WO2013175262A1

WO2013175262A1 - Method of finite element post - processing for a structural system involving fasteners

Info

Publication number: WO2013175262A1
Application number: PCT/IB2012/001603
Authority: WO
Inventors: Odnel Sadi CALME; François RIBOUR
Original assignee: Ingeliance Technologies
Priority date: 2012-05-25
Filing date: 2012-05-25
Publication date: 2013-11-28

Abstract

The present invention relates to a method of post-processing a structural system comprising load-bearing members and fasteners, said load-bearing members being joined by said fasteners according to a fasteners pattern and under constraint by superimposed parts carrying a junction load, the method comprising: - reading 91 a finite element model file, - reading 92 a fasteners pattern file, - representing 93 finite element model and fasteners pattern in order to define a finite element cut carrying the junction load, - associating 94 each fastener with some elements of the finite element cut in accordance with an interest zone sizing process, - distributing 95 the loads of each element of the finite element cut among its associated fasteners by respecting the loads conservation rules, and -computing 96 the loads applied on each fastener, in order to assess the junction strength of the structural system with said fasteners.

Description

METHOD OF FINITE ELEMENT POST - PROCESSING FOR A STRUCTURAL SYSTEM INVOLVING FASTENERS

Field of the invention

This invention belongs to the field of industrial implementations of materials and structures mechanics. It is useful, especially in design offices involved in stress activities, in order to quantify the criticality of structural mechanical behavior, wherever a hazardous level is reached by the failure risks to manage.

Balance loads between two load-bearing members attached to each other interact through the interfacial junction shared by both load-bearing members. A junction may be continuous (adhesive tape, bonding line, welding line, stitching) or discrete (nails, rivets, clips, bonding spots, welding spots, etc.). The invention technical domain is more specifically the one of technical resources aiming at productivity enhancement when calculating a fastener loads in a discrete junction.

The invention relates more particularly to a method of post-processing a structural system comprising at least two load-bearing members and at least one fastener, said load-bearing members being at least partially superimposed on each other by a part of them and said fastener joining under constraint said load-bearing members by their superimposed parts between which a junction is defined which carries a junction load.

The following definitions should be used to interpret the scope of the here claimed and described method.

We mean by fastener, any kind of discrete (or punctual) structural joint: rivet, screw, bolt, spot-weld spot ...

We mean by fasteners pattern, a geometrical arrangement of a set of fasteners linking at least two load-bearing members of a structural system.

We mean by node (or grid point) of a Finite Element Model, a point localized in space (with respect to other nodes) and at which variables of the Finite Element problem are defined. Variables attached to a node are principally: displacements (including translations and /or rotations) and loads (forces and/or moments).

We mean by element or finite element of a Finite Element Model, a constitutive elementary geometry delimited by nodes and lines connecting these nodes, and associated to physical properties ruling/which rule nodal variables (forces, moments, temperatures, displacements, rotations ...).

We mean by mesh (or grid) of a Finite Element Model, a set of elements with their delimitating nodes.

We mean by internal loads in a Finite Element Model, loads that are inside the structure and correspond to the loads applied by the finite elements on the nodes. The nodal loads are balanced when equilibrium is found: total loads (external plus internal) are equal to zero at each node.

We mean by node force balance (or Grid Point Force Balance, GPFB), a list of all loads applied to the nodes of the Finite Element Model. This list consists of external loads (applied loads or reactions due to frozen degrees of freedom) and internal loads (applied by all types of elements or constrained degrees of freedom connected to the node). Considering a spatial location, we mean by torsor, a description of forces and moments applied at this location. A torsor can be nodal in a finite element analysis; in this case, coordinates of the considered node and GPFB loads from the chosen side of a finite element cut containing this node are constitutive of the desired torsor.

We mean by junction load, a load carried by the structural junction. The junction load may be expressed in several forms: either a unique torsor at a node when the junction is modeled by a unique node, or a set of loads (torsors) given at each node of a set of nodes which represents the junction.

We mean by finite element cut, a set of nodes and elements allowing calculating the junction load from the GPFB of the Finite Element Model.

