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~ -

A LIMITED INVESTIGATION OF THEBENDING OF 24S-T ALUMINUM ALLOY ANGLE BEAMS

IN THE PLASTIC RANGE

U)

BY

BRUCE G. WOOLPERTdc H. NORMAN ABRAMSON

HARRY A. WILLIAMS

TECHNICAL REPORT NO. 9

AIR FORCES CONTRACT W33-038 AC- 16697

DEPARTMENT OF CIVIL'ENGINEERING

STANFORD UNIVERSITYSTANFORD, CALIFORNIAL

I 2~fJIUTIONI Appro.po-

APRIL 1,1949tn

A LIMITED INVESTITION OF THE BR!NDIN OF

2.S-T AL I M ALLOT ANGLE BFAS IN TN PLASTIC RAuO'GE

Brum 0. Woolpert, H. Non-wn Abramom

and Harr'y A., W~fl4lnm

Technical Report No. 9

Air Forces Gontract '33-O38 Ao-16697

Civil Engieering %pjartw~nt,Stanford University.3anford, Califo'rna

April 15, 1949

, /

. ... ~. a-r

1.

ABST~RACT

An earlier instigation revealed the riecossity for further -tudy of the

bending of unsymetrical beams in tha plastic range, inmch as twist occars

and has a marked effect.

Angle cross-section beams of alnjium allory 24 ST material ware arabjected

to bending in the plastic range with the plane of loading at angles of O°, 30o,

60P, and 76o-071 to the minor principal axis of the croassection. The moment

was applied by means of an eccentric load, with a relatively long moment arm.

For a simplified approach, it was found that Cowsone'a method of plastic

bending analysis and an analysis based on an exponential relationship between

stress and strain both. gave reasonable correlation between theoretical and ex-

perimental bending mwents in the plastic range beyond the yield strength.

Both methods are approximato since they assume that the neutral axis does not

rotate or translate and the cross-section does not rotate. The exponential

method is not valid in the range below the yield strength.

Other factors concerned with the general behavior of angle beams sub-

jected to plastic bending are also considered.

1

- -- . . -.,- '. .- .. - • - : > _- + -. " "

2.

Pure bending in the plastic range shen the applied loads are parallel

to a principal plane of bending has been studied by a number of investigators

(references 1 to 6). rWhen the loads are not parallel to a principal plane,

the complexity of the problem increases many fold. The comiplexity is further

increased when the beam cross-section is not symmetrical with respect to a

principal axis. Goodier's general solution of the problem for open sections,

(roference 9), indicates the difficulties encountered even under elastic con-

ditions. It ia seen at once that pure bending, as it is usually undersood,

does not prevail except under very restricted conditions.* If the applied

torsion and axial thrust are eliminated from Ocodiers solution, it is evident

that a torsion couple must still act at any cross-section except the mid-point

of the length unless all of the deflection is parallel to a principal plane.

This couple produces twist and it is evident that the relationship between

components causing normal stress must vary along ,he length of the be%.

Hence, the neutral axis must rotate also.

If plastic action prevails to any large extent, the various parts or an

open section must be relatively heavy and the usual assumptions fr shear

flow might be questioned. The major obstacle to the solution of the plastic

bending problem results, however, ftri the non-linear relationship between

stress and strain. This leads to even more complex relationships between

moment and stress and eliminates the use of superposition. Fxcept in special

cases, pronounced curvature results in considerable rotation.

It is evident that a rigorous solution of the problem of plastic bending

oP unsymmetrical beams must be developed from the concepts of pure plasticity.

*This Ceneral subject is discussed further in reference 1n.

- -- -- ~-

3.

Phillips presents tie exact theorj of c.nbined pure bending and axial force

for a prismatical bar utdng the equations of Us theoz7 of plastic deformations.

The solution of the differential equatlons is extreely difficult. Hence, it

appears vise, for the benefit of the designers to seek approximate but practical

solutions which il give reasonably accurate rasults.

In a previous investigation, by one of the authors (reference 7), of plas-

tic bending when loads weres not parallel to a principal plans, some exploratory

tests of angle bams mre made. A modification of Cozsone'a approach (reference

5) gave a resisting moment at the mid-span which was less than 15 per cent larger

than the applied end moment until plastic instability occurred at which time the

per cent difference increased rapidly. These experiments were made with 1/h" by

1" by 2"1 angle bews of 75 S-O aluminum alloy with an 8" clear span. Tests were

made with the plane of loading at various angles from 0 to 90 with the 2" leg.

The present investigation is an extension of the previous work and is con-

fined to angle beams of the swe cross-section. In planning the program it was

decided to increasa the clear span to 241" in order to accentuate rotation and

to apply the end moents ts reans of a small eccentric load with a relatively

large moment arm.* The primax purpose is to determine to vhat extent a rela-

tionship between bending m=ment at the mid-span and outer fiber stress wil

agree ith the applied end moment if a simplified theoretical approach is used

in the analysis. Hence, the resisting moment is computed by a modification of

oatone's graphical method used in refermnce 5 and by an analytical method

based on an exponential relationdhip between stress and strain. BoVa methods

neglect rotation and other variables in the interests of simplicity. Data on

rotation of the cross-section and movement of the neutral axis axv also pre-

sented to add information regardinG the general behavior of the tested beams,

*As used in this repert, rotation refers to twist and not to changes in end

slope of the longitudinal axis.

