Welcome back to prestressed concrete structures. This is the first lecture of Module 5 on analysis

and design for shear and torsion.In this lecture, first we shall study about the stresses in

an uncracked beam. Then we shall learn about the types of cracks generated in reinforced

concrete and in prestressed concrete beam. Then we shall learn about the components of

shear resistance, the modes of failure due to shear and then we shall learn about the

effect of prestressing force in a shear type of failure.The analysis of reinforced concrete

and prestressed concrete members for shear is more difficult compared to the analysis

for axial load or flexure.

The analysis for axial load and flexure are based on the following principles of mechanics:

equilibrium of internal and external forces, compatibility of strains in concrete and steel,

and constitutive relationships of materials. When we studied these two actions earlier,

we based our equations on these three principles and the behaviour of the member under each

of this action was well defined. We were able to plot the complete load versus deformation

behaviour for a member under axial load or for a member under flexure, but for shear

this is relatively much more difficult.

The conventional analysis for shear is based on equilibrium of forces by a simple equation.

The compatibility of strains is not considered. The constitutive relationships (relating stress

and strain) of the materials, concrete or steel are not used. The strength of each material

corresponds to the ultimate strength. The approach that we followed for axial load and

flexure was more rational compared to the approach that we shall follow in shear. For

shear, we shall consider only the equilibrium of forces, we are not considering the compatibility

of strains, the concrete and steel; neither shall we use the constitutive relationships

which relate the complete stress-strain curve of each material. Rather, we shall use only

the ultimate strength of a material under shear to get the ultimate shear strength capacity

of a member.

The strength of concrete under shear although based on test results is empirical in nature.

The equation that we shall use for the shear strength of concrete is not related with any

stress-strain curve. That equation is just based on the strength the concrete has and

this is based on test results.

Shear stresses generate in beams due to bending or twisting. The two types of shear stress

are called flexural shear stress and torsional shear stress respectively. In this module,

the analysis for shear refers to flexural shear stress. The torsional shear stress will

be covered under analysis for torsion. The shear that we shall consider at this beginning

is the shear due to flexure. The shear due to torsion will be covered in the later part

of this module.To understand flexural shear stress, the behaviour of a simply supported

beam under uniformly distributed load, without prestressing will be explained first. The

presentation will be in the following sequence.

First, we shall study the stresses in an uncracked beam. An uncracked beam is considered as a

homogeneous beam, where the effect of steel is negligible. Second, the type of cracks

generated due to the combination of flexure and shear will be discussed. Next, we shall

move on to components of shear resistance and the modes of failure due to shear. Finally,

we shall see the effect of prestressing force in the shear resistance of the concrete member.

First, the stresses in uncracked beam. Here we are seeing a simply supported beam, subjected

to uniformly distributed load. We are considering two points in the beam: one lies at the neutral

axis and the other lies close to the tension face. For such a beam, from basic structural

analysis, we know that the shear force varies along the span it varies linearly, the moment

also varies along the span and it varies along a parabolic curve.If we look at the stresses

at any point of the beam, the variation of normal stress along the depth is linear, it

is compressive at the top and tensile at the bottom. The variation of a shear stress at

a section along the depth is parabolic in nature, where the shear stress is zero at

the top and bottom, and maximum at the neutral axis. Thus the two types of stresses, the

normal stress and the shear stress in a beam under flexure have their maximum values at

different locations.At a particular section, the flexural stresses are maximum at the top

or at the bottom and minimum at the neutral axis. Whereas, the shear stress is maximum

at the neutral axis and is minimum at the two edges.

Under a general loading, the shear force and the moment vary along the length. The normal

stress and the shear stress vary along the length, as well as along the depth. What we

had seen in the previous graph is that the shear force and the moment vary along the

length. If I look at a particular section, the shear stress varies along the depth as

well as the flexural stress. Thus, if we look the variation of the shear or the flexural

stress point by point, it varies along the span as well as it varies along the depth.

The combination of the normal and shear stresses generate a two-dimensional stress field at

a point. In the analysis for shear, we are entering into a new concept which is the two-dimensional

stress fields, that means there is normal stress in one direction and there is shear

stress. This combination of normal and shear stresses generate a two-dimensional stress

field.