We mean by load path, structural items (fasteners, ropes, belts, lugs...) that are sufficient and necessary to carry the junction load, the considered items can work in series as well as in parallel. Any group of necessary structural items, the failure of which is likely to totally release the junction load, can be focused on to retrieve the said junction load by GPFB analysis of a finite element cut. Generally, the most simplified Finite Element Model is relevant for structural analysis as long as fully descriptive of all significant existing load paths in the modeled structure.

We mean by interest zone, a closed spatial domain around a specified fastener, inside which are applied the torsors considered relevant to assess the load of the fastener.

We mean by interest sphere, an interest zone the shape of which is a sphere centered on the fastener.

We mean by interest distance of an interest sphere, the radius of the interest sphere.

We mean by lever arm length of a lever arm of momentum transfer, the distance between a point at which a moment is applied and a point at which the applied moment balances a force.

We mean by a beam element, a mathematical construct used to model beam-type structures. The beam is a common structure used in engineering. The frame of a steel building, the frame of street lights can all be described by beams. For these reasons the beam element is frequently used in finite element analyses. The typical one-dimensional beam element assumes all loading is introduced at its extremities. To introduce intermediate loads along a modeled beam, multiple beam elements are used in series.

Background of the invention

Three possible input data scales allow to calculate the junction load, or equivalently the fasteners loads, in discrete junctions. These three possible input data scales are illustrated by figures 1 to 3 respectively, when applied to an example in which input data are representative of a fuselage junction and for which the calculation consist in seeking for fasteners loads inside the fuselage junction.

The first possible input data scale, illustrated by figure 1 , can be called the 'total junction load torsor'. In the above mentioned example, the input data consist at least in one single torsor, defining the normal and transverse forces and further the bending and twisting moments at the fuselage junction, the fuselage being regarded as a beam. The second possible input data scale, illustrated by figure 2, can be called 'several partial junction load torsors'. In the above mentioned example, the input data consist at least in a group of torsors defined at each frame intersection in the junction plane of the two fuselage load-bearing members (or slices). These torsors are typically nodal torsors derived from a Finite Element Model, the mesh size of which corresponds to frame intersection pitch.

The third possible input data scale, illustrated by figure 3, can be called 'full detail of each junction load torsor at each fastener'. In the above mentioned example, the input data consist in all local junction balance loads between the two fuselage slices, with these loads being defined as numerous torsors, each one associated to a unique fastener. This is typically obtained by a Finite Element Model , the refinement of which allows to have for each fastener at least one distinct representative node.

Using input data of the first scale, classical load computation methods (for example the analysis methods of an eccentrically loaded bolt group which are described by Bruhn, Niu, the American Institute of Steel Construction, etc) can be used to redistribute, in function of fasteners spatial locations, the unique torsor known at input data location. The US Patent Application numbered 5,844,232 of Buder is an advanced mean of performing fastener load analysis of a junction on the basis of a unique node chosen by the user. The selected nodal torsor is used to load a bolt group containing the junction fasteners, and a fail-safe analysis is automatically performed. For each fastener of the junction, envelope loads considering all possibilities to have one missing fastener in the junction are retrieved.

Using input data of the second scale, although some partial torsors are known at some node locations of the Finite Element Model, the fasteners number is still barely the same as the nodes number. Existing post-treatment tools (such as FLUX FASTENER tool of STREAME ® software) are dedicated to solve this discrepancy by projecting the known torsors onto a virtual junction line. Projected torsors are translated into load fluxes (i.e. local amounts of load per unit length), and then shared between fasteners in function of the local fastener pitch. The product of the fastener pitch by the fastener flux gives the load applied to each fastener. More complex actions are needed to solve multiple fasteners rows problems, especially in case of fasteners misalignment issues.

Using input data of the third scale, the torsor is known at each fastener owing to at least one distinct representative node from the Finite Element Model. The loads are computed by simple expression of the significant torsor components corresponding to the applicable sizing criteria.

In course of design development, the need for multiple configurations assessment leads to frequent modifications of known torsors number, frequent changes in fasteners number, location and orientation. Late after production of the genuine design, following a manufacturing defect or an event during service cycle, the same kind of local modifications do occur. Thus, whichever is the beginning configuration, conditions for using input data of the second scale are systematically met, soon or late.

A come back to the first scale needs gathering all known torsors into one, which may be fastidious and source of errors. For instance, in case of complex junction geometry, the unique bolt group kinematical assumptions may strongly increase the uncertainness level of the obtained fasteners loads distribution. To fight this, the user can subdivide the junction in several partial junctions; each one associated to a chosen nodal torsor. Then several independent bolt group analyses can be performed, at the cost of time spent to subdivide the problem.

