untitled BRITISH STANDARD BS EN 1993 1 7 2007 Incorporating corrigendum April 2009 Eurocode 3 — Design of steel structures — Part 1 7 Plated structures subject to out of plane loading ICS 91 010 30; 9[.]
Scope
EN 1993-1-7 outlines essential design guidelines for the structural integrity of both unstiffened and stiffened plates used in plated structures, including silos, tanks, and containers, subjected to out-of-plane loads.
It is intended to be used in conjunction with EN 1993-1-1 and the relevant application standards
This document outlines the design values for resistances, indicating that the partial factor for resistances can be sourced from the National Annexes of applicable standards It also provides recommended values as specified in the relevant application standards.
(3) This Standard is concerned with the requirements for design against the ultimate limit state of:
The overall equilibrium of the structure, including aspects such as sliding, uplifting, and overturning, is not addressed in this Standard but is covered in EN 1993-1-1 For specific applications, additional considerations can be found in the relevant application parts of EN 1993.
This Standard outlines rules for plate segments in plated structures, which can be either stiffened or unstiffened These segments may consist of individual plates or components of a larger plated structure and are subjected to out-of-plane loading.
(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
Wind loading and the flow of bulk solids should be considered as quasi-static actions It is essential to account for dynamic effects related to fatigue in accordance with EN 1993-1-9 In this context, the stress resultants resulting from dynamic behavior are addressed as quasi-static.
Normative references
This European Standard includes provisions from other publications, which are referenced throughout the text and listed subsequently For dated references, any amendments or revisions apply only when incorporated into this Standard In the case of undated references, the latest edition of the cited publication is applicable.
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 1.6: Strength and stability of shell structures
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
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:
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
Transverse stiffener (trough or closed)
Longitudinal stiffeners (open or closed)
Figure 1.1: Components of a plated structure
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
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:
– longitudinal if its longitudinal direction is in the main direction of load transfer of the member of which it forms a part
– transverse if its longitudinal direction is perpendicular to the main direction of load transfer of the member of which it forms a part
Plate with transverse and/or longitudinal stiffeners
Unstiffened plate surrounded by stiffeners or, on a web, by flanges and/or stiffeners or, on a flange, by webs and/or stiffeners
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
A failure mode in the ultimate limit state where failure of the plate occurs due to tension
Where repeated yielding is caused by cycles of loading and unloading
Where the structure looses its stability under compression and/or shear
Where cyclic loading causes cracking or failure
The load applied normal to the middle surface of a plate segment
Forces acting parallel to the surface of a plate segment arise from in-plane effects, such as temperature and friction, as well as from global loads applied to the plated structure.
Symbols
(1) In addition to those given in EN 1990 and EN 1993-1-1, the following symbols are used:
Membrane stresses in a rectangular plate are characterized by three key components: the membrane normal stress in the x-direction, denoted as \$\sigma_{mx}\$, which arises from the membrane normal stress resultant per unit width \$n_x\$; the membrane normal stress in the y-direction, represented as \$\sigma_{my}\$, resulting from the membrane normal stress resultant per unit width \$n_y\$; and the membrane shear stress, indicated as \$\tau_{mxy}\$, which is due to the membrane shear stress resultant per unit width \$n_{xy}\$.
Bending and shear stresses in rectangular plates arise from various forces, as illustrated in Figure 1.3 The stress in the x-direction, denoted as \$\sigma_{bx}\$, results from the bending moment per unit width \$m_x\$ Similarly, the stress in the y-direction, \$\sigma_{by}\$, is caused by the bending moment per unit width \$m_y\$ Additionally, the shear stress \$\tau_{bxy}\$ is attributed to the twisting moment per unit width \$m_{xy}\$, while \$\tau_{bxz}\$ and \$\tau_{byz}\$ represent shear stresses due to transverse shear forces \$q_x\$ and \$q_y\$ associated with bending, respectively.
