Primary and secondary reinforcements in reinforced concrete corbels.
Rezaei, Mehdi ; Osman, Siti Aminah ; Shanmugam, Nandivaram E. 等
Introduction
A report on the 1964 Alaskan earthquake indicated that a
substantial number of precast concrete structures distressed as a result
of insufficient attention to their connections (Clottey 1977).
Connections in precast concrete construction, particularly of primary
members, form a critical part of the load-carrying and transfer
mechanism. Most of the simple precast concrete connections are assembled
with corbels, used extensively for beam-column connections. Corbels are
projections from faces of columns and they behave like short
cantilevered deep beams. Because of the usually low shear
span-to-effective depth ratio the loads are transferred predominantly
through shear. The principal parameters influencing the structural
response of reinforced concrete corbels are type (monotonic or cyclic)
and direction (vertical or horizontal) of external loads, shear
span-to-depth ratio, strength of concrete, shape and dimensions of
corbels, grade and arrangements of longitudinal and transverse steel
reinforcements (Kriz, Raths 1965; Mattock et al. 1976; Yong, Balaguru
1994; Fattuhi 1994). Depending on the combination of these parameters,
the ultimate loads and failure mode can change; in the worst case shear
brittle failure (diagonal splitting) occurs, while in the best case
ductile flexural failure mode is observed. Provision of secondary
reinforcements reduces crack widths, improves ductility, changes failure
mode from diagonal splitting to ductile (Fattuhi 1995).
Two types of reinforcement--primary and secondary--are designed in
order to avoid the sudden failure of a plain concrete corbel. Primary
reinforcements are mostly attributed to the cause of ductile failure and
secondary reinforcements are accentuated when either combination of
vertical and horizontal loading or dynamic loading is involved.
Secondary reinforcements are normally used to improve the shear capacity
and reduce the likelihood of sudden failure. However, the contribution
of secondary reinforcements has been shown to be variable when corbels
are subjected either to combined vertical and horizontal loading or
dynamic loading. Furthermore, most corbels without secondary
reinforcement fail in shear that displays little or no ductility
(Fattuhi 1995). Therefore, it is expected that the ratios of primary and
secondary reinforcements influence the ultimate load carrying-capacity
of corbels.
Kriz and Raths (1965) carried out 195 tests on corbels of which 121
were subjected to vertical loads, and 71 to combined vertical and
horizontal loads. The variables included in the tests were: size and
shape of the corbel, amount of primary reinforcement, concrete strength,
ratio of shear span-to-effective depth and ratio of horizontal force to
the vertical force. They proposed an empirical formula to calculate the
vertical load at ultimate capacity of the corbel. Mast (1965) introduced
the shear-friction concept to the design of corbels. Mast's (1965)
objective was to develop a simple rational approach based on a physical
model, which could be used to design a number of different types of
concrete connections. The approach assumes a number of possible failure
planes and then such reinforcement is chosen that failure along these
planes is prevented. Mast (1965) applied his design method to the test
data reported by Kriz and Raths (1965). He considered only those
specimens having shear span-to-effective depth (a/b) of less than 0.7,
and where the steel has yielded. Tests showed that the shear-friction
hypothesis gives a good strength prediction for both vertical loading
only and combined vertical and horizontal loading. Mattock (1976) tested
28 reinforced concrete corbels subjected to vertical and horizontal
loads. The variables included: the ratio of shear span-to-effective
depth, the ratio of vertical load to horizontal load, the amount of
steel reinforcements, the concrete strength and the type of aggregate.
They also developed an empirical expression for the strength of corbels
without stirrups. They concluded that the minimum amount of stirrups
should always be provided.
The PCI Design Handbook (2004) is basically based on Mattock's
design method, modified slightly in capacity reduction factor. The
strut-and-tie method is considered as a basic tool for analysis and
design of reinforced concrete structures and has been incorporated in
different codes of practice. The stress trajectories or load path
methods are used to generate strut-and-tie models. However, the models
produced by these methods are not unique. Tests of 28 deep beams were
carried out by Zielinski and Rigotti (1995) in order to define the
maximum carrying capacity and rational reinforcement ratio for
structures such as deep beam, corbels and dapped-end beam. Strut-and-tie
method was used to analyse the test results. They proposed that the
maximum usable amount of reinforcement can be obtained by [rho] =
0.424[f'.sub.c]/[f.sub.y] for beams reinforced by horizontal
reinforcement and by [rho] = 0.85[f'.sub.c]/[f.sub.y] for beams
reinforced by inclined reinforcement. Hwang et al. (2000) proposed a
softened strut-and-tie model for determining the shear strength of
corbels associated with failure of the compression strut. The precision
of the analytical model was gauged by some experimental results from
previous researchers. A new model for determining the shear strength of
reinforced concrete corbels or brackets was proposed by Russo et al.
