Strength Prediction of Corbels Using Strut-and-Tie Model Analysis
International Journal of Concrete Structures and Materials
Strength Prediction of Corbels Using Strut-and-Tie Model Analysis
A strut-and-tie based method intended for determining the load-carrying capacity of reinforced concrete (RC) corbels is presented in this paper. In addition to the normal strut-and-tie 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 load-carrying 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. Load-carrying 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 shear-friction based approach implemented by the ACI 318-11, the strut-and-tie provisions in different codes (American, Australian, Canadian, Eurocode and New Zealand).
corbels; load-carrying capacity; shear strength; strut-and-tie model
Reinforced concrete (RC) corbels, defined as short cantilevers
jutting out from walls or columns having a shear span-to-depth
ratio, av/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 (D-region), 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 318-11 code (ACI Committee 318 2011) requires
corbels having shear span-to-depth ratio, av/d, less than 2 to
be designed using the strut-and-tie method and those with
shear-to-span ratio less than 1 to be designed either using
strut-and-tie method, or by the closely related traditional
ACI design method based on shear-friction approach.
However, the shear-friction hypothesis has little correlation
with the observed failure phenomenon of concrete crushing
in the diagonal strut (Hwang et al. 2000b).
Strut-and-tie models (STM) have been generally
recognized as an acceptable rational design approach for D-region
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 A23.3-04); Eurocode 2 (2004) and New Zealand code
(NZS 3101-1)] 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
In the current paper, a strut-and-tie based method intended
for determining the load-carrying capacity of corbels is
presented. In addition to the normal strut-and-tie 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), beam-column joints (Hwang and Lee 1999,
2000, 2002), dapped-end 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 strut-and-tie based method is
developed for calculating the load-carrying capacity of
reinforced concrete corbels. The proposed method is based on an
Fig. 1 Geometry and strut-and-tie model with forces acting on corbel.
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 load-carrying 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 load-carrying
capacity of 455 corbels that have been tested experimentally.
The findings illustrate that the strut-and-tie 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
3. Compatibility-Based Strut-and-Tie Model
Approach for Corbels
Strut-and-tie 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 strut-and-tie 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 strut-and 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
out-ofbalance 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
strut-and-tie model. For corbels with short span-to-depth
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 full-length
horizontal tie to balance the thrust of the inclined struts
(Fig. 1). The corbel is loaded by the vertical force Vcv
applied at the distance av from the column face and it is
assumed that the horizontal outward load, Nu, 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 /, 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 av 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 = Es/Ec, and the flexural reinforcement ratio qf
is assumed to be given by:
where An is the cross-sectional area of principal
reinforcement used to resist the applied outward load,
taken as An = Nu/fys, where fys is the yielding strength of
the principal reinforcement, As and Ash are the
crosssectional areas of the principal tensile reinforcement and
horizontal web reinforcement, respectively, and X 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
318-11, that Es = 200 GPa and Ec 4700pffiffficffi0 (MPa), it
The diagonal strut is assumed to have a bottle-shaped form.
That is, it spreads laterally along its length. The lateral spreading
of the bottle-shaped strut introduces tensile force transverse to
the strut, Fst. 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 318-11. The considered
strutand-tie model leads to the following equilibrium equations:
where Dc, Hc are the compressive forces in the diagonal and
horizontal concrete struts, respectively; and Fst is the
where r2d; r2c; fsh; fsv and fs are the uniaxial stresses that are
obtained from the constitutive relations of each member.
bursting tensile force in the tie of the strut-and-tie model.
Because the bursting force Fst represents a quarter of the
compressive force of the diagonal strut, Dc, the horizontal
and vertical components, Fsh and Fsv, of the tie force can be
obtained from equilibrium as follows:
4.1 Secant Stiffness Formulation
The proposed procedure is based on a compatibilitybased
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 Ad, Ac are the cross-sectional areas of the diagonal and
horizontal concrete struts; Ash, Asv and As are the
crosssectional areas of horizontal, vertical and longitudinal steel
ties; Ed; Ec; Esh; Esv and Es, 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):
Fig. 2 Constitutive relations and secant moduli used in analysis procedures for a concrete in compression, b reinforcing steel; and
c compatibility conditions for diagonally cracked concrete.
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 strut-and-tie 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
6. Reinforcing Steel
The stressstrain relationship of steel is assumed to be
linear up to yielding, followed by a yield plateau (Fig. 2b).
This elasticperfectly plastic type of stressstrain
relationship is represented mathematically by:
where Es is the elastic modulus of the steel bars; fs and es are
the average tensile stress and strain of the reinforcing bars,
respectively; and fy and ey are the yield stress and strain of
the bars, respectively.
where r2 is the average principal stress of concrete in the 2
direction; w is the softening coefficient; fc0 is the compressive
strength of a standard concrete cylinder in unit of MPa; e2
and e1 are the average principal strains in the 2 and 1
directions, respectively; and e0 is the concrete cylinder strain
corresponding to the cylinder strength fc0, which can be
defined approximately as:
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, e1, 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
7. Compatibility Condition
In the proposed approach, the normal tensile strains in the
horizontal and vertical web steel, eh and ev and the principal
compressive and tensile strain in concrete strut, e2 and e1,
have a simple relationship that satisfies the compatibility
condition of Mohrs circle (Fig. 2c):
Adjust [ ]assumed
Fig. 3 Flow chart showing solution algorithm.
