cven9806 lecture 2 - serviceability 2009

8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009

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CVEN9806 Prestressed Concrete 3

Prestressing force P and the cable profile are based on

Serviceability Requirements:

- Crack Free Design limit stresses

- Control Deflections

- Camber

- Axial Shortening

For Crack Control need to specify Stress Limits.


Short Term: sG Q P + +

Long Term: lG Q P + +

- Dead Load - Live Load

- Short Term Live Load Factor

- Long Term Live Load Factor

s

L

G Q


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time

loading

G

sQ

lQ


Type of Live Load Short-Term Factors Long-Term Factor l

Floors

Domestic 0.7 0.4

Offices 0.7 0.4Parking 0.7 0.4

Retail 0.7 0.4

Storage 1.0 0.6

Other As per storage, unless

Assessed otherwise

As per storage, unless

Assessed otherwise

Roofs

Trafficable 0.7 0.4

Non-Trafficable 0.7 0.0


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Table 2.4.2 AS3600

Notes:

1. In flat slabs, the deflection to which the above limits apply is the theoretical deflection of the line diagram representing the

idealized frame defined in Clause 7.5.2

1. Deflection limits given may not safeguard against ponding

2. For cantilevers, the values of/Lefgiven in this table apply only if the rotation at the support is included in the calculation of .


0.2% proofstress

True curve

Approx. curve

Strain

Stress

fpy

fpu


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0.8 fpu pretensioned tendons

0.85 fpu stress-relieved post-

tensioned tendons

0.75 fpu not stress-relieved post-tensioned tendons



During the stressing operation, immediate losses can

occur by:

elastic contraction of the concrete, friction along the ducts and slip and deformation in the end anchors.

Estimate ~ 8% Pi ~ 0.92 Pj

Pj

jacking prestress

Pi

initial prestress


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inelastic creep and shrinkage strains at the level of thebounded steel.

stress relaxation in the tendons.

Pi

initial prestress

Pe

effective prestress

Estimate ~ 15-20% Pe ~ 0.8 Pi


The initial stress level in prestressing steel after transfer is usually high,

often in the range 60-75% of the tensile strength of the material.

If a tendon is stretched and held at a constant length (constant strain), the

development of creep strain in the steel is exhibited as a loss of elastic strain,

and hence a loss of stress.

Relaxation in steel is highly dependent on the stress level

and increases at an increasing rate as the stress level increases.


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-

+

(a)

-

+

(b) (c)

-

(d)

-

(e)

-

Transfer Full Service Load Condition Ultimate Load

Condition


-

+

Prestress Cable

Immediately after application of Prestress

Pi - initial prestress

Pi


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-

+ -

+

-

Prestress Cable

After full service loads and Time Dependent

Losses

Pe - effective prestress

Pe


Concrete Compressive Stress Limit: Fci= 0.5 f ci Large nonlinear creep strains.

Large prestress losses Safety against brittle failure (should check strength at

transfer)

Tensile Stress Limit: Fti =

advisable especially in unreinforced zones

cracks may not close completely (local spalling)

if cracking allowed add conventional steel

'0.25 cif


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Concrete Compressive Stress Limit: Fc= 0.45 f c usually not enforced.

prevents large creep strains

Tensile Stress Limit: Ft =

upper limit some cracking due to shrinkage, may need

added reo.

' '0.25 to 0.5c cf f

No Limits needed unless Cracking

To be avoided


Fti tensile stress limit at transfer

Ft tensile stress limit under fullload

Fci compressive stress limit at transfer

Fc compressive stress limit under fullload


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TRANSFER

FULL

SERVICELOADS

These limits are not explicitly suggested in AS3600

but are generally used for most prestressed designs.

