7/29/2019 8 Lentur 2

1/47

BETON PRATEGANGTKS - 4023

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

Jurusan Teknik Sipil

Fakultas Teknik

Universitas Brawijaya

Session 10:

Members Analysisunder Flexure(Part II)

7/29/2019 8 Lentur 2

2/47

IntroductionThe analysis of flexural members under service loads

involves the calculation of the following quantities:

Cracking moment.

Location of kern points. Location of pressure line.

7/29/2019 8 Lentur 2

3/47

Cracking MomentThe 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, theevaluated cracking moment is an estimation and

important in the analysis of prestressed members.

7/29/2019 8 Lentur 2

4/47

Cracking Moment(contd)Based on the allowable tensile stress the prestressmembers are classified into three types as:

1. Type 1: Full Prestressing, when the level ofprestressing is such that no tensile stress is allowedin concrete under service loads.

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

3. Type 3: Partial Prestressing, when the level ofprestressing is such that under tensile stresses dueto service loads, the crack width is within theallowable limit.

7/29/2019 8 Lentur 2

5/47

Cracking Moment(contd)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 thestress at the edge due to service loads is less than the

modulus of rupture.

7/29/2019 8 Lentur 2

6/47

Cracking Moment(contd)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 andmeasured 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)

7/29/2019 8 Lentur 2

7/47

Cracking Moment(contd)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

7/29/2019 8 Lentur 2

8/47

Cracking Moment(contd)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 andmaterial properties and prestressing variables.

(Eq. 14)

7/29/2019 8 Lentur 2

9/47

Kern PointsWhen the resultant compression (C) is located within aspecific zone of a section of a beam, tensile stresses arenot generated. This zone is called the kern zone of asection. For a section symmetric about a vertical axis,

the kern zone is within the levels of the upper and lowerkern points. When the resultant compression (C ) underservice loads is located at the upper kern point, thestress at the bottom edge is zero. Similarly, when Cattransfer of prestress is located at the bottom kern point,

the stress at the upper edge is zero. The levels of theupper and lower kern points from CGC are denoted asktand kb, respectively.

7/29/2019 8 Lentur 2

10/47

Kern Points(contd)Based on the stress concept, the stress at the bottomedge corresponding to C at the upper kern point , isequated to zero. Fig. 10 shows the location ofCand theresultant stress profile when compression is at upper

kern point.

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

7/29/2019 8 Lentur 2

11/47

Kern Points(contd)The value ofktcan be calculated by equating the stressat the bottom to zero as follows:

(Eq. 15)

Eq. 15 expresses the location of upper kern point (kt) interms of the section properties. Here, r is the radius ofgyration and yb is the distance of the bottom edge fromCGC.

7/29/2019 8 Lentur 2

12/47

Kern Points(contd)Similar to the calculation ofkt, the location of the bottomkern point, kb can be calculated by equating the stress atthe top edge to zero. Fig. 11 shows the location of Cand the resultant stress profile when compression is at

lower kern point.

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

7/29/2019 8 Lentur 2

13/47

Kern Points(contd)The value ofktcan be calculated by equating the stressat the top to zero as follows:

(Eq. 16)

Eq. 16 expresses the location of lower kern point (kb)and ytis the distance of the top edge from CGC.

C ki t i

7/29/2019 8 Lentur 2

14/47

Cracking oment usingKern 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. Cis

located above kt to cause a tensile stress fcr at thebottom. The incremental moment is fcrI/yb. Fig. 12 shows

the shift in Coutside the kern to cause cracking and the

corresponding stress profiles.

7/29/2019 8 Lentur 2

15/47

Cracking Moment using

Kern Points(contd)

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

7/29/2019 8 Lentur 2

16/47

Cracking Moment using

Kern Points(contd)

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 sumofktand z, the shift ofCoutside the kern.

(Eq. 17)

7/29/2019 8 Lentur 2

17/47

Cracking Moment using

Kern Points(contd)

Substituting C = Pe, kt = r2/yb and r

2 = I/A, Eq. 17

becomes same as the previous expression of Mcr (Eq.

14).

(Eq. 18)

7/29/2019 8 Lentur 2

18/47

Pressure LinesThe pressure line in a beam is the locus of the resultant

compression (C) along the length. It is also called the

thrust line orC-line. It is used to check whether Cat

transfer and under service loads is falling within the kernzone of the section. The eccentricity of the pressure line

(ec) from CGC should be less than kb orktto ensure Cin

the kern zone.

