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Jurnal kekuatan gaya geser pd penulangan T beam tanpa sengkang

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ACI Structural Journal/July-August 2007503
DISCUSSION
The authors have presented an interesting paper on theshear strength of reinforced concrete T-beams withouttransverse reinforcement. However, the discusser would liketo offer the following comments:1. The authors have mentioned the basic outline of aderivation of Eq. (1), but Eq. (1) appears to be based on theneutral axis (NA) located at the center of the beam in atypical homogeneous rectangular concrete beam. Theauthors’ Eq. (2) was a simplification of Eq. (1) based on theexperimental database of reinforced concrete beams, whichis inconsistent with the Rankine’s failure criteria of a plainhomogeneous concrete beam.Based on
f
t
=
6
(or 0.1
f
c
′
) and assuming the Rankine’sfailure criteria of plain concrete, and by considering variousstrength ratios of flexural stress
σ
m
versus concrete compressivestress
f
c
′
(
σ
m
/
f
′
c
= 14.2%,
10
the flexural stress
σ
m
of a plainhomogeneous concrete beam equals to 114.2%
10
of the tensilestrength of plain concrete
f
t
.
Based on the aforementionedassumptions, the discusser arrived at the authors’ Eq. (2)without considering the experimental database of reinforcedconcrete beams.Another simplified approach is that Eq. (2) can also bederived from the current ACI Building Code
9
(that is,authors’ Eq. (5)) by assuming an average depth of NA equals0.4
d
11,12
and by substituting
c
= 0.4
d
in the authors’ Eq. (5),which would result in authors’ Eq. (2). Based on the afore-mentioned two approaches, the discusser believes that thereis no need to have a reinforced concrete beam database, thatis, Fig. 1 and 2. Is this consistent with the shear strength of reinforced concrete T-beams without transverse reinforce-ment plain concrete?2. The authors’ concept on shear funnel (Fig. 8 and 10) issomewhat unclear. Please note that there is no reinforcementwithin the compression and/or flange area. Based on Fig. 8,considering a simplified approach, a portion of the cross-sectional area above NA in the T-beam could be convertedinto an equivalent rectangular section, but not the entiresection of the T-beam when a shear force is computed. Thediscusser has computed over 100 specimens of T-beamsfrom Reference 1 by assuming the flange depth as one unitand the overall depth and web width were transferred into theflange depth units with varying depths of NA (that is, NAwas assumed within the flange and within the web of theT-beams) and found that approximately 20% of the cross-sectional area increases above NA as compared with itsequivalent rectangular section and approximately 10% of thecross-sectional area increases to its equivalent rectangularsection, if the entire beam was compared with the rectangularsection. These values are somewhat inconsistent in theauthors’ Table 2.
REFERENCES
10. Kato, K.,
Concrete Engineering Data Book
, Nihon University,Koriyama-City, Fukushima Prefecture, Japan, 2000.11. Eurocode No. 2, “Design of Concrete Structures, Part 1: GeneralRules and Rules of Buildings,” ENV 1992-1-1, Commission of the EuropeanCommunities,1991.12. British Standard Institution, “Structural Use of Concrete, Part 1:Code of Practice for Design and Construction,” BS 8110:Part 1:1997, London,UK, 1997.
