ultimate strength of reinforced concrete in american design practice

8/18/2019 ULTIMATE STRENGTH OF REINFORCED CONCRETE IN AMERICAN DESIGN PRACTICE

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PORTLAND CEMENT ASSOCIATION

RESEARCH AND DEVELOPMENT LABORATORIES

ULTIMATE STRENGTH OF

REINFORCED

CONCRETE IN

AMERICAN DESIGN PRACTICE

By Eivind Hognestad

Authorized Reprint From

Proceedings of a Symposium on the Strength of

Concrete Structures, London, May, 1956


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Bulletins Published by the

Development Department

Research and Development Division

d the

Portland Cement Association

D1

—“Influence of Soil Volume Change and Vegetation on Highway Engf.

neering,”

by E, J.

FELT.

Reprinted from Twent Sixth Annuat Highuxw Conference of the Universit?j

of Coiorado, May 12S2.

D2 -“Nature of Bond in Pre-Tensioned Prestressed Concrete,”

by

JACK R.

J ANNEY,

Reprinted from JournfIl of the American Concrete Institute (May, 1954);

proceedings, 30, 717 (12S4).

D2A—Discussion of the papw “Nature of Bond in Pre-Tensioned Prestressed

Concrete,”

by P. W.

ABELES, K. HAJNAL-KONYI, N. W. HANSON

and

Author, JACK R. JANNEY.

Reprinted from Journal of the American Concrete Institute (December,

Part 2, 1954): Proceedings, SO, 73S-1 (1254).

D3

_f ~Investigationof M~isture.Vo]ume Stability of Concrete Masonry

Units,” by

JOSEPH

J. SRJDELER,March, 1955.

D4

—“A Method for Determining the Moisture Condition of Hardened Con-

crete in Terms of Relative Humidity,”

by

CARL A. MENZEL.

~le ~ted froml Proceedings, American Soctetv For Testing M@tU’iaiS, 55

D5 —“Factors Influencing Physical Properties of Soil-Cement Mixtures,”

by EARL J.

FELT.

Reprinted from~ Bulletin 108 of the IIigfwav Research Board, p.

123 19S5).

D6 -“Concrete Stress Distribution in Ultimate Strength Design,” by E.

HOONESTAO, N. W. HANSON and D. MCHENRY.

Reprinted from JonrnaI of the American Concrete Institute (December,

1955); Proceedings, 52, 455 (19S6).

D?

-“Ultimate Flexural Strength of Prestressed and Conventionally

Rein-

forced Concrete Beams,” by J. It. JANNEY, E. HOONESTAD and D. Mc-

HENRY.

Reprinted from Journal of the American Concrete Institite (FebruaxT, 19S0;

Proceedings, S2, S01 (12SS).


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SYMPOSIUM ON THE STRENGTH OF CONCRETE STRUCTURES

LONDON MAY 1956

Sessicm E: Paper 1

ULTIMATE STRENGTH OF

REIN FORCE(D CONCRETE IN

AMERICAN IDESIGN PRACTICE

by Eivind Hognestad,

Dr. techn.

Portland Cement Association, U.S.A.

SUMJ4AR Y

Ultimate strength design procedures for reinforced concrete were recom-

mended in an October 1955 report of a joint committee of the American

Society of Civil Engineers and the American Concrete Institute. This paper

discusses the background for and contents of that report, which represents

a signj?cant stage in the development of an American design practice based

on ultimate strength by inelastic action.

Introduction

The past fifty years have been a period of rapid growth and develop-

ment in the use of reinforced concrete as a structural material throughout

the world. The production of Portland cement in the United States rose

twenty-five fold from about 2 million long tons in 1900 to over 50 million

tons in 1955. Similarly, the U.S. production of reinforcing stee[ increased

from a small amount to about 1”8 million tons.

Introduction of new design procedures for reinforced concrete must be

considered with this background of great progress and expansion. Though

the classical straight-line theor:y was evolved when reinforced concrete was

in its infancy some 60 years ago, it has served us well; and it certainly

cannot be put aside on the basis that it has led to unreasonable or unsafe

designs.

On the other hand, through half a century of practical experience and

laboratory experimentation, our knowledge regarding the strength and

behaviour of structural concrete has been vastly improved. To some extent,

such improvements of knowledge have been utilized in design practice by

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periodic adjustments and modifications of the straight-line theory. In this

manner the original sitnplicity of an elastic theory based on a few funda-

mental assumptions has largely been lost.

It is primarily to facilitate further progress, therefore, that many of us

feel that the time has come to introduce a new theory of reinforced con-

crete design based on the actual inelastic properties of concrete and steel.

Such a new theory is needed to realize the full future benefits of such

highly important developments in the field of structural concrete as high-

strength reinforcement,, prestressing, and precasting.

