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ACI Structural Journal/July-August 2013 681
Title no. 110-S56
ACI STRUCTURAL JOURNAL TECHNICAL PAPER
ACI Structural Journal, V. 110, No. 4, July-August 2013.MS No.
S-2011-270.R1 received August 29, 2011, and reviewed under
Institute
publication policies. Copyright 2013, American Concrete
Institute. All rights reserved, including the making of copies
unless permission is obtained from the copyright proprietors.
Pertinent discussion including authors closure, if any, will be
published in the May-June 2014 ACI Structural Journal if the
discussion is received by January 1, 2014.
On the Probable Moment Strength of Reinforced Concrete Columnsby
Jos I. Restrepo and Mario E. Rodriguez
The probable moment strength (or flexural overstrength, as it is
also known) is the theoretical maximum flexural strength that can
be calculated for the critical section of a member, with or without
axial load, subjected to bending in a given direction. In ACI 318,
this strength is needed to capacity-design beams, columns of
special-moment frames, and columns not designated as part of the
seismic-resisting system. Supported on a column database, this
paper provides evidence that the current method prescribed by ACI
318 to calculate this strength has a clear nonconservative bias and
explains the reasons for this. To improve predictability, the
authors propose a very simple, statistically calibrated mechanics
model for determining the probable moment strength of rectan-gular
and circular columns. An extension of the concept is made for
computing the probable moment strength of rectangular columns
subjected to bending along the two principal axes.
Keywords: biaxial bending; capacity design; codes; confinement
plastic hinges; long-term concrete strength; probable moment
strength; reinforced concrete columns; seismic design.
INTRODUCTIONThe probable moment strength (or flexural
overstrength,
as it is also termed by other codes1-3 and textbooks4) is the
theoretical maximum flexural strength that can be calculated for
the critical section of a member, with or without axial load,
subjected to bending in a given direction. The prob-able moment
strength is needed to calculate design forces to capacity protect
any member where plastic hinges may develop, particularly if the
kinematics of the mechanism of inelastic deformation indicates so.
Examples of the former are the bases of first-level columns in
buildings and building columns not designated as part of the
seismic-resisting system framing into strong beams. For instance,
in ACI 318-11,5 the probable moment strength is needed to calculate
the design shear forces of beams of special-moment frames. This is
done to capacity protect these members by reducing the potential
for shear failure during a rare but intense earth-quake. Moreover,
ACI 318-115 specifies that all columns of special-moment frames in
buildings and columns not desig-nated as part of the
seismic-resisting system be capacity designed. Furthermore, this
code specifies that when plastic hinges will likely develop in
columns, the design shear force has to be determined using the
column end probable moment strengths, regardless of the shear
forces obtained from the structural analysis. Other codes1-3 have
similar requirements.
In ACI 318, the probable moment strength is calculated using a
simplified theory for flexure, where an elasto-plastic
stress-strain relationship is assumed for the steel reinforce-ment,
a rectangular stress block is assumed for concrete in compression,
and strain compatibility is enforced, accepting the hypothesis that
plain sections before bending remain plane after bending. In this
analysis, the yield strength of the
reinforcement is made equal to 1.25fy, where fy is the
speci-fied yield strength of the reinforcement.
The ACI 318 approach does not account for the likely increase in
the concrete compressive strength over the speci-fied strength in
the computation of the probable moment strength. The compressive
strength of concrete batched, delivered to a construction site, and
placed in a member following accepted quality control procedures
should be similar toif not greater thanthe specified strength at
the specified date, typically at 28 days. However, most concrete
types continue to gain significant strength over time,6-8 even in a
dry environment9 or in harsh environments subjected to
freezing-and-thawing cycles.10,11 The presence of passive
confinement, by way of closely spaced hoops, also causes an
additional strength increase. Moreover, the presence of an elastic
member, such as a footing or beam-column joint, at the framing end
of a member results in additional local concrete strength
gain.12-15 This is because this elastic element effec-tively
confines the compressed concrete by preventing it from expanding
transversely. The greatest manifestation of this local effect is
the reduction in concrete cover spalling at the member end and a
shift of the critical section away from the end.16,17 In lightly
axially loaded columns, a signifi-cant increase in the concrete
compressive strength has only a minor influence on the probable
moment strength. For this reason, the increase in the concrete
compressive strength can be ignored in calculations. However, as
the axial load increases, the probable moment strength becomes more
sensitive to the compressive strength of the concrete. In the
context of capacity design, an underestimation of the prob-able
moment strength can result in a reduction of the defor-mation
capacity of a hinging column, as the intended ductile mode of
response may be hampered by the development of another behavioral
mode associated with reduced ductility.
RESEARCH SIGNIFICANCEACI 318-115 specifies that columns in
special-moment
frames shall be capacity-designed. To achieve this objective
when hinging is likely to occur in the columns, this code requires
the computation of the probable moment strength at the column ends.
This paper shows that the current approach in ACI 318 for computing
the probable moment strength has a clear nonconservative bias. To
improve predictability, the authors propose a very simple,
statistically calibrated mechanics model for determining the
probable moment
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682 ACI Structural Journal/July-August 2013
Jos I. Restrepo, FACI, is a Professor of structural engineering
at the University of California at San Diego, San Diego, CA. He
received his BS in civil engineering from the Universidad de
Medelln, Medelln, Colombia, and his PhD from the Univer-sity of
Canterbury, Christchurch, New Zealand. He is a member of Joint
ACI-ASCE Committee 550, Precast Concrete Structures, and is also a
past recipient of the Chester Paul Siess Award for Excellence in
Structural Research. His research interests include reinforced and
precast/prestressed concrete, particularly seismic design.
Mario E. Rodriguez, FACI, is a Professor of structural
engineering at the Univer-sidad Nacional Autnoma de Mxico (UNAM),
Mexico City, Mexico. He received his BS in civil engineering from
the Universidad Nacional de Ingeniera, Lima, Peru, and his PhD from
UNAM. He is a member of ACI Subcommittee 318-C, Safety,
Service-ability, and Analysis. His research interests include
seismic design and evaluation of reinforced concrete
structures.
strength of columns built with Grade 60 to 80 reinforce-ment and
normal-strength concrete and covering the entire range of axial
compressive loads allowed by ACI 318. An extension of the model is
made for computing the probable moment strength of biaxially loaded
rectangular columns.
MOMENT STRENGTH DEFINITIONSThis paper uses the following six
moment strength defini-
tions: 1) nominal moment strength Mn calculated with the simple
flexure theory stated in ACI 318 using the specified concrete
compressive strength fc and the specified longitu-dinal steel
reinforcement yield strength fy; 2) ideal moment strength Mi
calculated with the simple flexure theory stated in ACI 318 using
the mean concrete compressive strength f c, which could account for
the additional strength gained through age, and the mean steel
reinforcement yield strength fy; 3) probable moment strength Mpr,
which is the maximum moment of resistance that can be calculated at
a column end. Moment Mpr may be computed from one of several
flexure theories with mean strengths f c and fy and considering the
effect of work and cyclic hardening in the reinforcement; 4)
critical section probable moment strength Mpr, which is the maximum
moment of resistance that can be calculated at the critical section
of the column if away from the column end; 5) credible moment
strength Mcd, which is the maximum moment of resistance that can be
calculated at a column end. Moment Mcd may be determined from one
of several flexure theories with the measured concrete compressive
strength fc and the measured steel reinforcement yield strength fy
and considering the effect of work and cyclic hardening in the
reinforcement; and 6) maximum moment strength MMAX, which is the
maximum bending moment resisted at a critical column end in a
reversed cyclic load test. This moment is computed accounting for
bending induced by the applied lateral force and the axial force
when it induces the P-D moment.