NOTATION

a constant dependent upon the materia3. for the exponential equation

a distance to point on neutral axis, inchesc4ai change in ac inches

b length of short leg, inches

c general symbol for d tances from neutral aes, inches

c., c distance frm neutral axis to outer compressive and tensile fibers.,

respectiveLy, inches

d, g symbols for particular crossecitoa dimsnsions. inches

a general symbol for strain. inches per inch

0 e 0 eouter fiber compressive and tensile strain, respectively, inches per inch

S t1 to *12 strains determined experimentally, inches per inch

fc' ft stress determined frm cmpression and tensio,- tests, respectively, psi

f , f proportion limit stress tuid yield strength, respectively, from averagep 7

stress strain curve, pal

fm outer fiber bending stress, psi

Pmc' outer fiber compressive and tensile stress, respectively., psime tfo Oorone's intercept stress, psi

' j t intercept streas for compressive and tensile area, respectively, pal

kc, kt Gossone's beam section constants

K constant in exponential equation, psi

Xn, y D .distanoe to point on rotated neutral awxs, inches

E Young' s modul~is of elasticity, psi

Ix X IV moments and product of inertia of cross-section with respect to x

andy axes, incheS -I

X ly mo-.ent of insrt.s -4 It respect to prncipal axis, inches 4

I i moment of inertia of c=mpressive and tensile areas, respectively,

with respect to aeutral axis, inchos

K f plastic stress function based on average strssa-strain cu,-ve, psi

Kfc 3Kt plastic stress function baeqd on ca-pression and tensile stress-strain

curves, respectively, psi

L moment arm to centroid at zid-span, inches

At experimental bending mwment, Inch-pounda

'Ath bending mament c-mputed analytically, inch-pounds

Mt M" be 4ing mcments computed awiytically Cor compressive and tensileth th

areas, respectively, inch-pounds

n pomr to zhich e and p are raised in tho exponnti l equation

P dynamo=eter load, pounds

( p( statical moment of compressive and tensile aroas, respectively, inches 3

r distance from neutral axia to an olement in the beam cross-section used

in the derivation of the exponential equation, inches

t thickness of angle beam leg, inches

AT Aac sin y, inches

a angle between principal axis and neutral axis, degrees

Aa change in angle a, degrees

a angle between plane of loading and y axis, degrees

0 angle betueen plane of loading and normal to neutral axis, dagreoe

p radius of curvature, inches

y angle between x-axis and neutral axis, degrees

(

6.

'TEST SPMEN ,:D I PiAPUTUS

The test specimens mere macined to the dimensions shomn in Fig. I from

two ten-foot long 24 S.T alhmium! alloy bars. Those bars were I by 2 inchas

in cross-section and zere designated bar L and N. Photographs of the

specimens after testing ame ab.ozn in ng. 2.

Considerable difference betgen the stress-otrain curves in tension and

compression exists for 24 S-T alminu-m alloy. Wt iimmuch as this material

is coamonly used in practice, it nas decided to use it rather than ec-e other

material which might have tension and coprassion curves that are in botter

agreement. ,l analytical computations erse based upon an averago stress-

strain curve. Tension and compression samples frm a number of bars, including

U and N, gave stress-strain data as Indicated in igs. 3 and 4. The curves

of . gi 5 were selected as being representative of the material in bars M and

N and imre used for all coputationm.

Three tension and three cempression specimens vere taken from each end

of several bare of the sar4le sbipment. The tension specimens vere 8 inches

long over all and were machined to 1/8 inch by 1/2 inch at the reduced section.

The compression spocimens were 15/16 inch by 1 inch by 3 inches in length and

might be referred to as the block type of compression specimen.

Loading Apparatus

The arrangement of the loading apparatus used for bending thi anele beams

Is shown in :'igs. 6 and 7. The tmo lover arms nachined from 3/h inch steel

plate were supported at one end by cajle3 from an overhead beam and at the other

end by ball bearing rollers.

.2,

1.

Tracks for the rollersa wre attached to a heavy table inediately below

the suspended arms and so an~anged as to constrain the movement of the rore-

end of the ,ams in a path perpendicular to the center line of the apparatus.

The ams at the cable end were provdd with fittings which securely

held the specimen in a position perpendicular to ths two arms and parallel to

the tracks, thus forming a rigid fraze. The specimen fittings were further

designed, by means of a movable disk uhlch could be secured with a retaining

ring, so that the specimen could be rotated about its lengthuise axis to pro-

vide any angle of bending desired. The load ras applied through cables at-

tached to the arms as ahomn. It is obvious from the loa(ing arrangement that

pure bending sa only approilmated., due to the presenco of a alight axial load.

Strain Apparatuia

Strain readings ere taken with an SR-4 type indicator which is shown to

the left of the bending apparatus in ig. 6. The strain uere mea&ured at

various points apound the center cross-section by SR-4 type A-12 electrical

strain gages cnuented to the specImen. A typical arrangement of strain gages

around the center cross-section is shown in Fig. 8.

(

8.

Each specimen "sas measarde with a tiacrmeter and electric strain gages

rare cemented to the specLmn along previ -isly scribed lines, The seecien

was then placed in the bending apparatus and carefully adjusted to the de-

sLred angle of rotation. A pro2m1r- check on the operation of the entire

systen was then made by bendin the baem until the maxlimhm stress was approxi

mately tw-t rds of the proporton<Q 31Z't and reducing the recorded data to

conf im~ the proper operation of zal. 27\2onn!t9 of thesyt.

iThe normal testng procedure .W to apply an inc-ement of load, stop the

testing machines and tixuediate~y rcwd th stratn indicator readings for al-

gages. The latter step was parti larly important in the plastic range be-

cause of creep.

As the plastic range was reached, the curvature of the beam decreased

the moment am. This decrease was masured by means of a plumb bob attached

near the centroid of ths beam and hanging along side a scale measuring to the

nearest 0.02 inch.

Specimens wore removed tten the reduction in moment wnn was sufficient

to tend to prevent an, increase in bending moment when additional load was

applied. Generally, one or more of the electric strain gages bad become in-

operative at this point, and the maximm outer fiber strain was on the order

of 2 per cent.

Rotations were measured by means of an arm attached near the beam centroid.

A protractor and plumb bob at the end of this ann enabled the rotations to be

read directly to the nearest 0.2 degree.

