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DEEP
BEAM

Introduction:

Deep beam is a very important structural element in various types of concrete
structures such as pile cap, transfer girder, panel beams, foundation walls of
rectangular tanks, bins, shear walls, folded plate of roof structures etc. A
beam having much greater depth than that of a normal beam in relation to its
span, while the thickness in the perpendicular direction is much smaller than
both span and depth, is usually termed as a deep beam. ACI-ASCE Committee
426(44) classified a beam with shear span-to-depth ratio, a/d less than 1.0 as
a deep beam and a beam with a/d exceeding 2.5 as an ordinary shallow beam. Any
beam in between these two limits is categorized as a short beam.

The design of reinforcement concrete deep beam is a subject of considerable interest in structural engineering practice. Despite wide structural applications or deep beams, only a few national codes include their design. For example, the British Standards BS8110 for structural use of concrete explicitly states that for the design of deep beams. Reference should be made to the specialist literature. Similarly, the Eurocode EC2 simply states that “it does not apply, however, to deep beams”. Currently the main design documents for deep beams are the ACI Code ACI 318-95, the Canadian Code CAN-A23.3-M84 and the CIRIA Guide of the U.K. Among the three design documents, the CIRIA Guide is the most comprehensive document on the subject.

The behavior of deep beams is significantly different from that of shallow beams, thus, they require special consideration in analysis, design and detailing of reinforcement. The load carrying capacity of deep beam is controlled by shear and shear strength of deep beam is significantly greater than those predicted by usual equations for shallow beam, because of a special capacity to redistribute internal stresses before failure.

Three procedures are currently available for the design of deep beams:

· Empirical design methods

· Two or three dimensional analysis either linear or nonlinear

· By strut-and-tie model based on compression field theory.

Equilibrium truss model for deep beams:

In strut-and-tie model, the member is idealized as a series of tension ties, concrete struts, loads and supports interconnected at nodes to form a truss. Equilibrium truss models assume or require that:

1. Equilibrium must be satisfied

2. The
concrete only resists compression and has an effective compressive strength f_{ce}
= nf_{c}’ where the efficiency
factor, n is usually taken less than
1.0

3. Steel required to resist all tensile forces.

4. The centroids of each truss member and the lines of action of all externally applied loads at a joint must coincide.

5. Failure of the equilibrium truss model occurs when a concrete compression member crushes or when a sufficient number of steel tension members reach yield to produce a mechanism.

These assumptions satisfy the lower-bound theorem of plasticity, which states that ¾

If an equilibrium distribution of stresses can be found which balances the applied load and is everywhere below yield or at yield, the structure will not collapse or will just be at the point of collapse. Since the structure can carry at least this applied load, it is a lower bound to the load carrying capacity of the structure.

Components of equilibrium truss model:

Equilibrium truss models are made up of five
building blocks:

1.
Struts: Struts are concrete compression members in uniaxial
compression with a uniform stress of f_{ce} at the ultimate load. The
struts have a finite width and thickness, which depend on the force in the
member and the permissible stress level. The end faces of a strut are principal
stress faces and must be perpendicular to the longitudinal axis of the strut.

2.
Ties: Ties are steel tension members permitted to reach and
sustain the yield stress f_{y}. Collapse of the beam does not occur
when a single tension member reaches yield unless this is sufficient to convert
the truss in a mechanism.

3.
Joints: Joints are accommodated by “hydrostatic stress
elements”. The concrete within these elements has both in-plane principal
stresses equal to f_{ce}. These elements may have a uniaxial
compression stress of f_{ce} action of each face or facet of the
element within the plane of the beam. (These elements do not have true
hydrostatic stress since the stresses on the free faces will not be equal to f_{ce}.)

4. Compression fans: They occur under point loads and over supports where a number of minor compression struts fan out, distributing the load or reaction to a number of stirrups.

5. Compression fields: They occur when parallel minor compression struts transmit force from one stirrup to another.

__Historical Background__[1]:__
__

The investigation of the stresses in a deep beam is not a
new subject. It started at the end of 19^{th} century. Most widely know
results were published in 1932 by Dischinger who solved the problem of a
continuous deep beam subjected to periodic loading by representing the load in
terms of a Fourier series. He constructed a stress function satisfying the
boundary conditions and a governing differential equation in terms of
hyperbolic functions. To handle a deep beam of finite span, Dischinger
suggested that the results of the continuous deep beam be used. His solution
for a finite beam did not satisfy the free end boundary conditions because the
finite beams was being supported by sharing stressed and the finite span did
not satisfy the condition of zero bending stress at its free ends.

In the United States, Chow (1951) undertook the investigation of a single span deep beam. He used the solution of a continuous deep beam, which satisfied all but one of the boundary conditions, as his stress function. He then constructed another stress function via the principle of least work, which negated the residual boundary stresses. Chow checked his result by solving the governing differential equation by the method of finite differences. Although, Chow claimed that the principle of least work yielded more accurate results than the method of finite difference, he reverted to the method of finite differences in his next study of a single span beam subjected to five loading cases and three depth-to-span ratios. Although Chow was able to predict the general behavior of finite deep beams, he could not produce accurate results at all cross sections of the beam.

Slightly prior to chow’s paper on single span beams, Uhlmann (1952) investigated a simply supported beam by solving the governing differential equations using Richardson’s method of successive approximations. Uhlmann computed the stress trajectories for a number of cases, and was thereby successful in illustrating the fundamental difference between the load carrying action of a deep beam and a shallow beam.

In 1956, Kitchen and Archer utilized the method of least work to construct a stress function, but it was different from Chow’s work in that it did not depend on the solution of the continuous beam. Their bending stress values agreed fairly well with the results of Chow, but their shear stress results were not in good agreement.

In a study of the effects of loaded length vs. clear span, Geer (1960) used the method of finite differences but with a much finer computational grid than that of Chow. His results show that the distribution of the load can significantly affect the normal stress characteristics and the maximum compressive bending stresses were found to exist below the mid-height of the beam. Unfortunately, no data for the distribution of the shear stresses was given.

