Continuum mechanics of beam-like structures using one-dimensional finite element based on Serendipity Lagrange cross-sectional discretisation, Mayank Patni, Prof. Paul Weaver, Dr Alberto Pirrera Bristol Composite Institute University of Bristol United Kingdom www.bris.ac.uk/composites

Motivation 2/15 Perform detailed analysis of complex beam structures by enriching the kinematic field Build a mathematical model that has the following properties Hierarchical Local mesh control Capture detail 3D stress fields Computationally efficient Numerically stable

Outline 3/15 1) Brief description of Unified Formulation 2) SL expansion model 1D polynomials 2D polynomials Displacement field 3) Convergence and stability 4) Square cross section beam 5) Thin Box - Buckling 6) Conclusions and acknowledgements

Carrera Unified Formulation 4/15 Beam Structure Cross Section kinematics Beam-like structure Beam nodes and cross section Describe beam-axis and cross section separately Beam axis kinematics Taylor (TE) - Hierarchical - Euler-Bernoulli - Timoshenko - Equivalent Single layer Lagrange (L) - Nodes - Local mesh - Layer wise 1D Lagrange functions B4 element 3D model Taylor displacement field u kinematics Coefficients/unknowns Expansion functions Lagrange displacement field u kinematics

Finite Element Formulation 5/15 Principle of Virtual Displacements Variation of internal, external and inertial work Stiffness Matrix Force Vector Expansion functions dependency Displacement field compact form Fundamental Nucleus and Assembly Nucleus 3x3 Index for Cross Section Index for Beam Axis C.U.F - F.E. formulation

Serendipity Lagrange Element 6/15 Serendipity Lagrange (SL) main idea + = Hierarchical polynomial 4 node Lagrange element (L4) What we want to achieve? Taylor-like hierarchical model Local expansion control Ability to mesh complicated regions Reduce effort (mainly time) in re-meshing Capture localised stress fields and increase accuracy Improve numerical stability Reduce number of degrees of freedom (DOFs) How? Build expansion functions Functions defined inside L4 2 Step process: 1D high order polynomials 2D high order polynomials

1D High-Order Polynomials 7/15 Polynomial p order N Equal spacing Equal spacing Equal spacing Requirements Explicit form High order polynomial Defined in [-1,1] At least two zeros Zeros equally spaced Requirements are needed for continuity and completeness N=2 N=3 N=4 N=5

2D High-Order Expansion 8/15 node sides 2D Polynomial L order N inside Requirements High order 2D polynomial Defined in [-1,1] x [-1,1] 3 sets of polynomials one that vanish at all nodes except one one that vanish at three sides except one one that vanishes at all sides (except inside) Requirements are needed for: continuity and Completeness of vector space basis Set 1 Set 2 Set 3 index 1D polynomial

SL Displacement Field 9/15 Displacement field Order 1 SL expansion functions Order 2 Expansion functions order 1 SL expansion functions order 2 unknowns

Convergence rate and stability 10/15 Square cross section beam Beam cross section, BCs & force Displacement, normal and shear stress at tip centre. Comparison with TE and LE Comparison with conventional expansions Fast convergence rate for displ. & stress Improvement relative to LE expansion Similar performance with TE but much more stable as order increases Reciprocal of conditioning number vrs order

Square Cross Section Beam 11/15 Boundary layer effect Good agreement Slower Convergence/ More mesh Through thickness shear at Mid span Through thickness shear/normal Several locations DOFs Solid ~500,000 DOFs SL ~ 12,900 ~3% DOFs!! Able to capture the correct gradient Mesh size Order 8 8 8 8

Thin-Box 12/15 8L4 elements Minimum SL cross section mesh Linear Buckling Analysis Using 3D stresses Solid FEM Shell FEM SL Reduces also time! (~5-10%) ~2.5% DOFs!! Critical load and DOFs Normal stress along length. SL order 3 Non linear analysis - Post Buckling

Summary SL Expansion Model 13/15

Conclusions and acknowledgments 14/15 SL model retains the advantages of both TE and LE models without retaining their disadvantages Hierarchical model and local mesh control Reduce effort in re-meshing Mesh complicated regions Effectively capturing localised 3D stress fields Improves numerical stability Reduces significantly the number of degrees of freedom compared with 3D Finite Element model - The H2020 Marie Sklodowska-Curie European Training Network is gratefully acknowledge. - Many thanks to Bristol Composite Institute and POLITO research groups who were involved in the development of the method.

15/15 Thank you for your attention