Figure 1.3: Normal and shear stresses due to bending
In a plate, there are typically eight stress resultants at any given point However, the shear stresses \$\tau_{bxz}\$ and \$\tau_{byz}\$ caused by loads \$q_x\$ and \$q_y\$ are often negligible compared to other stress components, allowing them to be disregarded in design considerations.
In the context of plate mechanics, several Greek lower case letters are used to represent key parameters: \( \alpha \) denotes the aspect ratio of a plate segment (a/b), \( \epsilon \) signifies strain, and \( \alpha_R \) refers to the load amplification factor The reduction factor for plate buckling is represented by \( \rho \), while \( \sigma_i \) indicates normal stress in direction \( i \), as illustrated in Figures 1.2 and 1.3 Additionally, shear stress is denoted by \( \tau \), and Poisson's ratio is represented by \( \nu \) The partial factor is indicated by \( \gamma_M \).
In the context of plate segments, the following parameters are defined: \( a \) represents the length, while \( b \) denotes the width, as illustrated in Figures 1.4 and 1.5 The yield stress or 0.2% proof stress for materials exhibiting a non-linear stress-strain curve is indicated by \( f_{yk} \) Additionally, \( n_i \) refers to the membrane normal force in the direction \( i \) measured in kN/m, and \( n_{xy} \) signifies the membrane shear force in kN/m The bending moment is represented by \( m \) in kNm/m, while \( q_z \) indicates the transverse shear force in the z direction, also measured in kN/m Lastly, \( t \) denotes the thickness of the plate segment, as shown in Figures 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 1.5: Dimensions and axes of stiffened plate segments; stiffeners may be open or closed stiffeners
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.
Principles of limit state design
(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
Plastic collapse occurs when a structural component experiences significant plastic deformations, leading to the formation of a plastic mechanism The load at which plastic collapse happens is typically determined using a mechanism derived from small deflection theory.
Cyclic plasticity represents the critical condition where repeated loading and unloading cycles induce yielding in tension, compression, or both at the same location, resulting in continuous plastic work on the structure This alternating yielding can cause local cracking due to the material's diminished energy absorption capacity, thereby imposing a low cycle fatigue limitation The stresses related to this limit state arise from a combination of various actions and the structural compatibility conditions.
Buckling occurs when a structure experiences significant displacements due to instability from compressive and/or shear stresses in the plate, ultimately resulting in its inability to withstand increased 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
(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.
Actions
(1) The characteristic values of actions should be determined from the appropriate parts of EN 1991.
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
(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
(1) For durability see section 4 of EN 1993-1-1
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
Stress resultants in the plate
(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
Yield line analysis is applicable in the ultimate limit state when in-plane compression or shear is below 10% of the corresponding resistance The bending resistance in a yield line should be considered accordingly.
(1) Boundary conditions assumed in analyses should be appropriate to the limit states considered
In the design calculations for a plated structure, it is essential to document the boundary conditions assumed for stiffeners in each individual plate segment within the drawings and project specifications.
5.2.3 Design models for plated structures
(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
A linear bending theory operates under small-deflection assumptions, establishing a proportional relationship between loads and deformations This theory is applicable when in-plane compression or shear does not exceed 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
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
Amplitudes for geometrical imperfections in imperfect geometries are selected to ensure that the calculated results are reliable when compared to test specimens manufactured according to EN 1090-2 tolerances Consequently, these amplitudes typically differ from the specified tolerances.
The internal stresses of an individual plate segment in a plated structure can be determined using relevant design actions and appropriate design formulas, as outlined in section 5.2.3.1.
Annex B and Annex C present tabulated values for rectangular unstiffened plates subjected to transverse loading For the design of circular plates, refer to the formulas outlined in EN 1993-1-6 Additional design formulas may be applicable, provided they meet the reliability standards specified in the guidelines.