AUTHORS’ CLOSURE
The authors thank the discusser for his interest in thispaper. The comments are addressed in the same order aspresented by the discusser.The detailed derivation of Eq. (1) and its simplificationinto Eq. (2) are presented in Reference 2 of the paper. Thisderivation was not based on the neutral axis located at thecenter of a beam, but rather based on the location of theneutral axis as calculated based on a cracked section analysis.The discusser is referred to Reference 2 for further clarification.As noted in Reference 2, Eq. (1) was derived consideringthat failure initiates when the principal stress in the compressionzone reaches the tensile strength of concrete
f
t
. It was shownthat this equation could be simplified for an assumed tensilestrength (6) and considering the flexural stress
σ
m
. Theexperimental results, however, were considered so that acomplete perspective of the performance of the simplifiedexpression could be accessed.The discusser notes that Eq. (2) can be derived from theACI code. It appears that the discusser is referring to ACIEq. (11-3) rather than (11-5). Perhaps a better view is thatEq. (11-3) is a subset of Eq. (2). For
k
= 0.4, Eq. (2) simplifiesit to 2
b
w
d
. The neutral axis depth,
c
=
kd
varies accordingto the flexural reinforcement ratio
ρ
and the modular ratio
n
.Therefore, Eq. (2) accounts for the reinforcement ratio andthe concrete compressive strength, whereas ACI 318 Eq. (11-3)is insensitive except with respect to its inclusion in the term .Unfortunately, the discusser’s question “Is this consistentwith the shear strength of reinforced concrete T-beamswithout transverse reinforcement plain concrete?” is unclearand cannot be addressed.The results presented in Fig. 10 are based on an angledapproach using a 45-degree angle. Simplification can beachieved using an effective flange width approach. Based onthe area achieved from the 45-degree shear funnel, an effectiveoverhanging flange width of 0.5 times the flange depth oneach side of the web can be considered for shear. If theneutral axis falls within the thickness of the flange, thiseffective width approach is conservative. It should be notedthat in either the shear funnel or equivalent flange widthapproach, the neutral axis depth should be computed usingan effective flange width that is based on flexural behavior
f
c
′
f
c
′
f
c
′
f
c
′
Disc. 103-S67/From the Sept.-Oct. 2006
ACI Structural Journal
, p. 656
Shear Strength of Reinforced Concrete T-Beams without Transverse Reinforcement.
Paper by A. KorayTureyen, Tyler S. Wolf, and Robert J. Frosch
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.
504ACI Structural Journal/July-August 2007
and that is different from the flange width considered effectivefor shear.Table 2 presents a statistical comparison of the performanceof the various design methods considering the ratio of
V
test
/
V
calc
. Therefore, it is unclear what inconsistencies thediscusser is referring to. However, as emphasized in thepaper, for the evaluation of the shear area when the flangeswere ignored, the neutral axis depth was calculated ignoringthe flanges while the shear funnel approach computed theneutral axis depth with the flanges considered. This mayexplain the perceived inconsistencies in the discusser’s anal-ysis if he was directly comparing the results provided inTable 2. Regardless, the main premise is that additionalshear area beyond that bounded by the web can be consideredas effective in shear transfer. The percentage of flange areaconsidered will vary depending on the section considered andthe location of the neutral axis.
Disc. 103-S71/From the Sept.-Oct. 2006
ACI Structural Journal
, p. 693
Shear Strength of Reinforced Concrete T-Beams.
Paper by Ionanis P. Zararis, Maria K. Karaveziroglou, andProdromos D. Zararis
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.
The authors have presented an interesting concept in theirpaper on shear strength of reinforced concrete T-beams.However, the discusser would like to offer the followingcomments:1. The authors have considered
ε
co
= 0.002 and
f
ct
=0.30
f
′
c
2/3
(Eq. (4)) based on Reference 10, but no considerationwas given to the depth of compression zone equals 0.8
c
value as suggested in Reference 10. Also, the authors havenot thoroughly explained the assumption of 0.667
c
value (inthe Appendix) other than the test result values versus theirtheoretical values. Please note that BS 8110:Part 1:1997
15
considers the depth of compression zone equal a value of 0.9
c
.2. Based on Eq. (7) and Fig. 7, the authors assumption fora 45-degree projection angle from web to flange appears tobe inconsistent with Fig. 6(b) and other researchers. The45-degree projection angle was a simplified assumptionbased on the depth of compression equals the depth of flange, that is, the neutral axis (NA) is located at the interfaceof the bottom of flange and the top of web.3. In conclusion, the authors’ statement “An increase of stirrups does not give any advantage to T-beams over therectangular beams” is a little confusing without thoroughexplanation, because the authors have converted a T-beaminto a rectangular beam with
