DX012

PORTLAND CEMENT ASSOCIATIONRESEARCH AND DEVELOPMENT LABORATORIES

ULTIMATE STRENGTH OF

REINFORCEDCONCRETE IN

AMERICAN DESIGN PRACTICE

By Eivind Hognestad

Authorized Reprint From

Proceedings of a Symposium on the Strength of

Concrete Structures, London, May, 1956

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?jof Coiorado, May 12S2.

D2 -“Nature of Bond in Pre-Tensioned Prestressed Concrete,” by JACK R.

JANNEY,

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 PrestressedConcrete,” by P. W. ABELES, K. HAJNAL-KONYI, N. W. HANSON andAuthor, 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).

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 basedon 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 rosetwenty-five fold from about 2 million long tons in 1900 to over 50 milliontons in 1955. Similarly, the U.S. production of reinforcing stee[ increasedfrom a small amount to about 1”8 million tons.

Introduction of new design procedures for reinforced concrete must beconsidered with this background of great progress and expansion. Thoughthe classical straight-line theor:y was evolved when reinforced concrete wasin its infancy some 60 years ago, it has served us well; and it certainlycannot be put aside on the basis that it has led to unreasonable or unsafedesigns.

On the other hand, through half a century of practical experience andlaboratory experimentation, our knowledge regarding the strength andbehaviour of structural concrete has been vastly improved. To some extent,such improvements of knowledge have been utilized in design practice by

I

periodic adjustments and modifications of the straight-line theory. In thismanner 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 suchhighly 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 ofknowledge regarding the properties of all engineering materials. New and

improved concepts of structural behaviour and design have thereforebecome significant in the practice of civil engineering. These concepts are

identified by rather reeent additions to engineering terminology such asrheology, plasticity, inelastic behaviour, plastic analysis, limit strength,and many others. Definitions for these terms vary to some extent betweencountries as well as between groups concerned with the various materials.It is necessary, therefore, to define common American word usage inconnexion 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 orprestressed structural concrete cross-sections subject to simple bending,

axial load, shear, bond., or combinations thereof. Ultima~e strength designdoes not necessarily involve an inelastic theory of structures. Evaluationof external moments and forces that act in indeterminate structural frame-works by virtue of dead and live loads may be carried out either by thetheory of elastic displacements or by limit design.

Limit design

Limit design indicates a design method involving an inelastic theory ofstructures in which readjustments in the relative magnitude of bending

moments at various sections due to non-linear relationships between loadsand moments at high loads are recognized. Limit design does not by

definition necessarily involve a final design of sections on an inelasticbasis.

Yield line theory

Yield line theory indicates a theory of reinforced concrete slab structuresbased 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 structuralconcrete has been devoted to ultimate strength design. A term indicating,a combination of ultimate strength design, limit design and yield linetheory therefore still remains tc}be adopted. Perhaps the most importantaspect of ultimate strength design is that it represents a significant steptoward a broader consideration of inelastic behaviour in design.

AMERICAN DESIGN SPECIFICATIONS

Two groups have made important contributions to the development ofreinforced concrete design specifications in the United States—the Joint(Committees on Standard Specifications for Concrete and ReinforcedConcrete, and committees of the American Concrete Institute(1).

The Joint Committees have consisted of delegates from the AmericanConcrete Institute (ACI), American Institute of Architects (AIA),American Railway Engineering Association (AREA), American Societyof Civil Engineers (ASCE), American Society for Testing Materials

(ASTM), and the Portland Cement Association (PCA). The first, secondand third Joint Committees were organized in 1904, 1919 and 1930, andsubmitted final reports in 1916, 1924 and 1940 respectively. These reports,which were milestones on the road of progress and had a strong effecton American concrete usage, were submitted to the constituent organiza-tions. The sections concerning reinforced concrete design were written inthe form of a recommended practice rather than a design code, so that itwas possible to give a broad reflection of the state of the art as representedby the best practice of the day.

