Prestress LN5 Black and White
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Transcript of Prestress LN5 Black and White
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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
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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
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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.]
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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:
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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
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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
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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 . . .
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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
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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.
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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)
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So which one do we use?
How about the easiest
one?
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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? . . .
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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
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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:
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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? . . .
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Answer: Not with any confidence.
But there is an empirical method available in
AS3600:
For span/depth ratios
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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.)
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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:
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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