Strengths of Struts and Nodal Zones for Strut-and-Tie Model Design of Reinforced Concrete Corbels

The strut-and-tie model (STM) method is useful for the limit state design of reinforced concrete (RC) corbels. However, for the rational design of RC corbels, designers must accurately determine the strengths of concrete struts and nodal zones to check the strength conditions of a selected STM and the anchorage of reinforcing bars in nodal zones. In this study, the authors suggested a numerical process for determining the strengths of concrete struts and nodal zones in RC corbel STMs. The technique incorporates the state of two-dimensional (2-D) stresses at the strut and nodal zone locations, 2-D failure envelope of concrete, deviation angle between the strut orientation and compressive principal stress trajectory, and the effect of concrete confinement by reinforcing bars. The authors also proposed the strength equations of struts and nodal zones that apply to the typical determinate and indeterminate STMs of RC corbels. The authors considered the effects of the shear span-to-effective depth ratio, the horizontal-to-vertical load ratio, and the primary tensile and horizontal shear reinforcement ratios in developing the strength equations. The authors predicted the failure strengths of 391 RC corbels tested to examine the appropriateness of the proposed numerical process and strength equations. The predicted failure strength compares very well with experimental results, proving that the rational analysis and design of RC corbels are possible by using the present study's strut and nodal zone strengths.


Introduction
RC corbel is a representative structural member with static discontinuities due to the concentrated vertical and horizontal loads. The primary design codes, including FIB [1], AASHTO [2], ACI 318 [3], have accepted the STM method for the design of RC corbels. The STM method promotes a better understanding of the load transfer mechanism, thus improving a designer's ability to handle unusual design circumstances. However, to employ the method in practice, a structural designer must determine the strengths of struts and nodal zones to verify their loadcarrying capacities.
Many researchers have conducted experimental and analytical studies on the strength of concrete struts and nodal zones. The readers can find various suggested strength equations and values from previous researches [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. However, the suggested equations and values are not consistent and general because they base on structural concretes with specific loading and geometric conditions. Yun & Ramirez [16] proposed a numerical method for determining In Section 2 of this paper, the authors introduced three types of statically determinate and indeterminate corbel STMs that designers employ in practice. In addition, the authors introduced the strength equations/values of struts and nodal zones suggested by the current design codes and the present study. In Section 3, the authors predicted the failure strengths of RC corbels by using three types of corbel STMs to evaluate the properness of the existing and proposed strength values of concrete struts and nodal zones. The authors provided a detailed STM procedure for estimating the failure strength of corbels with tables and figures. In Section 4, the authors concluded the research results of this paper.

Strut-and-Tie Model
Although the current design codes stipulate the use of the STM method for the limit state design of a corbel, they do not recommend STMs for RC corbels specifically. Three types of simple determinate and indeterminate STMs are general for corbel analysis and design. Most books and publications dealing with disturbed regions, including Bergmeister et al. [10], ACI 445 [17], Collins & Mitchel [20], Nawy [21], and Wight [22] introduced a determinate STM representing an arch mechanism, as shown in Figure 1(a). The arch mechanism transfers an external load to a column directly by an inclined concrete strut. KCI [19] introduced a determinate STM representing a truss mechanism, shown in Figure 1(b). The truss mechanism transfers an external load to a column by inclined concrete struts and a horizontal steel tie. Chae & Yun [18] proposed a first-order statically indeterminate STM for RC corbels. The proposed model, defined as a combined mechanism, is shown in Figure 1(c). As the axial forces in the indeterminate STM depend on the struts and ties' axial stiffness, they suggested a load distribution ratio that transforms the indeterminate STM into a determinate one: where the parameters A, B, and C are functions of the shear span-to-effective depth ratio, the compressive strength of the concrete, the primary tensile reinforcement ratio, the minimum flexural reinforcement, and the horizontal-tovertical load ratio. Details of parameters are described in the reference.

