Probabilistic approach to determine the increased shear capacity in reinforced concrete slabs under a concentrated load. Concept v. 12-11-2012

Delft University of Technology Report nr. 25.5-12-15 Faculty of Civil Engineering and Geosciences

Department of Design & Construction – Concrete Structures

November 12, 2012

Probabilistic approach to determine the increased shear capacity in

reinforced concrete slabs under a concentrated load

CONCEPT v. 12-11-2012

Author:

Delft University of Technology Report nr. 25.5-12-15 Faculty of Civil Engineering and Geosciences

Department of Design & Construction – Concrete Structures

November 12, 2012

Probabilistic approach to determine the increased shear capacity in

reinforced concrete slabs under a concentrated load

CONCEPT v. 12-11-2012

Author:

Ir. E. Lantsoght © 2012

Delft University of Technology

Faculty of Civil Engineering and Geosciences

Department of Design & Construction – Concrete Structures Stevinlaboratorium Postbus 5048 2600 GA Delft Telephone 015 2783990/4578 Telefax 015 2785895/7438 AUTEURSRECHTEN

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Table of Contents

1. Introduction ... 4

2. Overview of methods and approaches ... 5

2.1 Probabilistic methods ... 5

2.2 Possible approaches... 6

2.2.1 Comparison to test data ... 6

2.2.2 Modified approach for comparison to test data ... 8

2.2.3 Design assisted by testing ... 10

3. Determination of distributions ... 14

3.1 Geometric properties ... 14

3.2 Material properties ... 14

3.3 Experimental force to predicted force ... 16

3.3.1 Lognormal distribution ... 16

3.3.2 Frechet distribution ... 16

3.3.3 Generalized extreme value distribution ... 18

3.3.4 Gumbel distribution ... 19

3.3.5 Beta distribution ... 20

3.3.6 Distribution from histogram ... 21

3.4 Overview ... 24

4. Results ... 27

4.1 Determination of enhancement factor by hand ... 27

4.1.1 Initial approach ... 27

4.1.2 Design assisted by testing: EN 1990:2002... 30

4.2 Monte Carlo simulation ... 32

4.2.1 Comparison to test data ... 32

4.2.2 Overview of results ... 38

5. Extension of Eurocode shear provisions ... 40

5.1 Influence of concrete compressive strength ... 40

5.2 Influence of type of reinforcement ... 41

5.3 Reduced support length ... 42

5.4 Verification with second series of experiments ... 43

5.5 Extension for av/dl > 2,5 ... 46

6. Discussion and Assumptions ... 48

7. Conclusions ... 49

1. Introduction

This report contains the approaches and assumptions which are used to extend the formulation of the shear capacity as given in Eurocode 2, taking into account the larger shear capacity in slabs as compared to beams. As the formulation from Eurocode 2 is based on a statistical analysis, the extension should be based on a similar statistical analysis – taking into account the distribution of the ratio between the experimental value and the value calculated according to the beam shear formula from EN 1992-1-1:2005.

The report reflects the followed process. First a small review of the literature is carried out and reported. From this review, a few possible methods are selected. Next, the assumed distributions of the considered random variables are determined. With this knowledge calculations are carried out. Initially hand calculations are given and, consequently, Monte Carlo simulations. Lastly, the assumptions and points of discussion and improvement are summarized.

Initially, the enhancement factor for slabs under concentrated loads is determined based on a normal distribution (in: “Analyserapport”). The ratio between the experimental value and the calculated value (assuming mean material properties and no safety factors) according to EN 1992-1-1:2005 is used as a starting point to define the additional capacity from transverse load redistribution. Initially, these values were considered normally distributed and all other random variables were considered constant. Subsequently, the characteristic value of the enhancement was found as a 5% lower bound (average minus 1,64 * standard deviation). However, it is found that the left tail of the distribution does not match a normal distribution very well. Also, it is doubted whether this initial approach is according to the philosophy of the Eurocodes and if an additional increase in the enhancement factor can be found by studying the variability of the random variables under consideration.

2. Overview of methods and approaches

2.1 Probabilistic methods

The scope of this research is the shear capacity of slab bridges under concentrated traffic loads. An overview of relevant literature is cited in this paragraph. Ellingwood (1979a, b) showed that for reinforced concrete design from ACI 318, the reliability index for shear is less than for flexure, despite the desirability for beam and slab failures to occur in the ductile flexural rather than the relatively brittle shear mode. Ellingwood and Galambos (1982) determined which factors should be taken into account to increase the reliability index from 2,3 with the considered provisions to an index of 3. Nowak (1995) gives the background for the probabilistic development of a live load model for bridges.

For a reliability analysis, three levels of probabilistic approaches can be followed, with increasing accuracy. The following methods (CUR 190) are considered level III approaches for solving the reliability function and the joint probability density function fR,S(R,S) of the strength R and the load S:

- finding the exact solution of the integral,

- using numerical integration techniques to expand the integral in a series, - using a Monte Carlo simulation (with or without importance sampling). The expression for the reliability index β in this case is (Rackwitz, 2000):

1

(1 P_F)

β _{= Φ}− _{− (1)}

with:

Φ the standard normal distribution,

PF the probability of failure, defined as the chance that the limit state is

exceeded.

A powerful combination between Monte Carlo simulations and finite element calculations is described by Schlune et al. (2011). Ellingwood (1994) however points out that: “The use of increasingly complex reliability analysis methods and computerization reduce or eliminate insight in the role of uncertainty in safety that might be more easily obtained from simpler methods.”

To reduce the required numbers of samples in a Monte Carlo simulation, importance sampling can be used. Waarts (2000) lists five methods of importance sampling:

- decrease variance of unimportant variable

- truncate distribution function of variables with known importance to the limit state function

- apply weighting functions - skip unimportant variables - shift variables.