9o

MIAL!'1'CIAL PROCEMYRE

Cozn I -ethod

Frwt reference 7, Cozzom'- maethad, as pro.osed in reference may be

modified for an angle baamn if the rotation of the beam in the plastic range

is ne&lected and tf* nutra aris is asslmed to remain stationary. Neglecting

these factors, an apwroxtiate solution may be made by dividing the cross-

section into a copression area and a tension area. The resisting moment of

the compression area is obtained ty ass.uzing the symmetrical cross-section

shozn in Fig. 1(b). The resisting moment of this beam is computed and divided

by two. This is equivalent to computing Ic for one-half the cross-section

"meonly. Then k* and Vie moment becomescC

-- r* (kc - ('1)

A similar approach for the tenzion area (F-ig. lc)) results in the

equation'Nt Eft).c

MrF o (k- 1) (2)th c.t 0 ot

The final restraining moment is

thth th

The relationship betneen stress, strain, and intercept stress (f) based

on an average stress-strain curve is presented in Fig. 5.

w. (

100

Exoonential Method

An exponential relationship between stress and strain may be formulated

which will closely ap roi:cate the stresi-strain curve in the rnge of higher

strains. This relationship takes tha fon

fe - en K (h)

But it must be noted that then stress equals zero, stmain is not zero althugh

it may be approached through proper selection of the constant K.

a 01 then equation (4) becomesP

n-I5)r

P

In the case of angle beans,, an expresion for bending moment may be found

as follows,

n~ ,n+') (do-c11h co-7(~l (n CU 92 8

+ t a (6)

where all notation is referred to Fig. 9. A rather complete derivation of

equation (6) may be found in the appendix.l'

C~aison of Bending !Aagontwa

It is easier to compare experimental and computed bending moments in the

plastic range than to compare outer fiber stresses; hence., the general pro-

cedure of this report has been to take the experimental strain corresponding

to a given load, determine the necessary stress values from a representative

stress-strain curve and then insert these In the appropriate formula to com-

pute the so-called theoretical bending moment.

The ratio of Uth/Ut has been taken as the criterion for comparison of

bending moments.

P. %41

1!.

RESITS AND :DISOUSSION

In order to avoid needless repetition, it was dcidsd to present detailed

information on a single typical angle beam, namely .4,. 6 = 00, and to pre-

sent only the important test results for the remaindear of the specimens tesedo

Angle beam '-L has been omitted from this report inasmuch as its pur-

pose was only to provide a check on the raliability of the bending and strain

measuring apparatus.

Strain Measurements

Typical Load and strAin data are presented in table 3. where the parti-

cular strain gage locations are shown in ?ig. 8(b).

In general strain rnadings agreed reasonably ell for two specimens tested

under the same conditions. ith omse exceptions, the difference was not more

than 5 per cent in the elastic range and 10 per cent in the plastic range. The

latter might be partially explained by the non-uniformity of the material itself

and by the difficulty in locating SR-4 gages exactly the same on duplicate

specimens. Such discrepancies, hoesver, did not greatly affect the computed

bending maent because even large dL.fferences in outer fiber strain did not

affect the stress values zhen wor dng on the higher range of the stnrss-strain

curve, where the slope is small.

Maximum Strains and Neutral Axis Locations

Maximum ccwpressive strain (oc)p maximum tensile strain (e ), locationof neutral axis intersection vith the outside of the short lag (xn), and loca-

tion of neutral axis intersection with the outside of the long leg (yn) were

all four by a graphical method ubich assumes a linear variation of strain across

/

_ .:- - ..'- --

12.

each face of the beam. A sample solution illustrating the ethod is given in

Fig. 10. The procedure was to lay out to scale a leg of the beam with proper

gage locations and for a given load plot the st-ains for each gage perpendi-

cular to the leg. A straight line connecting all strains for a particular

bending moment was drawn, from *Aich ec (or et) and z (or yn could be

read directly. Tables 1, 2, h, 6, 7, and 8 present this data for all beams.

One would expect the outer fiber strains to become less exact when deter-

mined as described above as the rotation and warping increases. flovver, such

errors would have little effect on bending moment in the plastic range.

The xn and y values are approximations only because these distances

are quite sensitive to small change in strain. -vTen a gage was near the

neutral axis and the strains were relatively small, the effect of errors in

the strain was quite pronounced. Heomever, the results when considered col-

lectivoly indicate definite trends.

Theoretical Va xErrimental iBending Uoents

Theoretical bending moments were computed from equations (3) and (6).

The per cent differences between theoretical and experimental bending moments

were computed from the expression (Mth/"t 1 1) x 100 per cent. A computation

table for theoretical bending moment by both Cozsone's method and the expo-

nential method is presented for a typical angle beam in table 4. Tables 1 and

2 and 5 through 8 present the values of M th by both methods for all beams,

as well as the equation used. These values are presented graphically in Figs.

11 through 13. For comparative purposes, results for different angles of

loading are shown on the same figares. The elastic, partially-elastic, and

plastic ranges are shown on these graphs by indicating on the moment scale the

(I

- - - '- ,. --. -

13.

approximate values of the experimental moment at which the proportional limit,

f and the 0.2 per cent offset yield stren1 #t, f . were reached in the

outer fibers of the ba= in question.

An inspection of Figs. 11 through 13 shows that the theoretical bending

moment was generally within approximately 10 per cent of the experimental

moment in the plastic range beyond the yleld strength for both Coszone's method

and the exponential method. The differences in the range below the yield strength

were approximately the same for Coszone's method but very much greater for the

exponential method, which is to be expected in view of the fact that the expo-

nential relationship between stress and strain should not be used below the

yIeld strength. In certain cases where the twist was large, the differences (

were slightly greater.

It will be noted that the maximum applied roment is larger for one beam

than for the other at both the O° and 300 positions. However, the difference

is only about 7 per cent which might be explained by a difference in the

.materials and by the difficulty in exactly duplicating test procedure.

Beam M-5 which was tested at 609 gave results consistent with other speci-

mens. It is seen from Fig. 1 that the neutral axis makes an angle of less than

20 with the principal axis. There was little rotation of the cross-section.

