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Masters Theses Student Theses and Dissertations

1966

Comparison of plastic stress-strain material properties as Comparison of plastic stress-strain material properties as

determined from conventional experiments determined from conventional experiments

Wen Mo Chen

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Recommended Citation Recommended Citation Chen, Wen Mo, "Comparison of plastic stress-strain material properties as determined from conventional experiments" (1966). Masters Theses. 2965. https://scholarsmine.mst.edu/masters_theses/2965

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COMPARISON OF PLASTIC STRESS - STRAIN 1'4ATERIAL PROPERriES

AS DEI'ERMINED FROM CO'NVENI'IONAL EXPERIMENTS

BY

WEN MO CHEN- !9~f-

A

THESIS

submitted to the f a culty of

'rHE UNIVERSI'rY OF MISSOURI AT ROLLA

in partial fulfillment of the requirements for the

Degree of

MASrER OF SCIENCE IN ENGINEERING MECHANICS

Rolla , r1issouri

~~~~~--~~~~

1966

Approved by

(advisor)

1

I. PURPOSE AND OBJECTIVES

In practice, the stress-strain curves of materials,

as obtained from different loading condition~are generally

not the same. However, in theory, it can be shown that by

appropriate selection of stress and strain related para­

meters, the stress-strain curves obtained from a single

material, subjected to tension, compression, torsion and

bending loads, will coincide. Like materials have been

tested under the aforementioned loading conditions, and

these observations have been used to construct the appro­

priate stress-strain diagrams. It is the primary purpose

of this study to present and discuss the relative agreement

between the theoretical and experimental results.

ABSl'RAC-r

In problems of engineering design, it is usually

assumed that the stress-strain diagrams of a given

material, as obtained from different loading conditions,

are the same. In fact, by using the classical theory of

plasticity, it can be shown that, with an appropriate

selection of stress and strain related parameters, the

diagrams should coincide.

rhree different materials - magnesium, aluminum and

steel, have been tested under tension, compression,

torsion and bending loads. Stress-strain diagrams were

constructefl from these observations. This thesis

presents an analysis of the relative agreement between

i

the theoretical and experimental results of these materials.

1 :i

AC f~\J OWL EDG EifcE\TT

rhe author wishes to express his sincere appreciation

to his advisor, Dr. Robert L. Davis, Assistant Professor

of Engineering Mechanics, for hls assistance in the comple­

tion of this study.

The author would also like to take this opportunity

to thank all the other professors and his friends vrho have

sincerely instructed and helped him to overcome the diffi­

culties that he has encountered.

111

TABLE OF CONrENTS

ABST RAGr •••••••••••••••••••••••••••••••••••••••••••••• 1 .

ACKN'OWLEDG MENT ••••••••••••••••••.••••••• , ••••••••••••• 11

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • .••••• 111 TABLE OF CONTENTS

LIST OF FIGURES • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 1v

LIST OF TABLES • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• vi

NOT AI' IONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• vii

I. PURPOSE AND OBJECI' IVES •••••• , , •• , , • , •••••••• , , • , 1

II.

III,

INrRODUGr ION •••••••••.••..•••••••••••.•••••••. , • 2

THEORY OF STRESS-STRAIN CURVES OBTAINED FROM V~~IOUS

LOADING CONDITIONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 4

III-1.

III-2.

III-3.

Stress-Strain Curves for Tension and

Compression •••••• · ••••••• , •••..•••••••••• 4

Stress-Strain Curves for Torsion ••••••.• 8

Stress-Strain Curves for Bending •••••••• 16

I I I - 4. Summary • • • • • • • • • · • • • • • • • • • • • • • • • • • • • • • • • • 2 2

III-5. Stress-Strain Curves for Biaxial Tens1on.25

IV. EXPERIMENTAL FACILITIES AND PROCEDURES •••••••••• 29

v. RESULTS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 39

VI. SUMMfu~Y AND CONCLUSIONS • • • • • • • • • • • • • • • • • • • • • • • •• 58

REFEREN'CE •••••••••••••••••••••••.••.••.•••••••.••••••• 6 0

SELECTED BIBLIOGRAPHY • • • • • • • • • • • • • • • • • • • • • • • • • • • .•.•.. 61

VITA ••••.•••••••••••••••••••••••••••••••••••••••••••••• 6 2

APPENDIX I. SAMPLE RESULTS ••••••••••••••••••••••••••• 6 3

iv

LIST OF FIGURES

Fig. III-1 Effective stress-effective strain diagram 6

Fig.

Fig.

Fig.

III-2 Torsion of solid circular shaft

III-3 Shear stress-shear strain curve

III-4 Torque-angle of twist curve for circular

bar

Fig. III-5 Beam in pure bending

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Flg.

III-6 Element taken from deformed beam

III-7 Deformed bar in pure bending

III-8 Pressure-loaded membrane

III-9 Element taken from deformed membrane

III-10 Equivalent states of stress

IV-1 'rensile testing apparatus

IV-2

IV-3

IV-4

V-1

V-2

V-3

V-4

V-5

Compressive testing apparatus

Torsional testing apparatus

Bending testing apparatus

Effective stress-effective strain rela•

tions for tension

Effective stress-effective strain rela­

tions for compression

Experimental torque-angle of twist curve

for circular bars

Experimental moment-curvature curve for

bending

Elastic portion of stress-strain curves

for aluminum

11

13

15

17

19

19

2.5

26

27

30

31

33

34

40

41

42

43

44

Fig.' V-6 Elastic portion of stress-strain curves for

magnesium 45

Fig. V-7 Elastic portion of stress-strain curves for

steel 46

Fig. V-8 Fractured tensile specimens

Fig. V-9 Fractured compressive specimens

Fig. V-10 Necking in tensile steel specimen

Fig. V-11 Fractured torsional specimens

Fig. V-12 Experimental comparison of plastic stress-

48

49

50

51

strain properties of aluminum 54

Fig. V-13 Experimental comparison of plastic stress-

strain properties of steel

Fig. V-14 Experimental comparison of plastic stress­

strain properties of magnesium

Fig. A-1 Experimental torque-angle of twist curve

for aluminum bar

Fig.

Fig.

Fig.

A-2

A-3

A-4

External fiber strains for aluminum beam

Experimental moment-curvature curve for

bending

Beam curvature-strain curve for aluminum

55

56

67

70

7;1

72

v

vi

LIST OF TABLES

Table III-1 Stress-strain parameters for constructing theoretically identical curves 24

Table V-1 Elastic value'of Poisson's ratio 53

Table A...;l Plastic tensile data for aluminum 65

Table A-2 Plastic torsional data for aluminum 68

Table A-3 Plastic bending data for aluminum 74

Symbol

d

Et

E c

~i

F

G

ho ' h

I

Ip

k

ko

f.o t J.

Po

ro ' r

to ' t

T

w

w

wp

Vii

NOTATIONS

Definition

beam depth

modulus of elasticity for tension

modulus of elasticity for compres-sion