Level II methods are based on first and second order reliability approaches (Rackwitz, 2011), in which the reliability index is determined as the distance from the origin to the failure space. The expressions are developed around the design point, which is the location with the greatest joint probability density.

A level I method uses the partial safety factors as defined in the codes.

2.2 Possible approaches

In a classical approach, the limit state is defined as a relation between the acting loads and the occurring resistance. However, the comparison with experimental results (Lantsoght, 2011) does not correspond to the general design case in which a combination of loading is applied to the structure. Therefore, other methods for assessment need to be sought.

2.2.1 Comparison to test data

In this section, an example of a possible approach from the literature is given, in which statistical considerations from experimental results are used to determine the required safety factors. Yura, Galambos and Ravindra (1978) used a probabilistic approach (detailed in Ravindra and Galambos, 1978) to determine the overall safety factor φ required for steel beams in bending. This safety factor was determined as:

(

)

exp 0, 55 m R n R V R φ = − β (2) 2 2 2 R M F P V = V +V +V (3) with

Rn the nominal strength predicted by the formula used in design (Rd in

Eurocode notations),

β the safety index, taken as 3,0 for beams,

VM the expected variation in materials strength,

VF the expected variation in fabrication,

VP the expected variation in strength analysis.

As the strength of these beams is determined based on their bending moment capacity, Eq. (2) was rewritten as:

(

)

exp 0, 55 m R n M V M φ = − β (4)

Here, Mm is the mean elastic critical moment of a beam in place, which is in the

elastic range:

(

)

2 2 Test Capacity Prediction y w b m y y b z b E I C C M EI GJ K L K L

π

π

  = × +   (5) with:

Lb the unbraced length of the beam,

E the elastic modulus,

G the elastic shear modulus,

Iy the minor axis moment of inertia,

J the St. Venant torsion constant,

Cw the warping moment of inertia,

Cb the loading coefficient,

Ky,z the effective length factors that account for lateral and torsional end

Fig. 2.1.: Histogram of the ratio between the buckling loads and their theoretical predictions for 185 Elastic Lateral-Torsional Buckling Tests of Beams (Yura,

Galambos and Ravindra, 1978).

As shown in Fig. 2.1, the mean value of Test Capacity

Prediction is 1,03. The handbook properties are assumed to be mean values. The value of Mn is determined from the

design office formula, with Ky = Kz = 1:

(

)

2 2 y w b n y y b z b E I C C M EI GJ K L K L

π

π

= + (6)

The same procedure was followed in the inelastic and plastic range, for different loading conditions, to determine and recommend a factor φ for a design with a reliability index of β.

2.2.2 Modified approach for comparison to test data

This method is based on the approach taken by Yura, Galambos and Ravindra (1978) for the determination of the required φ factor for the bending moment of steel beams. The limit state is defined as:

g= −R S (7)

and failure occurs when g < 0, or when the load exceeds the resistance. This can be expressed as seeking the chance that the experimental resistance is smaller than the design shear resistance:

{

}

f d

For the comparison with the test data, the following expressions are used for R and Rd

(replacing S in this case) based on the shear capacity expression from EN 1992-1-1:2005 §6.2.2 (6):

(

)

1/3 , 1 100 Rd c d ly ck w l c C R k

ρ

f b d

γ

α

= (9)

(

)

1/ 3 , , Test 1 100

Prediction Rd c test ly cmean w l

R C k ρ f b d

α

= × (10)

In which:

CRd,c 0,18 as default value for EN 1992-1-1:2005,

γc 1,5 for concrete,

k the size factor,

s ly

l

A bd

ρ = the ratio of longitudinal steel,

As the area of longitudinal steel,

b the member width,

dl the effective depth to the longitudinal reinforcement,

fck the characteristic concrete cylinder compressive strength,

2 v

l

a d

α = takes into account direct load transfer in the vicinity of the support,

av the clear shear span, the face-to-face distance between the load and the

support,

bw the web width, not to exceed the effective width in shear, which is

determined here based on the French load spreading method, Fig. 4.1, Test

Prediction the ratio of the tested value to the predicted value according to EN 1992-1-1:2005,

CRd,c,test 0,15 for the comparison with test data, as given by König and Fischer

(1995),

fcmean the mean cylinder concrete compressive strength.

The expression for the limit state is then:

(

)

1/ 3 _,

(

)

1/ 3 , , Test 100 100 Prediction Rd c Rd c test ly cmean ly ck w l c C k g C

ρ

f

ρ

f b d

γ

α

  = −    (11)

(

)

1/ 3 _,

(

)

1/ 3 , , Test 100 100 0 Prediction Rd c Rd c test ly cmean ly ck c C C

ρ

f

ρ

f

γ

  − <     (12)

In Equation (12), the following random variables can be distinguished: - Test/Prediction

- b

- dl

- fcmean

- fck

2.2.3 Design assisted by testing

EN 1990:2002, Annex D8.2.2. describes the standard evaluation procedure for the statistical determination of resistance models. This procedure assumes that all variables follow either a normal or a lognormal distribution.

The first step consists of developing a design model for the theoretical resistance rt

determined by the resistance function, covering all relevant basic variables X that effect the resistance at the relevant limit state:

( )

t rt

r =g X (13)

This resistance function could be the EN 1992-1-1:2005 formulation including an enhancement factor for slabs under concentrated loads.

In the next step, the experimental and theoretical values are compared. The actual measured properties are substituted into the resistance function to obtain the theoretical values rti to form the basis for a comparison with the experimental values

rei from the tests. The points representing pairs of corresponding values (rti, rei) should

be plotted on a diagram, Fig. 2.2. If the resistance function is exact and complete, then all points lie on the line with θ = 45°. In practice, the points will show some scatter, but the causes of any systematic deviation from that line should be investigated to check whether this indicates errors in the test procedures or in the resistance function.

