MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam

Transcription

1 MODELING OF CONCRETE MATERIALS AND STRUCTURES Class Meeting #1: Fundamentals Kaspar Willam University of Colorado at Boulder Notation: Direct and indicial tensor formulations Fundamentals: Stress and Strain (Voigt format) Scope: Linear Elasticity and Triaxial Failure Criteria

2 NOTATION 3-D Tensor Format: 1. Scalar Fields: Temperature T (x, t) = T (x i, t) where i = 1, 2, 3 2. Vector Fields: Displacement u(x, t) = u i (x i, t) where i = 1, 2, nd Order Tensor Fields: Stress σ(x, t) = σ ij (x i, t) where i, j = 1, 2, 3 Strain ɛ(x, t) = ɛ ij (x i, t) where i, j = 1, 2, 3 Direct Notation: σ(x, t) = σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 and ɛ(x, t) = ɛ 11 ɛ 12 ɛ 13 ɛ 21 ɛ 22 ɛ 23 ɛ 31 ɛ 32 ɛ 33

3 DIRECT TENSOR NOTATION Elastic Stiffness: E(x, t) = E ijkl (x i, t) Elastic Stress-Strain Relation: σ = E : ɛ such that σ ij = E ijkl ɛ kl Matrix Format of 2-D Stress-Strain Relationship: σ 11 E 1111 E 1122 E 1112 E 1121 σ 22 σ 12 = E 2211 E 2222 E 2212 E 2221 E 1211 E 1222 E 1212 E 1221 σ 21 E 2111 E 2122 E 2112 E 2121 When ɛ 12 = ɛ 21 and σ 12 = σ 21, the stress-strain relationship may be compacted into [3 3] matrix form since E 1112 = E 1121 and E 1211 = E 2111 : σ 11 E 1111 E 1122 E 1112 ɛ 11 σ 22 = E 2211 E 2222 E 2212 ɛ 22 σ 12 E 1211 E 1222 E ɛ 12 ɛ 11 ɛ 22 ɛ 12 ɛ 21

4 Linear Elastic Relationship: σ 11 σ 22 σ 33 σ 12 = σ 23 σ 31 VOIGT NOTATION [1924] Triclinic materials exhibit 21 elastic moduli. E 1111 E 1122 E 1133 E 1112 E 1123 E 1131 E 2211 E 2222 E 2233 E 2212 E 2223 E 2231 E 3311 E 3322 E 3333 E 3312 E 3323 E 3331 E 1211 E 1222 E 1233 E 1212 E 1223 E 1231 E 2311 E 2322 E 2333 E 2312 E 2323 E 2331 E 3111 E 3122 E 3133 E 3112 E 3123 E 3131 ɛ 11 ɛ 22 ɛ 33 2ɛ 12 2ɛ 23 2ɛ 31 Summary of Voigt Notation: 1. Stress: if σ ij = σ ji then [σ] 6 1 = [σ 11, σ 22, σ 33, σ 12, σ 23, σ 31 ] t 2. Strain: if ɛ ij = ɛ ji then [ɛ] 6 1 = [ɛ 11, ɛ 22, ɛ 33, 2ɛ 12, 2ɛ 23, 2ɛ 31 ] t 3. Elasticity Matrix: if E = E t or E ijkl = E klij then [E] Elastic Strainenergy: W = 1 2 ɛ : E : ɛ = 1 2 ɛ ije ijkl ɛ kl = 1 2 [ɛ]t [E][ɛ]

5 CLASSES OF ELASTIC SYMMETRY 1. Monoclinic Materials: One plane of symmetry with 13 elastic moduli E 1111 E 1122 E E 1131 E 2211 E 2222 E E 2231 [E] = E 3311 E 3322 E E E 1212 E E 2312 E E 3111 E 3122 E E Orthotropic Materials: Two planes of symmetry with 9 elastic moduli E 1111 E 1122 E E 2211 E 2222 E [E] = E 3311 E 3322 E E E E 3131

6 CLASSES OF ELASTIC SYMMETRY 3. Transversely Isotropic Materials: One additional plane of isotropy and 5 elastic moduli E 1111 = E 2222, E 1133 = E 2233, E 3131 = E 2323, E 1212 = 0.5[E 1111 E 1122 ]. [E] = E 1111 E 1122 E E 2211 E 2222 E E 3311 E 3322 E E E E Cubic Materials Two additional planes of isotropy and 3 elastic moduli E 1111 = E 2222 = E 3333, E 1122 = E 1133 = E 2233, E 3131 = E 2323 = E [E] = E 1111 E 1122 E E 2211 E 2222 E E 3311 E 3322 E E E E 3131

7 ISOTROPIC ELASTICITY Infinite number of planes with equal material properties and 2 elastic moduli E 1111 = E 2222 = E 3333, E 1122 = E 1133 = E 2233, E 1212 = E 3131 = E 2323 = 0.5[E 1111 E 1122 ]. [E] = E 1111 E 1122 E E 2211 E 2222 E E 3311 E 3322 E E E E 3131 Lamé format of isotropic elastic stress-strain relationship: where Λ = σ = E : ɛ where E = Λ GI and σ = Λ(trɛ)1 + 2Gɛ Eν (1 2ν)(1+ν) and G = [E] = E 2(1+ν) Λ + 2G Λ Λ Λ Λ + 2G Λ 0 Λ Λ Λ + 2G G 0 G G

