Evaluation of Punching Shear Strength of Reinforced Concrete SlabsBased on DatabaseSumio Hamada', QiuningYang2 and Mingjie Mao3

AbstractAlthough various studies related to the punching shear strength of slabs have been published, databases related topunching shear strength are small in size. A database of 313 specimens has been structured through the present study,compared to the 114-data compiled by Kakuta et al. (1974) and the 138 data compiled by Gardner et al. (1996).In this study, six equations for the punching shear strength prescribed in specifications were evaluated based on the da-tabase. This paper includes a discussion of the parameters of punching shear strength in JSCE and All specifications. Asimplified strength equation is also proposed from the database.

Test data related to punching shear strength of slabs havebeen randomly presented allover the world. More than ahundred data were collected by Kakuta et al. (1974) andGardner et al. (1996), including their own test data.Unfortunately, Gardner missed Japanese and Swedishpapers written in Japanese and Swedish, respectively.Kakuta et al. (1974) and Gardner et al. (1996) also pro-posed their own equations for the evaluation of punchingshear strength. These equations present a small variationbetween tested and evaluated values.

Research related to the punching shear strength ofslabs was initiated from an experimental report by Talbotet al. (1913). Kinnunen et al. (1960) presented the firstpaper on the punching shear strength of reinforced con-crete (RC) slabs, and went on to publish more than 50papers. Although Kinnunen et al. (1960) considered thedowel strength and the membrane resistance, their focuslater moved to the yielding line theory against shearresistance, which was applied to the basis of Eurocode 2equations at that time. These papers were, however,mostly published in the form of technical reports of theSweden Royal College of Engineering, and said papershave not been referred by other researchers. This hasbeen reported by Holmgren et al. (2000) on the punchingshear strength of slabs. Many researchers have proposedtheir equations based on concrete shear strength. Theparameters are the area of the shear failure surface abovetension reinforcement, size of loading area, and concrete

'Professor, Graduate School of Civil and EnvironmentalEngineering, Yamaguchi University, Japan.E-mail:shamada@yamaguchi-u.ac.jp2Doctoral student, Graduate School of Civil andEnvironmental Engineering, Yamaguchi University,Japan.3Lecturer, Environmental and Safety EngineeringDepartment, Polytechnic University of Jiangsu, China.

strengths and amount of tension reinforcement. Most ofthe strength equations vary widely due to limited data,because researchers tend to use only their own test data.The strength equations provided in specifications aremostly based on specific equations, and such equationsare not always agreeable to the test data due to the lack ofcollected data. This requires more data acquisitionedfrom various researchers. Although available papers arestill limited, more than 300 test data including Japanesetest data conducted by Kakuta et al. (1974), Matsui et al.(1984), and Hamada et al. (200 I), have been collected.

The aims of the present study are to create database forthe punching shear strength of RC slabs, to evaluate theequations proposed in previous research and in specifi-cations, and to propose a rational strength equation. Themember strength factors based on the evaluated equationand failure probability are also determined in this study.

2. Database and evaluation of designstrength

2.1 Strength equationThe equations for the punching shear strength of RCslabs are given in Table I. Table 2 lists the main pa-rameters of six equations. The IS\ 1/3rd

, 1/2nd, and 2/3rd

powers were applied to the concrete strength. This im-plies that these powers are a relationship between theconcrete compressive strength and the shearing strengthdeveloped from the punching shear ofRC slabs.

The ratio of tension reinforcement is used in equationsin a form from cubic root to linearity. All and ACI do notprovide any parameter of ratio of tension reinforcement.

The effective depth of the slab is a significant factor inthe evaluation of punching shear strength and this isapplied in design and the most proposed conventionalequations. The failure area in the specifications is relatedto the square of the effective depth.

Tn the Building Code of ACI (1983), a simple equa-tion is applied for the design. The equation is developedbased on the classical shear strength equation. The con-

crete Standard Specification of JSCE (2002) was revisedin 1986 and 2002, and the equation was modified fromthe 'equation of Kakuta et al. (1974). The allowablestress design method is a simple alternative method.

