Chapter 4 - Gross Plastic Deformation Design Check (GPD-DC)

7/18/2019 Chapter 4 - Gross Plastic Deformation Design Check (GPD-DC)

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Chapter

G r o s s P l a s t i c D e f o r m a t i o n D e s i g n C he c k

G P D - D C )

4 1 I n t r o d u c t i o n

T h e G r o s s P l a s t i c D e f o r m a t i o n D e s i g n C h e c k ( G P D - D C ) d e a l s w i t h

9 t h e m a i n f a il u r e m o d e g r o s s p la s t i c d e f o r m a t i o n a n d

9 t h e s e c o n d a r y f a i l u re m o d e e x c e s s i v e lo c a l s tr a in ,

w i t h r e g a r d t o

9 s h o r t - t i m e r e s p o n s e , i. e . ex c l u d i n g c r eep e f f ec ts , an d

9 s i n g le m o n o t o n i c a p p l i c a t i o n o f a c t io n s .

F i g s. 4 .1 a - d s h o w f o u r t y p e s o f s u c h r e s p o n s e s o f re a l s t r u c tu r e s m a d e o f re a l

ma t e r i a l s . F i g . 4 . 1 a s h o w s a l o ad v e r s u s d i s p l acemen t cu r v e t h a t i s t y p i ca l f o r ex -

p e r i men t a l r e s u l t s w i t h t en s i l e b a r s an d t r u s s e s s u b j ec t ed t o ax i a l f o r ce s a t amb i en t

t emp er a t u r e s , o r r e s u l t s f o r cy l i n d r i ca l o r s p h e r i ca l s h e l l s u n d e r i n t e r n a l p r e s s u r e ,

m a d e o f m i l d s te e l o r a n y o t h e r m a t e r ia l w i t h p r o n o u n c e d y i e ld p o i n t a t th e t e s t in g

t emp er a t u r e .

W h e n t h e m o n o t o n i c a l l y i n c r e a s in g l o a d r e a c h e s t h e u n r e s tr i c te d p l a s ti c f lo w l o a d

t h e r a t e o f ch an g e b eco mes ze r o , o r n ea r l y ze r o , b u t t h e s t r u c t u r e can s t i l l c a r r y

l a r g e r l o ad s , u p t o th e u l t i ma t e l o ad . Th e i n c r e a s e i n l o ad ca r r y i n g cap ac i t y i s m a i n l y

d u e t o t h e s t r a i n h a r d en i n g o f t h e ma t e r i a l , p o s s i b l y en h an ced b y g eo me t r i c e f f ec t s .

A l i m i t an a l y s i s mo d e l o f th e s t r u c t u re w i t h y i e l d l im i t eq u a l t o t h e y i e l d s t r en g t h

o f it s ma t e r ia l , w i ll s h o w a l o ad v e rs u s d i s p l ace m en t d i ag r am s i mi l a r t o th e o n e s h o w n

i n F ig . 4 . 2 - a l im i t an a l y s is m o d e l w i t h t h e ap p r o p r i a t e y i e l d l im i t w i ll , in t h is c a s e ,

b e a r ea s o n ab l y g o o d m o d e l f o r t h e d e t e r mi n a t i o n o f th e u n r e s t r ic t ed p l a s t ic f l o w l o ad .

F i g s . 4 . 1 b s h o w t w o l o a d v e r s u s d i s p l a c e m e n t c u r v e s t y p i c a l f o r e x p e r i m e n t a l

r e s u l t s

9 w i t h m a t e r i a l s w i t h o u t p r o n o u n c e d y i e l d p o i n t a t t h e t e s t t e m p e r a t u r e b u t w i t h

s t r o n g h a r d e n i n g ,

9 w i t h s t r u c t u r e s a n d l o a d s f o r w h i c h d e f o r m a t i o n h a s a s t r o n g p o s i t i v e e f f e c t ,

t hu s c o n t r ib u t i n g m a r k e d l y to t h e i m p r o v e m e n t o f t h e l o a d c a r r y in g c a p a c i t y o f

t h e s t ru c t u re , e .g . n o t to o t h i n - w a l l e d d i s h e d e n d s u n d e r i n t e r n a l p r e s s u r e ,

6 Pr e s s u r e Ve s se l De s ig n : Th e Dir e c t Ro u te

9 w i t h s t r u c t u r e s a n d l o a d s w h e r e g r o w t h o f p l a s t i fi e d z o n e s a n d / o r n e w l y p l a st i-

f le d z o n e s c o n t r ib u t e m a r k e d l y t o t h e in c r e a s e o f t h e l o a d c a r r y i n g c a p a c i t y o f

t h e s t r u c t u r e , e . g . f l a t c i r c u l a r e n d s w i t h p r o n o u n c e d d i a p h r a g m e f f e c t .

C o m p a r e d w i t h t h e r e s p o n s e s h o w n i n F i g . 4 .1 a , t h e r e i s, in t h is c a s e , n o p l a t e a u

( o f u n r e s t r i c t e d p l a s t i c f l o w ) a n d t h e s h a r p b e n d i n th e l o a d v e r s u s d i s p l a c e m e n t

a) Displacement

b) Displacement Displacement

c) Displacement

d) Displacement

Figure 4 .1 : (a) Load versus displacem ent curve, type 1: Three -bar t russ made o f mi ld

s teel [120] ; (b- l ) Load versus displacement curves , type 2: Uni formly loaded, s imply

suppor ted beam , I -shaped cross section, ma de of mi ld s tee l [ 120]; (b-2) Load versus dis-

placem ent curve, type 2: Dished end, internal pressure , mad e o f mi ld s teel [ 121 ] ; (c) Loa d

versus d isplaceme nt curve, type 3: Post buckl ing of a cylindr ical shel l under tors ion [53];

(d) Load v ersus displacem ent curve, type 4: Three-bar t russ mad e o f a lum inium [ 120].

Gross Plastic Deformation Design Check GPD-DC) 57

curve a t the unres t r i c t ed p las t i c f low load i s miss ing due to the absence o f a p ro -

n o u n ced y i e l d p o in t , a ma rk ed h a rd en i n g , o r a p o s it i v e e f f ect o f d e fo rma t i o n .

In t h is c a s e , a l i m i t an a l y s i s m o d e l o f th e s t r u c t u r e w i l l n o t b e a g o o d s i mu -

l a t i o n mo d e l , b u t s t i l l c an b e a r ea s o n ab l y g o o d d es i g n mo d e l , p o s s i b l y a h i g h l y

co n s e rv a t i v e o n e .

