Effect of point loads on reinforced concrete slabs without ... · PDF fileconcrete slabs without transverse reinforcement 1 Effect of point loads on reinforced concrete slabs without

Extended Abstract R. Terras – Effect of point loads on reinforced concrete slabs without transverse reinforcement

1

Effect of point loads on reinforced concrete slabs without transverse reinforcement R. Terras The study of shear transfer in reinforced concrete slabs without transverse reinforcement has begun since the development of this material. According to how the internal shear forces are distributed inside the slab, this phenomenon is categorized into shear and punching shear. In a simple way, when the paths of principal shear forces in a reinforced concrete slab are parallel, the shear transfer is simply referred as shear. When these forces develop radially around a support, then shear transfer is referred as punching shear. On the present work, several analysis on reinforced concrete slabs without transverse reinforcement subjected to point loads are undertaken, with the goal of comparing the application of different methods of estimating the shear/punching shear strength. Shear transfer mechanisms The ASCE-ACI 426 committee, has identified, in 1973 [1], four shear transfer mechanisms: The first mechanism is the development of shear stresses in the uncracked concrete element. This is a relatively unimportant mechanism given the low concrete tensile stress. The second mechanism is by the commonly referred aggregate interlock, or crack friction. This mechanism occurs as a consequence of the crack roughness, which opposes shear stresses. This shear transfer mechanism is also referred in [2], where Muttoni and Schwartz demonstrate the possible shear transfer mechanisms using strut and tie models. This article would later constitute the basis of the CSCT.

Figure 1 – Shear transfer mechanisms according to [2]. a) cantilever action b) aggregate interlock; c) dowel action. (Adapted from [3])

The third mechanism is by dowel action of the longitudinal reinforcement, i.e., when this type of reinforcement also transmits forces in the

transverse direction. The importance of this shear transfer mechanism increases with the amount of longitudinal reinforcement, particularly when the concrete elements present more than one layer of longitudinal reinforcement. The fourth mechanism is the direct support, which occurs when the forces are directly transmitted to the supports. This mechanism is related to the shear span to depth ratio 𝑎/𝑑, i.e., its importance increases for lower 𝑎/𝑑 values. In 1964, following his research on reinforced concrete beams, Kani presented a diagram concerning the resistance of the reinforced concrete beams to the span to depth ratio 𝑎/𝑑 where he has identified that the minimum resistance would occur at 𝑎/𝑑 values of 2.5. In his research, Kani admitted the existence of a shear transfer mechanism which turns the reinforced concrete beam into a comb-like structure, where the comb teeth are taken as cantilevers supported on the upper compression zone and loaded horizontally by the longitudinal reinforcement. Further research has been made in the following years considering this shear transfer mechanism, including [2]. In 1988, the ASCE-ACI 445 committee has also suggested that residual tensile stresses also transfer shear stresses, particularly in inclined cracks, less than 0.05 mm wide. Revision of the current representative models Simplified Modified Compression Field Theory [4] The development of the compression field theory, CFT, was a significant step toward a rational theory for shear. Unlike the traditional models, the theory uses the strain conditions in the web to determine the inclination 𝜃 of the diagonal compressive stresses. The found relationship was that tan! 𝜃 = (𝜀! + 𝜀!)/(𝜀! + 𝜀!) . Because 𝜀! is usually much smaller than 𝜀!, the angle 𝜃 can be considerably less than 45 degrees, which increases the predicted shear strength of the web.

a) b) c)


2

Figure 2– Average strains on a cracked concrete element (adapted from [5])

To study the relation between the diagonal compression stress, 𝑓! , and the diagonal compression strain, 𝜀! Vecchio and Collins [5] tested 30 reinforced concrete elements in an innovative testing machine. They found that 𝑓! is a function of 𝜀! as well as 𝜀!. They also found that even after extensive diagonal cracking, tensile stresses still existed in the concrete between the cracks. When the CFT were modified to account for the average principal tensile stresses in the cracked concrete, 𝑓! , the equilibrium, geometric, and constitutive relationships of the MCFT were obtained. Since solving the 15 equations of the MCFT could be a very tedious process, a simplified version was developed. According to Bentz et al., the shear strength of a concrete element can be expressed as: 𝑣 = 𝑣! + 𝑣! = 𝛽 𝑓!! + 𝜌!𝑓! 𝑐𝑜𝑡 𝜃 1

Where the first term of the expression concerns to the strength given by the concrete, while the second concerns the strength given by the transverse reinforcement. The variable β is calculated by: 𝛽 = !.!