A come back to the third scale means updating the detailed Finite Element Model, which is a costly action and likely to slow down the design activities. Although the second scale conditions are the more frequently encountered, available tools working at this second scale for Finite Element Models post-treatment are still likely to be improved.

Indeed, methods based on projection of fasteners loads and locations on an arbitrary line (such as 2010 SP1 R1 version of STREAME ® software FLUX FASTENER tool) allow the user to calculate separately balanced loads in each fastener of a lineal junction. Results given by the subjacent algorithms depend on the fasteners selection sequence. Thus, user must select Finite Element Model nodes and associated fasteners following a precise selection sequence. Nodal forces are projected on fasteners using approximate formulae assuming a lineal or quasi-lineal fasteners pattern. Consequently, in some particular cases in which relative locations of nodes and fasteners are not compatible, calculation fails owing to geometrical exceptions. Beside this technical limitation, it is relevant to point out the time constraint imposed by the need for the user to select numerous Finite Element Model items with respect to a precise selection sequence. With maintaining or improving calculation performance and precision, the main aim pursued by the here proposed method consists in:

- reducing and simplifying post-treatment tasks by implementing a method for which any selection sequence of representative nodes and fasteners is equivalent to one another,

- getting rid of geometrical exceptions by proposing a universal formulation covering all possible spatial configurations of nodes and fasteners (i.e. all input data scales), and

- obtaining a result in accordance with state-of-the-art kinematic assumptions for flight structures junctions and with respect to forces and moments balances.

Summary of the invention

This aim is achieved by a method of post-processing a structural system comprising at least two load-bearing members and at least one fastener, said load-bearing members being at least partially superimposed on each other and said fastener joining according to a fasteners pattern and under constraint said load-bearing members by their superimposed parts between which a junction is defined which carries a junction load, the method comprising:

- reading a finite element model file comprising at least two associated sets of data: a first set of data comprising at least spatial position of each node of a set of nodes distributed on a finite element mesh, with this latter modeling said at least two load-bearing members, and

a second set of data comprising a node force balance consisting in a list of loads applied to each node of the finite element mesh,

- reading a fasteners pattern file containing the description of at least the fasteners pattern according to which said two load-bearing members are joined and comprising at least spatial position of at least one modeled fastener, this latter modeling said at least one fastener,

spatial positions of said nodes and at least one modeled fastener being defined in a same coordinate system,

- in the coordinate system, representing nodes and said at least one modeled fastener in order to define a finite element cut consisting in a subset of selected nodes which is fully representative of a load path carrying the junction load,

- associating each modeled fastener with at least one selected node of said subset in accordance with an interest zone sizing process which consists in defining for each modeled fastener a zone of interest to which the modeled fastener belongs, with each zone of interest including at least one selected node and each selected node belonging to at least one zone of interest, in such a way to ensure that each selected node is associated with at least one modeled fastener and that each modeled fastener is associated with at least one selected node, and

- in accordance with the interest zone sizing process, distributing the loads of each selected node among its at least one associated modeled fastener by using a distribution method which respects the loads conservation rules like force balance and moment balance, and computing the loads applied on each modeled fastener, this post-processing method allowing to assess the junction strength of the structural system with said fasteners pattern.

According to this method, and whatever is the representative scale of the finite element model, there is no need to associate the selected node(s) with the modeled fastener(s) according to a particular selection sequence, since associating the selected node(s) with the modeled fastener(s) is done according to the interest zone sizing process which is independent from any selection sequence.

In a particular embodiment, the interest zone sizing process is an interest sphere sizing process, with each zone of interest being a sphere of interest, with each sphere of interest being centered on its associated modeled fastener and with the radius of each sphere defining an interest distance for each modeled fastener.

According to this particular embodiment, the method is advantageously easy to implement in an automatic manner, i.e. by help of a computer software rendering automatic at least the associating step.

In a first aspect of the preceding particular embodiment, the interest sphere sizing process is an iterative process which consists in adjusting the interest sphere size of each modeled fastener by iteratively enlarging it from an initial sphere size having an initial interest radius as interest distance.

