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Principle of prestressingPrinciple of prestressingp p gp p g

BasicsBasicsSecondary MomentSecondary Momentyy

Force Diagram Force Diagram -- ExtensionsExtensions

By Max MEYER, VSLBy Max MEYER, VSL--TCAATCAA

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BasicsBasicsBasicsBasics


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••static systemstatic system••external loading:selfweight gexternal loading:selfweight gg g gg g g••prestressing: profile and prestressing: profile and forceforce

••crosscross--sectionsection••external static forcesexternal static forces••on concrete acting forceson concrete acting forceson concrete acting forceson concrete acting forcesdue to prestressingdue to prestressing••fibre stressesfibre stresses


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Together Load case prestressingLoad case prestressingLoad case prestressingLoad case prestressing•• Prestressing creates an internal stress state, Prestressing creates an internal stress state,

which can, if suitably arranged, effectively which can, if suitably arranged, effectively compensate fully or partially for stresses compensate fully or partially for stresses induced into structure by external loading.induced into structure by external loading.

•• Prestressing limits deformation by actively Prestressing limits deformation by actively counterbalancing part of the external load and counterbalancing part of the external load and by limiting cracking of concreteby limiting cracking of concrete

•• Prestressing allows use of high strength Prestressing allows use of high strength tensile steel with 4x higher strength than tensile steel with 4x higher strength than


passive reinforcementpassive reinforcement

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How to estimate required How to estimate required prestressing for a simple beamprestressing for a simple beam

••External moment MExternal moment MP t i b li filP t i b li fil••Prestressing: parabolic profile, Prestressing: parabolic profile, eccentricity e at midspaneccentricity e at midspan••Rectangular cross section: b x hRectangular cross section: b x h


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••Cross sectional area: A = b x hCross sectional area: A = b x h

••Section modulus: S = (b x hSection modulus: S = (b x h22)/6)/6••Section modulus: S = (b x hSection modulus: S = (b x h22)/6)/6

••Core value: k = S / A = h/6Core value: k = S / A = h/6

••Permissable tensile stresses = zero Permissable tensile stresses = zero


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••Basic formula for flexural bottom fibre stress atBasic formula for flexural bottom fibre stress atmidspan:midspan:pp

fbot = M / Sbot fbot = M / Sbot -- P x (1 / A + e / Sbot)P x (1 / A + e / Sbot)

••Required prestressing for zero tensile stress atRequired prestressing for zero tensile stress atmidspan:midspan:

P = M / Sbot x { A x Sbot / (Sbot+A x e) } =P = M / Sbot x { A x Sbot / (Sbot+A x e) } == M x { 1 / (Sbot / A + e) } == M x { 1 / (Sbot / A + e) } ={ ( ) }{ ( ) }= M x { 1 / (k + e) } = M x { 1 / (k + e) }


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Example:Example:

b 5b 5b = .5mb = .5mh = 1.0mh = 1.0me = .15m e = .15m l = 20.0m l = 20.0m --> Mg = {.5 x 1.0 x 25 x 20> Mg = {.5 x 1.0 x 25 x 2022} / 8 = 625 kNm} / 8 = 625 kNmk = h / 6 = .167mk = h / 6 = .167m

P = 625 x 1 / {.167 + .15} = 1972 kN P = 625 x 1 / {.167 + .15} = 1972 kN

b f t d ith P 115 kN 5” t db f t d ith P 115 kN 5” t dnumber of strands: with Pav=115 kN per .5” strandnumber of strands: with Pav=115 kN per .5” strand--> 1972 kN / 115 kN = 17.1 strands> 1972 kN / 115 kN = 17.1 strands


P / A = 1972 / {1.0 x .5} = 3.94 MPa P / A = 1972 / {1.0 x .5} = 3.94 MPa

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PT layout IPT layout I

PT layout IIPT layout II

PT layout IIIPT layout IIIyy


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Which PT layout gives biggest upwards deflection?Which PT layout gives biggest upwards deflection?


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Uplift of profile III > uplift of profile I > uplift of profile IIUplift of profile III > uplift of profile I > uplift of profile II


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Why do we need high strength steel for prestressing ?Why do we need high strength steel for prestressing ?Why do we have to stress this high strength steel?Why do we have to stress this high strength steel?Why do we have to stress this high strength steel? Why do we have to stress this high strength steel?

••Prismatic concretePrismatic concreteb 40 lb 40 lbeam, 40 m long, beam, 40 m long, centrically prestressedcentrically prestressedwith prestressing force Pwith prestressing force Pp gp g••P/A= 6 MpaP/A= 6 Mpa••Modulus of elasticity: Ec=Modulus of elasticity: Ec=30 x 10E3 Mpa30 x 10E3 Mpa30 x 10E3 Mpa30 x 10E3 Mpa••Creep value: 2Creep value: 2••Shrinkage value: 150x10EShrinkage value: 150x10E--66


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Creep and shrinkage of concrete:Creep and shrinkage of concrete:

Delta Epsilon=6/30x10EDelta Epsilon=6/30x10E--3x2 + 150x10E3x2 + 150x10E--6 = 550x10E6 = 550x10E--6 6


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Case I: Use normal reinforcement with fy = 460 Mpa Case I: Use normal reinforcement with fy = 460 Mpa and Es = 195 x 10E3 Mpa stressed to 75% x fy=and Es = 195 x 10E3 Mpa stressed to 75% x fy=and Es 195 x 10E3 Mpa stressed to 75% x fyand Es 195 x 10E3 Mpa stressed to 75% x fy345 Mpa 345 Mpa -->>Force loss in % of initial force:Force loss in % of initial force:{550 195 10E{550 195 10E 3}/345 31 1%3}/345 31 1%{550 x 195 x 10E{550 x 195 x 10E--3}/345 = 31.1% 3}/345 = 31.1%

