An optimum indeterminate strut-and-tie model for reinforced concrete corbels

Abstract

The failure behavior of a reinforced concrete corbel is complicated due to the shear span-to-effective depth ratio, reinforcement patterns, load conditions, and material properties. In this study, an optimum first-order indeterminate strut-and-tie model that reflects all characteristics of the failure behavior is proposed for the rational design of reinforced concrete corbels with a shear span-to-effective depth ratio of less than 1.0. A load distribution ratio that transforms the indeterminate strut-and-tie model into a determinate model is also developed to help structural designers design reinforced concrete corbels using the strut-and-tie model methods of current design codes. For the development of the load distribution ratio, a material nonlinear finite element analysis of the proposed first-order indeterminate strut-and-tie model was conducted repeatedly by changing the combination of primary design variables of the corbels. To examine the validity of our results, the ultimate strengths of 294 reinforced concrete corbels tested to failure by other investigators were predicted using the proposed strut-and-tie model with the load distribution ratio, the existing determinate strut-and-tie models representing arch and truss load transfer mechanisms, and the American Concrete Institute 318 conventional design method based on a shear friction theory. The ultimate strengths predicted by the proposed strut-and-tie model agreed fairly well with the experimental results. The ratio of the experimental strength to the predicted strength (and coefficient of variation) was 1.09 (28.0%), implying that the proposed strut-and-tie model can represent the load transfer mechanisms of corbels most appropriately.

Keywords

corbel load distribution ratio reinforced concrete strut-and-tie model ultimate strength

Introduction

Reinforced concrete corbels are used extensively in precast concrete construction to support primary beams and girders. Because of the prevalence of precast concrete, the design of corbels has become increasingly important. Many experimental studies have been conducted (Abdulhaleem et al., 2018; Campione, 2009; Fattuhi, 1994; Fattuhi and Hughes, 1989; Foster et al., 1996; Gulsan and Shaikhan, 2018; Kriz and Raths, 1965; Kurtoglu et al., 2017; Mattock et al., 1976; Yong and Balaguru, 1994). The experimental studies proved that the ultimate strength of reinforced concrete corbels is influenced mostly by the shear span-to-effective depth ratio, the compressive strength of the concrete, the patterns and amount of reinforcing bars, and the geometrical shape. The studies also proved that the corbels tend to fail in several modes. The most common modes are yielding of the main tensile reinforcement, the crushing or splitting of a compression strut, shearing failure along the column face, and localized bearing failure under a loading plate.

Several methods for the analysis and design of corbels by considering the failure modes and the effects of primary design variables have been introduced. These include the plasticity-based method (Jensen, 1979, 1982), the shear friction model methods (American Concrete Institute (ACI) 318, 2014; Hermansen and Cowan, 1974; Kriz and Raths, 1965; Mast, 1968; Shaikh, 1978), and the truss analogies and strut-and-tie model (STM) methods (Ali and White, 2001; Campione, 2009; Hagberg, 1983; Hwang et al., 2000; Rogowsky and MacGrogor, 1983; Russo et al., 2006; Solanki and Sabnis, 1987; Yun et al., 2007). Among these, the STM method has proven to be effective for the ultimate strength design of structural concrete having geometric and/or static discontinuities. For that reason, the major design codes including Canadian Standards Association (CSA, 2004), European Committee for Standardization (EC2, 2004), Building and Civil Engineering Standards Committee (2008), Comite Euro-International du Beton (2010), Korean Concrete Institute (KCI, 2012), ACI 318 (2014) and American Association of State Highway and Transportation Officials (AASHTO, 2018) have accepted the STM method. However, even if the basic concepts of the method and the strength of the struts, ties, and nodal zones are provided in current design codes, specific and practical standards regarding the STM construction are not suggested. The STMs must be constructed based on the designer’s experience and subjectivity. Hence, different design results can be obtained under the same circumstances, and the reliability of the design results can be greatly reduced if a designer is deficient in understanding the structural behavior and load transfer mechanism. In addition, the use of simple determinate STMs for the simplicity of structural analysis and satisfaction of the lower bound theorem of plasticity is recommended in current design codes. However, idealizing an STM as a simple determinate truss structure by simplifying complex internal stress trajectories causes difficulties in reflecting the actual structural behavior. Also, simple determinate STMs limit the effective reinforcement details by inducing difficulties in terms of practical horizontal and/or vertical placements of reinforcing bars.

In this study, an optimum first-order statically indeterminate STM for the rational design of reinforced concrete corbels with shear span-to-effective depth ratio of less than 1.0 is proposed. The proposed model not only overcomes the limitations of existing STM methods of current design codes from the viewpoint of the STM construction, but also reflects all characteristics of ultimate strength and the complicated structural behavior of corbels. In addition, a load distribution ratio by conducting numerous finite element material nonlinear analyses of the proposed indeterminate STM with changeable primary design variables is developed. The load distribution ratio helps structural designers design reinforced concrete corbels using the STM methods of current design codes since it provides a reasonable basis to transform the indeterminate STM into a determinate model. To examine the validity of the proposed STM and load distribution ratio, the ultimate strength of 294 reinforced concrete corbels tested to failure by other investigators was predicted using the ACI 318 STM method associated with the proposed indeterminate STM and load distribution ratio. For comparison, the ultimate strength of the corbels was also predicted using the ACI 318 conventional design method and the existing determinate STMs representing arch and truss load transfer mechanisms.