According to this first aspect, the method allows to associate, with each modeled fastener, only the selected nodes which are the more relevant for computing the loads of said modeled fastener.

In a second aspect of the preceding particular embodiment, the iterative enlargement of the interest sphere size consists in incrementing, iteration by iteration, the interest radius by an amount equal to the initial interest radius.

In a third aspect of the preceding particular embodiment, the initial interest radius of the interest sphere of each modeled fastener is equal to a head diameter of each corresponding fastener. According to this third aspect, the initial sphere size is sufficiently small to ensure that not all the selected nodes of the finite element cut are associated with all the modeled fasteners at the first iteration of the iterative interest sphere sizing process.

In an another particular embodiment, with the loads of each node consisting in a torsor, the appropriate distribution method consists in that the torsor of each selected node contributes to loads applied on each of its at least one associated modeled fastener in a determined amount, in such a way that, per selected node, the sum of the determined amounts in which the torsor of said selected node contributes to loads applied on all of its at least one associated modeled fastener is equal to or lower than the torsor of said selected node.

In a first aspect of the preceding particular embodiment, said determined amounts in which the torsor of a determined selected node contributes to loads applied on all of its at least one associated modeled fastener depend on the distances between the selected node and each of its at least one associated modeled fastener.

In a second aspect of the preceding particular embodiment, said appropriate distribution method consists in a bolt group method.

In a third aspect of the preceding particular embodiment, when a lever arm length, that is the distance between a centroid of modeled fasteners associated with a determined selected node and one of its associated modeled fasteners, is less than a predetermined threshold value, the corresponding moment constituting a part of the torsor of the selected node is not transferred to said one of its associated modeled fasteners.

According to this latter aspect, when the length of a lever arm of momentum transfer is considered too short, i.e. less than the radius of the fastener, the method allows, in a good approximation, to avoid mathematical singularities, here division by zero, by not transferring the corresponding moment.

In a fourth aspect of the preceding particular embodiment, when more than one selected node is in the interest zone of a single modeled fastener, computing the loads applied on the modeled fastener consists in calculating the sum of determined amounts of loads in which the torsors of the selected nodes belonging to the interest zone of the modeled fastener contribute to the loads applied on this latter.

According to this another particular embodiment, the method allows a matrix formulation of the here above mentioned sums, which leads to improved readability and commodity of use.

According to a particular implementation, said finite element model file corresponds to the result of a given finite element model modeling the structure comprising said at least two load-bearing members with an initial fasteners pattern, and wherein said fasteners pattern file is representative of a modified fasteners pattern in comparison with the initial fasteners pattern.

According to this particular implementation of the method, there is no need to update the pre-existing finite element model whatever is its representative scale, in order to compute consequences of a change of the fasteners pattern. This advantage is useful not only in the optimization process, which consists for instance in taking into account the design loops of the fasteners pattern, but also in helping to the decision process, which consists in deciding whether or not the structural system can be securely used in its foreseen use conditions, by rendering these processes at least partially automated.

According to an aspect of the preceding particular implementation, modification of said initial fasteners pattern consists in subtracting, adding, displacing and/or changing the kind of one or several fasteners.

According to an another particular implementation, the part of the finite element mesh which surrounds the finite element cut changes between two implementations of the method according to the present invention.

According to this latter particular implementation of the method, using an updated finite element model, input and results, with the unchanged fasteners pattern, allows the method to assess the effect, on the junction strength, of a modification of the total structure represented by the finite element model. This advantage is useful al least in the optimization process which can then be at least partially automated. Thus, the method according to the present invention allows iterative looping taken into account the design loops of a structural product, said design loops leading to iterative changes of the finite element model.

The invention also relates to a computer program product comprising portions of program code for processing and/or aiding to process steps of the method according to the present invention, when said program is executed on a computer.

Depending on the structural system, a particular embodiment may be preferred as easier to adapt or as giving a better result. Aspects of these particular embodiments may be combined or modified as appropriate or desired, however.

