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Domain |
Explanation |
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Topics |
- Weak formulation
- FE concepts
- Shear locking & shear blocking
- Instability
- Material nonlinearity
- Geometrical nonlinear
- Dynamics
- Structural & geotechnical
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FEM idea |
- Physical phenomenon à ODE/PDE à transformation (PVW + shape functions) à algebraic equations à variables: primary (deformation, strains, essential) & secondary (forces, stresses, natural) à computer solution à responses
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FEM approach |
- 3 approaches:
- Exact: continuum mechanics / dynamics
- Weak form: principle of virtual work
- Energy-based approach
In structural analysis ...
Given: structural layout, material, geometry + loads (static/dynamic, point/distributed)
Response: 3 forms
- Primary (essential) responses: displacements, strains (& their derivatives: velocity (KE), acc (inertia), etc.)
- Secondary (natural) responses: forces, stresses (& their derivatives)
- Energy responses: various forms (but relative in nature (Einstein et al 1910)
to derive e responses ... structural analysis approach to derive these differ into three methods:
- Strong form: exact, continuum mechanics (dynamics)
- Weak form: approximate, variational mechanics (dynamics), convergence towards exact @ infinitum
- Energy form: account for all work (internal + external)
in weak form (championed by FEM) ... there are 3 types:
1) Forces/stresses in equilibrium
2) Displacements/strains in compatibility
3) Energy balance in ext. & int. work
For existence of solution ... any 2 of the above 3 types must be satisfied:
- (1) + (3): virtual force assumed --> compatibility imposed: Force method
- (2) + (3): virtual displacement assumed --> equilibrium imposed: Displacement method (Stiffness method & D'Alembert are special applications)
Hence, it can be seen that whichever force or displacement comes ... there is always a way to analyse given energy basis ... thus, dilemma resolved ... & our Dynamics equilibrium (M,C,K) is the result of D'Alembert which is Displacement method ... with equilibrium imposed, compatibility assumed
Equilibrium condition: forces & stresses
Compatibility condition: displacements & strains
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FEM basic steps |
- Continuum
- Discretisation
- Availability of equilibrium behaviour
- Trial function construction
- Assembly
- Impose B.C./I.C.
- Solution
- Post-processing: derive primary & secondary variables
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FEM 3 general step |
- Galerkin’s method
- Gaussian integration
- Response derivation
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Trial functions |
- Basic requirement for trial function for FEM:
- Ensure convergence:
- Compatibility: continuity
- Domain (element)
- Boundary (inter-element)
- Completeness:
- Existing (existence)
- Complete low order: else no convergence
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Continuity of trial functions |
- Continuum-based concept:
- C0
: zeroth-order polynomial continuous
- C1
: 1st-order polynomial continuous
- Aim: move C1 requirement to C0 requirement
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Shear locking/blocking |
- Blocking the stiffness contribution of (flexure, bending energy) by (shear effects) due to inappropriate selection of trial functions
- Evident from the formulation of the stiffness matrix from Principle of Virtual Work
- From a shear factor
 , or of similar form
- Especially shear locking in the following:
- Thin beams: shear locking
- Thin plates: shear locking
- Thin shells: membrane & volume locking
- Shear contribution >> bending contribution
- Corrections required: Iso-parametric concepts & elements + Degenerate concepts & beam/plate/shell elements
- Consistent interpolation element:
- Reduced (selective) integration element: reduced Gauss points needed for full Gauss integration
- Assumed strain field
: mixed interpolation, but Full Gaussian integration required à with trial functions for both displacements (normal FEM) & strain (imposed)
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Rank, spurious singular modes & patch test |
- Rank of an element:
- Proper rank = order of stiffness (DOFs) – no. of rigid body modes
- Rank deficiency = proper rank – actual order of stiffness (DOF)
- If rank deficiency>0: must use more Gauss integration points
- Causes of rank deficiency:
- Reduced integration: spurious singular (zero energy) modes
- Statics: @ or near singularity
- Dynamics: weak instability grows with time (spurious oscillations)
- Notes:
- Check rank of matrix: eigenvalue analysis or patch test
- Rank deficient element may not cause global rank deficiency (due to mutual strengthening & non-excitation of zero energy modes)
- Spurious singular modes SSM:
- Hour-glass mode
- Motion that is not rigid body displacement & produces zero strain à SSM
- Causes: reduced integration
- Zero strain: SMM is communicable à by group of element behaviour
- Patch test:
- Uses: check soundness of element
- Check completeness & stability à i.e. if rigid body moves with zero actual strain, yet incomplete element gives non-zero strain
- Check corrections in implementation & programming
