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EUROPEAN STANDARD
NORME EUROPÉENNE
EUROPÄISCHE NORM

EN 1993-1-7

April 2007

ICS 91.010.30; 91.080.10

Supersedes EN V 1993-1-7:1999
Incorporating corrigendum April 2009

English Version

Eurocode 3 - Design of steel structures - Part 1–7: Plated structures subject to out of plane loading

Eurocode 3 - Calcul des structures en acier - Partie 1–7: Résistance et stabilité des structures en plaques planes chargées hors de leur plan       Eurocode 3 - Bemessung und Konstruktion von Stahlbauten - Teil 1–7: Plattenförmige Bauteile mit Querbelastung

This European Standard was approved by CEN on 12 June 2006.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the CEN Management Centre or to any CEN member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CEN member into its own language and notified to the CEN Management Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.

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© 2007 CEN All rights of exploitation in any form and by any means reserved worldwide for CEN national Members.

Ref. No. EN 1993–1–7:2007: E

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Content

Page
Foreword 3
1 General 4
  1.1 Scope 4
  1.2 Normative references 4
  1.3 Terms and definitions 5
  1.4 Symbols 6
2 Basis of design 9
  2.1 Requirements 9
  2.2 Principles of limit state design 9
  2.3 Actions 9
  2.4 Design assisted by testing 10
3 Material properties 10
4 Durability 10
5 Structural analysis 10
  5.1 General 10
  5.2 Stress resultants in the plate 10
6 Ultimate limit state 15
  6.1 General 15
  6.2 Plastic limit 15
  6.3 Cyclic plasticity 16
  6.4 Buckling resistance 17
7 Fatigue 18
8 Serviceability limit state 18
  8.1 General 18
  8.2 Out of plane deflection 18
  8.3 Excessive vibrations 18
Annex A [informative] – Types of analysis for the design of plated structures 19
  A.l General 19
  A.2 Linear elastic plate analysis (LA) 19
  A.3 Geometrically nonlinear analysis (GNA) 19
  A.4 Materially nonlinear analysis (MNA) 20
  A.5 Geometrically and materially nonlinear analysis (GMNA) 20
  A.6 Geometrically nonlinear analysis elastic with imperfections included (GNIA) 20
  A.7 Geometrically and materially nonlinear analysis with imperfections included (GMNIA) 20
Annex B [informative] – Internal stresses of unstiffened rectangular plates from small deflection theory 21
  B.l General 21
  B.2 Symbols 21
  B.3 Uniformly distributed loading 21
  B.4 Central patch loading 24
Annex C [informative] – Internal stresses of unstiffened rectangular plates from large deflection theory 26
  C.1 General 26
  C.2 Symbols 26
  C.3 Uniformly distributed loading on the total surface of the place 26
  C.4 Central patch loading 32
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Foreword

Foreword

This European Standard EN 1993-1-7, Eurocode 3: Design of steel structures: Part 1–7 Plated structures subject to out of plane loading, has been prepared by Technical Committee CEN/TC250 « Structural Eurocodes », the Secretariat of which is held by BSI. CEN/TC250 is responsible for all Structural Eurocodes.

This European Standard shall be given the status of a National Standard, either by publication of an identical text or by endorsement, at the latest by October 2007, and conflicting National Standards shall be withdrawn at latest by March 2010.

This Eurocode supersedes ENV 1993-1-7.

According to the CEN-CENELEC Internal Regulations, the National Standard Organizations of the following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.

National annex for EN 1993-1-7

This standard gives alternative procedures, values and recommendations with notes indicating where national choices may have to be made. The National Standard implementing EN 1993-1-7 should have a National Annex containing all Nationally Determined Parameters to be used for the design of steel structures to be constructed in the relevant country.

National choice is allowed in EN 1993-1-7 through:

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1 General

1.1 Scope

  1. P EN 1993-1-7 provides basic design rules for the structural design of unstiffened and stiffened plates which form part of plated structures such as silos, tanks or containers, that are loaded by out of plane actions. It is intended to be used in conjunction with EN 1993-1-1 and the relevant application standards.
  2. This document defines the design values of the resistances: the partial factor for resistances may be taken from National Annexes of the relevant application standards. Recommended values are given in the relevant application standards.
  3. This Standard is concerned with the requirements for design against the ultimate limit state of:
  4. Overall equilibrium of the structure (sliding, uplifting, overturning) is not included in this Standard, but is treated in EN 1993-1-1. Special considerations for specific applications may be found in the relevant applications parts of EN 1993.
  5. The rules in this Standard refer to plate segments in plated structures which may be stiffened or unstiffened. These plate segments may be individual plates or parts of a plated structure. They are loaded by out of plane actions.
  6. For the verification of unstiffened and stiffened plated structures loaded only by in-plane effects see EN 1993-1-5. In EN 1993-1-7 rules for the interaction between the effects of inplane and out of plane loading are given.
  7. For the design rules for cold formed members and sheeting see EN 1993-1-3.
  8. The temperature range within which the rules of this Standard are allowed to be applied are defined in the relevant application parts of EN 1993.
  9. The rules in this Standard refer to structures constructed in compliance with the execution specification of EN 1090-2.
  10. Wind loading and bulk solids flow should be treated as quasi-static actions. For fatigue, the dynamic effects must be taken into account according to EN 1993-1-9. The stress resultants arising from the dynamic behaviour are treated in this part as quasi-static.

1.2 Normative references

  1. This European Standard incorporates, by dated or undated reference, provisions from other publications. These normative references are cited at the appropriate places in the text and the publications are listed hereafter. For dated references, subsequent amendments to or revisions of any of these publications apply to this European Standard only when incorporated in it by amendment or revision. For undated references the latest edition of the publication referred to applies.
    EN 1993 Eurocode 3: Design of steel structures:
    Part 1.1: General rules and rules for buildings
    Part 1.3: Cold-formed members and sheeting
    Part 1.4: Stainless steels
    Part 1.5: Plated structural elements 4
    Part 1.6: Strength and stability of shell structures
    Part 1.8 : Design of joints
    Part 1.9: Fatigue strength of steel structures
    Part 1.10: Selection of steel for fracture toughness and through-thickness properties
    Part 1.12: Additional rules for the extension of EN 1993 up to steel grades S700
    Part 4.1: Silos
    Part 4.2: Tanks

1.3 Terms and definitions

  1. The rules in EN 1990, clause 1.5 apply.
  2. The following terms and definitions are supplementary to those used in EN 1993-1-1:

1.3.1 Structural forms and geometry

1.3.1.1 Plated structure

A structure that is built up from nominally flat plates which are joined together. The plates may be stiffened or unstiffened, see Figure 1.1.

Figure 1.1 : Components of a plated structure

Figure 1.1 : Components of a plated structure

1.3.1.2 Plate segment

A plate segment is a flat plate which may be unstiffened or stiffened. A plate segment should be regarded as an individual part of a plated structure.

1.3.1.3 Stiffener

A plate or a section attached to the plate with the purpose of preventing buckling of the plate or reinforcing it against local loads. A stiffener is denoted:

5
1.3.1.4 Stiffened plate

Plate with transverse and/or longitudinal stiffeners.

1.3.1.5 Sub-panel

Unstiffened plate surrounded by stiffeners or, on a web, by flanges and/or stiffeners or, on a flange, by webs and/or stiffeners.

1.3.2 Terminology

1.3.2.1 Plastic collapse

A failure mode at the ultimate limit state where the structure loses its ability to resist increased loading due to the development of a plastic mechanism.

1.3.2.2 Tensile rupture

A failure mode in the ultimate limit state where failure of the plate occurs due to tension.

1.3.2.3 Cyclic plasticity

Where repeated yielding is caused by cycles of loading and unloading.

1.3.2.4 Buckling

Where the structure looses its stability under compression and/or shear.