DEFINITIONS

In recent years good progress has been made in the development of

knowledge regarding the properties of all engineering materials. New and

improved concepts of structural behaviour and design have therefore

become significant in the practice of civil engineering. These concepts are

identified by rather reeent additions to engineering terminology such as

rheology, plasticity, inelastic behaviour, plastic analysis, limit strength,

and many others. Definitions for these terms vary to some extent between

countries as well as between groups concerned with the various materials.

It is necessary, therefore, to define common American word usage in

connexion with structural concrete design.

Ultimate strenglh design

Ultimate strength design indicates a method of structural design based

on the ultimate strength by inelastic action of conventionally reinforced or

prestressed structural concrete cross-sections subject to simple bending,

axial load, shear, bond., or combinations thereof. Ultima~e strength design

does not necessarily involve an inelastic theory of structures. Evaluation

of external moments and forces that act in indeterminate structural frame-

works by virtue of dead and live loads may be carried out either by the

theory of elastic displacements or by limit design.

Limit design

Limit design indicates a design method involving an inelastic theory of

structures in which readjustments in the relative magnitude of bending

moments at various sections due to non-linear relationships between loads

and moments at high loads are recognized. Limit design does not by

definition necessarily involve a final design of sections on an inelastic

basis.

Yield line theory

Yield line theory indicates a theory of reinforced concrete slab structures

based on inelastic behaviour occurring in a pattern of yield lines, the

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location of which depends on loading and boundary conditions. Final

design of sections does not necessarily involve inelastic action.

So far, most American work regarding inelastic behaviour of structural

concrete has been devoted to ultimate strength design. A term indicating

,a combination of ultimate strength design, limit design and yield line

theory therefore still remains tc}be adopted. Perhaps the most important

aspect of ultimate strength design is that it represents a significant step

toward a broader consideration of inelastic behaviour in design.

AMERICAN DESIGN SPECIFICATIONS

Two groups have made important contributions to the development of

reinforced concrete design specifications in the United States—the Joint

(Committees on Standard Specifications for Concrete and Reinforced

Concrete, and committees of the American Concrete Institute(1).

The Joint Committees have consisted of delegates from the American

Concrete Institute (ACI), American Institute of Architects (AIA),

American Railway Engineering Association (AREA), American Society

of Civil Engineers (ASCE), American Society for Testing Materials

(ASTM), and the Portland Cement Association (PCA). The first, second

and third Joint Committees were organized in 1904, 1919 and 1930, and

submitted final reports in 1916, 1924 and 1940 respectively. These reports,

which were milestones on the road of progress and had a strong effect

on American concrete usage, were submitted to the constituent organiza-

tions. The sections concerning reinforced concrete design were written in

the form of a recommended practice rather than a design code, so that it

was possible to give a broad reflection of the state of the art as represented

by the best practice of the day.

The first committee on reinforced concrete of the ACI, then the

National Association of Cement Users (NACU), was the Committee on

Laws and Ordinances. The first report of this committee appeared in 1909

and was essentially based on what has later become known as ultimate

strength design. The report was later revised to introduce the concepts of

the straight-line theory, allowable stresses, and service loads; and it was

then adopted as “ Standard Building Regulations for Reinforced Con-

crete “ in 1910. Later a Committee on Reinforced Concrete and Building

Laws was formed, sponsoring ‘


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numerous cities and municipalities throughout the United States. Many

agencies of the U.S. Government also refer to the ACJ Code, though

minor adjustments are often made to suit their particular needs.

In the field of bridge design and construction, specifications have been

developed and periodically revised by the American Association of State

Highway Officials and by the Ameriean Railway Engineering Association.

NOTATION

The letter symbols used are generally defined where they are first

introduced; they are also listed below for convenient reference.

Loads and load factors

B = effect of basic load consisting of dead load plus volume change

due to elastic and inelastic actions, shrinkage and temperature

E =

effect of earthquake forces

Fb

= ultimate strength for balanced condition given by equation (I 5)

FO =

ultimate strength of concentrically loaded column given by

equation (12)

FU =

ultimate strength of eccentrically loaded member

Fu’ =

maximum axial load on long member given by equation (21)

