Discussion and recommendations

The performed research is based on the extended beam model . Consequently, this model defines the outcome of this research. That is why, the three assumptions for the design of the extended beam model are reviewed.

1. In section 2.1.4 is assumed that the stress-strain relation representing the tension stiffening effect is applicable for a simply supported beam subjected to two point loads. The effect of tensile stress-strain relation on the shear capacity, however, is minimal. Therefore, the shear capacity of other load types is most likely also possible. This only concerns determined structures.

2. The definition of the effective shear height is based on an research of Pruijssers (1986), which investigated reinforced concrete beams, thus without steel fibre reinforcement.

Section 4.1.2 illustrated that the residual tensile strength influences the effective shear height. However, whether steel fibre reinforcement influences the definition of the effective shear height is unknown.

3. The shear stress flow is assumed parabolic. As a result, the contribution of the tensile zone is overestimated, which is illustrated by figure 2.15. The magnitude of this overestimation is unknown. However, the overestimation is expected to be limited because the extended beam model provides an quite accurate prediction of the shear capacity.

Although the main hypothesis only concerns the contribution of the steel fibre reinforcement, three other parameters are also investigated. Paragraph 4.1 explains how and why the residual tensile strength, the reinforcement ratio, shear span to effective depth ratio, and the concrete strength effect the shear capacity. Not only the residual tensile strength but also the other three parameters influences the effective shear height. Since the contribution of steel fibre reinforcement to the shear capacity strongly depends on the effective shear height, this contribution is effected by the

reinforcement ratio, the shear span to depth ratio and the concrete strength. This shows an complex interference between parameters, which can more thoroughly be investigated.

This research considers a beam which is simply supported and has normal strength concrete, a residual tensile strength lower than the critical strength, and a shear span to depth ratio larger than 2.5. Whether the conclusions are also valid for other boundary condition should be investigated.

The extended beam model can be used to analyse beams or slabs of a determined structure. Other material properties can be inserted in the extended beam model. However, the model currently calculates several material and structural properties from a few inserted properties. For instance, the tensile concrete strength is calculated from the compressive concrete strength. These internal calculation possibly have to be adjusted. Inserting a residual tensile strength higher than the critical strength is possible but this significantly increases the calculation time of the tension stiffening module. Furthermore, the tension stiffening effect disappears in this case.

Paragraph 3.3 explained that the beam model is only suitable for a shear span to effective depth ratio larger than 2.5. Beams with a smaller ratio develop a multi-axial stress condition, which is not incorporated in the extended beam model. When an extra normal stress, in y-direction, would be defined, it could be possible to implement the multi-axial stress condition. However, this could be very complex and other models might be more suitable to define complex stress conditions.

References

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Bruggeling, A.S.G. and Bruijn, W.A. de, 1986. Theorie en praktijk van het gewapend beton.

’s Hertogenbosch: Vereniging Nederlandse Cementindustrie.

European standard with Dutch national annex, 2011. NEN-EN 1992-1-1, eurocode 2: design of

concrete structures, Part 1: general rules and rules for buildings. Nederlands normalisatie-instituut [online] Available through: BRIS bv <http://www.briswarenhuis.nl> [August 2013].

Gribniak, V., Kaklauskas, G., Kwan, A.K.H., Bacinskas, D. and Ulbinas, D., 2012. Deriving stress-strain relationships for steel fibre concrete in tension from tests of beams with ordinary reinforcement.

Engineering structures, 42, pp. 387-395.

Hamelink, S.A., 1989. Ontwikkeling van een trekstaaf-analogie (toepassing op een buigligger van gewapend beton). Master. Eindhoven University of Technology.

Hordijk, D., 1991. Local approach to fatigue of concrete. PhD. Delft University of Technology.

Kani, G.N.J., 1967. How safe are our large reinforced concrete beams? ACI Journal, 64(4), pp. 128-141.

Kani, G.N.J., 1966. How Basic facts concerning shear failure. ACI Journal, 63(6), pp. 675-692.

Kooiman, A.G., 2000. Modelling steel fibre reinforced concrete for structural design. PhD. Delft University of Technology.

