleich,6 Timoshenko and Gere4
and Ko11brunner and Meister. l7
Goodier's simplified equations have been solved exactly by
C 1 18 , 19 d · I b ThO. 1· 20 D b k· 21 du ver an approx~mate y y ur 1mann, a rows ~ an
22Prawel and Lee.
331.1
Recently, Birnstiel, Harstead and Leu23
have reported that
Goodier's equations are not applicable at larger loads to elastic
problems such as those selected by Culver. This is due to the
fact that as the value of rotation of the column cross section
becomes larger the error in Goodier's approximation becomes con-
siderable.
Analytical studies of inelastic columns loaded eccentrically
in a plane, of symmetry have been investigated thoroughly and well
understood since Von K~rm~n's efforts in the early part of this
24 25 .century.' References to the earl~er work of the development
are given by Bleich. 6 An up-to-date summary of the research dealing
-5
with the inelastic behavior of wide-flange columns that are sufficiently
braced in the lateral direction will appear in the revised Commentary
1 · .. S 1 26 'on P ast1c Des~gn 1n tee '.
The inelastic behavior of beam-columns bent out of the plane
of the applied moment and twist at the same time is generally too
.complicated for detailed solutions even for the case of a double
symmetrical section. Galambos defined failure of th'e column as
the load at which the column begins to deflect laterally accompanied
by twisting. 27 This criterion does enable solutions to be obtained
for the critical load. But the resulting value'is a conservative
estimate of the ultimate load. Nevertheless, good correlation of
this estimate with test results was observed.
The inelastic behavior of columns under biaxial loading has
not been studied thoroughly, but a number of solutions have been
Kloppel
obtained. Pinadzhyan studied the ultimate load carrying capacity of
1 " h H h d "d b" "l 1 dO 28co umns w~t -s ape cross sect~on un er ~ax~a oa 1ng.
and Winkelmann have conducted experimental and analytical studies for
isolated steel columns of channel and H-shaped cross sections under
" 29biaxial load1ngo The solution was based on assuming polynomial
expressions for the lateral displacements of the column axis and
satisfying equilibrium at a sufficient number of points so that the
coefficients of a. power series solution are determined. Tw'ist ·was
found later in a separate operation. A numerical procedure for the
analysis of this problem was developed. However, the procedure is not
described in their paper and the details are not available. An inter-
action formula for the maximum load carrying capacity of such columns
was proposed. It should be noted that the warping strains that result
. from the twisting of the cross se~tion of the column were neglected.
Therefore, the results reported by Kloppel and Winkelmann are not exact
but Btill provide useful information.
The inelastic behavior of isolated H-columns under biaxial load
was studied extensively by a team working with Birnstiel at New York
University. Birnstiel and Michalos30
following the related work of
31Johnston presented a general procedure for determining the ultimate
load carrying capacity of columns under biaxial loading. Warping
strains due to nonuniform twist were considered. However, their procedure
requires successive trials and corrections and needs considerable
computational effort for a solution. In a more recent work, Harstead
succeeded in reducing the laborious trial and correction procedure
to a few cycles by solving a system of linear equations for the
corrections at each station along the column. 32 More recently, the
331.1 -7
New York University team has conducted experiments on isolated H-columns
b - d b- - lId- 33su Jecte to 18X1B 08 1ng. The results of these tests and the
effect of warping restraint at column ends on the ultimate load carrying
capacity of the column and the effect of residual thermal strains on the
behavior of the column are examined and compared by Birnstiel, Harstead
23and Leu. The agreement between the numerical and experimental results
appears to be satisfactory."
Smith considered the biaxially loaded columns in terms of the
equations of three-dimensional elasticity taking into account the
nonlinear effects of the end tractions on a bilinearly-elastic
34column. Finite-difference approximations to the governing partial
differential equations are used and a general procedure for the
numerical step-by-step integration of these equations is presented_
A few solutions were obtained for a Solid rectangular column.
The analytical work at New York University assumes that the
strain increment at any point on the cross section is linearly related
to curvature increments in bending and twisting. The results of their
analysis are therefore not exact in comparison with the most general
formulation by Smith. However, Birnstiel's approach is an engineering
approximation.
Ellis, Jury and Kirk have reported on experimental and theore-
tical studies of thin-walled box sections subjected to biaxial
1 d · 35oa 1ng. In their theory, the plane section is assumed to remain
plane after bending and warping of the section is neglected, The
failure criterion of overlapping shapes which was ,used to find the
maximum load carrying capacity of col~ns in a ~ingle plane,36,37,38
-331.1 -8
was extended" to the three-dimensional biaxial loading case. The
numerical results were found to agree well with small scale tests.
