Bauschinger adaptation of rigid, workhardening trusses

MECH. RES. COMM.

BAUSCHINGER

W. Prager Savognin, (Received

Vol.l,

ADAPTATION

253-256,

OF

1974.

RIGID,

P e r g a m o n Press.

WORKHARDENING

Printed in USA

TRUSSES

Switzerland 27 A u g u s t

1974;

accepted

as ready for p r i n t

5 September

1974)

Introduction

An elastic, p e r f e c t l y p l a s t i c structure may adapt itself to a set of alternative loadings, w h i c h it m u s t carry r e p e a t e d l y and in a r b i t r a r y sequence, in such a m a n n e r that no further p l a s t i c d e f o r m a t i o n occurs after c o m p l e t e adaptation. This k i n d of adaptation, w h i c h is k n o w n as shakedown, has b e e n the subject of m a n y p u b l i c a t i o n s , of w h i c h we m e n t i o n only the p i o n e e r i n g ones by Gr~ning {i}, Bleich {2}, and Melan {3}. The p r e s e n t note deals with another k i n d of adaptation. The structure, w h i c h is here s u p p o s e d to consist of a rigid, k i n e m a t i c a l l y h a r d e n i n g material, is again s u b j e c t to a set of a l t e r n a t i v e loadings, w h i c h it must carry any number of times and in a r b i t r a r y sequence. W i t h o u t tracing the p l a s t i c deformations c a u s e d by the s u c c e s s i v e loadings, we w i s h to a s c e r t a i n w h e t h e r the s t r u c t u r e will adapt itself to these loadings in the sense that plastic deform a t i o n w i l l e v e n t u a l l y cease. If it does occur, this k i n d of a d a p t a t i o n is b r o u g h t a b o u t by the B a u s c h i n g e r effect; it w i l l t h e r e f o r e be c a l l e d B a u s c h i n ger adaptation.

B a u s c h i n g e r A d a p t a t i o n of a Simple Truss

To i l l u s t r a t e B a u s c h i n g e r a d a p t a t i o n by a simple example,

c o n s i d e r a plane

truss c o n s i s t i n g of bars that join the loaded joint A to fixed joints B.i , ( i = i, 2,

.., n ). The a l t e r n a t i v e loads Q

, ( ~ = i, 2,

.., ~ ) are to be

r e p e a t e d l y a p p l i e d to the joint A in a r b i t r a r y sequence. The rigid, w o r k h a r d e n i n g b e h a v i o r of the typical b a r B.A is c o m p l e t e l y descri1 bed by the c o n t r i b u t i o n of this bar to the internal p o w e r of d i s s i p a t i o n D of the truss w h e n the joint A has u n d e r g o n e the d i s p l a c e m e n t s p e c i f i e d by the column v e c t o r u and moves w i t h the v e l o c i t y s p e c i f i e d by the column v e c t o r v. Sci en ti fi c Communi ca t i on

253

254

W.

We d e n o t e

the

unit

column

vector

PRAGER

of the

Voi.l , No.%,.I:

direction

from

B.

to

A

by

and

b

1

D(U,V)

Here

Y.

: }]i ~" Y i l b ~ vl

is the

common

+ c ( u ' b ( b) : V )1l

magnitude

set

1

(i)

}

of the

tensile

and

compressive

yield

forces

1

in the v i r g i n

state

of w o r k h a r d e n i n g , b~v

are

the

of the b a r

B.A, l

c > 0 is a c o n s t a n t

and the prime indicates

elongation

I.

i

a n d rate

of

this

bar

of e l o n g a t i o n

< 0. T h i s

means

to D thus

that

Note t h a t

u'b.

U. of the b a r

B.A.

1

is

(Y.+ cl.)U, 1

U.

the

transposition.

i

tribution

specifying

the

tensile

1

and

for

rate l

and

The

con-

1

~ > 0 and -(Y.-

1

1

compressive

cl.)U,

1

yield

forces

l

for

l

of the b a r

1

B.A

experienced

t h a t has

the permanent elongation

t.

t

a r e g i v e n by

1

T

=

i

Y.