2.2 Parameters in specification equationsStrength equations for the RC slab have been providedin various specifications. The design equations at thetime of allowable stress were expressed in a very simpleform based on the punching shearing stress, dividing thestress of shearing force by the shearing area. The shear-ing area was evaluated by the mean perimeter multipliedby the effective depth, unlike the maximum shearingstress in a beam, which is determined by the momentarm length. The allowable shearing stress was obtainedfrom the punching shear tests. A recent specificationformula in the ACI specification for the Building Codeprovides a similar equation of the previous allowablestress design method.

Table 1 gives the equations prescribed in designspecifications published by associations, and their pa-

rameters are given in Table 2. The parameters, however,do not always have the same meaning. For example, thepower of a parameter may differ among equations. Thepower implies the sensitivity of the parameter, which isobtained from the derivative with respect to the pa-rameter. Taking for example the concrete strength ofif) 1/3 and if)2/3 and two strengths of 20 MPa and 40 MPa,the influence of the strength to punching shear strengthratio between these strengths is obtained as(40)13/(20)J3= 1.26 and (40)23/(20)23= 1.59. This im-plies that 2/3'd power is higher sensitivity than 1I3'dpower. Problems related to concrete strength occur indifferent compressive testing methods such as differ-ences in test specimen size and curing method. At least

- European compression test specimens are different fromAmerican and Japanese test specimens. The curingmethods for testing slabs differ among researchers. Veryoften curing of the compression test specimen differsfrom the curing of testing slabs. Testing slab specimenshave been often cured in the laboratory or in the field,whereas compression test specimens are cured in water

Organization Calculation equation Parameters

AJJ(1991) V = 1.5bJ . (7/ 8)d . Is b : Column perimeter of the loading plate

Ipcd = 0.2.flb0 : Column perimeter of the loading plate +4d

lSCE V = ~d~ p~JpcdU pd 1 rb but bJ : Column perimeter of the loading plate +nd(2002)

~d ='J.hld,~p =VlOOP'~r =1+1/(1+0.25bld) c : Diameter of circular plated : Effective slab depth

V=0.083(2+4/~c).[l·bo ·d I; : Compressive strength of concrete

ACI V = 0.083(usd 1 bo +2).[l.bo·dj, :Working stress of concrete

( 1983) p: Ratio of tension reinforcement

V = 0.33.fl.bo·d up: Perimeter located at distance of d/2 from

The least value out of3 equations the'loading plate faceEurocode2

V=dn(b+3dXl035k(1.2+40p)/;23V : Punching shear strength

(1991) a. s: 40 for interior columns, 30 for edge columns

V = 0.79ud(400 1 dt4 (lOOp. V; 10.78)/25)1/3and 20 for corner columns

BS ~ c : Ratio of the longer side length to shorter one( 1985) U = 4(c + 3d): Circular plate u = 4(b + 3d): Square plate of the loading plate

( ')1/3 ~ = 1+ (200 1 d)l /2rb = 1.3

CEB-FIPV=0.18wd·~100p·fc 11.5 k=1.6-d but d (m)

( 1990) w=n(c+4d) : Circular plate w=4(b+nd): Square Iplate

Equation Concrete strengthSlab dimension shape Ratio of tension Steel Dimensions of loading plate

Effective depth Cover reinforcement strength Side Perimeter ShapeAIl 1/2nd power 0 x x x 0 0 x

lSCE l/2nd power 0 x 1/3rd power x x 0 xACI 1st power 0 x x x 0 0 x

Eurocode2 213rd power 0 x 1st power x 0 x xBS Il3rd power 0 x l/3rd power x 0 0 0

CEB-FIP Il3rd power 0 x Il3rd power x 0 0 0

or moisture conditions. The perimeter of the loadingarea is included in the BS code and JSCE specification.