In 1972, Holmes & Mason also used the method of least work to solve the problem of a single span deep beam supported by a parabolic shearing force applied at the vertical edges. This loading condition does not take into account the high bearing pressures normally associated with the reactions. Their results did not differ from the results obtained by shallow beam theory by as much as might be expected for beams of deep proportions.

In 1964, Schleeh published a landmark paper in German in which he presented his results in terms of tables which contained correction factors to ordinary bending theory so that the actual stress distribution could be found by adding the correction factors to the results of ordinary beam theory. The correction terms to the shallow beam theory were worked out in terms of a few standard loading configurations. By superpositioning those standard loading configurations, the results for most loading condition encountered in practice can be obtained. A drawback of Schleeh’s tables is that the solutions are not unique. In some cases, by decomposing the applied load into different standard loading configurations, different answers were obtained. Furthermore, the tables limit the evaluation of the stresses to a finite number of specific locations.

A reinforced concrete member may be subjected to four types of actions; namely, axial load (compression & tension), bending, shear and torsion. The first two types of actions, axial loads and bending, are quite well understood; and the empirical design methods are essentially same for different nations. In contrast, the last two types of actions, shear and torsion, are not well understood; and the empirical design methods used in the codes and specifications are very different around the world.

In the past, there were two basic approaches used to analyze shear and torsion problems in reinforced concrete; namely, the mechanism method and the truss model method. The mechanism method is the basis of the current shear and torsion provisions in the ACI Building Code. By fitting the mechanism method and the test results, the ACI method becomes empirical or at best semi-empirical. From a theoretical point view, this method cannot satisfy the compatibility condition and in certain cases, even the equilibrium condition.

It is generally agreed by the researchers in recent years
that the truss model theory provides a more promising way to treat shear and
torsion. The original truss model concept was first proposed to treat shear
problems by Ritter and Morsch at the turn of the 20^{th} century. It
was extended to treat torsion problems by Rausch in 1929. These theories hold
that a concrete element reinforced with orthogonal steel bars and subjected to
shear stresses would develop diagonal cracks at an angle inclined to the steel
bars. These cracks would develop separate the concrete into a series of
diagonal concrete struts, which are assumed to resist axial compression.
Together with the steel bars, which are assumed to take only axial tension,
they form a truss action to resist the applied shear stresses. For simplicity,
concrete struts are assumed to be inclined at 45° to the steel bars.
Consequently, these theories are known as the 45° truss model.

The rudimentary truss model of Ritter, Morsch & Rausch is very elegant and the equations derived from the equilibrium conditions are simple. Unfortunately, the prediction from these equations did not agree with the test results. For the case of pure tension, the theory may overestimate the test strengths by 30%. For the case of low-rise shearwalls, the overestimation may exceed 50%.

To improve the prediction of the truss model, the theory had undergone three major developments. The first important developments the generalization of angle of inclination of the concrete struts by Lampart & Thurlimation. They assumed that the angle of inclination might deviate from 45°. On this basis, three basic equilibrium equations had been derived, which could explain why longitudinal & transverse steel with different percentages could both yield at failure. Their theory was known as the variable-angle truss model. Since plasticity was assumed at failure, it could also be called the plasticity truss model. The second development was the derivation of the compatibility equation by Collins to determine the angle of inclination of the concrete struts. Since this angle was assumed to coincide with the angle of inclination of the principal compression stress and strain, this theory was known as compression field theory. It can be better could as compatibility truss model, because the average strain condition should satisfy Mohr’s strain circle in this theory. The third development was the discovery of the softening of concrete struts by Robinson & Demorix and the quantification of this phenomenon by Vecchio & Collins. Vecchio & Collins proposed a softened stress-strain curve, in which the effect depended on the ratio of the two principal strains.

__A. Alshegeir and J. A. Ramirez (1992)__[2] presented an evaluation of
prestressed concrete deep beams using the strut-and-tie model. Strut-and-tie
systems reflecting actual support and loading conditions were developed for
three pre-tensioned deep beams tested to failure. The strut-and-tie approach
was used to illustrate the effects of prestressing, concrete compressive
strength and reinforcement detailing on the behavior and strength of these
members. They concluded that the presence of prestressing in deep members
contributes to delay inclined web cracking, but it is not necessarily the
critical factor in the inclination of the failure diagonal crack. The failure
inclined crack for deep members will span between the point load and the
support reaction. Flexure shear type cracking will seldom control the failure
of this type of member. The degree of prestressing also affects the tensile
force demand on the main longitudinal tie of the strut-and-tie model. The
contribution of transverse reinforcement to the shear strength of deep beams is
not as significant as in more slender members. However, stirrups contribute to
the shear strength of deep beams through aggregate interlock mechanism by
controlling the width of the main diagonal cracks.

__A. Akhtaruzzaman and A. Hasnat (1989)__[3] published the results of
the tests of twenty-six concrete deep beams with and without a transverse
opening tested to failure under torsion with varying span-depth ratio, concrete
strength and size and location of the opening. The presence of an opening
significantly reduced torsional strength. For span-depth ratios greater than
3.0, the torsional strength of beams remained practically constant. It
increased significantly, however, as the span-depth ratio decreased. The crack
inclinations on the beam surfaces followed a similar trend and were also influenced
by concrete strength and web opening size.

__R. Narayanan and I.Y.S. Darwish (1988)__[4] reported the results of
tests on twelve reinforced concrete deep beams including eleven containing
steel fibers, provided to act as web reinforcement. Three parameters were
varied in the study, namely, the volume fraction of fibers, shear span-to-depth
ratio and the concrete compressive strength. Based on the test results the
authors drew following conclusions:

1. The inclusion of steel fibers in concrete deep beams resulted in enhanced stiffness and increased spall resistance at all stages of loading up to failure and reduced crack widths.

2. Deep beams of fiber concrete under shear loading develop zones of approximately zero stress in many cases over a height of 0.15 to 0.3h.

3. In general, the primary cause of failure was diagonal cracking, which led to the splitting of the beam along the diagonal cracks.