b
ef
web width in lieu of
b
w
webwidth. Let’s consider beam pairs from Table 1: Beam PairTA11-TA12 of Reference 2; Beam Pair T2-T3 and BeamPair T15-T16 of Reference 4; Beam Pair T3a-T3b of Reference 5; and Beam Pair A00-A75 of Reference 7. Thesebeam pairs all have test parameters such as concrete strength
f
c
′
, longitudinal reinforcement
ρ
%, and shear span-to-depth ratio
a
/
d
approximately identical, except for theshear reinforcement
ρ
v
f
vy;
but the shear strength increaseswith an increase in shear reinforcement
ρ
v
f
vy
. This meansthe shear reinforcement
ρ
v
f
vy
does have some influence onthe shear strength.4. The authors’ Eq. (9) and the calculated values of
A
′
s
of the depth of compression block in Fig. A (of the authors’Appendix) are unclear. It appears that the authors haveconsidered a routine rectangular beam with compressionreinforcement but have not considered the reinforcementwithin the flange width when a T-beam section wasconverted into a rectangular beam section above NA. Thereinforcement in the flange would improve the value of
c
(depth of NA) as well as the value of
V
cr
in Eq. (8) and
V
u
inEq. (10).5. The discusser has calculated all T-beams exceptBeam ET1, which is a rectangular beam from References 1 and2, as outlined in the authors’ Table 1, by considering thereinforcement in the flange width and by using authors’Eq. (10) for a calculation of NA,
c
, and then
V
cr
and
V
u
werecalculated. Based on this concept, a mean value of
V
u,exp
/
V
u,th
of 1.006 and a standard deviation value of 0.05 were found.It was also noticed from Table 1
1,2,4-8
that the thinner webwidth with higher reinforcement ratios (both longitudinaland shear reinforcement ratios) do not have any advantage overwider web width with lower reinforcement ratio in T-beams.
REFERENCES
15. British Standard Institution, “Structural Use of Concrete, Part 1:Code of Practice for Design and Construction,” BS 8110:Part 1:1997,London, UK, 1997.
AUTHORS’ CLOSURE
The authors would like to thank the discusser for hisinterest in the paper and his kind comments. The authorswould like to reply to his comments in the order they are asked.In the case of rectangular or T-section beams, the truedistribution of stresses in the compressive zone is usuallyreplaced for simplification by an equivalent rectangularstress block. In the ultimate limit state, that is, when thecompressive strain in concrete at extreme fiber is
ε
c
= 0.0035,the true distribution of stresses in the compressive zonefollows a parabola-rectangular diagram. Then, the compressiveforce of concrete, as a resultant of stresses, is
F
c
= 0.81
bcf
c
′
.Thus, the equivalent rectangular stress distribution has anapproximate height equal to 0.8
c
. In this case, however, theauthors choose a state where the strain of concrete at extremefiber is
ε
co
= 0.002. This strain corresponds in a true, exactlyparabolic distribution of stresses in the compressive zone. Inthis case, the corresponding compressive force of concrete is
F
c
= 0.667
bcf
c
′
. Thus, the equivalent rectangular stressdistribution (shown in Fig. A) has a height equal to 0.667
c
.There has never been made a 45-degree projection angle bythe authors. As it is written in the text of the paper, the failureoccurs due to a splitting of concrete that takes place in thecompression zone of the T-beam. Taking into account Fig. 2
ACI Structural Journal/July-August 2007505
and 6, the splitting takes place in an inclined area, the projectionof which, on a cross section of the beam, is approximatelydefined from the shaded part of the section in Fig. 7.Equation (7), giving the effective width, results simply fromthe area of this shaded part of cross section of the T-beam.This statement means that the contribution of stirrups in theshear strength is the same for T-beams and rectangular beams,as it results from the second part of Eq. (10). The increase inthe strength of the beams that the discusser has mentioned isdue to an increase of the first part of Eq. (10).The compression reinforcement
A
s
′
within the flangewidth has been considered and takes part in Eq. (9) with theratio
ρ′
=
A
s
′
b
w
d
. Nevertheless, an increase of
A
s
′
does notincrease the shear strength of a beam; on the contrary, itdecreases the shear strength, exactly because the reinforcement
A
s
′
improves the value of
c
. Equations (8) and (10) show thata decrease of the depth
c
decreases the strength. This hasbeen observed both in T-beams and rectangular test beams.The compression reinforcement
A
′
s
has not been consideredin the calculations of Table 1, because of the lack of dataregarding this reinforcement for all the test beams. As itresults from the discusser’s calculations, however, the smallratios of
ρ′
have only a small effect on the shear strength.
Disc. 103-S76/From the Sept.-Oct. 2006
ACI Structural Journal
, p. 736
Effect of Reinforced Concrete Members Prone to Shear Deformations: Part I—Effect of Confinement.