It was then decided to test beam N-7 in such a position that the neutral axis

coincided with the principal axis. The results showed negligible rotation of

the cross-section although the rotation and translation of the neutral axia

were consistent with other specimens. However, the difference between the cAv-

puted and experimental bending momont appears out of line with the results for

the 600 beam, U-5. Since there was insufficient time to repeat the experiment,

no explanation can be offered 'or the discrepancy. V(I

C~m oition of te N ora Ai

Figures 14 through 16 show the rotation and translation of the neutral

axis for the various beam~s. The values of Scx and AT 0may be computedC

from the expressions

6a (90 0 - ai) - tan' (x ).130580 (7)

Yn

aa 0.261 + 0.760-n _x (-)cyn n

T %a sin (a - 13°58) (9)

These expressions are derived from simple geometrical relationships, referred

to Fig. 1. The angle bc in considered positive when the neutral axis rotates

clockwise. Aa is positive if it represents an increase in a • . henC C

AT 0 a sin a is positive, it indicates that the neutral axis is translating

toward the compression side of the beam. It should bs noted that all of these

changes are with respect to the xes of the cross-section. Obviously they are

intimately related to the cross-section rotation as will be discussed in the

rext section.

In general, rotations were all counterclockwise and were of the order of

Magnitude of 3 to h degrees for all beans. The transations were all positive

and of an order of magnitude of 0.01 inches except for angle beam N-7 which

was approximately 0.02 inches.

The same information is presented in another manner in Figs. 17 through

19 bV plotting the values of xn and yn against applied moment. since xn

and y were obtained by graphical methods, the values are approximate only.

(

i1;.

Curves have been indicated to show general trends, M;hen the curves move in

the same direction, translation is indicated; when they move in opposite

directions and cross over, -ota:,Uin is indicated.

Behavior of Resisting Counle at Mid-SM -a Gross-3ection Rotates

The torsion couple which causes the beam to rotate must be equal to zero

at the mid-span because of symetry. Hence, the only couple acting at this

section is the conventional bending moment couple. Also, equilibrium condi-

tions demand that it be equal in magnitude and opposite in direction to the

external couple. It must also lie in a plane which is parallel to that of

the latter.

The location of the resisting couple for beam !-h is shown in Fig. 20 for

three stages of loading. The resultant tensile and compressive Corces are Ft

and %,0 respectively, and are located approximately to scale. The forces

were evaluated and located by graphical integration. The difference in magni-

tudes was as roalows:

!xternal ResistingFig. 20 Mw=ent Ft Fc AM Cotiple(in-kcips) (hips) (kips) (inches) (in-kips)

(a) 6.1 5.0 h.7 1. 24 5.S2

(b) 11.90 10.3 10.6 1.00 iO.45

(c) 13.03 13.7 13.2 0.86 n,55

It is seen that the porcentage differawcw remains essentially constant

with increasing load. Some error is introduced by the graphical solution,

and also b the fack that the true external moment couple does not remain In

a horizontal plane but rotates slightly as the beam deflects vertically.

Hence, its magnitude becomes increasingly larger than FL and the rotation

16.

is counterclockwise in Fig. 20. It will be noted that the resisting couple

rotates in the same direction.

The sketches of Fig. 20 also show that the neutral axis rotates consider-

ably less, with respect to the cross-section, than the latter rotates with re-

spect to its original position. This difference depends on the shape of the

cross-section and on the degree of plasticity. In this case, when the left

hand tip of the tension log becomes stressed beyond tbe proportional limit, a

"softening" results %nd the resultant,, F., moves toward the neutral axis and

away from the "soft" area. Vn the stres3 at the outer compressive corner

moves into the plastic range, a similar shift is made by F c Since the re-

sisting couple must remain parallel to the plane of the external couple, the (

neutral axis rotation must be adjusted with the cross-section rotation so that

the stress distribution satisfies aquilibrium conditions.

Deviation From Pure Beami

As mentioned earlier, the bending apparatus used only roughiy approximated

pure bending. Actually. an axial stress having a maximum value of approximately

1000 p.s.i. existed. (Quite often in practice combined loadings of this type

are encountered; therefore, the investigation of this report more rearly ap-

proaches practical applications. In contrast, a rigid method of analysis for

the loading used is considerably more difficult than for pure bending. However,

it has been shown that an analysis which assumes pure bending gave f&drly con-

sistent revilts none the less. Since rotation was neglected, the agreement

resulted to a certain extent from the fact that errors were partially, compen.

sating and moment was not particularly sensitive to strain variations in the

plastic range.

OF

17.

As a matter of interest, strain gages wore placed on beaa M-5 as shown

in Fig. 8(a). The results are shown in table no. 6 where it will be noted

that the moment am has been corrected for the different degrees of curva-

ture existing at stations B and C.

Inspection of the data shows that strains uere essentially constant at

the three stations. This is partially accounted for by the fact that while

the resultant m,-ment couple increased from A to C; the bending couple about

the neutral axis remained approximately constant after the tension component

vas subtracted. flowever, the xn and Y. values show that the neutral axis

location was difrerent at each station as might be expected.

i i

! 1 €'

1 :' -- I

18.

CONOLUSIOn S

Since rotation of the beoav cross-wection and of the neutral axis were

neglected in the analysis, arq conclusion must be confined to tho particu-

lar beam shape used in the experiments. If the analysis is extended to

shapes having essentially different proprtions from the one reported in

this investigation, the differences between a theoretically computed bending

moment and the actual moment might easily be saveral times as large as these

reported. This statement also applies if another type of loading is used.

The method should be extended to other material with caution.

The following conclusions can be dramn from the results of this par-

ticular investigation;

1. A modification of Coazone's method of plastic bending analysis gave

approximate but reasonable results. For the particular material and Iconfiguration investigated, the correlation between computed and ex-

perimental bending moments in the semi-plastic and plastic ranges was

within 10 per cent.