The first committee on reinforced concrete of the ACI, then theNational Association of Cement Users (NACU), was the Committee onLaws and Ordinances. The first report of this committee appeared in 1909

and was essentially based on what has later become known as ultimatestrength design. The report was later revised to introduce the concepts ofthe straight-line theory, allowable stresses, and service loads; and it wasthen adopted as “ Standard Building Regulations for Reinforced Con-crete “ in 1910. Later a Committee on Reinforced Concrete and BuildingLaws was formed, sponsoring ‘<Standard Building Regulations for theUse of Reinforced Concrete “ in 1920. Committees E-1 and 501 foIlowed,

sponsoring tentative regulations in 1928 and 1936 respectively. The threelast revisions of” The ACI Building Code “ were sponsored by Committee318 and were adopted in 1941, 1947 and 1951. A new proposed revisionwas published in December 1955 ‘2).

The last three ACI codes were approved verbatim as American Standardby the American Standards Association. The codes have also been incor-porated verbatim or adopted by reference in the general building codes of

3

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 StateHighway Officials and by the Ameriean Railway Engineering Association.

NOTATION

The letter symbols used are generally defined where they are firstintroduced; they are also listed below for convenient reference.

Loads and load factors

B = effect of basic load consisting of dead load plus volume changedue to elastic and inelastic actions, shrinkage and temperature

E = effect of earthquake forcesFb = ultimate strength for balanced condition given by equation (I 5)

FO = ultimate strength of concentrically loaded column given byequation (12)

FU = ultimate strength of eccentrically loaded memberFu’ = maximum axial load on long member given by equation (21)

K = load factor equal to 2.0 for columns and members subject tocombined bending and axial load, and equal to 1”8 for beamsand girders subject to bending

L = effect of live load plus impact

MU = ultimate resisting momentU = ultimate strength capacity of sectionW = 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 reinforcementarea of compressive reinfcwcementsteel area to develop compressive strength of overhanging flangein T beams, defined by equation (11)area of tensile reinforcementwidth of a rectangular section, or overall width of flange inT beamswidth of web in T beamsn.dl = depth to neutral axistotal diameter of circular sectiondiameter of ci rcle circumscribing the longitudinal reinforcementin circular sectioneffective depth to centroid of tensile reinforcement

4

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)

Ar’ = ratio of compressive reinforcement = ~

bd,

A,,rt = ratio of total reinforcement = —

AC

A ,trW =—

b’d,

t = flange thickness in T section, or total depth ofsection

Properties of’ materials

ECU =

=W =

f=>U

f.Y ‘“

k, =

k, =

mu =

mu’ =

maximum strain in concrete at ultimate strength0“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 ultimatestrength

ratio of depth to resultant of compressive stress and depth toneutral axis at ultimate strength

0“85 U,y,

mu—l

= cfya—

ucy/

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

5

Report of ASCE-ACI Joint Committeeon Ultimate Strength Design

Advancement in the field of structural design and analysis must ofnecessity proceed with extreme caution and deliberation. This has been

true of the recommendations in the report of the ASCE-ACI JointCommittee on Ultimate Strength Design which culminates over ten yearsof continuous study of the subject. The joint committee was formed as asub-committee of the ASCE Committee on Masonry and Reinforced

Concrete under the chairmanship of the late A, J. Boase in 1944. Itimmediately commenced a comprehensive study of the adequacy of variousultimate 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 thesponsorship of the Reinforced Concrete Research Council of theEngineering Foundation.

In 1949 L. H. Corning was made chairman of the sub-committee. Atthis 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 wasmade a joint committee o:f AC1 and ASCE and designated as Committee327 by ACI.

Hand in hand with the :studies made on ultimate strength formulae, thejoint committee has investigated the question of overload factors in termsof the practice prevailing in countries where design by ultimate strength isin practical use, and of the factors of safety implied in conventionalstraight-line design methods.

During the annual convention of ACI in 1952, the joint committeesponsored a symposium on ultimate strength design(3). This provided anopportunity for public discussion of such topics as reasons for changingdesign method, fundamental concepts of ultimate strength design, reviewof research, practical design, and overload factors.