El Darwish and Johnston studied the particular shape composed
of four corner angles with lacing on four sides.39
The four angles
were considered as point areas in their analysis and torsion was
neglected.
Experimental studies on small frameworks in which the columns
were subjected to biaxial loading have also been conducted at the
University of Cambridge in England. The results of these tests
and the effect of various end conditions on the ultimate load
carrying capacity of the biaxial1y loaded columns were examined in
40the book by Baker, Horne and Heyman.
29As an extension of the work of Kloppe! and Winkelmann,
Birnstiel and Michalos30
and Harstead32
to the case of an elastically
restrained column, Milner2
appears to be the first to report the
theoretical and experimental study of restrained and biaxially
loaded H-sections columns. In his procedure, the governing differ-
ential equations of equilibrium are first expressed in term of finite
differences and a numerical integration procedure is used for the
solution. His main purpose was to investigate the effect of the
irreversible nature of plastic strains and also the effect of the
residual stress upon the elastic-plastic behavior of the elastically
restrained H~section columns subjected to biaxial loading.
Milner's results indicated that the effect of unloading after
yielding occured in a biaxially loaded column was to strengthen the
column rather than weaken it. The effect of the order of load ap-
331.1
plication upon the failure load was observed to be significant.
Also, the effect of residual stress is less when the loading is
eccentric and decreases as the eccentricity increases so that this
effect may be small for restrained columns.
A different analytical procedure for determining the maximum
load carrying capacity of H-columns loaded biaxially was suggested
-9
b R· 41y l.ngo. The column is first assumed to be sufficeintly braced
in the lateral directions so that the entire section at the mid-
height of the column can reach its plastic capacity. Successive
reduction of the fully plastic stress distribution at the mid-height
is then introduced because of the instability effect and an iterative
method is used until equilibrium based on the deformed configuration
is satisfied.
A 1 hI · f v k' .. 42,43,44 h· hva UB e extenSl.on 0 Jeze s approx1mat10ns, W 1C
gave an analytical solution for eccentrically loaded steel columns
based upon the perfectly plastic idealization ,for the steel as well
as by assuming the sine curve shape of the deflected column axis,
was given by Sharma and Gaylord. 45 ,46 Jezek's theory proves useful
in obtaining approximate results in a simple manner for columns
subjected to biaxial loading. However" the analysis neglects the
effect of shifting of the shear center, which in general, moves
as yielding of the section grows, as does the center of rotation.
No analytical expressions which define the relation between the
slenderness ratio of the column and the ultimate load carrying
capacity of the column as was obtained by Jezek for the case of
eccentrically loaded column were obtained, but a large number of
numerical solutions for the maximum load carrying capacity of wide~
331.1
flange steel columns under biaxial loading are given in the form of
-10
interaction curves and are compared with existing interaction equations.
The approximation is seen to be satisfactory.
A different approach to the complex problem of biaxially loaded
columns, extending the simple plastic hinge concept by taking into
account the effect of axial force and biaxial moments on the fully
47plastic moment, was made by Pfrang'and Toland.. They construct,
· 41 d h d · d 45 h· - fl· - 1as R1ngo an at ers 1, t e 1nteract1on sur ~ces re at1ng aX1a
force and bending moments acting in two perpendicular directions
of the wide-flange section under the condition that the entire
section will be fully plastic. This approach is exact only for
columns of zero length or for columns with sufficient lateral bracing
but will not be applicable for the plastic analysis of space frames
in which column length is relatively long and the effect of the
geometrical change of the 'column on the ultimate strength of the
column becomes appreciable,
All the papers reviewed are based on an equilibrium approach
(stress solution) and yield a lower bound solution to the problem
d - h 11 k 1- - h f 1 - - 48,49 Aaccor 1ng to t e we - nown ~m1t t eorems 0 p ast~cl.ty.. n
excellent review for various interaction curves under different
1loading combinations is contained in the book by Hodge.
In Sections 3, 4 and 5, a brief account will be given of the
nature of the biaxially loaded column problem, of the procedure
developed by Birnstiel, Michalos and Harstead,30,32 and the sim
plified method presented by Sharma and Gaylord. 45 ,46 Section 6
331.1 -11
will deal with the interaction surface concept by applying upper
and lower bounds technique of limit analysis. The tests on biaxially
loaded columns by various investigators is summarized in Section 7.
Finally the future development of 'the solutions of the biaxially
loaded columns will be discussed in Section -8.