Strength Equations of Design Codes
EC2 [13] proposed the following strength equation for struts and nodal zones: where is the design value of the cylinder strength of concrete ′ in MPa. equals / , where is the characteristic value of the cylinder strength of concrete in MPa, (=0.85~1.0) is the coefficient accounting for long term effects on the compressive strength, and (=1.5) is the partial safety factor for the concrete. is 1.0 for the strut in a uniaxial compression stress state and for the nodal zone bounded by struts, bearing area, or both (CCC-nodal zone), 0.85 for the nodal zone anchoring one tie (CCT-nodal zone), 0.6 for the for the strut subjected to tensile stresses transverse to the strut, and 0.75 for the nodal zone anchoring two or more ties (CTT-nodal zone). is 1.0 for the strut in a uniaxial compression stress state, 1 − /250 for the strut subjected to tensile stresses transverse to the strut and for all nodal zones.
FIB [1] also proposed Equation 2, where is 1.0 for the strut in undisturbed uniaxial compression stress states, for the strut in regions with transverse compression, and for CCT-nodal zone. Besides, is 0.75 for the strut with cracks parallel to the direction of compression and tension reinforcement perpendicular to this and for CCT-and CTT-nodal zone, and 0.55 for the strut with reinforcement running obliquely. equals (30/ ) / . The code limits the value of to 1.0, 0.8, and 0.55 for the three cases of struts, as mentioned above, and 1.0 for all nodal zones. DIN [15] proposed Equation 2, too, where is 1.0 for the strut in uncracked compressive zones and CCC-nodal zone and 0.75 for the strut with cracks parallel to the direction of compression and CCT-and CTT-nodal zone. equals 0.4 + 0.6(0.16 /(2200 )) for lightweight concrete and 1.0 for normal weight concrete, where and are the mean values of the axial tensile strength of concrete and the characteristic tensile strength of the reinforcement. CSA [14] proposed the equation for the strength of concrete struts as follows: Where and are the tensile strain of a steel tie crossing a concrete strut and the smallest angle between the steel tie and the concrete strut, respectively. The codes suggested 0.85 , 0.75 , and 0.65 as the strengths of CCC-, CCT-, and CTT-nodal zone, respectively. Here, equals lesser of 2.0 and / ( = area under the bearing device, = area of the lower base of the largest frustum of a pyramid, cone, or tapered wedge contained wholly within the support and having its upper base equal to the loaded area) if a bearing or loading plate connects to the nodal zone. is zero if a bearing or loading plate does not connect to the nodal zone.
Lately, ACI 318 [3] proposed the equation for the strength of concrete struts and nodal zones by considering the shape of a concrete strut, the amount of reinforcement confining the strut, and the influence of concrete confinement effect as follows: where is a strut confinement modification factor. is lesser of / and 2.0 for strut whose end of a strut connected to a node that includes a bearing surface. is a factor that accounts for the effects of cracks and confining reinforcement. Detailed explanations, values, and related equations about for strut strength are given in the reference. is 1.0, 0.8, and 0.6 for CCC-, CCT-, and CTT-nodal zone, respectively.