searchers. More than 300 data were obtained for thepresent study, the largest number in this field, as shownin Table 3. These data include various specimen sizes,concrete strengths, loading areas and areas of rein-forcement. Since some reports do not provide the coverthickness and maximum aggregate size, the table doesnot include these data. The maximum aggregate sizeinfluences the shear strength of the beam, and the

2.3 Database of punching shear strength ofslabsThe collection of data from around the world presentsdifficulties. Very often the collected data applied in re-search are limited to the reports available to the re-

-Side of loading

Dimensions Concrete(eo)

Slab d Perimeter PtestSlab No Researchers strength thickness (em) of loading plate

of slab (em) (N/mm2) (em) plate (em) a (em) b (em) (kN)

I 15.6 0.6 - 27.1 1164.22 Graf 150xl50 -15.7 -l.l --

-47.3 120.0 30.0 30.0 -1662.13 Elstner 183xl83 12.7 1.2

15.011.4 101.6 25.4 25.4 301.8

-27 ( 1956) -50.5 -7.0 -11.8 -142.4 -35.6 -35.6 -578.628 Seordelis \83x 183

19.42.5 10.8 132.0 33.0 33.0 466.5

29 ( 1958) -27.9-- -484.1

30 Moe(1961) 183xl83 20.5 0.1 15.2 11.4 60.8 15.2 15.2 311.6-42 -26.1 -\.5 -122 -30.5 -30.5 -432.2

43 Yitzhaki ipl16 12.3 0.5 7.8 42.2 10.5 10.5 98.0-57 ( 1966) ~!p171 -28.7 -2.0 -- -10.9 -118 -29.5 -29.5 -306.7

#58 Gardner ip38.1 13.2 0.7 10.0 7.3 36.0 9.0 9.0 107.4-76 ( 1990) ~!p112 -52.1 -5.0 -15.2 -12.3 -72.0 -18 -18 -356.6

*77 Mowrer 122xl22 15.5 1.17.6 5.1 40.6 10.2 10.2 79.2

-95 (1967) -53.8 -2.2 -162.6 -40.6 -40.6 -184.696 Bazant ip40.64 47.9 2.6 10.2 7.8 360.0 102.0 102.0 228.9

-98 ( 1987) -52.9 -4.0 -309.699 Menetrey 16.1 0.2 15.0 12.1 42.5 10.6 10.6 107.0

-III (1996)--- -40.6 -2.0 -45.0 -39.7 -159.5 -39.9 -39.9 -1370.0

112 Kinnunen152xl52 23.9 0.4 15.0 10.0 17.7 4.4 4.4 183.0

-136 -183x183 -31.9 -2.3 -24.0 -20.1 -106.4 -26.6 -26.6 -558.0137 Pralong 180xl80 31.6 0.7 18.0 16.2 106.3 26.6 26.6 366.0

138 Regan150xl50 31.4 0.8 11.8 19.2 4.8 4.8 170.0

-147 -180x180 -43.6 -1.0 -- -20 -88.8 -22.2 -22.2 -825.0148 Van

29.0 0.15 15.0 11.4 88.8 22.2 22.2 375.0-150

----36.5 -255.2 -63.8 -63.8 -592.0

151 Corley 213x213 18.7 0.15 14.6 11.1 81.2 20.3 20.3 243.8-155 ( 1968) -25.5 . -101.6 -25.4 -25.4 -334

#156 Marzouk 150xl50 30.0 0.5 9.0 7.0 60.0 15.0 15.0 178.0-172 (199\ ) -80.0 -2.4 -15.0 -12.0 -120.0 -30.0 -30.0 -560.0

#173 Akiyama 21.7 0.3 4.0 39.2 9.8 9.8 62.7-196 ( 1982) --- -24.6 -3 -- -9.5 -348.0 -87.0 -87.0 -981.5

197 Umehara 1l0xii022.8 0.2 12.0 8.5

120.0 30.0 30.0202.9

-201 ( 1992) -40.1 -0.8 -15.0 -11.5 -402.8*202 Higashi- 148xl00 33.5 1.3 6.0 4.4 28.0 10.0 4.0 96.7

