Comparison of Analysis Procedures for Two-way slabs

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Title no.85-553

Comparison

of Analysis

Procedures

for Two-way

slabs

@ffi

by

Mary

Theresa

Cano

and

Richard

E. Klingner

Two-way

reinforced

concrete

slabs

act with

columns and walls to

Jorm

structural

systems

Jor

resisting

graviry

and loteral

loads. Cur-

rent

analysis

approaches

for

such

systems usually involve

finite

ele-

ments or

equivalent

frames

(effective

beam widths or equivalent

Jrame

properties).

Each approach

has advantages

and

disadvantages.

As currently

used, neither

is

completely

suitoble

for

anolyzing two-

way

slgQ

systems

under combined

gravity

and lateral loads.

Thi*development,

advantdges,

and limitatiorc

of each approach arg.*

discussed,

with

emphosis

on

the

equivalent

frame

approach.

AkY

equivalenl

frame

analysis

method

is

proposed

that iyvolves

explicit

modeling

of

attached

lronsverse

torsional membersQateral

deflec-

tions calculated

by various

slab onalysis

methods

aie compared with

publishedlgcerimental

results

Jor

a multistory

slab system under lot-

eral loads.'Slab

moments

calculated

by variow

slab analysis methods

ore

compared

with

each.o\her

for

idealized

flat-plate

ond two-way

slab-on-beam

structurei;lihe

explicit

transyerse

torsionol

member

method

is

found

to

give

good

results

for

drifts

and

slab-biam actibrls

and is recommended

for

analysis

and

design

of tio-way

slab systems

under

combined

gravity

and lateral

loads.

Keywords:

concrete

slabs;

lateral

pressure;

reinforced

concrete;

structural

anal-

ysis;

structural

design;

two-way

slabs.

A

common

structural

engineering

problem

is

the

de-

sign of two-way,

reinforced

concrete

slab systems-flat

plates,

flat

slabs,

and two-way

slabs

on

beams.

Starting

with

a

trial

slab

thickness

based on deflection

or

punching

shear

considerations,

the

designer

must

pro-

vide

for

satisfactory

strength

and

stiffness under com-

binations

of

gravity

and lateral

loads.

This

rbquires

that

actions within

the slab

system

be

computed.

Designers

differ

as

to

the

best procedures

for

this.l

Available

approaches

include

those

involving

finite

ele-

ments

and those

involving

equivalent

frames.

These

approaches

can

produce

widely

differing

results,

and

each has

advantages

and

disadvantages.

Thus,

it is

sometimes

difficult

for

a

designer

to select

an appro-

priate

analysis

method

and

interpret

the

results

for

de-

OBJECTIVES

AND

SCOPE

The

objectives

of this

paper

are:

l.

To review

available

analysis

approaches for

two-

way

slabs.

ACI Structural

Journal

/

November-December

1988

2. To

discuss co_mpulel:aided

m_ejhoels

based

on each

approach.

3.

To

compare

_numerr_qa les,qllq

from

each

method.

4.

To recommend

analysis

methods

for

two-way

slab

systems.

This

study

concerns

two-way

slab

systems

of

rein-

forced

concrete under

gravity

and

lateral

loads.

It

is

limited

to

analysis methods

that

are

readily

adaptable

to

computer-aided solutions.

Methods

based

on the

equivalent frame

concept

are

emphasized;

yield-line

methods2

and

strip

design

methods3

are

not

covered.

Discussion

is

confined

to obtaining

design moments;

steel

placement

and minimum

reinforcement

require-

ments

are

not

addressed.

ANALYSIS

APPROACHES

FOR TWO.WAY

SLAB

SYSTEMS

Behavior

of

two-way

slab systems

under

gravity

and

lateral

loads is complex.

Unlike

planar

frames,

in which

beam

moments

are transferred

directly

to columns,

slab

moments

are

transferred

indirectly,

due

to

the.t_ots,ion"g

fleailihly-,of

the

slab. Also,

slab

moments

from

gravity

Ioads

can

"leak"

from

loaded

to unloaded

spans;

this

must

be accounted

foi in

inatviii.

fhJ

need

to model

torsional

flexibility

and moment leakag6.

has

given

rise

to two

main

analysis approaches

for

two-way

slabs:

those

involving finite

elements,

and

those involving

equivalent

frames

(using

effective

slab

widths

or equiv-

alent frame properties).

Finite

element approach

Slab

behavioral

modes

can be modeled

directly

using

finite

element methods,

typically

involving

plate

bend-

ing

elements.a,s Because

many

elements

are usually

re-

quired

to achieve.good

results,

finite

element

ap-

proaches

are

expensive

for

large

structures.

Also,

the

_

Received-Sept.

10,

1987,

and

reviewed

under Institute

publication

policies.

Copyright

@

1988,

American

Concrete

Institute.

All righti

reserved,

iricluding

the

making

of co-pies unless.permission

is obtained

from

the

copyright

propri-

etors.

Pertinent

discussion

will

be

published

in

the

September-Octbbei

ISES

ACt

Structural

Journal if re'ceived

by

May

l,

1989.

597



ACI member

Mary

Theresa

Cano receiyed

a

BS in Architecturul

Engineering

and an

MS in

Civil

Engineering

from

the IJniversity

of

Texas

at Austin.

Ms.

Cano has worked

as

a

design engineer,

and is currently

an

engineer

wilh

the

Bridge

Division

of

the

Texos State

Department

of

Highways and

public

Trans_

portation,

Austin.

ACI

member

Richard

E. Klingner

is

an

associate

professor

o

civil

engineering,

The

University

of

Texas

at Austin.

He

is a member

of

ACI

Committees

531,

Concrete

Masonry

Struclures;

349,

Concrete Nuclear Structwes;

and

joint

ACI-

ASCE

Committee

442, Response

of

Concrete

Buildings to

Loterol Forces.

applicability

of

linear

elastic

analysis

is

questionable

when

calculated

slab

stresses exceed cracking

values.

For these reasons,

direct

use of finite element

ap-

proaches

is

not

discussed further here.

Equivalent

frame approach

To

reduce

the complexity and

cost

often

associated

with

finite

element analyses,

the

equivalent

frame

ap-

proach

can be

used

indirectly

to

compute

equivalent

beam widths

or

equivalent frame

properties.

In

this

ap-

proach,

a

three-dimensional slab

structure is

idealized

as two independent

sets

of

parallel planar

frames,

crossing each

other

(usually

at right

angles).

This

gen-

eral classification should

not

be

confused

with

the

spe-

cific

analysis method known

as

the

ACI

equivalent

frame method,

to

be discussed

later in

this

paper.

Two familiar

examples

of

the equivalent

frame

ap-

proach

are

the effective

beam

width

and the

transverse

torqional member procedures..