F i g . 4 .1 c s h o w s a l o ad v e r s u s d i s p l acemen t cu rv e t h a t i s t y p i ca l f o r ex p e r i -

m e n t a l r e s u l t s w i t h d e f o r m a t i o n - w e a k e n i n g , w h e r e d e f o r m a t i o n d u e t o t h e a p -

p l i ed ac t i o n h a s a d e t r i men t a l e f f ec t o n t h e u n re s t r i c t ed p l a s t i c f l o w l o ad .

E x a m p l e s a r e

~ ins tab i l i ty p rob lems , e .g . cy l indr i ca l , spher i ca l , and con ica l she l l s , o r d i shed

en d s , u n d e r ex t e rn a l p r e ss u re , o r b eam s an d cy l i n d r i ca l s h e ll s u n d e r ax i a l co m-

press ive load ing ,

9 ca s e s w h e re d e fo rm a t i o n h a s a n eg a t iv e , w eak en i n g e f f ec t o n t h e s t r u c tu r e ' s c a r-

ry ing capac i ty , e .g . nozz les in cy l indr i ca l she l l s under t ransverse moments , noz-

z les in cy l indr i ca l she l l s under ax ia l compress ive fo rces , bends under cu rva tu re

i n c r ea s i n g mo men t s , cy l i n d r i ca l s h e l l s w i t h o u t -o f - ro u n d n es s o r p eak i n g u n d e r

ex te rna l p ressu re .

The response i s sens i t ive to in i t i a l s t res ses , o f t en h igh ly sens i t ive to in i t i a l de-

fo rmat ions and to in i t i a l dev ia t ions f rom the idea l geomet ry , and o f t en s t rong ly

d ep en d en t o n k i n ema t i c b o u n d a ry co n d i t i o n s .

In th i s case , a l imi t ana lys i s model o f the s t ruc tu re i s ne i ther su i t ab le as a s im-

u la t ion model nor as a des ign model . A des ign model , t o be su i t ab le in cases o f

s u ch a b eh av i o u r , mu s t a t l e a s t i n c l u d e s eco n d -o rd e r t h eo ry , p r e f e r ab l y a l s o n o n -

l inear k inemat ic re l a t ions .

F i g . 4 .1 d s h o w s a l o ad v e r s u s d i s p l acemen t cu rv e w h i ch i s t y p i ca l f o r ex p e r i -

men t a l r e s u l t s w i t h s t r u c t u r e s mad e o f a ma t e r i a l w i t h o u t p ro n o u n ced y i e l d p o i n t

and wi th ac t ions resu l t ing in s t res s s t a t es wi thou t a marked re -d i s t r ibu t ion dur ing

load ing , e .g . s t a t i ca l ly de te rmina te t russes , o r cy l indr i ca l and spher i ca l she l l s

u n d e r i n t e rn a l p r e s s u re . Fo r a mo n o t o n i ca l l y i n c r ea s i n g ac t i o n t h e u l t i ma t e l o ad

wi l l be reached asympto t i ca l ly , o r , qu i t e o f t en , a f t e r a f in i t e d i sp lacement , and the

r a te o f ch an g e v a r ie s f r eq u e n t l y m o n o t o n i ca l ly .

Desp i t e the s imi la r i ty o f the load versus d i sp lacement cu rve o f th i s type and the

one of a l imi t analys is model , see Fig . 4 .2 , a l imi t analys is model wi l l , in th is case,

n o t b e a g o o d s i mu l a t io n m o d e l - f o r a r e a s o n ab l y g o o d s i mu l a ti o n , t h e y i e ld

s t reng th o f the l imi t ana lys is m odel has to be ad jus ted o r ca l ib ra ted , to a va lue som e-

whe re be twe en a p roo f s tres s and the u lt imate s t reng th o f the mater i a l, ob ta ined w i th

the usua l tens ion t es t spec imen , depend ing on the s t ruc tu re and the ac t ion .

N ev e r t h e l e s s, i n t h is c a s e a l i m i t an a ly s i s mo d e l m ay b e a r ea s o n ab l y g o o d d e -

s ign model , wi th a y ie ld s t reng th based on the 0 .2% (o r 1%) p roof s t res s spec i f i ed

8 Pr e s s u r e Ve ss e l De s ig n : Th e Dir e c t Ro u te

isplacement

Figure 4.2: Load ve rsus displacemen t curve, limit analysis model: Flat end to cylinder

connection with relief groove [3].

in the mater i a l s t andards . Very l ike ly , such a des ign model i s h igh ly conserva t ive ,

neg lec t ing to ta lly the sa fe ty ) marg in due to harden ing o f the rea l mater i a l .

F ig . 4 .2 shows a load versus d i sp lacement cu rve tha t i s typ ica l fo r the response

of l imi t ana lys i s model s to ac t ions wi thou t ins t ab i l i ty e f fec t s and wi thou t d i s -

p l acemen t w eak en i n g . Fo r a mo n o t o n i ca l l y i n c r ea s i n g ac t i o n t h e u l t i ma t e l o ad i s

reach ed asym pto t i ca l ly , o r a f te r a f in it e d i sp laceme nt , a nd s t ra ins a re usua l ly smal l

up to loads very close to the u l t imate load.

In th i s case , the response o f l imi t ana lys i s model s to monoton ica l ly increas ing

act ions i s insensi t ive to in i t ia l s t resses and fai r ly insensi t ive to in i t ia l deformat ions

and in i t ia l dev ia t ions f rom the per fec t geom et ry . The l imi t ana lys i s load i t se l f i s in -

dep ende n t o f in it ia l s t res ses , o f in i ti a l defo rm at ions , and o f the ac t ion h i sto ry , bu t

m ay d epen d on in it ia l dev ia t ions f rom the per fec t geom et ry , i f t hese dev ia t ions a re

non- loca l s t ruc tu ra l per tu rba t ion sources .

4 2 P r o c e d u r e

The GPD-DC can be seen as an inves t iga t ion o f the capac i ty o f the s t ruc tu re to

carry safely al l of the to-be considered s ta tes of act ions . As such, the des ign check

encompasses al l act ion s ta tes in the des ign domain and deals wi th al l load cases that

cor respond to the ver t i ces o f the des ign domain . In cases o f defo rmat ion-weaken-

ing , i. e. i n cases wh ere d efo rm at ion d ecreases the load car ry ing capac i ty o f the

s t ructure , addi t ional des ign checks , as descr ibed in the ins tabi l i ty des ign check

spec i f i c Chap te r 6 , a re requ i red . I f in such cases non-s ta t ionary thermal s t res ses

in f luence the resu l t , add i t iona l load cases wi th charac te r i s t i c t ime-dependen t t em-

pera tu re func t ions ) a re requ i red .

