TECHN. VÜEGTUiGBOUV

Kinaalfl:

2 3 iïiri.i :oA Note No.90

THE COLLEGE OF AERONAUTICS

CRANFIELD

SOME VALUES OF THE REDUCTION FACTOR FOR PLASTIC BUCKLING OF PLATES IN SHEAR

by

B. EMSLIE

TECHNISCHE HOGESCHOOL VUEGTUIGBOUWKUNDE

K»naBlitraat 10 - DElfT

MOTE NO. 90.

January. 1959.

1-H.-E_._0 _q,L L_E_G E OF A E R O H A U T I G S

9JLLILLJJLk2.

Some Values of the Reduction Factor for Plastic

Buckling of Plates in Shear

-by -

Betty Emslie, B Ji.

In this note values are ohtained for the plasticity reduction factor

for plates in shear, "based on incremental plasticity theory. These

values are coinpared with other theoretical results and vd.th some

experimental results,

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GOMiCTTTS

Suimiary

Notation

Introduction

The Approximate Solution of the Equation for Plastic Shear Buckling of an Iiifinite Plate

The Approximate Solution of the Equation for Plastic Shear Buckling of a Square Plate

Comparison of Some Plasticity Reduction Factors

Discussion of Results

References

Table 1

Table 2

Figures

- 3 -

. NOTATION

X. , X , x^ ReotangulajT c a r t e s i a n c o o r d i n a t e s . Ox. cjrid Ox„ a re

axes i n the middle sur face of the p l a t e and Ox, i s

normal t o the p l a t e .

V, • Displacement i n the x , d i r e c t i o n .

S S t r e s s r e s u l t a n t s e-pplied to the edges of tlie p l a t e .

D F l e x u r a l r i g i d i t y of the p l a t e .

"b T/idth of long p l a t e .

a Side of square p l a t e .

s Ijcngth of h a l f vra-ves of the "buckled p l a t e .

a Slope of the nodal l i n o s .

E Young' s Modulus

G Shear Modulus

G_ Tangent Shear Modulus

G Secant Shear Modulus

s

V Po i s son ' s R a t i o

2 G^ (1 + i )

m Material characteristic in the expression f f. m

e = = + e (-ir) (See R.Ae.S Data Sheet 00.02.04) ^ R ±j

T? Plasticity redijction factor

_ actual "buckling, sti-ess elastic buckling stress

^, Buckling shear stress lb/in

> 4 -

Introduction

The plastic buckling stress of a plate is usually obtained by multiplying the elastic buckling stress by a factor V , the plasticity reduction factor.

D. H, Smith, Ref. 3, obtained some experimental values of V for plates in shear and the aim of this note is to calculate theoretical values of H , based on incremental 'theory, for coiiipaxison. Other tlieoretical valines of H , based on total strain theory and some suggested approximations for H ,

S/G and T/G are also shovm in Figures 3 aj d 4»

" • The Approximate Sqlution of the Equation for Plastic Shear Bucklinj? of an Iiifihite Pla.te

Consider equation (14), Ref,1,

bff/a^V, \2 /a^V^V 9 % 9^V, LfirJ^ - t^) / 9^V, xa"

s = - ID -^

o o \ 1

9 x^ 9 x^

9V,

T' E

9x„ dx. dx-, 1 2

Following Timoshenko, l e t the deflection be

2 ^ V, = A sin -^— sin "• 3 b s (x^ - «x^)

> 9 X 9x -, ''1^2 ".dx dx

...(1.1)

...(1,2)

With

\ =

the deflection (1 ,2) becomes

b

b"

s b

. . . ( 1 . 3 )

V3 = A

and equation (I .1) becomes

ir 3 - ^ s±n '"' ^ s in % { ^^ " "-^ p) . . . ( 1 . 4 )

l"^ 1

S = -D o o

2b^

- 5 -

^ 'if A v-ré; ^^^ Qjt 9-|H + 2M(I - . ) 3^V,

2/V 1

3V, 9V,

0 0 1 2 i ? dS

1 2

vAiere {2 = 2^G^(1 + ^

. . . ( 1 . 5 )