-

..-_

.ljfegqjve

b-eam-width

procedurA.;-The

effective

beam-width procedure

was

developed

for

analyzing

two-way

slab

systems

subjected

tojglgt4l,lggdq,and

has

been used

primarily

for flat

slabs

and fla-t

plates.

This

method

incorporates

the

effects

of

slab torsional flexi-

bility,

but,aet

_BgirrgnifQkece.

an

errectiv;

;id;h-

fac-

tor

cy

is

obtained

such

that

a slab

of

effective

width

culr,

subjected

to uniform

support

rotation

0, would

have

a

total

support

moment

equal

to

that of

the

original

slab

(width

lr,

varying

0). Once

the effective

beam

width

is

determined,

a

conventional planar

frame

analysis

is

carried

out.

Effective

beam

widths

so derived

depend

on

the

as-

sumed

stiffnesses

of

the columns

and

of

the

beam-col-

umn

connection

regions.

Typical

of

the

results

of

such

methods

are the effective

widths obtained

by Khan

and

Sbarounis.6

Though

strictly

applicable

only

to slabs

with

boundary

conditions

and cracking

consistent

with

the

assumptions

of

their

original derivations,

such

re-

sults

are

qften

used

for

a wide-r4 ge__glcases.

-Tronsverse

tirs|onat

mem_uei

iiiiii,ii.*rn"

t

urt-

verse

torsionai

mem6er

pro..dur.

was developed

fol-

lowing

extensive

testing

of two-way

slabs.T-e

Those

por-

tions

of

the

slab attached

to

the columns and trans-

verse

to

the

direction

of

the

span

(plus

the

transverse

beams,

if

any),

are

assumed

to

act as transverse

tor-

sional

members,

transferring

moments

from

slabs to

columns.

These

transverse members

are assumed

rigid

except

in

torsion.

Moment

transfer is

treated as occur-

ring directly

over

the column

width

c, and along

the

torsional

members.

The

rotational

stiffness

of the

joint

598

is

determined

as

a function

of the

torsional

stiffnesses

of

the transverse members

on

each

sid-e of the

joint

and

of

the

flexural

stiffnesses

of

the columns

above and

be-

low

the

joint.

DESIGN METHODS BASED

ON THE

TRANSVERSE TORSIONAL

MEMBER

PROCEDURE

The

transverse

torsional

member procedure

accounts

both

for

slab

torsional

flexibility

"rrd

-o.n"n'i*ffi["

and

hai

bbtir indorpoi6lefi

nto'seneiai

i[eCi]iC

d.esiin

methods.

Two

of

these

are

the ACI

equivalent

frame

method

(ACI

EFM),,o

and

the

extended

equivalent

frame

method.rr-r3

A

new method,

termed

the

explicit

transverse

torsional

member

method,ta

is

also

pre-

sented.

In

all such

methods,

member

actions

are com-

puted,

distributed

to

column

and middle

strips,

and

then

used

for

slab design.

AC equivatent

frame method

The

ACI

EFM'o

first

requires

thar the

building

be

idealized

as a series

of equivalent

planar

frames (Fig.

1).

The actual three-dimensional frame

is

assumed

to

be

composed

o{_{gb: ggm;

(horizontal

elements

with

flexural

stiffness

I(,)

supported

on an

assemblage

of

columns (vertical

elements

with

flexural

stiffness

K")

ana

qransyerse

torsional

members

(horizontal

elements

with

torsional

stiffn6ss

K).'

?iie'equivalent planar

frame

is

composed

of

slab-beams

(horizontal

elements

with

flexural

stiffness

.&',)

supported

by

equivalent

columns

(vertical

elements

with

flexural

stiffness

_r(,",

defined

as

follows)

(1/K*):

(t/DK)

+

(t/K,)

(t)

This notation

conforms

to that

of

ACI3l8-83.'o

The

flexibility

of

the

equivalent

column

is the

sum of

the

flexibilities

of the

actual columns

and

attached trans-

verse

torsional

members.

Required member

stiffnesses

K,,

K",

and

K,

are

defined

as follows.to

Torsional

member

stiffness

K,-The transverse

tor-

sional

member concept

was

first

proposed

by Corley.T

Moment

transferred

frorn slab

to

column was

origi-

nally

assumed

to

be

uniformly distributed

across

the

width

of

the slab.

Jirsa later

proposed8

a

triangular

moment

variation

(maximum

intensity

over

the

col-

umn,

zero

at each edge

of

the

equivalent

frame).

The

corresponding

torsional

stiffness

K,

was

then

obtained

by

approximate

procedurese

K,:

DgEC/lrU

-

@r/l)13

A)

For

slabs

with

beams,

K,

lBq.(2)l

is increased

by the

factor

I"u/1,, using the notation

of ACI

318-83.t0

Eq.

(2)

for

K,

was

intended

to apply

to cracked

slabse

and

was

calibrated

using

the

results

of

gravity-load

tests.e,15'r6

Column

stiffness

&-K"

is independent

of K,

and

is

calculated

conventionally, using

the actual

column

mo-

ment

of

inertia

between

the

slabs,

and

an infinite mo-

ment

of inertia within

the

slabs.

ACI

Structural

Journal

/

November-December

1g88



-n_

zl

/z

(a)

Definition

of

equivalent

frame-plan

(b)

Members

of three-dimensionol

structure,

Detait

A

(c)

Members

of equivalent

frame,

Detail

A

Fig.

|-Member

con/igurations

assumed in

ACI

equivalent

frame

method

i

z/

/2

vrcw

K

Slab

stiffness

K.-Slab

stiffness

is

calculated

conven-

including

the effects

of

column capitals

and

panels.

The moment

of

inertia

of that

portion

be-

the

ienter

of

column

and face

of column,

or

capital

is

then

incieased

by

ihe factor

l/(l

cr/lr)2,

both

to

match

test results

and to account

for

increased

flexural

stiffness

of

the

slab-column

con-

region.

For

flat

pldtes

under

uniformly

distributed gravity

slab moments calculated

by the

ACI

EFM

were

to

differ

from

measured

values

by at most

15

at

interior

columns,

but by

much more

than

at,qlleri,or

columns.e

For.two-way

slabs

on beams,

l0

to

20

percent

were observed

at

some

Despite

discrepancies

in

the

distribution

of

the

ACI

EFM provides

sufficient

flexural

to

resist

the

total

factored

static moment

for

equivalent

frame

span.

_F.g_B,a

dgsigner,s

viewpoint,

the

ACI

EFM

has

three

First,

rtime-consuming

computations

are

required

for

stiffnesses

K,,

K,,

K ,

and

K .

Structural

Journal

/

November_December

19gg

Second,

ES.