Gross Plast ic Deformation Design Check GPD -DC) 59

Fro m t h i s p o i n t o f v i ew , th e G PD -D C p ro ced u re can b e s u m m ar i zed a s f o l lo w s :

~ t h e g ro ss p l a st ic d e fo rma t i o n ch eck d ea l s w i t h a ll lo ad ca s e s t h a t co r r e s p o n d t o

the ver t i ces o f the des ign domain ;

9 o f these load cases som e can be e l imina ted , need no spec i f i c inves t iga t ion be-

cau s e t h e i n v es t i g a t i o n r e s u l t i s en co mp as s ed b y an y o f t h o s e fo r t h e n o n -

e l imina ted load cases ;

9 fo r al l o f the rem ain in g load cases the chec ks sha l l be per fo rm ed as spec i f i ed in

th i s chap te r ; and

9 t h e s e p ro p e r ch eck s a r e i n v es ti g a t io n s a s to w h e t h e r t he d e s i g n mo d e l s c an ca r ry

the des ign ac t ions o f the load cases , wi th spec i f i ca l ly l imi ted s t ruc tu ra l s t ra ins .

Fo r d e fo rma t i o n -w ea k en i n g l o ad ca s e s ad d i t io n a l i n v es t i g a ti o n s a r e r eq u i red . I f,

i n an y o f t h e s e d e fo rm a t i o n -w eak en i n g l o ad ca s e s , n o n - s t a t i o n a ry t h e rma l s t r e ss e s

poss ib i l i ty in f luence the resu l t , t hen a t l eas t one add i t iona l load case wi th charac-

t e r i s t i c t ime-dependen t t empera tu re i s requ i red . For these (add i t iona l ) load cases

o n l y t h e ad d i t i o n a l i n v es t i g a t i o n s , a s d e s c r i b ed fo r t h e s e d e fo rma t i o n -w eak en i n g

l o ad ca s e s , a r e req u i r ed i f t h e re s u l t s f o r a l l o w ab l e d e s i g n ac t i o n s o b v i o u s l y

e n c o m p a s s t h e c o r r e sp o n d i n g o n e s o f t h e u s u a l G P D - D C s . I n ca s e o f d o u b t b o t h

the checks a re requ i red - the model s d i f fe r no t on ly in the geom et r i c non- l inear i ty ,

b u t a l s o i n t h e i n i t i a l co n d i t i o n s an d t h e ad d i t i o n a l co n s i d e r a t i o n o f t h e rma l

effects , i f any.

Th e fo l l o w i n g s ec t io n s (o f t h is ch ap t e r ) d ea l s o l e l y w i t h t h e i n v es t ig a t i o n s w i t h

r e s p ec t t o th e f a i lu r e mo d es g ro s s p la s t i c d e fo rm a t i o n an d ex ces s i v e l o ca l s t r a in s ,

r eq u i r ed fo r a l l ( n o n -e l i mi n a t ed ) l o ad ca s e s , w i t h o u t an d w i t h d e fo rma t i o n -

w e a k e n i n g . T h e a d d i t i o n a l i n v e s t i g a t i o n s r e q u i r e d f o r d e f o r m a t i o n - w e a k e n i n g

l o ad ca s e s a r e d ea l t w i t h in Ch ap t e r 6 - t h e b eh av i o u r o f t h e r eq u i r ed g eo m e t r i -

ca l l y n o n - l i n ea r d e s i g n mo d e l s i s s i mi l a r t o t h a t o f t h e d e s i g n mo d e l s f o r b u ck -

l ing load cases , and o f t en inc ludes buck l ing .

4 3 D e s i g n M o d e l s

A l l d e s i g n mo d e l s u s ed i n t h e i n v es t i g a t i o n s d i s cu s s ed h e r e may b e s t r e s s -

co n cen t r a t i o n - f r ee mo d e l s .

I n t h e mo d e l l i n g o f th e s t r e s s - co n cen t r a t i o n - f ree g eo me t ry , i n p r in c i p l e, ma t e -

r ia l is to b e r e m o v e d - ad d i n g o f ma t e r i a l r eq u i re s j u s t if i c a ti o n .

Fo r c l ad co m p o n e n t s t h e fo l l o w i n g ap p li e s:

9 S t ruc tu ra l s t reng th ma y be a t t r ibu ted to c l add ing on ly in the case o f in teg ra l ly

b o n d e d t y p e , an d b y t h e ag reem en t o f t h e p a r ti e s co n ce rn ed .

9 The nom ina l face o f the c l add ing i s to be used as the su r face o f the p ressure

app l i ca t ion .

60 Pressure Vessel Design: The Direc t Route

9 A p r e s s u r e c o r r e c t i o n m a y b e r e q u i r e d in c a s e s o f f in i te e l e m e n t m o d e l s w i t h

s h e l l e l em en t s i f p r e s s u r e i s (i n t h e s o f t w a r e ) a p p l i ed a t t h e cen t r o i d a l s u r f aces

o f th e e l em en t s an d n o t o n t h e i r a c t u a l s u r f aces . A p r e s s u r e co r r ec t i o n is r e -

q u i r e d i f c l a d d i n g i s n o t i n c l u d e d i n th e m o d e l . I n b o t h c a s e s t h e c o r r e c t io n f a c -

t o r i s g i v en b y t h e r a t i o o f t h e ( i n f i n i t e s i ma l ) a r ea s o f t h e s u r f aces o n w h i ch

p r e s s u r e i s a p p l i e d - a c t u a l l y a n d i n t h e f i n i t e e l e m e n t m o d e l .

A l l o f t h e d e s i g n m o d e l s a r e li m i t a n a l y s is m o d e l s , u s i n g

9 f i r s t -o rd er t heory ,

l i n ea r - e l a s t i c i d ea l - p l a s t i c co n s t i t u t i v e l aw s ,

* T r e s c a ' s y i e l d c o n d i t i o n a n d a s s o c i a t e d f lo w r u l e,

9 s p e c i f ie d m a t e r i a l a n d t e m p e r a t u r e - d e p e n d e n t y i e l d s t re s s g iv e n b y th e s p e c if ie d

d e s i g n v a l u e o f t h e m a t e r i a l s t r e n g t h p a r a m e t e r .

