Investigation on the shear strength of SFRC beams with small opening in web using nonlinear finite element method
Making a traverse opening in concrete beams in order to accommodate utility services through the member instead of below or above of that, Sometimes may be necessary. It is obvious that inclusions of an opening in a beam decrease its flexural and shear strengths. Fabricated steel bars are usually used to increase the capacity of the opening section, but details of reinforcements around the opening are dense and complex resulting in laborious pouring and setup process.
The goal of this study was to investigate the possibility of using steel fibers in concrete mixture instead of complex reinforcement detailing order to strengthen opening section. Nonlinear finite element method was employed to investigate the behavior of steel fiber reinforced concrete beams. The numerical models were validated by comparison with experimental measurements tested by other investigators and then used to study the influence of fiber length, fiber aspect ratio and fiber content on the shear performance of SFRC slender beams with opening. Finally, it was concluded that the predicted shear strength enhancement is considerably influenced by the use of steel fibers in concrete mixture but the effect of fiber length and fiber aspect ratio wasn’t significant.
Keywords: Shear strength, opening, steel fiber, RC beam, Finite element method.
Traverse openings through beams with different shapes and sizes are often provided for essential services and accessibility and are generally located close to the supports where shear is dominant. Among different shapes, circular and rectangular openings are the most common ones in practice . With regard to the size, openings are classified as either large or small. When the opening is small enough to maintain the beam-type behavior or, in other words, if the usual beam theory applies then the opening may be termed as small. When beam-type behavior ceases to exist due to the provision of openings, then the opening may be classified as a large opening . Openings that are circular, square or nearly square in shape maybe considered as small openings provided that the depth (or diameter) of opening is in a realistic proportion to the beam size, say, about less than 40 % of the overall beam depth.
According to Somes and Corley  when a small opening is introduced in the web of a beam, unreinforced in shear, the mode of failure remains essentially the same as that of a solid beam, However, because opening represents a source of weakness, the failure plane always passes through the opening, except when the opening is very close to the support.
Two types of diagonal tension failure are possible for beams containing a small opening, beam type failure and frame type failure and they require separate treatment for a complete design. In frame type failure, two independent diagonal cracks form in each of the chord members. The applied shear maybe be distributed between the two chord members in proportion to their cross-sectional area. Knowing the internal forces, each member can be independently designed for shear following the usual procedure for conventional solid beams .
For beams without shear reinforcement containing small openings (beam type failure), Mansur  proposed that the effective depth, d, in ACI simplified equation  be replaced by the net depth, (d-d0), irrespective of vertical and horizontal location of an opening, where d0 is diameter/depth of an opening.
SFRC can be regarded as composite material formed by a brittle concrete matrix with short dispersed fibers that can debond and slip from the matrix . .Fibers are added to inhibit a propagation of cracks in concrete which occur due to its low tensile strength. Fiber reinforced concrete specimens, even those with a small fiber volume fraction, retain post-cracking ability to carry loads . Among different types, steel fibers are most the common in concrete applications.
Dipti R. Sahoo and others  studied the Behavior of Steel Fiber-Reinforced Concrete Deep Beams with Large Opening. They concluded that replacement of conventional reinforcing bars with deformed steel fibers at a volume of 1.5% can be a feasible alternative to the current practice.
The reinforcement detailing around an opening can be complex, in other hand, Eliminating shear reinforcement in RC structures can potentially reduce the congestion of reinforcing bars and construction costs . This paper presents the performance of SFRC slender beams with small opening under shear failure. The final goal of this study is to investigate the possibility of using steel fibers in concrete mixture in order to increase opening section shear capacity instead of using complex reinforcement scheme around an opening.
To do this, a FE model for RC and SFRC beams is described first and then verified. At last, validated FE model is used to examine the effect of fiber length, fiber aspect ratio and volumetric percentage on the predicted shear strength of the beams. The results of this study show that using steel fibers in the mixture of concrete is an effective strengthening method to enhance the performance of concrete beams with opening.
