Strength Prediction of Corbels Using StrutandTie Model Analysis
International Journal of Concrete Structures and Materials volume 9, pages 255–266 (2015)
Abstract
A strutandtie based method intended for determining the loadcarrying capacity of reinforced concrete (RC) corbels is presented in this paper. In addition to the normal strutandtie force equilibrium requirements, the proposed model is based on secant stiffness formulation, incorporating strain compatibility and constitutive laws of cracked RC. The proposed method evaluates the loadcarrying capacity as limited by the failure modes associated with nodal crushing, yielding of the longitudinal principal reinforcement, as well as crushing or splitting of the diagonal strut. Loadcarrying capacity predictions obtained from the proposed analysis method are in a better agreement with corbel test results of a comprehensive database, comprising 455 test results, compiled from the available literature, than other existing models for corbels. This method is illustrated to provide more accurate estimates of behaviour and capacity than the shearfriction based approach implemented by the ACI 31811, the strutandtie provisions in different codes (American, Australian, Canadian, Eurocode and New Zealand).
1 Introduction
Reinforced concrete (RC) corbels, defined as short cantilevers jutting out from walls or columns having a shear spantodepth ratio, a _{ v }/d, normally less than 1, are commonly used to support prefabricated beams or floors at building joints, allowing, at the same time, the force transmission to the vertical structural members in precast concrete construction. Corbels are primarily designed to resist vertical loads and horizontal actions owing to restrained shrinkage, thermal deformation and creep of the supported beam and/or breaking of a bridge crane. They are becoming a common feature in building construction with the increasing use of precast concrete. Owing to their geometric proportions, corbels are commonly classified as a discontinuity region (Dregion), where the strain distribution over their crosssection depth is nonlinear, even in the elastic stage (MacGregor and Wight 2009), and their strength is predominantly controlled by shear rather than flexure (Yang and Ashour 2012).
The ACI 31811 code (ACI Committee 318 2011) requires corbels having shear spantodepth ratio, a _{ v }/d, less than 2 to be designed using the strutandtie method and those with sheartospan ratio less than 1 to be designed either using strutandtie method, or by the closely related traditional ACI design method based on shearfriction approach. However, the shearfriction hypothesis has little correlation with the observed failure phenomenon of concrete crushing in the diagonal strut (Hwang et al. 2000b).
Strutandtie models (STM) have been generally recognized as an acceptable rational design approach for Dregion members including deep beams and corbels (Schlaich et al. 1987). In addition, most current design codes [ACI Committee 318; Australian code AS 3600 (2009); Canadian code (CSA A23.304); Eurocode 2 (2004) and New Zealand code (NZS 31011)] have recommended the STM approach as a design tool for RC corbels. However, shear capacity of corbels evaluated from STMs and available formulae and computing procedures showed substantial scatter when compared to experimental results (Hwang et al. 2000b; Ali and White 2001; Russo et al. 2006). A rational design procedure to produce safe and economic corbels is therefore required.
In the current paper, a strutandtie based method intended for determining the loadcarrying capacity of corbels is presented. In addition to the normal strutandtie force equilibrium requirements, the proposed model accounts for strain compatibility and constitutive laws of cracked reinforced concrete, and uses a secant stiffness formulation. A similar approach was used previously to calculate the shear capacity of of squat walls (Hwang et al. 2001), deep beams (Hwang and Lee 2000), beamcolumn joints (Hwang and Lee 1999, 2000, 2002), dappedend beams (Lu et al. 2003), and corbels (Hwang et al. 2000a), while using a statically indeterminate truss for modeling the flow of forces and an approximate estimation of members stiffness in evaluating the capacity.
2 Research Significance
In the present study, a strutandtie based method is developed for calculating the loadcarrying capacity of reinforced concrete corbels. The proposed method is based on an iterative, secant stiffness formulation and employs constitutive laws for cracked reinforced concrete, while considering strain compatibility. The secant stiffness formulation approach has previously been implemented in nonlinear finite element procedures to predict the nonlinear response of reinforced concrete membrane elements (Vecchio 1989), as well as to estimate the loadcarrying capacity of deep beams (Park and Kuchma 2007). The method accounts for the failure modes due to crushing of the nodal compression zone at the top of the diagonal strut, yielding of the longitudinal reinforcement, as well as that of strut crushing or splitting. This method is used successfully to predict the loadcarrying capacity of 455 corbels that have been tested experimentally. The findings illustrate that the strutandtie model proposed by different code provisions provide conservative and scattered estimates of the strength of corbels, which should be expected since these provisions were developed for the design of all forms of discontinuity regions and not explicitly for corbels.
3 CompatibilityBased StrutandTie Model Approach for Corbels