Proposal of STM and load distribution ratio

STM

The construction of an STM that reflects the true load transfer mechanism of structural concrete is the most important element for the application of STM methods in current design codes. Although a reinforced concrete corbel is one of the representative concrete members with disturbed regions, the current design codes do not recommend an STM for the practical design of corbels. Multiple books including Collins and Mitchell (2001), Wight et al. (2011), and Nawy et al. (2011) have introduced a determinate STM representing an arch load transfer mechanism, shown in Figure 1(a), in which an external load is directly transferred to a column by an inclined concrete strut. Yun and Ramirez (1994) and KCI (2013) introduced a determinate STM representing a horizontal truss mechanism, shown in Figure 1(b), in which an external load is transferred to the column by the combination of inclined concrete struts and a horizontal steel tie. In this study, a first-order statically indeterminate STM for reinforced concrete corbels with a shear span-to-effective depth ratio of less than 1.0 is proposed. The proposed model, defined as a combination of arch and horizontal truss mechanisms, is shown in Figure 1(c).

Figure 1.

Strut-and-tie models for reinforced concrete corbels: (a) arch mechanism, (b) horizontal truss mechanism, and (c) combined mechanism.

Load distribution ratio

The cross-sectional forces of struts and ties in the determinate STMs shown in Figure 1(a) and (b) are determined, regardless of the axial stiffness of the struts and ties. However, the cross-sectional forces of struts and ties in the indeterminate STM shown in Figure 1(c) depend on the axial stiffness of struts and ties. Therefore, the axial stiffness of struts and ties must be determined appropriately. The axial stiffness can be determined numerically using an iterative technique that requires the following steps. Step A: Assume a value for the axial stiffness of every strut and tie. Step B: Determine the cross-sectional forces of struts and ties by conducting a linear elastic finite element analysis of the indeterminate STM. Step C: Calculate the cross-sectional areas of the struts and ties by dividing the cross-sectional forces of the struts and ties by their corresponding effective strength values. Obtain the axial stiffness by multiplying the cross-sectional areas of the struts and ties by their modulus of elasticity. The elastic moduli of concrete and steel are taken as the moduli of the strut and tie. Step D: Return to Step A if the difference in the axial stiffness determined at the previous and current iterative steps exceeds a tolerance limit. The proposed procedure, which requires a finite element analysis of the indeterminate STM repeatedly, is not practical if a computer program developed for this purpose is not provided.

In this study, a load distribution ratio that transforms the first-order indeterminate STM shown in Figure 1(c) into a determinate one is developed to avoid the aforementioned iterative procedure. The load distribution ratio is defined as $F_{E} / P$ , where P and $F_{E}$ are the vertically applied external load and the cross-sectional force of steel tie E in Figure 1(c), respectively. The load distribution ratio helps structural designers design reinforced concrete corbels using the proposed indeterminate STM along with the provisions of the current design codes. In the development of the load distribution ratio, the effects of the primary design variables, including the shear span-to-effective depth ratio $a / d$ , the main tensile reinforcement ratio $ρ$ , the cylinder strength of concrete $f_{c}'$ , and the horizontal-to-vertical load ratio $N_{u} / P$ , are considered. The algorithm used for developing the load distribution ratio is shown in Figure 2. Details of the algorithm are explained in the following steps.

Step 1: Determine the primary design variables including $a / d$ , $f_{c}'$ , and $ρ$ . Assume a maximum vertical load $P_{max}$ in a corbel. To verify whether the assumed maximum load is a true value, define the maximum available cross-sectional areas of concrete struts B, C, D, F, and G in Figure 1(c). For the definition of the maximum available areas of the struts, the required width of loading plate $l_{b, 1}$ for $P_{max}$ and the required width of non-hydrostatic nodal zone 2 $l_{b, 2}$ for the cross-sectional force R of concrete strut I as shown in Figure 3 are determined in order to satisfy the ACI 318 (2014) strength requirement for non-hydrostatic nodal zones, which are expressed as follows

l_{b, 1} = \frac{P_{max}}{0.68 f_{c}' b}

(1a)

l_{b, 2} = \frac{R}{0.85 f_{c}' b}

(1b)

where b is the thickness of the corbel. The maximum available width of horizontal concrete strut H (which is the same as the depth of the equivalent rectangular stress block) is determined as follows

w_{H} = \frac{f_{y} A_{s}}{0.85 f_{c}' b}

(2)

In equation (2), $A_{s}$ is the cross-sectional area of steel tie A ( $= ρ bd$ , where d is the effective depth of the corbel) and $f_{y}$ is the yield strength of the steel. The width of steel tie can vary depending on the distribution of tie reinforcement. As the reinforcing bars in tie A are generally in one layer, the width of tie A, $w_{A}$ , is taken to be twice the distance from the center of the main tensile reinforcement to the top of the corbel. Then, the maximum available widths of inclined concrete struts C, D, and F placed at the shear span are determined as follows

w_{C} = w_{A} \cos θ_{2} + l_{b . 1} \sin θ_{2}

(3)

w_{D} = min (w_{H} \cos θ_{1} + l_{b, 2} \sin θ_{1}, w_{A} \cos θ_{1} + l_{b, 1} \sin θ_{1})