T r e s c a ' s y i e l d c o n d i t i o n h a s b e e n c h o s e n f o r c a l i b r a t io n p u r p o s e s - t h e r e s u lt s

o f t h e m a x i m u m a d m i s s i b l e i n t e r n a l p r e s s u r e o f s u f f i c i e n t l y l o n g , c l o s e d c y l i n d r i -

c a l a n d o f s p h e r i c a l s h e ll s a c c o r d i n g t o D B F a n d t o t h e G P D - D C s h o u l d a g r e e . O r ,

i n o t h e r w o r d s : T h e f o r m u l a e g i v e n i n t h e D B F s e c t i o n o f t h e s t a n d a r d f o r c y l i n -

d r i ca l an d s p h e r i ca l s h e l l s a r e b a s ed o n t h e l i m i t an a l y s i s p r e s s u r e f o r T r e s ca ' s

y i e l d c o n d i t i o n . C o n s e q u e n t l y , t h e v e r y s a m e y i e l d c o n d i t i o n , i . e . T r e s c a ' s y i e l d

c o n d i t i o n , w a s s p e c i f i e d i n t h e G P D - D C , i n o r d e r n o t to f a v o u r D B A f o r t h e se s i m -

p l e g e o m e t r i e s , f o r w h i c h s u f f i c i e n t e x p e r i e n c e w i t h t h e r e s u l t s a c c o r d i n g t o t h e

D B F a p p r o a c h e x i s ts . U s a g e o f M i s e s ' y i e l d c o n d i t io n w o u l d r e s u l t i n t h i n n e r

c l o s e d c y l i n d r ic a l s h e l l s in c a s e s w h e r e g r o s s p l a s ti c d e f o r m a t i o n i s t h e d o m i n a n t ,

d e c i s i v e f a i l u r e m o d e , w i t h o u t a n y c o n v i n c i n g a r g u m e n t .

A n o t h e r r e a s o n f o r th e c h o i c e o f T r e s c a ' s y i e l d c o n d i t io n h a s b e e n t h e w i s h f o r

r e s u l t s t h a t a r e c o n s e r v a t i v e c o m p a r e d t o e x p e r i m e n t a l o n e s .

N e v e r t h e l e s s , M i s e s ' y i e ld c o n d i t i o n m a y b e u s e d i n s t e a d o f T r e s c a ' s , b u t t h e n

t h e s p ec i f ied y i e l d s t re s s , s p ec i f i ed f o r T r e s ca ' s y i e l d co n d i t i o n , h a s t o b e mu l t i p l i ed

b y V ~ / 2 i n o r d e r t o o b t a i n t h e y i e l d s t re s s t o b e u s ed w i t h M i s e s ' y i e l d co n d i t i o n .

U s a g e o f M i s e s ' y i e l d c o n d i t io n m a y b e n e c e s s a r y b e c a u s e o f s o ft w a r e o r n u -

m e r i c a l p r o b l e m s r e l a t e d t o T r e s c a ' s y i e l d c o n d i t i o n . R o u t i n e s w i t h M i s e s ' y i e l d

c o n d i t i o n a r e m u c h f a s te r th a n r o u t i n e s w i t h T r e s c a ' s - u s a g e o f M i s e s ' y i e l d c o n -

d i t io n i s r e c o m m e n d e d f o r fi rs t tr ia l s a n d f o r n o n - c r i ti c a l c h e c k s .

T h e y i e l d s tr e s s e s u s e d i n th e d e s i g n m o d e l s - t h e d e s i g n v a l u e s o f th e m a t e r i a l

s t r e n g t h p a r a m e t e r - a r e o b t a i n e d b y d i v i s io n o f t h e r e l e v a n t m a t e r i a l s tr e n g t h p a -

r a m e t e r

b y t h e r e l ev an t p a r t i a l s a f e t y f ac t o r

T h e v a l u e s to b e u s e d a r e g i v e n

i n th e f o l l o w i n g t w o t a b l e s , T a b l e 4 .1 f o r n o r m a l o p e r a t i n g l o a d c a s e s a n d T a b le

4 .2 f o r t e s t i n g l o ad ca s e s . F o r ex cep t i o n a l l o ad ca s e s , t h e v a l u es f o r t h e r e l ev an t

m a t e r i a l s t r e n g t h p a r a m e t e r s a r e g i v e n b y t h o s e f o r n o r m a l o p e r a t i n g l o a d c a s e s ,

G r o s s P l a s ti c D e f o r m a t i o n D e s i g n C h e c k G P D - D C ) 6 1

Table 4.1: R M and ~'R for no rm al op erating load cases

Material

Ferritic steel a

R p o . 2 / t

o r R P O 2 / t -< 0 . 8

R m / 2 O

1.5625 ( R p O 2 / t

R m / 2 O

otherwise

Austenitic steel e p l . O / t

w ith 30 >-- A s < 35

Au sten itic steel 2.5Rpl.0

with A 5 > 35 Rpl.~

m R m / t

for R p l ~ --<0.4

R m / t

for 0.4 R p l ~ ~ 0.5

R m / t

for R P l ~ > 0.5

Steel castings R p o . 2 / t

RP~ <--

R m / 2 O

2 R p o . 2 1 t

R m / 2 O

otherwise

a S t ee l o t h e r t h a n a u s t e n i t i c s t e e l w i t h A 5 > 3 0 a n d s t e e l c a s t i n g s .

i .e . b y t h o se i n T a b le 4 . 1 , a n d t h e p a r t ia l s a f e t y f a c t o r s a r e to b e a g r e e d b y t h e p a r -

t i e s c o n c e r n e d , b u t m u s t n o t b e sm a l l e r t h a n t h o se f o r t e s t i n g l o a d c a se s , g i v e n i n

Table 4 .2 .

F o r t h e v a l u e s o f t h e m a t e r i a l s t r e n g t h p a r a m e t e r s , s e e a lso t h e d e f i n i ti o n o f t h e

d e s i g n v a l u e o f t h e m a t e r i a l s t r e n g t h p a r a m e t e r , i t i s c o m m o n p r a c t i c e t o u se t h e

m i n i m u m v a l u e s s p e c i f ie d i n th e m a t e r i a l s t a n d a r d s .

F o r t h e r e f e r e n c e te m p e r a t u r e f o r t h e d e t e r m i n a t i o n o f t e m p e r a t u r e - d e p e n d e n t

m a t e r i a l s tr e n g t h p a r a m e t e r s , a t e m p e r a t u r e n o t le s s t h a n th e m a x i m u m m e t a l t e m -

p e r a t u r e o f th e l o a d c a s e is to b e u s e d . T h is r e f e r e n c e t e m p e r a t u r e m a y b e ( c h o -

se n ) sp a c e i n d e p e n d e n t , o r sp a c e d e p e n d e n t . I n t h e f i r s t c a se , t h e ( c h o se n ) v a l u e

m u s t n o t b e s m a l l er th a n t h e m a x i m u m c a l c u l a ti o n t e m p e r a t u r e i n a n y p o i n t o f t h e

c o n s i d e r e d m o d e l , i n t h e s e c o n d t h e r e f e r e n c e t e m p e r a t u r e i n e a c h p o i n t m u s t n o t

b e sm a l l e r t h a n t h e c a l c u l a t i o n t e m p e r a t u r e i n t h i s p o i n t . I t i s c o m m o n p r a c t i c e t o

6 2 P r e s s u r e V e ss e l D e s i g n : T h e D i r e c t R o u t e

Table 4.2:

and ?'R for tes ting load cases

Material

Ferritic steel b

ReH r Rpo.2

Austenitic steel

wi th 30% - A 5 < 35%

R P l~

Austenit ic steel Rp~.O

with A 5 _> 35% 1.05 for Re1~ < 0.525

2.0Rel.o

Steel castings

otherwise

aValues fo r

are for the t est temperature.

bSteel other than austenitic steel w ith A5 > 30% and steel castings.

u se , i n t h i s l a t t e r c a se , t h e s t a t i o n a r y r e su l t o f t h e ( n u m e r i c a l ) t e m p e r a t u r e c a l c u -

l a t ion fo r the load case d i rec t ly .

