Ultimate strength design of longitudinally stiffened plate panels.pdf

7/24/2019 Ultimate strength design of longitudinally stiffened plate panels.pdf

1/132

Lehigh University

Lehigh Preserve

F$3 Lab*a*2 R+* C$0$' ad E0$*a' E$$

1967

Ulimae srengh design of longi!dinall# si$enedplae panels "ih large b/, A!g!s 1967

J. F. Voja

A. Osapenko

F*''* #$ ad add$$*a' *& a: #+://+0.'#$#.d/-c$0$'-0$*a'-!$3-'ab-+*

5$ Tc#$ca' R+* $ b*# * 2* !* ! ad *+ acc b2 # C$0$' ad E0$*a' E$$ a L#$# P0. I #a b acc+d

!* $c'$* $ F$3 Lab*a*2 R+* b2 a a#*$3d ad$$a* *! L#$# P0. F* * $!*a$*, +'a c*ac

+0@'#$#.d.

Rc*dd C$a$*V*%a, J. F. ad Oa+&*, A., "U'$a # d$ *! '*$d$a''2 $4d +'a +a' $# 'a b/, A 1967" (1967).Fritz Laboratory Reports. Pa+ 1666.#+://+0.'#$#.d/-c$0$'-0$*a'-!$3-'ab-+*/1666
http://preserve.lehigh.edu/?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]:[email protected]://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1666?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1666?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F1666&utm_medium=PDF&utm_campaign=PDFCoverPages


2/132

A

r ? T Z

. A B C : ; ~ A T O ~ Y _ .

. ~ , . . q ~ . . L . : A - ~ ~ . . R

_ . . ; ; ? ~ L t : 9

C / L ? 7 ~ 4 r r . A ? , . 6 ' ; o G P ' ~ , . y .o..e:-s/6 N o .c- : : O / V G / r - ~ ~ ' - : / V ' 4 L ~ Y

y

.......

__

/ ; : ~ ; ; c ; ~ / V O

~ ~ A : ? ? ; . :

~ ~ , . y . Z ~ - ~ / ~ ~ . 4 R t : P C ~ . . f .

0 0 5 ~ ~ V e : ~

~

A k./s C 7 ~ ? ~ : 1 ~

; _ _ _ _

.---- -_-------

Cd;?

k

/rc S,;?7 bt?

-6

r.-? C

t:?Y e o/C

Io 7? ?

k

. r - ? r e : r f r - ? 7 i P : ; / ~ .

t ? Y t : - - ~

_, _ _ ~ _ _

__

. - -- -..._

__

-_..

__. ._

..

_._. _

~ .v

40.> -1;;

6 a v ~ V= ~ . 2 . .

~ . s

/f

= .3

s

6


3/132

. 5 : r ~ P s v ~ 4Qn

At . ffl._2 =-

.

( :ft:?, 6

; .

~ : .2:?t

S

.....

:,. :z

~ t diO

~

a 9 ' ~

( ~ ; ; J ' . r 2 /.; ~ = / . 1 . 2 / / ~

~

~

- = - . . . ~ ~

a

-3 sr;.9. ~ / /:d :y:J:

JI .

9/ ; . 2

1..

~

~

.

,/..:J,2/

/.7 .

2

-

~

6 .3

/.7 .

2....

0:..r4.;; .

e

/ ; ; ~ / / I ~

.4

~ #

/ .9

.2 / / ~

.2 ;-.2

~

5 .2

/ ~

1 'c:::::

7 . . ? ~ 12

--------_.-----_P._--_

.-

---

I

..#;r . . S )/ :

) .

~ O ~

I

r /tIJ.,

If: - ( ~ ~ 6 / 1 .

.. O . .;JI ... / ~

I

L .

; J ( ~

~

e:;J

. . ; ; ~ *

I ~ ~ Zl :

a ;?

==- 6:

. 4'

I ~ r . , .

J

i I

,..::;'AG.- -

__

. ~ ~ . = _ C - ? 7 c ; : v v

I te / I

rCev > ~ 1/-}

7-

oft /O

< : ~ t I ? ~ . s

A l & 4 r - . / ~ ~ e ~ t : : c r ~ o ; ; J ~ rl? ~ ~ c : . ~ ' Z P t J ' . s

~ d ~

r:::l?

~ . c;;- qt:1 G 2

; : , ~

If f

.1:'G8

a7t6/ ..

/4

aGS

9.9,,0

~ 7 . . s n


4/132

Built Up Members

in Plast ic Design

ULTIM TE STRENGTH

ESIGN

OF

LONGITU IN LLY

STIFFENE

PL TE P NELS

WITH

L RGE

t

by

Joseph F. Vojta

and

Alexis Ostapenko

This work

has been

carried

out as

part

of

an investigation sponsored by

the Department

of

the Navy with funds furnished by the Naval Ship

Engineering

Center Contract Nobs 94092.

Reproduction of this report in

whole or

in

part is permitted

for

any

purpose

of the United

States

Government

Qualified

requesters

may

obtain copies

of

this report from the Defense Documentation Center.

Fritz Engineering Laboratory

Department of

Civil Engineering

Lehigh University

Bethlehem Pennsylvania

August

967

Fritz Engineering Laboratory Report No 248.18


5/132

248. 18

T LE OF CONTENTS

STR CT

1 . INTRODUCTION

2 .

DEVELOPMENT OF

THE

M THO FOR THE DETERMIN TION

OF ULTIM TE

STRENGTH

2 1 Moment Curvature

Relationships

2 1 1 Cross Section

and Loads

2 1 2

Residual

Stresses

2 1 3

C r i t i c a l

Buckling

Str e ss

2 1 4 Pla te B uc kl in g A ct io n

2 1 5

S tres s S train

Diagrams

2 1 6 Equations

Governing

the Moment-

Curvature

Relationships

2 1 7

Computation

Procedure

i

5

5

7

9

11

12

26

2 .2 Numerical I nt eg ra ti on to Determine

Ultimate

Strength 27

2 2 1 Derivation o f Equilibrium

Equations

29

2 2 2

Discussion

of

th e

r o ~ d u r e

33

3 .

NUMERIC L

RESULTS OF THE N LYSIS 39

3 1

Moment Curvature Computations 4

3 1 1

Moment Capacity

of

th e Cross Section 40

3 1 2 Moment Curvature Curves 41

3 .2 Numerical Integl ation

3 2 1

Typical

Ultimate Strength Plots

3 2 2

E ffects o f

Geometric

Parameters

3 .3 Comparison With

Test Results

4 .

ULTIM TE

STRENGTH

DESIGN

CURVES

4 1

Development

o f the

Design

Curves

4 1 1

Simply Supported

Ends

4 1 2 .Fixed Ends

42

42

43

45

47

47

50

54


6/132

248 18

4 2

The Use

of the

Design Curves

4 2 1

Schematic Example

4 2 2

Outline

of

Steps to

Follow

4 2 3 Numerical Examples

4 2 4 Topics Concern ing the

Use

of

the

Design

Curves

4 2 5 Errors Involved in the Use of the

Design Curves

i i

56

57

59

6

64

66

5

CONCLUSIONS

N

RECOMMENDATIONS

FOR FUTURE RESEARCH

68

5 1

Conclusions 68

5 2

Recommendations

for Future Research 69

6 NOMENCLATURE 7

7 TABLES N FIGURES

76

8

APPENDIX

8 1

Numerical

Integration

for Ultimate

Strength

8 1 1

Subscripting

8 1 2

n i t i a l

Values

8 1 3

Firs t

Segment

8 1 4

Other Segments

8 1 5

End

Conditions

A

Pinned

Ends

B

Fixed Ends

8 1 6

Ultimate Condition

9

REFERENCES

111

112

2

4-

6

6

7

7

12


7/132

248 18

STR CT

i i i

This report

presents

the resul t s of an

ana lyt ica l

inves t i -

gation of the ult imate

str en gth o f

longi tudinal ly s tif fe ned p la te

panels

with plate

width

to

thickness

ra t ios such tha t

the

plate

buckles

before

the

ult imate

strength

of

the

panel

is

reached

The loading

conditions

considered

are axia l

loads a t the ends

and

a

simultaneous

uniformly distributed

load

applied

l a tera l ly .