8 Elastic Compliance Relationship: ISOTROPIC ELASTICITY ɛ = C : σ where C = ν E G I and ɛ = ν E (trσ) G σ Matrix Format of 3-D Strain-Stress Relationship: C 1111 C 1122 C C 2211 C 2222 C [C] = [E 1 ] = C 3311 C 3322 C C C C 3131 Poisson s ratio: ν = ɛ lat ɛ axial 1 ν ν [C] = [E 1 ] = 1 ν 1 ν 0 ν ν 1 E 2[1 + ν] 0 2[1 + ν] 2[1 + ν]

9 ISOTROPIC ELASTICITY Volumetric-Deviatoric Split: ɛ = e (trɛ)1 and σ = s (trσ)1 (a) Volumetric Response: where K = Λ G = E 3(1 2ν) (trσ) = 3 K(trɛ) (b) Deviatoric Response: s = 2 G e where G = 3(1 2ν) 2(1+ν) K = E 2(1+ν) Canonical decoupling of strainenergy: W = 1 2 σ : ɛ = 1 2 K(trɛ)2 + Ge : e Constitutive Restrictions: 0 < E < and 1 < ν < 0.5.

10 FAILURE HYPOTHESES 1. Isotropic Strength Criteria: F iso (σ) = F iso (R t σ R) (a) Principal Format: F iso = F iso (σ 1, σ 2, σ 3 ) = 0 (b) Invariant Format: F iso = F iso (I 1, I 2, I 3 ) σ = 0 where I 1 = (trσ); I 2 = 1 2 (trσ2 ); I 3 = 1 3 (trσ3 ) (c) Volumetric-Deviatoric Format: F iso = F iso (I 1, J 2, J 3 ) σ = 0 where J 1 = 0; J 2 = 1 2 (trs2 ); J 3 = 1 3 (trs3 ) = det s Examples: Rankine, Tresca, Mohr-Coulomb, von Mises, Drucker-Prager, Willam-Warnke, Ottosen, Gudehus, Lade.. 2. Anisotropic Strength Criteria: F aniso (σ : P : σ, q : σ) = 0 F aniso = F (σ 11, σ 22, σ 33, σ 12, σ 23, σ 31 ) = 0 Examples: Hill, Hoffman, Tsai-Wu..

11 GEOMETRIC FAILURE ENVELOPE Haigh-Westergaard Format of 3-Invariant Format: F iso = F iso (ξ, ρ, θ) σ = 0 - Hydrostatic Coordinate: ξ = 1 3 (trσ) - Deviatoric Coordinate ρ = s : s - Angle of Similarity: cos3θ = 27J3 2J Concrete Failure Envelope Compression Tension Chinn Compression Tension Mills Compression Richart Tension Compression Balmer ρ/f c ξ/f c

12 TRIAXIAL FAILURE ENVELOPE FOR CONCRETE 5-Parameter Willam-Warnke Model [1975], 3-Pars. Menétrey-Willam [1995] Triaxial Hardening-Softening Model: Kang-Willam [1999] F (ξ, ρ, θ) fail = ρr(θ, e) f c ρ 1 f c [ ξ ξo ξ 1 ξ o ] α = 0 Out-of Roundness: r(θ, e) = 2(1 e 2 )cosθ+(2e 1) 4(1 e 2 )cos 2 θ+(2e 1) 2 4(1 e 2 )cos 2 θ+5e 2 4e Failure Envelope of Triaxial Concrete Model Tensile Meridian High Conf. Extension High Conf. Compression Uniaxial Tension Uniaxial Compression Vertex in Equitriaxial Tens. σ 2 /f c Evolution of Deviatoric Section with ξ σ 1 /f c σ 3 /f c ρ/f c 0.0 ξ o 2.0 σ 3 /f c σ 2 /f c 4.0 Compressive Meridian ξ/f c σ 1 /f c

13 ALTERNATIVE FAILURE HYPOTHESES 2. Strain-Based Deformation Criteria: F (ɛ) = 0 Isotropic Case: F iso (ɛ) = F iso (ɛ 1, ɛ 2, ɛ 3 ) = 0 or F iso (I 1, I 2, I 3 ) ɛ = 0 or F iso (I 1, J 2, J 3 ) ɛ = 0 Examples: St. Venant Energetic Failure Criteria: F (W ) = 0 Isotropic Case: F W (σ : ɛ) = 0 or F Wvol (trσ trɛ) = 0 or F Wdev (s : e) = 0 Examples: Beltrami, Huber...

14 CONCLUDING REMARKS Main Lessons from Class #1: Voigt Notation for Minor Symmetries: Matrix Form of Elastic Stiffness and Compliance Canonical Form of Linear Elastic Behavior: Decoupling of Volumetric-Deviatoric K-G Moduli Concrete Failure: Strong Interaction of all Three Invariants