These and other aspects of the invention will be apparent from and elucidated with reference to the embodiment given as example only and described hereafter where:

- Figure 1 shows a finite element mesh modeling two load-bearing members of a structural system, with said finite element mesh being defined at said first input data scale,

- Figure 2 shows a finite element mesh modeling the same two load-bearing members than those showed on figure 1 , but with said finite element mesh being defined at said second input data scale,

- Figure 3 shows a finite element mesh modeling the same two load-bearing members than those showed on figures 1 and 2, but with said finite element mesh being defined at said third input data scale,

- Figure 4a shows the finite element mesh as showed on figure 2 on which selected nodes are identified and numbered,

- Figure 4b shows the two load-bearings members of the structural system modeled as shown on figure 4a, with a particular fasteners pattern,

- Figure 5a shows a superimposition of Figures 4a and 4b and illustrates the interest zone sizing process according to the present method,

- Figure 5b shows a table representing, in a matrix form, the result of the associating step according to the present method when applied to the case shown on figure 5a,

- Figures 6a and 6b show the same finite element mesh with an initial fasteners pattern and with a modified fasteners pattern respectively, - Figures 7a and 7b show the same fasteners pattern with an initial finite element mesh defined at said second input data scale and with a modified finite element mesh, here a finite element mesh defined at said third input data scale, respectively,

- Figure 8 shows three times the same finite element mesh with an initial fasteners pattern and two different modified fasteners patterns, and

- Figure 9 shows a flowchart of the method according to the present invention. Detailed description

The present method is dedicated to the post-processing of a structural system 100, as illustrated on figure 4b, which comprises at least two load-bearing members 101 , 102 and at least one fastener 1 , 2, 14. More particularly, said load-bearing members 101 , 102 are at least partially superimposed on each other and said at least one fastener 1 , 2, 14 joins according to a fasteners pattern and under constraint said load-bearing members by their superimposed parts. Moreover, we define, between said superimposed parts, a junction which carries a junction load.

The fact that the two load-bearing members are joined under constraint means that the junction load is a non zero load, including loads occurring when the superimposed parts are pressed the one on the other. Said parts are more particularly local surfaces of the two load-bearing members which are planar or at least complementary the one from each other.

The first of said two load-bearing members is for instance a polycarbonate structural element 101 and the second of said two load-bearing members is for instance a machined aluminum plate 102, as illustrated on figure 4b.

The method comprises a first step consisting in reading 91 a finite element model file comprising at least two associated sets of data.

As illustrated in an example by table 3 below, a first set of data comprises spatial position of each node 201 , 202, 209 of a set of nodes distributed on a finite element mesh 300, with said finite element mesh 300 modeling said at least two load-bearing members 101 , 102.

As illustrated in an example by table 4 below, a second set of data comprises a node force balance consisting in a list of loads applied to each node 201 , 202, 209 of the finite element mesh 300.

Other sets of data may make part of a finite element model file such as the temperature at each node, the displacement at each node, etc.

The method further comprises a second step consisting in reading 92 a fasteners pattern file. Said fasteners pattern file contains the description of at least the fasteners pattern according to which said two load-bearing members 101 , 102 are joined. As illustrated in an example by table 1 below, the fasteners pattern file more particularly comprises spatial position of each modeled fastener 401 , 402, 414, with this latter modeling one of said at least one fastener 1 , 2, 14. For instance, the fasteners pattern file may also comprise data representative of kind, orientation, and head diameter of each fastener.

Said first and second steps can be implemented in any order. Spatial positions of said nodes 201 , 202, 209 of the finite element mesh 300 and at least one modeled fastener 401 , 402, 414 are defined in a same coordinate system CS0, which is for instance a Cartesian coordinate system.

The method further comprises a third step consisting in representing 93, in the coordinate system CS0, nodes 201 , 202, 209 and said at least one modeled fastener 401 , 402, ... , 414. This step consists for instance in superimposing figure 4a with figure 4b as illustrated on figure 5a. This step allows an operator to define a finite element cut 310 by selecting some nodes among nodes 201 , 202, 209 of the finite element mesh 300, that is a subset of selected nodes 201 , 202, 208. This selection has to be done in order that the finite element cut 310 is fully representative of a load path carrying the junction load. It should be noted that this step remains to the responsibility of the operator, albeit the representing step may be computer-aided.

Figures 1 to 3 illustrate for each input data scale how is defined the corresponding finite element cut 310.

On figure 1 , the plate 102 is modeled with a one-dimensional finite element cut 310, as a beam element between two nodes, and the polycarbonate structural element 101 is modeled with two-dimensional mesh elements. The interface load is applied to the first beam element node correspondingly located at a lug centre. The junction load retrieved at its second node which connects the polycarbonate structural element 101 balances the interface load.