(2006). The model was based on the equilibrium conditions of the
strut-and-tie mechanism.
By adding artificial substances, e.g. fibre to plain concrete, the
compressive and/or tensile strength of concrete would increase. Tests on
corbels made of high-strength concrete and various secondary
reinforcements such as monofilament polypropylene fibres, steel fibres
and plastic mesh were carried out by many researchers (Fattuhi 1987,
1990; Yong et al. 1985; Abdul-Wahab 1989; Saafi, Toutanji 1997; Al-Shawi
et al. 1999; Bourget et al. 2000; Ahmad, Shah 2008; Yousif 2008; Aziz,
Othman 2009). The test results have shown that corbels reinforced with
steel fibres sustained smaller crack widths, achieved high strengths and
failed in a gradual and controlled manner. Campione et al. (2005)
examined, through experimental studies, the flexural behaviour of
corbels made of plain or fibrous concrete with steel bars. They observed
that the presence of a higher steel reinforcement may not allow the
complete yielding of the primary reinforcement and it results in a
brittle failure due to the crushing of the compressed regions.
Campione (2009a) proposed a strut-and-tie macromodel to determine
the flexural response of fibre reinforced concrete (FRC) corbels,
reinforced with longitudinal bars and subjected to vertical load. The
simplified flexural response of corbels was in good agreement with
experimental results reported by other researchers. The model was
extended by Campione (2009b) to predict the flexural behaviour of
corbels made of plain and fibrous concrete. The main focus of his study
was to determine the load-deflection curves of corbels subjected to both
vertical and horizontal loads.
Due to the fact that corbels may need to get repaid after
constructing, a bridge corbel was strengthened by steel plates (Bi-hai
et al. 2009). The results showed that it is feasible to strengthen
corbels with vertical-bounded steel plates. The possibility of using
CFRP (Carbon Fiber Reinforced Polymers) laminates to strengthen corbels
was studied (Ahmad et al. 2010). The behaviour of high-strength steel
fibre reinforced concrete corbels (HSSFRC) was studied by Abdul-Razzak
and Mohammed Ali (2011a). They proposed new material constitutive
relationships by means of a regression analysis of experimental data,
which were employed to formulate the material finite element (FE)
models. Later, more sophisticated FE models for reinforced concrete
corbels were developed (Abdul-Razzak, Mohammed Ali 2011b; Syroka et al.
2011).
Artificial neural network may be interested to analytical codes to
predict the outputs as a function of input parameters after training
according to the experimental or numerical data. An artificial network
model was developed to measure the ultimate shear strength of steel
fibrous reinforced concrete corbels (SFRC) without shear reinforcement
(Kumar, Barai 2010). The models give satisfactory predictions of the
ultimate shear strength when compared with available test results and
some existing models. In addition, a parametric study was carried out.
The effect of the primary reinforcement in the SFRC corbels was the most
sensitive and a slight increase of primary reinforcement up to 1.3% was
observed, which is significant in the shear capacity of corbels. The
shear strength was almost constant beyond 1.3%.
In accordance with ACI 318-05 (2008), the cross sectional area of
secondary reinforcement shall not be less than 0.04
([f'.sub.c]/[f.sub.y]). A similar requirement is also given by the
PCI (Precast/Prestressed Concrete Institute). Despite the number of
studies carried out on corbels only a few of them have addressed the
issues of varying the ratios of primary or secondary reinforcements.
This paper is, therefore, concerned with corbels in which ratios of
primary or secondary reinforcements are varied. Sixteen reinforced
concrete corbels were designed in accordance with the PCI. These corbels
were analysed under vertical or horizontal loading using the FE
modelling with the software package LUSAS (2006). Since no limit to
ratios of primary and secondary reinforcements is found, this paper aims
at determining, based on the results, optimal ratios that could
establish some guidelines on the provision of reinforcements.
1. FE analysis of corbels
1.1. Modelling
With the development of high-powered computers, together with
state-of-the-art FE software and user-friendly graphical interfaces,
three-dimensional (3-D) FE analysis has become a popular choice to
predict the behaviour of structural elements. FE software LUSAS version
14.1 (2006) has been used in this study. Steel is assumed to behave as
an elastic-perfectly plastic material in both tension and compression.