modulus of elasticity for biaxial tension

arbitrary function

modulus of rigidity

initial and current heights of the compressive specimen

centroidal moment of inertia of the cross section

polar moment of inertia of the circular bar

yield shear stress

maximum yield shear stress

initial and current lengths, respec­tively

internal pressure

initial and current radii, respec• tively

initial and current thickness, respectively

twisting moment

work done per unit volume

total work done

total plastic work done per unit volume

Symbol

X ' y

Yt y

c

ere

v-

~ ' Q"'h

' e

E'T

£e

€p

Ec ' -e

('

~

tf:>,

p

e

cr2 ,and

<rr '

and

viii

NO~TIONS (continued)

Definition

Worizontal and vertical coordinates defining a point in the beam

tensile yield stress

compressive yield stress

elastic stress

effective stress

principal stresses

hoop, radial and thickness stresses, respectively

natural strain

total strain

elastic strain

plastic strain

unit strains on the top and bottom surfaces of the, beam;·. respectively

effective strain

radius of curvature

slope of the tangent to the elastic line of the bent bar

angle by which a generatrix of the cylinder is titled after twist

Poisson's ratio

shearing strains at the outside radius and at any radius, respectively

angle of twist per unit length

Symbol

ix

NOTATIONS (continued)

Definition

~Q~p, radial and thickness strain, respt9ct1vely

principal strains

I I. I tiT RODUCT I ON

When a material is subjected to external forces its

behavior depends not only upon the magnitudes of the

2

forces and the strength and the shape of the material

itself, but also upon the manner in which the forces are

applied. Metals, for example, generally exhibit rela­

tively high tenacity and are therefore better suited to

resisting tensile loads. Brittle materials, such as

mortar, concrete, brick and cast iron, have tensile

stren3ths that· are low compared ~v-ith their compressive

strengths, and are principally employed to resist com­

pressive forces. Again, the tensile strene;th of 't'lood is

relatively high, but it cannot always be used in structural

members because of its low shear resistance, which causes

failure before the full tensile resistance of a member

can be developed.

Before an engineer designs a structure, a stress

analysis must be made. It is usually assumed that the

material follows Hookets Law, and the modulus of elasticity

of the material is known. But for selecting the safe

dimension of a structure, this knowledge is not sufficient.

The designer must know the behavior of the material beyond

the proportional limit under various stress conditions.

Information of this type can be obtained only by experi­

mental investigationso Material-testing laboratories are

equipped with test machines for performing tensile, com­

pressive, torsional and bending tests of a specimen that

3

~.Arill p-rovi(1e adc1ect. information on material beha\rior.

In plastic design, it is generally assumed that the

stress-strain curves for tension, compression and bending

are identical. Thus, when an engineer designs a structure,

the property of the material to be used is obtained from a

simple tension test, an~ these results represent the basis

of deslgn regar~less of the nature of loa~ing or state of

st:ress in the structure. In practice, the behavior of a

~~i -;ren rnat erial, when subjected to various loari i ng cond1-

tto:'1s, i.s not the same, and these deviations must be

accounte~ for. this fact is extremely important in plastic

~nalysis, and is discussed further in Section V.

4

III. ·rHEORY OF 8rRESS-STRAIN CURVE~3 OHrAI"NED

FROM VARIOUS LOADING CONDITIONS

l.. Stre:3s-s:~rain curves for Tension and compression.

In securing stress-strain diagrams for a given

material, the three usual types of tests taken are the

tension of a rod, the compression of a short cylindri-

cal block, 3.nd the twisting of a thin-1-,ralled tube.

The results of such tests are represented by plotting

the mean stress acting in the current cross-sectional

area, against some measurement of the total strain.

For an ideal tensile test, the volume of the

tensile specimen is practically constant

nr a A = nr2 R 0 ~.

that is,

where

r a 0

r and rare the initial and current radii, 0

respectively.

£0

and ~ are the initial and current length,

respectively ..

(a)

The total work (W) done by the tensile force is

W = Jnr 0

2 Q"'edf + J nr2 YtdR

where ~e is the elastic stress, and Yt is the tensile

yield stress. Comparing the elastic work done '1J'ith

the plastic work, the elastic work is so small that it

can be neglected. Thus, the total work done is con­

sidered to be

5

W = } 11' r 2y t dR • (b)

Combining equations (a) and (b), the work done per

unit volume (w) is

11 dJ w = Yt T .

'J.. The natural strain (€) is defined as

dE = slj-Hence, the work done_per unit volume is just the area

under the true stress-natural strain curve.

Now, the amount of recoverable elastic work done

is approximately equal to the shaded area (As) shown

in Figure III-1, which can be expressed mathematically

as

where Et is the modulus of elasticity for tension.

Thus, the total plastic work done per unit volume

is

J J1 Yt2

Wp = dwp = t.Yt~- 2Et •

According to theory of the "equivalent of plastic

work", the yield stress Yt is related to the plastic

work, thus

(!)

(IJ Q)

H .+) (/)

6

----+--d€ = ¥ --.J. E- Natural Strain

Fig. III-1

EFFECr IVE S'rRESS-EFFEGL' IVE SrRAii\: DIAGRA?1

1*

where the symbol F represents some arbitrary function.

If Yt is plotted against 1n+ - Yt , the argument of o Et

7

F is the area under the curve up to the stress level Yt.

Similarly, for simple compression, the compressive

yield stress is

y = F ( Ih~ _dh - y~ ) c e--n- 2E

h c

Y0

is the compressive yield stress

h and h are the initial and current heights of the 0

specimen, respectively

E0

is the modulus of elasticity for compression.

It can be shown that the

function of 1n-L1- - Yt as Yc o Et

true stress Yt is the same

is of ln~ - ~ • Thus, c

theoretically, the stress-strain curves for simple tension

and simple compression will coincide

against ln-t- - ~ , and Yc against A• Et

if Yt is plotted

ln~ - ~ • That c

is, the two curves coincide only when true stress is

plotted against the natural strain, but not when plotted

against the engineering strains J. J..f.. and hofi h

·U· Superscript numbers refer to references listed on page

60.

8

2. Stres3-8train Curves for Torsion

It is very interesting to find that, 3fter selec-

tton of the app~opriate stress anrl stratn relate\-l_

paramete~s, the stress-strain curves for torsion coin-

ci~e with that for tension. ~his fact is demonstrate~

in ~he fo11oT.Aring paragraphs.