331.1 -12
3. BASIC CONCEPTS
The solution of the biaxially loaded column requires consid
eration of geometry or compatibility, equilibrium or dynamics, and
of the relation between stress and strain. Compatibility and
equilibrium are independent of material properties and hence valid'
for elastic and elastic-plastic columns. The differentiating feature
is the relation between stress and strain. The extreme difficulty
in obtaining an exact plastic analysis of columns under biaxial
loading even with the aid of digital co~puters is due mainly to
the fact that the stress strain relation in the plastic range is far
more complicated than the Hooke's law for linearly elastic materials.
As was discussed in the previous sections, plastic behavior
is extremely path dependent and almost always requires step-by-step
solutions that follow the history of ~oading. They are further
complicated by the fact that the elastic-plastic boundary is moving
and the stress-strain relationship for loading and unloading are
different. Even without this complication, there are no solutions
available for columns under biaxial loading that consider non-linear
elasticity.
It is apparent that an exact elastic-plastic 'solution of the
biaxially loaded column is unlikely. Drastic simplifications and
idealizations are essential for a reasonable approximate solution.
The geometry or compatibility of the column, the stress-strain
relations and the equations of equilibrium must be idealized to
accomplish a solution..
331.1 -13
dashed curve ~n Fig. 3).
For example, the material may be idealized as perfectly plastic.
This ignores work-hardening as symbolized by the stress-strain curve
for simple tension (Fig. 3). This idealization is reasonable for
materials with sharply defined yield strength such as mild structural
steel. However, it may be interpreted as an approximation to the
work-hardening material by using 8 suitably chosen yield stress (see
50As remarked by Lee, the adoption of this
ideal material must not be thought of as neglecting work-hardening,
but rather as averaging its effect over the field of flow. Chen
51and Santathadaporn have shown that idealized material applied to
eccentrically loaded columns which are assumed to fail by excessive
bending in the plane of the applied moments results in good agree
ment with the more precise analysis of Von Karman,24,25 which utilizes
the real stress-strain curve.
The difficulty of an exact analysis of even a perfectly 'plastic
column under biaxial loading has led to the approximate formulation
of equilibrium equat~ons in terms of generalized stresses and strain-
rates, such as force and moment resultants and rates of extension
and curvature of the column section. In addition, the traction
bpundary conditions are not specified in detail, at least for end
loading conditions. Normally they are expressed in terms of the
stress resultants and displacements. Except for Smith's three-
dimensional elasticity formulation, an approximate formulation of
the equilibrium equations was usually adopted in the past.
Moreover, as a consequence of such an approximate formulation
of equilibrium, there is one fundamental assumption about the strain
331.1
distribution of the section, that is, the plane cross sections re-
-14
main plane after loading for each of the thin-wall flat plates of
which the column is composed. As remarked by Bleich,6 this assumption
appear to be justified because St. Venant's theory of torsion indicates
that the longitudinal center line of the cross section of a thin
flat plate remains in a plane during torsion. Warping can only
vary slightly across the plate because of its small thickness, and
the entire cross section must remain approximately plane. This
assumption enables one to obtain the stress distribution over the
cross sections of the column for a given stress-strain curve pro-
vided the curvatures at each station along the column is known.
Other assumptions and idealizations are also made. The column
is assumed perfectly straight, the load is static and is applied
along the center line, and the material is homogeneous and free 'of
initial stress. None of these are strictly true. Considerable
care then is required in attempting to correlate the theoretical
analysis with the results of experimental data.
If an axial load is applied with an eccentricity in a plan~
of symmetry, a column will deflect but remain untwisted at loads
less than the buckling load. However, if a column is loaded with
biaxial eccentricity, it will usually deflect and twist at any load
as illustrated in Fig. 4. Typical load versus mid-height displace-
ments of the column for an elastic-plastic material are shown in
Fig. 5. The essential feature of the column due to space action
is that the lateral displacement is always accompanied by a rotation
of the column sections. The importance of this twisting lies in the
331.1
fact that the ultimate load carrying capacity of such columns,
especially columns with open thin-walled sections that have sIDall
-15
rigidity.
torsional rigidity, may be less than the maximum load carrying capacity
for in-plane loading.
PekOz and Winter have noted that the twisting of the column
section under any axial load that is not in a plane of symmetry may
52be·explained by considering a simple physical model (Fig. 6).
The biaxial load can be decomposed into four components as
shown in this figure. The first three are statically equivalent to
an axial force 4P and bending moments M and M about the twox y
principal axis of the .section. However, these three equivalent
systems do not produce the biaxial load 4P. It is necessary to
consider a fourth system which produces zero axial and bending moment
resultants on the section. This fourth system causes the column
to warp or twist. The twisting effect is small for columns with
solid or closed wall sections, but it is significant for columns
with open thin-walled sections 'because of their small torsional
23Computations by Birnstiel, Harstead and Leu on the
biaxially loaded elastic H-column show clearly that the twisting
effect becomes ~o large that even the elastic analysis has sub-
stantial error introduced by Goodier's approximate formulation.
