Nonlinearity

Autodesk Nastran for Inventor: Unlocking Nonlinear Analysis Tony Abbey FRAeS Owner FETraining Join the conversation #AU2017

Moving into the nonlinear world with FEA  “The world is nonlinear”.  That’s right, but in FE analysis we like to keep things as simple as possible and ignore the nonlinearity if we can  A lot of the time we can get away with this, but sometimes we can’t..  After a quick dabble in nonlinear FEA a lot of engineers understand the reason for the coyness  Nonlinear analysis is tough to do effectively and efficientlyit is a steep learning curve!

Steps in Linear Analysis  Set up the model  Mesh the part  Apply Material and Physical Properties  Apply Loads (100%) and Boundary Conditions  Run analysis  Job Solution  Assembly of and solution of stiffness Matrix  Compute displacements, strains, stresses (and other results)  View results

Steps in Non-Linear Analysis  Set up the model  Mesh, Apply Material and Physical Properties  [1] Apply Load Increment (10%) and Boundary Conditions  Run analysis  Job Solution  [2] Assembly of and solution of stiffness Matrix  Check for convergence – if none then  Update stiffness and displacement, loop back to [2]  If convergence is ok – loop back to [1]  Stop when load reaches 100%

Steps in Non-Linear Analysis

Types of non-linearity  Geometric (large Displacement)  Buckling  Follower Forces  Material  Contact  Boundary Conditions

Geometric Non-linear Analysis

Geometric (large Displacement) Nonlinearity  Tent walls deflected inwards under the wind pressure.  Internal balancing membrane loads develop in the tent walls as shape changes  The undeformed, flat, initial state, can’t resist the wind pressure  If loads can only balance in a deformed configuration geometric nonlinearity

Geometric (large Displacement) Nonlinearity  Simple example  Rigid rod  Linear spring  Linear solution (small angle theory)

 Nonlinear Solution

Geometric (large Displacement) Nonlinearity  Solutions:  Linear  Nonlinear • The linear solution – just stays linear! • A force of 5E5 N applied at the tip. • The rotation is ‘round the flagpole.’

Geometric (large Displacement) Nonlinearity  Key settings  Nonlinear Static Analysis

 Large Displacements on  AKA – Geometric nonlinearity

Geometric (large Displacement) Nonlinearity  Demo – Model of a stool  Constant pressure distribution across the seat of the stool  Base of the stool assume fixed to ground  Will deflections be small enough to allow a linear solution?  Or do we need - geometric nonlinearity?  What metric can we apply to decide?

Geometric (large Displacement) Nonlinearity  Conclusions  Linear solution gives enormous deflections (real scale shown)  Stresses are due to bending and shear only  Nonlinear solution gives realistic deflections – edges roll and stiffen  Membranes stresses can develop and significantly stiffen

Buckling Analysis

Buckling Analysis  Linear Buckling  Displacement is small, linear perturbation  User Interface:  Static Solution to obtain differential stiffness  Normal Modes Solution to find Eigen Value

Buckling Analysis  Non-Linear Buckling  Large Displacement effects  Can be combined with Material nonlinearity  Can use linear or nonlinear prestress  Initiation load or imperfections usually required

Buckling Analysis  Transition from linear to nonlinear  Euler (linear) buckling at high slenderness ratio  Compressive yield for very short slenderness ratio  Intermediate zone  Linear assumptions increasingly non-

Buckling Analysis  Limitations of linear buckling  Non-conservative for non-slender structures  Only buckling shape is found  No way to assess level of deflection  Stresses are meaningless - only arise from deflected shape, ignore initial load  Useful as a check on Critical Buckling

Buckling Analysis Demo  Tall cylinder  Axial and side load  Constrained at base  Linear Buckling Analysis  Non-Linear Buckling Analysis  Arc Length method

Material Nonlinearity

NL Material Analysis  When the stress level in a component exceeds the yield point, the material in the effected zone starts to go plastic  The presence of plasticity means the material is following a nonlinear stress strain curve

NL Material Analysis  The input to FEA can rationalized by:  an elastic linear section, with material stiffness E  plus a plastic nonlinear part which can be a constant slope (the two slopes are described as a bi-linear fit)  a varying slope defined by a data table

NL Material Analysis  A fine mesh, with good shaped elements is needed  Coarse mesh – bad result  Poor mesh – trips plastic zone in wrong place  Often have to re-think linear static mesh

NL Material Tips  Non-linear Material Input  Yield Stress  Yield Criteria  Type of non-linear material  Reconfirm yield and criteria!  Tangent Modulus  Hardening Rule

NL Material Analysis Demo  Plate with hole  Constrained on LH edge  Loaded at RH edge  Bi-Linear Material model  Looking for plastic strain growth

NL Material Analysis Demo  Results  Nonlinear loaddeflection response  Hits yield at 50%

 Plastic strain growth

NL Material Analysis Demo  Results  Below yield distribution is constant

 Above yield distribution changes 60%

20% 80% 40%

100%

Contact Analysis

Contact Analysis  Slave Nodes penetrate master regions  (single sided contact)  nonsymmetric  Slave Nodes penetrate master region, Master Nodes penetrate slave regions  (double sided contact)  symmetric

Contact Analysis

 Penalty stiffness method  Uses a ‘fake’ spring stiffness to develop a reaction between contacts  Allows penetration of surfaces through each other  Can be tricky to control  Lagrange multiplier method  Solves for unknown balancing forces to hold contacts together  Can be unstable  Augmented Lagrange Multiplier

Contact Analysis  General Contact  Contact can be considered as ‘touching’ or ’making/breaking’ – when in contact a continuous load path is formed  General relative movement of contacts is allowed  Separation is permitted if criterion is met  Option - Sliding only permitted in plane of faces  Option - No lateral sliding permitted ( rough contact)  Option – Frictional effects (good for stability)

Contact Analysis  Glued (bonded) Contacts  Contacts are permanent  Can be used in linear analysis to bond dissimilar meshes  Can be used in nonlinear analysis with general contact  Great for debugging a nonlinear model

Contact Analysis Demo  Plate forming  Stiff dies top and bottom  Flexible plate  Contact surfaces on both sides of plate  General contact  Geometric nonlinearity as well!  Enforced displacement

Contact Analysis Demo  Plate forming  Contact surface selection  Master Surfaces  Slave Surfaces  Contact Type  Surface to Surface contact  With separation  Symmetric

Thanks for watching! Contact: [email protected] LinkedIn – Tony Abbey Activities: Digital Engineering – Tony Abbey author portal (60 articles) NAFEMS e-learning – next nonlinear course starts next Tuesday https://www.nafems.org/events/nafems/2017/el209/

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