4.
Both shear cracking load V_{fo} and ultimate shear
load V_{uo} were influenced by the fiber factor F (Improvement in
concrete tensile strength and ductility due to the inclusion of steel fibers
are thought to be dependent upon three factors. These are the fiber volume
fraction, r_{f} in the mix,
fiber aspect ratio l_{f}/d_{f} and the extent of bond between
the fiber and the matrix. The influence of these three factors has been
incorporated into a combined parameter called fiber factor F and given by ¾ F =
(l_{f}/d_{f}) r_{f}b, where b is
the bond factor that accounts for differing bond characteristics of the
fiber.), the a/h ratio, and the concrete strength.

__D. M. Rogowsky, J. G. MacGregor and S. Y. Ong
(1986)__[5] reported results of tests
on seven simply supported and seventeen two-span deep beams. Two main types of
behavior were observed. Beams without stirrups or with minimum stirrups
approached tied-arch action at failure. This was true regardless of the amount
of horizontal web reinforcement present. These failures were sudden with little
or no plastic deformation. On the other hand, beams with large amount of
stirrups failed in a ductile manner. The horizontal web reinforcement had very
little effect on the strength. The ACI Building Code overestimated the strength
of continuous deep beams and those having horizontal shear reinforcement.

__B. S. Maxwell and J. E. Breen (2000)__[6] reported the performance of
four deep beams each of a geometric discontinuity in the form of a large
opening. The beams were simply supported and tested using a point load. The
beams were designed using the strut-and-tie model. The specimens performed very
well as each one performed just as the strut-and-tie model theory predicted.
Each specimen resisted the factored design load with deflections at
approximately L/400 and even less at service load levels. This investigation
successfully verified the theoretical ideas of the strut-and-tie model. the
authors made the following specific conclusions from this study:

5. Strut-and-tie models provide lower bound design solutions that are valid, conservative and reliable.

6. Using two strut-and-tie models for the design of one specimen, each proportioned to carry half of the factored design load, provided a beneficial redundancy that yielded a stiffer and stronger deep beam.

7. The performance of the deep beam specimens at the factored design load was very good. Each specimen was much stronger than the factored design load, and at this load, there was very little physical damage.

8. The performance of the deep beams at the service load was excellent. None of the specimens developed visible cracks at service loads and deflections were very small.

9. Each specimen experienced different failure mechanisms. The strut-and-tie model successfully allows for an explanation of each failure mode. Difference between failure patterns could lead to better and more efficient future designs.

__S. J. Hwang, W. Y. Lu and H. J. Lee (2000)__[7] proposed a softened
strut-and-tie model for determining the shear strengths of deep beams. The
proposed model originates from the strut-and-tie concept and satisfies
equilibrium, compatibility and constitutive laws of cracked reinforced concrete.
The word ‘softened’ emphasizes the importance of the compression softening
phenomenon, which means that cracked reinforced concrete in compression
exhibits lower strength and stiffness than uniaxially compressed concrete. To
predict the shear strength of deep beams, the softened truss model was
originally developed by Mau and Hsu (1987). In the softened truss model, the
state of stresses in the web shear element is assumed to be uniform and the
flow of compressive stresses is idealized by a series of parallel compressive
struts. The internal stress flow of the beam web, however, is highly disturbed
by the presence of the top load and the bottom support reaction. In this
disturbed region, it is inappropriate to assume that the shear stress is
uniform. Therefore, the strut-and-tie model is believed to be a better choice
in modeling the flow of the forces of the deep beam, with compressive stresses
in the concrete and tension ties representing the reinforcing steel. The shear
strength prediction of the proposed model and the empirical formulae of the ACI
318-95 Code are compared with the collected experimental data of 123 deep
beams. The comparison shows that the performance of the softened strut-and-tie
model is better than the ACI Code approach for all the parameters under
comparison. The parameters included the ratios of horizontal and vertical
reinforcement, concrete strength and the shear span-to-depth ratio. According
to the proposed model, it is found that the ACI 318-95 Code empirical equations
underestimate the contribution of concrete while overestimating the
contribution of web reinforcement or the shear strength of deep beams.

__A. F. Ashour and G. Rishi (2000)__[8] reported the test results
of 16 reinforced concrete two-span continuous deep beams with web openings. All
test specimens had the same geometry and main longitudinal top and bottom
reinforcement. The main parameters considered were the size and position of the
web openings and web reinforcement arrangement. Two modes of failure were
observed, depending on the location of web openings, regardless of the web
opening size and the amount and type of web reinforcement. For beams having web
openings within interior shear spans, the failure is developed by diagonal
cracks between the web opening corners and the edges of the load and central
support plates. For beams having web openings within interior shear spans, the
mode of failure is characterized by major diagonal cracks within interior and
exterior spans. The diagonal cracks that occurred in the interior shear spans
extended to join the edges of the load and central support plates, and at the
same time, the diagonal cracks that formed at the web opening corners
propagated both ways towards the edges of the load and end support plates. The
following conclusions were made:

1. Web opening within interior shear span had more reduction on the beam capacity than those within exterior shear spans.

2. The vertical web reinforcement had more influence on the shear capacity of the beams than the horizontal web reinforcement.

3. Support reactions were influenced by the size and location of web openings.

4. Continuous deep beams having small web openings within exterior shear spans showed the closest behavior to that of their companion solid beams.

__S. Teng, W. Ma and F. Wang (2000)__[9] presented an experimental
investigation involving 12 reinforced concrete deep beams subjected to fatigue
loading. Three different arrangements of web reinforcement were considered,
namely, without web reinforcement, with vertical web reinforcement only and
with inclined web reinforcement. The tests revealed that the arrangements of
web reinforcement have significant influence on the structural response of deep
beams under fatigue loading. With load repetitions, the midspan deflection and
maximum crack width increased, while the shear strength decreased. The authors
concluded that:

1. Web reinforcement were effective in restraining the development of cracks and deflections under static as well as fatigue loading, with the inclined web reinforcement being the most effective. The beams with inclined web reinforcements also had the high failure strength.

2. With an increasing number of load cycles, especially during the first few load cycles, new cracks formed and existing cracks extended, the stiffness of a deep beam gradually decreased, the deflection and crack width increased, and the ultimate fatigue shear strength decreased.

3. The wider the fatigue load range, the shorter the fatigue life of the beam.

4. The relevant ACI equations can be applied to deep beams under fatigue or repeated loading once the properties of the concrete and the reinforcement are adjusted to take account of the effect of fatigue loading.