Paper by Suraphong Powanusorn and Joseph M. Bracci
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.
Though the authors have presented an interesting concepton shear deformations in their paper, they have not fullyexplained all necessary assumptions other than the use of Mander et al.’s methodology. Also, the authors have notprovided the details as outlined by Mander et al. (1988).Without a detailed explanation and information, particularlyof the test specimens supplemented by the associatedassumptions, it is very difficult to verify the author’s resultsas well as published results available elsewhere; therefore, thediscusser has the following comments:1. The discusser has tried to understand the authors’methodology, and has described the authors’ methodologyto the best as follows. In the following concept, there areseveral assumptions that were neither mentioned by Manderet al. (1988), nor by the authors.The authors’ Eq. (12) is expressed aswherein which
ε
1
=
ε
s
+ (
ε
s
+ 0.002)cot
2
α
Ferguson (1964) suggested that the stress in steel developsfrom 1.15
f
y
to 1.20
f
y
. Therefore, an average value of 1.175
f
y
was considered. That is,
ε
s
= 1.175
f
y
/
E
s
, where
E
s
= 29,000 ksi.Furthermore, it was assumed that the tensile strain iscausing approximately a 35-degree skew angle crack to thestrut’s axis. The 35 degrees falls within the range from 25 to45 degrees, and this angle is consistent with Cusson andPaultre (1994) and Fig. 5 and 13 of Ferguson (1964):
ε
1
=0.0024 + (0.0024 + 0.002) cot
2
35 degrees = 0.0114. Now,
ε
cc
=
ε
co
[1 +
R
((
f
′
cc
/
f
′
co
) – 1)].Based on the test results of Mander et al. (1988) andScott et al. (1980),
f
′
cc
/
f
′
co
≈
1.75 and
ε
co
≈
0.002 (Richartet al. 1928).In the previous equation, the
R
value varies from 3 to 6(Park and Paulay 1990). Based on the authors’ Fig. 1 and 2,the transverse reinforcement details with respect to the longi-tudinal reinforcement,
R
= 5, as suggested by the authors intheir Eq. (14) appears to be on the low side. Therefore,
R
= 6 was appropriate and was assumed in the aforementionedequation by the discusser. That is,
ε
cc
= 0.002 [1 + 6((1.75) – 1)]= 0.011.Based on the Mander et al. (1988) and Scott et al. (1980)test results,
ε
cc
≈
0.0115.Based on an average value of
ε
cc
= 0.01125 Also, based on an average value of
ε
cc
= 0.01125 and
ε
c
≈
0.0048 was chosen due to lateral expansion (biaxial tension-compression)x =
ε
c
/
ε
cc
= 0.0048/0.01125 = 0.425
E
sec
=
f
′
co
/
ε
cc
= 1.75
f
′
co
/3.52
ε
co
≈
0.5
E
c
Now,
r
= = = 2.0
E
c
Now
σ
c
=
σ
c
β
f
′
cc
xr r
1
–
x
r
+----------------------=
β
10.8 0.34
ε
1
ε
cc
⁄ ( )
+---------------------------------------------
1
≤
=
β
10.8 0.34
ε
1
ε
cc
⁄ ( )
+---------------------------------------------
1
≤
=
E
c
E
c
E
sec
–---------------------
E
c
E
c
0.5
E
c
–-------------------------
β
f
′
cc
xr r
1
–
x
r
+----------------------
506ACI Structural Journal/July-August 2007
Substituting
β
,
x
, and
r
values in the previous equationBecause
f
′
cc
≈
1.75
f
c
′σ
c
= 1.1008
f
c
′
or
≈
1.10
f
c
′
This means approximately 10% compressive stressincreases due to the confinement. This value is consistentwith Vecchio’s (1992) concept as well as the authors’ testsresults as shown in Tables 1 through 3.Based on Vecchio’s study (Vecchio 1992), an averagestress in shear panels was increased by approximately 5.6%,while an average stress in shearwalls was increased byapproximately 13.4%, that is, an overall average valueincreased in stress would be 9.5%. Is this consistent with themethodology/concept/logic used in this paper?2. Based on Fig. 1(a), the authors have considered asymmetrical loading case, but the symmetrical loading casemay not be the case for all structures in the practice. Becauseasymmetrical loading conditions would create unbalancedloading, it would require some additional reinforcement pertruss analogy in the dark area, as shown by the authors inFig. 9(a) and (b), depending on the unbalanced load due tothe asymmetrical loading condition.3. It is unclear how the theoretical values stated in Tables 1through 3 were calculated. Was any correction for variabledepth considered? Or was a uniform depth considered?Though the authors stated the advantage of overlapping stirrupsversus single stirrups, the effectiveness of stirrups ascompared with the longitudinal reinforcement was unclearfrom Table 1 through 3.4. The discusser would like to point out that because theshear strength and shear deformations relate to the strengthof concrete, a simplified method proposed by Muttoni(2003) could be extended to the authors’ specimens.5. Using the aforementioned concept outlined in thisdiscussion and Muttoni’s (2003) methodology, the discusserhas also analyzed other test specimens available in the literatureelsewhere (Rodrigues and Muttoni 2004; Fukui et al. 2001;Ferguson 1964). The results are found to be in good agreementwith the test results. Due to brevity, the results are notincluded in the discussion.
