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BETON PRATEGANGTKS - 4023

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

Jurusan Teknik Sipil

Fakultas Teknik

Universitas Brawijaya

Session 9:

Members Analysisunder Flexure(Part I)

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Geometric Properties

A prestressed member may also have non-prestressedreinforcement to carry the forces. This type of membersis called partially prestressed members. The commonlyused geometric properties of a prestressed member with

non-prestressed reinforcement are defined as follows: A = gross cross-sectional area

Ac = area of concrete

As = area of non-prestressed reinforcement

Ap = area of prestressing tendons At = transformed area of the section

= Ac+ (Es/ Ec) As+ (Ep/ Ec) Ap

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Geometric Properties(contd)Fig. 1 shows the commonly used areas of a prestressedmember with non-prestressed reinforcement.

Fig. 1 Transformation Areas

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Introduction

The analysis of members refers to the evaluation of the

following conditions:

1. Permissible prestress based on allowable stresses at

transfer.2. Stresses underservice loads. These are compared

with allowable stresses under service conditions.

3. Ultimate strength. This is compared with the demand

underfactored loads.4. The entire loads versus deformation behaviour.

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Introduction(contd)The analysis of members under flexure considers the

following assumptions:

1. Plane sections remain plane untill failure (known as

Bernoullis hypothesis).2. Perfect bond between concrete and prestressing

steel for bonded tendons.

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Introduction(contd)The analysis of behavior involves three principles of

mechanics:

1. Equilibrium of internal forces with the external

loads. The compression in concrete (C) is equal tothe tension in the tendon (T). The couple of C andT are equal to the moment due to external loads.

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Introduction(contd)

2. Compatibility of the strains in concrete and insteel for bonded tendons. The formulation also

involves the assumption of plane section

remaining plane after bending and a perfect bondbetween the two materials. For unbonded tendons,

the compatibility is in terms of total deformation.

3. Constitutive relationships relating the stressesand the strains in the materials. The relationships

are developed based on the material properties.

(Collins & Mitchell, Prestressed Concrete Structures)

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Variation of Internal ForcesIn reinforced concrete members under flexure, the

values of compression in concrete (C) and tension in thesteel (T) increase with increasing external load. The

change in the lever arm (z) is not large.In prestressed concrete members under flexure, at

transfer of prestress Cis located close to T. The coupleofCand Tbalance only the self weight. At service loads,C shifts up and the lever arm (z) gets large. Thevariation ofCorTis not appreciable.

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Variation of Internal Forces(contd)

Fig. 2 explains this difference schematically for a simplysupported beam under uniform load.

Fig. 2 Variations of internal forces and lever arms

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Variation of Internal Forces(contd)

For the reinforced concrete memberC2 is substantiallylarge than C1, but z2 is close to z1. For the prestressedconcrete member C2 is close to C1, but z2 is

substantially large than z1, where:C1, T1 = compression and tension at transfer due to self weight

C2, T2 = compression and tension under service loads

w1 = self weight

w2 = service loadsz1 = lever arm at transfer

z2 = lever arm under service loads

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Analysis at Transfer and at ServiceThe analyses at transfer and under service loads are

similar. Hence, they are presented together. A

prestressed member usually remains uncracked under

service loads. The concrete and steel are treated aselastic materials. The principle of superposition is

applied. The increase in stress in the prestressing steel

due to bending is neglected.

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Analysis at Transfer and at Service(contd)

There are three approaches to analyse a prestressed

member at transfer and under service loads. These

approaches are based on the following concepts:

a. Based on stress concept.b. Based on force concept.

c. Based on load balancing concept.

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Based on Stress Concept

In the approach based on stress concept, the stresses

at the edges of the section under the internal forces in

concrete are calculated. The stress concept is used tocompare the calculated stresses with the allowable

stresses.

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Fig. 3 shows a simply supported beam under auniformly distributed load (UDL) and prestressed withconstant eccentricity (e) along its length.

