Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof.

Devdas Menon

3.3 Analysis of Members under Flexure (Part II)

This section covers the following topics.
Cracking Moment
Kern Points
Pressure Line

Introduction
The analysis of flexural members under service loads involves the calculation of the
following quantities.
a) Cracking moment.
b) Location of kern points.
c) Location of pressure line.

The following material explains each one of them.

3.2.1 Cracking Moment

The cracking moment (Mcr) is defined as the moment due to external loads at which the
first crack occurs in a prestressed flexural member. Considering the variability in stress
at the occurrence of the first crack, the evaluated cracking moment is an estimate.
Nevertheless, the evaluation of cracking moment is important in the analysis of
prestressed members.

Based on the allowable tensile stress the prestress members are classified into three
types as per IS:1343 - 1980. The types are explained in Section 1.2, Advantages and
Types of Prestressing. For Type 1 (full prestressing) and Type 2 (limited prestressing)
members, cracking is not allowed under service loads. Hence, it is imperative to check
that the cracking moment is greater than the moment due to service loads. This is
satisfied when the stress at the edge due to service loads is less than the modulus of
rupture.

The modulus of rupture is the stress at the bottom edge of a simply supported beam
corresponding to the cracking moment (Mcr). The modulus of rupture is a measure of
the flexural tensile strength of concrete. It is measured by testing beams under 2 point

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

loading (also called 4 point loading including the reactions or middle third loading). The
modulus of rupture (fcr) is expressed in terms of the characteristic compressive strength
(fck) of concrete by the following equation (IS:456 - 2000). Here, fcr and fck are in N/mm2.

fcr = 0.7 fck (3-3.1)

The following sketch shows the internal forces and the resultant stress profile at the
instant of cracking.

Mcr
CGC
yb Pe
fcr
Internal forces in Resultant stress
concrete profile
Figure 3-3.1 Internal forces and resultant stress profile at cracking

The stress at the edge can be calculated based on the stress concept as follows. The
cracking moment (Mcr) can be evaluated by transposing the terms.

Pe Pe ey b Mcr y b
- - + = fcr
A I I
Mcr y b P P ey
or, = fcr + e + e b
I A I
f I PI
or, Mcr = cr + e + Pe e (3-3.2)
y b Ay b

The above equation expresses Mcr in terms of the section and material properties and
prestressing variables.

3.2.2 Kern Points

When the resultant compression (C) is located within a specific zone of a section of a
beam, tensile stresses are not generated. This zone is called the kern zone of a section.
For a section symmetric about a vertical axis, the kern zone is within the levels of the
upper and lower kern points. When the resultant compression (C) under service loads
is located at the upper kern point, the stress at the bottom edge is zero. Similarly, when
C at transfer of prestress is located at the bottom kern point, the stress at the upper

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

edge is zero. The levels of the upper and lower kern points from CGC are denoted as kt
and kb, respectively.

Based on the stress concept, the stress at the bottom edge corresponding to C at the
upper kern point, is equated to zero. The following sketch shows the location of C and
the resultant stress profile.

CGC C kt
yb

Location of Resultant stress

resultant profile
compression
Figure 3-3.2 Resultant stress profile when compression is at upper kern point

The value of kt can be calculated by equating the stress at the bottom to zero as follows.

C Ckt y b
- + =0
A I
C Ck y
or, - + t 2 b = 0
A Ar
r2
or, kt = (3-3.3)
yb

The above equation expresses the location of upper kern point in terms of the section
properties. Here, r is the radius of gyration and yb is the distance of the bottom edge
from CGC.

Similar to the calculation of kt, the location of the bottom kern point can be calculated by
equating the stress at the top edge to zero. The following sketch shows the location of
C and the resultant stress profile.
yt
CGC C kb

Location of Resultant stress

resultant profile
compression
Figure 3-3.3 Resultant stress profile when compression is at lower kern point

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

C Ck b y t
- + =0
A I
C Ck y
or, - + b 2 t = 0
A Ar
r2 (3-3.4)
or, k b =
yt

Here, yt is the distance of the top edge from CGC.

Cracking Moment using Kern Points

The kern points can be used to determine the cracking moment (Mcr). The cracking
moment is slightly greater than the moment causing zero stress at the bottom. C is
located above kt to cause a tensile stress fcr at the bottom. The incremental moment is
fcr I/yb. The following sketch shows the shift in C outside the kern to cause cracking and
the corresponding stress profiles.
z
C
ec kt
CGC + =
e
CGS

fcr fcr
Location of
Resultant stress
resultant
profile
compression
Figure 3-3.4 Resultant stress profile at cracking of the bottom edge

The cracking moment can be expressed as the product of the compression and the
lever arm. The lever arm is the sum of the eccentricity of the CGS (e) and the
eccentricity of the compression (ec). The later is the sum of kt and z, the shift of C
outside the kern.

