combined load analysis of unbraced frames 1967
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Lehigh University
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1967
Combined load analysis of unbraced frames, 1967 J. H. Daniels
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Plast ic
Design of Mul ti -S to ry Frames
COM INED LO D
N LYS S
OF
UN R CED FR M S
by
J
Hartley
Daniels
This
work has been
carried
out as a part of an
investigation
ent i t led
Plast ic
Design
of
Multi-Story Frames
tT
with funds
furnished by an
American
Iron
and
Steel ns t i tu te Doctoral
Fellowship.
Fritz
Engineering Laboratory
Department of
Civil Engineering
Lehigh University
Bethlehem Pennsylvania
July 1967
Fritz
Engineering
Laboratory Report
No
338.2
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KNOWLEDGMENTS
The
author wishes
to
acknowledge
th e
s p eci al
advice
an d a ssi st a n c e
t h a t
he
received from Profes s or
John Wo Fisher
who
supervised
this
d i s s e r t a t i o n . Sincere thanks
are
also ex
tended
t o
each member of the s p eci al committee which
directed
the a u t h o r s
d o c t o r a l
work. The committee comprises: Profes
sors Lynn
So Beedle
Chairman John
W
Fisher Supervisor
Le Wu
Lu
Vo Vo Latshaw
an d
D
A
VanHorn.
The author is
p art i cu l arl y indebted
to h is wife and
family
fo r
the i r
patience
kindness an d
encouragement
during
the l as t
three
years an d to Profes s or Le-Wu Lu
fo r
th e
s p r ~ -
t i o n and advice received
from
hi m during
th e
e a r l y
development
o f
t h i s work.
The work
des cribed i n this d i s s e r t a t i o n
was conducted
as p a r t
of
a
g e n e r a l i nv e st ig a ti on i nt o th e
p l a s t i c design of
m u l t i - st o r y frames a t F r i t z Engineering
Laboratory
Department
of
C i v i l Engineering
Lehigh University.
Dr.
Lynn S. Beedle
is
Acting Head of
the
C i v i l Engineering
Department
and
Director
of
th e
Laboratory.
This
i n v es t i g at i o n was sponsored
j o i n t l y
by the
Welding R e se ar c h C o un ci l
an d
th e
Department
of the
vy
with
funds
furnished
by
the American nst i tute of
S t e e l Construction
American
Iron
an d S t e e l
ns t i tu te
Naval
Ships Systems Command
an d
Naval F a c i l i t i e s
Engineering
Command
Technical
guidance
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iv
was
provided
by the
Lehigh Project
Subcommittee of the
Struc
tu ra l
Steel
Committee of the
Welding
Research Council, under
the chairmanship of
Dr . T. R
Higgins.
Their
support is
ac
knowledged.
The
author grateful ly
acknowledges.
the assistance of
the
American
Iron and Steel ns t i tu te who provided a doctoral
fellowship so
that
a year of fu l l time
study
could be devoted
to th is work.
Sincere appreciation is
also extended
to the author s
many colleagues for the i r many crit icisms
and
suggestions.
Special thanks are due
to
Professor G Co Driscoll , J r who en
couraged th e early development of th is work and permitted
i t
to be introduced
a t the
Summer Conference
on
P last ic Design
of
MUlti-Story Frames
held in September 1965 at Lehigh Uni
vers i ty .
Appreciation
i s
also
extended
to
Professors
A
Ostapenko
and
T Galambos,
and to
Drs.
M
McNamee
Eo
Yarimci,
P.
Fo Adams
W
C Hansell and B
P.
Parikh.
The manuscript was typed by Mrs.
D
Eversley and
Mrs.
D
Kroohs and the i l lustrat ions
were
prepared by Miss. S.
D
Gubich
and Mr J
M
Gera,
Jr
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1
T BLE OF
CONTENTS
BSTR CT
INTRODUCTION
1 1 The Basic
Design Process
1 2 f f ~ t s
and
Instabi l i ty
1 3 Preliminary Design
Methods
1 4 Sway Analysis Methods
1 5
Purpose
and
Scope
of the
Dissertat ion
1 6 Definitions
and Assumptions
v
12
14
16
2
22
1 6 1
6 2
1 6 3
6 4
Frame
Layout
Loading Conditions
Secondary
Failures
Materials
24
6
6
2
3
1 6 5
Application
of the Sway Sub-
assemblage Method
1 7
Summary of the Dissertat ion
THE
SW Y SUB SSEMBL GES IN N
UNBR CED
MULTI STORY FR ME
2 1 The
Role of Subassemblages
2 2
Possible Plast ic Hinge
Locations
and
Failure Mechanisms
2 3 Sign
Convention
2 4 The One Story Assemblage at
Level
n
2 5 The Half Story Assemblage a t
Level
n
2 6
The Sway
Subassemblages
at
Level
n
THE
RESTR INED OLUMN IN A
SW Y
SUB SSEMBL GE
3 1 Nature of the Restraint
27
8
3
3
31
35
35
41
44
7
47
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4
5
3 2
Equations
of Equilibrium and
Compatibility
3 2 Moment Rotation Relationship
3 4 Load Deflection Equation for
Constant
Restraint Sti ffness
3 5 Load Deflection Behavior for Constant
Restraint St i f fness ·
3 6 Load Deflection
Behavior
for
Variable
Restraint
Stiffness
3 6 1 Constant Zero Restraint
Sti ffness
3 6 2 Constant
Constant Restraint
Stiffness
3 7
Design
Charts
RESTR INING COEFFICIENTS
OF STEEL BE MS
ND
COLUMNS
4 1
In i t i a l
Restra in t
4 2
In i t i a l
Restra in t Coeff icients
4 3
Reduced Restraint Coeff icients
4 3 1 Plast ic Hinges Outside the Sway
Subassemblage
4 3
Plast ic
Hinges
Within
tlle
Sway
Subassemblage
RESTR INING
CH R CTERISTICS
OF COMPOSITE BE MS
ND
STEEL
COLUMNS
5 1 Flexural Behavior
Under Combined
Loads
5 2
In i t i a l
Restra in t
5 3 I n i t i a l Restra int Coeff icients
5 3 1
Slope Deflection
Coefficients
5 4
Reduced
Restraint Coeff icients
5 5
Ultimate Strength Behavior of
Composite
eams Under Combined Loads
vi
48
5
52
54
58
60
61
67
70
70
72
81
8
83
85
85
87
8
91
95
97
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6
LO D DEFLECTION BEH VIOR OF A
STORY
v ii
101
6 1 Load Deflection Curve of a Sway Sub
assemblage
101
6 1 1
Evaluation of M in a Sway Subassemblage
With
Stee l
Beams 101
6 1 2
Evaluation
of
M
in
a
Sway Subassemblage
With Composite Beams 106
6 2
Construction of
a
Typical Load Deflection
Curve
108
6 3 Load Deflection
Curve
of
a
Story
7
FUTURE RESE RCH
7 1
Analytical
Studies
7 2
Experimental Studies
8 SUMM RY ND CONCLUSIONS
9
NOMENC L TURE
10
FIGURES
11 REFERENCES
12
VIT
113
114
116
119
122
124
144
148
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LIST
O IGUR S
19 Construction
of
Load Deflection Curve
for
a Story
15
Derivation
of
Slope Deflection Coeff icients
16
Distr ibut ion of Bending
Moments
in
the
Sway
Subassemblage
17
Construction
of
Load Deflection
Curve
Steel
eams
18
Construction
of Load Deflection
Curve
Composite
eams
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STR CT
n
approximate
method
of
analysis
which is
suitable
for the
combined
load
analysis
of an
unbraced
frame on a maxi-
mum strength
basis
is
presented in
this
disser ta t ion.
i s
called
the
sway subassemblage method of
analysis
and is
part ic-
ularly useful
for developing
the
la tera l load
versus
sway de-
f lect ion curve of a story
in
the
middle and
lower
stor ies of an
unbraced
frame.
t has been assumed
tha t
a pre liminary des ign of the
frame has
been
made
~ r e f e r l y
by the moment
balancing method.
