Lecture 6 Lecture 6 Flexural Design Flexural Design Dr. Hazim - - PDF document

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The Hashem ite University Departm ent of Civil Engineering

Lecture Lecture 6 6 – – Flexural Design Flexural Design

Dr Hazim Dwairi Dr Hazim Dwairi

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“Every Design is Essentially an “Every Design is Essentially an Analysis.” Analysis.” -

• Nawy

Nawy

• Stages at which stresses are estimated

Stages at which stresses are estimated

I i i l P I i i l P

– Initial Prestress

Initial Prestress

– Self

Self-

• weight application

weight application

– Superimposed dead load

Superimposed dead load

– Decompression in steel

Decompression in steel

– Service load limit

Service load limit Ultimate load state Ultimate load state

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– Ultimate load state

Ultimate load state

• According to current practice PS members

According to current practice PS members are proportioned using allowable stress are proportioned using allowable stress design (ASD) design (ASD)

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• Cross

Cross-

• section dimensions, Prestress force, and

section dimensions, Prestress force, and eccentricity are selected to keep stress within eccentricity are selected to keep stress within specified limits. specified limits.

• Beams designed this way must satisfy deflection

Beams designed this way must satisfy deflection requirement and other load combinations must requirement and other load combinations must requirement and other load combinations must requirement and other load combinations must be checked be checked

• Basic flexure theory assumptions

Basic flexure theory assumptions

– Plane section before bending remain plane after

Plane section before bending remain plane after bending (i.e. small deflections) bending (i.e. small deflections)

– Material is elastic

Material is elastic

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Material is elastic Material is elastic

– Effect of transformed section is neglected

Effect of transformed section is neglected

– Section is

Section is uncracked uncracked

– No variation of PS force along the beam

No variation of PS force along the beam

– Effect of small curvature is neglected

Effect of small curvature is neglected

Equal

Flexural Stress Distribution Flexural Stress Distribution Throughout Load History Throughout Load History

T C C C β3f’c

(a) Beam section (a) Beam section

Tw C C C or zero T (a) (b) (c) (d) (e) (f)

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(a) Beam section (a) Beam section (b) Initial stressing stage (b) Initial stressing stage (c) self (c) self-

• weight and effective

weight and effective prestress prestress (d) Full D.L. + (d) Full D.L. + P Peff

eff

(e) Full service load + (e) Full service load + P Peff

eff

(f) Ultimate limit state for under (f) Ultimate limit state for under-

• reinforced beam

reinforced beam

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Maxim um Fiber Stresses Maxim um Fiber Stresses

t L SD

S M M +

t

f Δ

t D

S M

c

f

ti

f

1 Pi Stresses 4 ct cb cgc

i

2 Pi + MD Stresses 3 Pe + MD Stresses 4 Pe + MD + MSD + ML Stresses

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1 2 3

b L SD

S M M +

b

f Δ

b D

S M

ci

f

t

f

Load Deflection Curve Load Deflection Curve

Load

Service load limit Ultimate Steel yielding Full dead load Balanced Decompression First cracking cgs (f=0) fcr

• verload

Service load limit

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Δ Δo ΔD ΔL ΔPe

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Selection of Geom etric Properties Selection of Geom etric Properties

• Select the min. section

Select the min. section moduli moduli S St & & S Sb that that ti f t li it t t f l di ti f t li it t t f l di satisfy stress limits at stage of loadings: satisfy stress limits at stage of loadings:

support at SS for 5 . 25 .

' ' ci ti ci ti

f f OR f f = =

' '

6 . 45 .

c c c c

f f OR f f = =

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'

6 .

ci ci

f f =

met is deflection term

• Long

if 5 .