(4)

w_{F} = w_{H} \cos θ_{2} + l_{b . 2} \sin θ_{2}

(5)

where $θ_{1}$ and $θ_{2}$ are the angles between the inclined concrete struts (D and F) and the horizontal axis, as shown in Figure 1(c). Note that the width $w_{D}$ should not overlap with the non-hydrostatic nodal zone areas at nodes 3 and 4. The maximum available widths of the concrete struts are shown in Figure 3. The maximum available areas of the concrete struts are obtained by multiplying their widths by the thickness of the corbel.

Step 2: Determine the effective strengths of the concrete struts and steel tie. The yield strength of steel is taken as the effective strength of the steel tie. The effective strengths of the concrete struts are determined using Yun’s (2005) numerical method, which determines the effective strength of any type of concrete strut accurately by incorporating the results of a two-dimensional (2D) linear elastic finite element analysis of an unreinforced concrete corbel. In Yun’s method, the effects of the principal stresses associated with the tensile strains of reinforcing bars crossing a strut, the longitudinal length of a strut, the deviation angle between the strut orientation and the compressive principal stress flow, the compressive strength of the concrete, and the degree of concrete confinement due to reinforcing bars are considered.

Step 3: By using iterative steps (A) to (D), determine the axial stiffness of the struts and ties. Then, determine the cross-sectional forces of the struts and ties by conducting a finite element material nonlinear analysis of the indeterminate STM. For the material nonlinear analysis, the tangent modulus of elasticity of a concrete strut is obtained by differentiating Pang and Hsu’s (1995) stress–strain relationship for concrete with the compressive strain of a concrete strut $ε_{c}$ as expressed in equation (6)

\begin{matrix} {E_{c}}^{t} = E_{c} [1 - \frac{ε_{c}}{ζ ε_{0}}] for ε_{c} / ζ ε_{0} ⩽ 1 \\ {E_{c}}^{t} = - E_{c} [\frac{ε_{c} / ζ ε_{0} - 1}{{(2 / ζ - 1)}^{2}}] for ε_{c} / ζ ε_{0} > 1 \end{matrix}

(6)

where $ζ$ (= $f_{cs} / f_{c}'$ , where $f_{cs}$ is the effective strength of concrete strut as determined in Step 2) is the softening coefficient of the concrete, and $ε_{0}$ (= $2 f_{c}' / E_{c}$ , where $E_{c}$ is the initial modulus of elasticity of concrete) is the compressive strain corresponding to the compressive strength of concrete. The tangent modulus of elasticity of a steel tie is obtained by assuming a bilinear stress–strain relationship for steel as follows

\begin{matrix} {E_{s}}^{t} = E_{s} for ε_{s} ⩽ ε_{y} \\ {E_{s}}^{t} = 0.001 E_{s} for ε_{s} > ε_{y} \end{matrix}

(7)

where $E_{s}$ is the initial modulus of elasticity of the steel.

Step 4: Check whether the norm of the cross-sectional forces of struts and ties between the previous and current iterative steps is within a tolerance limit of 0.001. If the norm exceeds the tolerance limit, return to Step 2. Then, along with the external load, apply the cross-sectional forces of the steel ties (as the updated confining forces) to the finite element model of the unreinforced concrete corbel. If the norm does not exceed the tolerance limit, check whether the difference between the required and the maximum available cross-sectional areas of the struts is within a tolerance limit. If the difference does not exceed the tolerance limit, determine the load distribution ratio as defined in Figure 2. Otherwise, return to Step 1 and modify the maximum applied load, $P_{\max}$ .

Figure 2.

Algorithm for determining the load distribution ratio of the indeterminate strut-and-tie model.

Figure 3.

Maximum available cross-sectional areas of concrete struts.

According to the algorithm shown in Figure 2, the load distribution ratio was determined for the corbel shown in Figure 4. In the corbel, the primary design variables $a / d$ , $ρ / ρ_{min}$ , $f_{c}'$ , and $N_{u} / P$ vary in the ranges of 0.2–1.0, 1.0–5.0, 20–70 MPa, and 0–1, respectively. Here, $ρ_{min}$ is defined as $0.04 f_{c}' / f_{y}$ . An example of the load distribution ratios, determined at given design conditions, is shown in Figure 5. The load distribution ratios are arranged by the shear span-to-effective depth ratio and show a similar tendency to those suggested by Chae (2012) for the indeterminate STM (representing a combined arch and vertical truss load transfer mechanism) of reinforced concrete deep beams. In addition, as the ratio $a / d$ decreases, the load distribution ratio decreases, and the vertical load transferred by the arch mechanism increases. This phenomenon can be intuitively predicted for corbels. Furthermore, the load distribution ratios change nonlinearly according to the primary design variables. This implies that the load carrying capacities of the arch and the horizontal truss load transfer mechanisms of the indeterminate STM are influenced in a complex manner by the primary design variables, and that the structural behavior of corbels is complicated.