Even greater error can be expected for the inelastic case of biax-
ially loaded columns.
Three typical solutions which range from an almost exact to a
rather crude 8p'proximation are summarized hereafter.
331.1 -16
4 . BIAXIALLY LOADED COLUMNS OF H SECTION
THE METHOD OF BIRNSTIEL, MICHALOS AND HARSTEAD30 ,32
The analytical procedure is based upon the general assumptions
described in the preceding section which discusses the basic concepts
of columns biaxially loaded beyond the proportional limit. However,
the procedure may be modified to account for almost any stress-
strain relationship. Residual thermal strains can also be considered
in the analysis.
Successive equilibrium positions of the biaxially loaded column
are determined from each incremental increase in the deformations
dtThe axial strain €, the bending curvatures ---2 and
.2 dztwisting curvature dB
2 are chosen to be the controllingdz
(see Fig. 4). Since the small increments of curvatures
of the column.
d~--2' and thedzdeformations
and axial shortening are unknown along 'the column, a trial-and-error
procedure must be followed to obtain each equilibrium position of the
column corresponding to each increment of deformations. The total
strain increment distribution over the section is assumed to be the
sum of the small increments of strain caused by the bending curvatures,
the twisting curvature and the axial shortening. The total strain
at each point of the section is then obtained by superimposing the
total strain increment on the previously established strain cor-
responding to an equilibrium configuration. Curvature and strain
relationships are assumed to vary linearly over the section for each
small increment.
331.1 -17
The superposition of each incremental strain distribution results
in a change in stre~s distribution over the section and also an ad-
ditional deflection of the column. The initially assumed incremental
deformations along the column must therefore be adjusted such that
equilibrium still exists between the internal and external tractions.
This requires successive trial-and-error estimates of the deformations
until· convergence is attained and hence the new equilibrium position is
established. The maximum load capacity is then obtained from a plot
of the mid-height displacements VB. load relationship.
In this analysis the effect of shearing stress on the plastic
yielding of the material is neglected so that only the normal stress
in the axial direction of the column is considered. Also the shear
modulus of the yielded portion of the cross-sectional area is disre-
garded. The coefficient of torsional rigidity is ,assumed to be reduced
directly proportional to the ratio 'of the remaining elastic area to the
total cross-sectional area.
The numerical results obtained at New York University indicate
that warping restraints of the end cross sections of isolated H-columns
subjected to biaxial bendin&, increases the maximum load carrying
capacity of such columns. For a particular l4~43 section with
e = 0.5 in., e = 5.0 in., and ~/r = 117, the increase was aboutx y
12 percent (see Fig. 7).
The residual stress resulting from the manufacturing process
has a significant influence on the load-deformation response as well
331.1
as the maximum load carrying capacity of columns subjected to biaxial
loading (see Fig. 8). The presence of the assumed.residual stress
reduced the maximum load carrying capacity by 8 per cent.
-18
331.1
5. 'APPROXIMATE SOLUTION OF THE BIAXIAL"LY LOADED COLUMN PROBLEM
-19
The labor involved in the numerical determination of the ultimate
load carrying capacity of biaxially loaded colurrms based upon Hexact"
-theory makes its practical use unlikely. Therefore, simplifications
,and idealizations are essential if simple design formula are developed.
~ 42 43 44 .The approximate solutions obtained by Jezek ' , for eccentr~cally
loaded columns of rectangular cross section, indicates the possibility
of extending the Je~ek's concept to the problem of biaxial1y loaded
column.
According to Jezek's concept, the solution of the syste~ of govern-
ing differential equations can be simplified drastically by establishing
equilibrium only at mid-height of the column, by idealizing the material
as elastic-perfectly plastic (Fig. 3), and by assuming the deflected
shape of the column axis as the half wave of a sine curve. The solution
based upon these assu~ptions leads to analytical ~xpressions for the ul-
timate load-carrying capacity of eccentrically loaded columns' of rectan-
gular cross section. A comparison of the results of the approximate
method 'with results obtained from the "exact" theory'and the results of
tests indicates that the idealization and, simplification are reasonable.
Studies on the degree of approximation 'involved for the eccentrically
loaded col~mn problem as influenced by I the use of the idealized stress-
strain diagram and as influenced by the use of the half wave of a sine
curve for the deflected column have been critically reviewed recently
, 51by Chen and Santathadaporn.