Fig. 2.2.: re – rt diagram

In a third step, the mean value of the correction factor b is to be determined. For this purpose, the probabilistic model of the resistance r is given proportional to b the least squares best-fit to the slope :

t r=brδ (14) 2 e t t r r b r =

∑

∑

(15)

The mean value of the theoretical resistance function, calculated using the mean values X of the base variables, can be obtained from: _m

( )

( )

m t m rt m

r =br X

δ

=bg X

δ

(16)

In a fourth step, the coefficient of variation of the errors is estimated. The error term δi

for each experimental value rei should be determined as:

ei i ti r br δ = (17)

From these values of δi an estimated value for the coefficient of variation of the errors

Vδ can be determined based on ∆ =i ln

( )

δ

i , which has an estimated value: 1 1 n i i n ₋ ∆ =

_∑

∆ (18)

The variance is:

(

)

2 2 1 1 1 n i i s n ∆ = = ∆ − ∆ −

∑

(19) Then,

2

es 1

V_δ = ∆− ₍₂₀₎

In the fifth step, the compatibility of the test population with the assumptions made in the resistance function is analyzed. In this step, it can be decided to split up the data into different subsets.

In the 6th step, the coefficients of variation VXi of the basic variables are determined. If

it can be shown that the test population is fully representative of the variation in reality, then the coefficients of variation VXi of the basic variables in the resistance

function may be determined from the test data. However, since this is not generally the case, the coefficients of variation VXi will normally need to be determined on the

basis of some prior knowledge.

In a 7th step, the characteristic value rk of the resistance is determined. If the resistance

function for j basic variables is a product function, the formulas in D8.2.2.7(1) can be applied. For a more complex resistance function like the shear design formula:

(

1,...,

)

t rt j

r=brδ =bg X X δ (21)

the mean value E(r) may be obtained from:

( )

rt

(

( 1),..., ( j)

)

rt

( )

m

E r =bg E X E X =bg X (22)

and the coefficient of variation Vrt may be obtained from:

( )

( )

( )

2 2 2 2 1 1 j rt _rt rt i i m m rt rt i VAR g X _g V g X g X ₌ X

σ

  _∂    = ≅ ×   ∂  

∑

(23)

If the number of tests is limited (n < 100) allowance should be made in the distribution of ∆ for statistical uncertainties. The distribution should be considered as a central t-distribution with the parameters ∆, V∆ and n. The characteristic resistance

rk should be obtained from:

( )

(

2

)

exp 0, 5 m k rt rt rt n r =bg X −k_∞

α

Q −k

α

_{δ δ}Q − Q (24) with:

(

2

)

ln( ) ln 1 rt rt rt Q =

σ

= V + (25)

(

2

)

ln( ) ln 1 Q_δ =

σ

_δ = V_δ + (26)

(

2

)

ln( )r ln r 1 Q=

σ

= V + (27) rt rt Q Q

α

= (28)

Q Q δ δ

α

= (29) where:

kn the characteristic fractile factor from Table 2.1 for the case VX

unknown,

k∞ = 1,64; the value of kn for n → ∞,

αrt the weighing factor for Qrt,

αδ the weighing factor for Qδ.

Table 2.1: Values of kn for the 5% characteristic value (Table D1 from EN

1990:2002).

If a large number of tests (n ≥ 100) is available, the characteristic resistance rk may be

obtained from:

( )

(

2

)

exp 0, 5 m k rt r =bg X −k Q_∞ − Q (30)

3. Determination of distributions

In this section the distributions which are used in the Monte Carlo simulations are introduced. All distributions are based either on experimental data from the series of slab test (Lantsoght, 2011) or based on the guidelines of the JCSS Probabilistic Model Code.

3.1 Geometric properties

The value of As is considered a deterministic value of 6597mm2. The width b and the

effective depth dl have a normal distribution with a coefficient of variation of 2,5%

(Vrouwenvelder, Holicky and Markova, 2002). For the width b the mean is 2,5m and the standard deviation is 0,063m. For the effective depth dl the mean is 0,265m and

the standard deviation 0,00663m.

3.2 Material properties

The concrete compressive strength is determined based on the cube compressive strengths measured at 28 days, for C28/35 and C55/65. According to the JCSS Probabilistic Model Code, the cube concrete compressive strength follows a lognormal distribution.

The sample sizes are 36 cubes for C28/35 and 14 cubes for C55/65. The distribution for C28/35 is shown in Fig. 3.1 and the distribution for C55/65 in Fig. 3.2. Assuming a lognormal distribution, the constants defining the distribution are for C28/35:

λ = 3,7624 the mean value of the natural logarithm of the cube compressive strength of the concrete,

ε = 0,0838 the associated standard deviation. For C55/65 these constants are:

λ = 4,2925 the mean value of the natural logarithm of the cube compressive strength of the concrete,

0 2 4 6 8 10 12 36 38 40 42 44 46 48 50 52 More Bin F re q u e n c y 0,00% 10,00% 20,00% 30,00% 40,00% 50,00% 60,00% 70,00% 80,00% 90,00% 100,00% Frequency Cumulative %

Fig. 3.1.: Distribution of cube compressive strength for C28/35.

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 67 69 71 73 75 77 More Bin F re q u e n c y 0,00% 10,00% 20,00% 30,00% 40,00% 50,00% 60,00% 70,00% 80,00% 90,00% 100,00% Frequency Cumulative %

Fig. 3.2.: Distribution of cube compressive strength for C55/65.

The distribution of the characteristic concrete strength fck is also assumed to be

3.3 Experimental force to predicted force

For the comparison between the test results and the predicted shear force according to EN 1992-1-1:2005 taking into account the French method of horizontal load distribution, the values given in the report containing the data analysis (Lantsoght, 2012) are used. To ensure that all experimental results belong to the same population, the results, the results of slabs with concrete C35/45, on line supports and with ribbed bars are used. To find distributions other than the normal distribution, the following input parameters were used:

µ = 2,0231 the mean value of the experimental shear force to the predicted shear force,

σ = 0,2592 the associated standard deviation,

m = 2,025 the median of the experimental over predicted values, γ1 = 0,0982 the skewness of the distribution,

γ2 = 0,4830 the kurtosis of the distribution,

λ = 0,697 the mean value of the natural logarithm of the quotient of the experimental and predicted shear force,

ε = 0,130 the associated standard deviation.