The pre liminary des ign should p ~ o v i d e not only·
the
tenta t ive
beam and column sizes but also the
dist r ibut ion
of
axial
forces
in
the columns
corresponding to
ei ther
the
m ximum
l te r l
load
capacity
of the frame or the plast ic mechanism load.
The method is based
on
the concept
of
sway
subassem-
blages and uses direct ly the resul ts
of
previous
research on the ·
strength and b eh av io r o f restrained columns permitted
to
sway
In t he a na ly si s a story with known member sizes is subdivided
into
a number of sway subassemblages each consisting of a
re-
strained
column and ei ther one or two
adjacent restraining
beams.
The restraining beams
together
with assumed re l is t ic boundary
conditions consti tute
the
restraining
system. The
restraining
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-2-
system represents the rotat ional
restr in t provided
by the s tee l
columns
in
the story as well as the.
steel
or composite steel-
concrete
beams
n analysis
of
each
sway
subassemblage
is
made
to determine i t s load-deflection behav io r using
specially pre
pared design charts .
The
resul t ing load-deflection curves
of
l l the sway subassemblages in a story are then combined
to
de
termine
the complete
load-deflection curve
of
the
story.
This
curve may
be
obtained up to
and beyond
the deflect ion correspond
ing
to
the
maximum
load
and mechanism
load
capacit ies.
The adequacy
of the preliminary
design
may be deter
mined on the
basis of strength
maximum
or
mechanism
load for
example
or
deflection
working load, maximum
load or
mechanism
load
for example .
The sway subassemblage method
of
analysis
accounts for
the
reduction
in
strength of
a
frame due to
P
effects .
t
also
considers plast i f icat ion of the
columns including
residual
s t resses , as well
as plast ic
hinges in
the beams A recent
pi
lo t
study on
the
ultimate strength behavior of composite beams
under
combined
loads has provided experimental
evidence
th t
a
combination of plast ic analysis and u lt ima te s tr eng th
theory
may
be
used
for
the
design
of
such
beams.
The
apprOXimate
sequence
of
formation
of
plast ic hinges in
a story
may also
be
determined
from the analysis.
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3
he sw y subassemblage method as
developed in
th is is-
sertat ion
does
not consider unbraced frames
with
signi f icant ly
large
in i t i l
sw y def lec t ions
under
factored
gravity
lo ad s a lone .
he effect of ifferent i l column shortening on the
strength
nd
deflection
of
the frame is also neglected.
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INTRO U TION
This
disser ta t ion
co ns id er s th e
theoretical
develop-
ment
of
an
approximate method of analysis for unbraced multi-
story
frames which a re s ub je cte d
to
combined loads. Throughout
the
disser ta t ion the method of a nalysis
wil l
be re fe rr ed to as
the sway
subassemblage
method. The
discussion
in
this Chapter
wil l center in l t ia l ly on
the
required steps in the basic
design
process
for unbraced
frames and on
w the se step s
are
carried
out. The
available
methods for executing
these
steps wil l then
be
discussed.
I t wil l be
shown
that
a
need exists for the fur-
ther qevelopment of
methods
of analysis.
The
purpose
and
scope
of
this disser ta t ion
wi l l then
be
presented. The
chapter wil l
conclude with a d is cu ss io n o f the res t r ic t ions r eg ar di ng t he
frame considered
in this
disser ta t ion
the applicable loading
conditions the frame material and the
restr ic t ions
regarding
the applicat ion of the
sway
subassemblage method.
The ~ i c
Design
Process
The
direct
design
of
an
unbraced multi story
frame
for the combined load condition i s a problem
of great complexity
and
is
vir tual ly im.possible
to
perform with tTexactnesstf
for
t a l l
frames. For this
reason a
large
number
of approximate
methods
4
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1 .1
-5 -
have
been
developed which supposedly g iv e r ea so na bl y good de
signs and t
th e
same
time
simplify
the
work
involved.
I d e a lly a d i r e c t ~ s i n method would be
used
to de
termine
th e f i n a l exactf member sizes s t a r t i n g
only w ith
a
knowledge of
frame
geometry
loading
conditions
an d
material
properties and working towards c e r ta in s t r e n g t h s t i ffn es s an d
economic
cr i te r i Pl as t i c
design methods
approach th is
id e a l.
However the complex
i n t e r r e l a t i o n
between s t r e n g t h
an d
s t i f f -
ness
in
problems
concerning unbraced
m Ulti-sto ry
frames
demands
a
method which would y ie ld
a
d i r e c t solution
to a
non-linear
problem. Such a method has no t y et be en f ou nd .
From
p r a c t i c a l
standpoint th e
complete
design o f
an unbraced
frame
requires
me th od s w hi ch
w i l l
achieve a
solu
t i o n step
by
s t e p from
a
gr adually
converging
t r i a l - a n d
e r r o r
procedure.
S uc h me th od s
would
r e q u i r e
three
w e ll d ef in ed
steps to
c om pl et e e ac h
t r i l d es ig n c yc le :
Step 1 .
Step
3.
The preliminary design; th e
s e l e c t i o n
o f
t e n t a t i v e
beam and column siz e s.
The analysis;
th e determination
of th e
adequacy
of
th e members selected
in
Step
based
on strength a n d s t i f f n e s s .
The
revision;
the revision of one or more
members based on
th e r e s u l t s of
th e
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1 .1
6
analysis
or
on
o th er fa ct or s such
as economy y revision
const i tu tes
another
prel iminary design.
Because of the tedious uneconomical and
time
con
suming
work involved
in
such a t r ia l and error procedure i t is
necessary to develop methods which wil l
produce
f ina l designs
in only one or two cycles. This imposes considerable demands
on preliminary
design methods
which
must
determine
the member
sizes
with
a relat ively
high
degree
of
accuracy_
he demands
placed
on the methods of
analysis
depend to a considerable de
gree
on
the design
philosophy;
tha t is whether allowable s t ress
or maximum strength
cr i te r ia are employed.
Analyses for
strength
and st i ffness
a t
working
loads
can
be considerably
less involved than those for
maximum
strength.
Very l i t t l e
a t
tention i s usually given to
rat ional
means of making th e neces
sary
revisions. he designer often re l ies only on his
design
experience
and intui t ive abi l i ty . Quite often
i f
t he i nd ic at ed
revisions are
re la t ively minor
the resul t ing frame consti tutes
the f ina l
design
further
analysis
deemed unnecessary
because
of the conservative nature
of
o th er f ac to rs not
considered
in
the
analysis such
as
the
s t i ffening effects of cladding and in
t e r ior
part i t ions.
Present design
procedures for
unbraced multi s tory
frames are based on the
allowable
s t ress concept. That i s the
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1 .1
-7-
members
selected
are
considered adequate for strength
i f
under
working loads, the allowable st resses are not exceeded
anywhere
in
the frame. Although many design procedures are used which
vary
from
one office to another,
a
complete design based on the
above three
s teps
is rarely
made This is
due to the complexity
of
performing
an
exact
e las t ic analysis . Most methods of
anal
ysis
are
baseo on approximations and are
derived
from
an
assumed
frame
behavior
under
load.