Strength of Present Study
In the present study, the authors introduce a numerical method to determine 2-D concrete struts and nodal zones' strength levels. The procedure of the method is as follows.
Stage 1: After constructing a finite element model for a concrete member's disturbed region, determine the principal stresses ( and ) and principal angles of the finite elements by conducting a linear elastic finite element analysis. Then, construct an STM that describes a proper load transfer mechanism of the region by considering the principal stress trajectories and patterns of reinforcement. Lastly, collect the finite elements close to the centreline of each concrete strut.
Then, obtain the concrete strut's strength by averaging the principal failure strengths of the finite elements (that located near along the longitudinal axis of the strut) in the standard deviation range. The reason for taking the average in the standard deviation range is to minimize the effect of the variation of the stresses of the finite elements placed along the strut's longitudinal axis. The nodal zone's strength in the direction of the strut's longitudinal axis equals the average of the principal failure strengths of the finite elements located at the nodal zone area.
Stage 4: Conduct a structural analysis of the STM using concrete struts' strengths determined in Stage 3. Then, along with the external forces, apply the steel ties' axial forces to the finite element model of the disturbed region and iterate the procedure from Stage 1. This iteration continues until the difference between the newly and previously determined axial forces of the steel ties reaches a tolerance limit. Figure 2 shows the iterative procedure for considering the confinement effect by reinforcing bars. The procedure mentioned above requires considerable effort and iterative finite element analyses of the unreinforced concrete member and STM. Thus, the authors determined the strengths of concrete struts for the three types of corbel STMs shown in Figure 1 using the above procedure, considering numerous combinations of design variables. By using a curve-fitting process, the authors developed the strength coefficient of the concrete strut, (= / ), as follows: where is the tensile reinforcement ratio defined by / , is the horizontal shear reinforcement ratio defined by / , and / is the shear span-to-effective depth ratio of the corbel. Here, and are the areas of the primary tensile and horizontal shear reinforcing bars that will be (or were) placed be in a corbel, respectively. (= ( × / + )/ , = 0.9 , and is the yield strength of the steel) and (=( × / )/ ) are the areas of the primary tensile and horizontal shear reinforcing bars that resist the external vertical force and horizontal force . The strength coefficient is valid for the corbels with ranges of 0.4 ≤ ≤ 1.0, 0 ≤ ≤ 1.0, and / ≤ 1.40. The parameters ( = 1~4) for the strength coefficient of strut are as follows: Figure 3 shows the strength coefficients and the parameter for struts , , , and . Using the same procedure, the authors also determined the strength of the nodal zone at node 1 (nodal zone 1) of the corbel STMs showed in Figure 1. By using a curve-fitting process, the authors developed the strength coefficient of nodal zone 1, , in the directions of struts , , and as follows: In Equation 8, the maximum value of / is limited to 0.20. The nodal zone at node 2 (nodal zone 2) of the corbel is the CCC-node confined by all the struts framing into the node. Thus, the authors recommend 0.95 for the strength coefficient.

Verifications
Using the three types of STMs shown in Figure 1, the authors predicted 380 RC corbels' failure strengths to examine the properness of the existing and proposed strength values of concrete struts and nodal zones. The corbels were tested by Kriz [42], Romanichen and Souza [43], Al-Talqany and Alhussainy [44], and Shakir [45]. Table 1 shows the specifications and ranges of the primary design variables of the corbels. The references above provide the test setups, rebar details, failure patterns, and additional details.