__K. H. Tan, F. K. Dong and L. W. Weng (1998)__[10]
reviewed the shear strength equations of the ACI Building code, the Canadian
Code and the UK’s CIRIA Guide for high strength concrete deep beams. A modified
form of the CIRIA equation was proposed and its predicted values were compared
with experimented values and those obtained from the ACI code, the Canadian Code
and the original CIRIA Guide. The comparison involves a total of 233 deep
beams, out of which 57 specimens were tested in-house. The study revealed that
the modified CIRIA equation gives the smallest coefficient of variation &
standard deviation among the methods considered. Based on the test results and
their comparison with the design equations the following concessions were made:

The Proposed formula:

1. The proposed equation gives the lowest stances deviation for all three strength categories. The COV & the SD are also the smallest among the four design methods considered, indicating the consistency of the formula.

2.
In addition the proposed equation has extended the validity of
CIRIA equation form 0 £ a/d £ 0.9 to 0 £ a/d £ 2.5 i.e. it now covers both deep and short beams
design. The equation can also be used for designing deep beams with concrete
strength f'_{c} in range of 12 to 90 MPa.

3. The UK CIRIA Guide-2

4. CIRIA’s predictions are not conservative for high-strength concrete deep beams and for a/d exceeding 0.9. The guide tends to overestimate the beneficial effect of main longitudinal steel, horizontal & vertical web reinforcement on shear strength.

5. The ACI Building Code:

6. Among the three national codes considered the ACI codes strength predications give the most conservative estimates of shear strength of deep beams. The predications can be very conservative at the low end of a/d ratio and this conservation reduces with increasing a/d.

7. The effect of steel ratio (r) on the ultimate shear strength of high-strength concrete deep beams is also underestimated by the code.

8.
The beneficial effect of horizontal web reinforcement on
ultimate shear strength is overestimated. This is due to the critical threshold
being set at l_{n}/d = 5. It has been shown that with the threshold
fixed at l_{n}/d = 2.5, the unconservatism is removed.

9. The Canadian Code:

10. The conservation of the Canadian Code increases with increasing a/d ratio. As for the effect of concrete strength, it is slightly overestimated by the Code.

11. Since the Canadian Code does not take the contributions from web reinforcement into account, the method is safe for beam specimens with horizontal & vertical web reinforcements.

__S. J. Foster and R. I.
Gilbert (1998)__[11]
reported the test results of 16 high-strength concrete (HSC) deep beams.
Variables considered in the investigation were shear span-to-depth ratio,
concrete strength and the provision of secondary reinforcement. The results of
the experiments were compared with the CIRIA Guide-2, ACI 318 and the Plastic
truss model of Rogowsky and MacGregor with the efficiency factor of Warwick and
Foster. In this study, the Plastic truss model was categorized into three types
covering the full range of non-flexural members. The CIRIA design model, on the
other hand, is limited to a small range of shear span to depth ratios. Both
CIRIA & ACI models gave conservative results for the expected failures
loads. Of the three design methods used, the plastic truss model is clearly the
most rational and the simplest to apply. The plastic truss model gave
predicated to experimental ratios closer to unity then either the CIRIA Guide
of ACI method, albeit, with a slightly higher standard deviation.

__K. H Tan, F. K Kong, S. Teng
and L. W. Weng (1997)__[12]
reported the results of an experimental investigation on the behavior and
ultimate shear strength of 18 high strength concrete deep beams. Observation
were mode on mid-span deflections, crack widths, failure modes & ultimate
strengths. From the study following conclusions were made: -

1. It is evident that web reinforcement can play an important role for HSC deep beams. The most favorable pattern is the orthogonal web reinforcement, it is the most effective in increasing the beam stiffness, restricting the diagonal crack width development and in increasing the ultimate shear resistance.

2. For deep beams with a/h ³ 1.0, the vertical web steel has greater effect on restraining the diagonal crack width and increasing the ultimate shear resistance of HSC deep beams than the horizontal web steel of the same steel ratio.

3. The web steel contribution of high strength deformed bears is significantly greater than that of lower strength plain mild steel bars.

4. The CIRIA Gude-2 gives unconservative predications for specimens with high percentage of horizontal web bars. The guide also overestimates the concrete contribution from high strength concrete.

5. The Canadian Code gave conservative predications for most of the specimens as the method does not take the web steel contribution into account.

__F. K Kong, S. Teng, A. Singh and K. H. Tan (1996)__[13]
reported test results of 24 lightweight concrete deep beams. The tests were
carried out to investigate how their behavior was affected in the ultimate load
range by the embedment lengths for the end anchorage of the main tension reinforcement.
This investigation indicated that the favorable effect of normal pressure due
to the support reaction and the compression strut, on bond strength could be
exploited in deep beam design. Following conclusions were made:

1. When normal pressure can be secured, such as that exists across the supports of simply supported deep beams, it can be taken into account to allow some reduction in the required embedment length for end-anchorage of the main tension reinforcement.

2. It seemed that due to the presence of normal pressure, an embedment length of 17.5d equivalent to an ACI standard hook is reasonable and sufficient to provide full anchorage for tension reinforcement in deep beams.

3. Among the three sets of design theories, the pleasure theory, gave the most conservative predictions on the ultimate strength of light-weight concrete deep beams, followed by the bearing theory and then the shear theory.

4. For the Measure design of deep beams, all tension steel lying within the bottom h/3 of the bean could be considered the main tension reinforcement.

5. For predicting the shear strength of lightweight concrete deep beams, both ACI and the CIRA Guide-2 methods are equally good, but the latter can be used for a wider range of web reinforcement types.

__S. Teng, F.K Kong, S.P. Poh, L.W. Guan and K.H. Tan
(1996)__ [14] reported
the results of an experiment in which 18 prestressed & non-prestressed
concrete deep beams were tested to failure, strengthened and then retested to
failure for a second time. On the occurrence of the first failures, the failed
shear spans of the beams were strengthened by using steel clamping units that
acted as external stirrups. This investigation yielded that following general
conclusions: -

1) After the introduction of the clamping unit, many of the damaged deep beams underwent change in their internal force transfer mechanism. This beneficial change is true for both prestressed and non-prestressed deep beams and it makes the beams cable of carrying more loads than the original beams.