2. An exponential relationship between stress and strain also gave reason-

able results. For the particular materiW investigated, the correlation

between computed and experimental bending moments in the plastic range

beyond the yield strength was within 10 per cent. The method is not

valid for the range below the yield strength. Simplification of the

method appears to be possible, but required further study.

3. Rotation of the cross-section was the primary factor limiting the ulti-

mate strength of the beame. This rotation cannot be included in a

simplified anlysis such as was used in ths investigation.

- (

19.

4. It is suggested that if further Investigatiors of the general problem

of this report are to be carried out., a larger testing machine be

used in order to carry the tests as ar into the plastic range as

possible. Further suggestions are: (1) an apparatus for providing

a more nearly pure bending condition than the apparatus of this re-

port be used, and (2) that rectangular specimers be used with test

sections machined to provide a variety of cross-section shapes and

proportions, as vell as lengths of test sections.

I

Nt

I - -~ -

-1- -

20.

1. Timosiiqnko, S.,Strength of Materials" Pert iI 2nd Edition, Chapto VIII;

D. Van Nostrand Co,., Inc., New York, 19 UI.

2. Beilschmidt, J. I..,"The Stresses Develooed in Sections Sbjected to Bending

MomentV Journal o the Royal Aeronautical society, Vol. ;LVI, No. 379, July

1942, ?ages 161-182.

3. Osgood, W. R.,"Plastic Bendng Jour. Aero. Sciences, Vol. 11, No. 3, July

1943, Page 213.

4. Rappleyea, F. A. and Eastman, E. J.,,"Flexural Strength in the Plastic Range

of Rectangular Iagnasium xtrusions' presented at the 12th Annual Meeting

of Inst. of the Aero. Sciences, New Yorc, January 1944.

5. Cozsone, F. P.,"Bending Strength in the Plastic Range" Jour. Aero Sciences,

Vol. 10, No. 5s P. 137, mAay 1943.6. Williams, U. A.,'Pure Bending in the Plastic Range" Jour. Aero. Sciences,

Vol. ih, No. 8. P. 4557, Au ust 1947.

7. Williams, H. A.,"1n Investigation of Pure Bending in the Plastic Range fhen

Loads Are Not Parallel to a Principal Plane",(Performed under contract to the

N.A.C.A.; a-5 yet unpublished.)

8. Goodier, J. N.,"The Buclling of Compressed Bars by Torsion and Flexure,"

Corne.ll University Experiment Station, Bull. No. 27, December 191l.

9. Goodier, J. Ni.,"Flexural-Torsional Buckling of Dam of Open Section Under

Bending, Eccentric Thruat or Torsional Loads" Come! University Fxperitent,

Station, Bull. No. 28, January 1942.

10. Davis, D. 3o,"Ebipirical Equations and Nomograpvy Ist Edition; McGraw-Hill

Book CO., Inc., New York, 1943, *ages fl-i.

(

21.

i. WlliaIs, 11, ., "Stabil:1yW of iectangular and I iection Bewas in

the Plastic fange", Technical Report No. 8, Air Forces Contract

W33-038 Ac-6697, June 1, 1949.

12. Phillips, A., "Pare iiending with Axial Force in the Theory of Plastic

Deformations", Stanford University Techical Report No. 3, J4anuary 1949.

J

l ,,

- ___

22.

APPEMdIX

Derivation of-the i ntiql iqation for Theoretical Bending Roment of an

n 3e&M

A number of equations relating stress to strain beyond the proportional

limit have been suggested. Several of these xere Lnvest-igated by the authors

with the view of developing a suitable flexure formula for an angle cross-

section. MIany of the expressions lend themselves quite readily to a simple

formula or a rectangular beu w-ith loads in a principal plane but become

quite complicated for mare general solutions.

It was noticed that when the product fe was rlotted against e for a

number of materials, the outer portion of the curve was of tho fom

fo a an

It was noticed furthor that this expression could be modified by conventional

methods (reference 10) so that it also roughly approximated the straight line

portion of the stress-strain curve by writing it in the form

fe3 aen -n

or r a - - K/ (1)

It is evident that this curve does not pass through the origin when o ,, 0;

however, K can be adjusted so that it passes quite close to the origin.

Using this equation as a basis, approximate expressions can be derived fo~r an

angle beam.

it no warping of cross-sections occurs (a12 notation referred to Fig. 9)

rP

andr() (2)

P

23.

onsidering the area above the neutral axis, i.e. the compression area.

c %-rOuter triangles U2 fre where CU U

fd cu

-n rdl dr

d (3)

Considering the area below the neutraL axis, i.e. the tension area, Ii

Ln,,,1

b-leg: n a le- KP1 (7)

The total resisting mmnn is tho sum otf equation~s (s), (6) end (7) or

t~ nf)l)(d Cu c ga (d a g

to3 /-a_( cu 9 it ()

C t

h log:

2l.

Dscussion of th nEonenti lethod

Figur 21 presents curves which may be used for rapidly deteraining

values of p n- c, and an+o for known values of p and c. Although

this figure is limited to aluminum alloy 2h S-T material and a particular

value of n, it is relatively simple to construct similar figures for other

materials after the proper values for n and K have been determined.

Values of n and X for materials similar to the one of tt* present

investigation are not difficult to determine, by the method of reference 10,

when applied to the stress-strain curve of the material in question.

An exploratory investigation aimed at siplifying the exponential method

would appear to indicate that for the portion of the stress-strain curve beyond

the yield strength, equation (8) may be modified by setting 9 a 0. The effect

is to increase slightly the value of the compouted bending mment. Again, such

a generality applies only to the particular conditions of this report.

*Inasmuch as the unmoditied exponential equation (eq. 8) usually gives values

for the computed bending moment whieh a less than the axperimental mrtent,

the modification just mentioned would apparently tenr; to bring the emixpted (and experimental moments into closer ageemont.