Th e re su l t s o f the inves t iga t ions a re insens i t ive to the used va lues fo r the ma te r i a l

p a r a m e t e r s f o r t h e li n e a r- e la s t ic r e g i m e - m o d u l u s o f e l a st ic i ty a n d P o i s so n ' s r a t io ;

in many cases these re su l t s a re even independent f rom these ma te r i a l pa ramete r s .

T a k i n g t h i s i n t o a c c o u n t , a n d a l so f o r s i m p l i c i t y , t h e m a t e r i a l p a r a m e t e r s t o b e

u se d f o r t h e l i n e a r - e l a s t i c r e g i m e a r e sp e c i f i e d su c h t h a t t h e y c a n b e u se d u n -

c h a n g e d i n t h e r m a l s t r e s s p r o b l e m s i n v o l v i n g t e m p e r a t u r e c h a n g e s f r o m a m b i e n t

t o t h e m a x i m u m c h a r a c t e r i s t i c t e m p e r a t u r e o f t h e lo a d c a se c o n s i d e r e d :

9 T h e r e f e r e n c e t e m p e r a t u r e t* E f o r t h e d e t e r m i n a t i o n o f th e t e m p e r a t u r e -

d e p e n d e n t m o d u l u s o f e l a s t i c it y s h a l l n o t b e l es s th a n

t * z - 0 . 7 5 t ' m 4 + 5 K ( 4 .1 )

w i t h t * E a n d t ' m 4 i n ~ a n d w h e r e t ' m 4 i s t h e r e f e r e n c e t e m p e r a t u r e f o r t h e d e-

t e r m i n a t i o n o f t h e m a t e r i a l s t re n g t h p a r a m e t e r d i s c u s se d i n t h e p r e c e d i n g p a r a -

g r a p h . T h e c o r r e s p o n d i n g m o d u l u s o f e l a s ti c it y i s to b e d e te r m i n e d u s in g A n n e x

O o f E N 1 3 4 4 5 -3 .

9 For Po i sso n ' s r a t io the va lue 0 .3 is spec i f i ed ( fo r the e l a s t i c r eg im e) .

T h e su m m a n d i n t h e e q u a t i o n f o r t h e r e f e r e n c e t e m p e r a t u r e , E q u a t i o n ( 4 .1 ) , i s

n o t a t e m p e r a t u r e m a r g i n , i n t r o d u c e d f o r s a f e t y r e a so n s , b u t i s t h e c o n t r i b u t i o n

f r o m a m b i e n t t e m p e r a t u r e , t a k e n t o b e 2 0 ~

Gross Plast ic Deforma t ion Design Check GP D-D C) 63

T h e d e s i g n m o d e l s e n s u r e u n i q u e r e s u l t s w i t h r e g a r d t o s a f e a c t i o n s , a n d t h e r e -

s p o n s e o f th e m o d e l s t o a g i v e n a c t i o n h i s t o r y i s u n i q u e i n t e r m s o f th e s t r es s h i s t o r y

( b u t n o t n e c e s s a r i l y i n t e r m s o f t h e s t r a in a n d d i s p l a c e m e n t h i s t o r i e s ) [ 4 0 ], p. 2 5 3 .

4 4 D e s i g n V a l u e s o f A c t i o n s

T h e d e s i g n v a l u e s o f t h e a c ti o n s , t o b e c o n s i d e r e d i n th i s ( p a rt o f th e ) d e s i g n c h e c k ,

a r e g iv e n b y t h e p r o d u c t o f t h e ir c h a r a c t e r is t i c v a l u e s a n d t h e r e l e v a n t p a r t i a l s a f e t y

f a c t o rs , g i v e n i n T a b l e 4 . 3 f o r n o r m a l o p e r a t i n g l o a d c a s e s a n d i n T a b l e 4 . 4 f o r t e s t i n g

l o a d c a s e s . T h e p a r t i a l s a f e t y f a c t o r s f o r e x c e p t i o n a l l o a d c a s e s a r e e q u a l t o u n i t y .

F o r m a l l y , t h e t e m p e r a t u r e d i s t r i b u t i o n i n a s t r u c t u r e i s t h e e f f e c t o f t h e t e m p e r -

a t u r e a c ti o n . T h i s i s o n e o f t h e r e a s o n s f o r t h e i n c l u s i o n o f t h e c o r r e s p o n d i n g

p a r t i a l s a fe t y f a c to r s f o r t e m p e r a t u r e i n t h e s e t a b l e s , f o r n o r m a l o p e r a t i n g a n d f o r

t e s t i n g l o a d c a s e s. A n o t h e r r e a s o n i s t h e f o l l o w i n g o n e :

T h e t e m p e r a t u r e ( d i s t r ib u t i o n ) i n th e s t r u c t u r e i s t a k e n i n t o a c c o u n t i n t h e s e i n -

v e s t i g a t io n s v i a th e t e m p e r a t u r e d e p e n d e n c e o f t h e m a t e r i a l s t r e n g th p a r a m e t e r s ,

T a b l e 4 . 3 : P a r t i a l s a f e ty f a c to r s f o r a c t i o n s a n d n o r m a l o p e r a t i n g l o a d c a s e s

Ac t ion Cond i t ion Pa r t i a l sa fe ty fac to r

P e r m a n e n t a F o r a c t i o n s w i t h a n u n f a v o u r a b l e e f f e c t ~ c = 1 .2

Pe rm an en t a For ac t ion s wi th a f avo urab le e f fec t ~/G = 0 .8

V a r i a b l e a F o r u n b o u n d e d v a r i a b l e a c t i o n s ? e = 1 .5

V a r i a b l e a F o r b o u n d e d v a r i a b l e a c t i o n s a n d l i mi t v a l u e s ? Q = 1 .0

Pres sure Fo r ac t ion s wi tho u t a na tu ra l l im i t ) 1~ = 1 .2

P r e s s u r e F o r a c t i o n s w i t h a n a t u r a l l i mi t , e . g . v a c u u m ? p = 1 .0

Te m pera tu re ? T = 1 .0

D i s p l a c e m e n t ? D = 1 .0

aOther than pressure, temperature and displacement.

