Shear Strength Reduction Factor of Prestressed Hollow-Core

Research ArticleShear Strength Reduction Factor of Prestressed Hollow-CoreSlab Units Based on the Reliability Approach

Hae-Chang Cho1 Min-Kook Park1 Hyunjin Ju1 Jae-Yuel Oh1

Young-Hun Oh2 and Kang Su Kim1

1Department of Architectural Engineering University of Seoul 163 Seoulsiripdae-ro Dongdaemun-guSeoul 02504 Republic of Korea2Department of Medical Space Design amp Management Konyang University 158 Gwanjeodong-ro Seo-guDaejeon 35365 Republic of Korea

Correspondence should be addressed to Kang Su Kim kangkimuosackr

Received 10 February 2017 Revised 9 May 2017 Accepted 13 June 2017 Published 17 July 2017

Academic Editor Paulo M S T De Castro

Copyright copy 2017 Hae-Chang Cho et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study investigated the shear design equations for prestressed hollow-core (PHC) slabs and examined the suitability of strengthreduction factors based on the structural reliability theoryThe reliability indexes were calculated for the shear strength equations ofPHC slabs specified in several national design codes and those proposed in previous studies In addition the appropriate strengthreduction factors for the shear strength equations to ensure the target reliability index were calculated The results of the reliabilityindex analysis on the ACI318-08 equation showed that the shear strengths of the members with the heights of more than 315mmwere evaluated to be excessively safe whereas some members with low depths did not satisfy the target reliability index

1 Introduction

The prestressed hollow-core (PHC) slabs are made in theform of hollow sections with a circular or elliptical shape asshown in Figure 1 to make them lighter and maximize thesectional efficiency The prestress is commonly introducedto the PHC slabs by the placing of prestressing tendonswhich leads them to have excellent flexural resistance Inaddition they have been widely used as a lightweight floorstructural system in Europe and North America due totheir excellent economic efficiency and constructability [1ndash3]However since the PHC slab members are typically formedby extruder machines that cast concrete while moving inthe longitudinal direction of the members it is very difficultto place shear reinforcements which are to be placed inthe vertical direction Moreover the PHC slabs have thinwebs due to the hollow-cores inside and thus their shearstrengths are typically dominated by the web-shear strengthat the ends of the members The method of enhancing theirshear strength is truly very limited compared to what can

be done to enhance their flexural strengths for which theirmember strengths are governed by the web-shear strengthin most cases [2 3] In particular the previous experimentalstudies [1 3] reported that for the PHC slabs with the depthsof greater than 315mm (125 in) the shear strengths weresomewhat overestimated by the web-shear strength specifiedin ACI318-05 On that basis the shear provision in ACI318-08 was substantially revised to reduce the shear strength ofthe PHC slabs greater than 315mm depth with no minimumshear reinforcement by half [3ndash6] As mentioned by Im [7]and Palmer and Schultz [6 8] however the revised shearprovision can provide very conservative shear strengths forthe PHC slabs thicker than 315mm and it can thus serveas a limiting factor for their wide applications Thereforemore detailed analyses and deep examinations on the shearstrengths of the PHC slabs are necessary In the authorsrsquoprevious study [9] the shear strength equations in the currentdesign codes [4 5 10ndash13] and by a research group [14] wereexamined in detail comparing to the experimental results ofthe PHC slab members collected from literature Then they

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 8280317 11 pageshttpsdoiorg10115520178280317

2 Advances in Materials Science and Engineering

Figure 1 Various types of prestressed hollow-core slabs

proposed an equation to calculate the web-shear strengths ofthe PHC slabs which is simple but provides a proper marginof safety

In the design codes based on the ultimate strengthdesign approach the design strength (120601119877119899) is calculated bymultiplying the nominal strength (119877119899) of themember and thestrength reduction factor (120601) that is less than 1 dependingon the importance and failure mode of the member to takeinto account the variability of materials and dimensions theconstruction errors the uncertainties of code equations andso onMacgregor et al [15] determined the strength reductionfactor (120601) by performing structural reliability analyses on theexperimental results available at that time In the currentACI 318 design code (2014) [16] however the shear strengthreduction factor is 075 for both reinforced concrete (RC) andprestressed concrete (PSC) members even though the con-struction errors and the uncertainties of the code equationsfor the RC and PSC members are not the same In particularLee et al [9] reported that the web-shear strength equationfor PSC members overestimated the shear strengths of thePHC slabs with the heights of less than 315mm (thin PHCslabs hereafter) as well as those of greater than 315mm (thickPHC slabs hereafter) Thus it is necessary to reexaminethe current reduction factor for the shear strengths of PHCslabs In this regard this study examined the suitability ofthe shear strength reduction factor of 075 specified in thecurrent ACI 318 design code (2014) based on the structuralreliability analysis on the experimental results of the PHCslabs collected from literature including the authorsrsquo previousstudy

2 Web-Shear Strength

The shear failures of PSC members without shear reinforce-ments are generally divided into the web-shear and flexural-shear failures [20ndash22] The shear failure of PHC slabs thathave a thin web due to the hollow-cores in the section isdominated by web-shear strength in most cases [3 6ndash9 23ndash25] In most design codes [4 5 11] two assumptions are

x

y

Critical sectionCritical point

Centroid axis

Tendon

ℎ2

lb

lc

lt

xy

x

Figure 2 Description of transfer length (119897119905) bearing length (119897119887) andcritical section (119897119888)

introduced to simplify the calculations of web-shear strengthas shown in Figure 2 the critical section is located at a certaindistance (119897119888) away from the support and the vertical normalstress (120590119910) at the centroid of the section is negligible Thenthe shear strength (120591119909119910) and the longitudinal stress (120590119909) atthe centroid axis of the critical section can be represented asfollows

120590119909 = 120572119891119901119890119860119901119904119860119888 (1)

120591119909119910 = 119878119887119908119868119892119881119910 (2)

where 120572 is the ratio of the prestress at the critical sectionto the fully effective prestress of the strands which can beestimated as the distance (119897119888) from the member end to thecritical section divided by the transfer length (119897119905) as shownin Figure 2 Also 119891119901119890 is the effective prestress 119860119901119904 and 119860119888are the cross-sectional area of the strands and the memberrespectively 119878 is the moment of area 119868119892 is the moment ofinertia of gross section 119887119908 is the sum of the web width and119881119910 is the shear force acting on the section According to thetheory of elasticity [26] the principal tensile stress (1205901) at thecritical section can be represented as follows

1205901 = 1205901199092 + radic12059011990924 + 1205911199091199102 (3)

The point at which the principal tensile stress (1205901) onthe web reaches the tensile strength of the concrete (119891119888119905) isconsidered as the shear cracking strength of the PSCmemberThe shear cracking strength and the shear strength of themember can be different but they are assumed to be the samebecause their difference is very small especiallywhenno shearreinforcements are placed Therefore by substituting (1) and(2) into 120590119909 and 120591119909119910 in (3) respectively the web-shear strength(119881119888119908) can be represented as follows

119881119888119908 = 119868119892119887119908119878 radic1198911198881199052 + 120572119891119901119890119891119888119905119860119901119904119860119888 (4)

Advances in Materials Science and Engineering 3

In Eurocode 2 [11] and FIP recommendations [12] (4) isused to calculate the web-shear strength of PSC membersEurocode 2 also presents the web-shear strength (119881119877119889119888) ofPSC members in a similar form as follows