The idealised stress-strain curve used in the numerical analysis is
shown in Figure 1. The material properties of steel were specified using
the elastic and the metal plasticity with plastic options. LUSAS
requires input of the Young's modulus, E, Poisson's ratio, u,
and yield stress of steel, [[sigma].sub.y].
Solid elements in LUSAS library are capable of predicting the
non-linear behaviour concrete. The element characteristic is able to
describe elastic, isotropic, plastic and multi-crack concrete behaviour.
Concrete is a quasibrittle material and has different behaviour in
compression and tension. The multi-crack concrete with crushing material
model is based on a multi-surface plasticity approach to represent the
non-linear behaviour of concrete in both tension and compression. The
model simulates directional softening and crushing in compression using
the same yield functions. Cracks in tension are assumed to form when the
major principal stress reaches the tensile strength, after which a
permanent crack plane is formed. Multiple cracks can form at
non-orthogonal directions to one another. The model simulates non-linear
behaviour in compression with hardening and softening functions applied
to the local yield surfaces. In tension zones permanent crack planes
result in directional loss of strength, whereas in compression zones the
planes are not permanent but rather may rotate and result in an
isotropic loss of strength. In both tension and compression unloading
from the yield surface is assumed to be elastic.
[FIGURE 1 OMITTED]
The tensile strength of concrete is typically 8-15% of the
compressive strength; stress-strain relationship for concrete assumed in
the analyses is shown in Figure 2.
It is necessary to assess the accuracy of the proposed FE modelling
so that the modelling could be used for the analyses of corbels
considered in the present study. Seven specimens tested by Foster et al.
(1996) were considered in this study to assess the accuracy of the
modelling by LUSAS. All specimens were modelled and analysed with LUSAS
and the results presented in Figure 3. It is clear from the figure that
FE results are close to the corresponding experimental values, maximum
deviation being 15%. The difference could be attributed to the
assumptions used for the modelling in which the properties of some of
the materials and dimensions not given in the paper were assumed. Since
LUSAS has been found to predict results with acceptable accuracy the
software package was used for further analyses.
1.2. Analyses
The dimensions along with the loading applied on corbels, typical
of those examined, are shown in Figure 4.
The corbel cantilevering on either side of the column was 254x305
mm in cross section and 965 mm long. The column was reinforced with four
16-mm-diameter longitudinal bars and 9-mm-diameter stirrups spaced at
216 mm centre to centre as shown in the figure. The reinforcement
details for all the corbels analysed herein are listed in Table 1.
Each of the corbels is identified in the text by designation such
as PV-0.15 and PH-0.15. In the designation of the specimens, P refers to
corbels associated with primary reinforcement, S, corbels associated
with secondary reinforcement, V, corbels subjected only to vertical
loading, and H, corbels subjected only to horizontal loading. For
example, PH-0.15 refers to the corbel in Series P with 0.15% primary
reinforcement (cross-sectional area of primary reinforcement over cross
section of the corbel), and also subjected only to horizontal loading.
Also, four corbels without either primary or secondary reinforcements
were analysed. In Series PV and PH, the cross-sectional area of primary
reinforcement was varied from 0 to 565.8 [mm.sup.2], and the area of
secondary reinforcement kept constant. On the other hand, in Series SV
and SH, the steel area of primary reinforcement was kept constant while
the area of secondary reinforcement varied from 0 to 257.2 [mm.sup.2].
For all models, the material properties such as Young's modulus of
26x[10.sup.3] and 209x[10.sup.3] N/[mm.sup.2], Poisson's ratio, 0.2
and 0.3 for concrete and steel, respectively were kept the same. The
compressive strength of concrete and the yield stress of steel were
assumed 30 MPa and 420 MPa, respectively.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
In the analyses of all models, top and bottom of the column were
assumed fixed and the load applied incrementally on a bearing pad made
of steel as shown in Figure 4. Thickness of the bearing pad of 15 mm was
so chosen that crushing of the concrete was prevented effectively. In
the analyses, corbels were subjected either to vertical or horizontal
loading as shown in Figure 4. The loadings are representative of beam
reactions in the vertical direction and, horizontal force due to surge
in the case of dynamic loading. Except the ratio of primary to secondary
reinforcements and direction of applied load, details of all the models
were kept the same. Taking advantage of the symmetry in geometry,
loading and support conditions only a quarter of the models was analysed
in order to reduce the computational time. Mesh size of 30x30 mm was
chosen based on convergence studies carried out to determine the optimal
mesh that gives a relatively accurate solution and one that takes low
computational time. In LUSAS, to set the incremental loading (automatic
or manual) for a non-linear analysis, specifies how an automatic
solution is to proceed and when it should terminate. There are two ways
to apply load, named attained load and control displacement. The
attained load usually takes more time, therefore, control displacement
was used in this study. A typical FE model is shown in Figure 5.