Figure 4.

Numerical analysis model of corbel for determining load distribution ratio.

Figure 5.

Example of load distribution ratio associated with primary design variables ( $ρ / ρ_{min} = 1.0$ , $N_{u} / P = 0.2$ ).

In this study, we developed an equation for the load distribution ratio, $α$ (= $F_{E} / P \times 100$ ). The equation can be directly applied to the design of a corbel. The proposed equation is

α (%) = β (a / d - η)^{2} + γ

(8)

where the parameters $η$ , $β$ , and $γ$ characterize the variation of the load distribution ratio according to primary design variables and are defined as follows

\begin{matrix} β = {\begin{matrix} - 92 + 32 κ_{n} + (11 - 10 κ_{n}) ρ / ρ_{min} - (0.3 + 0.5 κ_{n}) f_{c}' for a / d ⩽ η \\ - 25 + 3 ρ / ρ_{min} - (2 - 0.1 f_{c}') κ_{n} for a / d > η \end{matrix} \\ η = 0.36 + 0.08 κ_{n} + 0.02 (1 - κ_{n}) ρ / ρ_{min} + 0.002 (1 + κ_{n}) f_{c}' \\ γ = 18.9 + 2.1 κ_{n} - (0.2 + 0.1 κ_{n}) ρ / ρ_{min} + 0.13 f_{c}' \end{matrix}

(9)

In equation (9), $κ_{n}$ (= $N_{u} / P$ ) is the horizontal-to-vertical load ratio.

Verification

To examine the appropriateness of the proposed first-order indeterminate STM and load distribution ratio, the ultimate strengths of 294 reinforced concrete corbels, tested by Kriz and Raths (1965), Mattock et al. (1976), Fattuhi and Hughes (1989), Yong and Balaguru (1994), Fattuhi (1994), and Foster et al. (1996), were predicted using the three types of STMs shown in Figure 1. The specifications and ranges of the primary design variables of the corbels are given in Table 1. The test setups, rebar details, failure patterns, and additional details are given in the aforementioned references.

Table 1.

Specification of reinforced concrete corbels tested to failure.

Investigators	Number of corbels	b (mm)	c (mm)	d (mm)	h (mm)	$f_{c}'$ (MPa)	$f_{y}$ (MPa)	$a / d$	$ρ / ρ_{min}$
Kriz and Raths (1965)	185	203–406	152–610	307–1059	457–660	16.9–42.2	293–498	0.11–0.62	0.601–6.675
Mattock et al. (1976)	28	152	152–330	221–231	254	23.8–30.7	321–447	0.22–1.02	2.956–9.615
Fattuhi and Hughes (1989)	12	152	150	103–135	148	47.9–57.4	491–558	0.66–0.86	1.961–2.977
Yong and Balaguru (1994)	14	254	356	305	406	39.2–79.5	420	0.25–0.75	0.902–3.154
Fattuhi (1994)	25	150–155	150	210–231	248–251	25.9–31.3	427–454	0.43–1.04	1.651–5.542
Foster et al. (1996)	30	125–150	300–550	450–740	600–800	45.0–105	415–495	0.30–1.00	0.678–8.189
Total	294	125–406	150–610	103–1059	148–800	16.9–105	293–558	0.11–1.04	0.601–9.615

b, c, d, h: width, overhanging length, effective depth, height of corbel; $f_{c}'$ : compressive strength of concrete; $f_{y}$ : yield strength of steel; $a / d$ : shear span-to-effective depth ratio; $ρ$ : flexural reinforcement ratio; $ρ_{min}$ : minimum flexural reinforcement ratio.

Strength prediction using determinate STMs

The procedure for predicting the ultimate strengths of corbels using the determinate STMs of arch and truss load transfer mechanisms is illustrated with corbel PG1, which is one of the corbels tested by Foster et al. (1996). The geometrical shape and reinforcement details of the corbel are shown in Figure 6. The shear span-to-effective depth ratio $a / d$ , main tensile reinforcement ratio $ρ / ρ_{min}$ , compressive strength of the concrete, yield strengths of the main tensile and horizontal shear reinforcing bars, and the width of loading plate are 0.60, 6.04, 45.0 MPa, 415 and 490 MPa, and 100 mm, respectively.

Figure 6.

Geometrical shape and Rebar details of corbel PG1 (from Foster et al., 1996).

The two determinate STMs for corbel PG1 are shown in Figure 8(a) and (b). In the models, steel ties A and E were placed at the centroid of the main tensile and horizontal shear reinforcing bars, respectively. The horizontal concrete struts crossing the column were placed to fit the lower boundary lines of their widths to the bottom end of the corbel. The widths of the horizontal concrete struts were taken as the depth of an equivalent rectangular stress block (= $A_{s} f_{y} / (0.85 f_{c}' b)$ , where $A_{s}$ is the cross-sectional area of the main tensile reinforcement and b = 102 mm). The locations of the vertical concrete struts placed at the column depend on the horizontal coordinate of node 2. In this study, as the ultimate strength determined by the STMs changes according to the horizontal coordinate of node 2, three horizontal coordinates of node 2 (located at 20 mm, 40 mm, and 60 mm away from the left face of the column) were considered.