Read corbel dimensions, section
properties, loading plate dimensions
and material properties
Vcv =Vcv + Vcv
Calc. Ds, Hc, Fsh, Fsv Eqs.(6-10)
Calc. Ad, Ac, Ash, Asv Eqs.(25-26)
is acting stress < allawable stresses
Fig. 4 Histograms of geometric and material properties of 455 reinforced concrete corbels.
governed by yielding of all reinforcement crossing the crack
direction. To guard against this, a limiting value of the
principal tensile strain, e1, was proposed. Thus, the value of
tensile strain, eh, in Eq. (24) is limited by the yielding strain,
eyh, after yielding, or the value of eh 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 ev is conservatively taken as 0.002 in
8. Effective Depth of Concrete Struts
Diagonal struts frequently are wider at mid-length than at
their ends because strut stresses is greater at mid-length 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 bottle-shaped 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, Wd, was assumed equal to (Park and
where av/2 should not be less than the loading plate width,
Wp, 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 Wc 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 Nu is the applied horizontal tension load (negative for
tension), Ac 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 load-carrying capacity of the corbels. The algorithm
in Fig. 3 starts with a selection of the vertical corbel shear
force Vcv and can be proceeded as outlined in following
According to the member forces Dc, Hc, Fsh, and Fsv,
calculated from Eqs. (6) to (10), the values of the strains
in concrete struts and steel reinforcements are estimated
for the selected Vcv 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;
Using the state of strain in each member, the normal
stresses are determined from the stressstrain relations
of Eqs. (21a, 21b) through (23a, 23b);
Fig. 5 Selected strut and tie models.
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
5. The stresses in the diagonal and horizontal struts, r2d
and r2c, are compared to their capacity. The capacity of
the diagonal strut can be estimated from
vcv1 0:85 bsfc0, where bs = 0.6 as suggested by the
ACI 318-11(American Concrete Institute 2011) for
bottle-shaped strut with web reinforcements not
satisfying the minimum reinforcement requirements, while
the capacity of the horizontal strut is taken as
vcv2 0:85 fc0.
6. The stresses on the on nodes vertical back face and
node-to-strut interface are compared to nominal
strengths due to crushing, assumed equal to Vcv4
0:85 fc0 and Vcvs5 0:68 fc0 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 Vcv; 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 wide-ranging research into a
single database provides the ability to examine code
provisions as well as develop new models for use in design.
Strength ratio (vtest/vcalc)
Shear-friction based model
Fig. 6 Effect of concrete strength, fc, on shear strength predictions by means of: a proposed strut-and-tie model approach; and
b the shear-friction model used by the ACI 318-11 for 357 reinforced concrete corbels.
Aimed at verifying the accuracy of the proposed
compatibility-based strut-and-tie 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:
Abdul-Wahab (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
span-to-overall 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
Fig. 7 Effect of shear span-to-depth ratio, av/d, on shear strength predictions by means of: a proposed strut-and-tie model
approach; and b the shear-friction model used by the ACI 318-11 for 357 reinforced concrete corbels.
Fig. 8 Variation of ratio of measured-to-calculated strength by means of: a proposed strut-and-tie model approach; b ACI 318-11;
c AS 3600; d CSA A23.3-04; e Eurocode 2; and NZS 3101-1 with shear span-to-depth ratio av/d.
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
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 A23.3-04), European (Eurocode 2), and New Zealand
(NZS 3101-1) 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 well-defined
strength mechanism that generally occurs in the formation of
a strut-and-tie resistant mechanism. Several other methods
are available to estimate the shear capacity of corbels,
including empirical equations (Fattuhi 1994), shear-friction
approach (Hermansen and Cowan 1974, Mattock 1976), and
strut-and-tie models(Solanki and Sabnis 1987; Siao 1994;
Hwang et al. 2000b; Russo et al. 2006) including plastic
truss models(Campione et al. 2007). The load-carrying
capacity of the 455 corbels was calculated using the proposed
analysis method, the shear-friction based approach provided
by the ACI 318-11(American Concrete Institute 2011) and
the strut-and-tie model proposed by different code
provisions [ACI Committee 318; Australian code AS 3600;
Canadian code (CSA A23.3-04); Eurocode 2 and New
Zealand code (NZS 3101-1)].
