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Slide 2

Limit Analysis Theorems

The Upper Bound Theorem

A load for which a failure mechanism can be found that satisfies the flowrule is greater than or equal to the yield load.

The Lower Bound Theorem

A load for which a statically admissible stress distribution can be foundthat satisfies the yield condition is less than or equal to the yield load.

The Uniqueness Theorem

The lowest upper bound and the highest lower bound coincide, and

constitute the complete solution for the yield load.

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Slide 3

Limit Analysis: Gvozdev 1936

Gvozdev, A.A, Opredelenie velichinyrazrushayushchei nagruzki dlya statischeskineopredelimykh sistem, preterpevayushchikhplasticheskie deformatsii,Svornik trudov konferentsii po plasticheskimdeformatsiyam 1936, Akademia Nauk SSSR,Moscow-Leningrad, 1938, pp 19-30

English translation:The Determination of the Value of theCollapse Load for Statically IndeterminateSystems Undergoing Plastic Deformation,International Journal of MechanicalSciences, Vol 1, 1960, pp 322-333

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Slide 5

Yield Line Theory: Gvozdev 1939

Gvozdev, A.A, Obosnovanie 33

norm proektirovaniyazhelezobetonnykh konstruktsii(Comments to 33 of the designstandard for reinforced concrete

structures),Stroitelnaya Promyshlenmost,Vol 17, No 3, 1939, pp 51-58

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Slide 6

Yield Line Theory: Johansen 1931 -

Johansen, K.W., Beregning afkrydsarmerede jernbetonpladersbrudmoment,Bygningsstatiske Meddelelser, Vol 3, No 1,

1931, pp 1-18

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Slide 7

Yield Line Theory: Johansen 1931 -

Ingerslev, A., Om en elementrberegningsmetode af

krydsarmerede plader,Ingeniren, Vol 30, No 69, 1921,pp 507-515. (See also:The Strength of Rectangular Slabs,

The Structural Engineer, JournalIStructE, Vol 1, No 1, 1923,pp 3-14)

Johansen, K.W., Beregning af krydsarmerede jernbetonpladersbrudmoment,Bygningsstatiske Meddelelser, Vol 3, No 1, 1931, pp 1-18

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Slide 8

Yield Line Theory: Johansen 1931 -

Johansen, K.W., Beregning af krydsarmerede jernbetonpladersbrudmomentBygningsstatiske Meddelelser, Vol 3, No 1, 1931, pp 1-18

Johansen, K.W., Bruchmomente der Kreuzweise bewehrtenPlatten,Memoirs, International Association for Bridge and Structural

Enginering (IABSE), Vol 1, 1932, pp 277-296Johansen, K.W., BrudlinieteorierGjellerup, Copenhagen, 1943, 189 pp

Johansen, K.W., Yield-Line Theory,Cement and Concrete Association, London, 1962

Johansen, K.W., Yield-Line Formulae for Slabs,Cement and Concrete Association, London, 1972

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Slide 9

Yield Line Theory vs Limit Analysis

Johansen, K.W., Yield-Line Theory,Cement and Concrete Association, London, 1962

Recent Developments in Yield-Line Theory,MCR Special Publication, Cement and Concrete Association,London, 1965 (Jones, Kemp, Morley, Nielsen, Wood)

Prager, W., The General Theory of Limit Design,Proc 8th International Congress of Theoretical and AppliedMechanics 1952, Vol II, 1955, pp 65-72

Nielsen, M.P., Limit Analysis of Reinforced Concrete Slabs,Acta Polytechnica Scandinavica, Civil Engineering and BuildingConstruction Series, No 26, 1964, 167 pp

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Slide 10

Yield Line Theory vs Limit Analysis

Recent Developments in Yield-Line Theory,MCR Special Publication, Cement and

Concrete Association, London, 1965

such a criterion is useless within

the strict framework of limitanalysis, which must develop itsown idealised criteria of yield.Until yield-line theory and limit

analysis employ the samecriterion of yield, they must gotheir own separate ways

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Slide 11

Concrete Plasticity: Slabs

Nielsen, M.P., Limit Analysis of Reinforced Concrete Slabs,Acta Polytechnica Scandinavica, Civil Engineering andBuilding Construction Series, No 26, 1964, 167 pp

Yield conditionOrthotropic slabs

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Slide 12

Concrete Plasticity: Slabs

02)())(( + FxyMxyMFyMyMFxMxM 0>n

0

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Slide 13

Concrete Plasticity: Walls (Discs, Disks)

Nielsen, M.P., On the Strength of Reinforced Concrete Discs,Acta Polytechnica Scandinavica, Civil Engineering and Building

Construction Series, No 70, 1971, 261 pp

02)())(( +Fxy

NxyNFyNyNFx

NxN

02))(( +++ xyNchfyNchfxN

0>n

0

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Slide 14

Concrete Plasticity: Shells

Moment Axial Force Interaction

Linearised Interaction Curve

Generalised yield line

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Slide 15

Concrete Plasticity: Beam Shear (w/ Stirrups)

Failure Mechanisms

Rotation Translation

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Slide 16

Coulomb Failure Criterion

= c -tan

Coulomb, C.A., Essai sur une application des rgles de maximis& minimis quelques problmes statique, relatifs alarchitecture, Mmoires de Mathmatique & de Physiqueprsents a lAcadmie Royale des Sciences, 7, 1773, pp 343-382. (English translation:Note on an Application of the Rules of

Maximum and Minimum to some Statical Problems, Relevant toArchitecture, In Heyman, J.,Coulombs Memoir on Statics: AnEssay in the History of Civil Engineering, Cambridge UniversityPress, 1972, 212 pp.)