Strength Prediction Using Determinate STMs
The authors used corbel PG2, one of the corbels tested by Foster et al. [32], to illustrate the procedure for predicting the corbel's failure strength by using determinate STMs. Figure 4 shows the geometrical shape and reinforcement details of the corbel. The shear span-to-effective depth ratio / , the    The authors determined the failure strength of corbel PG2 by using the STM methods of current design codes, requiring the verification of STM elements' load-carrying capacities, including concrete struts, steel ties, and nodal zones. The authors examined the load-carrying capacity by comparing the element's available area with its required area. The available areas of concrete struts and the nodal zone boundaries were determined by the ACI-ASCE 445 approach [46], which considers the STM's geometric shape and the loading and bearing plates' sizes. The steel tie's available area is the same as the area of reinforcing bars placed within the effective width of a steel tie. Figure 6 shows the available widths (areas) of the struts and ties of the determinate STMs. For cases in which two or three struts share one boundary of the nodal zone shown in Figures 6(b) and (c), the authors used the KCI [19] procedure (illustrated in Figure 6(d) for a nodal zone under a loading plate) to determine the available widths of the struts. The authors obtained the required areas of the struts and ties under an experimental failure load by dividing the axial forces (shown in Figure 5) by their strengths. The authors also determined the required areas at the boundaries of a nodal zone by dividing the axial forces of the struts and ties framing the nodal zone by the nodal zone's strength.
The authors accounted for the tension-stiffening effect in the strength predictions by increasing the available area of horizontal steel tie A to the following amount: where , is the cross-sectional area of concrete at the location of steel tie , is the yield strength of the reinforcing bars in the tie, and is the concrete tensile stress suggested by Lee et al. [47]: In Equation 10, and are the reinforcement ratio and the tie's rebar diameter, respectively. Table 2 illustrates a detailed procedure for predicting the failure strength of corbel PG2 with the arch mechanism STM, proposed strut strengths, and Bergmeister et al.'s [10] nodal zone strength. As shown in the table, concrete strut D failed at a load of 888.3 kN (84.6% of the experimental failure load), and nodal zone 1 did not fail at this load. Thus, 84.6% of the failure load proved to be its failure strength. The nodal zone 2 did not fail in all corbels in tests; therefore, the authors did not verify the strength of nodal zone 2. The authors predicted the corbels' failure strengths with variable horizontal coordinates of node 2 using different strength values of concrete struts and nodal zone 1. Table 3     The authors speculate that the more conservative results by the design codes come from using the improper (lower) strengths of struts and nodal zones. The average strength coefficients of the inclined strut (and CCT nodal zone) in the arch mechanism STM by EC2, DIN, FIB, CSA, and ACI 318 are 0.51(0.72), 0.64(0.64), 0.44(0.61), 0.76(0.76), and 0.56(0.69), respectively. Whereas the average strength coefficient by the present study is 0.85(1.03). The differences in the strength coefficients between the design codes and the present study are mainly due to the different ways of considering the confinement effects by reinforcing bars. Besides, the expected failure strengths of 146 corbels (with horizontal shear reinforcing bars) are reasonably well by the truss mechanism STM with the proposed strengths. The same inference about the strength analysis results applies to the truss mechanism STM. The average strength coefficients of the inclined struts B, C, F, and G (and CCT nodal zone) in the truss mechanism STM in Figure 5 11(0.91-1.05), respectively. These outcomes imply that the proposed strut and nodal zone strengths are appropriate for analyzing and designing a corbel with the determinate STMs.

