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Discontinuous Modelling of Structural Masonry and Quasi Brittle Media

NIEeS Workshop Cambridge 3-4 April 2003. Discontinuous Modelling of Structural Masonry and Quasi Brittle Media. N. Bi ć ani ć , C. J. Pearce C. Stirling, C. Davie Computational Mechanics Group Department of Civil Engineering University of Glasgow.

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Discontinuous Modelling of Structural Masonry and Quasi Brittle Media

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  1. NIEeS Workshop Cambridge 3-4 April 2003 Discontinuous Modelling of Structural Masonry and Quasi Brittle Media N. Bićanić, C. J. Pearce C. Stirling, C. Davie Computational Mechanics Group Department of Civil Engineering University of Glasgow

  2. Scale of Discontinuties – Macro/Meso/Micro etc Scale of Observation Multi-Scale, Homogenisation Material Heterogenity Continuum Physics RV - Representative Volume

  3. Discontinuous Modelling • of Structural Masonry • (macro discontinuities- existing • and emerging - fracturing associated • with an internal length scale) • Flow through Jointed Rock • (multifield, existing discontinuities) • Lattice Modelling of • Hydraulic Fracturing • (emerging macro discontinuities, • softening, internal length scale)

  4. Structural and Historic Masonry Computational Failure Predictions Nonlinear FE Analysis, including Interface Elements DDA Analysis, with Simply Deformable Blocks. Interface Failure DEM Analysis -Rigid Disk Particles -Spring and Cluster Model

  5. Discontinuous Modelling of Masonry FE + Joint (Interface) Elements DEM + Interface Law DDA + Interface Law Block Spring Interface – Fracture Energy Controlled Softening Block Deformability (Low, High, Arbitrary) Block Fracturing (Through Block) Contact Detection (Search Algorithms) Contact Kinematics, Contact Kinetics

  6. 3D Block Spring Model Li, Vance

  7. Discrete Cracking of Masonry Walls (Alfaiate, de Almeida) FE with Multi-Surface Plasticity for Interface -u Fracture Energy Control Masonry Units Elastic

  8. Rigid or deformable blocks Linear and non-linear force-displacement for discontinuities Library of material models for deformable blocks and for discontinuities Lemos 3DEC large displacements (slip and opening) along distinct surfaces in a discontinuous medium

  9. Benchmarks - Masonry Walls on Inclined Plane Pagnoni, Nistico DECICE

  10. Block Deformability Representation Discontinuous Deformation Analysis DDA Lowest order DDA, constant stress field per block of arbitrary shape

  11. Higher Order DDA

  12. Contact Constraints Penalty Format Lagrangian Augmented Lagrangian Mohr Coulomb with Tension Cut-Off

  13. Time Stepping Scheme Dynamic Relaxation Solver Shi (DDA) Newmark Implicit, unconditionally stable scheme

  14. Poor stress representation for Low Order DDA Improvements Block FE Submeshing Block Fracturing Rock Bolts Higher Order DDA Numerical Manifold DDA recast as a special case of Numerical Manifold, Partition of Unity Method

  15. Manifold Method, Partition of Unity Simplex Integration

  16. Masonry Arch Influence of Backfill on the Collapse Load Hinge Hinge Hinge Hinge DDA Analysis of Edinburgh Arch Bridge (φ = 200 – Failure Mode at 31.8kN/m)

  17. φ =20o φ =28o φ =36o 0.8 Pfail 17 kN/m 1.0 Pfail 21 kN/m 1.20 Pfail 25.5 kN/m 1.50 Pfail 31.8 kN/m

  18. Contact Predictor/corrector Steps INITIAL GEOMETRY At start of time step blocks are not in contact. PREDICTOR STEP Displacements of blocks are calculated for current time step. Overlaps are used to detect possible contacts and the nature of pre-existing contacts. CORRECTOR STEP Stiff springs are placed between blocks in contact and friction forces applied to satisfy interface criteria. Multiple contact predictor-corrector steps may be required to determine location and nature of all contacts in a complex system. These are the iterations in DDA.

