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7/24/2019 Ultimate strength design of longitudinally stiffened plate panels.pdf
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Lehigh University
Lehigh Preserve
F$3 Lab*a*2 R+* C$0$' ad E0$*a' E$$
1967
Ulimae srengh design of longi!dinall# si$enedplae panels "ih large b/, A!g!s 1967
J. F. Voja
A. Osapenko
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Rc*dd C$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=PDFCoverPages7/24/2019 Ultimate strength design of longitudinally stiffened plate panels.pdf
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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
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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
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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
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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
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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.
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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.
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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 .
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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
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248.18
2. DEVELOPMENT OF THE
METHOD
FOR THE
DETERMINATION OF
ULTIMATE STRENGTH
he
ultimate s treng th o f l ongi tudina lly
s ti ff ened p la te
-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
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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
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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.
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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)
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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
s ti ff ened p la te
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
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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
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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
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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
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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.
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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
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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
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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
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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
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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.
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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)
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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
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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.
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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 ,
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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
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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
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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.
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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-
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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
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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
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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)
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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.
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>
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
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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,
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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,
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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
s ti ff ened p la te
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
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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 .
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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.
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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.
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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.
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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) .
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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.
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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.
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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.
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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
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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