Influential parameters in the shear strength of RC beams without stirrups

abstract: Experimental investigations have been commonly used to improve the existing knowledge about structural design, presenting accurate conclusions regarding structure behavior. However, considering the limitations of experimental studies, such as restricted amount of strain gauges, unexpected pathologies, difficulties to foresee the correct position of cracks, or simply saving in costs, the use of numerical analysis can present some innovative approaches to understand the process of failure in concrete elements, presenting easier results where experimental programs can hardly report. This study presents the numerical analysis of eight beams experimented by Sherwood [26], with two different sizes and variable aggregate sizes, seeking to understand the influence of coarse aggregate size in shear strength of beams without stirrups. The numerical approach was used to derive the influence of each internal shear mechanism and to identify the specific amount of dowel force, shear transfer by uncracked compression zone and aggregate interlock portion. The results showed that fair results can be obtained by 2D smeared crack approaches, enabling the identification of major aggregate interlock portion in beams with bigger coarse aggregate sizes. Comparing the size effect in beams allowed us to conclude that a higher contribution of aggregate interlock contribution can be obtained in large beams, with almost 64% of the total contribution, whereas smaller beams had only 43%. To evaluate the accuracy of the studied mechanisms, the results of Sherwood [26] were compared with the standards Model Code [23], CSA-A.23.3 [24], NBR 6118 [38], and ACI 318 [39]. Regarding the prediction of element rupture by concrete internal mechanisms, the normative instructions Model Code [23] and CSA A.23.3 [24] were the closest to the experimental results.


INTRODUCTION
Several studies have attempted to understand the shear behavior of reinforced concrete (RC) beams, and various approaches to predict its strength have been proposed.Some important studies, primarily proposed by Ritter and Die Bauweise [1] and Mörsch [2], present the strength of concrete beams with stirrups as an analogy for a trussed system, wherein the steel and concrete stresses must be maintained under a factored capacity limit.This limit is considered as a function of the concrete and steel limits, assuming that only the stirrups resist the formation of cracks.
Without shear reinforcement, the strength mechanism of concrete typically relies solely on the nonlinear behavior of the concrete used, considering that the cracks can transmit normal and parallel forces to their formation site.These results in mechanical responses help avoid the failure of the structural element to a certain extent.Figure 1 shows the shear capacity mechanisms of reinforced concrete beams, where Vcz is the shear strength of the un-cracked concrete in the compression zone [3]; V agg is the aggregate interlock interface shear transfer, also referred to as crack friction [4]- [10]; and V dowel is the dowel action due to the longitudinal reinforcement bars [3], [11]- [16].Considering that the design codes are based on empirical equations, employing simple models -as the strut and tie models -on the structural members without transverse reinforcement results in an unsafe design.Bazant and Kazemi [17], Kani [18], and Taylor [19] tested beams based on empirical equations that are not applicable for practical situations.
Many studies have been conducted in the last 40 years on the internal capacity of concrete to resist shear strength, promotion owed to the continuing need to understand the real effect of these parameters on structural elements.To this end, several theoretical approaches have been proposed such as the modified compression filed theory (MCFT) by Vecchio and Collins [20], and Bentz et al. [21], and the critical shear crack theory (CSCT), developed by Muttoni and Fernández Ruiz [22].MCFT and CSCT consider the influence of aggregate size, size effect, and strain on RC shear capacity.These methodologies compose the basis of recent design codes, such as Model Code [23] and CSA-A23.3 [24].
In the 1970s, Walraven [25] presented an experimental method to understand the influence of aggregate interlock mechanism on shear strength and found that raising coarse aggregate size resulted in increased major shear strength.Paulay et al. [14] compared longitudinal bars of different diameters, crossing a discontinuous surface on an experimental block, allowing for an investigation on the influence of the dowel force on shear strength.The study related higher shear strengths to larger bar diameters acting like dowels.
Despite the importance of these studies, no clear conclusions can be drawn regarding the influence of each mechanism on the total shear strength of RC beams.Other experimental studies such as those conducted by Sherwood [26], Daluga et al. [27], and Yang and Ashour [28] found that the influence of the coarse aggregate size could result in an increased shear capacity.However, to obtain a clearer understanding of stress and strain distribution inside RC beams, numerical modeling tools are still necessary.
Belbachir et al. [29] conducted an experimental study on three sizes of geometrically similar concrete beams without transverse reinforcements.The shear span-to-depth ratio (a/d) was set to 2.5.For the beams tested with this ratio, the aggregate interlock mechanism was not completely activated at the failure load, especially for smaller beams.This shear transfer mechanism exhibits size dependence, which is primarily related to the crack shape.
To contribute to the understanding of the shear strength mechanism in beams without stirrups, this study presents a numerical non-linear modeling for eight beams experimentally tested by Sherwood [26].
This study is motivated by the need to develop numerical approaches to analyze structural elements using computational tools and understand the role of each internal shear capacity mechanism on the strength of RC beams without stirrups.Performing numerical simulation of the experimental cases typically involves two main phases: calibrating the model; and analyzing internal cracks, strains, and stress (typically limited to obtaining measurements from sensors or strain gauges).
This study aims to deepen the studies on the phenomena of shear strength of reinforced concrete beams without stirrups, regarding the internal mechanisms of the concrete to resist the shear force of beams, usually called Vc in technical standards.
In these design standards, V c is usually used as a shear strength contribution portion and is considered in the determination of shear reinforcement.The greater the portion of shear strength force by internal mechanisms, the smaller the portion of shear force that an engineer, instructed by technical standards will calculate and apply as a form of transverse reinforcement ratio.