[FIGURE 5 OMITTED]
2. Results and discussion
Analysis was carried out on each of the models until failure of the
specimen was achieved. Extensive results for each of the models were
obtained in the form of load-deflection plots. However, selected results
are presented as shown in Figures 6-9. Displacements under the applied
load are plotted on the horizontal axis and the corresponding load on
the vertical axis. Discussion of these results are presented in the
following sections.
2.1. Results for PV series
The results obtained for the PV series models are summarised in
Table 2 and the corresponding load-deflection plots shown in Figure 6.
Only the ratio of cross section in respect of primary reinforcement was
varied in this series. Addition of primary reinforcement to corbels
appears to have enhanced the ductility and toughness with the degree of
enhancement evident in corbels reinforced with a lower ratio of primary
reinforcement. It can be seen from Table 2 that ultimate shear capacity
increases significantly with increase in the ratio of primary
reinforcement.
However, this increase is sustainable only up to the reinforcement
ratio of 0.4%, beyond which the increase in ultimate load is marginal.
For example, the ultimate shear values for the corbel PV-0.2 and PV-0.4
are 667 kN and 930 kN, respectively; compared to the corresponding value
in respect of PV-0.0, increase by 145% and 242% for 0.2% and 0.4%
primary reinforcement ratios, respectively. The corbel PV-0.5, however,
could carry until failure, only at 970 kN, a very small increase if
compared to PV-0.4. The results show that any further increase in
reinforcement beyond 0.4% does not yield any improvement in the load
resistance characteristics. The corbel PV-0 failed catastrophically in a
brittle manner as expected. It can be seen from Figure 6 that the curves
had either two or three segments. The first segment terminated at the
occurrence of the first crack after which the strains increased rapidly
even for a small load increment. The crack starts from the inner edge of
the bearing pad and runs into corbel-column face diagonally. The strain
of reinforcement bars is usually measured in order to find out whether
the steel yields or not. This is followed quickly by the initiation of
the second (primary) crack, the load-strain response becomes relatively
linear until approximately 90% of the ultimate load. The first segment
of the curves is steep followed by a kink and rapid increase of crack
width coincident with the formation of the primary crack.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
2.2. Results for SV series
In this series, the cross-sectional area of secondary reinforcement
varies from 0.0% to 0.45% and the models are analysed under the action
of vertical loadings. The load is increased until failure with the
behaviour of the corbels and their deflection recorded at selected
interval of loading. Results thus obtained from the analyses are
summarised in Table 3 and the load-deflection curves for the models are
shown in Figure 7.
It is noted from the analyses that in all models flexural cracks
that start at or near the junction of the tension face of the corbel and
face of the column appeared first. The results show clearly that
presence of additional secondary reinforcement resulted in an increase
in load-carrying capacity as well as in ductility of corbels. Corbels
SV-0.0 containing no secondary reinforcement failed suddenly in an
explosive manner. Increase in load-carrying capacity of corbels is found
to be significant for corbels with the percentage of secondary
reinforcement up to 0.3%. For example, as shown in Table 3, the ultimate
shear values for SV-0.0 and SV-0.3 are 530 kN and 770 kN, respectively,
which is an increase of 45%. This increase in ultimate shear becomes
gradually insignificant for the ratio of secondary reinforcement beyond
0.3. The load deflection curves, as shown in Figure 7, are similar to
those shown for PV series with stages of almost linear behaviour. The
load-deflection plots for corbels with reinforcement ratios 0.3% and
above lie very close to each other, indicating that the influence of
reinforcement ratio beyond 0.3% on ultimate shear capacity is
negligible.
2.3. Results for PH series
Nine corbels, in which the ratio of primary reinforcement varied
from 0.0% to 0.55%, were analysed under horizontal loading in this
series. The analyses of the models in this series were similar to those
in series PV, except the loading, which was horizontal in this case. The
results obtained from the analyses are presented in the form of
load-deflection plots as shown in Figure 8 and summary of results given
in Table 4.
It can be seen from the figure and the table that the corbel PH-0.0
failed at the lowest load; this is expected since no primary
reinforcement was provided in the corbel. The load-deflection plot in
respect of the corbel PH-0.0 does not show brittle behaviour as observed
in the corbel PV-0.0. The shape and area beneath the load deflection
curves are often used as indicators of ductility and toughness,
respectively. It can be seen from the figure that the corbels with
primary reinforcement ratio of 0.3% and above display similar behaviour,
almost the same value of ultimate shear capacity and the same ductility.