(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
(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: τ σ σ σ σ eq , Ed = σ 2 x, E d + 2 y, E d - x, E d y, E d + 3 2 xy, E d (5.2) where
Ed xy, Ed xy, Ed xy, ± τ and n x,Ed, n y,Ed, n xy,Ed, m x,Ed, m y,Ed and m xy,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
To determine the internal stresses of a plated structure, a numerical analysis based on a materially linear approach should be conducted The maximum equivalent Von Mises stress, denoted as \$\sigma_{eq,Ed}\$, must be calculated for the appropriate 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
, , + , , , + 3 , eq Ed x Ed y Ed x Ed y Ed xy Ed σ = σ σ −σ ⋅σ τ (5.3) 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:
– deviations from the nominal geometric shape of the plate (initial deformation, out of plane deflections);
– irregularities of welds (minor eccentricities);
– residual stresses because of rolling, pressing , welding, straightening;
Initial equivalent geometric imperfections of a perfect plate must consider both geometrical and material flaws The shape of these imperfections should be based on the corresponding buckling mode.
The initial equivalent geometric imperfection amplitude, denoted as e0, for a rectangular plate segment can be determined through numerical calibrations using test results from representative test pieces This process is based on the plate buckling curve outlined in EN 1993-1-5.
The reduction factor for plate buckling, denoted as \$\rho\$, is defined in section 4.4 of EN 1993-1-5, where it is specified that the aspect ratio \$\alpha\$ must be less than 2 The geometric properties of the plate are represented by \$a\$ and \$b\$, as illustrated in Figure 5.1, while \$t\$ indicates the thickness of the plate.
λp is the relative slenderness of the plate, see EN 1993-1-5
Figure 5.1: Initial equivalent geometric bow imperfection e 0 of a plate segment
(6) As a conservative assumption the amplitude may be taken as e 0 = 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
(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
An unstiffened rectangular plate subjected to out-of-plane loads can be effectively represented as an equivalent beam aligned with the primary direction of load transfer, provided certain conditions are met.
– the aspect ratio a/b of the plate is greater than 2;
– the plate is subjected to out of plane distributed loads which may be either linear or vary linearly;
– the strength, stability and stiffness of the frame or beam on which the plate segment is supported fulfil the assumed boundary conditions of the equivalent beam b a e 0
(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
When the initial deflections caused by out-of-plane loads resemble the buckling mode of a plate under in-plane compression forces, it is essential to consider the interaction between these two phenomena.
(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
(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
When calculating the cross-sectional area \( A_i \) of the cooperating plate for an individual member \( i \) of the grillage, it is essential to consider the effects of shear lag This is achieved by applying a reduction factor \( \beta \).
For a member \( i \) of the grillage aligned with the direction of in-plane compression forces, it is essential to calculate the cross-sectional area \( A_i \) while considering the effective width of the neighboring subpanels, as dictated by plate buckling guidelines in 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:
The equation for the area \( A_i \) is given by \[A_i = [\rho_c (A_{L,eff} + \Sigma \rho_{pan,i} b_{pan,i} t_{pan,i})] \beta \kappa\]where \( A_{L,eff} \) represents the effective area of the stiffener, accounting for local plate buckling The term \( \rho_c \) denotes the reduction factor due to global plate buckling of the stiffened plate segment, as specified in section 4.5.4(1) of EN 1993-1-5 Additionally, \( \rho_{pan,i} \) is the reduction factor related to local plate buckling of subpanel \( i \), defined in section 4.4(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
(3) For partial factor γM of connections of plated structures see EN 1993-1-8.
Plastic limit
In a plated structure, it is essential that the design stress, denoted as \$\sigma_{eq,Ed}\$, adheres to the condition \$\sigma_{eq,Ed} \leq \sigma_{eq,Rd}\$ Here, \$\sigma_{eq,Ed}\$ represents the maximum value of the Von Mises equivalent stress as outlined in section 5.2.3.
In an elastic design, the resistance of a plate segment to plastic collapse or tensile rupture under combined axial forces and bending is characterized by the Von Mises equivalent stress, denoted as \$\sigma_{eq,Rd}\$, which is calculated using the formula: \$\sigma_{eq,Rd} = \frac{f_{yk}}{\gamma_{M0}}\$ (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
In a materially nonlinear analysis utilizing a design stress-strain relationship defined by \( f_{yd} (= \frac{f_y}{\gamma_{M0}}) \), the plated structure must be subjected to a load arrangement \( F_{Ed} \) derived from the design action values The load can be incrementally increased to ascertain the load amplification factor \( \alpha_R \) at the plastic limit state \( F_{Rd} \).