EXPERIMENTAL PROGRAM
To investigate the influence of each internal strength mechanism on the total shear strength, the first step was to find an experimental program capable of standardizing many parameters and allowing for comparisons between the several shearinfluencing factors.Based on these requirements, Sherwood [26] reported different campaigns of RC beams without stirrups, using different sizes of coarse aggregate and other similar attributes as premises.These methods, paired with an effective registration of the principal parameters experimentally obtained, make the study a useful reference for numerical analysis.
Table 1 shows the main characteristics of the specimens used as reference for the numerical analysis, where bw is the cross-section width, d the effective depth, L the effective beam length, d ag the aggregate diameter, f c the concrete compressive strength, and  + is the cross-section of longitudinal tensile reinforcement.The aggregate diameter was used as an input parameter as recommended by the Model Code [23], such as: tensile strength, elasticity modulus, and Poisson coefficient.Sherwood [26] considered two categories of beam sizes and types in the experiment: eight beams of size 122 × 363 × 1,800 mm, called the Small (S) Series and, eight beams of size 295 × 1,510 × 9,000 mm, called the Large (L) Series.Figure 2 shows the characteristics of the beams analyzed in this study.The samples possess different concrete strengths.Four different sizes of coarse aggregate are used, with the major aggregate size varying from 9.5 mm to 51 mm.The specimens were not reinforced with stirrups and all experiments were loaded until shear failure.
Reinforcement yield stresses were experimentally evaluated and recorded.Table 2 shows the yield values and nominal areas.