On figure 2, the plate 102 and the polycarbonate structural element 101 are both modeled with two-dimensional elements, so as projections of some nodes of both superimposed parts onto the junction plane are coincidental. These nodes are linked by one-dimensional beam elements representing the fasteners and comprising the finite element cut 310. However, the number and locations of the beam elements are actually different from the number and locations of the modeled fasteners.

On figure 3, the plate 102 and the polycarbonate structural element 101 are both modeled with two-dimensional elements, so as projections of some nodes of both superimposed parts onto the junction plane are coincidental with the fasteners projections. These nodes are linked by one-dimensional beam elements representing the fasteners and comprising the finite element cut 310. Moreover, the number and locations of the beam elements are exactly the same as the number and locations of the fasteners.

The method further comprises a fourth step consisting in associating 94 each modeled fastener 401 , 402, 414 with at least one selected node of said subset. This step is done in accordance with an interest zone sizing process. This latter consists in defining for each modeled fastener a zone of interest 501 , 502, 514 to which the modeled fastener 401 , 402, ... , 414 belongs. Said definition is done in such a way that each zone of interest 501 , 502, 514 includes at least one selected node and each selected node belongs to at least one zone of interest 501 , 502, 514. It is thus ensured that each selected node 201 , 202, 208 is associated with at least one modeled fastener 401 , 402, ... , 414 and that each modeled fastener 401 , 402, 414 is associated with at least one selected node 201 , 202, 208.

A man skilled in the art should understand that the interest zone sizing process is independent from any selection sequence and thus there is no more need to associate the selected node(s) with the modeled fastener(s) according to a particular selection sequence. According to prior art, it is under the responsibility of the operator to select the nodes of the finite element cut in an appropriate order which furthermore can have an influence on the result of the fasteners loads calculation. This step implemented according to prior art is cumbersome, time consuming and source of errors. It is thus advantageously avoided owing to the present method . Moreover, the associating step according to the present method may be made totally automated as described below.

Indeed, according to a particular embodiment, the interest zone sizing process is more particularly an interest sphere sizing process.

With respect to this particular embodiment, a man skilled in the art should understand that said previously generically considered interest zone can be of all forms and all dimensions according to the local finite element mesh geometry. Each generic interest zone can always be considered as included in an interest sphere of a determined interest radius as previously defined.

According to another example, said zone of interest can advantageously be different from a sphere in a case where selected nodes densities are not the same in all directions of space. For instance, in a two-dimensional finite element model having two generating directions, if the selected nodes density in the first direction is twice the selected nodes density in the second direction, it can be advantageous to choose an interest zone having an ellipsoidal shape, with a minor axis in the first direction and a mayor axis in the second direction. In this way, the number of selected nodes which belong to the ellipsoidal zone may be the same in both directions.

Nonetheless, an interest zone having spherical shape is preferred because, according to this particular embodiment, the associating step of the present method is advantageously easy to implement in an automatic manner whatever is the dimensionality, the particular geometry or the input data scale of the finite element mesh, as described here below.

The particular embodiment according to which the interest zone sizing process is an interest sphere sizing process is illustrated in two dimensions on figure 4a and 5a.

Figure 5a illustrates the same finite element mesh than the one illustrated on figure 2 for which we already observe discrepancies between nodes of the finite element mesh and modelled fasteners. Selected nodes 201 , 202, 208 and modeled fasteners 401 , 402, 414 are numbered for the purpose of the present detailed description, but this numbering is arbitrary and has no influence on the computation result(s) of loads applied on fasteners. According to the operator, the eight numbered nodes are relevant to perform the load path and thus are selected in order to define the finite element cut 310. The node force balance of these selected nodes balances loads from eight beam elements representing the fasteners and loads from the mesh elements representing the plate 102.

According to the case illustrated on figure 5a, each zone of interest 501 , 502, ... , 514 is a disc of interest which is included in a sphere of interest having the same center and radius. Each sphere of interest is centered on its associated modeled fastener and the radius of each disc defining an interest distance, that is the radius of the disc, for each modeled fastener 401 , 402, ... , 414.