Boo1<e's l9.1AJ for three dimensional stresses 1 n tl)e

plast tc range is TITi t ten as

€, = (!. [ G'; - P. t t-~-. +- crl ) ]

t& = ~ [ ~ )A l v-, -t Gi ) J (IIT-1) 2

tJ = ~ [ 0": -.l )A\<1, +'~)]

:~rhcrc p. ls Poisson's ratio, a no. f is, in general, a

function of the strains, and. can be netermined expert-

mentally. ·rhe quantities r, , 9; , and 'I; are the

principal stresses, an~ ~ , f~, ~0~ ~ are the

principal strains.

rhe equation of volvme constancy is wrttten in

terms of the total _strains as

c t + Gz -t ~ J = o • (III-2)

If equations (III-1) and (III-2) are combined, it

can be sho~vn that Poisson's ratio JJ. remains constant

and equal to one half in the plastic range.

I:·Jow, it is desired to find some relationships

between shear stress and normal stress, and shear

strain and normal strain. According to the Von Mises

yield criterlon

where

t~1US

Yt is the tensile yield stress

k is the yield shear stress

9

(III-4)

~v1a1cing use of the known relations of the material

constants gives

= Et _Et G 2(l+JA)- 3

where

f" is the modulus '_T

= Yt _ k 3e· -T

of rigidity

Ft is the modulus of elasticity for tension

r is the shearing strain at any radius r

Combining the above equations gives

10

J'>~ov~r, consider and isotropic cylinder of unit gage

length and outside radius r0

, subjected to a plastic

torsional strain. (See Figure III-2) rorsion theory

assurr.es that the diameter of the bar remains constant

du_ring the loading process, thus, for incompressihle

materials, the gage length also remains fixed. At

any radius r, the shear strain is ( = tan~1 • And, for

a given angle of twist per unit lenzth e

or

a= .L r (c)

11Jhere r; is the shear strain at the outside rad.i u.s,

and ¢1 is angle by which a generatrix of the cylinrler

is tilted after twist.

Since the total strain (~) is the sum of the

s t~rains 1 n the elastic ( Ee) and plastic ( tp) ranges,

then the plastic strain can be expressed as

k t = tr- € = Y Yt = re -G

p e 7J-E; n . (III-5)

From equations (III-4) and (III-5), it is seen that

the stress-strain curves for the torsion of a circular k

shaft, constructed by plotting fjk versus r8 - G tT

will coincide with the usual curves for tension.

11

Fig. III-2 TORSION OF SOLID CIRCULAR SHAFr

12

Referring again to Figure III-2, the resisting

torque T, acting on a given cross section, is ~·rritten as

T = {o 1<(2 'lfr dr)r (III-6) Jo • Substituting equation (c) into equation (III-G) gives

Te' 2 TT

(III-7)

Thus, the t"t"Jisting nonent T is proportional to the

second monent of the shaded area under the stress-strain

Clu~ve for shear' lr = f ( r) ' 'ShOT·m in FigUJ."e III-3. If the

nom.ent-tl·Jist curve, T = f(e), is lmo't'm from observation

of a torsion test, the un.kno1~m material stress-strain

curve for shear, k = f(Y), may be determined from equation

(III-7). Differentiating equation (III-7) with respect

to e

S~nce (is independent of 9, the first tern on the ri8ht

side is zero, thus

~eo This result is obtained by employing the Leibnitz

formula given on page 20.

13

k = r ( r)

ro K- Strain

Fig. III-3 SHEAR S'I'RESS-SHEAR STRAIN CURVE

14

If the T-e curve is lmo~m, the 1-c- Y curve can be

derived. Note that k = .....21:_ is the usual simnle 0 21fT:- -

e:x:pression for torque due to a maximu:n yield shear stress

of k 0 in the fully plastic condition of a non-~;ork­

hardening material.

From Figure III-4, the slope at point B on the T-9

curve is given by

or

dT _ BC d9 - na-BC = etr.

Hence, the shear stress existing at the outside

cylindrical surface, corresponding to an angle of t1'J'ist

e' is

lc0 = 2 Tf~j ( 3 BN + 3c).

0

(III-8)

Therefore, by dra't'ring tangents to points on the

T-e diagram, the lc-r' diagram can be derived •. A more com­

plete discussion of this procedure is given in Appendix

I, 't~here a sample problem has been solved.

15

( e -r)

---- --- c

N 0

9 - Angle of T1AT1st per Unit Lensth

Fig. III-4 TORQUE-ANGLE OF 1'v!IS1' CURVE FOB

CI;tCULAR BAtt

3. 3tress-Stra1n Curves for Ben:ling

In an ideal bending test, it is senerally assu:neo

that the extreme fibers on the tensile ann compressive

sides of the beam are under the sole influence of ten­

sile and co~pressive stresses, respectively. In other

words, theoretically, the stress-strain curves for the

extreme fibers on both sides of the beam are the same

as those ob~aine~ from direct tensile and compresscve

tests. Althou13h yleldine; of the extreme fibe-r- is

marked 1,~r the supportin.:?; effect of the less hic;hly

stressed fibers near the neutral axis, it will be

shown later that, for certain materials, this effect is

small.

Consider a long prismatic beam ha.vine; a rectangu­

lar cross section of width b and depth d subjecten to

tt<Jo equal and symmetrically spaced concentrated loads

as shO\rin in Figure III-5. It is assumed that a uni­

axial state of stress exists at all points of the beam

with the longitudinal stress being the only nonvanish­

ing principal stress along the pure bending portion.

If t1·Jo adjacent yertical lines m-m and n-n are clravm

along the side of the beam, experiment shows that

these lines remain essentially straisht cl~1r1ng bending,

and rotate so as to remain perpendicular to the

longitudinal fibers of the beam.

F

1- a ----,

F

F

m n ,~ a ---1

X

m n fi'

y

(a)

Loading of beam

1--rc--f

TF=112 d

l I I j neutral

_I 11

(b)

Cross section

(c)

stress distribution

Fig. III-.5 BEAJ.I IN PURE BENDING ~ -..J

T'he theory of bendins ls baserl on the assumption 't11n~

not only such lines as m-m remain strai3ht, but thqt

the entire transverse section of the heam, ori~inally

plane, remains plane and normal to the longitudinal

fibers of the beam after ben~ing.

From Figure III-6 it is seen that

(d)

and.

~vhere f is the radius of curvature, d is the bean

depth and Ec and Et are the unit strains on the top

and bottom surfaces, respecti vel;y. !,urther, from

Figure III-7

(e)

1vhere ¢ is the slope of the tangent to the elastic

line of the bent 't>ar ..fo is t~e ,gage length. Thus

¢ = _Jo__..._€_c~+......--€_t ..... ) 2d •

(III-9)

rhe equilibrium equations for any section of the

beam are

I 6'"dA = 0 'A

(f)

and

I VydA = M (g)

19

Fig. III-6 ELEMENr TAKEN FRON ~EFORMED BEAIVI

,-----F ~ i -1 F

Fig. III-7 DEFORMED BEAM IN PURE BENDING

20

where the nomenclature is

A = cross sectional area of beam

M = resistinB moment at a given cross section

If it is assumed that the relation between stress

and strain is

(h)

then ( i)

(i""t = f(t"t) •

Combinin3 equations, (d), (h), and (i))and

substituting these values into equation (f) gives

f+9; f(f) df.= 0 • )_lc

( j )

Differentiating equation (j) with respect to ~ , and

following the generalized Leibnitz formula*, gives

(III-10) •