3.3.1 Lognormal distribution

The lognormal distribution uses the following expressions for the probability density function and the cumulative distribution function:

For the Monte Carlo simulation in Matlab, the built-in function for drawing random numbers from a lognormal distribution is used. This function is lognrnd

( )

λ ε

, with λ and ε as found from the data.

3.3.2 Frechet distribution

The constants u and k which determine the Frechet distribution are determined from the given mean µ and the given standard deviation σ:

These constants then lead to the Frechet probability density function:

Likewise, the cumulative distribution function becomes:

For application in a Monte Carlo simulation, this cumulative distribution function has to be inverted. For uniformly distributed random variables ui, the following inversion

( )

( )

( )

(

)

( )

(

)

1/ 1/ exp ln ln ln k i X i k i i k i i i k i u u F x x u u x u u x u x u  _{ }    = = _−  _       ⇔ = −    ⇔ − = ⇔ = −

The random variables xi now are drawn from a Frechet distribution, for uniformly

distributed ui numbers.

3.3.3 Generalized extreme value distribution

The generalized extreme value distribution is determined by three constants: a location constant, a scale constant and a shape constant.

These constants then lead to the probability density function of the generalized extreme value distribution:

For the Monte Carlo simulation in Matlab, the built-in function for drawing random numbers from a generalized extreme value distribution can be used. This function is

(

)

gevrnd K, ,

σ µ

with:

K the shape factor,

σ the scale factor,

µ the location.

Knowing the statistical properties, these values can be determined in MathCad. If the distribution of Test

Predictionwere a genuine Frechet distribution, then the value of the location µ needs to be 0 and the shape factor larger than 0.

3.3.4 Gumbel distribution

The Gumbel distribution (or Extreme value type I distribution) is defined by two parameters: the mode u of the distribution and the measure of the dispersion, α.

These constants can then be used to define the probability density function and the cumulative distribution function of the Gumbel distribution:

3.3.5 Beta distribution

The four-parameter beta distribution has the great flexibility in being able to be fitted to observed data. The four parameters which define the distributions are a and b which describe the intervals for the general beta distribution and q and r which define the shape of the distribution. These 4 parameters can be identified with the mean, standard deviation, skewness and kurtosis of the distribution:

The probability density function and cumulative distribution function are then described with these four parameters and the use of the of the beta function:

3.3.6 Distribution from histogram

Based on the histogram obtained in Excel, the distribution of the data can also be drawn. The input parameters are the different bins in which the data are ordered, and the frequency per bin:

From these input data, the probability density function can be assembled, using linear interpolation in the bin intervals:

The resulting cumulative distribution is:

This density function is also shown in Fig. 3.3 and the cumulative distribution function is shown in Fig. 3.4. Here, a coarser bin is used to avoid zero values in the density function.

Fig. 3.3: Resulting probability density function from histogram of experimental to calculated (Test/Predicted) values.

Fig. 3.4: Resulting cumulative distribution function from histogram of experimental to calculated (Test/Predicted) values.

Since the distribution from the histogram is stepwise continuous, it is not possible to be directly inverted and used in a Monte Carlo simulation.

3.4 Overview

An overview of the different probability density functions discussed in the previous sections, is shown in Fig. 3.5. This plot shows that the measured distribution is, at its left tail, most closely simulated by a lognormal or Beta distribution. However, the Beta distribution slightly underestimates the left tail of the real distribution. On the other hand, the lognormal distribution overestimates the left tail of the real distribution. The peak value of the real distribution is more towards the right than of any of the studied distributions. However, the results of the left tail and the low values in the distribution are of most interest in this study. The generalized extreme value distribution shows a peak which is too high in frequency and an underestimation of the left tail of the distribution. Similar conclusions can be drawn for the Frechet and Gumbel distributions. The normal distribution overestimates the lowest values in the

left tail of the distribution. Based on the cumulative distribution functions of the chosen distributions, the 5% lowest value can be found, as well as the 50% value (median), Table 3.1.

Fig. 3.5: Probability density function of the experimental over the calculated value (Test/Predicted), for a normal distribution, lognormal distribution, Frechet distribution, the measured distribution, the generalized extreme value distribution, the

Fig. 3.6: Probability density function of the experimental over the calculated value (Test/Predicted), for a normal distribution, lognormal distribution, Frechet distribution, the measured distribution, the generalized extreme value distribution and

Beta distribution.

Table 3.1.: 5% lower percentile and median of the experimental to calculated value (Test/predicted), for different distributions.

Distribution 5% 50% data 1,637 2,100 normal 1,597 2,023 lognormal 1,621 2,008 frechet 1,720 1,970 GEV 1,813 1,958 Beta 1,649 1,995

4. Results

4.1 Determination of enhancement factor by hand

4.1.1 Initial approach

The method from Yura, Galambos and Ravindra (1978) is followed in a first attempt to determine the enhancement factor. The factor is defined as:

(

)

exp m R n R V R

ξ

= −

αβ

(31)

with Rm the strength as obtained in test results and Rn the strength as predicted by the

design formula. The value for β in EN1990:2002 equals 4,3 for a 50 years reference period and an RC3 class structure, Table 4.1. For the level “repair” from NEN 8700:2011 and an RC3 class structure, β equals 3,8. For structures constructed before 2012, this value is lowered to β = 3,6 (Steenbergen et al., 2012). For the level “disapproval” from NEN 8700:2011 and an RC3 class structure, β equals 3,3, lowered to β = 3,1 for structures constructed before 2012 (Steenbergen et al., 2012). It is recommended to carry out the assessment at the “repair” level.