In 1955 the committee completed its assignment “ to evaluate andcorrelate theories and data bearing on ultimate strength design procedureswith a view to establishing them as accepted practice “. A final report wassubmitted to ASCE and AC1(4’5). It is the principal purpose of this paperto discuss the contents of that report, and to present the author’s opinionsand interpretations regard ing the report.

NATURE OF THE REPO!RT

The joint committee report presents recommendations and formulae forultimate strength design (of reinforced. concrete structures together withbasic supporting and explanatory data. The report is confined to designof 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 didnot 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 serviceloads 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 atvarious sections are usually due to different load arrangements. Becauseof moment redistribution at high loads, therefore, the maximum loadcapacity of the indetermini~te structure may considerably exceed thatindicated by the capacity at a single section. Accordingly, a combinationof ultimate strength design of sections and elastic structural analysis maybe conservative in some cases, but it is not at all unreasonable.

The joint committee report as published by ASCE(4) consists of a briefsection on historical background, and the essence of the report appearsunder the heading “ Recommendations for design “ Three appendixesdeal with substantiating test data, design aids, and derivation of formulae.The report ends with a selected bibliography. The AC I publication (sJdoes not contain the appendixes concerning test data and derivation offormulae.

LOAD FACTORS

Consideration was given by the joint committee to the circumstancethat ultimate strength design may be carried out in two ways. Momentsand forces acting at various sections may be evaluated for service loads,Sections may then be designed by “ deducted “ or “ allowable “ ultimatestrength equations, in which chosen safety factors are incorporated.Another alternative is LOmultiply the, service loads by chosen load factorsbefore the cross-section forces are evaluated. The design of sections thentakes place by equations expressing actual ultimate strengths.

The joint committee chose to follow the second alternative, principallybecause ultimate strength equations are essentially factual in nature, whilethe choice of load factors to a considerable extent is a matter of engineeringjudgment. By keeping load factors and strength equations separated, thereport should be conveniently useful even to specification-writing bodiesthat find it necessary for special applications to change the numericalvalues of the load factors recommended by the joint committee. Further-more, it is believed to be wise for a designer clearly and unmistakably tokeep 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 ofcarrying service loads with ,ample safety against an increase in live loadbeyond 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 excessivecracking. The committee found that these criteria are satisfied by thefollowing 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 sectionB ==effect of basic loald consisting of dead load plus volume changes

dueto elastic and inelastic actions, shrinkage, and temperatureL = effect ofliveload plus impactK = load factor equal to 2“0 for columns and members subject to

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

( )U=K B+ I!+; . . . . . . . . . . . . . . . . . . . .

( )U==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 asspacing and cover of reinforcement, and special considerations regardingthe various typical bui[ding elements. A reference is therefore made tothe ACI Building Code in all matters not otherwise provided for in thecommittee report.

It is required” that bending moments should be taken into account incalculating the ultimate strength of compression members. Analysis ofindeterminate structures should be carried out by the theory of elasticdisplacements, though approximate coefficients such as those recom-mended in the ACI code are acceptable for the usual types of buildingsIn structures such as arches, the effect of shortening of the arch axis,

8

temperature, shrinkage, and secondary moments due to deflexion shouldbe 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 concretewill 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 assumedin design, and the average of any three consecutive tests should not beless than the assumed design strength. In this manner, the design concretestrength is not an average strength; with a reasonable probability it is aminimum strength. Similarly, through the general reference to the AC1code, the joint committee assumed that design values for the yield pointof reinforcing steel are minimum values. Accordingly, the ultimate strengthdesign equations should express an average and not a minimum relation-ship between ultimate strengths of the various reinforced concrete membersas observed in tests and the cclrresponding compressive strengths.

BASIC ASSUMPTIONS FOR IJLTIMATE STRENGTH

After a thorough study of many ultimate strength theories presented inEurope as well as in America, the committee recommended that thecalculation of ultimate strengtlh be based on the following assumptions.