An appropriate combination of equations (d), (e), (~),

an~ (j) results in

~H£ 2 ~r:: f(£)Ede, ( k)

If equation (k) is differentiated. with respect to ¢ ,

again proceeding according to the generalized Leibnitz

formula, then

* The generalized Leibnitz formulas is written as

rb Jb db d 1"\J/(x,tl.)dx = a arJ!/11) dx+f(b,•l)dcl; -f(a,c() i!~ ,

L! d ( M02) = ~ cl~ · ~) .X.. - 'fl

0

Usir![!, the equatior1s (III-9) and (III-10), the

s~rain increments can be written as

2cl <r"+-elf v

d~ = -ro-c (lc +o-t

dE._ 2r1 Vc d¢ = J.o \.I (Jc+ o-t .

The desired moment equation is now in the form

G"c tt rr- + f'r'

"C "t (III-11)

rhe right side of this equation is known if,

from observed test data, the bending moment M is a

21

known function of the slope ¢ • From the observe~

d €.c strains Et and G

0, the ratio (i € t , ann_, from equation

( III-10)') the quotient Vt may be determined for rc 7arious val11es of the observAd angle ~ . -rhese

expressions permit the calculation of the stresses

~ ani <Jt farthest from the neutral axis as functions

of the strains or, in other words, the construction

of the stress-strain diagrams for tension and com-

pression from bending tests.

NorE:

In the case of beams with circular cross sections,

22

equatic~ (f) becomes

(III-12)

where r0

is the ra~ius of the circular cross sec­

tton. Accor~inc to equation (e), f ~epen~s on

¢. If Aquatton (IIJ-12) is ~ifferenttated with

respect to ¢, it can be seen that the first ter~

of the rtght si~e of the ~e~eraltze~ Lethnttz

for11ula 0oes not 1ranish. That ts, tt is not 3.S

conve~:tent-, to find the relationship betwPen a-c

and ~t for heams of circular cross section. Con-

siderable effort has been put forth by this

author in an attempt to solve the beam proble~

for circular cross section; however, since the

completion of this thesis does not Qepend upon

the knowledge of circular bar behavior, this

problem has been left as a future thesis project.

~he reason for pursuing this problem was to Make

possible the selection of tension, co~presston,

torsion, and bending specimens from a single

piece of round bar stock.

4. Summary

From the preceding discussion of the theoretical

stress-strain curves for tension, compression, torsion

and bending, it is concluded that by appropriate

23

selection of stress and strair. related parameters,

these four curves will coincide. rable III-1 shows

the necessary quantities for stress and strain that

~1st be used in constructing these graphs for various

loading conditions.

Parameters

Stress Strain

Loading Conditions

Simple rension Yt 1 yt ln 1:; - Et

Simple Compression Yc ho 'r .l.c ln11 - rr

~..~c

e k Si11ple Torsion /3 k r -G

n

rt J G'""t ln J;' - Eb

Pure Bending

v-c ho ere lnT - E b

Table III-1

81'"9-ESS-STRAIN PARAMErERS FOR

CO~JSrRUGl'ING ~rHEORET I CALLY IDENTICAL CURVE.·)

25

5. Stress-Strain Curves for Biaxial Tension

In addition to the four types of loads discussed

previously, it is interesting to note that the stress­

strain curves of a metal rnembrane subjected to bi­

axial tension can also be reduced to an identical dia-

gram.

Pressure

Fig. III-8 PRESSURE-LOADED HEHBRANE

Figure III-8 shows a circular membrane clamped

around its periphery ~ri th fluid pressure acting on one

side. It is assumed that the thickness-to-diameter

ratio of the membrane is so small that bending and shear­

ing stresses can be neglected. From the equilibrium

condition of the membrane element shown in Figure III-9

p 0

z

Fig. III-9 ELEHENT TAKEN FROH DEFORI•·TED I·1EHBRANE

Po r Pof Oh = 2t sin)\ = 2t

where the nomenclature is

~h = hoop stress, psi

p 0 = internal pressure, psi

t = current thickness, in

r = radius of curvature, in

26

From symmetry' the hoop strain sh at the pole is

equal to the radial strain €r• The thickness strain et is obtained from the equation of incompressibility,

equation (III-2), as

to = -lnt

where t is the initial thickness, inches. 0

2?

Considering the stress system at the pole for a

thin membrane, the stress normal to the sheet is neg-

ligible, and the material at the pole is therefore sub­

jected to a balanced bi-axial t·ensile stress. That is,

where the symbology is

'"t = thickness stress, psi

r:'-1 = hoop stress, psi

<rr = radial stress, psi

Thus ~t can be neglected. Since at the pole, the hoop

stress and radial stress are equal, the following diagram

can be used to express the state of stress.

o-h = crh+

O"r

Fig. III-10 EQUIVALIDJT STATES OF STRESS

Since it is assumed that a hydrostatic pressure has

no effect on yielding, the system is therefore equivalent

to a simple compressive stress ~h acting normal to the

sheet.

28

The equations for effective stress i' and effect

-strain e are

(III-13)1

It can be shown from equations (III-13), that the hoop

stress <r11 and thiclmess strain Et are just the effect­

ive stress and effective strain, respectively. If the

quanti ties lrh and ·ln ~ _ ~h. (~i is modulus of elas-o bl

ticity for biaxial tension) are chosen as the ordinate

and abscissa, respectively, the resulting stress-strain

curve will coincide with the curves described in Table

III-1.

IV. EXPERIHENTAL FACILITIES AND PROCEDURES

!Three different materials - magnesium (AZ 6l.A)

aluminum (2~ST-4),and (AISl Cl018)steel have been tested

under tension, compression, torsion and bending loads.

All of the test specimens, for a given ~terial, are cut

from the same bar. The subject testing materials were

29

purchased in the form of a 12-foot section of bar, having

a 1~-inch by 3/4-inch rectangular cross section. The

bending test specimens were made by cutting a section of

appropriate length from the original bar. The tension,

compression, and torsion specimens were also made from

the original bar, but they required that a machining

operation be made.

A. Objective:

To obtain experimental stress-strain curves for

tension, compression, torsion and bending.

B. Equipment Necessary:

1. Tension Test (shown in Figure IV-1)

Testing machine with grips to fit specimen

Two inch dial gage (calibrated with incre­ments of 0.0001")

Pair of dividers

One inch micrometer

Gage length punch

2. Compression Test (shown in FigureiV-2)

Testing Machine

30

Fig. IV-1 TENSILE TESTING APPARATUS

31

Fig . I - 2 CO· PRE SI' E r ES _I J PPARATUS

3'") f_~

Baldwin-Lima-Hamilton portable Model 120 strain indicator and the Model 225 Switching and Balancing Unit (this instrument has been designed and calibrated especially to work with resistance strain gages having an initial resistance of 120 ohms)

I~1 i cr ome t er s

Lubricant (graphite)

Bearing plates (these plates are essentially rigid in comparison to the compression speci­mens)