(2)

(for

the

equivalent

column

stiffness

K*)

was

developed for

gravityJoad

analyses

only.

The

ACI

EFM

can

be

applied

correctly

to lateral-load

cases

only

if

the slab-beam

stiffnesses

K,

are

reduced

for

the

effects

of cracking.to

Third,

the

ACI

EFM is

strictly

applicable

only to

single-story

substructures.

The

stiffnesses

K

of

the

equivalent

columns

above

and below

a

joint

depend

on

the

stiffnesses

of the

attached

torsional

members

fram-

ing

into

the

joint

and also on the

stiffnesses

of

the

col-

umns

above and

below

the

joint.

Based

on

such calcu-

lations,

a single stiffness

(K*),

is

assigned

to the

equiv-

alent

column at

a

given

level.

In

applying

the ACI

EFM

to multistory

structures,

however,

analogous

calcula-

tions

are

performed

for

each

level;

on the next

higher

level,

the

same

equivalent

column

can

have

a different

stiffness

(rQr.

This

discrepancy

can be

avoided

by

ap-

plying

the

ACI

EFM to

single-story

substructures

only.

Extended

equivalent

frame methods

In

rqspgqse

to

these disadvantages,

the

ACI

EFM

was

extended

by Vanderbilt

and others,

in

the

form

of

599



DISTR IBUTE

(a) Extended equivalent columnmethod (three-dimensionalmethod)

(b) Extended equivalent columnmethod using nrneuz (two-di-mensional method)

(c) Extended equivalent staLmethod (three-dimensionalmodel)

DISTRIBUTE KI

*ffic l-__&__l

[t,*Ktr]

-[^-+ftJIG, ILEF;

(d) Extended equivalent slabmethod using nrn.e,tta (two-di-mensional model)

FLEXURALMEMBER

TORSIONALMEMBERS(e) Special member

used in EFRAME

Fig. 2-Extended equvalent frame methods (Vonderbilttr)

the extended equivalent column K"" and. extendedequivalent slab K* methoals.il-r3

The extended equivalent column method tFig. 2(a)luses conventional beam elements (no attached torsionalmembers at ends). Column elements incorporate theflexural flexibilities of the columns, plus the torsionalflexibilities of the attached torsional members [Fig.2(e)1, distributed to column elements above and belowthe joint in proportion to the flexural stiffnesses of

those columns[Fig. 2(b)].Tfre equivaient slab method IFig. 2(c)l uses conven_ tional column elements (no attached torsional membersat ends). Slab-beam elements incorporate the flexuralflexibility of the beams and slabs, plus the torsionalflexibilities of the attached torsional members [Fig.2(e)1, distributed to slab-beam elements on each siOe ofthe joint in proportion to the flexural stiffnesses ofthose slab-beams [Fig. 2(d)].

Both methods reduce the slab system to a planarframe which is then analyzed conventionally. Both in_ clude the effects of slab torsional flexibility under lat_ eral and gravity loads. Because the equivalent

slab600

method cannot reproduce the effects of moment leak_ age under gravity loads, it should be used for cases in_ volving lateral loads only.tt

While removing some of the disadvantages of theACI EFM, these methods require a special computerprogramrT to handle the equivalent beam and slab ele_ ments. Also, hand computation is required to distrib_ ute the torsional member flexibilities to the columnsabove and below a joint, or to slab-beams on either side

of a joint.

Explicit transverse torsional member methodTo eliminate these drawbacks, a modification of the

preceding models, termed the explicit transverse tor_ sional member method,ra is proposed. As shown in Fig.3, conventional columns are connected indirectly bytwo conventional slab-beam elements, each with halfthe stiffness of the actual slab_beam. The indirect con_ ggUqll, made using explicit transverse Gi"fi;;"r_ bers, permits the mo-deling of moment feaEag_e qllgellqq_ql9b-loilio-riitnixuiliji.wnilJtneresurtinsf rameis

;onpi#;; thiils-n6t a ierious comptication. Speciat_ ACI Structural Journal / November-December 19gg



purpose

programs

such

as ETAqs,18

available

in

micro_

computer

as

well

as mainfiame

versions,

are widely

used

for

analyzing

three-dimensional

structures.

Be-

cause

the

transverse

torsional

members

are

present

only

in

the analytical

model,

their lengths

are

arbitrary,

pro-

vided

that their

torsional

stiffnesses are

consistently

de-

fined,

as

explained later.

The

explicit

transverse

torsional

member method

has

several

advantages.

Structural modeling

is

simple

and

direct, requiring very few

hand computations. Also,

computed

member

actions

in the

slab-beams and

trans-

verse

torsional

members

can

be used directly for

design

of

slabs

and spandrels,

respectively.

Finally,

this

method

can

even

be used

for

true

three-dimensional

analysis

of

slab

systems

under combined

gravity

and

lateral

loads.

Two

sets

of

equivalent frames,

each

run-

ning

parallel

to

one

of

the building,s

two

principal

plan

orientations,

can

be combined

to form

a

single

three-

dimensional

model.

This

single

model

can be used to

calculate

actions

in

all

members

(slabs,

columns,

and

spandrels)

under

as many

combinations

of

gravity

and

lateral

loads

as

desired.

COMPUTER

APPLICATION

OF

SLAB

ANALYSIS

METHODS

Computer

application

of the

preceding

methods

re-

quires

that

each

equivalent

frame's

geometry,

material

characteristics,

and member properties

be

defined.

For

all

except

the explicit

transverse

torsional

member

method,

a

planar

frame

is used.

The

secant

modulus

of

concrete

is usually

computed

according

to

ACI

318-

83.t0

Procedures

for

calculating

member

properties

are

summarized

in Tables

I

through

3

and

are

discussed in

the

following.

EXPLICIT

TORSIONAL

MEMBERS

Fig.

3-Three-dimensional

model

of

equivolent

frame

using

explicit transyerse

torsional

member

methodta

Effective width

method

For

computer

analysis,

column stiffnesses

are com-

puted

conventionally and

beam

stiffnesses

are

com-

puted

using the effective width /r. Cracking,

if

present,

should generally be accounted

for

separately.

Most

ef-

fective

width methods

do not

address

this isiue.

In

this

faper,

additional

stiffness

reduction factors

of 0.33 and

0.70 were used

for

flat

plates

and two-way

slabs

on

beams, respectively.

ACI

equivatent frame method

For

computer analysis, the

equivalent

stiffness K

of

each

column is

computed

from

the

joints

at

each

end of

the column. The

column inertia

is

set so

that

the col-

umn's rotational

end stiffness

coefficient

4EJ /h

equals

the

average of

the

two

Ku

values.