Strutandtie modelling is a generalisation of the truss analogy in which a structural continuum is transformed into a discrete truss with compressive forces being resisted by concrete and tensile forces by reinforcement. The method is based on the lower bound theorem of plasticity. Consequently, there are an unlimited number of possible solutions with only some having sufficient ductility for the assumed stress distribution to develop. In the proposed approach, a simple and statically determinate strutandtie load path is proposed to model the force transferring within the corbel as shown in Fig. 1. Statically determinant model requires no knowledge of the member stiffness which makes it simple to calculate member forces using simple statics rules. The proposed strutand tie models assumes that the corbel resists the loads by compressive struts feeding directly into the column, and a tension tie is required to resist the outofbalance forces at the loading point.
4 Equilibrium Conditions
Figure 1 presents loads acting on a corbel and the proposed force transferring mechanisms in view of the proposed strutandtie model. For corbels with short spantodepth ratios, a large portion of the applied vertical shear force is directly transferred to the supporting columns or walls through inclined strut, with the formation of a fulllength horizontal tie to balance the thrust of the inclined struts (Fig. 1). The corbel is loaded by the vertical force V _{ cv } applied at the distance a _{ v } from the column face and it is assumed that the horizontal outward load, N _{ u }, is directly applied at the centroid of the principal tensile reinforcement and the effect of shifting is neglected for simplicity (Hwang et al. 2000b). The angle between the compressed diagonal concrete strut and the horizontal direction \( \phi \), can be defined as (Russo et al. 2006):
where Z is the distance of the lever arm from the centroid of the principal tension steel to the resultant compressive force and a _{ v } is the shear span. According to linear bending theory, the lever arm Z of a singly reinforced rectangular section can be estimated as (Hwang et al. 2000a):
where d is the effective depth of the corbel; kd is the depth of the neutral axis of the cross section; and coefficient k can be defined as (Hwang et al. 2000b):
in which n is the ratio of the elastic moduli of steel and concrete, n = E _{ s }/E _{ c }, and the flexural reinforcement ratio \( \rho_{f} \) is assumed to be given by:
where A _{ n } is the crosssectional area of principal reinforcement used to resist the applied outward load, taken as A _{ n } = N _{ u }/f _{ ys }, where f _{ ys } is the yielding strength of the principal reinforcement, A _{ s } and A _{ sh } are the crosssectional areas of the principal tensile reinforcement and horizontal web reinforcement, respectively, and \( \Omega \) is an efficiency factor representing the contribution of the web horizontal reinforcement, assumed equal to 0.2 (He et al. 2012). The value of n is obtained by assuming, from ACI 31811, that E _{ s } = 200 GPa and \( E_{c} \; = \;4700\sqrt {f_{c}^{\prime } } \) (MPa), it follows that:
The diagonal strut is assumed to have a bottleshaped form. That is, it spreads laterally along its length. The lateral spreading of the bottleshaped strut introduces tensile force transverse to the strut, F _{ st }. The tensile force could potentially cause cracking along the length of the strut resulting in a premature failure. Hence, transverse skin reinforcement should be provided in order to control the cracking. The strut compressive force is assumed to spread at a 2:1 slope (longitudinal: transverse direction) as suggested by the ACI 31811. The considered strutandtie model leads to the following equilibrium equations:
where D _{ c }, H _{ c } are the compressive forces in the diagonal and horizontal concrete struts, respectively; and F _{ st } is the bursting tensile force in the tie of the strutandtie model. Because the bursting force F _{ st } represents a quarter of the compressive force of the diagonal strut, D _{ c }, the horizontal and vertical components, F _{ sh } and F _{ sv }, of the tie force can be obtained from equilibrium as follows:
4.1 Secant Stiffness Formulation
The proposed procedure is based on a compatibility—based iterative, secant stiffness formulation and employs constitutive relations for cracked concrete and reinforcement. The secant stiffness approach was used to calculate the normal strains in the horizontal concrete strut, diagonal concrete strut, horizontal web steel, vertical web steel, and the longitudinal steel tie according to the following equations:
where A _{ d }, A _{ c } are the crosssectional areas of the diagonal and horizontal concrete struts; A _{ sh }, A _{ sv } and A _{ s } are the crosssectional areas of horizontal, vertical and longitudinal steel ties; \( \overline{E}_{d} ,\,\overline{E}_{c} ,\,\overline{E}_{sh} ,\,\overline{E}_{sv} \,{\text{and}}\,\overline{E}_{s} \), are the corresponding secant moduli. Given compatible stress and strain fields, secant moduli can be defined for the concrete and reinforcement (shown in Fig. 2). Secant moduli can be estimated by (Park and Kuchma 2007):
where \( \sigma_{2d} ,\,\sigma_{2c} ,\,f_{sh} ,\,f_{sv} \,{\text{and}}\,\,f_{s} \) are the uniaxial stresses that are obtained from the constitutive relations of each member.
5 Constitutive Relationships of Concrete and Steel
5.1 Softened Concrete in Compression
Cracked reinforced concrete in compression has been observed to exhibit lower strength and stiffness compared with uniaxially compressed concrete, see Fig. 2a. This phenomenon of strength and stiffness reduction is commonly referred to as compression softening. Applying this softening effect to the strutandtie model, it is recognized that the tensile straining perpendicular to the strut will reduce the capacity of the concrete strut to resist compressive stresses. The stress in the concrete is determined from the strains according to the following equations (Vecchio and Collins 1993):