(3) The result of the numerical analysis should satisfy the condition:
F Ed≤ F Rd (6.3) where F Rd = αR F Ed αR is the load amplification factor for the loads F Ed for reaching the ultimate limit state
6.2.3 Supplementary rules for the design by simplified design methods
When designing an unstiffened plate as an equivalent beam, it is essential to verify its cross-sectional resistance against both in-plane and out-of-plane loading effects, following the design guidelines outlined in EN 1993-1-1.
When modeling a stiffened plate segment as a grillage, it is essential to verify the cross-section resistance and buckling resistance of each member for both in-plane and out-of-plane loading effects This assessment should be conducted using the interaction formula outlined in EN 1993-1-1, section 6.3.3.
When designing a stiffened plate segment as an equivalent beam, it is essential to verify both the cross-section resistance and the buckling resistance This assessment must account for the combined effects of in-plane and out-of-plane loading, utilizing the interaction formula outlined 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.
Cyclic plasticity
(1) At every point in a plated structure the design stress range ∆σEd should satisfy the condition:
∆σEd≤∆σRd (6.4) where ∆σEd is the largest value of the Von Mises equivalent stress range
∆ eq, Ed = 2 x, Ed + 2 y, Ed - x, Ed y, Ed +3 2 Ed 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 ∆σRd
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
The total accumulated Von Mises equivalent strain, denoted as \$\epsilon_{eq,Ed}\$, must be evaluated at the conclusion of the structure's design life through an analysis that incorporates all loading cycles.
The total accumulated Von Mises equivalent plastic strain, denoted as \$\epsilon_{eq,Ed}\$, can be calculated using the formula \$\epsilon_{eq,Ed} = m \Delta\epsilon_{eq,Ed}\$, where \$m\$ represents the number of cycles within the design life.
∆ εeq,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
(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
NOTE 1: The National Annex may choose the value of n eq The value n eq = 25 is recommended
NOTE 2: For the numerical value of γM0 see 1.1(2)
Buckling resistance
(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
(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 F Ed should satisfy the condition:
(2) The plate buckling resistance F Rd of a plated structure is defined as:
F Rd = k F Rk/γM1 (6.9) where F Rk 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)
The characteristic buckling resistance, denoted as \$F_{Rk}\$, must be determined from a load-deformation curve specific to the structure's relevant point, considering the appropriate combination of design actions \$F_{Ed}\$ Furthermore, the analysis should incorporate the imperfections outlined in section 5.2.3.2.
(4) The characteristic buckling resistance F Rk is defined by either of the two following criterion:
– maximum load of the load-deformation-curve (limit load);
– maximum tolerable deformation in the load deformation curve before reaching the bifurcation load or the limit load, if relevant
(5) The reliability of the numerically determined critical buckling resistance should be checked:
(a) either by calculating other plate buckling cases, for which characteristic buckling resistance values
F Rk values are determined under similar imperfection assumptions, ensuring consistency across check cases These cases should align in their buckling control parameters, including non-dimensional plate slenderness, post-buckling behavior, imperfection sensitivity, and material behavior.
(b) or by comparison of calculated values with test results F Rk,known
(6) Depending on the results of the reliability checks a calibration factor k should be evaluated from: k = F Rk,known,check / F Rk.check (6.10) where F Rk,known,check as follows from prior knowledge;
F Rk.check are the results of the numerical calculations
6.4.3 Supplementary rules for the design by simplified design methods
To assess the buckling resistance of a stiffened plate segment, it can be subdivided into subpanels and equivalent effective stiffeners as outlined in section 5.2.3.4 The design rules provided in EN 1993-1-5 can then be applied Additionally, the lateral buckling of free stiffener-flanges should be evaluated accordingly.
(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
(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
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.
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.
