Advanced FEACoded in Matlab

JL Mantari, PhD

Contents• Plates theories

• Accuracy problem

• New theories

• FEA simple applications

• FEA layerwise

• FEA applications

Plate theories Existing theories

CPT (Classical plate theory), based on the assumptions of Kirchhoff’s plate theory.

FSDT (First order shear deformation theory), assumes constant transverse shear deformation and violates free surface boundary conditions.

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HSDT (Higher order shear deformation theory), assumes adequate transverse shear deformation and comply free surface boundary conditions.

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Plate theories Existing theories

ESL (Equivalent single layer) theories: CPT (Classical plate theory),FSDT (First shear deformation theory),HSDT (Higher order shear deformation theory),Others

Plate theories Existing theories

Layerwise theories:FSDTHSDTOthers

Plate theories Existing theories

Plate theories Accuracy problem

For unstiffened and stiffened ship type structures, HSDTs should be used because the deformation field is close to the deformation in real situation.

Variation of central deflection (mm) with varying geometric configurations

HSDTs

Plate theories Well-know HSDTs

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Plate theories New GHSDTs

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GHSDT1

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FEA Simple applications

Plates special case of shells. Beams special case of plates. Then, curved beams are special case of shells!

Beams Plates

Shells FGP

FEA bases A unified theory

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,

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Assume the following displacement field to begin the derivation of the unified theory I.

Laminate analysis. Steps for the validation of the theory

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,

.

Displacement field.Elasticity theory to get strains.Law of Hook to get stresses.The principle of virtual work and the variational theory to get,A set of differential equations that govern the plate and finally,Solve it with an appropriated solution (closed-form).

D E L T A S

Laminate analysis. Finite element method

,

,

.

Displacement field.Elasticity theory to get strains.Law of Hook to get stresses.The principle of virtual work, then use the discretization.Apply boundary conditions, and finallySolve it!

D E L T A S

1. When ESL FE codes are implemented by using: FSDT

A (Cº) FEM code can be implemented Requires arbitrary shear correction factors No comply free surface boundary condition

HSDT A (C ) FEM need to be implemented Normally non-conforming FEM are implementedA (Cº) FEM code can be also achieved

2. FEM using FSDT and HSDT have been explored since 1960s.

3. Layerwise FE codes do not need shear correction factors as some well-know ESL HSDT.

4. A new generalized layerwise HSDT is presented in this thesis.

Laminate analysis. Existing theories (FEM)

Laminate analysis. Trigonometric layerwise HSDT(FEM)

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Laminate analysis. Generalized layerwise HSDT

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Laminate analysis. Reducing DOFs

Shear continuity imposed

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Laminate analysis. Energy principle (FEM)

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Laminate analysis. Applications

Beams Plates

Stiffened plates FGP