REVIEW OF PREVIOUS WORKIn 1985, Ang et al.18 compiled a database
of rectangular
and circular columns tested at the University of Canterbury, New
Zealand, computed the MMAX/Mi ratios, and empiri-cally fitted a
relationship for Mpr/Mi that was then modified by Paulay and
Priestley4 for Mpr/Mn. In 1998, Mander et al.19 performed a series
of parametric monotonic moment-curvature analyses and obtained
results that enabled the development of Mpr/Mn charts, including
credible upper and lower bounds, for rectangular and circular
columns and derived approximate equations to calculate the axial
load-moment pairs.
In 2001, Presland et al.16 developed approximate solutions in
closed form for Mn and Mpr and calculated the differences
between Mpr and the values of MMAX collected in the data-base.
Using a regression analysis, Presland et al.16 concluded that the
presence of an elastic member adjacent to the end of a hinging
column shifts the critical section of the column a distance between
0.5 and 1.0 times the depth of the neutral axis. They also proposed
that Mpr should be calculated from Mpr using a geometrical
correction term to account for the shift of the critical section
away from the column end.
COLUMN DATABASEThe research work presented in this paper makes
exten-
sive use of the PEER column performance database.20 This
database was audited, corrected where appropriate, and also
enhanced with test data for rectangular columns14,17,21-23 and
circular columns.24 The database includes 35 rectangular columns,
which are all square but one, with a minimum cross-section
dimension of 350 mm (29.2 in.) and 30 columns with circular or
octagonal cross sections of depths greater than 305 mm (12 in.),
hereafter called circular columns (the relevant properties of the
rectangular and circular columns are listed in Tables A-1 and A-2
found in Appendix A*). All columns had the transverse reinforcement
spaced at maximum six times the longitudinal bar diameter, except
one rectangular column, whose transverse reinforcement was 6.25
times the longitudinal bar diameter. These columns were all tested
quasi-statically with a reversed cyclic loading protocol and under
constant axial load. All columns devel-oped flexural plastic hinges
at an end adjacent to an elastic member. The database contains a
somewhat narrow range of concrete strengths. For example, 66% of
the rectangular columns have 27.4 MPa fc 43.3 MPa (4.0 ksi fc 6.3
ksi) and two-thirds of the circular column sections have 28.2 MPa
fc 38.1 MPa (4.1 ksi fc 5.5 ksi).
As far as the grade of the reinforcement in the data-base is
concerned, 27% of the rectangular columns incor-porate Grade 275
MPa (40 ksi) longitudinal reinforce-ment, 63% incorporate Grade 420
MPa (60 ksi), and 12% incorporate Grade 500 (nominally 75 ksi)
longitudinal reinforcement. Of the circular columns, 23%
incorporate Grade 275 MPa (40 ksi) longitudinal reinforcement, 70%
incorporate Grade 420 MPa (60 ksi) longitudinal reinforce-ment, and
7% incorporate Grade 500 (nominally 75 ksi) longitudinal
reinforcement. All Grade 420 MPa (60 ksi) reinforcement meets the
requirements set for ASTM A706/A706M-09b25 reinforcement for the
ultimate tensile strength when this strength was reported.
The longitudinal reinforcement ratio rl of the rectangular
columns ranges between 1.3 and 3.3%. Seventy percent of the
rectangular columns have 1.5% rl 1.8%, which is a rather narrow
range, but the authors note that 12% of the columns have rl 3.0%.
In circular columns, ratio rl ranges from 0.8 to 5.2%, 30% of the
columns have a ratio 1.9% rl 2.6%, and one-sixth have rl 3.0%.
Sixty-six percent of the rectangular columns in the database have
Ash/Ash,ACI < 1, where Ash is the total cross-sectional area of
transverse reinforcement within spacing s and perpendicular to
dimen-sion b in a rectangular column and Ash,ACI is the amount of
Ash specified by ACI 318-11.5 Only 30% of the circular columns in
the database have rs/rs,ACI < 1, where rs is the
*The Appendix is available at www.concrete.org in PDF format as
an addendum to the published paper. It is also available in hard
copy from ACI headquarters for a fee equal to the cost of
reproduction plus handling at the time of the request.
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ACI Structural Journal/July-August 2013 683
ratio of the volume of transverse reinforcement to the total
volume of core confined by this reinforcement and rs,ACI is the
amount of rs specified by ACI 318-11.5 Further examina-tion of the
test data shows that the ratio between the trans-verse
reinforcement provided to that required by ACI 318 is largely
uncorrelated with the axial load ratio P/Ag fc, where P is the test
column axial load and Ag is the gross area of the concrete section.
This is statistically relevant because the current provisions for
confinement in ACI 318 are not made a function of the axial load
ratio, as they are in other codes.1 A correlation of the
confinement reinforcement with the axial load ratio could have
introduced a small bias in the statistical analysis that will be
described later in this paper.
The database has columns with a wide range of axial load ratios
P/Ag fc varying from near-zero axial load to P/Ag fc = 0.74. The
rectangular column database has a fairly uniform distribution to
the axial load ratio. However, the circular column database is
somewhat biased because 53% of the columns have axial load ratios
P/Ag fc < 0.112. This is largely because of the large number of
tests conducted on circular bridge columns.
The column database also contains other useful informa-tion
(supplementary information for the rectangular and circular columns
is listed in Tables A-3 and A-4, respec-tively. Refer to Appendix
A). The column aspect ratio M/Vh, where M and V are the moment and
shear at the column end induced only by the applied lateral force,
varies between 2.2 and 6.9 for rectangular columns, while it varies
between 2 and 10 for circular columns.
It is also interesting to examine the drift ratio Qr,MAX when
columns listed in the database reached MMAX. Twenty-two out of 35or
63%of the rectangular columns reached MMAX at Qr,MAX 2%. A drift
ratio Qr = 2% could be thought of as a reasonable ratio for the
demand in hinging columns during the design earthquake. Except for
one rectangular column, the ratio between the moment resisted at
the column base at a drift ratio Qr = 2%, M2%, and MMAX was greater
than 0.9 for the remaining 12 columns. This column had M2%/MMAX =
0.88 at Qr = 2%, but by Qr = 4%, the moment of resistance had
reached 0.98MMAX. Contrary to the responses of the rect-angular
columns, only five out of the 30 circular columnsthat is,
17%reached MMAX at Qr,MAX 2%. However, the moment of resistance M2%
of 23 circular columnsthat is, 77%was M2%/MMAX > 0.9 at Qr = 2%.
Only two circular columns displayed M2% < 0.9MMAX. By Qr = 4%,
the moment of resistance in these columns was at least 0.93MMAX.
These results indicate that if MMAX is not reached before Qr = 2%,
the moment of resistance M2% is only slightly smaller than MMAX
because little hardening occurs in the response of the columns past
Qr = 2%.