119881119877119889119888 = 119868 sdot 119887119908119878 radic1198911198881199051198892 + 120572119897120590119888119901119891119888119905119889 (5)

where 119891119888119905119889 is 12057211988811990507119891119888119905119898120574119888 Here 120572119888119905 is a coefficient to takeaccount of long term effects on the tensile strength andunfavorable effects depending on the patterns of appliedloads and the recommended value is 10 in most cases Also120574119888 is the partial safety factor for concrete and 15 was used119891119888119905119898 is the mean value of axial tensile strength of concretewhere 03(1198911015840119888 )23 was applied when the compressive strengthof concrete (1198911015840119888 ) is less than 50MPa and 212 ln(1+(1198911015840119888+8)10)when1198911015840119888 exceeds 50MPa In addition 120572119897 is 1198971199091198971199011199052 which shallnot be greater than 10 where 119897119909 is the distance of the criticalsection from the starting point of the transfer length (equalsto 119897119888) and 1198971199011199052 is the upper bound value of the transfer length(equal to 119897119905) which is calculated to be 1205721120572211988911988711989111990111989412057811990111205781119891119888119905119889Here 1205721 is 125 1205722 is 019 1205781199011 is 32 1205781 is 10 119889119887 is the diameterof strands and 119891119901119894 is the stress of strands right after release120590119888119901 is the concrete compressive stress at the centroid axis dueto axial loading andor prestressing (equals to 119891119901119888)

Walraven and Mercx (1983) proposed the web-shearstrength (119881119888119908) of PHC slabs by applying the overall reductionfactor of 075 to Eurocode 2 equation as follows

119881119888119908 = 075119868119887119908119878 radic1198911198881199052 + 120572119897119890120590119888119901119891119888119905 (6)

where 120572119897119890 is as follows120572119897119890 = 1 minus [(119897119905 minus 119897119887)119897119905 ]2 (7)

and 119891119888119905 is 005119891119888119906 + 1 and 119897119887 is the bearing lengthThe FIP recommendation [12] suggests the web-shear

strength (11988111987711988912) as follows11988111987711988912 = 119868119887119908119878 radic1198911198881199051198892 + 09120572120590119888119901119891119888119905119889 (8)

In (5) compared to (8) the reduction factor of 09 wasadditionally considered Also 03(1198911015840119888 )23 presented in theCEB-FIP Model Code [12] is used for 119891119888119905119889 and the value120572119897119890 presented by Walraven and Mercx [14] in (7) is alsoapplied for 120572 In JSCE (2007) the web-shear strength (119881119908119888119889)of the prestressed member without shear reinforcements ispresented as follows

119881119908119888119889 = 119891119908119888119889119887119908119889120574119887 (9)

where 119891119908119888119889 is 125radic1198911015840119888 120574119887 is the member factor that isgenerally taken to the value of 13 and 119889 is the effectivemember depth

ACI318-08 and AASHTO-LRFD simplified method [10]assumes the shear stress distribution as the mean shear stress(120591119909119910119886) as follows

120591119909119910119886 = 119881119910119887119908119889119901 (10)

Accordingly the web-shear strength (119881119888119908) is presented inACI318-08 (same as ACI318-14) and AASHTO-LRFD simpli-fied method respectively as follows

119881119888119908 = (029radic1198911015840119888 + 03119891119901119888) 119887119908119889119901 + 119881119901 (11)

119881119888119908 = (016radic1198911015840119888 + 03119891119901119888) 119887119908119889V + 119881119901 (12)

where 119889119901 is the distance from the extreme compressionfiber to the centroid of prestressing steel 119889V is the effectiveshear depth and 119881119901 is the vertical component of prestressingforce However since most of the PHC slabs without shearreinforcements fail in shear soon or right after diagonal cracksoccur in the web mostly having no flexural cracks at the endregion it is not suitable to calculate the web-shear strengthby assuming the shear stress as the mean stress value of acracked sectionTherefore in the authorsrsquo previous study [9]they considered the shear stress distribution as a parabolicshape and proposed the web-shear strength (119881119888119908) as follows

119881119888119908 = (029120582radic1198911015840119888 + 03119891119901119888) 119887119908119868119878 + 119881119901 (13)

where the web-shear strength presented in ACI318-05 waslowered as much as the ratio of the average to the parabolicshear stress that is [119868119892(119878119889119901)] In addition a more simplifiedversion of (13) was also proposed by applying the mean valueof 119868119892(119878119889119901) of the collected specimens as 076 as follows

119881119888119908 = (022radic1198911015840119888 + 023119891119901119888) 119887119908119889119901 (14)

Even in the authorsrsquo previous study (Lee et al 2014) theyrecommended reducing the shear strength calculated by (14)of the PHC slabmembers havingmore than 500mmdepth byhalf to ensure a proper margin of safety

As shown in Table 1 a total of 155 experimental datasets were collected from Lee et al [9] and Im [7] in orderto validate the accuracy and reliability of the shear strengthequations of design codes or previous studies Table 3 presentsthe strength ratios (119881119888119908Cal119881119888119908Test) of the shear strengthscalculated by design codes and previous researchers to thetest results on the specimens collected in this study Notethat no material or strength reduction factor was applied inthe calculation only to check the accuracies of their shearstrengths It seems that the web-shear strength equation ofACI318-05 provided a good accuracy with a mean of 0955and a COV (coefficient of variation) of 0251 Lee et al[9] reported however that the shear strength by ACI318-05provided unsafe results for a large number of specimens asshown in Figure 3(a) and a similar observationwas alsomadein the cases of FIP and JSCE As shown in Figure 3(b) the


Table1Dim

ensio

nsandmaterialpropertieso

fshear

testdatacollected

from

literature[79]

Authors

Specim

ens

name

Void

type

Geometry

Prestre

ssreinforcem

ent

Loading

119891 119888119896 (MPa)

119887 119908 (mm)

119889 119901 (mm)

119860 119892 (mm2)

119868sdot119887119908S

(mm3)

119860 119901119904 (mm2)

119891 119901119906 (MPa)

119891 119901119888 (MPa)

119886119889 (mdash)

L (m)

119881 test (kN)

WalravenandMercx

(1983)

mdashC

NC

640

2500sim2

940

2250sim2

600

171000sim199

000

59601sim6

0490

4700sim

9400

18600

27sim6

317sim67

60

1816sim28

63

Becker

andBu

ettner

(1985)

mdashNC

414

3378sim43

181700sim2

159

1406

45sim165

806

43658sim4

4528

4490

sim10322

17225

29sim5

335sim6

746sim10

41153sim2

719

Pajari

(2005)

mdashC

410sim72

62150sim3

350

2000sim5

000

11900

0sim300

000

36333sim12

5833

3720sim

19530

mdash22sim8

324sim4

640sim11

0800sim52

80

TNOBu

ildingand

Con

structio

nsRe

search

(2005)

[17]

mdashIC

599sim113

92410sim44

902550sim4

000

172000sim26

1000

49073sim11

4927

8537sim

16844

mdash42sim10

425

40sim6

02240sim6

520

Bertagno

liand

Mancini(200

9)mdash

I557sim65

72150sim4

440

1510sim49

70116

000sim2

9300

041425sim13

0089

2491sim

1440

318600

24sim8

023sim3

041sim7

01570sim714

0

Celal

(2011)

mdashC

629sim67

92190

sim2830

1580sim2

550

132887sim180

414

34246sim4

2359

5025sim

8883

18600

54sim6

930sim3

8mdash

1630sim2

970

Rahm

anetal

(2010)

mdashC

400

3157sim3

254

2120sim2

620

151812sim179

115

47304sim7

0400

5922sim

7896

19600

40sim4

723sim2

8mdash

1985sim2

989

Im (2015)