' ' c t c t

f f OR f f = =

(a) At Transfer (B) At Service

Selection of Geom etric Properties Selection of Geom etric Properties

• Stresses at transfer:

Stresses at transfer:

(1) .......... 1

2

≤ − ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ − − =

ti t D t i t

f M ec P f force ng prestressi initial P (2) .......... 1 ( )

i 2 2

≡ ≤ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ⎟ ⎠ ⎜ ⎝

ci b D b c i b ti t c

f S M r ec A P f f S r A f ⎟ ⎞ ⎜ ⎛

t

M ec P

• Effective stresses after losses:

Effective stresses after losses:

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losses after force ng prestressi effective P 1 1

i 2 2

≡ ≤ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ≤ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − =

c b D b c e b t t D c e t

f S M r ec A P f f S M r ec A P f

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Selection of Geom etric Properties Selection of Geom etric Properties

• Service load final stresses:

Service load final stresses:

(3) 1

T t e t

f M ec P f ≤ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ − (4) .......... 1 (3) .......... 1

2 2 t b T b c e b c t T c e

f S M r ec A P f f S r A f ≤ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ≤ − ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − =

• Where:

Where: M = M = M + M + M + M + M

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MT

T = M

= MD + M + MSD

SD + M

+ ML P Pi

i = initial

= initial prestress prestress Pe = effective = effective prestress prestress after losses after losses

Selection of Geom etric Properties Selection of Geom etric Properties

• Decompression stage is when the stress

Decompression stage is when the stress at the at the cgs cgs is equal to zero. The change in is equal to zero. The change in at the at the cgs cgs is equal to zero. The change in is equal to zero. The change in the concrete stress due to decompression the concrete stress due to decompression is: is:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + =

2 2

1 r e A P f

c e decomp

• For variable tendon eccentricity:

For variable tendon eccentricity:

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For variable tendon eccentricity: For variable tendon eccentricity:

assume the effective assume the effective prestress prestress P Pe=γPi

i

i.e. loss of i.e. loss of prestress prestress = P = Pi

i – P

Pe = ( = (1 1-

• γ) P

) Pi

i

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Selection of Geom etric Properties Selection of Geom etric Properties

t D ti t c i

S M f r ec A P + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − (5) .......... 1 Eq(1) from

2 c t L SD D t c i c t T t c e

f S M M M r ec A P f S M r ec A P ⎞ ⎛ + + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − : Eq(5) using

• 1
• 1

Eq(3) from

2 2

γ

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t L SD D c ti c t L SD D t D ti

S M M M f f f S M M M S M f + + − = − + + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ) 1 (

• γ

γ γ

Selection of Geom etric Properties Selection of Geom etric Properties

c ti L SD D t

f f M M M S − + + − ≥ ∴ ) 1 ( γ γ

ti b t ci t L SD D b

f f S c f f M M M S − − + + − ≥ ∴ : e Furthermor ) 1 ( : similarly γ γ γ

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b t b t b t t b b t ci t c ti t b

S S S h c S S S h c h c c f f f f S S c c + = + = = + − = = ; ; 1 γ γ

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Required Eccentricity Required Eccentricity

• At critical section, usually

At critical section, usually midspan midspan, , eccentricity can be determined using eccentricity can be determined using eccentricity can be determined using eccentricity can be determined using concrete concrete centroidal centroidal stress under initial stress under initial conditions: conditions:

( )

D t cgc ti c

P M P S f f e + − =

ti

f

cgc

f

t

c h

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( )

c cgc i i i cgc ti c

A f P P P f f =

ci

f

b

c

Beam with Constant Eccentricity Beam with Constant Eccentricity

• If the PS force and the eccentricity are

If the PS force and the eccentricity are kept constant along the span, as is often kept constant along the span, as is often kept constant along the span, as is often kept constant along the span, as is often convenient in PS construction, the stress convenient in PS construction, the stress limits will most likely be exceeded at limits will most likely be exceeded at several point in the span, especially at several point in the span, especially at supports. supports. C t i lt ti il bl f C t i lt ti il bl f

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• Certain alternatives are available for

Certain alternatives are available for reducing excessive stresses at supports, reducing excessive stresses at supports, as follows: as follows:

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1. . Debonding Debonding

12 cables

• 2. Raised Tendons

. Raised Tendons

4 cables sheathed without grouting

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L/4

• 3. Supplem entary

. Supplem entary nonprestressed nonprestressed steel steel

• Minimum section

Minimum section moduli moduli values are: values are:

c ti L SD D t

f f M M M S γ − + + ≥ : and

ci t L SD D b

f f M M M S γ − + + ≥ : and

• Required eccentricity at critical section:

Required eccentricity at critical section:

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( )

i t cgc ti c

P S f f e − =

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Shape Selection Shape Selection

• A unique feature PS concrete design is the

A unique feature PS concrete design is the freedom to select cross freedom to select cross-

• sectional properties to

sectional properties to it i l i t t h d it i l i t t h d suit special requirement at hand suit special requirement at hand

• In steel structures, choices are limited to

In steel structures, choices are limited to standarized standarized shapes in shapes in tember tember, rectangular , rectangular sections are almost always used. Since mid sections are almost always used. Since mid-

• span moment normally controls PS design, the

span moment normally controls PS design, the larger the mid larger the mid span eccentricity the smaller is span eccentricity the smaller is

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larger the mid larger the mid-span eccentricity, the smaller is span eccentricity, the smaller is the needed PS force, and the design is more the needed PS force, and the design is more

• economic. in this case a large top flange is
• economic. in this case a large top flange is

needed, resulting in T or I sections. needed, resulting in T or I sections.

• For short span beams, rectangular sections may

For short span beams, rectangular sections may provide the most economical section because provide the most economical section because forming costs are minimized for longer spans, forming costs are minimized for longer spans, the more efficient flanged sections are preferred. the more efficient flanged sections are preferred.

• I

I – – sections are used as floor beams with sections are used as floor beams with composite slab topping in long composite slab topping in long-

• span parking

span parking structures. structures. I and T I and T-

• sections are commonly used for bridge

sections are commonly used for bridge structures. structures.

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Double T Double T-

• sections are widely used in floor

sections are widely used in floor systems in building and parking structures, systems in building and parking structures, because of the large compressive area available because of the large compressive area available in slab. in slab.

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• Large hollow box girders are used in very large

Large hollow box girders are used in very large span segmental bridge construction. These span segmental bridge construction. These girder have high girder have high torsional torsional strength and strength and strength/weight ratio. strength/weight ratio.

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Typical Span Typical Span-

• Depth Ratio

Depth Ratio

Type Span/Depth Ratio I-Beam and single T-beam 24 - 36 g Double T-beams 30 - 40 Bridge Girders 25 - 30 One way Solid Slabs 35 - 50 One way Hollowcore Slabs 40 - 50

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For long spans with high self weight to superimposed load ratio, the bottom flange may be eliminated all together .

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Concrete Protection and Tendon Concrete Protection and Tendon Spacing Spacing

• ACI

ACI 7 7. .7 7 imposes the minimum cover distance for imposes the minimum cover distance for PS concrete member PS concrete member PS concrete member. PS concrete member.

• For post

For post-

• tensioned members, the cover

tensioned members, the cover requirements apply to the ducts and metal and requirements apply to the ducts and metal and fitting. fitting.

• If the member is designed for a service load

If the member is designed for a service load tension in excess of cracks in concrete tension in excess of cracks in concrete

'

5 f

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tension in excess of , cracks in concrete tension in excess of , cracks in concrete are likely, and the cover requirements must be are likely, and the cover requirements must be increased by increased by 50 50%. %.

5 .

c

f

Concrete Protection and Tendon Concrete Protection and Tendon Spacing Spacing

• At the mid

At the mid-

• span and any elsewhere than at the

span and any elsewhere than at the ends spacing between bars and strands is the ends spacing between bars and strands is the ends, spacing between bars and strands is the ends, spacing between bars and strands is the larger of d larger of db

b and

and 25 25 mm. mm.

• At the ends of the

At the ends of the pretensioned pretensioned members, members, spacing is increased for proper bond, S ≥: spacing is increased for proper bond, S ≥:

– 4d

db for wires for wires

– 3db for strands

for strands

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3db for strands for strands

• Elsewhere, bundling of no more than four

Elsewhere, bundling of no more than four tendons or bars is permitted. tendons or bars is permitted.