Usual ly , St eps
and
2
are
combined
into
one
operat ion.
The
well-known
porta l
and
canti lever
methods
and variat ions
of
them) for
wind
loads
are
examples.
2
Approxi
mate methods are also used to
determine the dis t r ibut ion
of
moments
under gravity loads and to c alc ula te the
wind
induced
sway
deflections a t working loads.
l
From the
point of view of
the
three s teps r equi red
in
the basic
design
p ro ce ss , th es e
procedures
const i tute
nothirig
more
than
pre limina ry des ign
methods
coupled
with an
approximate
working
load
sway
analysis.
~ o w v r many years of design
ex
perience,
bui l t
upon observed
frame
behavior have
established
procedures
which resu l t
in
sat is factory
structures
from the
point
of
view of
serviceabi l i ty .
The
r ecent int roduc tion
of
electronic compu ta tion has reduced
many of
the approximations
involved
but
has
not changed the basis
for
the des ign a llowable -
s t ress .
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1 .1
8
With the successful
application o f
plast ic design me
thods
to
the
design
of low
building frames 3
and
to
the
design
of
braced
multi story
frames
4
S 6
much
n t r s t
has
been
aroused
in the possibi l i ty
of
extending
these
methods to
the
design o f
unbraced multi-story frames. Plast ic design methods employ the
concept of the m x mum or plast ic strength of
structure
as
the
basis for design. hey are
founded
on
the
unique duct i l i ty ex
hibited by the structural steels nd on
the
abil i ty of
struc
tures to r ed is tr ibu te int ernal forces
moments
as
plas t i f i ca-
tioD occurs. hey usually result in more ef f i c ient
use of
m -
ter ia l more uniform
factor of
safety, nd
what
is
equally
important
re la t ively simple
des ign procedures.
Investigations are also
presently
directed
towards
extending
plast ic design methods to composite
steel-concrete
structures.
7
The resul t ing
methods
would require simple rules
for
determining the
ultimate
moment
of
resistance as well
as
the
rotation capacity
of
cross-section
when the slab is in ten-
sian
or compression .
The current specifications
8
for the design of
s t l
structures l imit application
of
plast ic design
to
one- nd two
story
rigid frames nd to be ms
in
mult i -story
frames
where
the
columns have
been
designed by the
allowable stress
method.
This
l imitation
is
consequence of the well-known assumptions
of the
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1.1 -10-
the
P6 effect Because moments resul t
from the
gravity loads,
the t o t a l wind shear in
a
story
wil l be unchanged. Thus, is
clear
tha t
for
a
given value
of
combined
ult imate
loads
working
load times
the load factor the required shear cap acity wil l not
only be
a function
of the plas t ic
strength
of i t s
members
but
also
of
the
sidesway
s t i f fness of
each
story
of
the
frame.
The
s ig ni fi canc e o f
moments
has
not
generally been
recognized designers using the allowable s t ress methods
for
3
reasons:
1.
The
moments
are
relat ively
small
a t
working
loads,
and a t f i r s t
yield l lO
2.
Indirect ly the P6 effec t has been accounted
for
in design
specificat ions
by
the
use
of an
8
effec t ive
length
concept,
and,
3. The
s ti ff en in g e ff e ct
of the ex ter io r cladding
and
the in te r ior
part i t ions has,
in
the
past ,
reduced
the
sidesway d ef le cti on s o f the frames
in a
building
to lower
values
than
the calcu-
la t ions
have ind{cated;
the Ph moments therefore
are even
smaller .
With the trend in
modern
building designs
towards
l ight cur ta in -wal l cons t ruc tion, larger
areas
of glass and
re-
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1.1
11
movable
in te r ior par t i t ions
the
bracing
effect from these
sources
is
becoming small
o r u n re li ab le .
Consequently the bare s t ruc
tur l frame must
then
r es i s t
the
to t l
combined
loads.
Three
conditions
are
therefore imposed on unbraced frames
which
are de
signed by pl s t ic methods:
1.
They must be able to res is t the combined working
loads within accep table sidesway deflect ion
limi
t
1 11
a lons
2. They must be able to res is t
the
combined ultimate
loads and
3. For effic ient
use o f m ate ria l
the
s he ar c ap ac it y
should not great ly
exceed the required shear ~
pacity under the combined ultimate
loads.
From the previous
discussion
i t is
apparent
that con
di t ion wil l depend on the v i l bi l i ty of a suitable method of
calculat ing working load deflect ions . Conditions 2 and 3 wil l
require a suitable method for calculat ing the maximum shear ca
pacity
of the frame
which in turn is
a function of the sidesway
deflect ion corresponding to the m x mum shear capacity. In
effect an analysis procedure is
required
which wil l predict the
complete load deflection
behavior
of the
frame
t le s t unt i l
the maximum s he ar c ap ac ity has been
reached
and
preferably
be
yond. In keeping with one of the
advantages
o f·t he p la st ic
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1 .2
-12-
methods
stated ear l ie r
the method should be relat ively simple
to
apply. I t y also be approximate so long as gives rea
sonably
dependable
resu l t s .
1.2 Effects
and Ins tabi l i ty
References
10
and
3
contain
extensive
discus
sions of P6 effec ts and
ins tabi l i ty . They
also include
many
addi t ional references.
A
fur ther elaborat ion wil l
not be
a t -
tempted
in
th is dissertat ion.
However
in
order
to
bring
into
s ha rp er p er sp ec tiv e th e
importance
of sidesway
deflections
and
to
give
background
for l a t e r development
a
br ie f review wil l
be
necessary.
Figure l
i l lus t ra tes .
the P6 effec ts and ins tabi l i ty
under
combined loads. It is assumed that an unbraced
multi
story
frame i s
subjected
to
proport ional g ravi ty
and wind
loads.
Consider a story in
the
middle or lower regions of the
frame.
The story height
wil l
be taken as h and
the relat ive
sidesway
def lect ion between the top
and
the bottom of the story is assumed
to
be
I t is
also assumed tha t the story
wil l
eventually
de
velop a plas t ic mechanism. I t is fur ther supposed th at in sta -
bi l i ty
does
not
occur
in another story
prior the formation
of
th is collapse mechanism and that
the
material
exhibi ts
elas
t i c e las to p las t ic
perfect ly
plas t ic moment-curvature behavior.
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1.2
-13-
f
P6
effects were absent, the
wind
load
H
Fig. 1
would
at ta in the value predicted
by
the simple plas t ic theory.
The
load-deflection
curve
for
the
story
would
then
be
curve
O-a-b. This
curve
would be
l inear
from point 0 un t i l_ f i r s t
yielding a t point a.
The actual
load deflect ion
curve for the
story
is
shownias curve O-c-d.
This
curve wil l be non-linear
from
point
o
and
wil l reach a peak value a t point c. The plas t ic mechanism
of
the
story
wil l
form a t
point
d
where
the
load-deflection
curve
in tersects the
second-order r ig id plas t ic mechanism curve
for
the story. is
character is t ic
of those s tor ies in un-
braced multi-story frames
where
the member
sizes
are control led
by the
combined
loading condition that the maximum shear capa-
ci ty
point
c
in
Fig. 1 wil l be attained prior to the forma-
. . 10 12 14
t ion of
the
collapse mechanlsm.