1.3.2.5 Fatigue

Where cyclic loading causes cracking or failure.

1.3.3 Actions

1.3.3.1 Out of plane loading

The load applied normal to the middle surface of a plate segment.

1.3.3.2 In-plane forces

Forces applied parallel to the surface of the plate segment. They are induced by in-plane effects (for example temperature and friction effects) or by global loads applied at the plated structure.

1.4 Symbols

  1. In addition to those given in EN 1990 and EN 1993-1-1, the following symbols are used:
  2. Membrane stresses in rectangular plate, see Figure 1.2:
    σmx is the membrane normal stress in the x-direction due to membrane normal stress resultant per unit width nx;
    σmy is the membrane normal stress in the y-direction due to membrane normal stress resultant per unit width ny;
    τmxy is the membrane shear stress due to membrane shear stress resultant per unit width nxy.
    6

    Figure 1.2: Membrane stresses

    Figure 1.2: Membrane stresses

  3. Bending and shear stresses in rectangular plates due to bending, see Figure l.3:
    σbx is the stress in the x-direction due to bending moment per unit width mx;
    σby is the stress in the y-direction due to bending moment per unit width my;
    τbxy is the shear stress due to the twisting moment per unit width mxy;
    τbxz is the shear stress due to transverse shear forces per unit width qx associated with bending;
    τbyz is the shear stress due to transverse shear forces qy associated with bending.

    Figure 1.3: Normal and shear stresses due to bending

    Figure 1.3: Normal and shear stresses due to bending

    NOTE: In general, there are eight stress resultants in a plate at any point. The shear stresses τbxzand τbyz due to qx and qy are in most practical cases insignificant compared to the other components of stress, and therefore they may normally be disregarded for the design.

  4. Greek lower case letters:
    α aspect ratio of a plate segment (a/b);
    ε strain;
    αR load amplification factor;
    ρ reduction factor for plate buckling;
    σi Normal stress in the direction i, see Figure 1.2 and Figure 1.3;7
    τ Shear stress, see Figure l .2 and Figure l .3;
    v Poisson’s ratio;
    γM partial factor.
  5. Latin upper case letter:
    E Modulus of elasticity
  6. Latin lower case letters:
    a length of a plate segment, see Figure 1.4 and Figure 1.5;
    b width of a plate segment, see Figure 1.4 and Figure 1.5;
    fyk yield stress or 0,2% proof stress for material with non linear stress-strain curve;
    ni membrane normal force in the direction i [kN/m];
    nxy membrane shear force [kN/m]
    m bending moment [kNm/m];
    qz transverse shear force in the z direction [kN/m];
    t thickness of a plate segment, see figure 1.4 and 1.5.

    NOTE: Symbols and notations which are not listed above are explained in the text where they first appear.

    Figure 1.4: Dimensions and axes of unstiffened plate segments

    Figure 1.4: Dimensions and axes of unstiffened plate segments

    Figure 1.5: Dimensions and axes of stiffened plate segments; stiffeners may be open or closed stiffeners

    Figure 1.5: Dimensions and axes of stiffened plate segments; stiffeners may be open or closed stiffeners

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2 Basis of design

2.1 Requirements

  1. P The basis of design shall be in accordance with EN 1990.
  2. P The following ultimate limit states shall be checked for a plated structure:
  3. The design of a plated structure should satisfy the serviceability requirements set out in the appropriate application standards.

2.2 Principles of limit state design

2.2.1 General

  1. P The principles for ultimate limit state given in section 2 of EN 1993-1-1 and EN 1993-1-6 shall also be applied to plated structures.

2.2.2 Plastic collapse

  1. Plastic collapse is defined as the condition in which a part of the structure develops excessive plastic deformations, associated with development of a plastic mechanism. The plastic collapse load is usually derived from a mechanism based on small deflection theory.

2.2.3 Cyclic plasticity

  1. Cyclic plasticity should be taken as the limit condition for repeated cycles of loading and unloading produce yielding in tension or in compression or both at the same point, thus causing plastic work to be repeatedly done on the structure. This alternative yielding may lead to local cracking by exhaustion of the material’s energy absorption capacity, and is thus a low cycle fatigue restriction. The stresses which are associated with this limit state develop under a combination of all actions and the compatibility conditions for the structure.

2.2.4 Buckling

  1. Buckling should be taken as the condition in which all or parts of the structure develop large displacements, caused by instability under compressive and/or shear stresses in the plate. It leads eventually to inability to sustain an increase in the stress resultants.
  2. Local plate buckling, see EN 1993-1-5.
  3. For flexural, lateral torsional and distortional stability of stiffeners, see EN 1993-1-5

2.2.5 Fatigue

  1. Fatigue should be taken as the limit condition caused by the development and / or growth of cracks by repeated cycles of increasing and decreasing stresses.

2.3 Actions

  1. The characteristic values of actions should be determined from the appropriate parts of EN 1991.
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2.4 Design assisted by testing

  1. For design assisted by testing reference should be made to section 2.5 of EN 1993-1-1 and where relevant, Section 9 of EN 1993-1-3.

3 Material properties

  1. This Standard covers the design of plated structures fabricated from steel material conforming to the product standards listed in EN 1993-1-1 and EN 1993-1-12.
  2. The material properties of cold formed members and sheeting should be obtained from EN 1993-1-3.
  3. The material properties of stainless steels should be obtained from EN 1993-1-4.

4 Durability

  1. For durability see section 4 of EN 1993-1-1.

5 Structural analysis

5.1 General

  1. P The models used for calculations shall be appropriate for predicting the structural behaviour and the limit states considered.
  2. If the boundary conditions can be conservatively defined, i.e. restrained or unrestrained, a plated structure may be subdivided into individual plate segments that may be analysed independently.
  3. P The overall stability of the complete structure shall be checked following the relevant parts of EN 1993.

5.2 Stress resultants in the plate

5.2.1 General

  1. The calculation model and basic assumptions for determining internal stresses or stress resultants should correspond to the assumed structural response for the ultimate limit state loading.
  2. Structural models may be simplified such that it can be shown that the simplifications used will give conservative estimates of the effects of actions.
  3. Elastic global analysis should generally be used for plated structures. Where fatigue is likely to occur, plastic global analysis should not be used.
  4. Possible deviations from the assumed directions or positions of actions should be considered.
  5. Yield line analysis may be used in the ultimate limit state when inplane compression or shear is less than 10% of the corresponding resistance. The bending resistance in a yield line should be taken as

Image

5.2.2 Plate boundary conditions

  1. Boundary conditions assumed in analyses should be appropriate to the limit states considered. 10
  2. P If a plated structure is subdivided into individual plate segments the boundary conditions assumed for stiffeners in individual plate segments in the design calculations shall be recorded in the drawings and project specification.

5.2.3 Design models for plated structures

5.2.3.1 General
  1. The internal stresses of a plate segment should be determined as follows:
  2. The design methods given in (1) should take into account a linear or non linear bending theory for plates as appropriate.
  3. A linear bending theory is based on small-deflection assumptions and relates loads to deformations in a proportional manner. This may be used if inplane compression or shear is less than 10% of the corresponding resistance.
  4. A non-linear bending theory is based on large-deflection assumptions and the effects of deformation on equilibrium are taken into account.
  5. The design models given in (1) may be based on the types of analysis given in Table 5.1.
    Table 5.1 : Types of analysis
    Type of analysis Bending theory Material law Plate geometry
    Linear elastic plate analysis (LA) linear linear perfect
    Geometrically non-linear elastic analysis (GNA) non-linear linear perfect
    Materially non-linear analysis (MNA) linear non-linear perfect
    Geometrically and materially non-linear analysis (GMNA) non-linear non-linear perfect
    Geometrically non-linear elastic analysis with imperfections (GNIA) non-linear linear imperfect
    Geometrically and materially non-linear analysis with imperfections (GMNIA) non-linear non-linear imperfect

    NOTE 1: A definition of the different types of analysis is given in Annex A.