+

1

As p l a s t i c

c

%.

,

1

C.

=

l

deformation

-(

occurs,

Y.

-

I

c

I

(2)

)

i

the y i e l d

interval

(C ,T ) thus 1

length

2 Y

but

undergoes

the

displacement

c I. in a c c o r d a n c e

1

square

a n d the

column

the

be

with

the

concept

hardening.

root

u = MU

will

the

1

of k i n e m a t i c The

retains

1

the p o s i t i v e

definite

matrix

~i b.b~ll w i l l

be

denoted

by M,

vectors v- = M v

,

called

reduced

of

the

,

b

= M -i b

i

reduced

directional

~

,

i

displacement

vector

= M -I Q

~

(3)

and velocity

of the b a r

B A,

and

vectors

of the

the r e d u c e d

~-th

joint

A,

l o a d vec-

1

tot. the

In t e r m s truss

of t h e s e

reduced

vectors,

the

internal

power

of d i s s i p a t i o n

is

D u,v) = Xi { Yil i i} + eu'v . It

is r e a d i l y

properties function

of

gin

verified the

that,

vectors

truss,

that

namely

X i]biv] -.-

vectors

±Y b.

function function,

of p o i n t s

as s p e c i f i c a t i o n s

, represents , a n d D(0,v)

value

we

shall

in this

u = 0. T h e the

line

itself

of u,

of a c o n v e x

the

function

domain.

regard

in a l o a d p l a n e ,

of d i r e c t i o n s

is the case

<4)

for a f i x e d

supporting

(4) as s u p p o r t i n g

as p o s i t i o n tors

of

and

the

the

In i n t e r p r e t i n g reduced

reduced

plane.

We

first

contribution

of

the b a r

segment

represents

whose the

endpoints

linear

(4) has

the the

load vectors

velocity

consider

vec-

the vir-

B . A to D ( 0 , ~ ) , l h a v e the p o s i t i o n

combination

of

the

li-

ii

ne s e g m e n t s

corresponding

to the b a r s

of the

truss

( see,

for

instance,

Pra-

Vol.l,

No.5/6

B A U S C H I N G E R A D A P T A T I O N OF T R U S S E S

get {4}). This is a polygon, w h i c h has,

255

for each bar of the truss, a p a i r of

o p p o s i t e sides that have the same d i r e c t i o n and length as the line s e g m e n t c o r r e s p o n d i n g to this bar.

For the virgin truss,

this p o l y g o n is the y i e l d

locus in the sense that any i n t e r i o r p o i n t represents,

by its p o s i t i o n vector,

a r e d u c e d load that cannot cause p l a s t i c deformation, while any e x t e r i o r p o i n t represents

a r e d u c e d load that exceeds the l o a d - c a r r y i n g c a p a c i t y of the truss

An i n t e r i o r p o i n t of a side of the y i e l d locus represents, by it p o s i t i o n vector, a r e d u c e d load under w h i c h only the bar c o r r e s p o n d i n g to this side is not ready to yield,

and which will therefore cause the loaded joint to move per-

p e n d i c u l a r l y to this bar.

If this is the bar B.A, we have b~v = 0 and hence 1 1 b~v = 0. Under the influence of a r e d u c e d load r e p r e s e n t e d by an interior 1

p o i n t of a side of the y i e l d locus,

the joint A will therefore assume a redu-

ced v e l o c i t y v that is normal to this side. By e x a m i n i n g the signs of the forces in the y i e l d i n g bars, one readily v e r i f i e s that v has the d i r e c t i o n of the e x t e r i o r normal. A vertex of the y i e l d locus r e p r e s e n t s a reduced load under which only yield.

one or both of the bars c o r r e s p o n d i n g to the a d j a c e n t sides may

Accordingly,

v may then be any n o n n e g a t i v e

linear c o m b i n a t i o n of the

unit vectors along the e x t e r i o r normals to these sides. For u'~ 0, the s u p p o r t i n g f u n c t i o n

(4) represents the p o l y g o n that is o b t a i n e d

from the virgin y i e l d locus by the t r a n s l a t i o n

cu , in a c c o r d a n c e with the

c o n c e p t of k i n e m a t i c hardening.