Kotsovos, M.D., 1987a. Consideration of triaxial stress conditions in design: a necessity. ACI Structural Journal, 84 (3), pp. 266-273.

Kotsovos, M.D., 1987b. Shear failure of reinforced concrete beams. Engineering Structures, 9 (1), pp.

32-38.

Kotsovos, M.D., 1988. Compressive force path concept: Basis for reinforced concrete ultimate limit state design. ACI Structural Journal, 85(1), pp.68-75.

Kotsovos, M.D. and Pavlovic, M.N., 1999. Ultimate limit-state design of concrete structures, a new approach. Londen: Thomas Telford Ltd.

Marinus, P.C.P., 2013. Onderzoeksproject beton, non-ferro wapening als alternatief voor staal in gewapend betonconstructies. Master. Eindhoven University of Technology

Narayanan, R. and Darwish, I.Y.S., 1987. Use of steel fibres as shear reinforcement. ACI Structural Journal, 84(3), pp. 216-227.

Paus, T.J., 2014. Steel fibre reinforced concrete in flat slaps, by using the finite difference method for design analysis. Master. Eindhoven University of Technology.

Pruijssers, A.F., 1986. Shear resistance of beams based on the effective shear depth. Delft: Stevin-reports.

Salet,T.A.M., 1991. Structural analysis of sandwich beams composed of reinforced concrete faces and a foamed concrete core. PhD. Eindhoven University of Technology.

Vandewalle, L. and Dupont, D., 2002. Dwarskrachtcapaciteit van staalvezel-betonbalken. Cement, 2002(8), pp. 92-96.

Vandewalle, L., 2003. RILEM TC 162-TDF: ‘Test and design methods for steel fibre reinforced concrete’ σ-ε-design method final recommendation. Materials and Structures, 36, pp. 560-567.

Walraven, J., (2011). Van exotisch naar volwassen product. Cement, 2011(3), pp. 4-9.

Walraven, J.C. and Reinhardt, H.W., (1981). Theory and experiments on the mechanical behaviour of cracks in plain and reinforced concrete subjected to shear loading. HERON, 26 (1A).

Appendices

A. Functioning beam model 72

A.1 Flow chart calculation moment-curvature diagram (multi-layer model) 72 A.2 Flow chart calculation deflection diagram (multi-layer model and moment-area method) 73

B. Determination post-cracking behaviour of SFRC 75

C. Tension stiffening effect 77

C.1 Bond theory 77

C.2 Reinforced concrete bar 78

C.3 Impact area 80

D. Flow charts additional modules beam model 81

D.1 Flow charts tension stiffening module 81

D.2 Flow chart shear module 83

D.3 Flow chart load module 84

D.4 Flow chart load-deflection module 85

E. Material properties and calculation settings 86

E.1 Inserted material properties 86

E.2 Applied calculation settings 88

E.3 Inserted compressive behaviour 89

E.4 Critical concrete stress 90

E.5 Influence number of layers, segments and load steps 92

F. Additional diagram verification contribution tensile zone 96

G. Combination diagrams effect parameters 100

A. Functioning beam model

A.1 Flow chart calculation moment-curvature diagram (multi-layer model)

A.2 Flow chart calculation deflection diagram (multi-layer model and moment-area method)

B. Determination post-cracking behaviour of SFRC

To determine the post-cracking behaviour of a SFRC-mixture, a three-point-bending test is proposes by the ‘International Union of Laboratories and Experts in Construction Materials, Systems and Structures’ Technical Committee 162-TDF (RILEM TC 162-TDF). Among others the RILEM TC 162-TDF conducted research on steel fibre reinforced concrete.

To execute the three-point-bending test, RILEM TC 162-TDF prescribes a small beam with a cross-section of 150mm by 150 mm and a span of 500mm (fig. B.1). The specimen is supported by two rolling pins and subjected to a point load, which transferred to the specimen by a rolling pin. In the middle of the beam a notch of 25mm is made because this guaranties a crack location in the middle.

