Short Summary for FEM Computational Modelling

7/24/2019 Short Summary for FEM Computational Modelling

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Oral examination on case-study: issues like selection element types, material models, loads, supports, checks, outpinterpretation will be raised, with possibly a few links to exercises, lectures too

Short SummaryMotivation for FEM: increasing complexity: large span, wide cantilevers, high rise, underground soil-structure, new materials

structures, hybrids, non-cartesian free-form structures, shells etc.

nonlinear finite element analysisStructural reassessments of existing infrastructureSeismic assessments of buildings in the province of Groningen

Nonlinear FEM: reveals stress redistributions and capacity beyond elastic design stage

Inputselection of element types, meshes, constitutive models, material parameters, boundary conditions, tyings, loadings,

Analysis control procedures

Output interpretation, presentation, checks, judgement

Displacement method: Displacements and rotations at the nodes are considered as the fundamental unknownsTypically, Displacements continuous over element boundaries. Strains and stresses are not!

Physically nonlinear: plasticity, crack & crush models, nonlinear engineering stress-strain relations, interface and beddingmodels,

Geometrically nonlinear: deformations are so large that they alter the orientation of forces and moments

Running a FEA job1. Preprocessing Via pre-processors (FX+): model preparation and checking (element types, generate mesh, check

connectivities, Choose units, dimensions, Material (isotropic linear, or nonlinear), Schematize loads, supports


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2. Analysis and checking: Check log-file with warnings, error messages; Check condition stiffness matrix, ratio diagonterms; Check convergence for nonlinear analysis

3.

Post-processing and checking: Plot displacements, (principal) stresses and strains, history data. Check equilibriumtotal load versus reactions. Quantitative interpretation & Check hand calculation

Top-5 errors1. Inconsistent use of units, dimensions: e.g. gravity, density, length2. Inadequate constraints (supports, tyings) rigid body motion, mechanism,3. mixing up local/global axes,4. It doesnt converge (divergence). divergence failure!, check solution procedures, constitutive models, softening

An overview of structural finite element typesWe only focused on element types used for structural mechanics applications, relating forces (moments) to displacements(rotations)

Displacement-based FEMthe displacements (& rotations) at the nodes are considered as the fundamental unknowns. Typically: The displacements arecontinuous over the elements. The strains and stresses are not! 2 super slides are thus:

FEM wants to relate displacement at nodes to theinternal forces at the nopde. Note that the DoF isdetermined by the displacement/rotation.Stiffness matrix is an important aspect

Displacement at node

Differentiated to get strain =

(kinematic relation)

Next is material model linking stress& strain =

Both stress and strain are atintegration point

Internal forces obtained fromIntegration = over wholeelement. Internal force @ node

Now look at an example of an element. Follow the reasoning well


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Unin the figure is displacement and rnisinternal forcesDoF is 8x3 = 24. We have internal forcecorresponding to each DoF.

Stress and strain are 6x1 matrix

D which is the material model is 6 X 6matrix

Stiffness matrix F=KU has to be enoughfor each DoF. Hence 24X24 matrix

At the end, it still led to F=Ku

An integration point is the point within an element at which integrals are evaluated numerically. The stresses at the integrationpoints are the most accurate. These points are chosen in such a way that the results for a particular numerical integrationscheme are the most accurate. Depending on the integration scheme used the location of these points will vary. finite elementdisplacements are most accurate at the nodes. However, derived values (i.e. stresses and strains) tend to be most accurate athe integration points (and sometimes least accurate at the nodes). They occur inside the element and may not be the higheststress in the area. i think it's because FEA calculates stresses at the integration points of the elements and extrapolates these the nodes, and averages across adjacent elementsOn a (somewhat) related note, folks often compare elemental strains with averaged nodal strains to determine whether theirmesh is adequate. If there is a significant difference, mesh refinement is required.

Read the full slide on elements

Interface elements:Rather than a strain an interface elements uses relative displacements u: E.g. representing an opening or A sliding. Typicaapplications include Discrete cracking, Bond slip of reinforcement bars. No-tension and elastic beddings

Connectivity & Compatibility of elements: Requires common nodes of neighboring elements Requires similar (displacemeinterpolation

How does the finite element calculate the internal forces? (numerical integrated, analytical)


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The figure show a FEM vs analyticalresult

See the result. Numericallyintegrated at the intregration point.Stress at the integration point is

accurate.