At any point in the beam, the state of two-dimensional stresses can be expressed in terms of the

principal stresses. The principal stresses are the stresses in the planes where there

is no shear stress. The Mohrâ€™s circle of stress is helpful to understand the state

of stress. These concepts were covered in our basic structural analysis; we shall review

this again in the light of analysis for shear.

Before cracking, the stress carried by steel is negligible. Hence, we can neglect the presence

of steel in the concrete beam. When the principal tensile stress exceeds the cracking stress,

the concrete cracks and there is redistribution of stresses between concrete and steel. It

is easier to understand the state of stress before the cracking of the member, which we

can derive by analyzing the beam by elastic analysis. After the cracking of the member,

the state of stress is more complicated because there is redistribution of stresses between

concrete and steel. With increasing load, the concrete enters into its non-linear behaviour

and the steel also after yielding enters into its non-linear behaviour.

For a point in the neutral axis, which we are representing by the Element 1, the shear

stress is maximum and the normal stress is zero. That means, when we are considering

this small element at the neutral axis, the shear stress v is maximum and the shear stress

generates a two-dimensional stress load.

Whereas, there is no normal stress at the neutral axis. From this state of stress if

we draw the principal stresses, the shear stress generates a biaxial tensile compressive

stress field. Sigma1 is the tensile stress and sigma2 is the principal compressive stress.

The principal tensile stress sigma1 is inclined at 45 degree to the neutral axis. Thus, if

we try to understand the state of stress with the help of a Mohrâ€™s circle, we see that

for the vertical and the horizontal surfaces, the state of stress is the top and the bottom

of the Mohrâ€™s circle where there is shear stress v but there is no normal stress. At

a surface which is inclined at 45 degrees, we have the principal stresses; in one side,

there is the principal tensile stress and in the perpendicular side, we have the principal

compressive stress. The magnitude of sigma1 and sigma2 are same under a pure shear.

Since, the shear force is maximum near the supports, cracks due to shear occur around

the neutral axis near the supports and perpendicular to sigma1. As I said, if you look point by

point along the beam, first, the shear force is maximum near the supports, and next the

shear stress varies along the depth and that is maximum at the neutral axis.

Hence, the cracking due to shear we expect to occur first near the supports and around

the neutral axis. These cracks will be normal to the principal tensile stress sigma1. Thus,

the understanding of the inclination of the principal tensile stress helps us to understand

the inclination of the cracks due to shear. In this sketch, we find that under this pure

shear since, sigma1 is inclined at 45 degree to the neutral axis, the cracking which occurs

perpendicular to sigma1 is also inclined at 45 degrees to the neutral axis, this is the

inclination of the crack due to pure shear.

For a point near the bottom edge, which is element number 2, the normal stress is maximum

and the shear stress is close to zero; that is, below the neutral axis, the normal stress

is tensile, the shear stress is small. The principal stresses are as follows. Here, the

inclination of sigma1 is less than 45 degree to the neutral axis. The value of sigma1 is

much larger than the magnitude of sigma2, which is the principal compressive stress.

If we plot the Mohrâ€™s circle we find that the state of stress in the horizontal and

the vertical surfaces can be denoted by these two points. For the vertical surface the state

of stress is given as f, v it has both the normal stress and the shear stress. Whereas,

for the horizontal surface it has only the shear stress and the point lies in the vertical

axis. The principal tensile stress sigma1 is almost parallel to the bottom edge, that

means the more we go close to the bottom edge sigma1 becomes more and more parallel to the

bottom edge; the value of alpha is much smaller than 45 degrees.

Since, the moment is maximum at the mid span, cracks due to flexure occur near mid span

and perpendicular to sigma1. It starts from the bottom of the beam and it gradually goes

up. If we see the state of stress, the inclination of the crack is perpendicular to sigma1. Since

sigma1 is almost parallel to the bottom edge, the cracks will be starting as perpendicular

to the bottom edge. In this figure, we see that the crack is almost perpendicular to

the horizontal axis, and when we come to the bottom edge of the surface, this cracks becomes

perpendicular to the bottom edge. With this understanding of the state of stress for an

uncracked beam, we are now moving on to understand the types of crack that generates in a beam

under flexure.

