Universitatea Politehnica TimişoaraFacultatea de ConstrucţiiDepartamentul de Construcţii Metalice şi Mecanica Construcţiilor

ĂPLĂCI CURBE SUBŢIRI

- CURS 8 -

EN 1993-1-6EN 1993 1 6 Proiectarea structurilor din oţelRezistenţa şi stabilitatea PCS (3)

Conf.dr.ing Adrian CIUTINA

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Buckling-relevant boundary conditionsFor the buckling limit state, special attention should be paid to the

g ( )

g , p pboundary conditions which are relevant to the incremental displacements of buckling.

examples of boundary conditions for limit state LS3conditions for limit state LS3

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Buckling-relevant geometrical tolerancesthe following tolerance limits should be observed if LS3 is one of the

g ( )

g S3ultimate limit states to be considered:

The geometric tolerances given Each of the imperfection types here are those that are known to have a large impact on the safety of the structure.

should be classified separately: the lowest class should then govern the entire design.

The fabrication tolerance quality class should be chosen as Class A, Class B or Class C (the description of each class relates only to the strength evaluation)strength evaluation).

Sample imperfection measurements should be undertaken on the unloaded structure (except for self weight) and, where possible, with the operational boundary conditionsboundary conditions.If the measurements of geometrical imperfections do not satisfy the geometrical tolerances limits, any correction steps, such as by straightening should be investigated and decided individuallystraightening, should be investigated and decided individually.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Out-of-roundness toleranceThe out-of-roundness should be assessed in terms

g ( )

of the parameter Ur:Where:

d is the maximum measured internal diameterdmax is the maximum measured internal diameter,dmin is the minimum measured internal diameter,dnom is the nominal internal diameter.

The out-of-roundness parameter Urshould satisfy the condition:

Values of the out-of-roundness tolerance parameter Ur,max

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Accidental eccentricity toleranceAt joints in shell walls perpendicular to membrane compressive forces,

g ( )

j p p p ,the accidental eccentricity should be evaluated from the measurable total eccentricity etot and the intended offset eint from:Where:Where:

etot is the eccentricity between the middle surfaces of the joined plates,eint is the intended offset between the middle surfaces of the joined plates.

Values of the maximum permitted accidental eccentricities

The accidental eccentricity ea should also be assessed in terms of the accidental eccentricity parameter Ue given by:Where:Where:tave is the mean thickness of the thinner and thicker plates at the joint.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Accidental eccentricity tolerance

g ( )

Accidental eccentricity and intended offset atand intended offset at a joint

The accidental eccentricity parameter U should satisfy the condition:The accidental eccentricity parameter Ue should satisfy the condition:

Values for accidental eccentricity tolerancesValues for accidental eccentricity tolerances

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)g ( )

Dimple tolerancesA dimple measurement gauge should be used in every position both p g g y pmeridional and circumferential directions.The depth ∆w0 of initial dimples in the shell wall should be measured using gaugesusing gauges.The depth of initial dimples should be assessed in terms of the dimple parameters U0x, U0θ, U0w given by:

Th l f th di l t U U U h ld ti f th

Where:lg is the gauge length

The value of the dimple parameters U0x, U0θ, U0w should satisfy the conditions:

Values for dimple tolerance parameter UValues for dimple tolerance parameter U0,max

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)g ( )

Measurement ofDimple tolerances

Measurement of depths ∆w0 of initial dimples

Interface flatness toleranceWhere another structure

ti l t h llcontinuously supports a shell (such as a foundation), its deviation from flatness at the interface should not include a local slope in the circumferential direction greater than βθ.g βθ

The value βθ = 0,1% = 0,001 radians is recommendedradians is recommended.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Stress designDesign values of stresses

g ( )

gThe design values of stresses σx,Ed, σθ,Ed and τxθ,Ed should be taken as

the key values of compressive and shear membrane stresses obtained from linear shell analysis (LA) Under purely axisymmetric conditions offrom linear shell analysis (LA). Under purely axisymmetric conditions of loading and support, and in other simple load cases, membrane theory may generally be used.

The key values of membrane stresses should be taken as the maximum value of each stress at that axial coordinate in the structure, unless specific provisions are given in Annex D. For basic loading cases the g gmembrane stresses may be taken from Annex A or Annex C.

Design resistance (buckling strength)Design resistance (buckling strength)The design buckling stresses should be obtained from:

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)g ( )

Design resistance (buckling strength)The buckling reduction factors χx, χθ and χt should be determined as a g χx, χθ χt

function of the relative slenderness of the shell λ − from:α is the elastic imperfection

reduction factorreduction factorβ is the plastic range factorη is the interaction exponentλ i h h li i l iλ0 is the squash limit relative

slendernessThe values of α, β, η and λ0 are found in Annex D.

The value of the plastic limit relativeThe value of the plastic limit relative slenderness λp should be determined from:

The relative shell slenderness parameters for different stress components should be determined from:

The critical buckling stresses σ σ and τ should be obtained The critical buckling stresses σx,Rcr, σθ,Rcr and τxθ,Rcr should be obtained by means of the relevant expressions in Annex D.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)g ( )

Stress limitation (buckling strength verification)The buckling limit state should be represented by limiting the design g p y g g

values of membrane stresses.The influence of bending stresses on the buckling strength may be

neglected provided they arise as a result of boundary compatibility effectsneglected provided they arise as a result of boundary compatibility effects. OBS: In the case of bending stresses from local loads or from thermal gradients, special consideration should be given.

Depending on the loading and stressing situation, one or more of the following checks should be carried out:

If more than one buckling-relevant membrane stress components are present the following interaction check should be carried out:

the buckling interactionthe buckling interaction parameters kx, kθ , kτ and ki aregiven in Annex D.