Finite element model
Here, FE model used to simulate the behavior of the beams is briefly described. A 2D nonlinear FE model is developed using ABAQUS . Taking advantage of symmetry about mid-span plane, just a half of the beam is included in the model as a plane stress problem. The vertical load is applied through displacement increments and to avoid convergence difficulties, the explicit dynamics solver ABAQUS/EXPLICIT is employed to perform the nonlinear analyses. In the following subsections, constitutive models, element types and modeling procedures used in this research are described.
. Concrete in tension
Linear tensile stress-tensile strain relationship is considered for Conventional concrete in tension up to the value of the concrete tensile strength. The post-peak Behavior is modeled as below :
Where _ is fracture energy, _ is crack width and _ is concrete tensile strength. By adding fibers to a concrete mix, the objective is to bridge discrete cracks providing for some control of the fracture process and increase the fracture energy . To realistically evaluate the tensile stress response of SFRC members, the tensile stress due to the tension-softening effect of the concrete matrix should be added to that attained by steel fibers  as shown in Fig.1 and Eq.(
The model proposed by reference  is used to calculate the tensile stress attained by fibers.
In above equations _ can be assumed to be 0.5, _ is volumetric percentage of fibers,_, _,_ and _ equal to 0.67, 0.76, 0.01 and 0.1 respectively, the frictional bond strength (_ ) and the mechanical anchorage strength (_) are assumed to be _and _ respectively.
. Concrete in compression
The compressive strength, strain corresponding to the failure load and material ductility increase with increasing fiber volume, but the key role of steel fibers is to reduce the rate of strength loss after the peak Stress. In this paper, the behavior of SFRC in compression is modeled as below :
Compressive strength of SFRC:
Where _ and _ are the compressive stress and strain corresponding to the failure in plain concrete respectively and RI (reinforcing index) equals to:
In Eq. (10), _ is fiber weight fraction, _is the fiber length and _is fiber diameter. The following equation is used to define stress- strain curve :
. Steel reinforcement
Steel is assumed identical in tension and compression and an elastic–perfectly Plastic material. Elastic modulus of 200000 (Mpa) and Poisson’s ratio of 0.3 are used for the steel reinforcement.
Bond between reinforcement and surrounding concrete
Steel–to-concrete bond is the many-faceted phenomenon which allows longitudinal force to be transferred from the reinforcement to the surrounding concrete in RC or PC structures. Due to this force transfer, the force in reinforcing bar changes along its length, as does the force in concrete embedment .
Under monotonic loading, two types of bond failures are typical. The first is direct pull-out of the bar, which occurs when large confinement is provided to the bar. The second type of failure is a splitting of the Concrete cover when the cover or confinement is insufficient to obtain a pull-out failure . According to the Model Code , concrete is considered well confined when this ratio is not less than five. In this paper, second type of failure is considered.
In the direction parallel to the steel bar-to-concrete interface, the properties of the connector elements are defined using the model proposed by reference . According to this model the relation between local bond stress (_) and slip (_) of reinforcing bars is defined as below:
Where _ is the clear distance between the lugs. The maximum shear stress equals to:
In Eq. (18) _ and _ are clear cover and bar diameter respectively. In SFRC, the post- splitting bond resistance can be calculated as following:
For_ , shear stress decrease linearly from (_) to (_), and for _ has a constant value equal to_ .
Figure 2-Local bond stress-slip relationship
To model concrete response, the damaged plasticity model for concrete available in the ABAQUS material library is adopted. Concrete damaged plasticity model provides a general capability for modeling concrete and other quasi brittle materials in all types of structures (beams, trusses, shells, and solids) . The model requires that the elastic behavior of the material be isotropic and linear. It assumes that the main two failure mechanisms are compressive crushing and tensile cracking of the concrete material and cracking initiates at points where the tensile equivalent plastic strain is greater than zero, _, and the maximum principal plastic strain is positive.
