I. INTRODUCTION
1. Historical Review
The subject of strength of materials has from the earliest develop-
ments, due to the very nature of the problems involved, been a science
of empirical character. This is particularly so in the case of concrete
and reinforced concrete. The first attempts to establish a mathematical
design procedure for reinforced concrete were all aimed at an agreement
with load carrying capacities determined in tests. In modern terms,
several of these early theories may be referred to as the inelastic or
ultimate theories.
A rational analysis of simple reinforced concrete slabs subject to
bending was first published by M. Koenen in 1886(1)*, and many other
theories were published shortly after.(2' 3, 4, 5, 6 7, 8. 9) Some characteristic
basic assumptions of these early theories are reviewed in Fig. 1.
Navier's theory of bending, based on Bernoulli's assumption regard-
ing plane sections remaining plane and Hooke's Law, was well known in
the 1890's. Thus, it is a reasonable fact that the Coignet-Thdesco theory
(Fig. 1) became generally accepted almost immediately after its publi-
cation in 1894.(3) The majority of the authors referred to in Fig. 1
conceded that this latter theory was accurate enough for design purposes
and had great advantages in its mathematical simplicity. Hence, the
"straight-line" or "standard" theory became established at the turn of
the century, and a very rapid development in the use of reinforced
concrete as a construction material followed.
The scientific studies of reinforced concrete were continued on a
steadily broadening scope through the work of such men as Bach,
Considere, Emperger, Graf and Morsch. In this country, the researches
of Talbot and Withey may have made the most important contri-
butions. Much experimental work was done in these first decades of our
century, but few new ideas of importance regarding the basic assump-
tions involved in reinforced concrete design were developed.
The standard theory, on the other hand, became so widely used
that the approximative character of this theory was forgotten, and
applications beyond its range of validity resulted. When beams were
designed with an allowable concrete compressive stress, fc, equal to
*Parenthesized superscript numbers refer to the bibliography at the end of this bulletin.
ILLINOIS ENGINEERING EXPERIMENT STATION
Coignet a Thdesco (/894)
SIKoenen (/886) P. Neumann (/890) (Stndd rer)
(Standard Theory)
J Melan (1896) R. v. Thu/lie (1897) W Ritter (/899) A. Ostenfeld (/102)
Fig. 1. Early Assumptions in Flexural Analysis
0.325 times the strength of 6- by 12-in. cylinders, fe', it was at times
erroneously concluded that the safety factor against a compression fail-
ure was near three. In the period from 1920 to 1930, Slater, Zipprodt
and Lyse made valuable contributions in pointing out that the safety
factor in the case mentioned above generally is considerably larger than
that indicated by the ratio f/'/fe, thus re-emphasizing the actual inelastic
behavior of concrete.(16' 24)
Another important development took place in the decade from 1920
to 1930. Before that time, bending stresses were generally neglected in
the design of concrete building columns, and such stresses were assumed
to be provided for, together with other effects, in an all-inclusive factor
of safety. This neglect was mainly due to the lack of suitable methods
for structural analysis of monolithic structures. Two major methods
were developed to meet this lack of information. The Slope-Deflection
Method appeared in 1918,(15a) and the Moment Distribution Method
followed.(23a, 35a) The moments of inertia used in these methods have
generally been based on the uncracked section. Recent extensive tests
of structures(6", 69a) have shown that this application of P. Neumann's
60-year-old theory (Fig. 1) is satisfactory for the structural analysis
of indeterminate reinforced concrete structures at working loads if
proper values of the modulus ratio are chosen.
About 1900, centrically loaded reinforced concrete columns were
generally designed after the following formula for the allowable load:
Ao
by
I"
f,
IA pt~
Y-A
le
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
P = fA, + fA,g (1)
where f, and f were allowable stresses, and A, and A,t were areas of con-
crete and steel, respectively. Later the standard theory was used, and
the transformed area formula resulted
P = Acfc [1 + (n - 1) p,t] (2)
in which p,t is the ratio of effective longitudinal reinforcement area to
the gross area of concrete, and n is the modular ratio.
In 1921, McMillan made a study of column test data(7i which
showed that building columns under load may develop steel stresses, due
to plastic action, considerably higher than those predicted by the stand-
ard theory, Eq. (2). This study led to the ACI Column Investigation in
the 1930's which was carried out by Lyse, Slater, and Richart. Through
their work, rational equations for the strength of centrically loaded rein-
forced concrete columns were developed.(23, 25, 26, 27, 28, 30, 31, 32, 33, 36, 40, 43)
The ultimate load of tied columns and the yield point of spiral columns
were expressed as
P = 0.85f,'A, + fpA,t. (3)
For the ultimate load of spiral columns, the following equation was
developed
P = 0.85f 'Acore + fypAst + 2.0pspAcoefp (4)
in which
fyp = yield point stress of longitudinal reinforcement
fsp = useful limit stress of spiral reinforcement, generally assumed as
the stress at a strain of 0.005
Acore = area of the column core (out-to-out of spiral)
p,p = ratio of volume of spiral reinforcement to the volume of
concrete core.
Following the publication of the results of this investigation, an inelastic
design formula of the same form as Eq. (1) became much used in this
country as well as in several countries abroad. The basic formula of the
1947 ACI Building Code(93) may be written
P = 0.225fi'A, + 0.4fpA.t. (5)
This design equation is, however, characterized by a larger "factor of
safety" for the concrete than for the steel. Furthermore, the inelastic
design is used only for columns loaded centrically or with small eccen-
tricities. Beams and columns with large eccentricities are still being
ILLINOIS ENGINEERING EXPERIMENT STATION
designed after the standard theory. In beams, however, the inelastic
properties of concrete are recognized to some extent by allowing the
effectiveness of compression reinforcement in resisting bending to be
taken as twice the value obtained from the standard theory.
Another milestone of progress in the theory of reinforced concrete
was passed in 1931 when Emperger wrote a critical study of the modular
ratio and the allowable stresses.(29) This paper initiated intense studies
of the ultimate strength of reinforced concrete beams in bending, which
soon spread over the world. A large number of ultimate theories, some
of which are described in detail in Section 10, were developed.
In recent years it has been claimed repeatedly that our knowledge
of the entire field of reinforced concrete design has advanced so far that
a transition to inelasticity rather than elasticity and to ultimate loads
rather than working stresses is necessary in order to continue progress.
It has also been argued that equal factors of safety should be used for
concrete and steel, and that different safety factors should be used for
live and dead loads. A transition to ultimate design has been made in
some countries such as USSR and Brazil, and several European authors
have claimed that the ultimate theories are "ripe for the specification
form now."
There are two major phases involved in an ultimate design of rein-
forced concrete structures: (1) the structural analysis of indeterminate
structures, and (2) the process of dimensioning sections. In the past, the
major interest has been focused on the dimensioning problem. Hence,
the ultimate design methods adopted in USSR and Brazil, while speci-
fying dimensioning procedures based on ultimate loads, have still main-
tained the theory of elasticity for most purposes of structural analysis.
In recent years, some studies have been devoted to the inelastic anal-
ysis of reinforced concrete structures. Examples of such studies are
the Fracture Line Theory for slabs,(99a) the Stringer Theory for cylin-
drical shells,(11ob) and the Method of Partial Restraints for continuous
beams.(39) The present Danish specifications for reinforced concrete
structures(11oa) permit an inelastic analysis of indeterminate structures,
while the dimensioning methods are based on the standard theory.
In this country it is generally felt, however, that the ultimate dimen-
sioning theory is worthy of primary interest. For concentrically loaded
columns, and beams failing in bending, this theory has been rather well
established as the result of a large number of tests. Our knowledge of
some other types of members, among them eccentrically loaded columns,
is incomplete because tests made on such members of large dimensions
are too few to be conclusive.
Thus, the investigation reported herein was undertaken in order
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
to throw new light on the behavior of reinforced concrete members
subject to combined bending and axial load. It is the purpose of this
bulletin to describe observations regarding the basic behavior of such
members and to express this behavior, as far as possible, in mathemati-
cal terms. Hence, the present bulletin deals with two major problems.
First, simple approximate expressions for ultimate capacities suitable
for design purposes are studied. Secondly, a general theory is developed
with the aim of predicting behavior of eccentrically loaded reinforced
concrete members from the smallest loads through the entire range of
loading to failure, including the mode of failure. Only by expressing
this general behavior in mathematical terms can we determine appropri-
ate factors of safety under various conditions, develop equations suit-
able for design, study variables not included in existing test data, and
advance our knowledge pertaining to the inelastic theory of reinforced
concrete structures.
2. Outline of Tests
The total number of 120 test specimens was divided into four groups,
three of tied columns and one of spirally reinforced columns. An out-
line of the tests is given in Table 1, which indicates that the major vari-
ables were: amount of reinforcement, concrete quality and eccentricity
of load.
Table 1
Outline of Tests*
Total
Group Col. Eccentricity, Col. Longitudinal Concrete No. of
No. No.f in. Size Reinforcement Quality Columns
1 0 Tied Tens. 4% in.rd.
2 2.5 Col. A. =1.24 sq in.
I 3 5.0 10 in. Compr. 2 % in.rd. A,B,C 30
4 7.5 square A,' =0.22 sq in.
5 12.5
6 0 Tied Tens. = Compr.
7 2.5 Col. 4 Yin.rd.
II 8 5.0 10 in. A. =A.'=1.24 sq in. A,B,C 30
9 7.5 square
10 12.5
11 0 Tied Tens. =Compr.
12 2.5 Col. 4% in.rd.
III 13 5.0 10in. .A, =A,'=2.40 sq in. A,B,C 30
14 7.5 square
15 12.5
16 0 Spiral 8 s in.rd.
17 3.0 Col. A, =4.80 sq in.
IV 18 6.0 12 in. A,B,C 30
19 9.0 round
20 15.0
* The columns were made with intermediate grade Hi-Bond longitudinal bars and drawn wire spirals.
Groups I to III were made with plain M-in. ties at 8-in. spacing. The spiral size varied with concrete strength:
Concrete A, USSWG No. 1; Concrete B, No. 3; Concrete C, No. 7; all at 1%-in. pitch. The three grades of
concrete were: A =5000, B =3500, and C =2000 p.s.i. Two companion specimens were made throughout.
The total column length was 7.5 times the least lateral dimension.
? Columns are completely designated by a capital letter, a numeral, and a small letter. The capital
letter - A, B or C - indicates the grade of concrete; the numeral- 1 through 20- indicates the column
number; the small letter- a or b- indicates one of two companion specimens. Thus B-4a indicates 3500
P.s.i. concrete, column No. 4, the first companion specimen.
ILLINOIS ENGINEERING EXPERIMENT STATION
The tied columns were all 10 in. square with a total length of 6 ft
3 in., while the spirally reinforced columns were 12 in. round and 7 ft
6 in. long.
All specimens were tested in 15 to 20 increments of load to failure,
the total testing time being about an hour. Strains in the reinforcement
and on the concrete surface as well as deflections were measured after
each increment of load.
3. Acknowledgment
The investigation reported herein was carried out at the Talbot
Laboratory in the University of Illinois Engineering Experiment Sta-
tion, under the auspices of the Engineering Foundation through the
Reinforced Concrete Research Council, and was supported by the
Portland Cement Association and the Concrete Reinforcing Steel Insti-
tute. The research program was under the general administrative guid-
ance of Dean W. L. Everitt, director of the Station, and Professor F. B.
Seely, head of the Department of Theoretical and Applied Mechanics.
Credit for initiating the investigation must be given to the ASCE
Sub-committee on Ultimate Load Design and especially to the late A. J.
Boase who planned the tests in cooperation with the late Professor
F. E. Richart. The tests were carried out under the general supervision
of Professor Richart and the Council, consisting of the following:
Chairman
R. F. BLANKS, U. S. Bureau of Reclamation
Secretary
J. M. GARRELTS, Columbia University
Special Adviser
B. A. BAKHMETEFF, representing the Engineering Foundation
Members
RAYMOND ARCHIBALD, U. S. Bureau of Public Roads
J. R. AYERS, Bureau of Yards and Docks
R. L. BLOOR, Corps of Engineers
L. H. CORNING, Portland Cement Association
A. E. CUMMINGS, ASCE Research Committee
0. W. IRVIN, Rail Steel Bar Association
H. D. JOLLY, Concrete Reinforcing Steel Institute
DOUGLAS McHENRY, U. S. Bureau of Reclamation
C. T. MORRIS, Ohio State University
D. E. PARSONS, National Bureau of Standards
HARRY POSNER, American Railway Engineering Association
F. E. RICHART, University of Illinois
E. J. RUBLE, Association of American Railroads
The steel for the spirals used in Group IV was furnished by the
American Steel and Wire Company, Chicago. These spirals were fabri-
cated by the Ceco Steel Products Corporation, Chicago.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
The manuscript of this bulletin was critically studied by C. P. Siess
and I. M. Viest of the University staff and by a review committee of
the Council consisting of L. H. Corning, chairman, R. Archibald and
E. J. Ruble. Their helpful comments and criticisms are gratefully
acknowledged.
4. Notation
The letter symbols used in this bulletin are generally defined when
they are first introduced. The most common symbols are listed below
for convenient reference.
A = area
Ac = concrete gross area
Acore = area of column core (out-to-out of spiral)
A, = area of tension reinforcement
A,' = area of compression reinforcement
A,, = total area of longitudinal reinforcement
a = depth of a rectangular stress-block in the concrete
b = width of a rectangular member
C = total internal compressive force in concrete
c = distance from neutral axis to compression edge of member
c/w = cement-water ratio by weight
D = diameter
d = distance from centroid of tension reinforcement to compression edge of member
d' = distance between centroids of tension and compression reinforcements
Ec = modulus of elasticity of concrete
E, = modulus of elasticity of reinforcing steel
Eeý = secant modulus computed from deflections with moment of inertia of concrete
only
Ect = secant modulus computed from deflections with moment of inertia of trans-
formed section
e = eccentricity with respect to mid-depth of section
e' = eccentricity with respect to centroid of tension reinforcement
Ae = increase in eccentricity due to deflection
fe = compressive stress in concrete; also allowable compressive stress
fe' = compressive strength of 6- by 12-in. cylinders or prisms of similar dimensions
fe.' = compressive strength of cubes
fe' = compressive strength of concrete in flexure
fea = tensile strength of concrete
f, = stress in tension reinforcement; also allowable stress in reinforcement
J,' = stress in compression reinforcement
fyý = yield point of reinforcement, especially tension reinforcement
fvp' = yield point of compression reinforcement
fp = useful limit stress of spiral reinforcement, generally assumed as the stress at a
strain of 0.005
I = moment of inertia
Ic = moment of inertia of concrete
I, = moment of inertia of steel
k = c/d = ratio indicating position of neutral axis
kc, k2 = coefficients related to magnitude and position of internal compressive force in
concrete
L = length
M = bending moment
ILLINOIS ENGINEERING EXPERIMENT STATION
Mo = moment of all internal compressive forces about centroid of tension reinforce-
ment. for balanced section
m = inelastic "modular ratio"
n = E,/Ee = modular ratio
P = load; also ultimate load
Po = ultimate load of concentrically loaded column
p = A,/bd
p' = A,'/bd
p., = A,t/A, = ratio of total reinforcement area to gross area of concrete
p,p = ratio of volume of spiral reinforcement to volume of concrete core
T = internal force in tension reinforcement
t = depth of section
V = coefficient of variation
x = distance to centroid of an area
Z = section modulus
z = internal moment arm
a = k2/ki; also a coefficient
= Jensen's plasticity ratio
S = deflection
e = strain
ec = strain in concrete
so = compressive strain in concrete corresponding to maximum stress
e. = ultimate concrete strain in flexure
e, = strain in reinforcement
, = volume strain
a = standard deviation
% = percent (hundredths)
%o = per mill (thousandths)
Columns are designated by a capital letter, a numeral, and a small letter. The capital
letter-A, B or C-indicates the grade of concrete; the numeral-1 through 20-indicates
the column number; the small letter-a or b-indicates one of two companion specimens.
Thus B-4a indicates 3500 p.s.i. concrete, column No. 4, the first companion specimen.
II. MATERIALS, FABRICATION AND TEST METHODS
5. Materials
a. Cement
Lehigh Portland Cement Type I was used throughout the tests. The
cement was purchased in paper bags in two lots from a local dealer and
stored under proper conditions.
b. Fine and Coarse Aggregates
The fine aggregate used was a Wabash River torpedo sand having
an average fineness modulus of about three. The coarse aggregate was a
Wabash River gravel of 1-in. maximum size. Both aggregates have been
in use at this laboratory for years and they passed the usual specification
tests. Aggregate sieve analyses are given in Table 2. The specific grav-
ities were 2.65 and 2.70 for sand and gravel, respectively. The absorp-
tion of both fine and coarse aggregate was about one percent by weight
of surface-dry aggregate.
Table 2
Sieve Analysis of Aggregates*
Percentage Retained on Sieve No.
Kind of 1 Y2 1 Y4 V2 N 4 8 16 30 50 100 Fineness
Aggregate in. in. in. in. in. Modulus
Sand ... ... .... .... 0 2.4 11.3 28.3 64.6 91.0 98.4 2.96
Gravel 0 3.8 34.3 64.9 91.8 98.7 100 100 100 100 100 7.25
* Average of four lots of each aggregate.
The origin of these aggregates is a glacial outwash, mainly of the
Wisconsin glaciation. The major constituents of the gravel were lime-
stone and dolomite; minor quantities of quartz, granite, gneiss, etc.,
were present. The sand consisted mainly of quartz with a character
similar to the gravel in the coarser fractions.
c. Concrete Mixtures
The three concrete mixtures used were designed to have 28-day
cylinder strengths of about 2000, 3500 and 5000 p.s.i. The second lot
of cement, however, gave about 20 percent higher strengths than the
first. The mixtures were therefore adjusted slightly as the tests pro-
ceeded. The average properties of the mixtures are given in Table 3,
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 3
Concrete Mixtures
Mix Average Percent Cement, Water, c/
Concrete (By Slump, Sand, lbs per lbs per (By Wt.) (By Wt.)
Weight)* in. (By Wt.) cu yd cu yd
A 1:2.2:3.1 5.9 41.5 594 323 1.83 0.55
B 1:3.1:4.3 6.3 42.0 450 311 1.45 0.69
C 1:4.8:6.4 6.6 43.0 308 313 0.98 1.02
* Surface dry basis.
and the relations between strength, c/w and curing procedures are
given in Fig. 2. The strengths of 6- by 12-in. control cylinders are
listed with the results of the column tests.
All concrete was mixed in a non-tilting drum mixer of 6.5-cu ft capac-
ity, and was placed in the forms by means of vibration.
d. Reinforcing Steel
Three sizes of deformed bars were used as longitudinal reinforcement
in these tests: % in., Y8 in., and Y7 in. For the Ys- and 78-in. sizes, the
Hi-Bond bar, which is representative of modern deformed bars meeting
ASTM Designation A305-49, was chosen in intermediate grade billet
steel, and the quantity necessary for the complete investigation was
purchased from a commercial firm and received in one shipment. The
laboratory's stock of old deformed bars was used for the 3%-in. com-
pressive reinforcement in Group I. The ties for the square columns were
manufactured by hand from the laboratory's stock of %4-in. plain bars.
The drawn wire spirals were received as gifts. Properties of the rein-
forcement as determined from tension tests are given in Table 4. Since
no uncoiled wire was furnished with the spirals, no strain measurements
were made during the tension tests of the wires.
Electric SR-4 gages were used to measure strain in the longitudinal
steel of all columns. In order to attach such gages to the Hi-Bond bars,
the lugs were removed on one side of the bars over a length of about 2 in.
By testing pairs of 2-ft specimens cut from the same reinforcing bar and
attaching SR-4, A-11 gages to one specimen of each pair, it was found
that yield point and ultimate strength were little affected by the local
removal of lugs, as indicated in Table 4. The modulus of elasticity as
measured by a mechanical gage over an 8-in. length was also practically
unaffected. The local strains as measured by the SR-4, A-11, 1-in. gages
were, however, 10 to 15 percent larger than the corresponding strains
measured over an 8-in. length, as shown in Fig. 3a and 3b. This differ-
ence is believed to be due mainly to the eccentricity introduced by the
removal of lugs. Thus, all strains measured by SR-4, A-11 gages during
the column tests were corrected to an 8-in. length by using Fig. 3.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Cement-Water Ratio by Weight, (c/w)
Fig. 2. Strength Relations for Concrete
The properties of the deformations of the Hi-Bond bars are given in
Table 5. The details of the lug measurements and the computation of
the various significant properties are presented in an earlier paper by
C. P. Siess and the writer.<114
6. Types of Specimens
Tests of 120 reinforced concrete columns are reported herein, of
which 90 were square tied columns and 30 cylindrical spiral columns.
Since the investigation was confined to the combined stress problem,
the specimens were purposely kept fairly short, so that the results
would not be confused by the occurrence of buckling failures. The two
types of columns with four types of longitudinal reinforcement are
shown in Fig. 4. Cylinders, 6 by 12 in. were used as auxiliary specimens.
7. Fabrication and Curing
The reinforcement was assembled into a unit or cage before it was
placed in the forms for casting. The longitudinal steel, which was placed
inside the ties or spirals and securely wired to them, was carried straight
ILLINOIS ENGINEERING EXPERIMENT STATION
C)
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Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
0 5 /0 /S ZO
Strain in Thousandth
25 30 35
hs (%o)
Fig. 3. Stress-Strain Relations for Reinforcement
through the entire length of the columns. Spacing blocks were used to
provide accurate spacing of these bars. Additional reinforcement was
provided in the brackets or capitals, the purpose of which was to transfer
the eccentric load and its moment to the prismatic shaft of the columns.
For the small eccentricities this was achieved with a light reinforcement
ILLINOIS ENGINEERING EXPERIMENT STATION
a
0
=
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4E'
0,
4, 0
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Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
bent around the longitudinal bars as shown to the left in Fig. 5a. In the
cases of large eccentricities, however, a heavy reinforcement welded to
the longitudinal bars was necessary to prevent tension, diagonal tension,
or bond failures in the capitals. Such a reinforcement unit is shown to
the right in Fig. 5a. Except for the top capital of one column which failed
in bond, all capitals proved to be satisfactory, as they were perfectly
sound after the prismatic part of the columns had failed.
The columns were cast in a vertical position. Some previous investi-
gators(4461,94) cast their specimens horizontally in order to avoid a dif-
ferential in concrete quality along the column length. Such a differential
»/,, «--z- ?-s T,)
/4,,
hi5±
Grouvp 1T
8-2'0
Group I Group, Groupg, L7
4-i", 2-3i' 8-1f 8-10f
Fig. 4. Types of Specimens
will exist in a vertically cast column because the concrete in the bottom
part generally will be compacted better than in the top part. Also, even
a moderate water gain or bleeding will decrease the c/w ratio and hence
the strength of the concrete in the upper part. Horizontal casting, on
the other hand, will cause a strength differential across the cross section
of the columns.
Furthermore, the orientation of the members during casting will de-
termine whether the longitudinal stresses during testing will be parallel
or perpendicular to the direction of casting. Since all columns tested at
8 ,
m~I
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 5. Types of Forms
this laboratory in earlier investigations(22, 43 95) were cast vertically,
since most columns in practice are cast vertical or inclined, and since it
is very difficult to cast cylindrical columns horizontally, all columns de-
scribed herein were cast and tested in a vertical position.