331.1 -20
A similar approximation of the deflected shape of a column axis
has been made by Kloppel and Winkelmann29 and Westergaard and Osgood.53
•It was found that the error involved is acceptable for all practical
purposes.
The extension of Jezek's concept to biaxially loaded columns. proves
to be equally successful. Sharma45 ppplied this concept and assumed
as well an elastic-perfectly plastic idealization for the material
and sinusoidal variations for the lateral displacements and rotation
of the cross section of the deflected axis of the column. The solution
of the equations of equilibrium were simplified by only considering
equilibrium at mid-height of the column between the applied force
at the ends of the column and the internal resistance of the mid-height
of the column.
In order to compute the internal moments and axial force at the
mid-height of the column from the assumed deflected shape, Sharma makes
the fundamental assumption that plane sections remain plane after loading
for each of the thin-walled flanges and the web. Once the deflected
shape is assumed, the strain and stress distributions can be determined.
The internal moments and axial force of the mid-height of the biaxially
loaded column can then be evaluated.
The equilibrium consideration at mid-height of the column yields
-a system of four equations of equilibrium in terms of the values of2· 2
duo dvoa~ial strain €, two bending curvatures ---2 ' ---2 ' and twisting curvaturedf3 dz dz--1' (the subscript "0" is used to denote the mid-height). The fourthdzequation which relates the twisting deformation of the column to the
twisting moment is satisfied identica~ly since from symmetry the twisting
331.1 -21
moment is zero at mid-height. Hence a fourth equation must be sub-
stituted. This can be provided by noting that the rate of change of
twisting resistance must equal the rate of change of the twisting
moment resulting from P. The four equations of equilibrium are then
solved by Sharma using iteration for various pattern of stress distri-
bution at the mid-height.
The comparison of Sharma's results with the more precise solution
of Birnstiel and Michalos30
indicates that Sharma's approximate solution
tends to overestimate the predicted ultimate load slightly. The difference
for a particular 12W79 section with e = 6 in., e = 12 in. and t = 15x y
feet was found to be about 3.3 per cent.
Two points should be noted. First, despite the drastic idealization,
the analytical expression is still so complex that numerical results
are presented in the form of an interaction curve. Secondly, the shifting
of the shear center due to the plastification of the mid-height is neglec-
ted~ Hence the amount of inaccuracy is unknown even though the com-
parison for the particular example cited was good.
331.1 -22
6 • THE INTERACTION SURFACE RELATING P, Mx
AND My
In simple plastic theory, bending usually predominates so that
the concept of a simple plastic hinge is sufficient in most cases.
Should there be axial force in addition, the reduction of the fully
plastic moment at the hinge section can be easily taken care of by
interaction curves. For example, if normal force P and bending moment
M alone are considered, it is a 'simple matter to construct an inter-
action curve relating P and Munder the condition that the entire
section will be fully plastic. In those problems where the shear
force V is important, an interaction curve, relating V and M has
1 54 55 56also been useful." However, as noted by Drucker, inter-
action curves are not unique and the manner of loading of the entire
beam is important. Nevertheless, it seems clear that an interaction
curve can provide an engineering solution that is reasonable.
It is natural to expect that columns under biaxial loading could
be considered in a similar manner. If normal force P and bending
moments M and M acting in two perpendicular directions are con-x y
sidered, the construction of an interaction surface relating limiting
values of P, M , and M for perfectly plastic material would be mostx y
desirable for various shapes of cross section in common use. Un-
fortunately, such an approach is not valid for compression members
where the geometry change affects equilibrium. The limiting theorems
of plastic analysis upon which the construction of interaction surface
331.1
must be based, assume that the influence of the deformation on the
equilibrium can be neglected. The present plastic analysis and de-
sign procedures do not consi~er biaxial loading of the columns at
*all and hence have an unknown amount of inaccuracy when applied
to space frame analysis and design. The concept of the interaction
surface approach, which is correct only for columns of zero length,
can be considered as a first step in extending planar structures
-23
analysis and design to more realistic space frame analysis and design.
There is very extensive literature on the application of upper
and lower ·bounds of limit analysis of plasticity to obtain inter-
action curves or surfaces for various loading combinations and various
shapes of cross section. These previous solutions are discussed
1by Hodge, but not much has been ,reported on the interaction surface
that relates P, M and M for the commonly used wide-flange sections.x y
Pfrang and Toland have assumed various stress distributions· over the
wide-flange section that satisfy yield conditions (simple tension) and
47equilibrium of axial force P and bending moments M and M. Theyx y
then calculated several interaction curves relating P, M and Mx y
for wide-flange sections. It was not shown that the stress fields
so assumed could be possi~le to associate with velocity fields.