The Uniaxial tension and compression stress behavior, Poisson’s ratio, Young’s modulus, the angle of dilation _, the eccentricity_, the ratio of equibiaxial to uniaxial compressive stress_and the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian at initial yield _ are needed to calibrate the model. In this paper, tensile and compressive behavior, compressive strength and elastic module (the slope of a line from origin through the point corresponding to_ ) were modified in order to introduce the effect of steel fibers as described in subsections 2.1 and 2.2. The adopted values for mentioned parameters are presented in Table 1 (all within the range encountered in literature).
Table 1- assumed values for defining concrete
In order to model the bond behavior between internal longitudinal bars and concrete, beam and longitudinal bars are meshed in a way that they have coincident nodes. Then connector element is defined between two nodes. In the direction normal to the bars, no relative displacement is allowed between the nodes and the behavior of connector element parallel to the bars is defined based on the model described in subsection 4.2.
Due to the symmetry of the geometry, loadings and boundary conditions, only one-half of the beams is modeled using symmetry boundary conditions in mid-span planes. Concrete beam is simulated using Plane stress 4 nodes elements and 2 nodes truss elements were employed to represent the longitudinal Reinforcement and shear stirrups.
Two sets of experimentally tested RC and SFRC beams without stirrups failing in shear are used for model validation. The first set comprised the three beams (control beam, HE-50-0.5, HE-50-0.75) Tested by reference . The beam specimens were 200 x 300 x 2400 mm in size with a reinforcement ratio of 1.7% and were subjected to 4-point bending (Fig. 4) under the constant shear span to the effective depth ratio a/D = 3. The concrete used had a maximum aggregate size of 10 mm and characteristic compressive strength of 40 MPa and Standard Grade 400 deformed steel bars were used for the longitudinal reinforcement with a specified yield strength of 400 MPa. 0.5% and 0.75% hooked end steel fibers of 50 mm length and 1 mm diameter were used in HE-50-0.5 and HE-50-0.75 beams respectively. The properties of concrete assumed in numerical concrete are summarized in Table 2. Using Lower limit of tensile strength led to closer agreement with test results.
Figure 4 - Schematic of beam dimensions and test setup 
Table 2- assumed concrete characteristics 
The experimental and numerical load-deflection curves of the beams are depicted in Fig. 5. This figure clearly demonstrates that there is an appropriate agreement between the experimental and FE-predicted load-deflection behavior from initial loading up to beam failure.
Figure 5 – experimental and numerical load – deflection curves, a: control beam, b: S-HE-50-0.5 beam, c: S-HE-50-0.75 beam
The second set included the three RC beams tested by reference . Beam ‘‘Con-s1” was the control beam without an opening. Beams ‘‘C-con” and ‘‘R-con” were provided with circular and square holes without any strengthening. The circular opening had the diameter of 150 mm and the square opening was 150 mm _ 150 mm in size. The average compressive strength of concrete at was 34.1 MPa. The properties of steel bars are given in Table 3.
Table 3- steel bars properties
of the beams are shown in Fig.6.
Figure 6 - Dimension of beams and reinforcement details 
Fig.7 and Fig. 8 depicts the comparison between experimental and numerical load - deflection curves and crack patterns comprising control beam (Con-s1) and beams with opening (C-con and R-con). Comparison indicates that results from the FE models are matched with those from the experiment.
The FE model had predicted/experimental shear strength ratio of 0.94, 1.06 and 1.12 for Con-s1, C-con and R-con beams respectively.
Figure 7- experimental and numerical load- deflection curves, a: solid beam, b: beams with opening
Based on the demonstrated accuracy of the developed FE model, a numerical parametric study was executed to investigate the shear behavior of SFRC beams with small circular opening. The variable parameters included the fiber length (35-50-60), fiber aspect ratio (46.6-50-60-63.6-66.6-80) and volumetric percentage of fibers (0.5-1-1.5 %). the Details of analytical specimens are shown in Fig. 9. Totally, 12 beams with shear span to effective depth ratio (a/d) of 3 and longitudinal reinforcement ratio of 1.1 % are analyzed. Details of the beams and Material properties assumed in the analysis are reported in table 4 and Table 5 respectively.