The ultimate strength of corbel PG1 was determined using the STM methods of current design codes, requiring verifications of the load carrying capacities of all the elements including struts, ties, and nodal zones. The load carrying capacity of an element of the STM was examined by comparing the maximum available cross-sectional area of the element with its required cross-sectional area. The maximum available areas of concrete struts and nodal zone boundaries were determined by the ACI-ASCE Subcommittee 445 (2010) approach, which considers the geometrical shape of the selected STM and the size of the loading and bearing plates. The cross-sectional area of reinforcing bars placed within the effective width of a steel tie was taken as the maximum available area of the steel tie. The maximum available widths (areas) of the struts and ties of the determinate STMs are shown in Figure 8. In the STMs, the horizontal coordinate of node 2 is located at distance of 5% of the column width. As the thickness of the corbel is constant, the cross-sectional areas of the concrete struts and nodal zone boundaries were obtained by multiplying the cross-sectional widths and the thickness together. For the case in which two struts share one nodal zone boundary as shown in Figure 8(b), the maximum available widths of the struts were determined by the procedure illustrated in the STM design example book of KCI (2013).

The required cross-sectional areas of the struts and ties under the experimental failure load of 674 kN were determined by dividing the cross-sectional forces (shown in Figure 7) by their effective strengths. The required cross-sectional areas at the boundaries of a nodal zone were determined by dividing the cross-sectional forces of the struts and ties framing the nodal zone by the effective strength of the nodal zone. The effective strengths of the struts determined using the provisions of Building and Civil Engineering Standards Committee (2008), ACI 318 (2014), AASHTO (2018), and Yun’s (2005) numerical method were used. These values are listed in Table 2. The yield strength of the reinforcement was taken as the effective strength of the steel tie. For the corbel whose experimental failure strength $P_{u}$ is greater than the experimental yield strength $P_{y}$ , the modified strength shown in equation (10) was taken as follows

f_{y, mod} = \frac{P_{u} a + N_{uu} (h - d)}{P_{y} a + N_{uy} (h - d)} f_{y}

(10)

where h is the height of the corbel, and $N_{uy}$ and $N_{uu}$ are the horizontal tensile forces when the corbel is subjected to the experimental yield and failure strengths, respectively. The effective strength of the nodal zone connected to the loading plate was determined using the following equation proposed by Bergmeister et al. (1993)

f_{cn} = ν_{e} f_{c}' (A / A_{b})^{0.5} ⩽ 2.5 f_{c}'

(11)

where $ν_{e}$ is 0.8 for $f_{c}' ⩽ 28$ MPa, $0.9 - 0.25 f_{c}' / 70$ for $28 < f_{c}' < 70$ MPa, and 0.65 for $f_{c}' ⩾ 70$ MPa. A and $A_{b}$ are the areas of bearing plate and the confined concrete concentric with (and geometrically similar to) the bearing plate, respectively. The ratio $A / A_{b}$ should be less than 4.0. The effective strength for the nodal zone without the bearing plate was determined by multiplying $ν_{e}$ and $f_{c}'$ .

Figure 7.

Strut-and-tie models for corbel PG1: (a) arch mechanism, (b) horizontal truss mechanism, and (c) combined mechanism.

Table 2.

Coefficients of effective strengths of struts in strut-and-tie models for corbel PG1.

Effective strut strength
Model	Strut number	Building and Civil Engineering Standards Committee (2008)	American Concrete Institute 318 (2014)	American Association of State Highway and Transportation Officials (2018)	Yun (2005)
Arch mechanism model	D	0.64	0.64	0.79	0.86
Horizontal truss mechanism model	B	0.64	0.64	0.85	0.73
	C	0.64	0.51	0.45	0.75
	F	0.64	0.51	0.45	0.95
	G	0.64	0.64	0.85	0.95

Tables 3 and 4 show a detailed procedure for predicting the ultimate strength of corbel PG1 using the determinate STM shown in Figure 1(a) with Yun’s (2005) effective strength values of concrete struts. Concrete strut D failed at a load of 542.6 kN (80.5% of the experimental failure load), and the nodal zone at node 1 did not fail at that load. Thus, 80.5% of the experimental failure load was predicted as its ultimate strength. As the nodal zone at node 2 did not fail in all corbels in the tests, the strength of the nodal zone was not verified in this study. Similarly, the ultimate strengths of the other corbels were predicted using different effective strength values of concrete struts and the variable horizontal coordinate of node 2. The strength prediction results are listed in Tables 5(a) and (b). Note that the ultimate strengths were predicted very conservatively using the determinate STM representing the horizontal truss mechanism. This is because a small number of horizontal shear reinforcing bars in the corbel specimens were used, causing the yielding of horizontal steel tie E at low levels of applied load.

Table 3.

Prediction of ultimate strength of corbel PG1 by determinate STM representing arch load transfer mechanism: Strength verification of struts and tie.