(1) As ultimate strength is approached, stresses and strains are notproportional, and the distribution of compressive stress in sections subjectto bending is non-linear. The diagram c~f compressive concrete stressdistribution may be assumed a rectangle, trapezoid, parabola, or any othershape which results in ultimate strength in reasonable agreement withcomprehensive 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 limitingequations are in good agreement with comprehensive tests of reinforcedconcrete, and calculated ultimate strengths based on a chosen stressdistribution should therefore not exceed these given limits.(2) Plane sections normal to the axis remain plane after bending. Whendeformed reinforcing bars are used, this assumption has been verifiedeven for high loads by numerous tests to failure of eccentrically loadedcolumns 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 assumptionleads to results in good agreement with tests. For very small percentagesof 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 ofthe cylinder strength has been found in tests of horizontally cast mem-bers’7’. In vertically cast members such as columns, however, due to watergain resulting in a lower strength near the top, and due to effects of sizeand shape, a maximum stress of 8.5’~ of the cylinder strength has beenobserved ‘8’9’. Since some effect of size and shape probably also is presentin large beams, and since the concrete near the top of beams as well ascolumns may be somewhat weaker than control cylinders, it seemsreasonable in all cases to use an 8.50/0stress.(6) Stress in tensile and compressive reinforcement at ultimate strengthis 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 nowin use throughout our country. It is also possible, to some extent, tocontrol cracking by other variables than steel stress.

RECTANGULAR BEAh4S

To establish limiting equations for ultimate strength in the variouscases, the joint committee chose a theoretical approach originated byF. Sttissi of Switzerland in 1932 and based on the general properties ofthe stress distribution shown in Figure 1. The properties of the stress

l-- b--l

l!!!!!!Th:J

c—.

As,● ***

Figure 1: Flexwd analysis.

block are given by the stress factor O“85kl (German: Volligkeitsgrad) andthe centroid factor k2 (German: Schwerpunktsbeiwert). Equilibrium offorces 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 bysolving 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 reinforcementyield point stress of reinforcement (limited to 60,000 lb/in2)effective depth lto centroid of tensile reinforcement

stress factor, ratio of average compressive stress to uCYlcentroid factor,, ratio of depth to resultant of compressivestress and depth to neutral axiscompressive strength of 6x 12 in. cylinders at 28 days

A,,,ratio of reinforcement = —–

bdl

width of beamThe 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 equationsof equilibrium from measured properties of the materials steel andconcrete.

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 tests1

by the statistical method c)f least squares, and the value of 0“593 was0.5

found. A value of— = 0.59 was suggested by C. S. Whitney over ten0“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 exceedthat given by

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

which can be re-stated as

Mu—=q(l-– 0.59q) . . . . . . . . . . . . . . . . ..(4a)

z,%.,,

rfyin 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 — cE$U= Ecu-— . . . . . . . . . . . . . . . . . . . . . . (5)

c

Combining equations (1) and (5) and seeking the balanced reinforcementratio for which f,. = fY, we obtain

Ecu U@

‘b=0”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 alimiting 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 per1,000 lb/in2 concrete strength in excess of 5,000 lb/inz. Such reductionfor high concrete strength is desirable on the basis of several experimentalstudies that indicate a decrease of the stress factor, 0.85k,, with increasingconcrete strength (7,11’.

When the ratio of reinforcement exceeds that given by equation (7),compression reinforcement must be provided. For this case, the jointcommittee recommended that the resisting ultimate moment should notexceed

MU = (AS*— AJj;dl 1 — 0.59(r -– r’)~ 1+&fJd,—d,)~(8)Ury(

in which (r — r’) should not exceed the value given by equation (7), andA$C = area of compressive reinforcement

Ar’ = ratio of compressive reinforcement = ~

bd,

d2 = effective depth to centroid of compressive reinforcementFor 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 isf,

little difference between designs resulting from ultimate strength andstraight-line procedures. The major changes suggested by the committeetherefore concern a more efficient use of reinforcement with yield pointsover 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 suchreinforcement, while the ultimate strength design method as outlined maylead 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 practicallyfeasible, to utilize more fully the strength of the concrete compressionzone.

