prestress ln5 black and white

8/6/2019 Prestress LN5 Black and White

1/19

University of Western AustraliaSchool of Civil and Resource Engineering 2004

5. Prestressed Concrete :

Ultimate bending strength

of beams

Introduction

Bending strength with bonded tendons

Ultimate strength without non-tensioned steel

Ultimate strength with non-tensioned steel

Bending strength with unbonded tendons

Bending strength at transfer


2/19

INTRODUCTION

Prestressing has great advantages at working load, where

deflections and cracking are controlled -serviceability limitstates.

But we must also satisfy thesafety limit state. This means:

f Muo >= M* everywhere along the beam.So we need a method of estimating Muo.

Ductility limits must be observed, just as for reinforced

concrete :

Lower: Muo >= 1.2 [ Z(f cf+ P/A) + Pe ]

Upper: ku


3/19

STRENGTH WITH BONDED TENDONS

In bonded tendon construction, the tendon is connected, either directly or

indirectly, to the adjacent concrete. So the strain in the tendon is alwaysequal to the strain in the adjacent concrete. For example:

in grouted, internal post-tensioned construction, and

in pre-tensioned construction.

Grouted duct, with4 strand tendon

Concrete cast

around stressed

strands

In these cases, the Bernoulli/Navier

postulate is valid, and we use this in

estimating the ultimate bending strength.

[See later discussion for unbondedtendons, where Bernoulli/Navier does

not apply. The ultimate bending

strength is rather different.]


4/19

BMD

M crcracked range

TT - dT

dT

The importance of bond :

Bonding allows the

force to change along

the tendon.

Prestressed

beam, crackedin bending.

Ultimate bending strength:


5/19

Section Curvature k

AppliedMoment Mtot

prestressonly

prestress andself-weight

Moment / Curvature Diagram at a Section

balanced

(equiv.load)

de-compression

moment Mo

crackingmoment Mcr

post-crackingcurvature

ultimate

moment Muo

ultimatecurvature ku

This is our

focus today

ULTIMATE STRENGTH WITHOUTNON-TENSIONED STEEL


6/19

Ultimate Bending Strength Without non-tensioned steel:

tendon d p

0.003

epu

SECTION STRAIN

dn = ku d p

gku d p = x

T p

C

STRESS AND

FORCES

0.85 f c

Note how similar this is to ultimate strength in reinforced concrete.

At ultimate moment, a rectangular stress block may be adopted, just as for

reinforced concrete. The block is defined by an ultimate concrete strain of

0.003, and a uniform stress of 0.85 f c.

So Muo = C z = Tp z

More generally, and with the same result:

Muo = Tp dp - C (dp - z)

BUT how do we

estimate Tp? . . .

ku z


7/19

There are three methods available.

Each uses a different approach to estimating spu ,the tensile stress in the tendon at which peak

bending moment is achieved:

Method 1 . Very simple, usually

conservative.

Use spu = fpyand equate C to Tp = spu Ap

Method 2 . Trial and error, following a

known stress/strain curve.

Select spuso that C = Tp = spuAp

Method 3. Empirical formula for spu.(See AS3600 cl. 8.1.5)

Use sputo calculate Tp, then use C = Tp.

sp

epspu = fpy

sp

epSelect spu

sp

epCalculate spu

Lets check them out . . .


8/19

Method 1 . spu = fpy

x = gdn 0.85 f c

Tp

C

Tp = Apt fpy

But C = 0.85 f c b x So x = Tp / (0.85 f c b)

dp - x/2dn =ku d

Then Muo = Tp ( dp - x/2)

Also ku d = x /g to check that ku


9/19

Method 2 . Selectspu ,eputo lie on stress/strain line.

epu

epeeceept

0.85 f c0.003

KNOWN STRESS/STRAIN

CURVE FOR TENDON

1. Select dn, and calculate epu from

epu = epe + ece + ept

2. Estimate spu = 0.85 f c b gdn/ Ap

3. Plot on curve.

4. Adjust and repeat until spu epu lies on

curve. Then adopt this value ofspu

dn

x =g dn

dp - x/2

Point 1

Point 2

Point 3 o.k.

C

Tp

Then Tp = spu Ap

Proceed as for Method 1.


10/19

Method 3 . Estimatespu from empirical formula.

