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BIAXIAL FLEXURE IN MODEL STEEL COLUMNS

by

Satoru Niimoto

A report submitted in fulrillment or the requirements

IOn Course C.E. 404 "Structural Research"

Fritz Engineering LaboratoryDepartment or Civil Engineering

and Mechanics

Lehigh UniversityBethlehem, Penna. "

June, 1955

Fritz Laboratory Report No. 205A.16

..

TABLE OF CONTENTS

Page

I 0 ABSTRACT 1

II a INTRODUCTION 2

III. THEORETICAL ANALYSIS 4

IV. DESCRI PTI ON OF EXPERIMENTAL PROGRAM AND TEST ·7RESULTS

Va DISCUSSION A1m SUMMARY 13

VI. ACKNOWLEDmilEJ\TTS 13

VII a NOMENCLATURE 14

VIII. REFERENCES 16..IX. FIGURES 17

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I. ABSTRACT

This is a study of the stability of rectangular

model steel colurmns whose axes of end rotation were controlled

under biaxially eccentric compressive loads~ Columns were

tested to failure to study the effect of this variable. The

ultimate strengths of these columns, experimentally obtained,

are compared with analrtical values as computed by a virtual(3), (4)

displacement methodo

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-2

II. INTRODUCTION

The problem of biaxial flexure occur~ in structures

in very complex forms. In most cases the complexity of their

boundary conditions render an exact analytical solution impos-

sible. Top chord members of bridges subjected to lateral wind

forces and vertical loads, and corner columns of tier bUildings

are two examples of structural members sUbjected to this. type

of flexure. End conditions have been simplified to facilitate

the theoretical analysis in this study of biaxial flexure in

model steel columns./ / (2)

Historically, Von Karman first solved the stability

problem of eccentrically loaded pin-ended columns. He assumed a

strain distribution pattern across the section and froci the

stress-strain curve of the material computed the average axial

stress in the column section. From considerations of stress

equilibrium and moment he expressed curvature as a ·function of

axial load, eccentricity, and center line deflection.

yll =...1:- , where y" = f (y, P, e)-S

Karman and Chwalla solved this differential equation

by graphical or numerical integration. The solution for y

defines the shape of the center line deflection for any value

of average stress P, eccentricity, and column length. TheA

condition for stability is then defined as:

= 0

-3

Later investigators advanced simplified analytical

used an idealized stress-strain

assumed a partial cosine curve for( 2)

Jezekcenter line deflection.

I /methods to approximate the exact solution of Karman and Chwallao

(11)Westergaard and Osgood

curve, one perfectby elastic up to the yield stress then per-

fectly plastic. Center line deflection was approximated by a

half sine curve. His analytical solutions were in good agree-.

ment with test results for materials exhibiting a sharply defined

yield stress level.(3 )

A recent paper by Ro L. Ketter describes a virtual

displacement method of determining the stability of beam columns

in the elastic and inelastic ranges. The point of indifferent

equilibrium is defined as follows:

"When external moments at the most cri ticalsection increase at a rate equal to the rate ofincrease in resistance moment supplied by the in­ternal stress system at that same section, themember becomes unstable."

This can be written in the form of an equation as follows:

~MIi:¢ ext

~Ml= t:¢ int

due to a prescribed stress

where, 6. MI f (P, /:),.y) and~¢ ext =

stiffness at the most stressed

b.MI .~¢ int =

section

the internal bending

distribution patterno The mechanics of.this method are similar( 10)

to those developed by Ros and Brunner.

III. THEORETICAL ANALYSIS

The following assumptions are made for this approxi-

mate solution:

1. Twisting will have little influence on the column strength,

and therefore the member can be considered as having a neutral

axis parallel to the direction about which rotation is allowed.

2. Material properties will be idealized as perfectly elastic up

to a yield point and then perfectly plastic.

3. Plane sections'will remain plane.

4. Ultlmate column strength is dependent on the stress-pattern

at the center line section of the member.

5. The member will deform in a sine configuration in a plane

perpendicular to the axis about which end rotation is allowed.

Elastic column behavior is described by:

-P .yll = lr.f (y T e)

and for' a column bent in single curvature, center line deflec-

tion is given by:

'I\. e (1 - cos u)~ = , wherecos u

Considering the actual Ultimate carrying capa~ity of

such columns, the upper limit of column strength is bounded by

the Simple Plastic Theory. A set of plastic stress patterns are

assumed and their corresponding axial loads and moments computed.