The types and formation of cracks depends on the span-to-depth ratio of the beam and

the loading. In the following figures, the formation of cracks for a beam with large

span-to-depth ratio and uniformly distributed loading is shown.

The formation of cracks in a reinforced concrete beam is inherently variant in nature. Wherever,

there is a weak point in the concrete beam the crack propagates through that point. Instead

of going into the detailed analysis of a crack formation, we are trying to have a visual

representation of the growth of cracks, which will help us to understand, the mechanism

of crack formation and finally the failure of concrete due to shear. For a long beam,

the span to the depth ratio is large. For such a beam under a uniformly distributed

load, the cracks start due to flexure and it starts from the bottom of the beam near

the mid span. In this sketch we find that the first cracks that are initiated are near

the middle of the beam and it starts from the bottom. Since the moment is maximum near

the middle and the flexural stresses are tensile and maximum at the bottom, the cracks generate

from the bottom surface. These cracks are called flexural cracks because they have generated

due to the maximum value of the moment.

As we go on increasing the load, we find the more number of flexural cracks and those cracks

have increased in size. We also find the formation of some flexural shear cracks; cracks are

generating due to flexure, but then they are propagating due to the effect of shear those

are called flexure shear cracks. Such types of cracks are initially perpendicular to the

bottom edge, but as they increase, they get inclined to the neutral axis. We also observe

some web shear cracks, which occur due to the large value of shear near the supports.

Near the supports, the moment is very small and these cracks form at the neutral axis

because the shear stress is maximum at the neutral axis. There is no flexural stress

at the neutral axis. Moreover the flexural stress near the supports is very small because

the moment is very small. Hence the cracks that are initiated by shear alone are near

the supports and they are inclined to the neutral axis at an angle of 45 degree. If

we load the beam further then we find a complete picture of the types of crack that is generated

in the beams.

In the middle region, we have the flexural cracks which are perpendicular to the axis

of the beam and they start from the bottom and propagate upwards. On the two sides of

the flexural cracks, we have flexure shear cracks they start at the bottom, initially

they are perpendicular but then they gradually get inclined due to the effect of shear. These

types of cracks are at transition between the web shear cracks and the flexural cracks.

Near the supports of the beam we find web shear cracks, it generates near the neutral

axis of the beam and they are inclined to the neutral axis. These are the three types

of cracks that are generated in a long span beam under uniformly distributed load.

The crack pattern can be predicted from the principal stress trajectories. In this figure,

the solid lines give the trajectories of the compressive stress and the dotted lines give

the trajectories of the tensile stress. As we know that the cracks form perpendicular

to the tensile stress, or they are parallel to the compressive stress. Hence, the formation

of the cracks follows the stress trajectories of the compressive stress. We find that near

the middle span the cracks are perpendicular, whereas when we go closer to the support the

cracks at the neutral axis are inclined. Thus, the crack pattern in a reinforced concrete

beam can be explained by the help of the trajectories of the principal stresses of a homogeneous

beam.

For a simply supported beam, under a uniformly distributed load without prestressing three

types of cracks are identified. The flexural cracks: these cracks form at the bottom near

the mid span and propagate upwards. Web shear cracks: these cracks form near the neutral

axis close to the support and propagate inclined to the beam axis. Flexure shear cracks: these

cracks form at the bottom due to flexure and propagate due to both flexure and shear. Next,

we move on to understand the components of shear resistance in a reinforced concrete

beam.

This is studied based on the internal forces at a flexure shear crack, the components are

as follows. Thus, we are seeing the free body of a section of reinforced concrete beam.

This section has been drawn as per the direction of a flexural shear crack, it is perpendicular

to the bottom edge but as we go up this crack is getting inclined to the neutral axis. What

we observe is that the shear stress is resistant by several components in the beam. The first

component is the component of the shear resistance due to the concrete under compression.

Above the tip of the crack, the concrete is uncracked and this part of the concrete can

carry shear just like a homogeneous material. Next, since the crack traverses the stirrups,

the stirrups carry shear after the cracking of concrete and that is represented by Vs.