Because

r( is

strictly

the

stiffness

of

a

joint

rather than a column, this

pro-

Table

1-Modeling

idealizations

used in

computer.aided analysis

methods

for

two.way

stabs-

Slab

analysis

method

of

member

SIab-beams

Columns

Attached

torsional

members

Effective

width

method

(Khan

and

Sbarounis6)

Modeled

with

effective

width

factor

cu

Modeled

normally

Not

modeled

ACI

equivalent

frame

method8

Modeled

normally

Column

stiffness is

com-

bined

with

attached

tor-

sional member

stiffness

to

give

equivalent

column

stiffness.

If

column

stiff-

ness

differs

as

determined

from

each

end,

average

properties

are used

Attached

torsional

mem-

ber

stiffness

is

combined

with

column stiffness

to

give

equivalent

column

stiffness

Extended

equivalent

column

method

(Vanderbilt,,)

Modeled

normally

Special

column

element

with

ends

incorporating

torsional flexibilities

of

attached

torsional

mem-

bers

Torsional stiffness

of

at-

tached

torsional

member

is modeled

using special

column

elements

with

torsionally flexible

ends

Extended

equivalent

slab

method

(Vanderbilt )

Special beam

elements

with

ends

incorporating

torsional

fl

exibilities

of

attached

torsional

members

Explicit transverse

torsional member

method'4

Modeled

normally

Modeled

normally

Modeled

normally

ACI

Structural Journal

November-December

1 988



*Does

not exhibit moment

leakage

under

gravity

loads.

I

a

=

0.44

for flat

plate

example

of

this

paper.

I

Not recommended

for

gravityJoad

analyses

using

[

<

Iu.

Table

3

-

Computation

of

member

properties

for

analyses

of

two.way

slabs

on beams

*Does

not exhibit moment

leakage

under

gravity

loads.

I

I"nt

:

0,42 Ib

for

two-way

slab

on beami example

of

this

paper,

I

I*,

:

0.70 Io for

two-way

slab

on

beams

example

of this

paper.

r

Not recommended for graviryJoad

analyses using

.{j

< 1r.-

cedure

cannot

be applied

consistently

to

multistory

structures.

Other required

member

properties

are the

beam

area

A,

length

Z,

shear

area

Au, and moment

of

inertia

1u.

Beam

properties

are

based

on the

gross

sec-

tion,

using

the full

slab

width

/r.

Effects

of cracking

need

not

be

considered

explicitly for

gravity

loads,

be-

cause the formula

for K""

was calibrated using

cracked

test

specimens.T,e

However,

this

may

not be sufficient

for

a

gravity-loaded

slab

previously

cracked by lateral

loads.

For

lateral load

cases,

slab-beam

cracking

should

be considered

by reducing

the slab-beam's moment of

inertia

by

a reduction

factor

(usually

0.25

to

0.33

for

flat

slabs

or flat

plates'o).

Extended

equivalent

column

methodll

For

gravity

loads,

gross

member

properties

are

used

as in

the

preceding.

Transverse

torsional

member

stiff-

nesses

are distributed

to special

torsional

end members

above

and below

each

joint

in

proportion

to

relative

column

stiffnesses. This

process

can

be carried

out

au-

tomatically by the computer

program

EFRAME.T'7

Re-

duction of slab-beam stiffnesses

for'gravity-load cases

is

not

recommendedlt

because

it results

in

erroneous

moments at exterior

columns.

For

lateral

load cases,

602

the slab-beam

stiffness should

be

multiplied by

a re-

duction factor

of 0.33

for

flat slabs or flat

plates.r0'rr

Extended equivalent slab methodir

As noted, this model should not

be

used

for

gravity

loads.tr

Gross member

properties

are used

as

in the

preceding.

Transverse torsional

member

stiffnesses

are

distributed

to special

torsional

end members

on

each

side of

a

joint

in

proportion

to relative

slab-beam

stiff-

nesses. When lateral

loads

are involved,

the

slab-beam

stiffness should be

multiplied

by a

reduction

factor

of

0.33

for

flat

slabs

or flat

plates.to,tt

Explicit

transverse

torsional

member

method

Gross member properties

are

used

for

slab_beams

and columns.

Area,

moment

of

inertia,

and

shear area

are

calculated

conventionally.

For

computer

input,

the

torsional

stiffness

,I

of

the

transverse

torsional mem_

bers is calculated

by

the

following

procedure

K,

=

D9Ec/lrll-(cr/tr)13

[Eq.

(2),

repeated]

. Using

an

arbitrary

length

Z

for

the

torsional

members,

ACI

Structural

Journal

/

November_December

lggg

Table

2-computation

of member

properties

for analyses

of

flat

plates

Method

GravityJoad

analvsis

LateralJoad

analysis

Uncracked

,

Cracked

Uncracked

Cracked

Effective

width

(Khan

and

Sbarounisu)

Should not

be used*

Should

not be

used*

May

be used

I

--

ot r|

May

be used

Ij

--

0.33 a

lul

Extended equivalent

column

method

(Vanderbilt")

May

be

used

Il=1"

Should not

be

usedl

May

be used

Il=Io

May be used

ri,

:

0.33

Io

Extended

equivalent

slab method

(Vanderbiltrr)

Should not

be used"

Should not

be

used"'t

May

be used

Il=Io

May be used

Il,

=

0.33 I'

Explicit

transverse

torsional member

methodr4

May

be used

Il=ro

K,'

=

K,

May

be

used

r;

--

0.33

rb

Ki

=

0.33 K,

May

be used

Il=Io

K,'

=

K,

May

be used

I;

--

0.33

r"

K:

--

0.33 K,

Method GravityJoad

analysis

Lateral-load

analysis

Uncracked

Cracked

Uncracked

Cracked

Effective width

(ACI

T-beam width'o)

Should

not

be

used*

Should

not

be

used*

May

be used

Il=Io

May

be used

Il

=

I,nf

Extended

equivalent

column

method

(Vanderbilt")

May

be

used

Il

--

Io

Ii

=L

May

be

used

Il

=

I*,'

I

=

0.75

I,

May

be used

II

:

IO

l--L

May

be used

Il

=

Iq|

I

=

0.75

I"

Extended equivalent

slab

method

(Vanderbiltrr)

May

be used

Ii=1"

Should

not

be

used"'5

May

be used

Il=Io

Should not

be

useds

Explicit

transverse

torsional member

method'a

May

be used

Il=Io

Ki

__

K,

r =L

May

be

used

Il

:

I"r,t

Ki

=

o.33

K,

|

=

0.'1s

L

May

be used

Il

--

Iu

Ki=K,

I::1"

May

be used

Il

=

I"tt.t

Kl

=

0.33

K,

I

=

0.75

I,



c

2'-

9"

l'-9"

Tv p.