K = load factor equal to 2.0 for columns and members subject to

combined bending and axial load, and equal to 1”8 for beams

and girders subject to bending

L = effect of live load plus impact

MU = ultimate resisting moment

U =

ultimate strength capacity of section

W =

effect of wind load

Cross-sectional properties

AC =

A, ==

A,C =

A,f =

A,, =

b=

b, =

;=

D, =

d, =

gross area of concrete section

total area of IIongitudinal reinforcement

area of compressive reinfcwcement

steel area to develop compressive strength of overhanging flange

in T beams, defined by equation (11)

area of tensile reinforcement

width of a rectangular section, or overall width of flange in

T beams

width of web in T beams

n.dl = depth to neutral axis

total diameter of circular section

diameter of circle circumscribing the longitudinal reinforcement

in circular section

effective depth to centroid of tensile reinforcement

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d, =

effective depth to centroid of compressive reinforcement

e= eccentricity of axial load measured from the centroid of tensile

reinforcement

e’ = eccentricity of axial load measured from plastic centroid of

section

e~’ = eccentricity of loacl

Fb

measured from plastic centroid of section

L =

unsupported Iengtlh of an axially loaded member

nudl =

depth to neutral axis at ultimate strength

A,,

r

=

ratio of tensile reinforcement .= —

bdl

rb =-

ratio of balanced tensile reinforcement defined by equation (6)

A

r’ =

ratio of compressive reinforcement = ~

bd,

A,,

rt =

ratio of total reinforcement = —

AC

t

rW =—

b’d,

t

= flange thickness in T section, or total depth of

section

Properties of’materials

ECU =

=W =

f=

U

f

Y ‘“

k, =

k, =

mu =

mu’ =

maximum strain in concrete at ultimate strength

0“003)

strain in tensile reinforcement at ultimate strength

stress in tensile reinforcement at ultimate strength

:ctangular

limited to

yield point stress of reinforcement (limited to 60,000 lb/in2)

ratio of average compressive stress to 0.85

u.+,

at ultimate

strength

ratio of depth to resultant of compressive stress and depth to

neutral axis at ultimate strength

0“85

U y

mu—l

=

cfy

a—

ucy/

U.yl =

compressive strength of 6

x

12 in. cylinders at 28 days

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Report of ASCE-ACI Joint Committee

on Ultimate Strength Design

Advancement in the field of structural design and analysis must of

necessity proceed with extreme caution and deliberation. This has been

true of the recommendations in the report of the ASCE-ACI Joint

Committee on Ultimate Strength Design which culminates over ten years

of continuous study of the subject. The joint committee was formed as a

sub-committee of the ASCE Committee on Masonry and Reinforced

Concrete under the chairmanship of the late A, J. Boase in 1944. It

immediately commenced a comprehensive study of the adequacy of various

ultimate strength theories and design formulae. As a result of its studies,

it initiated extensive series of both short-time and sustained load tests on

eccentrically loaded columns. These tests have been completed under the

sponsorship of the Reinforced Concrete Research Council of the

Engineering Foundation.

In 1949 L. H. Corning was made chairman of the sub-committee. At

this time, the sub-committee further recommended an extensive test pro-

gramme on the shear resistance of reinforced concrete members. Extensions

of this investigation are still in progress. In 1952 the sub-committee was

made a joint committee o:f AC1 and ASCE and designated as Committee

327 by ACI.

Hand in hand with the :studies made on ultimate strength formulae, the

joint committee has investigated the question of overload factors in terms

of the practice prevailing in countries where design by ultimate strength is

in practical use, and of the factors of safety implied in conventional

straight-line design methods.

During the annual convention of ACI in 1952, the joint committee

sponsored a symposium on ultimate strength design(3). This provided an

opportunity for public discussion of such topics as reasons for changing

design method, fundamental concepts of ultimate strength design, review

of research, practical design, and overload factors.

In 1955 the committee completed its assignment “ to evaluate and

correlate theories and data bearing on ultimate strength design procedures

with a view to establishing them as accepted practice “. A final report was

submitted to ASCE and AC1 4’5). It is the principal purpose of this paper

to discuss the contents of that report, and to present the author’s opinions

and interpretations regard ing the report.

NATURE OF THE REPO RT

The joint committee report presents recommendations and formulae for

ultimate strength design (of reinforced. concrete structures together with

basic supporting and explanatory data. The report is confined to design

of cross-sections; it does not deal with evaluation of external moments

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and forces. The committee recognized limit design as important but did

not recommend practical use thereof at the present time.

The report is based on the assumption, therefore, that structural analysis

will be carried out by the theory of elastic displacements. On the basis of

this assumption stresses will remain within the elastic limits under service

loads when proper load factors are used. For statically determinate mem-

bers, the ultimate capacity equals the computed capacity. For indeter-

minate structures, it is important to note that the maximum moments at

various sections are usually due to different load arrangements. Because

of moment redistribution at high loads, therefore, the maximum load

capacity of the indetermini~te structure may considerably exceed that

indicated by the capacity at a single section. Accordingly, a combination

of ultimate strength design of sections and elastic structural analysis may

be conservative in some cases, but it is not at all unreasonable.

The joint committee report as published by ASCE(4) consists of a brief

section on historical background, and the essence of the report appears

under the heading “ Recommendations for design “ Three appendixes

deal with substantiating test data, design aids, and derivation of formulae.