The major e lement

of the panels

is a plate having a large width-

thickness ra t io bit such tha t the plate buckles before the

attainment

of the

ultimate

axia l load

The analysis is performed by so lv ing equi libr ium equations

numerically with

a d ig i t a l

computer Non linear

ef fects

non-

symmetrical cross section and ine las t ic behavior are

considered

in

the

analysis A

comparison between analy ti ca l r e sul ts and t e s t

resul ts

shows

tha t the analyt ical method can

accurately predict

the u ltima te lo ad

Information

from

the

numerical analysis

has been organized

in the

form

of

design

curves for s t ee l

with a yield

s tress

of

7 ksi and a t

greater

than

about

fo r

the main plate element

Optimum

cross

sections

can

be

readi ly obtained through the

use

of

these

d es ig n cur ve s


8/132

248.18 1

1. INTRODU TION

ship

hul l

is essential ly a hollow box girder composed

of

plates

stiffened

by a grid of longitudinal and transverse members

Fig.

1.1. The

deck

and bottom

plating

are

stiffened panels which

serve

as

flanges

of

this hollow girder. The principal function

of the

panels is

to res is t the

longitudinal forces

induced by

bending of the

ship

under

wave act ion Fig.

1.2.

The longitudinal

framing resis ts the bending action and also

in cr ea se s th e

buckling

strength

of

the

outer

plating

by subdividing

i t

into a

series

of

subpanels.

Because of these advantages many ships use this system

of longitudinal

framing.

This report

deals

with the analysis and design of the longi-

tudinally

stiffened plate panels which serve as

the

bottom

plating. t considers the panels under the severe loading

condition

of

axial compression due to

bending

of the hul l

combined with a

uniformly distributed l ter l

load.

The conventional method used for

design

of the longitudinally

s ti ff ened p la te

panels is

based on

el s t ic

considerations.

n

allowable

st ress

is

chosen

as

some

function

of

the

yield

st ress

of

the

material

and is kept below th e buck ling stress of the plate

elements.

major disadvantage

of

the

method

is

that the

f i r s t

yield is not con siste ntly re la te d to the

ultimate

failure load.


9/132

248.18

The

design curves

and the method

of

analysis presented in

this

report

are

based

on the

ultimate

fai lure load.

This

approach provides

a more

logical cr i ter ion as well as

a

simpler

analysis .

-2

In 965 Kondo presented a

theore t ic l el s t ic p l s t ic

analysis

for the

beam column

capacity o f l ong it ud inal ly

stiffened

plate

panels subjected to an a xia l th ru st and a

uniformly

distr ibuted

l ~

l te r l

loading.

His

analysis

was

for

sections

having

plate

elements

that did not buckle.

Unlike columns where fai lure is often

instantaneous with

b u c k l i n ~ plate panels can

sustain

axia l

loads

well in excess

of

the

buckling

load.

This

depends

largely

on the s t ruc tur l action of

i t s main plate element,

and

the correct p re dic tio n o f the fai lure

load

depends

on

the

knowledge

of the

e ~ v i o r

of

th i s

plate

component.

The his tory

of the

s t bi l i ty of plates under edge compression

dates

back

to

1891, when Bryan presented

the an aly sis fo r

a simply

supported rectangular plate

acted upon

by

a

uniformly distr ibuted

compressive

load on two opposite

edges of

the plate.** Davidson

studied plate action

in

the

P9stbuckling

range. 6

recommended

that Koiter1s

equation be used

to determine

the

postbuckling

The

numbers in parentheses refer to the

l i s t

o f re fe re nc es ,

Chapter 9.

**References

and

7

give br ief discussions of the history and

development of plate s t bi l i ty

theory.


10/132

248. 18

-3

behavior o f

long

re c t a n g u l a r p l a t e s

subjected

to

edge compression

as th ey a re used in s ti ff en ed p la te

p a n e l s .

In 1965

Tsui j i

a p p l i e d Davidson s f indings an d revised Kondo s method o f a na ly sis

fo r

use

on s e c t i o n s having p l a t e s

that

buckle

bef or e

at ta ining

th e u ltim ate l o a d ~ T h i s method o f

a n a l y s i s

w s

used

in this r e p o r t .

In th e

analysis

is

assumed

that th e

s t ress s t ra in

re la t ion-

ship in th e

p l a t e af te r

buckling can be pr esented by th e

average

stress

v s .

edge

s train

curve

fo r

f l a t

plate

he

ultimate

s t r e n g t h

o f th e p l a t e

p a n e l

is

th u s

obtained

with o u t

complex

a n a l y s i s

o f s t r e s s e s in th e plate he problem is

th u s

reduced

to

an ultimate

str e n g th a n a l y s i s o f

p l a t e panels

which c o n s i s t

o f

s t i f fener

an d p l a t e

having

di f fe rent s t ructura l

p r o p e r t i e s ;

th e

s t i f fener follows an

e las t i c p las t i c

s t ress s t ra in curve while th e

p l a t e

is governed by

K o i t e r s

equation

in

the post-buckling r ange.

he

a n a l y s i s becomes complex because i t ut i l izes

n o n -l i n e a r

moment-curvature

r e l a t i o n s h i p an d

deals

with

an

unsymmetric

cr oss

se c tio n . However th e required i te ra t ion procedure is r e a d ily

handled

through th e use o f d ig i t a l computer.

Since

design would re q u i re r epeated

use

o f th e method o f

a n a l y s i s becomes u s e fu l to

have

design curves

which

would

give

s u i t a b l e

cross

s e c t i o n s without d e ta ile d a n a l y s i s fo r each

s e c t i o n . For th is reason

design

curves a re p re se nte d

in

t h i s

r e p o r t . he

curves

also i l l u s t ra t e th e feas ib i l i ty o f

preparing

such curves fo r v a r i o u s combinations

o f

m a t e r i a l p r o p e r t i e s .


11/132

248 18

4

Chapter 2 presents a discussion of the method of analysis

with i t s development and use Chapter 3

discusses

resul ts of the

use of

the method

and compares this t o exper imen ta l t s t r sul ts

The development and use of the design charts i s presented in

Chapter

4

The

Appendix contains detai led information concerning

some

specif ic

topics in

the

analyt ical procedure


12/132

248.18

2. DEVELOPMENT OF THE

METHOD

FOR THE

DETERMINATION OF

ULTIMATE STRENGTH

he

ultimate s treng th o f l ongi tudina lly


-5

panels is determined by an

analysis

which considers ~ panel

as

a beam column

with

a cross sec tion conta in ing a plate element that

buckles. he

analysis

consists of two main parts; the calculation

of

a

moment-curvature

relationship

for

a given axial load,

m - ~ - P ,

which

evaluates

the

response

of

the

cross

section to

the

given loading conditions, and a numerical integration

of small

segments to determine the maximum lengths that the panel can

have under these loads.

he

development of

the moment curvature relationships is

presented in Sec.

2.1.

he numerical integration procedure which

util izes these

m - ~ - P

curves is presented in Sec. 2.2.

2.1 Moment-Curvature

Relationships

ondo

developed M - ~ relationships for constant

axial loads

for the

case where

the cross section

contained plate

elements

that have width-thickness

ratios, bi t ,

suff iciently low, such

that

the plate would not buckle

unti l af ter the

ultimate

load

is

reached. l) Davidson

investigated the

action

of the

plate when

buckling

did occur. 6) Tsu iji us ed th is

information

to

develop a

method to find the M relationships for cross sections where the

main

element

was a plate having a

large b i t

value and

which buckled

before

fai lure. 3)

This method,

with further

improvements

is


13/132

248.18

u t i l i z e d here

-6

Moment-curvature

rel at i o n s h i p s

fo r

co n s t an t

a x i a l

thrust

depend

on th e magnitude o f th e

a x i a l load,

the moment the

m a t er ia l p r op e rt ie s of th e member

namely

the yie ld s t r e s s e s o f

the p l at e and s t i f fener th e dimensions o f th e cross s ect i o n , an d

the

magnitude and

d i s t r i b u t i o n

o f

th e r esid u al stresses

2 . 1 . 1 Cross

Section

and

Loads

The

analyzed cross

se c tion is composed of a

p late

and a

s e r i e s

o f equally spaced

tee

st i ffeners

as

shown in F ig. 2 . 1 . The

l oa ds c on si de re d are a compressive a x i a l load,

p I ,

a ctin g in to the

plane o f the paper an d

a

uniformly d i s t r i b u t e d l a te ra l load, q, on

the p l a t e sid e of

the

cross se c tion causing compresssion

in th e

p late

at

the

c e nte r of

the span during bending, F ig. 2 . 2 ) .

