Stress Concentration Factors in T-Head

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A D

A M R A R 66-36

STRESS

CONCENTRATION

FACTORS

N

T-HEADS

o

I

T E C H N I C A L

R E P O R T

b y

F R A N C I S

.

A R A T T A

N O V E M B E R

9 66

C

istribut

o

n of

this

document is

unlimited

mm

U .

.

R M Y

M A T E R I A L S

R E S E A R C H

A G E N C Y

W A T E R T Q W N ,

M A S S A C H U S E T T S

0 2 17 2

CO

H



L » - • *

s r c

°

U

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S E C , I

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W

n l.m

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AB U

c«/mii»B;un

G O O S

ß , S T .

tt«L

*/«

S K C I A l f

/

Mention

of

any

trade

names

or

manufacturers

in

this

report

shall not be construed as

advertising nor as an

official

indorsement or approval

of such products or

companies

by

the United States Government.

Th e findings

in

this

report

are

not to

be

construed

an

an

official

Department of the Army

position,

unless so

designated by other authorized documents.

i

DISPOSITION INSTRUCTIONS

Destroy his

eport

hen

it

is

o

onger

eeded.

D o ot

return

it

to he

riginator.

T-*— « I 'Tnt i

,

'>t^ *~-^*~

-

*mr



STRES S

CONCENTRATION

FACTORS

IN

T-HEADS

Technical

Report

M R A R 6-36

by

Francis

I. Baratt a

November 1966

D/A

roject

IN54271&D387

A M C M S ode

5547.12.^2700

Projectile XM

474/573

Subtask 35425

Distribution of

this

document

i s unlimited.

U.

S.

ARMY

MATERIALS

ESEARCH

AGENCY

WATERTOWN,

MASSACHUSETT S

0

217

2

V

tm*m*mvmm

J T m



U . S

4

A R M Y

ATERIALS ESEARCH

G E N C Y

STRESS CONCENTRATION

FACTORS

IN

T-HEADS

A B S T R A C T

Two simple formulae are presented which predict stress concentration

factors applicable to a two-dimensional s y m r r . *tric

a

l T-head configuration.

This configuration consists of a deep head

joined

to a

shank

by fillet

radii.

The

independent

equations

that predict

stress concentration

factors

fo r

the

same

geometry

are

derived

fo r two different

leading

conditions.

In one

instance,

a tensile force is applied

to

the shank end of the T-shape.

Equilibrium

of

forces is attained

by

supporting

the

bottom edge

of the head

section,

resulting in

the

shank section

being

pulled in

tension.

In the

second instance, a compressive

load

is appliec to the top edge

of

the head

section while the

configuration

is again

supported

at the bottom edge.

Thus,

only

the head section is stressed

an d

r , >

a

compressive manner.

Because the analysis is no t exact,

th e

magnitudes

of

the

stress

con-

centration

factors

resulting

from the

predictive

equations

appear

to

be

overly

conservative

at

some

ranges

of

the geometry parameter

ratios.

There-

fore, an arbitrary

limit

of

application ,

s

it

is

termed

in the

text, is

recommended when using these equations.

Again, because

of the inexactness

of

the

analysis, experimental stress

concentration

factors

are

indirectly

obtained

for

the first

loading

condi-

tion and lirectly obtained fo r the

second

loading condition

mentioned

above.

These

data

were

obtained

fo r

several

geometric

atios

of

the

T-head

config-

uration

ai d

compared

to

the

corresponding predicted

values.

It

W4 s

found

that

the

formulae could

be

utilized, with engineering ac-

curacy,

witiin

a

certain

range of the two

pertirent geometry

ratios.

Beyond

these

ranges, the

error became excessive,

but

conservative.

. .



CONTKNTS

age

ABSTRACT

INTRODUCTION

FORMULATION

.......

LIMIT

07

APPLICATION

EXPERIMENTAL STUDY

.... ............

RESULTS WD

DISCUSSION

2

SUMMARY

OF

CONCLUSIONS

..... 5

ACKNOWLEDGMENT 5

REFERENCES 6

l ii

i«n

_.



I N T R O D U C T I O N

A brief literature survey is presented

of past work relating to stress

concentration

factors

in

two-dimensional

T-head

configurations.

The

results

of

the

survey are pertinent to subsequent

discussions

Hetenyi

wa s

one

of

the

first

experimenters

to test

a two-dimensional T-head configuration. He

stressed the body in a similar

but

slightly

different manner

than

that

shown

in Figure la. The results were applicable for geometries which included

sev-

eral

h/ d

ratios

but

only

on e

d/R ratio.

Twenty years later Hetenyi presented

additional results which were an extension of

Reference

1 .

Heywood

8

further

applied an empirical formula to

the

initial results of Hetenyi His objec-

tive

was to extrapolate

Hetenyi's data by including the

effect

of d/ R fo r

T-heads

of various

h/ d ratios.

