Lecture Lecture 3 3 Basic Concepts Basic Concepts Dr. Hazim - - PDF document

Prestressed Concrete Hashemite University

• Dr. Hazim Dwairi

1

The Hashem ite University Departm ent of Civil Engineering

Lecture Lecture 3 3 – – Basic Concepts Basic Concepts

Dr Hazim Dwairi Dr Hazim Dwairi

Prestressed Concrete Prestressed Concrete

• Dr. Hazim Dwairi
• Dr. Hazim Dwairi

The Hashemite University The Hashemite University

• Dr. Hazim Dwairi
• Dr. Hazim Dwairi

(i i) Com bined Load Concepts ) Com bined Load Concepts

• PS beam is assumed to be homogenous and

PS beam is assumed to be homogenous and elastic Stress in this beam : elastic Stress in this beam :

• elastic. Stress in this beam :
• elastic. Stress in this beam :

I y M I y e P A P f

ext c

± ± − = ) . (

P.e

Prestressed Concrete Prestressed Concrete

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cgc cgs

P e

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2 A A – – No prestress (self No prestress (self-

• weight only)

weight only)

t

f

• Consider rectangular section, simply supported

Consider rectangular section, simply supported beam: beam:

b

f

h/2 h/2 b L/2 +

2

l

L/2

Prestressed Concrete Prestressed Concrete

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2 2 3 2 2

4 3 12 2 8 8 ; weight

• self

Let bh wl bh h wl f f wl M w

b t CL

= ± = = = =

B B – – Eccentric prestress + self Eccentric prestress + self-

• weight

weight

cgc

t

f

e b g

+

cgs

Prestress Prestress

Pi Pi

Prestressed Concrete Prestressed Concrete

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• +

=

• b

f

Prestress Prestress ONLY ONLY

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≡ ) . ( ) . ( losses without force prestress initial c e P P f c e P P f P If

i i b i i t i

Prestress ONLY Prestress ONLY ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ + − ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ − = ≡ − − = + − = . 1 ; . 1 then , gyration

• f

radius if ) ( ; ) ( c e P f c e P f A I r I A f I A f

b i b t i t c g g i c i b g i c i t

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⎟ ⎟ ⎠ ⎜ ⎜ ⎝ + = ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − =

2 2

1 ; 1 r A f r A f

c i c i D t i t

M c e P f ⎟ ⎞ ⎜ ⎛ − − . 1 M moment a causes weight self beam If

D

Prestress + Self Prestress + Self-

• weight

weight

b t b D b c i b t D c i t

S S S M r c e A P f S M r c e A P f + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = fibers bottom and at top moduli section are , . 1 . 1

2 2

• +

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b g b t g t

c I S c I S S S = = , fibers bottom and at top moduli section are ,

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4 C C – – Eccentric prestress + self Eccentric prestress + self-

• weight + Live Load

weight + Live Load

cgc w e b g cgs Pe Pe

• Subsequent to erection and installation of the floor or

Subsequent to erection and installation of the floor or deck, live loads act on the structure, causing a deck, live loads act on the structure, causing a i d t M i d t M Th f ll i t it f h Th f ll i t it f h

Prestressed Concrete Prestressed Concrete

superimposed moment M superimposed moment Ms. The full intensity of such . The full intensity of such loads normally occurs after the building is complete and loads normally occurs after the building is complete and in full use. Thus, some time in full use. Thus, some time-

• dependent losses in

dependent losses in prestress have already taken place and prestress have already taken place and P Pe should be should be considered in calculations. considered in calculations.

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C C – – Eccentric prestress + self Eccentric prestress + self-

• weight + Live Load

weight + Live Load MT = M = MD

D + M

+ MSD

SD +M

+ML

• MD = moment due to self

= moment due to self-

• weight.

weight.

• MSD

SD = moment due to superimposed dead load

= moment due to superimposed dead load

• ML = moment due to live load including impact

= moment due to live load including impact and seismic and seismic

t T t e t

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

2

. 1

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b T b c e b c

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

2

. 1

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Loading Stages Loading Stages

(1 1) Casting: No concrete stresses ) Casting: No concrete stresses (2 2) Stressing: P ) Stressing: Pi

i effect

effect

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Loading Stages Loading Stages

(3 3) Transfer (temporary load): Prestress + self ) Transfer (temporary load): Prestress + self weight weight weight weight (4 4) Loading: PS + D + SD + L ) Loading: PS + D + SD + L

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(ii) Internal Couple Concept (ii) Internal Couple Concept (C (C-

• Line Method)

Line Method)

• Consider a simply supported beam

Consider a simply supported beam prestressd prestressd with draped tendon: with draped tendon: with draped tendon: with draped tendon:

C-Line cgc cgs A A e e’