The

ends

C and D

ca n

e i t h e r

be

both fixed

o r

b o th s im p ly -s u pp or te d

in

the a na lysis t h a t

follows.

An

ide a liz e d

cross se c tion is used fo r the

a na lysis

F ig. 2 . 3 ) .

The

a x i a l load act i n g

on

the

ide a liz e d

cross s ect i o n is

termed P

and is a

por tion of the to ta l a x i a l l o a d P l

S im ila r ly th e en d

moments m

C

and

m

D

a re

a portion o f the

to ta l panel

moment

mI .

The

en d

moments

are

assumed

equal an d

are c a lle d

m

In the

ide a liz e d

cross se c tion the

areas of the

p l a t e

and

s t i f fener flange

are

assumed to be

concentrated

about a

h o r i z o n t a l l in e through the i r


14/132

248.18

mid-thicknesses.* The s t i f fener

depth,

d, is assumed to be

approximately

equal

to

the

distance from

the

mid-thickness of

the plate to

the

mid-thickness of the s t i f fener flange.

2.1.2

Residual Stresses

n

idealized

residual

stress

distribution in

the

plate is

assumed

as

shown

in Fig.

2.4.** The

tensi le residual stress

is

assumed to

be equal

to the yield stress of the plate . The zone

containing

this

stress

is

assumed

to h v

a

width

c,

and

to

be

-7

centered about the

connection to

the stiffener.***

The compressive

residual stress

zone spans the remainder of

the plate width.

For

equilibrium,

where

cr

r

=

the

plate compressive residual

s t ress .

2.1

c

b

=

the width

of

the tensile

yield stress

zone.

=

the

plate width, center-to-center of s t i f feners .

= the yield stress

of

the plate .

For l t r convenience this is non-dimensionalized to

c

cr

c

b

=

cr

r

yp

cr

c

cr

c

2.2

The

stress

is assumed

constant

through

the thickness of the plate .

** References and show

that

this assumption closely approximates

the true

measured

distr ibut ion.

~ : P o i n t s

TT

denote

both edges of

the

tensi le residual stress

zone. These

points are

very important

in

th e stress-s train

diagram

discussed in

Sec. 2.1.5.


15/132

248.18

where

c

is

the cr i t i ca l

buckling stress

discussed

below

-8

No r es idua l s tr es se s are

assumed to act

in

the

s t i f fener .

ondo showed that these

stresses

have

had

a

negligible ef fec t . l

2.1.3 Cri t ica l Buckling Stress

y

assuming that the

plate

in one

subpanel between

two

s tiffe ne rs ) is not affected by the adjacentsubpanels ,

as

in the

case of antisymmetric

buckling, th e

plate

is essential ly

simply-

supported at the

st iffeners .

The cr i t i ca l

buckling

st ress

is

then

defined

as:

1

2.3)

where E

is the

modulus of

elast ic i ty , is

PoissonTs

ra t io ,

and k

is

the plate buckling coefficient which

depends on

the length-to-

width

rat io

of the

plate .

The

rat io

.vb for longitudinally

stiffened panels

is usually

larger than

3 and the

corresponding

k

value

is

approximately

equal

to 4.

The cr i t i ca l

buckling

st ress

then becomes:

TT

2

E

1

c

=

2

b/t)2

1 ~

which

is a

function

of b it

for a

given

material.

2.4)


16/132

248.18

-9

Equation

2.4

is

used

throughout the analysis when defining

the

cr i t ica l

buckling stress of the

plate.

This equation

is

appli

cable only to

plates

subjected

to

a uniform

compressive

s t ress

The

residual stress pattern is not uniform, but since

the tensi le

stress zone is narrow

in

comparison with the

total

width, only a

small

error is

involved in

assuming that the compressive residual

stress crris uniform

over

the

ful l width.

Because the narrow tensi le

zones

have only

a small influence on

the

buckling

of the

plate,

the cr i t ica l buckling stress

is

assumed

to

be reduced by an

amount

equal

to

r

2.1.4

Plate

Buckl ing Act ion

In

a typical longitudinally


panel, the plate

element in

Fig. 2.1 is

restrained

by

the

st i ffeners

in the x

direction. I t

is

otherwise conside red as simply supported at the

st iffeners . When loaded with

increasing compressive

stresses

with

no

residual

stresSes present, the

plate

will behave

in

the

manner shown

in Yig.

2.5.

1)

For low stresses the stress

dis tr ibution

is uniform

across the

width

of the subpanel, Fig.

2.5a).

This

is true unt i l the s t ress

cr

reaches the

cr i t ica l

buckling

stress

c

2) The plate .begins

to

buckle at

cR

When

the average

stress

is increased above

c

some

of

the

added stress

is

transferred

from the center of the plate

to

the


17/132

248.18

edges

a t

the junction with th e s t i f fener ,

F ig .2 .5b . The

s t r e s s e s

a t the

edges then

continue to

increase

above

the

st ress in th e

middle unt i l

th e

maximum pla te

stress is

reached.

3)

With

only

a

small

e r r o r th e maximum condition

occurs

when

th e edge

s t r e s s

reaches

the

value o f

the

pla te yield stress ,

cr

yp

F ig . 2 . 5 c). The

p l a t e

has

then

reached

i t s

maximum

st ress

and

w i l l

take

no

more load.

-10

4) A ft e r

reaching i ts

maximum

s t ress , the average p l a t e

s t ress ,

cr

av g

is assumed

to remain

constant

and

equal

to

cr

max

.

Davidson

s ta te s t h a t

K oite r s

equation

appears to a cc ur ate ly

describe

the

pla te

action in t he p os t- bu ck li ng range, s t ep s

2

an d 3 . 6)

K oite r s equation

0.45

1 .2

~ 0 . 6

c

0.2

0.65

EO

c

-0.2

::R

2.5)

defines the

a t

the

cr

average

pla te

s t ress , -2-, as a function o f

th e

s t ra in ,

ocR

connection of

the

s t i f fener .

hen

r e s i d u a l s t r e s s e s

a re in clu de d, th e

r e s i d u a l s t ress

p at t ern

in Fig. 2 .4

is used.

The

pla te

a c tio n fo r

th is

case

is

shown in

Fig.

2.6.

Here the important

str e sse S

are a t p o in ts A


18/132

248.18

a t the

edges o f the t e n s i l e

res idual

s t r e s s

zone._ The maximum

-11

compressive

membrane s tr e ss w ill occur here and Fig. 2.6c shows

the assumed

ultimate

condition when

th e s tres s es a t

points T AT

cr

e

, are equal to the yield

s t r e s s of the p l a t e .

The s t re s s d is tr ib u ti o n is unknown, but by assuming

t h a t

i t

is parabolic

th e ultimate

condition is found by solving Eq. 2.5

simultaneously

with Eq. 2.6.

0.6

crp

)

e )

= 2 -

cr

c

max

cR

0.2

-0.2

- 0.65

+ 0 . 4 5

~

2.5)

cR cR

ewhere

2 YP}

r

3

l E ~

-lJ 2- .

cR

t

cr

c

max

=

2

c

3 1 - 1

is

s t i l l

th e

stra in a t

th e

s t i f f e n e r .