Budd Metalfilm strain gages (post yield, high elongation, type HE-141 gages having a gage factor of 2.02 to.s ~and an initial resis­tance of 120 ::!: 0. 2 ohms)

). Torsion Test (shown in Figure IV-3)

Testing machine

Angular-measurement device

Micrometers

Pair of dividers

Gage length punch

4~-Bending Test (shown in Figure IV-4)

Testing machine

Balwin-Lima-Hamilton portable Model 120 strain indicator and the Model 225 Switching and Balancing Unit

Budd Metalfilm strain gages (these gages are identical to the ones described in paragraph 2 of this Section)

Ruler

Two hard- material rollers

c. Specimen Requirements:

1. Tension

3

F ig ~ IV- 3 rORSION ES'ri G APPARATUS

34

Fi 0 . IV-4 BENDI G TE TI G APP R U

35

a. rhe central portion of the length is

usually of smaller cross section than

the end portions in order to cause

failure at a section where the stresses

are not affected by the gripping device.

b. The specimen must be symmetrical with

respect to the longitudinal axis through­

out its length in order to avoid bending

during application of the load.

2. Compression:

a. Compressive specimemare li~ited to such

a length that colu~n action is not a

factor.

b. In order to get uniformly distributed

stresses over the cross section, the

ends of the specimen must be flat and

normal to the longitudinal axis of the

specimen. It is necessary to put a

good lubricant between the compression

plates and the ends of the specimen to

prevent the development of shear stresses

which will disrupt the assumed uniform

normal stress distribution.

). rorsion:

a. In order to prevent the occurrance of

complicated stress distributions, speci­

men of circular cross section were used.

b. The specimen must be symmetrical with

respect to the longitudinal axis through­

out its len::th.

'-~. Bend.ing:

a. The beam under test ~ust be so propor­

tioned that it will not fall in shear

or by lateral deflection before it

reaches its ultimate flexural stren~th.

To prevent a shear failure, the

specimen must not be too short with

respect to the beam depth. The test

specimen were made ln acco~dance with

AS'-rT{~ designation: E lh-39.

b. For results 1_n 1vhich the measured per­

formance is independent of the thickness

of the specimen, rectangular specimens

1-1ith -r,•ridths at least one and one-half

times the thickness must be used. The

edges must be smoothed such that

local1Zefl fracture will not take place.

D. Proc ed11re ~

1. Tension Test"

Using a 2-inch punch, two points, symmetrical

with respect to the specimen center, were

made, and the dial gage mounted in position.

The specimen, and dial gage, were then

37

mounted in the testin:; machlne, and a check

made to see that ~he ga~e and gripping de­

vices functioned properly. Making sure that

the testing machine initially registered zero

load, the tension test was performed by

measuring and documenting the diameter of

the mid-point of the specimen, the specimen

elonzation as recorded on the ~ial gage, and

the corresponding machine load. This pro­

cedure was repeated. at predetermined load.

increments until fracture of the specimen.

2. Compression Test~

Before testing, the en~s of the speciwen

and the face of the bearing plates were

cleaned with acetone and then coated with

graphite. Next, the strain indicator was

balanced and its initial reading recor~ed.

r·he diameter of the specimen and correspond­

i~g strain at increasing loads were recorded.

The aliznm.ent and centering of the heads of

the testing machine, the bearing plates, and

the axis of the specimen were preserved

throughout the test.

3. rorsion Test,

Before testing, the gripping device of the

torsion testing machine was cleaned and then

the specimen and angular measurement device

38

mounte1. Doth the ~orque and an~ular

meo<n:.rer.:1Gnt device TJC?re zcroen and 0hecked

to sec that they were functioninG properly.

~he specimen ~ia~eter, D~~le of twist 9,

[.uv:l :nasni-t·uce of :~orque for predetermined

incTernents of torque w8re recorded.

4. 3endin~ ~est,

In orclcr to prod.uce pure benrlin:;, tv·ro

hard-mat crlal rol1ers \·'rere set at equStl dis-

-;·~as 111oun.ter.l on th8sc '~No rollers, and a con-

csnt.rated loa.(l, suppli~0 by t~1e CO'";'lp:resslon

machine, applie~ at the center of the top,

as sho1·v-n in :Pi cure IV-'+. I'he strain ea3es

located on the top and bottom sides of the

beam ar6 connected to individual channels of

the switching and balancing unit. Before

testins,hoth channels were balanced an~ the

i~itial readings recorded. A loan was

applied to the beam and held constant. 1he

strain 1n~icator was balanced and a new

stra1.n readtng obtained from both the tension

and compression cages. This process was

continued at predetermined load increments.

39

V. RESULTS

Three materials - magnesium, aluminum, and steel -

have been tested under tension, compression, torsion, and

bending. Figures V-1 through V-4 are constructed from

these experimental observations. It is evident that init-

ially the relationship between load and deformation is

essentially linear. However, on further straining beyond

the elastic limit, the relation between load and deforma­

tion is no longer linear, and plastic flow has started.

In order to predict the modulus of elasticity and modulus

of rigidity, Figures V-5, V-6 and V-7 have been construct­

ed for tension and compression. Figures V-3 and V-4 can be

used to determine the modulus of elasticity for bending

and the modulus of rigidity for torsion, respectively.

where

Thus,

Within the elastic limit, for bending,

M is the applied moment

I is the centroidal moment of inertia of the cross section

10

is the gage length (-h-"l ~ is the modulus of elasticity for bending

x and y are the horizontal and vertical coordinates defining a point in the beam

E = ~ t- = 0.593 ~ b 2I . 'P

-..... fJl ~ -(J.l

en

80

60

~ 40 +' (I)

G)

:> ..... +' () Q)

~ ft-t 2 0 ~

0

0 200 400 oOO 800

Effective strain (in/in)-104

El Steel

0 Aluminum

8 Magnesium

1000 1200

Fig. V-1 EFFECriVE STRESS-EFFEGriVE STl1AilJ llELATIOKS FOR TENSION

1400

+=" 0

-.,-4 (1J

~ .........

Cll til G) J.t ..p fll ., t>

.,-4

..p C) G>

"" ft..t f%1

70

60

40

20 ... -

0 0

0 Steel

0 Aluminum

A Magnesium

8 8 5 Effective strain {in/in)

Fig. V-2 EFFECTIVE STRESS-EFFECTIVE STRAIN RELATIONS FOR COMPRESSION -+="' J--1

-a or!\ I P-4 .,...,.

.!14 ..._,

CD ~ o' J-1 0 8

~

},J.

3

2

1

0

~A~ EJ Steel

0 Aluminum.

8 Magnesium

0--- - - 200 4-00 bOO '8"00

e -Angle of twist per unit length (rad/in) 103

Fig. V-3 EXPERIMENTAL TORQUE-ANGLE OF TWIST DATA FOR CIRCULAR BARS

1000

~ l\)

-s:: oM I

p,. ...... ~ -.p

~ a 0 ~

I :e:

30

22

18

12

- ---EJ Steel

0 Aluminum .. . ,---6_

A Hagnesium

0~--~--~--~--~--~--~--~--~----~--~--~--~--~--~--~----0 22.5 45.0 3