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• C-
• line or center of pressure locates the concrete

line or center of pressure locates the concrete compressive force compressive force C for a given load level for a given load level

• Zero external load (self

Zero external load (self-

• weight neglected)

weight neglected) – – Hypothetical case: Hypothetical case:

Lever arm = 0 Moment = M =0

cgc cgs

T C

• Loaded Condition (including self

Loaded Condition (including self-

• weight):

weight):

C R=0

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cgc cgs

T C Lever arm = e + e’ = M/P

e’ e

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(iii) Equivalent Load Concept (iii) Equivalent Load Concept (Load Balancing) (Load Balancing)

• Consider a simply supported beam

Consider a simply supported beam prestressd prestressd with draped tendon the profile of which is with draped tendon the profile of which is with draped tendon, the profile of which is with draped tendon, the profile of which is assumed parabolic: assumed parabolic:

Note the cable ends cgc cgs emax

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g

2 max max 2

4 2 / , L) (x & ) ( L e a L x at e e aL b c x at e c bx ax e − = ⇒ = = − = = ⇒ = = = + + =

( ) ( )

− = − − = ∴ : by given is alone ng prestressi to due concrete

• n

moment section, any at Now 4

2 max

x L x L e x L ax e

( ) ( )

− − = = − − = − = 8 2 4 4 . : by given is alone

2 2 max 2 max

P M d x L L Pe dx dM V x L x L e P e C M

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↑ = − = − = Constant 8

2 max 2 2

L Pe dx M d we

Equivalent Load

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• Hence, the parabolic tendon profile gives an

Hence, the parabolic tendon profile gives an equivalent uniformly distributed load on the equivalent uniformly distributed load on the concrete over the length of the tendon: concrete over the length of the tendon:

Pe de P P P

max

4 tan sin ≈ θ θ Pemax 4

2 max

8 L Pe we = P P ≈ θ cos L dx P P P

max

tan sin = = ≈ θ θ L

P

• N t

th t th f th ti l & h i t l N t th t th f th ti l & h i t l

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• Note that the sum of the vertical & horizontal

Note that the sum of the vertical & horizontal forces is zero, since the beam must be in forces is zero, since the beam must be in equilibrium under the action of equilibrium under the action of prestressing prestressing. .

• The effect of

The effect of prestressing prestressing and applied load on and applied load on the concrete may be simulated as follows: the concrete may be simulated as follows:

8P 2 L we 2 L we

w

2 max

8 L Pe we =

P P

2 wL 2 wL L w

M t d

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2 wL

P

2 L we

P

e e

w M to due =

g net c e net

I y M A P f w w w ± − = − =

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• Load balancing method of design was first

Load balancing method of design was first proposed by T.Y. Lin and is described in detail proposed by T.Y. Lin and is described in detail

• n page
• n page 16

16 of the textbook by

• f the textbook by Nawy
• Nawy. See also

. See also pp. pp.488 488 in Collins and Mitchell in relation to slab in Collins and Mitchell in relation to slab design. design.

• Equivalent loads may be used to input the effect

Equivalent loads may be used to input the effect

• f
• f prestressing

prestressing in the form of load into computer in the form of load into computer programs for analysis of statically indeterminate programs for analysis of statically indeterminate

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structures. structures.

(a) Linear Profile (a) Linear Profile

cgc e θ

L Pe 2 L Pe 2

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P P

L Pe 4

• Use this profile to support concentrated loads

Use this profile to support concentrated loads

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(b) Constant Profile (b) Constant Profile

cgc e

P P

Pe Pe

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P P

• Use this profile to support uniform moment

Use this profile to support uniform moment

(c) Mixed Profile (c) Mixed Profile

cgc e1 e3 e2

P P

1 2 1

) ( L e e P +

2 3 2

) ( L e e P +

L1 L2

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P P

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + +

2 3 2 1 2 1

L e e L e e P

• End moments are due to end eccentricity

End moments are due to end eccentricity

3

Pe

2

Pe

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• To avoid tension it’s necessary to reduce the

To avoid tension it’s necessary to reduce the eccentricity so that the eccentricity so that the centroid centroid of the

• f the

prestressing prestressing steel at the ends of the beam is steel at the ends of the beam is within the middle third for a rectangular section. within the middle third for a rectangular section. This is achieved by using harped or blanketed This is achieved by using harped or blanketed This is achieved by using harped or blanketed This is achieved by using harped or blanketed strands in strands in pretensioned pretensioned beams and draped beams and draped tendons in post tendons in post-

• tensioned beams to maintain

tensioned beams to maintain ‘e ‘emax

max’ at mid

’ at mid-

• span and smaller ‘e’ at ends.

span and smaller ‘e’ at ends.

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Pretensioned Harped at two hold down points Post-tensioned Draped parabolic tendons to suit typical bending moment diagrams