The CMOD of this notch is measured by a device during the loading. From this test a load-CMOD diagram is obtained (fig. B.2) in which five points are indicated. The first point represents the Limit Of Proportionality (LOP or L), which is the end of the linear elastic branch. The second till fifth point represent respectively a CMOD of 0.5, 1.5, 2.5, and 3.5 mm.

As mentioned before, this force-CMOD relation can be converted to a stress-strain relation. The RILEM TC 162-TDF proposes a stress-strain relation (fig. B.3)which is obtained in two steps. First, the forces of each point are converted to stresses

;

In case of the residual tensile stresses f_{R i;}the use of the elastic section modulus is noticeable because the behaviour is not elastic after cracking.

Next, only three stresses are used: the stress at cracking f_{fct L;} , the first residual tensile stress f_R;1, and the forth residual tensile stress f_R;4. Thus, the second and third residual tensile stresses, f_R;2 and f_R;3, are omitted. f_R;1and f_R;4are multiplied with respectively the factors 0.45 and 0.37 to correct the use of the elastic section modulus instead of the plastic section modulus. Furthermore, the CMOD’s corresponding to the three stresses f_{fct L;} , f_R;1,and f_R;4are replaced by three strains.

The stress-strain diagram derived from the load-CMOD relation in accordance with RILEM TC 152-TDF (2003) is shown in figure B.3.

Figure B.1: Three-point-bending test to determine the flexural behaviour of a SFRC-mixture according to the RILEM TC 162-TDF. In red the device that measures the CMOD during loading. Measurements in mm.

Figure B.2: Load-CMOD diagram of a three-point bending test (Vandewalle, 2003)

Figure B.3: Stress-strain diagram derived from the load-CMOD diagram according to RILEM (Vandewalle, 2003)

a size-factor

C. Tension stiffening effect

The first paragraph explains the bond theory. After that, the second paragraph describes how the bond theory is used to derive a post-cracking stress-strain relation for a reinforced concrete bar subjected to a tensile load. Finally, the third paragraph clarifies the concrete area which is affected by the bond between concrete and reinforcement bar.

C.1 Bond theory

The bond between concrete and reinforcement is explained with a tension bar. Figure C.1 shows the impact of a crack on the traction between the steel rebar and surrounding concrete and the impact on the stress in the steel and concrete. At the crack, which is located at the right, the tensile force is entirely carried by the rebar (N_cr N_s). A transition length away from the crack, x=0, the bond between the concrete and rebar is entirely intact; as a result, the tensile force is carried by both the steel and the concrete (N_cr N_sN_c). Along the transition length the situation gradually changes.

This change is defined by the relative displacement of the steel and the traction-separation relation between the concrete and rebar.

Figure C-.C.1: Along the transition length from top to bottom: bond between concrete and rebar; displacement of the rebar relative to the concrete; traction between concrete and rebar; stress in the steel rebar; stress in the concrete.

Right: relation between the traction and relative displacement. (Bruggeling and Bruijn, 1986).

C.2 Reinforced concrete bar

This bond theory can be applied on a reinforced concrete bar. When the cracking moment is reached in a reinforced bar, the development of cracks can be described by two phases. During the first phase the crack pattern develops until the crack distance l_cr is twice the transition length l_t(fig. C.2).

Because the concrete stress at the end of transition length is equal to the cracking stress, secondary cracks will develop between the primary cracks (fig. C.3). As a result, the transition lengths of adjacent cracks overlap. For second generation cracks the steel stress along the beam is defines by superposition of the steel stresses of each transition length. Due to the increase of the steel stress, the concrete stresses decrease. The strains in the rebar and concrete change similar to the stresses.

However, the real crack pattern shows a erratic course due to stochastic effects. As a result, the minimum crack distance occurs when two transition lengths completely overlap (l_cr l_t) and the maximum crack distance occurs when two transition lengths do not overlap (l_cr 2l_t).

The contribution of the concrete between cracks to the bearing of the tensile load results in a stiffer reaction of the reinforced bar compared to an uncovered rebar. This contribution of the concrete can be included in definition of the tensile behaviour of concrete through the addition of a post-cracking branch to the tensile stress-strain relation of concrete. Bruggeling and Bruijn (1986) propose a bilinear post-cracking branch (fig. C.4).