Nishihara an d

Fujii

4

obtained, by elasticity

theory,

the

stress distribution in a two-dimensional

bolt head. Although

the

results

of

this

reference yield

the

desired stress concentration

factor,

he

formulae

are

complicated

an d

applicable over a limited range.

V

The

objective of

this

report

is

to

obtain

an

expression

for

the

stress

concentration factor associated with

the

configuration and loading

shown

in

Figure

la.

Th e

formula

is

developed because:

a .

he

empirical

expression

given

in Reference 3

is

based

on

the

limited

data

of

Reference 1 ;

an d

b.

he

formulae of Reference 4 are not amenable to simple computations

an d are inaccurate

at some

ratios

of

d/R of

practical

interest.

Formulae

are

obtained

by

superposition

of

various

load-

ing

cases of Figure 1 which

are guided

by experimental

observation. Two expressions,

which

yield

yalues

of the


factors

applicable

to

Figures

la

and

lc,

esult from this analysis.

These

formulae

increase

the

usable

range

of

existing

data

and are

easy

to apply.

However,

because

of

the

appar-

ent

conservativeness of

the

resulting equations at some

ranges

of

the geometry param-

eters,

a

limit

of

application

is suggested in the text.

Also,

ince

these

formulae

are

not

exact,

experimental

veri-

fication is provided.

[Ul i

/

w

-d

-

n

h-t

w

L-

±_.i__i _

J

m

*a

* a

( a )

( b ) c

F i g u r e

SUPERPOSED T-HEADS

L O A D I N G S

.

.

• •



F O R M U L A T I O N

Photographs

of

models

loaded

similarly

to

those

shown

in

Figures lb and

lc

are

presented in

Figures

2a an d

2b .

In particular, note

the isochromatic

fringe

distribution

at

the fillets. An

experimental

determination of

the

location of

these fringes

at

the

fillet

periphery

for both

loading

cases

revealed that for this model which ha d a large D/d ratio:

a . he stress gradient at and near the maximum stress site was

relatively small; nd

b. hese regions for the

two

loading cases overlapped.

fTTT

r *

m

I D

a . PPLIED LOAD 2 6 0 POUNDS . PPLIED

LOAD 5 8 0

POUNDS

Figure

.

ISOCHROMATIC

PATTERN

Case I A , D/d =

4.0,

d/R

=

10.0

As

the

D/d ratio was reduced,

while retaining a constant d/ R ratio, he

above

statements

became

less

correct.

Even

so,

t

shall

be

assumed

hereafter

that the maximum stress

for

the two loading cases occur at

the

same location

and

can

be added

or

superimposed. The

limitation on

such

an

assumption will

subsequently

be determined. However, he

maximum

principal stresses at the

fillets

of

Figures

lb

and

lc

are superimposed

to

obtain

an

approximate

rela-

tionship

for

the maximum

principal

stresses

at

the fillets

of Figure

la.

This

relationship

is given

by

the

following formula:

r

f

r

f

a

+

fe-

rn

where

fj

t

C T

f

a»

nc

*

i

c

are

tne

max

i

m

u

m

principal

stresses

at

the

fillets

of Figures la, b, and

lc.



The

stress

concentration

factors

for the loading

cases

shown

i n

Figure

are

defined

as

follows:

r

fT

l

iT

=

~n

K

fa

and

t

( 2 )

° f c

kfc

v57d)

where

aj

and

c r

a

are

the

uniformly

applied

stresses.

The sign convention adopted

for the

above stress

concentration

factors

is as

follows:

a . ositive when the

examined

stress is

the

same sign

(tension

or

compression) as the

nominal

stress

to which it is compared; and

b . egative

when

the examined

stress

is the

opposite

sign

to

the

nominal

stress.

Substitution

of

Equations

2

into

Equation

1

[note

(X j

a

cr ^

(D/d)]

results

in

a

relationship among the

three

stress concentration factors, given

by

the

following:

k

fT~

f

a

k

f

C

( 3 )

Thus,

it

is seen that the three concentrators associated

with

Figure

are

directly superposable, if it

can be

assumed

that

aximum stresses are

coincident.

It

remains

to

obtain

a

relationship

between the

stress

concentrators,

k£

a

and kf

c

,

to

reduce

the

number

of variables

in Equation

3

to one.

Accomplishment

of this

is realized by equating

the

loading

state

represented

by

Figure l b , for which

the

stress concentration factor

kf

a

is experimentally

well known,

5

to

the

sum

of

the

two

loading states

shown

in

Figures 3b

f e n d

3c.

Figure

3b can,

in

turn, be reduced

to

the

same loading

cases

as

those

shown

in

Figures

lb

and

l c ,

in which

it

is

again assumed

that

the points

of

maximum

stress

of

the two

cases

coincide.

