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

Prestressed Concrete Hashemite University The Hashem ite University Departm ent of Civil Engineering Lecture Lecture 3 3 – – Basic Concepts Basic Concepts Dr. Hazim Dwairi Dr Hazim Dwairi Dr Hazim Dwairi Dr. Hazim Dwairi Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete (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 : − P ( P . e ) y M y = ± ± ext f A I I c P.e P cgc e cgs Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 1

Prestressed Concrete Hashemite University Consider rectangular section, simply supported Consider rectangular section, simply supported beam: beam: A A – – No prestress (self No prestress (self- -weight only) weight only) t f - h/2 h/2 L/2 L/2 + b b f 2 2 wl l = = Let self - weight w ; M CL 8 2 2 wl h 12 3 wl = = ± = t b f f 3 2 8 2 bh 4 bh Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete B B – – Eccentric prestress + self Eccentric prestress + self- -weight weight cgc g e P i P i cgs b t f + Prestress Prestress Prestress Prestress ONLY ONLY + = - - - b f Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 2

Prestressed Concrete Hashemite University Prestress ONLY Prestress ONLY ≡ If P initial prestress force without losses i P ( ( P . e ) ) c P ( ( P . e ) ) c = − + = − − t t b b i i i i i i i i f f ; f f A I A I c g c g I ≡ = g if r radius of gyration , then A c ⎛ ⎞ ⎛ + ⎞ − − t b P e . c P e . c ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ = − = + t b i i i i f f ⎜ ⎜ 1 1 ⎟ ⎟ ; ; f f ⎜ ⎜ 1 1 ⎟ ⎟ 2 2 ⎝ ⎠ ⎝ ⎠ A r A r c c Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Prestress + Self- Prestress + Self -weight weight − If beam self weight causes a moment M D − ⎛ ⎛ − ⎞ ⎞ t P P e e . . c c M M = ⎜ ⎜ ⎟ ⎟ − t t i i D D f f 1 1 ⎜ ⎟ 2 t ⎝ ⎠ A r S c ⎛ + ⎞ + − - b P e . c M ⎜ ⎟ = + b i D f 1 ⎜ ⎟ 2 b ⎝ ⎠ A r S c t b S S , , S S are are section section moduli moduli at top at top and and bottom bottom fibers fibers I I = = t b g g S , S c c t b Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 3

Prestressed Concrete Hashemite University C C – – Eccentric prestress + self Eccentric prestress + self- -weight + Live Load weight + Live Load w cgc g e P e P e cgs b • 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 superimposed moment M superimposed moment M s . The full intensity of such i i d d t M t M . The full intensity of such Th Th f ll i t f ll i t it it f f h h 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 P prestress have already taken place and P e should be should be considered in calculations. considered in calculations. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete C – C – Eccentric prestress + self Eccentric prestress + self- -weight + Live Load weight + Live Load M T = M = M D D + M + M SD SD +M +M L • M D = moment due to self = moment due to self- -weight. weight. • M SD SD = moment due to superimposed dead load = moment due to superimposed dead load • M L = moment due to live load including impact = moment due to live load including impact and seismic and seismic − ⎛ − ⎞ t P e . c M ⎜ ⎟ = − t e T f 1 ⎜ ⎟ 2 t ⎝ ⎝ ⎠ ⎠ A A r r S S c ⎛ + ⎞ − b P e . c M ⎜ ⎟ = + b e T f 1 ⎜ ⎟ 2 b ⎝ ⎠ A r S c Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 4

Prestressed Concrete Hashemite University Loading Stages Loading Stages (1 1) Casting: No concrete stresses ) Casting: No concrete stresses (2 2) Stressing: P ) Stressing: P i i effect effect Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete 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 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 5

Prestressed Concrete Hashemite University (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 A e’ cgc e cgs A • 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 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete • Zero external load (self Zero external load (self- -weight neglected) weight neglected) – – Hypothetical case: Hypothetical case: Lever arm = 0 Moment = M =0 cgc cgs R=0 T C • Loaded Condition (including self Loaded Condition (including self- -weight): weight): C C Lever arm = e + e’ e’ = M/P cgc cgs e T Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 6

Prestressed Concrete Hashemite University (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 e max cgs g = + + 2 e ax bx c = = = ⇒ = = − e 0 at ( x 0 ) & (x L) c 0 , b aL − 4 e = = ⇒ = max e e at x L / 2 a max 2 L Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete ( ) 4 e ( ) ∴ = − − = − max e ax L x x L x 2 L Now at any section, moment on concrete due to prestressi ng alone alone is is given given by by : : 4 e ( ) = − = − − max M C . e P x L x 2 L − dM 4 Pe ( ) = = − max V L 2 x 2 dx L − 2 2 d d M M 8 8 P Pe = − = = ↑ max w e Constant 2 2 dx L Equivalent Load Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 7

Prestressed Concrete Hashemite University • 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: 4 Pe max de 4 Pe θ θ ≈ ≈ θ θ = = max max P P sin sin P P tan tan P P L dx L 8 Pe w e = max 2 L θ ≈ P P cos P • Note that the sum of the vertical & horizontal • N t N t Note that the sum of the vertical & horizontal th t th th t th f th f th ti ti l & h l & h i i t l t l forces is zero, since the beam must be in forces is zero, since the beam must be in equilibrium under the action of prestressing equilibrium under the action of prestressing. . Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete • 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: w w e L w e L 2 2 8 8 P Pe w e = max 2 L P P wL wL 2 2 = w e w L L = − M M d due to t w w w w e e net e 2 − P M y P = ± net f P A I c g wL 2 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 8