2.6)

2.1.5 S tres s -S train

Diagrams

In

t he d et er mi na ti on

o f the

m-0

relations hips ,

s t r e s s ~ s t r a i n

diagrams for

the plate

and fo r the s t i f f e n e r are

very

important.

U t i l i z i n g

the pla te

buckling

a c tion,

th e

diagrams

chosen are

shown

in Fig.

2.7.

The s t r e s s - s t r a i n diagram

fo r

the pla te

has

3 steps

for

the

compression range, Fig.

2.

7a). The diagram is l i n e a r from th e

o rig in to th e cr i t ica l

buckling s t r e s s

which is reached

a t

point T.

K o i t e r s e qu at io n t he n d efin es the n o n -l i n ear

portion

u n t i l

yielding


19/132

248.18 -12

begins a t

points T T of

the cross section.

This

occurs for an

average plate stress that

never

exceeds

and is designated by

yp

point

Won the stress-s train curve.

For

higher values

of strain

the s tress remains

constant

and equal to crp)max

The tension range

of

the

stress-strain

diagram for the plate

is fully

elast ic plast ic as

shown

in Fig.

2.7a.

The stiffener follows an elast ic-plast ic

relationship

between

stress and strain

for

both

tension

and

compression,

Fig. 2.7b.

2.1.6 Equations Governing the Moment-Curvature Relationships

Governing

the moment-curvature relationships is a series of

equations

for various

conditions

of

plate

buckling action and

yielding

of th e various component

elements

of the cross section.

The equations will now be presented s epara te ly f or

positive

and

negative bending.

Assumptions

The following assumptions are made for

this

analysis:

1

The

st ress s t ra in

relationships for

the st i f fener

and

plate

are as described in Sec. 2.1.5 and as

shown in Fig. 2.7.

2 No local

instabi l i ty

takes place in

the

st iffeners

prior to the ultimate fai lure.

3

The structural action and

the

loading are

ident ical in

a ll

subpanels between adjacent

st iffeners of

the

section. The idealized


20/132

248.18

cross section is

thus

representat ive of i ts

portion of the

to t l cross section.

-13

4

5

S tresses in the plate

and

st iffener flange are

constant

through

thei r thickness.

The

l t r l

loading

is

applied

uniformly over

the

plate side of the panel and the moment and

axial

loading

are applied

at

the

centroid

of the

cross

section.

6 The analysis cons,iders

the

uniformly distributed

l t r l

loading

to act

as concentrated

line loads

t the st iffeners .

In other

words the effect of

plate bending between st i ffeners

is

neglected.

7

Shear

deformation is neglected.

8 The

dis tr ibution

of residual stresses does not

change

over the

length.

9 Sections that are plane before deformation remain

plane f t r deformation.

1

The plate

panels

are

in i t i l ly

f l t and

there are

no

in i t i l deformations.

11 The

axial

thrust

is constant along the st iffener ,

for

m relationships only

12

Plate bending

and buckling

causes

no change in the

geometry

of

the

cross section.


21/132

248.18

13

No strain reversal occurs befo re the ultimate

panel

load is reached.

14 Strain

hardening effects are not considered. When

the

moment

reaches

99

per cent of i ts

maximum

value

the moment-curvature curve is

assumed

to have

a

f la t

plateau

o f constan t

moment for

a l l

curvatures.

B Sign Convention

The

following

sign

convention

is used

for

the

moment-

curvature

relat ionships:

1

2

Compressive stresses

and

compressive axial loads are

considered

positive .

Bending

moments and

curvatures causing

compression

in the plate are posit ive.

3

Plate

and

st iffener stress

and

strain

values

must

have

correct

signs

when

substi tuted into the

equations.

However,

positive

values

are always to

be

used for ,c r ,c r

R and

ys

yp c p

max

C Positive Bending

A general

s tr es s d is tr ibu ti on in the idealized cross

section

for positive

bending

is

shown

in Fig.

2.8. The

axial force

obtained

from

t hi s d is tr ibu ti on i s :

-

f e

s

~ -

g e

E -

w w

ys

2 .7


22/132

248.18

where

P =

the

axial

load for the idealized cross

section.

A ,Af,A

=

are the areas from the s tandardized cross

w

p

section of

the web flange, plate , and total ,

respectively.

es,ee

= the strains

in

the s t i f fener

flange and

in the

plate

at the

connection

to the s t i f fener ,

respectively.

p

=

the average

stress

in the

plate

ys

=

the yield stress of

the

s t i f fener

= the

plate

compressive residual

stress

r

-15

.

f

f ,g

= the non-dimensional depths of yield

penetration

measured

from the flange and plate,

respectively

The moment about the

X-axis i s :

1 1

m

=

-2 .;d3-ad E e

-e A

- f d a

- -3

fd

cr

+Ee A

e s w

2

ys

s w

where

d

=

the

depth

of the s t i f fener .

=

the

non-dimensional

distance

from

the

plate to

the

centroid of the cross section. This can be written

1 A

W

A

f

as =

2

A

A

2.8

2 .9


23/132

248.18

From the geometry

i t is possible

to express relationships

between the plate and flange

strains,

th e curva tu re , and the

yield penetration distances.

-16

s

= - 0d

e

gd

=

d

e-ys

)

ee-e

s

fd

= d [ 1 -

~ s

e s

(2.10

(2.11)

(2.12)

Since the equations

are

more

conviently util ized

in non-

dimensional

form, the

equations will be

rewritten.

The

non-

dimensional forms of

Eqs.

2.7

to

2.12

are:

P

e

s

A

2 ~ -

f

A

w

~

cr

yS

A f c r y S + ~

=

A 2

+

PcR e

cR

ecR ecR A e

cR

cr

cR

A

crcR

e

cR

A

p

(cr

p

Jr

_

2

A

w

_ cr

yS

(2.13

~

+

- - -

2

cr

cR

cr

cR

e

cR

A . e

cR

cr

cR

e A

w

1 1 cr A

___ _ +

- f

l -a - - f YS s ~

cR

A 2 3

cr

cR

eCRrA

s

cr

yS

+ ( l-a) - - - +

e

cR

cr

cR

1 1

-

2 g

(0

-

39

(2.14)

(2 .15


24/132

248.18

g =

2 _ ~ 1

cR cR T

cR

-17

2.16

f

1 -

~ +)

cR cR

cR

where

2 .17

cR

=

the strain

O cR

cR

=

E

corresponding to the r i t i l buckling stress

2.18

~

the

curvature

corresponding

to the moment m R

. c

m

cR

the

ri t i l

moment

m

cR

O cRSpL

S

pL

the

section modulus with

respect to

the plate

SpL

I

2.19

2.20

CR

=

the axial

load

which

causes

buckling

in

the

plate.

P

CR

AO cR

S L

~ d

the

section modulus

in non-dimensional

form

2.21

2.22

From

Eqs.

2.13 to

2.17 fo r p os itiv e bending, the relationships

for various strain

states can be calculated and put

into

the forms

shown

in Table

lao

These

values

were l l computed

by

using the


25/132


26/132

248. 18

-19

Using steps

1 ), 2 ),

and

3 ), above, the a x i a l

load and moment

can

be written as

P _ s A

w

2_i

P

cR

- cR

cR

cR

2 . 2 3 )

m

[ 1

) (

s

m

cR

= SpL) [ 2 r R

Ad

y

s u b s t i t u t i n g

Eqs.

2 .9

and 2. 15

these ca n

be rewritten

as

P

=

P

cR

~

2 . 2 5 )

E

=

_ _

l l_

x 2

m

cR

SpL)L

2 3 cR

Ad .

which are in

th e

form

presented

in Table la o

2.26)

The numerical inte gr a tion p ro ce du re u se s the

equations o f

Table la

to

find th e m ~ relations hips .

The moment

equations

are

solved

d irectly while the a x i a l load equations

are

rearranged

and

used to solve fo r ~ / ~ c R

The value of

the ultimate

bending moment fo r the c ro ss s ec ti on

is required

by th e

numerical inte gr a tion

procedure.

This moment

determines

the

l i m i t

of

the load

c ar ry in g c ap ac it y

under

constant

a x i a l load.