ACI 318-115 PROCEDURE FOR COMPUTING PROBABLE MOMENT STRENGTH
This section presents a critical review of the provisions
contained in ACI 318-115 for calculating moment Mpr in columns. To
this end, moments Mcd were calculated with the ACI 318 procedure
for each column listed in the data-base. Values of Mcd rather than
Mpr were computed because strengths fc and fy were reported, thus
resulting in the best possible prediction of the ACI 318 procedure
for MMAX. Values of Mcd were calculated using a magnified yield
strength lh fy. Because fy was known, the 1.25 magnification factor
could not be used in the calculations because such a factor already
accounts for the measured-to-specified yield
strength ratio. Thus, only an allowance for overstrength caused
by work and cyclic hardening had to be made. Calcu-lations were
made with lh fy, where factor lh accounts for overstrength due to
hardening in the steel only. For consis-tency, a value lh = 1.15
was used, which is the same value derived for this factor in the
following section through an error minimization procedure. Figure 1
plots the MMAX/Mcd ratios (Tables A-5 and A-6 in Appendix A list
the individual values of MMAX/Mcd calculated for rectangular and
circular columns, respectively) computed versus the axial load
ratio P/Ag fc. This plot shows a clear nonconservative bias in the
ACI 318 procedure. While this procedure results in a very good
prediction of MMAX of columns with axial loads approaching zero,
the prediction becomes poor as the axial load increases, with
values of MMAX being underestimated for all columns with P/Ag fc
0.09. For example, the average ratios MMAX/Mcd for the five test
columns with axial load ratios clustered at approximately 0.4
(refer to Fig. 1) is 1.29.
Another way to visualize the bias in the ACI 318 proce-dure is
to plot points
,c cf f
MAX2
g
MPA bh
for six rectangular test columns together with the credible
moment strength-axial load interaction diagram computed using
average values for the material strengths and assuming lh = 1.15
(refer to Fig. 2). It just so happens that the six test columns
have very similar material strengths fc and fy,
Fig. 1Ratio MMAX/Mcd computed as per ACI 318 versus axial load
ratio of all test columns.
Fig. 2Comparison of axial load-credible moment interac-tion
diagram computed using ACI 318 method with test data.
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684 ACI Structural Journal/July-August 2013
with the ACI 318 procedure are less than the test column values
of MMAX for a given axial load ratio. In other words, the ACI 318
procedure underestimates the values of MMAX. Any underestimation of
the value of MMAX by the procedure cannot be attributed to
significant moment gain in the test columns caused by an increase
in the concrete core compres-sive strength as a result of excessive
confinement. The differ-ence between Mcd and MMAX is largely due to
the confine-ment of the concrete provided by the elastic reinforced
concrete element framing with the column at the region where the
maximum bending moment occurs, as pointed out by others.12,16 Such
a confinement effect is not captured by the ACI 318 procedure.
Moreover, it can be shown that when Mcd is calculated with the ACI
318 procedure, the magni-fied yield strength of the reinforcement
lh fy is not attained in any of the layers if the columns are
subjected to moderate or high axial loads. In the particular
example, no yielding of the reinforcement is observed at a moderate
axial load ratio of 0.3. Contrary to what is observed during
computation with the ACI 318 procedure, one would expect that at
MMAX, the longitudinal reinforcement in both extreme layers would
be strained well into the work-hardening region. This will be
discussed in more detail in the following section.
ALTERNATIVE APPROACH FOR CALCULATING CREDIBLE MOMENT STRENGTH OF
COLUMNS
Definitions and assumptionsThis section presents a simple
formulation to calculate the
credible moment strength of the critical section at the end of a
well-tied column, in which transverse reinforcement has been
detailed to prevent premature buckling of the longitu-dinal
reinforcement. The formulation is equally applicable to
symmetrically reinforced rectangular and circular columns.
Figure 3 shows an elevation of a symmetrically circular or
rectangular reinforced concrete column bending about a principal
axis and subjected to an axial force P when MMAX is attained. The
internal forces shown at the lower end of the column add to the
moment of resistance that must balance MMAX. When the moment of
resistance is calculated with any of the various flexure theories,
this moment becomes Mcd. Because of approximations made in this
theory, the ratio MMAX/Mcd = 1 should only be possible
statistically when: 1) the mean value in a large population nears
1; 2) the theory displays negligible bias with respect to the main
variables; and 3) the dispersion is small.
In Fig. 3, Cs is the compressive force resisted by the layer of
longitudinal reinforcementmarked Bars Athat are closest to the
extreme fiber in compression; Ts is the tensile force resisted by
the layer of longitudinal reinforcementmarked Bars Bthat are
closest to the extreme fiber in tension; Ti is the force resisted
by the entire inner column longitudinal reinforcement (shown as
Bars C); and Cc is the force resisted by the concrete in
compression. Force Cc is located a distance xc from the extreme
compressed fiber. The first assumption made herein is that forces
Cs and Ts are equal and opposite, implying that, for equilibrium,
force Cc = P + Ti. Figure 4 shows a visual justification for this
assump-tion. Figure 4(a) depicts the strain profiles for the
service load and two seismic load cases. These seismic load cases
indicate that the column has undergone one large curva-ture
reversal. Figure 4(b) plots stress-strain relationships that are
consistent with the strain history experienced by bars marked A and
B. Low-amplitude curvature rever-sals causing strain reversals have
been omitted from these
Fig. 3Applied and internal forces of resistance in
symmet-rically reinforced column.
Fig. 4Effect of large-amplitude strain reversals in column
section.
ratio rl is the same, the columns have the same cross section,
and the location of the longitudinal reinforcement is prac-tically
identical. Furthermore, the transverse reinforcement ratio provided
in these columns is less than that required by ACI 318. Figure 2
shows that the Mcd values computed
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ACI Structural Journal/July-August 2013 685
figures for clarity. Under service load and after creep and
shrinkage have taken place, these bars remain well below the yield
point and most often in compression (refer to points marked (1) in
Fig. 4). The column section reaches its maximum moment strength
when Bars A and B experi-ence hardeningrefer to points marked
(3)after a large earthquake-induced curvature reversal has occurred
(refer to points marked (2)). At points marked (3), the tensile and
compressive stresses in the extreme bars are comparatively similar
(refer to Fig. 4(b)), for which the compressive and tensile steel
reinforcement hardening factors lc and lt can be assumed to be
equal. The conceptual behavior described previously and illustrated
in Fig. 4 can be generalized for various other neutral axis depths
without altering the conclu-sion just reached. This finding is
surprisingly different from values calculated from a conventional
flexure analysis and even from the more sophisticated monotonic
moment-curva-ture analyses, which are unable to capture the cyclic
hard-ening phenomena. The second assumption made herein is that
force Ti (refer to Fig. 3) always acts in tension. This assumption
is strictly correct when the neutral axis depth in the column is
shallow. When the neutral axis depth approaches or exceeds the
column middepth, the resultant force in these inner bars (shown in
tension in Fig. 3) will eventually become compressive.
Consequently, the assump-tion made of force Ti always being in
tension force will evidently become erroneous and the probable
moment given by Eq. (2) will present a bias at high axial load
ratios. An analysis of the error, not presented in this paper,
indicates that Eq. (2) could overpredict the probable moment by
less than 10% when P/Ag fc 0.5. When the axial compressive load
ratio nears the limit imposed in ACI 318 for columns with tie
reinforcement, the probable moment could be over-predicted by as
much as 27% when rl = 0.04, fy = 515 MPa (75 ksi), and fc = 30 MPa
(4.4 ksi). However, when rl 0.02, fy = 414 MPa (60 ksi), and fc =
30 MPa (4.4 ksi), the prob-able moment is overpredicted by less
than 13%.