Any such value obtained for the interaction curve is therefore not
necessarily a good lower bound nor is known when it is an upper
bound according to the general theorems of limit analysis.
58Santathadaporn and Chen obtained lower'~and upper bound values for
P, M and M by maximizing an arbitrary assumed stress field for thex y.
* In Great Britain the recommended practice for column design alwaysconsiders the-effect of biaxial bending. 57
331.1
lower bound solution and by minimizing an assumed pattern of velo
cities for the upper bound solution. Typical interaction curves
for wide-flange sections are shown in Fig. 9.
-24
331.1 -25
7 . TESTS ON BIAXIALLY WADED COLUMNS
Kloppel and Winkelmann 29 conducted a series of.tests on 74
rolled steel wide-flange columns. The end conditions were such that
the columns were essentially pinned against rotation, and warping
wa~ restrained by heavy end plates. The specimens investigated
were sections IP 16/16/0 (16 em x 16 em) and I~10/l0/0 (10 cm x 10 em),
for various biaxial eccentricities. In the series of tests with
section IP 16/16/0, the strong axis slenderness ratio waS maintained
at 34 while the weak axis slenderness ratio varies from 57 to 114.
The successively more slender sections were obtained by trimming the
tips of the flanges. Similar tests were carried out for the section
IP 16/16/0 with the strong axis slenderness ratio maintained -at 49
while the weak axis slenderness ratio varied from 83 to 121. Figure
10 summarizes these tests in outline form and the details were given
59in report by Galambos. Reasonable agreement' was found between the
observed column strength and the values calculated from Sharma's
· 1. 45approx1mate so ut1ons.
Chubkin 60 reported on another series of extensive tests. Two
hundred eighty-one steel columns were tested with various conditions
of eccentricity (axial, uniaxial, biaxial) and end conditions (warping
restrained and warping free). In one series, -the specimens were 1-
type (9.4 em x 18 cm) rolled sections with the weak axis slenderness
ratios 50, 100 and 150 respectively. Ip the other series, 12 cm x
12 cm welded buil~-up H-shapes were used with weak axis slenderness
331.1 -26
ratios of 50 and 100 respectively. The details were also summarized
5Yby Galambos. Interaction curves obtained by Sharma and the test
results are compared in Fig. 11. The observed values are in reason-
able agreement with the calculated results.
Recently conducted tests on 16 H-shaped columns provide further
33experimental information on biaxially loaded columns. Some of these
tests were permitted to warp at both end cross sections. The other
test specimens were loaded through heavy end plates, and thus warping
was restrained. The tests were made on columns of various H-shapes
made of three types of steel A7,.A36 and V65. The biaxial eccen-
tricities varied from-O.89 in. to 2.38 in. for e and 1.87 in. tox
3.21 in. for e .y
Analytical studies were also made-using the procedure outlined
in Section 4 and assuming an elastic-perfectly plastic material be-,
havior. The results of the experimental and analytical study are
compared in Fig. 12. A reasonable agreement is observed.
Milner reported on tests of 10 H-shaped elastically restrained
columns which were also restrained against sway.2 The experimental
work was concentrated on a series of exploratory tests, in which
significant parameters.were varied. The specimens are shown dia-
grammatically in Fig. 13. Loads were applied to the beams by means
of turnbuckles. In the first three tests, major axis beam loads
and axial load were applied. In the second seven tests, both the
major and minor axis beam loads and axial load were applied, In
all tests, beam loads were first applied and then held constant
331.1 -27
and the column was finally loaded to failur~ by axial force. The
loading path for the second series of tests is shown as path O-D-A in
in Fig. 2.
Milner's numerical solutions of the maximum load carrying
capacity of his experimental subassemblages were within 8 percent of
his' test results, except for one case, where the difference was 11
percent.
331'.1 -28
8. FurURE DEVELOPMENTS OF THEORY OF BIAXrALLY LOADED COLUMNS
The review of the theory of biaxially loaded columns outlined
in this paper indicates that a reasonable approximate solution has
been achieved. Comparison of the numerical solutions with results
obtained fr6m tests proves that the idealizations needed for the
analysis are justified. However, the labor involved in the numerical
calculations limits 'its practical use. Therefore, numerous tables
and diagrams for various shapes of column cross sections must be
prepared before simple design formula can be developed.\
Most of the numerical procedures and solutions for biaxially
loaded columns 'discussed in this paper d~a1 only with isolated column
under proportional loading with like eccentricities of loading at
both ends of the column (see Fig. Ib). Furthermore, the end con-
ditions are usually simply supported against translation with warping
either permitted or completed restrained. Columns in an actual
building framework usually are connected with other members and
frequently subjected to twisting moment in addition to the biaxial
loading and the order of load application is rather random. There
are only a few studies on th~ effects of elastic end restraints and
h d f 1 d 1 · · 2 ,40t e or er 0 oa app 1cat10n.