Table 4.1: Recommended minimum values for reliability index β (ultimate limit states) (Table B2 from EN 1990:2002).

While the value for α was taken as 0,55 in the approach by Yura, Galambos and Ravindra (1978), the value for αR equals 0,80 and αS is 0,70 in EN 1990:2002 (CUR

190). The coefficient of variation is taken as:

2 2 2

R M F P

V = V +V +V (32)

VF 0,05 as used by Yura, Galambos and Ravindra (1978) for the

fabrication of a steel specimen; assumed 0,10 for the fabrication of a concrete specimen,

VM 0,17 for concrete, as given in de JCSS probabilistic model code

examples by Vrouwenvelder, Holicky and Markova (2002),

VP 0,14 as determined from the data analysis (Lantsoght, 2012) as the

coefficient of variation of the ratio between the experimental result and the design shear capacity for the results of slabs S1 to S6.

The resulting coefficient of variation is V_R = 0,172+0,142+0,102 =0, 242 . The resistance from test results can be expressed as:

(

)

1/ 3 , , Test 1 100 Prediction m Rd c test ly cm w l R C k

ρ

f b d

α

= × (33)

The design resistance as prescribed by EN 1992-1-1:2005 for the shear capacity is:

(

)

1/ 3 , 1 100 n Rd c ly ck w l R C k

ρ

f b d

α

= (34) in which:

CRd,c 0,18/γc as default value for EN 1992-1-1:2005,

γc 1,5 for concrete,

k the size factor,

s ly

l

A bd

ρ

= the longitudinal reinforcement ratio,

As the area of main flexural longitudinal steel,

b the full member width,

dl the effective depth to the longitudinal reinforcement,

fck the characteristic concrete cylinder compressive strength,

2 v

l

a d

α

= takes into account direct load transfer in the vicinity of the support,

av the clear shear span, the face-to-face distance between the load and the

support, Fig. 4.2,

bw the web width, not to exceed the effective width in shear, which is

determined here based on the French load spreading method, Fig. 4.1. Test

Prediction the mean value of the tested value to the design value according to EN 1992-1-1:2005,

CRd,c,test 0,15 for the comparison with test data, as given by König and Fischer

(1995),

fcmean the mean cylinder concrete compressive strength.

load

(b)

support bef

f

Fig. 4.1: Top view of slab with horizontal load spreading between the concentrated load and the support and resulting effective width assuming 45º horizontal load

spreading from the far corners of the load.

Fig. 4.2: Loads near supports: beam with direct support: Figure 6.4 from EN 1992-1-1:2005. Thus, 1/3 8 0,15 2, 023 2, 750 0,12 m ck n ck R f R f  ₊  = ×   =  

For C28/35, the value of fck is 28MPa, which leads to an enhancement factor ξ of:

(

)

(

)

exp 2, 750 exp 0,80 3, 6 0, 242 1, 370 m R n R V R

ξ

= −

αβ

= − × × =

For C55/65, the value of fck = 55MPa, which leads to m

n

R

R =2,552 and ξ = 1,270.

For assessment at the “disapproval” level, with a reliability index of 3,1 the following enhancement factors are found: ξ = 1,509 for C28/35 and ξ = 1,400 for C55/65.

The disadvantage of this method is that it heavily relies on the previously mentioned assumptions with regard to the coefficient of variation VR and assumes normally

distributed random variables.

The sensitivity to the assumptions with regard to the material properties are highlighted in this paragraph. Using the statistical properties of the results of the cube compressive tests, the 5% lower bound of the distribution is found to be 37,9MPa for the cube compressive strength, which gives a cylinder compressive strength fck =

0,82*37,9MPa = 31,1MPa. The average cube compressive strength is found to be 43,20MPa, which gives a cylinder compressive strength of 35,42MPa. The coefficient of variation on the compressive strength of the concrete is found to be VM = 0,085. As

a result, a value VR = 0,192 can then be found. Moreover,

1/3 0,15 35, 42 2, 023 2, 641 0,12 31,1 m n R R   = ×   =  

This leads to an enhancement factor ξ of:

(

)

(

)

exp 2, 641exp 0,80 3, 6 0,192 1, 519 m R n R V R ξ = −αβ = − × × =

Or, assuming the disapproval level: ξ = 1,640 for C28/35

The resulting enhancement factor is thus quite sensitive to the assumptions with regard to the material parameters.

4.1.2 Design assisted by testing: EN 1990:2002

Using the approach as described in section 2.3, the design formula for shear taking into account an enhancement factor for concentrated loads on slabs can be evaluated. The considered experiments are carried out on specimens S1 to S6. For comparison with EN 1992-1-1:2005 with CRd,c = 0,15 and taking the French method for load

spreading into account, the following value for b is obtained for re the experimental

value and rt the design value:

2 2, 7919 e t t r r b r =

∑

=

∑

Next, for all tests, the value of δi is determined:

ei i ti r i br δ = ∀ and ∆ =_i ln

( )

δ

_i

1 1 0, 02457 n i i n ₌ ∆ =

∑

∆ = − for n = 14 Then,

(

)

2 2 1 1 0, 01969 1 n i i s n ∆ = = ∆ − ∆ = −

∑

( )

2 exp 1 0,14103 V_δ = s_δ − = Next VXi is determined. For the shear capacity:

(

)

1/3

(

)

, , 1 100 , , Rd c Rd c ly ck w l l ck V C k ρ f b d f d b f α = =

If Vfck=0,17 and Vd,b=0,025 as in the JCSS probabilistic model code and the examples

by Vrouwenvelder, Holicky and Markova (2002), then:

2 2 2 2 2 1 0, 025 0, 025 0,17 j Rt xi i V V = =

_∑

= + + such that VRt = 0,174.