The ascending branch

The descending branch
where σ_{2} is the average principal stress of concrete in the 2 direction; ψ is the softening coefficient; \( f_{c}^{\prime } \) is the compressive strength of a standard concrete cylinder in unit of MPa; ε_{2} and ε_{1} are the average principal strains in the 2 and 1 directions, respectively; and ε_{0} is the concrete cylinder strain corresponding to the cylinder strength \( f_{c}^{\prime } \), which can be defined approximately as:
6 Reinforcing Steel
The stress–strain relationship of steel is assumed to be linear up to yielding, followed by a yield plateau (Fig. 2b). This elastic–perfectly plastic type of stress–strain relationship is represented mathematically by:
where E _{ s } is the elastic modulus of the steel bars; f _{ s } and ε_{ s } are the average tensile stress and strain of the reinforcing bars, respectively; and f _{ y } and ε_{ y } are the yield stress and strain of the bars, respectively.
7 Compatibility Condition
In the proposed approach, the normal tensile strains in the horizontal and vertical web steel, ε_{ h } and ε_{ v } and the principal compressive and tensile strain in concrete strut, ε_{2} and ε_{1}, have a simple relationship that satisfies the compatibility condition of Mohr’s circle (Fig. 2c):
The compatibility equation employed in this paper is the first strain invariant. Equation (24) is used to estimate the value of the principal tensile strain, ε_{1}, which is directly related to the extent of softening of the concrete, as per Eq. (24). Hwang and Lee (2000) pointed out that the used concrete softening model tended to overestimate the softening effect in situations where behaviour was governed by yielding of all reinforcement crossing the crack direction. To guard against this, a limiting value of the principal tensile strain, ε_{1}, was proposed. Thus, the value of tensile strain, ε_{ h }, in Eq. (24) is limited by the yielding strain, ε_{ yh }, after yielding, or the value of ε_{ h } is set to a yielding strain of 0.002 for the corbels not detailed with a horizontal shear reinforcement. Since all the corbels considered in the current study were not provided with vertical shear reinforcement, the tensile strain ε_{ v } is conservatively taken as 0.002 in Eq. (24).
8 Effective Depth of Concrete Struts
Diagonal struts frequently are wider at midlength than at their ends because strut stresses is greater at midlength than at the ends of the strut. The curved, dashed outlines of the strut in Fig. 1 represent the effective boundaries of the diagonal strut. In the proposed model, the bottleshaped strut is idealized as the prismatic struts shown by the straight, solidline boundaries of the struts in Fig. 1. The effective depth of the diagonal strut, W _{ d }, was assumed equal to (Park and Kuchma 2007):
where a _{ v }/2 should not be less than the loading plate width, W _{ p }, and kd is the depth of the compression zone at the section. The horizontal bottom strut was assumed to have a uniform prismatic cross section over its length with effective depth W _{ c } which is presumed equal to the depth of the neutral axis (He et al. 2012):
9 Dimensions of Nodal Zone
Following the suggestion of Paulay and Priestley (1992), the effective width of the bottom node of the horizontal concrete strut was approximated by the depth of the flexural compression zone of the elastic column as:
where N _{ u } is the applied horizontal tension load (negative for tension), A _{ c } is the gross sectional area of corbel, and h is the corbel overall depth, see Fig. 1. The effective width of the top and bottom nodes in the face of the diagonal concrete strut was taken as:
10 Proposed Solution Procedures
The failure modes associated with nodal crushing, yielding of the principal tensile reinforcement, and crushing or splitting of the diagonal strut were used to evaluate the ultimate loadcarrying capacity of the corbels. The algorithm in Fig. 3 starts with a selection of the vertical corbel shear force V _{ cv } and can be proceeded as outlined in following major steps:

1.
According to the member forces D _{ c }, H _{ c }, F _{ sh }, and F _{ sv }, calculated from Eqs. (6) to (10), the values of the strains in concrete struts and steel reinforcements are estimated for the selected V _{ cv } using Eqs. (11) through (15). In initiating the analysis, an initial estimate of the material secant stiffness can be made by assuming linear elastic values. Alternatively, the stiffness determined in a previous analysis can be used as the starting values;

2.
Using the state of strain in each member, the normal stresses are determined from the stress–strain relations of Eqs. (21a, 21b) through (23a, 23b);

3.
The secant moduli for each member are then calculated by Eqs. (16) through (20) using the strain and stresses values calculated in the previous step;

4.
If the differences between the secant moduli in step 3 and those assumed in Eqs. (11) through (15) are larger than the specified tolerance, then the assumed secant moduli are considered incorrect and must be revised until convergence;

5.
The stresses in the diagonal and horizontal struts, σ _{2d} and σ _{2c}, are compared to their capacity. The capacity of the diagonal strut can be estimated from \( v_{cv1} \; = \;0.85\;\beta_{s} f_{c}^{{\prime }} \), where β _{s} = 0.6 as suggested by the ACI 31811(American Concrete Institute 2011) for bottleshaped strut with web reinforcements not satisfying the minimum reinforcement requirements, while the capacity of the horizontal strut is taken as \( v_{cv2} \; = \;0.85\;f_{c}^{{\prime }} \).

6.
The stresses on the on nodes’ vertical back face and nodetostrut interface are compared to nominal strengths due to crushing, assumed equal to \( V_{cv4} \; = \;0.85\,f_{c}^{{\prime }} \) and \( V_{cvs5} \; = \;0.68\,f_{c}^{{\prime }} \) for nodal zones bounded by compressive struts (node A) and nodal zones crossed by tension tie reinforcement in one direction (node B) respectively, refer to Fig. 1;

7.
If the acting stress determined in Steps 5 and 6 is less than the allowable stress, iteration continues from Step 1 by increasing the value of V _{ cv }; and