T SECTIONS

If the neutral axis falls within the flange of a T beam, the equations forrectangular beams are applicable with r cc~mputed as for a beam with awidth equal to the overall flange width. The depth to the neutral axis, c,may be estimated by solving equation (1):

1 r~dc=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 developthe compressive strength of the overhanging portion of the fldnge andanother part, (A,f — A,f), that will develop the compressive strength of aa portion of the web. Assuming a uniform stress of 0“85uCYZin the flange,the joint committee recommended:

[ 1Mu=(A,,—Atf)jji,1 — 0.59(r~ — rf)$ + A,f~(dl — ~“5t). .(10)CY

in which A,f is the steel area necessary to develop the compressive strengthof the overhanging flange:

A,~=O”85@- lY)~l . . . . . . . . . . . . . . . . ..(11)h

andt = flange thicknessb = overall width of flangeb’ = width of web

r =2:dlAS,

‘w = b’dlA,f

rf=—b’dl

In equation (10) the value of (rW— r~) should not exceed that given byequation (7).

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CONCENTRICALLY LOADED SHORT COLUMNS

The joint committee recognized that the strength of concentricallyloaded short columns is given by

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

AC = gross area of concrete sectionA, = total area oflcmgitudinal reinforcementIt 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 measuredfrom the centroidal axis equaI to 0.05 times the depth of the columnsection; for tied columns O.10 times the depth was recommended.

This recommendation involves a change from present practice whichlimits the allowable load for a tied column to 80% of that for a spirallyreinforced column. This change seems reasonable since in practice veryfew columns are trul~y concentrically loaded, and recent tests(g) have

indicated that for columns with even a small eccentricity of load, nosecond maximum Ioad is developed due to spiral action.

ECCENTRIC LOAD, F~ECTANGULAR SECTION

The uItimate strength of members subject to combined bending andaxial 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 asfollows (Figure 2):

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

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

in whichFU = ultimate eccentric axial load

14

e = eccentricity of the axial load measured from the centroid oftensile reinforcement

Au = stress in tensile reinforcement which equals ~Y when tensioncontrols ultimate strength, but is smaller than ~Y when com-

pression controlsnudl = c = depth to neutral axis at ultimate strengthIn the above equations, the joint committee recommended that k,

should not be taken as less than ~kl, and hl should not be greater than0“85 for UCY1~ 5,000 lb/in2. The coefficient ().85 should be reduced at therate 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), ultimatestrength is controlled by tensicm and .fiU = fY. Taking into account theconcrete area displaced by the compression reinforcement and solvingequations (13) and (14) for the ultimate strength, we then obtain:

1 ()\

FU = 0.85uCYlbd1 <r’m.’ — rmu + 1 —~( 1’

in which

f,m ——u – 0“85uCY1

mU’ =mU—lA,,

r =—bdl

Ar’ .&

bdl

For symmetrical reinforcement or for members without compressionreinforcement, the general equation (16) is simplified considerably.

When FU exceeds the value of F~ giV(?JI by equation ( 15), ultimatestrength is controlled by compression; ,f,U is less than fY and must bedetermined by the strain equation (5). Solution of equations (5), ( 13) and( 14) involves a cubic equation which is further complicated when theneutral axis is outside the section. For this case, therefore, the jointcommittee recommended two approximate solutions that have been foundto be in good agreement with results of extensive tests of reinforced con-crete eccentrically loaded columns(g).

15

A linear relationship between axial load and moment may be assumed

for values of FUbetween Fb as given by equation (15) and the concentricultimate load Fo given by equation (12). For this range the ultimatestrength may therefore be computed by

FOFU =—

()

. . . . . . . . . . . . . . . . . . . (17)

1+ ;—15eb’

in which

e’ == eccentricity measured from plastic centroid of sectione~’ = 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”18d, — d, 1

in which

t = 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 majorchange from present American practice. Even though the principle of theaddition law as expressed by equation ( 12) is recognized in present designcodes for small eccentricities, the safety factor with respect to ultimatestrength may vary frc~m near one to over four, depending on the com-bination of variables involved. By the proposed ultimate strength designprocedures, a much more uniform safety factor will be obtained. It shouldalso be noted that l.he mathematical equations involved are greatlysimplified as compared to a modified straight-line theory.