ACKNOWLEDGMENTS
The discusser gratefully appreciates S. Unjoh, Leader, EarthquakeEngineering Team, Public Works Research Institute, Tokyo, Japan; A.Muttoni, Institut de Structures, Ecole Polytechnique Fédérale de Lausanne,Lausanne, Switzerland; and N. Pippin and A. Wards, TTI, Texas A&MUniversity, College Station, Tex., for providing publications related to theshear strength of beams.
REFERENCES
Fukui, J.; Shirato, M.; and Umebara, T., 2001, “Study of Shear Capacityof Deep Beams and Footing,”
Technical Memorandum
No. 3841, PublicWorks Research Institute, Tokyo, Japan. (in Japanese) Cusson, D., and Paultre, P., 1994, “High Strength Concrete ColumnsConfined by Rectangular Ties,”
Journal of Structural Engineering
, ASCE,V. 120, No. 3, Mar., pp. 783-804.Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Observed Stressand Strain Behavior of Confined Concrete,”
Journal of Structural Engineering
,ASCE, V. 114, No. 8, Aug., pp. 1827-1849.Muttoni, A., 2003, “Schubfestigkeit und Durchstanzen von Platten ohneQuerkraftbewehrung,”
Beton und Stahlbetonbau
, V. 98, No. 2, Feb., pp. 74-84.Park, R., and Paulay, T., 1990, “Bridge Design and Research Seminar:V. I, Strength and Ductility of Concrete Substructures of Bridges,”
RR Bulletin
84, Transit New Zealand, Wellington, New Zealand.Richart, F. E.; Brandtzaeg, A.; and Brown, R. L., 1928, “A Study of Failureof Concrete under Combined Compressive Stresses,”
Bulletin
185, Universityof Illinois Engineering Experimental Station, Champaign, Ill.Rodrigues, R. V., and Muttoni, A., 2004, “Influence des DéformationsPlastiques de l’Armature de Flexion sur la Résistance a l’Effort Trenchantdes Pouters sans étriers: Rappart d’essai,” Laboratoire de Construction enBéton (IS-BETON), Istitut de Structures, Ecole Polytechnique Fédérale deLausanne, Oct.Scott, B. D.; Park, R.; and Priestley, M. J. N., 1980, “Stress-StrainRelationships for Confined Concrete: Rectangular Sections,”
Research Report
80-6, Department of Civil Engineering, University of Canterbury,Christchurch, New Zealand, Feb.
AUTHORS’ CLOSURE
The authors would like to express a sincere gratitude to thediscusser for comments that give the authors an opportunityto clarify certain issues in the article. The authors’ responseto the discusser is as follows:
General
The purpose of the article under discussion was to presentan alternative method that incorporates the effects of confinement into the constitutive equations of the ModifiedCompression Field Theory (MCFT), first proposed byVecchio and Collins (1986). In essence, the extension of theMCFT proposed by the authors is based on two-dimensionalstress and strain analysis. All necessary assumptions werestated at the beginning of the article under the sectionProposed analytical model.