ST200

C600

2470

1750

121590

378395

5048

18600

50

300

78738

ST200R

C600

2470

1750

121590

378395

5048

18600

50

300

781718

ST265

NC

600

2570

2300

151120

528534

5048

18600

40

300

781113

ST265R

NC

600

2570

2300

151120

528534

5048

18600

40

300

782084

ST4N

-1NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2685

ST4N

-2NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2506

ST4N

-3NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2833

ST5N

-1NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4175

ST5N

-2NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4448

ST5N

-3NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

3596

Total155

specim

ens

CfvoidtypeN

C(non

circular)C(circ

ular)andI(Ishape)119891119901119906specifiedtensile

strengthof

prestre

ssingste

el119891119901119888com

pressiv

estre

ssin

concreteatthec

entro

idof

thes

ectio

ndu

etoeffectiv

eprestr

ess


Undesirable side

Ove

resti

mat

ed si

deU

nder

estim

ated

side

200 400 600 8000Vcw＝Ｆ(ACI318-05) (kN)

0

200

400

600

800

V(k

Ncw

Tes

t)

00

05

10

15

20

Vcw

＝ＦV

cwT

est

200 300 400 500 600100Member height (mm)

ℎ gt 315ＧＧℎ lt 315ＧＧ

(a) ACI318-05

Ove

resti

mat

ed si

deU

nder

estim

ated

sideUndesirable side


0

200

400

600

800

Vcw

４ＭＮ

(kN

)

200 400 600 8000Vcw＝Ｆ(ACI318-08) (kN)

00

05

10

15

20Vcw

＝ＦV

cw４ＭＮ

200 300 400 500 600100Member height (mm)

(b) ACI318-08

Figure 3 Shear strength ratios by design codes of ACI318-05 and ACI318-08

ACI318-08 revised fromACI318-05 provided very conserva-tive estimation on the shear strength of the PHC slabs havingmore than 315mm depth Compared to these shear strengthequations the proposed equations by Walraven and Mercx[14] and Lee et al [9] showed a better accuracy as will beexamined in detail later in the section on the shear strengthreduction factor for PHC slabs In particular the equationproposed by Lee et al [9] is fairly easy to apply in practicaldesign because of its simple form

3 Strength Reduction Factor

Many researches have been conducted to ensure the struc-tural safety by considering the uncertainties of structuralmaterials and members [18 27] which required a lot oftime and efforts since they are based on the vast amountof experimental results and statistical data Macgregor [27]collected the experimental results of reinforced concreteand prestressed concrete members based on which they

performed statistical analysis to investigate the uncertaintiesof the materials and members and derived the relationshipbetween the strength reduction factor and the load factorbased on the predetermined target reliability index (120573119905) Thetarget reliability is an index set to ensure a certain level ofsafety In general the probabilistic approaches to evaluate thestructural safety by applying the structural reliability theory[28 29] are divided into three types that is Level I Level IIand Level III [30] Level III is the most accurate approachin which numerous random values are generated througha Monte Carlo simulation based on the mean and varianceof each random variable and then substituted into the limitstate function to directly calculate the number of failure casesThis method requires a significantly large number of trialswhich is very disadvantageous Level II is an approximateapproach that estimates the failure probability through thereliabilitymethods such as the First-OrderReliabilityMethod(FORM) and the Second-Order Reliability Method (SORM)The limit state function can be linearized by the FORM and


the reliability index is then estimated in a simple mannerThe limit state function can be approximated in a quadraticform by the SORM and the probability of failure is thenestimated at the quadratic failure surfaceThesemethods havean advantage in that the time required for analysis is relativelyshorter than Level III Level I is a method that determinesthe safety factor so that the reliability index (120573) based onthe design strength exceeds the target reliability (120573119905) [31]The target reliability can be determined by considering thesafety margin and the consequences of failure Macgregor[27] performed statistical analysis of the relationship betweenthe strength reduction factor and the load factor based onwhich he determined120573119905 as 30 to satisfy the failure probabilityof 013 and decided the load factors for the combination ofdead loads live loads and snow loads as follows

119880 = 12119863 + 16119871119880 = 12119863 + 16119878119880 = 14119863

(15)

where 119880 is the factored load 119863 is the dead load 119871 is the liveload and 119878 is the snow load In addition when a wind loadand earthquake load are applied 120573119905 was set to 25 and 20respectively and the load combinations were determined asfollows

119880 = 12119863 + 13119882 + 05119871119880 = 12119863 + 15119864 + (05119871 or 02119878) 119880 = 09119863 minus (13119882 or 15119864)

(16)

where 119882 is the wind load and 119864 is the earthquake loadThe failure probability varies greatly depending not only onthe variability of the loads but also on the failure modes ofthe members For the case of shear the differences betweenthe actual failure loads and the member strengths calculatedby code equations are relatively large and thus it should bedesigned to be safer compared to the case of flexure There-fore Macgregor [27] proposed different strength reductionfactors according to failure modes so as to satisfy 120573119905 withthe load factors In ACI318-83 the strength reduction factorswere 085 for flexure 070 for shear and torsion and 070 and065 for the spiral and the tied columns under compressionrespectively which were then revised a few times later up tothe current version of ACI318-14

In recent years however the uncertainties of the materialand member strengths have been reduced with the develop-ment of material production and processing technology andstructural design and construction technology Accordinglythe design of concrete members by applying the strengthreduction factors based on previous statistical data on thematerials andmembers can lead to becoming excessively con-servative which may cause restraining economic efficiencyNowak and Szerszen [18 19] conducted a research on thestrength reduction factors by setting 120573119905 to 35 based on therecent statistics data that is mean variance and coefficientof variation of material and proposed the strength reductionfactors of 09 and 085 for flexural and shear respectively

which are equal to or greater than those of 09 and 075presented in ACI318-14 This is as previously mentionedbecause the uncertainties are reduced due to a decrease inthe variation of the material and member strengths and anenhanced accuracy of their strength estimationThus a morereasonable strength reduction factor which is on the safe sidewas proposed by setting higher 120573119905

In accordance with the recent researches the targetreliability for shear in PHC slabs was set to 35 in this studyThe strength reduction factor for the shear and torsion inRC members is 075 in ACI318-14 which applies equally toPHC slabs As mentioned previously because PHC slabs aremostlymanufactured in a factory the quality ofmaterials andmembers is consistently good and the uncertainties of themember strengths are relatively small compared to typical RCmembers In addition the PHC slab members have hollow-cores in the section and relatively small depths compared tothe typical PSC beam members and thus they have differentdeviations of the material and member strengths from thosePSC beams Therefore a detailed analysis is necessary tocheck the suitability of 075 as the strength reduction factorfor the shear design of PHC slabs when they are designedby applying the shear strength equations used for the generalPSC members

31 Shear Failure Probability of PHC Slabs In order to inves-tigate whether it is appropriate to apply the shear strengthreduction factor of 075 in calculating the shear strengthsof PHC slabs the reliability index (120573) was estimated for 155data pieces collected in this study based on the First-OrderReliability Method [30] For the load combination of 12119863 +16119871 the nominal strength (120601119877119899) should be designed to beequal to or greater than the required strength (11987612119863+16119871)which can be expressed as follows

120601119877119899 ge 11987612119863+16119871 (17)

where the safety factor (SF) can be defined as follows

SF = (12119863 + 16119871) 120601119863 + 119871 (18)

Considering 119863 + 119871 in (18) as the load at shear failure ofthe specimen the limit state function (119892) can be defined asfollows

