NUMERICAL METHODS IN APPLIED STRUCTURAL MECHANICS Lecture notes: Prof. Maurício V. Donadon

Non-linear static problems

Introduction

Sources of nonlinearities in structural analysis Geometrical non-linearity Non-linear material behaviour Non-linear boundary conditions

Geometrical nonlinearities Normal strain-displacement relationships

Geometrical nonlinearities Shear strain-displacement relationships

Geometrical nonlinearities

Nonlinear material behaviour ELASTIC ELASTO-PLASTIC MICROCRACKING ELASTIC +

Nonlinear material behaviour

Nonlinear material behaviour

Nonlinear boundary conditions Transient boundary problems: Boundary conditions change during the analysis!!!

Nonlinear boundary conditions

Solution methods for non-linear static problems

Solution methods Incremental solutions Iterative solutions Combined incremental/iterative solutions Arc-length method Quasi-static solutions

General form for a static problem Example: Linear/Nonlinear Spring K = K0 f(x) Fe

Example: Linear/Nonlinear Spring

General form for a static problem Example: Linear/Nonlinear Spring Trivial solution: Displacement control Non-trivial solution: Load control which is commonly used in structural analyses!!!

Incremental solution

INCREMENTAL SOLUTION METHOD BASED ON THE EULER METHOD

THE EULER METHOD ALGORITHM

Example: Nonlinear Spring

Example: Nonlinear Spring

Example: Nonlinear Spring

Iterative solution

ITERATIVE SOLUTION BASED ON THE NEWTON RAPHSON METHOD

THE NEWTON RAPHSON ALGORITHM

Example: Nonlinear Spring

Example: Nonlinear Spring

Example: Nonlinear Spring

Combined incremental/iterative solutions

COMBINED INCREMENTAL/ITERATIVE SOLUTIONS

INCREMENTAL/ITERATIVE SOLUTION ALGORITHM

Example: Nonlinear Spring

Example: Nonlinear Spring

Example: Nonlinear Spring

Arc-length Method

Highly non-linear structural responses snap-through snap-back

Arc-length method: Constraint equation ψ: scale factor, scale forces to the same order of magnitude of the displacements qef: Fixed external load level vector

Arc-length method: Residual force Equations to be solved simultaneously

Arc-length method Subscript “o” refers to old (previous) iteration Subscript “n” refers to current iteration

Equations to be solved simultaneously Augmented stiffness matrix Arc-length method Equations to be solved simultaneously Augmented stiffness matrix

Arc-length method Error function computation:

Spherical Arc-length method The computational cost associated with the inversion of the augmented stiffness matrix during the iterations is very high because the augmented stiffness matrix is neither symmetric nor banded! Better solution: Spherical Arc-length!!!!

Spherical Arc-length method Instead of solving both constraint and equilibrium equations simultaneously, one may assume displacement control at single point. Thus, by assuming displacement control the residual forces can be written as follow,

Spherical Arc-length method δpt must be computed for the initial predictor step based on the Forward-Euler tangential predictor and it is fixed because Kt does not change during the iterations

Forward-Euler Tangential Predictor Spherical Arc-length method Forward-Euler Tangential Predictor Substituting into the constraint equation we obtain,

Spherical Arc-length method

Spherical Arc-length method δλ can be found by substituting the previous equation into the constraint equation, which leads to the following quadratic equation,

Spherical Arc-length method with, Which can be solved for δλ.The choice of the root will be based on the cylindrical Arc-length method

Spherical Arc-length method The great advantage of the spherical arc-length over the standard arc-length method is that the former only requires the factorisation of the banded symmetric tangent stiffness matrix. Therefore it avoids the use of the augmented stiffness matrix, which is neither symmetric nor banded!!!

Cylindrical Arc-length method The cylindrical Arc-length consists of setting to zero the scaling factor ψ. In practice, this parameter is not known a priori. Moreover, setting to zero the scaling factor simplifies the choice of the appropriated root of the quadratic equation used to compute δλ, that is,

Cylindrical Arc-length method Choosing the root Solution 1 Solution 2

Cylindrical Arc-length method Choosing the root

Cylindrical Arc-length method Choosing the root Having computed δλ, update displacement and load parameter

Convergence criterion Cylindrical Arc-length method Convergence criterion

Quasi-static solutions

Example: Nonlinear Spring DYNAMIC RELAXATION Example: Nonlinear Spring K = K0 f(x) M Fe(t) C

DYNAMIC RELAXATION Central difference method Critical time step computation

DYNAMIC RELAXATION Central difference method Displacement field Velocity field

DYNAMIC RELAXATION Damping definition Critical damping Rayleigh damping

EXPLICIT TIME INTEGRATION ALGORITHM Initial conditions, v0, σ0, n=0, t=0, compute M Compute acceleration an = M-1Fe,n Update nodal velocities: vn+1/2 = vn+1/2-α + αΔtan α = 1/2 if n=0 α = 1 if n>0 Update nodal displacements: un+1 = un+ Δtvn+1/2 Compute strains Compute stresses Compute internal forces Compute residual force vector: Fi - Fe Update counter and time: n = n+1, t = t+Δt If simulation not complete go to step 2

Example: Nonlinear Spring

Example: Nonlinear Spring

Example: Nonlinear Spring – 1.0 N/s

Example: Nonlinear Spring – 0.1 N/s

Example: Nonlinear Spring – 0.01 N/s

Example: Nonlinear Spring – Damping effect Thigh=0.12 Tlow=0.22

Example: Nonlinear Spring – Damping effect

Example: Nonlinear Spring – Damping effect

Example: Nonlinear Spring – Damping effect

Example: Nonlinear Spring – Damping effect

Over damping effects in dynamic relaxation Over damping MUST BE AVOIDED in dynamic relaxation methods! Special care must be taken with over damping Over damping increases artificially the internal energy of the system!!!!