combined load analysis of unbraced frames 1967

8/17/2019 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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http://preserve.lehigh.edu/?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F351&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%2F351&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F351&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%2F351&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%2F351&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]:[email protected]://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/351?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F351&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/351?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F351&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%2F351&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%2F351&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/engr-civil-environmental?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F351&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%2F351&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://preserve.lehigh.edu/?utm_source=preserve.lehigh.edu%2Fengr-civil-environmental-fritz-lab-reports%2F351&utm_medium=PDF&utm_campaign=PDFCoverPages


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3/156

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


4/156

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


5/156

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


6/156

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


7/156

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


8/156

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


9/156

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


10/156

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


11/156

-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.


12/156

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.


13/156

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


14/156

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


15/156

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


16/156

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 .


17/156

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


18/156


19/156

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-


20/156

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


21/156

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.


22/156

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


23/156

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


24/156

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.


25/156

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 ,


26/156

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


27/156

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-


28/156


29/156

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.


30/156

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.


31/156

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 .


32/156

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.


33/156

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


34/156

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-


35/156

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.


36/156

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


37/156


38/156

1 7

29

assumptions used in the

analysis

Chapter discusses

the fu-

ture

research

required and Chapter summarizes

the resul ts

of

this study


39/156

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 -


40/156

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


41/156

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


42/156

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


43/156

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 .


44/156

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


45/156

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


46/156

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


47/156

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


48/156

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


49/156

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.


50/156

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


51/156

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

combined load analysis of unbraced frames 1967

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