2) Performance or strength of damaged deep beams can be restored to the fullest as long as their damaged /failure mode is the diagonal splitting shear failure. This types for failure mode is the most common for deep beams with little to moderate web reinforcement.

3) After clamping, the nominal slope of the load-deflection curve of the strengthened deep beam was approximately 15% less steep compared to that of the original undamaged beam. Obviously this was because the beam had diagonal cracking and thus the stiffness of the beam was much reduced.

4) The presence or amount of web reinforcement does not have much effect on the strength of the clamped shear span. The shear force in the failure shear span is basically carried by the clamping unit. So, the clamping unit should be designed to carry the full shear force without any contribution form the cone.

5) The best location for the clamping unit is in the middle of the failed shear span for a straight diagonal crack, closer to the support for an upwardly curved diagonal crack. When anchorage failure is likely, the clamping unit should be placed as close to the support as possible.

6) Shear strengths of all the clamped deep beams are higher than those predicted by ACI method or mostly higher than those anticipated by CIRIA method, indicating that the capacity of the strengthened deep beams is quite conservative with respect to the design methods.

__A. T. C. Goh (1995)__[15]
investigated the feasibility of using neural networks to evaluate the ultimate
strength of deep reinforcement concrete beams in shear. A neural network is an
information processing system whose architecture essentially mines the
biological system of the brain. The neural network is particularly useful for
evaluating systems with a multitude of nonlinear variables such as the strength
of concrete, the beam geometry and the steel reinforcement in the beam. No
predefined mathematical relationship among the variables is assumed. Instead
the neural network learns by example patterns obtained from published
experimental data of cone beams tested to failure. The neural network predictions were more reliable than
predications using conversational methods. A neural network consists of a
number of interconnected processing units, commonly known as neurodes or
neurons. Each neurode receives an input signal from neurodes to which it is
connected. Each of these connections has a numerical weightiness associated
with it. These weights determine the
mature & strength of the influence between^{ }the interconnected
neurodes. The signals from each input are then processed through a weighted sum
of the inputs. The processed output signal is then transmitted to another
number via a transfer function. A typical transfer function is the sigmoid
transfer function. The sigmoid function modulates the weighted sum of the
inputs sot the output approaches unity when the input gets larger and
approaches unity when the input gets larger and approaches zero when the input
gets smaller. Each neurode is connected to all the neurodes in the next layer.
There is an input layer where data is presented to the neural network and an
output layer that holds the response of the network to the input. It is the
intermediate layers, also known as hidden layers, which enable these networks
to represent and compute complicated associations between^{ }patterns.
Currently, there is no rule for determining the optical member of hidden
layers, except through experimentation. A single hidden layer has been found to
be satisfactory for many problems, including this study.

__J.E Barry and H. Ainso (1983)__[16]
used the multiple Fourier technique to compare the stress fields in single span
deep beams due to uniform loading at the top edge and at the bottom edge. The
method involves the superposition of three stress functions. The first stress
function is used to the boundary conditions on the upper & lower edges of
the beam. The second & third stress functions are used to satisfy the
boundary conditions on the vertical edges of the beam. This approach allows to
satisfy all the required boundary conditions. Contour maps of the stress field
reveal the existence of regions of pure tension & pure compression. These
regions indicate proneness to spalling, bursting or crushing.

__Ali and Habib (1992)__[17]
presented a comprehensive experimental study on the deflection and stress
distribution characteristics in deep beam under uniformly distributed load.
They investigated the effect of L/d ratio, variation
of web reinforcement on stress distribution and deflection of deep beams. The
authors stated that the available theories for stress computation did not
predict the stress in concrete, in flexural steel or in web steel of a deep
beam accurately. The mid-span deflection could not be predicted accurately
using the deflection computation methods usually used for ordinary shallow
beams. The smaller the span/depth ratio the more pronounced is the deviation of
stress patterns from that of Bernoulli and Navier. Design of deep beams based
on the linear distribution of bending stresses as used in shallow beams may be
seriously in error, since the simple theory of flexure does not take into
account the effect of normal pressure on the top and bottom edges of the beam
caused by the load and reactions. Mid-span deflections were measured and
compared with the theoretical values calculated from deflection formulas for
ordinary shallow beam theory with both cracked and uncracked sections. Efforts
were made in the test program because in order to observe the effects of the
variation of horizontal and vertical web reinforcement on the deflection and
the stress distribution in the concrete masses, in the flexural steel and in
the horizontal and vertical web reinforcements. Brick aggregates were also used
in the test program because it is widely used in developing countries like
Bangladesh. After investigation the authors concluded that before the
initiation of cracks, deflection of beams with L/d = 2 was fairly accurately
predicted by the ordinary shallow beam theory using uncracked sections.
However, for beams with L/d = 1, this approach grossly underestimated the
actual deflection of beams. On the other hand, after initiation of cracks,
ordinary shallow beam theory with cracked sections, predicts the deflections of
beams fairly accurately in both the test series (L/d = 1 and L/d = 2). Both
ordinary shallow beam theory and Holmes and Meson’s approach did not predict
the stresses in vertical and horizontal web reinforcements properly. Flexural
steel stresses were predicted fairly accurately using the ordinary shallow beam
theory with cracked section only when the load level is closed to the ultimate
load capacity of the beam. Extreme fiber concrete stresses values for the beam
with L/d = 1. However, it predicts fairly accurately the stress in deep beams
having L/d = 2. On the other hand both ordinary shallow beam theory and Holmes
and Meson’s approach are inadequate for computation of stresses in concrete at
various locations.

