3-Flexural Analysis prestress concrete ppt

PRESTRESSED CONCRETE
1
Flexural Analysis (Lec-3)
Dr. Qasim Shaukat Khan
Associate Professor
Civil Engineering Department
UET Lahore
Email: qasimkhan@uet.edu.pk

PRESTRESSED CONCRETE
(Revision)
2
PRESTRESSED CONCRETE is a particular form of
reinforced concrete, which involves the application of an initial
compressive load (Pre-loading before the application of
Service Loads) on a structure to reduce or eliminate the
internal tensile forces / stresses and there by control or
eliminate cracking.
The compressive force is imposed and sustained by highly
tensioned steel reinforcement reacting on the concrete.
The concept of Prestressing of concrete is to introduce
sufficient axial precompression in beams so all tension in the
concrete was eliminated in the member at service load.

PRE-TENSIONED
CONCRETE (Revision)
3
1
Tendons are
stressed between
supports
PRECASTING PROCEDURE
The figure above illustrates the three stages requiredfor
pretensioning a concretemember.
Concrete cast
and cured
2
Tendons released
and prestress
transferred.
3

POST-TENSIONED
CONCRETE (Revision)
4
The three stages of post-tensioned concrete are
shown above.
hollow duct
1. Concrete
cast and
cured uplift forces
TENSILE
FORCE
COMPRESSIVE
FORCE
2. Tendons stressed
and prestress
transferred
dead end
live end
3.Tendons
anchored and
duct grouted

FLEXURALANALYSIS (Revision)
5
Both Analysis and Design of Prestressed Concrete may require
the consideration of the following load stages:
1. Initial Prestress, immediately after transfer, when (𝑃𝑖) alone
may act on the concrete.
2. Initial Prestress plus self-weight of the member.
3. Initial Prestress plus full Dead Load.
4. Effective Prestress, (𝑃𝑒), after losses, plus service loads
consisting of full dead load and expected live loads.
5. Ultimate load, when the expected service loads are increased by
load factors and the member is about to fail.
At and Below, the Service Load, both Concrete and Steel
Stresses are usually within the Elastic Range.

LOSS OF PRESTRESS- TIME DEPENDENT
LOSSES (Revision)
6
The prestressing force in a prestressed member gradually
decreases with time.
This loss of prestress is mainly caused by inelastic creep and
shrinkage strains, which develop with time in the concrete at the
level of the bounded steel.
In addition to creep and shrinkage losses, a gradual loss of
prestress occurs owing to stress relaxation in the tendons.
The combined effect of the three time-dependent losses usually
reduces the initial prestressing force by between 10-25 %.
The initial stress level in prestressing steel after transfer is usually
high, often in the range 60-75% of the tensile strength of the
material. At such stress levels, high-strength steel creeps.

LOSS OF PRESTRESS- TIME DEPENDENT
LOSSES & IMMEDIATE LOSSES (Revision)
7
If a tendon is stretched and held at a constant length (constant
strain), the development of creep strain in the steel is exhibited as
a loss of elastic strain, and hence a loss of stress. This loss of
stress in a specimen is subjected to constant strain is known as
relaxation.
Relaxation in steel is highly dependent on the stress level and
increases at an increasing rate as the stress level increases.
Immediate Losses: During the stressing operation, immediate
losses can occur by elastic contraction of concrete, by friction
along the cables and by slip and deformation in the end anchors.
The calculation of losses is an important step in the design process.
Typical immediate losses are around 6% to 8%

8
When the jacking force is first applied and the strand is stretched
between abutments, the steel stress is 𝒇𝒑𝒋
Upon the transfer of force to the concrete member, there is an
immediate reduction of stress to the initial stress level 𝒇𝒑𝒊 due to the
elastic shortening of concrete.
At the same time, the self weight of member is caused to act as the
beam cambers upward. It is assumed that all time dependent losses
occur prior to superimposed loading, so that the stress is further
reduced to an effective prestress level 𝒇𝒑𝒆 .
As the superimposed dead and live loads are added, there is a slight
increase in steel stress. Assuming that perfect bond is maintained
between steel and concrete, this increase must be 𝒏𝒑 times the increase
in the stress in concrete at the level of steel. The change is no more
than about 3-4% of the initial stress and is usually ignored in
calculations.