Excessive vibrations
Excessive vibrations are defined as the threshold at which a plated structure may fail due to fatigue from these vibrations, or when serviceability limits are reached.
NOTE: For limiting values of slenderness to prevent excessive vibrations see application standard
Annex A [informative] – Types of analysis for the design of plated structures
General
(1) The internal stresses of stiffened and unstiffened plates may be determined with the following types of analysis:
– GMNA: Geometrically and materially nonlinear analysis;
– GNIA: Geometrically nonlinear analysis elastic with imperfections included;
– GMNIA: Geometrically and materially nonlinear analysis with imperfections included.
Linear elastic plate analysis (LA)
Linear elastic analysis models the behavior of thin plate structures using plate bending theory, which assumes perfect geometry This linearity arises from the principles of linear elastic material law and the theory of small deflections.
(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):
Geometrically nonlinear analysis (GNA)
Geometrically nonlinear elastic analysis utilizes the principles of plate bending theory for ideal structures, incorporating both linear elastic material laws and 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
The large deflection theory considers the interplay between flexural and membrane actions, leading to non-linear variations in deflections and stresses as the out-of-plane pressure increases.
(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)
(A.2b) where f is the Airy´s stress function
Materially nonlinear analysis (MNA)
The materially nonlinear analysis utilizes plate bending theory for ideal structures, assuming small deflections, while also considering the material's nonlinear behavior.
Geometrically and materially nonlinear analysis (GMNA)
The analysis of geometrically and materially nonlinear behavior is grounded in the plate bending theory of ideal structures, incorporating the principles of nonlinear large deflection theory and the nonlinear elasto-plastic material law.
Geometrically nonlinear analysis elastic with imperfections included (GNIA)
The geometrically nonlinear analysis that incorporates imperfections is equivalent to the GNA analysis defined in section A.3 In this approach, the geometrical model utilizes a geometrically imperfect structure, such as applying a predeformation to the plate that is influenced by the corresponding buckling mode.
The GNIA analysis is essential for evaluating plated structures subjected to significant compression or shear stresses caused by in-plane effects This method provides insights into the elastic buckling resistance of actual imperfect plated structures.
Geometrically and materially nonlinear analysis with imperfections included (GMNIA)
The analysis that incorporates both geometric and material nonlinearity, along with imperfections, is equivalent to the GMNA analysis outlined in section A.5 In this approach, the geometric model utilizes a geometrically imperfect structure, such as a pre-deformation applied to the plate, which is influenced by the corresponding buckling mode.
The GMNIA analysis is essential for evaluating dominating compression or shear stresses in a plate caused by in-plane effects It provides insights into the elasto-plastic buckling resistance of the actual imperfect structure.
Annex B [informative] – Internal stresses of unstiffened rectangular plates from small deflection theory
General
This annex presents design formulas for calculating internal stresses in unstiffened rectangular plates, utilizing the small deflection theory It is important to note that the design formulas provided do not consider the effects of membrane forces.
(2) Design formulae are provided for the following load cases:
– uniformly distributed loading on the entire plate, see B.3;
– central patch loading distributed uniformly over a patch area, see B.4
The deflection \( w \) and bending stresses \( \sigma_{bx} \) and \( \sigma_{by} \) in a plate segment can be determined using the coefficients provided in the tables of sections B.3 and B.4, which assume a Poisson's ratio \( \nu \) of 0.3.
Symbols
The article defines key symbols related to load design: \( q_{Ed} \) represents the design value of the distributed load, while \( p_{Ed} \) denotes the design value of the patch loading Additionally, \( a \) refers to the shorter side of the plate, \( b \) indicates the longer side, and \( t \) signifies the thickness of the plate.
The Elastic modulus (E) is a key parameter in determining the mechanical properties of materials The coefficient for plate deflection (k_w) is tailored to the specific boundary conditions outlined in the data tables Additionally, the coefficients for bending stress, k_σbx and k_σby, correspond to the respective bending stresses σbx and σby, also aligned with the specified boundary conditions in the data tables.