8.
The predicted strength employed in the proposed analysis method is the minimum value of the nominal strengths computed from the different failure modes, which are crushing of the horizontal and diagonal concrete strut, crushing of the compression zone, and yielding of principal tensile reinforcement.
11 Experimental Verification
11.1 Experimental Results Database
Combining the results of wideranging research into a single database provides the ability to examine code provisions as well as develop new models for use in design. Aimed at verifying the accuracy of the proposed compatibilitybased strutandtie method and assessing the performance of code provision that are used in concrete corbels design, a database with relevant information from tests was constructed. The database contains the results of tests of 550 reinforced concrete corbels collected from (in chronological: AbdulWahab (1989); Alameer (2004); Bourget et al. (2001); Chakrabarti et al. (1989); Clottey (1977); Fattuhi (1987); Fattuhi (1994); Fattuhi (1990); Foster et al. (1996); Hermansen and Cowan (1974); Kriz and Raths (1965); Lu et al. (2009); (Mattock 1976); Yong and Balaguru (1994) and Yong et al. (1985).
Several possible failure modes of corbels have been identified from past experimental testing, including shearing along the interface between the column and the corbel, yielding of the principal reinforcement and crushing or splitting of the compression strut (Russo et al. 2006). Premature failure modes, such as anchorage failure of principal reinforcement and bearing failure under loading plate, would be avoided by correctly designing the corbel details (ACI Committee 318 2011). The results of corbels that were reported to have failed prematurely and those with insufficient information on the test setup and material properties were excluded from the database, leaving only 455 results in the database. Fig 4 presents summary information associated with different parameters, in the form of histograms, on the 455 RC corbels considered in this study. The test specimens in the database were made of plain and fibrous concrete having a relatively low compressive strength of 14.5 MPa and very high compressive strength of 132 MPa. The shear spantooverall depth ratio of corbels ranged from 0.11 to 1.69. The primary tension reinforcements were anchored using a structural weld to transverse bars, bending to form a horizontal loop, or using headed bars. The main longitudinal reinforcement ratio varied between 0.1 and 6.5 %, whereas the horizontal shear reinforcement ratio varied from 0 to 3.05 %. All the corbel specimens included in the database had no vertical shear reinforcement. The horizontal load to yield force of main longitudinal reinforcement ratio ranged from 0 to 1.56. The corbel thickness ranged from 51 to 600 mm and overall thickness varied between 140 and 1140 mm.
11.2 Code Provisions and Analytical Models
Although several methods to compute the strength of RC corbels are adopted in design codes around the world, little is known about the accuracy and conservativeness of design procedures based on different rationales. American (ACI Committee 318 2011), Australian (AS 3600), Canadian (CSA A23.304), European (Eurocode 2), and New Zealand (NZS 31011) code recommendations include special provisions for corbels design. The main aim of the recommendations is to give practical design rules to avoid brittle shear failure ensuring the development of a welldefined strength mechanism that generally occurs in the formation of a strutandtie resistant mechanism. Several other methods are available to estimate the shear capacity of corbels, including empirical equations (Fattuhi 1994), shearfriction approach (Hermansen and Cowan 1974, Mattock 1976), and strutandtie models(Solanki and Sabnis 1987; Siao 1994; Hwang et al. 2000b; Russo et al. 2006) including plastic truss models(Campione et al. 2007). The loadcarrying capacity of the 455 corbels was calculated using the proposed analysis method, the shearfriction based approach provided by the ACI 31811(American Concrete Institute 2011) and the strutandtie model proposed by different code provisions [ACI Committee 318; Australian code AS 3600; Canadian code (CSA A23.304); Eurocode 2 and New Zealand code (NZS 31011)].