Since this theory presupposes columns of negligible lengths, the

computed values of axial load and moment capacity are greater

than those influenced by the length parameter'. A non-dimensional

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pplot of.:-

Pyand M­

Myi 8 g:i. ven in Pig 0 1 for the consi dGred cross-

section using this methodo

The ultimate strength of a biaxially compressed column

may be predicted from a E- , L collapse curveo In order to ob­Py. r

tain the desired curve, yield penetrations were assumed for the

most stressed column section (Figo 2). Compatible load, moment,

and curvature quantities were computed. A non-dimensional plot¢

of these relations!!... , E- , and ¢. is shown in Figo 3. FromMy Py Y

these curves another set of auxiliary curves were plotted for

and :y

at constant ratios (Fig. 3 and Figo 4).

Next, two sets of curves (Fig. 5 and Figo 6) repre­

senting tangent values of the above auxiliary curves were plottedo

of a section at a given R- ratio.Py

values of ~ fIfj.¢ My int

moand M quantities, -- may beMy My

and reading corresponding¢. ¢Y

These tan§snt values, then, represent a measure of the internal

bending stiffness ~ • ¢Y6.¢ M

Y int(Eq. 10, Ref. 3). By selecting suitable

computed. The equation,

(Eq. 14, Ref. 3),

relates the compatible end moment mo with the moment and unit

rotation quantWies at the ':most stressed center line section

(Mo and ~o) for a gi van value of' internal stiffness L 0 SincePe

-6

Pel the Euler load is a function

a family of curves corresponding

( Fig 0 7).

of thep

to -­Py

length parameter ~,L

and r may be computed

tricity

It can be shown that :o;l~y equals :;. the eccen~-• . y ec Ilio P

ratio. A l1ne drawn at a slope ~ proportional to --M --r Py

I P I L curves at the critical loadPy r

of curves may thus be drawn. Fig. 8

the ~My

family

will then intersect

ratios f-. AnotherPy

is such a presentation of critical load ratios for constant ec-

centricity ratios. The critical load ratio ~ is then read fromy

these collapse curves for given eccentricity ratios and alender-

ness ratios.

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IV. DESCRIFTION OF EXPERIMENTAL PROGRAM AND TEST RESULTS

Figure A shows the parts and assembled view of the end

fixtures 0 The rectangular bearing block is of mild steel. High

carbon steel is used for the principal load carrying pieces.

The circular insert acconmodates a maximum three quarter inch

square section. Smaller sections are fitted by shims. The axis

of end rotation is perpendicular to the screw axis. Orientation

of sections with respect to the plane of applied eccentricity is

indicated by the pointer attached to the circular insert. Move-

ment of the rocker, therefore eccentricity, is controlled by the

knobs threaded onto the screw. In this study the strong axis of

three quarter by one half inch columns was oriented at 30 degrees

to the plane of applied eccentricity, and eccentricity was set

for an ec of one halfo '?

Figure B is a view of the alignment apparatus in the

60,000 lb. hydraUlic testing machine. The bottom circular wedge

plates were levelled and the columns were first geometrically

aligned. Load was transmitted through a spherical seating block

attached to the movable crosshead. Final alignment for a given

eccentricity was made to check computed elastic curves and sYm-

metry of bending. Figure C Shows the adjustable frame for three

Ames dial gagesoDeflection readings are referred to the column

axis and are independent of end eccentricity. Twist was observed

with transit and,an optical mirror that reflected a scale mounted

on the transit telescope.

-8

Tension and column specimens were annealed at l1400 F

for one and a half hours at temperature then oven cooled. Three

quarter by one half inch tension coupons were milled to one half

inch square sections for two inch ga@S length. A Huggenberger

gage, set at one inch ga6e lencth, was used to record elastic

load deflection measurements. The tested coupons exhibited un­

usual variation for Young's elastic modulus. Table 1 is a summary

of the tension tests.

Fi ve" colll.'11Il specimens, whose measurements are summarized

in Table 2, were selected for straightness and freedom from ini­

tial twist. These annealed specimens were milled to length.

Computed elastic load deflection curves and experimental points

are plotted in Fig. E. Fig. F is a summary of the five experi­

mental load deflection curves.

Fig. A

TABLE 1.

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v. DISCUSSION AND SUIIlIMARY

The five column tests, although far from conclusive,

seem to indicate that this method of analysis is conservative.