The crack surface is not smooth, it is an irregular jagged surface. There due to the

interlocking of the aggregates we have a component called aggregate interlock and that is denoted

as Va.We have the dowel action, that means whenever a longitudinal bar is tried to be

shifted along with a direction perpendicular to the axis, then it generates some resistance

which is called the dowel action. This action generates when the longitudinal bar is properly

inserted within the concrete and the bond has not deteriorated. The dowel action also

provides some resistance to shear. For a prestressed concrete beam, we have another component,

which is the vertical component of the prestressing force for an inclined tendon and that is represented

as Vp. Thus, the value of Vp depends on the inclination of the tendons.

To summarize, the components of the shear resistance are as follows: we had Vcz, which

is the shear carried by uncracked concrete, Va is the shear resistance due to aggregate

interlock, Vd is the shear resistance due to dowel action, Vs is the shear carried by

stirrups and Vp is the vertical component of prestressing force in inclined tendons.

The magnitude and the relative value of each component change with increasing load. Once

we have identified these components, we have to understand their individual contribution

in the shear resistance of the concrete.The individual contribution varies as the load

is increased. In the initial stage after the cracking, the aggregate interlock and the

dowel actions are high, but as the load is increased, as the cracks open up, the aggregate

interlock gets reduced. As the bond between the steel and the concrete gets disrupted

near the cracks the dowel action gets reduced. The zone of concrete under compression gets

reduced with the propagation of the cracks, thus Vcz gets reduced. Finally, it is the

Vs which increases and resists the external shear. Next, we are studying the modes of

failure due to shear.

For beams with low span-to-depth ratio or inadequate shear reinforcement, the failure

can be due to shear. A failure due to shear is sudden as compared to a failure due to

flexure. Earlier, when we studied the behaviour of a beam under flexure, we had studied the

moment curvature curve and we had studied the ductility. We understood that after the

steel yields, the section is able to resist the strength with some large deformation.

That is called ductility in the moment versus curvature curve of a beam. Unlike the behaviour

in flexure, the failure of shear is quite sudden, it is a brittle failure as compared

to the flexural failure. The cracks open up suddenly and the beams tend to fail in a brittle

mode.

Five modes of failure due to shear are identified. The occurrence of a mode of failure depends

on the span-to-depth ratio, loading, cross-section of the beam, amount and anchorage of reinforcement.

The modes of failure are explained next.

The first one is the diagonal tension failure: In this mode an inclined crack propagates

rapidly due to inadequate shear reinforcement. That means, if the shear reinforcement is

not adequate enough, the cracks that have formed due to shear or flexural shear they

will propagate through the depth of the beam quickly. We see that the right part of the

beam has got literally separated from left part. This is an instance of a failure due

to shear and this type of failure is called the diagonal tension failure.

The second one is the shear compression failure: There is crushing of the concrete near the

compression flange, above the tip of the inclined crack. If the concrete near the top flange

is not strong enough, then as the inclined crack propagates there will be crushing of

the concrete at the top of this crack. This will also be affected by the amount of moment

at that particular section. When there is crushing of the concrete at the top of a flexural

shear crack, that type of failure is called a shear compression failure.

The third type of failure is the shear tension failure. Due to inadequate anchorage of the

longitudinal bars, the diagonal cracks propagate horizontally along the bars. If the longitudinal

bar is not properly anchored at the support, then the diagonal cracks once it hits the

level of the longitudinal bars the crack propagates horizontally towards the supports and the

bond between the concrete and steel gets disrupted. It leads to an anchorage failure and this

type of failure is called the shear tension failure. Thus, the shear tension failure leads

to a separation of the longitudinal bars, with the concrete and which leads to an anchorage

failure of the longitudinal bars.

The fourth type of failure is the web crushing failure: The concrete in the web crushes due

to inadequate web thickness. Thus, this type of failure happens in an I girder if the width

of the web is small. Since, there is a principal compression in an inclined direction if the

web is not strong enough then it will crush due to the principal compression.

The fifth type of failure - the arch rib failure - is observed in deep beams, where the span-to-depth

ratio is small, the web may buckle and subsequently crush. There can be anchorage failure or failure

of the bearing. This type of arch rib failure is characteristic of a deep beam where the

span-to-depth ratio is small. There can be crushing of the concrete in the web, there

can be anchorage failure and also crushing of the concrete near the bearing. This type

of failure due to the arch action within the deep beam is called an arch rib failure.