An increase in horizontal load due to an increase in primary
reinforcement ratio is, however, not appreciable.
2.4. Results for SH series
In this series consisting of eight corbels with secondary
reinforcement ratio varied from 0.0% to 0.5% were analysed under
horizontal loading only. The results obtained from the analyses are
given in Figure 9 in the form of load-deflection curves and summarised
in Table 5. The figure shows that the curves for all models coincide
indicating that the variation in reinforcement ratio does not affect the
behaviour in any respect. The results in the figure and in the table
clearly show that the effect of increase in reinforcement ratio has
insignificant effect on the ultimate shear load.
3. Evaluation of PCI equation
The PCI recommends equations based on friction theory for
reinforced concrete corbels. According to PCI, a corbel is theoretically
capable of carrying loads of any magnitude. In accordance with the PCI,
As, cross-sectional area of primary reinforcement is taken as the larger
of the two values given by Eqns (1) and (2):
[A.sub.s] = 1/[phi][f.sub.y] [[v.sub.u] (a/b) + [N.sub.u] (h/d)];
(1)
[A.sub.s] = [1/[phi][f.sub.y] [[2/3] [Vu/[[mu].sub.e]] +
[N.sub.u]], (2)
in which: a is the eccentricity of applied load; d--depth of
primary reinforcement; h--height of corbel; [[mu].sub.e]--effective
shear-friction coefficient as defined in section 11.7.3 of ACI 318-05
(2008); [phi]--strength reduction factor equal to 0.75, [f.sub.y]--yield
strength of tension steel; [V.sub.u]--factored vertical load applied and
[N.sub.u]--factored horizontal load applied.
The minimum primary steel required in tension is given as:
[A.sub.s,min] = 0.04 [[f'.sub.c]/[f.sub.y]] bd, (3)
where: b is width of corbel and [f.sub.c]--concrete strength in
compression.
The required cross-sectional area of the secondary reinforcement is
calculated as:
[A.sub.n] = [[N.sub.U]/[phi][f.sub.y]]; (4)
[A.sub.h] [greater than or equal to] 0.5 ([A.sub.s] - [A.sub.n]),
(5)
in which: [A.sub.h] is cross-sectional area of secondary
reinforcement, distributed within the upper 2/3 of the corbel depth. In
order to evaluate the above equations, a number of corbels were designed
using the PCI equations. The resulting reinforcement ratios, primary or
secondary and the corresponding horizontal or vertical loading are
listed in Table 6.
All the dimensions and material strengths were kept the same as
those corbels analysed using the FE method in the current studies. It
can be seen from the table that primary reinforcements have significant
influence on the behaviour of corbels. It could change the mode of
failure from catastrophic to flexure, in which reinforcements yield
before concrete crushes. The results presented for the corbels in PV
series and SV series showed significant difference in behaviour due to
different reinforcement ratios that would lead to change the failure
mode. The change in ductility is more apparent in the case of PV Series.
Concrete corbels containing neither primary nor secondary reinforcement
failed in brittle manner with a noticeable difference in ultimate load.
Results from the PH and SH series showed that the presence of primary or
secondary reinforcements did not enhance significantly either the
ductility or the load-carrying capacity.
It is clear from the results given in Table 6 that the primary and
secondary reinforcement ratios increase as only the vertical load is
increased. Similar observations could be made when only the horizontal
load is increased but, with moderate increase in the secondary
reinforcement ratio. It was shown earlier in the case of PV series that
the increase in ratio of primary reinforcement beyond 0.4% does not
affect the load-carrying capacity. Also, it was noted in series SV that
an increase in the secondary reinforcement ratio of up to 0.3 enhances
the load-carrying capacity of corbels and has no significant effect
thereafter. But the PCI does not provide any limitation on the
reinforcement ratio, primary or secondary. In accordance with the PCI
equations, both primary and secondary reinforcement ratios increase
notably when the corbel was subjected only to vertical loading. When a
corbel was subjected only to horizontal loading, on the other hand, the
primary reinforcement ratio increased significantly; the corresponding
increase in the secondary reinforcement ratio is not as significant as
that of the primary ratio. It should be noted that this study was made
for only width/ height, d/d, ratio of corbels equal to 0.64, and hence
it is necessary to carry out further studies covering a wider range of
parameters.