A view of the two types of steel forms is given in Fig. 5a and 5b. It
can be seen that the columns were cast in pairs in forms built from
plates, shapes, and split pipes bolted together. The lateral dimensions of
the prismatic parts of the columns were found to be true to ±0.1 in.
at the time of testing. Spacers were provided to keep the reinforcement
units centered in the forms with a 1-in. cover over the longitudinal bars
of the tied columns, and a 1-in. cover over the spirals of the cylindrical
columns. Since a heavy external vibrator was attached to the forms to
compact the concrete, this spacing was found to vary about ± in.
The forms were carefully leveled and plumbed.
In all columns a reasonably fluid consistency of concrete was used
to facilitate its placing around the heavy capital reinforcements and to
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
provide a smooth surface which is desirable for the application of electric
strain gages. The forms were removed the day after the columns were
cast, and the columns were stored in a fog room at 75-80 deg F for seven
days after casting. They were then stored in the air of the laboratory
until testing took place at an age of 28 days. This curing process is
reasonably similar to practical conditions, and it was also chosen be-
cause, as yet, no methods are known of applying electric gages to a wet
concrete surface.
In all cases 9 to 12 cylinders (6 by 12 in.) were made, stored and
tested with each pair of columns.
8. Attachment of Electric SR-4 Gages
Electric gages were used throughout these tests to measure strains in
the reinforcement as well as on the concrete surface. The manner of at-
taching the SR-4 gages to the reinforcement varied in the tests. Attach-
ment of gages to bars to be embedded in fresh concrete requires careful
waterproofing of the gages and leads. For compression gages in the
square columns and for all gages in the spiral columns this was consid-
ered necessary. However, for tension measurements in the square col-
umns, it was considered satisfactory, and very much simpler, to provide
core holes in the concrete during casting and to attach the gages to the
reinforcement after the concrete had hardened and cured. The core holes
may have slightly influenced the formation of tension cracks; but such
cracks would form in any event before the steel stresses became large.
Even though the errors introduced by the attachment of gages are
small, it was decided to avoid attaching all gages at the same cross sec-
tion of the columns. Thus, for the compression as well as the tension side
of the square columns, one gage was attached at midheight of the col-
umn and the other two gages 8 in. above or below midheight. In the
spiral columns gages were attached to all longitudinal bars alternately
at midheight and 12.5 in. above midheight.
Due to the heterogeneous character of concrete, strains measured
over a short length on a concrete surface cannot be expected to yield
satisfactory results. Thus SR-4, A-9 gages with 6-in. gage lengths were
used for measurements of strains in the concrete. In some preliminary
tests these gages were checked against an 8-in. Berry gage, and the
agreement was very good. Generally about five gages were used to deter-
mine the strain distribution across the cross section. The majority of
these gages were placed where the columns were expected to fail, near
the upper quarterpoint of the column shaft. One SR-4, A-9 gage was
also used to measure lateral strain on the compression face of the column
in order to obtain information regarding volume strains.
ILLINOIS ENGINEERING EXPERIMENT STATION
The electric gages gave very satisfactory service throughout these
tests and their easy operation facilitated the testing considerably. A de-
tailed description of the methods involved in the use of these gages as
applied to reinforced concrete is given in an earlier publication by I. M.
Viest and the writer.(111)
Fig. 6. Test of Cylindrical Column
9. Testing Procedure
All columns were tested in a 3,000,000-lb capacity Southwark-Emery
hydraulic testing machine. A view of the machine during the test of a
cylindrical column is shown in Fig. 6. Load was applied through "knife
edges" at the top and bottom support of all columns, except the concen-
trically loaded spiral columns which were tested with "flat ends." A
sketch of the testing equipment with knife edges is given in Fig. 7 which
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Movable Head
insertead Olurli lest
Fig. 7. Testing Arrangement
shows a square column. The bottom knife edge rested on a wheeled car-
riage, thus permitting easy placing, centering, and removal of the col-
umns in the testing machine. The upper knife edge and base plate were
suspended from the movable head of the machine. Two auxiliary tension
rods were attached to each capital of the square columns, because it was
found impracticable to reinforce these capitals sufficiently with only
embedded bar reinforcement. A deflection bridge, carrying five dial
indicators, furnished deflection readings in the plane of eccentricity.
The arrangement for testing the spirally reinforced columns was
very similar; however no auxiliary rods were used. The concentrically
loaded spiral columns were tested with flat ends.
All columns were carefully centered and plumbed in the testing
machine. However, since the knife edges were 2 in. thick and since no
optical instruments were used to center the columns, the actual eccen-
tricities of loads during testing probably deviated up to ±+ 1 in. from
28 ILLINOIS ENGINEERING EXPERIMENT STATION
the nominal values. Initial readings of all gages were taken immedi-
ately before application of any load. Then a load of about 3.5 kips,
which corresponds to the weight of the spherically seated block in the
movable head of the testing machine, was applied. During this loading,
the block was allowed to adjust itself by movements in the spherical
bearing. Before further load was applied, however, this block was fixed
by wedges and the temporary jacks under the column were removed.
Loading proceeded to failure in 15 to 20 increments, the time between
the first and last increments being about one hour. After each increment
of load, the machine was operated in small increments until the immedi-
ate creep had taken place and the load came to rest at a reading about
one percent over the nominal load. Then, all gages were read in a cyclic
manner, returning to the first gage for control purposes. After all gages
were read, the load had generally dropped to about one percent under
the nominal load. This may have been caused by creep of the specimen
and by a slight leaking of oil from the pressure cylinders of the machine.
III. INELASTIC FLEXURAL THEORIES
10. Historical Development of Inelastic Dimensioning Theories
It was mentioned in the introduction to this bulletin that a great
number of theories relating to the flexural analysis of reinforced concrete
have been published since M. Koenen's basic note in 1886.(1 Generally,
it has been agreed that the "standard" theory, using the concepts of a
transformed section and a modular ratio, is sufficiently accurate for
estimating stresses and for dimensioning sections with reference to
safe loads.
Many authors have argued, however, that the safety and economy
of reinforced concrete structures are not controlled satisfactorily by this
"classical elastic theory" and that a consideration of the plastic
character of concrete near ultimate loads is necessary to achieve such
control. In order to review the most important inelastic theories pre-
sented before 1950, it is convenient to group the basic types of assump-
tions as follows:
1. Distribution and magnitude of the compressive stresses in the
concrete
2a. The ultimate strain in the concrete, or
2b. The limiting depth of the neutral axis
3. Tension stresses in the concrete
4. Bernoulli's hypothesis regarding a linear distribution of strains
5. The stress-strain relation of the reinforcing steel
6. The absence of a general slip between concrete and steel.
Most earlier investigators have assumed that the concrete resists
no tension, that Bernoulli's hypothesis is valid, and that no slip occurs
between concrete and steel. The stress-strain relation for mild steel rein-
forcement has generally been assumed trapezoidal with the yield level
at the yield-point stress. These assumptions will apply to all theories
discussed below unless a statement to the contrary is made.
The modes of failure of reinforced concrete members subject to flex-
ure either without or combined with axial load are generally character-
ized by one of the following five groups:
1. Failure by excessive compressive strain in the concrete before the
tension steel reaches yielding - compression failure
ILLINOIS ENGINEERING EXPERIMENT STATION
2. A failure initiated by yielding of the tension steel at the yield point
of the member, with a resulting movement of the neutral axis which leads
to excessive compressive strain in the concrete - tension failure
3. A balanced condition between (1) and (2) where the tension rein-
forcement reaches yielding exactly at the ultimate load, at which the
concrete also fails at the compression edge - balanced failure
4. Compression failure of the concrete with tension steel stresses
greater than the yield point
5. A brittle mode of failure caused by rupture of the tension steel
immediately after the formation of tension cracks in the concrete.
The majority of previous theories consider compression and tension
failures only, thus neglecting the effect of strain hardening in the ten-
sion steel as unreliable for practical purposes. The brittle mode of failure
takes place only for very small reinforcement percentages and is there-
fore of secondary importance for practical purposes.
All inelastic theories advanced with reference to reinforced concrete
subject to bending have been limited to the uni-axial state of stress.
Thus, two equations of equilibrium and one compatibility equation
involving strain relationships are available for a flexural analysis. All
three equations have generally been used by past investigators in studies
of compression failures and yield points of members. The compatibility
equation then served the purpose of determining the tension steel stress
for compression failures and the concrete stress for yield points. When
tension failures are discussed, however, the tension steel stress is gener-
ally assumed to be at the yield point and the stress-block in the concrete
will be that corresponding to failure of the concrete. Thus, the compati-
bility equation need not be used, nor is it necessary to assume an ulti-
mate strain in the concrete, the validity of Bernoulli's hypothesis, and
the absence of slip.
a. E. Suenson, 1912(11)
The writer believes that Suenson originated the use of the rec-
tangular stress-block, Fig. 8, which has been much used later, directly
or indirectly. Suenson's analysis covers the case of tension failures
in a rectangular beam or slab only. Therefore no compatibility equation
is needed.
An equation for dimensioning the reinforcement of a slab was devel-
oped in the following manner (Fig. 8).
From the equilibrium of forces fc"ab = A,fJ
(6)
and the equilibrium of moments M = A,f, d - )
2(6
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
SSaenson (19/2) G. v Kazinczy 1/933) Tension Compression
E. B/iner (/935) A. Brand/zaea (/935J) ailure Failure
H F.Michielsen (/836) C.S. Whitney (1937) L.J. Mdensch (1914)
Tension compress/on
Failure Failure F Stussi (1932)
H/ Kempton Dyson (1922) R. Sa/,ier (1/936)
f .
*oLf
Tension Compression
Failure Failure
S. Steuermann (1933)
f.Gebaer (/934)
S- Af
J. Me/lan (1936)
VP. Jensen (1943)
Specifications (1938) A. Guerrin (/941)
Tension Compression
Failure Failure
Sv Emp/erqer (1936)
c, 094 f,
i7 Cubic
co _ R Parabo/o
SSine
R Chambaud (1949)
Fig. 8. Assumptions in Flexural Analysis
C. Schreyer (1933)
0 Baumann (/934)
A. Brand/zaeg (1935)
E. Bi/fner (1935)
. Chambaud (1949)
R i
ILLINOIS ENGINEERING EXPERIMENT STATION
a 2M
can be obtained - 1 - 1 - 2-
(6 cont.)
M
f. -
Az
These equations can be written in the form well known in later years
- pf 1- p (7)
bd
and the compressive strength in flexure, fo", has been given various
values. Suenson put fc" equal to the cube strength, ft,'.
b. L. J. Mensch, 1914(11)
The theory presented by Mensch covers tension as well as com-
pression failures of rectangular beams with or without compression rein-
forcement. He criticized the building codes of that time and pointed out
that the assumptions of the standard theory do not agree with the test
results at high loads.
The expressions for ultimate moments were developed without the
use of a compatibility equation, and no assumptions regarding ultimate
strains, Bernoulli's hypothesis and absence of slip needed to be made.
For the case of tension failure, Ritter's second degree parabola was as-
sumed as stress-block (Fig. 8) and the computations made were very
similar to those of Suenson.
In the case of compression failures (Fig. 8) Mensch assumed as
a limiting condition that the neutral axis reached the centroid of the
tension steel. Hence
M 1
- f=C.
bd2 2.4
Mensch realized that this limiting condition was too extreme, however,
and he suggested for "balanced reinforcement"
M 1
S- 2.6 f'. (8)
c. H. Kempton Dyson, 1922(18)
This theory, like that of Mensch, makes no use of a compatibility
equation. Kempton Dyson assumed an elliptical stress-block as shown
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
in Fig. 8. In the case of tension failure, the difference from other
theories is small. For compression failure
M 1
- = - fc" . (9)
bd - 2.21
The effect of compression reinforcement was also discussed, and a
design equation of the same form as Eq. (1) was recommended for
centrically loaded columns.
d. F. Stiissi, 1982 (34)
This paper discusses rectangular beams with tension reinforcement
only. Compression and tension failures were subjected to a mathemati-
cal analysis, and strain hardening as well as brittle fractures was men-
tioned in connection with very small reinforcement percentages. How-
ever, Stiissi's approach to the analysis of ultimate moments in such
beams was so general that several later theories may be considered as
refinements and improvements of his work. Hence, it is convenient to
refer to these approaches as the StUssi type of ultimate theory. The as-
sumptions made are shown in Fig. 8. An arbitrary form of the con-
crete stress-block was assumed, which, for rectangular sections, may be
characterized by the constants ki and k2, the compressive strength in
flexure, fe", and the ultimate strain in flexure, e.. These four quantities
have been derived differently by the various authors who have con-
tributed to ultimate theories of the Stussi type. One or two of the
following means have been used:
1. Tests of beams in bending
2. Tests of concrete in concentric compression
3. Studies of stress-strain relations of concrete in flexure
Sttissi used stress-strain relations derived from concentric compression
of prisms to determine his constants. He found ki = 0.70 to 0.77, k2 =
0.39 to 0.41 and E. = 2.0 to 2.5 per mill. The compressive strength in
flexure f/', was assumed equal to the prism strength, f/. It should be
noted that Stiissi did not recognize that larger strains may be developed
in bending than those corresponding to the ultimate load of a prism.
Hence, his values for ki, k2 and eu are smaller than the values reported
in.later years.
With the assumptions shown in Fig. 8, and fi"= fc' the following
equations were written
Equilibrium of forces: AJ,f = kibcf,' (10a)
Equilibrium of moments: M = kibcfc' (d - k2c).
(10b)
ILLINOIS ENGINEERING EXPERIMENT STATION
For tension failures, Sttissi further assumed the steel stress, f, to
be at the yield level, f,,. Hence the following equation for the ultimate
moment was found
M , k, f
b-= pf -- p . (11)
For the ratio k2/ki, the value 0.55 was recommended as accurate enough
for practical purposes.
For compression failures, Stiissi made the usual assumptions re-
garding linear distribution of strain, etc., and he used the following
compatibility equation, Fig. 8
d-c
, = E,e = E,8-- . (12)
c
Solving Eqs. (12) and (10a) for the position of the neutral axis results
in the expression
d /pm , pm / \2 pm (13)
c/d P -- 2 (13)
ki \ 2ki / 2k,
where m = Eeu/f,/. By substituting Eq. (13) into Eq. (10b), an equa-
tion for the ultimate moment may be found.
Stiissi concluded his studies by showing that the standard theory
leads to safety factors ranging from about 2.3 to 4.1.
e. C. Schreyer, 19330 )
This theory of ultimate moments for reinforced concrete beams,
which is of the Stiissi type, is based on stress-strain relations for con-
crete in compression obtained by tests of cubes, Fig. 8. A hyperbolic
stress-strain relation was derived
0.63
-c = X 10-3. (14)
1.1- L
fcul
The ultimate strain, e. = 6.3 per mill, was found to be independent of
the concrete strength.
Schreyer's expressions for the ultimate moments, which are alge-
braically rather involved, were derived from the assumptions mentioned
above by means of Eqs. (10) to (13).
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
f. S. Steuermann, 1983(38)
Steuermann's theory which was introduced in the Russian specifi-
cations, Mjassochladstroj, in 1932 is based on the assumptions shown in
Fig. 8. In the case of tension failures, some tension was assumed to be
taken by the concrete in beams and slabs, but not in T-beams. The
tensile stress, fct, which was considered an equivalent stress rather than
the tensile strength of the concrete, was generally assumed as 0.10ft,'.
For compression failures, a parabolic stress-block was assumed with
the neutral axis at a depth 3d/4. This stress-block was in turn converted
into an equivalent triangular block, and the following formula for the
ultimate moment resulted
M 1
-= fe'. (15)
bdl 3
It should be noted that Mr. Steuermann's assumptions for tension
failures are very similar to those made by J. Melan in 1896, Fig. 1.
g. G. v. Kazinczy, 19338(9)
Ultimate moments of rectangular beams failing in tension were de-
rived on the basis of a rectangular stress-block, Fig. 8. The ultimate
concrete stress, fe", was assumed equal to the prism strength, fe', and
the steel stress f, = fm,. Hence, Eq. (7) was derived in the form
= pf 1 -p f). (16)
A major part of Kazinczy's paper was devoted to an analysis of
test results regarding ten 2-span beams. Three of these beams were
reinforced in accordance with the theory of elasticity, while five beams
were under-reinforced and two were over-reinforced over the center
support. It is believed that this paper initiated the study of the plastic
theory of structures as applied to continuous reinforced concrete beams.
h. F. Gebauer, 1934(42)
Another theory for rectangular beams failing in tension, taking
tensile stresses in the concrete into consideration, was developed by
Gebauer with the assumptions shown in Fig. 8. The tensile stresses in
the concrete were considered due to shrinkage of the concrete sur-
rounding the steel.
i. 0. Baumann, 1934(44)
This report is not a direct contribution to the inelastic flexural
analysis of reinforced concrete, since the subject of Baumann's study
ILLINOIS ENGINEERING EXPERIMENT STATION
was the buckling of reinforced concrete columns subject to centric or
eccentric loads. For the purpose of such a study, however, a relation be-
tween moments and rotations of reinforced concrete members in the
range from zero to ultimate rotations was necessary. Hence, the as-
sumptions shown in Fig. 8 were made.
The stress-strain relation of concrete in flexure was assumed to
follow the virgin curves of centric compression, until the maximum
stress, f,", was reached at a strain, eo. A parabola was found to be a satis-
factory approximation for such virgin curves, and fe" was assumed equal
to the prism strength, fe'. However, Baumann found, through tests of
eccentrically loaded prisms, that the ultimate strain in flexure, 4E, was
larger than the value co corresponding to centric compression. For a con-
crete with about 3500 p.s.i. cylinder strength, he found to = 1.8 per mill
and e. = 2.5 to 3.3 per mill. This important observation had already
been reported by Talbot in 1904-1906;(10, 11) but Baumann seems to have
made an independent rediscovery.
Baumann's study of buckling was verified by tests of reinforced
columns with two concrete qualities. The values of Eo and aE were deter-
mined in auxiliary tests. No functional relationships between concrete
strength and these strains were therefore developed.
j. E. Bittner, 1935(4S8, 53)
Bittner's inelastic theory is of the Stiissi type. He assumed a stress-
block very similar to Baumann's, Fig. 8. It should be noted, however,
that Eo was assumed equal 1.5 per mill regardless of the concrete strength.
Hence, eo is not related to the modulus of elasticity, Ec, as determined
in compression tests. It was further assumed in the analysis of com-
pression failures (Eqs. 10, 12 and 13) that E. = 3.0, 5.0 and 7.0 per
mill. No particular value of e. was recommended.
For tension failures, a rectangular stress-block with fc' as the ulti-
mate compressive stress was used, and Eq. (16) was developed.
k. A. Brandtzsag, 1935(51 56, 57, 61)
These studies represent the first complete analysis of the ultimate
capacity of rectangular sections with or without compression reinforce-
ment subject to bending as well as bending combined with axial load.
The analysis of compression failures in beams is of the Stilssi type,
based on the assumptions shown in Fig. 8. Brandtzaeg improved and
generalized on Baumann's stress-block by determining an empirical
relation between the ultimate concrete strain in flexure, e., and the
compressive strength of concrete. Substituting f,' = 0.85fc,,' and 1
p.s.i. = 0.07 kg per sq cm, this equation may be written as follows:
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
fu = 6.88 - 0.77 X 10-3. (17)
1000)
After determining the value of co as a function of concrete strength
and choosing the constant stress in the plastic range, fc", equal the prism
strength, fe', Brandtzeg introduced a "plasticity ratio" 71 = </eo, as a
function of concrete strength. Equations (10), (12) and (13) were derived,
introducing ki and k2 as functions of the plasticity ratio, q. Equations
(10a) and (10b) were also extended to include consideration of compres-
sion reinforcement and combined bending and axial load.
While this discussion of compression failures led to rather involved
mathematical formulas, very simple expressions for tension failures were
derived by extending Suenson's formula based on a rectangular stress-
block, and Eq. (16) was derived for tension failures of beams.
For tension failures of rectangular sections subject to combined
bending and axial load, Brandtzsg(51) developed an equation very simi-
lar to that used by the writer later in this paper, Eq. (39).
Brandtzag verified his theory with tests of 20 beams, 13 eccentri-
cally loaded columns and many auxiliary specimens. He further used
this theory to develop a modified standard procedure which influenced
the 1939 Norwegian Standard Specifications for design of reinforced
concrete structures3") to a considerable extent. In this modified stand-
ard theory, a fairly uniform factor of safety is achieved by introducing
an allowable compressive stress for the concrete which is a function of
the relative eccentricity of the load. Furthermore, the "modular ratio"
for compression reinforcement is related to fyp/fo' while that for tension
reinforcement is related to E,/Ec.
1. F. v. Emperger, 1936(59)
Emperger, who was one of Europe's outstanding concrete engineers,
initiated the thought-provoking discussions on ultimate theories in
Beton und Eisen through his paper in October 1931(2) which was written
just before his 70th birthday. In October 1936(69) he reviewed five years'
discussions, and concluded that the ultimate analysis of reinforced con-
crete beams may be carried out with sufficient accuracy through the as-
sumptions shown in Fig. 8. It should be noticed that the assumptions
for tension failures, which do not satisfy statics, are very similar to
Koenen's basic paper of 1886.(1)
m. R. Saliger, 1936(, 66, 96. 99)
A thorough study of rectangular beams was presented in Saliger's
original paper.(60) All five modes of failure mentioned in the introduc-
tion of Section 10 were considered.
ILLINOIS ENGINEERING EXPERIMENT STATION
The approach used in the development of ultimate moment equa-
tions was essentially the same as Stiissi's. The approximation was made,
however, that k2 = j/2k, which is equivalent to replacing the curved
stress-block by a rectangular block with a depth kic. Saliger further as-
sumed fZ" = f,' and developed Eq. (16). Thus, it was possible to deter-
mine k, by observing the position of the neutral axis in beam tests,
without any specific assumptions regarding the stress-strain relation in
flexure. Values from 0.90 to 0.94 were found.
In discussing compression failures of beams, Saliger emphasized
that his tests had shown values of the ultimate strain in flexure, eu,
from three to seven per mill. The following relation between eu and f/'
was suggested in his most recent paper(99)
e. = 1.75f/P X 10-6 (18)
where f,' is in p.s.i.
n. C. S. Whitney, 1937(6, 69, 74, 97)
This author is perhaps better known in this country than any other
spokesman of ultimate theories for reinforced concrete. His theories for
compression failures are not of the Stfissi type as no assumptions re-
garding ultimate strains, Bernoulli's hypothesis or the absence of slip
were made.
Tension failures of beams were analyzed assuming a rectangular
stress-block, Fig. 8, with fc" = 0.85fe'. Thereby the following formula
was developed
b- 2 pfi, 1 - p (19)
bd2 2 0.85 f<'
which is Eq. (7) with f, = f, and fe" = 0.85 f'.
For the study of compression failures of beams no compatibility
equation was used. A limiting value of a/d was assumed on the basis of
values computed from test results. A value of a/d = 0.537 was found,
which led to the following formula
M 1
- = - fc. (20)
bd2 3
This equation is very similar to Steuermann's Eq. (15) which was de-
rived from entirely different assumptions. Beams with compression
reinforcement were analyzed by adding the full yield stress of the
compression reinforcement times its moment arm to the right side of
Eq. (20).
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
On a semi-empirical basis the following equation for compression
failures in combined bending and axial load was developed for rec-
tangular sections
2A8'f,, btf' (21)
P = + (21)
2e 3te
-- +1 --+ 1.178
d' d'
This equation reduces to the ultimate load for a centrically loaded col-
umn, Eq. (3), for e = 0 and to the ultimate moment for a beam when
e = o. The equation is valid, however, only for rectangular sections
with symmetrical reinforcement at the compression and tension faces.