NUMERICAL MODELS
A numerical model was developed to accurately simulate the experimental program.The problem was split into several partial problems and determined the internal relationship based their contour connectivity and characteristics.This is called the finite-element method (FEM).The FEM model must consider many effects to model the RC beams correctly and derive the basic conditions for the interaction between steel and concrete and the gradual formation of cracks.
The models tested in the laboratory were reproduced using DIANA 10.1.The discretization of the mesh in the S-Series used quadratic finite elements of approximately 26 × 26 mm, which is approximately 1/12 of the total beam height (330 mm), which has been by Feenstra et al. [30], Vecchio and Shim [31], and Pimentel et al. [32].In the L-Series, the mesh proportion contained quadratic finite elements of approximately 75 × 75 mm, on the order of 1/20 of the total beam height (1,510 mm). Figure 3 shows the boundary conditions, load application points, and mesh sizes used in the FEM simulations.The models were constructed with 2D plane stress elements using square and triangular formats to represent the exact position of loads, restrictions, and longitudinal reinforcements.The longitudinal bars employ stiff beam elements for longitudinal and transverse displacements, particularly to act as dowels when the crack occur near the reinforcement.
The beams were configured to present two vertical restrictions at the supports, in addition to the horizontal restriction at the load point.The load was applied with a continuous displacement of 0.05 mm during each load step.The modified Newton-Raphson method was used to reach the convergence, with an internal energy-based criterion and a tolerance of 0.001.To accelerate the convergence, the line search and arc-length method were included in the model, according to the recommendations described in DIANA [33].
Table 3 shows the main parameters used to simulate the concrete and steel materials used in this study.These parameters were obtained with numerous analyses, aiming to ensure consistency between numerical models and the experimental program.The data obtained by Sheerwood [26] were used as values for fc, the other properties were obtained based on the Model Code [23] recommendations, such as tensile strength of concrete, (f ct ), elasticity modulus of concrete (), and concrete Poisson coefficient ().Based on these values, calibrations were performed for each parameter and the ideal calibrated model was chosen.
The evaluation included 4 to 5 energy ranges from fracture to traction.The initial value was adopted as recommended by the CEB/FIP Model Code [34] with a maximum reduction of 40% for L-Series beams and up to 30% for S-Series beams.
Based on the parametric analysis performed, for the models studied, the increase in fracture energy proportionally reduced the load capacity of the analyzed beams, presenting reductions of 30-40% of the fracture energy in traction to better approach the experimental result.According to the CEB/FIP Model Code [34], the fracture energy can be estimated from Equation 1, which depends on the fracture energy base value, as shown in Table 4.
Table 3. Parametrization of concrete and steel at beams type S-and L-Series.Where: Exp. (1) Exp. (1) Hordijk Exp. (1) Exp. ( The results were compared not only based on the load-displacement behavior but also regarding cracking and longitudinal reinforcement strain.These results confirm the reliability of the proposed numerical models and enable indirect observation of the shear behavior.

Load and displacement
Figure 4 shows the variation of load with displacement for the S-Series numerical models for concrete with aggregates size ranging from 9.5 mm to 51 mm, comparing them with the reference models used by Sherwood [26].
The 9.5 mm and 19 mm models showed greater rigidity behavior at the beginning of the curve, especially in the initial linear behavior of the parameterized concrete.When the specimens began to crack, they lost their rigidity and developed a similar behavior as that of three experimental reference models, obtaining an ultimate load close to the experimental value.
This rigid behavior is even more apparent in the numerical models for specimens with 38 mm and 51 mm aggregates.However, a trend like the experimental curve was observed throughout the development of the analysis.A varying load level, with a sequential increase in displacement, may lead to shear failure and the samples may become brittle like the experimental behaviors reported by Sherwood [26].
For the four numerical analyses conducted in the L-Series, the equivalence conditions between the numerical and experimental models showed similar characteristics, as shown in Figure 5.The choice of elasticity module at the beginning of loading did not show discrepancies.The variation between the behaviors of beams with different aggregate diameters can be understood by using the results for beams with 9.5 mm and 51 mm aggregates, which represent the two extreme aggregate sizes used in the analysis.Their concrete strengths show a difference of less than 2.0 MPa for experimental strength of concrete.
In the S-Series models, the ultimate load, ultimate displacement level, and ductility remained almost constant.The strength gain was more apparent for the L-Series, in the ultimate loads of both S and L-Series, with a difference of approximately 100 kN between the average values.The L-Series models exhibited similar stiffness and showed no major variations at the rupture.