!!!"##!!∙ !"##!"""!!!"

2

And the angle of inclination the cracks, θ, is given by the simplified expression [4]: 𝜃 = 29º + 7000𝜀! 0.88 + !!"

!"##≤ 75º 3

Model Code 2010 [6] The recently published Model Code 2010 uses a formulation based on the MCFT to calculate the shear strength of reinforced concrete elements. The shear strength of a concrete is given by the following expression:

𝑉!" = 𝑉!",! + 𝑉!",!

= 𝑘! ∙ 𝑓! ∙ 𝑧 ∙ 𝑏!,!"+ 𝜌!𝑓! 𝑐𝑜𝑡 𝜃

4

Where 𝑘! is, for the level of aproximation III: 𝑘! =

0.41 + 1500 ∙ 𝜀!

∙1300

1000 + 𝑘!" ∙ 𝑧 5

The authors also suggest the following expression in order to calculate the longitudinal strain: 𝜀! =

12𝐸!𝐴!

𝑀!"

𝑧+ 𝑉!" + 𝑁!"

12∓∆𝑒𝑧

6

Critical Shear Crack Theory [7] Aurelio Muttoni, originally developed the critical shear crack theory in 1985, at the Technical University of Zurich. This theory was based on the hypothesis that the shear strength of concrete slabs without transverse reinforcement depends on the width and roughness of a crack that propagates through an inclined strut that carries shear [8] [9]. According to [8], taking the slab’s rotation nearby the column as the degree of freedom, the shear strength is reduced with the increase of the rotation. A detailed analysis of this phenomenon [10] shows that the contribution of the concrete’s tensile strength is essential for small rotation, when for moderate to high rotation, the aggregate interlocking of the aggregates carries most of the shear forces. The CSCT employs a failure criterion, in order to calculate the shear strength. To be able to use of a failure criterion, the engineer must evaluate the load-rotation curve. On the present work, the load-rotation curve was assessed using the finite element method. According to the CSCT, the shear strength on reinforced concrete slabs without transverse reinforcement can be determined based on the crack width at a critical region, located at 0.5 ∙ 𝑑 from the face of the support. With the assumption that the crack width is proportional to the longitudinal strain and the effective depth, the shear strength is given by: 𝜏 =

𝑉!𝑏 ∙ 𝑑

=𝜏!

0.9 + 2.3 ∙ 𝜀 ∙ 𝑑 ∙ 𝑘!" 7

With the longitudinal strain given by: 𝜀 =

𝑚!

𝑑 ∙ 𝜌 ∙ 𝐸! ∙ (𝑑 − 𝑥/3)∙0.6 ∙ 𝑑 − 𝑥𝑑 − 𝑥

8

Where 𝑥 is the height of the compression zone, given by:

Ʌɂ2

ɂx

ɂz

ɂ1

z

x


3

𝑥 = 𝑑 ∙ 𝜌 ∙𝐸!𝐸!∙ 1 +

2 ∙ 𝐸!𝜌 ∙ 𝐸!

− 1 9

The plot of the shear failure criterion, along with the experimental results it is based on, is displayed on Figure 3. For reinforced concrete slabs without transverse reinforcement, the CSCT assumes the punching strength as a function of the deformations on the critical region. According to [2], the punching shear strength is related to the opening of the critical shear crack, and therefore, the punching shear strength, which can be expressed as: 𝜏! =

𝑉!𝑢 ∙ 𝑑

=𝜏!

0.4 + 0.125 ∙ 𝜃 ∙ 𝑑 ∙ 𝑘!" 10

Where 𝑢 is the control perimeter, located at a distance of 0.5 ∙ 𝑑 from the face of the load/support.

Figure 3 – Comparison between the failure criterion and several experimental results of beams subjected to point loads (adapted from [7])

The plot of the punching shear failure criterion, along with the experimental results it is based on, is displayed on Figure 4.