For tension failures of such eccentrically loaded rectangular sections,
Mr. Whitney developed equations very similar to those developed by
Brandtzoeg and the writer, Eq. (39), by assuming a rectangular stress-
block in the concrete.
Finally, equations for rectangular and cylindrical columns with
round cores were developed on a semi-empirical basis. These equations
are discussed in Section 15.
o. USSR Specifications OST 90003, 1938(9', 109)
These Russian specifications were based on an ultimate design pro-
cedure. The stress-block in the concrete was assumed as a cubic parabola
(Fig. 8). Thus the following equation was developed for tension failures
of rectangular beams
S= pf,, 1 - P (22)
b1 1.89fZ"
and for compression failures of beams
M
- = 0.4f/". (23)
b&
In Eqs. (22) and (23), fc" is specified as a fraction of the ultimate strength
of 8-in. cubes, varying from 0.8 to 1.0.
For axially loaded columns the following formula is used
P = fcA, + fy A8t (24)
where f/ = 4f1"/5.
Eccentrically loaded columns are analyzed on the basis of assump-
tions similar to those involved in Eqs. (22) and (23). Compression rein-
forcement is generally considered effective with the full yield-point stress.
ILLINOIS ENGINEERING EXPERIMENT STATION
p. K. C. Cox, 1941(71)
Mr. Cox reported tests of 110 rectangular beams. He developed
equations for tension and compression failures with and without com-
pression reinforcement based on a rectangular stress-block (Fig. 8),
choosing fe" = fe'. For tension failures Eq. (16) was developed. For
compression failures he determined an empirical value for the critical
reinforcement
I '
p,= 0.47-- (25)
which leads to
M 1
- = - f'. (26)
bd2 2.76
Thus, Mr. Cox's Eq. (26) is intermediate between Mensch's value of
1/2.6 and Whitney's 1/3.
q. V. P. Jensen, 1943(, 81)
Jensen wrote one of the most complete studies of rectangular beams
reinforced only in tension that have been published. His analysis, which
is of the Sttissi type, was based on a trapezoidal stress-block in the con-
crete with a maximum stress fe" = f,' (Fig. 8). Contrary to Melan's
assumptions, (8) however, the properties of Jensen's trapezoid were re-
lated to the cylinder strength
Eo = P- (27a)
Ec
30,000,000
E - = (27b)
S 10,000
5+-
Furthermore
F e (1 - 3) = Eo (27c)
in which the plasticity ratio, 3, was defined by
1
-2 = (27d)
1 +(± ('
+ \ 4000 /
These assumptions led to the following ultimate moment
M 1 pf\
- pf, 1 (28)
bd2 N f,'
-where N is a function of f only.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Jensen then made the simplifying approximation that N = 2, which
is the equivalent of using a rectangular stress-block with depth kic
since Stiissi's constants ki and k2 (Fig. 8) corresponding to N= 2 are
related by k2 = Y2k1. Thus, Jensen's equation for tension failure was
reduced to Eq. (16).
For compression failures, a compatibility equation was used, in-
volving Eq. (27) and a trapezoidal stress-strain relation for the steel.
Studies of the effects of strain hardening in the tension reinforcement
were also made.
r. R. Chambaud, 1949(02, 108)
The flexure of beams with rectangular cross section and tension re-
inforcement only, was considered. The assumptions made were, how-
ever, of such a nature that the generalization of the theories involved is
a matter of algebra only. In both his two papers dealing with ultimate
theories for reinforced concrete beams, Chambaud developed theories of
the Stiissi type, assuming a constant compressive ultimate strain for the
concrete, e. = 3.6 per mill. The correctness of this value was verified
by tests.
Mr. Chambaud's first paper'102) presents a stress-block in the con-
crete very similar to those of Baumann, Brandtzaeg and Bittner (Fig.
8). The initial modulus of elasticity, Ec, was determined from 5.5-in.
cubes tested perpendicular to the direction of casting with cardboard
pieces inserted between the bearing plates of the testing machine and the
cubes. The maximum stress in flexure, fc", was assumed equal the com-
pressive strength of the cubes mentioned above.
The second paper(108s presents a stress-block as indicated in Fig. 8,
the modulus of elasticity, Ec, and the maximum concrete stress, fc",
both being assumed as above.
s. Concluding Remarks
The ultimate moments derived in the theories discussed above differ
relatively little in the case of tension failures of beams. The majority
of the authors mentioned presented equations which may be derived
from the following expression
M k2 fy,
= pfI, 1- p . (29)
bd kI fe"/
The ultimate compressive stress in flexure, fc", has been given values
from 0.85 f/' to f,.' = 1.18 fe'. The ratio k2/ki has varied from 0.50,
which corresponds to a rectangular stress distribution, to 0.67 for a
triangular distribution. In practical cases, the ratio pf~,/fc" is generally
ILLINOIS ENGINEERING EXPERIMENT STATION
of the order of 0.2. Hence, the extreme variation of Eq. (29) is from
about 0.85 to 0.92 times pf,,,, or approximately eight percent. This vari-
ation is of the same order of magnitude as the experimental scatter in
well-controlled laboratory tests. Hence, any reasonable equation such
as Eq. (16) or (19), and even a fixed value of 7d/8 for the internal
moment arm, seem to be satisfactory for design purposes, when the
member fails in tension and the percentage of reinforcement is small.
For compression failures, theories of the Sttissi type lead to Eq.
(10) and (13). By proper choice of ki, k2 and e., this system of equations
will give accurate results, and the theory is further flexible enough to
permit extension beyond the symmetrical bending of a rectangular
section. It may also be noted that, in the standard theory, ki = 0.5 and
the modular ratio n replaces m. Hence, Eq. (13) will reduce to the well-
known expression
c/d = V 2pn + (pn)2 - pn. (30)
On the other hand, equations of the form
M
-bd = af' (31)
bd2
have great advantages in their simplicity. Values of a from 1/2.6 to 1/3
have been proposed.
All theories which include studies of compression reinforcement
have proposed that such reinforcement should be considered effective
with its full yield-point stress.
Some applications of ultimate theories to combined bending and
axial load have been made, most of which are referred to in detail in
Sections 14 and 15.
A number of authors not referred to above have made valuable
contributions to the ultimate theories of reinforced concrete without
claiming that their assumptions were new. This is especially so in recent
years since every set of assumptions logically possible seems to have
been tried in the past. These authors are referred to in the bibliography
of this bulletin.
Numerous articles and several books have pointed out the shortcom-
ings of the standard theory and a transition to ultimate load design has
been recommended, at times emphatically. Safety factors to be used in
such ultimate design have been subject to considerable discussion. The
majority of authors seem to agree that, contrary to our present design
methods, equal safety factors for steel and concrete should be used. It
has often been recommended, however, that the various ultimate equa-
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
tions be entered with the minimum, not the average, concrete strength
specified for a certain structure. It has also been repeatedly recom-
mended that a smaller factor of safety be used for dead load than for
live load.
11. Basic Assumptions in Writer's Flexural Analysis
A study of the many assumptions that have been made in the past as
applied to the results of the tests reported herein showed that the ap-
proach used by Stiissi (Fig. 8) is satisfactory for the purpose of analyz-
ing the behavior of reinforced concrete members subject to combined
bending and axial load.
a. Distribution and Magnitude off Compressive Stresses in Concrete
The stress-strain relation of concrete subjected to concentric com-
pression has been the subject of many tests and considerable discussion
in the past. A great number of mathematical expressions for this relation
have been developed, most of which consider the range from zero to the
maximum stress only, since the final collapse of a compressive specimen
generally was believed to take place very shortly after the maximum
stress was reached. Such stress-strain relations were also applied to bend-
ing by assuming a linear distribution of strains in the compression zone
of a beam. At the maximum load the extreme "fibers" in a beam were as-
sumed to be subject to a maximum stress and a corresponding ultimate
strain which were both similar to those determined from the simple
compression test. Talbot(10, 11) recognized, however, that an ultimate
strain can be developed in flexure which is greater than the strain corre-
sponding to the maximum stress in concentric compression. Since most
early investigators removed strain-measuring instruments before the
ultimate load was reached in order to avoid damage, and since the re-
searches in the first three decades of this century were concerned with
conditions under working loads mainly, this important observation
seems to have been little known until it was rediscovered by 0. Baumann
in 1934.(44)
It has later been shown that such sudden failures as have been ob-
served in concentric compression often are properties of the testing
machine rather than of the test specimen, and stress-strain relations for
concrete in compression have been reported(59, 60 92, 104) which have been
obtained in such a manner that loads and strains could be observed be-
yond the maximum load. An example of such diagrams is given in Fig. 9.
It is recognized that the application of such a stress-strain relation in
a flexural analysis has been questioned.('13) A plain concrete specimen
which has been strained beyond the ultimate load in concentric com-
pression is generally badly cracked and the response to load is highly
ILLINOIS ENGINEERING EXPERIMENT STATION
sensitive to time. Furthermore, the application to bending of stress
strain relations obtained in concentric compression rests on the assump-
tion that the stress which occurs as a response to strain is independent
of the space gradient of strain. This assumption is one of expediency
only. Its justification has never been proved, since measurements of
stresses rather than strains are difficult indeed.
It has been observed, however, that considerably larger strains may
be developed in bending than in concentric compression before cracks
appear on the concrete surface. The highly strained outer fibers seem to
Strain in Thousandths, (%o)
U.S. Bureau of Reclamation
Fig. 9. Tests of 3- by 6-in. Cylinders
be able to yield and thus transfer stress to less strained fibers closer to
the neutral axis and to compression steel. It is reasonable to assume,
therefore, that the general characteristics of the diagrams in Fig. 9 are
applicable to flexure. Hence the relation shown in Fig. 10 was adopted.
The maximum stress in flexure, fc", has generally been expressed in
terms of the strength of cylinders, prisms or cubes tested in concentric
compression. If such compressive specimens were cut from the actual
structural elements, it is reasonable to expect that the relation between
f"' and f0' or fu' would be largely dependent on the effects of size and
shape. (19, 45) However, auxiliary compression specimens are generally
cast in separate forms, and it can therefore not be expected that the
effects of degree of compaction during casting, curing conditions, and
possible later drying conditions will be the same for the compression
specimens as for the larger structural elements. It must also be recog-
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
nized that the quality of concrete in beams and columns cast from the
same concrete mixture may differ, since beams generally are cast in a
horizontal position while columns are cast vertically.
In later years it has been attempted to measure concrete strength in
structural elements directly by means of indentation tests(01) or wave-
velocity methods.(03) Since such methods are still in the experimental
stage, however, the strength of 6- by 12-in. cylinders, fc', has been
adopted as a measure of concrete strength in the present tests. The
maximum stress in flexure, fc", corresponding to the column specimens
was chosen equal 0.85 fc'. This value was found as an average in numer-
ous tests of vertically-cast concentrically loaded columns tested with flat
ends.(43) Effects of size and shape of the columns as well as of the casting
position is therefore included in the factor 0.85.
It is believed that the initial, curved part of the stress-strain diagram
in Fig. 10 is fairly similar to the relation in direct compression. Since
the important factors in a flexural analysis related to the stress-strain
S-raA'/, c',
Fig. 10. Assumed Stress-Strain Diagram in Flexure
relation are k,, k2 and e., small variations in the initial part of the stress-
strain diagram are of minor importance. Auxiliary tests of 6- by 12-in.
cylinders showed that Ritter's parabola is a good approximation when
expressed in the following form
f =f[2 : -( . (32)
With eo = 2fc"/Ec this may be written as
f =eE,1 - 4".
\ 4f"
ILLINOIS ENGINEERING EXPERIMENT STATION
Cy/ina/er Strength, f,, in /b. per sq. in.
Fig. 11. Modulus of Elasticity of Concrete
The initial modulus of elasticity, Ec, was determined from cylinder tests,
Fig. 11. A satisfactory agreement was found with Inge Lyse's equation
Ec = 1,800,000 + 460f/'. (34)
It was decided, however, to use f' =f," in Eq. (34) in order to determine
the value of Ec corresponding to the column specimens. Hence, for a col-
umn with cylinder strength f,' = 4000 p.s.i., fe" is 3400 p.s.i., and the
modulus of elasticity, E,, is (1,800,000+460 - 3400) p.s.i.
The ultimate strain, e., was determined from tests of eccentrically
loaded columns reported herein. Such strain measurements are, how-
ever, difficult to interpret. With one exception, all columns were cast
and tested in the same vertical position, and it was observed that all col-
umns failed in compression in the upper half. One column was turned
upside down before testing, and only this column failed in the lower
half. This weakness of the upper portion of vertically-cast columns,
which the writer believes is due to a water gain with resulting lowered
c/w ratio in the upper part and a better compaction in the lower part of
the specimens, was also observed and discussed by Slater and Lyse.(251
~'Fa//ure
Crack
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
I
I
t==
I
0 /0 Z.O 3.0 4.0
Strain in Thousandths, (%o)
,Strains a, P =17/S 7 iks
(Pmax. =8/ k=,0s)
Co/vu
-SR-4, A
I- SR-4, A-I
I
Fig. 12. Distribution of Strains over Height of Column
In order to throw some light on the distribution of strains along the
length of the column shaft, an extra column specimen was prepared and
tested with a large number of SR-4 gages attached to the compression
face. The distribution of strains near the ultimate load is given in Fig.
12. It should be noticed that a general trend of increasing strains exists
from the bottom towards the top of the column shaft. Near failure two
major compression cracks extending horizontally across the compression
face were developed as indicated in Fig. 12. As the final failure took
place, the shaded area shown in the figure and marked "Failure" was ex-
truded. Unfortunately, both major compression cracks occurred be-
tween two 1-in. gages. Nevertheless it appears that the strain at a given
high load is not a well-defined quantity. At least three different quan-
tities of strain may be considered: (1) The average strain over the entire
length of the compression face, (2) A strain measured over a reasonably
long gage length (6 in.) in the failure region, and (3) A local strain meas-
ured over a short gage length (1 in.) at a compression crack. These three
quantities will increase in value in the order listed above, and it is
believed that near failure No. 3 will be at least twice as large as No. 1.
L
m
>
\
'n B-2c
=4220*ps/
9 Gag'e, 6:-1/.
c Gage, 6-in.
6,7q__1__ 7
ILLINOIS ENGINEERING EXPERIMENT STATION
It is felt, however, that No. 2 is the most reasonable value to be con-
sidered in a flexural analysis. Hence two SR-4, A-9 gages with 6-in. gage
length were attached to the compression face of all columns, one near
the center and one near the upper quarter point.
Strains measured in the failure region in this manner are shown in
Fig. 13 for the columns C-1 to C-5. Similar curves were prepared for all
other columns. It should be noted that while the ultimate load was ob-
served in all cases, the last increment of strain could generally not be
observed. Thus, the dotted lines in the last increment of all curves in
Strain in Thousandths, (%o)
Fig. 13. Load-Strain Curves for Columns C-1 to C-5
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Fig. 13 represent an extrapolation of strain to a known ultimate load.
Ultimate strains obtained in this manner are presented in Fig. 14 for
all square columns failing in compression before or shortly after the ten-
sion steel reached yielding.
The experimental findings in Fig. 14 are compared with relations be-
tween f/' and c. given by previous investigators.
Cy/nder StreFngh, f., in /b per sq. In.
Fig. 14. Ultimate Strains
A. Brandtzeg(51, 56, 57, 1) developed the empirical Eq. (17). R. Sali-
ger(60) reported values of ultimate strain from three to seven per mill
and recommended Eq. (18).(99) V. P. Jensen(so' 81) assumed
(1 -- ) E(
where the plasticity ratio, #, and the modulus of elasticity, Ec, were as-
sumed as functions of f' (Eq. 27). M. Ro(78) gave the following equation
for the ultimate strain
fu = (3.5 + 2860/fZ') 10-3.
(35)
ILLINOIS ENGINEERING EXPERIMENT STATION
R. Chambaud reported some carefully made tests in 1949.(102, 108) In
beam tests using electric strain gages, he found a constant value f = 3.6
per mill for cube strengths ranging from about 2000 to 7000 p.s.i.
It appears from Fig. 14 that the findings of these earlier investigators
are very different. This is understandable, since the test values obtained
probably are sensitive to time effects, the gage length used, and the loca-
tion of the gage with respect to compression cracks.
A considerable scatter is also present in the test results presented in
Fig. 14. Since the ultimate strength of reinforced concrete members is
rather insensitive to variations in e., however, a constant average value
4u = 3.8 per mill was considered satisfactory for the present analysis.
Table 6
Values of k, and kJ
Writer's Theory Assumptions Jensen's Modified Theory Jensen's Original Theory
of Fig. 10 f.' =0.85f/ f.' =A/
/ k k a ki k2 a ki k2 a
0 0.925 0.513 0.555 1.000 0.500 0.500 1.000 0.500 0.500
1000 0.873 0.481 0.551 0.978 0.490 0.501 0.970 0.485 0.501
2000 0.835 0.459 0.550 0.923 0.463 0.502 0.900 0.452 0.502
3000 0.808 0.444 0.550 0.856 0.431 0.503 0.820 0.417 0.508
4000 0.786 0.432 0.550 0.790 0.404 0.512 0.750 0.389 0.518
5000 0.770 0.423 0.550 0.735 0.383 0.521 0.695 0.370 0.531
6000 0.758 0.417 0.550 0.691 0.368 0.532 0.654 0.358 0.547
The descending part of the stress-strain diagram (Fig. 10) was as-
sumed linear, and the value of Af," =0.15 f," was found to give the best
agreement with the ultimate loads of columns. Thus the slope of the de-
scending branch in Fig. 10 is intermediate between Chambaud's assump-
tion (Fig. 8) and the results of compression tests (Fig. 9).
The constants ki and k2 (Fig. 10) are presented in Table 6 as a
function of the cylinder strength, fe'. Values of the corresponding con-
stants as determined from V. P. Jensen's assumptions are also entered
in this table.
b. Tensile Stresses in Concrete
It is recognized that the concrete near the tensile reinforcement will,
to a certain extent, relieve steel stresses between cracks.(60, 90) The writer
also believes that concrete near the neutral axis develops some tensile
stresses. It is felt, however, that such increases in ultimate flexural
strength of reinforced concrete members as may be due to the concrete
carrying longitudinal tension are generally small, and are further un-
reliable for practical purposes.
F. Gebauer(42) (Fig. 8) and others have pointed out that a precom-
pression due to shrinkage of the surrounding concrete will exist in the
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
tensile reinforcement. Thus, an equivalent tension in the concrete should
be considered. The strains which correspond to a precompression of
this nature have generally been observed with reference or zero read-
ings taken after the cement had set. In this manner, precompression
stresses as large as 10,000 p.s.i. have been found. In the present tests,
measurements of such compressive stresses in the reinforcement were
made by means of electric SR-4 gages attached to the steel and em-
bedded in the concrete. Reference readings could thereby be taken im-
mediately after the concrete was placed and the steel strains could be
followed throughout the period of setting, moist curing and final drying
in the air of the laboratory. Variations in steel strains were found very
similar to those reported by F. R. Beyer.(107) The total compressive
stresses after 28 days, when the specimens were tested, varied from a
few to about 500 p.s.i. only.
For these reasons it was assumed in the present analysis that no
tensile stresses exist in the concrete.
c. Bernoulli's Hypothesis
The hypothesis of linear distributions of strains in bending has some-
times been questioned when applied to combined bending and axial
load. The numerous strain measurements made in these tests provide
some evidence on this question.
Figures 15a, b and c and 16a, b and c show typical distributions
of strains across a section near the failure region. Some representative
examples of compression and tension failures as well as those near bal-
anced sections are given. Some deviation from linearity must be ex-
pected due to inaccuracies in individual strain measurements and to
small errors in location of gage lines. It is evident from the figures, how-
ever, that a reasonable agreement exists between strains measured in the
reinforcement and on the concrete surface. Furthermore, the departures
from linearity appear to be inconsistent, indicating accidental or local
rather than systematic variations. Hence it is assumed that Bernoulli's
hypothesis is valid.
d. Absence of Slip
Bond was of minor importance in these tests since theoretically no
shear existed in the prismatic part of the columns in which failure took
place. Even so, local bond stresses and local slips must exist near tension
cracks. Such slips were believed to be small, however, since a modern
type of deformed bar was used as longitudinal reinforcement. It is
reasonable to assume, therefore, that no general slip exists between con-
crete and reinforcing steel.
52 ILLINOIS ENGINEERING EXPERIMENT STATION
II -i
tQ .-
s^-°3
(s'-pufsnol) '°% U 1--wS
ft*
r
0
u
41
0
0
w;
U)
C
0
.0
C
I
U)
0)
U-
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
(s9z1Pu1snoc/_4) 1 °. u uq1 /S
ILLINOIS ENGINEERING EXPERIMENT STATION
e. Stress-Strain Relation for Reinforcing Steel
This relation was assumed as the usual trapezoid with a yield level
at the yield-point stress. The moduli of elasticity indicated in Table 4
were used, as all steel stresses in this bulletin are referred to the nominal
area of reinforcing bars. It was further assumed that the stress-strain
relations in compression and tension are equal.
Figure 3a and b indicates that such a trapezoidal relation is an ex-
cellent approximation up to a strain of about 1.5 percent. None of the
specimens reported herein appeared to be reinforced lightly enough
to develop steel strains in the strain hardening range above 1.5 percent
before the maximum load was reached.
CW
S8(b)
^- e=3.8%0
4'^ (a)
Fig. 17. Flexural Analysis of Rectangular Columns
IV. PRESENTATION AND ANALYSIS OF TEST RESULTS
12. Development of Equations for Ultimate Loads of Rectangular Sections
a. Tension Failures
A rectangular section with reinforcement at opposite faces is con-
sidered loaded in the plane of symmetry as shown in Fig. 17. The basic
assumptions of the analysis were discussed in Section 10. Since it is as-
sumed that the ultimate strain of the concrete is ae = 3.8 per mill while
the yield-point strain of the steel generally is about one to two per mill,
both tension and compression reinforcement may be assumed to be
strained over the yield point at failure. Using the notation of Fig. 17a,
and taking moments about the tension steel
Pe' = kif,"bc (d - k2c) + A,'d'fy,'. (36)
Equilibrium of forces gives
P = kif,"bc + A.'f},' - AJ,. (37)
For tension failure f. =f,,, and solving for c/d, we obtain with a = k/ki =
0.55 (Table 6)
c 1 e'
ki- = -- - ---1
d 2a d
+ - 1 + 4a p p - p (38)
d I f," d + k
A ' A,
in which p' = , p - and fe" = 0.85f,'. Substituting Eq. (38) in
bd bd
Eq. (37) gives
P = f"bd j' p fP + -( - 1y ' ]]
2- 1 24a fp ' + p - . (39)
ILLINOIS ENGINEERING EXPERIMENT STATION
Brandtzxeg, Saliger, Whitney, Jensen and others have assumed a= Y
which corresponds to an equivalent rectangular stress-block. When this
value, a = Y2, is substituted in Eq. (39) the resulting expression for the
ultimate load is very similar to those developed by A. Brandtzag and
C. S. Whitney.
Equation (39) may be simplified for symmetrical reinforcement,
when pf,, = pfp,',
f '"bd e' e' 2 fp d'
P = 1 + -- 1 + 4ap (40)
2a d d fc" d
When there is no compression reinforcement, p'= 0 and the ultimate
load is
P = f,"bd - p + -( 1
f,' 2a d
- -1) 4ap (41)
When e'/d is large (>1.5), the two equations above do not give
satisfactory slide-rule accuracy. This can be improved by multiplying
and dividing by + (- - 1) +V . . . . . Furthermore, putting
a = Y results in an error less than one percent if e' is large. For sym-
metrical reinforcement, 2A,,,
P = (42)
(e' - d) + (e' - d)2 + 2A-
and when there is no compression steel
(e' + d)- (e' - d) + 2Af, b
P = Aj,* ,f e' (43)
(e' - d) + (e'-d)2 +2A.- b
In addition when e' = co (pure bending), Eq. (42) becomes M = A,fypd',
while Eq. (43) reduces to Eq. (16).