-204 Yama (1998) -43.4 -6.5 -4.7 -46.0 -13.0 -10.0 -104.3205 Suehiro 51.2 0.6 15.0 12.0 354.0 89.0 89.0 267.5

-207 (1989) --- -52.3 -1.8 -20.0 -16.2 -563.5208 Nagai(1993) 200x200 19.6 1.0 15.0 12.0 101.2 25.3 25.3 362.6

209 -392210 Hamada 100xl00 20.4 0.4 10.0 7.5 40.0 10.0 10.0 144.1

-246 I (200 I ,2005) -150x150 -60 -3.4 -15.0 -12.0 -120.0 -30.0 -30.0 -699.7247 Kakuta 65x65 15.1 0.5 10.0 7.5 20.0 5.0 5.0 115.4

-293 (1974) -220x220 -51.9 -3.4 -15.0 -12.0 -120.0 -30.0 -30.0 -735.0#294 Matsui 23.8 0.8 7.1 5.8 39.0 15.0 4.5 100.0

-311 (1984)---

-46.8 -1.3 -22 -18.1 -180.0 -60.0 -30.0 -989.8312 Yamada 200x200

21.6 0.6 20.0 16.3 120.0 30.0 30.0441.0

313 ( 1992) -26 -1.5 -658.0* :All are deleted, #Partly deleted,

Graf (Kakuta, 1974; Ishibasi, 1984.) , Kinnunen, Pralong, Regan and Van (Bhatt, 2000. Menetrey, 1996.)

punching shear strength of thin slabs may be influencedby the maximum aggregate size.

The punching shear strength evaluation has been ob-tained from test results and not theoretical results, be-cause pure theoretical estimation is based on the yieldline theory, which is related to the bending strength ofslabs, and because the finite element method cannot beapplied in the design method.

Estimated values obtained from the conventionalequations provide larger variation for tested data ofpunching shear strength, compared with bendingstrength of beams. There are two variables in the esti-mated equation, namely the ratio of experimental dataand estimated data, r\ = Viesl N eq, and the ratio of dif-ference between estimated data from experimental dataand estimated data, r2 =( V1esl - Veq)/ Veq. There is a re-lationship between them, which is r, = 1- r2. The nota-tion rj is herein called error ratio. Two probability func-tions, the Gauss function and the Weibull function, havebeen often applied in the evaluation of the error estima-tion of the strength of materials (Ang and Tang 1977).These equations are expressed as follows.

1 x (t_m)2E = r;::;- f exp - dts-v2rr. -<Xl 2s

{ (X - Jl JX}F =1-exp - p

The Weibull function has 3 parameters, which are noteasy to determine from experimental data. In order tofacilitate the determination of these parameters, the fol-lowing assumptions were applied.(1) A value of 0.5 of the Weibull function coincides

with the mean value m of the Gauss function.(2) Selected points of the Weibull function, which are

standard deviation, s, and double of s, coincide withthe equivalent values of the Gauss function.

These assumptions provide the following equations.

F:,,(O) =0.5

F:v (O-s) = 0.15866

F:v (0 - 2s) = 0.02275

Solving these equations, the values of fl. fJ and l/a areobtained as functions of the standard deviation, s, andmean value, m.

X=4.52

parameters, such as the maximum likelihood methodproposed by Blum (Hamada et al. 1971), exist, theyrequire very complicated computations.

2.4 Discussions for previous equationsTable 4 provides the mean value of rl, Viesl Neq-, thestandard deviation, and the coefficient of variance. Themean value of r, provided by BS is near unity, that is,the estimated values are close to experimental values.The equations of ACI, Eurocode2 and CEB-FIP provideconservative values. The coefficients of variation in BSand CEB-FIP are small, less than 20%. The mean valuefor AU exceeds 2.0, and the coefficient of variation ofthis equation is 32.1 %, which implies that the precision

- of the equations is not satisfactory. The value estimatedby JSCE is close to 1.2, and the coefficient of variationof this equation is 23.1 %.