Strains at the flexural reinforcement
The behavior of the longitudinal reinforcement of the model, referring to the axial deformations observed under the equilibrium torque of beams subjected to bending moment, can be observed for the analyzed beams, as shown in Figures 6 and 7.The values were compared with those obtained by strain gauges fixed to the reinforcements in the center of the span of the evaluated beams.The numerical and experimental comparison of the reinforcements sought to understand the deformations verified by the structural response of the model, while ensuring that no equivalence between the curves (Figures 6 and 7) by any resistant phenomena, such as bending failures, and that the ultimate shear load that occurred in the experimental tests was preserved.Based on the results, it can be noted that despite the numerical models tending to behave more rigidly than the respective experimental values, good approximations were numerically registered, indicating an effective modeling of the deformation and rupture conditions of the experimental reference model.

Internal shear strength mechanisms of RC beams without stirrups
The behaviors and derivations of the shear strength plots based on the internal concrete mechanisms in the immediate moments preceding the rupture were obtained from two distinct surveys.The first survey involved identifying a compressed strip of concrete just above the critical crack, and the second involved obtaining the shear force acting on the longitudinal reinforcement, identified as the dowel force.Figure 10 shows the horizontal compression stresses occurring in the beam with 9.5 mm aggregates for the S and L-Series beams, alongside the shear profile distributed in the area above the crack in two reference sections, A and B. Section A is in the vicinity of the load, close to the top of the critical shear crack, and section B is at a distance d from the center of the span of the beam, adopted as the authors' choice.In these studies, the chosen reference lines become relevant due to easily discriminating the influence of each of the plots at different distances from the supports, avoiding inconsistent readings on the real influence of each mechanism on the total shear force.
In the numerical analysis procedure, the steps for obtaining the shear plots were performed using the following criteria.Firstly, the intact and non-cracked regions of the compression zone above the critical crack are identified with a compression stress diagram in the horizontal direction of the stress map σ.The end of the compressed zone is delimited using the correct coordinates of the point where the compressive stresses begin to be replaced by the tensile stresses.This premise is important since active shear stresses where the crack has occurred can also be observed from the shear distributions transferred by the beam body in the distributed crack mode.Thus, when the compression zone is properly delimited, the numerical procedure guarantees better precision and efficiency in the derivation of the shear strength plots.
Based on the distribution of shear stresses τ obtained in sections A and B of the S and L-Series, one can conclude that the closer to the peak of the critical crack the stress is measured, the smaller is the compression zone, adding to the intensity of the shear stresses transmitted above the crack.Unlike the S-series, the larger scale beams of the L-series showed a lower shear stress intensity transferred through the compression zone, even with external loading intensities considerably higher than that of the S-Series.At points near the center of the span, the crushing stresses are also more concentrated, reaching almost 50% of the maximum concrete compressive strength (f c ).For section B, the maximum stresses are less intense, which is distributed over a larger compressed area above the crack.
Based on the analysis of Table 5, the shear force transferred by each of the complementary concrete mechanisms was quantified for the reference sections of the S-and L-series beams.In this study, the procedure consists of identifying the shear portion transferred by the compression zone V cz and the dowel action V dowel of the analyzed beams, and then subtracting the terms following the relation , where V exp is the experimental shear force, calculating the mean contribution to the total shear force by aggregate interlock mechanism in the reference sections A and B.
Comparing the values obtained for each strength phenomenon, V agg increased gradually as the maximum diameter of the aggregate in the mixture increased, as shown in the last column of Table 5, starting with %V agg .This behavior was also discussed by Queiroz [35] and MacGregor and Wight [36], in which they concluded that the shear force by aggregate interlock increases as the coarse aggregate increases its diameter, consequently, the rugosity of the crack surfaces also increases.In this way, a greater shear stress is transferred across the cracks.The results of the S-series indicated that the aggregate interlock mechanism contributed to 34-46% of the shear force capacity.The predominant mechanism of shear capacity for this series was V cz .The effect of the aggregate interlock mechanism is inversely proportional to the portion of the compression zone, which gradually decreases with an increase in the aggregate size.For the L-series, this mechanism is dominated by more than half of the total shear strength force (54-73%).
Observing the stress map σ of the compression zone, the rupture was unrelated to the crushing of the compression zone in all the models presented since stress levels were below the strength limit of concrete in each d sample.
These results confirm the conclusions proposed by Walraven and Reinhardt [37], who found that, in addition to concrete strength, the interlocking of aggregates plays an important role in the shear strength of beams.Comparing the 9.5 mm and 38 mm aggregate models, when using concrete with a lower strength, the use of aggregates with larger diameters could guarantee similar breaking loads.
During the experiments for understanding the influence of dowel action, difficulties were found in predicting the influence of the position of the bars on shear force.In distributed crack models, the deformation caused by dowel action is distributed over the width of the finite elements, not defining a perfect cut line on the bar.Therefore, shear force peaks of different intensities occur in numerous positions along the longitudinal beam.Table 6 shows the maximum shear force obtained in other sections of the analyzed beams.According to the results presented, the largest numerically evaluated beams, such as those represented by the Lseries, did not benefit greatly from the strength mechanism of the dowel action, whose maximum contribution to the total shear strength was only 5%.