Fig. 3A simply supported beam under UDL

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Fig. 4 shows the internal forces in concrete at a section

and the corresponding stress profiles. The first stress isdue to the compression P, the second is due to theeccentricity of the compression, and the third is due tothe moment. The moment is due to self weight attransfer, and due to service loads at service.

Fig. 4 Stress profiles at a section due to internal forces

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The resultant stress at a distance y from the CGC isgiven by the principle of superposition as Eq. 1. For acurved tendon, P can be substituted by its horizontal

component. But the effect of the refinement is negligible.

(Eq. 1)

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Based on Force Concept

The approach based on force concept is analogous to

the study of reinforced concrete. The tension in

prestressing steel (T) and the resultant compression inconcrete (C) are considered to balance the externalloads. This approach is used to determine the

dimensions of a section and to check the service load

capacity. Of course, the stresses in concrete calculated

by this approach are same as those calculated based on

stress concept. The stresses at the extreme edges are

compared with the allowable stresses.

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Fig. 5 shows the internal forces in the section.

Fig. 5 Internal forces at a section

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The equilibrium equations are as follows:

(Eq. 2)

(Eq. 3)

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The resultant stress in concrete at distance y from theCGC is given as follows:

(Eq. 4)

Substituting C= Pand Cec= MPe, the expression ofstress becomes same as that given by the stress

concept as Eq. 5.

(Eq. 5)

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Based on Load Balancing Concept

The approach based on load balancing concept is used

for a member with curved or harped tendons and in the

analysis of indeterminate continuous beams. Themoment, upward thrust and upward deflection (camber)

due to the prestress in the tendons are calculated. The

upward thrust balances part of the superimposed load.

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The expressions for three profiles of tendons in simply

supported beams are give such as:

a. For a Parabolic Tendon

b. For Singly Harped Tendonc. For Doubly Harped Tendon

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a. Parabolic Tendon

Fig. 6 Simply supported beam with parabolic tendon

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The moment at the centre due to the uniform upward

thrust (wup) is given by Eq. 6.

(Eq. 6)

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The moment at the centre from the prestressing force is

given as M= Pe. The expression ofwupis calculated byequating the two expressions of M. The upward

deflection (

) can be calculated from wup based onelastic analysis.

(Eq. 7)

(Eq. 8)

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b. Singly Harped Tendon

Fig. 7 Simply supported beam with singly harped tendon

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The moment at the centre due to the upward thrust

(Wup) is given by the following equation. It is equated tothe moment due to the eccentricity of the tendon. As

before, the upward thrust and the deflection can becalculated.

(Eq. 9)

(Eq. 10)

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c. Doubly Harped Tendon

Fig. 8 Simply supported beam with doubly harped tendon

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The moment at the centre due to the upward thrusts

(Wup) is given by the following equation. It is equated tothe moment due to the eccentricity of the tendon. As

before, the upward thrust and the deflection can becalculated.

(Eq. 11)

(Eq. 12)

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Example 1

A concrete beam prestressed with a parabolic tendon is

shown in the figure. The prestressing force applied is

1620 kN. The uniformly distributed load includes the selfweight. Compute the extreme fibre stress at the mid-

span by applying the three concepts. Draw the stress

distribution across the section at mid-span.

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Solution

a. Stress conceptArea of concrete,

Moment of inertia,

Bending moment at mid-span,

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Solution(contd)

Top fibre stress,

Bottom fibre stress,

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Solution(contd)

b. Force concept

Applied moment,

Lever arm,

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Solution(contd)

Eccentricity ofC,

Top fibre stress,

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Solution(contd)


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Solution(contd)

c. Load balancing methodEffective upward load,

Residual load,

Residual bending moment,

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Solution(contd)

Residual bending stress,

Top fibre stress,

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Solution(contd)


The resultant stress

distribution at mid-span:

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Thanks for Your Attention

andSuccess for Your Study!

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