__Ali
and Habib (1992)__[18] presented another research
paper depending on the test results as previously discussed. They found that the
amount of web reinforcements influenced only the diagonal cracking load not the
ultimate load capacity. Diagonal cracks developed first in relatively deeper
beams (L/d = 1) and flexural cracks developed first in the shallower deep beams
(L/d = 2). ACI Code underestimates the diagonal cracking shear stress for deep
beams when subjected to UDL. However, the upper limit of the shear stress
causing diagonal cracks set by ACI Code is in conformity with the test results,
although somewhat conservative. On the other hand, the Cosio and Siess (1960)
equation is not conservative for calculating for diagonal shear stress for deep
beams having lower L/d ratios. Vertical web reinforcement is more effective in
resisting diagonal cracking in deep beams than is horizontal web reinforcement.
The upper limit of 8Öf_{c}¢ for the ultimate shear stress suggested by
the ACI Code was fairly conservative estimate for RC deep beams when adequate
web reinforcement was provided. Moreover, the upper limit of 6Öf_{c}¢ for the contribution of
concrete in ultimate shear was also conservative for RC deep beams when
subjected to UDL. Shear compression failure (by crushing of concrete) was
observed in relatively shallow beams (L/d = 2) whereas in the deeper beams (L/d
= 1) diagonal tension failure was predominant.

__Mansur M. A., Lee Y. F., Tan K.H. and Lee S. L (1991)__[19] experimentally investigated
eight reinforced concrete continuous deep beams, each containing a large
transverse opening. The beams are rectangular in cross-section and all contain the
same amount and arrangement of longitudinal reinforcement. The number of spans,
the size of opening and its location along the span were considered as major
variables. Final failure of the beam occurred by the formation of a mechanism
i.e. the formation of hinge and the two opening ends represent the most
vulnerable locations for the development of plastic hinges. Besides early
cracking, the strength and stiffness of the beam decreased with an increase
either in the length or depth of opening. Similarly, openings located in the
high moment region produce larger deflections and result in early collapse of
the beam. The authors observed that the location of opening had very little
influence on cracking load but affects the load deflection response. Similarly,
the opening location had no influence on the mode of failure.

__M.
D. Kotsovos (1984)__[20] performed a finite element
analysis of under- and over-reinforced concrete beams subjected to two-point
loading. The analysis indicated that placing shear reinforcement within the
middle span rather than the shear span results in a significant improvement of
both load carrying capacity and the ductility of the beams. It was shown that
this behavior is due to collapse of the beams by splitting of the compressive
zone of the middle span rather than crushing of the loading point region as in
generally thought. Splitting is caused by tensile stresses that develop in the
compressive zone from the interaction of adjacent concrete elements subjected
to different states of stress. It was found that collapse of the beams always
occurs before the compressive strength of concrete was exceeded anywhere within
the beams. Even in the compressive zone concrete fails under combined
compressive & tensile stresses, the tensile stresses result from the
interaction of adjacent concrete elements subjected to different states of
stress.

__Q. Q. Liang, Y. M. Xie and G. P. Steven (2000)__[21]
presented a performance based evolutionary topology optimization method for
automatically generating optimal strut-and-tie models in reinforced concrete
structures with displacement constraints. In the proposed approach, the element
virtual strain energy was calculated for element removal, while a performance
index was used to monitor the evolutionary optimization process. By
systematically removing elements that have the least contribution to the
stiffness from the discretized concrete member, the load transfer mechanism in
the member is gradually characterized by the remaining elements. The optimal
topology of the strut-and-tie model is determined from the performance index
history, based on the optimization criterion of minimizing the weight of the
structure while the constrained displacements are within acceptable limits. The
method can also be applied for finding optimal strut-and-tie models in
prestressed concrete structures and reinforced concrete shear walls. The steps
of evolutionary optimization procedure are as follows:

Step 1: Model the concrete member with fine finite elements.

Step 2: Analyze the concrete member for real loads and virtual unit loads.

Step 3: calculate the performance index.

Step 4: Calculate the virtual strain energy for each member.

Step 5: Delete a small number of elements with lower virtual strain energy.

Step 6: Repeat steps 2 through 5 until the performance index is less than unity.

Based on this study following conclusions were drawn:

1. For very slender concrete beams, the optimal topologies obtained by the topology optimization method are continuum-like structures in which strut-and-tie model actions are difficult to be defined.

2. For deep beam loaded at the bottom, the vertical and inclined reinforcement should be provided to transfer the loads to the compressive arch with sufficient anchorage, but not necessarily to the top of the deep beam, depending on the span-to-depth ratio of the beam.

3. When openings intercept the natural load paths, the load in to be rerouted around the openings where inclined tensile ties join the upper and lower struts. It is important to provide inclined reinforcement at top and bottom of the opening. This inclined reinforcement is efficient for crack control and for increasing the ultimate load capacity of the deep beam.

4. For reinforcement concrete beam with span-to-depth ratio ³ 3.0, the inclined reinforcements bent up from bottom steel bars sre most efficient in resisting shear in the shear spans.

5. In the structural idealization of corbels, the column that joins the corbel should be considered together with the corbel in developing the strut-and-tie model.

__K. H. Tan, S. Teng, F. K. Kong and H. Y. Lu (1997)__[22]
presented test results of 22 reinforced concrete deep beams with cylindrical
compressive strengths f'_{c} generally exceeding 55 MPa tested under
two-point symmetric top loading. Based on the main steel ratio r_{w} the beams were categorized into
four groups with r_{w} = 2.00,
2.58, 4.08 and 5.80 percent. Web reinforcement comprising 10mm diameter plain
MS stirrups at 300mm centers was provided for all specimens, giving a vertical
web steel ratio r_{v }of 0.48%.
The beams were tested for different shear span-to-total depth ratios, ranging
from 0.25 to 2.5. The authors claimed based on the experiment that in beams
with low a/h, arch action dominates the behavior, a high main steel ratio means
a more effective tension tie which leads to a stiffer tied-arch. Thus,
increasing r_{w} for beams with
low a/h enhances the deep beam stiffness. However, with increasing a/h, the
arch action decreases and instead regular beam action dominates. Two types of
crack width were recorded in the tests: flexural cracks and diagonal cracks.
The development of flexural cracks was more rapid in specimens with high a/d.
Towards failure, the flexural cracks reduced their crack widths somewhat,
probably due to redistribution of internal forces. Diagonal cracks usually
occurred in both shear spans of the beam, after the formation of flexural
cracks in the mid-span region. Unlike flexural cracks, some diagonal cracks
were very wide immediately after their formations, typically in the order
between 0.1 and 0.2mm. They grew rapidly with increasing applied loads. The
higher the r_{w} the lower the
diagonal crack development. It was found that the ACI Code was conservative for
all beams with a/d £
1.5. But for a/d > 1.5, there were four cases in which ACI Code slightly
overestimated the shear strength. The Canadian Code predictions of the test
results show that good agreement is obtained for beams with 0.5 £ a/d £ 2.0.
Undue conservatism sets in as a/d increases. The greatest scatter was found in
the CIRIA Guide predictions, with the highest standard deviation of 37.2%
against 26.3% for ACI Code and 29.7% for CSA Code. Clearly, CIRIA Guide-2 is
not suitable for deep beams with f'_{c} ³ 55 MPa and with r_{w} ³ 2.0%. The authors
specifically drew following conclusions:

1. The transition point between HSC deep beams and shallow beams (in terms of load carrying capacities) is around a/d of 1.50. From medium to low strength concrete beams, it was reported to occur between 2.0 and 2.5.

