Plasticitetsteori for Betonkonstruktioner Mikael W Bræstrup M.Sc., Ph.D. Senior Engineer
[email protected]
Limit Analysis Theorems The Upper Bound Theorem A load for which a failure mechanism can be found that satisfies the flow rule is greater than or equal to the yield load. The Lower Bound Theorem A load for which a statically admissible stress distribution can be found that 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.
Slide 2
Limit Analysis: Gvozdev 1936 Gvozdev, A.A, Opredelenie velichiny razrushayushchei nagruzki dlya statischeski neopredelimykh sistem, preterpevayushchikh plasticheskie deformatsii, deformatsii, Svornik trudov konferentsii po plasticheskim deformatsiyam 1936, Akademia Nauk SSSR, Moscow-Leningrad, 1938, pp 19-30 English translation: The Determination of the Value of the Collapse Load for Statically Indeterminate Systems Undergoing Plastic Deformation, Deformation, International International Journal of Mechanical Sciences, Vol 1, 1960, pp 322-333
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Limit Analysis: The Prager School 1948 Hill, R., The Mathematical Theory of Plasticity, Clarendon, Oxford, 1950, 356 pp Hodge, P.G, & Prager W., A Variational Principle for Plastic Materials with Strainhard Strainhardening, ening, Journal of Mathematics and Physics, Vol 27, No 1, 1948, pp 1-10 Drucker, D.C., Some Implications of Work Hardening and Ideal Plasticity, Quarterly of Applied Mathematics, Vol 7, 1950, pp 411-418 Drucker, D.C., Prager, W. & Greenberg, H.J., Extended Limit Analysis Theorems for Continuous Media, Quarterly of Applied Mathematics, Vol 9, 1952, pp 381-389 Slide 4
Yield Line Theory: Gvozdev 1939 Gvozdev, A.A, Obosnovanie § 33 norm proektirovaniya zhelezobetonnykh konstruktsii (Comments to § 33 of the design standard for reinforced concrete structures), Stroitelnaya Promyshlenmost, Vol 17, No 3, 1939, pp 51-58
Slide 5
Yield Line Theory: Johansen 1931 -
Johansen, K.W., Beregning af krydsarmerede jernbetonpladers brudmoment, Bygningsstatiske Meddelelser, Vol 3, No 1, 1931, pp 1-18 Slide 6
Yield Line Theory: Johansen 1931 Ingerslev, A., Om en elementær beregningsmetode af krydsarmerede plader, Ingeniøren, Vol 30, No 69, 1921, pp 507-515. (See also: The Strength of Rectangular Slabs, The Structural Engineer, Journal IStructE, Vol 1, No 1, 1923, pp 3-14)
Johansen, K.W., Beregning af krydsarmerede jernbetonpladers brudmoment, Bygningsstatiske Meddelelser, Vol 3, No 1, 1931, pp 1-18 Slide 7
Yield Line Theory: Johansen 1931 Johansen, K.W., Beregning af krydsarmerede jernbetonpladers brudmoment Bygningsstatiske Meddelelser, Vol 3, No 1, 1931, pp 1-18 Johansen, K.W., Bruchmomente der Kreuzweise bewehrten Platten, Memoirs, International Association for Bridge and Structural Enginering (IABSE), Vol 1, 1932, pp 277-296 Johansen, K.W., Brudlinieteorier Gjellerup, 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 Slide 8
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 Applied Mechanics 1952, Vol II, 1955, pp 65-72 Nielsen, M.P., Limit Analysis of Reinforced Concrete Slabs , Acta Polytechnica Scandinavica, Civil Engineering and Building Construction Series, No 26, 1964, 167 pp Slide 9
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 limit analysis, which must develop its own idealised criteria of yield. Until yield-line theory and limit analysis employ the same criterion of yield, they must go their own separate ways’
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Concrete Plasticity: Slabs Yield condition Orthotropic slabs
Nielsen, M.P., Limit Analysis of Reinforced Concrete Slabs , Acta Polytechnica Scandinavica, Civil Engineering and Slide 11
Concrete Plasticity: Slabs Bi-conical yield surface, arbitrary reinforcement − ( M x − M )( M y − M ) + ( M xy − M )2 ≤ 0 Fx Fy Fxy )2 ≤ 0 − ( M x + M ' )( M y + M ' ) + ( M xy + M ' Fx Fy Fxy
θ n > 0 θ n < 0
(MFx, MFy, MFxy) (-M’ Fx, -M’ Fy, -M’ Fxy) Slide 12
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
Bi-conical yield surface, arbitrary reinforcement )2 ≤ 0 − ( N x − N )( N y − N ) + ( N xy − N Fx Fy Fxy
− ( N x + hf c )( N y + hf c ) + N xy2 ≤ 0 Slide 13
ε n > 0 ε n < 0
Concrete Plasticity: Shells Moment – Axial Force Interaction
Generalised yield line
Linearised Interaction Curve
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Concrete Plasticity: Beam Shear (w/ Stirrups)
Failure Mechanisms
Rotation
Translation Slide 15
Coulomb Failure Criterion
= c - σ tanφ
Coulomb, C.A., Essai sur une application des régles de maximis & minimis á quelques problèmes statique, relatifs a l’architecture, Mémoires de Mathématique & de Physique présentés a l’Académie Royale des Sciences, 7, 1773, pp 343382. (English translation:Note on an Application of the Rules of Maximum and Minimum to some Statical Problems, Relevant to Architecture, In Heyman, J.,Coulomb’s Memoir on Statics: An Essay in the History of Civil Engineering, Cambridge University Slide 16 Press, 1972, 212 pp.)