Point 1 and 2 are respectively defined by the average concrete stress and steel strain for the maximum cracks distance and the minimum crack distance. Point B is based on the logical assumption that the average crack distance is 1.5 times the transition length:

;min ;max

Considering the above definition of the average crack distance, the average steel strain along the bar is

And the average concrete stress along the bar is

1 2

For point C the logical assumption is made that concrete does not further contribute to the stiffness when the reinforcement yields.

Figure C.2: First generation cracks in a tension bar (top); concrete stress along the length (mid); steel stress along the length (bottom); in red the averages stress in the bar.

Figure C.3: Second generation cracks in a tension bar (top); concrete stress along the length (mid); steel stress along the length (bottom); in light grey the stresses related to each transition length; the black curve is the actual stress defined by superposition; in red the averages stress in the bar.

Figure C.4: Tensile stress-strain relation of concrete including tension stiffening (red). A: cracking point; B: decrease of concrete contribution after cracking; C: ending of tension stiffening at yielding; 1: average concrete stress and steel strain for first generation cracks; 2: average concrete stress and steel strain for second generation cracks.

C.3 Impact area

Finally, the impact area, which is the concrete area that contributes to the bond with the rebar, has to be known to calculate the transition length. This area is equal to the concrete cross-section in case of an axial tensile load; however, in case of bending not the whole tensile zone contributes. The impact area in this case is defined with the effective concrete height h_{c eff,} and the width. The horizontal equilibrium of the compressive force (eq. C-4) and the tensile force (eq. C-5) is used to formulate the effective height (eq. C-6).

, ,

D. Flow charts additional modules beam model

D.1 Flow charts tension stiffening module

First generation cracks

Second generation cracks

D.2 Flow chart shear module

D.3 Flow chart load module

D.4 Flow chart load-deflection module

Load-deflection diagram

E. Material properties and calculation settings

The first paragraph describes which material properties are inserted in the beam model. All calculations are performed with these material properties unless states otherwise. The applied calculations settings are specified in paragraph E.2. The underlying research of the compressive behaviour and the critical concrete strength is presented in respectively the third and fourth paragraph. Finally, paragraph E.5 contains the research regarding the influence of the number of layers, load steps and segments.

E.1 Inserted material properties Steel fibre reinforced concrete

The tensile behaviour of a SFR-RC beam is presented in section 3.2.1. The compressive behaviour and linear-elastic tensile behaviour is based on Eurocode 2 (European standard with Dutch national annex, 2011). This research is aimed at normal strength concrete and the real material behaviour;

therefore, safety factors are omitted and mean values are applied. The European Standard formulates the mean compressive and tensile strength and Young’s modulus as

(MPa)

Figure E.1 shows the applied stress-strain relation for steel SFRC. The compressive behaviour is described by a bilinear stress-strain relation. This relation is a non-conservative an realistic assumption based on the parabolic-linear relation of the European standard (2011). Details of this assumption can be found in paragraph E.3. A parabolic behaviour would be more realistic than a bilinear behaviour; however, this results in higher calculation time because an equilibrium is harder to gain (paragraph E.5).

The critical concrete strength _cr is not equal to the mean tensile strength because the tensile strength decreases with the load time. For comparison with experiments, the load time is assumed rapidly (first cracks appear after a load time of a few minutes) resulting in a reduction of 0.7:

cr 0.7fctm



 _{( E-4 )}

The influence of the load time and the reduction for the critical strength is described paragraph E.4.

The applied reduction of 0.7 is in good agreement with experimental results (paragraph E.4).

Steel reinforcement bars

Figure E.2 shows the applied behaviour of the reinforcement steel. The stress-strain relation is elastic-plastic and uses the mean yielding strength:

if

with _s ^y

s

f



E

An ascending branch after yielding would be a more realistic approach; however, the ultimate strength is mostly not given in the references. An ultimate steel strain is not entered because failure of the concrete is assumed. The relations between the reinforcement ratio _s, the reinforcement area A_s, the diameter of a reinforcement bar , and the number of reinforcement bars n_s is described as

s s

A 



bd and

0.25

s

s

A



n

  

Figure E.1: The bilinear stress-strain relation assumed for compressive behaviour of concrete.