The objective of design for shear is to avoid shear failure. The beam should fail in flexure

at its ultimate flexural strength. Hence, each mode of failure is addressed in the design

for shear. The design involves not only the design of

the stirrups but also limiting the average shear stress in concrete, providing adequate

thickness of the web and adequate development length of the longitudinal bars.

Once we have understood the modes of failure, we can understand the objectives of the shear

design. The shear design includes not only the design of stirrups, but it includes other

detailing requirements which checks the different modes of shear failure. We have to have adequate

thickness of the web, we have to have adequate anchorage of the longitudinal bars, and we

have to have the stress in the concrete limited, because if the concrete crushes then it will

lead to a sudden shear failure. As the shear failure is brittle, we do not allow shear

failure before the concrete reaches its final strength. The objective of the design is that

a beam should fail in flexure at its ultimate strength and it should not fail in shear before

it reaches its ultimate strength.

Next, we are studying about the effect of prestressing force. In presence of prestressing

force, the flexural cracking occurs at a higher load. For Type 1 and Type 2 sections, there

is no flexural crack under service loads. This is evident from the typical moment versus

curvature curve for a prestressed section.

We had seen this curves earlier that for a reinforced concrete beam, the moment versus

curvature curve passes through the origin, after cracking there is redistribution of

stresses that is increase in curvature. Then we observe non-linearity in the behaviour

due to the non-linearity of concrete. Once the steel yields then we observe the non-linearity

of the steel as well and gradually the curve tapers off to the ultimate strength. If we

have prestressing in the beam, the curve gets shifted from the origin, which means that

for zero curvature we need to have an external moment. In other words, in absence of any

moment there is a negative curvature which is the cambering effect due to prestressing.

As we increase the load, the linear behaviour is up to a much higher level. Usually the

prestressed concrete members are designed such that it does not crack under service

loads. This type of sections has a cracking load which is higher than the service load.

After cracking, we observe the non-linearity in the behaviour and the strength tapers off

to the ultimate strength. These two curves have been drawn for a reinforced concrete

and a prestressed concrete section which have equivalent flexural strength.

In presence of prestressing force, the web shear cracks also generate under higher load.

Thus, what we observe is when ever there is a prestressing force the flexural cracks,

the web shear cracks, both of them occur at a higher load. With increase in the load beyond

the cracking load, the cracks generate in a similar sequence; for a long beam, we will

observe flexural cracks to occur first, then we shall observe some flexural cracks to change

to a flexural shear cracks, then we may observe web shear cracks near the supports. The sequence

is similar but the inclination of the flexural shear and the web shear cracks are reduced,

depending on the amount of prestressing and the profile of the tendon.

Let us understand the effect of prestressing force for a beam with a concentric prestressing

force Pe. This is a simply supported beam with a uniformly distributed load subjected

to a concentric prestressing force Pe. We shall observe the state of stress for an element

which is at the neutral axis. For a point at the neutral axis which is represented by

Element 1, there is normal stress due to the prestressing force which is compressive and

denoted as minus fpe. This is the main difference with reinforced concrete that in a reinforced

concrete at the neutral axis we did not have any normal stress it was the case of pure

shear stress. For a prestressed concrete beam, we are not only having a shear stress at the

neutral axis, but we are also having a compressive stress due to the prestressing force.

Due to this, the state of stress is such that the principal tensile stress, sigma1 is inclined

to the neutral axis at an angle greater than 45 degree. The principal tensile stress sigma1

is much smaller, compared to the principal compressive stress sigma2. If we plot the

Mohrâ€™s circle for this state of stress, then this point corresponds to the vertical

surface which has the shear stress and in the normal stress which is negative minus

fpe. The point lying in the vertical axis is the horizontal surface, which has only

the shear stress. We find that the magnitude of sigma1 the principal tensile stress is

much smaller, than the magnitude of sigma2 which is the principal compressive stress.