Curvature (rad.) 12 x 10 6?.5 90.0

Fig. V-4 HOiviENT-CURVATURE DATA FOR BENDING +=" VJ

" ..... fl.l ~ ........,

Ol Ol Q)

k .f.) fiJ

~ ..... ,., Q) Q)

s::· .....

a

50

40

30

20

10

0 0 5 10 1.5

/ /

/

/

/ /

/

A"

/

/. /

/

/

/ /

/

// /""' 0

6 0 Tension (Et = 10.6 x 10 psi)

// /

~ Compression (E0 = 10.7 x 106 psi)

Bending (Eb = 10.9 x 106 psi)

(See Fig. V-4 for data)

20 25 30 35 4 Engineering strain (in/in) 10

Fig. V-5 ELASTIC PORTION OF STRESS-STRAIN CURVES FOR ALUMINUM

40

~ +:-

"' ..... Ul ,!4 -

CD Ul G)

b l7J

t() ~ ..... ~ Q) .s a

20

16

12

8

4

/

/ /

/ /

/ / A

/ /

/

/ /

/

/

/ /

/

0

8

/ /

/ /

/ /

/

/ /

/ /

/

/ /

/ /

/

Tension (Et = 6.27 x 106 psi) 6 Compression (E0 = 6.25 x 10 psi)

6 Bending (Eb = 6.8 x 10 psi)

(See Fig. V-4 for data)

12 16 2R 24 Engineering strain (in/in) 10

Fig. V-6 ELASTIC PORTION OF STRESS-STRAIN CURVES FOR ~~GNESIUM

32 i="

\..}'\

30

25

-.-4 til 20 ~ -tfl tfl CD ~

-1-) ~5 tfl

b01 ~ .-4 ~ (I)

~ ll() ...... bO

Gi $

0 0 20

/

40

/ , /

/ /

/ /

/ /

/ /

/

/

/ /

"::' /

/

/ /

""' /

/ /

/ /

/ ./

/

0 JY/

6 0 Tension (Et=30.6 x 10 psi)

6 A Compression (Ec=29 x 10 psi)

60

6 Bending (Eb=28.5 x 10 psi)

(See Fig. V-4 for data)

80 100

Engineering strain (in/in) 105

Fig. V-7 ELASTIC PORTION OF STRESS-STRAIN CURVES FOR STEEL ~ 0\

47

Hence the slope of the straight line portion of the M

'rersus f/J curve, multiplied by the constant 0.593, gives

the desired modulus, Eb.

Similarly, for torsion,

(V-3)

where IP is the polar moment of inertia of the circular

cross section.

of the T versus

Thus, the slope of the straight line section

e curve, multiplied by the factor 1/IP,

gives a value for the modulus of rigidity, G.

It is evident that the magnesium tensile specimens

(right · specimens in Figures V-8 and V-9) fractured along

a 45° plane, which is indicative of a shear fracture. Fig­

ure V-10 shows that a tensile steel specimen will neck down

locally before fracture. The stress distribution over the

smallest section of the neck is no longer uniaxial, but tr1-

axial. So once necking down occurs, the stress-strain dia-

gram has no meaning for simple tension.

The general types of observations and records of test

in torsion are similar to those of tension and compression.

However, the resulting shear fracture is quite distinct from

either the tension or compression fracture. A solid rod of

magnesium that fails during a torsion test will break along

a plane normal to the axis of the rod, as shown in the center

specimen of Figure V-11. For the steel bar (specimen on the

left in Figure V-11), the fracture is silky in texture, and

left side - steel

center - aluminum

right side - magnesium

Fig. V- 8 TENSILE FRACTURED SPEtiHENS

48

left side - steel

center - aluni num

right side - ma~~esium

FiG . V- 9 COHPRESSIVE FRACTURED SPECII1Ei·JS

50

Fi g . V-1 0 TF.CKING IN TE, SILE S EEL SPECIM~

Fie; . V-1

left s e - steel

center - magnesium

right side - a uminum

L'OR IOJ FRAGrURED PECIME S

51

52

the axis about which the final twisting takes place can be

seen. Since the surface of the break may not be quite

smooth, the outer portions, acting like cams, push the piece

apart in the direction of its longitudinal axis. The center

portion of the specimen that is not yet broken by shear is

possibly broken in tension by this cam action.

Elastic stress-strain diagrams for the three different

materials are obtained from various loading conditions and

are shown in Figures V-5, V-6 and V-7. For aluminum, the

modulus of elasticity in bending is reasonably close to that

in tension. However, for magnesium and steel, the modulus

of elasticity in flexure tends to be slightly below those for

tension and compression. Since some error is involved in

setting up the bending test, slight amounts of shear can be

present which will induce shearing strains, thereby causing

an increase in the observed strain over that due to fiber

strains alone.

Substituting the values of the moduli of elasticity and

rigidity into the equation

E G = -2-=-( 1-+-/A.~)

Poisson's ratio for all three materials can be found. The

results are shown in Table V-1.

53

material Hagnesium Aluminum Steel

Poisson ratio

J1 max 0.485 0.378 0.404

A min 0.365 0.352 0.307

Table V-1 ELASTIC VALUES OF POISSON'S RATIO

From this table, it can be seen that in the elastic range,

2o1sson•s ratio is always below one half.

The stress diagrams shown in Figures V-12, V-13 and .

V-14 are constructed according to the stress and strain

parameters shown in Table III-1. A sample proble~, showing

how these stress-strain curves are constructed, is presented,

with discussion, in Appendix I. From these stress-strain

diagrams, it is seen that the torsion tests produce a dis­

tinctively different curve (especially for the soft materials­

aluminum and magnesium). The reason for this is that in

simple torsion, the principal axes of stress and strain do

rotate relative to the element during the loading process.

A more complete discussion of this is given in the next

section. Another explanation is that the center portion of

a cylinder of originally soft ductile metal, after it has

been subjected to a plastic torque, is permanently strained

-r-i I

H H H

<D r-i ,D m

8

<D Q)

Cll --.,..., ttl ~ -F-f Q) _,_, Q) a F-f m A

s::: .,..., oS F-f +> fJ)

100

80

60 1- . ---------- 0

a(i)=€fii.a + +

ld' ~+' ~ A Tension

40 p-r.l' 0 Compression

I + Torsion

J D Bending (compression)

20 ~ () Bending (tension)

0 100 200 300 400 500 600 700 800

Strain parameter (See Table III-I) (in/in) x 104

Fi~. V-12 EXPERIT,IE!JTAL COHPARISON OF PlASTIC STRESS-STRAIN PROPERTIES OF ALUI\1INUI-1 V'\ .{:'-

-100 oM (f.)

~ --r-l I

H 80 H H

Q)

r-l ,D m 8 (1) 60 (l) (/') -H (1) .p (1)

s ~ hO H ~ p.

(f.) (f.) (1)

H .p (I) 20

co 20

8 Tension

0 Compression

+ Torsion

8 Bending (compression}

()Bending (tension)

~0 bO Rn 100 1~0

Strain parameter (See fable III-1) (in/iD) x

F1 rr V-13 Ti'"'rn'P.ni,.·"'E~fT'/\L ,..,o~.';P""li~O"·~ Qf•' pT."'~)T'..,..L(1 ·":'>·'"D"7C;";-S,DIIJ··.r "PT')()"C'~"{T'I"'i'S OT;l "':'.,.,~T.i'T w• ,_J..._·:-r:-"_1.. J..lJl\:-.t'\ '-.I t_._i..l..-- _, ...... _._~l _ ...., . --~_ .. ~..._ ... _ 1.'-\. ... . L _. J.._j ••.• · • ) __ :._J --~

\)'\

\....1'\

-H I

H H H

Q)

:a ~ Q) Q) Cll --.,-i {Jl

~ -J...t Q)

-1-) Q)

~ J...t as A

~ .,.., as J...t +l Cll

20

16

12

8 ~

~~

' 4 ~

' 0

0

Fig. V-14

8

.~· 6 Tension

0 Compression

+ Torsion

D B~nding (compression)

0 Bending (tension)

4 8 12 16 20 44 28 32 Strain parameter (See Table III-I) 10

EXPERIMENTAL CONPARISON OF PlASTIC STRESS-STRAIN PROPERTIES OF NAGNESIUH \....1'\ ()".

57

in the axial direction. Thus the assumed distribution of