Mcr = C ( e + ec )
= C ( e + kt + z )
fcr I (3-3.5)
or, Mcr = C ( e + kt ) +
yb

Substituting C = Pe, kt = r2/yb and r2 = I/A, the above equation becomes same as the
previous expression of Mcr.

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

r 2 fcr I
Mcr =Pe +e+
yb yb
f I PI
or, Mcr = cr + e +Pee (3-3.6)
yb Ayb

3.2.3 Pressure Line

The pressure line in a beam is the locus of the resultant compression (C) along the
length. It is also called the thrust line or C-line. It is used to check whether C at
transfer and under service loads is falling within the kern zone of the section. The
eccentricity of the pressure line (ec) from CGC should be less than kb or kt to ensure C
in the kern zone.

The pressure line can be located from the lever arm (z) and eccentricity of CGS (e) as
follows. The lever arm is the distance by which C shifts away from T due to the moment.
Subtracting e from z provides the eccentricity of C (ec) with respect to CGC. The
variation of ec along length of the beam provides the pressure line.

M
z=
C
ec = z - e (3-3.7)

A positive value of ec implies that C acts above the CGC and vice-versa. If ec is
negative and the numerical value is greater than kb (that is |ec| > kb), C lies below the
lower kern point and tension is generated at the top of the member.
If ec> kt, then C lies above the upper kern point and tension is generated at the bottom
of the member.

Pressure Line at Transfer

The pressure line is calculated from the moment due to the self weight. The following
sketch shows that the pressure line for a simply supported beam gets shifted from the
CGS with increasing moment towards the centre of the span.

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

Pressure line CL

CGC kt
CGS kb

Figure 3-3.4 Pressure line at transfer

Pressure Line under Service Loads

The pressure line is calculated from the moment due to the service loads. The following
sketch shows that the pressure line for a simply supported beam gets further shifted
from the CGS at the centre of the span with increased moment under service condition.

Pressure line
CL

CGC kt
CGS kb

Figure 3-3.4 Pressure line under service loads

Limiting Zone
For fully prestressed members (Type 1), tension is not allowed under service conditions.
If tension is also not allowed at transfer, C always lies within the kern zone. The limiting
zone is defined as the zone for placing the CGS of the tendons such that C always lies
within the kern zone.

For limited prestressed members (Type 2 and Type 3), tension is allowed at transfer
and under service conditions. The limiting zone is defined as the zone for placing the
CGS such that the tensile stresses in the extreme edges are within the allowable values.

The following figure shows the limiting zone (as the shaded region) for a simply
supported beam subjected to uniformly distributed load.

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

CL

Locus of emin
CGC

Locus of emax
Figure 3-3.4 Limiting zone for a simply supported beam

The determination of limiting zone is given in Section 4.4, Design of Sections for Flexure
(Part III).

Example 3-3.1

For the post-tensioned beam with a flanged section as shown, the profile of the
CGS is parabolic, with no eccentricity at the ends. The live load moment due to
service loads at mid-span (MLL) is 648 kNm. The prestress after transfer (P0) is
1600 kN. Assume 15% loss at service. Grade of concrete is M30.

CGC

18.0m
500

200

150
1000 Values in mm.

CGS
150 200

250
Cross-section at mid-span

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

Evaluate the following quantities.

a) Kern levels
b) Cracking moment
c) Location of pressure line at mid-span at transfer and at service.
d) The stresses at the top and bottom fibres at transfer and at service.

Compare the stresses with the following allowable stresses at transfer and at
service.
For compression, fcc,all = 18.0 N/mm2
For tension, fct,all = 1.5 N/mm2.

Solution

Calculation of geometric properties

The section is divided into three rectangles for the computation of the geometric
properties. The centroid of each rectangle is located from the soffit.