Fig. 10 and depicts the load -deflection of the “B-Control” and “B-Circle” beams. Both beams experienced brittle shear failure. The maximum load of beam B-Control was 176.8 kN while that of beam “B-Circle” was 135.5 kN. The capacity, drops 23.3% in beam ‘‘B-Circle” Compared to the control beam without opening.
The predicted crack patterns for these two beams are also shown in Fig. 11. In both beams, the main diagonal crack first starts in approximately distance d from the loading point and then propagate toward support and loading point simultaneously.
. Effect of fibers volumetric percentage
The effect of the steel fibers content is examined in this sub-section. Table 6 summarizes maximum applied load, corresponding deflection and the Percent increase in shear strength over that of the “B-circle” beam for each beam specimen analyzed. The load-deflection relationships for the various beams are also shown in Fig. 12.
Figure 12- effect of fiber content
It can be seen that the beams reinforced with 0.5 % steel fibers resisted the load approximately equal to the solid beam and beams reinforced with 1.0 % and 1.5 % steel fibers resisted over 19 % and 32 % more load than that of solid beam respectively.
It can be observed in Fig. 12 that all of the SFRC beams failed in shear, but had greater deflection at maximum load and more ductile failure than that of the beams made of plain concrete. The predicted crack patterns are shown in Fig. 13. As it can be seen, there are deeper and closer flexural cracks in SFRC beams in the pure bending region. In the shear span, as the volumetric percentage of fibers increases, more number of cracks pass through the opening.
Figure 13 – predicted crack patterns, a: B-circle beam, b: B-50-1-0.5 beam, c: B-50-1-1 beam, d: B-50-1-1.5 beam
. Effect of aspect ratio
The effect of fiber length and fiber aspect ratio is studied by modeling beams having fibers of 35 to 60 (mm) length and 0.55 to 1 (mm) diameter. The volumetric percentage was considered constant and equal to 0.5 %.
Fig. 14 presents the load-deflection diagram and influence of fiber length and aspect ratio on the predicted shear strength, respectively.
These figures demonstrate that the effect of fiber length and aspect ratio isn’t significant. The predicted shear strength increase about 10 % and 4 % for the SFRC beams when the fiber aspect ratio and fiber length is increased from 46 to 63 and 35 (mm) to 60 (mm) respectively.
Figure 14 – a: load – deflection curves, b: effect of fiber length and fiber aspect ratio on the shear strength
In this paper, a FE model for RC & SFRC beams was developed and validated using published experiments from the literature. A parametric study was conducted to investigate the influence of fiber length, fiber aspect ratio, and fiber volumetric parentage on the predicted behavior of RC and SFRC beams without stirrups with circular penning. The following conclusions can be drawn from non-linear FE analyses:
Introducing a circular opening of 0.4H diameter decreased the shear strength approximately 24 % relative to solid beam.
Strengthening the opening section using steel bars can be time-consuming and labor-intensive due to the complex detailing of reinforcing bars. Replacement of conventional reinforcing bars with hooked steel fibers was studied. The results showed that beams with 0.5 % fiber volume fraction can resist the load approximately equal to solid beam.
Using more amounts of fibers in the mixture of concrete considerably increased the capacity of the B-circle beam. In average, 74 % increase in capacity for 1.5 % fiber volume fraction and 55% increase in capacity for 1% fiber volume fraction was achieved. In addition to strength, SFRC beams had more ductile failure mode. Hence, replacement of reinforcing bars with steel fibers seems to be a feasible alternative.
Despite the fiber volumetric percentage, fiber length and fiber aspect ratio had no significant effect in the range studied in this paper (length, 35 to 60 mm, aspect ratio 47 to 80). Having 0.5 % fiber content, in the maximum case 10 % increase in the capacity was observed.
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