1

FLOOR 7'

x 6'-0"

x t7a"

(

typicol

)

Fig.

4-NRC

model

used

for

drift

comporisons

the

torsional

stiffness

,I,

of

each

is

then

calculated

that

K,i:

J;G/L

Therefore

J.

:

K,;

L/G

Fig.

S-Loteral

drift of

transverse

frome,

NRC

model

was

not

available,

the

extended

equivalent

column

and

extended

equivalent

slab methods

were

implemented

using

the

ETABS

program.

Calculated

results,o

using

ETABS

were

within

5

percent

of

published

results,

ob_

tained

by Vanderbilt

using

EFRAME,,T

for

the'same

structure.

Shearing

deformations

were

neglected

throughout.

Computation

of

member

propertie-s

is

de_

scribed

in

detail

in

Appendix

B

of

Reference

14.

The results,

shown

in

Fig.

5,

indicate

that

for

this

example

the

e

ff

ectivg-.yi"*

[r.

-g-re1,

o d

gi

ves

very

.accur

ate

lalelal

dri_fls,

'ffililtTfiiriee

other

rnlit

"ar,

giving

very

slmllar

answers,

all

overestimate

drift

by

as much

as 20

percent.

In

evaluating

these

results,

it

should

be noted

that

only the

effective

width

method

was

developed

as_

suming

an uncracked

slab.

A

typical

slab

structuie

would

be more

likely

to

have

some

cracking,

and its

lateral

drifts

would

be

closer

to

the values

;ffiffi;

;;

either

of the

equivalent

frame

methods.

ihe

effective

widih

method

and

the

explicit

transverse

torsional

member

method

were

judged

much

easier

to

implement

than either

of

the

extended

equivalent

frame

methods.

While

this

observation

probably

would

have

changed

slightly

had

the

EFRAME

program

been

available,

it

is

advantageous

for

a method

to

require

only

standard

analysis

programs.

COMPARISON

OF

DIFFERENT

ANALYSIS

METHODS:

IDEALIZED

FLAT.PLATE

To

compare

the

accuracy

and

convenience

of

differ-

ent

slab

analysis

methods,

the

same

four

methods

were

used

to

compute

slab

moments

in

an idealized

two_story

flat-plate

frame:

a.

Effective wldth

method (Khan and

Sbarounis)6

603

Meosured

v

I

*

Iv

v

o.oo5

0.o

I

DRTFT

(in)

Comouled

EFE'WIDTH

(o:

O.5l)

EXPI-ICIT.

Kec,

ETABS

Kes,

ETABS

o

,ors

(3)

(4)

Effects

of

cracking

need

not

be

considered

explicitly

for

gravity

loads,

because

the

ACI

expression

for

K,

is

con_

sistent

with

some

cracking.T,e

However,

this may

not

be

sufficient

for

a

gravity-loaded

slab previously

cracked

by lateral

loads.

When

Iateral

loads

are

present,

sla6_

beam

cracking

should

be considerea

Uy

muttiptying

the

slab-beam's

moment

of

inertia

by.

a

reduction

factor

of

0.33

for

flat

slabs

or flat

plates.'o

Comparison

of different

methods:

Lateral

drift

calculation

To

compare

the

accuracy

and

ease

of

use

of

the

preceding

methods,

each was

used

to

compute

the

de_

flections

of

a

small-scale,

multistory

flat

plate

test

specimen,re,2o

cited

also

by

Vanderbilt.i

This

specimen,

tested

under the

auspices

of

the Canadian

National

Re_

yargh

Council (NRC),

is

referred

to

here

as

the

NRC

Model,

and

is

shown

in Fig.

4.

Measured

lateral

deflec_

tions

of

a transverse

frame

of

the

NRC

model

were

compared

with

those

computed

by

the

following

four

methods:

a.

Effective

width

method (Khan

and

Sbarounis)6

b.

Explicit

transverse

torsional

member

methodra

c..

Extended

equivalent

column

method (Vanderbilt)lr

d,

Extended

equivalent

slab

method (Vanderbill;rr

-

Because

cracking

was

not

observed

in

the

NRC

Model,2o

all member properties

were

calculated

neglect_

ing

cracking.

Since

the

computer program

EFRAMET,T?

ACI

Structural

Journal

/ November-December

lggg



INTERIOR

EOUIVALENT

Slob thickness 8"

All

col.

24"x24"

Fig.

6-Three-dimensional

view

of

ideolized two-story

flot-plate

example

b. Explicit

transverse

torsional

member

methodra

c. Extended

equivalent column method

(Vanderbilt)"

d. Extended

equivalent

slab

method

(Vanderbilt)"

As

shown

in Fig.

6, the idealized

example

frame

has

20-ftbay

widths,

a

uniform

l2-ft.

story

height, and24-

in.

square

columns. The

slab thickness

of 8

in.

was

se-

Iected

based

on the

shear and moment

transfer

and

de-

flection

provisions

of ACI

318-83,'0

assuming a

dead

load

of

self-weight

plus

15

lb/ft2,

and a live load

of 50

lb,/ft'?.

The frame was

analyzed

for

gravity

loads

and

also for

lateral loads

of 20lb/ft'z.

Member

properties

were

calculated

as

shown in Table 2.

Joints were

con-

sidered

ri

gid,

sh

e

aring

4eloflgalLb

B *y-e,f

9 t9g 99q94,

and member

actions were

computed at member

faces.

As

noted

previously,

the ef_fectjve

wid_1h and

9a

_ en{g{

equivalent

glab

m91 q{s_.

cannot model moment

leakage,

and hence

should not

be used for

gravity-load

analyses.

In

analyzing slab systems

for

combined

grav-

ity

and lateral loads,

the

preceding

two

methods can

only

be used

if

the

analysis

is

split into two

parts:

either

of the

preceding

two methods

is

used

for

the lateral-

load

portion

of the

analysis, and other

methods

(such

as

the extended_equjvalent

qolumn

or

the explicit trans-

verse

torsional

member methods)

are used

for

the grav-

ityJoad

portion.

The

results are

then combined man-

ually.

This

process

is referred

to

here

as

a

"two-model"

analysis.

The

other

two methods

(extended

equivalent

column

and explicit

transverse

torsional member meth-

ods) can

be

used

for

gravityJoad

as

well

as

lateral-load

.analyses.

Using

a

single model,

results

for

different

load

cases

can be computed

and combined automati-

cally.

Slab moment

results for

gravity

loading

Results

for

gravity

loading

are shown

in Table 4. The

extended

equivalent

column method, which

is.almost

identical

to

the

ACI

EFM,rt

is

used

as

the

standard

of

comparison

for

gravity

loading. For gravity

loading

of

an

uncracked structure,

the effective

width

method

and

the extended

equivalent

slab method give

expectedly

poor

results.