Uniformly distributed loading
(1) The deflection w of a plate segment which is loaded by uniformly distributed loading may be calculated as follows:
NOTE: Expression (B.1) is only valid where w is small compared with t
(1) The bending stresses σbx and σby in a plate segment may be determined with the following equations: t k q 2
(2) For a plate segment the equivalent stress may be calculated with the bending stresses given in (1) as follows: σ σ σ σ σ 2 bx, d by, d d by,
The stress points specified in the data tables are situated along the center lines or boundaries, resulting in zero bending shear stresses (\$τ_b\$) due to symmetry or the established boundary conditions.
B.3.3 Coefficients k for uniformly distributed loadings
All edges are rigidly supported and rotationally free b/a k w1 k σbx1 k σby1
All edges are rigidly supported and rotationally fixed b/a k w1 k σbx1 k σby1 k σbx2
Three edges are rigidly supported and rotationally free and one edge is rigidly supported and rotationally fixed b/a k w1 k σbx1 k σby1 k σbx4
Two edges are rigidly supported and rotationally free and two edges are rigidly supported and rotationally fixed b/a k w1 k σbx1 k σby1 k σbx4
Two opposite short edges are clamped, the other two edges are simply supported b/a k w1 k σbx1 k σby1 k σby3
Two opposite long edges are clamped, the other two edges are simply supported b/a k w1 k σbx1 k σby1 k σbx2
Central patch loading
(1) The deflection w of a plate segment which is loaded by a central patch loading may be calculated as follows:
(1) The bending stresses σbx and σby in a plate segment may be determined by the following formulas: t k σbx p 2 Ed
(2) For a plate segment the equivalent stress may be calculated with the bending stresses given in (1) as follows: σ σ σ σ σ 2 bx, d by, d d by,
All edges are rigidly supported and rotationally free
Annex C [informative] – Internal stresses of unstiffened rectangular plates from large deflection theory
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:
– uniformly distributed loading on the entire plate, see C.3;
– central patch loading distributed uniformly over the patch area, see C.4
The bending and membrane stresses in a plate, along with the deflection \( w \), can be determined using the coefficients provided in the tables of sections C.3 and C.4, which assume a Poisson's ratio \( \nu \) of 0.3.
Symbols
The article defines key symbols related to load design: \( q_{Ed} \) represents the design value of the load uniformly distributed across the total surface, while \( p_{Ed} \) denotes the design value of the patch loading uniformly distributed over the dimensions \( u \times v \) Additionally, \( a \) refers to the shorter side of the plate, \( b \) indicates the longer side, and \( t \) signifies the thickness of the plate.
The MBC membrane boundary conditions include several coefficients that are crucial for analyzing plate behavior The coefficient \( k_w \) relates to the deflection of the plate based on specified boundary conditions Additionally, \( k_{\sigma bx} \) and \( k_{\sigma by} \) correspond to the bending stresses \( \sigma bx \) and \( \sigma by \), respectively, under the same boundary conditions Furthermore, \( k_{\sigma mx} \) and \( k_{\sigma my} \) represent the membrane stresses \( \sigma mx \) and \( \sigma my \), also aligned with the specified boundary conditions.
Uniformly distributed loading on the total surface of the place
(1) The deflection w of a plate segment which is loaded by uniformly distributed loading may be calculated as follows:
(1) The bending stresses σbx and σby in a plate segment may be determined with the following equations: t q a = k σ 2
(2) The membrane stresses σmx and σmy in a plate segment may be determined as follows: t q a = k σ 2
At the loaded surface of a plate, the total stresses are determined by combining the bending and membrane stresses The equations for these calculations are given by \$\sigma_{x,Ed} = -\sigma_{bx,Ed} + \sigma_{mx,Ed}\$ and \$\sigma_{y,Ed} = -\sigma_{by,Ed} + \sigma_{my,Ed}\$.