Figure 4 – Comparison between the failure criterion and experimental results of slabs subjected to punching shear (adapted from [7])

ACI 318-11 [11] For concrete slabs without transverse reinforcement, the ACI 318-11 shear strength calculation method consists of checking the shear stress at the critical region, located at a distance of 0.5 ∙ 𝑑 from the face of the load/support. The general expression is: 𝑉! = 0.17 ∙ 𝑓!! ∙ 𝑏!,!"# ∙ 𝑑 11

The determination of the punching shear strength of a reinforced concrete slab without transverse reinforcement is similar to previous method. The critical region consists of a control perimeter, located at a distance of 0.5 ∙ 𝑑 from face of the load/support. The general expression is: 𝑉! =

13∙ 𝑏!,!"# ∙ 𝑑 ∙ 𝑓!! 12

Eurocode 2 (EN 1992-1-1) [12] The shear strength calculation method present on the Eurocode 2 is similar to the equation presented on the Model Code 90 [13]. Unlike the formulation of the ACI 318-11, the following expression takes into account the size effect: 𝑉! = 0.18 ∙ 𝑏!,!"! ∙ 𝑑 ∙ 𝑘 ∙ (100 ∙ 𝜌 ∙ 𝑓!)!/!

≥ 𝑣!"# ∙ 𝑏!,!"! ∙ 𝑑 13

The punching shear strength calculation method is based on the previous formulation, with some adjustments to the control perimeter in order to take into account the experimental tests that it is based on. The proposed expression is: 𝑉! = 0.18 ∙ 𝑏!,!"! ∙ 𝑑 ∙ 𝑘 ∙ (100 ∙ 𝜌 ∙ 𝑓!)!/!

≥ 𝑣!"# ∙ 𝑏!,!"! ∙ 𝑑 14

Where 𝑏! is the control perimeter, taken at a distance of 2𝑑 from the face of the load/support. Numerical Analysis The application of the different calculation methods for predicting the shear strength was compared with the results of the experimental campaigns performed at Hamburg, Lausanne, and Zurich Technical Universities. Experimental tests by Latte – TUHH The geometry and reinforcement of the tests performed by Latte, S. at the Technical University of Hamburg-Harburg are presented on Figure 5 to

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20

ɂ�� [mm]

ɒɒc

Ʌ�� [mm]0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10 11

Ʌ

ɒɒc


4

Figure 7. The properties of the materials as well as the characteristics of the loading are summarized on Table 1 to Table 4. Experimental tests by Vaz Rodrigues – EPFL The geometry and reinforcement of the tests performed by Vaz Rodrigues, R. at the Technical University of Lausanne are presented on Figure 8 to Figure 9. The properties of the materials as well as the characteristics of the loading are summarized on Table 5 to Table 8.

Experimental tests by Jäger – ETHZ The geometry and reinforcement of the tests performed by Jäger, T. at the Swiss Federal Institute of Technology Zurich are presented on Figure 10 to Figure 11. The properties of the materials as well as the characteristics of the loading are summarized on Table 9 to Table 12.

Figure 5 – Geometry and loading layout, VK1 (bottom) & VK2 (top) – [mm]

Figure 6 – Geometry and loading layout, VK3 (bottom) & VK4 (top) – [mm]

VK1 VK2 VK3 VK4

Figure 7 – Schematic top & bottom reinforcement layout on the VK1, VK2, VK3 and VK4 slab – [mm]


5

Table 1 – Concrete cover and reinforcement of the tests carried out by Latte

Test

Concrete cover 𝑐!

[mm]

Top longitudinal reinforcement Bottom longitudinal

reinforcement (Cantilever) Bottom longitudinal

reinforcement (Deck) 𝜙

[mm] 𝑎!"