cc fF'45.0==== compression

citi fF'25.0==== tension

ct fF'5.0==== tension

'0.5c i ciF f= compression

AS3600 Clause 8.1.4.2


If Cracking Permitted - CHECK CRACK WIDTHS

Increment in Steel Stress near the Tension Face< 200 MPa as the load increases from its valuewhen the extreme tensile fibre is at zero stressto the full short-term service load

c/c spacing 200 mm for beams; 500 mm for slabs


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C A

ye

yt

yb

-

+

SECTION

STRESSES

ResultantDue to Pi(prestress force)

++++

I

eyP

A

P ii

Due to Mo(w, applied load)

I

yMo

8

2wLMo ====

-

+ t

b

=+

Mo

-

+


(((( ))))ti

t

oii FZ

MeP

A

P

++++====

ti

totii

t

FI

yM

I

eyP

A

P++++====

Axial

stress

Bending

(due to Pi)

Bending

(due to Mo)

tensile

stress limit

at transfer

t

o

t

iti

Z

M

Z

Ae

A

PF

1

8

2wLMo ====

TENSILE STRESS LIMIT AT TRANSFER

(1)


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tbTbii

b FI

yM

I

eyRP

A

RP++++====

Axial

stress

Bending

(due to Pi)

Bending

(due to MT)

Bottomfibrestress

b

T

b

it

Z

M

Z

Ae

A

RPF ++++

++++ 1

TENSILE STRESS LIMIT

(3)b

by

IZ ====


i i t T t

t c

R P R P e y M y

FA I I = +

Axial

stress

Bending

(due to Pi)

Bending

(due to MT)

Topfibrestress

1i Tct t

R P MA eF

A Z Z

tt

y

IZ ====

COMPRESSIVE STRESS LIMIT

(4)

due to Pi due to w


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Equations (1) (4) can be rearranged to express 1/Pi

as a linear function ofe.

Equation (1) gives:

++++

t

i

t

oti

Z

AeP

Z

MFA 1

or(((( ))))toti

t

i ZMFA

ZAe

P /

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++++

++++

t

ty

IZ ====

b

by

IZ ====


Ift

tZ

A====

b

bZ

A====

(((( ))))

ttc

t

i MAF

eR

P

++++

11(4)

(((( ))))tbt

b

i MAFeR

P

++++++++ 11(3)

obci

b

i MAF

e

P

++++

++++

11(2)

otti

t

i MAFe

P

++++ 11(1)

Fti , Fci - TRANSFER Ft , Fc FULL LOADING


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-1/b 1/t Eccentricity, e

1/Pi Equation 1

Equation 3

Equation 2

Equation 4

acceptable region

cF

ciF

tiF

tFminimumiP

maxe

In order to minimise prestressing costs,

the smallest possible value ofPi would generally be selected.

eLimit


b b

c i b o t b t

R

A F M A F M

=

+ +

( ) t obt c i

M R MZ

F R F

If section too small line 2 will lie above line 3

When slopes are equal:


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otti

t

i MAF

e

P

++++

11(1)

(((( ))))

tbt

b

i MAF

eR

P

++++

++++

11(3)=

1L i m i t

t b

ae

a

+=

( )ti t o

t b t

R A F M a

A F M

+=

+

Equation 1 = 3


When the prestressing force and eccentricity are determined for the

critical section, the location of the cable at every section

along the member must be specified.

For a member which has been designed to be uncracked throughout,

the tendons must be located so that

the stress limits are observed on every section.

At any section, Equations (1) (4) may be used to establish a range of

values for eccentricity which satisfy the selected stress limits.

IfMo and MT are the moments caused by the external loads at transfer and

under full service loads, respectively, and Pi and Pe are the corresponding

prestressing forces at the same section, the extreme fibre stresses must satisfy

Equations (1) - (4).


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After Pi and Pe have been determined at the critical sections,

the friction losses along the member are estimated and the corresponding

prestressing forces at intermediate sections are calculated.

At each intermediate section, the maximum eccentricity that will satisfy

both stress limits at transfer is obtained from either Equation (1) or (2).