Additional studies of these aspects are needed. As a first
step, it is logical to extend the techniques that were developed
for isolated columns under biaxial loading to columns subjected
to twisting moment in addition to the biaxial loading in (Fig. 14).
331.1
A detailed study of subassemblage shown in Fig. 15 should provide
insight into the behavior of columns 'as part of a framework.
-29
As noted previously, plastic action is extremely path dependent,
and the calculations of deformational response must always follow the
history of loading. Further study of the effect of' the irreversible
nature of plastic strains upon the behavior of columns subjected
to·bi~xial loading with and without elastic restraints should provide
information on the loading conditions encountered in actual structures.
As noted by Birnstiel, Harstead and Leu, the predicted values
of twist of the isolated H-columns ~nder biaxial loading were much
smaller than the observed values for most columns at loads near
23ultimate. Part of the discrepancy probably results from initial
twist of the test specimens. The study of the effect of initial
twist upon the behavior of a biaxially loaded column is desirable.
An initial e~centricity or twisting is probable in actual structure
as well as in laboratory test specimens.
'331.1 -30
9. CONCLUSION
A general understa~ding of the elastic-plastic behavior of an
isolated, pinned-end column subjected to biaxial loading has been
achieved through the efforts of many research workers in this country
and abroad. However,- additional studies on restrained columns with
biaxial loading and on sway and non-sway subassemblages are needed.
-.-
It is probable that the procedures now available for designing
columns subjected to uniaxial bending can be modified to include
the effect of the additional bending moment. 6l
331.1 -31
10 • ACKNOWLEDGMENTS
This study is a part of the general investigati,on on "Space
Frame with Biaxial Loading in Columns", currently ~eing carried out
at Fritz Engineering Laboratory, Lehigh University. Professor L.
S. Beedle is Director of the Laboratory and Professor D. A. VanHorn
is· Chairman of the Civil Engineering Department. This investigation
is sponsQred jointly by the Welding Research Council and the De
partment of the Navy, with funds fur~ished by the American Iron and
Steel Institute, Naval Ship Systems~ Co~mand, and Naval Facilities
Engineering Command.
Technical guidance for the project is provided by the Lehigh
Project Subcommittee of ,the Structural Steel Committee of the Welding
Research Council. Dr. T. R. Higgins is Chairman of the Lehigh Project
Subcommittee. This study is also guided by the special Task Group
in the Subcommittee cons~sting of Professor Charles Birnstiel '(New
York University), Professor E. H. Gaylord (University of Illinois),
Dr., T. R. Higgins (AISC), Professor B. G. Johnston (University of
Michigan), and Dr. N. Perrone (Office of Naval Research).
The authors wish to thank Professors L. W. Lu and J. W. Fisher
who reviewed the manuscript and made many valuable suggestions and
corrections.
The manuscript was typed by Miss K. Philbin and the drawings
were prepared by Mr. R. N. Sopko and his staff. Their care and
cooperation are appreciated.
331.1
p
B
C 15-----0+-----41... A(v)
( i i )
(vi i)
D
Radial Loading
Mx
-34
Fig. 2 Various Loading Paths
331.1
z
y
u
DeformedPosition
y(b)
x(0)
Fig. 4 Isolated H-Column Under Biaxial Loading
-36
OriginalPosition
v
331.1
PCOLUMN
LOAD
-37
DISPLACEMENTS U,VROTATION f3
Fig. 5 Load-Displacements or Load-Rotation Curve forColumn Subjected to Biaxial Loading
331.1
p p
-38
p
p
+
p p
Axial Force 4P Bending Mx
p p p
p p
+ +
Bending My Bending Causes Warping
Fig. 6 Decomposition of a Biaxial Loading(reference No. 52)
331.1 -39
1601-----+----+----+----+--~-___+_-___4_-_+_-___f
...-----
14VF43 ex=.5inl
ey=5in.I00l--+-~lI------tfl-----+----+----I-----+---i
- Warping Permitted--- Warping Restrained
.08 rod.-2.8 in.-.6 in.
