Introduction to Finite Element Analysis for Skeletal Structures Dr. Nick A Alexander © 2006 Department of Civil Engineering University of Bristol 22 November, 2018

General Structural Problem Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol General Structural Problem For the following structure we want to compute Bending moments, shear forces, axial forces Bending, shear and axial stresses and strains Deflections

Finite Element Analysis (FEA), The basic idea Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Finite Element Analysis (FEA), The basic idea Complex structures systems are often too complicated to simply derive relationships between applied loads, deflections and internal stresses. Hence large structures are divided up into many individual finite elements; that have a much simpler structural form, e.g. a beam or column The relationship between load, displacement, stresses and strains in a finite element can be determined Thus, it is computationally possible for a complex structure to be modelled by assembling (aggregation) many individual finite elements. The aggregation process must satisfy equilibrium and continuity.

Modelling Idealisation Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Modelling Idealisation Nodes connection points (not necessarily hinges) Loads Elements Restraints

For a finite element, we need to derive the relationship between Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol For a finite element, we need to derive the relationship between External Loads Deflections/deformations Internal stresses and strains

Deriving force-displacement relationship for general finite element. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Deriving force-displacement relationship for general finite element. Five basic steps Conjecture a displacement function Use nodal boundary conditions Derive strain–displacement relationship Derive stress–displacement relationship Use principle of Virtual Work

Bar Element example u1 u2 Deformed shape f2 f1 x Node (a hinge) Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example u1 u2 Deformed shape f1 f2 Element Node (a hinge) x

Bar Element example Conjecture a displacement function u(x) x Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Conjecture a displacement function u(x) x

Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Express u(x) in terms of nodal displacements by using boundary conditions. Deformed shape u(0) = u1 u(L) = u2

Bar Element example Sub (2) into (1) Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Sub (2) into (1) Displacement polynomial that satisfies boundary conditions

Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Derive strain-displacement relationship by using mechanics theory Axial Strain

Bar Element example Derive stress-displacement relationship by using Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Derive stress-displacement relationship by using elasticity theory Axial Stress Elastic Modulus

Bar Element example Use principle of Virtual Work Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Use principle of Virtual Work Work = Stress x Strain x Volume Bar cross-sectional area A Internal work External work

Bar Element example Equate internal and external work Stiffness matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Equate internal and external work Stiffness matrix

Bar Element example Resultant stiffness matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Bar Element example Resultant stiffness matrix

Assembling issue (1); Element coords. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (1); Element coords. Element axes are not all the same. So there is a need for a coordinate transformation u v local axes u v local axes X Y Z Global axes

Assembling issue (1); Element coords. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (1); Element coords. for forces FX FZ fu fv q Fx1 Fz1 Fx2 Fz2 fu1 fv1 fv2 fu2 q (8)

Assembling issue (1); Element coords. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (1); Element coords. Similarly for displacement X1 Z1 X2 Z2 u1 v1 v2 u2 q (9)

Assembling issue (1); Element coords. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (1); Element coords. Element force-displacement in global coordinate Element nodal forces and displacements in local coordinates Local coordinates Element stiffness matrix in local coordinates (extended) Global coordinates Element nodal forces and displacements in global coordinates Element stiffness matrix in global coordinates

Assembling issue (1); Element coords. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (1); Element coords. Element stiffness matrix in global coordinates (10)

Assembling issue (2); Structure matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (2); Structure matrix Element and nodal numbering Node number 1 2 3 -45 90 P [kN] Element number 1 2 3 L m

Assembling issue (2); Structure matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (2); Structure matrix Create Structure stiffness matrix from element stiffness matrices 1,1 1,3 3,1 3,3

Assembling issue (2); Structure matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (2); Structure matrix 2,2 2,3 3,2 3,3

Assembling issue (2); Structure matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (2); Structure matrix 1,1 2,1 2,2 2,1

Assembling issue (2); Structure matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (2); Structure matrix a b i ith Element ith Element in Structure Stiffness Matrix with Node numbers a and b ith Element’s Stiffness Matrix with Node numbers a and b

Assembling issue (2); Structure matrix Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (2); Structure matrix vector of nodal forces vector of nodal displacements Structure stiffness matrix

Assembling issue (3); Supports Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (3); Supports How to deal with the problem of supports (restraints) ? These are nodes where the displacements are known, zero in the perfectly rigid support case. unknown support Reactions R known support Displacements Ds known applied nodal loads P unknown nodal Displacements D

Assembling issue (3); Supports Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Assembling issue (3); Supports This system of equations needs to be partitioned to determine the unknown nodal displacements and support reactions. Solve system of algebraic equations key numerical process Determine nodal displacements Determine support reactions

Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Finishing off; calculate element internal stresses, strains and actions. Calc. nodal displacements in local coordinates, eqn (9) Calc. element strains, equation (4) For Element 1 For Element 1 Negative sign indicates Compression

Finishing off; calculate element stresses, strains and actions. Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Finishing off; calculate element stresses, strains and actions. Calc. element stresses, equation (5) Calc. element axial forces For Element 1 Negative sign indicates Compression For Element 1

Summary of FEA * * key numerical process Divide complex structure into Dr Nick. A. Alexander, Department of Civil Engineering, University of Bristol Summary of FEA Divide complex structure into many finite elements connected at nodes Compute Element Stiffness Matrices (global Coords.) Assemble Structure Stiffness Matrix and applied load vector Introduce Supports (partitioning) Compute element internal stresses, strains and actions Compute element nodal displacements in local coords * * key numerical process Solve partitioned system determine reactions and nodal displacements