1
yt

+ CGC
2
900 Values in mm.
y yb
500
3
100

Area of the section

Area of 1 = A1 = 500 200

= 100,000 mm2

Area of 2 = A2 = 600 150

= 90,000 mm2

Area of 3 = A3 = 250 200

= 50,000 mm2

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

A = A1 + A2 + A3
= 240,000 mm2

Location of CGC from the soffit

A1 900 + A2 500 + A3 100

y=
A
= 583.3 mm

Therefore,
y b = 583.3 mm
y t = 1000.0 - 583.3
= 416.7 mm

Eccentricity of CGS at mid-span

e = y -150
= 583.3 -150
= 433.3 mm

Moment of inertia of 1 about axis through CGC

1
I1 = 500 2003 + A1 (900 - 583.3)2
12
= 1.0361010 mm4

Moment of inertia of 2 about axis through CGC

1
I2 = 150 6003 + A2 (583.3 - 500)2
12
= 3.32109 mm4

Moment of inertia of 3 about axis through CGC

1
I3 = 250 2003 + A3 (583.3 -100)2
12
= 1.1841010 mm4

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

Moment of inertia of the section

I = I1 + I2 + I3
= (1.036 + 0.336 +1.184)1010
= 2.5521010 mm4

Square of the radius of gyration

I
r2 =
A
2.5521010
=
240,000
= 1.063105 mm2

a) Kern levels of the section

r2 r2
kt = kb =
yb yt
1.063105 1.063 105
= =
583.3 416.7
= 182.2mm = 255.1mm

CGC 182.2
+
255.1 Kern zone

CGS

Values in mm.

Calculation of moment due to self weight (MDL).

2
m2
w DL = 24.0 kN/m3 240,000 mm2 3
1
10 mm
2

= 5.76 kN/m

w DL L2
MDL =
8
5.76 18.02
=
8
= 233.3 kNm

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

b) Calculation of cracking moment

Modulus of rupture fcr = 0.7 fck

= 0.7 30
= 3.83kN/mm2

fcr I PeI
Mcr = + + Pe e
y b Ay b
3.83 2.5521010 0.851600103 2.5521010
= +
583.3 240103 583.3
+ 0.81600 103 433.3 Nmm
= 167.6 + 247.9 + 554.6
= 970.1 kNm

Live load moment corresponding to cracking

MLL cr = 970.1- 233.3

= 736.8 kNm

Since the given live load moment (648.0 kNm) is less than the above value, the section
is uncracked.
The moment of inertia of the gross section can be used for computation of
stresses.

c) Calculation of location of pressure line at mid-span

At transfer
MDL
z=
C
233.3103 ec = z - e
=
1600 = 145.8 - 433.3
= 145.8 mm = - 287.5mm

Since ec is negative, the pressure line is below CGC.

Since the magnitude of ec is greater than kb, there is tension at the top.

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

CGC
+
287.5 Kern zone
x
Location of pressure line
CGS

Value in mm.
At service
MDL+LL
z=
C
ec = z - e
(233.3 + 648.0)103
= = 648.0 - 433.3
0.851600
= 214.7mm
= 648.0 mm

Since ec is positive, the pressure line is above CGC.

Since the magnitude of ec is greater than kt, there is tension at the bottom.

x
Location of pressure line
214.7
CGC
+
Kern zone

CGS

Value in mm.
d) Calculation of stresses
The stress is given as follows.
P Pey My
f =-
A I I
M
CGC + + =
P

P/A Pey/ I My/ I Resultant

stress
profile
Calculation of stresses at transfer (P = P0)

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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

P0 1600 103
=-
A 240 103
= -6.67 N/mm2
Stress at the top fibre

P0ey t 1600103 433.3 416.7

=
I 2.5521010
= 11.32 N/mm2

MDL y t 233.3106 416.7

=-
I 2.5521010
= -3.81N/mm2

fc = - 6.67 +11.32 - 3.81

t

= 0.84 N/mm2
Stress at the bottom fibre

P0ey b 1600103 433.3 583.3

=-
I 2.5521010
= -15.85 N/mm2

MDL y b 233.3106 583.3

=
I 2.5521010
= 5.33 N/mm2

fc = - 6.67 -15.85 + 5.33

b

= - 17.19 N/mm2

Calculation of stresses at service (P = Pe)

Pe P
= 0.85 0
A A
= - 5.67 N/mm2

Stress at the top fibre

Pf ey t
= 0.85 11.32
I
= 9.62

MLL y t 648.0106 416.7

=-
I 2.5521010
= - 10.58 N/mm2
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Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas Menon

fc = - 5.67 + 9.62 - 3.81-10.58

t

= -10.44 N/mm2

Stress at the bottom fibre

Pf ey b
= - 0.85 15.85
I
= - 13.47 N/mm2

MLL y b 648.0106 583.3

=
I 2.5521010
= 14.81N/mm2

fc = - 5.67 - 13.47 + 5.33 +14.81

b

= 1.0 N/mm2

The stress profiles are shown.

0.84 10.44

Numeric values
in N/mm2.

17.19 1.0
At transfer At service

The allowable stresses are as follows.

For compression, fc,comp = 18.0 N/mm2
For tension, fc,tens = 1.5 N/mm2.

Thus, the stresses are within the allowable limits.

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