    NOTE 2: The type of analysis appropriate to a structure should be stated in the project specification.

    NOTE 3: The use of a model with perfect geometry implies that geometrical imperfections are either not relevant or included through other design provisions.

    NOTE 4: Amplitudes for geometrical imperfections for imperfect geometries are chosen such that in comparisons with results from tests using test specimens fabricated with tolerances according to EN 1090-2 the calculative results are reliable, therefore these amplitudes in general differ from the tolerances given in EN 1090-2.

5.2.3.2 Use of standard formulas
  1. For an individual plate segment of a plated structure the internal stresses may be calculated for the relevant combination of design actions with appropriate design formulae based on the types of analysis given in 5.2.3.1. 11

    NOTE: Annex B and Annex C provide tabulated values for rectangular unstiffened plates which are loaded transversely. For circular plates design formulas are given in EN 1993-1-6. Further design formulas may be used, if the reliability of the design formulas is in accordance with the requirements given in EN 1991-1.

  2. In case of a two dimensional stress field resulting from a membrane theory analysis the equivalent Von Mises stress σeq.Ed may be determined by

    Image

  3. In case of a two dimensional stress field resulting from an elastic plate theory the equivalent Von Mises stress σeq.Ed may be determined, as follows:

    Image

    where

    Image

    and nx,Ed, ny,Ed, nxy,Ed, my,Ed and mxy,Ed are defined in 1.4(1) and (2).

    NOTE: The above expressions give a simplified conservative equivalent stress for design

5.2.3.3 Use of a global analysis: numerical analysis
  1. If the internal stresses of a plated structure are determined by a numerical analysis which is based on a materially linear analysis, the maximum equivalent Von Mises stress σeq,Ed of the plated structure should be calculated for the relevant combination of design actions.
  2. The equivalent Von Mises stress σeq,Ed is defined by the stress components which occurred at one point in the plated structure.

    Image

    where σx,Ed and σy,Ed are positive in case of tension.

  3. If a numerical analysis is used for the verification of buckling, the effects of imperfections should be taken into account. These imperfections may be:
    1. geometrical imperfections:
      • – deviations from the nominal geometric shape of the plate (initial deformation, out of plane deflections);
      • – irregularities of welds (minor eccentricities);
      • – deviations from nominal thickness.
    2. material imperfections:
      • – residual stresses because of rolling, pressing, welding, straightening;
      • – non-homogeneities and anisotropics. 12
  4. The geometrical and material imperfections should be taken into account by an initial equivalent geometric imperfection of the perfect plate. The shape of the initial equivalent geometric imperfection should be derived from the relevant buckling mode.
  5. The amplitude of the initial equivalent geometric imperfection e0 of a rectangular plate segment may be derived by numerical calibrations with test results from test pieces that may be considered as representative for fabrication from the plate buckling curve of EN 1993-1-5, as follows:

    Image

    where

    Image

    ρ is the reduction factor for plate buckling as defined in 4.4 of EN 1993-1-5;
    a,b are geometric properties of the plate, see Figure 5.1;
    t is the thickness of the plate;
    α is the aspect ratio Image;
    Image is the relative slenderness of the plate, see EN 1993-1-5.

    Figure 5.1: Initial equivalent geometric bow imperfection e0 of a plate segment

    Figure 5.1: Initial equivalent geometric bow imperfection e0 of a plate segment

  6. As a conservative assumption the amplitude may be taken as e0 = a/200 where b ≤ a.
  7. The pattern of the equivalent geometric imperfections should, if relevant, be adapted to the constructional detailing and to imperfections expected from fabricating or manufacturing.
  8. P In all cases the reliability of a numerical analysis shall be checked with known results from tests or compared analysis.
5.2.3.4 Use of simplified design methods
5.2.3.4.1 General
  1. The internal forces or stresses of a plated structure loaded by out of plane loads and in-plane loads may be determined using a simplified design model that gives conservative estimates.
  2. Therefore the plated structure may be subdivided into individual plate segments, which may be stiffened or unstiffened.
5.2.3.4.2 Unstiffened plate segments
  1. An unstiffened rectangular plate under out of plane loads may be modeled as an equivalent beam in the direction of the dominant load transfer, if the following conditions are fulfilled:
  2. The internal forces and moments of the equivalent beam should be determined using an elastic or plastic analysis as defined in EN 1993-1-1.
  3. If the first order deflections due to the out of plane loads is similar to the (plate) buckling mode due to the in plane compression forces, the interaction between both phenomena need to be taken into account.
  4. In cases where the situation as described in (3) is present the interaction formula specified in EN 1993-1-1, section 6.3.3 may be applied to the equivalent beam.
5.2.3.4.3 Stiffened plate segments
  1. A stiffened plate or a stiffened plate segment may be modeled as a grillage if it is regularly stiffened in the transverse and longitudinal direction.
  2. In determining the cross-sectional area Ai of the cooperating plate of an individual member i of the grillage the effects of shear lag should be taken into account by the reduction factor β according to EN 1993-1-5.
  3. For a member i of the grillage which is arranged in parallel to the direction of inplane compression forces, the cross-sectional area Ai should also be determined taking account of the effective width of the adjacent subpanels due to plate buckling according to EN 1993-1-5.
  4. The interaction between shear lag effects and plate buckling effects, see Figure 5.2, should be considered by the effective area Ai from the following equation:

    Image

    where

    AL,eff is the effective area of the stiffener considering to local plate buckling of the stiffener;
    ρc is the reduction factor due to global plate buckling of the stiffened plate segment, as defined in 4.5.4(1) of EN 1993-1-5;
    ρpan,i is the reduction factor due to local plate buckling of the snbpanel i, as defined in 4.4(1) of EN 1993-1-5;
    bpan,i is the width of the subpanel i, as defined in 4.5.1(3) of EN 1993-1-5;
    tpan,i is the thickness of the subpanel i;
    β is the effective width factor for the effect of shear lag, see 3.2.1 of EN 1993-1-5;
    k is the ratio defined in 3.3 of EN 1993-1-5.

    Figure 5.2: Definition of the cross-section Ai

    Figure 5.2: Definition of the cross-section Ai

  5. The verification of a member i of the grillage may be performed using the interaction formula in EN 1993-1-I, section 6.3.3 taking into account the following loading conditions:
  6. If the stiffeners of a plate or a plate segment are only arranged in parallel to the direction of inplane compression forces, the stiffened plate may be modeled as an equivalent beam on elastic springs, see EN 1993-1-5.
  7. If the stiffeners of a stiffened plate segment are positioned in the transverse direction to the compression forces, the interaction between the compression forces and bending moments in the unstiffened plate segments between the stiffeners should be verified according to 5.2.3.4.2(4).
  8. The longitudinal stiffeners should fulfill the requirements given in section 9 of EN 1993-1-5.
  9. The transverse stiffeners should fulfill the requirements given in section 9 of EN 1993-1-5.

6 Ultimate limit state

6.1 General

  1. P All parts of a plated structure shall be so proportioned that the basic design requirements for ultimate limit states given in section 2 are satisfied.
  2. For the partial factor γM for resistance of plated structures see the relevant application parts of EN 1993.
  3. For partial factor γM of connections of plated structures see EN 1993-1-8.

6.2 Plastic limit

6.2.1 General

  1. At every point in a plated structure the design stress σeq,Ed should satisfy the condition:

    σeq,Edσeq,Rd     (6.1)

    where σeq,Ed is the largest value of Von Mises equivalent stress as defined in 5.2.3.

  2. In an elastic design the resistance of a plate segment against plastic collapse or tensile rupture under combined axial forces and bending is defined by the Von Mises equivalent stress σeq,Rd as:

    σeq,Rd = fyk / γM0     (6.2)

    NOTE: For the numerical value of γM0 see 1.1 (2).