Conclusions
The following conclusions can be drawn based on the results:
1) The failure mode of corbels with neither secondary reinforcement
nor primary reinforcement was brittle and explosive when corbel was
subjected to only vertical loading;
2) The ultimate load of a corbel was increased by increase in
percentage of primary reinforcement steel, nevertheless, it is mostly
pronounced for lower ratios of the main reinforcement;
3) The load-carrying capacities of corbels are considerably
enhanced by the addition of secondary reinforcements when a corbel was
subjected to only vertical loading. The enhancement is noticeable until
the percentage of secondary reinforcement reached to 0.3%;
4) The PCI equations do not specify any limits on the ratio of
primary and secondary reinforcement. The study showed that some limits
should be introduced in the code.
doi: 10.3846/13923730.2013.801896
Received 6 Dec 2011; accepted 2 Feb 2012
References
Abdul-Razzak, A. A.; Mohammed Ali, A. A. 2011a. Modelling and
numerical simulation of high strength fibre reinforced concrete corbels,
Applied Mathematical Modelling 35(6): 2901-2915.
http://dx.doi.org/10.1016/j.apm.2010.11.073
Abdul-Razzak, A. A.; Mohammed Ali, A. A. 2011b. Influence of
cracked concrete models on the nonlinear analysis of high strength steel
fiber reinforced concrete corbels, Composite Structures 93(9):
2277-2287. http://dx.doi.org/10.1016/j.compstruct.2011.03.016
Abdul-Wahab, H. M. S. 1989. Strength of reinforced concrete corbels
with fibers, ACI Structural Journal 86(1): 60-66.
ACI 318-05. 2008. Shear and torsion. Part 11.8: provision for
brackets and corbel. American Concrete Institute (ACI), Detroit. 188 p.
Ahmad, S.; Shah, A. 2008. Evaluation of shear strength of high
strength concrete corbels using strut-and-tie model, The Arabian Journal
for Science and Engineering 34: 27-35.
Ahmad, S.; Shah, A.; Nawaz, A.; Salimullah, K. 2010. Shear
strengthening of corbels with carbon fiber reinforced polymers (CFRP),
Materiales de Construccsion 60: 79-97.
http://dx.doi.org/10.3989/mc.2010.50009
Al-Shawi, F. A. N.; Mangat, P. S.; Halabi, W. 1999. A simplified
design approach to corbels made with high strength concrete, Materials
and Structures 32: 579-583. http://dx.doi.org/10.1007/BF02480492
Aziz, O. Q.; Othman, Z. S. 2009. Ultimate shear strength of
reinforced high-strength concrete corbels subjected to vertical loading,
Al-Rafidain Engineering 18(1): 1-6.
Bi-hai, L.; Yuan-han, W.; Hong-Guang, C.; Yu, M. 2009.
Reinforcement and numerical analysis on the corbel of a half-through
arch bridge, Journal of Chongqing University 8(1): 57-62.
Bourget, M.; Delmas, Y.; Toutlemonde, F. 2000. Experimental study
of the behaviour of reinforced high-strength concrete short corbel,
Materials and Structures 34: 155-165. http://dx.doi.org/10.1617/13620
Campione, G. 2009a. Flexural response of FRC corbels, Cement &
Concrete Composites 31(3): 204-210.
http://dx.doi.org/10.1016/j.cemconcomp.2009.01.006
Campione, G. 2009b. Performance of steel fibrous reinforced
concrete corbels subjected to vertical and horizontal loads, Journal of
Structural Engineering 135(5): 519-529.
http://dx.doi.org/10.1061/(ASCE)0733-9445(2009)135:5 (519)
Campione, G.; Mendola, L. L.; Papia, M. 2005. Flexural behavior of
concrete corbels containing steel fibers or wrapped with FRP sheets,
Materials and Structures 38: 617-625. http://dx.doi.org/10.1617/14210
Clottey, C. 1977. Performance of lightweight concrete corbels
subjected to static and repeated load: PhD Thesis. Oklahoma State
University, Stillwater, OK.
Fattuhi, N. I. 1987. SFRC corbel tests, ACI Structural Journal
84(2): 119-123.
Fattuhi, N. I. 1990. Column-load effect on reinforced concrete
corbels, Journal of Structural Engineering 116(1): 188-197.
http://dx.doi.org/10.1061/(ASCE)0733-9445(1990)116:1 (188)
Fattuhi, N. I. 1994. Strength of FRC corbels in flexure, Journal
ofStructural Engineering 120(2): 360-377.
http://dx.doi.org/10.1061/(ASCE)0733-9445(1994)120:2 (360)
Fattuhi, N. I. 1995. Reinforced corbel made with high-strength
concrete and various secondary reinforcement, ACI Structural Journal
101(4): 345-368.