The

extended equivalent

column method

and the

explicit

transverse

torsional

member method

give good

results.

To u_sq

q

$Letj

flq :p_ el_e-q94_elfor ggavity

g ygll.as

laleral loading,

the

slabs

shquld

be

qraq[ed.

However,

when

the

slabs alone

are

cracked,

both

extended equiv-

Table

4

-

Comparison

of slab

moments,

idealized

flat

plate

example

Load

case and analysis

method

Slab

moments

at Level

1,

kip-in.

Exterior Interior

Interior

Exterior

column

column column

column

Gravity

pattern

loads, uncracked

(1.4D

+ l.7L)

Effective width

methodu

Extended

equivalent

column

method'l

Extended

equivalent slab method"

Explicit

transverse torsional mbmber

method,o

1629

850

lt97

854

r699

1980

665

1995

1675

t'176

762

1790

Cravity

pattern

loads, cracked slabs

Effective width

method"

Extended equivalent

column method"

-Extended

equivalent slab method'r

Explicit

transverse torsional member

method'o

661

292

553

299.

683

866

t46

881

6?5

700

257

7t3

Gravity

pattern

loads, crack slabs,

and torsional

members

Explicit

transverse torsional

member

method,ra

Ki

:

0.33 K,

882

2041 1787

Lateral loads,

cracked slabs

Effective width

methodu

Extended

equivalent column methodl

Extended equivalent

slab method'l

Explicit

transverse

torsional member

method,la

Ki

=

K,

-

3'1

-50

-49

-49

3'7

45

43

44

-37

_40

-40

-39

37

40

40

39

37

50

49

49

-37

-45

-43

-44

Lateral

loads, cracked slabs

+ torsional

members

Explicit

transverse

torsional member

method,'o K:

:

0.33 K,

-38

31

-23

23

-31

-16

Combined loads

0.75

(1.4D

+

l.7L

+ l.7W)

Two-model

method

(Extended

column

uncracked

+ effective

width

cracked)

Extended

equivalent column method,

cracked

Explicit

transverse

torsional member

method,'o

K,'

:

0.33 K,

52'1

882

571

I

550

t446

t534

t268

1396

123'7

1316

1318

1365

1420

6s9

t35't

982

14'71

647

604

ACI

Structural

Journal

/ November-December

1988



alent

frame

methods

and

the

explicit

transverse

tor_

sional

member

method give

erroneous

results

fo1

grav;

ity-load

cas-es.

As

shown

in

Table

4,

decreasinl

the

slab-beamstif

f

nes5_i4qgagqstherl"b-";;*;;;:

p

o it

s,

lelh

er

t.

hAqtqigAfqll

eA

_4,rt,

;r

a

u

e

ex

_

p6ctEd-ihe-rea;n-iilahE

is

ifiaTiri"

.;iffi*

"*

onliiictrid

to

the

torsional

members

rather

than

to

the

slab-beams.

When

the

latter

are

made

more

flexible,

the

increased

relative

stiffness

of the

torsiorrt-;;;;;;$ii

causes

support moments

to

increase.il,ra

The

solution

to

this

problem

is

to include

the

effects

of

cracking

in

the

transverse

torsional

members

as

well

as

the

slabs.

This

modification

is

easy

to

carry

out with

the

explicit

transverse

torsional

*.-L.,

method.

Table

4

shows

the results

for

Ki

:

0.33

K,.

-Bggu-lts

are

close

to

those

of

the extended

equivalent

cotrrr--metnoO

Cuncraakeil

case)

:'

'

-

Slab

moment

results

for

laterat

loading

Lateral

loading

results

are also

shown

in

Table

4.

When

slabs

alone

are

cracked,

all four

meth.ods give

similar

results.

When

transverse

torsional

members

are

also

crackedln

the

case

of

the explicit

transverse

tor_

sional

member

method,

moments

are

decreased

slightly.

Slab

moment

resutts

for

combined

loading

(gravity

plus

laterat)

Combined

loading

results

are also

shown

in

Table

4.

Using

the

"two-model',

procedure,

gravity_load

mo_

ments

calculated

using

the

extended

equivalent

column

method (uncracked

members)

are

combined

with

lat_

eral-load

moments

calculated

using

the

extended

equiv8

alen _

qla _me1ho,{

(crackbd

slabs).

Using

tfre

t,single_

model"'procedure,

gravity-

and iateral_ltad

moments

are

calculated

using

the

extended

equivalent

column

method,

as

well

as

the

explicit

transverse

torsional

member

method.

Cracking

is

taken

into

account

for

the

slab-beams

in

each

case.(Ii

=

0.33 1r).

Because

the

ex_

tended

equivalent

column

rirettrOa

does

not permit

easy

incorporation

of

cracking

of

the torsional

members,

ii

gives

erroneous

results

for

the combined

load

case.

The

explicit

transverse

torsional

riremlier

method,

on

the

.

other

hand, gives

results

close

to

those

of

the

two_

model

method.

COMPARISON

OF

DIFFERENT

ANALYSIS

METHODS:

tDEAL|ZED

TWO.WAY

SLAB

ON

BEAMS

To

compare

the

accuracy

and

convenience

of

differ_

ent

slab

analysis

methods,

the

four

methods

were

used

to

compute

slab-beam

moments

in

an idealized

two_

story

frame

made

up

of

two-way

slabs

on

beams:

a. Effective

width

method

(ACI

effective

width for

T-beams).Io

b.

Explicit

transverse

torsional

member

method.ra

-

.-c.

Extended

equivalent

column

method (Vander_

bilt).'r

d.

Extended

equivalent

slab

method (Vanderbilt).il

-

As

shown

in

Fig.

7,

the

idealized

frame

has

two_way

.sl1b1

on

beams,

20-ft

spans,

a uniform

l2_ft

story

height,

and

l6-in. sqrar.

columns.

The

slab

thickness

ACI

Structural

Journal

/

November_December

lggg

Fig.

7-Plan

view

of

idealized

two-story,

two_woy

slab

with

beams

example

of

6

in. was

selected

based

on

the deflection

provisions

of ACI

318-83,'0

assuming

a

dead

load

of

self_weight

plus

15

lb/ft2,

and

a live

load

of

SO

lb/ftr.

The

frame

was

analyzed

for

gravity

loads,

afid

also

for

lateral

Ioads

of 20

lb/ftr.

Joints

were

considered

rigid,

shear_

ing

deformations

were

neglected,

and

member

actions

were

computed at member

faces.

The

frames

were

analyzed

using

both

two_model

and

single-model

procedures.