2.5 Effect of concrete strengthThe punching shear strength of RC slabs is not based oncompressive strength but tensile strength, and the rela-tionship between the tensile or shearing strength and thecompressive strength is provided by various researchers.This strength, however, differs between the parts wherethe principal stress is compressive and the tensile re-gions. Although different strengths exist region by re-gion, this strength is proposed in the form of the 1/3rd

,

1/2nd and 2/3rd powers of compressive strength. Sensi-tivity, h of the punching shear strength related to a pa-rameter, x, is given in the following equation.

One second and 1/3rd powers of concrete strengthcause different strength ratios, and these sensitivitiescan be evaluated from the following equations.

VI = K1f;'/3 R,

Differentiating these equations, the following equa-tions can be obtained.

av, = K ~ /-213 R8/; '3 c I

Equation MeanStandard Coefficient ofdeviation variation (%)

AIl 2.05 0.66 32.1lSCE 1.20 0.28 23.1ACI 1.41 0.36 25.4

Eurocode2 2.11 0.53 25.2BS 1.05 0.20 19.7

CEB-FIP 1.46 0.29 19.7

where it is assumed that fc' = 30 N/mm2, VI = V2 = I,

KJRJ = 0.322, and K]R] = 0.1036. These are developedas

avl/O/; = 0.0111,

aV2 10/; = 0.0222

Presumably the sensitivity of Eq. (12) is half that ofEq. (13). This also implies that 2/3rd power of the con-crete strength is more sensitive than 1I3rd power of theconcrete strength. This sensitivity is fairly effective forfinding the power of the parameter. Some formulas limitthe application of the maximum concrete strength.However, most formulas do not necessarily limit therange of concrete strength. Based on the relationshipbetween the strength ratio and concrete strength, con-crete strength may largely fit 1/3 rd power in 1/3 rd, 1/2nd

,

2/3rd and I sl powers.The evaluated values for the All equation and test

values are shown in Fig. 1. The figure implies that thevalues calculated by the All equation are estimatedlower values than tested values, and that these valuesare scattered. The relationships for evaluation by the AlJequation between the error coefficient and the concretecompressive strength, effective depth, and ratio of ten-sion reinforcement are shown in Figs. 2, 3 and 4. In Fig.2 of the range of error coefficient, the values are be-tween 0.5 and 3.0, and this figure does not exhibit aclear tendency between the punching shear strength andconcrete strength due to large variation. Figure 3 showsthe relationship between the error coefficient and effec-tive depth. This relationship also shows a large variationand the tendency for larger depth to have lower punch-ing shear strength. In Fig. 4, the error of punching shearstrength increases proportionally with increases in theratio of tension reinforcement. Evaluated values of theJSCE equation and test values are shown in Fig. 5. Re-lationships for the JSCE equation between the error co-efficient and the concrete compressive strength, effec-tive depth, ratio of tension reinforcement are shown inFigs. 6, 7 and 8. From Figs. 6, 7 and 8, variations of theerror coefficients provided by JSCE are not large.

The substantial slab dimensions, when a slab is sub-jected to vertical loads, are the ratio of span length tothickness or the difference in span length and loadlength. These ratios affect the failure mode of punchingshear. The depth itself is an important factor affectingthe punching shear strength. The size of aggregate maybe affected by the slab thickness and amount of tensionreinforcement, which is called size effect in shearstrength. Many equations include the size effect, whichis related to the depth. This may be due to the dowelaction of coarse aggregates. Slabs with smaller aggre-gates may resist through only the bond between aggre-

1200.0 0

1000.0 00

00

0 0800.0Z 00 0

c ~go 0~ 600.0'" ~'¥co (J)'">

400.000

200.0

0.00.0 200.0 400.0 6000 8000 1000.0 1200.0

Veq (kN)

Fig. 1 ~elationships between Veq obtained from AIJequation and Vtest.