6.2.2 Supplementary rules for the design by global analysis

  1. If a numerical analysis is based on materially linear analysis the resistance against plastic collapse or tensile rupture should be checked for the requirement given in 6.2.1.
  2. If a materially nonlinear analysis is based on a design stress-strain relationship with fyd, (=fy/γM0) the plated structure should be subject to a load arrangement FEd that is taken from the design values of actions, and the load may be incrementally increased to determine the load amplification factor αR of the plastic limit state FRd.
  3. The result of the numerical analysis should satisfy the condition:

    FEdFRd     (6.3)

    where FRd = αR FEd

    15
    αR is the load amplification factor for the loads FEd for reaching the ultimate limit state.

6.2.3 Supplementary rules for the design by simplified design methods

6.2.3.1 Unstiffened plates
  1. If an unstiffened plate is designed as an equivalent beam, its cross-sectional resistance should be checked for the combination of inplane loading and out of plane loading effects with the design rules given in EN 1993-1-1.
6.2.3.2 Stiffened plates
  1. If a stiffened plate segment is modeled as a grillage as described in section 5.2.3.4 the cross-section resistance and the buckling resistance of the individual members i of the grillage should be checked for the combination of inplane and out of plane loading effects using the interaction formula in EN 1993-1-1, section 6.3.3.
  2. If a stiffened plate segment is designed as an equivalent beam as described in section 5.2.3.4 the cross-section resistance and the buckling resistance of the equivalent beam should be checked for the combination of inplane and out of plane loading effects using the interaction formula in EN 1993-1-1, section 6.3.3.
  3. The stress resultants or stresses of a subpanel should be verified against tensile rupture or plastic collapse with the design rules given in 5.2.3.2, 5.2.3.3 or 5.2.3.4.

6.3 Cyclic plasticity

6.3.1 General

  1. At every point in a plated structure the design stress range ΔσEd should satisfy the condition:

    ΔσEd ≤ ΔσRd     (6.4)

    where ΔσEd is the largest value of the Von Mises equivalent stress range

    Image

    at the relevant point of the plate segment due to the relevant combination of design actions.

  2. In a materially linear design the resistance of a plate segment against cyclic plasticity / low cycle fatigue may be verified by the Von Mises stress range limitation ΔσRd.

    ΔσRd = 2,0fyk/γM0     (6.5)

    NOTE: For the numerical value of γM0 see 1.1 (2).

6.3.2 Supplementary rules for the design by global analysis

  1. Where a materially nonlinear computer analysis is carried out, the plate should be subject to the design values of the actions.
  2. The total accumulated Von Mises equivalent strain εeq,Ed at the end of the design life of the structure should be assessed using an analysis that models all cycles of loading.
  3. Unless a more refined analysis is carried out the total accumulated Von Mises equivalent plastic strain εeq,Ed may be determined from:

    εeq,Ed = m Δεeq,Ed     (6.6)

    where:

    m is the number of cycles in the design life;
    Δεeq,Ed is the largest increment in the Von Mises plastic strain during one complete load cycle at any point in the structure occurring after the third cycle.
    16
  4. Unless a more sophisticated low cycle fatigue assessment is undertaken, the design value of the total accumulated Von Mises equivalent plastic strain εeq.Ed should satisfy the condition

    Image

    NOTE 1: The National Annex may choose the value of neq. The value neq = 25 is recommended.

    NOTE 2: For the numerical value of γM0 see 1.1 (2)

6.4 Buckling resistance

6.4.1 General

  1. If a plate segment of a plated structure is loaded by in-plane compression or shear, its resistance to plate buckling should be verified with the design rules given in EN 1993-1-5.
  2. Flexural, lateral torsional or distortional stability of the stiffness should be verified according to EN 1993-1-5, see also 5.2.3.4 (8) and (9)
  3. For the interaction between the effects of in-plane and out of plane loading, see section 5.

6.4.2 Supplementary rules for the design by global analysis.

  1. If the plate buckling resistance for combined in plane and out of plane loading is checked by a numerical analysis, the design actions FEd should satisfy the condition:

    FEdFRd     (6.8)

  2. The plate buckling resistance FRd of a plated structure is defined as:

    FRd = k Frk/γm1     (6.9)

    where

    FRk is the characteristic buckling resistance of the plated structure
    k is the calibration factor, see (6).

    NOTE: For the numerical value of γM1 see 1.1(2).

  3. The characteristic buckling resistance FRk should be derived from a load-deformation curve which is calculated for the relevant point of the structure taking into account the relevant combination of design actions FEd. In addition, the analysis should take into account the imperfections as described in 5.2.3.2.
  4. The characteristic buckling resistance FRk is defined by either of the two following criterion:
  5. The reliability of the numerically determined critical buckling resistance should be checked:
    1. either by calculating other plate buckling cases, for which characteristic buckling resistance values FRk,known are known, with the same basically similar imperfection assumptions. The check cases should be similar in their buckling controlling parameters (e.g. non-dimensional plate slenderness, post buckling behaviour, imperfection-sensitivity, material behaviour);
    2. or by comparison of calculated values with test results FRk,known.
  6. Depending on the results of the reliability checks a calibration factor k should be evaluated from:

    k = FRk.known.check / FRk.check     (6.10)

    17

    where

    FRk.known.check as follows from prior knowledge;
    FRk.check are the results of the numerical calculations.

6.4.3 Supplementary rules for the design by simplified design methods

  1. If a stiffened plate segment is subdivided into subpanels and equivalent effective stiffeners as described in section 5.2.3.4 the buckling resistance of the stiffened plate segment may be checked with the design rules given in EN 1993-1-5. Lateral buckling of free stiffener-flanges may be checked according to EN 1993-1-1, section 6.3.3.
  2. The buckling resistance of the equivalent effective stiffener which is defined in section 5.2.3.4 of the plate may be checked with the design rules given in EN 1993-1-1.

7 Fatigue

  1. For plated structures the requirements for fatigue should be obtained from the relevant application standard of EN 1993.
  2. The fatigue assessment should be carried out according to the procedure given in EN 1993-1-9.

8 Serviceability limit state

8.1 General

  1. The principles for serviceability limit state given in section 7 of EN 1993-1-1 should also be applied to plated structures.
  2. For plated structures especially the limit state criteria given in 8.2 and 8.3 should be verified.

8.2 Out of plane deflection

  1. The limit of the out of plane deflection w should be defined as the condition in which the effective use of a plate segment is ended.

    NOTE For limiting values of out of plane deflection w see application standard.

8.3 Excessive vibrations

  1. Excessive vibrations should be defined as the limit condition in which either the failure of a plated structure occurs by fatigue caused by excessive vibrations of the plate or serviceability limits apply.

    NOTE: For limiting values of slenderness to prevent excessive vibrations see application standard.

18

Annex A – Types of analysis for the design of plated structures

[informative]

A.1 General

  1. The internal stresses of stiffened and unstiffened plates may be determined with the following types of analysis:

A.2 Linear elastic plate analysis (LA)

  1. The linear elastic analysis models the behaviour of thin plate structures on the basis of the plate bending theory, related to the perfect geometry of the plate. The linearity of the theory results from the assumptions of the linear elastic material law and the linear small deflection theory.
  2. The LA analysis satisfies the equilibrium as well as the compatibility of the deflections. The stresses and deformations vary linear with the out of plane loading.
  3. As an example for the LA analysis the following fourth-order partial differential equation is given for an isotropic thin plate that subject only to a out of plane load p(x,y):

    Image

    where

    Image

A.3 Geometrically nonlinear analysis (GNA)

  1. The geometrically nonlinear elastic analysis is based on the principles of the plate bending theory of the perfect structure using the linear elastic material law and the nonlinear, large deflection theory.
  2. The GNA analysis satisfies the equilibrium as well as the compatibility of the deflections under consideration of the deformation of the structure.
  3. The large deflection theory takes into account the interaction between flexural and membrane actions. The deflections and stresses vary in a non linear manner with the magnitude of the out of plane pressure.
  4. As an example for the GNA analysis the following fourth-order partial differential equation system is given for an isotropic thin plate subjected only to a out of plane load p(x,y).