Therefore, the

stress

states

rep-

resented by the loading

cases

shown

in

Figures

3b and 3 c are superposable

and result i n

the

following

formula:

r

f

a

~

a

f b

a

f c »

( 4 )

where

cr in

the

maximum

principal

stress at

the

fillet

of the configuration

and

loading

öhown in Figure

3b.



iilllWIIIMgBWa

wmimmmmmm

[ I I I ]

I

t»^i; 4 *- 4 — -i --*

*—

A_i._J.__i — J.

*a b

c

(a)

» > ) c)

F i g u r e

3 . S U P E R P O S E D

L O A D I N G S

O F T - H E A D

N

T E N S I O N

I t w a s indicated i n Reference 1 1 t h a t i f t h e stress loading

c r > p

could

have been applied

t o

a l l

horizontal

surfaces

o f t h e configuration shown i n

Figure

3 b

a s well a s

o n

t h e

fillet r a d i i , then t h e stress concentrator

would

b e exactly equal t o

o n e .

I t was

considered

impracticable

t o

accomplish

this

loading exactly

b y

experimental

means.

However,

i t was

found that

a s

long

a s

t h e

fillet

radius

was small

relative t o

t h e

neck width

d t h e stress a t

t h e

fillet

c r f b

tended t o

b e

equal

t o

t h e

applied

stress c r j

when

t h e stress

load-

ing and configuration

were

t h e

same

a s

that shown i n Figure 3 b . Thus,

Equation 4 c a n b e reduced

t o

t h e

following:

c r £

a

_

< Jj o - £

c

(R/d b e small).

( 5 )

Defining

the stress concentration factors

k f

a

a n d

kf

c

i n

terms

of t h e

applied stress

shown

i n Figures 3 a a n d 3 c ,

w e

h a v e :

and

f a

f c

=

*

r

f

a

f c

c r

c

(

D

/d).

Substitution

of

t h e

above

into

Equation

5

and

consideration

of

t h e

equilibrium

of forces o n t h e body

shown i n Figure 3 c

yields:

;

fa

(D/d-^-l>

L

fc

=

( 6 )



V

•

o r

[

f

c

f a

-

D/d

d/R

-

( 7 )

(For

physically

significant

geometry

D-d-2R>0

and

D/d-l-2/d/R>0).

Substitution

of Equation 7 into

Equation

3

f o r

k f - r finally results i n

t h e

following

formula:

k

f

a W* - d T R ) -

1

c

f

T

D/d

( 8 )

d 7 R

A s

previously

indicated,

experimental data

are

available

for

the stress

concentration factor kf

a

for

various

D/d and

d/R

ratios.

However, a n

empirical

formula

i s

given

by Heywood

i n

Reference 3

f o r

this

concentrator based

on

t h e

experimental data available.

Heywood's equation

becomes

(using the

nomen-

clature i n this report):

. _

[W 1 ) d / R

I

0

6

5

k*

a

% 1 +

I

9)

[ . 2 ( 2 . 8

D / d

- 2 ) J

f a

This empirical relationship

can be

utilized t o

obtain

formulae

f o r

the

stress

concentrators

kfr

and kf

c

, which are

dependent

upon only the geometry

ratios D/d

and d / R . Direct substitution

of Equation

9

into Equations

7

and 8

results i n

t h e

following

relationships:

1

c

fT

[ ( D

/ d

1 )

d / R ] ° '

65

[ . 2 ( 2 . 8

D / d

-

2 ) J

(D/d

d / R

)- 1

2 _

d/ R

/d T75- -

(10)

and

f c

D / d - 1 )

d / R

]

2 ( 2 . 8

D/d-2)J

0.65

D/ d

2

d/ R

(ii;

-

The reader i s cautioned that Equations 8 and

1 0

are invalid

f o r

small

values of h/d

( s e e

Figure

1 )

since bending of the flanges

of

t h e

head

i s

precluded

from the

analysis.

Hetenyi

2

indicates

that when h/d i s equal t o

3 . 0 o r greater, t h e

head of

t h e T can b e

considered infinitely

deep,

thus

eliminating

t h e

existence

of

bending

stresses

a t

t h e

fillet.

A

more useful definition of

t h e


factor

f o r

the c o n -

figuration

shown

i n Figure

3 c

i s

kj • o r ,

/a

,

and

from

Equation

2

w e

see

I ,



that

kf

c

*

k£

c

D/d.

Substitution

of

this

latter

relationship

into

Equation

11

results

in the following:

,65

f c

*

D

Hp/d

)

/R

l

-

'

1

L

2

<

2

-

8

°/

d

2 ) J

-äTR

1

(12)

LIMIT O F

P P L I C A TI O N

The stress

concentration factors,

kf-r and

K £

c

, described

by Equations

1 0

and

12,

are

shown

plotted

as

a

function

of

the

constant parameter

D/d and

var-

iable

d/R

in

Figures

4a

and

4 b .

here

are

distinct

minimum

values

for

each

curve

de-

scribed

by D/d

in

these figures. As previously indicated,

t

would seem that

as d/R is decreased, kfj and kf

c

should

also decrease (which they do up to a

point)

and

asymptotically

approach

a

minimum

value

(which they do

not). It

is

evident

that this behavior

is

no t

correct

and

occurs

as

a result of

inexactness

of

the

equations

which in turn

could be

cauaeu by:

a .

he

approximate

nature

of

the observation

that

the maximum

principal

stresses

occur at

the

same fillet locations;

and

b .

he

violation

of th*

estriction that the

radius

R be small

compared

to

the

width d.