3.0

2.5

2.0

c~ 1.5

~8 1.0

g 0.5w

0.0

-0.5

-1.00.0

o~og, 0

10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0

Concrete strength (N/mm2)

Fig. 2 Relationships between concrete strength and errorcoefficient for AIJ equation.

3.0

2.5 ~% 82.0 ~ 0

~ 0 01.5 0-

~00

1.0§o

8 <§ 0 0

g 0.5 ~w 0 00 00.0 0

o 0-0.5

-1.0

0 50 100 150 200 250 300 350 400 450 500Effective depth (mm)

Fig. 3 Relationships between effective depth and errorcoefficient for AIJ equation.

gates and these with large aggregates resist through thebond between aggregates and the dowel action of ag-gregates. The dowel action may be larger for slabs witha larger amount of reinforcement. Therefore, the sizeeffect relates not only to the slab depth but also to themaximum aggregate size and the amount of reinforce-ment. Tables 5 and 6 provide the mean values and coef-ficients of variation for the ratios of tested strength and

210

3.0

2.5

'2.0

1.5

1.0

0.0o 0

-0.5

-1.0

0.0

00--8- g= 0_

000 () 0

/60 ~ fJ8dJ 0

9o 5''00

0

8 8go~0 0o 8 0.a...-

CD(\To 0 0800 Q0

if 08 0

0

Fig. 4 Relationships between ratio of tension reinforce-ment and error coefficient for AIJ equation.

1200.0 0

1000.0 0 0

00

800.0Z 0

c1;) 600.0 0<ll;;:

400.0

200.0

000.0 200.0 400.0 6000 8000 10000 1200.0

Veq (kN)

Fig. 5 Relationships between Veq obtained from JSCEequation and Vtest.

2.0

1.5

c 1.0.~ 0.58g

0.0w

-0.5

-1.00.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0

Concrete strength (N/mm2)

Fig. 6 Relationships between concrete strength and errorcoefficient for JSCE equation.

evaluated strength for slab thicknesses under 100 mmand over 100 mm. These data exhibit larger coefficientsof variation for all the equations. That is to say, slabswith thickness under 100 mm may provide worse esti-mation.

3. Proposed rational equation for punchingshear strength

Concrete strength in the equations for punching shearstrength has been evaluated by failure area, concrete

2.0

1.5

c 1.0.~ 0.58

0

] 0.0

0 0-05

-1.0

0 100 200 300 400 500

Fig. 7 Relationships between effective depth and errorcoefficient for JSCE equation.

2.0

1.5

0c 1.0.1 0.5 0

g §w 0.0

00

-0.5

-1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Fig. 8 Relationships between ratio of tension reinforce-ment and error coefficient for JSCE equation.

Equation MeanStandard Coefficient ofdeviation variation (%)

All 2.17 0.88 40.4

JSCE 1.30 0.36 27.3ACI 1.66 0.42 25.3

Eurocode2 2.43 0.69 28.4

BS 1.07 0.29 27.1CEB-FIP 1.45 0.39 27.1

Equation MeanStandard Coefficient ofdeviation variation (%)

All 2.03 0.62 30.4

JSCE 1.19 0.26 22.0ACI 1.38 0.33 24.2

Eurocode2 2.07 0.49 23.7as 1.05 0.19 18.0

CEB-FIP 1.46 0.26 18.1

strength and ratio of tension reinforcement. The equa-tion is required to be a simple form with all the parame-ters, such as failure area including slab thickness, con-crete strength and ratio of tension reinforcement. AI-

though the conventional equation was the power ofcompressive strength of the concrete, 1/3rd power isobtained by an iterative least square method. Similarly1/4 th power is also obtained for the ratio of tension re-inforcement.