T a b l e 4 .4 : P a r t i a l s a f e t y f a c t o r s f o r a c t i o n s a n d t e s t i n g l o a d c a s e s

Ac t ion Cond i t ion Pa r t ia l sa fe ty fac to r

P e r m a n e n t a F o r a c t i o n s w i t h a n u n f a v o u r a b l e e f f e c t ~/G = 1 .2

P e r m a n e n t a F o r a c t i o n s w i t h a f a v o u r a b l e e f f e c t ? c = 0 . 8

Pr es su re ~ 1~ = 1.0

Te m per a tu re ? T = 1 .0

D i s p l a c e m e n t ? b = 1 .0

a O t he r t h a n p r e s s u re , t e m p e r a t u r e a n d d i s p l a c e m e n t .

6 Pressure Vessel Design: The Direct Route

an d i n g en e r a l , f o r ma l l y , t h e r ma l s t r e s s e s n eed n o t b e t ak en i n t o co n s i d e r a t i o n : I n

a p u r e l i m i t a n a l y s i s a p p r o a c h t h e r m a l s t r e s s e s n e e d n o t b e t a k e n i n t o a c c o u n t -

t h e l im i t a n a l y s i s a c t i o n i s i n d e p e n d e n t o f t h e r m a l s t re s s e s , b e c a u s e t h e r m a l

s t r e s s e s a r e s e l f - s t r e s s e s . Th e ( s t r u c t u r a l ) s t r a i n - l i m i t i n g r eq u i r emen t i n t h e p r i n c i -

p l e c h a n g e s t h i s ( s i m p l e ) p i c t u r e : T h e r e a r e ( b a d d e s i g n ) c a s e s w h e r e s t r a i n s d u e

t o t h e r m a l s t r e s s e s m a y b e c o m e t o o l a r g e t o b e n e g l e c t e d i n t h i s d e s i g n c h e c k .

T y p i c a l e x a m p l e s a r e t e m p e r a t u r e - i n d u c e d f o r c e s a t n o z z l e s d u e t o r e s t r a i n e d

d i s p l a c e m e n t a t a n c h o r ( f ix ) p o i n t s o f p i p i n g a t t a c h e d t o v e s s e l s , f o rc e s w h i c h c a n

r e s u l t i n n o n - n eg l i g i b l e s t r a i n s i n t h e n o zz l e t o s h e l l r eg i o n . Q u i t e o f t en , t h e s e

p i p i n g r e a c t i o n s a r e d e t e r m i n e d i n p i p i n g a n a l y s e s a s s u m i n g t o t a l re s t ra i n t o f t h e

p i p i n g a t t h e n o zz l e s . Th i s ap p r o ach u s u a l l y l e ad s t o o v e r l y l a r g e r eac t i o n s , an d ,

t h u s , t o o v e r l y l a r g e s t r a i n s . A l t h o u g h n o t r eq u i r ed i n t h e s t an d a r d , i t i s , i n s u ch

c a s e s , r e c o m m e n d e d t o in c l u d e t e m p e r a t u r e i n th e l o a d c a s e s, a n d t o i n c lu d e , a t t h e

s ame t i me , r e l ev an t p a r t s o f t h e p i p i n g i n t h e mo d e l s . A l t e r n a t i v e l y , i n t h e p i p i n g

a n a l y s e s a p p r o p r i a t e l y f l e x i b l e s u p p o r t s a t t h e n o z z l e s c a n b e u s e d . A s a w a r n i n g

n o t e : O n l y t h e t h e r ma l s t r e s s - r e l a t ed s t r a i n s h av e t o b e co n s i d e r ed i n t h e s t r a i n -

l i m i t i n g r e q u i r e m e n t , a n d n o t t h e s t r a i n s d u e t o t h e r m a l e x p a n s i o n , f i a t , w i t h t h e

c o e f f ic i e n t o f l i n e a r th e r m a l e x p a n s i o n / 3 .

A n a l o g o u s l y , t h e p a r t ia l s a f e t y f a c to r s f o r im p o s e d d i s p l a c e m e n t s h a v e b e e n in -

c l u d ed h e r e i n t h e s e t ab l e s , w h e r eas n o n e a r e g i v en i n t h e s t an d a r d .

F o r w i n d , f o r s n o w , an d f o r ea r t h q u ak e ac t i o n s , co u n t r y - s p ec i f i c d a t a , i . e . v a l -

u e s s p ec i f i ed i n r e l ev an t r eg i o n a l co d es , may b e u s ed , b u t t h e i r u s ag e s h o u l d b e ac -

c o m p a n i e d b y a c h e c k o f t h e ( c h a r a c t e r i s t i c ) v a l u e s u s e d s u c h t h a t t h e o v e r a l l

s a f e t y m a r g i n s a r e m a i n t a i n e d .

T a k i n g i n to a c c o u n t t h e f a c t th a t th e c o i n c i d e n t o c c u r r e n c e o f v a lu e s o f n o n -

co r r e l a t ed v a r i ab l e ac t i o n s n e a r t h e ac t i o n s ' ch a r ac t e r i s t i c v a l u es i s s ma l l i f o n e o f

t h e s e ac t i o n s h a s s t o ch as t i c p r o p e r t i e s , t h e d e s i g n v a l u es o f s t o ch as t i c v a r i ab l e

a c t i o n s , l i k e w i n d , s n o w , e a r t h q u a k e , m a y b e m u l t i p l i e d b y a c o m b i n a t i o n f a c t o r ,

i f t h e s e s t o c h a s t ic a c t i o n s a r e c o m b i n e d , i n m u l t i p l e - a c t i o n l o a d c a s e s, w i t h

p r e s s u r e an d / o r a t l e a s t o n e o t h e r s t o ch as t i c ac t i o n .

I f o n l y a p a r t o f t h e p r e s s u r e i s s u b j ec t ed t o a n a t u r a l l i m i t , e . g . t h e p a r t o f t h e

p r e s s u r e t h a t co r r e s p o n d s t o a s ta t ic h ead , t h en , i n t h e d e t e r m i n a t i o n o f t h e d e s i g n

v a l u e o f p r e s s u r e , t h i s p a r t ma y b e m u l t i p l i ed b y t h e p a r ti a l s a f e t y f ac t o r 1 .0 an d

the o ther par t by 1 .2 .

