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BETON PRATEGANG TKS - 4023

Dr.Eng. Achfas Zacoeb, ST., MT.

Jurusan Teknik Sipil

Fakultas Teknik

Universitas Brawijaya

Session 10:

Members Analysis under Flexure (Part II)

Introduction

The analysis of flexural members under service loads

involves the calculation of the following quantities:

Cracking moment.

Location of kern points.

Location of pressure line.

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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 estimation and

important in the analysis of prestressed members.

Cracking Moment (cont’d) Based on the allowable tensile stress the prestress members are classified into three types as:

1. Type 1: Full Prestressing, when the level of prestressing is such that no tensile stress is allowed in concrete under service loads.

2. Type 2: Limited Prestressing, when the level of prestressing is such that the tensile stress under service loads is within the cracking stress of concrete.

3. Type 3: Partial Prestressing, when the level of prestressing is such that under tensile stresses due to service loads, the crack width is within the allowable limit.

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Cracking Moment (cont’d) For Type 1 and Type 2 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.

Cracking Moment (cont’d) 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 and

measured by testing beams under 4 point loading

including the reaction. The modulus of rupture (fcr) is

expressed in terms of the characteristic compressive

strength (fck) of concrete in N/mm2.

(Eq. 13)

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Cracking Moment (cont’d) FIg. 9 shows the internal forces and the resultant stress

profile at the instant of cracking.

Fig. 9 Internal forces and resultant stress profile at cracking

Cracking Moment (cont’d) The stress at the edge can be calculated based on the

stress concept and the cracking moment (Mcr) can be

evaluated by transposing the terms as Eq. 14. This

equation expresses Mcr in terms of the section and

material properties and prestressing variables.

(Eq. 14)

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

Kern Points (cont’d) Based on the stress concept, the stress at the bottom edge corresponding to C at the upper kern point , is equated to zero. Fig. 10 shows the location of C and the resultant stress profile when compression is at upper kern point.

Fig. 10 Resultant stress profile (compression at upper kern point)

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Kern Points (cont’d) The value of kt can be calculated by equating the stress at the bottom to zero as follows: (Eq. 15)

Eq. 15 expresses the location of upper kern point (kt) in terms of the section properties. Here, r is the radius of gyration and yb is the distance of the bottom edge from CGC.

Kern Points (cont’d) Similar to the calculation of kt, the location of the bottom kern point, kb can be calculated by equating the stress at the top edge to zero. Fig. 11 shows the location of C and the resultant stress profile when compression is at lower kern point.

Fig. 11 Resultant stress profile (compression at lower kern point)

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Kern Points (cont’d) The value of kt can be calculated by equating the stress at the top to zero as follows:

(Eq. 16) Eq. 16 expresses the location of lower kern point (kb) and 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 fcrI/yb. Fig. 12 shows

the shift in C outside the kern to cause cracking and the

corresponding stress profiles.

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Cracking Moment using

Kern Points (cont’d)

Fig. 12 Resultant stress profile at cracking of the bottom edge

Cracking Moment using

Kern Points (cont’d)

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.

(Eq. 17)

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Cracking Moment using

Kern Points (cont’d)

Substituting C = Pe, kt = r2/yb and r2 = I/A, Eq. 17

becomes same as the previous expression of Mcr (Eq.

14).

(Eq. 18)

Pressure Lines 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.

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Pressure Lines (cont’d) 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.

(Eq. 19)

Pressure Lines (cont’d) 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.

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Pressure Lines (cont’d) Pressure Line at Transfer

The pressure line is calculated from the moment due to the self weight. Fig. 13 shows that the pressure line for a simply supported beam gets shifted from the CGS with increasing moment towards the centre of the span.

Fig. 13 Pressure line at transfer

Pressure Lines (cont’d) Pressure Line under Service Loads

The pressure line is calculated from the moment due to the service loads. Fig. 14 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.

Fig. 14 Pressure line under service loads

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Pressure Lines (cont’d) 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.

Pressure Lines (cont’d) For limited and partially 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.

Fig. 15 Limiting zone for a simply supported beam

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Problem

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.

Grade of concrete is 30 MPa. Assume 15% loss at

service.

For the post-tensioned beam with a

flanged section as shown, the profile of

CGS is parabolic,

Question

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 = – 18.0 N/mm2

For tension, fct = 1.5 N/mm2

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Solution

Calculation of geometric properties

The section is divided into three rectangles for the computation of the geometric properties.

Area of the section, A:

A1 = 500X200 = 100,000 mm2

A2 = 600X150 = 90,000 mm2

A3 = 250X200 = 50,000 mm2

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

Solution (cont’d)

Location of CGC from the bottom side,

Therefore,

Eccentricity of CGS at mid-span,

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Solution (cont’d)

Moment of inertia, I1 about axis through CGC:

Moment of inertia, I2 about axis through CGC:

Moment of inertia, I3 about axis through CGC:

Solution (cont’d)

Moment of inertia of the section:

Square of the radius of gyration:

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Solution (cont’d)

a. Kern levels of the section

Solution (cont’d)

b. Calculation of cracking moment

Moment due to self weight (MDL):

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Solution (cont’d)

Modulus of rupture (fcr):

Moment of cracking (Mcr):

Solution (cont’d)

Live load moment corresponding to cracking (MLL cr):

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.

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Solution (cont’d)

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

At transfer

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.

Solution (cont’d)

At transfer

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Solution (cont’d)

At service

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.

Solution (cont’d)

At service

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Solution (cont’d)

d. Calculation of stresses

The stress is given as follows:

Solution (cont’d)

Calculation of stresses at transfer (P = P0):

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Solution (cont’d)

Stress at the top fibre (at transfer):

Solution (cont’d)

Stress at the bottom fibre (at transfer):

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Solution (cont’d)

Calculation of stresses at service (P = Pe):

Solution (cont’d)

Stress at the top fibre (at service):

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Solution (cont’d)

Stress at the bottom fibre (at service):

Solution (cont’d)

The stress profiles:

The allowable stresses are as follows:

For compression, fcc = – 18.0 N/mm2

For tension, fct = 1.5 N/mm2.

Thus, the stresses are within the allowable limits.

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Thanks for Your Attention and

Success for Your Study!