Plasticitetsteori for
Betonkonstruktioner
Mikael W Bræstrup M.Sc., Ph.D.
Senior Engineer
mwb@ramboll.com
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,
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,
International Journal of Mechanical
Sciences, Vol 1, 1960, pp 322-333
Slide 3
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 Strainhardening,
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’
Slide 10
Concrete Plasticity: Slabs
Yield condition
Orthotropic slabs
Nielsen, M.P., Limit Analysis of Reinforced Concrete Slabs,
Acta Polytechnica Scandinavica, Civil Engineering
and
Slide 11
Building Construction Series, No 26, 1964, 167 pp
Concrete Plasticity: Slabs
Bi-conical yield surface, arbitrary reinforcement
− (M x − M
)(M y − M
) + (M xy − M
)2 ≤ 0
Fx
Fy
Fxy
− (M x + M ' )(M y + M ' ) + (M xy + M '
)2 ≤ 0
Fx
Fy
Fxy
θn > 0
θn < 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
− ( N x − N )( N y − N ) + ( N xy − N
)2 ≤ 0 ε n > 0
Fx
Fy
Fxy
− ( N x + hf c )( N y + hf c ) + N xy2 ≤ 0 ε n < 0
Slide 13
Concrete Plasticity: Shells
Moment – Axial Force Interaction
Generalised yield line
Linearised Interaction Curve
Slide 14
Concrete Plasticity: Beam Shear (w/ Stirrups)
Failure Mechanisms
Rotation
Translation
Slide 15
Coulomb Failure Criterion
τ = 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
fc = 2c√k
k = (1 + sinφ)/(1 - sinφ)
Coulomb Friction
Rankine Separation
τ = c - σ tanφ
σ = ft
Slide 17
Concrete Yield Surface
tanφ = 0.75
ft ≈ 0
fc = ν fcyl
fc
Plane Stress, ft = 0:
Square Yield Locus
Slide 18
Shear Crack (Yield Line)
Stresses:
σn = -½ fc (1 – sinα)
τnt = ½ fc cos α
σ2 = - fc
Dissipation:
α
τnt
n
v
Dc = ½ fc (1 – sin α) v
σn
σ1 = 0
Slide 19
Beams with Shear Reinforcement
Upper Bound Solution:
V = rfy bhcot β + ½ fc (1 – cos β) bh/ sin β
Optimal yield line inclination:
cot β = (½ fc - rfy)/ [rfy(fc – rfy)]1/2
Slide 20
≥0
Beams with Shear Reinforcement
Plasticity Solution
(Web Crushing Criterion)
θ fc
cot β = (½ fc - rfy)/ [rfy(fc – rfy)]1/2
V = bh [rfy (fc – rfy)]1/2 for
V = ½ bhfc
for
θ = β/2
≥0
rfy ≤ ½ fc
rfy ≥ ½ fc
Slide 21
Beams with Shear Reinforcement
V/bh
Plasticity Solution
(Web Crushing Criterion)
½ fc
β
V = bh[rfy(fc – rfy)]1/2
V = ½bhfc
for rfy ≤ ½ fc
for rfy ≥ ½ fc
rfy
θ
½ fc
θ = β/2
cot β = (½ fc - rfy)/ [rfy (fc – rfy)]1/2 ≥ 0
Slide 22
Beams with Shear Reinforcement
V/bhfcyl
ρ= 2.8%
fc = 0.86 fcyl
fcyl = 0.8 fcube
rfy/fcyl
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/bhfcyl
ρ = 6.0%
fc = 0.74 fcyl
rfy/fcyl
Slide 24
Beams with Shear Reinforcement
fc
Failure Mechanism
Slide 25
Beams without Shear Reinforcement
Failure Mechanism
Slide 26
Beams without Shear Reinforcement
Upper Bound Solution
V = - Ty cos(α + β) + ½ fc (1 – sinα) bh/ sin β
Slide 27
Beams without Shear Reinforcement
Plasticity Solution
cotβ = a/h
V = ½([(bafc)2+4Ty(bhfc-Ty )]1/2 - bafc)
for Ty ≤ ½bhfc
V = ½bfc([a2+h2]1/2 - a)
for Ty ≥ ½bhfc
Slide 28
Beams without Shear Reinforcement
V/bhfcyl
Shear Failure
Flexural Failure
Φ=Ty/bhfcyl
ν = fc/fcyl
V/bhfcy
l
Slide 29
Beams without Shear Reinforcement
V/bhfcyl
V/bhfcyl
Slide 30
Beams without Shear Reinforcement
Stress Distribution
fc
Shear failure
Flexural failure
fc
fc
V = ½([(bafc)2+4Ty(bhfc-Ty )]1/2 - bafc) for
V = ½bfc([a2+h2]1/2 - a)
Ty ≤½bhfc
for Ty ≥½bhfc
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
fc
fc
fc
Slide 33
Beams without Shear Reinforcement
Slide 34
Shear in Construction Joints
Failure in joint: Plane strain
Failure outside joint: Plane stress
Slide 35
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
Slide 36
Punching Shear in Slabs
Axisymmetric failure: Plane strain
Slide 37
Punching Shear in Slabs
Optimal failure surface generatrix: Catenary
ft = fc/400
Hess, U., Udtrækning
af Indstøbte Inserts,
DIA-B, Rapport No 75:54
1975, 25 pp
Slide 38
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
ft = 0
Code approach
Slide 40
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
Concrete Plasticity: Further Reading
Braestrup, M.W.,Shear Strength Prediction – Plastic Method,
Reinforced Concrete Deep Beams (ed F.K. Kong), Blackie and Son,
London, 1990, Ch 8, pp 182-203
Braestrup, M.W., Concrete Plasticity – The Copenhagen Shear Group
1973-79, Bygningsstatiske Meddelelser, Vol 65, Nos 2,3,4, 1994, pp
33-87
Braestrup, M.W., Punching Shear Revisited: Impact of the Plasticity
Approach, Bygningsstatiske Meddelelser, Vol 72, No 1, 2001, pp 1-26
Braestrup, M.W., Plastic Analysis and Design of Structural Concrete,
Second International fib Congress, Napoli 5 – 8 June, Proceedings
Vol 1, 2006, pp 490-491 + CD-ROM, 12 pp
Braestrup, M.W., Yield Line Theory and Concrete Plasticity, Magazine
of Concrete Research, Vol. 60, No. 8, October 2008, pp 549-553
Braestrup, M.W., Structural Concrete Beam Shear – Still a Riddle?,
ACI Special Publication (ed A. Belarbi & al), Farmington
Slide 42
Hills, Michigan, 2009, SP-265-15, pp 327-343