F o r c h e c k i n g t h e c o m p l e t e n e s s o f th e l o a d c a s e s p e c i f ic a t i o n s w i th r e g a r d to th i s

G P D - D C , t h e f o l lo w i n g c o m b i n a t i o n r u le s m a y b e u s ed :

9 a ll p e r m a n e n t a c t i o n s m u s t o c c u r in e v e r y l o a d ca s e ,

9 each p r e s s u r e ac t i o n m u s t b e co m b i n ed w i th th e mo s t u n f av o u r ab l e v a r i ab le act io n ,

9 e a c h p r e s s u r e a c t i o n m u s t b e c o m b i n e d w i t h th e r e le v a n t s u m o f v a ri a b le a c ti o n s,

Gross Plastic Deformation Design Check GPD-DC) 65

9 s p e c i f ic a t i o n s f o r c o m b i n a t i o n o f a c t io n s i n re l e v a n t r e g i o n a l c o d e s ( o n e n v i-

r o n m e n t a l a c t io n s ) m a y b e u s e d, a n d

9 f a v o u r a b l e v a r i a b l e a c t i o n s, i n c l u d i n g p r e s s u r e , m u s t n o t b e c o n s i d e r e d .

I n c a s e s o f " n o r m a l " s t a n d a r d h y d r a u l i c te s ts , a s sp e c i f i e d in E N 1 3 4 4 5 - 5 , i. e.

h y d r a u l i c t e s t s w i t h o u t s t a b i l i t y p r o b l e m s a n d w i t h n e g l i g i b l e a c t i o n s o t h e r t h a n

p r e s s u r e , t h e G P D - D C f o r t e s t i n g l o a d c a s e s i s , i n g e n e r a l , n o t r e q u i r e d . O n l y i n

c a s e s w h e r e t h e i n f l u e n c e o f t h e s t a t i c h e a d i n t h e t e s t s i t u a t i o n c a n n o t b e n e g -

l e c t e d c o m p a r e d w i t h t h a t o f t h e t e s t p r e s s u r e i s a G P D - r e l a t e d t e s t i n g l o a d c a s e

r eq u i r ed .

4 . 5 . T h e

rinciple

U s i n g t h e p r e c e d i n g s p e c i f i c a t i o n s f o r d e s i g n m o d e l s , S e c t i o n 4 . 3 , a n d d e s i g n v a l -

u e s o f a c t io n s , S e c t i o n 4 . 4 , t h e G P D - D C ' s p r i n c i p l e c a n b e s t a te d q u i te s i m p l y :

T h e d e s i g n v a l u e s o f a c t i o n s , o f a l l r e l e v a n t l o a d c a s e s , s h a l l b e c a r r i e d b y t h e

r e le v a n t d e s i g n m o d e l s w i th m a x i m u m a b s o l u te v a l u e s o f p r in c i p a l s t r u ct u ra l

s t r a i n s n o t e x c e e d i n g 5 % i n n o r m a l o p e r a t i n g l o a d c a s e s , a n d 7 % i n t e s t i n g l o a d

cas e s , f o r a s t r e s s - f r ee an d w e i g h t l e s s i n i t i a l s t a t e , an d f o r p r o p o r t i o n a l i n c r ea s e o f

a ll a c t io n s e x c e p t t e m p e r a t u r e .

F o r e x c e p t i o n a l l o a d c a s e s t h e s t r a i n l i m i t a t i o n d o e s n o t a p p l y .

Temp er a t u r e h a s a s p ec i a l r o l e i n t h i s p r i n c i p l e " I n g en e r a l , t h e r ma l s t r e s s e s

n e e d n o t to b e t a k e n in t o c o n s i d e r a t i o n ( s e e S e c t i o n 4 . 4 ), a n d t h e n t e m p e r a t u r e e n -

t e r s t h e i n v e s t i g a t i o n s o n l y v i a t h e t e m p e r a t u r e - d e p e n d e n t m a t e r i a l s t r e n g t h p a -

r a m e t e r s ( s e e S e c t i o n 4 . 3 ) . T h e r e f o r e , t e m p e r a t u r e h a s b e e n e x c l u d e d f r o m t h e

p r o p o r t i o n a l i n c r e a s e o f a c t i o n s , a n d t h e m a t e r i a l s t r e n g t h p a r a m e t e r s d o n o t v a r y

d u r i n g t h e i n c r ea s e o f t h e ( o t h e r ) a c t i o n s .

W i t h o u t t h e s t r a i n - l i m i t i n g r e q u i r e m e n t t h e n e c e s s a r y i n v e s t i g a t i o n s w e r e l i m i t

an a l y s i s in v es t i g a t io n s , an d t h e t h eo r em s o f li m i t an a l y s i s w o u l d t h en ap p l y , e. g . t h a t

9 t h e p r o b l e m o f t h e d e t e r m i n a t i o n o f li m i t a n a l y s is a c t i o n s h a s a u n i q u e a n s w e r ,

9 t h e r e s u l t s , i. e . t h e v a l u es o f t h e l i m i t an a l y s i s a c t i o n s , a r e i n d e p e n d en t o f i n i t ia l

c o n d i t i o n s ,

9 t h e r e s u l ts a r e in d e p e n d e n t o f a c t i o n p a t h s ,

9 t h e r e s u l t s a r e i n d e p e n d e n t o f t h e m a t e r i a l p a r a m e t e r s i n t h e l i n e a r - e l a s t i c

r e g i m e ,

9 t h e s e t o f s a f e ac t i o n s i s co n v ex , g i v en t h a t t h e r e l ev an t ch a r ac t e r i s t i c v a l u e o f

t h e m a t e r i a l s t re n g t h p a r a m e t e r i s a c o n v e x f u n c t i o n o f t e m p e r a t u r e , a n d

9 i f , f o r a g i v en ac t i o n , an y s t a t i c a l l y ad mi s s i b l e s t r e s s f i e l d can b e f o u n d w h i ch

is c o m p a t i b l e w i t h t h e r e l e v a n t y i e l d c o n d i t i o n , t h e n t h e a c t i o n i s a s a f e o n e .

Pr e s s u r e Ve s se l De s ig n : Th e Dir e c t Ro u te

Al l o f these theorems a re impor tan t in the app l i ca t ions .

The f i r s t one g ives the des igner conf idence : For a su i t ab ly se t up model , wi th

appropr ia t e in i t i a l and boundary cond i t ions , there i s one , and on ly one , so lu t ion ,

desp i t e the non- l inear i ty o f the des ign model owing to the non- l inear cons t i tu t ive

The second one t e l l s the des igner tha t the unavo idab le in i t i a l res idua l s t res ses

o w i n g t o man u fac t u r i n g p ro ces s e s can b e i g n o red .