Strength Prediction Using Indeterminate STM
By using the indeterminate STM shown in Figure 1(c), the authors examined the failure strengths of the corbels tested to failure. The same corbel, PG2, was used to illustrate the strength evaluation procedure. Figure 5(c) shows the selected indeterminate STM for corbel PB2. The authors located the steel ties A and E at the centroid of the primary tensile and horizontal shear reinforcing bars. The authors placed the horizontal concrete strut crossing the column to fit the lower boundary line of its width to the corbel's bottom end. The width of the horizontal concrete strut equals the depth of an equivalent rectangular stress block. As the failure strength changes according to the horizontal coordinate of node 2, the authors considered three different cases of horizontal coordinates of node 2, as in the determinate STMs.
The authors determined the failure strength of corbel PG2 by verifying the load-carrying capacities of STM elements, including concrete struts, steel ties, and nodal zones. The authors examined the load-carrying abilities by comparing the elements' available areas with their required areas. The authors obtained the available areas using the ACI-ASCE 445 approach [46], as in the determinate STMs. The authors considered the tension-stiffening effect at the location of steel tie A. Figure 6(c) shows the available widths (areas) of the struts and ties. The required areas of the struts and ties were obtained by dividing the axial forces shown in Figure 5(c) by their strengths. The authors calculated the elements' axial forces using the equilibrium equations at the nodes of determinate STM. Since the STM is an indeterminate truss structure, the authors used the load distribution ratio (%), defined in Equation 1, to transform the indeterminate truss structure into a determinate structure. Table 4 and Figure 7 explain a detailed procedure for evaluating the failure strength of corbel PG2 by using the indeterminate STM with the proposed strut strengths and Bergmeister et al.'s [10] nodal zone strength. Since the indeterminate STM reflects both the arch and truss load transfer mechanisms simultaneously, the authors determined the failure strength of the corbel following the sequential failure of both load transfer mechanisms. As illustrated in Table 4(a) and Figure 7(a), the first failure of the indeterminate STM occurred by the steel tie E's yielding at a load of 289.5 kN (27.6% of the experimental failure load). After the first failure, the indeterminate STM became determinate arch mechanism STM one, as shown in Figure 7(b). The arch mechanism STM could still transfer a fraction of the applied load to the column by steel tie A and inclined strut D. The authors added the remaining areas of struts B and C to the area of strut D at the boundary of nodal zone 1. As shown in Table 4(b) and Figure 7(c), the second failure of the STM occurred due to strut D's failure under an additional load of 709.7 kN (67.6% of the experimental failure load). After the second failure, the STM became an unstable truss structure that could not carry any additional load, as shown in Figure 7(d). At 999.2 kN (289.5+709.7 kN, 95.2% of the experimental failure load), which the indeterminate STM could carry to the utmost limit by the concrete struts and steel ties, the authors examined the strength of the nodal zone 1, as illustrated in Table 4(c) and Figure 7(e). Since nodal zone 1 did not fail, 95.2% of the experimental failure load becomes the corbel's failure strength. Since the nodal zones 2, 3, and 4 did not fail in all corbel tests, the authors did not verify those nodal zones' failure. Similarly, the authors predicted the corbels' failure strengths with variable horizontal coordinates of node 2 by using different strength values of concrete struts and nodal zone 1. : available width of concrete strut after the first failure (refer to Figure 7(b)).    0%), respectively. The proposed strut and nodal zone strengths with the indeterminate STM predicted the failure strengths most accurately, implying that the recommended strength values are adequate for corbels' design using the indeterminate STM. The more conservative results by the design codes come from using the lower strengths of struts and nodal zones. The average strength coefficients of the inclined strut D (and CCT nodal zone) in the combined mechanism STM in Figure 5 Tables 3 and 5, the indeterminate STM yields better results than the determinate STMs, indicating that corbels' failure strengths are affected by the type of STM used. Figure 8 compares some of the strength analysis results.

Conclusion
For the safe and economical design of RC corbels using the STM methods, designers must accurately determine the strength of concrete struts and nodal zones. In this study, the authors suggested a numerical process for determining the strengths of concrete struts and nodal zones in RC corbels. Besides, for practical application of the recommended process, the authors proposed the strut strength equations that apply to the typical determinate and indeterminate STMs of RC corbels. The authors considered the effects of the primary corbel design variables, including the shear span-to-effective depth ratio, the horizontal-to-vertical load ratio, and the primary tensile and horizontal shear reinforcement ratios in the development of the strength equations. The authors evaluated the validity of the proposed strut and nodal zone strengths by predicting tested 391 RC corbels' failure strengths. In the strength prediction, the authors employed three typical determinate and indeterminate corbel STMs. The average ratios of the calculated strength to experimental strength (and COV) by the strut and nodal zone strengths of EC2, DIN, FIB, CSA, ACI 318, and the present study are 0.74~0.77 (32.9~34.9%), 0.81~0.82 (31.1~32.0%), 0.64~0.65 (36.6~37.9%), 0.70~0.88 (29.5~39.0%), 0.68~0.85 (30.5~39.7%), and 0.94~0.99 (27.0~29.6%), respectively. The failure strengths predicted by the proposed strengths are consistent. They agree reasonably well with experimental results, indicating that the proposed strut and nodal zone strengths can be useful for RC corbels' rational STM design.

Data Availability Statement
The data presented in this study are available on request from the corresponding author.