Fo r

the

case

o f

positive bending

there

are

three

possible

cases

depending on

the lo ca tio n o f the neutral a xis.


27/132

248.18

1) When the ne ut ra l axis

is

located

in the s t i f f e n e r

web,

the

maximum positive bending moment

is

-20

... .....). =

~ 1 ( O ' _ ' )

m ( S T

2

eR

max

. Ad

~

O eR

2 . 2 7 )

where 1 is the

non-dimensional

distance from the p late to the

n eu tr al a xis ,

and

for

t h i s

case is

defined by

.1 =

2.28)

2) When the ne ut ra l axis

is in ,

o r above, the s t i f f e n e r

flange

m )

1

r:

O vs

O S)

. --- =

L l-O ) - - - -

meR max PL) O eR

0

eR

Ad

where

A

f

A

p

O ys _

0 pm

ax + O r

A - Q A O eR

O eR O eR

2.29)

2.30)

3 ) When

the

ne ut ra l axis is in , o r below, the p l a t e , the

maximum p o sitiv e bending moment is

m )

=

SlpL

[0

A

p

O ys O p _ O ru

m-- A

O eR

O eR O eR

eR max

Ad

where

2 . 3 1 )

=

GL

+ O yS) As

+

O r) Apl

P

R

O R A JR

A J A

e e e p

(2 .32)


28/132

248.18

D. Negative Bending

-21

A

general

stress

dis tr ibution

in the idealized cross section

for

negative bending s shown in Fig. 2.9. The

axia l

force and

moment

obtained

from

this

cross section are:

-

-2

f

E

-cr

A - 29 Ee+crys) A

,s

ys

w w

2.33

2.34

,

From

geometric

re

lationships

s

=

-

0d

e

gd

=

_ e ~ y ) d

see

e ]

fd

=

+ e _YS d

s e .

Non-dimensionalizing

Eqs.

2.33

to

2.37

P _

es

1 S _

ee

A

w

s

P

cR

- cR.- 2 e

cR

cR T - e

cR

2.35

2.36

2.37

2.38


29/132

248.18

-22

f

1 - 0 -

f

2 3

2 .39

2.40

1

2.41

f = -

2

-

e

cR

e

cR

1

s in the case fo r p os itiv e moments t hese equations are used

to

find

the

formulas

for various possible

strain

s ta tes Table

2b

con ta in s th e

equations

for

axial load

and

bending

moment

for

various

strain conditions

for

negative

bending.

These

values

were

a l l

computed using the

following

procedure:

1 ) When e

s

< e n o yielding

occurs

in the s t i f fener

yp

flange or in the web immediately adjacent to

i t

The

es O ys

term

-

does

not exist and is assumed to be

e

cR

O cR

equal to zero.


30/132

O r

T

ys no yielding occurs in

248.18

2

When e

ys

the web at the connection to the

pla te .

The term

e O ys )

does not

exis t and

is assumed to be

cR

O cR

equal

to zero.

-23

yielded

in

. .

0

.term --2.

O cR

3

When

- O r) cR the plate has nei ther

yp e

E

tension

nor buckled in compression. The

O r

)

is

se t

equal

to

R.

O yp

e c

4

)

When

cR

e

but the average

maximum

value.

Or

--E) , t h e

average

plate stress

has

pmax .

reached i t s maximum value and remains

constant .

Thus

p

=

0

pmax .

O r

When

e

T < - yp

the

plate has yielded in

tension.

Thus

= - .

v

yp

s an example consider Case 5 of Table lb For th is case the

s t i f fener yields a t the flange while the plate has

buckled but has

not

yet

reached

i t s

ultimate

average

s t ress .

This

s t ra in

state

is expressed

by

e+

O r

ys

ys

> R

-

< ys

c ,


31/132

248.18

-24

From steps

1

and

4 , above,

the axial load and moment

can be

written

as

A

_ e ~ _ _ s

cR

A

cR

2.43

)

O pKoiter

cR

A

A

W

l - ~ -

2.44

Substituting Eqs.

2.40

and

2.42 in

the

above, the

following

equations are obtai ned:

P

_

p

O pKoiter

:;)

A e

PeR

CR-

cR

cR

cR cR

A A;

2

:w

e

_

1

~

cR

cR

2 .45


32/132

248.18

-25

m

1

=

;

m

c

s ~ ~

[ ~

2.46

These are

presented in

Table lb .

s in the case for positive bending, the ultimate moments for

the cross section

are needed for the

computation of

the

m-0curves.

For negative bending only two cases are feasible, and they depend

on the location of the neu tr al a xis .

When the

neutral

axis is located in the s t i f fener

web

the

maximum

bending

moment

2 .47

1

~ ~

~

a]

.l .- =

m

c

max

where:T

the

non-dimensional

distance

from

the

plate to

the

neutral

axis, and for this case

defined

by

T =

[

s ~ _

A

C

pmax

- 2 J

ocR

1

2.48


33/132

24 8 8

2

When

the neutral axis is in , or

below, the

pl te

the maximum

bending

moment is

where

-26

2.49

2.50

2.1.7

Computation Procedure

The procedure for computing

the

~ curves for

constant x i l

th rus t

P, wil l be

described

here.

The procedure was programmed

in

FORTRAN language. A

br ief schematic flow diagram

is shown

in

Fig. 2.10.

The

steps

are

as

follows:

Compute

the cross-sect ional

parameters.

2 Compute the maximum average plate s t ress corresponding

to

the yield s tress t

points

All by simultaneously

solving Eqs.

2.5 and 2.6 for

cont inual ly inc reas ing

e

plate s t r ins

cR

3 Compute

the lo catio n o f the

neutral

axis

and

the

maximum

bending

moment for

posit ive

and

negative bending using

Eqs. 2.27 to

2.32

or Eq 2.47 to 2.50, respect ively;

A more

detai led

description

of this procedure as used by the

dig i t l computer

is contained

in Ref. 4.


34/132

248.18

4 . Assume a plate

s t ra in

5

Increment

the

plate

s t ra in

6 Determine the

configuration

of the

stresses

for the

strain s ta te

thus described. This entai ls

finding

what

portions have yielded

and whether

the plate has

buckled or reached i ts maximum

s t ress

-27

7

Calculate

the

curvature

and

the

c

corresponding

to

th is s tr ain

s ta te

m

moment

c

8

hen

the change in curva tur e exceeds some specified

value

go to step 9 . I f the value is not

exceeded

go

back

and re pe at step s 5 to 7 .

9

This is done for 100 sets of moment and corresponding

curvature for

both

posit ive

and

negative bending.

Approximately ~ minutes of

computation

time on a G 225

dig i ta l

computer

is

required to

compute

the

200 points

of the

m ~ curve.

The m ~ curve thus computed is ready for use in the numerical

integration procedure which is discussed

in

the following section.

2.2

Numerical Integration

to

Determine Ultimate Strength

The ultimate strength of longitudinally s ti ff ened p la te panels

under

combined axia l and

la teral

loading is determined by a

s tepwise numerica l integration procedure. This procedure uti l izes

the moment-curvature

curves for

constant axia l load the

deter-


35/132


36/132

248.18

2.2.1

Derivation

of

Equilibrium

Equations

-29

For

the

derivation

of equilibrium

equations

for small segments

of

the panel an assumption made in a dd itio n

to

the

assumptions

made

in computing the moment-curvature

relationships see

Sec.

2.1.6 ; the curvature change along each small segment

assumed

to

be l inear.

A small segment is shown in Fig. 2.12. The directions shown

are

considered

positive. Positive

moment

is

a moment

that

causes

compression

in the plate .

Curvature can be written as

the

rate of change of

slope w ith

respect to the length along

the

member ..

de

=

ds

where

e

=

the

slope.

s

=

the distance along the

centroid

of

the

panel.

=

the

curvature of the panel.