    Image

    Image

    19

    where f is the Airy’s stress function

    Image

A.4 Materially nonlinear analysis (MNA)

  1. The materially nonlinear analysis is based on the plate bending theory of the perfect structure with the assumption of small deflections - like in A.2 -, however, it takes into account the nonlinear behaviour of the material.

A.5 Geometrically and materially nonlinear analysis (GMNA)

  1. The geometrically and materially nonlinear analysis is based on the plate bending theory of the perfect structure with the assumptions of the nonlinear, large deflection theory and the nonlinear, elasto-plastic material law.

A.6 Geometrically nonlinear analysis elastic with imperfections included (GNIA)

  1. The geometrically nonlinear analysis with imperfections included is equivalent to the GNA analysis defined in A.3, however, the geometrical model used the geometrically imperfect structure, for instance a predeformation applies at the plate which is governed by the relevant buckling mode.
  2. The GNIA analysis is used in cases of dominating compression or shear stresses in some of the plated structures due to in-plane effects. It delivers the elastic buckling resistance of the “real” imperfect plated structure.

A.7 Geometrically and materially nonlinear analysis with imperfections included (GMNIA)

  1. The geometrically and materially nonlinear analysis with imperfections included is equivalent to the GMNA analysis defined in A.5, however, the geometrical model used the geometrically imperfect structure, for instance a pre-deformation applies at the plate which is governed by the relevant buckling mode.
  2. The GMNIA analysis is used in cases of dominating compression or shear stresses in a plate due to in-plane effects. It delivers the elasto-plastic buckling resistance of the “real” imperfect structure.
20

Annex B – Internal stresses of unstiffened rectangular plates from small deflection theory

[informative]

B.1 General

  1. This annex provides design formulae for the calculation of internal stresses of unstiffened rectangular plates based on the small deflection theory for plates. Therefore the effects of membrane forces are not taken into account in the design formulae given in this annex.
  2. Design formulae are provided for the following load cases:
  3. The deflection w of a plate segment and the bending stresses σbx and σby in a plate segment may be calculated with the coefficients given in the tables of section B.3 and B.4. The coefficients take into account a Poisson’s ratio v of 0,3.

B.2 Symbols

  1. The symbols used are:
    qEd is the design value of the distributed load;
    ρEd is the design value of the patch loading;
    a is the smaller side of the plate;
    b is the longer side of the plate;
    t is the thickness of the plate;
    E is the Elastic modulus;
    kw is the coefficient for the deflection of the plate appropriate to the boundary conditions of the plate specified in the data tables;
    kσbx is the coefficient for the bending stress σbx of the plate appropriate to the boundary conditions of the plate specified in the data tables;
    kσby is the coefficient for the bending stress σby of the plate appropriate to the boundary conditions of the plate specified in the data tables.

B.3 Uniformly distributed loading

B.3.1 Out of plane deflection

  1. The deflection w of a plate segment which is loaded by uniformly distributed loading may be calculated as follows:

    Image

    NOTE: Expression (B.1) is only valid where w is small compared with t.

B.3.2 Internal stresses

  1. The bending stresses σbx and σby in a plate segment may be determined with the following equations:

    Image

    21

    Image

  2. For a plate segment the equivalent stress may be calculated with the bending stresses given in (1) as follows:

    Image

    NOTE: The points for which the state of stress are defined in the data tables are located either on the centre lines or on the boundaries, so that due to symmetry or the postulated boundary conditions, the bending shear stresses τb are zero.

B.3.3 Coefficients k for uniformly distributed loadings

Table B.1 : Coefficients k
Image Loading:
Uniformly distributed loading
Boundary Conditions:
All edges are rigidly supported and rotationally free
b/a kw 1 kσbx 1 kσby 1
1,0 0,04434 0,286 0,286
1,5 0,08438 0,486 0,299
2,0 0,11070 0,609 0,278
3,0 0,13420 0,712 0,244
Table B.2: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary Conditions:
All edges are rigidly supported and rotationally fixed
b/a kw 1 kσbx 1 kσby 1 kσbx 2
1,0 0,01375 0,1360 0.1360 −0,308
1,5 0,02393 0,2180 0,1210 −0,454
2,0 0,02763 0,2450 0,0945 −0,498
3,0 0,02870 0,2480 0,0754 −0,505
22
Table B.3: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary Conditions:
Three edges are rigidly supported and rotationally free and one edge is rigidly supported and rotationally fixed.
b/a kw 1 kσbx 1 kσby 1 kσbx 4
1,5 0,04894 0,330 0,177 −0,639
2,0 0,05650 0,368 0,146 −0,705
Table B.4: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary Conditions:
Two edges are rigidly supported and rotationally free and two edges are rigidly supported and rotationally fixed.
b/a kw 1 kσbx 1 kσby 1 kσbx 4
1,0 0,02449 0,185 0,185 −0,375
1,.5 0,04411 0,302 0,180 −0,588
2,0 0,05421 0,355 0,152 −0,683
Table B.5: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary Conditions:
Two opposite short edges are clamped, the other two edges are simply supported.
b/a kw 1 kσbx 1 kσby 1 kσby 3
1,0 0,02089 0,145 0,197 −0,420
1,5 0,05803 0,348 0,274 −0,630
2,0 0,09222 0,519 0,284 −0,717
23
Table B.6: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary Conditions:
Two opposite long edges are clamped, the other two edges are simply supported.
b/a kw 1 kσbx 1 kσby 1 kσbx 2
1,5 0,02706 0,240 0,106 −0,495
2,0 0,02852 0,250 0,0848 −0,507

B.4 Central patch loading

B.4.1 Out of plane deflection

  1. The deflection w of a plate segment which is loaded by a central patch loading may be calculated as follows:

    Image

B.4.2 Internal stresses

  1. The bending stresses σbx and crby in a plate segment may be determined by the following formulas:

    Image

    Image

  2. For a plate segment the equivalent stress may be calculated with the bending stresses given in (1) as follows:

    Image

24

B.4.3 Coefficients k for patch loading

Table B.7: Coefficients k
Image Loading:
Central patch loading
Boundary Conditions:
All edges are rigidly supported and rotationally free
Parameters:
α = u/a
β = v/a
b/a α × β kw 1 kσbx 1 kσby 1
1 0,1 × 0,l 0,1254 1,72 1,72
  0,2 × 0,2 0,1210 1,.32 1,32
  0,3 × 0,3 0,1126 1,04 1,04
  0,2 × 0,3 0,1167 1,20 1,12
  0,2 × 0,4 0,1117 1,10 0,978
1,5 0,1 × 0,l 0,1664 1,92 1,70
  0,2 × 0,2 0,1616 1,51 1,29
  0,3 × 0,3 0,1528 1,22 1,01
  0,2 × 0,3 0,1577 1,39 1,09
  0,2 × 0,4 0,1532 1,29 0,953
2,0 0,1 × 0,l 0,1795 1,97 1,67
  0,2 × 0,2 0,1746 1,56 1,26
  0,3 × 0,3 0,1657 1,28 0,985
  0,2 × 0,3 0,1708 1,45 1,07
  0,2 × 0,4 0,1665 1,35 0,929
3,0 0,1 × 0,l 0,1840 1,99 1,66
  0,2 × 0,2 0,1791 1,58 1,25
  0,3 × 0,3 0,1701 1,30 0,975
  0,2 × 0,3 0,1753 1,47 1,06
  0,2 × 0,4 0,1711 1,37 0,918
25

Annex C – Internal stresses of unstiffened rectangular plates from large deflection theory

[informative]

C.1 General

  1. This annex provides design formulas for the calculation of internal stresses of unstiffened rectangular plates based on the large deflection theory for plates.
  2. The following loading conditions are considered:
  3. The bending and membrane stresses in a plate and the deflection w of a plate may be calculated with the coefficients given in the tables of section C.3 and C.4. The coefficients take into account a Poisson’s ratio v of 0,3.