shearing stresses does not represent the complete system of

stresses since a system of secondary normal stresses occur­

ring in the axial, and possibly also in the radial and

tangential directions, must be present. For this reason,

the effective strain equation (III-13) will not be satis­

fied with the actual large deformation of the torsional

specimen.

The stress-strain diagrams in compression always appear

below those determined from tension tests. The reason for

this is that lateral expansion at the end of the specimen

is prevented by the friction force that exists on the sur­

faces of contact between the specimen and the bearing plates,

thereby causing a complicated non-uniform stress distribu­

tion. When this occurs, the total plastic work done by the

triaxial stresses are greater than that due to the axial

stress alone. In tension, if the specimen satisfies the

requirements mentioned in section IV-C, the stress distri­

bution is always considered uniaxial until necking down

occurs.

58

VI S~1MARY AND CONCLUSIONS

This study shows that with the appropriate selection

of stress and strain related parameters the stress-strain

curves obtained from a single material subjected to tension,

compression, torsion and bending loads coincide. Further,

the stress-strain diagrams obtained from experimental ob­

servation are presented 1and these diagrams show the relative

agreement for magnesium, aluminum and steel.

With the exception of the torsional loads, the stress­

strain diagrams constructed from the various loading tests

show relatively little deviation for steel and aluminum

materials. The discrepancy in the torsion test results is

to be expected since the effective strain ~quation used in

this analysis is only valid if the following two conditions

are satisfied: {1) the ·orincipal axes of stress and strain

for a particle do not rotate with respect to the particle

during the process of straining; and, (2) the strain

increments are proportional to the total strains. Since

the torsion of a circular bar does not satisfy these con­

ditions, the experimental results will show some deviation

from the expected curve. A more appropriate torsional

analysis can be made by utilizing the incremental effect­

ive strain equation1 • Ho"t~ever, the results of this analysis

are valuable in that the magnitude of the error involved in

assuming the validity of equation (III-13) for torsion

problems is not fully discussed in the literature.

59

The stress-strain diagrams constructed from the

various load tests for magnesium have very little re­

semblance. This was suspected at the outset of this

program since the handbook data for magnesium indicates

that it's tensile and compressive properties are different.

Magnesium was purposely selected to illustrate the care

that must be exercised in applying the simplified equations

of plasticity to all materials.

In making a more critical review of the results achieved

for the steel and aluminum, it is seen that the aluminum

stress-strain data gives the closest correlation. Using

the extent of strain as measured in a tensile test as an in­

dex of ductility, it appears, from the results of this study,

that the materials that exhibit the greatest ductility are

most likely to satisfy the simplified laws of plasticity

previously presented.

Tensile tests are generally used as the basis of design

regardless of the nature of loading or state of stress in the

structure. Fortunately, \-the tensile stress-strain curves

are always in the highest position (see Figures V-12, V-13,

and V-14). Thus1 if an engineer uses tensile stress-strain

data for all design work, there is a degree of assurance

that this will be a conservative design •.

60

REFERENCES

1. P. B. Hell or and H. uohnson. Plasticity for l~!echanical Engineers. D. Van Nostrand Company,--rtd., Ne't·J York: 1962.

2. P. 1:1. Bridgman. Large Plastic Flow and Fracture. HcGraw­Hill Book Company, Ne1-.r York: 'i932.

J. Hugh Ford. Advanced Nechanics of ~Iaterials. Wiley and Sons, Inc., NeN York: 19SJ.

John

1.

2.

3.

SELECTED BIBLIOGRAPriT

A. Nadai~ Theor~ of Flow ~ Fracture £! Solids, Vol. 1. ~IcGraw-fiiT Book Company, Inc, New York: 1950.

P. B. ~!ellor and Ttl. Johnson. Plasticity for r1echanioal En~ineers. D. Van Nostrand Company,-rtd, New York: l9 2.

Hugh Ford. Advanced Nechanics of Haterials. and Sons, Inc., Ne~r York: -r9;.JJ.

John vliley

4. R. Hill. ~ Hathematical Theor;z of Plasticitl,• Oxford University Press, New York: 19)0.

5. Hoffman and Sachs. Theorl of Plasticitu. HcGraw-Hill Book Company, Inc., New York: 1953.

62

vrrA

The author of this thesis, Wen-mo Chen, son of Mr.

and Mrs. Mou-lin Chen,was born on June 20, 1939, at Yin-ko,

'raipei shen, TaiTN"an, Free China.

The author graduated from the Civil Engineerins

Department of Provincial Cheng Rune; University in 19'SJ.

After he received the B. s. dec;ree, he ser,red 1.11 the Chinese

Army for one year.

After leaving the Chinese Army, he stud.iefl Applier1

Mechanics at the Graduate School of Engineering, Colle3e

of Chinese Culture, for one year, and then transferefl to

the Department of Engineerinr; Mechanics, University of

Missouri at Rolla, for continued graduate study, in

September, 1965.

APPIDIDIX I

SANPLE RESULrS

As indicated earlier, in order to an3.lyzc ':.:1·; -:--~19.-

tive agreement of the stress-strain properties obtained

from basically different experi~ental tests of a given

materi9.l, some appropriate stre:3s anr1 stra:t·'l related parg,-

meters shoul·1 be selected and the correspondi~1.g dia.:ra~ns

constructed from experimental observations. These dia-

:;rams have been mentionec1 and cliscusse(l 1~1 previous

sections. Here, the aluminum specimeG is taken as an

example to sho't~J how these stress-strain ClJrvcs ~>Jere con-

structed for various loading conditions-tension, CO!n.pre~3-

sion, torsion and bending.

A. Const~1ction of stress-strain diagrams for tension.

1. Loads, current diameters and elongations (or

strains) are the quantities of observations.

2. In order to cle~ermine the modu1us of elastt-

city, S.t- the stres.s-strail1 dia(';rams in ~he .~

elas~ic ~an3e are constructed an~ the slope

is ca1.~u1ated. I'hts c.onstruction is shoi•rn

i 1" vi ·:)· ur e 'l- ~ -~ .l !~ • 1 _, •

). Pron the ::neasurefl~ diameter, the current area

of the specimen can be determine~. ~ext,

the correspon~lns load acting on the specimen

is divided by the current area, giving a

value of the true material stress (Yt)•

64

4. Knowing the initial gage len3th 10

, plus the

elongation at any load, the current length L

of the specimen can be obtained. The natu~al

strain is defined as being the natural lo~ar-

ithrn of the quotient, current length 'iividen

by initial len3th. Subtracting the ratio