The shearfriction based approach provided by the ACI 31811 is valid for corbels made from both normal and high strength concrete with spantodepth ratio less than unity. This procedure refers to two typical modes of failure: the first is the failure mode due to shear constraint occurring at the interface between column and corbel, and which occurs with very small shearspan ratios and reduced percentages of reinforcement. For a shear failure, the shear strength of a corbel is given by:
where ρ _{ vf } = (ρ _{ s } + ρ _{ h }) is the frictional reinforcement ratio; f _{ ys } is the yield strength of the friction reinforcement; ρ _{ s } and ρ _{ h } is the principal reinforcement ratio and horizontal web reinforcement ratio, respectively; μ is the coefficient of friction (taken as 1.4 for monolithic construction); and b is the corbel width. The second mode of failure is due to flexural yielding of the principal longitudinal reinforcement, and the carrying capacity can be estimated as:
where α is the horizontaltovertical loads ratio; and jd is the lever arm calculated by \( jd\; = \;d\;  \;\frac{{(A_{s} f_{ys} \;  \;Nu)}}{{0.88\,f_{c}^{{\prime }} b}} \). The corbel strength is taken as the minimum value of Eqs. (29) and (30). Moreover, the code imposes an upper limit on the loadcarrying capacity with a maximum value of V _{ cv } shall not exceed the smallest of \( 0.2f_{c}^{{\prime }} \,bd,\,(3.3\; + \;0.08\,f_{c}^{{\prime }} )\,bd\,{\text{and}}\,11\,bd. \)
Several code recommendations (CSA Committee A23.3 2004, NZS 3101 2006) specify the strutandtie models for the design of corbels, while only for corbels having shear spantodepth ratio greater than 1.0, the ACI 31811 recommends the use of a strutandtie model described in ACI 31811, Appendix A. However, it does not provide detailed guidance on strutandtie models for different cases. It is well known that in using the strutandtie model, the designer is free to select the form and dimensions of the loadresisting truss to transfer the applied forces to the supports. More than one strutandtie model is usually feasible and thus there is no unique design solution as there typically is with the use of the conventional sectional design procedures. The safety of the strutand tie model approach is highly dependent on the suitability of the assumption in lowerbound plasticity theory that the structure is adequately ductile to allow the load to be supported in the way chosen by the designer.
The experimental results are compared with predictions made on the basis of a simplified strutandtie model (STM) accounting for the main tie steel only, a refined strutandtie model accounting for the secondary crackcontrol reinforcement (Reineck 2003). These two models were reported to be very conservative and assume corbels failure are due to yielding of the tie and/or horizontal web reinforcement, thus prevent the assessment of codes provisions (Yang et al. 2012). Instead, two strutandtie models for a doublesided corbel and a single corbel projected from a column shown in Fig. 5, are proposed and it is assumed corbel failure is due to either crushing of the horizontal and diagonal concrete strut, crushing of the compression zone, or yielding of principal tensile reinforcement, similar to that assumed in the proposed strutandtie based method.
11.3 Comparison of LoadCarrying Capacity