At the unloaded surface of a plate, the total stresses can be calculated by combining the bending and membrane stresses Specifically, the total stress in the x-direction, σx,Ed, is the sum of the bending stress, σbx,Ed, and the membrane stress, σmx,Ed, as expressed in equation (C.8) Similarly, the total stress in the y-direction, σy,Ed, is the sum of the bending stress, σby,Ed, and the membrane stress, σmy,Ed, as given in equation (C.9).
(5) For a plate the equivalent stress σv,Ed may be calculated with the stresses given in (4) as follows: σ σ σ σ σ x, E d y, E d
The stress points defined in the data tables are situated on the center lines or boundaries, resulting in zero membrane shearing stresses (\$τ_m\$) and bending shear stresses (\$τ_b\$) due to symmetry or boundary conditions The algebraic sum of the relevant bending and membrane stresses at these points yields the maximum and minimum surface stress values.
C.3.3 Coefficients k for uniformly distributed loadings
FBC: All edges are simply supported MBC: Zero direct stresses, zero shear stresses
E t q a b/a Q k w1 k σbx1 k σby1 k σmx1 k σmy1 k σmy2
FBC: All edges are simply supported
MBC: All edges remain straight Zero average direct stresses, zero shear stresses
Q = b/a Q k w1 k σbx1 k σby1 k σmx1 k σmy1 k σmx2 k σmy2
FBC: All edges are clamped
MBC: Zero direct stresses, zero shear stresses
Q = b/a Q k w1 k σbx1 k σby1 k σmx1 k σmy1 k σbx2 k σmy2
FBC: All edges are clamped
MBC: All edges remain straight Zero average direct stresses, zero shear stresses
Q = b/a Q k w1 k σbx1 k σby1 k σmx1 k σmy1 k σbx2 k σmx2 k σmy2
Central patch loading
To determine the deflection \( w \) and the stresses in a plate subjected to a central patch loading \( p_{Ed} \), it is essential to use the appropriate formulas for a loading area measuring \( u \) in length and \( v \) in width.
(1) The bending stresses σbx and σby in a plate segment may be determined with the following equations:
(2) The membrane stresses σmx and σmy in a plate segment may be determined as follows:
At the loaded surface of a plate, the total stresses are determined by combining the bending and membrane stresses The equations for these calculations are given by \$\sigma_{x,Ed} = -\sigma_{bx,Ed} + \sigma_{mx,Ed}\$ and \$\sigma_{y,Ed} = -\sigma_{by,Ed} + \sigma_{my,Ed}\$.
At the unloaded surface of a plate, the total stresses can be calculated by combining the bending and membrane stresses, represented by the equations: \$\sigma_{x,Ed} = \sigma_{bx,Ed} + \sigma_{mx,Ed}\$ and \$\sigma_{y,Ed} = \sigma_{by,Ed} + \sigma_{my,Ed}\$.
(5) For a plate the equivalent stress σv,Ed may be calculated with the stresses given in (4) as follows: σ σ σ σ σ 2 x, d y, d d y,
The stress points defined in the data tables are situated on center lines or boundaries, resulting in zero membrane shearing stresses (\$τ_m\$) and bending shear stresses (\$τ_b\$) due to symmetry or specified boundary conditions The algebraic sum of the relevant bending and membrane stresses at these points yields the maximum and minimum surface stress values.
FBC: All edges are rigidly supported and rotationally free
MBC: Zero direct stresses, zero shear stresses
P= p Ed b/a = 1 α×β p k w1 k σbx1 k σby1 k σmx1 k σmy1
FBC: All edges are rigidly supported and rotationally free
MBC: Zero direct stresses, zero shear stresses
P= p Ed b/a = 1,5 α×β p k w1 k σbx1 k σby1 k σmx1 k σmy1
FBC: All edges are rigidly supported and rotationally free
MBC: Zero direct stresses, zero shear stresses
P= p Ed b/a = 2 α×β p k w1 k σbx1 k σby1 k σmx1 k σmy1
FBC: All edges are rigidly supported and rotationally free
MBC: Zero direct stresses, zero shear stresses
P= p Ed b/a = 2.5 α×β p k w1 k σbx1 k σby1 k σmx1 k σmy1