°>T =

s t r e s s a t

f i l l t t

J

T

=

applied

s t r e s s

a t

s h a n k

c r

a p p l i e d s t r e s s a t shoulders

h / d

>

3 . 0

4. 0

8.0 2 . 0

1 6 . 0

20.0

24.0

8.0

2.0

d / R

Figure

a.

S T R E S S

0 N C E M T R A T I 0 N

ACT OR

f



0 / d =

1 . 5

D / d

= 1 . 7 5

D / d =2.0

D / d

= 2 . 5

D / d

=3.0

D / d

=

4 . 0

0 / d

=5.0

D / d = o o

L E G E N D

1

m i

J l

(7

m i

k

Hilft

*t

fc

-c

stress

at illet

applied

tress

t ea d

applied

tress

t houlders

3.0

4.0

8.0

12.0 16.0

20.0

24.0 28.0 32.0

d /R

F i g u r e

4 b .

STRESS

CONCENTRATION

FACTOR

k f

c

Thus, the

accuracy

and application

of

the expressions beyond

the

mini-

mum

points

are questionable.

The

seemingly odd behavior indicated above

wi'l

provide convenient

practical

cut-off

points

for

each

curve

describing

kfj and kf

c

.

(The cut-off points are experimentally checked

for two

cases

and

are

discussed

later).

These

cut-off

points can

be

determined

by

t h « »

minimum values of the stress concentration factors defined by particular

values of d/ R as a function

of

D/d. These minimum

values

will provide

limits

on

the

use

of

Equations

10 and

12

and

shall

be

called

limits of

application .

The

limits

are analytically

determined

by simply

applying maximum-

minimum principles to Equations

10

and 1 2 . The limits of application of

Equation

10 describing

kfj

are

given

in

the

following:

d/ R >

D/ d

1/2

(1-1/n)

-/(l/n)(D/d)

1 /4 (1-1/n)

2

The

limits

of

application

of

Equation

12

describing

kj

c

are

given

by

2(1

n )

(13)

d/R

>

n(D/d-l)

(14)



where

n

=

0.65.

Both limits

have

been computed

and are

shown i n Figures

4a

and

4b

as

lines

connecting

the

minimum

point

in

each

curve

and are labeled

limit of application .

E X P E R I M E N T A L

S T U D Y

Because

of the approximations used

in

the previous section it

was neces-

sary

to establish

the

validity

of

the derived

equations.

Therefore,

the

pri-

mary

objective of the experimental program

was

to

determine

the order

of

magnitude

of

the

inherent error

associated with

the determination

of

kff

and

k } -

c

by

Equations 10

and

1 2 .

To accomplish

the

above in

the

simplest

manner,

the

stress

concentration

factors

kf

a

and

j -

c

were

experimentally

determined

for

four

cases

which

had

the various

geometry

ratios shown in Table I . The stress concentration

factor

k£j was then

obtained

from

these

data

for

each

of

the

four

cases

by

simple

superposition as

indicated

by

Equation 3 .

Table I . EXPERIMENTAL, SUPERPOSED, AND PREDICTED

DATA

C a s e

d/R

D / d

EXPERIMENTAL DATA SUPERPOSED DATA

PREDICTED DATA

h f .

> J «

k f c

k

fT

C o a b i n e d F a c t o r , E q u a t i o n 6

1

-;.»

P e r c e n t

d i

f f e r e n c e l

f.

+

(D

/

d

-d7R*

f

c

E

6

k

n»

P e r c e n t

difference

J A

1 0 . 0 4 . 0

2 . 3 7

« 2 . 0 5

- 0 . 5 1

2 . 8 8

0.94

2 . 8 6

-

0 . 7

. 1 . 9 6

-

4 . 4

I B

1 0 . 0

2 . 0 2 . 2 5 • 2 . 4 8 - 1 . 2 4 3 . 4 9 1 . 2 7

3 . 7 9

+

8 . 6

- 3 . 1 0 - 5 . 0

j

IC

1 0 . 0

1 . 5

1 . 9 S

^ 2 . 7 1 . 1 . 8 1 3 . 7 6

1 . 4 1

5 . 7 1

+ S 1 . 8

- 5 . 4 3

+

1 0 0

II

5 . 1

2 . 0

1 . 9 5

- 1 . 0 8

- 0 . 5 4

2 . 4 9

1 . 6 2

3 . 1 3

+ 2 5 . 7

- 2 . 6 5

+

1 4 5

O b t a i n e d

b y

anperpoaing

k f

a

a n d

k f

c

according

t o

Equation

.

t C o a p u t e d according t o

Equation

1 0 .

f C o a p u t e d according t o E q u a t i o n

1 2 .