9
EQUIVALENT RECTANGULAR STRESS BLOCK
To calculate the ultimate resisting moment of a prestressed concrete
beam, compressive resultant (C) which must be equal to tensile force
(T) and the internal lever arm at failure is required.
The shape of the stress-strain curve of concrete varies greatly. For
this reason, explicit equation cannot be written, the actual stress
distribution of concrete is replaced with a simplified version.

10
EQUIVALENT RECTANGULAR STRESS BLOCK
So, that correct value of compressive resultant (C) is obtained and
the force compressive resultant (C) acts at the correct level in the
beam.
Based on the analysis and experiment, that the actual distribution of
compressive stress in the beam can be replaced with an equivalent
rectangular stress distribution having a uniform stress intensity of
0.85𝒇𝒄
,
and depth a.
The relationship between the equivalent and actual stress block
depth is 𝜶 = 𝜷𝟏𝒄
The value of 𝜷𝟏 has been established experimentally as
𝜷𝟏 = 𝟎. 𝟖𝟓 − 𝟎. 𝟎𝟓( Τ
𝒇𝒄
′ − 𝟒𝟎𝟎𝟎 𝟏𝟎𝟎𝟎)
where 𝜷𝟏 is not to exceed 0.85 and not to be less than 0.65.

11
FLEXURAL STRENGTH BY STRAIN COMPATIBILITY
ANALYSIS

12
ANALYSIS
Strains and stresses in the concrete and steel at loading stages
are shown in Figure 3.11.
Strain distribution (1) results from the application of effective
prestress force (𝑷𝒆), acting alone, after all losses. At this stage,
the stress in the steel and the associated strain can be computed
as below
𝒇𝒑𝒆 =
𝑷𝒆
𝑨𝒑
∈𝒑𝒆=∈𝟏=
𝒇𝒑𝒆
𝑬𝒑
Next, an intermediate load stage (2) corresponding to
decompression of the concrete at the level of the steel centroid is
considered. Assuming, bond remains intact between the concrete
and steel, the increase in steel strain produced as loads pass from
Stage 1 to Stage 2 is the same as the decrease in concrete strain at
that level in the beam, given by following expressions

13
ANALYSIS
When the member is overloaded to the failure Stage (3), the
neutral axis is at a distance c below the top of the beam. The
increment of strain is
The total steel strain at failure (∈𝒑𝒔), is the sum of the three
strains
and the corresponding steel stress at failure is 𝒇𝒑𝒔 .
The depth of the compressive stress block at failure can be
found from the equilibrium requirement that C = T , for a beam
in which the compression zone is of constant width b,
0.85𝒇𝒄
′ 𝒂𝒃 = 𝑨𝒑𝒇𝒑𝒔
∈𝟐=
𝑷𝒆
𝑨𝑪𝑬𝒄
(𝟏 +
𝒆𝟐
𝒓𝟐)
∈𝟑=∈𝒄𝒖 (
𝒅𝒑 − 𝒄
𝒄
)
∈𝒑𝒔=∈𝟏 +∈𝟐 +∈𝟑

14
ANALYSIS
Solving this equation for the Stress block depth
The resisting moment at failure is the product of tensile /
compressive force and internal lever arm. For a member with
constant width compression zone, the nominal flexural
strength is
𝐚 =
𝑨𝒑 𝒇𝒑𝒔
𝟎. 𝟖𝟓𝒇𝒄
′
𝒃
= 𝜷𝟏 𝒄
𝑴𝒏 = 𝑨𝒑𝒇𝒑𝒔(𝒅𝒑 −
𝒂
𝟐
)
The Equations for the calculations of stress block depth and
nominal flexural strength cannot be used directly to calculate
the failure moment for a beam, because the steel stress 𝒇𝒑𝒔 at
failure is unknown. An iterative solution can be devised, which
is as follows