Summing forces and moments

about point

G gives:

2.51

tF

z

=

0

h +

qbds

p

sin

e h+dh

=

0

2.52

tF

=

0

v +

Qbds

p

cos e

v+dv

=

0

2.53

G

0

vdscos e hds

sin

e

qbds

ds

m+dm

= 0

m

-

p

2.54


37/132


38/132

248.18

-31

Integrat ing

from point i

to

point

i+l ,

and uti l iz ing Eqs.

2.55 to 2.57 , the slope .between those points is

ds

2 .62

For

_point

i+l this becomes

8.

1+

8

.

8. 1

=

8.

2

s.

1

1+ 1 1+

and the

horizontal

and

vert ica l forces and the moment are

2 .63

.

h. 1

1+

S

i+l

= h i + .

s . q b l - ~ a d sin

8ds

1

=

h.

+ bql1y.+

qbad cos

8. 1

-

cos

8.

1

J

1+

1

li l

v.

1 =

v.

+

s. q b l - ~ a d

cos 8

ds

1+

1

1

2 .64

l1z.

J

=

v.

+

qbl1z. - qbad

sin

8

i

+

l

-

sin

8.

2.65

1

J

1

f i + l v

fi l

. m

i

+

l

=

m.

cos 8

ds

s. h

sin 8

ds

1

S.

1 1

2 2

[

6z.

6y.

=

m

h.

l y

.

-v

.

l z

.

-

qb

_J

_

+

_ J__

ad6z.

1

1

J

1

J

2

2

J

s in8.

l - s in8

. + adl1y

cos81-coS8 .

]

2 .66

1+ 1

J 1+ 1

where

the z

and

y

components

l1z.,6y.

of 6s . ;

are

found

by

J J J

considering a l inear

var ia t ion in

curvature

J i+ l

_

~ i

+

~ i + l

2

=

cos8ds

l s .cos8. 3 sin

8. 6s. 2.67

s .

J

1 1

J

1


39/132

248.18

b.y.

J

-32

The a x i a l t h r u s t , n, i s

r e la te d to

h and

v

as follows:

n

=

h co s 8 -

v sin

8

2.69)

The

shortening

o f the member i s caused not

only

by

the

a x i a l

t h r u s t ,

but a lso by the s t r a i n

a t

the cen t ro i d .

2b.s.

t

=

t

J 7 - : : : r _ - - - : : ~ - . . . . ,

i+ l i ~ ~ e i + l - e i - 0 i + 1 - 0 i

}

where

e i

ei+l are

p l a t e

s t r a i n s

2.70)

Eqs. 2.62 t o

2.68 c o n s t i t u t e

the equilibrium equations

an d

form

the

b asis fo r th e n um eric al i n t e g r a t i o n procedure.

Eqs.

2.69

and 2.70 are also needed. For

convenience

a l l o f th ese equations

are non-dimensionalized.

r

8 =

8 .

+ iP. S - d -2 iP. l-iP.) S)

1 1

a

yp 1 1

2 . 7 1 )

8 .

1

1+

1 r

= 8. +

-2

iP. l-iP.) b --d

~

1 1+ 1 J

a yp

2.72)

b.Z.

J

b

y

J

i

i

+

l

) 2

= b Sj

cos8

i

- T +

- a S

j

)

y ~ s in

8

i

=

b.S.

s in 8

i

+

:i

+

i P ~ l

b.S

J

.)2

cos

8

i

v

gy p

Cf U

V.

1

= V.

+ QIR

[b Z

J

- ad

~ s i n 8

l - s i n 8 . ]

1+ 1 r yp 1+ 1

2 . 7 3 )

2.74)

2 . 7 5 )

2.76)


40/132

248.18

-33

M. 1=M.-V.6Z.-H.6Y.

ad

-QIR [ (6y.)2

+

2(6Z.)2_

6Z

. a

d

(sin8. l -s in8.

1+ 1 1 J 1 J r 2 J yp J

r

1+ 1

+ 6Y. ad

(cos8. l-COS9.)J

J r

yp

1+ 1

(2.77)

N . 1 = H lCes

8.

1 -

V.

1 r

d

Ie sin 8.

1

1+ 1+ 1+ 1+

a yp

1+

(2.78)

where

268.

J

L =

1

2 -

If:: .

+G.

1

. 1

. yp

1+

1 .1+ j

(2.79)

.

6 .

6Y =

Y =

r

mad

2

G

Ar

yp

QIR

=

qb ad H - h V

=

adv L - t

2

-

c rA

Ar 3 -

r

yp

Gyp

A yp

Gyp

e

i+l)

, r 1

A

f

A

w

2

G. 1 = - - ~

(1-2a)

A + 3 -a)--A

a

1+

y p ad

-

n

N =

GypA

2

.2.2

Discussion

of the

Procedure

This

section wil l contain

a

brief

discussion

of the n u m ~ r i l

integration procedure

to

determine the

ultimate

strength

of the

plate panels.


41/132

>

248.18

Since th e conditions

at both

ends are

ident ica l

and the

-34

la te ra l

load is

applied

uniformly, symmetry requires tha t

the

shear

force and

slope at the mid-span of the panel

are

zero.

Using th i s

as

a

s tar t ing

point the

integrat ion

need

only

be applied

to

one

hal f of

the panel

length. Thus,

by

assuming

a

mid-span

curvature

the

integrat ion

s ta r ts here

and

progresses segment by segment along

the member Fig.

2.13 unt i l i t

reaches the

pinned-end

condition

as

defined by

m =

0

and

then

the

f i x e d ~ e n d condit ion,

defined

by

2.80

8 = 0 2.81

Values

of deflect ions, length,

moment

e tc . are computed for each

case.

Throughout

the

procedure

Eqs.

2.71

to

2.77

are

used

to

main-

tain

equilibrium

for each

segment. Knowing the values

and

forces

a t

one end

i n i t i a l end and

assuming a

curvature,

~ i + l a

a t

the

other

end terminal end these equations are

solved

to find the

moment

by Eq. 2.77..

Knowing the moment, the corresponding curvature

can be obtained from the moment-curvature

curve

which is w non-

dimensionalized

to the

f o r m M ~

compared with

the

assumed one.

The

computed

curvature ~ i + l C

is

I f

the

difference

between

the

curvatures is large, ~ i + l c becomes

the new

assumed curvature and

the procedure is repeated. Final ly,

when the

difference converges

to

a

small value,

equilibrium

is

considered

to

be

fu l f i l led for the


42/132


43/132

248.18

(4) Compute the Z and Y

components

of the segment from

Eqs. 2.73 and 2.74.

(5) Knowing

the

ver t ica l and horizontal segments a t the

i n i t i a l

point and

the

values from step

(4) ,

compute

the moment a t th e term in al

end

of the new segment

by using Eq.

2.77.

(6)

From the

~ curves

of

Sec.

2.1 find the curvature

~ i + l C

corresponding

to

th is

moment.

-36

(7)

Check the difference between the assumed curvature

from

step

(3) and

the

computed

curvature

from step

(6) .

f the

absolute

value

of

th is difference

is

larger

than a certain specified amount

use

~ i + l c

as

a new

estimate of the term inal curvature and return to

step

(3).

f

the

difference

is

less

than the

specified

amount compute

9.

l H l and V 1 from

1 1 1

Eqs.

2.72, 2.76,

and

2.75,

sum L

Z,

and Y by

Eqs.

2.82, 2.83,

and

2.84. Then

le t

the terminal values

of

th is segment become i n i t i a l

values

for

the

next

segment,

and

go to step

(2)

and

begin

computations for

the next

point.

This pat tern

is

followed unt i l

the

terminal moment changes sign. When

th is

happens

proceed

to

step

(8)

in stead o f step

(2) .

(8)

y Newton s Method compute the

increment of

segment

length

corresponding to zero moment.

Compute

L, Z,


44/132

248.18

Y fo r

the panel to

the point

of

zero moment, and

by Eq.

2.71

find

the slope

8 corresponding

to t h i s

p o in t.

These are the pinned-end values for the

assumed

mid-span curvature.