Recently, covers are enlarged for protection againstsalt from corrosion of reinforcing bars. The authorstested several slabs with a larger cover, such as 50 mmand 70 mm. These results can be included in the pro-posed equation, which is expressed as

where V = punching shear strength, fe' = compres-sive strength of concrete, p = ratio of tension rein-forcement, up = average failure perimeter (=4r+2 7[ kch),h = the slab thickness, and kc = factor for larger coversover 30 mm. kc is expressed by the following equation,and if the cover is less than 30 mm, may be provided asunity 1.

where C = cover.The scope of Eq. (14) is the ranges between 37 mm

and 503 mm in slab thickness, 0.176% and 7.02% inratio of tension reinforcement, and 125 KN/m2 and 816KN/m2 in compressive strength of concrete. The rela-tionships between the error coefficient of Eq. (14) andcompressive strength of the concrete, slab thickness andratio of tension reinforcement are shown in Figs. 9, 10,nand 12. The range of error coefficient values shownin Fig. 9 is between -0.5 and 0.5, and this figure exhibitsa clear correlation between the punching shear strengthand concrete strength. Figures 10 and n show rela-tionships between the error coefficient and the slabthickness, and between the error coefficient and the ratioof tension reinforcement, respectively. The ranges oferror coefficient value for these parameters are between-0.5 and 0.5 as shown in Figs. 10 and 11. The error ofpunching shear strength increases proportionally withincreases in the ratio of tension reinforcement. Theevaluated values of the proposed equation and test val-ues are shown in Fig. 12. The evaluated values in thisfigure exhibit better agreement with the test results.

The mean, the standard deviation and the coefficientof variation of the strength ratio obtained from Eq. (14)are given in Table 7. The mean of strength is 1.01, andcoefficient of variation is 17.1% for the data excludingslabs thicker than 100 mm and 18.9% for all the data.These coefficients of variation are fairly small in spiteof the simplicity of the equation.

The variation in the error coefficient is also influ-enced by uncertainty, such as quality of structural con-crete materials, the position of reinforcement arrange-ment and slab size. The error coefficients evaluated un-der order statistics and Gauss error-function equation (1)are shown in Figs. 13, 14 and 15 from the All equation,the JSCE equation and the proposed equation.

2.0

1.5

~ 1.00

1 0

0.5

0

Jj 0.0

-0.5

-1.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0Concrete strength(N/mm2

)

Fig. 9 Relationships between concrete strength and errorcoefficient for proposed equation.

1.5

c1.0. 0

0 0

'".0.500

e 0cb0.0

00

-0.5 00

-1.0

100 200 300 400 500Slab thickness (mm)

Fig. 10 Relationships between slab thickness and errorcoefficient for proposed equation.

~ 1.0

~g 0.5

Fig. 11 Relationships between ratio of tension rein-forcement and error coefficient for proposed equation.

1200.0

1000.0

800.0Z2S~ 600.0'"";;:

400.0

200.0

200.0 400.0 600.0 800.0 1000.0 1200.0Yeq (KN)

Fig. 12 Relationships between Veq obtained from pro-posed equation and Vtest.

~l

212 s. Hamada, Q. Yang and M. Mao I Journal of Advanced Concrete Technology Vol. 6, No.1, 205-214, 2008

The member factor for the punching shear strength is ties of 5%, 2.5%, I% and 0.5%, respectively, based ondetermined from the probability of failure. The member the Gaussian distribution. The JSCE equation is 0.83 forfactor is a safety factor provided by uncertainty of the a failure probability of50%, and member factors of 1.18,evaluated equation, variation of member size and the 1.34, 1.52, 1.80 and 2.05 for failure probabilities of 10%,importance of the member. In this study, the member 5%, 2.5%, 1% and 0.5%, respectively. The proposedfactor is evaluated by failure probabilities such as 0.5%, equation provides 0.99 for a failure probability of 50%,1%, 2.5%, 5%, 10% and 50%. These equivalent member and member factors of 1.44 and 1.77 for failure prob-factors are given in Table 8. The AU equation overesti- abilities of 5% and I%, respectively. There is a smallmates at a failure probability of 50% and member fac- difference between the Gauss and Weibull functions for. tors are 1.03, 1.31, 1.92 and 2.81, for failure probabili- failure probabilities over 1%.