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Flexural Crack Control Flexural Crack Control

• Flexural tensile cracks may be limited or

Flexural tensile cracks may be limited or eliminated completely by eliminated completely by prestressing prestressing However However eliminated completely by eliminated completely by prestressing

• prestressing. However,

. However, partial partial prestressing prestressing has gained increasing has gained increasing popularity due to technical and economical popularity due to technical and economical reasons resulting in need for crack width control. reasons resulting in need for crack width control.

• No special provisions are included in the ACI

No special provisions are included in the ACI code for PS concrete. The provisions for regular code for PS concrete. The provisions for regular

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p g p g RC members are applicable. RC members are applicable.

Envelopes for Tendon Placem ent Envelopes for Tendon Placem ent

• There is an envelope within which the

There is an envelope within which the prestressing prestressing force can be applied with causing force can be applied with causing t il t ll bl t t il t ll bl t no tensile stresses or allowable stress no tensile stresses or allowable stress

c t t b t c i

A S c r k e r c e A P = = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − =

2 2

. 1

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c b b t c

A S c r k e Similarly = = =

2

,

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• In a similar manner, kern points can be

In a similar manner, kern points can be established to the right and the left. established to the right and the left.

Envelopes for Tendon Placem ent Envelopes for Tendon Placem ent

• To design the tendon along the span to develop

To design the tendon along the span to develop no tension or limited tension, a draped or harped no tension or limited tension, a draped or harped tendon should follow the shape of the bending tendon should follow the shape of the bending moment diagram. moment diagram.

• Draped tendons are used for uniformly distributed

Draped tendons are used for uniformly distributed

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loading loading

• Harped tendons are used for concentrated

Harped tendons are used for concentrated loading. loading.

• Lower

Lower cgs cgs envelope: envelope:

Envelopes for No Tension Envelopes for No Tension

M D

cgc cgs e kt kb amin C T

min min

a k e P M a

b b i D

+ = =

• Upper

Upper cgs cgs envelope: envelope:

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cgc cgs e kt kb amax C T

b t e T

k a e P M a − = =

max max

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• Additional eccentricity at the bottom:

Additional eccentricity at the bottom:

Envelopes for Lim iting Tension Envelopes for Lim iting Tension

b i ti

e P f =

'

ti

f

cgc cgs e kt kb e’b+amin C T

i t ti b t ti

P S f e S f =

'

• Upper

Upper cgs cgs envelope: envelope:

e’t

t ee

P f =

'

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cgc cgs e kt kb amax C T

t t

f

e b t t b t

P S f e S f = =

'

Permitted Tension No Tension

Tendon Profiles Tendon Profiles

amax e’t

No Tension

kt kb

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amin e’b

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Flexural Design of Com posite Flexural Design of Com posite Beam s Beam s

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(a) (a) Unshored Unshored Slab Case Slab Case

* Before casting the top slab “ no composite action”

SD D t e t

M M c e P f + ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ − = . 1

b SD D b c e b t c

S M M r c e A P f S r A f + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = − ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − =

2 2

. 1 1

* After top slab hardens “composite action”. New section moduli should be used.

Additional composite superimposed DL

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section moduli should be used.

b c L CSD b SD D b c e b t c L CSD t SD D t c e t

S M M S M M r c e A P f S M M S M M r c e A P f + − + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = + − + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − =

2 2

. 1 . 1

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(b) Fully Shored Slab Case (b) Fully Shored Slab Case

* Before casting the top slab & shoring

D t e t

M c e P f ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ − = . 1

b D b c e b t c

S M r c e A P f S r A f − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = − ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − =

2 2

. 1 1

* After top slab hardens “composite action”. New section moduli should be used.

Additional composite superimposed DL

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section moduli should be used.

b c L CSD SD b D b c e b t c L CSD SD t D t c e t

S M M M S M r c e A P f S M M M S M r c e A P f + + − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = + + − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − =

2 2

. 1 . 1

Effective Flange Width Effective Flange Width

• Only part of the slab contributes to the

Only part of the slab contributes to the stiffness increase in the composite stiffness increase in the composite stiffness increase in the composite stiffness increase in the composite section. section.