.04 .05 .06 .07-1.2 -1.6 -2.0 -2.4-.2 -.3 -.4 -.5
.02 .03-.4 -.8o -.1
.01o
80L----.J.-.--J.--.....L....------L---...L....-....L.----.L-------JL..---......I....-_----'
p' 0uv
Fig. 7 The Effect of Warping Restraint at Column Ends(reference No. 23)
1601----+----+---+----+-------t
180--~-____r_-_____,--~-_r__----o;__;_---......,
140J----+------I""'-----+----+---+----r--~--.,....----tp
(k ips)I 20 I-------I--------.-'+---I-~L.-c::....:......-+---~~----L-_-"---_--A.----t
ex=.5 in.ey=5 in.
IOO~~~~~~~~~~~~~~~ithout ResidualStress
--With Residual Stress80L---L------L....-.L.......L---&....------L..--L..----'--......I....-------a.-----a--.......
p 0 .01 .02 .03 .04 .05 .06 .07 .08 rod.u 0 -.4 - .8 -1.2 -1.6 -2.0 -2.4 -2.8 in.v 0 -.1 -.2 -.3 -.4 -.5 -.6 in.
Fig. 8 The Effect of Residual Stress(reference No. 23)
331.1 -4u
p
y
my
.8
.6
.4
o
II1.9III
.2 .4
.6
.6
---x
-12\AF31
--- 14 \AF 426
1.0
Fig. 9 Comparison of Interaction CurvesBetween Heavy and Light Sections
(reference No. 58)
331.1 -41
xxxx x x
Section 1P 16/16/0 Section IP 16/l!
..! = 34 4xxx
x x x x ...J. = 34rx . r x
I..!= 57 I .£ = 91ry ry
Number of Test 12 Number of Test 8
x x x xx
Section I P 16/:C Section IP 16/Nx x x xx
r~ =34.4 -:!. = 34r x
I .l:.. = 71 I L= 114ry ry
Number of Test 9 Number of Test 6
xx
Section IP 16/ II Section 110/10/0xxx
.l..=34x
xxx..i...= 48rx =ex rx
I .l:... = 85xxx
x...:!= 83ryry
Number of Test 7Number of Test 15
Section IPO/ IxIX -L= 49xx x rx
-.L= 102ry
Number of Test 8
Section IPIO/n
1: ~= 49xxxr x
.A.. =12 Iry
Number of Test 6
Fig. 10 Outline of Kloppel and Winkelman's Tests(x denotes the location for the variousbiaxial eccentricities used in their tests)
331.1 -42
p ex b-Py -22ry2 -
.5 Chubkin's Testey d
- I •2rx2 -
ey dx.4 --=2
2rx2
=I
.3
x.2 ey d
=22rx2
.1
o 20 40 60 80 100 120 140 160
..:!.ry
Fig. 11 The Comparison of Sharma's InteractionCurves with Chubkin's Tests
(reference No. 45)
~0
PCbc en
~ ~~ n~ ~
~" f 0 iY-V b1{j ! u
I. ~lf I~
A 0 1 Column No.4r
SW-25AI -., rex= 1.6611
ey=2.95"A 0 C
I I I
331.1
p(ki ps)
100
80
60
40
20
v 0
{j -0.005
-0.4
0.005u
-0.8
0.015o -0.4 -0.8 -1.2
-43
rad.in.
Fig. 12 Comparison of Birnstiel and HarsteadlsAnalytical Curves with Their Tests
(reference No. 23)
331.1
z
-45
y
Fig. 14 Isolated H-Column Subjected to TwistingMoment in Addition to Biaxial Loading
p
p
Fig. 15 Elastically Restrained Column Subjected to Biaxial Loading(Center Column)
331.1 -46
12 • NOTATION
b
d
ex
ey
t
= width of flange
depth of wide-flange section
= eccentri~ity of loading measured in the x direction
= eccentricity of lpading measured in the y direction
= length of the column
m = M 1Mx x .px
m = M 1My y py ..
M = moment about the x-axisx
M moment about the y-axisy
M = moment about the z-axisz
M = fully plastic moment about the x-axis when no axial loadpxor moment about the y-axis is acting.
M = fully plastic moment about the y-axis when no axial loadpy
or moment about the x-axis is acting.
p pipy
p = axial load
p = 'yield load when no moment is actingy
r = radius of gyration about the x-axisx
r = radius of gyration about the y-axisy
u displacement of the centroid in the x-direction
v = displacement of the centroid in the y-direction
x, y, z cartesian reference coordinates
.~ = angle of twist
e normal strain
~, 11, , = principal section axes after deformation
IT = normal stress
331.1 -41
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331.1 -48
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331.1 -49
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331.1 -50
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