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Slide 17

Modified Coulomb Failure Criterion

fc = 2ckk = (1 + sin)/(1 - sin)

Coulomb Friction = c -tanRankine Separation = ft

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Slide 18

Concrete Yield Surface

tan= 0.75ft 0fc = fcyl

fc

Plane Stress, ft = 0:

Square Yield Locus

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Slide 20

Beams with Shear Reinforcement

Upper Bound Solution:

V = rfy bhcot + fc (1 cos ) bh/ sin

Optimal yield line inclination:

cot = ( fc - rfy)/ [rfy(fc rfy)]1/2 0

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Slide 21

Beams with Shear Reinforcement

cot = ( fc - rfy)/ [rfy(fc rfy)]1/2 0

V = bh [rfy (fc rfy)]1/2 for rf y fc

V = bhfc

for rfy f

c

Plasticity Solution(Web Crushing Criterion)

fc = /2

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Slide 22

Beams with Shear Reinforcement

cot = ( fc - rfy)/ [rfy (fc rfy)]1/2

0

V/bh

rfy

fc

fc

= /2

V = bh[rfy(fc rfy)]1/2 for rfy fc

V = bhfc for rfy fc

Plasticity Solution(Web Crushing Criterion)

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Slide 23

Beams with Shear Reinforcement

Leonhardt, F., and Walther, R.,Schubversuche an Plattenbalkenmit unterschiedlicherSchubbewehrung, DeutscherAusschuss fr Stahlbeton,

Heft 156, 1963, 84 pp

= 2.8%

fc = 0.86 fcyl

fcyl = 0.8 fcube

V/bhfcyl

rfy

/fcyl

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Slide 24

Beams with Shear Reinforcement

= 6.0%fc = 0.74 fcyl

V/bhfcyl

rfy/fcyl

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Slide 25

Beams with Shear Reinforcement

Failure Mechanism

fc

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Slide 26

Beams without Shear Reinforcement

Failure Mechanism

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Slide 27

Beams without Shear Reinforcement

Upper Bound Solution

V = - Ty cos(+ ) + fc (1 sin) bh/ sin

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Slide 28

Beams without Shear Reinforcement

V = ([(bafc)2+4Ty(bhfc-Ty )]

1/2 - bafc) for Ty bhfc

V = bfc([a2

+h2

]1/2

- a) for Ty bhfc

cot = a/h

Plasticity Solution

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Slide 29

Beams without Shear Reinforcement

V/bhfcyl Shear FailureFlexural Failure

=Ty/bhfcyl = fc/fcyl

V/bhfcyl

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Slide 30

Beams without Shear Reinforcement

V/bhfcyl V/bhfcyl

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Slide 31

Beams without Shear Reinforcement

V = ([(bafc)2+4Ty(bhfc-Ty )]

1/2 - bafc) for Ty bhfc

V = bfc([a2+h2]1/2 - a) for Ty bhfc

Shear failureFlexural failure

fc

fc

fc

Stress Distribution

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Slide 32

Beams without Shear Reinforcement

Hyperbolic yield line

Jensen, J.F., Discussionof An Upper Bound Rigid-

Plastic Solution for theShear Failure of ConcreteBeams without ShearReinforcement by K.O.

Kemp & M.T. Safi,Magazine of ConcreteResearch, Vol 34,No 119, June 1982,

pp 96-104

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Slide 33

Beams without Shear Reinforcement

Hyperbolic yield lineBottom steel only

Reinforcement not yielding

fc

fc

fc

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Slide 34

Beams without Shear Reinforcement

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Slide 35

Shear in Construction Joints

Failure in joint: Plane strainFailure outside joint: Plane stress

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Slide 36

Shear in Construction Joints

Jensen, B.C., SomeApplications of PlasticAnalysis to Plain and

Reinforced Concrete,Institute of BuildingDesign, Report No 123,1977, 129 pp

Hofbeck, J.A. & al, ShearTransfer in ReinforcedConcrete,ACI Journal, Vol 66, No 2,Feb 1969, pp 119-128

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Slide 37

Punching Shear in Slabs

Axisymmetric failure: Plane strain

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Slide 38

Punching Shear in Slabs

Optimal failure surface generatrix: Catenary

Hess, U., Udtrkningaf Indstbte Inserts,DIA-B, Rapport No 75:541975, 25 pp

ft = fc/400

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Slide 39

Punching Shear in Slabs

Failure load prediction

Taylor, R. & Hayes, B., Some Tests on the Effect of EdgeRestraint on Punching Shear in Reinforced Concrete Slabs,Magazine of Concrete Research, Vol 17, No 50, pp 39-44

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Slide 40

Punching Shear in Slabs

Failure load prediction

Code approach

ft

= 0

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Slide 41

Concrete Plasticity: Overview

Nielsen, M.P., Limit Analysis and Concrete Plasticity,

2nd ed, CRC Press, Boca Raton, Florida, 1998

Braestrup, M.W. & Nielsen, M.P., Plastic Methods of Analysis andDesign,Handbook of Structural Concrete (ed F.K. Kong & al), Pitman,

London 1983, Ch 20, 54 pp

Beams and FramesSlabsWallsShellsBeam Shear (w/ & w/o stirrups)Joints

CorbelsTorsion

Punching ShearDome EffectAnchorageConcentrated Load

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