In a first aspect of the preceding particular embodiment, the interest sphere sizing process is an iterative process. This latter consists in adjusting the interest sphere size of each modeled fastener 401 , 402, 414 by iteratively enlarging it from an initial sphere size having an initial interest radius as interest distance.

In a third aspect of the preceding particular embodiment, the initial interest radius of the interest sphere of each modeled fastener 401 , 402, 414 is equal to a head diameter of each corresponding fastener 1 , 2, 14.

According to this third aspect, the initial sphere size is sufficiently small to ensure that not all the selected nodes of the finite element cut are associated with all the modeled fasteners at the first iteration of the iterative interest sphere sizing process. Moreover, the head diameter of the fasteners is the shortest expectable distance between two fasteners, such that no other modeled fastener than the one on which the initial interest sphere is centered is expected to belong to said initial interest sphere.

These three aspects of said particular embodiment can be considered solely or in combination without departing from the frame of the presently described method.

In order to illustrate in more details said iterative interest sphere sizing process, we present below an example of its implementation.

Be [Aij] a modeled fastener-to-selected node distances matrix, with Aij being the distance from the modeled fastener denoted by Ϊ to the selected node denoted by 'j'. For each modeled fastener, the interest distance Dint is initialized with the head diameter of its corresponding fastener.

Then we define an interest matrix [5ij] which is a function of Dint and [Aij] in such a way that:

- 5ij = 1 if Aij < Dint,

- 5ij = 0 otherwise.

According to the iterative interest sphere sizing process, the interest distance Dint is incremented while each selected node is not associated with at least one modeled fastener and each modeled fastener is not associated with at least one selected node. It means checking the sums of each column and each raw of interest matrix [5ij]. As soon as one sum is found equal to zero, the interest distance Dint is incremented by the fastener head diameter and then [5ij] is checked again. Incrementing ceases when there is no sum of raw and no sum of column equal to zero anymore.

Figure 5b shows the matrix [5ij] resulting from this implementation applied to the case illustrated on figure 5a.

It appears more clearly that this iterative interest sphere sizing process allows to associate loads of a group of j-indiced selected nodes to a group a i-indiced modeled fasteners without having to take care of the selection sequence in the calculation procedure and preferably in the here above described automated manner. The purpose consisting in reducing and simplifying post-treatment tasks of the structural system is reached. Indeed, the result of iterative interest sphere sizing process depends on comparisons between distances and is not dependent on selection sequence. Moreover several fastened junctions can be calculated at once from the selection of identified finite element cuts and fasteners in any order. This results in saved time.

The purpose consisting in getting rid of geometrical exceptions by proposing a universal formulation covering all possible spatial configurations of nodes and modeled fasteners (i.e. all input data scales) is reached. Indeed, the selected node- to-modeled fastener distances being measured in three dimensions, all possible spatial configurations are treated equivalently according to the iterative interest sphere sizing process.

Owing to the here above described interest sphere sizing process, the present method auto-adapts to the different input data scales without any specific intervention.

Notably and as detailed below, owing to the present method, there is no need to update the pre-existing finite element model whatever is its representative scale, in order to calculate consequences of shrinkage, addition, displacement or modification of one or several fasteners or modification of surrounding structure. Thus the economic shift of the present method is notably to increase update possibilities of fasteners loads calculation results without having to previously update the finite element model.

The present method further comprises a fifth step consisting in distributing 95 the loads of each selected node 201 , 202, 208 among its at least one associated modeled fastener 401 , 402, 414. This step is more particularly done by using a distribution method which respects the loads conservation rules like force balance and moment balance. Thus, the present method allows to obtain a result in accordance with state-of-the-art kinematic assumptions for flight structures junctions and with respect to forces and moments balances.

In an another particular embodiment, not exclusive but potentially complementary of the previous embodiment, we define the loads of each node 201 , 202, 209 as consisting in a torsor. The distribution method consists in that the torsor of each selected node 201 , 202, 208 contributes to loads applied on each of its at least one associated modeled fastener in a determined amount. Said determined amount is defined in such a way that, per selected node 201 , 202, 208, the sum of the determined amounts in which the torsor of the considered selected node contributes to loads applied on all of its at least one associated modeled fastener is equal to the torsor of said selected node.

It should be noted here, but we will come back later, that the sum of the determined amounts in which the torsor of the considered selected node contributes to loads applied on all of its at least one associated modeled fastener can also be lower than the torsor of said selected node.

