Projecto Avançado de Estruturas Navais

Mestrado em Engenharia Naval

Contents (1)• Structural Stability – basic principles

– Linear approach– Non-linear response

• Non-linear behavior of beam-columns and plates subjected to compressive and combined loading.– Influence of initial imperfections

• Buckling of plates and stiffened plates– Initial imperfections– Residual stresses

Contents (2)• Ultimate strength of plates and stiffened plates

– Concept of effective width– Reduction of strength due to imperfections and

welding• Residual strength and post buckling behavior of

components• Hull girder strength

– Linear and Non-linear behavior– Ultimate bending moment– Residual strength

Structural StabilityBasic principles

• Equilibrium stability concept– A) Stable– B) Unstable– C) Neutral

• Structural limit states– Ultimate limit

• Global Failure• Local Failure

– Serviceable limit

Structural Stability

• Stability and Instability– Ex: Euler column

• EI – Structural modulus

2

2EEIPl

sin xw x q

l

Equilibrium state

• A) Stable• B) Unstable• C) Stable

Transition between differentstates of equilibrium

Types of instability•Bifurcation•Snap-through

Euler columnP/PE=1+(q/l)2/8

Local Buckling

22

23 1crE t

b

2

231 18cr

qt

Cylindrical shell

2222 4

2 22

3 11

43 1 where cr

E t bb R t

Snap-through

Non linearityPhysical non-linearityGeometrical non-linearity

Bifurcation Instability

2

2

0 1 0

0

1

A

cr

M Pq Kql q

q

P Kl qP Kl

Snap-through Instability

0

2 sin

coscos

2 sin tan cos

: 0

vF

P K

ll

P Kl

dPSnapd

Initial Imperfections

Initial Imperfections (Example)

2112

P Kl qq

Material (Plasticity)

• Plasticity• Residual stresses

Stability Criteria

• Static criteria– Adjacent equilibrium criteria (bifurcation)– Energy criteria (Minimum Potential Energy)

Linear Analysis

• Discrete systems• Continuous systems

• Aproximate methods– Finite differences method– Engesser-Newmark method– Galerkin method (diferential equations)– Rayleigh-Ritz method (potential energy)– Finite element method

Basic concepts• Effective length

– Boundary conditions• Effective width

– Boundary conditions– Loading– Manufacturing

• Initial geometric imperfections– Residual stresses and welding procedure– Mounting process and restraining

• Residual stresses– Welding procedure– Restraining– Physical properties

• Loading of the component– Combined loading– Boundary conditions