* * L i a i i t

o f

application

c a a e

f o r k f

c

.

N O T E :

-J >

3 . 0

Cases

IC

and I I were

designed

to determine the

maximum

error allowed

by

the

limit of application. The

geometries

for these cases

were determined

according to

th e

limit

of

application

on

th e

concentrator

kf

c

, (Equation

14)

rather

than

kfr.

This was

done

because th e

minimum points

for

kf

c

as

shown

i n

Figure 4 b are

slightly

more limiting than the corresponding

curves

i n

Figure

4 a .

Thus,

for the same D/d value

th e

limiting d/R ratio

i s

greater

for

j -

than

for

krj,

and,

therefore, more restricted.



v

The

experimental data were

obtained by a photoelastic

study using

a

model

made

of

Homalite

100.

The

two

methods of

applying the

load

and

the

model dimensions

for

the

four

cases

investigated are shown

in

Figure

5 .

he

model

was compressively

stressed*

by the first method of

loading,

Figure

5a,

then

by

the

second

method,

Figure

5b.

The

photoelastic model

was

stressed

by

the first method to determine

the

factor kf

a

»

the

location

of the maximum

fringe

order, and

to

index

this

maximum stress

site

by

scribing

fiduciary

lines

on

the

model. At

this location

the

fringe order was

also

determined

when the

model was

leaded

by the

second

method.

Thus,

the

nominal

fringe

order

at

the shank

section

Nj,

the

maximum

fringe

order

at

the

fillet

N £

£t

and

at

this

same site

the

fringe order Nf

c

,

were

obtained for

each

case and

are shown plotted as functions of load

in

Figures 6 and 7 .

These

experimental

data were then

utilized to

obtain

the

stress

concentration

factors

kf

a

and

kf

c

,

from which

kfj

was indirectly

determined.

Figures

2a and 2b are photographs

of

the isochromatic fringe

patterns,

with a

light

background resulting

from

the two

loading

methods

chosen

as

representative of

typical patterns of

case

IA

(d/D

4.0

and

d/ R

10.0).

H

LLL

__

*h

« f.

( b )

C A S E

IA

IB

IC

n

R

0.079

0079

OXJT*

0 . 1 1 0

d

0.790

0.790

0.790

0.969

0

BJOOO

1.900 1 . 12 9

1 . 1 2 9

/

2

2

2

2

h

7

3

5

5

i

1/2

1 / 2 1 / 2

1 / 2

NOTES:

1 .

DECI MAL

T O L E R A N C E

IS

±

0.001

2 .

S H A D E D R E A S S H O W

C A R D B O A R D

PA D

L O C A T I O N S

F i g u r e

5 .

LOADING

A N D

MODEL PARAMETERS

•Pads

were

used

st

si l contact surfaces

to siaulate

constantly distributed

load.



mm

-

C A S E

IA

IB

Z C n

R

0.075

0.075

0.075

0 . 1 1 0

d

0.750

0.750 0.750

0.565

0

1 . 5 0 0

25

l 2 5

f t

2

2

2

2

h

T

5 5

5

t

1/2

1 / 2 °

1/2 1/2

H

N O T E S :

1 .

DECI MAL

T O L E R A N C E

IS

t

0 . 0 0 1

2 .

S H A O E O

R E A S S H O W

C A R O e O A R D

AD

L O C A T I O N S

- 1 0 0 - 2 0 0

- 3 0 0 - 4 0 0 - 5 0 0

P

A P P L I E D

LOAD-LB

F i g u r e 6 F R I N G E

ORDER

N

fa

A N D N

d

A S

A F U N C T I O N

O F

A P P L I E D

L O A D

1 0



\

-

7.0

6.0

5.0

5

UJ

CASE

IA

19

IC

n

R

0.075

04)75 OJOTS

0.110

d

0.750

0.750

0.750

0.365

0

3000

1.500 1.125 1.125

/

2 2 2

2

h

r

5

5

H

5

t

1 /2 1/2- 1/2 1/2

N O T E S :

I. O E C I M A L

O L E R A N C E

IS

± 0.001

2 S H A D E D R E A S S H O W

C A R 0 6 0 A R 0

AD

L O C A T i O N S

R d

f

in

Hfc

- 3 0 0 - i * 0 0 - 5 0 0

P

APPLIED

L O A D - L B

- 7 0 0

F i g u r e

7 .

F R I N G E

O R D E R

N

f

c

A S

A

F U N C T I O N

O F

A P P L I E D

L O A D

r

I

i

isri



R E S U L T S

A N D

D I S C U S S I O N

A

least-square

method which

incorporated the

data

shown

in Figures

6

and

7

was used

to

determine accurately

the

equation

of

each straight iine

designated b y

Nj,

N£

a>

and

Nf

c

for

all

cases.

A straight-line fit

for

each

curve

was corrected to

initiate at

th e

origin

of

th e plot. Residual stresses

and time edge effects present in th e

photoelastic model were

thus

compensated.

These

results were then used to determine

th e

desired experimental concentra-

tion

factors defined as:

f a

l i

l

fc

_Nfc

and

k

f

c

k

fc

(D/d).

The

stress

concentration

factors

for

each

case are shown

in

Table

I .

It

is

noted

that

in

th e

formulation

of

Equation

6 ,

it

was

necessary

to

make

use of:

a .

n

experimental

approximation,

i.e., it

was

assumed

that

th e

points

of

maximum

stress

of the

loading

cases shown

in

Figures

lb

and

l c

coincided;

b . n

experimental

fact, i.e.,

from

Reference 10 it was determined that

under idealized

conditions the

stress concentrator would be

equal

to 1 . 0 ; and

c . he restriction that th e

fillet radius R be small

compared

to

th e

small

width

d «

If

all of th e

above

were

true,

the

left side of

Equation

6 ,

which

is

termed

th e

Combined

Factor,

would

be

equal

to

unity

(1.0).

This

Combined

Factor

could be used as an index,

when

compared to 1.0, on th e relative accuracy of

Equations

10

and 1 2 , which predict kfr and k£

c

. The Combined Factor, for

which th e computations

were

based on

th e

experimentally determined concentra-

tors kf

a

and kf

c

,

as

well

as th e

predicted

values

of

kfp and

j «

c

,

and

th e

percent

difference

when compared

to experimental

values

ar e also

given

i n

Table

I .

Results

given i n

Table

indicate that

when d/R

is

constant (see

cases

I A , IB , and IC), the Combined Factor

approaches

th e idealized

value

of 1.0

with

increasing

D/d. Also, when D/d

is

constant but d/R

i s

increased, a s i n

12



F

3

*

7

*

:

x

cases I I

and

I B ,

the

Combined

Factor appears

to

approach

the idealized

value

of

1.0.

It

is

apparent

from

Table

I

that

as

the

Combined

Factor

approaches

1.0,

the percent differences for

both

kfj

and k£

c

become

small.

These

dif-

ferences are

based

on

a

comparison

of

experimental values of

krj

(and

super-

posed data) and j -

c

to those

given

by the

predictive

equations

1 0

and 1 2 .

Further

examination

of

these

relative

differences

reveals

that:

1 .

f - r

determined

by Equation

10

is

accurate,

compared

to

the experi-

mental value

(see

superposed data),

to

within

less

than

9. 0%

if

D/ d

2 2.0

and

d/ R >

10.0

(compare

cases IA

and

IB

to case IC).

2 . £

c

determined

by

Equation

1 2

is accurate to within

25.0%

if

D/d > 2.0 and d/ R £ .

10.0

when

related

to

the

experimental

data

(compare

cases IA and IB to

case

IC).

3 .

he two

cases,

IC

and I I ,

which

are at the

limit

of

application for

the

concentrator k£

c

, yieJd the maximum

differences, 1 0 0%

and

145%, r e s p e c t i v e l y .

However,

these

differences are

positive

and

are

considered

conservative.

Comparisons can

also

be

made

to

the datt given

in

References

1 and

2 by

Hetenyi

even

though

the

loading

configuration

used

by this

author,

shown

in

Figure

8a,

is different

from

that

shown in Figure

8b.

The

difference

between

these

two loading

states

is

shown

in

Figure £c.

It

is

readily apparent that

if

the fillet radius R

is

small, the

Hetenyi-type

stress

concentration

factor

termed

here

as

k £ j j

is

approximately equivalent to

k£j.

The

data from

Refer-

ences

and

2

are

presented

in

Table

II

as well

as

the predicted values of

the

stress concentration factor

k£j given

by Equation

1 0 ,

and

the

percent

error

when compared to k £ | f •

l

fH »

ittii/

*-

-•

v

Uiii

ÜÜ1/

Mill

+

V

(a)

(b)

Figure . H E T t N Y l YP E - H E A D S

(O

J3

Ü—

'--



Table

I I .

COMPARISON TO EXPERIMENTAL RESULTS O F T-HEADS

AVAILABLE

IN TOE LITERATURE

Ketenyi's

Results

- References 1

and 2

\w

d/R

0.0

d/R

*

3.33

d/R

10.0

d/R

«5.0

k

fH

k

n

«rror

k

fl

k

f

T*

X «rror

>>ff l

k

fT

#

«rror

k

f

k

fT

#

rror

s.o

2.5

2.0

l.S

4.10

4.47

S.00

| 6.0S

4.20

4.

S O

S.10

6.97

♦ 2.4

-

1.0

♦

2.0

+15.0

3.