AS3600 provides (with several qualifications) :

spu = fp (1 - k 1 k 2/g)where k 1 = 0.4 generally, or if fpy/fp >= 0.9 then k1 = 0.28

and k 2 = [Apt fp + (Ast - Asc) fsy] / (befdp f c)


11/19

So which one do we use?

How about the easiest

one?


12/19

We often find that, even though we have carefully selected the

prestress tendon for working load conditions, the ultimate

strength of the section is inadequate.

?

Dont worry!

The first thing to consider is the additionof some non-tensioned reinforcement, sayGrade 500N conventional rebar.

When properly placed and

anchored, the rebar provides

additional force at high

overload, and so increases theultimate moment.

How do we estimate this? . . .


13/19

tendon

rebar

d p d s

epu

d n = ku d

gku d = xC

Tp

Ts

0.003

SECTION STRAIN

x/2

STRESS ANDFORCES

Using the rectangular stress block as before: For a ductile section

(that is when the tendon plus rebar areas are not too large), Tp is

conservatively estimated as Apt fpy, and Ts as Ast fsy.

0.85 f c

The compression force C equilibrates the tension forces provided by the

tendon AND the rebar. So C = Apt fpy, + Ast fsy and therefore:

Muo = Tp dp + Ts ds - C x /2

How can we quickly size the rebar required for safety? . . .

ULTIMATE STRENGTHWITH NON-TENSIONED STEEL


14/19

Selecting non-tensioned steel (approximate):

d p d s

A st Apt

Apt fpy + A st fsy

A p fpy

A st fsy

We wish to select Ast required to satisfy fMuo >= M* :

This is an approximate method: Adopt the approximation x/2 = 0.15 ds.

x/2 = 0.15 dsapprox.

Then M uo = A pt fpy (d p - 0.15 d s ) + A st fsy (d s - 0.15 d s )

= A pt fpy (d p - 0.15 d s ) + A st fsy 0.85 d s

But M uo >= M* /f.So

A st >= [ M*/f- Apt fpy (dp - 0.15 ds) ] / [ fsy 0.85 ds ]Then check the ultimate strength of the section, and refine.

Ultimate Bending Strength With non-tensioned steel:


15/19

BENDING STRENGTH WITH

UNBONDED TENDONS

Unbonded tendons occur :

in conventional internally post-tensioned elementsprior to grouting, or

when grouting is not intended.

in externally post-tensioned elements - connection between tendon and

concrete occurs at ends of elements, and at harping points, if any.

Bernoulli / Navier postulate does not apply - tendon and concrete strain

independently.

Consider this beam:

Straight, unbonded tendon, stressed

and anchored at each end of beam

Clearly, the tendon

stress can respond

only to changes in the

overall extension of

the concrete.

How can we estimate spu? . . .


16/19

Answer: Not with any confidence.

But there is an empirical method available in

AS3600:

For span/depth ratios


17/19

BENDING STRENGTH AT TRANSFER

At transfer, using working loads, we check that sb = 1.15 Pjm where Pjm is the maximum jacking

force applied during stressing. (AS3600 cls. 3.3.1 and 8.1.4.2.)


18/19

1. Selection of tendon/rebar for given M*:

Select approximate line of T, the resultant of Tp and Ts ,bychoosing d.

Then Muo = T (d - x/2) approx. where x = 0.3 d approx.

But fMuo >= M*. So T = M* / (f 0.85 d)P is known from serviceability considerations. So select Apand Ast to satisfy strength and serviceability.

2. Influence of long. rebar on serviceability :

It is conservative to ignore this rebar.

Otherwise, use transformed section method , thus:

introduce equivalent concrete area at depth of each rebar layer;

calculate I, Ztop, Zbott, A, y;

then proceed as before, using these new properties.

Tp

Ts T

d

Two Tips for Designers:


19/19

SUMMARY

We mustalways check that a section has adequate

ultimate strength: f Muo >= M* with f = 0.8. This oftenrequires the introduction of non-tensioned steel.

Lower and upper ductility of a section must always be

checked, and the section, or stressing, or rebar adjusted if

necessary.

Three methods of estimating Muo are available for

elements with bonded tendons.

For elements with unbonded tendons, a different methodof estimating Muo is required.

Ensure that sections have adequate ultimate strength at

transfer. Ensuring that sa

prestress ln5 black and white

Documents