The characteristic resistance can then be expressed as:

( )

(

2

)

exp 0, 5 m k rt rt rt n r =bg X −k_∞α Q −k α_{δ δ}Q − Q with k∞ = 1,64

kn = 1,92 (for 10 samples, Vx unknown)

(

2

)

ln 1 0,1727 rt Rt Q = V + =

(

2

)

ln 1 0,1403 Q_δ = V_δ + =

(

) (

)

2 2 2 1 1 1 1 0, 0507 j r xi i V V_δ V =   = +  + − = 

∏



(

2

)

ln _r 1 0, 222 Q= V + = 0, 777 rt rt Q Q α = = 0, 631 Q Q δ δ α = =

As a result, the enhancement factor can be expressed as:

2 0,5 rt rt n k Q k Q Q b e α αδ δ

ξ

= × −∞ − −

2

1,64 0,777 0,173 1,92 0,631 0,140 0,5 0,222

2, 792 e 1,844

ξ

_{= ×} − × × − × × − × ₌

4.2 Monte Carlo simulation

4.2.1 Comparison to test data

The number of samples required for a Monte Carlo simulation equals (Waarts, 2000): 3

f N

P >

For the case where β = 4, the number of samples therefore is N > 105.

For assessment at the repair level, β = 3,6 and the required number of N > 18856. Therefore, the number of samples is taken as 105.

A Monte Carlo simulation is used in which the following random variables are considered:

1. Test

Predictionfor S1 to S6, having a lognormal distribution as determined in section 3

2. fck and fcmean having lognormal distributions for C28/35 as determined based

on the compressive tests

3. b and dl having normal distributions.

The limit state is defined according to Eq. (12). The initial code for Test

Predictionhaving a lognormal distribution and with a slab factor = 1, is shown here:

The code for Test

Predictionhaving a lognormal distribution and with a slab factor = 1,57 is shown next. After several trials with different slab factors, this factor is suggested as the optimal enhancement factor for Test

Fig. 4.3: Distribution of the quotient of the experimental to the calculated value Test

Prediction , based on a lognormal distribution, showing 10

5

samples

Fig. 4.5: Distribution of effective depth d.

Fig. 4.7: Distribution of characteristic concrete compressive strength for C28/35.

4.2.2 Overview of results

In Table 4.2, an overview is given of the results of the different simulations. The first column indicates which distribution was used for Test

Prediction. In the next column, the assumed enhancement factor is given. Then, the following abbreviations are used:

pf the calculated failure probability,

β the reliability index,

n the number of trials.

The target reliability index for the repair level is taken here as β = 3,6. However, since the loading is deterministic, the target reliability level should be taken as αβ = 0,8.3,6 = 2,88 as prescribed by EN 1990:2002 C.8b. It can be seen that the required enhancement factor is ξ = 1,76 for a lognormal distribution and ξ = 1,71 when using a normal distribution.

Table 4.2: Overview of results from simulations with different enhancement factors and different distributions

Distribution ξ pf β n Lognormal 1 0 ∞ 105 1,25 0 ∞ 105 1,5 6.10-5 3,8461 2.105 1,6 1,98.10-4 3,55 5.105 1,58 1,78.10-4 3,5707 5.105 1,55 8,2.10-5 3,7689 5.105 1,56 1,1.10-4 3,6949 5.105 1,57 1,3.10-4 3,6522 106 1,75 0,0013 3,0115 104 1,78 0,0022 2,8096 105 1,77 0,0023 2,8283 105 1,76 0,0020 2,8829 105 Normal 1,57 4,9.10-4 3,2962 2.105 1,4 6,5.10-5 3,8265 2.105 1,45 1,52.10-4 3,6119 5.105 1,75 0,0024 2,8202 104 1,73 0,0025 2,8057 105 1,71 0,0019 2,9027 105 1,72 0,0023 2,8338 105

If it would be decided to carry out the shear assessment of the existing bridges at the disapproval level, the requirement for the reliability index would be αβ = 0,8.3,1 = 2,48 for all bridges constructed before 2012. The results are given in Table 4.3. Based on a lognormal distribution it is found that for the basic cases of slabs under a

concentrated load near to the support, the shear capacity can be increased with a factor ξ = 1,86. The analysis, compared to Table 4.2 shows that for a smaller reliability level, the difference between using a lognormal and a normal distribution becomes smaller. The largest differences can be observed by the behavior in the tail of the distribution.

Table 4.3: Overview of results from simulations with different enhancement factors and different distributions to fulfill the requirements for the disapproval level of

assessment Distribution ξ pf β n Lognormal 1 0 ∞ 105 2 0,0266 1,933 104 1,8 0,0031 2,737 104 1,9 0,0087 2,378 104 1,87 0,0073 2,442 104 1,86 0,0065 2,486 105 1,85 0,0064 2,489 105 Normal 1,8 0,0055 2,5427 104 1,85 0,0067 2,470 105 1,84 0,0063 2,498 105

5. Extension of Eurocode shear provisions

5.1 Influence of concrete compressive strength

The enhancement factor is determined based on the results of slabs S1 to S6. As it is observed experimentally that the concrete compressive strength does not have a significant influence on the resistance of slabs under concentrated loads in shear and the formula from EN 1992-1-1:2005 takes into account the concrete compressive strength by a cube root relationship, it is necessary to reduce the enhancement factor for higher strength concrete.

The statistical properties of C55/65 are given in §3.2. The properties of Test

Predictionare: µ = 1,951 as average;

σ = 0,283 as standard deviation; λ = 0,659;

ε = 0,143.

These properties are used as input for Monte Carlo simulations assuming a lognormal distribution for Test

Prediction. The results are given in Table 5.1, showing that the enhancement factor is now ξ = 1,75 for the disapproval level and ξ = 1,64 at the repair level.

Table 5.1: Overview of results from simulations for slabs S7 – S10.