Plane of Couple

2.00 " X/"/ ,500

yN.A

Y(c)

SECTION PROPERTIES

Rectangula'r Axes Principal Axea",x = 0.2706 z, . , i i

].s 0.0456 - , =0.03081 2 " 0.0600

e 0 .I Q.O, Q,,, C, ___

0_____ 66033' 52=35' 0.03530 0.0385 0.0929 0.0862 0.670 0.744 ,

30* 83°37' 39=39" 0.0167 0.017:3 0.0635 0.0622 0.507 0.51860 • 88•13 14• 15' 0.0140 0.0177 0,05Sf 0.0558 0,457 0.572

76= 07' 900O0' -0•07 0.0127 0,0183 0.0570 0.0537 0.434 0,592j

FIG. I - ANGLE BEAM CROSS SECTION PROPERTIES (IK _K 1 __ __i , 0"t c

00 603' 503' 0033 0.38 0.9290.0620.60 0T430 33'3-9 .17007 .65002 .0 .1

60 81' 105 .101.17T00 8 .581.5 .7

76t7 900 0ce00171a 83007 .5T043 52

m- -3 k-4 -5 N-6 ti7

o M -* o * .6 * MU 0 76 'O

14'r

FIG., 2 PHOTOGRAPHS OF BEAMS

72

68

64

600

566 0 08 0 0 0 00

0 0 0

0008 0 000000 0

52 - o°0 00 00

8 0 8

00001848 0- -

(8- 4 0

c44 --- __

0

00

0 400 0

00

C 36 00

32 -s---(/) 00

t 28 -a--I-U)

0

24 8 _

0

0020 e

000

160

Material:D Aluminum Alloy 24 ST.

8 Results from eight coupons1/8" thick and 3/4" wide

at reduced section.

4

.01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11STRAIN (inches per inch (

FIG. 3 TENSION TEST RESULTS

72

0

68-0--0

0

0

00

00 00

8oo

00

52 - g 000

480

44 0

0b

V) COooo

00w 0

2 36 &0_ OD

,- %32 0 0

0W 0(n 0w 0

S280

000

24 --g________

0

82

0

00

12

0 Material:

Aluminum Alloy 24 ST.

Results from eight

coupons l1ox I.x3,,

0

1.01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .1

STRAIN Inches per inch

FIG. 4 COMPRESSION TEST RESULTS

.Aii li t-A .1 1_1-T

-A j 6K

7M

;sr

Elf_T:_F==1=_ T

_M 71

t - T'

1 ' 71 f A

ME

zT. ____f T_

tm:_ g=!

_F _

_ TE -

EMIEE..M.H.

T1.K. :N K RiR- "7

. ..... .... - - ttv

- - N4 P

P L fl-I H+

ar

7-, z _

flu ffih UH I

M3. tm gg 3iffBi 44 rl-, Am i4il i4l -1;-. =Hm +. j.MIN it.=.- = - :

Hi Ml#rut:

iiv" Zia u :&. R.r=-r N Am t P M ini hgfflw Nf

4 f4l; Pfilil' llit

M tillH M Off

L

L-A UP om zz+.tM #9 Em 'D =M rv .rj Uj -4HN

_. R. fl- N!M= _:._ * .. = iftH

M. R im FIE :3M uiffi 9:2 59 ffu 9 HIM fl -, .,= + . Ill"

-:+L; hHUM, HE f.M- Htv- iir-

N TO gr -A.14 44.

U gi

...........

_iGiH m HHH 4k

HH Tm

Mill

fnOURB 6

FULL VIEN OF PLASTIG BFNDOG APPARATUS-

! 1

(H,

- - --- -,-- ~

kv4

SYM

0Nq

000 000

l0 1

0 0

FIG. 7 -PLAN VIEW OFPLASTIC BENDING APPARATUS

A B C

9.0-12.0

24.0

(a) LAYOUT, ANGLE BEAM M-5, e= 600

5I.150 2

( N.A. (theo.)

~(b) CENTER CROSS SECTION,

ANGLE BEAM M - 4f 0 =0

.500

.150

FIG. 8 -LOCATION OF ELECTRIC STRA1I GAGES

C

-- - - --- - =

Plane at Couple

y

xI

For 24 ST AI. Specimens:o 91,4.50 psi

n=1.1228k : 25 psi

FIG. 9 -CROSS SECTION NOTATION AS USED IN THE

EXPONENTIAL SOLUTION FOR Mth

,U,

z/

A --

1 € 9

L - -- , .. ..

160__ _ _ _ _

140

120

S80

06

0

a,940

C 2

a)n

o MEASURED STRAIN

FIG. 10 -SAMPLE GRAPHICAL SOLUTION FOR

OBTAINING X., A ND e.; ANGLE BEAM M-4,

0 +100 0+0 0 0 00 00opt ~

-10 CozzonT's Met od

+100r 0 oLL 0- Exponent al Me hod _ _ _ _

3 4 5 6 7 8S +100

EXEIENA-EDIGMMNT to(i-is

0 0 0 r 00A^0000

Contie's Method

IIJ 2-3 5_ _ 6_ _ 7 8_

101

zi Exponetial Mthod

+U0 0_0 0

U -_ __-10 00 _ 0 0 0 0 0 6

0 f 1

_ I

0 2 4 6 8 10 12 14EXPERIMENTAL BENDING MOMENT, Mt - (inch-Id ps)

(b) ANGLE BEAM M-3 od 0

FIG. I I -%R/ DIFFERENCE BETWEEN EXPEIMENTAL ANDCOMPUTED BENDING MOMENTS FOR ANGLE BEAMS M-2 a M-3

LL 0

0 11o 0 0 000 0

-10 Gozz 0e's Me thod

+1I0+ 0 Expor ential Method

00W 0n- 0 & C n m Ww -10 __o_ 0UI.U-6 0 fp fy

0 2 4 6 8 10 12 14EXPERIMENTAL BENDING MOMENT, Mt - (inch-kips)