C.2 Symbols

  1. The symbols used are:
    qEd is the design value of the load uniformly distributed over the total surface;
    ρEd is the design value of the patch loading uniformly distributed over the surface u × v;
    a is the smaller side of the plate;
    b is the longer side of the plate;
    t is the thickness of the plate;
    E is the Elastic modulus;
    FBC    flexural boundary conditions;
    MBC    membrane boundary conditions;
    kw is the coefficient for the deflection of the plate appropriate to the boundary conditions specified in the data tables;
    kσbx is the coefficient for the bending stress σbx of the plate appropriate to the boundary conditions specified in of the plate in the data tables;
    kσby is the coefficient for the bending stress σby of the plate appropriate to the boundary conditions specified in the data tables;
    kσmx is the coefficient for the membrane stress σmx of the plate appropriate to the boundary conditions specified in the data tables;
    kσmy is the coefficient for the membrane stress σmy of the plate appropriate to the boundary conditions specified in the data tables.

C.3 Uniformly distributed loading on the total surface of the plate

C.3.1 Out of plane deflection

  1. The deflection w of a plate segment which is loaded by uniformly distributed loading may be calculated as follows:

    Image

26

C.3.2 Internal stresses

  1. The bending stresses σbx and σby in a plate segment may be determined with the following equations:

    Image

    Image

  2. The membrane stresses σmx and σmy in a plate segment may be determined as follows:

    Image

    Image

  3. At the loaded surface of a plate the total stresses are calculated with the bending and membrane stresses given in (1) and (2) as follows:

    σx,Ed = − σbx,Ed + σmx,Ed     (C.6)

    σy,Ed = − σby,Ed + σmy,Ed     (C.7)

  4. At the no-loaded surface of a plate the total stresses are determined with the bending and membrane stresses given in (1) and (2) as follows:

    σx,Ed = σbx,Ed + σmx,Ed     (C.8)

    σy,Ed = σby,Ed + σmy,Ed     (C.9)

  5. For a plate the equivalent stress σv,Ed may be calculated with the stresses given in (4) as follows:

    Image

    NOTE: The points for which the state of stress are defined in the data tables are located either on the centre lines or on the boundaries, so that due to symmetry or the postulated boundary conditions, membrane shearing stresses τm as well as bending shear stresses τb are zero. The algebraic sum of the appropriate bending and membrane stresses at the points considered in the data tables gives the values of maximum and minimum surface stresses at these points.