The important
concept
to gain
from Fig. is that the
P6 moments
induced by
the gravity
loads
acting through the
side-
sway
displacements cause
a s ignif icant reduction
in
the shear
capacity
of
the frame. Also fai lure
can
occur by ins tabi l i ty
rather than by the formation of
a fai lure
mechanism.
t
is quite
apparent
from this
in troduc to ry d i scuss ion
that any
design
method . fo r unbraced
mUlti-story
frames subjected
to combined loading which is
based
on the maximum shear c apac it y
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1.3
-14-
o f
each st o r y o f the frame
must
s t isf c tori ly account f o r
effects
o be a complete design method
i t
must
enable
a de
sig n er to
execute each
of the
3
s t e p s
in
the
b a si c
design pro
cess
Art. 1.1) in o rd e r to arri v e
t
a f in l design. Such a
method would be of g r e a t value i f
i t
also
allowed
a
f in l
de
s ign to be
made
within only
one o r two cycles.
1.3
Preliminary
Design Methods
ny
method
which
r e s u l t s in
a
d i s t r i b u t i o n of
t r i l
beam and column
si z e s throughout
an
unbraced m u l t i - st o r y frame
co n s t i t u t es a preliminary design. A wide v ari et y of
methods
s t isfy
t h i s condition. They vary in complexity from nothing
more
than
educated
guesses
based
on extens ive
experience and
intui t ive
bi l i ty to more involved
techniques
r e l y i n g upon only
a
knowledge
o f
th e
frame
geometry,
m a t e r i a l p r o p e r t i e s
and
l oa di ng c on d it io n s as well as an assumed d i s t r i b u t i o n of forces
within
the
frame.
From this
p o i n t
o f view
any
o f
the methods
employing t h e a l l o w a b l e s t r e s s concept would be adequate fo r the
preliminary
design st e p
in
a 3 - st e p design
process
based
on
maximum
shear capacity.
However
re l t ively
simple
p re lim in ar y d es ig n p ro
cedures have recently been developed which
c on sid er th e
i n e l a s
t ic
behavior
o f
frames. The most
successful
are pl s t ic moment
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1 .3
lS
d i s t r i b u t i o n
15
,3 8 an d p l a s t i c
moment
balancing.
lO
,1 6
Pl as t i c
moment d i s t ri b u t i o n i s in some r e s p e c t s ,
s i m i l a r
to elas t ic
moment
d i s t r i b u t i o n .
Pl as t i c moment
balancing
is
d escrib ed
in
Ref.
10 as a Tlre·fined
formulation
o f e qu il ib ri um fo r unbraced
mUlti-story frames
u
where tTthe
refinements
are
formulated
to
co n sid er
th e in flu en ce on frame , s t at i cs of. sway
d e fl e c t i o n ,
f ini te j o i n t
s i z e ,
an d p l a s t i c
g i rd e r
mechanisms or r e s t r i c t e d
g ir d e r hinge patterns
• Both
of
these methods employ e q u ili-
brium
conditions, b u t in
a
d if f e r e n t
manner.
P l a s t i c moment balancing o r p la stic moment dis t r i
bution)
is
id e ally su ite d fo r
th e p re li m in a ry d es ig n
o f
un -
braced frames because
ca n
include an
approximate P
e f f e c t .
n in i t i a l swaydeflectiori estimate is made
an d
th en the r e s u l t -
. 10
16
i moments a re in clu de d when equilibr ium is e s t a b l l s h e d .
The in i t i a l sway
d e fl e c t i o n estimate can be
made
guessed)
ei t h er o f two
ways; 1 )
on the basis
o f
the
expected
sway de-
f le c tio n
corresponding
to th e formation o f a m ech ani sm in each
s t o r y , or
2)
on th e b as is
o f th e e x p ~ t e
sway de f lection a t
th e
maximum
sh ear capacity of each story.
Either way a sway
analysis should be performed to verify
the in i t i a l sway est i
mate.. In
a dd itio n, th e sway
d e fl e c t i o n
should be calculated a t
working
loads
to
complete
th e
analysis.
The
need
fo r
a
r e la
tiv e ly simple method of
predicting
a l l
three
sway
deflections
is
therefore
indicated.
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1.4
-16-
The moment
balancing
method described in Ref. 10
is
also well suited to
perform par t
of the l a s t
step
in the basic
design
process,
Step 3
Art. 1.1) .
I f a new sway deflect ion
est imate i s ~ n d i c t e d the
resul t ing revised
P6 moments, when
added to the moments produced by the combined loads alone, dic
tate
direct ly
the rev ised beam
sect ions.
The
revised
column
sizes
are then dictated by the formulation o f equ il ib ri um , and
by conside ra ti on o f s tabi l i ty as before.
The remaining par t of the las t
step in
the basic de
sign process is
mainly
concerned with economy
Although
a f inal
design y be
made on
the
basis
of
strength and s t i f fness
y not be
the most
economical. The moment
balancing
method can
not
in i t se l f
lead
to the
most
economical frame although some
steps
in this direct ion have
been indicated
in
Ref.
10.
A preliminary design by moment
balancing
y also be
extended to
unbraced multi-story
frames
uti l iz ing composite
beams.
Reference
1
suggests
t hi s po ss ib il it y Such an exten
sion requires only the knowledge of the plas t ic moment capacity
of
composite beams
which are
subjected
to
posi t ive and
negative
end moments well as transverse
loads.
1.4 Sway Analysis Methods
The importance of
performing
a sway analysis
as
a
major
step in the complete design, on a maximum strength basis ,
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1 .4
17
of an unbraced mUlti story frame subjected to combined loads
has now
been
established. The
obvious
question to be considered
is :
Are
there
re la t ively
simple
methods
available
which
wil l
predict the sway d ef le ctio n o f each story
of
such a frame
a t
working loads
a t
design
ultimate
loads
a t
maximum
s he ar capa
ci ty and a t the· formation of a mechanism? Par t i cu la r ly methods
are
required
which are
suitable
for use in those stor ies where
the member
sizes are
controlled by
the
combined
loading
case.
n
introduction
to
the
l i tera ture on
the
analysis
and
design
of unbraced multi story frames m y
be found in Refs. 1
and
13.
There
is
unanimous agreement that sway induced effects
must be
considered. There
is
a lso conside rable agreement
t ha t
an TtexactTt analysis which
is
suitable for design purposes
is not
in
s ight .
Surprisingly l i t t l e has been
done
to provide
an
approximate analysis which would predict
the
load deflection be
havior
of
a
story
in an
unbraced
multi s tory
frame.
Much research has been devoted to frames of the order·
of
three stor ies or less .
Either
c om pa ti bi lit y a na ly si s o r
a
.second order e las t ic p las t ic analysis is used to obtain
the load
deflect ion curves.
The
compatibi li ty analys is
i s suff ic ient ly
involved tha t has not been applied to
other
than
very
simple
13
frames.
Par ikh 17 app li ed
a
second order elast ic plast ic
anal
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1 .4
- 1 8 -
y si s
to unbraced mUlti-s tory frames up to 25 s tor ies in
h e i g h t .
Although plas t i f icat ion o f
beams and
columns
was
part ia l ly
in .
eluded the
e f f e c t s
of
stra in
hardening
were
n o t
considered.
However
the
analys is
did
c on sid er th e
formation
o f
plas t ic
hinges the e f f e c t of
a x i a l
loads
on the plas t ic moment
capacity
and
s t i f fness o f the columns a x i a l s hortening an d instabi l i ty
o f the columns including
res i d u al
s t res s es and
the
P6
effect .
The
loads
were assumed
proportional
for th e combined
load
cas e.