S u b s t i t u t i n g (1 .4 ) i n equa t ion ( 1 . 5 ) ,

^ [ ( ^ + « ' ) T ^ + 6« +A%(« ^1)(2. +2/i (1-v)) ]...(1. 6) 2b'

By p u t t i n g -^^ = O = ?-=^ f o r s t a t i o n a r y va lues of S, the

follovTing tvro equa t ions are ob ta ined ,

^ «^H*f^ =^ 3a^ - V ) ^ + 6 - ^ '

2 y + 2/ i (1. - v )

2 y + 2M (1 - f )

. . . (1.7)

JL 1

The solutions of equations (1.7) and (l .6) are tabulated in Table 1 and plotted in Figure 1.

TECHNISCHE HOGESCHOC VLIEGTUIGQOUVVKUNDE

K»ni^!«(-..5l 10 - D."LPT

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2 • The _ ApiProximate g o l u t i o n of__^the^quatj^qn.^or P l a s t i c Shear Buckljja^ ^qf a Square P l a t e

For the squaie p l a t e , follov/ing Timoshenko, l e t the d e f l e c t i o n "be

V, V . m w X. . n 'T x„ ^ a s i n 1 s m 2 mpl n=1 ran . . . ( 2 . 1 )

Substituting (2.1) in equation (1.1),

S = - D v*

64 a

2 2 a ^ m n mn 2 2 2 2

m^ + n^ + m^ n^ \2v + 2M (I - i )

a a m n p q ranpq/ii Z\/ Z Z\ ^^ (P - m ) ( n - q )

. . . (2 .2)

The system of constants a , a , to make S a minimum, are '' ran* pq ' '

s e l e c t e d as i n Ref. 2 . This p rocess l eads t o the folla\fdng s e t of e q u a t i o n s ,

(2 + P)y

Vs 0

0

0

1é(

79 2 + F)y

-Vs

-Vs

^V25

0 0 0

-VS -V5 ^^25

(82 +9P)y 0 0

0 (82+9F)y 0

0 0 81(2+F)y

^ ^ 1

1 ^22

^ 3

"31

"33

= o~l 0

0

0

0 L. -J

vdiere y = -32 •

?r~D

a^S

. . . (2.3)

and F = 2 v + 2 / i ( l - v ) .

For the s e t of equa t i ons (2 .3 ) t o have a s o l u t i o n o t h e r than 3-. . = a.

= . . . = a , , = 0 , the deterai inant of the c o e f f i c i e n t s must van i sh i and 3J)

hence the c r i t i c a l v a l u e s of S can be found.

These r e s u l t s a re t a b u l a t e d i n Table 2 and p l o t t e d i n Figure 2 .

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3 . S3HR,^-^^i^l; .- .S£-^")^. -?-*- -g-' i.pi," ."y! R' .4''-'°' i .R? .i.- . P..' .y. .

I n F igu re s 3 aiid 4 the va lues of the p l a s t i c i t y r educ t ion f a c t o r ob ta ined i n paragraphs 1 and 2 are conxpared -viriLth : -

( i ) Bi^La^rd's t h e o r e t i c a l f a c t o r s based on t o t a l s t r a i n p l a s t i c i t y t h e o i y .

G G ( i i ) Values of S/G and T / G based on the Mises-Hencky

d i s t o r t i o n energy h y p o t h e s i s .

( i i i ) Exper imental va lues ob ta ined by D.H.Smith,

A l l the p l a s t i c i t y r educ t ion f a c t o r s have been c a l c u l a t e d f o r aluminixan a l l o y DTD.é03 ( . 2 ^ proof s t r e s s i n the 48,000 I b / i n ^ r e g i o n ; m = 16 i n the s t r e s s - s t r a i n exj^ression) , the m a t e r i a l used by Smith i n h i s t e s t s .