In a first aspect of the preceding particular embodiment, said determined amounts in which the torsor of a determined selected node contributes to loads applied on all of its at least one associated modeled fastener depend on the distances Aij between the selected node 201 , 202, 208 and each of its at least one associated modeled fastener 401 , 402, ... , 414.

In a second aspect of the preceding particular embodiment, said appropriate distribution method consists in a bolt group method. More particularly and as mentioned above, loads computation methods such as the Bruhn's bolt group method or the Niu's bolt group method can be used. This is advantageous because the reliability of these methods, which are already and largely known and used by the global structural analysis community no longer needs to be demonstrated.

This is for taken into account of said approximation that it has been noted earlier that the sum of the determined amounts in which the torsor of the considered selected node contributes to loads applied on all of its at least one associated modeled fastener can also be lower than the torsor of said selected node. Indeed, when the moment of the torsor is not transferred in the above specific conditions, the torsor is thus not entirely distributed. Nonetheless, the difference is negligible in good approximation.

A fourth aspect of the preceding particular embodiment allows to foresee the occurable case according to which more than one selected node 201 , 202, 208 is in the interest zone 501 , 502, 514 of a single modeled fastener. In such a case, computing the loads applied on the modeled fastener 401 , 402, 414 consists in calculating the sum of determined amounts of loads in which the torsors of the selected nodes belonging to the interest zone of the modeled fastener contribute to the loads applied on this latter.

According to this another particular embodiment, the method allows a matrix formulation of the here above mentioned sums, that is principally the sum of determined amounts of loads in which the torsors of the selected nodes belonging to the interest zone of the modeled fastener contribute to the loads applied on this latter. Said matrix formulation leads to improved readability and commodity of use. This technical feature and its induced advantage will be more clearly understood by the help of the hereafter detailed mathematical algorithm presenting the present method in an analytical manner.

Then the method further comprises a sixth step consisting in computing 96 the loads applied on each modeled fastener 401 , 402, 414. The result of said computing 96 step is given in Table 5 below for a determined example presented below. Thus, the present post-processing method allows to assess the junction strength of the structural system 100 with said fasteners pattern.

Figure 9 shows a flowchart of the method according to the present invention.

According to a particular implementation of the present method, said finite element model file corresponds to the result of a given finite element model modeling the structure 100 which comprises said at least two load-bearing members 101 , 102. Said result has been obtained with an initial fasteners pattern. Said initial fasteners pattern is different from the fasteners pattern of the reading 92 fasteners pattern file. Indeed this latter is representative of a modified fasteners pattern in comparison with the initial fasteners pattern. According to this particular implementation of the method, there is no need to update the pre-existing finite element model whatever is its representative scale, in order to compute consequences of a change of the fasteners pattern. This advantage is useful not only in the optimization process, which consists for instance in taking into account the design loops of the fasteners pattern, but also in helping to the decision process, which consists in deciding whether or not the structural system can be securely used in its foreseen use conditions, by rendering these processes at least partially automated.

Figures 6a and 6b illustrate such a case. Figure 6a shows the initial fasteners pattern with which the finite element model has been first obtained. Said initial fasteners pattern includes fourteen fasteners. Figure 6b shows the fasteners pattern represented in the reading 92 fasteners pattern file. In the instance shown on figure 6b, said fasteners pattern includes only nine fasteners, because of subtraction of modeled fasteners numbered 401 , 405, 407, 41 1 and 413, with respect to the initial fasteners pattern. Said subtraction can be representative of real shrinkages of fasteners of the structural system. And the method allows to assess the junction strength of the structural system 100 with the modified fasteners pattern, without requiring to modify the finite element model .

Owing to the present method which desynchronizes the numerical resolution of the finite element problem and the fasteners loads computation, the need to modify the finite element model in order to obtain its new solution (with all induced management about calculation model configuration) is advantageously avoided when the fasteners pattern is modified.

According to an another particular implementation, the part of the finite element mesh 300 which surrounds the finite element cut 310 changes between two implementations of the method according to the present invention.

According to this latter particular implementation of the method, using an updated finite element model, input and results, with the unchanged fasteners pattern, allows the method to assess the effect, on the junction strength, of a modification of the total structure represented by the finite element model . This advantage is useful al least in the optimization process which can then be at least partially automated. