Comments on FEA Minimum total potential energy principle

Finite element method - tutorial no. 3

Martin NESLADEK

Faculty of mechanical engineering, CTU in Prague

10th November 2015

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Comments on FEA Minimum total potential energy principle

Comments on problems analysed by FEM

Accuracy of the FEM solution is influenced by these majorfactors:

Level of spatial idealisation of geometry (e.g. plane stress or strainstate has to be chosen a priori if 2D problem is considered)Accuracy of material properties and suitability of the selectedmaterial modelBoundary conditions - accuracy and level of idealisation of appliedloads and kinematic conditionsInclusion of details of structure (chamfers, fillet radii, threads,spline coupling teeth, ...)Element type (linear vs. quadratic)Element distortion (regularity of mesh)Mesh density

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Comments on FEA Minimum total potential energy principle

Inclusion or exclusion of details of structure

Excluding a detail, which is a real weak point in terms of structural integrity ofcomponent, from the simulation model may lead to improper design of the part

Example: removing the fillet radius of the part from tutorial no. 2

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Comments on FEA Minimum total potential energy principle

Inclusion or exclusion of details of structure

In general, from the simulation model we cannot exclude detailswhich may be weak points, or those design components thatinfluence stress-strain state in the weak point directly

Otherwise, it is good practice to make geometry of simulationmodels as simple as possible to eliminate extensive model mesh

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Comments on FEA Minimum total potential energy principle

Influence of element typeTwo basic types are usually implemented in commercial codes:

“Linear” element - distribution of displacements on the domain ofelement is governed by linear shape functions

“Quadratic” element - quadratic shape functions instead of linearare applied

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Comments on FEA Minimum total potential energy principle

Influence of element type

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Comments on FEA Minimum total potential energy principle

Influence of element type

Mesh formed by linear elements converges more slowly than thequadratic type

So the linear elements hardly capture local high stress gradientsaccurately enough

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Comments on FEA Minimum total potential energy principle

Influence of element distortion

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Comments on FEA Minimum total potential energy principle

Influence of element distortion

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Comments on FEA Minimum total potential energy principle

Influence of element distortion

Elements with high aspect ratio of edges and excessively sharpcorner angles should be avoided to prevent numericalinaccuracies

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Comments on FEA Minimum total potential energy principle

Influence of mesh density

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Comments on FEA Minimum total potential energy principle

Minimum total potential energy principle(MTPEP)

How is MTPEP related to FEM?1 Total potential energy of solids implicitly contains equilibrium

equations of external and internal forces. The MTPEP isa strategy for finding the solution that satisfies the equilibrium.

2 FEM for structural analysis may be derived by using MTPEP

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Comments on FEA Minimum total potential energy principle

Basic terminology

Total potential energy of elastic body:

Π = U + W

U . . . strain energyW . . . potential energy of external forces

The above expression is a functional

note.: the functional is transformation of a function to a realnumber

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Comments on FEA Minimum total potential energy principle

Basic terminology

Minimum total potential energy principleDisplacement field u(x , y , z), which is a solution of a structuralanalysis problem, minimises the functional of the total potentialenergy Π

Admissible displacement fieldOnly the so-called admissible displacement field u(x , y , z) maysatisfy the equilibrium. “Admissibility” of a displacement fieldmeans that boundary conditions are met and that the firstderivative of the field is continuous.

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Comments on FEA Minimum total potential energy principle

Basic terminology

F

ϕ(x)v(x)

x

v

The Bernoulli-Euler differential equation of beam deflection

v ′′(x) =d2v(x)

dx2 =−M(x)

E · J

Strain energy of a Bernoulli-Euler beam

U =

∫ l

0

M2(x)

2E · Jdx

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