Strut number	$ν_{s}$	$f_{c}'$ (MPa)	$f_{cs}$ (MPa)	$F_{u}$ (kN)	$w_{req' d}$ (mm)	$w_{pro' d}$ (mm)	$w_{pro' d} / w_{req' d}$	Safety
D	0.86	45.0	38.6	840.9	145.4	117.1	0.805	×
Tie number	$ν_{t}$	$f_{y}$ (MPa)	$f_{st}$ (MPa)	$F_{u}$ (kN)	$A_{req' d}$ (mm²)	$A_{pro' d}$ (mm²)	$A_{pro' d} / A_{req' d}$	Safety
A	1.00	415.0	415	502.8	1111.6	1965.0	1.622	○

$ν_{s}$ and $ν_{t}$ : coefficients of effective strengths of strut and tie; effective strength of concrete strut $f_{cs} = ν_{s} f_{c}'$ ; effective strength of steel tie $f_{st} = ν_{t} f_{y}$ ; $F_{u}$ : cross-sectional force under experimental failure load of 674 kN; $w_{req' d} = F_{u} / (b f_{cs})$ ; $w_{pro' d}$ : maximum available width of concrete strut (refer to Figure 8(a)). The bold number in table are the numbers mentioned in the text.

Table 4.

Prediction of ultimate strength of corbel PG1 by determinate STM representing arch load transfer mechanism: Strength verification of nodal zones.

Node number	Node type	$ν_{e}$	A, $A_{b}$ (mm²)	$ν_{n}$	$f_{c}'$ (MPa)	$f_{cn}$ (MPa)	$F_{u}$ (kN)		$w_{req' d}$ (mm)	$w_{pro' d}$ (mm)	$\frac{w_{pro' d}}{w_{req' d}}$	Safety
1	CCT	0.74	500 × 150, 100 ×150	1.48	45.0	66.5	P	542.6	54.4	100.0	1.839	○
							Strut D	677.0	53.8	223.6	4.158	○
							Tie A	404.8	40.6	200.0	4.931	○

C: compression; T: tension.

$ν_{n}$ ( $= ν_{e} \sqrt{A / A_{b}}$ , $A / A_{b} ⩽ 4$ ): coefficient of effective strength of nodal zone; effective strength of nodal zone $f_{cn} = ν_{n} f_{c}'$ ; $F_{u}$ : cross-sectional force at nodal zone boundary under 80.5% of experimental failure load; P: applied load (= 80.5% of experimental failure load); $w_{pro' d}$ : maximum available width of nodal zone boundary (refer to Figure 8(a)).

Table 5.

Strength ratio $P_{test} / P_{cal}$ obtained by determinate and indeterminate strut-and-tie models.

Model & results
	Building and Civil Engineering Standards Committee (2008)	American Concrete Institute 318 (2014)	American Association of State Highway and Transportation Officials (2018)	Yun (2005)
(a) Determinate STM representing arch load transfer mechanism
(A)	2.18	2.60	1.69	1.64
(B)	1.43	1.69	1.17	1.16
(C)	1.19	1.37	1.04	1.05
Mean (294)*	1.49	1.74	1.22	1.22
COV (%)	37.6	41.7	30.9	30.8
(b) Determinate STM representing horizontal truss load transfer mechanism
(A)	2.45	2.45	2.75	2.39
(B)	2.48	2.48	2.67	2.45
(C)	2.64	2.64	2.73	2.62
Mean (84)*	2.50	2.50	2.68	2.47
COV (%)	49.3	49.3	44.1	50.8
(c) Indeterminate STM representing combined mechanism of present study
(A)	1.42	1.44	1.32	1.14
(B)	1.21	1.37	1.15	1.06
(C)	1.14	1.15	1.13	1.08
Mean (84)*	1.23	1.24	1.18	1.09
COV (%)	32.3	28.0	26.8	28.0

number of corbels; COV: coefficient of variation.

(A), (B), (C): strut-tie models whose horizontal coordinates of node 2 are located at distances of 5%, 10%, and 15% of column width from the left face of column, respectively.

Strength prediction using indeterminate STM

A detailed procedure for predicting the ultimate strength of corbels using the first-order indeterminate STM shown in Figure 1(c) is illustrated using the same corbel, PG1. The indeterminate STM for the corbel is shown in Figure 7(c). The locations of steel ties and concrete struts are the same as those of the determinate STMs shown in Figure 8(a) and (b). The width of the horizontal concrete strut placed at the column was taken as the depth of an equivalent rectangular stress block.

Figure 8.

Maximum available widths (areas) of struts, ties, and non-hydrostatic nodal zones in strut-and-tie models of corbel PG1: (a) arch mechanism, (b) horizontal truss mechanism, and (c) combined mechanism.

The ultimate strength of the corbel was determined by verifying the load carrying capacities of all the elements of the indeterminate STM. The load carrying capacity of an element of the STM was examined by comparing the available and required cross-sectional areas of the element. The maximum available areas of concrete struts, steel ties, and nodal zones, as shown in Figure 8(c), were determined by the ACI-ASCE Subcommittee 445 (2010) approach, as in the cases of the determinate STMs. The required cross-sectional areas of the struts and ties under the experimental failure load were determined by dividing the cross-sectional forces (shown in Figure 7) by their effective strengths. The required cross-sectional areas at the boundaries of nodal zones and the effective strengths of the struts and ties were determined in a manner similar to the cases of the determinate STMs.