27

C.3.3 Coefficients k for uniformly distributed loadings

Table C.1: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary conditions:
FBC: All edges are simply supported.
MBC: Zero direct stresses, zero shear stresses
Parameters:
Image
b/a Q kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1 kσmy 2
1,0 20 0,0396 0,2431 0,2431 0,0302 0,0302 −0,0589
  40 0,0334 0,1893 0,1893 0,0403 0,0403 −0,0841
  120 0,0214 0,0961 0,0961 0,0411 0,0411 −0,1024
  200 0,0166 0,0658 0,0658 0,0372 0,0372 −0,1004
  300 0,0135 0,0480 0,0480 0,0335 0,0335 −0,0958
  400 0,0116 0,0383 0,0383 0,0306 0,0306 −0,0915
1,5 20 0,0685 0,3713 0,2156 0,0243 0,0694 −0,1244
  40 0,0546 0,2770 0,1546 0,0238 0,0822 −0,1492
  120 0,0332 0,1448 0,0807 0,0170 0,0789 −0,1468
  200 0,0257 0,1001 0,0583 0,0141 0,0715 −0,1363
  300 0,0207 0,0724 0,0440 0,0126 0,0646 −0,1271
  400 0,0176 0,0569 0,0359 0,0117 0,0595 −0,1205
2,0 20 0,0921 0,4909 0,2166 0.0085 0,0801 −0,1346
  40 0,0746 0,3837 0,1687 0,0079 0,0984 −0,1657
  120 0,0462 0,2138 0,0959 0,0073 0,0992 −0,1707
  200 0,0356 0,1516 0,0695 0,0067 0,0914 −0,1610
  300 0,0287 0,1121 0,0528 0,0061 0,0840 −0,1510
  400 0,0245 0,0883 0,0428 0,0061 0,0781 −0,1434
28
Table C.2: Coefficients k
Image Loading:
Uniformly distributed loading:
Boundary conditions:
FBC: All edges are simply supported.
MBC: All edges remain straight. Zero average direct stresses, zero shear stresses
Parameters:
Image
b/a Q kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1 kσmx 2 kσmy 2
1 20 0,0369 0,2291 0,2291 0,0315 0,0315 0,0352 −0.0343
  40 0,0293 0,1727 0,1727 0,0383 0,0383 0,0455 −0,0429
  120 0,0170 0,0887 0,0887 0,0360 0,0360 0,0478 −0,0423
  200 0,0126 0,0621 0,0621 0,0317 0,0317 0,0443 −0,0380
  300 0,0099 0,0466 0,0466 0,0280 0,0280 0,0403 −0,0337
  400 0,0082 0,0383 0,0383 0,0255 0,0255 0,0372 −0,0309
1,5 20 0,0554 0,3023 0,1612 0,0617 0,0287 0,0705 −0,0296
  40 0,0400 0.2114 0,1002 0,0583 0,0284 0,0710 −0,0293
  120 0,0214 0,1079 0,0428 0,0418 0.0224 0,0559 −0,0224
  200 0,0157 0,0778 0,0296 0,0345 0,0191 0.0471 −0,0188
  300 0,0122 0,0603 0,0224 0.0296 0,0167 0,0408 −0,0161
  400 0,0103 0,0505 0,0188 0,0267 0,0152 0,0369 −0,0147
2 20 0,0621 0,3234 0,1109 0,0627 0.0142 0,0719 −0.0142
  40 0,0438 0,2229 0,0689 0,0530 0,0120 0,0639 −0,0120
  120 0,0234 0,1163 0,0336 0,0365 0,0086 0,0457 −0,0083
  200 0,0172 0,0847 0,0247 0,0305 0.0075 0,0384 −0,0067
  300 0,0135 0,0658 0,0195 0,0268 0,0067 0,0335 −0.0058
  400 0,0113 0,0548 0,0164 0,0244 0,0064 0,0305 −0,0050
3 20 0,0686 0,3510 0,1022 0.0477 0,0020 0,0506 −0,0007
  40 0,0490 0,2471 0,0725 0,0420 0,0020 0,0441 0,0000
  120 0,0267 0,1317 0,0390 0,0320 0,0027 0,0335 0,0010
  200 0,0196 0,0954 0,0283 0,0271 0,0044 0,0285 0,0027
  300 0,0153 0,0733 0,0217 0,0242 0,0059 0,0256 0,0044
  400 0,0127 0,0605 0,0178 0,0221 0,0066 0,0235 0,0051
29
Table C.3: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary conditions:
FBC: All edges are clamped.
MBC: Zero direct stresses, zero shear stresses
Parameters:
Image
b/a Q kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1 kσmx 2 kσmy 2
1 20 0,0136 0,1336 0,1336 0,0061 0,0061 −0,3062 −0,0073
  40 0,0131 0,1268 0,1268 0,0113 0,0113 −0,3006 −0,0137
  120 0,0108 0,0933 0,0933 0,0212 0,0212 −0,2720 −0,0286
  200 0,0092 0,0711 0,0711 0,0233 0,0233 −0,2486 −0,0347
  300 0,0078 0,0547 0,0547 0,0233 0,0233 −0,2273 −0,0383
  400 0,0069 0,0446 0,0446 0,0226 0,0226 −0,2113 −0,0399
1,5 20 0,0234 0,2117 0,1162 0,0061 0,0133 −0,4472 −0,0181
  40 0,0222 0,1964 0,1050 0,0098 0,0234 −0,4299 −0,0322
  120 0,0173 0,1406 0,0696 0,0124 0,0385 −0,3591 −0,0559
  200 0,0144 0,1103 0,0537 0,0116 0,0415 −0,3160 −0,0620
  300 0,0122 0,0879 0,0430 0,0105 0,0416 −0,2815 −0,0636
  400 0,0107 0,0737 0,0364 0,0098 0,0409 −0,2583 −0,0635
2 20 0,0273 0,2418 0,0932 0,0010 0,0108 −0,4935 −0,0150
  40 0,0265 0,2330 0,0897 0,0017 0,0198 −0,4816 −0,0277
  120 0,0223 0,1901 0,0740 0,0032 0,0392 −0,4223 −0,0551
  200 0,0192 0,1578 0,0621 0,0039 0,0456 −0,3780 −0,0647
  300 0,0165 0,1306 0,0518 0,0042 0,0483 −0,3396 −0,0690
  400 0,0147 0,1120 0,0446 0,0044 0,0487 −0,3132 −0,0702
3 20 0,0288 0,2492 0,0767 −0,0015 0,0027 −0,5065 −0,0033
  40 0,0290 0,2517 0,0795 −0,0022 0,0066 −0,5095 −0,0084
  120 0,0281 0,2440 0,0812 −0,0010 0,0247 −0,4984 −0,0331
  200 0,0260 0,2230 0,0750 0,0000 0,0368 −0,4702 −0,0497
  250 0,0247 0,2096 0,0707 0,0002 0,0415 −0,4520 −0,0564
30
Table C.4: Coefficients k
Image Loading:
Uniformly distributed loading
Boundary conditions:
FBC: All edges are clamped.
MBC: All edges remain straight. Zero average direct stresses, zero shear stresses
Parameters:
Image
b/a Q kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1 kσbx 2 kσmx 2 kσmy 2
1 20 0,0136 0,1333 0,1333 0,0065 0,0065 −0,3058 0,0031 −0,0055
  40 0,0130 0,1258 0,1258 0,0118 0.0118 −0,3000 0,0059 −0,0103
  120 0,0105 0,0908 0.0908 0,0216 0,0216 −0,2704 0,0123 −0,0202
  200 0,0087 0.0688 0,0688 0,0234 0,0234 −0,2473 0,0151 −0,0233
  300 0,0073 0,0528 0,0528 0,0231 0,0231 −0,2267 0,0169 −0,0244
  400 0,0063 0,0430 0,0430 0,0223 0,0223 −0,2119 0,0176 −0,0246
1,5 20 0,0230 0,2064 0.1125 0.0137 0,0097 −0,4431 0,0118 −0,0082
  40 0,0210 0,1833 0,0957 0,0218 0,0155 −0,4195 0,0200 −0,0133
  120 0,0149 0,1175 0,0532 0,0275 0,0202 −0,3441 0,0295 −0,0185
  200 0,0118 0,0876 0,0369 0,0259 0,0195 −0,3028 0.0304 −0,0182
  300 0,0096 0.0678 0,0275 0,0238 0,0180 −0,2710 0,0300 −0,0173
  400 0,0083 0,0562 0,0221 0,0220 0,0168 −0,2492 0,0291 −0,0163
2 20 0,0262 0,2288 0,0853 0,0140 0,0060 −0,4811 0,0149 −0,0052
  40 0,0234 0,1994 0,0701 0,0206 0,0086 −0,4492 0,0234 −0,0077
  120 0,0162 0,1276 0,0404 0,0238 0,0094 −0,3611 0,0299 −0,0086
  200 0,0129 0,0963 0,0296 0,0223 0,0085 −0,3162 0,0289 −0,0079
  300 0,0105 0,0752 0,0230 0,0208 0,0077 −0,2824 0,0274 −0,0072
  400 0,0090 0,0627 0,0190 0,0196 0,0071 −0,2600 0,0259 −0,0066
3 20 0,0272 0,2331 0,0700 0,0102 0,0010 −0,4878 0,0111 −0,0008
  40 0,0247 0,2071 0,0615 0,0149 0,0011 −0,4575 0,0167 −0,0009
  120 0,0177 0,1396 0,0413 0.0186 0,0009 −0.3727 0,0202 −0,0005
  200 0,0143 0,1074 0,0319 0,0184 0,0009 −0,3272 0,0197 −0,0003
  300 0,0117 0,0848 0,0251 0,0176 0,0008 −0,2924 0,0192 −0,0002
  400 0,0101 0,0709 0,0210 0,0169 0,0008 −0,2687 0.0182 0,0000
31

C.4 Central patch loading

C.4.1 General

  1. The deflection w and the Stresses should be determined with the formulas provided for a plate which is loaded by a central patch loading ρEd, distributed over an area of u long and v wide:

    Image

C.4.2 Internal stresses

  1. The bending stresses σbx and σby in a plate segment may be determined with the following equations:

    Image

    Image

  2. The membrane stresses σmx and σmy in a plate segment may be determined as follows:

    Image

    Image

  3. At the loaded surface of a plate the total stresses are calculated with the bending and membrane stresses given in (1) and (2) as follows:

    σx,Ed = − σbx,Ed + σmx,Ed     (C.16)

    σy,Ed = − σby,Ed + σmy,Ed     (C.17)

  4. At the no-loaded surface of a plate the total stresses are determined with the bending and membrane stresses given in (1) and (2) as follows:

    σx,Ed = σbx,Ed + σmx,Ed     (C.18)

    σy,Ed = σby,Ed + σmy,Ed     (C.19)

  5. For a plate the equivalent stress σv,Ed may be calculated with the stresses given in (4) as follows:

    Image

    NOTE: The points for which the state of stress are defined in the data tables are located either on the centre lines or on the boundaries, so that due to symmetry or the postulated boundary conditions, membrane shearing stresses τm as well as bending shear stresses τb are zero. The algebraic sum of the appropriate bending and membrane stresses at the points considered in the data tables gives the values of maximum and minimum surface stresses at these points.