7/29/2019 8 Lentur 2

19/47

Pressure Lines(contd)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 Tdue to the

moment. Subtracting e from zprovides the eccentricityofC(ec) with respect to CGC. The variation ofecalong

length of the beam provides the pressure line.

(Eq. 19)

7/29/2019 8 Lentur 2

20/47

Pressure Lines(contd)A positive value of ec implies that C acts above the

CGC and vice-versa. Ifecis 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.

7/29/2019 8 Lentur 2

21/47

Pressure Lines(contd)Pressure Line at Transfer

The pressure line is calculated from the moment due tothe self weight. Fig. 13 shows that the pressure line fora simply supported beam gets shifted from the CGS with

increasing moment towards the centre of the span.

Fig. 13 Pressure line at transfer

7/29/2019 8 Lentur 2

22/47

Pressure Lines(contd)Pressure Line under Service Loads

The pressure line is calculated from the moment due tothe service loads. Fig. 14 shows that the pressure linefor a simply supported beam gets further shifted from

the CGS at the centre of the span with increasedmoment under service condition.

Fig. 14 Pressure line under service loads

7/29/2019 8 Lentur 2

23/47

Pressure Lines(contd)Limiting Zone

For fully prestressed members (Type 1), tension is notallowed under service conditions. If tension is also notallowed at transfer, Calways lies within the kern zone.

The limiting zone is defined as the zone for placing theCGS of the tendons such that Calways lies within thekern zone.

7/29/2019 8 Lentur 2

24/47

Pressure Lines(contd)For limited and partially prestressed members (Type 2and Type 3), tension is allowed at transfer and underservice conditions. The limiting zone is defined as thezone 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

7/29/2019 8 Lentur 2

25/47

Example 1

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,

7/29/2019 8 Lentur 2

26/47

Question

Evaluate the following quantities:

a. Kern levels

b. Cracking moment

c. Location of pressure line at mid-span at transfer andat service.

d. The stresses at the top and bottom fibres at transferand at service.

Compare the stresses with the following allowable

stresses at transfer and at service. For compression, fcc= 18.0 N/mm

2

For tension, fct= 1.5 N/mm2

7/29/2019 8 Lentur 2

27/47

Solution

Calculation of geometric properties

The section is divided intothree rectangles for the

computation of the geometricproperties.

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

7/29/2019 8 Lentur 2

28/47

Solution(contd)

Location of CGC from the bottom side,

Therefore,

Eccentricity of CGS at mid-span,

7/29/2019 8 Lentur 2

29/47

Solution(contd)

Moment of inertia, I1 about axis through CGC:

Moment of inertia, I2 about axis through CGC:

Moment of inertia, I3 about axis through CGC:

7/29/2019 8 Lentur 2

30/47

Solution(contd)

Moment of inertia of the section:

Square of the radius of gyration:

7/29/2019 8 Lentur 2

31/47

Solution(contd)

a. Kern levels of the section

7/29/2019 8 Lentur 2

32/47

Solution(contd)

b. Calculation of cracking moment

Moment due to self weight (MDL):

7/29/2019 8 Lentur 2

33/47

Solution(contd)

Modulus of rupture (fcr):

Moment of cracking (Mcr):

7/29/2019 8 Lentur 2

34/47

Solution(contd)

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 forcomputation of stresses.

7/29/2019 8 Lentur 2

35/47

Solution(contd)

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.

7/29/2019 8 Lentur 2

36/47

Solution(contd)

At transfer

7/29/2019 8 Lentur 2

37/47

Solution(contd)

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.

7/29/2019 8 Lentur 2

38/47

Solution(contd)

At service

7/29/2019 8 Lentur 2

39/47

Solution(contd)

d. Calculation of stresses

The stress is given as follows:

7/29/2019 8 Lentur 2

40/47

Solution(contd)

Calculation of stresses at transfer (P= P0):

7/29/2019 8 Lentur 2

41/47

Solution(contd)

Stress at the top fibre (at transfer):

7/29/2019 8 Lentur 2

42/47

Solution(contd)

Stress at the bottom fibre (at transfer):

7/29/2019 8 Lentur 2

43/47

Solution(contd)

Calculation of stresses at service (P= Pe):

7/29/2019 8 Lentur 2

44/47

Solution(contd)

Stress at the top fibre (at service):

7/29/2019 8 Lentur 2

45/47

Solution(contd)

Stress at the bottom fibre (at service):

7/29/2019 8 Lentur 2

46/47

Solution(contd)

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.

7/29/2019 8 Lentur 2

47/47

Thanks for Your Attention

andSuccess for Your Study!