4 . D iscuss ion of R e s u l t s

The c a l c u l a t i o n s i n t h i s note f o r aluminium a l l o y DTD.603 show as u s u a l , see Benthem Ref. 4 , t h a t incrementa l theory p r e d i c t s va lues of 7? t h a t a re too h ig l i . Although Smi th ' s t e s t i^esults a re no t c o n c l u s i v e , the gene ra l t r e n d i s more i n agreement ¥d.th t h e o r e t i c a l va lues of V based on t o t a l s t r a i n t h e o r y .

5 . ReferCTLGCs

1 . Henip, T/.S.

2 . Timoshenko, S,

3 . Smith, D.H.

4 . Benthem, J . P .

5 . Kri-vetslqy, A ,

6 . B a r r e t t , A . J ,

7 . B a r r e t t , A . J .

" P l a s t i c Buckl ing of a P l a t e i n Shear" , Col lege of Aeronaut ics Note No. 64 , May 1957.

"Theoiy of E l a s t i c S t a b i l i t y " , 1936.

" I n v 3 s t i g a t i o n i n t o the P l a s t i c Buckling of P l a t e s i n Shear" , College of Aeronaut ics T h e s i s , June 1957.

"On the Buckl ing of Bars and P l a t e s i n the P l a s t i c Range". P t . I I N,A.C.A, T,M.1392 March 1956.

" P l a s t i c i t y C o e f f i c i e n t s f o r the P l a s t i c Buckling of P l a t e s and S h e l l s " . J o u r n a l of the Aeronau t i ca l Sc i ences , June 1955.

"The Es t ima t ion of Shear ?feb Buckling S t r e s s e s " , Royal Ae ronau t i ca l Soc ie ty Draf t Data Sheet O2 .O3 .S . I I .

"Note on Non-Elas t i c Shear Buck l ing" , Royal Ae ronau t i ca l Soc ie ly Dra f t Data Sheet 0 2 . 0 3 . S . I 2 .

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TABLE 1

ti

I 1.0

' 0.8

0.6 1

1 0.4

0.2

0

a

0.707

0.686

0.665

0,643

0,622

0,600

X

1,23

1.20

1.17

1,14

1 .11

1,08

[ '

A. D '"'

5.67

5.40

5.14

4.87

4.59

4.31

SDTJ^/' Y)t^ ^ ^=A ,0

1,0

0.955 !

0.909

0.861

0.812

0.762

TABLE 2

u

1,0

0,8

0,6

0,4

0,2

0

y

3.274 X 10"^

3.493 X 10"^

3.745 X 10~^

4.039 X 10"^

4.385 X 10"^

4.798 X 10'^

jyir

9,h2

8.83

8.24

7.64

7.03

6.43

/ &

1.0

0.937

0.874

0.810

0.747

0.682

;i = 1 .0

V |D/'1/ b ' v= '/, r lO

V . 0 3 • " » " •

„^ acrfu »1 I-w

o 0

0-8

0 7

J / r

E

A r

• • •

JÊ

/ ^

'^

1/ r /

O 0-2 0 4 0 6 OS lO

r FIG. I. INFINITE PLATE.

mi \^-\

lO

1*" lOïi 'J/ |DTT»j ft:\0

p . 2r^[\* vJ

0 9

O-8

O-7

0-6 O 0-2 0-4 0 6 0 8 lO

f

FIG.2. SQUARE PLATE

+ —

THEORETICAL

THEORETICAL

®$/G

EXPERIMENTAL

[INCREMENTAL THEORY)

[ BULAARO - KRIVETSKY)

lO

O S

0-6 ALUMINIUM ALLOY D.T.D. 603

0 4

0-2

O » -iqpoo

[lb/ in) 20;000 39000

FIG. 3. INFINITE PLATE.

[INCREMENTAL THEORY)

[OIJLAARD- KRIVETSKY)

10,000

FIG 4.

2QOOO

% ["> /'"] SQUARE PLATE.

30pOO