Derivation and calibrationWith the first assumption stated in
the previous section
and in the ideal scenario that MMAX = Mcd moment equilib-rium
about Point R in Fig. 3 results in
( )2cd s e i chM T h P T x = g + + (1)
Equation (1) can also be presented in terms of the total area of
longitudinal reinforcement Ast and fy. Assume that all the
reinforcement hardens by ratio lh. Then, Eq. (1) becomes
( ) 1 11 22 2
c ccd h st y e
x xM A f h Ph
h h
= l kg + k + (2a)
or in dimensionless form for rectangular columns
( )
2
11 22
12
ycd ch e
c c
c
g c
fM xhbh f f
xPhA f
= l r kg + k
+
(2b)
and in dimensionless form for circular columns
( )
3
11 24 2
14 2
ycd ch e
c c
c
g c
fM xhh f f
xPhA f
p = l r kg + k
p +
(2c)
where k is the ratio of the area of column longitudinal
reinforcement in one of the extreme layers to Ast; and ge is the
ratio between the distance between the centroid of the exterior
layer of barsthat is, of Bars A and B in Fig. 3to the column depth
h.
Equation (2) in any of its forms has two independent and
additive terms, and each term has a clear physical meaning. The
first term is the moment contributed to by the reinforce-ment and
the second term is the moment contribution due to axial load.
The following paragraphs discuss the evaluation of ratios k, ge,
lh, and xc/h, which are unknown so far.
Ratio k depends entirely on the way the longitudinal
reinforcement is distributed in the section. In a model of a column
cross section, the distribution should be sufficiently simple to
allow a clear distinction between outer and inner reinforcement
layers. The Eight and Twelve Equivalent Bar Modelswith three and
four equivalent bars per side, respectivelyare ideal models for
columns of rectangular section with longitudinal reinforcement
distributed along the faces. The values of ratio k for the Eight
and Twelve Equivalent Bar Rectangular Section Models are 3/8 and
1/3, respectively. The Six and Eight Equivalent Bar Models with
values of ratio k equal to 1/3 and 1/4, respectively, are ideal for
columns of circular section.
Ratio ge is a function of the equivalent bar diameter dbe of the
model cross section; the concrete cover to the hoop, cc; hoop
diameter dbh; the type of section and equivalent bar model used;
and the column depth h. Ratio ge is given by
( )11 2e be bh cd d ch g = z + + (3)
where
2 gbeb
Ad
nr
=
p
l (4)
and where nb is the number of bars in the model of the column
cross sectionthat is, nb is eight or 12 in a rectangular column and
six or eight in a circular column. Finally, z = 1 in rectan-gular
columns and z = cos(p/nb) in circular columns.
The authors made use of the column database to deter-mine ratios
lh and xc/h. Following the work of Presland et al.,16 it was
assumed that ratio xc/h varies linearly with the column axial load
ratio. An error minimization procedure for MAX cd MAXM M M was
performed to obtain optimum values for ratios lh and xc/h. This
procedure was carried out initially for the Eight Equivalent Bar
Model for rectan-gular columns and the Six Equivalent Bar Model for
circular columns. Minimization resulted in lh = 1.15 for the
rectan-
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686 ACI Structural Journal/July-August 2013
gular and circular columns and in the following two
relation-ships for ratio xc/h
0.34 0.07 cg c
x Ph A f
= +
(5a)
for rectangular columns and
0.32 0.10cg c
x Ph A f
= +
(5b)
for circular columns.Figure 5 plots histograms of the MMAX/Mcd
ratios deter-
mined for the Mcd calculated with the Eight Equivalent Bar Model
for rectangular columns and for that calculated with the Six
Equivalent Bar Model for circular columns. The distribution is
normal in both cases. The median and mean values calculated for the
MMAX/Mcd ratios are 0.995 and 0.998 for rectangular columns and
0.992 and 1.003 for circular columns, respectively. Very low
coefficients of vari-ation of 6.80% and 7.51%, which indicate
little dispersion from the mean, are found for the rectangular and
circular columns, respectively. This dispersion is comparable or
even lower than that reported by Mattock et al.26 in support of the
calibration of Whitneys stress block to calculate the nominal
moment strength of column sections in the develop-ment of the
ultimate strength theory.
Model sensitivityThe MMAX/Mcd ratios were also computed for Mcd
calcu-
lated with the Twelve Equivalent Bar Model for rectangular
columns and with the Eight Equivalent Bar Model for circular
columns (refer to Tables A-5 and A-6 in Appendix A for the
individual values of MMAX/Mcd for these two models and the
statistics). The MMAX/Mcd ratios and the median, mean, and
coefficients of variation are very similar to those computed with
the models with a fewer number of equivalent bars. Moreover,
residual analyses (not presented herein) of the MMAX/Mcd ratios
with P/Ag fc, with rl, and with strengths fc and fy show excellent
randomness. This indicates the appro-priateness of the model
calibrated.
Inspection of Eq. 2(b) shows that ratio Mcd/bh2 fc is most
sensitive to ratio ge when ratios rl and fy/ fc are high and ratio
P/Ag fc is low. Extreme values of ratio ge in rectangular columns
occur in small-sized columns with a large concrete cover and in
large-sized columns with the minimum concrete
cover. For such extremes, ratio ge can reach values as low as
0.685 or as high as 0.874 when using the Eight Equivalent Bar
Model. In rectangular columns of typical dimensions and average
concrete cover and hoop diameter, ratio ge fluctuates around 0.8
for the Eight Equivalent Bar Model. A sensitivity study shows that
when rl = 4%, fy/ fc=580/30, which can be considered at the high
end in most practical applications, and when ratio P/Ag fc = 0.1,
values of Mcd/bh2fc calculated from Eq. 2(b) with the extreme
values of ratio ge are only 0.9 and 1.06 times those obtained for
ratio ge = 0.8, indi-cating little sensitivity. As expected, little
sensitivity is also calculated for circular columns. This suggests
that Eq. (2) can be simplified by incorporating an average ratio
for ge as a constant. As discussed previously, an average value for
ratio ge is 0.8 for a rectangular column idealized with the Eight
Equivalent Bar Model, and an average value for ratio ge calculated
for a circular column idealized with the Six Equivalent Bar Model
is 0.69. Equation (2) can be further simplified if the values of
ratio k obtained from the Eight Equivalent Bar Rectangular Column
Model (k = 3/8) and for the Six Equivalent Bar Circular Column
Model (k = 1/3) and of ratio lh = 1.15 are also incorporated as
constants. The simplified equations are
2
1 1 11.15 0.34 2 2
ycd c c
c c g c
fM x xPh hbh f f A f
= r + + (6a)
for rectangular columns and
3
1 11.15 0.234 3 2
14 2
ycd c
c c
c
g c
fM xhh f f
xPhA f
p = r +
p +
(6b)
for circular columns, where ratio xc/h is given by Eq. 5(a) or
5(b), whichever is applicable.
Rectangular columns with bending along two principal axes
The derivations made in the previous section for the credible
moment of rectangular columns apply only for bending acting along
one of the two principal axes only. To the authors knowledge, only
four reversed cyclic load tests have been reported in the
literature on square or rectangular columns with bending applied
along an axis other than the principal. These tests are reported by
Zahn et al.27,28 Theoretical moment-curvature analysis, which
incorporated the strength enhancement due to the confine-ment of
the transverse reinforcementalso carried out by these
researchersindicates that the difference between the probable
moment strength square columns tested along the diagonal is only
marginally smaller than the probable moment strength of the same
column if loaded along a principal axis, except the extreme cases
of concentric axial tension of compression where no moment can be
resisted. Their experimental work also supported this finding. Such
a limited amount of test data suggests that the credible moment
strength along an axis can be obtained using the load contour
method of Bresler29 with the exponent set to 2that is, a circular
load contour
Fig. 5Histogram of moment ratio MMAX/Mcd with Mcd computed from
Eq. (2).