32

C.4.3 Coefficients k for patch loading

Table C.5: Coefficients k
Image Loading:
Central patch loading
Boundary conditions:
FBC: All edges are rigidly supported and rotationally free.
MBC:Zero direct stresses, zero shear stresses
Parameters:
α = u/a; β = v/a
Image
b/a = 1
α × β ρ kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1
0,1 × 0,l 10 0,1021 1,4586 1,4586 0,1548 0,1548
  20 0,0808 1,2143 1,2143 0,1926 0,1926
  60 0,0485 0,8273 0,8273 0,2047 0,2047
  100 0,0372 0,6742 0,6742 0,1978 0,1978
  150 0,0298 0,5693 0,5693 0,1892 0,1892
  200 0,0255 0,5005 0,5005 0,1823 0,1823
0,2 × 0,2 10 0,0998 1,0850 1,0850 0,1399 0,1399
  20 0,0795 0,8593 0,8593 0,1729 0.1729
  60 0,0478 0,5108 0,5108 0,1756 0,1756
  100 0,0364 0,3881 0,3881 0,1624 0,1624
  150 0,0293 0,3089 0,3089 0,1505 0,1505
  200 0,0249 0,2614 0,2614 0,1412 0,1412
0,3 × 0,3 10 0,0945 0,8507 0,8507 0,1144 0,1144
  20 0,0759 0,6614 0,6614 0,1425 0,1425
  60 0,0459 0,3702 0,3702 0,1425 0,1425
  100 0,0351 0,2704 0,2704 0,1300 0,1300
  150 0,0282 0,2101 0,2101 0,1186 0,1186
  200 0,0240 0,1747 0,1747 0,1102 0,1102
0,2 × 0,3 10 0,0971 0,9888 0,9128 0,1224 0,1288
  20 0,0776 0,7800 0,7101 0,1512 0,1602
  60 0,0468 0,4596 0,4021 0,1488 0,1624
  100 0,0358 0,3468 0,2957 0,1368 0,1512
  150 0,0287 0,2760 0,2307 0,1248 0,1389
  200 0,0245 0,2340 0,1926 0,1152 0,1310
0,2 × 0,4 10 0,0939 0,9119 0,7961 0,1078 0,1183
  20 0,0755 0,7216 0,6142 0,1320 0,1487
  60 0,0457 0,4235 0,3355 0,1287 0,1516
  100 0,0350 0,3201 0,2435 0,1166 0,1408
  150 0,0280 0,2541 0.1868 0,1045 0,1301
  200 0,0239 0,2156 0,1545 0,0968 0,1213
33
Table C.6: Coefficients k
Image Loading:
Central patch loading
Boundary conditions:
FBC: All edges are rigidly supported and rotationally free.
MBC: Zero direct stresses, zero shear stresses
Parameters:
α = u/a; β = v/a
Image
b/a = 1,5
α × β ρ kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1
0,1 × 0,1 10 0,1303 1,5782 1.3855 0,1517 0,1921
  20 0,1018 1,3056 1,1373 0,1786 0,2295
  60 0,0612 0,8986 0,7701 0,1824 0,2380
  100 0,0469 0,7411 0,6273 0,1747 0,2295
  150 0,0378 0,6298 0,5287 0,1670 0,2193
  200 0,0323 0,5568 0,4641 0,1594 0,2125
0,2 × 0,2 10 0,1281 1,1974 1,0049 0,1344 0,1780
  20 0,1007 0,9453 0,7766 0,1555 0,2116
  60 0,0605 0,5783 0,4554 0,1465 0,2103
  100 0,0462 0,4485 0,3457 0,1329 0,1974
  150 0,0372 0,3624 0,2748 0,1208 0,1845
  200 0,0317 0,3111 0,2322 0,1133 0,1742
0,3 × 0,3 10 0,1229 0,9589 0,7737 0,1074 0,1525
  20 0,0972 0,7405 0,5828 0,1232 0,1818
  60 0,0585 0,4282 0,3161 0,1110 0,1788
  100 0,0449 0,3221 0,2353 0,0988 0,1667
  150 0,0361 0,2550 0,1828 0,0878 0,1535
  200 0,0309 0,2147 0.1525 0,0805 0,1444
0,2 × 0,3 10 0,1260 1,1037 0,8360 0,1154 0,1657
  20 0,0994 0,8688 0,6322 0,1321 0,1984
  60 0,0598 0,5296 0,3553 0,1168 0,1973
  100 0,0459 0.4114 0,2649 0,1043 0,1853
  150 0,0369 0,3336 0,2082 0,0931 0,1722
  200 0,0314 0,2877 0,1755 0,0848 0,1624
0,2 × 0,4 10 0,1235 1,0294 0,7271 0,0993 0,1563
  20 0,0977 0,8101 0,5432 0,1109 0,1877
  60 0,0590 0,4954 0,2983 0,0955 0,1877
  100 0,0453 0,3857 0,2220 0,0826 0,1754
  150 0,0365 0,3148 0,1744 0,0722 0,1630
  200 0.0311 0,2722 0,1468 0,0658 0,1544
34

Table C.7: Coefficients k
Image Loading:
Central patch loading
Boundary conditions:
FBC: All edges are rigidly supported and rotationally free.
MBC: Zero direct stresses, zero shear stresses
Parameters:
α = u/a; β = v/a
Image
b/a = 2
α × β ρ kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1
0,1 × 0,l 10 0,1438 1,6351 1,3560 0,1517 0,1904
  20 0,1154 1,3692 1,1106 0,1773 0,2288
  60 0,0725 0,9633 0,7498 0,1753 0,2438
  100 0,0564 0,7979 0,6112 0,1675 0,2355
  150 0,0456 0,6797 0,5127 0,1596 0,2271
  200 0,0390 0,6028 0,4492 0,1517 0,2188
0,2 × 0,2 10 0,1414 1,2542 0,9752 0,1326 0,1751
  20 0,1138 1,0078 0,7510 0,1513 0,2104
  60 0,0716 0,6427 0.4410 0,1373 0,2167
  100 0,0555 0,5054 0,3339 0,1232 0,2054
  150 0,0449 0,4134 0.2646 0,1108 0,1928
  200 0,0384 0,3572 0,2230 0,1030 0,1827
0,3 × 0,3 10 0,1362 1,0227 0,7506 0,1062 0,1517
  20 0,1104 0,8090 0,5615 0,1190 0,1822
  60 0,0698 0,4941 0,3093 0,1024 0,1862
  100 0,0542 0,3789 0,2275 0,0883 0,1753
  150 0,0421 0,3046 0,1783 0,0794 0,1645
  200 0,0374 0.2586 0,1487 0,0717 0,1546
0,2 × 0,3 10 0,1395 1,1702 0.8164 0,1146 0,1231
  20 0,1129 0,9396 0,6153 0,1262 0,1990
  60 0,0712 0.6003 0,3488 0,1088 0,2044
  100 0,0553 0,4742 0,2611 0,0943 0,1947
  150 0,0447 0,3901 0,2065 0,0841 0,1830
  200 0,0383 0,3379 0,1744 0,0754 0,1733
0,2 × 0,4 10 0,1375 1,0976 0,7051 0,0959 0,1551
  20 0,1117 0,8829 0,5267 0,1053 0,1886
  60 0,0706 0,5670 0,2945 0,0851 0,1942
  100 0,0549 0,4496 0,2220 0,0729 0,1849
  150 0,0445 0,3713 0,1765 0,0635 0,1737
  200 0,0381 0,3227 0,1496 0,0554 0,1644
35
Table C.8: Coefficients k
Image Loading:
Central patch loading
Boundary conditions:
FBC: All edges are rigidly supported and rotationally free.
MBC: Zero direct stresses, zero shear stresses
Parameters:
α = u/a; β = v/a
Image
b/a = 2.5
α × β ρ kw 1 kσbx 1 kσby 1 kσmx 1 kσmy 1
0,1 × 0,l 10 0,1496 1,6636 1,3463 0,1552 0,1826
  20 0,1235 1,4109 1,1006 0,1811 0,2175
  60 0,0861 1,0428 0,7453 0,1811 0,2374
0,2 × 0,2 10 0,1470 1,2814 0,9650 0,1359 0,1688
  20 0,1218 1,0491 0,7400 0,1548 0,2000
  60 0,0849 0,7205 0,4363 0,1390 0,2088
0,3 × 0,3 10 0,1419 1,0504 0,7410 0,1092 0,1443
  20 0,1182 0,8489 0,5519 0.1222 0,1726
  60 0,0827 0,5681 0,3052 0,1014 0,1775
0,2 × 0,3 10 0,1455 1,1981 0,8056 0,1161 0,1579
  20 0,1210 0,9820 0,6053 0,1294 0,1876
  60 0,0847 0,6806 0,3487 0,1088 0,1982
0,2 × 0,4 10 0,1434 0,1126 0,6949 0,0986 0,1469
  20 0,1199 0,9261 0,5168 0,1069 0,1763
  60 0,0844 0,6480 0,2993 0,0849 0,1873
36 37