To obtain the cross-sectional forces of the struts and ties, the load distribution ratio $α$ (= $F_{E} / P \times 100$ , where $F_{E}$ is the cross-sectional force of horizontal steel tie E and P is the experimental failure load) was determined first using equation (8)

α (%) = β (a / d - η)^{2} + γ = - 6.88 (0.60 - 0.57)^{2} + 23.5 \approx 23.5

where

\begin{matrix} β = - 25 + 3 ρ / ρ_{min} = - 6.88 \\ η = 0.36 + 0.02 ρ / ρ_{min} + 0.002 f_{c}' = 0.57 \\ γ = 18.9 - (0.2) ρ / ρ_{min} + 0.13 f_{c}' = 23.5 \end{matrix}

In calculating parameters $β$ , $η$ , and $γ$ using equation (9), $κ_{n}$ was zero since the horizontal load $N_{u}$ was not applied in the test. After determining $F_{E}$ , the first-order indeterminate STM became a determinate model. Thus, the cross-sectional forces of the struts and ties were obtained by conducting a structural analysis of the determinate STM with applications of the equilibrium equations at the nodes.

The detailed procedure for predicting the ultimate strength of corbel PG1 using the indeterminate STM shown in Figure 1(c) with Yun’s (2005) effective strength values of concrete struts is given in Tables 6 to 8. Since the indeterminate STM reflects both the arch and horizontal truss load transfer mechanisms at the same time, the ultimate strength of the corbel was determined according to the sequential failure of both the load transfer mechanisms. As illustrated in Table 6, the first failure of the indeterminate STM occurred due to the yielding of horizontal tie E at a load of 375.4 kN (55.7% of the experimental failure load). After the first failure, the indeterminate STM became a determinate model that was still able to transfer a fraction of the applied load to the column by the steel tie A and the inclined strut D forming the arch load transfer mechanism. The remaining cross-sectional areas of struts B and C at the boundary of nodal zone 1 were transformed to the direction of strut D, and the transformed areas were added to the remaining cross-sectional area of strut D after the first failure. When an additional load of 320.5 kN (47.5% of the experimental failure load) was applied, the second failure of the STM occurred due to the failure of strut D, as shown in Table 7. After the second failure, the STM became an unstable truss structure that could not carry any additional load. At a load of 695.9 kN (375.4 + 320.5 kN, which is 103.2% of the experimental failure load), which the indeterminate STM could carry to the utmost limit by concrete struts and steel ties, the strength of the nodal zone at node 1 was examined as shown in Table 8. Since the nodal zone did not fail, 103.2% of the experimental failure load was determined to be the ultimate strength of the corbel. As the nodal zones at nodes 2, 3, and 4 did not fail in all corbels in the tests, the strength verifications of the nodal zones were excluded in this study. In the same way, the ultimate strengths of the other corbels were predicted using different effective strength values of concrete struts and the variable horizontal coordinate of node 2. The strength analysis results are shown in Table 5(c).

Table 6.

Prediction of ultimate strength of corbel PG1 by indeterminate STM representing combined load transfer mechanism: Strength verification of struts and ties at the first failure state.

Strut number	$ν_{s}$	$f_{c}'$ (MPa)	$f_{cs}$ (MPa)	$F_{u}$ (kN)	$w_{req' d}$ (mm)	$w_{pro' d}$ (mm)	$w_{pro' d} / w_{req' d}$	Safety
B	0.55	45.0	24.5	106.3	28.9	31.6	1.096	○
C	0.70	45.0	31.3	191.0	40.6	56.8	1.399	○
D	0.88	45.0	39.4	575.6	97.3	127.5	1.311	○
F	0.89	45.0	40.2	191.0	31.7	42.3	1.334	○
G	0.89	45.0	40.2	106.3	17.7	23.5	1.334	○
Tie number	$ν_{t}$	$f_{y}$ (MPa)	$f_{st}$ (MPa)	$F_{u}$ (kN)	$A_{req' d}$ (mm²)	$A_{pro' d}$ (mm²)	$A_{pro' d} / A_{req' d}$	Safety
A	1.00	415	415	502.8	1211.6	1965.0	1.622	○
E	1.00	420	420	158.6	377.7	210.4	0.557	×

$ν_{s}$ and $ν_{t}$ : coefficients of effective strengths of strut and tie; effective strength of concrete strut $f_{cs} = ν_{s} f_{c}'$ ; effective strength of steel tie $f_{st} = ν_{t} f_{y}$ ; $F_{u}$ : cross-sectional force under experimental failure load of 674 kN; $w_{req' d} = F_{u} / (b f_{cs})$ ; $w_{pro' d}$ : maximum available width of concrete strut (refer toFigure 8(c)). The bold number in table are the numbers mentioned in the text.

Table 7.

Prediction of ultimate strength of corbel PG1 by indeterminate STM representing combined load transfer mechanism: Strength verification of struts and ties at the second failure state.