-
ACI Structural Journal/July-August 2013 687
22,,
, ,
1cd ycd xcd xo cd yo
MMM M
+ = (7)
where Mcd,xo and Mcd,yo are the credible moment strengths along
the two principal axes. These moment strengths are evaluated with
either Eq. (2a) or (6a) or from the corre-sponding dimensionless
forms. Moments Mcd,x and Mcd,y are vector components along the two
principal axes of the skew-credible moment strength. For a square
column, Eq. (7) means that the credible moment strength is the same
along any axis of bending for a given axial load ratio. When Eq.
(7) is used to calculate Mcd with the Eight Equivalent Bar
Rectangular Section Model for the tests reported by Zahn et
al.,27,28 ratios MMAX/Mcd vary very narrowly between 0.98 and 1.01
(the relevant properties of these tests and the corresponding
MMAX/Mcd ratios are listed in Table A-7 in Appendix A).
EQUATION FOR PROBABLE MOMENT STRENGTH OF COLUMNS
Rectangular columns with bending along principal axis and
circular columns
The derivation and subsequent calibration of an equation for
calculating the credible moment strength for columns with
rectangular sections loaded along an arbitrary axis and of circular
columns can form the basis for the derivation of an equation for
computing the probable moment strength. For a large population, the
mean of the measured material strengths fc and fy become f c and
fy, respectively. Now, in the evaluation of the probable moment
strength, only the specified values for fc and fy are known. Hence,
relationships between the mean and specified strengths are needed
in the development of an equation for Mpr. ACI 318-115 makes 1.25 =
lh fy but provides no relationships for the concrete. Here, the
compressive strength of the concrete in a member to the test
cylinder strength is termed the hardening ratio lco = f c/fc. Such
a ratio captures: 1) the gain in strength that occurs over time;
and 2) the statistical variability inherent in the batching plant
and in construction. Part 1) of ratio lco depends on numerous
factors, such as the type of cement, type and amount of
cementitious materials present in the mixture, water-cement ratio
(w/c), and maturity of the concrete.6,9,10 The authors note that
ignoring an increase in the concrete compressive strengththat is,
assuming lco = 1is unconservative in capacity design because the
probable moment strength (required to capacity-protect other
elements or protect the columns against various unde-sirable modes
of failure) becomes underestimated.
A literature review found only three references reporting the
specified and long-term strength of concrete from a structure or a
field slab from which ratio lco could be calcu-lated
directly.7,11,30 Baweja et al.7 carried out comprehen-sive research
into the compressive strength of aged in-place concrete in 10 field
slabs-on-ground. The concrete tested was between 10 and 26 years
old. Ratio lco ranged between 1.48 and 2.75. Scanlon and
Mikhailovsky11 assessed the strength of the concrete in a
34-year-old bridge. Using the average strength of the concrete,
ratio lco = 1.78. Billings and Powell30 investigated the strength
of the concrete of a bridge that was nearly 30 years old. Using the
mean strength reported, ratio lco = 2.29. When the specified
strength is made equal to 0.87 times the measured 28-day strength,
the data reported by Atcin and Laplante10 on field
slabs-on-ground
gives lco = 1.64 on 4-year-old concrete and lco =1.72 on
6-year-old concrete. None of the concrete structures in the
aforementioned references contained silica fume.
Knowing that there are many factors that affect coef-ficient lco
and given the significant scarcity of data, a value for lco = 1.7
is recommended for normal-strength concrete 10 years of age or
older. Alternatively, ratio lco can be made a function of time by
modifying the strength gain expression proposed by Freisleben and
Pedersen.31 This equation takes the following form
65
2.8co
te +l =
(8)
with the age of concrete t measured in years. Equation (8) meets
the minimum requirement in ACI 318-115 that at 28 days, the
strength of the concrete obtained from field measurements should
not be less than 85% of the speci-fied strength and that the
maximum strength cannot exceed 2.8 times the specified strengtha
value that is arbitrary but seems reasonable. Figure 6 compares the
values of ratio lco computed with Eq. (8) and the available data.
Figure 6 shows that Eq. (8) gives a reasonable prediction of ratio
lco but, as expected, there is significant scatter about the values
predicted.
Building upon Eq. (6) and making the test axial load P equal to
the factored axial load Pu, the expression for Mpr becomes
2
1 1 11.25 0.34 2 2
pr y c u c
c g cc
M f x P xf h A f hbh f
= r + + (9a)
for rectangular columns and
3
1 11.25 0.234 3 2
14 2
pr y c
cc
u c
g c
M f xf hh f
P xA f h
p = r +
p +
(9b)
Fig. 6Comparison of ratio lco predicted with Eq. (8) and
obtained from field tests.
-
688 ACI Structural Journal/July-August 2013
for circular columns, where ratio xc/h is obtained from Eq. (5a)
and (5b) by substituting fc with lcofc
0.34 0.07 c uco g c
x Ph A f
= +l
(10a)
for rectangular columns and
0.32 0.10 c uco g c
x Ph A f
= +l
(10b)
for circular columns. Although Eq. (9a) and (9b) have been
checked against data of columns subjected to compression, they
could be used for predicting the probable moment strengths of
columns subjected to small axial tensionsay, up to Pu/Ag fc =
0.05.
Figure 7 plots the axial load-probable moment strength diagram
calculated for a rectangular column using Eq. (9a) for four
reinforcement ratios and for ratio lco varying from 1 to 2 at 0.25
intervals. The upper factored axial load ratios shown in this
figure are the maximum levels allowed in ACI 318-115 for
compression members with tie reinforcement. It is evident in this
figure that ratio lco has a negligible effect in lightly loaded
columns up to approxi-mately Pu/Ag fc = 0.15. Above this axial load
ratio, ratio lco gradually becomes important. When the axial load
limit is reached, the ratio between the credible moment strengths
calculated with lco = 2 to that calculated with lco = 1 is at least
1.25.
Rectangular columns with bending along two principal axes
For rectangular columns with bending acting along an axis
different from the two principals, Mpr is found building upon Eq.
(7)
2 2
, ,
, ,
1pr x pr ypr xo pr yo
M MM M
+ = (11)
where Mpr,xo and Mpr,yo are the probable moment strengths along
the two principal axes. These moments are evaluated with Eq. (9a),
with ratio xc/h calculated using Eq. (10a).
CONCLUSIONS1. This paper shows that the procedure specified
by
ACI 318 to calculate the probable moment strength of columns
underestimates the maximum moment capacity recorded in all tests of
a database of rectangular and circular columns with axial load
ratios greater than 0.09. A reason for the ACI 318 bias is the lack
of the procedure to capture the confinement provided by the elastic
member that frames to the column at the critical section. Another
reason is that for some moderate and high axial load ratios Pu/Ag
fc 0.3 and when using Grade 420 MPa (60 ksi) or higher, the ACI 318
procedure is unable to capture the work and cyclic hardening
phenomena expected in the column longi-tudinal reinforcement.
Calculations show that in such cases and because of strain
compatibility reasons, none of the reinforcement actually yields
when the axial load in the column is at least moderate.
2. To improve predictability, the authors proposed a very
simple, statistically calibrated mechanics model for determining
the probable moment strength of rectangular and circular columns.
Statistical analysis of measured maximum moment strengths and those
calculated from the proposed method give a very small dispersion
and a mean approaching unity.