[cm2/m] 𝜌! [%]

𝜙 [mm]

𝑎!" [cm2/m]

𝜌! [%]

𝜙 [mm]

𝑎!" [cm2/m]

𝜌! [%]

VK1 45 16@100 20.1 0.81 12@100 11.3 0.46 16@100 20.1 0.81 VK2 25 16@80 25.1 1.16 12@100 11.3 0.52 12@100 11.3 0.52 VK3 25 16@80 25.1 1.16 12@100 11.3 0.52 12@100 11.3 0.52 VK4 25 16@100 20.1 1.20 12@100 11.3 0.68 12@100 11.3 0.68 Table 2 – Concrete properties of the tests carried out by Latte

Test Age at test [days]

𝑓! [MPa]

𝑓!" [MPa]

𝐸! [GPa]

VK1 46 35.0 2.85 29.17 VK2 45 46.0 3.42 33.99 VK3 44 46.5 3.34 33.34 VK4 36 42.5 3.23 32.49

Table 3 – Reinforcing steel properties of the tests carried out by Latte

𝜙 [mm]

𝑓! [MPa]

𝑓! [MPa]

𝜀! [%]

𝑓!𝑓!

- 𝐸!

[GPa] 12 550 607 5.09 1.11 195 16 554 646 11.61 1.17 195

Table 4 – Loading properties of the tests carried out by Latte

Test 𝑑

[m]

Initial loading (End of the cantilever) Point loading

Failure load [kN]

𝑓! [kN/m]

𝑒 [m]

𝑎 [m]

𝑎/𝑑 -

VK1 0.247 32.1 1.5 0.71 2.87 690 VK2 0.217 22.5 1.5 0.71 3.27 678 VK3 0.217 22.5 1.5 0.71 3.27 677 VK4 0.167 - - 0.71 4.25 487


6

Figure 8 – Geometry and loading layout, DR1c & DR2c slabs – [mm]

DR1c

DR2c

Figure 9 – Schematic top & bottom reinforcement layout on the DR1c & DR2c slabs – [mm]

Table 5 – Concrete cover and reinforcement of the tests carried out by Vaz Rodrigues

Test


[mm]

Top longitudinal reinforcement Bottom longitudinal reinforcement

𝜙 [mm]

𝑎!" [cm2/m]

𝜌! [%]

𝜙 [mm]

𝑎!" [cm2/m]

𝜌! [%]

DR1c 30 16@75 25.13 0.78 12@150 7.54 0.22 DR2c 30 14@75 20.53 0.60 12@150 7.54 0.22


7

Table 6 – Concrete properties of the tests carried out by Vaz Rodrigues

Test Age at test [days]

𝑓! [MPa]

𝑓!" [MPa]

𝐸! [GPa]

DR1c 114 40.8 3.1 36.2 DR2c 112 42.4 3.1 37.5 Table 7 – Reinforcing steel properties of the tests carried out by Vaz Rodrigues

𝜙 [mm]

𝑓! [MPa]

𝑓! [MPa]

𝜀! [%]

𝑓!𝑓!

- 12 DR1c 541 629 9.05 1.16 12 DR2c 469* 580 5.19 1.24

14 505 591 11.1 1.17 16 499 600 10.7 1.20

* Offset yield stress at 0.2 % strain. Table 8 – Loading properties of the tests carried out by Vaz Rodrigues

Test 𝑑 [m]

Point loading Failure load

[kN] 𝑎

[m] 𝑎/𝑑

- DR1c 0.342 1.30 3.80 910.0 DR2c 0.343 1.30 3.79 719.4


8

Figure 10 – Geometry and loading layout of the B3V1 and B5V1 slabs – [mm]

B3V1 B5V1

Figure 11 – Schematic top reinforcement layout on the B3V1 and B5V1 slabs – [mm]

Table 9 – Concrete cover and reinforcement of the tests carried out by Jäger

Test


[mm]

Top longitudinal reinforcement Bottom longitudinal reinforcement

𝜙 [mm]

𝑎!" [cm2/m]

𝜌! [%]

𝜙 [mm]

𝑎!" [cm2/m]

𝜌! [%]

B3V1 30 30@200 + 30@200 70.69 1.75 - - - B5V1 30 30@150 47.12 1.06 - - - Table 10 – Concrete properties of the tests carried out by Jäger

Test Age at tet [days]

𝑓! [MPa]

𝑓!" [MPa]

𝐸! [GPa]

B3V1 276 53.7 3.9 36.0 B5V1 168 51.8 4.14 33.9 Table 11 – Reinforcing steel properties of the tests carried out by Jäger

𝜙 [mm]

𝑓! [MPa]

𝑓! [MPa]

𝜀! [%]

𝑓!𝑓!