The minimum eccentricity required to satisfy the tensile and compressive

stress limits under full loads is obtained from either Equation (3) or (4).

A permissible zone is thus established in which the line of action of the

resulting prestressing force must be located.


Permissible zone

Equation (3) and (4)Equation (1) and (2)


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When the prestress and eccentricity at the critical sections are

selected using the load-balancing approach,

the cable profile should match, as closely as practicable,

the bending moment diagram caused by the balanced load.

In this way, deflection will be minimised.

For cracked, partially prestressed members, Equations (1) and (2)are usually applicable and fix the maximum eccentrictity.

The cable profile should then be selected according to the loading

type and moment diagram.


A one-way slab is simply supported over a span of 12 m and is

to be designed to carry a service load of 7 kPa (kN/m2) in

addition to its own self-weight. The slab is post-tensioned by

regularly spaced tendons with parabolic profiles.The slab thickness D = 300 mm.

The material properties are:

MPa25' ====cif

MPa25300====ciE

MPa32' ====cf

MPa28600====cE

MPa1840====pf


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The prestressing force and eccentricity are to be determined to

satisfy the following concrete stress limits:

MPa25.12525.0 ========tiF

MPa5.12255.0 ========ciF

MPa41.13225.0 ========tF

MPa0.16325.0 ========cF

Example 1Example 1

(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)


At mid-span, the instantaneous and time-dependent losses are

taken to be 8% and 16%, respectively.

Slab self-weight(which is the only load other than the prestress at transfer):

kN/m2.73.024 ========sww (1 m wide strip)

and the moments at mid-span

both at transfer and under the full service load are:

kNm/m6.1298

122.7 2====

====oM

(((( ))))kNm/m6.255

8

122.70.7 2====

++++====TM

Example 1Example 1


0.94R =


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Cross-section properties:

/mmm10300 23====A /mmm10225046

====I

/mmm101536

============ bt ZZZ

02.0/ ============ ZAbt

mm5011

========bt

Example 1Example 1



( ) t obt c i

M R MZ

F R F

( )( )

( )

6 3 6

6 3 6 3

255.6 0.94 129.615 10 mm /m 10

1.41 0.94 12.5

15 10 mm /m 10.2 10 mm /m

b

b

Z

Z

=

=


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If 12.7 mm diameter strand is used with 30 mm concrete

cover, then

mm11436150max ====e

And from the Design Diagram, or Equation (3), the

corresponding minimum permissible value ofPi is found to be

610588.0

1 ====iP

kN/m1700==== iPand

Example 1Example 1



At the jacking point, the required prestressing force is

kN/m185092.0

1700========jP

(8% instantaneous losses)

Example 1Example 1



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From table 2.1, a 12.7 diameter 7-wire,

low-relaxation strand has a cross-sectional area of 100 mm2

and a minimum breaking load of 184 kN.

A flat duct containing four 12.7 mm strands

can therefore be stressed with a maximum jacking force of

kN626184485.0 ====For design purposes, the yield strength of stress-relieved wires

may be taken as 0.85 times the minimum tensile strength

(i.e. 0.85fp)

Example 1Example 1



The minimum number of cables

required in each metre width of slab is therefore:

96.2626/1850 ====

and the maximum spacing between cables is

mm33896.2/1000 ====

Therefore use 1-4 strand tendon every 330 mm.

Example 1Example 1



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Use the following design strategy:

Set the tensile stress to zero at top & the compressive stress to 0.6 fcp at btmat mid-span (include SW).

i i swtop

t t

i i swbtm

tm btm

P Pe M

A Z Z

P Pe M

A Z Z

= +

= +

Solve the two stress equations for P & e. Determine no. of cables and adjust the jacking load. Determine the Cable profile: e at midspan & zero at ends. (use straight cables) Distribute the cables along the span to maintain e

Show the cable layout through the span.

Pretensioned Beam Design Strategy

cven9806 lecture 2 - serviceability 2009

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