9)

Continue

the computation of steps 2 ) to 7

unt i l

the

s lope computed

in step 7)

-changes sig n . hen

-37

10)

t h i s

occurs us e Newto n s

Method to find

the

increment

of

segment

length to the zero slope. Calculate

L,

Z, Y

fo r the panel to the point of zero

slope

and

by using a parabolic in ter p o latio n find the moment

at the

point of

zero slope.

These

are the fixed-end

values

fo r the assumed mid-span curvature.

Increment the

mid-span

curvature by and repeat

o

steps 2)

to

9 unt i l one o f the to ta l

panel length

values

from

steps 8) o r 9)

is

lower than the

previous value.

Then go

to step 11).

11) For the

end

condition,simply-supported o r f i x e ~ t h t

corresponds to the decrease in length,

compute

the

length

value

corresponding to zero

slope

on

the

vs

p l o t .

This

is

done

by

a

parabolic

in ter p o latio n

using the

las t three computed values. The r e s u lt is

the

m ximum length

t h a t the

panel can have fo r

the

given

s e t

of

loads.

Using a

l i n e a r

in ter p o latio n

compute the mid-span curvature

corresponding

to

the

m ximum length. Using

a

parabolic i nt e rpol a t i on,


45/132

248.18

compute

the corresponding values of

chord

length,

mid-span curvature,

end moment and end slope.

Also find the

corresponding

axial force.

-38

12 Repeat

steps 2

to

10 unt i l

the u ltimate cond itio n

is

reached for the

end

condition

that remains. When

this occurs, compute the

values

described in step 11 .

This

numerical integration method for fin ding the ultimate

strength

of

longitudinally


panels has

been

programmed

in

FORTRAN

language

for the digi ta l computer.

Given

the

~

relat ionships,

and

using

a GE 225 computer, approximately

seconds of computer time

is

required to simultaneously find the

two

maximum

lengths that correspond to the pinned and

fixed

end

conditions.

This computer program

was used extensively for

computing the

design curves described in

Chapter

4


46/132

248.18

3. NUMERICAL

RESULTS

OF

THE

ANALYSIS

Numerous

computer

runs were

made

and the

m 0.

curves and

-39

maximum

lengths

computed and printed

out

for each

case.

The

major emphasis in these runs

was for

steels having

yield

stresses

of 34, 47, and 80

ksi

which correspond r espect ively to

the

MS-34

HTS-47 and BY-80

steels used

by

the

Navy. The runs were a ll made

for the

following

ranges of

parameters:

As

_

0.20 to

0.48

A

p

= 35 to

6

b = 60.to 110

t

Q

=

q

d / t

=

40

to

480

psi

3.1)

to 15

However extrapolations outs id e thes e ranges

can easily be

made

in

most

cases.

This

chapter

presents

a

brief

explanation

of

the

resul ts

for

typical cases.

The moment

curvature

curves

are

shown in Sec. 3.1,

and

the resul ts of

the

numerical

integration

are

discu ss ed in

Sec. 3.2. Sec.

3.3

compares the developed t heoret ica l r esu l ts

A l i s t ing of the input

and

results for these cases is

available

to

interested persons upon

request .


47/132

9

/

248.18

with

the results

of tests

performed

at Lehigh University and

t

the Naval Construction Research Establishment.

3.1 Moment-Curvature Computations

The computation

of moment-curvature

r el at ionsh ips cons is t

-40

mainly of

two

parts : the determination of the

moment

capacity of

the cross section,

and the computation of

the

moment and

corres-

ponding

curvature

for

the va rious strain

states .

3.1.1

oment Capacity

of the Cross

Section

Figure 3.1 shows a

typical

curve of

the

maximum externally

applied

moments that a

cross section can

withstand in positive and

in negative bending. The solid lin e represents the

case

with no

r

residual

stresses

while

th e do tted l ine is

for

= 0.15.

yp

This

figure

shows that the maximum positive bending moment

causing compress ion in the plate , occurs t an axial

load

other

than zerO

Here

i t occurs at

about

P/Pcr =

1.15.

Since

the cross

section is not symmetric

about i t s

bending axis the positive

bending moment increases unt i l the maximum plate stress is reached

and

thus

the moment capacity will

increase

u ntil th is point; f te r

wards

i t

decreases.

In addit ion, the maximum

axial

load corresponds to a moment

tha t is not

zero,

but rather

some

negative value. This is again

due

to the

non-symmetry

of the

cross

section.


48/132

248.18

-41

Figure 3.1 also shows the effect of r esidual s t resses .* The

m x mum positive moment is decreased while the m x mum negative

moment is increased in magnitude. The corresponding axial loads

dif fer from the case

with

no

residual stresses.

3.1.2

Moment-Curvature

Curves

Figure 3.2 shows

the

moment-curvature curves for the same

cross s ectio n fo r which the moment capacity was discussed.

1

The

curves

differ

for

positive

and

negative

bending

ranges. This is due to the.non-symmetry of the

cross section .

2 . For high axial loads the. moment is not zero for zero

curvature. This is exemplified by the axial load

=

1.80 for no

residual stresses. For axial

cr

loads

of

even

higher

magnitude

the

moment

tends

to

remain

negative

for

a l l values of curvature.

3

The m x mum

positive

moment

occurs

at a

non-zero

value

of axial

load.

This was

previously

discussed

but is bette r

seen

physically on this moment

curvature plot .

The reducing effect of the residual

stresses

is again pointed

out by t he not iceable differences in

the

m x mum

moments when

high

axial loads are reached; for example,

PIPer

= 1.60.

See Ref. 3 for

the effect

of

post buckling

on

the

moment

capacity

of

the

section, as

well

as

i ts effect

on

the

moment

curvature curves.


49/132

248.18

-42

i

An important assumption has been made in obta in ing

these

moment curvature plots, and

i t

is very important for the shape of

the curves presented here. When the

moment

reaches of i ts

maximum value, the moment is assumed to be essentially constant

for

a l l

curvatures.

The

actual structural

action and hence the

true shape of the

curves

is

yet unknown

in

this

range.

3.2

Numerical Integration

3.2.1

Typical

Ultimate

Strength

Plots

The

relat ion between the loading

and the maximum

length that

a

cross section can

sustain

can

be

displayed

by

an

ultimate

strength

plot of the axial load P P

R

vs.

the

maximum

slenderness

r a t i o t r

This

is shown in Fig. 3.3a for pinned ends and Fig. 3.3b

for

fixed ends.

The

cross section

used is

the

same as

in

Figs.

3.1

and 3.2. The solid l ines

shoW

the curves

for

no residual

stresses

while the

dotted. l ines are

for a

residual stress = O.lS.

a

yp

Separate curves are drawn for various

la teral

loads.

In the case of

high la teral

load,

Q

= q

d/t

= 18

or 320),

and low axial load, the curves are relat ively insensit ive to

changes

in axial

load. In

fact, an

increase

in

l a t e ra l

load

may sl ight ly

i n ~ r s

the

maximum

length

that

the cross

section

can

sustain

by

virtue of the changes in moment

capacity

for these axial loads.

In the

fixed

end case

the

negative moment at

the

ends

will

counter-

act this

to

some extent, and the maximum

t

continually

decreases

for increasing P P

values.


50/132


51/132

248.18

-44

Figures 3.3a

and 3.3b show plots fo r v ar io us v al ue s ofq d/ t

f q i s

held c on sta nt th e

e f f e c t of

th e

s t i f f e n e r

depth factor

d/t

can be seen. Fo r a given res idual

s t r e s s

th e

curves

a l l tend to

converge

to the same

maximum

a x i a l

load as

t /r

approaches

zero.

Figures 3.4a

and 3.4b show ultimate str e ngth plots

fo r various

b it r a t i o s and for

fixed

and

simply-supported

end conditions.

Curves fo r high b i t

r a t i o s

are cons is tently above those

with lower

r a t i o s .

The

spread

in

th e

curves

is

quite

la rg e,b ut fo r

a

given

t / r they vary

approximately

with the s t r e s s

r a t i o r R ra ise d to

yp c

some

power. Fo r instance, i f Fig.

3.4a

is r e plotte d when PIP R

1 c

is divid ed b ye

: :i2- 2,

and i f Fig.