3. An extension of the concept is made with the load contour
method proposed by Bresler29 for computing the probable moment
strength of rectangular columns subjected to bending along the two
principal axes.
4. A sensitivity analysis of the proposed method indi-cates that
the gain in compressive strength of the concrete over time has a
negligible increase in the probable moment strength of columns
subjected to axial load levels less than 0.15. This strength
increase becomes gradually more impor-tant as the axial load ratio
in the column increases and can reach at least 1.25 when the axial
compressive load ratio is at the limit of that permitted in ACI 318
for columns with tie reinforcement. It is recommended that an
allowance be made for the concrete strength increase over time of
1.7fc in capacity design calculations of columns.
ACKNOWLEDGMENTSThe authors would like to acknowledge M. Torres,
who helped with the
calibration of some of the equations presented in this paper; F.
J. Crisafulli, for his careful review; and the two anonymous
reviewers for their thoughtful and constructive comments.
NOTATIONAg = gross area of concrete sectionAsh = total
cross-sectional area of transverse reinforcement within
spacing s and perpendicular to dimension bAsh,ACI = amount of
Ash specified by ACI 318Ast = total area of column longitudinal
reinforcementb = cross-section widthCc = force resisted by concrete
in compressionCs = compressive force resisted by layer of
longitudinal reinforce-
ment closest to extreme fiber in compressioncc = clear cover of
reinforcementci = depth of neutral axis at moment Mndb =
longitudinal bar diameterdbe = equivalent bar diameterdbh = hoop
diameterfc = specified concrete compressive strengthf c = expected
concrete compressive strengthfc = measured concrete compressive
strengthfsu = measured ultimate tensile strength of steel
reinforcementfy = specified yield strength of reinforcementfy =
expected steel reinforcement yield strengthfy = measured steel
reinforcement yield strengthh = cross-section depth
Fig. 7Factored axial load-probable moment strength inter-action
diagram indicating influence of ratio lco on probable moment
strength Mpr.
-
ACI Structural Journal/July-August 2013 689
M = moment at column end induced only by applied lateral
forceM2% = moment resisted at column base at drift ratio Qr = 2%
Mcd = credible moment strengthMcd,x = component along x-axis of
skew-credible moment strengthMcd,xo = credible moment strength
along principal x-axisMcd,y = component along y-axis of
skew-credible moment strengthMcd,yo = credible moment strength
along principal y-axisMi = ideal moment strengthMMAX = maximum
moment strengthMn = nominal moment strengthMpr = probable moment
strengthMpr = critical section probable moment strengthMpr,x =
component along x-axis of skew-probable moment strengthMpr,xo =
probable moment strength along principal x-axisMpr,y = component
along y-axis of skew-probable moment strengthMpr,yo = probable
moment strength along principal y-axisnb = number of equivalent
bars in model of column cross sectionP = test axial force appliedPu
= factored axial loads = center-to-center spacing of transverse
reinforcementTi = tensile force resisted by entire inner column
longitudinal
reinforcementTs = tensile force resisted by layer of
longitudinal reinforcement
closest to extreme fiber in tensiont = time in yearsV = shear at
column end induced only by applied lateral forcexc = distance from
extreme compression fiber to point of applica-
tion of force Ccge = distance between centroid of exterior layer
of bars divided
by column depth hk = ratio of area of column longitudinal
reinforcement in one of
extreme layers to Astlc = compressive overstrength factorlco =
concrete strength-hardening factorlh = weighted average of
overstrength caused by work and cyclic
hardening of entire reinforcement in sectionlt = tensile
overstrength factorQr = drift ratioQr,MAX = drift ratio at which
column reaches its maximum moment of
resistance MMAXrl = longitudinal reinforcement ratiors = ratio
of volume of transverse reinforcement to total volume
of core confined by this reinforcementrs,ACI = ratio of volume
of transverse reinforcement to total volume
of core confined by this reinforcement that is specified by ACI
318
z = coefficient for defining parameter ge
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11. Scanlon, A., and Mikhailovsky, L., Strength Evaluation of an
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-
690 ACI Structural Journal/July-August 2013
NOTES:
-
ON THE PROBABLE MOMENT STRENGTH OF REINFORCED CONCRETE
COLUMNS
APPENDIX A
Relevant Properties of Columns
-
N Designation Ref. mm mm MPa MPa
1 TP005 20 400 400 5.38 36.8 363.0 NRx 0.0166 0.23 0.0272 TP001
20 400 400 5.38 35.9 363.0 NR 0.0166 0.23 0.0273 TP006 20 400 400
5.38 35.9 363.0 NR 0.0166 0.23 0.0274 TP002 20 400 400 5.38 35.7
363.0 NR 0.0166 0.23 0.0275 TP003 20 400 400 5.38 34.3 363.0 NR
0.0166 0.24 0.0296 TP004 20 400 400 5.38 33.2 363.0 NR 0.0166 0.25
0.0307 TSUNO-1 21 550 550 5.00 30.7 306.0 1.43 0.0135 0.46 0.0338
TANA90U5* 20 550 550 5.50 32.0 511.0 1.32 0.0125 0.80 0.1009
TANA90U6* 20 550 550 5.50 32.0 511.0 1.32 0.0125 0.80 0.10010
SOES86U1 20 400 400 5.31 46.5 446.0 1.57 0.0151 0.36 0.10011
TANA90U9* 20 600 400 3.33 26.9 432.0 1.56 0.0188 1.51 0.10012
SAATU6 20 350 350 2.60 37.3 437.0 NR 0.0327 1.00 0.13113 TANA90U1
20 400 400 4.00 25.6 474.0 1.52 0.0157 1.02 0.20014 ANG81U4 20 400
400 5.63 25.0 427.0 1.57 0.0151 1.22 0.21015 GILL79S1 20 550 550
3.33 23.1 375.0 1.69 0.0179 0.83 0.26016 LI-1 22 400 400 3.50 33.2
450.0 1.32 0.0157 1.51 0.28917 SATO-1 14 400 400 4.00 59.8 442.0
1.33 0.0314 1.03 0.30018 TANA90U7* 20 550 550 4.50 32.1 511.0 1.32
0.0125 0.98 0.30019 TANA90U8* 20 550 550 4.50 32.1 511.0 1.32
0.0125 0.98 0.30020 SOES86U2 20 400 400 4.88 44.0 446.0 1.57 0.0151
0.53 0.30021 SOES86U3 20 400 400 5.69 44.0 446.0 1.57 0.0151 0.35
0.30022 SOES86U4 20 400 400 5.88 40.0 446.0 1.57 0.0151 0.19
0.30023 ANG81U3 20 400 400 6.25 23.6 427.0 1.57 0.0151 1.94 0.38024
ZAHN86U8 20 400 400 5.75 40.1 440.0 1.53 0.0151 1.18 0.39025
S17-3UT 17 440 440 5.40 43.4 496.0 1.28 0.0125 1.08 0.49126 S24-2UT
17 610 610 4.30 43.4 503.0 1.32 0.0125 1.07 0.49227 LI-4 22 400 400
2.75 35.7 460.0 1.41 0.0157 1.79 0.50028 WAT89U5 20 400 400 5.06
41.0 474.0 1.34 0.0151 0.57 0.50029 WAT89U6 20 400 400 6.00 40.0
474.0 1.34 0.0151 0.29 0.50030 GILL79S4 20 550 550 2.58 23.5 375.0
1.69 0.0179 1.85 0.60031 SATYARNO-3 23 400 400 4.00 50.0 497.0 1.30
0.0314 0.79 0.60032 SATO-4 14 400 400 4.00 71.6 442.0 1.33 0.0314
0.86 0.60033 WAT89U7 20 400 400 6.00 42.0 474.0 1.34 0.0151 0.87
0.70034 WAT89U8 20 400 400 4.81 39.0 474.0 1.34 0.0151 0.63 0.70035
WAT89U9 20 400 400 3.25 40.0 474.0 1.34 0.0151 1.69 0.700
* P-Delta calculation modified from that reported in Ref. 20 to
match original reference
mm = 0.0394 in.