Figure E.2: Elastic-plastic behaviour assumed for the behaviour of reinforcement steel.

E.2 Applied calculation settings Four settings can be adjusted for a calculation:

1. the number of layers ‘i’

2. the number of segments ‘n’

3. the number of load steps

a. for the moment-curvature diagram ‘m’

b. for the load-deflection diagram ‘p’

The maximum number for these settings is 100. Paragraph E.5 contains an underlying research on the influence of the numbers. The applied calculation settings are affect by the following conclusions:

- The calculation time does not decrease in proportion to a decrease of layers.

- A very low number of layers (n=10) results in an inaccurate calculation.

- De calculation time and preciseness of the moment-curvature diagram is strongly influenced by the number of load steps.

- The load-deflection diagram is considered equally affected as the moment-curvature diagram.

- The number of segments logically influences the calculation time and accuracy due to the segmentation of the external moment.

- In case of one point load in the middle, the deflection is slightly underestimated for an even number and (strongly) overestimated for an uneven number.

In accordance with the underlying research, 50 layers, 50 segments, and 100 moment-curvature diagram load steps are applied. In case of a shear failure analysis, 100 layers are used to obtain a precise calculation of the stresses.

Figure E.3: stress-strain relations for compressive behaviour of concrete. Dark, mid tone and light blue: respectively equation E-6, E-7 and E-8 according to the European standard (2011).

E.3 Inserted compressive behaviour

The behaviour of concrete in compression can be specified by different stress-strain relations. The European standard (2011) specifies three non-linear options which are discussed in this paragraph.

The first one is a parabolic relation, which is described by the following stress-strain relation:

 

The second one is a parabolic-linear relation, describe by

if 0

The third one is a bilinear relation:

if 0

The parabolic relation (eq. E-6) should be suitable for short-term uni-axial loading; however, the crushing strain



_c1 and failure strain



_cu1are nominal and not real values. The parabolic-linear relation (eq. E-7) is prescribed for a cross sectional calculation with design (or characteristic) value of the compressive strength; therefore, this relation is not suitable for a realistic approach. The

European Standard (2011) allows other relations for a cross-sectional calculation if they are more conservative than the parabolic-linear relation. The bilinear relation (eq. E-8) is an example of a more conservative approach and is frequently used for ultimate limit strength calculations. Figure E.3 presents these three relations using the mean compressive strength as the maximum concrete stress.

The red curve in figure E.3 presents the applied bilinear stress-strain relation, which is defined by if 0

The applied stress-strain relation is a non-conservative approach of de parabolic-linear relationship;

it uses the mean compressive strength and the mean Young’s modulus. A smaller ultimate strain is assumed because experiments are a rapid load type resulting is a less ductile behaviour.

E.4 Critical concrete stress

When post-cracking behaviour is not taken into account, the cracking stress is assumed to be the mean tensile strength of the concrete f_ctm. However, post-cracking behaviour is taken into account and the tensile strength of concrete decreases with the load time, shown in figure E.4. That is why, a reduction of the mean tensile strength should be used to define the critical concrete stress.

Bruggeling and Bruijn (1986) distinguished three reduction factor to define the critical concrete stress:

1. Long-term load (load time is endless) 0.6 ,0

cr fctm





1. Slowly increased deformation (first cracks after a load time of a few days) 0.75 ,0

cr fctm





2. Rapidly increased deformation (first cracks after a load time of a few minutes) 0.9 ,0

cr fctm





,0

fctm Is the short-term mean tensile strength for concrete; however, nowadays only the mean

fctm Is the short-term mean tensile strength for concrete; however, nowadays only the mean

In document Eindhoven University of Technology MASTER Shear failure of reinforced concrete beams with steel fibre reinforcement Krings, H. (pagina 71-105)