(a) ANGLE BEAM M-4, -O°

A B C

o Cozzone's Method

a Exponential Method0

a 0 0 0 00 Sectior A 0, 0 nnlA

0 6 t A00 -10 A

0

+10 A 0 00

0 Sectio B 0 "

" -10 ___ 86

w

w 410w 0 0 0w Sectf C A"L 0A _,' -,

-10

0 1 2 3 4 5 6 7EXPERIMENTAL BENDING MOMENT, Mt - (inch-kips)

(b) ANGLE BEAM M-.5, 8= 60°

FIG. 12 - % DIFFERENCE BETWEEN EXPERIMENTAL AND

COMPUTED BENDING MOMENTS FOR ANGLE BEAMS M-4 & M-5

0

2 +10 -- _

0 0o 00 0

$ $EXEIET ExpoALBEiD l MT hod

0 00oL 0

10 o o oc

4-10

lp fy

2 3 4 5 6 7 8EXPERIMENTAL BENDING MOMENT, Mt - (ind'-kips)

(a) ANGLE BEAM N-6 0=300

0 +10 c (°Oo 0.

010 Coznones Me hod

w r, 0 0 (0 0000 ncr +10

lLi 0U. 0~Expoe ntial Mithod-I0

cfp fy

0 1 2 3 4 5 6 7EXPERIMENTAL BENDING MOMENT, Mt - (inch-kips)

'(b) ANGLE BEAM N-7, -&= 76"07'

FIG. 13 -%DIFFERENCE BETWEEN EXPERIMENTAL AND

COMPUTED BENDING MOMENTS FOR ANGLE BEAMS N-6 N N-7 !l2

__I__ _ _

+.01o0

I-.0

z

4?

01-

ICo

___0 __0---

eL>

ix 2 3 4 5 6 7 8EXPERIMENTAL BENDING MOMENT, Mt - (inch-kipo

0(a) ANGLE BEAM M-2, e= 30

0+a -.01 -o o

-0-

.01

I-

wCn82 __ oaoO

0-o 0

0 -

0

6w0 0

leod

0 2 4 6 8 10 12 14EXPERIMENTAL BENDING MOMENT, Mt - (inch-kips)

(b) ANGLE BEAM M-3, e= 00

FIG. 14 -CHANGE IN THE POSITION OF THE NEUTRAL (AXIS FOR ANGLE BEAMS M-2 & M-3

~ , a - a-

U,0)C-

0

<3 +.0100 0Q

zn

3 0-- 0

00

0

000 n o2

0

0 0 l 0

c 0 2 4 6 8 10 12 14EXPERIMENTAL BENDING MOMENT, Mt (inch-kips)

(a) ANGLE BEAM M-4, G=00

0)0

C.,

z

00

w 3o 0

"- -000

w 3

I-

2

, 3 I__ o

"" 0

200I 3

a: 0 1 2 3 4 5 6 7EXPERIMENTAL BENDING MOMENT, Mt (inch- kips )

Wb ANGLE BEAM M-5, 0=60"

FIG. 15-CHANGE IN THE POSITION OF THE NEUTRALAXIS FOR ANGLE BEAMS M-4 B M-5 i

6 A *,- - - i..

o .... 2

-9 +01

. 0 0 0 0

--.00

zI--

30

f 0 0 n

-E I60 2-

0 ___

I-2 3 4 5 6 7 8EXPERIMENTAL BENDING MOMENT, Mt - (inch-kips)

(a) ANGLE BEAM N-6, e = 30

0 0__-___ o+.02 0

(0 0

+-.01000 0

n- 0 !_ I__5 __

EXERMETA BENDINGMOMENT,_ t_-_(ich-kpI

00

3 0 T P I O H U

oFzo __

0 1 2 3 4 5 6 7EXPERIMENTAL BENDING MOMENT, Mt -(inch-kips)

(b) ANGLE BEAM N-7, 9-7e'07'

FIG. 16 - CHANGE IN THE POSITION OF THE NEUTRALAXIS FOR ANGLE BEAMS N-6 81 N-7(

0.

-- 6

z

z0 _oZ44 4

0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60

Xn (inches)

1.44 1.46 1.48 1.50 1.52 1.54 1.56 1.58Yn, ( inches)

(a) ANGLE BEAM M-2, 9=30*

14

12

10

8 /-0

wd 0z 4 n d"Xn

m

ta\d 2

0 __ __ __ _

0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90Xn (inches)

1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18y0 (inches)

(b) ANGLE BEAM M-3, G=0 °

FIG. 17 -NEUTRAL AXIS LOCATION FORANGLE BEAMS M-2 AND M-3.

14

12

8

z2E Yr - xn0

6 Cl

(D0 Izw 4a:xw I

2 II I

0

Q076 0.78 0.80 0.82 0.84 0.86 0.88 Q90

Xn (inches)

1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18Yn (inches)

(a) ANGLE BEAM M-4, e0 0 °

L 8

zw0

(D Y

z 0J /

a: 0X 0.40 Q42 Q44 0.46 0.48 050 052 054

Xn (inches)

1.66 1.70 1.74 1.78 1.82 1.86 1.90 1.94Yn (inches)

(b) ANGLE BEAM M-5, 0-600

FIG. !8 -NEUTRAL AXIS LOCATION FORANGLE BEAMS M-4 AND M-5

- I

1-

8

6

m 4IA0

0

4

L2

(30

°mI0

a:

x

0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60Xn. (inches)

1.45 1.47 1.49 1.51 1.53 1.55 1.57 1.59Yn (inches)

(a) ANGLE BEAM N-6, 0= 300

8 1>yh

g6---

z

0

4

z

I xn

a.:

x 0

0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50Xn (inches)

1.80 1.84 1.88 1.92 1.96 2.00 2.04 2.08Yn (inches)

(b) ANGLE BEAM N-7, =76 07'