The shear-friction based approach provided by the ACI
318-11 is valid for corbels made from both normal and high
Fig. 9 Variation of ratio of measured-to-calculated strength by means of: a proposed strut-and-tie model approach; b ACI 318-11; c
AS 3600; d CSA A23.3-04; e Eurocode 2; and NZS 3101-1 with concrete strength, fc:
strength concrete with span-to-depth 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 shear-span ratios and reduced percentages of
reinforcement. For a shear failure, the shear strength of a
corbel is given by:
where qvf = (qs ? qh) is the frictional reinforcement ratio;
fys is the yield strength of the friction reinforcement; qs and
qh is the principal reinforcement ratio and horizontal web
reinforcement ratio, respectively; l 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 a is the horizontal-to-vertical loads ratio; and jd is the
lever arm calculated by jd d A0sf:y8s8 fc0Nbu. 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 Vcv shall not
exceed the smallest of 0:2fc0 bd; 3:3 0:08 fc0
bd and 11 bd:
Several code recommendations (CSA Committee A23.3
2004, NZS 3101 2006) specify the strut-and-tie models for
the design of corbels, while only for corbels having shear
span-to-depth ratio greater than 1.0, the ACI 318-11
recommends the use of a strut-and-tie model described in ACI
318-11, Appendix A. However, it does not provide detailed
guidance on strut-and-tie models for different cases. It is
well known that in using the strut-and-tie 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 strut-and-tie 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 strut-and 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 strut-and-tie model (STM)
accounting for the main tie steel only, a refined strut-and-tie
model accounting for the secondary crack-control
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 strut-and-tie models for a double-sided
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 strut-and-tie based method.
11.3 Comparison of Load-Carrying Capacity
Very few studies found in the literature on the validity of
load-carrying capacity models of RC corbels in code
provisions including strut-and-tie models. Table 1 summarizes
the average, Avg, standard deviation, SD, and coefficient of
variation, CoV, of the ratio between measured and calculated
capacities, vtest/vcalc, of RC corbels considered, based on the
proposed strut-and-tie based method, the shear-friction based
approach provided by the ACI 318-11, the strut-and-tie
model proposed by five codes of practice examined (ACI
318-11; (c) AS 3600; (d) CSA A23.3-04; (e) Eurocode 2;
and NZS 3101-1). The distribution of average strength ratios
for the specimens in the database against the concrete
strength, fc0, and shear span-to-depth ratio, av/d, is shown in
Figs. 6, 7, 8 9, where Avg and CoV values are also reported.
For the comparison with the shear-friction based approach
provided by the ACI 318-11, only 357 corbels with shear
span-to-depth ratio less than unity have been taken into
account. Careful examination of the results shows that the
shear strength ratios, vtest/vcalc, using the shear-friction based
approach provides highly conservative and scattered
estimates of the strength of corbels over a wide range of
concrete strength and shear span-to-depth 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 shear-friction
based approach of ACI 318-11 in the 1980s, are clearly
unsafe. In particular, the shear-friction based approach is
unconservative in the prediction of the load-carrying
capacity for corbels with concrete strength less than 50 MPa,
(see Fig. 6b). By contrast, the calculated capacities by the
proposed strut-and tie based method are both accurate and
conservative with low scatter or trends for RC corbels with
shear-to-span depth ratios ranging from 0 to 1 (see Fig. 7a).
The selected strut-and-tie 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 span-to-depth
ratio, av/d, and concrete strength, fc0 on the load-carrying
capacity predictions of the strut-and-tie based method and
the five codes of practice examined for the double-sided
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 span-to-depth 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 load-carrying capacity of corbels with
concrete strength greater than 35 MPa and shear
span-todepth ratio greater than 0.3. The largest average of the shear
strength ratios, (vtest/vcalc) 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
strutand-tie perspective. Considering the width of the compiled
database, the obtained results are considered to be adequately
fair to suggest that the proposed strut-and-tie based method
provides a reliable and safe means of predicting the
loadcarrying capacity of reinforced concrete corbels.
12. Summary and Conclusions
A strut-and-tie based method intended for calculating the
load-carrying capacity of reinforced concrete corbels has
been presented. In addition to the normal strut-and-tie force
equilibrium requirements, the proposed model accounts for
strain compatibility and stressstrain 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 shear-friction
based approach provided by the ACI 318 formulas as well as
the strut-and-tie provisions in the American, Canadian, New
Zealand, Eurocode and Australian codes., the following
conclusions may be drawn:
1. The calculated load-carrying capacities by the proposed
method were both accurate and conservative with
limited scatter or trends for reinforced concrete corbels
with shear span-to-depth ratios ranging from 0.1 to 2
and made from normal or fibrous concrete.
2. The shear-friction 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
strut-and tie based method are both accurate and
conservative with low scatter or trends for RC corbels
with shear-to-span depth ratios ranging from 0 to 1.
3. The predictions by the proposed strut-and-tie based
method are adequately conservative and accurate to
conclude that it provides a safe and reliable means of
calculating the load-carrying capacity of concrete corbels.
4. Based on the conclusions drawn from this research, the
proposed strut-and-tie should be adopted in future
adjustments to code provisions and in the development
of design guidelines for all types of D-regions in
structural concrete. Furthermore, both experimental and
mathematical studies are still needed to investigate the
applicability and limitations of the proposed
strut-andtie method when applied to a wide range of D-regions.
The Author would like to thank Prof. Keun-Hyeok Yang,
Kyonggi University, South Korea for providing some
information on tests of corbels and assistance in populating
the corbels database.
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