The report ends with a selected bibliography. The AC I publication (sJ

does not contain the appendixes concerning test data and derivation of

formulae.

LOAD FACTORS

Consideration was given by the joint committee to the circumstance

that ultimate strength design may be carried out in two ways. Moments

and forces acting at various sections may be evaluated for service loads,

Sections may then be designed by “ deducted “ or “ allowable “ ultimate

strength equations, in which chosen safety factors are incorporated.

Another alternative is LOmultiply the, service loads by chosen load factors

before the cross-section forces are evaluated. The design of sections then

takes place by equations expressing actual ultimate strengths.

The joint committee chose to follow the second alternative, principally

because ultimate strength equations are essentially factual in nature, while

the choice of load factors to a considerable extent is a matter of engineering

judgment. By keeping load factors and strength equations separated, the

report should be conveniently useful even to specification-writing bodies

that find it necessary for special applications to change the numerical

values of the load factors recommended by the joint committee. Further-

more, it is believed to be wise for a designer clearly and unmistakably to

keep his load factors in view.

Two criteria were consic[ered as a basis for selecting load factors.

Members should be proport.ioned so that: (l) they should be capable of

carrying service loads with ,ample safety against an increase in live load

beyond that assumed in design and against other uncertainties; (2) the

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strains under service loacls should not be so large as to cause excessive

cracking. The committee found that these criteria are satisfied by the

following formulae.

(1) For structures in the design of which effects of wind and earthquake

forces can properly be neglected:

U=I”2B+2.4L . . . . . . . . . . . . . . . . . . . . . . . ...(1)

and

U= K(B -I- L), . . . . . . . . . . . . . . . . . . . . . . . . ...(2)

in which

U= ultimate strength capacity of section

B ==effect of basic loald consisting of dead load plus volume changes

dueto elastic and inelastic actions, shrinkage, and temperature

L = effect ofliveload plus impact

K = load factor equal to 2“0 for columns and members subject to

combined bending and axial load, and equal to 1”8 for beams and

girders subject to bending

(2) For those structures in which wind loading should be considered:

U~l”2B+ 2”4L+O”6W . . . . . . . . . . . . . . . . ..(la)

U=1”2B+ 0.6L+2”4W . . . . . . . . . . . . .

. . . ..(lb)

and

)

=K B+ I +; . . . . . . . . . . . . . . . . . . . .

)

==K B+:C+W . . . . . . . . . . . . . . . . . . . .

(2a)

(2b)

(3) Forstructures inthe design of which earthquake Ioading must be

considered, substitute for the effect of wind load, W, the effect of earth-

quake forces, E.

GENERAL REQUIREMENTS

The joint committee report does not deal with the many detailed require-

ments involved in reinforced concrete design and construction, such as

spacing and cover of reinforcement, and special considerations regarding

the various typical bui[ding elements. A reference is therefore made to

the ACI Building Code in all matters not otherwise provided for in the

committee report.

It is required” that bending moments should be taken into account in

calculating the ultimate strength of compression members. Analysis of

indeterminate structures should be carried out by the theory of elastic

displacements, though approximate coefficients such as those recom-

mended in the ACI code are acceptable for the usual types of buildings

In structures such as arches, the effect of shortening of the arch axis,

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temperature, shrinkage, and secondary moments due to deflexion should

be considered.

The committee report also calls attention to the need for checking

deflexion of members including effects of creep, especially for high

percentages of reinforcement.

In considering the recommended ultimate strength equations, it is im-

portant to note that the committee assumed that only controlled concrete

will be used in construction of structures designed by ultimate strength.

The quality of concrete should then be such that not more than one test

in ten has an average compressive strength less than the strength assumed

in design, and the average of any three consecutive tests should not be

less than the assumed design strength. In this manner, the design concrete

strength is not an average strength; with a reasonable probability it is a

minimum strength. Similarly, through the general reference to the AC1

code, the joint committee assumed that design values for the yield point

of reinforcing steel are minimum values. Accordingly, the ultimate strength

design equations should express an average and not a minimum relation-

ship between ultimate strengths of the various reinforced concrete members

as observed in tests and the cclrresponding compressive strengths.

BASIC ASSUMPTIONS FOR IJLTIMATE STRENGTH

After a thorough study of many ultimate strength theories presented in

Europe as well as in America, the committee recommended that the

calculation of ultimate strengtlh be based on the following assumptions.

(1) As ultimate strength is approached, stresses and strains are not

proportional, and the distribution of compressive stress in sections subject

to bending is non-linear. The diagram c~f compressive concrete stress

distribution may be assumed a rectangle, trapezoid, parabola, or any other

shape which results in ultimate strength in reasonable agreement with

comprehensive tests. In addition to this broad assumption, the joint

committee recommended a specific set of limiting equations for various

typical design cases as discussed in the following pages. These limiting

equations are in good agreement with comprehensive tests of reinforced

concrete, and calculated ultimate strengths based on a chosen stress

distribution should therefore not exceed these given limits.