15
ANALYSIS (ITERATION METHOD)
Assume a reasonable value for the steel stress, 𝒇𝒑𝒔 at failure,
and note from the steel stress-strain curve, the corresponding
failure strain ∈𝒑𝒔
Calculate the depth c to the actual neutral axis for that steel
stress, using the equation below based on horizontal equilibrium
Calculate the incremental strain ∈𝟏 from the equation
and add this to the prior strains indicated by the
𝐚 =
𝑨𝒑 𝒇𝒑𝒔
𝟎. 𝟖𝟓𝒇𝒄
′
𝒃
= 𝜷𝟏 𝒄
∈𝟑=∈𝒄𝒖 (
𝒅𝒑 − 𝒄
𝒄
)
∈𝒑𝒔=∈𝟏 +∈𝟐 +∈𝟑

16
If the failure strain ∈𝒑𝒔 so obtained differs significantly from
that assumed strain at the start, revise that assumption and
repeat the above steps until satisfactory agreement is obtained.
With both a and 𝒇𝒑𝒔 now known, calculate the ultimate flexural
moment using the equation
The suitability of the equivalent rectangular stress block with
uniform stress of 0.85 fc’, for determining the resisting moment
of I and T sections can reasonably be questioned as stress block
depth a is less than or at the most equal to the compression
flange thickness. However, comparison with the extensive
calculations based on stress distribution from actual stress-
strain curves indicate that the use of rectangular stress block for
I and T- sections introduce minor errors.
𝒂
𝟐
)

17
Using the Strain Compatibility method, find the ultimate
moment capacity for the I- beam. Normal density of concrete is
to be used, with compressive strength of 𝒇𝒄
′ = 4000 Psi (28 MPa)
and elastic modulus 𝑬𝒄= 3.61 x 106 Psi (24,890 MPa). The
ultimate strain capacity of the concrete is ∈𝒄𝒖 = 0.003 and
𝜷𝟏 =0.85. The beam is pretensioned, using seven ordinary
Grade 250 1/2” diameter seven wire strands, for which the
stress-strain is given. The effective prestress force 𝑷𝒆 = 144 kips
(641 kN)

18

19

20

21
▪ Although the stress-block depth exceeds the thickness of the
flange, it is about equal to the average thickness, refinement
to account for the actual shape of the compression zone would
have little effect on the results in this case.
• The steel increment strain ∈𝟐, caused by decompression of the
concrete is very small compared with ∈1, and ∈3. Neglecting
this strain ∈2 would have little influence on results.
• The strain at failure is close to that corresponding to the yield
stress. Hence, very little elongation of steel would occur
incase the beam is overloaded prior to the crushing of concrete.

22
FLEXURAL STRENGTH BY (ACI CODE EQUATIONS)
The Flexural strength of prestressed concrete beams can be
calculated using a strain compatibility analysis. Alternatively,
within certain limits, approximate equations for 𝒇𝒑𝒔 may be
used to compute the flexural strength.
In practice, a strain compatibility analysis is seldom required,
and the ACI Code approximate approach is used.
Steel Stress 𝒇𝒑𝒔 at Flexural Failure
The ACI Code contains approximate equations for 𝒇𝒑𝒔, the
stress in the prestressed steel when the beam fails, based on a
combination of test evidence and analysis.