X from the natural strg1n c;ives the de­

sired strain parameter.

5. The calculation of the above para~eters for

the plastic strain of an aluminum tensile

specimen is presented in Table A-1.

B. Construction of stress-strain diagrams for comp-

ression. The procedure of constructinc; stress-

strain diagrams for compression is the same as

that for tension and needs no further description

here.

c. Construction of stress-strain dia~rams for torsion.

1. 'C1.v-1st moment 9-nd anGle are the quantities of

o bs ervat 1 OYl.

2. In order to determine the modulus of rigidity,

a cu.rve of torque T versus angle of tt'fist 9,

in the elastic range, is constructed first.

rhen, the method described in Section· V,

equation V-JJ is employed.

J. rhe equation used for determining the shear

1\Taterial: Alumin,J.m

Sage length: Ro -- 1.80 inches

Hod.ulus of elasticity = 1o,soqooo psi

Formula:

Yield stress: y p t =-x-

Strain: 1n· J - ;t -r; ~t

p D Yt 104 (lb. ) (in) ....ll. e X lni ~ Yt lnJ;; ~ ...._ Loads Diameter (psi) j41 Elongation Et .;..Jt

19000 0.6858 51440 l.J-7. 6 123 67.() 0.0020

19500 0. () 81.~9 52930 h9. 0 175 97.0 o.oo'-f-8

20000 o.r5830 511-5 9 0 .50. s 242 133.6 0.00~3

20500 0.6820 56120 .52.0 313 172.0 0.0120

21000 0.6808 57690 53.5 383 210.5 0.0157

22050 O.h770 61260 c'? 7 .) ,) . 5R5 319.7 0.02h3

23000 0.6722 64810 6o.o 837 455.0 0.0395

23500 0.6097 66710 61.8 975 527.8 0. 0466

24000 0. 666 0 68890 63.8 114-5 616.7 0.0553

2L~5oo o.661l} 71309 66.0 1350 723.0 o. 0657

Table A-1 PLASTIC 'rENSILE DArA FOB ALUHINUM

66

stress-shear strain curve has been ~erive~;

eq ua t 1 on ( I I I- g ) , a n:l. i s r ew r 1 t t en here as

$lf,= /lra(3T+9~). 0

4. A curve of torque r versus angle of twist per

unit length 8 can be constructed from the

experimental data as shown in Fisure A-1.

By drawing tangents at points alonz the

curve, th 1 dT h ~ e s ope a:g- can e fo~Jnu. Substi-

tutinc; these values of I', e a·•.1(l the corres-

pondinr; slope) for a number of (1 ifferent points

into equation (III-8)) the yield shear stress

is determined. Further, for a '3ive:n 7alue

of e, the correspon~tng value of the shear

strain {can be calculaten by t.he equation

( = re = O.J44J8 •

Using this, the values of the strain para-

meter

re /f

k 0

can be calculated.

5. The calculation of the above stress-strain

parameter is presented in Table A-2.

D. Construction of stres:s-strain diagrar.rfs: for bend:i,.ng

1. Loads, and the strains occuring in the tensile

0

0

0 0 0 (V"'\

'-0 P<

0 0 ..:::t C\J

0 0 <X) ......

0 0 C\1 ~

0 0 \...0

67

(1.)

(\} . 0

--.: C\l . 0

,.... ~ .,....

c ............

(\i "C . m 0 ~

..c ~ bO s:: ,, C)

r--' ,.-{ . G +.:>

or-1 ~ ~

H Q)

~ (\) ,-I ~-~ . (/)

0 .,-1 ;;

-l-.:

c.-. 0

Cf.~ c (l)

·r-1 0 tO

s:i 4!

(I) .:::r 0 . 0

0

(:) 0:' ~ j::l

.... ... .. ' 1--.... , ,:: H "'~ "':"" ,..._, 1--

....::1 <.

n: c IJ,

~ r . <l! C'l

~=--~ (.]

H ~ ~ r~

~=-0

f:J H l) ~- ~ ,:t::..

~ I

w ~ 0' o:' 0 1=-J

H ct ~-, ... , .... ~~~ H r-:: r~

~ rx:l

rl I

<

. t.) ..,.., (il

Material: Aluminum

Radius: r 0 = O.J44J inch

Modulus of rigidity: ~ = ~92qooo psi

Formulas: ./3k = ~(Jr

re- ....k...

Points

1

2

J

4

7

Stress:

Strair1:

·11 .L

(in-lb)

1800

2105

2358

2hl0

2788

2925

2970

e (rad/in)

0.025

0.040

o. 060

0.100

0.160

0.240

0.320

,,.., \_A

dT de

29400

16125

9250

4538

2)42

829

1-t-OO

/3k

q.l200

46800

50600

55600

58tSoo

h0250

60650

rable A-2 PLASriC roRSIONAL DATA FOR ALUMINUM

6R

k r9- 'G

rr 0.078

0.402

0.758

1.521

2. 8Ao

4.268

5. 466

and compresslve oide of the beam, are the

qualities of observation.

2. In order to determine the modulus of elasti-

city Eb for bendin3, an elastic curve of

moment J'.T versus curvature 0 is constructen

first, and then the method described in Sec-

tion V equati~n (V-2) is applte~.

J • rr he a p p 1 :t c a "b 1 c c quat i 0 ns f 0 r rl c ~ e r:11 in 1 ng t h C!

stress-straill C11rve have b88<'1 rlc-r-i \TCrl pre-

(j;_ d ~-·-.) = v (A-1) CT"c (l€c

(A-2)

In order to find the relation be"SNeen a-t and

'b, 9. curve of €.c vers1ls Gt must be construct­

ed .:J.s sho1"rn in Fic:ure A-2. 'I'h.e results of

this figure indicate that

a--f-= 1.012 o-~ • (A-3) v J

In order to establish the value of the right

sirle of equa~ion (A-2), ~ curiTe of bending

moment versus curvature must be constructe~,

which is shown in Figure A-3. By drawing

tangents at incli vidual points alent: this

curve, and substituting the values of Ivl, ¢

c-1)

0 -~ ,.,.., s:: ~

' s:: ~

s:: .n «1 F-1 4-) til

(1)

r-1 01'-4 tl)

s:: (1)

.p (/)

I

\JJ+l

50

40

30

20 l

10

~Slope = ~ = 1.012 0.~t

€c - Compressive strain (in/in) x

Fig. A-2 ~~crERNAL FIBER s-rB.AIN3 FC'? A..Llfl.~I!:rTJtlf BEA.l\1

---.J 0

71

C'-p.

(~ .. ~ t'-•

1-: c:' ~: rrl (:()

(), (') c 0 r-~,

.--4 ~ <

C\.1 f-' ,...-! <t

r ~

rC !=..:· 1:0 t' ~ t-/

~ (l) c:t, H '-,

y

~ f)' +) ...... 1:0 t) ~ I H F-;:; 1--

r~

0 r--, .... , ::8 0 >::-,_._

s c ('""\ (\) I

~

. b'l _;

r=-.

--~--~--~~--~--~----~--~--~----L---~-=~0 0

100~~~--~~--~~--~~~--~~--~~--~----

80 ('1"'\

0 r-f

>< C\l ..-i

60 -'d cd J.-4 .........

C1> ~

40 E ~ ~ 0

I

'& 20 0 Compressive strain, €c = 0.504 0

8 Tensile strain, ~t = 0.497 0

0 0 10 2 0 3 0 40 50

Beam strain (in/in) x 103

Fig. A-4 BEAM CURVATURE-STRAIN DATA FOR ALUMINID·I

--..] C\)

73

and the corresponding slopes into equation

(A-2), the results can be combined with

equation (A-3) to find the values of ~t and

~0 • Further, by using the diagram of curva­

ture ) - versus tc and c. t, shown in Figure

A-4, the corresponding strains can be found.

The required strain parameters are then found

by subtracting the quantities ~ and ~· b b

Thus, the stress-strain curves can now be

constructed.

4. The calculation of the above stress-strain

parameter is presented in Table A-3.

74

Haterial: Aluminum

r-1od.u1us of elasticity = 1q9oo,ooo psi

Formulas:

For stress: G""G = 1175 X (2H +,.} u-;_ = 1. 012 ~

iJ

For strain: ac -6 {e-. cr = € 0.504 JC 10 fJ -~:) -r-= c c ~ I" ~b -~~

'Y = £t-~ == 0. l-J-<)7 X 10 ¢ - t Clt Eb

l\1 ~ d"·~'" 0" g~ l"l

Point (kip- dT ¥c ot OG

in) 12000 1/12000 (ksi) (ksi) 1000 1000

1 14.50 10~0 0.9 44- q . / L~5. 4 0.9 0.8

2 17.75 15·0 0 1-}c . ..) L~q o / • u 5o.'* ).0 2.8

3 19.6 20.0 0.27 52.6 53.3 5.3 5.0

4 20.6 25,0 0,10 51-}.7 55.3 7.:6 7.3

5 21. 1-t JO.O o. 068 55.2 55.9 10.1 9.8

r; 22 .. 5 !+-3. 8 0,0()2 56.2 5S.9 16.9 16.5

7 2).0 57.5 O.OJJ 56.3 57.0 2J,S 22.5

8 2],4 70.0 0,033 ss.o 58.7 30.0 29.3

9 2J.3 82.5 0.033 59.3 60.0 36.2 35.4

10 21}, 2 97.5 0.033 61.1 ~1.8 43 .. 6 42.7

11 25.0 120.0 O,OJJ 63.8 6 ).J- r:: ' •../ 5h. 6 5J.rS

12 2( 3 o. 157.5 0.033 68.2 69.0 73.0 72.0

Table A-3 PLASTIC BENDING DATA FOR ALUr1INUM

121641