- 𝐸!

[GPa] 30 533.6 613.7 12.47 1.15 209.8


9

Table 12 – Loading properties of the tests carried out by Jäger

Test 𝑑 [m]

Loading Failure load

[kN] 𝑎

[m] 𝑎/𝑑

- B3V1 0.405 1.6 3.95 1211 B5V1 0.435 1.6 3.68 1099 Variable depth concrete elements The majority of recent codes of practices make reference for a method of designing variable depth concrete elements. The general proposed expression is the following: 𝑉!" = 𝑉!"! − 𝑉!!" − 𝑉!" − 𝑉!" ≤ 𝑉!" 15

Where 𝑉!" is the design shear force, 𝑉!"! is the shear forces due to external loading, 𝑉!!" the inclined compression component, 𝑉!" the tensioned reinforcement component and 𝑉!" the pre-stress force component. For the present work, the last two components are not relevant. Therefore the design shear strength can be taken as: 𝑉!" ≥ 𝑉!"! − 𝑉!!" = 𝑉!"! −

𝑀!"

𝑧∙ 𝑡𝑎𝑛 𝛼 16

Stress fields For a better understanding of transmission of the shear stresses on the interior of the concrete slabs, the concept of stress fields was described and put in practice with the help of a routine developed by Vaz Rodrigues [3]. The stress field can be obtained from the expression 17, from the bending moment fields of the slab, respecting the boundary conditions as well as the equilibrium with the external forces. 𝑣! =

!"!

!"+ !"!"

!" ; 𝑣! =

!"!

!"+ !"!"

!" 17

The routine for the representation of the stress field was implemented on this work on the MATLAB software, by analysing the 𝑣! and 𝑣! at each node of the finite element analysis output. The representation of the stress fields can be helpful to assess which failure mode is to expect for each slab. The results are represented on Figure 12 to Figure 15.

Figure 12 – Stress field on the VK1 slab Figure 13 – Stress field on the VK3 slab

VK1

Shea

r

Punc

hing

She

ar

VK3Sh

ear

Punc

hing

She

ar


10

Figure 14 – Stress field on the DR1c slab Figure 15 – Stress field ond the B3V1 slab

From the analysis of the previous Figures, it can be verified that the principal shear forces develop radially around the point loading whereas at the support, such forces develop parallel to each other, which is consistent with shear behaviour. For this particular reason, the slabs were checked for punching shear on the region around the load and for shear on the region of the support. Non-linear model The model used to calculate the moment curvature diagram of the analysed slabs was the quadrilinear model, as defined in [8]. As it can be seen on Figure 16 this model takes into account the tension stiffening effect.

Figure 16 – Moment-Curvature diagram of a generic reinforced concrete slab

Results The obtained results are summarized on Figure 17 and on Table 13 and

Table 14 when considering shear failure. For the analysis considering punching shear failure, the results are summarized on Figure 18 and on Table 15 and Table 16

Figure 17 – Summary of the obtained results considering the failure by shear and taking into account the 𝑉!!" component.

DR1c

Punc

hing

She

ar

Shea

r

B3V1

Shear

EI0EI1

mcr

mR

m

ѯcr ѯ1 ѯy ѯ

Bilinear

Quadrilinear

ѯTS

VexpV’mod

TestsVK1 VK3 VK2 VK4 B3V1 B5V1 DR1c DR2c

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

MC 2010CSCTEC 2$&,��ï��


11

Figure 18 – Summary of the obtained results considering failure by punching shear in the region around the point load.