For large percentages of compression steel (p' > 2 percent) and
relatively small eccentricities, the error involved in neglecting a sub-
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
traction of the concrete area displaced by the compression steel may be
four to six percent on the unsafe side. This error may be corrected by
considering an effective yield point, fp,' - fe". For symmetrical rein-
forcement this leads to
P = fi"bd - p +- - - 1
C e' 2 /fyp d' e' - d' \
(- 1)2±4ap + . (44)
d Q` d d
b. Compression Failures
The stress in the tension steel at failure, being less than the yield
point, must be determined by geometrical considerations. Referring
to Fig. 17, we obtain the compatibility equation
d-c
f.= =uE- 5 fy,. (45)
c
N
i..
N
Dept/h to Neutral Axis, c, in /ncl7es
Fig. 18. Approximation Procedure
ILLINOIS ENGINEERING EXPERIMENT STATION
For compression failures, the error that is involved in neglecting a
subtraction in Eqs. (36) and (37) for the concrete displaced by the
compression reinforcement may be of the order of 4-6 percent on the
unsafe side. This error may be corrected by considering the effective
yield point of the compression steel as fp' - fc" in Eqs. (36) and (37).
Solving Eqs. (36), (37) and (45) for c and P yields a cubic equation
in c, the solution of which is very time-consuming. Therefore a pro-
cedure involving successive approximations was developed for the pur-
pose of the present analysis.
With known section dimensions, material constants and eccentricity
e', the problem is to find c and P. The approximation procedure is illus-
trated in Fig. 18 for a column of Group II with f,' = 6000 p.s.i. and e' =
6.42 in. An estimated value of c is substituted in Eqs. (36) and (45). The
resulting values of P and f, are introduced in Eq. (37) and a new value of
c results. This new c is carried back to Eqs. (36) and (45), etc. Solving
Eq. (37) for c involves a linear equation only. The computations of this
example are tabulated below:
For Group II: A. = A,' = 1.24 sq in., d = 8.67 in., d' = 7.34 in., b = 10 in.
Iffe' = 6000 p.s.i.,f," = 5100 p.s.i., ki = 0.758, k2 = 0.417
Corrected f,' = 43.6 - 5.1 = 38.5 ksi; eE, = 3.8 - 28.0 = 106.4 ksi
Eccentricity e' = 2.75 + 3.67 = 6.42 in.
c = 6.0 7.46 7.19 7.23 in.
kifc'bc = 232.0 288.4 278.0 279.5 kips
Eq. (36) kJf'bc(d-k2c) = 1431.4 1603.5 1576.3 1582.0 kips-in.
q. f A d' = 350.4 350.4 350.4 350.4 kips-in.
Pe' = 1781.8 1953.9 1926.7 1932.4 kips-in.
P = 277.5 304.3 300.1 301.0 kips
Eq. (45) f, = 47.3 17.3 21.9 21.2 ksi
A,(f,' - f.) = -10.9 26.3 20.6 21.5 kips
Eq. (37) P - A.(f,' -f,) = 288.4 278.0 279.5 279.5 kips
c = 7.46 7.19 7.23 7.23 in.
Solution: P = 301.0 kips, c = 7.23 in.
By differentiation it may be found that this procedure will con-
verge if
kif," 1 + d -2k2C - peE, -d > 0 (46)
which is generally the case.
If the left side of Eq. (46) is near or equal to zero, oscillations will
take place. In this case the average of the original and the new c should
be used in the next approximation cycle.
Finally, if the left side of Eq. (46) is negative, which may occur for
large values of p and e' with small values of f/', divergence will take
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
place. The above procedure should then be modified as follows: Eqs. (36)
and (45) are entered with a value c = cl, and c = c2 results from Eq. (37).
Then a value Ci+'y(c2-Ci), where 7 is a fraction, should be returned to
Eqs. (36) and (45). Convergence will result if a suitable value of 7 is
chosen by trial.
If the eccentricity, e, is very small, the neutral axis will fall outside
the section, Fig. 17b. The equilibrium equations must then be modi-
fied to
Pe'= kif,"bc(d - k2c) +A,'d'f '+ Abx (36a)
P= kzj"bc +A,'f,,' - Af, - Ab. (37a)
Equation (45) is still valid; and the quantities A and x may be found by
geometrical considerations. Hence, the approximation procedure out-
lined above may still be used.
c. Balanced Sections
A given section is considered balanced when it is loaded with such
an eccentricity that the tension steel reaches its yield point and the
concrete reaches its ultimate strain at the same load. When, for a given
eccentricity and concrete section, both limiting conditions are reached
at the same load, the reinforcement is referred to as being balanced. The
limiting condition between tension and compression failures, which is
defined as a balanced failure, may be referred to in terms of the ratio
k = c/d. For the balanced condition, this ratio is
Cb eu
kb =- (47)
d e. + fyI/E,
A tension failure will result if 0 < k < kb, and compression failure will take
place if kb<k< oo. The various balanced quantities, Pb,eb, Pb and pb
may then be computed from Eqs. (36) and (37), if the concrete section is
given. Any two of the four quantities may be chosen, and the remaining
two may be computed.
13. Test Results of Rectangular Sections, Groups I, II and III
a. General Behavior and Modes of Failure of Specimens
Two modes of failure prevailed in the tests reported herein: com-
pression failures and tension failures. Only a few columns had such a
combination of variables that the mode of failure was close enough to
balanced conditions to be difficult to observe. The observed modes of
failure of the individual test columns are listed in Tables 7, 8 and 9. The
agreement with predicted modes of failure is discussed in Section 13b.
ILLINOIS ENGINEERING EXPERIMENT STATION
The general behavior of the test columns may, perhaps, best be stud-
ied in the light of the assumptions made in Section 11. For the given sec-
tions and eccentricities (including deflections) the theoretical strains in
concrete and steel may be computed from the smallest loads throughout
the loading range to failure. This may be achieved by assuming a range
of values for the concrete strain ec at the compression face and computing
the corresponding values of ki and k2 in accordance with Fig. 10. For
example when e, = o, ki = 2/3 and ks = 3/8. When e < o, it may be
found by integration of Eq. (32) that
4 ---
- 1 -(- k = (48)
3e1
For fo <e e < c., the constants ki and k2 may be found by geometrical con-
siderations. Then the position of the neutral axis and the load corre-
sponding to the given section and eccentricity with the successive chosen
values of ec may be computed by an approximation procedure very
similar to that presented in Section 12b.
In the following discussion, the observed and predicted behavior of
eight typical specimens is presented.
Columns A-7b, C-8a and B-13a (Figs. 19, 20 and 21) are typical
examples of compression failures. Observed strains in the tension and
compression reinforcement are presented as an average of three SR-4,
A-11 gages. The strains on the compression face of the columns are gen-
erally given as measured by an SR-4, A-9 gage at and a short distance
away from the failure region. Center deflections of the prismatic part
of the column between the brackets are given as the center reading of the
deflection bridge minus the average of the readings at the top and bottom
of the column shaft (Fig. 7). The theoretical deflections were found as:
S = cL2/8c, where L = 50 in. The observed values of the position of the
neutral axis were obtained as indicated in Fig. 15.
It appears that Column A-7b (Fig. 19) deformed continuously under
increasing load with a slight movement of the neutral axis towards the
compression face until the yield point of the compression reinforcement
was reached. Then a discontinuity occurred in all theoretical curves pre-
sented in the figure as the compression steel continued to deform without
any increase in stress, and thus the concrete alone was carrying all
further increase in internal compressive force.
Columns C-8a and B-13a (Figs. 20 and 21) behaved in a very similar
manner. The neutral axis did, however, move slightly towards the tension
face of these columns before the compression steel reached yielding.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Inches k = c/d
Fig. 19. Behavior of Column A-7b
The mode of failure for all three columns mentioned above was a
typical compression failure. The final failure was caused by crushing of
the concrete at the compression face of the specimens after the compres-
sion steel reached yielding, while the tension steel did not reach yielding.
It should be noted that strains increased very rapidly with load when
the compression steel was yielding. Throughout these tests this phe-
nomenon was found to be more pronounced, the larger the contribution
of the compression reinforcement to the total internal compressive force.
This rapid increase of strains reflects important properties of eccentrically
loaded members. It is apparent that such a member is in a semi-neutral
state of equilibrium when failure takes place. The concrete is in a plastic
state in which the rate of loading and the duration of loads become
factors of considerable importance.
The presence of such a semi-neutral equilibrium has been observed
in the past and it has been stated that the concrete fails in compression
because the compression steel yields. Thus it has been claimed that a high
strength steel cannot be used effectively as a compression reinforcement
ILLINOIS ENGINEERING EXPERIMENT STATION
in eccentrically loaded members without a very high grade of concrete.
Since large deformations took place after the intermediate grade of steel
used in these tests reached yielding, however, the writer believes that a
reinforcement of hard grade can develop its full yield point even with a
concrete strength below 2000 p.s.i. Figure 20 shows an example of yielding
in an intermediate grade of compression steel being easily developed with
a concrete strength of 1820 p.s.i.
Volume changes of concrete due to centric and eccentric loads have
been studied in the past.(20, 21, 22, 35, 51) The volume compression of a
centrically loaded specimen may be expressed as
e,= e - 2E2 (49)
where e1 and E2 are the strains in longitudinal and lateral directions,
respectively. For a small element at the compression face of an eccen-
trically loaded member, however, a strain gradient is present and the
Inches k = c/d
Fig. 20. Behavior of Column C-8a
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
strains in the two lateral directions are probably not equal. Nevertheless,
Eq. (49) may be used as an approximation, £2 being the lateral strain at
the compression face. Numerous tests have shown that the volume of
concrete specimens decreases under increasing uni-axial compression,
whether a strain gradient is present or not, until a stress of 80 to 90
percent of the ultimate is reached. Then the volume starts increasing at
an increasing rate, so that a volume expansion is present at failure. The
load at which the derivative of volume strain with respect to load is zero
has been referred to as the critical load. It is believed that progressive
internal splitting of the concrete is initiated at the critical load. Such a
concept is basic in Brandtzweg's theory of failure for concrete.(20)
In the present tests of columns with a prismatic shaft 50 in. long it
was found that the lateral strains had to be measured in the region of
local compression failure if the volume strains were to be significant. One
lateral gage only was used for each specimen, and this gage could gener-
ally not be placed in the expected failure region, since the longitudinal
gages were already placed in this position. Therefore, the information
gained regarding lateral and volume strains was of qualitative value
Inches
Fig. 21. Behavior of Column B-13a
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 22. Detail of Compression Failure
only. Critical loads ranged from about 80 to 95 percent of the ultimate
loads, the percentage generally being higher the greater the force in the
compression steel with respect to the total internal compressive force.
The presence of a disintegration of the concrete was also visible on
the surface of the specimen shortly before failure. First, vertical tension
cracks appeared on the compression face of the columns as shown in
Fig. 22. At a slightly higher load compression cracks with a general
horizontal trend formed as shown in Fig. 22, and the layer of concrete
over the steel spalled off at a slightly higher load. Immediately after
such spalling, the compression reinforcement buckled between ties, as
shown in Fig. 23, and the load capacity of the members thereby dropped
to a fraction of the ultimate load.
It may be noted, therefore, that the compression mode of failure of
eccentrically loaded tied columns was found to be very similar to the
failure of such columns subjected to concentric compression. After failure
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Fig. 23. Columns B-6 to B-10 after Failure
ILLINOIS ENGINEERING EXPERIMENT STATION
had been initiated by crushing and spalling of the concrete at the com-
pression edge, the load fell off very quickly if further deformations were
applied tophe column.
The behavior of the columns which were loaded at mid-depth of the
section, e = 0, was found to be very dependent on the arrangement of the
reinforcement. In Group I an unsymmetrical reinforcement was used.
The plastic centroid of such a section may be defined as the point at
which the resultant of internal uniform stresses, computed with f, = fyp
and fc = 0.85 fe', cuts the section. If the load is applied exactly at this
plastic centroid, the column is truly centrically loaded. For unsymmetrical
reinforcement, the plastic centroid is not at mid-depth of the section, but
closer to the geometric centroid of the reinforcement. Hence, it is reason-
able that the behavior of Columns No. 1 was found to be very similar
to the compression failures already discussed, since these columns actually
were eccentrically loaded.
Columns No. 6 and 11 were symmetrically reinforced and therefore
concentrically loaded for e = 0. The behavior of these columns is dis-
cussed in Section 16.
O
r.1
't
13
Fig. 24. Behavior of Column C-5a
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Incm7es
Fig. 25. Behavior of Column B-14a
Ir=c/a
Column C-5a (Fig. 24) is an example of a column with an almost
balanced section. The tension steel, however, theoretically reached yield-
ing and caused some movement of the neutral axis towards the compres-
sion edge before the concrete reached its ultimate compressive strain.
Column B-14a (Fig. 25) gives an almost perfect balanced condition.
As loading proceeded, the first discontinuity was caused by yielding of
the compression steel. Yielding in the tension reinforcement followed,
and failure of the concrete in compression took place after a small in-
crease in deformation, but practically no increase in load.
Columns A-5a, A-15a and A-lOa (Figs. 26, 27 and 28) are typical
examples of tension failures. The specimens behaved much like those
failing in compression until the tension steel reached yielding. Then,
however, a neutral equilibrium even more pronounced than that for
compression failures was established, and it was practically impossible to
set the columns to rest under an applied load so that readings of strains
and deflections could be made. It was nevertheless noted that very large
ILLINOIS ENGINEERING EXPERIMENT STATION
deformations took place practically without any increase in load before
the concrete failed and the compression steel buckled. Such a large de-
formation is shown in Fig. 23 for Column B-10a, the test of which was
discontinued shortly after the concrete had failed in compression. As
predicted by theory, the columns failing in tension appeared to possess a
far higher degree of toughness than members failing in compression. This
property is important in many practical cases, for instance in buildings
subject to uneven settlements.
At failure the compression reinforcement was generally strained be-
yond the yield point for tension failures as well as compression failures.
It should be noted, however, that Column A-10a (Fig. 28), which had a
concrete of high grade and a medium reinforcement, failed in tension;
and the movement of the neutral axis appeared to take place at such a
high rate near the ultimate load that the compression reinforcement did
Inches k c/ad
Fig. 26. Behavior of Column A-5a
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
not reach yielding. The simplifying assumption made in Section 11, that
the compression reinforcement always reaches yielding before failure, is
therefore somewhat in error in this case. This error is fortunately not
important since the contribution of the compression steel to the ultimate
load is small as far as the ultimate load of such typical tension failures
are concerned.
It is felt that the observed and predicted strains, deflections and modes
of failure for all columns tested are in satisfactory agreement as indi-
cated by the eight typical examples given. The largest deviations are
found for the concrete strains, which is reasonable considering the cir-
cumstances shown in Fig. 12. The agreement between predicted and
observed strains in the reinforcement is generally excellent up to the
yield point, at which point a discontinuity is found in the theoretical
curves. In some cases, the measured strains do not follow the predicted
break. This is reasonable, since these measured strains are the average
of three gages and not a local observation in the failure region. Further-
more, the actual yield points of the individual reinforcing bars varied
from the average property assumed in the theoretical analysis.
Inches kr=c/d
Fig. 27. Behavior of Column A-15o
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 28. Behavior of Column A-lOa
The observed and predicted behavior of the test columns may also
be compared by studies of steel stresses and the position of the neutral
axis at failure. An example of such studies as applied to Group III is
presented in Figs. 29 and 30. The observed steel stresses in Fig. 29 were
obtained by multiplying the observed strains by E,. The theoretical values
were obtained by the writer's theory. It is felt that an agreement of
general nature exists between them. The relatively low measured steel
stresses for e = 2.73 may be interpreted as fairly good evidence of tensile
stresses in the concrete.
The observed positions of the neutral axis presented in Fig. 30 were
derived from graphs similar to those given in Fig. 15. As the theoretical
positions of the neutral axis are referred to the ultimate loads, while the
observed values refer to the last strain readings which were made at 90
to 95 percent of the ultimate, the direction of travel of the neutral axis
is indicated as observed in the tests. Two columns with e = 12.90 in.,
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
rYield .Point=43,600 lb. per sq. in.
50
.40
S30
' 20
I1.
c.^
.,
^- ^ / -
o e=S33in.
e = 2.73 .,
/000 2000 3000 4o00 .000 6000
C/indr rengh in er Streng per sI. l/.
Fig. 29. Tension Steel Stresses at Failure
A-15a and b, are not plotted in the figure because no strain readings were
made close enough to the ultimate to give significant information regard-
ing the position of the neutral axis at failure. A general agreement
between observed and predicted behavior is present in Fig. 30.
Hence it is believed that the behavior of tied reinforced concrete
members subject to combined bending and axial load is rather well rep-
resented through the assumptions made in Section 11.
The behavior of eccentrically loaded tied columns as observed in
these tests may be re-emphasized as follows:
1. The final failure of all columns was caused by crushing of the
concrete at an ultimate strain of about 3.8 per mill. After such crushing
had taken place, the compression reinforcement buckled between ties,
and the load capacity of the columns thereby suddenly dropped to a very
low value.
2. For moderate eccentricities it was found that the intermediate
grade compression steel used always reached the yield point before
failure of the concrete took place, even though the concrete strength
varied from about 1500 to 5500 p.s.i. This was also the case for eccentric-
ities as large as 1.25t if the concrete strength was low. For concrete
0
0
0
Group VZ
A,=A'= 2.40s$. n
0
ILLINOIS ENGINEERING EXPERIMENT STATION
strengths over about 3000 p.s.i. combined with eccentricities over 1.Ot,
however, yielding of the tension reinforcement generally caused such
rapid movements of the neutral axis that the compression steel did not
reach yielding before a complete failure had taken place by crushing of
the concrete.
3. It was observed that columns which failed in tension developed
much larger deflections before final crushing of the concrete took place
than did the columns failing in compression.
Cy/inder Strength in lb. per sq. In.
Fig. 30. Neutral Axis at Failure
4. Preceding the final failure of the concrete all columns were in a
semi-neutral equilibrium with the applied load. This semi-neutral state
of equilibrium was characterized by considerable increases in deforma-
tion for very small additions of load. Such relatively rapid increases in
deformation with load were initiated by yielding either of the tension or
the compression reinforcement, depending on the mode of failure.
b. Ultimate Loads
The essential information regarding the ultimate load capacities of
the test columns is given in Tables 7, 8 and 9. The measured ultimate
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
loads are also plotted in Figs. 31, 32 and 33 and compared with ultimate
loads predicted by means of the writer's, Jensen's and Whitney's theories.
The latter two theories are discussed in Sections 14a and b, respectively.
Theoretical relations between ultimate load, cylinder strength and
applied eccentricity were developed with these properties of the sections:
Group b = t A, A fyp E. f,,,' d d'
No. in. in.2 in.' p.s.i. p.s.i. X 10, p.s.i. in. in.
I 10 1.24 0.22 43 600 28 60 000 8.67 7.47
II 10 1.24 1.24 43 600 28 43 600 8.67 7.34
III 10 2.40 2.40 43 600 29 43 600 8.50 7.00
For all columns, a = 0.55, and the values of kI and k2 presented in
Table 6 were used.
Table 7
Results of Tests, Group I
A-la 5280 0 0.12 388 430 0.90 3.40
b 5660 0.37 0.14 441* 460 0.96 3.80
B-la 4250 0 0.12 343 348 0.99 4.90
b 4070 0 0.12 352 334 1.05 4.56
C-la 2270 0 0.13 222 199 1.11 3.60
b 2020 0 0.13 191 179 1.07 3.44
Av Cols. 1 0.13 1.01
A-2a 5280 2.5 0.22 239 232.5 1.03 3.34 C
b 5830 2.5 0.28 253 253.5 1.00 3.26 C
B-2a 4250 2.5 0.27 213 192.5 1.11 4.56 C
b 4070 2.5 0.24 190 186.0 1.02 4.60 C
C-2a 2270 2.5 0.27 118.5 114.0 1.04 3.00 C
b 1970 2.5 0.27 100.0 101.0 0.99 3.60 C
Av Cols. 2 0.26 1.03
A-3a 5660 5.0 0.32 133.5 147.0 0.91 3.00 T
b 5830 5.0 0.28 140.0 149.5 0.94 3.82 T
B-3a 4630 5.0 0.41 125.9 130.0 0.97 3.16 CT
b 4290 5.0 0.37 116.0 124.0 0.94 3.20 CT
C-3a 1880 5.0 0.28 60.5 67.0 0.90 3.80 C
b 1690 5.0 0.33 64.0 61.6 1.06 3.60 C
Av Cols. 3 0.33 0.95
A-4a 4810 7.5 0.45 84.5 83.3 1.01 .... T
b 5600 7.5 0.35 81.0 88.5 0.92 .... T
B-4a 3800 7.5 0.48 80.0 75.6 1.06 .... T
b 4290 7.5 0.52 81.0 79.5 1.02 .... T
C-4a 1690 7.5 0.32 50.5 47.5 1.06 3.12 C
b 1730 7.5 0.31 52.0 48.1 1.08 3.68 C
Av Cols. 4 0.40 1.03
A-5a 4810 12.5 0.40 48.2 44.6 1.08 .... T
b 5600 12.5 0.40 42.8 46.2 0.93 .... T
B-5a 4290 12.5 0.42 46.1 43.3 1.06 .... T
b 4590 12.5 0.45 45.5 44.1 1.03 .... T
C-5a 2310 12.5 0.34 39.0 35.7 1.09 4.40 CT
b 1770 12.5 0.34 32.8 32.0 1.02 3.68 CT
Av Cols. 5 0.39 1.04
Av Group I 1.012
* Error in eccentricity. P corrected from 401 to 441 kips.
t C, CT and T indicate compression, near balanced, and tension failure, respectively.
ILLINOIS ENGINEERING EXPERIMENT STATION
In the following discussion of ultimate loads the interest is aimed at
the properties of a column section, not at the properties of a column
with a certain length. Hence, the eccentricities used in predicting ultimate
loads are the initially applied eccentricities plus the deflections at failure
measured at mid-depth with respect to the knife edges. Such deflections,
Ae, are entered in Tables 7, 8 and 9. It may be noted that the indicated
values of Ae for tension failures probably are too small since they refer
to the beginning of the state of semi-neutral equilibrium preceding failure.
Fortunately, the columns which failed in tension had a large initial
eccentricity, so that errors in Ae are relatively insignificant.
Theoretical curves based on Eqs. (37) and (47) are shown in Figs.