119892 = SF sdot 119881119899 minus 119881test lt 0 (19)

where 119881119899 is the shear strength calculated by the codeequations or other proposed equations by researchers and119881test is the experimental value The shear strength (119881119899) ofthe PHC slab is a function of several random variables Asshown in (11) that is the ACI code equation the compressivestrength of concrete (1198911015840119888 ) the compressive stress in concreteat the centroid of the cross section (119891119901119888) and the shape of thePHC slab (119887119908 119889119901) are the random variables The moment ofinertia of gross section (119868119892) and section modulus (119876) can beadditional random variables in Eurocode (see (5)) The biasfactors and COVs for each random variable refer to the studyof Nowak and Szerszen [18] as shown in Table 2 Since 119891119901119888 is


Table 2 Statistical parameters for variables [18 19]

Parameters Nominal value(MPa) Bias factor COV Distribution

11989111988811989648 119 0115 Normal55 109 0090 Normal62 116 0100 Normal69 113 0115 Normal83 104 0105 Normal

119891119901119890 mdash 09 016 Normal119887119908 mdash 100 006 Normal119889119901 mdash 100 006 Normal

Table 3 Summary of shear strength ratios

Standards andauthors

Mean(119881119888119908Cal119881119888119908Test) Standard deviation COV

Eurocode 2 1111 0290 0260FIP recommendation 1185 0293 0247JSCE 1104 0291 0263AASHTO-LRFD 0600 0153 0255ACI 318-05 0955 0239 0251ACI 318-08 0735 0281 0383Walraven and Mercx 0913 0227 0249Lee et al(simplified equation) 0710 0175 0246

a function of the combinations of several random variablessuch as the effective prestress (119891119901119890) sectional area of tendon(119860119901119904) and sectional area of the PHC slab (119860119892 = 119887119908119889119901) thisstudy assumed a normal distribution of these variables (119891119901119890119887119908 119889119901) and then generated 10 times 106 random numbers foreach random variable through a Monte Carlo simulation todetermine the COV and bias factor of 119891119901119888 The First-OrderReliability Method (FORM) analysis was performed usingthe limit state function presented in (19) and the probabilitydistribution of the influencing variables on the shear strengthbased on which the reliability index (120573) was calculated for allthe specimens In case the limit state function is linear 120573 canbe estimated by calculating the minimum distance betweenthe origin and the limit state function in a plane where eachrandom variable (1198831 1198832) is changed to standard normaldistribution random variables (1198831015840111988310158402) as shown in Figure 4That is the point that orthogonally comes in contact with thelimit state function is the most probable failure point (MPP)and the distance between the origin and the MPP (120573) canbe expressed as a function of the mean and variance of eachrandom variable as follows [30]

120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)

When the limit state function is nonlinear it is very difficultto calculate the 120573 value and it can be thus simplified by the

first-order approximation The limit state function can beexpanded by the Taylor series as follows

119892 (1198831 1198832 119883119899) = 119892 (119909lowast1 119909lowast2 119909lowast1 )+ 119899sum119894=1

(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1

(119883119894 minus 119909lowast119894 ) (119883119895 minus 119909lowast119895 )(120597119892120597119883119894120597119883119895)

+ sdot sdot sdot

(21)

where 119892 is a limit state function119883119894 is a random variable and119909lowast119894 is a random variable at the MPP By the FORM the first-order term of (21) can be taken as follows

119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)

where 1198831015840119894 and 1199091015840lowast119894 are reduced variates and taken as followsrespectively

1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)

1199091015840lowast119894 = 119909lowast119894 minus 120583119883119894120590119883119894 (23b)


Most probable failure point

Limit state

Safe state

Failure state

Original limit state functionX

2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0

Figure 4 Space of reduced variates 11988310158401 and11988310158402

Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

Figure 5 Web-shear strength failure probability of PHC slabs inACI318-08

From the first-order approximation the mean value andvariance (120583119892 and 1205902119892) of the function 119892(119883) can be presentedas follows

120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)

As mentioned previously the reliability index can beexpressed as a function of the mean and standard deviationand thus the reliability index of FORM becomes

120573 = 120583119892120590119892 =minussum119899119894=1 1199091015840lowast119894 (1205971198921205971198831015840119894)radicsum119899119894=1 (1205971198921205971198831015840119894)2

(25)

Figure 5 shows the reliability index (120573) for 155 specimenswhich were calculated by applying the ACI318-08 equation

02

04

06

08

10

12

xya

xy

200 300 400 500 600100Member Height h (mm)

Figure 6 Ratio of average shear stress to parabolic shear stress

to the limit state function with a strength reduction factor of075 and a live load ratio to dead load of 05 The ACI318-08 equation showed significantly greater reliability indexvalues in the cases of the PHC members with more than315mm depth because it allows only half the shear strengthsof the members and it confirms that the shear strengthsby the ACI318-08 equation are excessively conservative forthose thickmembersTherefore the shear strength reductionfactor of 075 for the thick PHC slabs could result in a veryuneconomical design In addition there were many casesthat did not satisfy the target reliability (120573119905 = 35) in thePHC members with less than 315mm depth This is becausethe ACI318-08 equation is based on the assumption of theaverage shear stress distribution as shown in (10) and thusas shown in Figure 6 the shear strength is underestimatedby about 80 compared to the shear strength obtained byassuming the parabolic distribution of shear stress on theweb In contrast as shown in Figure 7 the shear strengthby Lee et alrsquos [9] and Walraven and Mercxrsquos [14] modelsin which the shear stress distribution of the PHC slab isconsidered to be a parabolicwas found to secure the reliabilityindex more than the target reliability (120573119905) for most PHC slabmembers including the members with more than 315mmdepth In Lee et alrsquos [9] and Walraven and Mercxrsquos [14]models however the reliability index values (120573) for most ofthe data were greater than 45 which means that the failureprobability is lower than 34 times 10minus4 Therefore if the PHCslabs are designed by these approaches the shear strengthreduction factor of 075 could result in a very uneconomicaldesign

32 Shear Strength Reduction Factor for PHC Slabs Thisstudy derived the relationship between the strength reductionfactor of the PHC slab and reliability index (120573) by using theshear strength estimation model proposed by the authorsrsquoprevious study [9] In order to consider various load con-ditions applied to the structures as shown in Figure 8 thereliability indexes (120573) were calculated according to the ratioof the dead load to the sum of dead load and live load (119863(119863+ 119871)) The load ratios 119863(119863 + 119871) ranged from 09 in which

Advances in Materials Science and Engineering 9Re

liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075

(a) Lee et al (2014)

Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075

(b) Walraven and Mercx (1983)

Figure 7 Reliability index values of PHC slabs

Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)

Figure 8 Reliability index for Lee et alrsquos model

the live load is very small to 009 where the live load is verylarge The reliability index (120573) is expressed as follows

120573 = 120583Lee minus 120583119863 minus 120583119871radic1205902Lee + 1205902119863 + 1205902119871 (26)

where 120583Lee and 120590Lee are the mean and variance of Lee et alrsquosmodel respectively In this study a normal distribution wasassumed for the variables used in Lee et alrsquos model and 10 times106 random numbers were then generated for each randomvariable through the Monte Carlo simulation to determinethe COV and bias factor (120582) of Lee et alrsquos modelThe sectionaland material properties of PHC slabs were referred from Imtests shown in Table 1 and their bias factors and COVs werereferred fromNowak and Szerszen [18] as shown inTable 2 In(26) 120583119863 and 120583119871 are themean values of the dead and live loadsrespectively and 1205902119863 and 1205902119871 are the variances of the dead andlive loads respectively The statistical data for the loads usedin this study is also based on the study byNowak and Szerszen[18] as presented in Table 4 The mean values of 120583Lee 120583119863 and120583119871 can be expressed as follows