(2) Plane sections normal to the axis remain plane after bending. When

deformed reinforcing bars are used,

this assumption has been verified

even for high loads by numerous tests to failure of eccentrically loaded

columns as well as of beams subject to bending only.

(3) Tensile strength in concrete is neglected in sections subject to bend-

ing. When normal percentages of reinforcement are used, this assumption

leads to results in good agreement with tests. For very small percentages

of reinforcement it is on the conservative side.

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4) Maximum concrete strain in ffexure is limited to 0.003. This is a safe

value; most strains observed in tests of reinforced concrete members fall

between 0“003 and 0“0015‘G).

(5) Maximum fibre stress is assumed not to exceed 85~0 of the com-

pressive strength of 6

x 12

in, cylinders, A maximum stress near 100~0 of

the cylinder strength has been found in tests of horizontally cast mem-

bers’7’. In vertically cast members such as columns, however, due to water

gain resulting in a lower strength near the top, and due to effects of size

and shape, a maximum stress of 8.5’~ of the cylinder strength has been

observed ‘8’9’.Since some effect of size and shape probably also is present

in large beams, and since the concrete near the top of beams as well as

columns may be somewhat weaker than control cylinders, it seems

reasonable in all cases to use an 8.50/0 stress.

(6) Stress in tensile and compressive reinforcement at ultimate strength

is assumed not to exceed the yield point of the steel used or 60,000 lb/in2,

whichever is smaller. The purpose of the 60,000 lb/in2 limit is, of course,

to avoid excessive cracking under service loads. This limit is conservative,

considering the high effectiveness of the bar deformations that are now

in use throughout our country. It is also possible, to some extent, to

control cracking by other variables than steel stress.

RECTANGULAR BEAh4S

To establish limiting equations for ultimate strength in the various

cases, the joint committee chose a theoretical approach originated by

F. Sttissi of Switzerland in 1932 and based on the general properties of

the stress distribution shown in Figure 1. The properties of the stress

l-- b l

l

h:

J

c

—.

As,

Figure 1: Flexwd analysis.

block are given by the stress factor O“85kl (German: Volligkeitsgrad) and

the centroid factor

k2

(German: Schwerpunktsbeiwert). Equilibrium of

forces and moments then gives:

-4s,j_,U=0.85 klu@c.. . . . . . . . . . . . . . . . ..(1)

,MU= 0.85kluCY@c(dl—I@ . . . . . . . . . . . . . . . . . .(2)

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When tension controls ultimate strength, the ultimate steel stress f,U

equals the yield point Y;, and the ultimate resisting moment obtained by

solving equations (1) and (2,) is given by:

/

in which

AS, =

d, =

O“85k1 =

k, =

ucy/ =

r

b=

k,

rfy

)

MU= A,lfydl I—-—— . . . . . . . . . . . . . . .

0.85k1 UCY,

}

(3)

area of tensile reinforcement

yield point stress of reinforcement (limited to 60,000 lb/in2)

effective depth

t

centroid of tensile reinforcement

stress factor, ratio of average compressive stress to uCYl

centroid factor,, ratio of depth to resultant of compressive

stress and depth to neutral axis

compressive strength of 6x 12 in. cylinders at 28 days

A,,,

ratio of reinforcement = —–

bdl

width of beam

The quantities kl and k2 are fundamental properties of concrete that

have been determined by direct tests of plain concrete specimens(T).

Equation (3) is then a fully rational equation developed by the equations

of equilibrium from measured properties of the materials steel and

concrete.

Equation (3) may also be developed on a more empirical basis by study-

ing the results of reinforced concrete beam tests. The author recently

determined the coefficient –—

~~k from published data on 364 beam tests

1

by the statistical method c)f least squares, and the value of 0“593 was

0.5

found. A value of— =

0.59 was suggested by C. S. Whitney over ten

0“85

years ago[lO), and this value is also in good agreement with the direct

tests of plain concrete(7’.

It is entirely reasonable, therefore, that the joint committee recom-

mended that the computed ultimate moment of beams should not exceed

that given by

)

W=A,,,fYdl 1-–0.59:/ . . . . . . . . . . . . . ...(4)

CY

which can be re-stated as

Mu

—=q(l-– 0.59q) . . . . . . . . . . . . . .

. . ..(4a)

z, .,,

rfy

in which q ==—.

Y

When compression controls ultimate strength, the steel stress at failure

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may be determined by considering linear distribution of strain (Figure 1):

dl — c

E

U

= Ecu-— . . . . . . . . . . . . . . . . . . . . . .(5)

c

Combining equations (1) and (5) and seeking the balanced reinforcement

ratio for which

f,. = fY, we

obtain

Ecu

U@

‘b= ”85k’k

h“”””””””’”””’”’”””

j-- +

Ecu

~s

6)

The maximum ratio of reinforcement in equation (4) should be some-

what less than the balanced ratio given by equation

(6).