1 MPa = 145 psix fsu not reported
Table A-1. Relevant properties of rectangular test columns.
'cf
yf
su
y
ff
'c g
P
f Ah b
b
sd
sh
sh ,ACI
AA
-
N Designation Ref. mm MPa MPa
1 KOWALSKIU2 20 457 4.00 34.2 565.0 1.23 0.0207 0.91 0.0412
KOWALSKIU1 20 457 4.00 32.7 565.0 1.23 0.0207 0.96 0.0433 RES-U1 24
914 2.53 64.1 426.0 1.67 0.0254 0.85 0.0634 NIST-F 20 1520 2.07
35.8 475.0 NRx 0.0200 1.46 0.0695 NIST-S 20 1520 1.26 34.3 475.0 NR
0.0200 3.21 0.0716 LEH1015* 20 610 2.00 31.0 462.0 1.36 0.0150 1.16
0.0727 LEH407* 20 610 2.00 31.0 462.0 1.36 0.0075 1.16 0.0728
LEH415* 20 610 2.00 31.0 462.0 1.36 0.0150 1.16 0.0729 LEH430* 20
610 2.00 31.0 462.0 1.36 0.0303 1.16 0.07210 LEH815* 20 610 2.00
31.0 462.0 1.36 0.0150 1.16 0.07211 KUN97A7 20 305 2.00 32.8 448.0
1.54 0.0200 1.06 0.09312 KUN97A8 20 305 2.00 32.8 448.0 1.54 0.0200
1.06 0.09313 KUN97A9 20 305 2.00 32.5 448.0 1.54 0.0200 1.07
0.09314 KUN97A10 20 305 2.00 27.0 448.0 1.54 0.0200 1.28 0.11215
KUN97A11 20 305 2.00 27.0 448.0 1.54 0.0200 1.28 0.11216 KUN97A12
20 305 2.00 27.0 448.0 1.54 0.0200 1.28 0.11217 WONG90U1 20 400
3.75 38.0 423.0 1.36 0.0320 0.96 0.19018 POT79N1 20 600 3.13 28.4
303.0 1.35 0.0256 0.67 0.23919 KOW96FL3 20 457 4.78 38.6 477.0 NR
0.0362 0.74 0.28120 VU98NH1 20 457 3.77 38.3 427.5 NR 0.0241 1.09
0.30721 VU98NH6 20 457 2.11 35.0 486.2 NR 0.0521 3.22 0.33322
POT79N5A 20 600 2.29 32.5 307.0 1.33 0.0256 1.93 0.36823 WONG90U3
20 400 3.75 37.0 475.0 1.32 0.0320 0.98 0.39024 POT79N4 20 600 2.92
32.9 303.0 1.35 0.0256 0.87 0.40725 WAT89U10 20 400 5.25 40.0 474.0
1.34 0.0192 0.51 0.52826 POT79N3 20 600 2.08 26.6 303.0 1.35 0.0243
1.07 0.57227 ANG81U2 20 400 3.44 28.5 308.0 1.51 0.0256 1.28
0.58928 ZAHN86U6 20 400 4.69 27.0 337.0 1.46 0.0243 1.66 0.61329
POT79N5B 20 600 2.29 32.5 307.0 1.33 0.0256 1.93 0.73730 WAT89U11
20 400 3.56 39.0 474.0 1.34 0.0192 1.09 0.739
* P-Delta calculation modified from that reported in Ref. 20 to
match original reference
mm = 0.0394 in.
1 MPa = 145 psix fsu not reported
Table A-2. Relevant properties of circular test columns.
'cf
yf
su
y
ff
'c g
P
f Ah s
s ,ACI
b
sd
h
-
N Designation c c +d bh
mm rad1 TP005 3.1 33.5 0.031 0.972 TP001 3.1 33.5 0.0173 TP006
3.1 33.5 0.081 0.924 TP002 3.1 33.5 0.0165 TP003 3.1 33.5 0.057
0.976 TP004 3.1 33.5 0.082 0.967 TSUNO-1 4.1 26.0 0.0178 TANA90U5
3.0 52.0 0.044 0.939 TANA90U6 3.0 52.0 0.026 0.98
10 SOES86U1 4.0 20.0 0.061 0.9711 TANA90U9 3.0 52.0 0.047 0.8812
SAATU6 2.9 32.4 0.090 0.9213 TANA90U1 4.0 52.0 0.01914 ANG81U4 4.0
32.5 0.036 1.0015 GILL79S1 2.2 50.0 0.028 1.0016 LI-1 4.1 30.0
0.01717 SATO-1 2.5 30.0 0.01618 TANA90U7 3.0 52.0 0.050 0.9819
TANA90U8 3.0 52.0 0.01520 SO S86 2 0 21 0 0 010
Table A-3. Supplementary rectangular column test data.
2%
MAX
MM
r ,MAXMVh
20 SOES86U2 4.0 21.0 0.01021 SOES86U3 4.0 20.0 0.00922 SOES86U4
4.0 19.0 0.01123 ANG81U3 4.0 36.5 0.01324 ZAHN86U8 4.0 23.0 0.02025
S17-3UT 6.9 36.5 0.022 0.9626 S24-2UT 5.0 50.8 0.02027 LI-4 4.1
30.0 0.01628 WAT89U5 4.0 21.0 0.01629 WAT89U6 4.0 19.0 0.01130
GILL79S4 2.2 50.0 0.01331 SATYARNO-3 4.0 27.4 0.01132 SATO-4 2.5
30.0 0.01833 WAT89U7 4.0 25.0 0.00834 WAT89U8 4.0 21.0 0.00835
WAT89U9 4.0 25.0 0.016 1 mm = 0.0394 in.
-
N Designation c c +d bh
mm rad
1 KOWALSKIU2 5.3 22.2 0.061 0.902 KOWALSKIU1 5.3 22.2 0.045
0.943 RES-U1 3.2 67.7 0.057 0.924 NIST-F 6.0 74.6 0.054 0.925
NIST-S 3.0 79.4 0.047 0.886 LEH1015 10.0 28.6 0.030 0.917 LEH407
4.0 28.6 0.031 1.008 LEH415 4.0 28.6 0.051 0.989 LEH430 4.0 28.6
0.074 0.92
10 LEH815 8.0 28.6 0.092 0.9111 KUN97A7 4.5 18.5 0.053 0.9612
KUN97A8 4.5 18.5 0.053 0.9413 KUN97A9 4.5 18.5 0.066 0.9414
KUN97A10 4.5 18.5 0.044 0.9415 KUN97A11 4.5 18.5 0.038 0.9216
KUN97A12 4.5 18.5 0.038 0.9717 WONG90U1 2.0 30.0 0.038 0.9418
POT79N1 2.0 35.0 0.037 0.9619 KOW96FL3 8.0 39.7 0.092 0.9120
VU98NH1 2.0 34.3 0.038 0.9921 VU98NH6 2.0 37.5 0.095 0.7922
POT79N5A 2.0 44.0 0.023 0.9923 WONG90U3 2.0 30.0 0.025 0.9824
POT79N4 2.0 35.0 0.027 1.0025 WAT89U10 4.0 25.0 0.020 1.0026
POT79N3 2.0 35.0 0.01227 ANG81U2 4.0 28.0 0.01228 ZAHN86U6 4.0 28.0
0.01929 POT79N5B 2.0 44.0 0.02030 WAT89U11 4.0 27.0 0.013 1 mm =
0.0394 in.