23
Code equations for 𝒇𝒑𝒔, can be applied only if 𝒇𝒑𝒆 is not less than
the 0.5𝒇𝒑𝒖 .
In their general form, they apply to prestressed members that
may also contain non-prestressed tension reinforcement,
compression reinforcement or both.
Members with bonded Tendons but without Tension or
Compression Rebars
For prestressed beams that do not contain supplementary bar
reinforcement or for the cases where the contribution of such
reinforcement to flexural strength can be neglected, the stress
in the tendons at beam failure can be found by following equation

24
𝑓𝑝𝑠 = 𝑓𝑝𝑢(1 −
𝛾𝑝
𝛽1
𝜌𝑝𝑓𝑝𝑢
𝑓′
𝑐
)
where, the prestressed reinforcement ratio is
𝜌𝑝 =
𝐴𝑝
𝑏𝑑𝑝
𝒅𝒑 is the effective depth to the centroid of the prestressing steel.
Based on the differences in the stress-strain properties of low-
relaxation wire and strand compared with ordinary steel, a
coefficient 𝜸𝒑 is introduced. Based on the strain compatibility
analysis 𝜸𝒑 is 0.40 for Τ
𝒇𝒑𝒚 𝒇𝒑𝒖 not less than 0.85 (ordinary stress
relieved tendons)

25
To ensure that prestressed concrete beams, if overloaded, shall have
a ductile response before failure, it is desirable to place an upper
limit on the tensile steel ratio, thereby ensuring that the steel will be
stressed at least to its yield stress, when the beam fails.
In prestressed concrete beams, based on the effective prestress
𝒇𝒑𝒆 of 0.60𝒇𝒑𝒖, reasonably typical of beams in practice, the upper
limit on the reinforcement ratio for rectangular sections with
prestressing steel only is
𝝎𝒑 = 𝟎. 𝟑𝟔𝜷𝟏
where 𝝎𝒑 =
𝝆𝒑𝒇𝒑𝒔
𝒇𝒄
′

26
The flexural strength of prestressed beam is computed according to
ACI Code, using similar methods as are used for ordinary
reinforced concrete members.
For beams with Rectangular cross-sections (I or T) beams in which
the stress block depth falls within the thickness of the compression
flange and containing no supplementary reinforcing bars on the
tension side, the nominal flexural strength as per ACI Code is
For flanged sections with the depth of the stress block greater than
the thickness of flange, the flexural strength equation is modified
to account for non-rectangular shape of compression flange.
𝒂
𝟐
)

27
𝑴𝒏 = 𝑨𝒑𝒘𝒇𝒑𝒔 𝒅𝒑 −
𝒂
𝟐
+ 𝑨𝒔𝒇𝒚 𝒅 − 𝒅𝒑 + 𝑨𝒑𝒇𝒇𝒑𝒔(𝒅𝒑 −
𝒉𝒇
𝟐
)
To determine which equation is to be considered, the stress block
depth 𝒂 , is computed assuming that rectangular beam analysis
applies (stress block depth is less than or equal to the flange)

28
Depth 𝒂, is then compared with the flange thickness 𝒉𝒇, If depth 𝒂
is greater than 𝒉𝒇 then flexural strength is computed using equation
below:
𝒂
𝟐
𝒉𝒇
𝟐
)
The total compression force in the beam section is divided into two
parts. The first, the compression in the overhanging portion of the
flange is equilibrated by a part of the total tension force
𝑨𝒑𝒇𝒇𝒑𝒔 = 𝟎. 𝟖𝟓𝒇𝒄
′ (𝒃 − 𝒃𝒘)𝒉𝒇
This flange force provides a resisting moment, with the internal
lever arm measured to the centroid of prestressing steel (𝒅𝒑 −
𝒉𝒇
𝟐
)

29
The remaining part of the total tension force 𝑨𝒑𝒘𝒇𝒑𝒔 + 𝑨𝒔𝒇𝒚 is
equilibrated by the compression in the beam web and is defined as
𝑨𝒑𝒘𝒇𝒑𝒔 = 𝑨𝒑𝒇𝒑𝒔 + 𝑨𝒔 𝒇𝒚 − 𝑨𝒑𝒇𝒇𝒑𝒔
The compression force in the web, which is equal to this partial
tension has an internal lever arm, measured to the centroid of the
prestressing steel
𝐚 =
𝑨𝒑𝒘 𝒇𝒑𝒔
𝟎. 𝟖𝟓𝒇𝒄
′
𝒃𝒘
The final contribution to the resisting moment is provided by the
non prestressed tension reinforcement, if any, contributing a force