Choosing a

limiting value of

r

equal to about 90’70 of

r~,

the joint committee recom-

mended that

r

should not exceed

r=o.40@’ . . . . . . . . . . . . . . . . . . . . . ...(7)

f

in which the coefficient 0.40 is to be reduced at the rate of 0“025 per

1,000 lb/in2 concrete strength in excess of 5,000 lb/inz. Such reduction

for high concrete strength is desirable on the basis of several experimental

studies that indicate a decrease of the stress factor,

0.85k,,

with increasing

concrete strength (7,11’.

When the ratio of reinforcement exceeds that given by equation (7),

compression reinforcement must be provided. For this case, the joint

committee recommended that the resisting ultimate moment should not

exceed

MU = (AS*— AJj;dl 1— 0.59(r -– r’)~

1

+ f J d,d, ) 8)

Ury(

in which (r — r’) should not exceed the value given by equation (7), and

A C=

area of compressive reinforcement

A

r’ =

ratio of compressive reinforcement = ~

bd,

d2 =

effective depth to centroid of compressive reinforcement

For beams with the usual amounts of reinforcement dictated by economy

and spacing of reinforcing bars,

r

is 0.15 to 0.25 times 4[, and there is

f,

little difference between designs resulting from ultimate strength and

straight-line procedures. The major changes suggested by the committee

therefore concern a more efficient use of reinforcement with yield points

over 40,000 lb/in*. In present American design codes based on straight-

line theory a ceiling allowable stress of 20,000 lb/inz is used for such

reinforcement, while the ultimate strength design method as outlined may

lead to the equivalent of a.n ailowable stress of 60,000/ 1.8 =. 33,300 lb/in2.

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As a second change it is made possible, when economically and practically

feasible, to utilize more fully the strength of the concrete compression

zone.

T SECTIONS

If the neutral axis falls within the flange of a T beam, the equations for

rectangular beams are applicable with r cc~mputed as for a beam with a

width equal to the overall flange width. The depth to the neutral axis, c,

may be estimated by solving equation (1):

1 r~d

c=n@~l=—’—

0“85k1uCYl 1“””””’””””””””””’”

.

.(9)

In this case, the joint committee recommended a conservative value of

c = l“30rfy4.

Ucyl

When c is greater than the depth of the flange, the tensile reinforcement,

A,,, may be considered subdivided into one part, A,f, that will develop

the compressive strength of the overhanging portion of the fldnge and

another part,

(A,f — A,f),

that will develop the compressive strength of a

a portion of the web. Assuming a uniform stress of 0“85uCYZn the flange,

the joint committee recommended:

[

1

u=A, ,At f ) j j i ,— 0.59(r~ — rf) + A,f~(dl — ~“5t). .(10)

CY

in which A,f is the steel area necessary to develop the compressive strength

of the overhanging flange:

A,~=O”85@- lY)~l . . . . . . . . . . .

. . . . . ..(11)

h

and

flange thickness

b = overall width of flange

b’ = width of web

r =2

:dl

AS,

‘w = b’dl

A,f

rf=—

b’dl

In equation (10) the value of (rW— r~) should not exceed that given by

equation (7).

13


16/22

CONCENTRICALLY LOADED SHORT COLUMNS

The joint committee recognized that the strength of concentrically

loaded short columns is given by

Fo=0.85uC,vlAC +A,~ . . . . . . . . . . . . . . . . . . ..(12)

in which

AC = gross area of concrete section

A, = total area oflcmgitudinal reinforcement

It wasrecommended, however, that all members subject to axial loads

should be designed for at least a minimum eccentricity. For spirally rein-

forced columns the cc)mmittee gave a minimum eccerrtricity measured

from the centroidal axis equaI to 0.05 times the depth of the column

section; for tied columns O.10 times the depth was recommended.

This recommendation involves a change from present practice which

limits the allowable load for a tied column to 80 of that for a spirally

reinforced column. This change seems reasonable since in practice very

few columns are trul~y concentrically loaded, and recent tests(g) have

indicated that for columns with even a small eccentricity of load, no

second maximum Ioad is developed due to spiral action.