Foster, S. J.; Powell, R. E.; Selim, H. S. 1996. Performance of
high-strength concrete corbels, ACI Structural Journal 93(5): 555-563.
Hwang, S.; Lu, W.; Lee, H. 2000. Shear strength prediction for
reinforced concrete corbels, ACI Structural Journal 97(4): 543-552.
Kriz, L. B.; Raths, C. H. 1965. Connections in precast concrete
structures: structures-strength of corbels, PCI Journal 10(1): 16-60.
Kumar, S.; Barai, S. V. 2010. Neural networks modeling of shear
strength of SFRC corbels without stirrups, Applied Soft Computing 10(1):
135-148. http://dx.doi.org/10.1016/j.asoc.2009.06.012
LUSAS. 2006. LUSAS Version 14.1, User's and theory manual.
Mast, R. F. 1965. Auxiliary reinforcement in concrete connections,
Proceedings of the American Society of Civil Engineering 94(6):
1485-1501.
Mattock, A. H. 1976. Design proposal for reinforced concrete
corbels, PCI Journal 21(3): 18-42.
Mattock, A. H.; Chen, K. C.; Soonswang, K. 1976. The behavior of
reinforced concrete corbels, PCI Journal 21(2): 18-42.
PCI. 2004. Design handbook: design of connection. Part 6.8:
concrete brackets and corbels. PCI Committee on Building Code, Chicago.
47 p.
Russo, G.; Venir, R.; Pauletta, M.; Somma, G. 2006. Reinforced
concrete corbels--shear strength model and design formula, ACI
Structural Journal 103(1): 3-7.
Saafi, M.; Toutanji, H. 1997. Design of high performance concrete
corbels, Materials and Structures 31: 612-622.
Syroka, E.; Bobinski, J.; Tejchman, J. 2011. FE analysis of
reinforced concrete corbels with enhanced continuum models, Finite
Elements in Analysis and Design 47(9): 1066-1078.
http://dx.doi.org/10.1016/j.finel.2011.03.022
Yong, Y.-K.; Balaguru, P. 1994. Behavior of reinforced
high-strength-concrete corbels, Journal of Structural Engineering ASCE
120(4): 1182-1201. http://dx.doi.org/10.1061/(ASCE)0733-9445(1994)120:4
(1182)
Yong, Y.; McCloskey, D. H.; Nawy, E. G. 1985. Reinforced corbels of
high-strength concrete, ACI Structural Journal 87: 197-212.
Yousif, A. R. 2008. Prediction of ultimate load capacity of
high-strength reinforced concrete corbels, Al-Rafidain Engineering
17(4): 12-27.
Zielinski, Z. A.; Rigotti, M. 1995. Tests on shear capacity of
reinforced concrete, Journal of Structural Engineering 121(11):
1660-1666. http://dx.doi.org/10.1061/(ASCE)0733-9445(1995)121:11 (1660)
Mehdi REZAEI, Siti Aminah OSMAN, Nandivaram E. SHANMUGAM
Department of Civil & Structural Engineering, The National
University of Malaysia (UKM), 43600 UKM Bangi, Selangor, Darul Ehsan,
Malaysia
Corresponding author: Mehdi Rezaei
E-mail: rezaei_mrc@yahoo.com
Mehdi REZAEI. A PhD candidate in UKM (the National University of
Malaysia). He has obtained his Master's degree from UKM in 2008. He
has published more than ten scientific papers in international journals
and conference proceedings.
Siti Aminah OSMAN. Senior lecturer at the Department of Civil and
Structural Engineering, the National University of Malaysia (UKM). She
graduated from Universiti Teknologi Malaysia in 1992 with BEng(Hons),
MSc in Structural Engineering from University of Bradford, UK (1995) and
PhD in Civil and Structural Engineering from the UKM (2006). She is a
member of the Board of Engineers Malaysia (BEM) and the Society of
Engineering Education Malaysia (SEEM). After graduating from her
undergraduate studies, she worked as a design engineer with Muhibbah
Engineering for 6 months and then joined the UKM as a tutor. She has
taught at undergraduate and graduate levels for more than 15 years. Her
research interest includes structural engineering, wind engineering,
structural dynamics and industrial building system (IBS) construction.
Nandivaram E. SHANMUGAM. A Professor at the Department of Civil and
Structural Engineering, the National University of Malaysia (UKM). He
obtained his PhD degree from the University of Wales (Cardiff) in 1978.