2. The failure mode is chiefly influenced by the a/h ratio; the effect of r is not significant. For a/h £ 0.25, the beams fail in bearing mode; for 0.25 £ a/h £ 1.00, the beams fail in shear compression mode; for 1.00 £ a/h £ 2.00, the beams fail in diagonal tension mode and at a/h = 2.50, the beams fail in shear-tension mode.

3. For a/d £ 1.5, increasing the main tension steel ratio will increase the load carrying capacities of HSC deep beams. But this beneficial effect is not as significant when a/d > 1.50.

4.
The addition of r_{w}
beyond 2.0% does not significantly enhance the ultimate shear strength of HSC
deep beams, apart from the particularly high value of 5.8%. Thus, r_{w} of 2.0% represents a practical
upper limit in maximizing the main steel to augment the shear strength V_{c}.

5.
ACI equation is conservative for the predictions of cracking
strengths of specimens with 1.23% £ r_{w} £ 5.8%
and with 74 MPa £
f'_{c} £
55 MPa. The multiplying term in ACI Eq. (11-30) tends to be unconservative as
a/d exceeds 1.50. This could be due to the transition point of deep to shallow
beam being fixed at a/d = 2.0. From this set of experimental data, the
transition point was found to occur at about 1.50. More test results with
higher f'_{c} are required before any modification can be made to the
multiplying term.

6.
It is confirmed that the 1994 Canadian Code is suitable for
deep beams with a wide range of r_{w}
and with a/d £
2.0. Beyond a/d of 2.0, undue conservatism sets in.

7.
The CIRIA Guide-2 predictions may not be conservative for
specimens with f'_{c} ³ 55 MPa and with r_{w
}³
2.0%. It is in dire need of revision.

__W. B. Siao (1995)__[23]
examined the use of strut-and-tie approach in predicting the shear strengths of
deep beams that are continuously punctured by web openings or have
span-to-depth ratios exceeding 1.0. This study gave a rationalized approach
toward the design of deep beams with openings of larger than normal
span-to-depth ratios or that are continuous as opposed to the simply supported
test specimens that were used in the overwhelming majority of experiments in
the past. From the investigation following conclusions were made:

1. The strut-and-tie approach can be used for the analysis of shear strengths of deep beams punctuated with web openings. It can be applied to beams with rectangular openings whose horizontal dimension varies from 0.25 to 1.0 times the clear span and vertical dimension ranges from 0.1 to 0.4 times the beam height. The opening must be located within the clear span region. The behavior of a punctuated beam is equated to that of two equivalent beams. However, when the opening is small, a more accurate prediction is obtained by strut-and-tie system straddles the opening. In such instances, the cross-sectional area of the refined strut-and-tie model cannot be reduced by more than 50% of its original area.

2. When a single-point load is applied in each span, the shear strength of a deep beam is reduced, as compared to when two-point load is applied per span. There is no difference between the shear strength of single- or double-span deep beams.

3. For very deep beams, i.e. h > 1.5L, the refined strut-and-tie approach is still valid by assuming that the effective depth of the beam is 1.1L.

__S. T. Mau and T. T. C. Hsu (1987)__[24]
developed a theoretical treatment of the shear behavior of deep beams. They
extended the softened truss model theory to deep beams, which was successfully
developed for low-rise shear walls and torsion. The theory includes an
effective transverse compressive stress acting on the web shear element and a
softened concrete stress-strain relationship for the concrete behavior.
Theoretical predictions of shear strength are compared to 64 test results.
Examination of the governing equations helps to identify three major factors
that affect the shear strength. They are the shear span-to-depth ratio,
transverse reinforcement index and the longitudinal reinforcement index. The
theory predicts that transverse reinforcement is effective in increasing the
shear strength of deep beams when the shear span-to-depth ratio is greater than
0.5 but is not effective when this ratio is less than 0.5. Within the shear
span, a beam can be separated into three elements – each with a different
function to resist the applied load. The top element with a thickness d',
including the concrete and the longitudinal compression steel, is to resist the
longitudinal compression resulting from the sectional moment. The middle
element, including both the top and bottom longitudinal steel, is to resist the
sectional shear. This is called web shear element. The height of the web shear
element, d_{v} is equal to (d-d'). The top and bottom longitudinal bars
are used to carry the flexural stresses as well as the longitudinal stresses
due to shear. For a deep beam with concentrated load on top, however, the top
load and the bottom support reaction create large compressive stresses
transverse to the horizontal beam axis. These transverse compression stresses
interact with the shear stresses to form a complicated stress field in the web.
Because of smaller a/d ratio, the effect of such a transverse compression
stress on the shear strength of the web is quite significant and should be ignored
as in the case of shallow beams. In fact such a transverse compressive stress
is the source of the arch action unique to deep beams. A close examination of
the governing equations revealed that the normalized maximum shear stress t_{max}/f'_{c} is mainly affected
by three non-dimensional parameters. They are shear span-to-depth ratio,
longitudinal reinforcement index, r_{l}f_{ly}/f'_{c}
and transverse reinforcement index, r_{t}f_{ty}/f'_{c}.
The maximum shear stress ratio generally decreases with increasing a/h ratio.
The rate of decrease is relatively large at the low index range of 0.1 to 0.3
and becomes gradually smaller at high index range. For large a/h ratios maximum
shear stress ratio increases with the increase of transverse steel index,
especially in the low range. For small a/h ratio (< 0.5), however, the
maximum shear stress ratio actually decreases slightly with the increase of
transverse reinforcement index.