FIG. 19 -NEUTRAL AXIS LOCATION FORANGLE BEAMS N-6 AND N-7

- ~ ------------ -- .-. .- ,- -- - . --- ~.w$-~~ --- - - - -- ,I0--~ -- ~.-- -

Am of CoupleTension N..CompressionArea N.A. Area

t! i__P Fc lane of CoupleA 0(= 00 a CKI=l 10

ec.n. 0.003 ilfjq

(b)

(a) m V: 51"18'

N.A. =:53 ° 51'

A OC 122t Fc

ecmax.r 0.017 in/in

(c)r 55" 21'

FIG. 20- LOCATION OF RESISTING COUPLE AT MID-SPANFOR THREE STAGES OF ROTATION

ANGLE BEAM M-4, e2 0*(

- -

B 10- - - -

796 - I

7

6

3 - - - - - - -- - - -

2 - -- - - -. 4 --

I.~II -1V---

-~.4 -- -

-. 1

0

.0

.059

S.048

C .0 - - - ---

0.05

.1 .2 .3 A4 .6 .6.7 .8.9 L 2 3 4 5 67T8 910

c (inches)

CL

1 2 3 4 56 7 8910 2 3 4 56 78 9 le 2 3 4 5 6 78 90

p (inches)

FIG. 21 -VALUES OF p AND c FOR USE IN THE EXPONENTIAL

EQUATION

10

00

04 0

AV 0 wSll00*9 zc

-00

00

14 A

$A 0

9-.,~ ~ ~ 0% NV '

00'

00000 0~* 000 000

RU 0

04 0~

o 9

0 99090*

90 .00. 4 C

-% -

A,11. I -r 00

43)ri41i 0 0*

*~ r -

M,0,0~L C. ca 000 ~

m8 H 0 IUU I

4430~~~~ 0C1A 0~~0-

(D-

-M~

4;0 0 0 C;0 * 0* s

.3 43

0* C38611H 4)4C ;r

oo8 F, *00 11 8 M0 U R -8?

W94 09 96* -* ~ac 0 0000HH 0%0%0% 0hp%

H~~ CM (

0 ~ 00'.

Table No. 3 - LOAD AND ST AIN DATAAnge Bau M.L

Total ~IcOa. ExPt - IMeasmmd strairiload m band.

mc.

P L 't 81 2 3 64 5 06 7(kips) (In) (in-kips) (+) (-) (+) (.) ) () (4)

0 20.oo 0 o 0 0 0 0 0 00.315 19.85 6.21 3.1 27.4 28.5 23.0 11.9 13.6 7.80.435 19.75 8.56 4-2 39.9 41.6 32.o 21.5 2o.4, 10.40.5b 19.67 9.81 b.o 47.8 50.2 38.5 25.7 25.1 12.1o.Z 15 19.60 10.72 4.o 56,.4 59.4 45.4 29.6 29.5 14.40.59 19.48 3I. ,5 3.5 66.3 70.3 53.2 34.7 35.2 16.9 f0.60 19-43 11.63 3.7 69.8 74.2 55.3 36.4 37.0 17.50.615 19.37 11.90 3.3 75.4 80.0 59.8 39.0 39.8 1.00.625 19.32 12.12 3.1 8o.4 86.0 64.2 41.9 43.0 2MO2o.635 19.25 12.23 3.4 83.5 89.0 66.0 h3.5 4.6 P0.70.645 19.22 12.38 3.0 87.9 94.0 69.6 15.3 46.9 21.8o.655 19.13 12.57 2.4 94.6 i0 75.2 48.6 50.7 23.40.67 19.08 12.79 1.8 103 i11 81.0 52,3 55.5 ..0.675 19.00 iz.87 1.2 111 119 86.0 55.8 59.6 26.20.685 18.88 12.96 0 n7 126 90.8 58.7 63.4 27.20.695 18.75 13.03 -2.0 128 139 98.2 63.0 70.1 28.5

.~r- ri 0

La.

434rqHHr - H 04

43 W4 MU\0 '~ 3 o\Ml0roe N1c 'V-co 6-46

0 09; 0. . . 0 6 0600 0 0

03~-% 430V4 0 %0 *W *Oz9- 004 0 0V \0 (n 04an\

%-0 00

+

I. -~ co LI-S HOO WA O''SNRO,"% %a A2 4 H

so~

o ~ ~ 4 0 0 4** ~ 0

'4~C (4.-

0-VS co 0fnO000 oUU%\c

fr.~ ~ .:V:~:i . ..caH-4- & -H~-44 '

g ng&*'8

1 ~ co-~' 04~cU~ e4:A Ii

QC'.,!:I H. S4 A~~ N N O S q , . q,,

a-r~- S' .0 (RS0429 RP

Table No. 5 - COMPARMSON OF TiHEORIMCIAL ANDEXPERIME14TAL BENDING UOMETSAne-ie y~em Mi-4

Expt. Cos son's Metho Exonential Methodbend.mom. ratio diff. ratio diff.

Ut ?th th/t Uth th/"t

0 0 - - 0 - -6.21 6.32 1.017 2 5.78 0.930 -78.56 8.86 1.035 4 8.01 0.936 -69.81 9.88 1.05 0 8.92 0.9,09 -9

10.72 10.73 1.000 0 9.75 0.910 -9u 4 .% 1.007 1 10.25 o.895 -n

1.63 1.73 1.008 1 10.55 0.907 -9n.9o 12.06 1.012 1 10.80 o.9o7 -912.12 12.31 1.01 2 n1.05 0.912 -912.23 12.h8 1.020 2 11.22 0.918 -.812.38 12.61 1.020 2 11.39 0.920 -812.-57 12.89 i.o2 2 n .59 0,923 -812.79 13.12 1.028 3 12.30 0.963 -412.87 13.28 1.032 3 12.55 0.977 -212.96 13.5 1.o5 4 12.72 0.982 -213.03 13.81 1.060 6 12.90 0.9100 -1

U'

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(n 4 (j I cc0% H -i ~j m c N N c(O H- H4ro 4 4 cm

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otoE- A 8 uN * aN *) 0*uN

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c 4c A44- -%~ 4~ q "

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