P a r i k h s
analysis
began
with
members
se l e c t e d
a
previous
preliminary
design and p re di ct ed t he l o a d - d e f l e c t i o n
curve fo r each s to ry o f the
frame. The
curves a l l
terminated
a t
the l oa d c or re sp on d in g to
th e
story with
the
s m a l l e s t shear ca
p a c i t y .
This was a consequence. o f the divergence occurring in
the
i tera t ive
c a l c u l a t i o n s
when a
st a b l e equilibrium could
n ot
be found
somewhere
in the frame.
a
resul t
the s hear capa
ci t y o f
the remaining s t o r i e s
was unknown
I f the h i g h e s t load
obtained
corresponded to
f a i l u r e of a
story
by
ins tabi l i ty
then
the mechanism
load and
d e f l e c t i o n for
t h a t
an d a l l
o t h e r
s t o r i e s
are unknown Although th e maximum s he ar c ap ac it y o f the frame
can be
determined the
method
is
not
s uit a b le fo r checking the
sway
es timates used in the
p r el im i n ar y d e si gn
except possibly
fo r
the
story
which
fai l ed
f i r s t .
The analysis in Ref. 17 re l ies on el ect ro n i c
computa-
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1.4
-20-
t ioned
to
remain
e las t ic when
t he c al cu la te d
fu l l
plas t i c
moments
and axia l t hr us t were applied.
Holmes
and
Gandhi
19
,20 introduced a two stage proce-
dure using an
analysis which
was
~ s i s t e by elec t ronic computa
t ion Columns and girders were proport ioned accord ing to
the
sim-
pIe plas t ic theory in the f i r s t stage and then
increased
i f neces-
sary in
the second
s tage to allow for sway
deflect ion and
ins ta
b i l i t y e f f ec t s
The
analysis in i t i a l ly
assumes
t ha t i nf le cti on
points occur
a t
mid-height
of
the
columns.
This assumption
is
then
l a te r modified
to account for unequal column
end moments .
The mechanism
condition
es tab l ishes the maximum frame capacity.
21
Stevens recognized
that the emphasis in design
must be placed not only on strength and safety
but
also on sa t i s
factory
deflect ion
behavior a t
working
loads.
suggested
tha t
design
methods must meet
two
conditions;
1 deformations
a t
working
loads must
be
less
than some
accep tabl e val ue ,
and 2
the
c r i t i ca l
l oad ing cond i ti on
must also
be less than
some
accept-
able value. suggested
tha t condition
2
must also
include
some
l imi t
on deformations a t the maximum load.
23
Gent
also suggested that
adequate
design procedures
must meet
s t rength ,
s tabi l i ty
and
def lect ion
cr i t e r i a
noted
that complete design methods which would consider a l l three cr i
t e r ia are a t
an
elementary stage of development.
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1.5
-21-
1.5
Purpose and Scope of the Dissertat ion
I t
i s
the purpose of
this
disser ta t ion
to develop
an
approximate analyt ica l
method which
wil l predict
the
load-de
f lec-
t ion curve of each story in
the middle
and lower stor ies of an
unbraced m l t i ~ t o r y
frame
Art . 1.6.5 .
The
method wil l be ap
plicable
to
the combined load condition only Art.
1 6 2
This
analy t ical
method wil l be part icu lar ly useful for
performing
Step 2
Art.
1.1
of
the
basic
design process. The
assumption
is
made
throughout
the
development tha t
a
preliminary design of
the frame Step
has
already been
made
preferably by the
mo-
. 10 16
ment balanclng method.
The
basic concepts of
the approximate
analyt ica l
method were previously
discussed
by
the
wri ter in Refs.
24
to
28. Because the method is :
based
on
the
idea of
sway
subassem
blages
29
and uses direct ly the resul ts of
recent
stud ies o f
re
strained columns permitted to sway30 the
method
is referred to
as
the sway
subassemblage
method. The
load-deflect ion
curve
of
a
story is
obtained
through the
app li ca ti on o f a semigraphical
analyt ica l procedure using specially-prepared design charts.
28
The sway subassemblage method
accounts
for the P6
effect
on both
the
columns and
the
story.
I t
also
considers
plast if icat ion
of
the columns
including residual
stresses,26 as
well
as plas t ic
hinges
in
the beams. The approximate
sequence
of
formation of
plas t ic hinges in
a
story y
be obtained
from the analysis.
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6
-22-
The sway
subassemblage method
y a l s o be extended to
develop
procedures fo r p er fo rm in g S t ep
3) R evision)
in
the
b a s i c
design
proces s .
Such
an
extension
wil l
n o t
be
included
in
th e
scope o f t h is d is se rt at ion
6
D e f i n i t i o n s an d Assumptions
6 rame Layout
Unbraced
m u l t i - st o r y
frames wil l be
defined in
th i s
disser tat ion as that c l a ss o f plane r e c t a n g u l a r frames, more
than on e s t o ry in
h e i g h t an d
o f on e
o r
more bays in wid th , which
derive the i r r e si st a n c e t o
i n - p l a n e
forces from the bending re -
si st a n c e o f
th e frame members
themselves.
Th ese fr am es may be
cons tructed o f s t ee l columns
and
ei ther
s t ee l beams o r
composite
st e e l - c o n c r e t e
beams. The
s t ee l
members y
be
ro l led o r welded
shapes such
as w id e- fla ng e
and I
s e c t i o n s
o r o th e r
s e c t i o n s
with
a s i m i l a r distribution o f m a t e r i a l over
th e
c r o s s - s e c t i o n .
Welded
hybrid shapes
may
a l s o be
used. The
s la b s o f composite
beams wil l b e a ss um e d to meet th e re q u ire me n ts o f
Ref.
and are
f u r t h e r assumed t o
be
continuous an d
continuously r e i n f o r c e d
a t
a l l in te r io r columns. The
s h e a r
connection
wi l l
be assumed to
e xt en d t hr o ug ho u t
th e len g th o f each beam. The span lengths o f
th e
beams wil l
be
taken as th e
d i s t a n c e
between th e c e n t r o i d a l
axes o f th e
columns._
The length
o f
each column
wil l
be e q u a l to
th e st o r y h e i g h t an d wil l be taken as
th e
d i s t a n c e between th e
c e n t r o i d a l axes o f
th e beams, w hich
wil l be assumed
c o - l i n e a r .
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1 6 1
made:
23
The following
addit ional assumptions
wil l also be
1. The
frames
are regular in geometry.
o
missing
columns or beams wil l
be
permitted
and
column
footings are assumed
to
be a t the same elevation.
2. The connections between
s tee l
beams
and s tee l
columns are r igid
and
m y
be
made by welding
or
bolt ing.
The jo in t is
assumed to transmit
the fu l l plas t ic
moment
capacity of the
beams
and the reduced
plas t ic
moment capacity
of
the
columns without
local
buckling or
excessive
dis tor t ion
3. o bracing or
cladding
is
used
in the plane of
. the
frame to
r es i s t
sway deflect ion.
4.
The
column
bases are assumed
to be
fixed in the
plane of
the
frame.
5. The minor axis of each member is assumed to l ie
in the plane of
the
frame
which
i s a plane
of
symmetry.
6. Biaxial bending
of
columns does not
occur.
7.
Axial
and
curvature
shortening
of beams and
columns wil l be neglected.
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1.6 .2
8.
9.
Concrete
cannot
res is t tens i le
forces.
Complete in teract ion
is
assumed
for
com-
posite
steel-concrete beams.
-24-
1 .6 .2 Loading Conditions
Unbraced multi-story frames may be subjected
to
two
types of s ta t ic loads; 1 gravity and
2
wind. Gravity loads
are assumed
to
be
ver t ica l uniformly
dis t r ibuted
or
concentrated
loads applied
to the
beams by
the f loor system.
These
loads con-
s i s t
of
both dead DL and
l ive
LL
loads.
Wind loads WL are
dis t r ibuted
horizontal
loads applied by the exter ior
wall system
and assumed to
be
concentrated
a t
the
exter ior
joints
of
the
frame. The s ta t ic loading condi tions
can
be
represented by the
following
cases:
1. + LL a l l beams
2. DL + LL some beams - -
checkerboard
3.
DL
LL a l l beams + W
4. DL
LL
some beams + W
Cases
1
and
2
const i tute the
gravity
load conditions
while
3
and
4 are the
combined load
condi t ions. The design
u l t i
mate values
of
load are
obtained by
mUltiplying the working loads
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1.6.2
-25-
by a load factor
LF .
Where indicated, load factors wil l
be
chosen in accordance with th ose
e sta bli sh ed i n Ref.
31 namely:
Gravity
load .conditions:
L
= 1.70
Combined
load
conditions:
L
=
1.30
All loads are assumed to be ei ther horizontal or ver
t i c a l
and
lyiDg in the
plane containing
the minor axes
of
the
members. Thus the loads and the· frame form a co-planar system.
This d is se rt at io n w i ll
consider
only
the
combined
load condit ions,
Cases
and
4. t is unlikely tha t pract i -
ca l
frame,
subjected to both gravity and wind
loads,
would be
loaded propor t ional ly . t is more l ikely tha t the
applied
grav
i ty
loads
wil l remain
vir tual ly
unchanged as the
wind loads are
applied. Therefore,
the ~ o l l o w n non-proportional
loading
se
quence
wil l
be as sumed:
1 The factored
uniformly
dis t r ibuted gravi ty loads,
1.30 w are
applied f i r s t
where w is the working
load
value.
2 The factored
wind
loads are then-app l ied , increas ing
monotonically from zero
to th e d esig n
Ultimate
load H
The
probabil i ty
that fu l l gravity l ive load plus wind
load wil l not
ac t
together
is
accounted by
l ive load
reduc-
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1.6 .4
-26-
t ion
factors
3 l
which a re app lic ab le only to
the
l ive gravity
loads. In general w wil l be di f fe ren t for each beam. In addi-
t ion
the
column
loads
wil l
not
l ikely
be
in
equilibrium
with
the t o t a l beam loads. This condition wil l be
acceptable
when
performing an analysis by
the
sway subassemblage method.
1.6.3 Secondary Failures
I t wil l be
assumed
tha t a l l secondary
fai lures
of
the
frame
are prevented.
These
include
la te ra l - tors iona l and
local
buckling of the s tee l sect ions as well as
spl i t t ing
and diagonal
tension fa i lures of
th e c oncrete
slab
and fai lure
of
the shear
connection.
These
fai lures may be
prevented by
adequate bracing
minimum
width-thickness
and depth-thickness ra t ios of projecting
s tee l
elements adequate slab
reinforcement and
a suff ic ien t
number of
shear connectors to develop
the
f lexura l
capacity
of
the member
1.6.4 t e r i ~ l s
Only ASTM A36 and A44l s teels are considered. The
design charts in
Ref.
28 have been
prepared
for
A s tee l
0 =
36 ksi
but can be
used with
l i t t l e modif icat ion
for A44
y
s tee l .
This res t r ic t ion may not
ult imately
be necessary
but
is
imposed
here in view of the
fact tha t
research into the
behavior
of
the
higher s t rength
s tee l s
is s t i l l in progress.
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1.6.5
-27-
The concrete slab in the case of composite beams m y
be of standard or l ightweight concrete and is assumed to
be
ade
quately
connected
to
the
s tee l
beams by
s tee l
shear
connectors
such as the headed
stud
type
1.6.5 Application of the Sway Subassemblage Method
The design
of
an unbraced mult i-story frame must con
sider
both the
gravity
load and
the
combined load condit ions.
The design must a ls o con sid er wind loads from both
direct ions .
t wil l be
found
that the gravi ty
load
conditions
wil l control
the
selection
of beam and column sizes for a
l imited
number of
s tor ies at the
top
of
the
frame.
l
The number of stor ies com-
prising this region is not def ini te and wil l depend on many fac
tors such as
frame
geometry
material propert ies load factors
and l ive load reduction
factors .
1
32
The number of stor ies
m y
also vary from one
bay
to another across the
frame.
The combined
load conditions
control the
selection of
beam and column sizes
in the middle and
lower s tor ies
of
the frame.
Between
the
re
gions
controlled by
the
gravity load
and combined load con di
t ions there wil l
be
a t rans i t ion zone where both m y govern in
anyone
story.
Figure 2 shows a t ypical d i s tr ibu t ion of the
three regions for a three-bay unbraced mult i-story frame.
t wil l
be assumed in this d i ss er ta ti on tha t the upper
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1 7
29
assumptions used in the
analysis
Chapter discusses
the fu-
ture
research
required and Chapter summarizes
the resul ts
of
this study
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2. THE SW Y SU SSEM L G S IN N
UNBR CED MULTI STORY
FR M
2.1 The Role of Subassemblages
subassemblage of a
multi-story frame
i s a limited
assemblage of beams and columns
analyzed
remote from the frame
the
behavior of which
under
r e a l i s t i c loads and boundary con-
d i t i o n s c a n be assumed to approximate
the
t rue behavior of
that
portion
of the
frame.
The
subassemblage
i s
usually
deter-
mined from the
point
of view of ease of analysis . The
case
for
the study
of
subassemblages has been clearly expressed in
Ref.
as follows:
1. nThe analysis and design of
an
ent i re multi
s tory
mUlti-bay
frame i s almost prohibi t ive
i f s t a b i l i t y and
deflect ion
effects are pre
dominant considerations; and
2.
n8ubassemb1ages can be used
in the a na ly sis
and
design
of individual members and of member
groups when conservative assumptions are made
for
end
conditions
n
•
The
use
of subassemblages to a s s i s t in the analysis and des.ign
of frames i s already widespread.
For
example:
1. The portal and canti lever methods
for
wind
analysis are subassemblage
methods;
a sub-
assemblage
being bounded by assumed points
- 3 0 -
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2.2
31
of inflect ion
above and
below
and
to
l e f t and
r ight of
a
jo in t
2. The determination
of
the effect ive
length
of
columns in braced and unbraced multi
story
frames
for use in
allowable
stress
design procedures
is
based on sub
bi
1
· 1 8
assem age
ana
YSlS
3.
A
proposed
method
of
designing
braced
mult i story frames by plas t ic
methods
is also based
on
the
use
of
sub
4 5 6 34 35
assemblages.
,
The configuration and extent of a subassemblage de
pends
on
the
design
p h i l o ~ o p h y the available
methods
of anal
ysis
and
whether
the
frame
is
braced
or
unbraced.
The
term
sway subassemblage
wil l define
a par t icular
c on figu ra ti on o f
subassemblage which wil l be
found
useful in p re dic tin g the
load
deflection
curve of
a
story in
an
unbraced mult i story frame.
This chapter
wil l be
concerned with developing
the
sway
sub
assemblages tha t w il l be used in the sway subassemblage method
of
analysis .
2.2 Possible
last ic Hinge
Locations
and Failure
Mechanisms
Consider the unbraced m U l t i s t o ~ y
frame
shown in
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2.2
-32-
Fig. 3 a .