31 to 33 representing the predicted balanced conditions of failure. All
combinations of cylinder strength and eccentricity giving ultimate loads
Table 8
Results of Tests, Group II
AV U01o. 0
A-7a 5240
b 5810
B-7a 4080
b 4040
C-7a 1970
b 1520
1.00
0.19 274* 273 1.00
0.26 284 294 0.97
0.25 256 229 1.12
0.24 248 227 1.09
0.28 141 146 0.97
0.27 12i 8 127 1 M0
Av Cols. 7 0.25 1.03
A-8a 5520 5.0 0.34 162 166.0 0.98
b 5810 5.0 0.40 152 169.6 0.90
B-8a 4700 5.0 0.35 156 154.0 1.01
b 4260 5.0 0.32 146 147.8 0.99
C-8a 1820 5.0 0.32 99 96.2 1.03
b 1820 5.0 0.39 99 96.2 1.03
C
C
C
C
T
T
T
CT
C
C
Av Cols. 8 0.35 0.99
A-9a 5100 7.5 0.37 89.0 95.6 0.93 .... T
b 5170 7.5 0.39 91.2 95.8 0.95 .... T
B-9a 4700 7.5 0.35 94.0 94.0 1.00 .... T
b 4370 7.5 0.32 89.5 92.3 0.97 3.36 T
C-9a 1880 7.5 0.38 73.0 72.6 1.01 4.32 CT
b 1730 7.5 0.35 65.5 70.6 0.93 5.52 CT
Av Cols. 9 0.36 0.97
A-10a 5100 12.5 0.28 46.1 46.8 0.98 .... T
b 5170 12.5 0.25 44.0 46.9 0.94 .... T
B-10a 4260 12.5 0.28 43.5 46.3 0.94 .... T
b 4370 12.5 0.29 44.0 46.4 0.95 .... T
C-10a 2300 12.5 0.35 44.5 43.2 1.02 .... T
b 1770 12.5 0.38 45.0 41.7 1.08 .... T
Av Cols. 10 0.31 0.99
Av Group II 0.992
* Error in eccentricity. P corrected from 229 to 274 kips.
t Be, C, CT and T indicate bearing, compression, near balanced and tension failure, respectively.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
above or below these lines are predicted to result in compression and
tension failures, respectively. The mode of failure of the various test
specimens was observed through measurements of strains and visual
inspection of the formation of cracks. Compression failures were char-
acterized by the formation of irregular compression cracks on the com-
pression face (Fig. 22) before the strains in the tension reinforcement
reached the yield strain. Tension failures, on the other hand, were char-
acterized by yielding in the tension reinforcement followed by large
deformations and considerable movement of the neutral axis before
crushing of the concrete took place. In intermediate cases, that is under
near balanced conditions, the mode of failure was difficult to observe since
the last strain readings generally were made shortly before the ultimate
load, and readings after the ultimate load are insignificant.
Table 9
Results of Tests, Group III
A-lla 4150
b 5050
B-11a 3870
b 4010
C-11a 2200
b 2070
.... 460 ... .. ... ... Be
.. 440 .. Be
0.08 500 494 1.01 .... C
0.10 485 501 0.97 .... C
.... 27 . .... Be
0.00 353 366 0.97 .... C
Av Cols. 11 0.98
A-12af 4150 2.5 0.20 315 286 1.10 .... C
bt 5050 2.5 0.22 325 321 1.01 .... C
B-12a 4300 2.5 0.22 303 292 1.04 3.48 C
b 4010 2.5 0.26 284 281 1.01 3.36 C
C-12a 2300 2.5 0.26 252 214 1.18 3.82 C
b 2200 2.5 0.22 230 210 1.10 4.48 C
Av Cols. 12 0.23 1.07
A-13at 5350 5.0 0.36 220 219.5 1.00 .... C
bt 4850 5.0 0.34 210 208.0 1.01 .... C
B-13a 3580 5.0 0.35 180 178.0 1.01 4.04 C
b 4290 5.0 0.34 206 195.0 1.06 4.12 C
C-13a 2300 5.0 0.33 151 146.5 1.03 4.36 C
b 2070 5.0 0.28 137 140.5 0.98 3.80 C
Av Cols. 13 0.33 1.02
A-14at 5350 7.5 0.37 142 150.0 0.95 .... T
b 5100 7.5 0.43 153 148.0 1.03 3.70 T
B-14a 3580 7.5 0.39 138.8 133.0 1.04 3.30 CT
b 4590 7.5 110 ..... .... .... Bond
C-14a 1950 7.5 0.34 115.5 105.5 1.09 4.30 C
b 2070 7.5 0.37 104.0 108.0 0.96 3.92 C
Av Cols. 14 0.38 1.01
A-15a 5100 12.5 0.42 88.0 81.0 1.09 .... T
b 4850 12.5 0.35 79.0 80.4 0.98 .... T
B-15a 3800 12.5 0.41 74.0 78.0 0.95 .... T
b 4630 12.5 0.42 84.5 80.0 1.06 .... T
C-15a 1950 12.5 0.39 72.5 72.1 1.01 3.40 C
b 2070 12.5 0.41 74.5 72.6 1.03 4.28 CT
Av Cols. 15 0.40 1.02
Av Group III 1.026
* Be, C, CT and T indicate bearing, compression, near balanced and tension failure, respectively.
t Strains were measured at midheight only. Hence, the ultimate strains are not significant.
ILLINOIS ENGINEERING EXPERIMENT STATION
- Writer's Theory-
--- odif/ied Jensen Theory
350 - Whitney
- Compression Fa/lure /
o- Aear Balanced ,
o- Tension Fa/ure /
Fig. 31. Ultimate Loads, Group I
K
'4
'3
t.
N.
6000 7000
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Such observed modes of failures are listed in Tables 7, 8 and 9. When
compared with the predicted modes in Figs. 31 to 33, it appears that all
failures observed as compression and tension failures were predicted
correctly; and failures observed as near balanced correspond to ultimate
loads near the theoretical dividing curve. Only one test specimen,
Column B-14a (Figs. 25 and 33) fell very close to a theoretical balanced
condition. Nevertheless, it is felt that the agreement between observed
and predicted modes of failure is very satisfactory.
The ultimate loads corresponding to the various eccentricities in each
group were computed by means of the equations developed in Section 11.
The average values of Ae for each curve in Figs. 31 to 33 were used, and
the curves were determined by establishing four to six points through
which a continuous curve could be drawn. The predicted ultimate loads
for the individual test columns, as presented in Tables 7 to 9, were
determined graphically from the figures, which for such purposes were
originally plotted to a much larger scale than presented herein. The theo-
retical equations used are referred to below:
Group I (Fig. 31) contains columns with a very light compression
reinforcement. Hence, no correction of fy,' was considered necessary. For
e = 0.13 in., Eqs. (36a), (37a) and (45) were used. For e = 2.76 in.,
and for the compression failures of e = 5.33, 7.90 and 12.89 in., Eqs. (36),
(37) and (45) were used. For all these compression failures, the mathe-
matical solutions were obtained by means of successive approximations.
For all tension failures, Eq. (39) was used.
Groups II and III (Figs. 32 and 33) contained sufficient compression
reinforcement to warrant a correction of f,p' to fy,' -fi". Equations (36),
(37) and (45) were used for all compression failures, and Eq. (44) was
used for tension failures.
It may be seen from Figs. 31 to 33 and the ratios between test and
calculated values of the ultimate loads in Tables 7 to 9, that the general
agreement between observed and predicted ultimate loads is very satis-
factory. It should be noted, however, that the slope of the descending
branch of the assumed stress-strain relation in Fig. 10 was so chosen as
to give such a general agreement. Hence, the average Ptest/Pcac ratio for
each group, 1.012, 0.992 and 1.026, may indicate a proper choice only.
A reliable expression for the accuracy of the theory which has been ad-
vanced can only be obtained through statistical studies of the dispersion
of the test results as compared to the various random errors in tests and
analysis. Such a statistical study is presented in Section 17.
The influence of some variables on the ultimate load may be studied
by means of Figs. 31 to 33. As may be expected, the ultimate load de-
creases for increasing eccentricity. The influence of conorete strength
ILLINOIS ENGINEERING EXPERIMENT STATION
varies with the mode of failure and the eccentricity. This influence is
small for very large eccentricities, which generally cause tension failures.
This is a reasonable result since the internal moment arm is large and the
changes in moment arm due to changes in concrete strength are compara-
tively small. For small eccentricities, however, considerable variations in
ultimate loads follow variations in concrete strength. For Group I (Fig.
31), which has a very small compression reinforcement, the effect of
concrete strength on ultimate loads approaches proportionality.
N
NJ
.t~.
N
c.yi/naer 5trength in lb. per sq. in.
Fig. 32. Ultimate Loads, Group II
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Some effects on ultimate loads of the amount and arrangement of
reinforcement may be studied by comparing the three groups. First, the
effects of amount of compression reinforcement, A,', may be studied as
the difference between Groups I and II. It appears that this effect is
largely dependent on the mode of failure. The effect of a change in A,'
from 0.22 to 1.24 sq in. is considerable for compression failures, the
40
3,
30,
26
20.
10
N.
z^ ,
o
Group IT
10-in. Sauare Columns /
As=As4"--n7. Bars X/
- Writer's Theory
--- Mod/fled Jensen Theory * /
-- Whifney .5+ 023 =2.73 in.
0 - *- Compression Fai/ure Y
o - Near Balanced /
o- Tension Failure ,/i
.' ____ /___ -
0 "7 e=-5.0+0.33
Comp. Tension C.
a l . Flure
S e=75.+0.38=7.88 in.
e =.26+0.40 /2.X0 In.
T:
I t I II
F_____ ______ ______ ___________________I
1000
Cylinder Strength in lb. per sq. in.
Fig. 33. Ultimate Loads, Group III
40 qv
3r/r/r; 6&&& /U V
ILLINOIS ENGINEERING EXPERIMENT STATION
increase in ultimate load being nearly constant and not a function of
concrete strength. For tension failures, however, especially for high
concrete strengths, the effect of an increase in compression reinforcement
is small. Such changes in compression reinforcement will not change the
curvatures and slopes of the line corresponding to balanced conditions
but displace this line in the direction of the load axis of the graphs.
The effects of changes in a symmetrical reinforcement from a total
of 2.48 to 4.80 sq in. may be found by comparing Groups II and III. It
is again apparent that the increase in ultimate load for compression
failures due to an increased total reinforcement is practically a constant
regardless of the concrete strength. For tension failures, the ultimate
loads seem to be nearly doubled by the increase in tension reinforcement
from 1.24 to 2.40 percent. The curve indicating balanced conditions is
very similar for the two groups, the small difference being caused by the
difference in d.
The discussions above give a qualitative indication of the effects of
some variables on ultimate loads. Since the primary purpose of this
bulletin is to establish ultimate load equations based on the present
tests, however, systematic studies of the effects of the many variables
influencing the strength and economy of eccentrically loaded tied mem-
bers was considered beyond the scope of this bulletin.
The preceding statements can be summarized:
1. The mode of failure of all test columns agreed very well with the
theoretical predictions based on the writer's theory.
2. A general agreement between the measured and calculated ultimate
loads appears to exist.
14. Other Ultimate Theories for Rectangular Sections Subject to
Combined Bending and Axial Load
a. Modification of Jensen's Theory
Jensen's studies of the ultimate strength of reinforced concrete
members(0' 81) were strictly limited to beams with tension reinforcement
only. Compression reinforcement in beams and the problem of combined
bending and axial load were not considered.
A method for designing the necessary reinforcement in rectangular
sections failing in tension under combined bending and axial load was
developed by the Structural Bureau of the PCA(86) using Jensen's basic
assumptions, Eqs. (27), in their original form. However, in place of
Jensen's definition that a tension failure takes place when the ultimate
strain, eu, is reached in the extreme concrete fiber, for design purposes
failure was considered to occur when the tension steel first reaches the
yield point. It was further assumed that the compression reinforcement,
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
for a balanced section, was effective with its full yield-point stress. A
similar method for ultimate design of beams was also developed.
These studies of extensions of Jensen's theory were continued in
the Committee on Masonry and Reinforced Concrete of the ASCE,
especially valuable contributions being made by the late A. J. Boase,
Chairman. The writer was fortunate to have access to some committee
correspondence related to these matters, which is referred to here as
Boase's method.
Case 1. Failure by Compression of Concrete; Part of the Section Is in
Tension (Fig. 34a). With 0 given by Jensen's Eq. (27d), ki and ks may
be expressed in terms of 0
ki = -- k2 - (50)
2 3(1 +( )
The following equilibrium equations may then be written with the
notation of Fig. 34a
P = L - k + p' p (51)
f'2 f fey
Pe' = fcbd2[ k(1 + k - p' + . (52)
2 1+# 3 - f," d (
The steel stresses, f, and f8', may be determined by compatibility
equations, assuming a linear distribution of strain,
f6' 1 - k
f, = Ee8 = n (53)
1- # k
fe" d'/d + k - 1
f E- = E,-,' = n - (54)
1-# k
in which n is given by Jensen's Eq. (27b) as n = 5 + 10,000/f,'.
The position of the neutral axis, as expressed by the ratio k, is a
parameter in Eqs. (51) to (54). The equations are valid within the
following limits:
In Eq. (53), f, 5 f1,, or
1
k > = kb. (55)
fyp (1 - P)
±1
If k is less than kb, a tension failure takes place. For k = kb, the sec-
tion is balanced. In Eq. (54), f,' < fyp', hence
ILLINOIS ENGINEERING EXPERIMENT STATION
1 - d'/d
k 1 d' = k,. (56)
1 - P fAl'
1
n fL"
When k is larger than k,, f,' is independent of k and equal to fyp'.
Finally the upper limit of k is given by
k 6 t/d. (57)
If k is larger than t/d, the neutral axis falls outside the section
giving rise to Case 2.
Case 2. Failure by Compression of Concrete; Entire Section in Com-
pression. From Fig. 34b are obtained the equations of equilibrium
P=f,"bd [ 2 k - L + p' f + p f (58)
2 2f"' (k- f (58)f
[1 + 1 _ 1 + 1 + 82 k
Pe' = fbd2 --+ k 1-i )
12 1+ 3
k-- k+2--3 p' - . (59)
2f " d d f," d
For these conditions k is generally larger than kc, and the stress in the
compression steel is therefore assumed equal flp'. The stresses f, and
f,, however, must be determined by means of a compatibility equation
k - t/d
fc = fc" -. (60)
k (1 - 0)
The tension steel stress is given by Eq. (53) with reversed sign since
the stress in this case is compressive.
Equations (58) and (59) are valid only if k > t/d. From Eq. (53)
we obtain another limit
1
k (61)
fli (1 - 3)
1-
nf,"
When k exceeds this value, f, is independent of k and equal to fyp.
The final limit is expressed by f, < fc", or
t
k < -- (62)
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
When k= t/3d, the concrete section is subject to a uniform stress. For
this value of k, Eq. (58) reduces to Eq. (3), and Eq. (59) combined with
(58) gives a position of the force P at the plastic centroid of the section.
The plastic centroid is the point at which the resultant of all internal
forces cuts the section, the uniform stress in the concrete and steel being
0.85 f' and the yield point, respectively. For symmetrical reinforcement,
the plastic centroid coincides with the geometric center of the section.
Fig. 34. Modification of Jensen's Theory
In Boase's method, equations very similar to Eqs. (50) to (62) were
used. The ultimate stress in the concrete, fe", was assumed equal to 0.85
fe', while Jensen's original theory for beams used f" = 1.0 f'. Thus, it was
recognized that the concrete in a column may be weaker than in a beam.
Nevertheless, n and 0 were computed from Jensen's Eqs. (27b) and
(27d) as a function of fe'. This writer has chosen, however, to introduce
f"= 0.85f/' in Eqs. (27b) and (27d), which is consistent with assuming
an ultimate stress 0.85 fe'. Hence
n = 5 + 10,000/0.85f,'
(63)
(64)
1 .85f'0 2
L 4000
/
ILLINOIS ENGINEERING EXPERIMENT STATION
Boase's method further used a compatibility equation (Eq. 54) for
the determination of the stress in the compression reinforcement, f/'.
Since it appears that the ultimate strains predicted by Jensen's theory
are too small (Fig. 14), the writer chose to consider the compression re-
inforcement effective with its full yield point, f/ =f,,', regardless of the
value of the ratio k.
The ultimate loads corresponding to the test specimens of the in-
vestigation reported herein could be computed by means of such suc-
cessive approximations as indicated in Fig. 18. The equations were
solved graphically, however, before the approximation procedure was
developed.
For each of the three groups of square columns, the relation between
the ultimate load, P, and eccentricity, e, was plotted in a Cartesian
coordinate system with the cylinder strength, f', as the parameter for
the family of curves. Such graphs were established by assuming suc-
cessive values of k and f', which, when introduced in the compatibility
and equilibrium equations, give the corresponding values of P and e.
Once the relations between P and e were known, the ultimate load
corresponding to a given section, concrete quality and eccentricity, could
be determined graphically.
Case 3a. Yield Point of Member. In Boase's method a tension failure
was considered to take place when the tension reinforcement reached the
yield point. This condition is generally referred to as the yield point of
the member. In this case, the horizontal part of the stress-block in Fig.
34a, is xc which is shorter than 0c, since by definition of a tension failure
the tension steel reaches yielding before the ultimate concrete strain
is reached. Hence, since f, =fp,
Snfo" (1 - k)
x = 1-- . (65)
fy, k
Since n is given by Eq. (27b) and f' may be determined by Eq. (54), x
may be entered in the equilibrium Eqs. (51) and (52) replacing the value
0. A relation between P and e may be established by choosing successive
values of k.
The upper limit for k in this case is the value kb defined by Eq. (55).
A lower limit for k is given by x = 0, or
1
k > - (66)
S+1
When k is under this limit, a triangular distribution of stress with ki=
1/2 and k2 = 1/3 will exist. Hence, the failure load may be expressed by
Eq. (39) if a = 2/3 is introduced.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Case 3b. Tension Failure. The writer chose to follow Jensen's original
definition of tension failure which refers to a condition in which the ulti-
mate concrete strain is reached while the tension reinforcement is yield-
ing under a constant stress, f,,.
In this case, the ultimate load may be found from Eq. (39) with a =
k2/ki, k2 and ki being defined by Eq. (50)
2(1 + P + 2)
a = (67)
3 (1 + #)2
This expression is Jensen's 1/N in Eq. (28), which he replaced by the
approximate value 0.5. The writer found that the errors due to this
approximation as applied to the test specimens in the present series
are less than one percent. Hence, Eq. (39) with a= 0.5 and f/= 0.85 f
was used.
This modification of Jensen's theory is compared to the test results
and the writer's theory in Figs. 31, 32 and 33.
The dividing curve between compression and tension failures was
obtained by substituting k=kb from Eq. (55), f.=f,~, f'=f,,,', and (3
from Eq. (64), into Eq. (51).
The curves for compression failures, which were referred to the total
eccentricity at failure, were obtained by graphic solution of Eqs. (51) to
(53) and (63) to (64). For e= 0.13 in Group I, Eqs. (58) to (60) were
used. The stress in the compression steel was assumed at the yield point
for all values of k and f,'; f' =fy,'. Such solutions were made in the early
stages of the investigation, and no corrections for the concrete displaced
by the compression steel were made.
The curves for tension failures were obtained by means of Eqs. (39)
and (40) with a = k2/ki = 0.5. In this case as well, no corrections of f,,'
were made.
Before this modification of Jensen's theory is compared with the
writer's theory, it is convenient to list the major differences between
the two:
1. From Fig. 14 it appears that Jensen assumed ultimate strains,
which for f/' = 3000-6000 p.s.i. were about half of the values
found by the writer.
2. Table 6 shows that for the modified Jensen theory, ki is larger
except for high concrete strengths, and k2 is smaller except for
low concrete strengths, than the values derived by the writer's
theory.
3. There is some difference in the assumed relationship between f/
and Ec, Fig. 11.
ILLINOIS ENGINEERING EXPERIMENT STATION
4. In the writer's theory, a correction was made for the concrete
displaced by the compression reinforcement, f/' =f,' - f. No
such correction was made in the other theory.
5. For tension failures, the writer used a= 0.55, while a= 0.50 was
assumed in the modified Jensen theory.
Comparing the ultimate loads predicted by the two theories by
means of Figs. 31 to 33, it is evident that the difference is small. The
same general trend exists for both theories, which is reasonable since
they both are of the Stiissi type.
The dividing curve between compression and tension failures is less
inclined for the modified Jensen theory. This is primarily caused by the
small values of e.; but the difference in ki has the same tilting effect.
It appears that the writer's curve gives the best agreement with the ob-
served modes of failure, but this is of minor importance since the Jensen
theory is intended for prediction of ultimate loads, not for prediction
of behavior.
The ultimate loads predicted by the modified Jensen theory are
generally higher than those predicted by the writer's theory. One reason
for this difference is that corrections for the concrete displaced by the
compression steel were made in the writer's theory. Other reasons de-
pend on the mode of failure of the columns for which predictions of ulti-
mate loads are made.
In the case of compression failures, it is evident that higher values
of ki and lower values of k2 will tend to give higher ultimate loads. It is
less obvious, however, that smaller values of e. will give higher ultimate
loads. This may be seen from Eqs. (36), (37) and (45) by comparing the
two theories with identical ki and k2 but different e.. For a given position
of the neutral axis, c, both theories will then give the same moment from
Eq. (36). The ultimate load which may be combined with a given mo-
ment is evidently a one-valued quantity for a given section. In Eq. (45),
however, the lower e. gives a lower f,, which in turn causes a higher P
in Eq. (37). The combined effects of these differences between the
writer's and the modified Jensen theory give a general trend of higher
loads for the latter theory.
In Group I, for e = 5.33 in., the difference in predicted ultimate loads
is also influenced by the difference in predicted mode of failure.
The curves for tension failures were obtained from Eqs. (39) and (40).
Hence, the difference in ultimate loads, which is smaller than the scatter
in the test results, is due only to the difference in a, and to the correction
f/ = f.,' - fo" made in the writer's theory.
It may be concluded that, within the scope of Groups I to III of the
present tests, the differences between the writer's and the modified
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Jensen theory are small as far as the accuracy of the prediction of ulti-
mate loads is concerned.
b. Whitney's Theory' 69 74 97)
Contrary to the writer's theory and the modified Jensen's theory,
the analysis developed by C. S. Whitney is not of the Sttissi type.
Whitney's equation for the ultimate load of rectangular sections failing
in compression under an axial load combined with bending (Eq. 21) was
developed on a semi-empirical basis. This equation was developed from
Eq. (20) by assuming that there is sufficient reinforcement to prevent
tension failure, and by computing the compressive ultimate moment
about the tension reinforcement. The expression for P was obtained as
2A,'f,' btf
P = 27 W (21a)
2e 3te 6dt - 3t2
+1 -+
d' d' 2d2
For small eccentricities, the neutral axis will be considerably further
away from the compression face than the position corresponding to
balanced conditions in a beam. Hence, the value a/d = 0.537 which leads
to Eq. (20), and thereby Eq. (20) itself, are not valid for small eccentric-
ities. Therefore, Whitney adjusted Eq. (21a) by making P approach
the proper value for a concentrically loaded column (Eq. 3) when e
approaches zero. This procedure led to Eq. (21), which is outstanding
in its mathematical simplicity. It should be noted, however, that this
equation is developed with the assumption that the reinforcement is
placed near the compression and tension faces, and that the equation
reduces to Eq. (3) for e = 0 only if 2A,' = A,t or, in other words, if the re-
inforcement is symmetrical, A,'= A,. If these conditions are not strictly
met, it nevertheless is believed that Eq. (21) will give a reasonably
good approximation.
Equation (21) is compared to the test results and the two other
theories in Figs. 31 to 33. Considering the simple form of the equation,
the agreement with the test results is satisfactory. It seems probable,
however, that Eq. (21) could be improved by adjusting the constants
in the two terms containing e.