He has taught at undergraduate and graduate levels for more than 45
years. He has published more than 200 scientific papers in international
journals and conference proceedings. He is a member of the editorial
board of the Journal of Constructional Steel Research, Thin-walled
Structures, the Journal of Structural Stability and Dynamics, the
International Journal of Steel Composite Structures, the International
Journal of Steel Structures and the IES Journal Part A: Civil and
Structural Engineering. He is a Chartered Engineer (CEng), Fellow of the
Institution of Structural Engineers, London, (FIStructE), Fellow of the
Royal Institution of Naval Architects (FRINA), Fellow of the American
Society of Civil Engineers (FASCE), Fellow of the Institution of
Engineers, Singapore (FIES) and Fellow of the Institution of Engineers,
India (FIEI). His research interests include steel plated structures,
steel-concrete composite construction, long-span structures and
connections, cold-formed steel structures, elastic and ultimate load
behaviour of steel structures, etc.
Table 1. Details of reinforcements in the corbels
Designation Secondary Primary
reinforcement reinforcement
([mm.sup.2]) ([mm.sup.2])
PV-0.0, PH-0.0 127.2 0.0
PV-0.15, PH-0.15 127.2 154.3
PV-0.20, PH-0.20 127.2 205.7
PV-0.30, PH-0.30 127.2 308.6
PV-0.35, PH-0.35 127.2 360.0
PV-0.40, PH-0.40 127.2 411.5
PV-0.45, PH-0.45 127.2 462.9
PV-0.50, PH-0.50 127.2 514.4
PV-0.55, PH-0.55 127.2 565.8
SV-0.0, SH-0.0 0.0 265.2
SV-0.15, SH-0.15 77.2 265.2
SV-0.20, SH-0.20 102.9 265.2
SV-0.30, SH-0.30 154.3 265.2
SV-0.40, SH-0.40 205.7 265.2
SV-0.45, SH-0.45 231.5 265.2
Table 2. Corbel series PV
Designation Ultimate Percentage Deflection at
shear increase of ultimate load
load (kN) ultimate load (mm)
PV-0.0 272 0 0.42
PV-0.15 557 105 0.45
PV-0.20 667 145 0.89
PV-0.30 736 171 0.94
PV-0.35 829 205 1.01
PV-0.40 930 242 0.91
PV-0.45 989 264 1.09
PV-0.50 970 257 1.28
PV-0.55 1029 278 1.66
Table 3. Corbel series SV
Designation Ultimate Percentage Deflection
shear increase in at ultimate
load (kN) ultimate load (mm)
load (%)
SV-0 530 0 0.58
SV-0.15 677 28 0.83
SV-0.20 724 36 0.85
SV-0.30 770 45 0.95
SV-0.40 876 65 0.99
SV-0.45 886 67 0.93
Table 4. Corbel series PH
Designation Ultimate Percentage Deflection
shear load (kN) increase of at ultimate
ultimate load (mm)
load (%)
PH-0 82 0 0.04
PH-0.15 110 35 0.22
PH-0.20 115 41 0.16
PH-0.30 121 49 0.13
PH-0.35 123 51 0.11
PH-0.40 123 51 0.1
PH-0.45 124 52 0.09
PH-0.50 125 53 0.09
PH-0.55 125 54 0.08
Table 5. Corbel series PH test results
Designation Ultimate Percentage Deflection
shear increase of at ultimate
load (kN) ultimate load (mm)
load (%)
SH-0 118 0 0.14
SH-0.15 118 0.2 0.14
SH-0.20 119 1.1 0.14
SH-0.30 119 1.2 0.14
SH-0.35 119 1.3 0.14
SH-0.40 119 1.4 0.14
SH-0.45 119 1.4 0.14
SH-0.50 120 1.5 0.15
Table 6. Details of reinforcements in corbels designed
as per PCI equations
Corbel Vertical Horizontal Ratio of Ratio of
designation loading loading primary secondary
(kN) (kN) reinforcement reinforcement
(%) (%)
1 700 0 0.4 0.2
2 800 0 0.46 0.23
3 900 0 0.52 0.26
4 1000 0 0.58 0.29
5 1100 0 0.63 0.32
6 1200 0 0.69 0.35
7 1300 0 0.75 0.37
8 1400 0 0.81 0.4
9 0 100 0.26 0.06
10 0 160 0.27 0.01
11 0 220 0.37 0.01
12 0 280 0.47 0.02
13 0 340 0.57 0.02
14 0 400 0.67 0.02
15 0 460 0.77 0.03
16 0 520 0.87 0.03