120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)

Table 4 Statistical parameters for loads [18 19]

Load component Bias COV DistributionDead load 103 008 NormalLive load 100 018 Normal

where120582Lee120582119863 and120582119871 are the bias factors of Lee et alrsquosmodeldead loads and live loads respectively By using (17) and theratios of dead load to dead load plus live load (119863(119863 + 119871))(26) can be rearranged as a function of 120583119863

When the dead load and live load are the same (ie119863(119863 + 119871) = 05) if the shear strength reduction factorof 075 is used the reliability index was about 40 whichis on the safe side For the strength reduction factor of 08the reliability index was about 35 which is almost equal tothe target reliability (120573119905) According to Nowak and Szerszen[18 19] the load ratio 119863(119863 + 119871) was reported to rangegenerally from 03 to 07 in beams from 03 to 06 in slabsand from 04 to 09 in columnsTherefore when the strengthreduction factor of 075 is applied for the cases of load ratiosranging from03 to 06 which is typical in slabs the reliabilityindex (120573) ranges from 390 to 407 which means being verysafe When the strength reduction factor of 08 is applied thereliability index (120573) ranges from 351 to 371 which is slightlygreater than the target reliability index (120573119905) This suggeststhat the use of the strength reduction factor of 08 makes itpossible to perform an economic design while satisfying thetarget reliability index As mentioned previously however alarge number of results of ACI318 equation do not satisfy thetarget reliability index (120573119905) for the members with less than315mm depth thus it is desirable to slightly lower the shearstrength reduction factor for the design of the PHC slabsbased on ACI318-08

4 Conclusion

This study evaluated the accuracy on the web-shear strengthequations of various codes and researchers by comparing tothe experimental results of PHC slabs failed in web-shear andexamined the shear strength reduction factors suitable for


the shear design of the PHC slabs The following conclusionswere obtained from this study

(1) The reliability analysis showed that the ACI318-08equation is excessively conservative for the PHCmembers with greater than 315mm depth the shearstrength reduction factor of 075 for the thick PHCslabs could result in a very uneconomical design

(2) The ACI318-08 equation did not satisfy the targetreliability (120573119905 = 35) for the PHC slabs with less than315mmdepth and thus it is desirable to slightly lowerthe shear strength reduction factor for the design ofthe PHC slabs based on ACI318-08

(3) The models by Walraven and Mercx and Lee et alprovided a good accuracy on the shear strengthsof PHC slabs but they provided very conservativeresults when the shear strength reduction factor of075 was used

(4) The reliability analysis of Lee et alrsquos shear strengthequation showed that the strength reduction factor of08 works well leading to an economic design whilesatisfying the target reliability index

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by Basic Science ResearchProgram through the National Research Foundationof Korea (NRF) funded by the Ministry of Education(2016R1D1A3B03932214)

References

[1] R J Becker and D R Buettner ldquoShear Tests of ExtrudedHollow-Core Slabsrdquo PCI Journal vol 30 no 2 pp 40ndash54 1985

[2] D R Buettner and R J Becker PCI Manual for the Designof Hollow-Core Slabs PrecastPrestressed Concrete InstituteILLINOIS Chicago 2nd edition 1998

[3] N M Hawkins and S K Ghosh ldquoShear strength of hollow-coreslabsrdquo PCI Journal vol 51 no 1 pp 110ndash115 2006

[4] ACI Committee 318 Building Code Requirements for StructuralConcrete (ACI 318-05) and Commentary American ConcreteCommittee Farmington Hills MI USA 2005

[5] ACI Committee 318 Ed Building Code Requirements forStructural Concrete (ACI 318-08) and Commentary AmericanConcrete Committee Farmington Hills MI USA 2008

[6] K D Palmer and A E Schultz ldquoFactors affecting web-shearcapacity of deep hollow-core unitsrdquo PCI Journal vol 55 no 2pp 123ndash146 2010

[7] J H ImWeb shear strength of prestressed hollow-core slab [PhDthesis] University of Seoul 2015

[8] K D Palmer and A E Schultz ldquoExperimental investigation ofthe web-shear strength of deep hollow-core unitsrdquo PCI Journalvol 56 no 4 pp 83ndash104 2011

[9] D H Lee M-K Park J-Y Oh K S Kim J-H Im and S-Y Seo ldquoWeb-shear capacity of prestressed hollow-core slabunit with consideration on the minimum shear reinforcementrequirementrdquoComputers andConcrete vol 14 no 3 pp 211ndash2312014

[10] AASHTOLRFDBridge Design Specifications American Associa-tion of State Highway and Transportation Officials WashingtonDC USA 4th edition 2007

[11] European Committee for Standardization (CEN) Eurocode 2Design of Concrete Structure Part 1-1 General Rules and Rulesfor Buildings EN 1992-1-12004 Brussels 2004

[12] FIP Recommendations Precast Prestressed Hollow Core FloorsFIP Commission on Prefabrication London UK 1988

[13] Concrete Committee of JSCE Ed Standard Specifications forConcrete Structures Design Japan Society of Civil EngineersTokyo Japan 2007

[14] J C Walraven and W P M Mercx ldquoThe Bearing Capacity ofPrestressed Hollow-Core Slabsrdquo Heron vol 28 no 3 pp 1ndash41983

[15] J G Macgregor S A Mirza and B Ellingwood ldquoStatisticalanalysis of resistance of reinforced and prestressed concretemembersrdquo ACI Journal vol 80 no 16 pp 167ndash176 1983

[16] ACI Committee 318 ldquoBuilding code requirements for rein-forced concrete (ACI 318-56) and commentaryrdquo AmericanConcrete Committee 2014

[17] TNOBuilding andConstructions Research TNOReport Stan-dard Shear Tests on Prestressed Hollow Core Slabs accordingto EN 1168 TNO Building and Constructions Research Hague2005

[18] A S Nowak and M M Szerszen ldquoCalibration of design codefor buildings (ACI 318) part 1 ndash statisticalmodels for resistancerdquoACI Structural Journal vol 100 no 41 pp 377ndash382 2003

[19] MM Szerszen andA SNowak ldquoCalibration of design code forbuildings (ACI 318) part 2 ndash reliability analysis and resistancefactorsrdquo ACI Structural Journal vol 100 no 42 pp 383ndash3912003

[20] M P Collins and D Mitchell Prestressed Concrete StructuresPrentice Hall 1991

[21] D A Kuchma N M Hawkins S-H Kim S Sun and K SKim ldquoSimplified shear provisions of theAASHTOLRFDbridgedesign specificationsrdquo PCI Journal vol 53 no 3 pp 53ndash732008

[22] D H Lee S-J Han and K S Kim ldquoDual potential capacitymodel for reinforced concrete beams subjected to shearrdquo Struc-tural Concrete vol 16 no 3 pp 443ndash456 2016

[23] G Bertagnoli and G Mancini ldquoFailure analysis of hollow-coreslabs tested in shearrdquo Structural Concrete vol 10 no 3 pp 139ndash152 2009

[24] M Pajari ldquoResistance of prestressed hollow core slabs againstweb shear failurerdquo Research Notes 2292 Technical ResearchCentre of Finland (VTT) Espoo Finland 2005

[25] M Pajari ldquoWeb shear failure in prestressed hollow core slabsrdquoJournal of Structural Engineering vol 42 no 4 pp 207ndash2172009