For
simpl ic i ty , only the cent roidal axes of the mem-
bers are
shown. The factored distributed
gravity
loads,
1.30
w
are
assumed
to
be
applied
f i r s t
where
w
is
the
working
load
value.
In
Fig.
3 a
the
subscr ipts ,
AB e tc .
re fe r
to
the
particu
la r
beam.
The
factored wind loads
are
then assumed
to
increase
monotonically from zero
to
the i r
design ultimate value H.
The
subscripts
1, 2, e tc .
re fe r to the
l eve l number. As
pre
viou sly discussed A rt. 1.6.2 th is
loading
sequence w ill l ik ely
be
more rea l i s t ic
than
proportional
loading
for pract ical
frames.
In
the middle
and lower s tor ies of
the frame,
the beam and col
umns
wil l l ikely be s t i f f enough tha t under the factored
gravity
loads alone
the
jo in t rotations wil l be negligibly
small.
on-
s equently , th e
beams
wil l behave as fixed-ended
for
gravity
loads. The
resul t ing
distribution
of bending
moments in
the
beams and columns in the vicini ty
of level
n has been shown
in
Fig.
3 b .
The i n i t i a l app li ca ti on o f w ~ loads introduces
addit ional
bending
moments such as
those
shown in Fig. 3 c .
When these two bending moment diagrams
are combined,
the bending
moments
a t
the leeward ends of the beams wil l increase
while
those
a t
the
windward
ends
wil l
decrease.
The
leeward
ends of
the beams
are
therefore the
potent ia l
locations for the f i r s t
plas t ic
hinges.
This
wil l
be
t rue
for
both
composite
and
non
composite beams. Similar ly, the bending moments a t the ends of
certain columns
wil l
in i t ia l ly increase.
Since
the columns in
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2 .2
-33-
th is
region o f a
frame wil l
l i k e l y
be bent
in a
double
curvature
c o n f i g u r a t i o n
17
t h e f i r s t plas t ic hinges
ca n
a l s o form a t
th e
ends
o f thes e
columns.
s th e
wind
loads
are in cr ea se d a dd i-
t i on al p la st ic
hinges wil l form
a t
th e
ends of o t h e r
columns
and
elsewhere
in
th e beams.
F i n a l l y ,
story wil l f a i l ei ther
y
instabi l i ty o r
by
th e
formation
o f a
mechanism.
Figures 3 d)
and 3 e) show two
possible fai lure
mechanisms which can
resul t .
In a weak-girder, s t r o n g column
d -
32
h 1
h
b
f f -
Slgn
t
e
co
umn
s t r e n g t s m y e
su
lClent
to
oree
a
plas t ic h in g e s to
develop in
th e beams. S i m i l a r l y ,
in a weak-
1 t b
d
32
th b
t
th
b f f t
o
umn S
rong
eam
eSlgn
e e rn s
reng
S
m y
e
su lClen
to force a l l plas t ic h in g e s to develop in
th e
columns.
The re -
s u i t i n g
mechanisms
m y be
c a l l e d
ei ther a
combined mechanism
Fig. 3 d) o r
a sway
mechanism Fig. 3 e) .
A
c om bi ne d m ec ha -
nism
m y also be formed with plas t ic hinges in
th e beams and.
th e
columns.
Such a
m ec ha ni sm w ou ld resul t i f
th e design was
between th e extreme cases c i t e d .
In
a well-proportioned
r e g u l a r frame, th e
member
s i z e s
in a region
containing
th e middle and lower s to rie s w ill l i k el y
i n c r e a s e a t a
re la t ively uniform
r a t e with
increasing
dis tance
from
the top o f the frame. Since the g r a v i t y within each
ba y
wil l
l ikely be cons tant in thes e regions o f
th e
frame, the
beams wil l increas e in
si z e du e
t o
th e
i n c r e a s i n g
wind
and P6
moments
which
must
be c a r r i e d
by
th e
lower s tor ies .
The
columns
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2.2
-34-
wil l increase in
size
due to the increasing
wind
and moments
as
well
as
the
accumulation
of the
gravity loads on
the beams.
Although
the
wind
loads
y
not
be
uniformly
dis t r ibuted
over
the height
of the frame,
and
the
sway index
wil l not
l ikely be
uniform for
each story,
the
variat ion
over a
l imited
number
of
s tor ies
could
be considered small for most frames. a resu l t
i f level n is within th is region Fig.
3
the
load deflect ion
behavior a t levels
n+l ,
n, and n l
could
be
expected
to be
nearly the
same.
Consequently, i t can be
assumed
tha t the be
havio r of the
frame in
the region of
these
several s to r ies
can
be represented by the behav io r o f a small assemblage which
con
ta ins level
TI
At th is point therefore , an
assemblage
of
beams
and
columns a t level n can
be
isolated from
the frame.
The
equi l i -
brium
and
compatibi l i ty
conditions
a t
the
boundary can be approxi
mated conservatively. There are two pr incip le approaches which
can be
followed
in choosing the
assemblage:
1. Isolate the beams
along
two adjacent levels
and the columns
between
those leve ls , o r
2.
Isolate the beams along one level and i so late
a
certain
length
of
each
column
above and below
t ha t level .
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2.4
-35-
t wil l be found easier to approximate
the boundary
conditions i f the second approach i s
used
because of
the
avai l
able
studies of restrained
columns
permitted
to
sway.3D The
result ing assemblage wil l then c on sist o f
the
beams a t level n
which
forms
the boundary
between
s tories m
and m l
Fig.
2),
plus a portion of each
column
in
these
two s tor ies
2.3
Sign Convent ion
The
sign convention
which
wil l
be
adopted
in
this
dissertat ion has
been
stated
in
Ref.
and y be
summarized
as follows:
1.
External
moments acting
a t a jo in t are
posi
t ive
when clockwise.
2. n te rna l moments acting
a t
a
jo in t are
posi
t ive
when
counterclockwise.
3. Moments
and
rotat ions a t the
ends
of members
are
posi t ive when clockwise,· an d
4.
Hori zont al shear is posit ive i f causes a
clockwise
moment about
th e opposite
jo in t
2.4
The One-Story Assemblage
a t
Level n
Parikh
17
studied the lo ca tio ns o f the
inflect ion
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2.4
-36-
points
in
the
columns in
s tor ies 5 and 15 of a regular 24
story
3 bay unbraced unsymmetrical s tee l
frame
Frame C
of
Ref.
5
These
s tor ies
were
immediately
below
levels 20
and
10 respective
ly . I f
equals the story height of Frame
C
then
the
following
resul ts
were obtained
in
the study:
Table
1
Location of Inflect ion
Points
Distance
Below
Level
Above
Story
Max
Min Average
5
0.500
h
0.398
h
0.440
15
0.485
0.441
0.461 h
Three observations
are
signif icant :
1. The
average position
of
the
inf lect ion
points
did not
vary
app reci ab ly i n 10
s tories
The
variat ion
was
only about
of
the story height.
2.
The maximum
variat ion
in the
pos itio n o f
the
inflection point
across
a story was
re la t ively
small; about
1
of the
story
height .
3. All of the inf lect ion points were
at or
above
mid-height of the s tor ies
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2.4
37
These
resul ts were not unexpected. Consider the
middle
and lower s tories of
t a l l
unbraced multi s tory frame
such as
the
one
discussed
above. Suppose
tha t the
frame
i s well
proportioned and behaves as suggested in Art . 2 2 The follow-
ing general statements m y then be made
he
gravity
load moments wil l
be
re la t ive ly
small
compared
with
the
plas t ic moment capa-
c i t ies of the members.