ECCENTRIC LOAD, CIRCULAR SECTION

The ultimate strength of members of circular cross-section subject tocombined bending and -axial load may be computed on the basis of theequations of equilibrium taking inelastic deformations into account. Thejoint committee also recommended use of a modification of the partiallyrational and partially empirical formul~ developed by C. S. Whitney (9.10).

16

When tension controls:

FU = 0“85uCYlD~{/(

0.85e’

)——0.38 2 + ‘%

D

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

When compression controls:

4fy _+Fu=—ACuCYl

. . . . . . . . . . (20)

:+ 119“6De’

(O%D + 0“67D$)2+ 1.18

s

in which

D = diameter of circular columnD, = diameter of circle circumscribing longitudinal reinforcement

A,rt=—

AC

LONG MEMBERS

For cases when the unsupported length., L, of an axially loaded memberis 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 elasticityused for sustained loads. A numerical procedure such as that recommendedin the report of ACI Committee 312 on Plain and Reinforced ConcreteArchestlzj may be used.

The maximum axial load may also be determined by

( )FU’=ZFO 1“6— O.04~ . . . . . . . . . . . . . . . . . .(21)t

in which F. is the concentric load capacity of the section with L/ t <15 asgiven by equation (12).

Equation (21 ) may be unduly conservative in some cases. Extensiveinvestigations of the effect c)f slenderness on the strength of reinforcedconcrete compression members are now in progress.

SHEAR AND BOND

Good progress has been made in recent years in studies regarding shear,diagonal tension, and bond. Further experimental investigations are underway, and special ACI-ASCE committees are working on these problems.The joint committee therefore made no recommendations regardingultimate strength design in these items at, the present time.

17

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 furtherdesign aids is in progress.

THE AC I BUILDING CC)DE

The joint committee report is an engineering report; it is not a buildingcode, it is not a standard specification. To gain widespread practicalapplication, therefore, ultimate strength design must be incorporated intodesign and building codes,. Ultimate strength design in America has now,in the opinion of many of us, been developed so far that extensive practicalexperience 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 incorporatesthe ultimate strength method of design as an alternative to tlhe straight-linetheory, and an abstract of the joint committee report on ultimate strengthdesign is given in an appendix. This proposed revision was unanimouslyadopted 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 beenplaced before the engineering profession. The future of ultimate strength

design in American practice is, therefore, now largely in the hands ofour engineers and architects practicing the science and art of reinforcedconcrete design and construction.

1.

2.

3,

4.

5.

REFERENCES

KEREKES, F. and REIID, H. B. Jr. Fifty years of development in building coderequirements for reittfoneed concrete. Journal oj” the American Concrete Institute.Vol. 25, No. 6. February 1954. pp. 441-470.Proposed revision of building code requirements for reinforced concrete (ACI318-51). Journal of the American Concrete Institute. Vol. 27, No. 4. December1955. 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. Journalof the Ainerlcan Concrete Institute. Voll. 23, No. 10. June 1952. pp. 797–900.Report of ASCE-ACI J,Dint Committee on ultimate strength design. Proceedingsof the American Society of Civil Engineers. Vol. 81, October 1955. Paper 809.

pp. 68.ACI-ASCE COMMITTEE 327. Ultimate strength design. Journal of the AmericanConcrete Institute. Vol. 27, No. 5. January 1956. pp. 505-524.

1!3

6. Discussion of a paper by E. Hognestad: Inelastic 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 stressdistribution 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 of

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

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

19

D8 —“Resurfacing and patching Concrete Pavement with Bonded Con-crete,” by EARL J. FEL!r.

lteprintedfrom Proceedings OJthe H{phuau Research Board, 35 (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 Jrournul of the Amertcan Concrete Institute (January, 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 ConcreteStructures, Lond,m, May, 1956.