Table 13 – Analysis results considering shear failure

Test Failure load [kN]

MC 2010 CSCT EN 1992-1-1 ACI 318-11 𝑉!"# [kN]

𝑉!"#! [kN]

𝑉!"# [kN]

𝑉!"#! [kN]

𝑉!"# [kN]

𝑉!"#! [kN]

𝑉!"# [kN]

𝑉!"#! [kN]

VK1 690 426.79 487.13 345.64 394.39 319.46 381.67 307.62 367.52 VK2 678 473.12 408.59 380.40 330.38 VK3 677 467.79 538.81 385.61 456.79 363.02 453.16 315.86 386.65 VK4 487 362.09 341.21 318.11 263.78 DR1c 910 761.03 897.94 750.52 879.99 776.60 992.35 836.37 1068.71 DR2c 719.4 694.63 819.62 679.33 793.24 716.63 916.94 850.42 1088.13 B3V1 1211 910.66 957.06 960.91 851.81 B5V1 1099 816.25 825.36 842.07 907.10 𝑉!"#! is the shear strength considering the 𝑉!!" component.

Table 14 – Correlation between the experimental results and the analysis results considering shear failure

Test MC 2010 CSCT EN 1992-1-1 ACI 318-11

𝑉!"#𝑉!"#

𝑉!"#𝑉!"#!

𝑉!"#𝑉!"#

𝑉!"#𝑉!"#!

𝑉!"#𝑉!"#

𝑉!"#𝑉!"#!

𝑉!"#𝑉!"#

𝑉!"#𝑉!"#!

VK1 1.62 1.42 2.00 1.75 2.16 1.81 2.24 1.88 VK2 1.43 1.66 1.78 2.05 VK3 1.45 1.26 1.76 1.48 1.86 1.49 2.14 1.75 VK4 1.34 1.43 1.53 1.85 DR1c 1.20 1.01 1.21 1.03 1.17 0.92 1.09 0.85 DR2c 1.04 0.88 1.06 0.91 1.00 0.78 0.85 0.66 B3V1 1.33 1.27 1.26 1.42 B5V1 1.35 1.33 1.31 1.21 𝑉!"#! is the shear strength considering the 𝑉!!" component.

VK1 VK3 VK2 VK4 DR1c DR2c0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

MC 2010CSCTEC 2$&,��ï��

VexpVmod

Tests


12

Table 15 – Analysis results considering punching shear failure

Test Failure load

[kN] MC 2010 CSCT EN 1992-1-1 ACI 318-11 𝑉!"# [kN]

𝑉!"# [kN]

𝑉!"# [kN]

𝑉!"# [kN]

VK1 690 438.72 523.86 231.69 406.91 VK2 678 564.82 650.09 339.78 562.26 VK3 677 463.44 528.43 232.50 400.90 VK4 487 412.08 469.33 286.72 417.01 DR1c 910 766.11 788.75 533.04 661.54 DR2c 719.4 717.01 729.06 540.05 677.40

Table 16 – Correlation between the experimental results and the analysis results considering punching shear failure

Test MC 2010 CSCT EN 1992-1-1 ACI 318-11 𝑉!"#𝑉!"#

𝑉!"#𝑉!"#

𝑉!"#𝑉!"#

𝑉!"#𝑉!"#

VK1 1.57 1.32 2.98 1.70 VK2 1.20 1.04 2.00 1.21 VK3 1.46 1.28 2.91 1.69 VK4 1.18 1.04 1.70 1.17 DR1c 1.19 1.15 1.71 1.38 DR2c 1.00 0.99 1.33 1.06 Conclusions This work has assessed and compared different methods for calculating the shear strength of reinforced concrete slabs without transverse reinforcement subjected to point loads. The main conclusions were:

1. When considering punching shear failure, the codes of practice EN 1992-1-1 [12] and ACI 318-11 [11] have shown relatively conservative results;

2. In average, the predicted failure loads when considering this type of failure, were 2.10 to 1.37 times higher than the experimental failure load when using these codes of practice;

3. For the results provided by the Model Code 2010 and the CSCT, the results were in average, 1.27 to 1.14 times higher than the experimental failure load;

4. Regarding the comparison of all the used proposals for predicting the punching shear failure load, it was verified that the CSCT provided the closest values to the experimental failure loads;

5. When considering shear failure, the discrepancy of the predicted shear failure loads between calculation methods is minor than when considering punching shear failure;

6. Taking into account the 𝑉!!" component provided, in some cases, less safe results, which suggests that it’s use shall be proven to be adequate for the design of elements similar to the analysed ones.