Response to discusser comments
1. The discusser demonstrates the application of Eq. (12)on the constitutive relationship of concrete in compressiontaken into account the effect of confinement given in thepaper with assumptions on a few parameters shown in theequation. It was concluded that the results from applyingEq. (12) led to an approximate 10% increase in compressivestrength of concrete, which was compared with a study byVecchio (1992) on shearwalls and panels and also by theauthors’ reinforced concrete (RC) bent cap tests. From theauthors’ point of view, however, the application of Eq. (12)alone to obtain an increase in strength is only part of thecomparative study. It is the force-deformation behavior thatis important for comparative purposes, especially formembers prone to shear deformations near ultimate loading.MCFT is generally developed on the basis of: 1) two-dimensional states of stress and strain; 2) the superpositionof stresses in the concrete and reinforcing steel as shown inEq. (1); and 3) the compatibility of strains in the concrete andreinforcing steel as shown in Eq. (2). The model can becategorized into the so-called rotating crack model to maintainthe coaxiality between the concrete principal stresses andprincipal directions. For two-dimensional states of stress andstrain, three components of stresses and strains, which are
ε
x
,
ε
y
, and
γ
xy
and
σ
x
,
σ
y
, and
τ
xy
, are required to define a stateof stress and strain at a given point within the member. Theconstitutive relationships under the context of MCFT,however, have been defined in the principal stress and straincomponents (
σ
1
,
σ
2
) and (
ε
1
,
ε
2
). The general state of stressand strain,
ε
x
,
ε
y
, and
γ
xy
and
σ
x
,
σ
y
, and
τ
xy
, are related to theprincipal stress and strain components (
σ
1
,
σ
2
) and (
ε
1
,
ε
2
)using Mohr’s circle of stress and strain. The concreteconstitutive equation in compression defined in the principalstress and strain directions are given in Eq. (4) through (8)and (11) through (13). The special emphasis of the article is
σ
c
0.8737
( )
f
′
cc
0.425
( )
2.0
( )
2.0 1
–
0.425 2.0
( )
+------------------------------------------------------------
0.629
f
′
cc
==
ACI Structural Journal/July-August 2007507
on the incorporation of the beneficial effects of lateralconfinement of the transverse reinforcement on the concretestress-strain relationship in the principal compressive directionusing an approach adopted by Mander et al. (1988) using thefive-parameter failure surface derived by Willam andWarnke (1974). Due to space limitations, the authors did notinclude the complete development of five-parameter failuresurface in the article. Interested readers should consult thesrcinal paper by Willam and Warnke (1974) or books byChen (1982), Chen and Han (1988), and Chen and Saleeb(1982) for further details.Regarding the discusser’s comments on the
R
value fordetermining the peak strain corresponding to the peak concrete stress, additional studies by the authors have shownthat the use of
R
= 6 led to only a marginal change in thestrength prediction.2. The MCFT was formulated on the basis of threefundamental principles of structural mechanics, which are:1) equilibrium; 2) compatibility; and 3) material constitutiverelationships. The rationality and generality of the MCFTshould make the theory applicable to any loading pattern.The case of unsymmetric loading, however, was not consideredin this work and would require further experimental andanalytical research to justify recommendations.3. To justify the proposed model, the authors implementedthe proposed model into a finite element code using a user-defined material subroutine. It is the results from FEM analysisthat are summarized in Tables 1 through 3.4 and 5. The authors agree with the discusser that the shearstrength and deformation are related to the compressivestrength of concrete and would like to look into furtherdetails on the article by Muttoni (2003).
REFERENCES
Chen, W.-F., 1982,
Plasticity in Reinforced Concrete
, McGraw-Hill, NewYork, 474 pp.Chen, W.-F., and Han, D. J., 1988,
Plasticity for Structural Engineers
,Springer-Verlag, New York, 606 pp.Chen, W.-F., and Saleeb, A. F., 1982, “Constitutive Equations for EngineeringMaterials,”
Elasticity and Modeling
, V. 1, John Wiley & Sons, New York. Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Theoretical Stress-Strain Model for Confined Concrete,”
Journal of Structural Engineering
,ASCE, V. 114, No. 8, pp. 1804-1826.Willam, K. J., and Warnke, E. P., 1974, “Constitutive Model for the TriaxialBehavior of Concrete,”
Concrete Structures Subjected to Triaxial Stresses
,Paper III-1, International Association of Bridge and Structural EngineersSeminar, Bergamo, Italy, pp. 1-30.

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