  19. Contact Predictor StepCalculating the Location of a Contact Inaccurate calculation of point of contact Pc. Can lead to solution oscillations and convergence problems. Traditional Approach (Shi, Jing) More Accurate Approach • Calculate the location of the contact point Pc at the exact time it occurred. • More accurate calculation of contact location results in faster convergence and reduces solution oscillations.

  20. Dry Brick Assembly Benchmark Problem

  21. Emerging Failure Mode Identification from the Eigenvalue Analysis of the Linearised K

  22. Changes in Fundamental Eigenvector identifying the eventual failure mode

  23. Modal Decomposition of Incremental Deformation Stage 1 Stage 2 Stage 3 l1 =3.72e3 a1=26.5 l1 =3.44e3 a1=45.8 l1 =0.0 a1=39.6 l2 =7.35e3 a2=47.9 l2 =7.23e3 a2=28.3 l2 =0.0 a2=36.8 l3 =1.90e4 a3=5.6 l3 =9.32e3 a3=9.2 l3 =0.0 a3=23.5 l4 =2.56e4 a4=6.5 l4 =1.89e4 a4=5.6 l4 =8.52e3 a4=0.0

  24. Four Hinges Arch Bridge Failure Bridgemill Arch Bridge – DDA model

  25. Bridgemill Arch DDA predictions

  26. Caledonian Canal (Telford) Fort Augustus Wall Collapse and Major Repair • Scottish Lighthouses 3D interlocking stones • Fractured Mudrock Seals

  27. DDA, Keyblock Theory Modelling of Discontinuous Rock Mass HYDRO-DDA Flow through Fractured Media Fractured Mudrock Seals Hydraulic Fracturing

  28. HYDRO (fixed mesh) HYDRO-DDA Interface HYDRO-DDA interface DDA Pressure Equivalent porosity

  29. Fixed fluid mesh DDA Mesh Configuration of discontinuities in solid domain Finite Element Mesh Densification along discontinuities

  30. Permeability mapping Simple permeability mapping , Directly proportional to equivalent porosity The essential behaviour can often be expressed successfully using the simple Kozeny-Carman Relation (through-going pipes) Equivalent porosity

  31. Staggered Approach between Solids & Fluid Domains DDDDA parallels Fluid flow calculation Permeability mapping Transform pressures into Nodal forces Fw(n) DDA mesh Fluid boundary conditions New permeability mapping New fluid flow calculation Transform pressures into Nodal forces Fw(n+1) New DDA calculation New DDA calculation

  32. Model Problem - SHEAR Changes in Flow Pattern though Fractured Seal Layer as a Result of Blocks Movement and Deformation (Changes in Mechanical Boundary Conditions)

  33. Steady State Fluid Fluxes and Effective Permeability for Different Overpressures Effective Permeability for Problems A and B

  34. Hydraulic Fracturing • Viscous fluid flow in the fracture • Leak off of fluid to the rock matrix • Elastic/plastic deformation of rock • Fracture propagation • Proppant transport

  35. Well Bore Stability Horizontal cross-section Hydraulic Fracture Vertical cross-section Lattice Modelling of Continua

  36. Plane Strain Equations Griffiths and Mustoe Lattice Models for Elastic Continua

  37. Viscous fluid flow in the fracture (PFM - Particle Fluid Model) • Leak off of fluid to the rock matrix • Elastic/plastic deformation of rock (PSM Particle Solid Model, Lattice, Softening Plasticity, Fracture Criterion) • Fracture propagation • Proppant transport (Distinct Particles)

  38. Fracture Energy Based Lattice Fracture Criterion A

  39. Fracture Energy Based Lattice Fracture Criterion B

  40. Fracture Energy Based Lattice Fracture Criterion C

  41. Benchmark Problems Lattice Fracture E = 1785MPa n = 0.25Gf = 200.0N/m ft = 0.5MPa  lattice size h = 0.5, 0.25, 0.1m Es = f(E, Gf, ft, h)

  42. PSM+PFM PSM+Proppant Particles

  43. Conclusions • Micro vs Macro • Discontinuous Modelling of Masonry • Multi-Physics – Flow through Discontinua • Lattice Modelling for Hydraulic Fracturing

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