The th i rd one i s o f impor tance espec ia l ly in the des ign s t age : The , in genera l

unknown, ac tua l ac t ion h i s to r i es a re un impor tan t , and the load case spec i f i ca t ions

can be s impl i f i ed .

Th e fo u r t h o n e a l l o w s fo r s i mp l e ap p ro x i ma t i o n s fo r th e s e p a r ame t e r s .

The f i f th one jus t i f i es the inves t iga t ion o f on ly load cases co r respo nd ing to the

ver t i ces o f the des ign domain .

T h e s ix th o n e, t h e l o w e r b o u n d t h e o r e m o f l i m i t a n a l y s i s t h e o r y , is t he b a si s

fo r many approx imat ion p rocedures and rough checks , and i t i s the reason fo r ne-

g lec t ing thermal s t res ses in these inves t iga t ions .

For va lues o f ac t ions c lose to the l imi t ana lys i s ac t ion va lues p rob lem s w i th nu-

mer ica l s t ab i l i ty a re qu i t e common; these p rob lems a re a lmos t a lways a nu i sance ,

requ i r ing o f t en a comple te res t a r t .

Therefo re , the s t ra in - l imi t ing requ i rement had been in t roduced ,

9 to avo id numer ica l ins t ab i l i ty p rob lems fo r des ign ac t ion va lues c lose to the

l imi t ana lys i s ac t ion va lues ,

9 to c rea te a un ique b reak-up po in t fo r the ca lcu la t ion , such tha t the resu l t does

no t depen d on the pa t i ence o f the des igner , no r on the com put in g power , bu t a l so

9 to encompass the fa i lu re mode excess ive loca l s t ra in , which i s impor tan t in

cases o f s t ra in concen t ra t ions .

The va lues o f the st ra in l imi t s a re the resu l t o f a com prom ise - l a rge enough to

a l low fo r resu l t s c lose to the l imi t ana lys i s ac t ions , and smal l enough to avo id nu-

merical ins tabi l i t ies and to cover the fai lure mode excess ive local s t ra ins safely .

They a re seen to be by fa r smal l enough fo r the su f f i c i en t ly duc t i l e mater i a l s con-

s i d e r ed - w i t h an e l o n g a t i o n a f t e r r u p t u r e n o t l e s s t h an 1 4 % . D i s cu s s i o n s h av e

show n tha t i t i s necessary to rem ind de s igners tha t these s t ra in l imi t s a re l imi ts fo r

t h e e f f ec t s i n t h e d e s i g n mo d e l s - w i t h d e s i g n ma t e r i a l s t r en g t h p a r ame t e r s - an d

that they apply for des ign values of act ions , i .e . the actual s t ra ins for character is -

t i c va lues o f ac t ions in the ac tua l s t ruc tu re a re in genera l much smal l e r than these

speci f ied l imi ts .

The spec i f i ca t ion o f l imi t s on s t ruc tu ra l s t ra ins a l lows fo r s impler FE-model s :

For she l l and p la te - type s t ruc tu res she l l and p la te FE-e lements can be used .

Gross Plast ic Deform ation Design Check GP D-D C) 7

B e c a u s e o f t h e in t r o d u c t io n o f th e s t r a in - l im i t i n g r e q u i r e m e n t s o m e o f th e

a bove - l i s t e d a dva n t a ge s a r e l o s t . T he r e s u l t s a r e s t i l l un i que , bu t t he y de pe nd , i n

p r i nc i p l e , on t he ( i n ge ne r a l unknow n) i n i t i a l c ond i t i ons a nd t he ( i n ge ne r a l un -

k n o w n ) a c t io n p a t h s .

Re s u l t s w i t h t he s t r a i n l i m i t a t i on a r e , i n ge ne r a l , c l o s e t o t he c o r r e s pond i ng

l i m i t a na l y s i s r e s u l t s , a nd qu i t e o f t e n t he s t r a i n - l i m i t i ng r e qu i r e m e n t i s no t

gov e r n i ng , t he r e s u l t s be i ng t he n i de n t ic a l . T he r e f o r e , t he in i t ia l c ond i t i ons a nd t he

a c t i on pa t h s ha ve be e n s pe c i f i e d qu i t e p r a gm a t i c a l l y : a s s i m p l e a s pos s i b l e , t o

a l lo w f o r e a s y p r e - p r o c e s s i n g a p p r o a c h e s .

Re s u l t s w i t h s t r a i n l i m i t a t i on be i ng c l o s e t o l i m i t a na l y s i s r e s u l t s , o f t e n i de n t i -

c a l, c onve x i t y o f t he s e t o f the l i m i t e d s t r a i n l im i t a c t i ons m a y b e a s s um e d , i s p r ob -

a b l y p r ova b l e , a nd t he r e f o r e t he inve s t i ga t i on o f l oa d c a s e s c o r r e s po nd i n g t o t he

ve r t i c e s i n t he de s i gn dom a i n s u f f i c e s .

4 6 Ap pl i ca tio n Ru le

T he ( on l y ) a pp l i c a t i on r u l e r e a ds :

I f it c a n b e s h o w n t h a t a n y l o w e r b o u n d l im i t v a l u e o f th e a c ti o n, d e t e r m i n e d

w i th t h e d es ig n m o d e l s p ec i f i ed in th e p r in c ip l e , i s r ea ch ed w i th o u t v i o la t io n

o f t h e s t r a in l im i t , t h e p r in c ip l e i s f u l f i l l ed i f t h e d e s ig n va lu e o f th e a c t i o n

d o e s n o t e x c e e d th a t l o w e r b o u n d v a l u e E N 1 3 4 4 5 - 3 A n n e x B ).

T he v a l i d it y o f th i s a pp l i c a t i on r u l e is obv i ous , i t h a s ne ve r t he l e s s be e n i nc l u de d

i n the s t a nda r d t o d r a w t he u s e r s a t t e n t i on t o t h is .

T h i s a p p l i c a t io n r u l e m a y b e u s e d i n c a s e s f o r w h i c h u p p e r b o u n d s o n t h e s tr u c -

t u r a l s t r a i n i n t he l i m i t a na l y s i s r e s u l t c a n be g i ve n a nd s uc h a n uppe r bound i s no t

l a r ge r t ha n t he s t r a i n l i m i t o f t he p r i nc i p l e . I t i s u s ua l l y u s e d i n c on j unc t i on w i t h

t h e l o w e r b o u n d l im i t a n a l y s is t h e o r e m .

4 7 E x a m p l e s

E x a m p l e s c a n b e f o u n d i n A n n e x E . 4 .

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