[26] A C Ugural and S K Fenster Advanced strength and appliedElasticity Prentice Hall 2003

[27] J G Macgregor ldquoLoad and Resistance Factors for ConcreteDesignrdquo ACI Journal vol 80 no 27 pp 279ndash287 1983

[28] M Mayer ldquoDie Sicherheit der Bauwerke und ihre Berechnungnach Grenzkraften anstatt nach zulassigen Spannungenrdquo JSpringer vol 66 1962


[29] A M Freudenthal ldquoThe Safety of Structurerdquo ASCE Transac-tions vol 112 pp 383ndash391 1947

[30] A H Ang and W H Tang Probability Concepts in EngineeringPlanning and Design Wiley 1990

[31] P Chetchotisak J Teerawong S Yindeesuk and J Song ldquoNewstrut-and-tie-models for shear strength prediction and designof RC deep beamsrdquo Computers and Concrete vol 14 no 1 pp19ndash40 2014

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of



Figure 1 Various types of prestressed hollow-core slabs

proposed an equation to calculate the web-shear strengths ofthe PHC slabs which is simple but provides a proper marginof safety

In the design codes based on the ultimate strengthdesign approach the design strength (120601119877119899) is calculated bymultiplying the nominal strength (119877119899) of themember and thestrength reduction factor (120601) that is less than 1 dependingon the importance and failure mode of the member to takeinto account the variability of materials and dimensions theconstruction errors the uncertainties of code equations andso onMacgregor et al [15] determined the strength reductionfactor (120601) by performing structural reliability analyses on theexperimental results available at that time In the currentACI 318 design code (2014) [16] however the shear strengthreduction factor is 075 for both reinforced concrete (RC) andprestressed concrete (PSC) members even though the con-struction errors and the uncertainties of the code equationsfor the RC and PSC members are not the same In particularLee et al [9] reported that the web-shear strength equationfor PSC members overestimated the shear strengths of thePHC slabs with the heights of less than 315mm (thin PHCslabs hereafter) as well as those of greater than 315mm (thickPHC slabs hereafter) Thus it is necessary to reexaminethe current reduction factor for the shear strengths of PHCslabs In this regard this study examined the suitability ofthe shear strength reduction factor of 075 specified in thecurrent ACI 318 design code (2014) based on the structuralreliability analysis on the experimental results of the PHCslabs collected from literature including the authorsrsquo previousstudy

2 Web-Shear Strength

The shear failures of PSC members without shear reinforce-ments are generally divided into the web-shear and flexural-shear failures [20ndash22] The shear failure of PHC slabs thathave a thin web due to the hollow-cores in the section isdominated by web-shear strength in most cases [3 6ndash9 23ndash25] In most design codes [4 5 11] two assumptions are

x

y

Critical sectionCritical point

Centroid axis

Tendon

ℎ2

lb

lc

lt

xy

x

Figure 2 Description of transfer length (119897119905) bearing length (119897119887) andcritical section (119897119888)

introduced to simplify the calculations of web-shear strengthas shown in Figure 2 the critical section is located at a certaindistance (119897119888) away from the support and the vertical normalstress (120590119910) at the centroid of the section is negligible Thenthe shear strength (120591119909119910) and the longitudinal stress (120590119909) atthe centroid axis of the critical section can be represented asfollows

120590119909 = 120572119891119901119890119860119901119904119860119888 (1)

120591119909119910 = 119878119887119908119868119892119881119910 (2)

where 120572 is the ratio of the prestress at the critical sectionto the fully effective prestress of the strands which can beestimated as the distance (119897119888) from the member end to thecritical section divided by the transfer length (119897119905) as shownin Figure 2 Also 119891119901119890 is the effective prestress 119860119901119904 and 119860119888are the cross-sectional area of the strands and the memberrespectively 119878 is the moment of area 119868119892 is the moment ofinertia of gross section 119887119908 is the sum of the web width and119881119910 is the shear force acting on the section According to thetheory of elasticity [26] the principal tensile stress (1205901) at thecritical section can be represented as follows

1205901 = 1205901199092 + radic12059011990924 + 1205911199091199102 (3)

The point at which the principal tensile stress (1205901) onthe web reaches the tensile strength of the concrete (119891119888119905) isconsidered as the shear cracking strength of the PSCmemberThe shear cracking strength and the shear strength of themember can be different but they are assumed to be the samebecause their difference is very small especiallywhenno shearreinforcements are placed Therefore by substituting (1) and(2) into 120590119909 and 120591119909119910 in (3) respectively the web-shear strength(119881119888119908) can be represented as follows

119881119888119908 = 119868119892119887119908119878 radic1198911198881199052 + 120572119891119901119890119891119888119905119860119901119904119860119888 (4)



119881119877119889119888 = 119868 sdot 119887119908119878 radic1198911198881199051198892 + 120572119897120590119888119901119891119888119905119889 (5)



119881119888119908 = 075119868119887119908119878 radic1198911198881199052 + 120572119897119890120590119888119901119891119888119905 (6)





119881119908119888119889 = 119891119908119888119889119887119908119889120574119887 (9)



120591119909119910119886 = 119881119910119887119908119889119901 (10)


119881119888119908 = (029radic1198911015840119888 + 03119891119901119888) 119887119908119889119901 + 119881119901 (11)

119881119888119908 = (016radic1198911015840119888 + 03119891119901119888) 119887119908119889V + 119881119901 (12)


119881119888119908 = (029120582radic1198911015840119888 + 03119891119901119888) 119887119908119868119878 + 119881119901 (13)


119881119888119908 = (022radic1198911015840119888 + 023119891119901119888) 119887119908119889119901 (14)




Table1Dim

ensio


fshear

testdatacollected

from

literature[79]

Authors

Specim

ens

name

Void

type

Geometry

Prestre

ssreinforcem

ent

Loading

119891 119888119896 (MPa)

119887 119908 (mm)

119889 119901 (mm)

119860 119892 (mm2)

119868sdot119887119908S

(mm3)

119860 119901119904 (mm2)

119891 119901119906 (MPa)

119891 119901119888 (MPa)

119886119889 (mdash)

L (m)

119881 test (kN)

WalravenandMercx

(1983)

mdashC

NC

640

2500sim2

940

2250sim2

600

171000sim199

000

59601sim6

0490

4700sim

9400

18600

27sim6

317sim67

60

1816sim28

63

Becker

andBu

ettner

(1985)

mdashNC

414

3378sim43

181700sim2

159

1406

45sim165

806

43658sim4

4528

4490

sim10322

17225

29sim5

335sim6

746sim10

41153sim2

719

Pajari

(2005)

mdashC

410sim72

62150sim3

350

2000sim5

000

11900

0sim300

000

36333sim12

5833

3720sim

19530

mdash22sim8

324sim4

640sim11

0800sim52

80

TNOBu

ildingand

Con

structio

nsRe

search

(2005)

[17]

mdashIC

599sim113

92410sim44

902550sim4

000

172000sim26

1000

49073sim11

4927

8537sim

16844

mdash42sim10

425

40sim6

02240sim6

520

Bertagno

liand

Mancini(200

9)mdash

I557sim65

72150sim4

440

1510sim49

70116

000sim2

9300

041425sim13

0089

2491sim

1440

318600

24sim8

023sim3

041sim7

01570sim714

0

Celal

(2011)

mdashC

629sim67

92190

sim2830

1580sim2

550

132887sim180

414

34246sim4

2359

5025sim

8883

18600

54sim6

930sim3

8mdash

1630sim2

970

Rahm

anetal

(2010)

mdashC

400

3157sim3

254

2120sim2

620

151812sim179

115

47304sim7

0400

5922sim

7896

19600

40sim4

723sim2

8mdash

1985sim2

989

Im (2015)