7. For a safe design, both shear failure and punching shear failure modes shall be checked;

8. When considering shear failure, the estimate of the failure load has provided (in average) higher shear strengths for elements at which the shear span to depth ratio of the slab !

! was smaller. Such

observation may indicate that the calculation methods should consider the effect of direct support (where shear forces are directly transmitted to the support) for values of !

! higher than 2.0.

Nevertheless, this hypothesis requires the analysis of more experimental results to support it, reason why future research on this matter is recommended.


13

Figure 19 – Comparison between the !

! values and !!"#

!!"#!

considering shear failure, for the CSCT and the MC2010

Notation 𝐸! Modulus of elasticity of the concrete 𝐸! Modulus of elasticity of the reinforcement 𝑎!" Longitudinal reinforcement area 𝑏! Control section for shear 𝑏! Control perimeter for punching shear 𝑐! Concrete cover 𝑑 Effective depth 𝑓!, 𝑓!! Compressive strength of the concrete 𝑓!" Tensile strength of the concrete 𝑓! Ultimate strength of the reinforcement 𝑓! Yield strength of the reinforcement 𝑓! Principal tensile stress 𝑓! Compressive diagonal stress 𝑘 Size effect factor k!" Coefficient for aggregate size 𝑚!" Cracking moment per unit width 𝑚! Nominal moment capacity per unit width s!" Cracking spacing 𝑢 Control perimeter 𝑥 Depth of the neutral axis 𝑧 Internal lever arm 𝛼 Inclination of the element’s face 𝜀, 𝜀! Longitudinal strain 𝜀! Ultimate tensile strain of the reinforcement 𝜀! Transverse tensile strain 𝜀! Principal tensile strain 𝜀! Diagonal compressive strain 𝜌!, 𝜌! Transverse reinforcement ratio 𝜌, 𝜌! Longitudinal reinforcement ratio 𝜃 Inclination of the diagonal compressive stresses 𝜙 Diameter of the reinforcement 𝜒!" Curvature at cracking

𝜒!" Decrease in curvature due to tension stiffening 𝜒! Yielding curvature 𝜒! Curvature in tangential direction References 1 ASCE-ACI Committee 426. The shear strenght

of reinforced concrete members. Journal of Structural Engineering. 1973;99(6):1091-1187.

2 Muttoni A, Schwartz J. Behaviour of beams and punching in slabs without shear reinforcement. IABSE Colloquium. 1991;62:703-708.

3 Vaz Rodrigues R. Shear strength of reinforced concrete bridge deck slabs. Thesis. Lausanne: EPFL; 2007.

4 Evan C. Bentz FJVMPC. Simplified Modified Compression Field Theory for Calculating Shear Strength of Reinforced Concrete Elements. ACI Structural Journal. 2006;103(4):614-624.

5 Vecchio FJ, Collins MP. The modified compression field theory for reinforced concrete elements subjected to shear. ACI Structural Jornal. 1986;83(22):219-231.

6 (fib) fidb. fib bulletin 66, Model Code 2010 - Final draft, Volume 2. 2010.

7 Muttoni A. Schubfestigkeit und durchstanzen von platten ohne querkraftbewerung. Beton und Stahlbetonbau. 2003;98(2):74-84.

8 Muttoni A. Punching shear strength of reinforced concrete slabs without transverse reinforcement. ACI Structural Journal. 2008;105(4):440-450.

9 Muttoni A, Fernández MR. Shear strength of members without transverse reinforcement as a function of the critical shear crack width. ACI Structural Journal. 2008;105(2):163-172.

10 Muttoni A, Fernandéz MR. MC2010: The Critical Shear Crack Theory as a mechanical model for punching shear design and its application to code provisions. fib Bulletin 57: Shear and punching shear in RC and FRC elements. 2010 31-60.

11 ACI Commitee 318-11. Building code requirements for structural concrete. Farmington Hills: ACI American Concrete Institute; 2005.

12 CEN. Eurocódigo 2 - Projecto de estruturas de betão. Parte 1-1: Regras gerais e regras para edifícios. Bruxelas 2004.

13 Comité Euro-International du Béton. Model Code 90. 1993.

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