The present simplified equation based on the database

Table 7 Analytical results obtained from proposedhas a considerably small coefficient of variation, andmost engineers can easily apply this equation to the de-

equation. sign of slabs.Standard CoeffieientOf -Data Mean deviation variation (%) 4. Conclusions

272 1.0 I 0.17 17.1 -313 1.0 I 0.19 18.9 Based on the results of this investigation, the following-

conclusions are drawn.(1) More than 300 test data of RC slabs conducted by

1.0 various researchers were collected, providing the~--0.9 /' largest database existing at present. Six equations.~ 0.8 G for punching shear strength in specifications were.2 0.7 JI evaluated based on this database.'0 06 , (2) The value of 1'1, the ratio of experimental data and>-

~ 0.5 J estimated data provided by AU, is over 2.0, and the~ 0.4 , coefficient of variation of this equation is 32.1%,~ 0.3 ,

which implies that the precision of the equations is~ 0.2f0.1

~not satisfactory. The value estimated by JSCE is

0.0~ 1.2, and the coefficient of variation of thisnear

0.0 1.0 2.0 3.0 4.0 5.0 equation is 23.1%. The ACl, Eurocode2 andStrenght ratio

CEB-FIP equations provide conservative values.Fig. 13 Normal distribution for AIJ equation. The coefficients of variation in BS and CEB-FlP

1.0~O'" 1.0 f""

0.9 0.90 0.8

.J§ 0.8

;0

" I"g t.2 0.7 1/ .2 0.7

I0 06

IIb 0.6

>- >-

~ 0.5 , ~ 0.5JK 0.4

I~ 0.4

I~ 0.3 ~ 0.3 I~ 0.2

A~ 0.2 I0.10.1 ..# ~

0.0 0.0

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

Strenght ratio Strenght ratio

Fig. 14 Normal distribution for JSCE equation. Fig. 15 Normal distribution for proposed equation.

Table 8 Failure probabilities and member factors.~

Member factor 0.5% 1% 2.5% 5% 10% 50%Gauss distribution (AU) 2.81 1.92 1.31 1.03 0.83 0.49Weibull distribution (AU) 2.47 1.84 1.31 1.03 0.83 0.49

~s distribution OSCE) 2.05 1.80 1.52 1.34 1.18 0.83Weibull distribution OSCE) 1.98 1.77 1.52 1.34 1.18 0.83Gauss distribution (proposed equation) 1.93 1.77 1.58 1.44 1.31 0.99Weibull distribution (proposed equation) 1.89 1.76 1.58 1.44 1.31 0.99

are small, less than 20%.(3) Slabs with a thickness under 100 mm may provide

worse estimation.(4) A simplified strength equation is also proposed

from the present database. The coefficient of varia-tion of the strength ratio evaluated from the equa-tion is smaller than that yielded by the All equationand the JSCE equation. The mean of strength is1.01, and the coefficient of variation is 17.1% forthe data excluding slab thicknesses of less thanlOOmm and 18.9% for all the data. These coefficientsof variation are fairly small in spite of the simplic-ity of the equation.

(5) The All equation overestimates at a failure prob-ability of 50% and member factors are 1.03, 1.31,1.92 and 2.81, for failure probabilities of 5%, 2.5%,1% and 0.5%, respectively, based on the Gaussiandistribution. The JSCE equation is 0.83 for a failureprobability of 50%, and member factors of 1.18,1.34, 1.52, 1.80 and 2.05 for failure probabilities of10%, 5%, 2.5%, 1% and 0.5%, respectively. Theproposed equation provides 0.99 for a failure prob-ability of 50%, and member factors of 1.44 and1.77 for failure probabilities of 5% and 1%, respec-tively. The difference between Gauss and Weibullfunctions for failure probabilities over 1% is small.

The present simplified equation based on the databasehas a considerably small coefficient of variation, andmost engineers can easily apply this equation to the de-sign of slabs.

AcknowledgementThe authors wish to express their gratitude and sincereappreciation to Kazuko Hamada, a former graduate stu-dent of Yamagchi University, for the collections oftested data presented in various journals.

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