For

the

effective

width

method,

widths

of

slab-beam

members

weiil

as

ildfined

bt

tfie

T-beam

width provisions

of

ACI

3t8-g3

(Refer_

ence 10).

Effects

of

slab-beam

cracking

were

accounted

for

using

an

average

effective

moment

of

inertia,

equal

in

this

case

to

abofi

0.42

times

the

gross

inertia.,.

For

the

qlh.qr-fngtg4s,

slab-beams

and

transverse

torsional

members

were

as

defined

by

the

ACI

equivalent

frame

method.10

Effects

of

slab-beam

cracking

were

ac_

counted

for

using

an

average

effective

moment

of in_

ertia,

equal

in

this

case

to

about

0.70 times

the

gross

inertia.ta

Column

stiffnesses

were

calculated

using

the

gross

inertia,

multiplied

by

a reduction

factor

equal

to

0.75

(Reference

l0).

procedures

for

calculating

mem_

ber properties

are

summarized,

in

Table

3.

Slab.beam

momenl

results

for

gravity

loading

These

results

are shown

in

Table

5.

The

extended

equivalent

column

method (uncracked

case),

almost

identical

to

the

ACI

EFM,I

is

used

as

the

standard

for

comparison

for

gravity

loading.

For gravity

loading

of

the uncracked

structure,

the

extended

equivalent

slab

method

gives expectedly

poor

results,

,hil"

th.

,*_

tended

equivalent

column

method

and

the

explicit

transverse

torsional

member

method give

almost

iden_

tical

results.

When

the

slab-beams

alone

were

cracked,

the

ACI

effective

width

method

and the

extended

equivalent

slab

method

were

much

less

accurate

than

the

other

methods.

Both

the,extended

equivalent

column

method

and

the

explicit

transverse

torsional

member

method

gave

slightly

high

results,

although

not

as far

off

as

those

of

the

preceding

flat-plate

example.

As

before,

this

problem

was resolved

by

using

thi

explicit

trans_

verse

torsional

member

method

with

cracked

torsional

members as

well

as

slab-beams.

Results

are very

close

605

4

IO x

2O

spondrels

l2

x 2O girders

lnterior

equivolent

frome

'

Slory height

=

l2'

Slob

lhickness

=

6"

All columns,

16"x

16"

l.2ao"

+

2+o"

l.

2ao"

+



I1l9J-_-comparison

ol

srab-bearn

moments,

ideatized

two.way

srab

on

Deams

example

to

those

given

by the

extended

equivalent

column

method (uncracked

case).

Slab-beam

moment

results

for

laterat loading

These

are also given

in

Table 5.

All

methods

ixcept

the

ACI

effective

width

method

give

satisfactory

re-

sults.

Unlike

the

preceding

flat-plate

example,

torsional

member cracking using the

explicit

transverse

torsional

member

method

does

not significantly

decrease

slab-

beam

moments

under lateral

load.

Slab.beam

moment

results for

combined loading

(gravity

plus

lateral)

These

results

are also

shown

in Table

5.

Using

the

"two-model"

procedure, gravity-load

moments

calcu-

lated using

the

extended

equivalent column

method (no

cracking)

are combined

with

lateral-load

moments

cal-

c u

I

at

ed u

s

i n

g

t h e ex.t

e

n

{

e-{

e

qqi_vp _e_n

-

Clab

_ir1g Lq*4'j

(cracked

slab-beams

and coiumns).

Using

tha

;;sin;i;-

model"

procedure,

gravity-

and lateral-load

moments

are calculated using

the

extended

equivalent column

method,

as

well

as the

explicit

transverse

torsional

member

method.

Slab-beam

and column cracking

are

considered

identicalty

in

both

cases,

and torsional

member

cracking

is considered

in

the

Iatter method.

As

shown

in Table

5, all

three methods give

acceptable re-

sults. The

single-model

procedures

are

much more

con-

venient.

SUMMARY

This

report

has focused

on the

following

analysis

methods

for

two-way

reinforced

concrete

slabs:

606

1.. Effective

width

methods,

exemplified

by

the

method

of Khan

and Sbarounis.6

2.

Transverse

torsional

member methods.

a.

ACI

equivalent frame

method.to

b.

Extended

equivalent

slab

method

(Vanderbiltlr).

c.

Extended

equivalent

column

method

(Vanderbiltrr).

d. Explicit

transverse

torsional

member

metfod.ra

The

ACI

equivalent

frame method

was

developed

and

calibrated

for

single-story

substructures

under

gravity

loads

and

cannot be

consistently

applied to

multistory

structures.,

In the

extended

equivalent col-

umn

and

extended

equivalent

slab niethods,

the

tor-

sional

stiffnesses

K,

of

the

transverse

torsional

mem-

bers

are

distributed

between

adjoining

columns

or slab-

beams,

respectively.

Both

methods

are used

with

a

spe-

cialized

computer

program.rT

The

explicit

transverse

torsional

member

method

accounts

directly

for

mo-

ment

leakage

and

slab

torsional

flexibility.

It

requires

only

conventional computer programs and can

be

spe-

cialized

to either

of

the extended methods.

Lateral

deflections

computed

by all

methods

were

checked

against results

obtained

from

an

uncracked,

scale-model,

flat-plate

structure.re,20

While

results

from

the effective

width

method

were

most

accurate, the

slightly

greater

drifts

predicted

by the

other

methods

might

be

more reasonable

in

a real

structure

with

some

cracking.

All

methods were considered

to

give

satisfac-

tory

deflection

calculations.

The

preceding

methods were

also compared

with

re-

spect

to

accuracy

and

convenience in

calculating

mo-

ACI

Structural

Journal

/

November-December

19gB

Load

case

and

analysis

method

Slab-beam

moments

at

Level

l, kip-in.

Exterior

Int.rioffi

column

column

column

column

Gravity

pattern

loads, uncracked

(1.4D

+

lr7L\

ACI

effective

T-beam

width

meihod,o.r4

'

Extended

equivalent column

methodri

Extended

equivalent

slab method',

Explicit

transverse

torsional

member

method,n

1457

196'7

948

20s7

1083 1584

952

20',12

814

881

513

895

Gravity

pattern

loads, cracked

slabs

ACI

effective T-beam

width

method,o',.