30
𝑨𝒔𝒇𝒚 acting at a distance 𝒅 − 𝒅𝒑 from the center of the moments
at the prestressing steel centroid.
The total resisting moment at failure is found by summing the
contributions of the three parts
𝒂
𝟐
𝒉𝒇
𝟐
)
Full Prestressing
Early in the development of prestressed concrete, the goal of
prestressing was complete elimination of concrete tensile stress at
service loads.

31
The concept of Full Prestressing was based on the idea that
homogenous material would remain uncracked and respond
elastically up to the maximum anticipated loading. In this design,
the limiting tensile stress in the concrete at full service load is zero.
Examples of Full Prestressing include tanks or reservoirs, where
leaks must be avoided, submerged structures or those structures
subjected to corrosive environment where maximum protection of
reinforcement must be ensured.
Partial Prestressing
In Partial Prestressing, flexural tension and usually some cracking
are permitted in the concrete at normal service load.

32
Partial Prestressing results in substantially improved performance,
reduced cost or both may be obtained by using the lesser amount of
prestress.
Fully Prestressed beams may exhibit an undesirable amount of
upward camber (upward displacement to counter downward
displacement produced by the gravity load) because of the
eccentric prestressing force. Moreover, the creep in concrete,
magnifies the upward displacement due to the prestress force but
has minor influence on the downward deflection due to live loads.
Full Prestressing may result in sudden brittle failure and may also
exhibit severe longitudinal shortening and substantial loss of
prestress.

33
Although, concrete cracking may be allowed at full service load, it
is noted that such full service load may be infrequently applied.
The typical load acting is likely to be dead load plus a small
fraction of the specified live load. Hence, a partially prestressed
beam may not be subjected to tensile stress under the usual
conditions of loading.
Cracks may form occasionally, when full service load is applied,
but these cracks will close completely when that load is removed.
These cracks are not objectionable in prestressed concrete
structures unlike that in ordinary reinforced concrete structures in
which flexural cracks are always formed.

35
Fig. (a) shows load deflection curves for under reinforced beams
with same steel area and concrete dimensions but with varying
amounts of prestress.
The dotted lines represent load-deflection curves based on the
flexural rigidity of the uncracked transformed section 𝑬𝒄𝑰𝒖𝒕 and
cracked transformed section 𝑬𝒄𝑰𝒄𝒕. The failure load is almost same
in all the cases.
Beam a, with zero prestress; responds linearly up to its cracking
load, after which load deflection curve is approximately linear and
parallel to 𝑬𝒄𝑰𝒄𝒕 line.

36
For beams b, c, d, with varying prestress; the load causing
cracking is higher than because of initial compression stresses are
superimposed in the tension zone.
In beam b, a partially prestressed beam; cracking may occur below
full service load.
In beam c, a fully prestressed beam; with zero tensile stress at
service load will crack only at a higher load reached.
In beam d, an over prestressed beam will fail suddenly in a brittle
manner.
In under-reinforced beams, a further change in the slope of load
deflection curve occurs before failure, as steel is stressed within its
inelastic range resulting in extensive cracking.

37
In over-reinforced beams, the degree of prestressing determines the
cracking load as in under-reinforced beams.
The effect of varying prestress is similar to that of under-
reinforced beams.
However, after cracking, the load deflection curves follow more or
less parallel to 𝑬𝒄𝑰𝒄𝒕 line all the way to the failure.
The failure load in over-reinforced beams was higher than that of
under-reinforced beams.
The failure is characterized by sudden brittle failure with less
warning than under-reinforced beam.

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