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

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.

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.

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

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-

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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Zealand, Wellington, New Zealand, 2006, 684 pp.2. EN 1998-2:2003, Eurocode 8: Design of Structures for Earthquake

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3. AASHTO, AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications, third edition, American Association of State Highway and Transportation Officials, Washington, DC, 2004.

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6. Neville, A., Properties of Concrete, John Wiley & Sons, Inc., New York, 1996, 846 pp.

7. Baweja, D.; Munn, R. L.; Roper, H.; and Sirivivatnanon, V., In Situ Assessments of Long-Term Performance of Plain and Blended Cement Concretes, Transactions of the Institution of Engineers, V. 34, No. 2, June 1992, pp. 115-127.

8. Washa, G. W., and Wendt, K. F., Fifty Year Properties of Concrete, ACI Journal, V. 72, No. 1, Jan. 1975, pp. 20-28.

9. Al-Khaiat, H., and Fattuhi, N., Long-Term Strength Development of Concrete in Arid Conditions, Cement and Concrete Composites, V. 23, No. 4-5, Aug.-Oct. 2001, pp. 363-373.

10. Atcin, P. C., and Laplante, P., Long-Term Compressive Strength of Silica-Fume Concrete, Journal of Materials in Civil Engineering, ASCE, V. 2, No. 3, 1990, pp. 164-170.

11. Scanlon, A., and Mikhailovsky, L., Strength Evaluation of an Existing Concrete Bridge Based on Core and Nondestructive Test Data, Canadian Journal of Civil Engineering, V. 14, 1987, pp. 145-154.

12. Park, R.; Priestley, M. J. N.; and Gill, W. D., Ductility of Square-Confined Concrete Columns, Journal of the Structural Division, ASCE, V. 108, No. ST4, Apr. 1982, pp. 929-949.

13. Dodd, L. L., and Cooke, N., The Dynamic Behavior of Reinforced Concrete Bridge Piers Subjected to New Zealand Seismicity, Research Report 92-4, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, 1992, 460 pp.

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15. Watson, S., and Park, R., Simulated Seismic Load Tests on Reinforced Concrete Columns, Journal of Structural Engineering, ASCE, V. 120, No. 6, June 1994, pp. 1825-1849.

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27. Zahn, F. A.; Park, R.; and Priestley, M. J. N., Strength and Ductility of Square Reinforced Concrete Column Sections Subjected to Biaxial Bending, ACI Structural Journal, V. 86, No. 2, Mar.-Apr. 1989, pp. 123-131.

28. Zahn, F. A.; Park, R.; and Priestley, M. J. N., Design of Reinforced Concrete Bridge Columns for Strength and Ductility, Research Report 86-7, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, Mar. 1986, 410 pp.

29. Bresler, B., Design Criteria for Reinforced Concrete Columns under Axial Load and Biaxial Bending, ACI Journal, V. 57, No. 11, Nov. 1960, pp. 481-490.

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31. Freisleben, H., and Pedersen, J., Curing of Concrete Struc-tures, Durable Concrete Structures Design Guide. Issue 166 of Bulletin dInformation, Comit Euro-International du Bton, second edition, Thomas Telford Ltd., London, UK, 1989, 112 pp.

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

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f b hmax

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