S O

3.6S

3.90

4.90

3.48

3.73

4.25

6.Ö

S

- .6

+

.2

♦

9.0

+24.0

3.10

3.02

3.30

4.70

3.08

3.30

3.79

5.71

. 1,0

♦ 9.3

+14.8

♦ 21.5

2.52

2.35

2.60

2.38

2.58

3.10

•

5.6

+

.8

♦ 19.2

*Coap«ted

according

to

Eqeatioa 10

(AH values

of

k

f

T

aro

within

the liait

of

application

fiten

by Eqaatioa 1 3 )

NOTE:

-y>3.0

A

comparison

of the results given

in

Table

II

indicates

that

as D/ d in-

creases,

with

d/ R remaining constant,

the.error

generally decreases.

This

is

because

the

points

of

maximum stress

for the

various

loading cases

tend

to coincide and the

stress

gradient

becomes

small

s

D/d

becomes

large.

The results

also

indicate

that,

generally,

Equation 10

becomes more accurate

as d/ R

increases,

with D/ d

remaining

constant.

This is

due

t o :

a . he

restriction

that

the

fillet

radius R be

small

compared

to

the

width

d

in the

analysis;

and

b . s R

becomes

small the

Hetenyi-type

concentration

becomes

equivalent

to kfj.

It is also seen

from

Table

I I

that as

the error

becomes

large, it

is

positive.

Further, if

one

is

interested in

accuracy

of

1 0%

or better, then

Equation 10 can

be used

when:

a .

/d

i s

equal

to

or greater than 2.5

and

d/ R lies between

5. 0

and

20.0; and

b . /d is

equal

to or greater

than

2.0

and

d/ R

lies between

13.33

and

20.0.

Analytically

computed results for the stress

concentration

factor

kfj as

a

function

of

d/R

when

D/d

*

2.325

re

iven in

Reference

4 .

These

data,

as well as those obtained

from

Equation 1 0 ,

can readily

be

compared

and

are shown

in Figure

9 .

It

is

seen

that

these

curves compare

to

within

a t

least

1 0%

of

each other

when

d/ R is

between 3 and

2 8 ;

beyond the

value of 2 8 ,

the

difference

is excessive.

As d/ R

increases,

kfj becomes more accurate.

On

the

other

hand, the

mapping

function

utilized i n Reference

4

becomes

in-

exact

for

large

values of

d/R.

This difference

i s

attributed to

the method

used

in

Reference

4

when

d/ R

>

2 8 .

However,

i t

i s expected

that when

d/ R i s

small, the

method

given

by

Reference

4

i s quite

accurate.

1 4





R E F E R E N C E S

1 .

ETENYI,

M .

om e

Applications

f

hotoelas

t

ic

i

t

y

in

urbine-Generator

Design. Trans.

ASME,

v . Sl

9

1939, p . A-151 to

A-155.

2 .

ETENYI, M .

tress Concentration

actors

or -Heads.

J .

Appl.

Mech.,

v .

81 ,

no.

3 ,

1959, p .

130-132.

3 .

EYWOOD,

R .

B .

Designing

by

Photoelasticity.

Chapman

& Hall

Ltd.,

1952, p . 178.

4 . ISHIHARA, T., and FUJII, T . tresses in

Bolt

Head.

Proc.

of

First

Japanese

National Congress for

Applied

Mechanics,

1951,

p .

145-150.

5 .

ROCHT,

M .

M.

actors

f

Stress

oncentration

Photoelastically

Determined.

ASME, V . 57,

1935,

p . A-67.

6 . ROCHT,

M . M. hotoelastic

tudies

in tress

oncentration.

Mechanical

Engineering, v . 58, 1936, p . 485-489.

7 .

ROCHT,

M .

M . ,

and LANDSBHtG,

D .

actors f

tress

oncentration in ars

with eep harp

rooves

nd Fillets in

ension. Proc. SESA,

v . VIII,

no. 2 ,

1951,

p . 148.

8 .

IMOSHENKO,

S., and

DIETZ,

W .

tress

oncentration roduced

y oles

and

Fillets. Trans. ASME, v .

4 7 ,

1925,

p . 199-237.

9 .

EIBEL, E . E .

tudies

in

Photoelastic

tress

etermination.

Trans.

ASME,

v .

5 6 , 1934,

p . 637-658.

1 0 .

ETERSON, R . E . tress oncentration Design Factors.

John

Wiley

&

Sons,

1953.

1 1 . ARATTA, F .

I . ,

and BLUHM,

J .

I .

n the

Nullification

f

Stress

Concentration actors

by

tress qualization. U .

S .

Army Materials

Research Agency,

AMRA TR 66-37,

November 1966.

Presented

at

the

June

1966

meeting

of

S.E.S.A.

in

Detroit.

1 6

4



U .

S . ARMY MATERIALS RESEARCH

AGENCY

WATERTOWN, MASSACHUSETTS

2172

TECHNICAL

REPORT DISTRIBUTION

Report No.:

AMRA

TR

66-36

November 1966

Title:

tress

oncentration

Factors

n

T-Heads

r

is

tribution

28

toer 966.