OBS: Where σx,Ed or σθ,Ed is tensile, its value should be taken as zero.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Design by global numerical analysis (MNA and LBA)Design resistance (buckling strength)

g ( )

es g es sta ce (buc g st e gt )The design buckling resistance shall be determined as a load factor R

applied to the design values of the combination of actions for the relevant load caseload case.

The design buckling resistance Rd should be obtained from the plastic reference resistance Rpl and the elastic critical buckling resistance Rcr, pcombining these to find the characteristic buckling resistance Rk. The partial factor γM1 should then be used to obtain the design resistance.

The plastic reference resistance Rplshould be obtained by materially non-linear analysis (MNA) as the plastic limit loadanalysis (MNA) as the plastic limit load under the applied combination of actions.

Where it is not possible to undertake a MNA the plastic reference Where it is not possible to undertake a MNA, the plastic reference resistance Rpl may be conservatively estimated from a LA.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Design resistance (buckling strength)The following steps should be followed in an MNA/LBA analysis

g ( )

g p / y

It has the same basis

The buckling strength verification is

It has the same basis as the traditional stress design buckling approachg g

performed through:buckling approach

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Design by global numerical GMNIA analysisDesign resistance (buckling strength)

g ( )

es g es sta ce (buc g st e gt )The design buckling resistance shall be determined as a load factor R

applied to the design values of the combination of actions for the relevant load caseload case.

The characteristic buckling resistance Rk should be found from the imperfect elastic-plastic critical buckling resistance RGMNIA adjusted by the calibration factor kGMNIA. The design buckling resistance Rd should then be found using the partial factor γM1.

To determine the imperfect elastic-plastic critical buckling resistance RGMNIA, a GMNIA analysis of the geometrically imperfect shell under the applied combination of actions should be carried out, accompanied by an pp , p yeigenvalue analysis to detect possible bifurcations in the load path.

The eigenvalue analysis should use the deformation theory of plasticity.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Design by global numerical GMNIA analysisDeveloped to exploit the full power of modern numerical analysis

g ( )

Application is more complex than for frame or plated structuresSeveral sequence of analysis:

LA f ll d b LBA t l t l ti iti l b kli i tLA followed by a LBA to evaluate elastic critical buckling resistanceGMNA to identify the elastic-plastic buckling resistance of the perfect structureGMNIA with different imperfection modes (the lowest value is selected)Check the precision of the GMNIA by comparison with test or other relevant data

Methodology:Action combinations causing compressive membrane stresses or shear membrane stressesmembrane stressesRk should be found from the imperfect elastic-plastic critical buckling resistance RGMNIA, adjusted by the calibration factor kGMNIA. Th d i b kli i t Rd h ld th b f d i th ti lThe design buckling resistance Rd should then be found using the partial factor γM1.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)

Design by global numerical GMNIA analysis

g ( )

C1: The maximum load factor on the load-deformation-curve (limit load);C2: The bifurcation load factorC3: The largest tolerableC3: The largest tolerable deformation

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)g ( )

Design by global numerical GMNIA analysisAllowances for imperfections:Allowances for imperfections:

geometric imperfections: pre-deformations, out of-roundness, irregularities at and near welds, thickness deviation, etcmaterial imperfections: residual stresses inhomogeneities anisotropiesmaterial imperfections: residual stresses, inhomogeneities, anisotropies

EN 1993-1-6 requires that imperfections are explicitly modelled numerically (not treated as small perturbations in geometry)

They are introduced by means of equivalent geometric imperfectionsThe form of the imperfections with the most unfavourable effect should

be consideredbe consideredThe amplitude of the imperfection form - dependent on the fabrication

tolerance quality class.

Usually, the designer take into account the imperfections through equivalent geometric imperfections which cover all other types of imperfections.

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Buckling limit state (LS3)g ( )

Design by global numerical GMNIA analysisAllowances for imperfections:Allowances for imperfections:

Fabrication tolerance quality class

Description Recommended value of Un1

Recommended value of Un2

Class A Excellent 0,010 0,010 Cl B Hi h 0 016 0 016Class B High 0,016 0,016 Class C Normal 0,025 0,025

( )0 0 1 0 2w max w wΔ = Δ Δ( )0,eq 0,eq,1 0,eq,2

0,eq,1 g 1

:

w max w , w

w n

where

l U

Δ = Δ Δ

Δ =

lg relevant gauge lengthst local shell wall thicknessni multiplier to achieve an appropriatetolerance level

0,eq,2 2w i nn tUΔ = Un1, Un2 dimple imperfection

EN 1993-1-6 DESIGN CHECKING OF SHELLS 8. Fatigue limit state (LS4)g ( )

GeneralThe anticipated action spectrum throughout the design life of the The anticipated action spectrum throughout the design life of the

structureRelevant action spectra - EN 1991 and appropriate parts of EN 1993F i d i h b i f l i l i (LA GNA)Fatigue design on the basis of an elastic analysis (LA or GNA)If a three dimensional finite element analysis is used, the notch effects

due to the local weld geometry should be eliminatedg yStress range limitation

In every verification of this limit state, the design stress range should satisfy the condition: As an alternative a satisfy the condition:Where:γFf is the partial factor for the fatigue loading

i th ti l f t f th f ti i t

As an alternative a cumulative damage assessment may be made using the γMf is the partial factor for the fatigue resistance

∆σE is the equivalent constant amplitude stress range of the design stress spectrum∆ is the fatigue strength stress range for the relevant

made using the Palmgren-Miner rule:

with∆σR is the fatigue strength stress range for the relevant

detail category and the number of cycles of the stress spectrum.