ST200

C600

2470

1750

121590

378395

5048

18600

50

300

78738

ST200R

C600

2470

1750

121590

378395

5048

18600

50

300

781718

ST265

NC

600

2570

2300

151120

528534

5048

18600

40

300

781113

ST265R

NC

600

2570

2300

151120

528534

5048

18600

40

300

782084

ST4N

-1NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2685

ST4N

-2NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2506

ST4N

-3NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2833

ST5N

-1NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4175

ST5N

-2NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4448

ST5N

-3NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

3596

Total155

specim

ens

CfvoidtypeN

C(non

circular)C(circ


strengthof

prestre

ssingste

el119891119901119888com

pressiv

estre

ssin

concreteatthec

entro

idof

thes

ectio

ndu

etoeffectiv

eprestr

ess


Undesirable side

Ove

resti

mat

ed si

deU

nder

estim

ated

side

200 400 600 8000Vcw＝Ｆ(ACI318-05) (kN)

0

200

400

600

800

V(k

Ncw

Tes

t)

00

05

10

15

20

Vcw

＝ＦV

cwT

est

200 300 400 500 600100Member height (mm)


(a) ACI318-05

Ove

resti

mat

ed si

deU

nder

estim

ated



0

200

400

600

800

Vcw

４ＭＮ

(kN

)

200 400 600 8000Vcw＝Ｆ(ACI318-08) (kN)

00

05

10

15

20Vcw

＝ＦV

cw４ＭＮ

200 300 400 500 600100Member height (mm)

(b) ACI318-08








119880 = 12119863 + 16119871119880 = 12119863 + 16119878119880 = 14119863

(15)


119880 = 12119863 + 13119882 + 05119871119880 = 12119863 + 15119864 + (05119871 or 02119878) 119880 = 09119863 minus (13119882 or 15119864)

(16)






120601119877119899 ge 11987612119863+16119871 (17)


SF = (12119863 + 16119871) 120601119863 + 119871 (18)














120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)




(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1


+ sdot sdot sdot

(21)


119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)


1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




119881119877119889119888 = 119868 sdot 119887119908119878 radic1198911198881199051198892 + 120572119897120590119888119901119891119888119905119889 (5)



119881119888119908 = 075119868119887119908119878 radic1198911198881199052 + 120572119897119890120590119888119901119891119888119905 (6)





119881119908119888119889 = 119891119908119888119889119887119908119889120574119887 (9)



120591119909119910119886 = 119881119910119887119908119889119901 (10)


119881119888119908 = (029radic1198911015840119888 + 03119891119901119888) 119887119908119889119901 + 119881119901 (11)

119881119888119908 = (016radic1198911015840119888 + 03119891119901119888) 119887119908119889V + 119881119901 (12)


119881119888119908 = (029120582radic1198911015840119888 + 03119891119901119888) 119887119908119868119878 + 119881119901 (13)


119881119888119908 = (022radic1198911015840119888 + 023119891119901119888) 119887119908119889119901 (14)




Table1Dim

ensio


fshear

testdatacollected

from

literature[79]

Authors

Specim

ens

name

Void

type

Geometry

Prestre

ssreinforcem

ent

Loading

119891 119888119896 (MPa)

119887 119908 (mm)

119889 119901 (mm)

119860 119892 (mm2)

119868sdot119887119908S

(mm3)

119860 119901119904 (mm2)

119891 119901119906 (MPa)

119891 119901119888 (MPa)

119886119889 (mdash)

L (m)

119881 test (kN)

WalravenandMercx

(1983)

mdashC

NC

640

2500sim2

940

2250sim2

600

171000sim199

000

59601sim6

0490

4700sim

9400

18600

27sim6

317sim67

60

1816sim28

63

Becker

andBu

ettner

(1985)

mdashNC

414

3378sim43

181700sim2

159

1406

45sim165

806

43658sim4

4528

4490

sim10322

17225

29sim5

335sim6

746sim10

41153sim2

719

Pajari

(2005)

mdashC

410sim72

62150sim3

350

2000sim5

000

11900

0sim300

000

36333sim12

5833

3720sim

19530

mdash22sim8

324sim4

640sim11

0800sim52

80

TNOBu

ildingand

Con

structio

nsRe

search

(2005)

[17]

mdashIC

599sim113

92410sim44

902550sim4

000

172000sim26

1000

49073sim11

4927

8537sim

16844

mdash42sim10

425

40sim6

02240sim6

520

Bertagno

liand

Mancini(200

9)mdash

I557sim65

72150sim4

440

1510sim49

70116

000sim2

9300

041425sim13

0089

2491sim

1440

318600

24sim8

023sim3

041sim7

01570sim714

0

Celal

(2011)

mdashC

629sim67

92190

sim2830

1580sim2

550

132887sim180

414

34246sim4

2359

5025sim

8883

18600

54sim6

930sim3

8mdash

1630sim2

970

Rahm

anetal

(2010)

mdashC

400

3157sim3

254

2120sim2

620

151812sim179

115

47304sim7

0400

5922sim

7896

19600

40sim4

723sim2

8mdash

1985sim2

989

Im (2015)

ST200

C600

2470

1750

121590

378395

5048

18600

50

300

78738

ST200R

C600

2470

1750

121590

378395

5048

18600

50

300

781718

ST265

NC

600

2570

2300

151120

528534

5048

18600

40

300

781113

ST265R

NC

600

2570

2300

151120

528534

5048

18600

40

300

782084

ST4N

-1NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2685

ST4N

-2NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2506

ST4N

-3NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2833

ST5N

-1NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4175

ST5N

-2NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4448

ST5N

-3NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

3596

Total155

specim

ens

CfvoidtypeN

C(non

circular)C(circ


strengthof

prestre

ssingste

el119891119901119888com

pressiv

estre

ssin

concreteatthec

entro

idof

thes

ectio

ndu

etoeffectiv

eprestr

ess


Undesirable side

Ove

resti

mat

ed si

deU

nder

estim

ated

side

200 400 600 8000Vcw＝Ｆ(ACI318-05) (kN)

0

200

400

600

800

V(k

Ncw

Tes

t)

00

05

10

15

20

Vcw

＝ＦV

cwT

est

200 300 400 500 600100Member height (mm)


(a) ACI318-05

Ove

resti

mat

ed si

deU

nder

estim

ated



0

200

400

600

800

Vcw

４ＭＮ

(kN

)

200 400 600 8000Vcw＝Ｆ(ACI318-08) (kN)

00

05

10

15

20Vcw

＝ＦV

cw４ＭＮ

200 300 400 500 600100Member height (mm)

(b) ACI318-08








119880 = 12119863 + 16119871119880 = 12119863 + 16119878119880 = 14119863

(15)


119880 = 12119863 + 13119882 + 05119871119880 = 12119863 + 15119864 + (05119871 or 02119878) 119880 = 09119863 minus (13119882 or 15119864)

(16)






120601119877119899 ge 11987612119863+16119871 (17)


SF = (12119863 + 16119871) 120601119863 + 119871 (18)














120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)




(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1


+ sdot sdot sdot

(21)


119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)


1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



Table1Dim

ensio


fshear

testdatacollected

from

literature[79]

Authors

Specim

ens

name

Void

type

Geometry

Prestre

ssreinforcem

ent

Loading

119891 119888119896 (MPa)