__S. C. Lee and W. B. Siao (1994)__[25]
investigated the effect of combined top and bottom loading simultaneously on
reinforced concrete deep beams. CIRIA Guide prescribes a maximum allowable
shear stress of 0.75 times that in a top-loaded beam for bottom-loaded beams
whose bottom-applied loading has been transferred to the top via vertical
stirrups. The factor 0.75 has not been adequately explained, it can be
attributed to prudence and conservatism. But the authors claimed that the
bottom loading did not impair the overall shear capacity of a deep beam if
sufficient vertical stirrups are provided.

__W. B. Siao (1993)__[26]
used strut-and-tie model to predict the shear strength of deep beams and pile
caps failing in diagonal splitting. Using test data from Subedi’s experimental
work, it was shown that when a/h > 1.4, the assumption of deep beam behavior
would result in over-predict of actual strength.

[1] J. E. Barry, H. Ainso, “Single-Span Deep Beam”, ASCE Journal of Structural Engineering, vol. 109, no. 3, March 1983, pp. 646-663

[2] A. Alshegeir and J. A. Ramirez, “Strut-tie approach in pretensioned beams”, ACI Structural Journal, v. 89, no. 3, May 1992, pp. 296-304

[3] A. Akhtaruzzaman and A. Hasnat, “Torsion in concrete deep beams with an opening”, ACI Structural Journal, v. 86, no. 1, January1989, pp. 20-25

[4] R. Narayanan and I.Y.S. Darwish, “Fiber concrete deep beams in shear”, ACI Structural Journal, v. 85, no. 2, March 1988, pp. 141-149

[5] D. M. Rogowsky, J. G. MacGregor and S. Y. Ong, ACI Structural Journal, 1986

[6] B. S. Maxwell and J. E. Breen, “Experimental evaluation of strut-and-tie model applied to deep beam with opening”, ACI Structural Journal, v. 97, no. 1, January 2000, pp. 142-148

[7] S. J. Hwang, W. Y. Lu and H. J. Lee, “Shear strength prediction of deep beams”, ACI Structural Journal, v. 97, no. 3, May 2000, pp. 367-376

[8] A. F. Ashour and G. Rishi, “Tests of reinforced concrete continuous deep beams with web openings”, ACI Structural Journal, v. 97, no. 3, May 2000, pp. 418-426

[9] S. Teng, W. Ma and F. Wang, “Shear strength of concrete deep beams under fatigue loading”, ACI Structural Journal, v. 97, no. 4, July 2000, pp. 572-580

[10] K. H. Tan, F. K. Dong and L. W. Weng, “High-strength reinforced concrete deep and short beams: Shear design equations in North America and UK practice”, ACI Structural Journal, v. 95, no. 3, May 1998, pp. 318-329

[11] S. J. Foster and R. I. Gilbert, “Experimental studies on high-strength concrete deep beams ”, ACI Structural Journal, v. 95, no. 4, July 1998, pp. 382-390

[12] K. H Tan, F. K Kong, S. Teng and L. W. Weng, “Effect of web reinforcement on high-strength concrete deep beams” , ACI Structural Journal, v. 94, no. 5, September 1997, pp. 572-582

[13] F. K Kong, S. Teng, A. Singh and K. H. Tan, ‘Effect of embedment length of tension reinforcement on the behavior of lightweight concrete deep beams”, ACI Structural Journal, v. 93, no. 1, January 1996, pp. -----------

[14] S. Teng, F.K Kong, S.P. Poh, L.W. Guan and K.H. Tan, “Performance of strengthened concrete deep beams predamaged in shear”, ACI Structural Journal, v. 93, no. 2, March 1996, pp. -------

[15] A. T. C. Goh, “Prediction of ultimate shear strength of deep beams using neural networks”, ACI Structural Journal, v. 92, no. 1, January 1995, pp. 28-32

[16] J.E Barry and H. Ainso, “Single span deep beams”, ASCE Journal, v. 109, no. 3, 1993, pp. 646-663

[17] Ali M. G. and Habib A., “Deflection and stress distribution in deep beams under uniformly distributed loads”, Journal of the Chinese Institute of Engineers, v. 15, no. 4, 1992, pp. 415-426

[18] Ali M. G. and Habib A., “Strength of deep reinforced concrete beams under uniformly distributed loads”, Proceedings of Nath Science Council ROC, v. 16, no. 5, September 1992, pp. 393-402

[19] Mansur M. A., Lee Y. F., Tan K.H. and Lee S. L, “Test on R. C. continuous beams with openings”, ASCE Journal, 1991

[20] M. D. Kotsovos, “Behavior of reinforced concrete beams with a shear span-to-depth ratio between 1.0 and 2.5”, ACI Journal, v. 81, no. 3, May-June 1984, pp. 279-286

[21] Q. Q. Liang, Y. M. Xie and G. P. Steven, “Topology optimization of strut-and-tie models in reinforced concrete structures using an evolutionary procedure”, ACI Structural Journal, v. 97, no. 2, March 2000, pp. 322-330

[22] K. H. Tan, S. Teng, F. K. Kong and H. Y. Lu, “Main tension steel in high strength concrete deep and short beams”, ACI Structural Journal, v. 94, no. 6, November 1997, pp. 752-768

[23] W. B. Siao, “Deep beams revisited”, ACI Structural Journal, v. 92, no. 1, January 1995, pp. 95-102

[24] S. T. Mau and T. T. C. Hsu, “Shear strength prediction for deep beams with web reinforcement”, ACI Structural Journal, v. 84, no. 6, November 1987, pp. 513-523

[25] S. C. Lee and W. B. Siao, “Shear strength of deep beams subjected to simultaneous top and bottom loading”, ACI Structural Journal, v. 91, no. 6, November 1994, pp. 663-665

[26] W. B. Siao, “Strut-and-tie model for shear behavior in deep beams and pile caps failing in diagonal splitting”, ACI Structural Journal, v. 90, no. 4, July 1993, pp. 356-363

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