Modified Coulomb Failure Criterion
f c = 2c√ k k = (1 + sinφ)/(1 - sinφ)
Coulomb Friction Rankine Separation
τ = c - σ tanφ
σ = f t
Slide 17
Concrete Yield Surface tanφ = 0.75 f t ≈ 0 f c = ν f cyl
f c
Plane Stress, f t = 0: Square Yield Locus Slide 18
Shear Crack (Yield Line) Stresses: σn = -½ f c (1 – sinα) τnt = ½ f c cos α
σ2 = - f c Dissipation:
α
Dc = ½ f c (1 – sin α) v n
τnt
v
σn σ1 = 0
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Beams with Shear Reinforcement
Upper Bound Solution: V = rf y bhcot β + ½ f c (1 – cos β) bh/ sin β Optimal yield line inclination: cot β = (½ f c - rf y)/ [rf y(f c – rf y)]1/2 Slide 20
≥ 0
Beams with Shear Reinforcement Plasticity Solution (Web Crushing Criterion)
θ
cot β = (½ f c - rf y)/ [rf y(f c – rf y)]1/2 V = bh [rf y (f c – rf y)]1/2 for V = ½ bhf c for
f c θ = β /2
≥ 0
rfy ≤ ½ f c rf y ≥ ½ f c Slide 21
Beams with Shear Reinforcement V/bh
Plasticity Solution (Web Crushing Criterion)
½ f c
β
V = bh[rf y(f c – rf y)]1/2 V = ½bhf c
for rf y ≤ ½ f c for rf y ≥ ½ f c rf y
½ f c
θ θ = β /2
cot β = (½ f c - rf y)/ [rf y (f c – rf y)]1/2 ≥ 0 Slide 22
Beams with Shear Reinforcement V/bhf cyl
ρ= 2.8%
f c = 0.86 f cyl f cyl = 0.8 f cube
rf y /f cyl
Leonhardt, F., and Walther, R., Schubversuche an Plattenbalken mit unterschiedlicher Schubbewehrung, Deutscher Ausschuss für Stahlbeton, Heft 156, 1963, 84 pp Slide 23
Beams with Shear Reinforcement
V/bhf cyl
ρ = 6.0%
f c = 0.74 f cyl
rf y /f cyl Slide 24
Beams with Shear Reinforcement
f c
Failure Mechanism
Slide 25
Beams without Shear Reinforcement
Failure Mechanism
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Beams without Shear Reinforcement
Upper Bound Solution V = - Ty cos(α + β) + ½ f c (1 – sinα) bh/ sin β Slide 27
Beams without Shear Reinforcement Plasticity Solution
cotβ = a/h V = ½([(baf c)2+4Ty(bhf c-Ty )]1/2 - bafc )
for Ty ≤ ½bhf c
V = ½bf c([a2+h2]1/2 - a)
for Ty ≥ ½bhf c Slide 28
Beams without Shear Reinforcement V/bhf cyl
Shear Failure Flexural Failure
Φ=Ty /bhf cyl ν = f /f c cyl
V/bhf cy l
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Beams without Shear Reinforcement V/bhf cyl
V/bhf cyl
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Beams without Shear Reinforcement
Stress Distribution f c
Shear failure Flexural failure
f c
f c
V = ½([(baf c)2+4Ty(bhf c-Ty )]1/2 - bafc ) for V = ½bf c([a2+h2]1/2 - a)
Ty ≤½bhf c
for Ty ≥½bhf c Slide 31
Beams without Shear Reinforcement Hyperbolic yield line
Jensen, J.F., Discussion of ’An Upper Bound RigidPlastic Solution for the Shear Failure of Concrete Beams without Shear Reinforcement’ by K.O. Kemp & M.T. Safi, Magazine of Concrete Research, Vol 34, No 119, June 1982, pp 96-104 Slide 32
Beams without Shear Reinforcement Hyperbolic yield line Bottom steel only Reinforcement not yielding f c f c
f c
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Beams without Shear Reinforcement
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Shear in Construction Joints Failure in joint: Plane strain Failure outside joint: Plane stress
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Shear in Construction Joints Jensen, B.C., Some Applications of Plastic Analysis to Plain and Reinforced Concrete, Institute of Building Design, Report No 123, 1977, 129 pp Hofbeck, J.A. & al, Shear Transfer in Reinforced Concrete, ACI Journal, Vol 66, No 2, Feb 1969, pp 119-128
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Punching Shear in Slabs
Axisymmetric failure: Plane strain
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Punching Shear in Slabs Optimal failure surface generatrix: Catenary
f t = f /400 c
Hess, U., Udtrækning af Indstøbte Inserts, DIA-B, Rapport No 75:54 1975, 25 pp
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Punching Shear in Slabs
Failure load prediction
Taylor, R. & Hayes, B., Some Tests on the Effect of Edge Restraint on Punching Shear in Reinforced Concrete Slabs, 39 Magazine of Concrete Research, Vol 17, NoSlide 50, pp 39-44
Punching Shear in Slabs Failure load prediction f t = 0
Code approach
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Concrete Plasticity: Overview •Beams and Frames •Slabs •Walls •Shells •Beam Shear (w/ & w/o stirrups) •Joints
•Corbels •Torsion •Punching Shear •Dome Effect •Anchorage •Concentrated Load
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 and Design, Handbook of Structural Concrete (ed F.K. Kong & al), Pitman, London 1983, Ch 20, 54 pp Slide 41