For the node however, the stress isextrapolated from the integrationpoint. Not as accurate obviously

See the impact of using nodalaveraging.Software can be instructed to usenodal average values forpostprocessing:

Visually more pleasingBut, information useful in judgingthe quality is lost! Remedy: P-refinement H-refinement

See. If we refine the mesh, we getbetter results. Even the differencewith nodal averaging become less.More integration points in therefined one.

P & H refinementA mesh is characterized by the localmesh size h and the order ofapproximation p. The h-version refinesmesh size (e.g. finer mesh) whilekeeping the order of polynomial fixed;thep-version uses a fixed mesh but

increasesp to improve accuracy (e.g.quadratic, cubic).additional stress points orintegration points

An introduction to nonlinear analysis


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Types of nonlinearity in structural mechanics

1. Physical/Material nonlinearity: Material properties are a function of State of stress or strain; (Deformation) history; Time,Temperature, Maturity etc. Example include Nonlinear elasticity, plasticity, cracking, creep viscoelastic,

2. Geometric nonlinearity: Deformation is large enough that equilibrium equations must be written with respect to the deformstructural geometry. Large Displacements effects. Loads may change direction or magnitude

3.

Contact nonlinearity: Gaps open or close; Contact areas changes; Possible sliding with frictional forces. In the civilengineering practice many contact problems could be simplified by using Interface elements with no-tension materialbehavior

Solution procedures

See Fig 1 above, if we use an incremental procedure, we could be monitoring better how the non-linearity affects. Importantdetails on the material behaviour (for instance) would be gained. However, we could be drifting from the true equilibrium path.

A purely incremental method usually leads to inaccurate solutions in nonlinear analysis, unless very small step sizes are usedan iterative process the errors that occur can be reduced successively.

Most commonly used: Incremental-Iterative Solution

In nonlinear Finite Element Analysis the relation between a force vector and displacement vector is no longer linear.For several reasons, discussed in Volume Material Libraryand 31.2,the relation becomes nonlinear and thedisplacements often depend on the displacements at earlier stages, e.g. in case of plastic material behavior. Just aswith a linear analysis, we want to calculate a displacement vector that equilibrates the internal and external forces. Ithe linear case, the solution vector could be calculated right away but in the nonlinear case it cannot. To determine tstate of equilibrium we not only make the problems discrete in space (with finite elements) but also in time (with

increments). To achieve equilibrium at the end of the increment, we can use an iterativesolution algorithm. Thecombination of both is called an incremental-iterativesolution procedure.
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Newton-Raphson

Within the class of Newton-Raphson methods, generally two subclasses are distinguished: the Regularand the ModifiedNewtRaphson method. In a Newton-Raphson method, the stiffness matrix Kirepresents the tangential stiffness of the structure: Thedifference between the Regular and the Modified Newton-Raphson method is the point at which the stiffness matrix is evaluate


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Regular Newton-Raphson.In the Regular Newton-Raphson iteration the stiffnessrelation (31.8)is evaluated every iteration [Fig.31.2]. Thismeans that the prediction of (31.7)is based on the lastknown or predicted situation, even if this is not anequilibrium state. The Regular Newton-Raphson method

yields a quadratic convergence characteristic, whichmeans that the method converges to the final solutionwithin only a few iterations.

See fewer iterations

Modified Newton-Raphson.The Modified Newton-Raphson method only evaluates thestiffness relation (31.8)at the start of the increment [Fig.31.3].This means that the prediction is always based on a convergedequilibrium state. Usually, Modified Newton-Raphson converges

slower to equilibrium than Regular Newton-Raphson. The

Modified Newton-Raphson method usually needs moreiterations, but every iteration is faster than in RegularNewton-Raphson.

See more iterations
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Load & Displacement control

Load control: increasing loads on nodes or elements, and monitoring displacement

Displacement control: Increasing prescribed displacement on the nodes, and monitoring the resulting reactio

forces

If both are possible: use displacement control!