List

approved

y

Picatinny

Arsenal,

telephone

conversation,

No. f

Copies

To

o

1

ffice

f

he

Director,

efense

esearch

nd

Engineering,

h e

entagon,

Washington,

.

. 20301

20

Commander, efense

ocumentation

Center,

ameron Station, uilding 5,

5010

uke treet,

lexandria, Virginia

22314

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Metals nformation

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atteile

emorial

nstitute,

Columbus , h io

43201

Chief

of

Research nd Development, epartment

f

he rmy,

Washington,

.

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A T T N :

Physical

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Division

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esearch Office

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Of

T o

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eronautics

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Center,

Huntsville,

labama

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1 ATTO: R-PSVE-M , r. . . Lucas

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.

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ilson

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Technology,

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ak Grove Drive,

asadena, California 91003

1 ATTN: M r.

oward

.

Martens,

Materials ection 51

Command ing

Officer,

. .

rm y

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esearch

gency,

Watertown, Massachusetts

02172

5 ATTN:

AMXMR-AT

1

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1

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1

M X M R - R X

1

uthor

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HZLhSSIHEL-

Seen

ity

Classification

DOCUMENT

CONTROL DAT

4

•

R&D

(SmcuTity

clmmmttlcmtlon of titlm.

ody

of mbmtrmct

mnd

ndexing

mnnotmlion

mumt

m

mntmtmd

mhmn

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vmrmll

rmpert

m clmmattimd)

1 .

ORIGINATING

ACTIVITY

Corporate

muthor

U.

S .

Army

Materials

Research

Agency

Watertown,

Massachusetts 2 17 2

2«

REPORT

SECURITV

CLASSIFICATION

Unclassified

2

6

GROUP

3 .

REPORT TITLE

STRESS

CONCENTRATION

FACTORS

IN

T-HEADS

4

DESCRIPTIVE NOTES

Typo

ot tmport mnd inelumivm dmtmm)

5

AUTHOR^)

(Lmmt

nmmm . int

nom«. nitiml)

Baratt

a,

Francis I .

«.

EPORT

DATE

November

966

«. CONTRACT

OR

GRANT

NO.

t, ROJECT NO.

D/A

N542718D387

C

AM C M S od e

547.12.62700

SuhtasV

5475

7« TOTAL

.O-

F

PAGES

- 1 £ _

76.

NO. F REFS

1 1

9a. ORIGINATOR'S

REPORT

NUMBER'S;

A M R A

R

6-36

f

6.

OTHER EPORT NOfS) Any otttmr

numbmrm

hmt mmv m

mmmimy imd

thim

mpott)

10. AVAILABILITY/LIMITATION

NOTICES

Distribution of

this document is

unlimited.

11 SUPPLEMENTARY

NOTES

\

v.

1 2

SPCKSOPING

ILITARY ACTIVITY

Picatinny

Arsenal

Dover, New

Jersey

07801

u . ABSTRACT wo

simple

formulae

ar e

presented w h i c h predict stress concentration rac-

tors applicable to a two-dimensional symmetrical T-head configuration.

his

config-

uration

corsists

of

a

deep head

joined

to a

shank

b y

fillet radii.

he independent

equations

that predict

stress

concentration

factors

for

th e

same geometry

ar e

de-

rived

for

two

different loading

conditions.

n

one

instance,

a

tensile force is

applied to

the

shank

end of th e

T-shape.

quilibrium of

forces i s attained b y

supporting the bottom

edge

of

the head section, resulting

in

th e shank section

being

pulled in

tension.

n the

second instance, a

conpressive load

is

applied to

the

top

edge

of

the

head

section

while

the

configuration

is

again

supported

at

the

bottom

edge.

hus, only the head section is stressed and

in

a compressive

manner.

Because the

analysis

is

no t exact, the

magnitudes

of

th e

stress

concentration

factors

resulting from

the

predictive

equations appear

to

b e

overly conservative

at

some

ranges

of

the

geometry

parameter

ratios.

herefore, an

arbitrary

limit

of

application , as it

is termed in

the text, is

recommended when

using these equations.

Again, because

of

the

inexactness of

the

analysis,

experimental

stress

concen-

tration

factors

are indirectly

obtained

for the

first

loading condition and directly

obtained

for

th e

second

loading

condition

mentioned

above. hese

data

were

obtainec

for

several

geometric ratios

jyf^

the

T-head configuration and

compared to the

cor-

responding

predicted values.

It was

found that

the

within

a

certain range of

the

th e error becomes excessive,

DD 'Jffl 1473

e

could

b e

utilized,

with

engineering

accuracy,

pertinent geometry

ratios.

eyond these

ranges,

conservative-'

UNCLASSIFIP

Security

Classic. j'.on



UNCLASSIFIED

TT

Security Classification

KEY O R O S

Experimental mechanics

Stresses

Stress concentrations

Stress intensity

Expefimental

stress

analysis

Photoelasticity

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Stress Concentration Factors in T-Head