ECCENTRIC LOAD, F~ECTANGULAR SECTION

The uItimate strength of members subject to combined bending and

axial load may be computed from tlhe usual two equations of equilibrium,

b

C

——

Figure 2:

Eccentric load analysis.

which, when the neutral axis is within the section, may be expressed as

follows (Figure 2):

FU = ().85klUcY~hnudl+ A.C~Y— ,4.,~,u. . . . . . . . . . . . . . (13)

Fue = 0085kluCY@Ud12(l— k2n.) + A,~L(dI -– dz). . . . . . . (14)

in which

FU =

ultimate eccentric axial load

14


17/22

e

= eccentricity of the axial load measured from the centroid of

tensile reinforcement

Au = stress in tensile reinforcement which equals ~Y when tension

controls ultimate strength, but is smaller than ~Y when com-

pression controls

nudl = c = depth to neutral axis at ultimate strength

In the above equations, the joint committee recommended that k,

should not be taken as less than

~kl,

and hl should not be greater than

0“85 for UCY1 5,000 lb/in2. The coefficient ().85 should be reduced at the

rate of 0“05 per 1,000 lblinz concrete strength in excess of 5,000 lb/in2.

By solving equations (5) and (,13) forj~U ==f, and ECU= 0.003, it is found

that the ultimate load for the balanced condition is given by

0.003E,

F~ = 0-85k1 ——

0“O03E, + f, )

UQ@dl + (Asc — z4Jfy . . . . . .

(15)

When FU is less than the value of Fb given by equation (15), ultimate

strength is controlled by tensicm and .fiU= fY. Taking into account the

concrete area displaced by the compression reinforcement and solving

equations (13) and (14) for the ultimate strength, we then obtain:

1

) \

FU = 0.85uCYlbd1


18/22

A linear relationship between axial load and moment may be assumed

for values of

FU

between

Fb

as given by equation (15) and the concentric

ultimate load Fo given by equation (12). For this range the ultimate

strength may therefore be computed by

FO

FU=—

()

. . . . . . . . . . . . . . . . . . .

(17)

1+ ;—15

eb’

in which

e’ == eccentricity measured from plastic centroid of section

e~’ = eccentricity of load Fb measured from plastic centroid of section

as computed bly solution of equations (14) and (15).

The plastic centroid of a section is computed with a “ modular ratio “

m“ = ~

For symmetrical reinforcement, the plastic centroid coin-

0.85uCY,“

tides with the geometric centroid.

The joint committee also recognized the equation developed by

C. S. Whitneytl”j for ultimate strength when compression controls:

A,Cu + - btu,,,

Fu= —

. ... . . . . . . . . . . .

(18)

e’

—“

;+1”18

d, — d,

1

in which

total depth of section.

Though we] 1 substantiated by test data, the methods presented above

for the ~esign of eccentrically loaded rectangular sections involve a major

change from present American practice. Even though the principle of the

addition law as expressed by equation (12) is recognized in present design

codes for small eccentricities, the safety factor with respect to ultimate

strength may vary frc~m near one to over four, depending on the com-

bination of variables involved. By the proposed ultimate strength design

procedures, a much more uniform safety factor will be obtained. It should

also be noted that l.he mathematical equations involved are greatly

simplified as compared to a modified straight-line theory.

ECCENTRIC LOAD, CIRCULAR SECTION

The ultimate strength of members of circular cross-section subject to

combined bending and -axial load may be computed on the basis of the

equations of equilibrium taking inelastic deformations into account. The

joint committee also recommended use of a modification of the partially

rational and partially empirical formul~ developed by C. S. Whitney (9.10).

16


19/22

When tension controls:

FU= 0“85uCYlD~

{/(

0.85e’

)

——0.38 2 + ‘%

D

-(

)

~-O.38 } . . . . . . . . . . . . ..(19)

When compression controls:

4fy _

Fu=—

ACuCYl

. . . . . . . . . .

(20)

:+ 11

9“6De’

(O%D + 0“67D )2

+ 1.18

s

in which

D =

diameter of circular column

D, =

diameter of circle circumscribing longitudinal reinforcement

A,

rt=—

AC

LONG MEMBERS

For cases when the unsupported length., L, of an axially loaded member

is greater than fifteen times its least lateral dimension, t, the joint com-

mittee recommended that the maximum axial load,

FM’,

should be deter-

mined by one of the following two methcjds.

The effect of slenderness on ultimate strength maybe taken into account

by stability determination with an apparent reduced modulus of elasticity

used for sustained loads. A numerical procedure such as that recommended

in the report of ACI Committee 312 on Plain and Reinforced Concrete

Archestlzj may be used.

The maximum axial load may also be determined by

)

U’=ZFO 1“6— O.04~ . . . . . . . . . . . . . . . .

. .(21)

t

in which F. is the concentric load capacity of the section with L/ t


20/22

Practical applications

After one becomes farniiiar with ultimate strength theory for reinforced

concrete, the design equations involved are considerably simpler to use

than those resulting from the straight-line theory, Further simplification

of routine design calculations is nevertheless desirable. The joint com-

mittee report’4 ) contains several charts intended to expedite the propor-

tioning of sections by ultimate strength theory. Development of further

design aids is in progress.