Force control- fails to overcome the topDisplacement controlovercomes it. It simply continues with load increment

Arc-length control: With automatic load increments and subsequentdecrements to overcome the top

Convergence criteria

Force convergence: The remaining force imbalance is a small fraction of the total applied force. Example: pu

stress relaxation, Force norm is the only choice

Displacement convergence: The last update of the displacement increment is a small fraction of the initial

displacement increment. Example: pure creep deformations, Displacement norm is the obvious choice

Energy convergence: The last update of the stored energy is a small fraction of the initial stored energy. The

Energy norm combines displacements and forces

Nonlinear springs and interfacesMotivation nonlinear springs and interfaces discrete/Discontinuous behavior:Bedding; Gapping, no-tension; Plastic hinges; Yield lines; Discrete cracking; Shear friction/slipping;Discrete crushing; Steel-concrete interface, bond-slip;Etc., all types of joint behavior

Essence: Deformations lumped into a hinge, a line or a plane. Often complies with true physicalbehavior, localization. If not: still a conceptually and numerically attractive method. Generic approach: nonly plastic hinges, yield lines, but also softening hinges, softening lines, brittle behavior, rotation capacissues


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Structural Interfaces: The structural interface elements describe the interface behavior in terms of a

relation between the normal and shear tractions and the normal and shear relative displacements acros

the interface. Typical applications for structural interface elements are elastic bedding, nonlinear-elastic

bedding (for instance no-tension bedding), discrete cracking, bond-slip along reinforcement, friction

between surfaces, joints in rock, masonry etc. Many types, usually place between 2 elements e.g. noda

interface element to be placed between two nodes. Line interface elements to be placed between

truss elements,Plane interface elements to be placed between faces of three-dimensional

elements etc

Where do we place them? How to predefine them

Mechanism based on engineering judgment, e.g.Plastic hinges in corners and at locations ofpoint loadsCracks in tensile zones of concreteYield line assumptions, based on insight inpossible mechanisms

At as many places as possible, e.g.:Discrete block/spring assemblies, particle/springmodelsAt all joints in masonry, rock or block systems,as mostly the joints are the weak places ofpreferential damageBlocks are then kept linear, while allnonlinearity is lumped into the springs orinterfaces

An alternative is remeshing: Start with continuum elements only; when strength is exceeded: split nodeand/or remesh, insert interfaces, map the old stress situation to the new mesh, continue and redo, follo


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the propagation of the crack or plastic zone. Cumbersome: most finite element programs prefer to havefixed mesh with fixed degrees of freedom

See example below:

tnnormal traction

(stresses)tt tangentialtraction (stresses)It could even becoupled with

shear tsIntegration schemes interface elements: Gauss integration along the line or in the plane


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Assignments:1.Verify Strong stiffness changes: full Newton-Raphson. Plot traction vs displacement

2. Study how moment is different between linear bedding and no-tension bedding


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Smeared cracks and reinforcement

The effect of cracking is spread over the area that belongs to anintegration point. Advantage: We dont need to know the cracklocation. it can occur anywhere in the mesh in any directionStress versus strain, continuum

While it was a single localized crack in interface (discrete), here wehave distributed cracking in concrete

Crack initiation in smearedA crack is initiated when the principal tensile stress 1 exceeds the value of the tensile strengthft. The direction of the crack is perpendicular to the direction of the principal tensile stress. Aftecrack initiation, the principal stress follows a tension-softening stress-strain diagram

Additional parameter: crack band width h over which the crackis smeared out.If its crushing, the 3 material parameters we need arecompressive strength, compressing crushing energy (both much

higher than tensile) and the shape of the diagram. Also crushingband width. Fracture energy in compression much higer

In the first diagram for discrete, see the high intial stiffness.. For smeared, initial stiffnes not as high. We

are smearing out the crack over 1 element hence the reason the thickness (h) come in. Also see the effeof element size (thickness) in the fracture energy. For discrete, thickness was not considered, the crack

was localized in a line

See the impact of tension stiffening effect in the above compared with noakowski. Concrete contributes steel, till the steel starts to yield

Modeling of reinforcementsModel reinforcement as truss element, connected to the nodes of the continuum elements. Embedded in thecontinuum elements. Strain is derived from nodal displacements of mother element. Embedded element dont havnodes. They depend on the concrete nodes.

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Short Summary for FEM Computational Modelling