Since sigma1 is inclined at an angle greater than 45 degrees, the crack which occurs perpendicular

to sigma1 or parallel to sigma2 is inclined at an angle which is lower than 45 degrees

in presence of the prestressing force.

Thus, for a prestressed concrete beam, the formation of the cracks can be explained by

the help of the state of stress. What we had seen was for a concentric prestressing force

is if the prestressing force is eccentric and if the profile of the tendon is parabolic,

then the analysis gets more involved, but the basic concept remains the same that due

to the effect of prestressing force the inclination of the cracks due to shear gets reduced. The

principal tensile stress is much smaller than the principal compressive stress. Hence, the

cracking occurs at a much higher level as compared to a reinforced concrete beam.

The crack pattern that occurs in a prestressed concrete beam is shown in this figure. Here

you can see that the flexural cracks are much smaller as compared to a reinforced concrete

beam. The number of cracks is small. The flexural shear cracks are more inclined, the web shear

cracks are also small they are more inclined to the neutral axis as compared to the cracks

in a reinforced concrete beam.

If we compare the cracking pattern in the two types of beams, what we observe is that

in a reinforced concrete beam, the flexural cracks are much more compared to a prestressed

concrete beam. The growth of the flexural cracks is more than in a prestressed concrete

beam for the same load. The flexural shear cracks are more inclined in a prestressed

concrete beam. The growth of the flexural shear cracks is less in a prestressed concrete

beam. The web shear cracks are at an angle of 45 degree in there reinforced concrete

beams, whereas for prestressed concrete beams the web shear cracks the inclination is smaller.

The amount of cracking throughout the beam in a prestressed concrete beam is much lower

as compared to a reinforced concrete beam. This is the big benefit of the prestressing,

that the cracking in the beam is much less as compared to a reinforced concrete beam.

After cracking, in presence of prestressing force, the length and crack width of a diagonal

crack are low. Thus, the aggregate interlock and zone of concrete under compression are

larger as compared to a non-prestressed beam under the same load. Hence, the shear strength

of concrete which is represented as Vc increases in presence of prestressing force. This is

accounted for in the expression of Vc. The main advantage of prestressing is being reflected

in this observation: since, the growth of cracks in a prestressed concrete beam is small,

we have higher aggregate interlock in a prestressed concrete beam. We have a larger concrete in

compression for a prestressed concrete beam. Both these effects increase the capacity of

the concrete to resist shear. Thus, in a prestressed concrete beam the capacity of concrete to

resist shear which is denoted as Vc is higher as compared to a reinforce concrete beam.In

addition, we also have the vertical component of the prestressing force if the tendon is

inclined. Thus, as a summation the shear capacity of a prestressed concrete beam is larger than

a reinforced concrete beam. This helps us to check the different modes of failure under

shear provided we do the other detailings properly and the beams are expected to reach

their ultimate flexural capacity before it fails under shear.

In todayâ€™s lecture, we studied the analysis for shear and we learned the important concept

of shear. First, we studied the stress condition in an uncracked beam. We understood this concept

from the basic structural analysis for a simply supported beam under uniformly distributed

loads. The shear force varies along the length of the member, the moment also varies along

the length of the member. The shear force is maximum near the supports and minimum near

the mid span whereas, the moment is minimum near the supports and maximum at the mid span.

If we look at the variation of stress along the depth, the flexural stresses we generate

normal stresses. They are maximum at the top and at the bottom, and minimum at the neutral

axis, whereas the shear stress is minimum at the top and the bottom, and maximum at

the neutral axis. If we see the variation of the normal and shear stress in a beam along

the length and along the depth, we are able to understand the formation of the cracks.

What we observe is that around the mid span, since the flexure is high the normal stresses

at the bottom of the beam leads to cracks, which starts from the bottom of the beam and

goes up, those are called the flexural cracks.

If we come near the supports the normal stresses are small since the moment is small, and the

shear stress is high near the neutral axis, also the shear force is high near the supports.

Hence, the cracking due to the shear occurs near the neutral axis. We understood the inclination

of the cracks from the state of stresses. For a point near the support at the neutral

axis it is under pure shear stress, and a pure shear stress is equivalent to a biaxial

stress field, where the tensile and the compressive principal stresses are equal in magnitude.