Deviations of 11.5 per cent and 11.2 per cent above predicted.I

strength were exhibited by columns whose slenderness ratios were

4306 and 6407 respectively. Load eccentricities were set Within.

the accuracy of the end fixtures. Elastic deformations for-these

specimens were symm~tric and checked computed deflections reason-

'ably well. Twist measurements at mid-length of the specimens were

confined within very narrow limits. The effect of strain harden-

ing may be discounted for the st'rains involved. A possible ex­

planation of the strength of these columns may be that their

yield stress level is highe'r than the value used. The unusual

scatter of the elastic modulus of the tension coupons is unex­

plained.

VI. ACKNOWLEDGJ\IfENTS

The author wishes to express his sincere appreciation

to Mro R. L. Ketter for his ready counsel and patient gUidance.

He was also responsible for the end fixtures used in this stUdy.

This project has been carried out at the Fritz En­

gineering Laboratory as a course in structural research, with

funds provided by Dean H. A. Neville through the Institute of

Research. The p~ogram was under the direction of Mr. R. L. Ketter.

A

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VII. NOMENCLATURE

Area of cross-section

Extreme fiber distance from the centroid of a section

End eccentricity of the applied axial load

Young's modulus of elasticity

Moment of inertia of the cross-section about an axisthrough the centroid parallel to the axis of allowed ro­tation

Length of test specimen

Bending moment

Bending moment at the center line section of a specimen

Bending moment at which first yield is reached in flexure

That part of the center line moment that is independentof deflection

Axial load

Euler buckling load

Axial load corresponding to yield stress level over entires·ection

Maximum load a member can carry

Radius of gyration of section corresponding to the momentof inertia I

Deflection in plane of applied moment

Second derivative of deflection with respect to longi­tudinal column axts

Incremental deflection due to increase In axial load

Unit normal stress

Lower yield stress level

¢ Curvature

¢y Curvature at initial yield

¢o Curvature at the center line

D. Center line deflection

1 Radius of curvature

-15

-16

VIII. REFERENCES

1. Beedle, L. S., Ready, J. A., Johnston, B. G., "Tests ofColumns Under Combined Thrust and Moment", Froc. ofthe Society for Experimental Stress Analysis, Vol.VIII, No.1, 1950.

2. Bleich, F., "Buckling Strength of Metal Structures ll, 1st ed. '

McGraw-Hill Book Co., Inc., 1952.

3. Ketter, R. L., IIA Virtual Displacement Method for Determin­ing Stabili ty of Beam-Columns Above the Elastic Limi til,Fritz Laboratory Report 205A.14, Progress Report 11,1954.

4., Ketter, R. L., Kaminsky, E. L., Beedle, Lo S., II Pl as ticDeformation of WF Beam Columns", Progress Report 10,205A.12, 1953 ASCE Proc. Separate No. 330.

Neal, B. Go, liThe Lateral r'nstabili ty of Yielded Mild-SteelBeams of Rectangular Cross Sections ll

, PhilosophicalTransactions of the Royal Society of London, Vol. 242,No. 846, January 1950, p. 197-242.

6. Niles, A. S., Newell, J. S., "Airplane Structures ll, 3rd ed.,

John Wiley & Sons, Inc., New York.

5.

70 Thiirlimann, B., "Deformations of and Stresses in Inl tiallyTwisted and Eccentrically Loaded Columns of Thin.JJl1alledOpen Cross Sections", Report 3 to Column ResearchCouncil and the Rhode Island Department of Public Roads,1953, Brown University.

8. Thfirl i mann, B., Drucker, D. C., Report 2 to Column ResearchCouncil and the Rhode Island Department of Public Roads,"Investigation of the AASHO and AREA Specifications forColumns Under Combined Thrust and Bending Moment", 1952,Brown University.

9. Timoshenko, S., IITheory of Elastic StaOility", McGraw-HillBook Co., Inc., New York, 1936.

10. Wlistlund, G., BergstrBm, S. G., "Buckling of Compressed SteelMembers", Trans. of the Royal Institute of Technology,Stockholm, Sweden, 1949.

11. Westergaard, H. M., Osgood, Wo R., "Strength of ,Columns",Trans. of the A.S.M.E., Vbls. 49, 50, APM-50-9, po 65(1928)"

12. Zickel, J., Drucker,D. C., Report 1 to Column Research Coun­cil, "Investigation of Interaction Formula; fa '1'bApril 1951, ,Brown University. --F + p- !l;

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