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

11. FIGURES

-32

331.1

z

p z p

-33

y y

p

(0 )

x p

( b)

x

Fig. 1 Column Under Biaxial Loading

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

(F

STRESS

--- -

-35

ESTRAIN

Fig. 3 Idealization as Elastic Perfectly Plastic

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

WI Major AxisTurnbuckle Load

Fig. 13 Mi1ner l s Tests(reference No.2)

-44

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

13 . REFERENCES

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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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-331.1 -52

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