Strut number	$ν_{s}$	$f_{c}'$ (MPa)	$f_{cs}$ (MPa)	$F_{u}$ (kN)	$w_{req' d}$ (mm)	$w_{pro' d}$ (mm)	$w_{pro' d} / w_{req' d}$	Safety
D	0.88	45.0	39.4	840.9	142.1	67.6	0.475	×
Tie number	$ν_{t}$	$f_{y}$ (MPa)	$f_{st}$ (MPa)	$F_{u}$ (kN)	$A_{req' d}$ (mm²)	$A_{pro' d}$ (mm²)	$A_{pro' d} / A_{req' d}$	Safety
A	1.00	415.0	415.0	502.8	1211.6	1290.0	1.065	○

$F_{u}$ : cross-sectional force under experimental failure load of 674 kN; $w_{pro' d}$ : maximum available width of concrete strut after the first failure. The bold number in table are the numbers mentioned in the text.

Table 8.

Prediction of ultimate strength of corbel PG1 by indeterminate STM representing combined load transfer mechanism: Strength verification of nodal zones.

Node number	Node type	$ν_{e}$	A, $A_{b}$ (mm²)	$ν_{n}$	$f_{c}'$ (MPa)	$f_{cn}$ (MPa)	$F_{u}$ (kN)		$w_{req' d}$ (mm)	$w_{pro' d}$ (mm)	$\frac{w_{pro' d}}{w_{req' d}}$	Safety
1	CCT	0.74	500 × 150, 100 × 150	1.48	45.0	66.5	P	695.9	69.7	100.0	1.434	○
							Strut B	59.2	69.0	223.6	3.242	○
							Strut C	106.4
							Strut D	720.5
							Tie A	519.2	52.0	200.0	3.845	○

C: compression; T: tension.

$ν_{n}$ ( $= ν_{e} \sqrt{A / A_{b}}$ ): coefficient of effective strength of nodal zone; effective strength of nodal zone $f_{cn} = ν_{n} f_{c}'$ ; $F_{u}$ : cross-sectional force under 103.2%(= 55.7 + 47.5) of experimental failure load; P: applied load (= 103.2% of experimental failure load); $w_{pro' d}$ : maximum available width of nodal zone boundary.

The ultimate strengths of the corbels, predicted using the three types of STMs representing the arch mechanism, the horizontal truss mechanism, and the combined mechanism of present study, are summarized in Table 5. The results are also plotted in Figure 9 along with those predicted by the ACI 318 (2014) conventional provisions based on a shear friction theory. The analysis results show that the ultimate strength values predicted using the proposed first-order indeterminate STM and load distribution ratio are more appropriate and consistent than those predicted using the determinate STMs and the provisions based on a shear friction theory, regardless of the effective strengths of the concrete struts. This outcome implies that the proposed indeterminate STM is one of the most appropriate load transfer mechanisms of the corbel, and the proposed load distribution ratio reflects properly the effects of primary design variables on the strength of corbels.

Figure 9.

Ultimate strengths predicted by determinate and indeterminate strut-and-tie models: (a) ACI 318 shear friction theory and arch mechanism STM, (b) truss mechanism STM, and (c) combined mechanism STM (present study).

Summary and conclusion

The failure behavior of reinforced concrete corbels is influenced by primary design variables, including the shear span-to-effective depth ratio, the quantity of main tensile and horizontal shear reinforcing bars, the vertical-to-horizontal load ratio, and the material properties of the steel and concrete used. To establish the STM methods of current design codes as rational methods, a proper STM reflecting the true load transfer mechanisms of corbels must be presented, and the primary design variables influencing the ultimate strength and behavior of corbels must be considered in the design process as well. In this study, for the rational designs of corbels with a shear span-to-effective depth ratio of less than 1.0, an optimum first-order statically indeterminate STM combined with an arch and horizontal truss load transfer mechanisms, as shown in Figure 1(c), is proposed. In addition, a load distribution ratio represented by equation (8), that transforms the indeterminate STM into a statically determinate truss structure, was developed by reflecting the effects of the primary design variables.

To examine the validity of the proposed indeterminate STM along with the load distribution ratio, the ultimate strengths of 294 reinforced concrete corbels tested to failure by other investigators were predicted using the ACI 318 STM method associated with the proposed indeterminate STM and the load distribution ratio. The strength analysis results were compared with those predicted using the existing determinate STMs representing arch and truss load transfer mechanisms, and the ACI 318 conventional design method based on a shear friction theory. The ranges of average ratios of the experimental strength to the predicted strength (and coefficient of variation) estimated by the existing determinate STMs representing arch and truss load transfer mechanisms were 1.22–1.74 (30.8%–41.7%) and 2.47–2.68 (44.1%–50.8%), respectively. The average ratio of the experimental strength to the predicted strength (and coefficient of variation) estimated by the ACI conventional design method was 1.99 (222.8%). On the other hand, the ultimate strength of the corbels was fairly well compared to the other methods by the proposed indeterminate STM. The range of average ratio of the experimental strength to the predicted strength (and coefficient of variation) was 1.09–1.24 (28.0%–32.3%). This implies that the proposed indeterminate STM can represent the load transfer mechanisms of corbels most appropriately. The results of this study could allow the use of an indeterminate STM with an appropriate load distribution ratio for the rational design of reinforced concrete corbels and provide a proper basis for structural design by reflecting the effects of primary design variables on the ultimate strength of corbels.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B06041177).

ORCID iD

Young Mook Yun

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