119887 119908 (mm)

119889 119901 (mm)

119860 119892 (mm2)

119868sdot119887119908S

(mm3)

119860 119901119904 (mm2)

119891 119901119906 (MPa)

119891 119901119888 (MPa)

119886119889 (mdash)

L (m)

119881 test (kN)

WalravenandMercx

(1983)

mdashC

NC

640

2500sim2

940

2250sim2

600

171000sim199

000

59601sim6

0490

4700sim

9400

18600

27sim6

317sim67

60

1816sim28

63

Becker

andBu

ettner

(1985)

mdashNC

414

3378sim43

181700sim2

159

1406

45sim165

806

43658sim4

4528

4490

sim10322

17225

29sim5

335sim6

746sim10

41153sim2

719

Pajari

(2005)

mdashC

410sim72

62150sim3

350

2000sim5

000

11900

0sim300

000

36333sim12

5833

3720sim

19530

mdash22sim8

324sim4

640sim11

0800sim52

80

TNOBu

ildingand

Con

structio

nsRe

search

(2005)

[17]

mdashIC

599sim113

92410sim44

902550sim4

000

172000sim26

1000

49073sim11

4927

8537sim

16844

mdash42sim10

425

40sim6

02240sim6

520

Bertagno

liand

Mancini(200

9)mdash

I557sim65

72150sim4

440

1510sim49

70116

000sim2

9300

041425sim13

0089

2491sim

1440

318600

24sim8

023sim3

041sim7

01570sim714

0

Celal

(2011)

mdashC

629sim67

92190

sim2830

1580sim2

550

132887sim180

414

34246sim4

2359

5025sim

8883

18600

54sim6

930sim3

8mdash

1630sim2

970

Rahm

anetal

(2010)

mdashC

400

3157sim3

254

2120sim2

620

151812sim179

115

47304sim7

0400

5922sim

7896

19600

40sim4

723sim2

8mdash

1985sim2

989

Im (2015)

ST200

C600

2470

1750

121590

378395

5048

18600

50

300

78738

ST200R

C600

2470

1750

121590

378395

5048

18600

50

300

781718

ST265

NC

600

2570

2300

151120

528534

5048

18600

40

300

781113

ST265R

NC

600

2570

2300

151120

528534

5048

18600

40

300

782084

ST4N

-1NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2685

ST4N

-2NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2506

ST4N

-3NC

600

2600

3600

212114

812018

8996

18600

51

250

64

2833

ST5N

-1NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4175

ST5N

-2NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

4448

ST5N

-3NC

600

2860

4550

255405

1118678

10970

18600

52

250

64

3596

Total155

specim

ens

CfvoidtypeN

C(non

circular)C(circ


strengthof

prestre

ssingste

el119891119901119888com

pressiv

estre

ssin

concreteatthec

entro

idof

thes

ectio

ndu

etoeffectiv

eprestr

ess


Undesirable side

Ove

resti

mat

ed si

deU

nder

estim

ated

side

200 400 600 8000Vcw＝Ｆ(ACI318-05) (kN)

0

200

400

600

800

V(k

Ncw

Tes

t)

00

05

10

15

20

Vcw

＝ＦV

cwT

est

200 300 400 500 600100Member height (mm)


(a) ACI318-05

Ove

resti

mat

ed si

deU

nder

estim

ated



0

200

400

600

800

Vcw

４ＭＮ

(kN

)

200 400 600 8000Vcw＝Ｆ(ACI318-08) (kN)

00

05

10

15

20Vcw

＝ＦV

cw４ＭＮ

200 300 400 500 600100Member height (mm)

(b) ACI318-08








119880 = 12119863 + 16119871119880 = 12119863 + 16119878119880 = 14119863

(15)


119880 = 12119863 + 13119882 + 05119871119880 = 12119863 + 15119864 + (05119871 or 02119878) 119880 = 09119863 minus (13119882 or 15119864)

(16)






120601119877119899 ge 11987612119863+16119871 (17)


SF = (12119863 + 16119871) 120601119863 + 119871 (18)














120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)




(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1


+ sdot sdot sdot

(21)


119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)


1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



Undesirable side

Ove

resti

mat

ed si

deU

nder

estim

ated

side

200 400 600 8000Vcw＝Ｆ(ACI318-05) (kN)

0

200

400

600

800

V(k

Ncw

Tes

t)

00

05

10

15

20

Vcw

＝ＦV

cwT

est

200 300 400 500 600100Member height (mm)


(a) ACI318-05

Ove

resti

mat

ed si

deU

nder

estim

ated



0

200

400

600

800

Vcw

４ＭＮ

(kN

)

200 400 600 8000Vcw＝Ｆ(ACI318-08) (kN)

00

05

10

15

20Vcw

＝ＦV

cw４ＭＮ

200 300 400 500 600100Member height (mm)

(b) ACI318-08








119880 = 12119863 + 16119871119880 = 12119863 + 16119878119880 = 14119863

(15)


119880 = 12119863 + 13119882 + 05119871119880 = 12119863 + 15119864 + (05119871 or 02119878) 119880 = 09119863 minus (13119882 or 15119864)

(16)






120601119877119899 ge 11987612119863+16119871 (17)


SF = (12119863 + 16119871) 120601119863 + 119871 (18)














120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)




(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1


+ sdot sdot sdot

(21)


119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)


1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




119880 = 12119863 + 16119871119880 = 12119863 + 16119878119880 = 14119863

(15)


119880 = 12119863 + 13119882 + 05119871119880 = 12119863 + 15119864 + (05119871 or 02119878) 119880 = 09119863 minus (13119882 or 15119864)

(16)






120601119877119899 ge 11987612119863+16119871 (17)


SF = (12119863 + 16119871) 120601119863 + 119871 (18)














120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)




(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1


+ sdot sdot sdot

(21)


119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)


1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of












120573 = 1205831198831 minus 1205831198832radic12059021198831 + 12059021198832

(20)




(119883119894 minus 119909lowast119894 ) ( 120597119892120597119883119894)

+ 119899sum119895=1

119899sum119894=1


+ sdot sdot sdot

(21)


119892 (1198831 1198832 119883119899) = 119899sum119894=1

(1198831015840119894 minus 1199091015840lowast119894 )( 1205971198921205971198831015840119894 ) (22)


1198831015840119894 = 119883119894 minus 120583119883119894120590119883119894 (23a)




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of




Limit state

Safe state

Failure state


2

X1

g = X1 minus X2

G gt 0

G = 0

G lt 0


Relia

bilit

y in

dex

beta

()

= 35

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14



120583119892 asymp minus 119899sum119894=1

1199091015840lowast119894 ( 1205971198921205971198831015840119894 ) (24a)

1205902119892 asymp 119899sum119894=1

( 1205971198921205971198831015840119894 )2 (24b)



(25)


02

04

06

08

10

12

xya

xy






liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of



liabi

lity

inde

x (

)

0

2

4

6

8

10

12

14

100 200 300 400 500 6000Depth (mm)

= 35

D(D + L) = 05

= 075


Relia

bilit

y in

dex

()

100 200 300 400 500 6000Depth (mm)

0

2

4

6

8

10

12

14

= 35

D(D + L) = 05

= 075



Relia

bilit

y in

dex

()

D = L

= 070 = 075 = 080 = 085 = 090

012345678

02 04 06 08 10D(D + L)





120583Lee = 120582Lee119877 (27a)

120583119863 = 120582119863119863 (27b)

120583119871 = 120582119871119871 (27c)





4 Conclusion










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of










Acknowledgments


References








































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of













CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of









CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


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