COMPARISON BETWEEN EXPERIMENTAL AND PREDICTED SHEAR STRENGTH FORCE
Understanding the methodologies used in the technical standards for predicting the shear strength mechanisms of concrete is a crucial part of studying shear in reinforced concrete beams.According to the Model Code [23], CSA-A.23.3 [24], NBR 6118 [38], and ACI 318 [39], the transverse reinforcements are usually dimensioned to resist all excess shear that the internal strength mechanisms of the concrete are unable to supply, as expressed in Equation 2.
where V u = total strength shear force (V u = V exp ); V s = shear strength force by transverse reinforcement; V c = shear strength force through internal concrete mechanisms.If V c is overestimated, that is, V c values obtained by numerical analysis is higher than the actual values in their respective experimental models, beams may lack relevant transversal reinforcements, passing excessive responsibility to concrete to resist shear.
To guarantee an analysis that fits the beam condition, the formulations contained in the design codes that estimate the shear strength of beams (as detailed in Appendix A) can be applied and verified.The numerical tests and analyses performed in reference by design standards used an L/h (beam length/height) ratio of 4.9 for the S-Series and of 5.3 for the L-Series.
The methods and limits applied in design standards are given below.• CSA-A.23.3 [24]: calculation for beam elements with L/h ≥ 2.0.
Table 7 shows the shear strength predictions for the eight beam models tested by Sherwood [26] using the complementary concrete mechanisms recommended by the studied standards.The safety coefficients specified in each standard were not considered in the analysis.
Based on the results presented in Table 7, it can be observed that all design standards presented favorable safety estimates for the S-series beams, except the Brazilian standard, which presented unfavorable results for the design.During the verification of the L-Series beams, the standards failed to predict the rupture load.The ACI 318 [39] standard presented more conservative results.Assessing the accuracy of the studied standards in predicting the experimental shear force V exp , Model Code [23] and CSA A.23.3 [24] proved the most capable in predicting rupture by the internal mechanisms of concrete.Model Code [23] presented the smallest variations, with a mean safety factor of 1.15 and a 19.6% variation coefficient.
For Model Code [23] and CSA A.23.3 [24], the importance of the influence of the aggregate diameter for the prediction of rupture by shear force can be inferred since they are based on MCFT.
Although NBR 6118 [38] showed fewer conservative results in the prediction of rupture for L-series beams, effective safety estimates were obtained for the S-series beams -those with a size range commonly used in most buildings -provided that all the recommended procedures and safety coefficients are applied.