For tension failures, Whitney developed equations on a rational
basis assuming a rectangular stress-block, that is a= 0.50. Thus, Eqs.
(39) to (43) were developed with a =0.50 and fe" =0.85 f'. Within the
range of eccentricities and concrete strengths for which tension failures
are predicted by the modified Jensen theory, Whitney's predictions will
therefore coincide with the modified Jensen theory as discussed above.
ILLINOIS ENGINEERING EXPERIMENT STATION
In Group II (Fig. 32) for e = 5.35 in. and f,' = 6000 p.s.i. both
Whitney's and the writer's theory predict tension failure. There is
some difference, however, in the value of the ultimate load, which is
due only to the difference in a (0.50 and 0.55) and the correction for
concrete displaced by the compression reinforcement, f,' = f,,,' - f,".
This difference in ultimate load may be estimated in this case from
Eq. (40). Since (e'/d-1) is small, P is nearly proportional to Vfyp/a,
which is V (43.6 - 5.1)/0.55 = 8.37 and / 43.6/0.50 = 9.33 for the
writer's and Whitney's theory, respectively. Hence, the writer's theory
leads to about a ten percent lower value of the ultimate load. This
percentage will decrease with increasing eccentricity.
It may be reemphasized that Whitney's Eq. (21) for compression
failure of members subject to combined bending and axial load is reason-
ably accurate within the scope of the present tests.
15. Studies of Ultimate Loads of Cylindrical Columns, Group IV
This group of tests included 30 cylindrical columns, all having a core
diameter of 10 in. and a shell thickness of 1 in. The shape and size of the
specimens are shown in Fig. 4.
The longitudinal reinforcement of all specimens consisted of eight
7/-in. bars, the ratio of longitudinal reinforcement to the gross area being
0.0425. Three grades of concrete and five eccentricities were involved as
outlined in Table 1.
The design of the drawn wire spirals indicated in Table 1 was based
on the following equation from the ACI Building Code: (93)
p, = 0.45 - - 1 (68)
Score /fsp
in which p,p is the ratio of the volume of spiral reinforcement to the
volume of the concrete core (out-to-out of spirals), and f, is the useful
limit stress of spiral reinforcement, generally taken as 60,000 p.s.i.
Equation (68) has been derived from Eqs. (3) and (4) of Section 1,
by equating the strength produced by the spiral reinforcement to the
strength of the concrete shell. Strictly applied, this procedure leads to a
coefficient of 0.425 in Eq. (68). Hence, the coefficient 0.45 as used should
result in a spiral slightly more effective than the concrete shell, so that
the characteristic behavior of spirally reinforced members may be
recognized.
Since the actual useful limit of the spirals used, the stress correspond-
ing to a strain of five per mill, probably was considerably more than
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
the 60,000 p.s.i. assumed, and since the concrete strengths varied some-
what from their designed values, the spirals did not conform strictly to
the requirements of Eq. (68). Nevertheless, the spirals used are believed
to represent designs resulting from the present ACI code fairly well.
The ultimate loads of such spirally reinforced cylindrical columns
subject to eccentric loads may be studied on the basis of the same general
assumptions which were made in Section 11. The spiral reinforcement is,
however, a new variable needing consideration. It is recognized that spiral
reinforcement contributes to the ultimate capacity of concentrically
loaded members as indioated in Eq. (4). On the other hand, large
deformations must take place and the concrete shell will generally spall
off before lateral restraint of the concrete core is developed through the
action of the spiral. In the case of eccentric loadings, such large deforma-
tions will generally increase the eccentricity considerably, and therefore
in most cases the spiral is not able to replace the strength of the shell
and later develop strength beyond the yield point of the member.
Defection in /nches
Fig. 35. Deflections of Spiral Columns
ILLINOIS ENGINEERING EXPERIMENT STATION
In the present tests, lateral deflections with respect to the knife
edges of the loading device were observed before as well as after failure
of the concrete shell. Some typical results of such measurements are
shown in Fig. 35, which indicates an absence of the second maximum
usually present in tests of concentrically loaded members. It is reasonable
to expect, therefore, that the ultimate strength of eccentrically loaded
columns with normal amounts of spiral reinforcement (Eq. 68) will
coincide with the yield point of such members, which is independent of
the amount of spiral reinforcement.
During the study of rectangular, tied columns (Fig. 17) it was found
that the internal compressive force in the concrete may be expressed as
C = kibcf,", where ki is given in Table 6. An equivalent uniform stress
distribution of intensity, fe", to a depth, kic, gives the same force, C. The
)I()
Fig. 36. Properties of a Circle Segment
Fig. 36. Properties of a Circle Segment
BUL. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Fig. 37. Analysis of Circular Sections
centroid of such an equivalent distribution, however, will be found at a
distance V2 kic from the compression face, while the distance used in
Fig. 17 and Table 6 is k2c =0.55kic.
In the case of cylindrical sections, an equivalent uniform stress dis-
tribution may be assumed over a segment with a rise kic. Strictly, the
values of k, involved should be a function of the depth of the neutral
axis, c, as well as the assumed stress distribution (Fig. 10). For the
purpose of the present analysis, however, the values of k, corresponding
to a rectangular section were used as an approximation. Furthermore, the
resultant compressive force in the concrete was assumed acting at the
centroid of the segment with a rise kic.
It is then convenient to study the properties of a segment. With
reference to Fig. 36, the following expressions apply
D2
Area: A = - (a - sin a cos a)
4
ILLINOIS ENGINEERING EXPERIMENT STATION
D sin' a
Centroid: x =
3 a - sin a cos a
Hence D2 (69)
-- - sin'a.
D 12A
These trigonometric equations are not suitable for a flexural analysis.
It is shown in Fig. 36, however, that the trigonometric equation for x/D
may be approximated by the straight line
x 2 4 2A
- =- + 0.293 .(-- (70)
D 37r 4 D2 )
This equation was developed and used by C. S. Whitney."4'
a. General Method of Analysis
The following method of analysis applies to tension as well as com-
pression failures. With reference to Fig. 37, the following equations are
available.
Equilibrium of forces: P = 0.85f/'A + EEAe, (71)
Equilibrium of moments: Pe = 0.85fA'Ax + E,EA,E,x, (72)
Compatibility equations: =
si = -- (D/2 + x,1 - c)
c
e2 = - (D/2 + x82 - c)
c (73)
EU
ei' = - (c - D/2 + x,,)
c
fu
E2' = - (c - D/2 + x,2).
C
Upon substitution in Eqs. (71) and (72) we have, due to the assumed
trapezoidal stress-strain relation for steel,
I l . (74)
E,
The solution of the equations above is rather complicated since trigo-
nometric expressions as well as a cubic equation in c are involved. Even
the approximation procedure used for rectangular columns (Fig. 18) is
rather time-consuming in the present case. Hence, Cartesian coordinate
graphs were developed relating the ultimate load of the test columns, P,
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
to the eccentricity, e, using the cylinder strength, f,', as the parameter
for the family of curves and the value from Eq. (70) as an approximate
value for the position of the resultant of compressive stresses in the
concrete. The ultimate loads corresponding to the four eccentricities
involved in Group IV could then be found from such graphs as a
function of fe'.
The curves relating P to e were found by assuming various values for
the rise, kic, of the circular segment representing the concrete compres-
sion zone subject to a uniform stress. With the rise known, most engi-
neering handbooks contain tables giving the area of the segment, A; the
position of the centroid may then be found from Eq. (70). Accordingly,
all quantities of Eqs. (71) and (72) may be found for a given section and
given materials if kic is chosen, and the values of P and e corresponding
to the chosen kic may be.found. By varying fc' as well as k1c, the family
of curves mentioned above may be established, and ultimate loads cor-
responding to given eccentricities may in turn be found graphically.
The results of such an analysis as applied to the eccentrically loaded
columns in Group IV are shown in full lines in Fig. 38, which gives the
ultimate load as a function of cylinder strength. It should be noted that
the eccentricities used in the analysis are the total eccentricities at fail-
ure; i.e., the nominal values plus the measured deflections with respect to
the knife edges of the loading device.
The curve shown in Fig. 38, which provides a division between tension
and compression failures, was developed by studying the limiting condi-
tions generally referred to as a balanced section. Such conditions exist
if, at the ultimate load, the tension steel reaches yielding and the concrete
reaches its ultimate strain simultaneously. Hence
Cb = ±^E D ( ±XB) (75)
cb - e f + x . (75)
The values of P which correspond to this position of the neutral axis may
then be found from Eqs. (70) and (71).
b. Test Results
The test results pertaining to failure of the specimens in Group IV
are given in Table 10. The ultimate loads are also plotted in Fig. 38 as a
function of cylinder strength with eccentricity as a parameter. Figure 39
shows a typical set of specimens after failure.
Four batches of concrete were used for casting every pair of columns,
three cylinders 6 by 12 in. being made from each batch. Hence, the
cylinder strengths entered in Table 10 each represent the average of
12 individual cylinders. The coefficient of variation, V, within each
94 ILLINOIS ENGINEERING EXPERIMENT STATION
such group of 12 cylinders has also been entered in the table. Using the
general methods of statistical analysis, the coefficient of variation of
the arithmetic mean of the 12 cylinders used should be near V/%/-2,
or about V/3. Since the four batches of concrete were placed on top
of each other in the forms rather than being thoroughly mixed, and
since some segregation and bleeding took place during compaction,
however, the principles of statistical analysis are not valid and the
given values of V must be regarded as indications only.
Concentrically Loaded Specimens. Columns 16 were loaded with flat
ends; i.e., the two 12-in. square steel blocks through which the specimens
Table 10
Results of Tests, Group IV
A-1a oiou 4. U .... 760 704 1.08 760 C
b 4640 4.0 0 0.01 693 655 1.06 770 2.60 C
B-16a 2990 7.4 0 0.03 515 497 1.03 644 2.40 C
b 3310 16.1 0 0.01 514 527 0.98 655 2.60 C
C-16a 1590 10.3 0 0.03 371 362 1.02 447 2.60 C
b 1420 7.4 0 0.02 365 345 1.06 398 2.70 C
Av Cols. 16 0.02 1.04 2.58
A-17a 5150 4.2 3.0 0.30 343 317 1.08 3.10 C
b 4640 4.0 3.0 0.29 283 298 0.95 3.10 C
B-17a 3620 9.0 3.0 0.34 253 258 0.98 3.20 C
b 3310 16.1 3.0 0.34 238 246 0.97 3.00 C
C-17a 1420 7.4 3.0 0.55 187 167 1.12 3.80 C
b 1600 10.3 3.0 0.50 179 175 1.02 4.04 C
Av Cola. 17 0.39 1.02 3.37
A-18a 5020 6.1 6.0 0.44 162 172 0.94 2.60 T
b 5000 6.2 6.0 0.50 171 171 1.00 3.42 T
B-18a 3380 7.3 6.0 0.42 140 144 0.97 3.00 T
b 3580 6.4 6.0 0.47 136 147 0.93 3.00 T
C-18a 1680 10.4 6.0 0.80 127 113 1.12 4.60 CT
b 1590 10.3 6.0 0.60 107 111 0.96 3.20 CT
Av Cols. 18 0.54 0.99 3.30
A-19a 5020 6.1 9.0 0.62 111.0 110.0 1.01 3.36 T
b 5310 7.1 9.0 0.62 114.3 112.0 1.02 3.40 T
B-19a 3380 7.3 9.0 0.54 98.5 97.5 1.01 3.56 T
b 3580 6.4 9.0 0.56 103.0 99.5 1.04 3.20 T
C-19a 1680 10.4 9.0 0.80 79.0 77.5 1.02 2.60 T
b 1630 8.9 9.0 0.80 79.0 77.0 1.03 5.00 T
Av Cols. 19 0.66 1.02 3.52
A-20a 5310 7.1 15.0 0.68 67.7 62.0 1.09 3.28 T
b 5000 6.2 15.0 0.58 63.5 61.5 1.03 3.00 T
B-20a 2990 7.4 15.0 0.75 57.5 55.5 1.04 2.40 T
b 3620 9.0 15.0 0.60 62.0 57.0 1.09 2.84 T
C-20a 1630 8.9 15.0 0.60 47.0 50.5 0.93 3.20 T
b 1600 8.6 15.0 0.72 47.0 50.0 0.94 3.92 T
Av Cols. 20 0.66 1.02 3.11
Av Group IV 1.018
The centrically loaded columns wereloaded with flat ends, the remaining columns through knife-edges.
loadThe test values of ultimate loads were recorded at failure of the concrete shell. The centrically
loaded columns did, however, generally develop a second and higher maximum load through the action
of the spiral. The calculated ultimate loads were found using Eq. (3) for columns 16, Eq. (70) to (74) for
t C, CT and T indicate compression, near balanced, and tension failure, respectively.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
I
K
I
Cy/linder Strength in lb. per sq. in.
Fig. 38. Ultimate Loads, Group IV
were loaded were locked against all rotations after application of about
10 kips. Column A-16a was loaded with knife edges to 700 kips, at which
load indications of failure in the knife edges were found. Hence, this
column was unloaded and reloaded to failure with flat ends. The deflec-
tions at midheight of the columns with respect to the ends, Ae, cor-
responding to the yield point of the members, have been entered in Table
10. It appears that the values of Ae are small; further, since the speci-
mens were restrained at the ends, they may be regarded as truly con-
centrically loaded.
ILLINOIS ENGINEERING EXPERIMENT STATION
The test values of ultimate loads for the concentrically loaded col-
umns in Table 10 refer to the loads at which the shell failed. Hence, a
comparison with Eq. (3) may be made. The resulting ratios between ob-
served and computed ultimate loads average 1.04. Ultimate strains of the
shells were of the order of 2.5 per mill, depending somewhat on the posi-
tion of the gages with respect to the region in which the shell failed first,
generally in the upper half of the specimens. After failure of the concrete
shell, the specimens continued to deform, thus entering the spiral range.
The second maximum load, which was reached shortly before fracture
of the spiral, was generally somewhat higher than the yield point. This
is reasonable considering the design of the spirals noted before. After
fracture of the spiral, the longitudinal reinforcement buckled as shown
for Column B-16b in Fig. 39. It may be concluded that the concentrically
loaded specimens behaved in accordance with earlier reports by F. E.
Richart(95) and others.
Eccentrically Loaded Specimens. These columns were loaded through
knife edges as indicated in Figs. 6 and 7. Accordingly, the deflections
at midheight of the specimens with respect to the knife edges, Ae, were
included as part of the eccentricity during the calculation of ultimate
loads.
The curve in Fig. 38 indicating the predicted division between tension
and compression failures is in general agreement with the observed modes
of failure. Thus, all Columns 17 failed in compression, Columns C-18
were near balanced conditions at failure, and the remaining columns
failed in tension. It should be noted, however, that no columns of concrete
qualities B or A were tested with such eccentricities that balanced con-
ditions were to be expected.
Before failure of the concrete shell took place, the general behavior
of these test columns with spiral reinforcement appeared to be very
similar to the tied columns discussed in Section 13. This may be seen by
studying Figs. 16a, b and c, which give strain measurements of typical
specimens. Column B-17a (Fig. 16a) is a typical example of a compres-
sion failure. The stresses on the tension side of the section were small
when the shell failed on the compression side as shown in Fig. 39. At
least the outer layer of compression steel, however, did reach yielding
before failure took place. For Column A-20b (Fig. 16b), which represents
a typical tension failure, these conditions were reversed. At least one
layer of tension steel was yielding when the shell failed, while the stresses
in the inner layer of compression steel were very small. Column C-18b
(Fig. 16c) represents a failure very near balanced conditions. The other
layer of tension reinforcement reached yielding slightly before the
concrete shell started spalling.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Fig. 39. Columns B-16 to B-20 after Failure
ILLINOIS ENGINEERING EXPERIMENT STATION
The spirally reinforced columns appeared to reach a stage of semi-
neutral equilibrium very similar to that described for tied columns, before
the shell failed. The ultimate strains of the shell (Table 10), however,
appear to be somewhat smaller than the values found for tied square
columns. Thus, the ultimate strain, e., appears to be dependent on the
Fig. 40. Column A-17a after Failure
shape of the section. The difference is small, however, and since the con-
stants k, and k2 are rather insensitive to changes in e., it is believed
that the use of c. = 3.8 per mill as determined for square columns will
not cause important errors in the estimate of ultimate loads.
The behavior of tied columns at high loads was found to depend on
the mode of failure. Members failing in tension were able to deform
considerably after the tension steel reached yielding and before the
concrete at the compression edge failed. For compression failures, such
large deformations were not present. Regardless of whether the concrete
in the compression zone failed before or after the tension steel reached
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
yielding, the compression reinforcement buckled between ties immedi-
ately after the concrete cover had spalled, and hence the load carrying
capacity of the members was suddenly reduced very seriously. For all
practical purposes the failure was complete.
The behavior of the spirally reinforced columns was different. When
the concrete shell spalled off on the compression side, the spiral rein-
forcement prevented buckling of the compression reinforcement as well
as failure of the concrete core. Hence, the failure was not complete: the
specimens were still able to carry fairly high loads and deform very con-
siderably, regardless of the amount of eccentricity. A typical example of
such behavior is shown in Fig. 35, which represents four very similar
specimens tested with different eccentricities. After failure of the concrete
shell, large deflections were developed, and the eccentricity of the load
thereby increased considerably. Hence, it is reasonable that no second
maximum was found. It should be noted in Fig. 35, however, that
Column B-17a, at a deflection of 3 in., had a total eccentricity of 6 in.,
and nevertheless carried a higher load than the maximum capacity of
Column B-18b which was loaded with an initial eccentricity of 6 in. A
similar comparison may be made for columns B-18b and B-19b. Hence,
it appears that the section, due to the action of the spiral reinforcement,
increased in strength after failure of the shell. The load carrying capacity
of the member, however, was generally impaired by the large deflections
present in the spiral range.
In the present tests, no attempts were made to measure strains in the
spiral. If gages had been attached to the spiral before casting, the
necessary waterproofing material would probably have upset the local
conditions by introducing bending in the spiral. If core holes had been
used to gain access to the spiral after casting, the shell area would have
been badly reduced. In earlier tests of eccentrically loaded spiral columns
without shells(95) it was found, however, that considerable stresses may
develop in the spiral near the compression edge while the spiral stresses
on the tension side of the section are very small.
It was further not found possible in the present tests to determine how
far the eccentrically loaded specimens could be strained before fracture
of the spiral. The great toughness of the members is, however, apparent
from Fig. 40. This column, A-17a, which was loaded with a 3-in. initial
eccentricity, lost its shell, with small stresses in the tension reinforcement,
at a load of 343 kips. In the condition shown in Fig. 40, the deflection
with respect to the knife edges was about 6 in., giving a total eccentricity
of about 9 in.; nevertheless the column carried about 150 kips, which is
about 40 percent more than the ultimate capacity of a similar column
ILLINOIS ENGINEERING EXPERIMENT STATION
loaded with an initial 9-in. eccentricity. After such large deformations,
the capitals of this and similar specimens were so heavily inclined that
it was feared the specimens might be ejected from the testing machine by
the shock caused by the fracture of the spiral. For reasons of safety, the
tests of all eccentrically loaded specimens were therefore discontinued
before such fracture took place.
Considering the general behavior described above, it appears reason-
able to recommend that the strength of eccentrically loaded spiral
columns, when designed as structural members, should be referred to the
yield point load at which the shell is lost. Such yield points as observed
in the present tests have been compared with the general method of
analysis as presented in Section 15a, in Table 10 and Fig. 38. It appears
that the theoretical ultimate loads on the average are somewhat on the
safe side. Since the control of the variables involved in the tests was
limited as discussed in Section 17, however, the over-all average of
1.8 percent on the safe side is very satisfying.
c. Whitney's Method of Analysis
For rectangular sections, C. S. Whitney(74 developed a formula for
ultimate loads in the case of compression failure, Eq. (21). However,
he also adjusted this formula so that when e is zero, P will be 2% times
the values given by the standard formula of the ACI code, Eq. (5).
The formulas for compression failures of square and circular sections
with a circular arrangement of the longitudinal steel given by Whitney
are both of the adjusted type mentioned above which reduce to 2% times,
Eq. (5). It would not be proper to compare such equations to ultimate
loads determined in tests. An equation for the ultimate load of cylindrical
columns failing in compression was developed, therefore, from Eq. (21)
with Whitney's assumptions.
In a column with diameter, D, the reinforcement is distributed
around the circumference of a circle with diameter, d. Equation (21)
can be adapted to this case by substituting for the value of d' the
equivalent steel distance, 0.67d. The effective depth of the concrete
section, t, for a rectangular member should be replaced by the value
0.8D. The following equation results
At f, AcfJ
P = + (76)
3e 9.6De
--d + 1 (08D .67d)2 1.18
d (0.8D + 0.67d)2
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
in which A,( is the total steel area and A. is the gross area of the section,
7rD2/4.
The results of Eq. (76), which reduces to Eq. (3) when e is zero, is
shown in Fig. 38 for e = 3.39 in. It appears that the ultimate loads are
somewhat conservative as compared to the test results and the more
exact theory. If 0.75d rather than 0.67d is substituted for d', however,
the following equation which practically coincides with the writer's
theory in Fig. 36 is found
P= + (77)
2.67e 9.6De
d (0.8d + 0.75d)2 + 118
Whitney also developed an equation for the ultimate load of cylin-
drical columns failing in tension. However, 0.09D was added to the
eccentricity, e, to allow for deflections. Since it has been chosen in the
present study to refer measured ultimate loads to the actual eccentricities
at failure, an equation for tension failures was developed without any
addition to the eccentricity; but otherwise with the same assumptions as
those used by Whitney.(74)
It was assumed that the total steel compression equals the total steel
tension. This is an approximation only, since the sum of forces in the
reinforcement will equal zero only if the neutral axis is at the center
line of the section, which is generally not the case. Assuming an equiva-
lent uniform stress-distribution as indicated in Fig. 37, this simplification,
however, reduces Eq. (71) to
P = 0.85f,'A. (71a)
It was further assumed that the centroid of the segment A may be
determined by means of Eq. (70), that four tenths of the total steel area
times the yield point is the effective force in the reinforcement on each
side, and that the effective moment arm of the reinforcement is 0.75d.
Then, equilibrium of moments about the effective centroid of the
tension steel gives
P (e + 0.375d) = 0.4At 0.75dfy, + 0.85f,'A(x + 0.375d) (78)
or
Pe = 0.3A,tdfy + P (0.442D - 0.586 0.85Df') (79)
For ultimate loads of cylindrical columns failing in tension this will
reduce to
ILLINOIS ENGINEERING EXPERIMENT STATION
r (/ 0.85e 2 p.t f,, d
P = 0.85D f 0 8- 5 - 0.377) + -t &
P =085D2f' D 2.5 0.85f,' D
- 0.5e 0.377)]. (80)
The results of this equation are compared to the test results and the
more accurate theory in Fig. 38. -It appears that the equation is very
conservative for the larger eccentricities if the concrete strength is over
3000 p.s.i. This result is reasonable because in these cases the neutral axis
was not at the center line of the section as assumed, but closer to the
compression edge.
An attempt was made to improve Eq. (80) by adjusting the constants
involved. It was not found possible to achieve any excellent approxima-
tion ,for all values of eccentricities and concrete strengths within the
range of tension failures by such means. An equation of the following
form is plotted in Fig. 38
P = 0.85D2f 0.85e - 0.325 2 f+, id
D 2 0.85f,' D
- ( 0.85e - 0.325)]. (81)
This equation appears to be more satisfactory than Eq. (80) even though
it is somewhat on the unsafe side for relatively small eccentricities. This
adjusted Eq. (81) as well as Eq. (77) fits the results of the present tests.