- Test:
- Element placed inside peripheral nodes
- Standard patch test: improve displacement for constant strain in peripheral nodes à check all element nodes: displacements must be equal
- Tests: bending, shear, twisting
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Non-linear analysis |
- Deflections:
- Small: linear FEM
- Large: must use non-linear FEM
- Objectives:
- Establish appropriate strain & stress measures suitable for large-deformation FEA
- Formulate NFE based on weak form (PVW) with Lagrangian approach
- Process:
- Effective strain measures: kinematics considerations
- Effective stress measures: kinetics considerations
- FE formulation using weak form: Principle of Virtual Work under Total Lagrangian (TL) or Updated Lagrangian (UL) approach
- Conceptual solution process:
modified Newton-Raphson iteration method, modified Riks algorithm, arc length method or path following technique
- Approaches:
- Lagrangian approach: from undeformed à deformed
- Eulerian approach:
- Fluid mechanics & moving continuum
- Impact & fast loadings
- Large deformations
- Meshless methods:
- Particles put together
- High impact velocity & large deformations
- For: numerical problems & where Jacobian is singular à cannot invert
- Formulations:
- Kinematic considerations:
- Initial (undeformed) configuration
- Reference configuration
- Current configuration
- Mapping:
- Deformation gradient: F (equivalent to Jacobian J)
- 2 approaches: in incremental steps
- Total Lagrangian (TL): from original configuration (Ref.) to next configuration
- Updated Lagrangian (UL): from current configuration (Ref.) to next configuration
- Noted: approach (procedure) + process of solution
- Strain measures:
- Criteria needed to satisfy:
- Should vanish for rigid body motion
- Should be path-independent
- Tensor:
- Order 0: scalar
- Order 1: vector
- Order 2: matrix
- Can be represented by: ()(plane)(direction) or by index notation
- Green strain tensor: E = ˝*[FTF - I]
- Symmetric
- Path-independent
- Vanish under rigid body motion (RBM)
- Rate of deformation gradient (at that instant/step): D = ˝*[L + LT], L=d(vel.)/dx
- Velocity gradient: L = D + W = (sym) + (skew-sym); W = ˝*[ L - LT]
- Procedure:
- Engineering stress: F/A0, where A0: undeformed area
- Not consistent with tensor
- Satisfy all requirements: conjugate with Green strain
- Cauchy stress: true stress F/A, A: actual deformed area
- Weak formulations
- General nonlinear FE
- Solution by incremental form: TL or UL (both equivalent) approaches
- Kinetic considerations:
- Stress measures: based on equilibrium of the tetrahedron blocks
- Stress tensor:
- Cauchy stress: true stress, based on deformed configuration
- Symmetric
- Not invariant under rigid body rotation (RBR)
- Engineering stress: based on undeformed configuration
- Not symmetric
- Not invariant under RBR
- Second Piola-Kirchhoff stress: no real physical equivalence, hence to convert to the above stresses
- Symmetric: good
- Invariant under RBR: good
- Criteria for rigid body motion:
 ; 
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Nonlinear formulations |
- Objectives: FE formulations & conceptual solution process
- Effective strain measure: Green strain tensor, E (from rate of deformation, D)
- : invariant under RBR
- Effective stress measure: 2nd Piola-Kirchhof stress, s (& Cauchy stress, s )
- : invariant under RBR
- Cauchy co-rotation stress:
- 2nd Piola-Kirchhoff stress: symmetric, invariant under RBR & conjugate of Green strain tensor
- Nonlinear FE: for large deformations & strains
- Use current configuration & next configuration
- Weak FE formulation of current configuration:
- Weighted integral: weighted function: penalty function or velocity of displacement
- Gauss divergence theorem: from 3-D to 2-D
- Weak form
- Virtual work principle
- Convert 2nd-order tensor into 0th-order tensor: scalar
- TL: w.r.t. to original undeformed configuration
- UL: w.r.t. to current configuration
- Updated Lagrangian:
- Governing equation: incremental
- PVW formulations
- Linearised UL for Piola-Kirchhoff stress
- Assumptions
- Definitions
- Imposed equilibrium: final governing equation
- dt: time increment within (t) and (t+1)
- Kt.(ui)dt = [KL + Kn](ui)dt = (Fi)t+dt – (Fs i)t
- Kt: tangent stiffness
 : linear stiffness – elastic  : nonlinear stiffness – inelastic; with  : 1x1 for 1-D, 4x4 for 2-D, 9x9 for 3-D - (Fi)t+dt: body forces & surface traction forces
- (Fs i)t
 : consistent nodal internal force vector due to stress already present (linear portion)
- Method of solution:
- Modified Newton-Raphson iteration method: large deformations, large strains, but only for increasing stiffness, but post-ultimate stiffness
- Initial time step
- At each time increment: calculate unbalanced force
- Apply at next time increment
- Until lower than specified tolerance
- Modified Riks algorithm, arc length method, path following technique:
- Loading must be proportional
- Enables decreasing stiffness
- Allows for varying time increment: linear ok with large increment size; nonlinear: use small increment size
- Notes:
- K: nonsingular, continuous à Newton-Raphson converges quadratically
- Modified Newton-Raphson: more cost-effective (same K)
- Use proper convergence criteria: displacement, loads or Fi
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