ξ pf β n 1,57 4,1.10-4 3,3460 2.105 1,5 1,94.10-4 3,5481 106 1,45 8,2.10-5 3,7689 5.105 1,48 1,3.10-4 3,6522 106 1,49 1,75.10-4 3,5752 106 1,7 0,0027 2,7822 104 1,8 0,0109 2,2938 104 1,75 0,0066 2,4772 105 1,77 0,0083 2,3968 105 1,67 0,0024 2,8188 105 1,66 0,0022 2,8494 105 1,65 0,0022 2,8437 105 1,64 0,0016 2,9537 105

As only two concrete classes are tested in this series of experiments, a linear dependence of the enhancement factor on the concrete compressive strength is assumed. This relation, based on linear interpolation, is:

( ) 1, 974 245 ck ck f f

ξ = − with fck in MPa for the disapproval level (35)

( ) 1,884 225 ck ck f f

ξ = − with fck in MPa for the repair level (36)

5.2 Influence of type of reinforcement

Slabs S11 to S14 are reinforced with plain bars. To study if the proposed enhancement factor can still be applied, the results of the experiments carried out on these slabs, are analyzed separately.

The properties of Test

Predictionfor S11 to S14 are: µ = 1,863 as average;

σ = 0,182 as standard deviation; λ = 0,618;

ε = 0,098.

These properties are used as input for Monte Carlo simulations assuming a lognormal distribution for Test

Prediction. The results are given in Table 5.2, showing that the enhancement factor is now ξ = 1,90 for the disapproval level and ξ = 1,82 for the repair level. It is interesting to note that a full statistical analysis shows a slightly higher shear capacity for slabs with plain bars as compared to slabs with deformed bars. These results differ from a regular analysis of the experimental results as compared to EN 1992-1-1:2005 (Lantsoght, 2012), and lead to similar conclusions as observed from experiments on beams with plain bars.

Table 5.2: Overview of results from simulations for slabs S11 – S14.

ξ pf β n 1,4 0 0 ∞ 1,5 0 0 ∞ 1,6 1.10-5 4,2649 105 1,65 4,5.10-5 3,9161 2.105 1,68 1,7.10-4 3,5827 5.105 1,67 1,02.10-4 3,7140 106 2 0,0222 2,0103 104 1,9 0,0059 2,5181 105

1,93 0,0115 2,2734 104 1,92 0,0086 2,3828 105 1,91 0,0075 2,4343 105 1,8 0,0015 2,9739 105 1,78 0,0013 3,0115 104 1,82 0,0018 2,9183 105 1,83 0,0022 2,8437 105

5.3 Reduced support length

Slabs S14 to S18 are supported by 3 elastomeric bearings of 280mm x 350mm per side, resulting in a total supported length of 1,05m per side only. Even though only 42% of the width is supported, the shear capacity is found to be less reduced. Assuming that only the supported width carries the load is thus too conservative. Therefore, a linear relationship to take into account the support length reduction is sought.

The properties of Test

Predictionfor S15 to S18 are: µ = 1,634 as average;

σ = 0,298 as standard deviation; λ = 0,475;

ε = 0,189.

These properties are used as input for Monte Carlo simulations assuming a lognormal distribution for Test

Prediction. The results are given in Table 5.3, showing that the enhancement factor is now ξ = 1,33 for the disapproval level and ξ = 1,23 for the repair level, indicating an important reduction of the shear capacity as the supported length is decreased.

Table 5.3: Overview of results from simulations for slabs S15 – S18.

ξ pf β n 1,4 0,0134 2,2138 105 1,2 0,0014 3 105 1,1 2.10-4 3,5401 105 1,08 1,775.10-4 3,5715 4.105 1,07 1,84.10-4 3,5720 5.105 1,05 1,2.10-4 3,6726 106 1,3 0,0049 2,5828 104 1,35 0,0078 2,4186 105 1,33 0,0062 2,4994 105

1,34 0,0072 2,4491 105 1,25 0,0029 2,7589 104 1,20 9.10-4 3,1214 104

1,23 0,0019 2,8976 105

As only two support layouts are tested in this series of experiments, a linear dependence between the supported length and the resulting effective width are used. This relation, based on linear interpolation, is:

sup sup

(l ) 0, 52l 0, 48

b b

λ = + for the repair level (37)

sup sup

(l ) 0, 49l 0, 51

b b

λ = + for the disapproval level (38)

with

lsup the support length;

b the full width of the slab.

The factor λ is then applied to the resulting effective width from the French load spreading method beff and reduces the effective width when not the full slab width is

supported.

5.4 Verification with second series of experiments

The proposed formula for the shear capacity of slabs under a concentrated load close to the support can be summarized as at the disapproval level:

(

)

1/3 sup , , , 100 1,974 0, 49 0,51 245 ck Rd c prop Rd c l ck eff l l f V C k f b d b

ρ

   =  −  +   _  (39)

The formula at the repair level is:

(

)

1/3 sup , , , 100 1,884 0,52 0, 48 225 ck Rd c prop Rd c l ck eff l l f V C k f b d b

ρ

   =  −  +   _{ }  (40)

of which the symbols are explained in §4.1.1.

First, the expression at the disapproval level is verified by using the data of the second series of experiments. In the second series of experiments, the slab is loaded with a concentrated load as well as a line load. The contribution of the concentrated load to the shear stress at the support is expressed as:

conc conc eff l V b d

τ

ξ

= with 1,974 0, 49 sup 0, 51 245 ck l f b

ξ

= −  +   _  (41)

with

Vconc the shear force at the support as a result of the concentrated load only;

beff the effective width resulting from the French load spreading method

with a lower bound of 4dl;

dl the effective depth;

lsup the support length (full width or length of bearings).

The contribution of the distributed loads and the line load to the shear stress at the support is then: line line l V bd τ = (42) with

Vline the shear force at the support as a result of all distributed loads;

b the full width;

dl the effective depth.