3.4b

is replotted when PIPeR is

0.675

divided by .2.

Figs.

3.5a and

3.5b

respectively

a re

obtained. The curves fo r various b i t r a t i o s have

thus been

brought

much clos er toge the r .

Ultimate

str e ngth curves are

also s u b s t a n t i a l l y

affected

by

the r a t i o o f s t i f f e n e r

area

to

plate area, th a t is A

I This is

s p

displayed by Figs.

3.6a

where

three

values

of As/Ap are

p lo tte d for

both end conditions Fo r low

values

ofP P

c r

h ig h v al ue s o f As/Ap

give

g r e a t e r

t / r

values

for both end conditions. The reverse is

true fo r

high

PIPeR

values

when

th e

ends

are

simply-supported

o r

when

b it

is

les s than

for th e

fixed end conditions. However

fo r fixed ends cases w ith b it g r e a t e r than

the curves

fo r

various

As/Ap tend toward co n v erg en ceat

t / r

=

O, Fig.

3.6b) .


52/132


53/132

248.18

-46

see Chapter 1). residual stress

measurements

were taken and

thus an assumed value of

or

3.0 psi . was used in the theoret ical

analysis.

The

so lid lin es

are for no residual stresses; the

-dotted l ines

are for

cases with

assumed

plate compressive

residual

stresses.

The

gri l lage

~ s t at N.C.R.E.

had

a l t r l

loading

of

psi .

The longitudinal panel s con ta ined plates

with

t

values

of

76.25

which was

considerably

higher

than

the

values

used

in the

t sts

at Lehigh

University. The panel ends had

very

small rotations

t

failure

signifying

that the

end conditions were probably

close

to ful l f ixi ty.

Figure

3.10 shows

that the panel

failed at a

load

that. was between the

fixed

and simply-supported

values

predicted

by

the

numerical analysis.

The r sult

is clo ser to

the fixed end value

and

hence

appears to correspond well with

the

theoret ical analysis.

The t s t specimens at Lehigh University were composed of

four

st i ffeners and a plate spanning

between them,

thus c reat ing three

subpanels. Boundary conditions imposed by

th is

section as compared

to the inf ini te length plate assumed

in

the numerical analysis could

conceivably account for the sl ight discrepancies that

exis t

between

the

predictions

and

the

actual

t st

resul ts .

Nevertheless,

i t

appears that the method of numerical analysis can accurately predict

the u ltimate

strength

of longitudinally s ti ff ened p la te

panels.


54/132

248.18

4. ULTIMATE STRENGTH DESIGN

CURVES

The method of

analysis p re sented in

Chapter 2

is

much

too

-47

cumbersome

to use for the design

of longitudinally

s tif fened p la te

panels. The ultimate strength design curves

presented in t is

chapter

provide a shorter method of design.

By using these

curves

the

designer can

rapidly determine the dimensions of a panel that

wi l l

just sustain

t he p rescribed loading conditions.

Given a se t

of loads,

material

parameters,

and

overal l

dimensions, and assuming some relative proportions of the cross

section,

the

design

curves

can

be

used

to

find

the

dimensions

of

the

required

cross section.

series

of different sets of relat ive

proportions

can be t r ied and the most advantageous

section

selected

from these.

Kondo presented a similar se t

of

ultimate

strength

design

curves for

plate panels

having no

pla tebuckl ing . l

This

chapter

describes

ultimate

strength

design

curves

for cases where the main

plate

element

of

the

panel

is in.the postbuckling range bi t 45

4.1

Development of

the Design

Curves

S te el p la te panels

having

s t i f fener

and

plate

yield

stress

values of 47 ksi were chosen as a basis

for

the design

curves.**

The

ranges

of the parameters used

are

given by Eq.

3.1.

These

Reference 5

contains

a resume

of

the

use of these design curves.

The yield

stress

value

of

47

ksi

corresponds to HTS 47 s t l

used

by the U

S.

Navy.


55/132

248.18

-48

design curves

presented for

one yield stress v lue d isp la y the

feasibil i ty of obtaining design

curves for

plate panels

with large

b/ t

Corresponding

design curves

for other yield

stress

v lues

can be

patterned

from these

curves

because of similar parameter

interact ions.

The problem of

organizing

the data from the

computer

runs for-

47 ksi was

complicated

by

i rregular

interaction of the parameters

as caused

by unsymmetry

of the

cross

section non-linear

diagrams

of stress-strain and moment-curvature and

buckling

of a component

member. These complications rendered simple formulations impossible.

The method

finally resorted to

was

virtually

one

of t r ia l-and-error .

Various

l ikely modification factors were t r ied unt i l one was found

that

null i f ied

the effect of

one or more

other

parameters.

This

was continually done unt i l

the

influence of the parameters could

be separated out. Then by

proper

arrangement of

the

null ifying

factors a logical

pattern

was

devised

to organ iz e th e in forma tion

into

a form that

is relatively easy to

use.

ow from loading geometry and material parameters

are

the

parameters

q

r E pI

t

and B where pI and B denote

. r yp ys

the

axial

load

and

width

respectively for

the

to ta l

p n l ~ h t

is

the

ful l

cross section . The unknown v lues

are the

relat ive

cross-

sect ional parameters As/A

p

Af/A

s

b/ t and d / t and some

dimensional v lue that w ill

fully

dimensionalize

the

cross section.


56/132


57/132

248.18

-50

The

design

curves are presented s ep ar ate ly fo r

fixed

and

s im p ly -su pp o rt e d e n ds ,

Figs.

4.9

and 4 .8 ,resp ectiv ely . Their

development

is quite s i m i l a r .

4.1.1

Simply-Supported Ends

The

e f f e c t

of

w

s found

to

be

give

a way

0

re late d to in such

O cR

d

Q = q F )

---p--- w i l l , fo r

each Q

Pctx

p

O cR

t h a t

a

plot of t

vs.

a

series

of curves t h a t

are

q uite clo sely banded.

together. By

representing

these curves as

one single curve we

have

a

graph t h a t

w i l l account fo r the

e f f e c t w. This

holds true fo r

p and

8 held

constant.

However since

and

P

P P

cR O cR

P w

2

B b l + p)

4.3)

t

can be seen t h a t

an

ultimate str e ngth p lo t

of t r

V 5.

P

would contain

the

unknown

d im e ns iq na l v pl ue b. Dividing th e

two

equations,

howe ver ha s

the de sire d

e f f e c t o f

cancelling the

b

terms.

Noting t h i s and

using

constant terms of

p = Po =

0.34 and

a = 8

= 0.60

t

was found

to

be

convenient to p l o t

S vs. N

o


58/132

248. 18

where

. t lr)

pt;

=

P

1+P7

=

8 l . 8

yP W

P

1

O cR

and

P

1+p/2

p w

2

.

N

= =

0.0003895

b

/yp

PCRO CR

- 5 1

4 . 5 )

4 . 6 )

in which O yp = 47. 0 k s i , E = 2 9 , 6 0 0 ,

J L

= 0.30 and P is in kips,

Band t are in inches.

This

p l o t

is

sl1own in Fig.

4 . 1 .

Here th e dots genote computer

r e s u l t s u si n g n um e ri ca l analysis

for

various

b I t

r a t i o s ,

while th e

l ines

p l o t

th e

approximate average.

value. Deviation

o f the

dots

from th e l i n e s shows th e e r r o r

involved

in the approximation.

This p l o t provides a

place

to e n t e r th e design curves.

By

s u b s t i t u t i n g

the known

values o f

P ,

t , an d

B and the

chosen

g eo me tr ic v a lu es o f y and w, a value o f S can be computed Using

Fig. 4 .1 the value o f N ca n be found which corresponds to a cross

section

having

y and

w as chosen

and

p = 0 . 3 4

and

e = 0 . 6 0 . *

The

othe r

portions o f th e

design curves

w i l l

then

modify

the

value

o f

N for various

values

o f p and e.

r th e d esig ne r w

Ultimate strength design of longitudinally stiffened plate panels.pdf

Documents