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Effective Flange Width Effective Flange Width

Edge Beam Intermediate Beam

* beff is the smallest of:

ACI bw + 6hf bw + 16hf bw + 1/2Lc bw + Lc bw + L/12 L/4 AASHTO bw + 6hf bw + 12hf bw + 1/2Lc bw + Lc bw + L/12 L/4

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* If the modulus of elasticity of the top slab Ect and of the precast beam Ec, then the effective flange width beff must be modified by the modular ration ‘n’

eff c ct m

b E E b =

End Zones and Developm ent End Zones and Developm ent Length Length

• The interlock or adhesion between the PS

The interlock or adhesion between the PS tendon circumference and the concrete over a tendon circumference and the concrete over a tendon circumference and the concrete over a tendon circumference and the concrete over a finite length of the tendon gradually transfers the finite length of the tendon gradually transfers the concentrated concentrated prestressing prestressing force to the entire force to the entire concrete section at planes away from the end concrete section at planes away from the end bock & towards the bock & towards the midspan midspan. .

• Minimum development length (ACI

Minimum development length (ACI318 318-05 05

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Minimum development length (ACI Minimum development length (ACI318 318 05 05, , section section 12 12. .9 9. .1 1) )

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MPa ; 7 21 .

b pe ps b pe d

d f f d f l Min ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

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End Zones and Developm ent End Zones and Developm ent Length Length

the

• ver which

distance the length transfer 21

t b pe

l d f = = = be should strand the

• ver which

length 150 additional the length bond flexural 7 losses. after prestress effective the develop to concrete the to bonded be should strand 50 21

f b pe ps b

) d ( l d f f ) d ( ≈ = = = − ≈

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develop. may , strength, nominal at steel PS in the stress a that so bonded be should strand the

• ver which

length 150

ps b

f ) d ( ≈

Transfer Zone in Transfer Zone in Pretensioned Pretensioned Beam s Beam s

h P l k f 138MP i i length beam ed pretension length fer pver trans stirrups

• f

area total BS & SI ; 021 . ≡ ≡ =

t t t s i t

f h l A l f h P A

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control crack for 138MPa stirrups in stress average = ≡

s

f

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Post Post-

• tensioned Anchorage Zone

tensioned Anchorage Zone

• Length of anchorage zone is at which the

Length of anchorage zone is at which the PS f t f i t li di t ib ti PS f t f i t li di t ib ti PS force transfer into a linear distribution PS force transfer into a linear distribution across the section depth and according to across the section depth and according to St.

• St. Venant’s

Venant’s principle is equal to ‘h’. principle is equal to ‘h’.

• This zone consists of:

This zone consists of:

– General zone: its length along span is ‘h’ General zone: its length along span is ‘h’

Prestressed Prestressed Concrete Concrete

General zone: its length along span is h General zone: its length along span is h – Local zone: it’s the insert prism of concrete Local zone: it’s the insert prism of concrete surrounding & immediately ahead of the surrounding & immediately ahead of the anchorage device. anchorage device.

• Dr. Hazim Dwairi
• Dr. Hazim Dwairi

The Hashemite University The Hashemite University

General Zone Design General Zone Design

• Confinement of the anchorage zone is

Confinement of the anchorage zone is i d t t b ti d litti i d t t b ti d litti required to prevent bursting and splitting required to prevent bursting and splitting due to high concentrated forces acting on due to high concentrated forces acting on the concrete section. the concrete section.

• Analysis methods:

Analysis methods:

– Linear elastic analysis including FEM Linear elastic analysis including FEM

Prestressed Prestressed Concrete Concrete

y g y g – Strut and Tie models Strut and Tie models – Approximate methods: applicable to rectangular cross Approximate methods: applicable to rectangular cross sections without discontinuities> sections without discontinuities>

• Dr. Hazim Dwairi
• Dr. Hazim Dwairi

The Hashemite University The Hashemite University

Prestressed Concrete Hashemite University

• Dr. Hazim Dwairi

20

e Pi A B C D cgc cgs y

i

Lt C D

x h

C T

cgc

Prestressed Prestressed Concrete Concrete

• Dr. Hazim Dwairi
• Dr. Hazim Dwairi

The Hashemite University The Hashemite University

Lt

Pi

Mmax shear Bursting crack