Their application beyond the scope of these tests (e.g., for lower rein-
forcement percentages) should be verified by comparison with the writer's
more rational procedure, Section 15a, and preferably with further tests.
d. Summary
The present tests of spiral columns were limited to only one percentage
of longitudinal reinforcement and an amount of spiral reinforcement
designed after the present ACI code. With these limitations, the observed
behavior of cylindrical spiral columns may be summarized as follows:
1. The concentrically loaded columns were found to behave as
observed in numerous previous tests.(95) The yield points of these columns
were found to be in good agreement with Eq. (3).
2. Two modes of failure prevailed in the tests of eccentrically loaded
columns, compression failures and tension failures. These modes of
failure, and the corresponding ultimate loads, were found to agree well
with the predictions of the writer's theory. The ultimate loads predicted
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
by Whitney's equations appeared to be rather conservative. Within the
scope of these tests, more accurate predictions were obtained by adjusting
the constants in Whitney's equations.'
3. The ultimate load of the eccentrically loaded columns was reached
at failure of the concrete shell.
4. After such failure of the concrete shell, the eccentrically loaded
columns developed extremely large deformations with relatively small
corresponding decreases in load capacity. This evidence of great tough-
ness was very pronounced, regardless of the amount of eccentricity and
the mode of failure.
16. Interaction Diagrams
A general and simple procedure for the approximate prediction of
ultimate loads of members failing in compression may be developed by
means of interaction diagrams.
p
Fig. 41. Interaction Diagram for Ideal Material
The basic principle of such interaction diagrams is that the strength
under one loading condition is affected by the presence of another super-
imposed loading condition. This principle has been made more general
by dealing with ratios rather than actual loads and stresses.(s85 105)
As an example illustrating combined bending and axial load, a
homogeneous and perfectly elastic material following Hooke's law is
assumed loaded with an eccentric force (Fig. 41). Then
P Pe
=- +
A Z (82)
P Pe
A Z
ILLINOIS ENGINEERING EXPERIMENT STATION
in which fc, ft, A and Z are compressive stress, tensile stress, area and
section modulus, respectively. Introducing
Po = Afo; MX = Zfco (83)
gives at failure
P M
- =1-
P0 M. (82a)
P fto M
Po f oO Mo
where fco and fto are compressive and tensile strength, respectively. An
interaction diagram based on these two equations is drawn in Fig. 41.
1~7
L
AS
As
I I,.
d
Fig. 42. Computation of Mo
For an isotropic material, fco = - fto, and the diagram is completed with
one line only from P/Po = 1 to M/Mo = 1. If the tensile strength is less
than the compressive strength, however, a diagram consisting of two
straight lines is obtained, as indicated in the figure for fto/lfo = - 0.4.
Combined bending and axial load in reinforced concrete members has
been studied by similar methods by Thomas(68) and Whitney.*74' It has
been claimed that Whitney's equation for compression failure of rec-
tangular sections (Eq. 21) represents a straight line when plotted in a
P versus Pe diagram. (74) The writer has found, however, that this is
strictly the case only when
2 3t d'
- = - or - = 0.535.
d' 1.178d2 d
It is nevertheless believed that a straight line in an interaction
diagram may represent a satisfactory approximation for compression
failures of any reinforced concrete section. In order to study the com-
pression failure of the column tests reported herein with the aid of such
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
a hypothesis, Po is defined by the ultimate load formula for a centrically
loaded column, Eq. (3). The eccentricity, e, is measured from the plastic
centroid of the transformed section computed with a plastic "modular
ratio" m=fyp/0.85 f,/. Since most columns are dimensioned in such a
manner that they would fail in tension if subjected to bending without
axial load, Mo is defined as the moment of all internal compressive forces
about the centroid of the tension steel for a balanced section. Thus, the
value of Mo for the section in Fig. 42 is computed from
Cb = -- d = - 3.8 d (84)
'e + e. 3.8 r + fI/E.
and
Mo = A,'fyp'd' + ki 0.85f/'bcb (d - k2Cb) (85)
where k, and k2 are the constants given in Table 6. For cylindrical
columns, Mo is computed in a similar manner with the aid of Fig. 37.
Sections of frames may at times be reinforced heavily enough to fail in
compression under bending without axial load. In this case it is believed
that Mo should be computed for the conditions at failure in bending with
the aid of the usual two equilibrium equations and the one compatability
equation, Eqs. (10) and (13).
An interaction diagram for all compression failures in these tests is
presented in Fig. 43. It should be noted that the measured deflections at
failure are included in the eccentricities in the same manner as in Figs. 31
to 33 and Fig. 38. The thin lines in the diagram represent a deviation
of measured ultimate loads, P, equal to - 10 percent from the theoretical
interaction line. Thus it appears that the straight line is a good approxi-
mation considering that the scatter in Fig. 43 is caused not only by
scatter of the test results but also by systematical errors introduced by
referring two shapes of section, four types of reinforcement, and concrete
strengths from about 1500 to 5500 p.s.i. to the same interaction line.
a. Columns with e = 0
The upper group of points in Fig. 43 represents columns loaded at
the mid-depth of the section. Thus the results of Group I with a non-
symmetrical arrangement of reinforcement are shown in the figure with
an eccentricity equal to the distance between the mid-depth and the
plastic centroid plus the deflection at failure.
Groups II and III were symmetrically reinforced and thus loaded
through knife edges at the centroid. In the past, it has been observed
that such columns generally show a strength about 10 to 15 percent
lower74, 95) than similar columns loaded with flat ends. The writer
believes that this reduction in strength is due to the fact that very few
ILLINOIS ENGINEERING EXPERIMENT STATION
concentrically loaded columns with hinged ends have been truly con-
centrically loaded. Small initial eccentricities due to errors in centering
the columns in the testing machine as well as inhomogeneities of the
concrete in the lateral direction will generally cause deflections and
thereby eccentricities which are large enough, when the plastic stage
of loading is reached, to explain the reduction in strength mentioned
above. The columns of Groups II and III were loaded through knife
edges %-in. thick as indicated in Fig. 7. Due to the unknown position of
the force within this %-in. width and errors in centering the columns,
the results of the concentrically loaded columns in Groups II and III were
plotted in Fig. 43 with an eccentricity of %-in. plus the measured
deflection at failure. It should be noted that, for concentrically loaded
columns with knife-edge loading, a centering error in any direction
will cause reductions in strength. For eccentrically loaded columns
with hinged ends, however, an error in centering will cause increases or
reductions in strength depending on the direction of the error. Further-
more, the ultimate loads of concentrically loaded members are far more
sensitive to such centering errors than eccentrically loaded columns.
Fig. 43. Interaction Diagram, All Compression Failures of Columns
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Hence, the writer believes there exists no reason to question the validity
of the test results of eccentrically loaded columns with hinged ends on
the basis that similar concentrically loaded columns show a reduction
in strength, as compared to columns with flat ends.
The concentrically loaded cylindrical columns of Group IV were
loaded with flat ends since their capacities at times were over the esti-
mated safe load on the knife edges. Thus, these columns are shown in
Fig. 43 as truly centrically loaded members.
b. Eccentrically Loaded Columns, e > 0
With the aid of the interaction diagram discussed herein, the ultimate
capacity of an eccentrically loaded column may be expressed as
Po
P = (86)
Po
1+--e
Mo
This equation is very similar to Richart's equation5")
P = -- (87)
a
1+C--e
kc2
in which C is a constant, a is the distance from the centroid to the
extreme edge of the section, and k is the radius of gyration of the section.
These two equations coincide if
Po k2
C = (88)
Mo a
Studying rectangular columns with d'= 0.8t, f,'= 2000, 4000 and
6000 p.s.i., and a symmetrical reinforcement of 1.0, 2.0, 3.0 and 4.0 per-
cent, it was found that C varied from 0.53 to 0.61. This result is in agree-
ment with the experimental findings of Fig. 10 in Bulletin 368. ()
The equations for ultimate loads of eccentrically loaded members
failing in tension (Eqs. 39 and 81) can be introduced in the interaction
diagram (Fig. 43) in a similar manner as the second straight line for an
ideal material (Fig. 41). An example is indicated in Fig. 43 for Group II
with cylinder strengths near 4500 p.s.i.
Such complete interaction diagrams with a compression as well as a
tension branch are believed to be most helpful in studying the effect of
various variables on the ultimate capacity of eccentricity loaded members.
ILLINOIS ENGINEERING EXPERIMENT STATION
The concepts of interaction curves also lead to very simple expressions
for the ultimate loads, Eq. (86). Nevertheless, it appears from Fig. 43
that the accuracy of Eq. (86) generally will be within ± 10 percent.
Hence, this equation should be suitable for practical purposes, at least
for preliminary design of sections in monolithic structures.
17. Errors in Tests and Analysis, Variation of Pte,t/Pcaic
The tests and the analysis reported herein are both subject to errors.
The components of variation in the tests were not perfectly controlled,
and the assumptions on which the analysis is based may not be strictly
correct.
The ratio between measured and calculated values of ultimate loads
as presented in Tables 7 to 10 derived by the writer's theory is believed
to be a convenient means of studying such errors with the aim of estab-
lishing some evidence regarding the reliability of the theoretical analysis.
There are a number of possible sources of error.
Of a total number of 120 column specimens, six failed locally in such
a manner that the observed ultimate loads were entirely irrelevant to
the problem of combined bending and axial load. For instance, Column
B-14b failed in tension and bond through the top bracket since, by an
error, the bracket reinforcement was not properly welded to the longi-
tudinal bars. These six local failures were caused by obvious errors and
the results, being entirely irrelevant, were discarded. The results of the
remaining 114 tests are believed to be significant, but are nevertheless
subject to various errors indicated in the following discussion.
1. All test pieces were manufactured in the laboratory, and the
strength of 6- by 12-in. cylinders was used as a measure of concrete
quality in the column specimens. From each batch of concrete three
cylinders were made, the average strength of which is believed to repre-
sent the true strength of the batch with a coefficient of variation of
about two to three percent. However, three to four batches were used
in casting every pair of columns, and these batches were stacked on
top of each other rather than being thoroughly mixed. Hence, the
significance of the average strength of all cylinders cast with a pair of
columns as well as the corresponding variance is questionable. The
strength of a column may possibly be more closely related to the batch
with lowest quality or to the batch located near the failure region of
the member. Since no better measure of concrete strength seems to
be available at the present time, an over-all average fi' was nevertheless
used as a measure of the quality of concrete used in casting the columns.
The coefficient of variation of this average f,', Vmean = V/V/ n - 1,
from Table 10 was probably of the order of one to four percent.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
The concrete strength developed in the columns, however, was not
solely dependent on the quality of concrete as expressed by the over-all
average f,'. Even though both columns and cylinders were compacted by
vibration, a longer period of vibration was generally necessary for the
columns than for the cylinders. Furthermore, the casting of the columns
was at times influenced by such irregularities as leaky forms and delays
between batches. Finally, the height of the columns was of the order of
6 to 7 ft as compared to the 12-in. height of the cylinders. Hence, it is
reasonable that a differential in strength, due to different degrees of
compaction, was present in the columns as indicated by the fact that all
columns failed in the upper half, and the intensity of this differential
may have varied from one pair of columns to another.
Four lots of aggregates and two lots of cement were used during the
manufacture of specimens. Even though these lots were very similar, the
relation between column and cylinder strength may have been influenced
by these step-wise changes of materials.
Columns and cylinders were cured in 100 percent moisture for seven
days. This period of curing was relatively well controlled. Before testing,
however, columns and cylinders were stored in the air of the laboratory
for 21 days. The tests were made over a period of about a year. Hence,
the temperature and relative humidity varied quite considerably, prob-
ably from 70 to 95 deg F and 30 to 95 percent. These changes may have
influenced the relation between cylinder and column strength.
It may be noted that the errors resulting from using the over-all aver-
age f,' discussed above will influence the measured ultimate loads of
columns failing in compression almost to the extent of proportionality.
Columns failing in tension, however, are relatively unaffected by changes
in fo'.
2. The concrete dimensions of the column sections were found to vary
up to _+ 0.1 in., which corresponds to a variation of ultimate loads of the
order of one to two percent.
3. The yield points of the longitudinal reinforcement, which was
received in one shipment, were determined by means of eight to ten
samples for each size of bar. It appears from Table 4 that deviations
from the average yield points of the order of five to eight percent was
present. Unfortunately the number of tension tests is entirely insufficient
to establish statistical trends. The ultimate loads of columns failing in
tension are almost proportional to the yield point of the tension reinforce-
ment; such columns will therefore be strongly influenced by variations
in such yield points. The ultimate loads of columns failing in com-
pression, however, will be independent of the yield point in the tension
steel and will be relatively little affected by variations in the quality of
compression steel.
ILLINOIS ENGINEERING EXPERIMENT STATION
4. The positions of the reinforcing bars in the sections probably
varied up to _± % in. from the nominal values. This error is not important
for compression failures. For tension failures, however, the variations of
the internal moment arm of the tension steel, and consequently varia-
tions of ultimate loads, may have been of the order of four to six percent.
The errors discussed above under (1) to (4) were related to the
manufacture of the specimens; another group of errors was introduced
through the testing procedure used.
5. The columns were leveled in the testing machine by means of a
plumb line. Hence, no columns were placed in the machine in a strictly
perpendicular position, and irregularities in the method of loading may
have been present.
6. For reasons of safety and stability, the spherically seated block
in the movable head of the testing machine had to be locked by wedges
before the jacks which served as a temporary support were removed
(Fig. 7). Therefore, the bending of the specimens under load did not take
place strictly in a plane of symmetry as was intended and assumed in the
analysis. Some evidence of such errors was found, as observed strains
at times were consistently larger on one side of the column than on the
opposite side.
7. The columns were placed in the testing machine with the desired
eccentricity of load by means of a plumb line and an inch rule. At least
twice during the tests, errors resulted from the use of a rule with Mo-ft
divisions or with the end broken off. Two such cases, Columns A-lb and
A-7a, were found by check measurements. The actual eccentricities were
recorded and the measured ultimate loads were corrected to the nominal
eccentricities.
Because the knife edges were % in. thick and because small errors
normally accompany the use of plumb lines and inch rules, the eccen-
tricities of the remaining columns probably deviated up to % in. from
the nominal values. The effect of such errors is relatively small for large
eccentricities. However, the influence on the ultimate loads of columns
with small eccentricities may have been of the order of three to five
percent.
8. The total time used in testing the columns to failure varied about
from 45 to 75 min. The level of load at which the last readings were
made varied about from 90 to 98 percent of the ultimate load. Finally,
the rate of applying load increments was not strictly controlled. Hence,
it is possible that uncontrolled time factors may have influenced the
ultimate loads.
9. The three testing machines used for the tests of reinforcement,
cylinders and columns, respectively, have been calibrated in the past.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
There is every reason to believe that the machines are all sensitive and
accurate enough to indicate actual loads with an error of less than one
percent within the loading ranges used.
The final group of sources of error are related to the theoretical
analysis of ultimate loads.
10. Some of the basic assumptions in the analysis, which were dis-
cussed in Section 11, may be in error. It was assumed, for instance, that
the maximum concrete stress in the columns subject to eccentric loads,
fc", equals 0.85 fe'. This constant 0.85 may be systematically too high or
too low. The ratio fc"/fe' may even not be a constant, but a function of f/
and/or the space-gradient of strain. It was further assumed that no
tensile stresses exist in the concrete and that strains vary linearly across
the section, both of which assumptions are not strictly true.
11. Variables which were not considered in the basic assumptions of
the analysis may be significant. For instance, the spacing of ties in the
square columns was 8 in. for %-in., %-in. and %-in. compression steel.
It is possible that the spacing of ties in terms of bar diameters is a
significant variable.
12. The ultimate loads were computed by means of a slide rule, and
some graphical solutions were made. The calculations are therefore
believed to be accurate only to about 0.5 percent.
13. Even if the manufacture of specimens, tests, and theoretical
analysis were perfect, concrete is a heterogeneous material, all properties
of which are subject to variations from statistical mean values.
The ratios between measured and computed loads may then be
studied in the light of these possible sources of errors. The total popu-
lation of 114 ratios was subjected to a statistical analysis, the results of
which are given in Table 11.
The arithmetic mean of the total population is 1.012. This figure alone,
however, cannot be used to indicate the reliability of the theoretical
analysis, since the slope of the assumed stress-strain relation beyond the
maximum stress in Fig. 10 was adjusted to fit the test results as well as
possible. It is significant, however, that the standard deviation of the
total population is 0.058 only. If the distribution is normal, this means
that about 92 percent of the observed loads were within -+- 10 percent of
1.01 times the predicted values, and 99 percent were within ± 15 percent.
A standard deviation of this order of magnitude may easily have been
caused by random errors from the sources (1), (3) and (13) alone.
It has not been proved, however, that the population is normal, and
systematic trends may be present. For compression failures, Table 11
gives a mean, xc, of 1.027 with a standard deviation, orc, of 0.059. For
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 11
Statistical Study of the Ratio Ptest/Pcalc
Variable Arithmetic Mean Standard Deviation Number
I 1.012 0.062 30
Group II 0.992 0.056 28
III 1.026 0.053 26
IV 1.018 0.052 30
A 0.993 0.057 36
Concrete B 1.016 0.049 39
C 1.024 0.063 39
0 1.013 0.060 19
Yt 1.038 0.059 24
Eccentricity Mt 0.986 0.053 24
4t 1.006 0.047 23
1 4t 1.015 0.057 24
Mode of Failure* C 1.027 0.059 58
T 0.996 0.052 56
All Columns 1.012 0.058 114
* Mode of failure as predicted by writer's theory-C: compression failure; T: tension failure.
tension failures, these statistics are XT = 0.996 and oT = 0.052, respec-
tively. In order to determine whether the difference in means, d= 0.031,
is statistically significant, it is assumed that the distribution within
the two groups is normal. The "t test" of significance,
t = d/a- (89)
where ad is the standard deviation of d, may then be used. Since both
groups are larger than 30, the variance of d may be expressed as
SZ(x, - j)2 + E(XT - )2 o,2 U 2
()== - = + (90)
ncnT nr n,
in which nc=58 and nr=56. With these figures, a0=0.010 and t=3.1.
Hence, a statistical probability of 516 to 1 exists that the difference,
d, is significant and not due to chance. This may indicate that the
value of fe" = 0.85fc' is chosen too low, since such a systematic error
in the analysis would increase the load ratio for compression failures
without any significant influence on tension failures.
In a similar manner it may be shown that the difference in means
between eccentricities of 1t and 1yt is significant with a probability of
727 to 1. This may be interpreted as evidence of tension stresses in
the concrete since negligible cracking occurred with eccentricities of %t
even though part of the section was stressed in tension. The increasing
trend of means for the three qualities of concrete is statistically insignifi-
cant, and no definite trend appears to be present for the four groups.
A trend of the variation of load ratios with time is possible since
temperature, humidity, and aggregates varied with time. In addition
different laboratory assistants worked on the project and it is possible
that the skill with which the tests were made improved with time.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
11111j§
NL
-I--
ts
NN -'^
':: CT)' *-
I'*- -
^ <0 ~ --- L.
'1 i^
N^^=^^
I I
U -c
^\ \ \\\\\ N\
__ _________ * - 0
-, ' '6
::zz 0
-^-i^ ^ <s -
iU
I I
-~
cywy
.zs
i r
\
\
ILLINOIS ENGINEERING EXPERIMENT STATION
Rae/o of PDst //P'CQc.
Fig. 45. Frequency Distribution of Pest/Pcalc
The ratios between measured and computed ultimate loads for all
columns were therefore plotted in Fig. 44 in the order that the columns
were cast. Neighboring blank or shaded blocks indicate columns which
were cast as pairs. There are six blocks without such a related neighbor
corresponding to the six discarded tests.
It may first be noted that 35 pairs of columns deviate in the same
direction while 19 pairs deviate in opposite directions. This may be
interpreted as meaning that errors in manufacturing the specimens were
more serious than errors in testing. Furthermore a slight trend is present
indicating low test values in the period just before May 10. This may
have been the effect either of a rainy spring, or of some property of the
second lot of aggregates. Otherwise, no systematic trend with time
appears to exist. Hence, the improvements in testing skill did not reduce
the scatter of the test results, though many time saving methods were
developed as the tests proceeded.
Finally Fig. 45 shows the frequency distribution of the load ratios for
all 114 columns. The class interval of one percent is rather small com-
pared to the population, and a "saw-tooth" variation results. Nevertheless
it appears that the results fall fairly close to the Gaussian curve of a
normal distribution.
Within the scope of the tests reported herein it may be concluded
that the predicted ultimate loads are, on the average, one percent on the
safe side. The test results appear to be distributed at random, since, for
all practical purposes, no systematic trends of variation were found. The
standard deviation of the ratio between measured and computed ultimate
loads was 0.058. Hence, 99 percent of the results of tests similar to those
reported herein may be expected to fall between 1.16 and 0.86 times the
values predicted by a flexural analysis based on the assumptions made
in Section 11.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
1 8. Studies of Deflections
The deflections of all columns tested were observed by means of a
deflection bridge carrying five dial indicators which was attached to the
tension face of the columns as indicated in Fig. 7. Some characteristic
examples of such measurements are presented in Figs. 46 to 49.
The deflection bridge was extended beyond the prismatic shaft of
the columns. Hence, the deflection at midheight with respect to the ends
of the prismatic shaft could be obtained by subtracting the average dial
reading at the ends of the shaft from the reading at midheight. The
deflection at midheight with respect to the knife edges, however, was
computed with the assumption that the brackets at the ends of the
columns were rigid.
The deflections at midheight with respect to the ends of the prismatic
shaft were discussed in Section 13a for the typical columns shown in
Figs. 19 to 21 and Figs. 24 to 28. It was shown that such deflections may
be predicted with a satisfactory accuracy throughout the range of loading
by means of the writer's theory. This is important, since a relation
between moment and curvature is a necessary basic assumption in
possible future studies of effects of L/d on the ultimate loads of con-
centrically and eccentrically loaded members.
The deflections at midheight with respect to the knife edges were
extrapolated to the ultimate load in order to estimate the actual eccen-
tricities at failure. Such deflections, Ae, are given in Tables 7 to 10.
In Figs. 46 to 49 it may be noted that at high loads the deflections of
the upper half of the columns are relatively larger than for the lower half.
This is another indication of a strength differential of the concrete due
to the method of casting, the fact that all columns failed in the upper
half being another indication. It may finally be noted in Figs. 46 to 49
that the deflection curve of the prismatic shafts of the columns is approxi-
mately a sine wave.
All equations for ultimate loads of eccentrically loaded members dis-
cussed in this report have been referred to the actual eccentricity at
failure. Hence, some consideration must be devoted to deflections if these
equations are applied to the design of structural members. The effects
of such deflections may be considered by several means:
1. The increased eccentricity may be accounted for in the safety
factor. This simple approach may be justified because the decreases in
ultimate load in monolithic structures due to deflections probably are
less than ten percent in the most common cases. Moments in a building
with flat slab floors, for instance, are often known with less accuracy
than ten percent.
