Prestressed Concrete Hashemite University The Hashem ite University Departm ent of Civil Engineering Lecture Lecture 8 8 Deflection and Cam ber Deflection and Cam ber Dr. Hazim Dwairi Dr. Hazim Dwairi Dr Hazim Dwairi Dr Hazim Dwairi Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Introduction Introduction • Prestressed concrete beams are more slender Prestressed concrete beams are more slender than R C beams high span/depth ratios; thus than R.C. beams, high span/depth ratios; thus, than R.C. beams, high span/depth ratios; thus, than R C beams high span/depth ratios; thus more deflection. more deflection. • Camber may be important. Camber may Camber may be important. Camber may increase, with concrete creep and with time. increase, with concrete creep and with time. � Bridge camber may cause pavement to be uneven, Bridge camber may cause pavement to be uneven, even dangerous. even dangerous. even dangerous. even dangerous. � Excessive roof camber may create drainage Excessive roof camber may create drainage problems. problems. � Excessive floor camber → partition cracking and Excessive floor camber → partition cracking and other non- other non -structural cracking. structural cracking. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 1
Prestressed Concrete Hashemite University Introduction Introduction • The total deflection is a resultant of the upward The total deflection is a resultant of the upward deflection due to prestressing force and deflection due to prestressing force and deflection due to prestressing force and deflection due to prestressing force and downward deflection due to the gravity loads. downward deflection due to the gravity loads. • Only the flexural deformation is considered and Only the flexural deformation is considered and any shear deformation is neglected in the any shear deformation is neglected in the calculation of deflection. calculation of deflection. • The deflection of a member is calculated at least • The deflection of a member is calculated at least The deflection of a member is calculated at least The deflection of a member is calculated at least for two cases: for two cases: � Short term deflection at transfer Short term deflection at transfer � Long term at service loading Long term at service loading Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Introduction Introduction • The short term deflection at transfer is due to the The short term deflection at transfer is due to the initial prestressing force and self initial prestressing force and self weight without initial prestressing force and self initial prestressing force and self-weight without weight without weight without the effect of creep and shrinkage of concrete. the effect of creep and shrinkage of concrete. • The long term deflection under service loads is The long term deflection under service loads is due to the effective prestressing force and the due to the effective prestressing force and the total gravity loads. total gravity loads. • The deflection of a flexural member is calculated • The deflection of a flexural member is calculated The deflection of a flexural member is calculated The deflection of a flexural member is calculated to satisfy a limit state of serviceability. to satisfy a limit state of serviceability. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 2
Prestressed Concrete Hashemite University Deflection due to Gravity Loads Deflection due to Gravity Loads • The methods of calculation of deflection are The methods of calculation of deflection are taught in structural analysis taught in structural analysis-I course. Such taught in structural analysis I course Such taught in structural analysis I course Such I course. Such methods used are: methods used are: � Double integration method Double integration method � Moment Moment- -area method area method � Conjugate beam method Conjugate beam method � Principle of virtual work � Principle of virtual work Principle of virtual work Principle of virtual work • Students are expected to review at least one of Students are expected to review at least one of the above mentioned methods. the above mentioned methods. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Deflection due to Gravity Loads Deflection due to Gravity Loads Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 3
Prestressed Concrete Hashemite University Deflection due to Gravity Loads Deflection due to Gravity Loads Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Deflection due to Prestressing Force Deflection due to Prestressing Force • The prestressing force causes a deflection only The prestressing force causes a deflection only if the CGS is eccentric to the CGC. if the CGS is eccentric to the CGC. if the CGS is eccentric to the CGC if the CGS is eccentric to the CGC • Deflection due to prestressing force is calculated Deflection due to prestressing force is calculated by the load by the load- -balancing method. balancing method. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 4
Prestressed Concrete Hashemite University Parabola 2 5 PeL Δ = 48 EI e L 2 1 PeL Δ = 12 12 EI EI e e L/2 L/2 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University − 2 2 ( 3 4 a ) PeL Δ = e 24 EI aL (1-2a)L aL 2 1 PeL Δ = 8 8 EI EI e e L Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 5
Prestressed Concrete Hashemite University Parabola e 1 e 2 e + e e + e 1 2 L 2 2 1 Pe L 5 Pe L Δ = + 1 2 8 EI 48 EI Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Mom ent of Inertia Mom ent of Inertia • Class U: ≤ ' Class U: f 0.62 f t c � Use gross section moment of inertia, I � Use gross section moment of inertia I Use gross section moment of inertia I Use gross section moment of inertia, I g • Class T: ≤ ≤ ' ' Class T: 0.62 f f f c t c � Use effective moment of inertia, I Use effective moment of inertia, I e • Class C: > ' Class C: f f t c � Use effective moment of inertia, I Use effective moment of inertia, I e Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 6
Prestressed Concrete Hashemite University Effective Mom ent of Inertia Effective Mom ent of Inertia 3 ⎛ ⎞ ( ) M = + − ≤ ⎜ cr ⎟ I I I I I e cr g cr g ⎝ ⎝ M M ⎠ ⎠ a ⎛ ⎞ − M f f = − ⎜ cr tl r ⎟ 1 M ⎝ f ⎠ a L ≡ Max. service unfactored live load moment M a ≡ f f total service load concrete stress total service load concrete stress tl tl ≡ modulus of rupture f r ≡ service live load concrete stress f L Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Cracked Mom ent of Inertia Cracked Mom ent of Inertia The PCI Approach: ) ( ) ( ) ) ( ( = + − ρ + ρ 2 2 2 2 I n A d n A d 1 1.6 n n cr p ps p s s p p s s E = ps n p E c E = s s n n s E c Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 7
Prestressed Concrete Hashemite University Long Long- -term Deflection term Deflection Approximate Method: Due to prestress: p Δ + Δ ⎛ ⎞ Δ = −Δ − ⎜ pi pe ⎟ C Final pe u To account for the ⎝ 2 ⎠ effect of creep on self weight P Δ = Δ e pe i P i D Due to prestress & Self weight: t t & S lf i ht Δ + Δ ⎛ ⎞ Δ − Δ − + + Δ pi pe ⎜ ⎟ = ( 1 ) C C Final pe u u D ⎝ 2 ⎠ Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Long Long- -term Deflection term Deflection Due to prestress, Self weight , sustained dead load & live load dead load & live load Δ + Δ ⎛ ⎞ Δ − Δ − pi pe + + Δ Δ Δ ⎜ ⎟ = ( 1 )( + )+ C C D SD L Final pe u u ⎝ 2 ⎠ Alternatively, use long-term multipliers from PCI (Table 4.8.2) Deflection limits in ACI (Table 9.5-b) PCI design aids 11.1.3 and 11.1.4 for typical elastic deflections Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Dr. Hazim Dwairi 8
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