C o n d i t i o n for A d a p t a t i o n

The points

that have the r e d u c e d load vectors Q

, ( e = i, 2,

.., v ) as po-

sition vectors will be called load points,

and the convex hull of these points

and the origin will be called load domain.

It will now be shown that B a u s c h i n -

ger a d a p t a t i o n will take place under an a r b i t r a r y sequence of reduced loads r e p r e s e n t e d by points of the load domain if the virgin y i e l d locus can be given a t r a n s l a t i o n u such that the d i s p l a c e d locus will contain the entire load domain. If there exists a t r a n s l a t i o n u of this kind, we have

{ xil i l]

O v

<51

for any reduced v e l o c i t y ~ and any reduced load Q r e p r e s e n t e d by a p o i n t of

256

W. PRAGER

the load domain.

If, on the other hand,

Vol.l,

No.~,,,(,

a p a r t i c u l a r reduced load Q* of this

k i n d exceeds the l o a d - c a r r y i n g capacity of the truss when the reduced displacement of the joint A is u*, there exists a positive

factor ¢ < i such that

the reduced load @ Q* is at the l o a d - c a r r y i n g capacity of the truss.

If v* #

0 is a r e d u c e d v e l o c i t y of the joint A that is c o m p a t i b l e with this load,

D(u*,v*) : Zi {

Yi

*

S u b t r a c t i n g this e q u a t i o n

+ cu*'v*

=

~ Q*'v*

from the i n e q u a l i t y

(6)

(5) w r i t t e n for v = v* and Q =

Q*, we find

c ( ~-

because %

~* )' ~* ~ ( - i -i

_ i ) ~ ~*' ~* > 0 ,

(v)

- i as well as the external power of d i s s i p a t i o n ~ Q*' v* are po-

sitive for v* # 0. Now,

the vector v* indicates the d i r e c t i o n in w h i c h the

y i e l d p o l y g o n will move under the influence of the load $ Q*. The inequality (7) thus shows that this m o t i o n will reduce the distance d b e t w e e n the points with the p o s i t i o n vectors cu* and cu. Accordingly,

a state at which none of

the loads r e p r e s e n t e d by points of the load domain exceeds the l o a d - c a r r y i n g capacity of the truss will e v e n t u a l l y be reached, of d or for d = 0. In other words,

the truss will adapt itself to the loads.

This a d a p t a t i o n theorem is analoguous A l t h o u g h it has, kind of truss,

either for a finite value

to the shakedown theorem of Melan {3}.

for the sake of brevity,

only b e e n p r o v e d here for a special

it is g e n e r a l l y v a l i d for plane and space trusses.

Expressed

in m e c h a n i c a l

terms that are free from reference

to the specific example dis-

cussed above,

the a d a p t a t i o n theorem states that B a u s c h i n g e r a d a p t a t i o n will

take place if there exists a test state of h a r d e n i n g such that none of the given states of loading exceeds the l o a d c a r r y i n g capacity of the h a r d e n e d structure. Note that the actual state of h a r d e n i n g r e a c h e d in B a u s c h i n g e r adaptation to a given p r o g r a m of loading may differ from the test state.

References

i. M. GrOning, Die T r a g f ~ h i g k e i t statisch u n b e s t i m m t e r T r a g w e r k e aus Stahl bei b e l i e b i g h ~ u f i g w i e d e r h o l t e r Belastung, Springer, Berlin (1926) 2. H. Bleich, B a u i n g e n i e u r 19/20, 261 (1932) 3. E. Melan, I n g e n i e u r - A r c h i v 9, 116 (1938) 4. W. Prager, A r c h i v e s of M e c h a n i c s 24, 827 (1972)