The crack occurs perpendicular to the principal tensile stresses and they are inclined at

an angle of 45 degree to the neutral axis.

Thus, the web shear cracks occur at an angle of 45 degree. In between the web shear cracks

and the flexural cracks, we have a transition region where we observe flexural shear cracks.

These cracks generate due to the flexural stresses from the bottom of the beam, but

they propagate and get inclined to the neutral axis due to the effect of shear. Hence, these

are called flexure shear cracks. The crack pattern can be explained from the principal

stress trajectories of a homogeneous beam. After the concrete cracks, there is redistribution

of stresses and we understood the components of shear resistance from the free body diagram

of a concrete beam adjacent to a flexural shear crack.

Among the components of the shear resistance, first we have the component for the uncracked

portion of the concrete above the deep of the crack. Next, we have the aggregate interlock

effect which occurs because the cracks are not smooth. Then we have the contribution

of the stirrups which is represented as Vs. We also have observed that the longitudinal

bars provide some resistance due to the dowel action, this component of the shear is represented

as Vd. For a prestressed concrete beam we have a vertical component of the prestressing

force in an inclined tendon and that is represented as Vp.

Now, the relative contributions of these various components, in the shear resistance varies

as the load is increased. For smaller loads after the cracking, the aggregate interlock,

the contribution of the concrete under compression and the dowel actions is high. As the load

is increased as the cracks propagate and the cracks open up, the aggregate interlock reduces,

the zone of concrete under compression reduces, the bond between the concrete and the steel

gets deteriorated and hence the dowel action gets reduced. The shear stress carried by

the stirrups gets increased till the yielding of the stirrups. Hence, what we find that

at ultimate strength, the summation of the dowel action, the aggregate interlock, and

the concrete under compression, drops down to a minimum value. The contribution of the

stirrups attains a maximum value and stays constant at the value where the stirrups start

to yield. This is the basis of developing the equations for shear resistance.

Now from the components of shear resistance, we move on to the modes of failure due to

shear: First, we had seen the diagonal tension failure, that if there is inadequate stirrups

then the diagonal cracks propagate through the depth of the beam quite rapidly, and this

will lead to a sudden failure of the beam. The second, we found was a shear compression

failure, where at the tip of a diagonal shear crack, the concrete under compression can

crush; this will also be affected due to the flexural compression and we need to have adequate

concrete at the top to resists this shear compression.We found another type of failure

which is a shear tension failure. If the longitudinal bars are not properly anchored near the support,

then the diagonal crack after it reaches the level of steel, propagates horizontally, disturbs

the bond between the steel and the concrete and the steel bar tries to pull out from the

ends, which leads to a anchorage failure. This type of failure is referred to as the

shear tension failure.

The fourth type of failure, we have seen was the web crushing failure. This occurs for

beams with thin webs like an I girder, where due to the principal compressive stress the

concrete in the web may crush. The fifth type of failure is observed for

deep beams where the span-to-depth ratio is small. In a deep beam, there is an arch action

and due to the arch action, we may find a failure which is termed as arch rib failure,

which consists of the crushing of the concrete in the web, crushing of the concrete near

the bearing. It can also have an anchorage failure, this type of failure of course is

not observed for long span beams.Now next, we move on to the effects of prestressing

force. We found is that when there is a prestressing force there is additional normal stress at

the level of the neutral axis which is compressive, it influences the principal stresses and their

inclination. The principal tensile stress is much smaller in magnitude than the principal

compressive stress. The inclination of the cracking is lower than 45 degrees. Cracks

are much less in a prestressed concrete beam as compared to a reinforced concrete beam.

The cracks are also inclined in small angle compared to the reinforced concrete beam.

This has a direct benefit in the shear resistance of the section, the aggregate interlock is

sustained, the zone of concrete under compression is higher and the dowel action is also sustained.

Thus, the shear capacity of a concrete beam increases in presence of prestressing force.The

objective of shear design is to check the modes of shear failure, such that the beam

attains its flexural capacity before it fails in shear. It involves not only the design

of stirrups but also other detailing to check the different modes of failure. In our next

class, we shall move on to the formulation of the shear resistance from which we are

able to design for the shear in a prestressed concrete beam. Thank you.