Extended equivalent column method,,

Extended

equivalent

slab method,,

Explicit

transverse

torsional member

method14

1591

99s

1 103

1002

1908

2065

t'709

2080

805

8'74

605

889

Gravity

pattern

loads, cracked

slabs

+ torsional

members

Explicit

transverse

torsional

member

method,'o

K,'

=

0.70 K,

962 2089

r

896

Lateral

loads,

cracked slab-beams

+

columns

ACI

effective

T-beam

width

methodr0.,a

Extended equivalent

column

method,,

Extended

equivalent

slab method',

Explicit

transverse

torsional

member

method,'a

Ki

:

K,

-

105

100

-164

133

-

163

129

-164

133

-95

95

-

l0l

l0l

-

104

104

-

10t

101

-

100

-

133

-

129

-

133

105

t64

163

164

Lateral

loads, cracked

slabs

+ columns

+

torsional

members

Explicit

transverse

torsional

member

method,'a Ki

=

0.7O K,

163

131

-99

99

163

31

Combined

loads 0.75

(1.4D

+

t.7L

+

l.7Wl

Two-model

method

(extended

column

unoacked

+

effective

width

cracked)

Extended

equivalent column

method,'ciacked

Explicit

transverse

torsional

member

method,'o

K,'

:

0,70

K,

606

583

559

1643

r683

t700

l3r6

1506

130'7

1509

1324

1523

1443

817

t418

9lt

1437

885





l

ii

ir

ii

ri

ii

REFERENCES

l. Design

of

Reinforced

Concrete

Slabs '

questionnaire

distrib-

uted

by

ACI-ASCE

Committee

421'

1986'

i.iuri,

Robert,

and Gamble,

William

L

Reinforced

Concrete

Slabs,

John

Wiley

&

Sons,

New

York,

1980'

pp'

274-464'

3. Hillerborg,

Arne,

S/rrp

Method

of

Design'

Cement

and

Con-

crete

Association,

Wexham

Springs,

191 5'

256

pp'

4.

Pecknold,

David

A., Slab

Effective

Width

for Equivalent

Frame

Analysis,

ACI

JouRNAL,

Proceedings

V' 72'

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4 Apr'

1sls,

pp.

tl\-tsl

.

Also,

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Allen'

P'

leP' Darvall'

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g.

Ctover,

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D.

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10'

Oct'

1975,

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583-586.

S.

Btias,

ZiadM., Lateral

Stiffness

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Plate

Structures '

ACI

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80,

No.

1,

Jan'-Feb'

1983,

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50-54'

6. Khan,

Fazlur

R',

and

Sbarounis,

John

A', Interaction

of

Shear

Walls and

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V'

90'

ST3'

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June

1964,

pp.

285- 335.

7.

Corley,

W.

Gene; Sozen,

Mete

A.;

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Chester

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Frame

Analysis

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Concrete

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Research

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Civil

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University

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Urbana,

June

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James

O.;

Sozen,

Mete

A.;

and

Siess,

Chester

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June

1969,

PP.

l1l7-1137'

9.

Corley,

W.

Gene,

and Jirsa,

James

O., Equivalent

Frame

:

Length

of span

transverse

to

direction

of

the span

for

which moments

are being determined,

measured

center-to-

center of

supports

(ChaPter

13)

:

Effective width factor

=

Uniform

rotation

at column,

used

in

deriving

effective

width factor o

CONVERSION

FACTORS

I

ft

=

0.305

m

I in.

=

25.4

mm

I lb/ft2

=

4.882k9/m'

I kiP

:

4'448

kN

I ksi

=

6.895

MPa

I

Psi =

0.006895

MPa

Analysis

for

Slab

Desigp.

ACI

Jcnxrr'

hoceedings

Y

'

67'

No'

1 I

'

Nov.

1970,

pp.

875-8&l-

AIso,

Discussion

by

A'

C'

Eberhardt'

E'

S'

Hoffman,

Ti Huang,

J.

C.

Jofrirr'

Y' K'

Hanson'

and

Closure'

Pro-

ceedings

V.

68,

No.

5,

May

l97l'

pp'

397-4OI'

10'ACICommittee3lE'..B'JilditrgCodeRequirementsforRein.

forced

Concrete

(ACI

318-83) '

American

Concretelnstitute'

De-

troit,

1983,

111

pp, and Commeatary

on

Building

Code

Require-

ments

for

Reinforced

Concrete

(ACI

318{3)

155

pp'

11.

Vanderbilt,

M.

D., Equivaleot

Frame

Analysis

of

Unbraced

Reinforced

concrete

Buildingi

for

Static

Lateral

Loads,

structural

Research

Reporl

No. 36,

Ci;il

Engineering

Departmenr'

Colorado

State

University, Fort Collins, Colorado, July

1981'

,

-

iZ.

Vuna.rUitt,

M.

Daniel, pquivalent

Frame

Aoalysis

for

Lateral

Loads,

Proceedings,

ASCE,

V. 105,

ST10,

Oct'

1979'

pp' 1981-

1998.

Also,

Discussion

by ZiadM.

Elias,

V.

106,

ST7,

July

1980'

pp'

167l-1672,

and Closure,

V.

107, STl,

Jan.

1981,

p'

245'

13.

Vanderbilt,

M.

Daniel,

and

Corley,

W.

Gene, Frame

Analysis

of

Concrete

Buildings,

Concrete

Internotionol:

Design

& Construc-

tion,Y.5,

No.

12, Dec.

1983,

pp.33-43.

14.

Cano,

M.

T., Comparison

of

Analysis

Procedures

for

Two-

Way Slabs,

MS

Thesis,

Depaftment

of

Civil

Engineering,

Univer-

sity

of

Texas,

Austin,

Aug.

1984.

l5. Reinforced

Concrete

Floor

Slabs

-

Research

and

Design,

Bulletin

No.

20, Reinforced

Concrete

Research

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American

Society

of Civil

Engineers,

New

York,

1918,209

pp.

16. Guralnick,

Sidney

A., and

LaFraugh,

Robert

W., Laboratory

Study

of

a

45-foot

Square

Flat

Plate Structure,

ACI

JounNel,

Pro-

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Sept'

1963,

pp.

1107-1185.

17.

EFRAME,

computer

program' National

Information

Service

in

Earthquake

Engineering,

University

of

California,

Berkeley'

18.

Wilson,

E.

L.,

Hollings,

J. P.

and

Dovey,

H'

H', Erars:

Three-Dimensional

Analysis

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Building

Systems

(Extended Ver-

sion),

computer

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in

Earth-

quake Engineering,

BerkeleY,

1977.

19.

Zelman,

Maier

I.;

Heidebrecht,

Arthur

C.;

Tso,

W'

K'; and

Johnston,

William A', Practical

Problems

and Costs

of

Fabricating

Multi-story

Models,

Models

for

Concrete Struclures,

SP-24,

Amer-

ican Concrete

Institute,

Detroit,

1970'

pp.

159-185'

20.

Hartley, G.;

Rainer,

J. H;

and Ward,

H.

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namic Properties

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Apr. 1919,24pp'

608

ACI

Structural

Journal

/

November-December

19BB

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Comparison of Analysis Procedures for Two-way slabs

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