THE AC I BUILDING CC)DE

The joint committee report is an engineering report; it is not a building

code, it is not a standard specification. To gain widespread practical

application, therefore, ultimate strength design must be incorporated into

design and building codes,. Ultimate strength design in America has now,

in the opinion of many of us, been developed so far that extensive practical

experience is necessary to continue progress.

A proposed revision of the ACI Building Code was reported by ACI

Committee 318 in December 1955(Z). This proposed revision incorporates

the ultimate strength method of design as an alternative to tlhe straight-line

theory, and an abstract of the joint committee report on ultimate strength

design is given in an appendix. This proposed revision was unanimously

adopted by the 1956 convention of the ACI.

In this manner, after extensive scientific researches and a decade of

thorough committee work, the ultimate strength design method has been

placed before the engineering profession. The future of ultimate strength

design in American practice is, therefore, now largely in the hands of

our engineers and architects practicing the science and art of reinforced

concrete design and construction.

1.

2.

3,

4.

5.

REFERENCES

KEREKES, F. and

REI I

H. B. Jr. Fifty years of development in building code

requirements for rei ttfoneed concrete. Journal oj”the American Concrete Inst itute.

Vol. 25, No. 6. February 1954.pp. 441-470.

Proposed revision of building code requirements for reinforced concrete (ACI

318-51).

Journal o f the American Concrete Institute.

Vol.

27, No. 4.

December

1955. pp. 401-445.

CORNING, L, H., ANDERSON, B. G,, HOGNESTAD, Ii, SIESS, C. P.,

REESE, R. C. and LIN, T. Y. Symposium on ultimate strength design.

Journal

of the Ainerlcan Concrete Institute. Voll. 23, No. 10. June 1952. pp. 797–900.

Report of ASCE-ACI J,Dint Committee on ultimate strength design. Proceedings

of the American Society of Civil Engineers.

Vol. 81, October 1955. Paper 809.

pp. 68.

ACI-ASCE COMMITTEE 327. Ultimate strength design.

Journal of the American

Concrete Institute. Vol. 27, No. 5. January 1956. pp. 505-524.

1 3


21/22

6. Discussion of a paper by E. Hognestad: Inelast ic behaviour in tests of eccentrically

loaded short reinforced concrete columns. Journal of the American Concrete

Instiiure.

Vol.

25, No. 4.

Dec[:mber 1953. Fig. G. p. 140/13.

7. HOGNESTAD, E., HANSC)N, N. W. and McHENRY, D. Concrete stress

distribution in ultimate strength design. Journal o~the American Concrete Institute.

Part 1. Vol.

27, No. 4.

December 1955. pp. 455-479.

8. RICHART, F. E. and BROWN, R. L. An investigation of reinforced concrete

columns. University of Illinois Engineering Experiment Station. June 1934.

Bulletin No. 267. pp. 91.

9. HOGNESTAD, E. A study Of combined bending and axial loud in reinforced

concrete members. University of Illinois Engineering Experiment Station.

November 1951. Bulletin No. 399. pp. 128.

10. WHITNEY, C. S. Plastic theory in reinforced concrete design. Transactions of

the American Society oj Civil Engineers. Vol. 107. 1942. pp. 25 1–282 . Discussion

pp. 282-326.

11. RUSCH, H. Versuche zw Festigkeit der Biegedruckzone. Deutscher Ausschuss

fur Stahlbeton. No. 120.1955. pp. 94.

12. Report of ACI Committee 31,2: Plain and reinforced concrete arches. Journal o f

the American Concrete Institute. Vol. 22,No. 9.May 1951. pp. 681-.69I.Discussion

Vol. 23, No. 4. December 1951. pp. 692/1-692/11.

19


22/22

D8

—“Resurfacing and patching Concrete Pavement with Bonded Con-

crete,”

by EARL J. FEL r.

lteprintedfrom Proceedings OJthe H{phuau Research Boar d 5 1956 ,

D9

—“Review of Data on Effect of Speed in Mechanical Testing of Con.

crete,”

by

DOUGLAS MCHENRY and J. J. SIIIDELER.

Reprinted from Speeiat Tectmical PubUcation No, 185, published by Ameri-

can Society for Testing Materials 1S5S).

DIO-’’Laboratory Investigation of Rigid Frame Failure,” by R. C.

ELST-

NER and E. HOGNESTAO.

Reprinted from Jrournulof the Amertcan Concrete

Institute

anuary, 1957);

proceedings, 53, 637 1957).

D12-’’Ultimate Strength of IReinforced Concrete in American Design Prac-

tice,” by EMND HOCNESTAD.

Reprinted from

Proceedings of a S mpos@n on the Strength of Concrete

Structures, Lond,m, May, 1956.

ultimate strength of reinforced concrete in american design practice

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