Case II: Use prestressing strands with fpu = 1860 MpaCase II: Use prestressing strands with fpu = 1860 Mpaand Ep = 195 x 10E3 Mpa stressed to 75% x fpu=and Ep = 195 x 10E3 Mpa stressed to 75% x fpu=1395 Mpa 1395 Mpa -->>ppForce loss in % of initial force:Force loss in % of initial force:{550 x 195 x 10E{550 x 195 x 10E--3}/1395 = 7.7%3}/1395 = 7.7%


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StressStress--strain diagramstrain diagramggfor deformed rebars and for deformed rebars and prestressing steelprestressing steel

Fle ral strains o er crossFle ral strains o er crossFlexural strains over crossFlexural strains over crosssection for beam reinforcedsection for beam reinforcedwith passive reinforcement,with passive reinforcement,non stressed prestressing non stressed prestressing steel and stressed prestressingsteel and stressed prestressingsteelsteel


steelsteel

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ConclusionsConclusions

UseUse ofof highhigh strengthstrength steelsteel isis aa mustmust inin orderorder toto•• UseUse ofof highhigh strengthstrength steelsteel isis aa mustmust inin orderorder totominimizeminimize forceforce lossloss duedue toto longlong termtermshorteningshortening ofof concreteconcreteshorteningshortening ofof concreteconcrete

•• IfIf highhigh strengthstrength steelsteel isis used,used, itit mustmust bebestressedstressed inin orderorder toto fullyfully utilizeutilize itsits ultimateultimatestressedstressed inin orderorder toto fullyfully utilizeutilize itsits ultimateultimatestrengthstrength


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Secondary momentSecondary momentSecondary momentSecondary moment


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External load (selfweight)External load (selfweight)

Prestressing layoutPrestressing layout

Loadcase prestressing modelled with externallyLoadcase prestressing modelled with externallyapplied anchor and deviation forcesapplied anchor and deviation forces

Deformation of individual spansDeformation of individual spans

Secondary moment due to prestressing momentSecondary moment due to prestressing moment


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Force diagram Force diagram -- extensionsextensionsgg


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Together Prestressing lossesPrestressing lossesPrestressing lossesPrestressing losses

•• Friction lossesFriction losses•• relaxation of PT steelrelaxation of PT steelrelaxation of PT steelrelaxation of PT steel•• elastic shortening, if more than 1 cable is elastic shortening, if more than 1 cable is

stressedstressedstressedstressed•• creep creep

shrinkageshrinkage•• shrinkageshrinkage


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Together Friction lossesFriction lossesFriction lossesFriction losses

Px = Po x 1/eE{Px = Po x 1/eE{ xx + k x l}+ k x l}


Px Po x 1/eE{Px Po x 1/eE{ x x k x l} k x l}

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Friction Friction parametersparameterspp


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Calculation ofCalculation ofCalculation of Calculation of summation alphasummation alpha


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Calculation of Calculation of summation alphasummation alphasummation alphasummation alphafor parabolic profilefor parabolic profile


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Wedge Wedge -- draw draw -- inin


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E t iE t iExtensionsExtensions


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Basic formula:Basic formula:ƒP(x) dxƒP(x) dx

l = l = ƒP(x) dxƒP(x) dx

Ep x ApEp x Ap

••Indirectly proportional to modulus of elasticity and areaIndirectly proportional to modulus of elasticity and areaof steel, which are both not guaranteed physical propertiesof steel, which are both not guaranteed physical properties••extension measurement can only serve as rough guide to extension measurement can only serve as rough guide to evaluate if a cable has been adequately stressed or notevaluate if a cable has been adequately stressed or notmeasured extensions may vary from theoretical extensionsmeasured extensions may vary from theoretical extensions••measured extensions may vary from theoretical extensionsmeasured extensions may vary from theoretical extensionsby +/by +/-- 11 %11 %••excessive measured extension has a theoretical limit excessive measured extension has a theoretical limit corresponding to zero force diagramcorresponding to zero force diagram


corresponding to zero force diagramcorresponding to zero force diagram

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A h f d i fA h f d i fAnchorage force = design force xAnchorage force = design force x(anchorage loss (anchorage loss correction factor)correction factor)

Applied pressure at jack inlet=Applied pressure at jack inlet=pp p jpp p janchorage force x (jack loss factor) /anchorage force x (jack loss factor) /piston areapiston area

Note: design force Po = jacking force as stated in BS 5400, cl. 6.7.1Note: design force Po = jacking force as stated in BS 5400, cl. 6.7.1


g j g ,g j g ,

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Assumption:Assumption:

Length lLength l

••stressing force correctstressing force correct••linear force diagram (actual andlinear force diagram (actual andtheoretical)theoretical)••total net extension is 25 % lowertotal net extension is 25 % lowerthan theoretical extensionthan theoretical extensioni fl l th f D lt < l/2i fl l th f D lt < l/2••influence length of Delta wc < l/2influence length of Delta wc < l/2

Question:Question:••By how much is tendon force at midspan lower than designBy how much is tendon force at midspan lower than designf i f d t i ?f i f d t i ?force in case of one end stressing?force in case of one end stressing?••By how much is tendon force at mispan lower than designBy how much is tendon force at mispan lower than designforce in case of two end stressing?force in case of two end stressing?


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50%50%25%25%

FORCEFORCE11--end stressingend stressing

LENGTHLENGTH

50%50%FORCEFORCE

22--end stressingend stressing


LENGTHLENGTH

principle of pt

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