Very few studies found in the literature on the validity of loadcarrying capacity models of RC corbels in code provisions including strutandtie models. Table 1 summarizes the average, Avg, standard deviation, SD, and coefficient of variation, CoV, of the ratio between measured and calculated capacities, v _{ test }/v _{ calc }, of RC corbels considered, based on the proposed strutandtie based method, the shearfriction based approach provided by the ACI 31811, the strutandtie model proposed by five codes of practice examined (ACI 31811; (c) AS 3600; (d) CSA A23.304; (e) Eurocode 2; and NZS 31011). The distribution of average strength ratios for the specimens in the database against the concrete strength, \( f_{c}^{{\prime }} \), and shear spantodepth ratio, a _{ v }/d, is shown in Figs. 6, 7, 8 9, where Avg and CoV values are also reported.
For the comparison with the shearfriction based approach provided by the ACI 31811, only 357 corbels with shear spantodepth ratio less than unity have been taken into account. Careful examination of the results shows that the shear strength ratios, v _{ test }/v _{ calc }, using the shearfriction based approach provides highly conservative and scattered estimates of the strength of corbels over a wide range of concrete strength and shear spantodepth ratio. The coefficient of variation is quite high, with a value of 52 %, thus a low 5 % fractile value is to be expected. Altogether, 51 tests exhibit unconservative estimations, which is remarkably more than the 5 % fractile of 18 tests. Therefore, the results from this new database, which is much larger and more comprehensive than that used to calibrate the shearfriction based approach of ACI 31811 in the 1980s, are clearly unsafe. In particular, the shearfriction based approach is unconservative in the prediction of the loadcarrying capacity for corbels with concrete strength less than 50 MPa, (see Fig. 6b). By contrast, the calculated capacities by the proposed strutand tie based method are both accurate and conservative with low scatter or trends for RC corbels with sheartospan depth ratios ranging from 0 to 1 (see Fig. 7a).
The selected strutandtie models shown in Fig. 5 produce results that are quite similar to each other, refer to Table 1. Figures 7 and 8 present the effect of shear spantodepth ratio, a _{ v }/d, and concrete strength, \( f_{c}^{\prime } \) on the loadcarrying capacity predictions of the strutandtie based method and the five codes of practice examined for the doublesided corbel model only, respectively. On the whole, the predictions of the proposed method are very consistent for a broad range of concrete strengths and shear spantodepth ratio, indicating that Avg, SD and COV are 1.39, 0.29, and 21 %, respectively. On the other hand, the overall performances of the five codes of practice examined are very similar, highly conservative and scattered. This conservatism may be attributed to the conservatism in the effective depth of the diagonal strut (Park and Kuchma 2007). They consistently underestimate the loadcarrying capacity of corbels with concrete strength greater than 35 MPa and shear spantodepth ratio greater than 0.3. The largest average of the shear strength ratios, (v _{ test }/v _{ calc }) of all STM models appear as specified in Eurocode 2. The size of this test database and the use of these five code provisions are enough to obtain valuable insight into the behaviour of RC corbels from a strutandtie perspective. Considering the width of the compiled database, the obtained results are considered to be adequately fair to suggest that the proposed strutandtie based method provides a reliable and safe means of predicting the loadcarrying capacity of reinforced concrete corbels.
12 Summary and Conclusions
A strutandtie based method intended for calculating the loadcarrying capacity of reinforced concrete corbels has been presented. In addition to the normal strutandtie force equilibrium requirements, the proposed model accounts for strain compatibility and stress–strain relationship of cracked reinforced concrete, and uses a secant stiffness formulation. Based on the available test results in the literature and their comparison with the proposed model and the shearfriction based approach provided by the ACI 318 formulas as well as the strutandtie provisions in the American, Canadian, New Zealand, Eurocode and Australian codes., the following conclusions may be drawn:

1.
The calculated loadcarrying capacities by the proposed method were both accurate and conservative with limited scatter or trends for reinforced concrete corbels with shear spantodepth ratios ranging from 0.1 to 2 and made from normal or fibrous concrete.

2.
The shearfriction based approach provided by the American code is highly conservative and scattered estimates of the strength of corbels over a wide range. By contrast, the calculated capacities by the proposed strutand tie based method are both accurate and conservative with low scatter or trends for RC corbels with sheartospan depth ratios ranging from 0 to 1.

3.
The predictions by the proposed strutandtie based method are adequately conservative and accurate to conclude that it provides a safe and reliable means of calculating the loadcarrying capacity of concrete corbels.

4.
Based on the conclusions drawn from this research, the proposed strutandtie should be adopted in future adjustments to code provisions and in the development of design guidelines for all types of Dregions in structural concrete. Furthermore, both experimental and mathematical studies are still needed to investigate the applicability and limitations of the proposed strutandtie method when applied to a wide range of Dregions.
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Acknowledgment
The Author would like to thank Prof. KeunHyeok Yang, Kyonggi University, South Korea for providing some information on tests of corbels and assistance in populating the corbels database.
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Kassem, W. Strength Prediction of Corbels Using StrutandTie Model Analysis. International Journal of Concrete Structures and Materials 9, 255–266 (2015). https://doi.org/10.1007/s400690150102y
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DOI: https://doi.org/10.1007/s400690150102y