For the considered case, the value of ξ equals 1,84. The comparison between the total shear stress at the support (τconc + τline) and the predicted shear stress at the support

(τ_EC2 =0,15 100

(

ρ_lf_cm

)

1/3) results in the following statistical properties for Test Prediction: µ = 1,308 as average;

σ = 0,232 as standard deviation; λ = 0,250;

ε = 0,204.

These values appear to be lower than the previous sets of values, but in this case, the enhancement factor of 1,84 is already taken into account. A set of 5.105 simulations with the assumptions of the described approach results in a probability of failure Pf =

4,724.10-3 and thus a reliability index β = 2,5954. The resulting reliability index is larger than the required β = 2,48. This value is deemed satisfactory and the method can be used.

This calculation is repeated for the slabs under a combination of loads from the second series, as this subset is assumed to be a better choice. The statistical properties for Test

Predictionare now: µ = 1,402 as average;

λ = 0,335; ε = 0,078.

A set of 5.105 simulations then results in a probability of failure Pf = 0 and thus a

reliability index β > 4,6.

Next, the results from the second series of experiments are also used to verify the formula for the repair level. The contribution of the concentrated load to the shear stress at the support is expressed as:

conc conc eff l V b d

τ

ξ

= with 1,884 0, 52 sup 0, 48 225 ck l f b

ξ

= −  +   _{ }  (43) with

Vconc the shear force at the support as a result of the concentrated load only;

beff the effective width resulting from the French load spreading method

with a lower bound of 4d;

dl the effective depth;

lsup the support length (full width or length of bearings).

The contribution of the distributed loads and the line load to the shear stress at the support is expressed as given in Eq. (42).

For the considered case, the value of ξ equals 1,74. The comparison between the total shear stress at the support (τconc + τline) and the predicted shear stress at the support

(τ_EC2 =0,15 100

(

ρ_lf_cm

)

1/3) results in the following statistical properties for Test Prediction: µ = 1,351 as average;

σ = 0,232 as standard deviation; λ = 0,284;

ε = 0,196.

A set of 5.105 simulations with the assumptions of the described approach results in a probability of failure Pf = 2,078.10-3 and thus a reliability index β = 2,8661. The

resulting reliability index is about the same as the required β = 2,88, and the method can be used. This calculation is also repeated for the slabs under a combination of loads from the second series, as this subset is assumed to be a better choice. The statistical properties for Test

Predictionare now: µ = 1,445 as average;

λ = 0,365; ε = 0,079.

A set of 5.105 simulations then results in a probability of failure Pf = 0 and thus a

reliability index β > 4,6. This is the result of the much smaller standard deviation.

5.5 Extension for a

v

/d

l

> 2,5

The results from the slab database are used to verify if an extension of this method is possible for concentrated loads with av/dl > 2,5. Three filters are applied onto the slab

database:

- results for av/dl > 2,5

- results for C20/25: the bounds for fc,mean are determined to be 20MPa – 36MPa

- beff < b to ensure that transverse load redistribution can be activated.

The variability of the width b and the effective depth d are not taken into account in this case, as all studied experiments from the slab database have a different layout. In total only 13 experiments satisfy the filter criteria. The results of the variability for the material and the comparison between the experimental results and the calculated value are summarized in Table 5.4, when using the factor for testing at the repair level.

Table 5.4: Statistical properties of material and Test

Prediction fc,mean fc,k Test Prediction, ξ Test Prediction µ 27,0 19,0 1,109 2,006 σ 6,2 6,2 0,647 1,195 λ 3,270 2,887 -0,050 0,537 ε 0,246 0,371 0,570 0,581

The results for the Monte Carlo simulation, taking into account the enhancement factor ξ, result in Pf = 0,3039 and β = 0,5132. When the enhancement factor is not

taken into account, Pf = 0,069 and β = 1,4830. This high failure probability is the

result of the large standard deviation on the results, which indicates that the subset resulting from filtering on 3 criteria from the database is not suitable for this type of calculations. When the results from this subset are plotted in a histogram, it becomes clear that the data from this subset do not form a distribution which can be studied. In fact, an even smaller subset should be extracted, but then the number of experiments

becomes insignificant to carry out a Monte Carlo simulation. Therefore, the enhancement factor cannot be used for slabs under concentrated loads with av > 2,5dl .

Fig. 5.1: Histogram of subset of data used to verify if the code extension proposal can be used for av > 2,5dl

6. Discussion and Assumptions

• The initial number of trials in the simulations was 104, but this number was increased when results needed to be studied in more detail. Simulations with 105trials were run, which was still within the bounds of the computing power of the used machine.

• The assumed distributions influence the results as the behavior in the left tail is dominant for the failure probability. A lognormal distribution is used as it is shown that this gives a conservative approach to the left tail. A beta distribution also closely parallels the real distribution, but for the beta distribution the left tail is unconservative.

• The following assumption has been used: f_ck = f_cmean−8MPa, as also used in EN 1992-1-1:2005.

• The distribution for test

prediction is determined per set of experiments that are studied.

• The required reliability index has now been chosen as β = 3,6. This reliability index is required for existing constructions built before 2012 to satisfy the “repair” level requirements.

7. Conclusions

This report summarizes the steps which have been taken to probabilistically determine the enhancement factor for slabs under a concentrated load close to the support failing in one-way shear. Several options were initially selected to try out. The determination with a Monte Carlo simulation allows the combination of different distributions, and is thus considered the most powerful solution. The resulting expression for the shear capacity of slabs under concentrated loads close to the support is:

(

)

1/3 sup , , , 100 1,884 0,52 0, 48 225 ck Rd c prop Rd c l ck eff l l f V C k f b d b

ρ

   =  −  +   _{ } 

So far, the results show that the slab factor is larger than initially determined in the analysis report as 1,25.

However, the results cannot be extended to the case of slabs under a concentrated load which is at av > 2,5d as the subset from the database has a too large standard

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