Prestressed Concrete Hashemite University The Hashem ite University Departm ent of Civil Engineering Lecture 6 Lecture 6 – – Flexural Design Flexural Design 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 Prestressed Concrete Concrete “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 – Initial Prestress Initial Prestress I i i l P I i i l P – 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 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) Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 1
Prestressed Concrete Hashemite University • 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 Material is elastic Material is elastic – Effect of transformed section is neglected Effect of transformed section is neglected – Section is Equal 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 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Prestressed Concrete Concrete Flexural Stress Distribution Flexural Stress Distribution Throughout Load History Throughout Load History β 3 f’ c C T C C T w C C C or zero T (f) (a) (b) (c) (e) (d) (a) Beam section (a) Beam section (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. + P (d) Full D.L. + P eff eff (e) Full service load + (e) Full service load + P P eff eff (f) Ultimate limit state for under (f) Ultimate limit state for under- -reinforced beam reinforced beam Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 2
Prestressed Concrete Hashemite University Maxim um Fiber Stresses Maxim um Fiber Stresses f f ti c M + M M D SD L t Δ S t f t S P i Stresses 1 i P i + M D Stresses 2 c t 3 P e + M D Stresses cgc P e + M D + M SD + M L Stresses 4 c b 4 3 2 1 Δ M + b f M M D f f SD L b S b S t ci Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Load Deflection Curve Load Deflection Curve Load Ultimate Steel yielding Service load limit overload First cracking Decompression f cr cgs (f=0) Balanced Full dead load Service load limit Δ Δ o Δ D Δ L Δ Pe Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 3
Prestressed Concrete Hashemite University Selection of Geom etric Properties Selection of Geom etric Properties S t & S b that • Select the min. section Select the min. section moduli moduli S & S that satisfy stress limits at stage of loadings: satisfy stress limits at stage of loadings: ti f ti f t t li li it it t t t t f l f l di di = ' = f 0 . 25 f OR ' f 0 . 45 f OR ti ci c c = ' = ' f 0 . 5 f for SS at support f 0 . 6 f ti ci c c = ' f 0 . 5 f OR t c = ' = ' f f if f 0 . 6 f t c ci ci Long - term deflection is met (a) At Transfer (B) At Service Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Selection of Geom etric Properties Selection of Geom etric Properties • Stresses at transfer: Stresses at transfer: − ⎛ − ⎞ t P ec M ⎜ ⎟ = − ≤ t i D f f ⎜ ⎜ 1 ⎟ ⎟ f f .......... ( ) (1) ti ti 2 2 t t ⎝ ⎠ A r S c − ⎛ + ⎞ b P ec M ⎜ ⎟ = − ≤ b i D f 1 f .......... (2) ⎜ ⎟ ci 2 b ⎝ ⎠ A r S c ≡ P initial prestressi ng force i • Effective stresses after losses: Effective stresses after losses: − ⎛ ⎛ − ⎞ ⎞ t P P ec ec M M ⎜ ⎜ ⎟ ⎟ = − ≤ t e D f 1 f ⎜ ⎟ t 2 t A ⎝ r ⎠ S c ⎛ + ⎞ − b P ec M = ⎜ ⎟ − ≤ b e D f 1 f ⎜ ⎟ c 2 b ⎝ ⎠ A r S c ≡ P effective prestressi ng force after losses i Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Prestressed Concrete Concrete Dr. Hazim Dwairi 4
Prestressed Concrete Hashemite University Selection of Geom etric Properties Selection of Geom etric Properties • Service load final stresses: Service load final stresses: ⎛ ⎞ − t P ec M = ⎜ ⎜ − ⎟ ⎟ − ≤ ≤ t e e T T f f 1 1 f f .......... (3) (3) ⎜ ⎜ ⎟ ⎟ 2 c t A ⎝ r ⎠ S c ⎛ + ⎞ − b P ec M ⎜ ⎟ = − ≤ b e T 1 .......... (4) f ⎜ ⎟ f t 2 b ⎝ ⎠ A r S c • Where: Where: � M T � M = M T = M = M D + M = M + M + M SD + M SD + M + M + M + M L � P P i i = initial = initial prestress prestress � P e = effective = effective prestress prestress after losses after losses Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Prestressed Concrete Concrete Selection of Geom etric Properties Selection of Geom etric Properties • Decompression stage is when the stress Decompression stage is when the stress at the cgs at the at the cgs at the cgs is equal to zero. The change in cgs is equal to zero. The change in 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 P e = ⎜ ⎟ e f 1 ⎜ ⎟ decomp 2 ⎝ ⎠ A r c • For variable tendon eccentricity: For variable tendon eccentricity: For variable tendon eccentricity: For variable tendon eccentricity: P e = γ P i assume the effective prestress assume the effective prestress P i - γ ) P i.e. loss of i.e. loss of prestress prestress = P = P i i – P P e = ( = (1 1- ) P i i Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Dr. Hazim Dwairi 5
Prestressed Concrete Hashemite University Selection of Geom etric Properties Selection of Geom etric Properties − ⎛ − ⎞ t P ec M ⎜ ⎟ = + i D from Eq(1) 1 f .......... (5) ⎜ ⎟ ti 2 t ⎝ ⎠ A r S c ⎛ − ⎞ − t P ec M ⎜ ⎟ = e T from Eq(3) 1 - f ⎜ ⎟ c 2 t ⎝ ⎠ A r S c − γ ⎛ − ⎞ + + t P ec M M M ⎜ ⎟ = i D SD L 1 - f ⎜ ⎟ c 2 t A ⎝ r ⎠ S c using Eq(5) : + + ⎛ ⎛ ⎞ ⎞ M M M M γ + = ⎜ ⎟ D D SD L f - f ti c ⎝ t ⎠ t S S − γ + + ( 1 ) M M M γ − = D SD L f f ti c t S Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Concrete Prestressed Concrete Selection of Geom etric Properties Selection of Geom etric Properties − γ + + ( 1 ) M M M ∴ ≥ t D SD L S γ − f f ti c similarly : − γ + + ( 1 ) M M M ∴ ≥ b D SD L S − γ f f t ci Furthermor e : γ γ − t b c c S S f f f f = = ti ti c − γ b t c S f f t ci + t b b t t b c c c S c S = = = 1 ; ; + + t b t b h h S S h S S Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Prestressed Prestressed Concrete Concrete Dr. Hazim Dwairi 6
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