CONCLUSIONS
Eight experimental RC beams without stirrups, separated into two series of different sizes, aggregate diameters, and compressive strength of the concrete, were studied by nonlinear numerical analysis using the DIANA software to understand the influences and contributions of the internal mechanisms of concrete on shear strength.The experimental results were compared with the estimates predicted by standards such as the Model Code [23], CSA-A.23.3 [24], NBR 6118 [38], and ACI 318 [39], to understand the ability of these codes in predicting the shear strength of beams by internal concrete mechanisms.
The nonlinear calculation method chosen in this study provided a satisfactory representation of the shearing behavior of the experimental models, showing good numerical-experimental approximations between the cracks pattern, ultimate load, strain in the longitudinal reinforcement, and the development of the deflection observed on the beam.It was concluded that the aggregate interlock phenomenon could be seen as the main form of failure for all the evaluated beams.This is due to the range of longitudinal stresses measured in the compression zone being unable to recognize the phenomenon of concrete crushing at the failure.Moreover, studying the rupture via the dowel effect of the reinforcement, which presented shear forces lower than a possible rupture, was impossible.
The outcomes suggest that, for a good approximation between the models studied, the G f values, initially adopted according to CEB/FIP Model Code [34], must range from 100% to 70% of the total value for the S-Series beams and from 100% to 60% of the total value for L-Series.Thus, the S10 model was 70%, the S20 model 85%, and the S40 model 73%.
For both S and L-Series, the percentages of influence of the aggregate interlock mechanism gradually increased as the diameter of the coarse aggregate of the mixture increased.Comparing the 9.5 mm and 51 mm extremes with the same characteristic strengths, an increase in the capacity of the model is evidenced.The strength force increased approximately by 10 kN in the S-Series samples and by 100 kN in the L-Series samples.
Based on the normative estimates, it is concluded that in the methodologies that incorporate the influence of the aggregate diameter on the shear strength in their design equations produced better predictions on the behavior of concrete without shear reinforcement, with less than 20% of variation from the comparative numerical-experimental model.
The results of the numerical tests, as well as in the experimental test carried out by Sheerwood [26], aimed to obtain a type of brittle failure in the beams that occurs by the formation of a typical critical crack.In the numerical model, some characteristics were verified to define this type of failure, such as: longitudinal reinforcement did not present high stress or close to the yield stress (eliminating the hypothesis of failure by bending); the compressed zone presented stresses, in any direction, always at levels significantly lower than the concrete strength; and the pattern of crack formation showed a typical typology of critical shear cracking.
Note that, the rupture moment was also assumed in the experimental tests by Sherwood [26] and showed correspondence with the numerical models, resulting in even more certainty about the model's rupture moment.

Figure 3 .
Figure 3. S and L series beams numerical model.

Figure 4 .
Figure 4. Load versus displacement curves for S-Series beams.

Figure 5 .
Figure 5. Load versus displacement curves for L-Series beams.

Figure 6 .
Figure 6.Load versus flexural reinforcement strain curves for S-Series beams.

Figure 7 .
Figure 7. Load versus flexural reinforcement strain curves for L-Series beams.

Figures 8
Figures 8 and 9 show cracking occurring in the S and L-Series beams, respectively, with their critical crack loads marked.Based on the figures below, the influence of the diameter of the aggregate can be understood by comparing the samples sized from 9.5 mm to 51 mm.The inclination of the slope gradually reduced as the nominal size of the aggregate increased for the S-Series, representing only minor changes in the L-Series cracks.

Figure 8 .
Figure 8. Numerical and experimental comparison for S-Series beams.

Figure 9 .
Figure 9. Numerical and experimental comparison for L-Series beams.

Figure 10 .
Figure 10.Compression and shear stresses and dowel effect for beams with 9.5 mm size aggregates.

Table 2 .
[26]rimental data on the steel bars used in the tests by Sherwood[26].Where, fy is the yield strength of reinforcement.
Figure 2. Beams of S-Series and L-Series of the experimental program (units in mm).Adapted by Sherwood [26].
refers to Exponential Tensile Softening.

Table 4 .
[34]rence value of fracture energy as a function of aggregate size by CEB/FIP Model Code[34].

Table 5 .
Plots of shear force obtained by numerical analysis of each of the internal mechanisms of resistant concrete.

Table 6 .
Maximum shear forces obtained by the dowel action on the results of numerical analysis.

Table 7 .
Comparison of analytical shear strength force with experimental results.