2. he r e la t iv e d is tr ib u ti o n
of the
s t i ffnesses
of the beams and ·columns a t a
jo in t
wi l l be
nearly
the
same for several consecutive joints
along one column.
3.
he dis t r ibut ion of column
s t i f fness above
and
below
a
jo in t
wil l
be
nearly the
same.
Under
t he se c on di ti on s
and
assuming
tha t
the jo in t
rotat ions
are
nearly zero
under
~ v i t y
loads alone the
in -
f lect ion
points in t h ~
columns
wil l
l ie
approximately
a t
mid-
height
of
each story ove and below level n. s wind loads
are applied the inf lect ion points must sh i f t upward to
account
for
the
greater
wind
shear
in
the
s tory
below.
I f
the
e l a s t i c
and inelas t ic behavior of several consecutive s to r ies
contain-
ing
leve l
~
are
nearly
the
same then
the
inf lect ion
points
must
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2.4
-38-
remain
at a
re la t ively
constant
positio n fo r a l l values of
mono-
tonically
increasing
wind
loads.
he
1
maximum
variat ion in the posit ion of the in
flection points
across
the
two
s tories of the frame studied in
Ref. 17
was
l ikely caused
by small
differences
in
the behavior
of the frame in the region of these
s tor ies
he dis t r ibut ion
of
moments
in s tories
and
15 corresponds to g en era l insta
bi l i ty near the top of the
frame
under the proportional gravi ty
and wind
loads.
This ins tab i l i ty load
was
less
than
the
maxi
mum
load
capaci t ies
of
s tor ies
and
15.
Although
a
consider
able number
of
plas t ic hinges had formed
in
story 15, only one
had developed in
story
5. he
pattern
of
plas t ic
hinges was
also s l ight ly
different
in
the s tories
above and
below
s tories
and
15.
f the design of
this
frame
were
to be improved
some-
what,
the
variat ion
in
the
posit ion
of the
inf lect ion
points
could be expected to decrease. t would also be expected
from
the ~ r v o u s
discussion that
the inf lect ion points would
remain
above mid-height in each s tory .
On the basis of
th i s
brief analysis,
wil l
be assumed
in
the following development tha t the point of inf lect ion in each
column above and below
level
~
Fig.
3 a
i s
a t
mid-height
of
the story. t wil l be
shown in
Art. 2.5 tha t this assumption
is
conservative providing
the
behavior of the frame reasonably
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2.4
-39-
approximates the idea l behavior
assumed
in
Art.
2.2. Consider
again the
frame
shown in Fig. 3 a . A one-story assemblage can
now
be i sola ted from
the
frame by passing horizontal cuts
through
the
assumed
inf lect ion
points above and below level n.
The resul t ing one-story assemblage i s shown
in
Fig. 4. Also
shown
in
th is figure
are
the forces acting on the members and
the resul t ing deformations. Center-to-center beam spans are
shown as L B
e tc . where
the subscripts re fe r to the bay
in
which
the
beam
occurs.
The
lengths
of the
half-s tory
columns
above
and
below level
are
designated
h
n
_
l
/2
and h /2.
The
n
to ta l
shear between levels
n
and
n l
is
n l ·
Similar ly ,
the
to ta l
shear between levels
n
and
n+l
i s
where
1
and H i s the concentrated wind load
a t
level n. The constants
n
A
B
--- e tc . define the distr ibut ion of the
to ta l
wind
sh ea r fo rc e
to each column in a story. I t wi l l be assumed
that
t he se const an ts have the same value in
each
story above
and
be-
low level n.
Referring again
to Fig. 4,
the sidesway displace-
ment of the top of
each column above
level n re la t ive to level n
is
assumed
to be equal
to
A 1/2. Sim ilarly , the sidesway
dis
placement
of each column below level
n
is
equal
to
/2. These
n
assumptions
are
a consequence
of
the previous discussion of the
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2 .5 40
behavior
of the frame in
the
region of level
n. The
axia l forces
in
the
columns above
level
n
are
designated
as
P 1
and
are
assumed
to
remain
constant for l l values of
For
a
part icu
l r column P I s calculated as the algebraic
sum
of
the
faI-
lowing loads:
1. The t o t l g ravity
loads
coming
·from
the
t r ibutary
length
of
each
beam
connected
to
the
column
above
l v l ~
taking
into
account the l ive load reductions
3l
for
the
columns.
2. The sh ea r fo rc es t the two
column faces
which come
from
each beam
connected to the
column
above level
n.
These forces are to
be
associated
with
the
design
ultimate
wind
moments
and
the moments in each beam
which were
assumed
in the
preliminary
d
·
10
eSlgn.
The
axia l loads
P
in the columns below level
n
are
also
to be calculated in
a
similar manner but they must include
the gravity
loads
and
shear
forces
from
the
beams
t
level
n.
The analysis of the
frame t l eve l
n has now been
re
duced to the analysis of the one story assemblage shown
in
Fig. 4.
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2.5
41
t wil l be assumed th t
the load def lection
behavior of
the
one
story region of the
frame
which is symmetrical about level
n
can
be
represented
by
the
load deflection
behavior
of
this
one
story assemblage.
t
th is
point i t
will
no
longer
be
neces
sary to consider th e an aly sis
of frame
as
a whole.
Instead the
n l y t i ~ l
method
wil l
consider
only
the
load deflection
be
havior of the one story assemblage.
2.5
The
Half Story Assemblage
t
Level
n
t f e x ~ c t
analysis of the
load deflection
behavior
of
the
one story
assemblage
would
s t i l l
be a
formidable problem
e s p e c i l ~ y
i f
the
assemblage
contained many
bays.
Even
i f such
an
analysis could
be
carried
out i t would
not l ikely
be
favored
by
designers. Furthermore
application
of the resul ts
to
the
frame
i t se l f would not
be
exact but would
reflect
the approxi
mations already used in developing the o ne sto ry assemblage.
For th is
reason an
approximate analysis of the one story
assem
blage
would
be satisfactory
providing
· i t
gave
dependable
resul ts
and
was
c o n s e r v a t i v e ~ t is proposed
th t
the
sway
subassem
blage method can
be
used
for such an approximate analysis .
Consider again
the
one story
assemblage
of
beams
and
columns
t
level n
which is
shown
in Fig. 4.
ssume
ow tha t
the shear
forces tH and ~ are replaced by shear
forces
n n
~ and ~ n · P revio us ly t he s hear f or ce s ~ r i l and ~ n were
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2.5 -42-
used
in
a
res t r ic ted
sense. That is the maximum value of
for
instance,
was equal
to th e sma lles t
value of
1. The maximum horizontal shear capac it y
ins tabil i ty or mechanism load) of the
f i r s t
story
in the
frame to
fa i l
or
2.
The sum
of
th e f ac to re d
wind
loads
for
the story immediately
below level
n.
A
similar
res tr ict ion was
placed on
the maximum
value
of
~
1. The shear f or ce s and
~ Q will
not be subjected
n n
to these restr ic t ions. The maximum values of these forces will
be
l imited
only by the
maximum
horizontal shear
capacity
of
the
one-story assemblage.
I t wil l now be necessary to find the relationship be-
tween the horizontal shear
forces,
DQ and the sway
deflections,
A Before proceeding further, i t is apparent that
the
one-
story
assemblage
in Fig. may be simplified. Each column above
level
applies a vert ical force, a horizontal shear force and
a bending moment to the i ~ i n t a t the base of the
column.
The
horizontal
shear f or ce s ~ l a t each jo in t
may
be combined
with
the applied force
Q