Table A-4. Suplementary circular column test data.
r ,MAX 2%MAX
MM
MVh
-
N DesignationACI
1 TP005 0.087 0.98 0.781 1.01 0.791 1.002 TP001 0.084 0.92 0.781
0.95 0.791 0.943 TP006 0.098 1.07 0.781 1.10 0.791 1.094 TP002
0.083 0.91 0.781 0.93 0.791 0.925 TP003 0.090 0.94 0.781 0.97 0.791
0.966 TP004 0.093 0.94 0.781 0.97 0.791 0.967 TSUNO-1 0.082 0.99
0.859 1.02 0.868 1.028 TANA90U5 0.140 1.15 0.766 1.09 0.774 1.099
TANA90U6 0.135 1.11 0.766 1.05 0.774 1.0410 SOES86U1 0.119 1.17
0.851 1.09 0.860 1.0911 TANA90U9 0.188 1.04 0.782 1.07 0.790 1.0612
SAATU6 0.214 1.13 0.743 0.99 0.756 0.9813 TANA90U1 0.176 1.08 0.690
0.93 0.699 0.9214 ANG81U4 0.183 1.08 0.788 0.97 0.797 0.97
8-bar model 12-bar model
max
cd
MM
e e 2
max
'c
M
f b hmax
cd
MM
max
cd
MM
MAX MAX2 '
cdc
M MTable A - 5. Ratios and calculated using various Mbh f
approaches for rectangular columns.
G8 U 0 83 08 0 88 0 9 0 9 0 915 GILL79S1 0.219 1.12 0.765 1.03
0.775 1.0316 LI-1 0.189 1.12 0.800 1.00 0.809 1.0017 SATO-1 0.200
1.23 0.779 1.01 0.792 1.0118 TANA90U7 0.200 1.32 0.766 1.09 0.774
1.1019 TANA90U8 0.203 1.35 0.766 1.11 0.774 1.1120 SOES86U2 0.171
1.15 0.846 1.01 0.855 1.0221 SOES86U3 0.168 1.13 0.851 1.00 0.860
1.0022 SOES86U4 0.181 1.17 0.856 1.02 0.865 1.0323 ANG81U3 0.226
1.24 0.768 0.99 0.777 0.9924 ZAHN86U8 0.169 1.12 0.836 0.89 0.845
0.9025 S17-3UT 0.193 1.49 0.789 1.02 0.798 1.0326 S24-2UT 0.176
1.34 0.789 0.93 0.797 0.9427 LI-4 0.198 1.28 0.800 0.92 0.809
0.9328 WAT89U5 0.204 1.41 0.846 0.99 0.855 1.0029 WAT89U6 0.205
1.39 0.856 0.98 0.865 0.9930 GILL79S4 0.229 1.41 0.765 0.92 0.775
0.9431 SATYARNO-3 0.217 1.41 0.792 0.83 0.805 0.8432 SATO-4 0.199
1.51 0.779 0.93 0.792 0.9433 WAT89U7 0.199 1.76 0.826 0.97 0.835
0.9934 WAT89U8 0.213 1.81 0.846 1.01 0.855 1.0235 WAT89U9 0.235
2.04 0.826 1.13 0.835 1.15
Median 1.152 0.995 0.999Mean 1.238 0.998 0.999CoV 20.84% 6.80%
6.70%
-
N DesignationACI
1 KOWALSKIU2 0.120 1.10 0.731 0.941 0.787 0.9272 KOWALSKIU1
0.119 1.05 0.731 0.895 0.787 0.8813 RES_U1 0.088 1.26 0.681 1.205
0.735 1.1934 NIST-F 0.108 1.11 0.731 0.987 0.787 0.9805 NIST-S
0.125 1.25 0.726 1.106 0.781 1.0976 LEH1015 0.085 0.95 0.742 0.878
0.797 0.8767 LEH407 0.062 1.06 0.754 1.037 0.809 1.0468 LEH415
0.099 1.11 0.742 1.021 0.797 1.0189 LEH430 0.165 1.14 0.723 0.964
0.780 0.951
10 LEH815 0.105 1.18 0.742 1.082 0.797 1.07911 KUN97A7 0.127
1.23 0.711 1.093 0.766 1.08712 KUN97A8 0.115 1.11 0.711 0.991 0.766
0.98613 KUN97A9 0.123 1.18 0.711 1.055 0.766 1.04914 KUN97A10 0.147
1.21 0.711 1.052 0.766 1.04815 KUN97A11 0.136 1.12 0.711 0.975
0.766 0.97116 KUN97A12 0.141 1.17 0.711 1.014 0.766 1.00917
WONG90U1 0.163 1.27 0.673 1.024 0.727 1.02318 POT79N1 0.145 1.17
0.708 0.993 0.764 1.00419 KOW96FL3 0.180 1.32 0.648 0.905 0.701
0.90620 VU98NH1 0.146 1.26 0.681 0.950 0.734 0.96321 VU98NH6 0.300
1.69 0.643 1.061 0.698 1.06022 POT79N5A 0.150 1.26 0.682 0.981
0.736 0.99823 WONG90U3 0.211 1.62 0.673 1.041 0.727 1.05424 POT79N4
0.141 1.19 0.708 0.895 0.764 0.91325 WAT89U10 0.156 1.58 0.709
0.957 0.763 0.97726 POT79N3 0.163 1.37 0.710 0.918 0.765 0.93727
ANG81U2 0.167 1.48 0.688 0.947 0.742 0.96528 ZAHN86U6 0.185 1.58
0.690 1.011 0.744 1.03029 POT79N5B 0.182 2.09 0.682 1.127 0.736
1.14030 WAT89U11 0.158 2.00 0.700 0.983 0.754 0.996
Median 1.221 0.992 1.001Mean 1.30 1.003 1.006CoV 20.66% 7.51%
7.30%
6-bar model 8-bar model
e e 3
max
'c
M
f hmax
cd
MM
max
cd
MM
max
cd
MM
MAX MAX2 '
cdc
M MTable A - 6. Ratios and calculated using various Mbh f
approaches for circular columns.
-
TestUnit c c +d bh
mm mm MPa MPa mm
1 400 400 5.25 36.2 423.0 1.61* 0.0153 0.98 23.0 0.230 0.160
0.836 0.982 400 400 4.06 28.8 423.0 1.61 0.0153 1.59 23.0 0.430
0.217 0.836 0.993 400 400 4.50 32.3 423.0 1.61 0.0153 1.08 23.0
0.230 0.172 0.836 1.004 400 400 3.44 27.0 423.0 1.61 0.0153 1.70
23.0 0.420 0.227 0.836 1.01
* See Reference 28 Mean 0.994
1 mm = 0.0394 in.
1 MPa = 145 psi
8barmodel
Table A-7. Relevant data of diagonally tested rectangular test
columns.
'cf
yf
su
y
ff 'c g
P
f Ah b sh
sh ,ACI
AA
b
sd e
2max
'c
M
f b hmax
cd
MM