ILLINOIS ENGINEERING EXPERIMENT STATION
Above: Fig. 46. Deflections, Columns A-12
Below: Fig. 47. Deflections, Columns B-7
10
Center Deflection in inc/es
Deflection in Inches
o
*I
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
Deflection in Inches
Center Deflectio
04 0.6
n in Inches Deflection in Inches
Above: Fig. 48. Deflections, Columns B-5
Below: Fig. 49. Deflections, Columns C-17
ILLINOIS ENGINEERING EXPERIMENT STATION
86
X
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Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
2. The reduction in ultimate load due to deflections may be accounted
for in a reduction factor for long columns which should preferably be a
function of L/d as well as the type of bending of the member.
3. The deflections near the ultimate capacity of a structure may be
estimated using the methods of the theory of elasticity and a reduced
modulus of elasticity for the concrete. The proper choice of such a re-
duced modulus is a matter of opinion, since the actual secant modulus
in a structure will generally vary from section to section depending on
the degree of strain in the various sections.
Some information regarding the minimum value of such a reduced
modulus may be established by means of the test results reported herein.
The prismatic shaft of the test columns was subject to a fairly constant
moment, Pe, and a deflection, 8, was developed over a length, L. Hence,
the common elastic theory gives
M L2
El 8
or
L2M
EI= 8 (91)
The moment of inertia of reinforced concrete members may be assumed
for the concrete section alone as
El = EcI, (92)
or for the transformed section as
El = Eic + E8,I. (93)
In Table 12 are presented for all eccentrically loaded test columns
values of Ecc and Ert as derived from Eq. (91) and Eq. (92) or (93)
with the observed values of M and 6 corresponding to the last readings
before failure; i.e., at 90 to 95 percent of the ultimate load. It appears
that both Ecc and Et are functions of fe'. However, the moduli, as pre-
sented in Table 12, also vary with the eccentricity and the amount of
reinforcement. It should also be noted that some scatter is caused by the
fact that the moduli are not referred to exactly the same percentage of
the ultimate load. The average values of Ec, were 2.56, 2.33 and 1.72
for concrete A, B and C, respectively. The corresponding values of Ec,
were 1.21, 1.00 and 0.37.
The writer believes that the most rational consideration of reductions
in ultimate loads due to deflections may be obtained through such
reduction factors as were mentioned under (2) above.
V. CONCLUSION
19. General Summary of Investigation
It was the object of the investigation reported herein to throw new
light on the behavior of reinforced concrete members subject to combined
bending and axial load.
A total of 120 eccentrically loaded column specimens were manu-
factured and tested to failure in short-time tests, the total time between
the first and last increment of load being about an hour. The test columns
were divided into four groups, each containing 30 specimens. Groups I, II
and III were 10-in. square tied columns with 1.46 to 4.8 percent rein-
forcement. Group IV was 12-in. cylindrical spiral columns with 4.25
percent reinforcement. Within each group, the concrete quality was
varied from about 1500 to 5500 p.s.i., and the eccentricity of loading
varied from 0 to 11/ times the lateral dimension of the columns. The
general behavior of the test columns was observed by measurements of
load, strains and deflections.
A critical study of the basic assumptions in inelastic flexural theories
for reinforced concrete was made. On the basis of such studies and of
phenomena observed in the present tests, a general inelastic flexural
theory was developed, by means of which the behavior of the test
columns as well as the ultimate loads could be predicted. The theory
developed by C. S. Whitney and a modification of V. P. Jensen's theory
were also compared to the measured ultimate loads.
20. Behavior and Mode of Failure of Test Columns
a. General Phenomena
Since the investigation was confined to the combined stress problem,
the specimens were purposely kept fairly short, so that the results would
not be confused by the occurrence of buckling failures.
Two modes of failure prevailed in the tests, compression failures and
tension failures. The compression failures were characterized by crushing
of the concrete at the compression face while stresses in the tension
reinforcement were less than the yield point. Tension failures, on the
other hand, were characterized by yielding in the tension reinforcement
followed by large deformations and considerable movements of the
neutral axis before crushing of the concrete took place.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
All columns were cast in a vertical position, and it was observed that
all columns failed in the upper half. This phenomenon was interpreted
as evidence of a variation in concrete strength from the bottom to the
top of the columns due to a difference in degree of compaction during
casting. Such a strength differential was also indicated by measured
strains and deflections.
b. Tied Columns, Groups I, II and III
The concentrically loaded tied columns were loaded through knife
edges. In agreement with the findings of earlier investigators, it was
found that the ultimate loads were 10 to 15 percent lower than for similar
columns tested with flat ends. This phenomenon is explained as being due
to the fact that it is impossible practically to obtain a truly concentric
application of load through knife edges.
The final failure of all eccentrically loaded tied columns was caused
by a crushing of the concrete at an ultimate strain of about 3.8 per mill.
After such crushing had taken place, the compression reinforcement
buckled between ties, and the load capacity of the columns thereby
dropped very considerably.
The intermediate grade compression reinforcement, in most cases,
yielded before failure of the concrete took place.
Preceding the final failure of the concrete, all eccentrically loaded
columns were in a semi-neutral equilibrium with the applied load. This
state of equilibrium was characterized by considerable increases in defor-
mation for very small additions of load.
It was observed that columns failing in tension developed much
larger deflections before final crushing of the concrete took place than
did the columns failing in compression.
A fairly linear distribution of strains across the section was found
to exist from the smallest loads to failure.
c. Spiral Columns, Group IV
The concentrically loaded spiral columns were loaded with flat ends.
The general behavior of these columns was in agreement with the findings
of earlier investigators. A maximum load was reached at failure of the
concrete shell. After considerable deformation, however, a second and
generally higher maximum load was developed through the action of the
spiral reinforcement.
The ultimate load of the eccentrically loaded columns, which were
loaded through knife edges, was reached at failure of the concrete shell;
no second maximum load was developed.
ILLINOIS ENGINEERING EXPERIMENT STATION
After failure of the shell, all eccentrically loaded spiral columns de-
veloped extremely large deflections without serious decreases in load
capacity, since the spirals prevented buckling of the compression rein-
forcement and crushing of the concrete core. Regardless of the amount
of eccentricity of load, the spiral columns appeared to possess great
toughness.
21. Inelastic Flexural Theories
The writer's theory rests on such general assumptions that the
behavior of the test columns may be predicted from the smallest loads to
failure, including the mode of failure. The average ratio of measured to
predicted ultimate loads for all test columns was 1.012, the standard
deviation of the ratio being 0.058.
Whitney's theory and a modification of Jensen's theory also give
satisfactory agreement with observed ultimate loads. However, behavior
at low loads and mode of failure may not be predicted with accuracy by
means of these theories.
The principles of interaction diagrams were used to develop a very
simple and fairly accurate method of predicting ultimate loads of
columns with small eccentricities.

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ILLINOIS ENGINEERING EXPERIMENT STATION
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Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
39. Kazinczy, G. v., "Die Plastizitat des Eisenbetons," Beton und Eisen, Vol. 32
1933 No. 5, March 1933, pp. 74-80.
40. Lyse, I., "Fifth Report on Column Tests at Lehigh University," ACI Journal,
1933 June 1933; Proceedings Vol. 29, pp. 433-42.
41. Gebauer, F., "Neue Balkenversuche des osterreichischen Eisenbetonauschusses.
1933 Vergleichsversuche mit St. 37, St. 55 und Isteg Stahl," Beton und Eisen, Vol. 32
No. 13, July 1933, pp. 204-07.
42. Gebauer, F., "Berechnung der Eisenbetonbalken unter Berticksichtigung der
1934 Schwindspannungen der Eisen," Beton und Eisen, Vol. 33 No. 9, May 1934, pp.
137-43.
43. Richart, F. E. and Brown, R. L., "An Investigation of Reinforced Concrete Col-
1934 umns," University of Illinois Engineering Experiment Station Bulletin No. 267,
June 1934. 91 pp.
44. Baumann, 0., "Die Knickung der Eisenbeton-Saulen," Eidg. Material-Prtifungs-
1934 anstalt an der E. T. H. in Zuirich, Bericht No. 89, Dec. 1934. 56 pp.
45. Blanks, R. F. and McNamara, C. C., "Mass Concrete Tests in Large Cylinders,"
1935 ACI Journal, Jan.-Feb. 1935; Proceedings Vol. 31, pp. 280-303.
46. Steuermann, M., "Wirdschaftliche Bemessung von einseitig bewehrten Eisen-
1935 betonbalken," Beton und Eisen, Vol. 34 No. 3, Feb. 1935, pp. 50-51.
47. Steuermann, M., "Bemessung der aussermittig belasteten Eisenbetonquer-
1935 schnitte," Der Bauingenieur, Vol. 16 Nos. 5 and 6, Feb. 1935, pp. 61-64.
48. Bittner, E., "Zur Kliirung der n-Frage bei Eisenbetonbalken," Beton und Eisen,
1935 Vol. 34 No. 14, July 1935, pp. 226-28.
49. Coppde, R., "Considerations sur le calcul et le s6curit6 des piAces flechies. Mo-
1935 ments de rupture," Proceedings, International Association for Bridge and Struc-
tural Engineering, Vol. 3, Zuirich, Sept. 1935, pp. 19-46.
50. Freudenthal, "Einfluss der Plastizitat des Betons auf die Bemessung ausser-
1935 mittig gedriickter Eisenbetonquerschnitte," Beton und Eisen, Vol. 34 No. 21,
Nov. 1935, pp. 335-38.
51. Brandtzeg, A., "Der Bruchspannungszustand und der Sicherheitsgrad von rech-
1935 teckigen Eisenbetonquerschnitten unter Biegung oder aussermittigem Druck,"
Norges Tekniske Hoiskole, Avhandlinger til 25-Arsjubileet 1935, pp. 667-764.
52. Gebauer, F., "Das alte n-Verfahren und die neuen n-freien Berechnungsweisen
1936 des Eisenbetonbalkens," Beton und Eisen, Vol. 35 No. 2, Jan. 1936, pp. 29-39.
53. Bittner, E., "Die Berechnung der Eisenbetonbalken nach Zustand III," Beton
1936 und Eisen, Vol. 35 No. 9, May 1936, pp. 143-46.
54. Michielsen, H. F., "Zur Klarung der n-Frage bei Eisenbetonbalken," Beton und
1936 Eisen, Vol. 35 No. 9, May 1936, pp. 146-49.
55. Glanville, W. H. and Thomas, F. G., "The Redistribution of Moments in Rein-
1936 forced Concrete Beams and Frames," Journal, Institution of Civil Engineers,
Paper No. 5061, London, June 1936, pp. 291-329.
56. Brandtzeg, A., "Die Bruchspannungen und die zuliissigen Randspannungen in
1936 rechteckigen Eisenbetonbalken," Beton und Eisen, Vol. 35 No. 13, July 1936,
pp. 219-22.
57. Brandtzaig, A., "Zulissige Betondruckspannungen in rechteckigen Eisenbeton-
1936 querschnitten bei aussermittigem Druck," Final Report, Second Congress, Inter-
national Association for Bridge and Structural Engineering, Berlin-Munich, Oct.
1936, pp. 121-35.
58. Melan, J., "Die Biegebruchspannungen des Eisenbetonbalkens," Beton und Ei-
1936 sen, Vol. 35 No. 19, Oct. 1936, pp. 315-17.
59. Emperger, F. v., "Der Beiwert n," Beton und Eisen, Vol. 35 No. 19, Oct. 1936,
1936 pp. 324-32.
ILLINOIS ENGINEERING EXPERIMENT STATION
60. Saliger, R., "Bruchzustand und Sicherheit im Eisenbetonbalken," Beton und Ei-
1936 sen, Vol. 35 Nos. 19 and 20, Oct. 1936, pp. 317-20 and 339-46.
61. Brandtzteg, A., "Forelesninger i jernbetong ved Norges tekniske h0iskole," Tapirs
1936 forlag, Trondheim, Norway, 1937.
62. Hajnal-Konyi, K., "The Modular Ratio-A New Method of Design Omitting m,"
1937 Concrete and Constructional Engineering, Vol. 32 Nos. 1, 2 and 3, Jan.-March
1937, pp. 11-26, 129-32 and 189-208.
63. Whitney, C. S., "Design of Reinforced Concrete Members Under Flexure and
1937 Combined Flexure and Direct Compression," ACI Journal, March-April 1937;
Proceedings Vol. 33, pp. 483-98.
64. Lyse, I., "Der Beiwert n im Eisenbetonbau," Beton und Eisen, Vol. 36 No. 7,
1937 April 1937, pp. 124-25.
65. Rog, M., "Versuche und Erfahrungen an ausgeftihrten Eisenbeton-Bauwerken in
1937 der Schweiz," Eidg. Material-Prilfungsanstalt, Bericht No. 99, Zirich, 1937.
405 pp.
66. Saliger, R., "Der bildsame Bereich im Eisenbetonbalken," Beton und Eisen, Vol.
1938 37 No. 1, Jan. 1938, pp. 15-20.
67. Richart, F. E. and Olson, T. A., "The Resistance of Reinforced Concrete Columns
1938 to Eccentric Loads," ACI Journal, March-April 1938; Proceedings Vol. 34, pp.
401-20.
68. Thomas, F. G., "Studies in Reinforced Concrete. VI. The Strength and Defor-
1938 mation of Reinforced Concrete Columns Under Combined Direct Stress and Bend-
ing," Department of Scientific and Industrial Research, Building Research Tech-
nical Paper No. 23, London, July 1938. 42 pp.
69. Whitney, C. S., "Eccentrically Loaded Reinforced Concrete Columns," Concrete
1938 and Constructional Engineering, Vol. 33 No. 11, Nov. 1938, pp. 549-61.
69a. Wilson, W. M., Kluge, R. W. and Coombe, J. V., "An Investigation of Rigid
1938 Frame Bridges, Part II," University of Illinois Engineering Experiment Station
Bulletin No. 308, Dec. 1938. 75 pp.
70. Glanville, W. H. and Thomas, F. G., "Studies in Reinforced Concrete. V. Moment
1939 Redistribution in Reinforced Concrete," Department of Scientific and Industrial
Research, Building Research Technical Paper No. 22, London, May 1939. 52 pp.
71. Habel, A., "Berechnung der Tragfahigkeit von Eisenbetonsiulen auf n-freier
1939 Grundlage," Beton und Eisen, Vol. 38 Nos. 13 and 15, July 1939, pp. 221 and 248.
72. Johnston, B. and Cox, K. C., "High Yield-Point Steel as Tension Reinforcement
1939 in Beams," ACI Journal, Sept. 1939; Proceedings Vol. 36, pp. 65-80.
73. "Regler for utf0relse av arbeider i armert betong, N.S. 427," Oslo, Oct. 1939.
1939 83 pp.
74. Whitney, C. S., "Plastic Theory in Reinforced Concrete Design," Proceedings
1940 ASCE, Dec. 1940; Transactions ASCE, Vol. 107, 1942, pp. 251-326.
75. Cox, K. C., "Tests of Reinforced Concrete Beams with Recommendations for
1941 Attaining Balanced Design," ACI Journal, Sept. 1941; Proceedings Vol. 38,
pp. 65-80.
76. Andersen, P., "The Resistance to Combined Flexure and Compression of Square
1941 Concrete Sections," University of Minnesota Engineering Experiment Station,
Technical Paper No. 29, 1941. 27 pp.
77. Guerrin, A., "Les th6ories nouvelles de la flexion dans les pikces en b6ton arm6,"
1941 Dunod, Paris, 1941. 450 pp.
78. Rog, M., "Festigkeit und Verformung von auf Biegung beanspruchten Eisen-
1942 betonbalken," Eidg. Material-Priifungsanstalt, Bericht No. 141, Zuirich, Oct.
1942.
Bul. 399. ECCENTRICALLY LOADED REINFORCED CONCRETE MEMBERS
79. Boase, A. J. and Morgan, C. E., "Balanced Design for Reinforced Concrete," ACI
1943 Journal, Feb. 1943; Proceedings Vol. 39, pp. 277-96.
80. Jensen, V.P., "The Plasticity Ratio of Concrete and its Effect on the Ultimate
1943 Strength of Beams," ACI Journal, June 1943; Proceedings Vol. 39, pp. 565-82.
81. Jensen, V. P., "Ultimate Strength of Reinforced Concrete Beams as Related to
1943 the Plasticity Ratio of Concrete," University of Illinois Engineering Experiment
Station Bulletin No. 345, June 1943. 60 pp.
82. Evans, R. Harding, "The Plastic Theories for the Ultimate Strength of Reinforced
1943 Concrete Beams," Journal of the Institution of Civil Engineers, Paper No. 5376,
London, Dec. 1943, pp. 98-121.
83. Bengtsson, B. A., "N&got om dimmensjoneringsmetoder for armerad betong,"
1944 Betong, Vol. 29 No. 2, Stockholm, 1944, pp. 83-97.
84. Aas-Jakobsen, A., "Jernbetongtversnittets dimmensionering," Betong, Vol. 29
1944 No. 4, Stockholm 1944, pp. 288-95.
85. Shanley, F. R., "Basic Structures," John Wiley and Sons, Inc., New York, 1944.
1944 392 pp.
86. Portland Cement Association, "Ultimate Design of Reinforced Concrete," Mod-
1944 ern Developments in Reinforced Concrete No. 11, 1944. 19 pp.
87. Granholm, H., "En ny berakningsmetod for armered betong," Transactions,
1944 Chalmers University of Technology, No. 38, Gothenburg, 1944. 88 pp.
88. Nylander, H., "Betongbalk belastad med excentrisk normalkraft, bereknad fran
1945 brottstadiet," Betong, Vol. 30 No. 4, Stockholm, 1945, pp. 286-304.
89. Aas-Jakobsen, A., "Jernbetongstavers deformasjoner og knekking etter n-fri be-
1945 regningsmetode," Tekniska Skrifter No. 114, Teknisk Tidskrifts F6rlag, Stock-
holm, 1945. 19 pp.
90. Bjuggren, U., "Den armerade betongens verkningssatt i sprickstadiet vid bojn-
1946 ing," Betong, Vol. 31 No. 3, Stockholm, 1946, pp. 182-211.
91. Richart, F. E., "The Structural Effectiveness of Protective Shells on Reinforced
1946 Concrete Columns," ACI Journal, Dec. 1946, Proceedings Vol. 43, pp. 353-63.
92. Ramaley, D. and McHenry, D., "Stress-Strain Curves for Concrete Strained Be-
1947 yond the Ultimate Load," Laboratory Report No. SP-12, U. S. Bureau of Rec-
lamation, March 1947. 22 pp.
93. "Building Code Requirements for Reinforced Concrete (ACI 318-47)," ACI
1947 Journal, Sept. 1947; Proceedings Vol. 44, pp. 1-64.
94. Hanson, R. and Rosenstrom, S., "Trykkfors6k med slanka betongpelare," Betong,
1947 Vol. 32 No. 3, Stockholm, 1947, pp. 247-62.
95. Richart, F. E., Draffin, J. 0., Olson, T. A. and Heitman, R. H., "The Effect of
1947 Eccentric Loading, Protective Shells, Slenderness Ratios, and Other Variables in
Reinforced Concrete Columns," University of Illinois Engineering Experiment
Station Bulletin No. 368, Nov. 1947. 128 pp.
96. Saliger, R., "Die neue Theorie des Stahlbetons auf Grund der Bildsamkeit im
1947 Bruchzustand," Second Edition, Franz Deuticke, Vienna, 1947. 110 pp.
97. Whitney, C. S., "Application of Plastic Theory to the Design of Modern Rein-
1948 forced Concrete Structures," Journal of the Boston Society of Civil Engineers,
Vol. 35 No. 1, Jan. 1948, pp. 30-53.
98. Aas-Jakobsen, A., "Gamle og nye dimmensjoneringsmetoder," Betong, Vol. 33
1948 No. 2, Stockholm, 1948, pp. 41-74.
99. Saliger, R., "Von Gegenwartstand der Stahlbetontheorie," Osterreichische Bau-
1948 zeitung, Vol. 3 No. 11, Nov. 1948, pp. 170-77.
99a. Johansen, K. W., "The Ultimate Strength of Reinforced Concrete Slabs," Final
1948 Report, Third Congress, International Association for Bridge and Structural
Engineering, Liege, 1948, pp. 565-70.
ILLINOIS ENGINEERING EXPERIMENT STATION
100. Brumer, M., "Comparative Designs of a Segmental Skewed Frame Concrete
1949 Bridge by the Straight Line and Plastic Theory Methods," ACI Journal, Jan.
1949; Proceedings Vol. 45, pp. 409-20.
101. Kleinlogel, A., "Zur Frage der zerst6rungsfreien Priifung von Beton im fertigem
1949 Bauwerk," Betonsteinzeitung, Wiesbaden, 1949, No. 2, pp. 17-21.
102. Chambaud, R., "Etude exp6rimentale de la flexion dans les pieces en beton arm6,"
1949 Annales de l'Institut Technique du Batiment et des Travaux Publics No. 61,
B6ton, Beton Arm6 No. 4, Paris, Feb. 1949. 36 pp.
103. Jones, R., "A Non-Destructive Method of Testing Concrete During Hardening,"
1949 Concrete and Constructional Engineering, Vol. 44 No. 4, April 1949, pp. 127-29.
104. Blanks, R. F. and McHenry, D., "Plastic Flow of Concrete Relieves High-Load
1949 Stress Concentrations," Civil Engineering, Vol. 19 No. 5, May 1949, pp. 320-22.
105. Shanley, F. R., "Applied Column Theory," Proceedings ASCE, Vol. 75 No. 6,
1949 June 1949, pp. 759-88.
106.
1949
107.
1949
108.
1949
109.
1949
110.
1949
ll0a.
1949
ll0b.
1949
111.
1950
112.
1950
113.
1950
114.
1950
Baker, A. L. L., "A Plastic Theory of Design for Ordinary Reinforced and Pre-
stressed Concrete Including Moment Re-Distribution in Continuous Members,"
Magazine of Concrete Research, Vol. 1 No. 2, London, June 1949, pp. 57-66.
Beyer, F. R., "Stresses in Reinforced Concrete Due to Volume Changes," ACI
Journal, June 1949; Proceedings Vol. 45, pp. 713-22.
Chambaud, R., "Theorie 6lasto-plastique de la flexion dans les poutres en b6ton
arme," Annales de l'Institut Technique du BAtiment et des Travaux Publics No.
101, Beton, B1ton Arme No. 10, Paris, Nov. 1949.
Hruban, K., "A Plastic Theory in the New Czechoslovakian Regulations," Con-
crete and Constructional Engineering, Vol. 44 No. 12, Dec. 1949, pp. 375-77.
Gebauer, F., "Die Plastizitatstheorie im Stahlbetonbau," Georg Fromme, Vienna,
1949. 184 pp.
Danish Standard No. 411, "Dansk Ingeni0rforenings Normer for Bygningskon-
struktioner, 2 Beton- og jernbetonkonstruktioner," June 1949. 60 pp.
Lundgren, H., "Cylindrical Shells," The Danish Technical Press, Copenhagen,
1949. 360 pp.
Hognestad, E. and Viest, I. M., "Some Applications of Electric SR-4 Gages in
Reinforced Concrete Research," ACI Journal, Feb. 1950; Proceedings Vol. 46,
pp. 445-54.
Lash, S. D. and Brison, J. W., "The Ultimate Strength of Reinforced Concrete
Beams," ACI Journal, Feb. 1950; Proceedings Vol. 46, pp. 457-72.
Hadley, H. N., "When Concrete Becomes Discrete," Civil Engineering, Vol. 20
No. 4, April 1950, pp. 29-31.
Hognestad, E. and Siess, C. P., "Effect of Entrained Air on Bond Between Con-
crete and Reinforcing Steel," ACI Journal, April 1950; Proceedings Vol. 46, pp.
649-67.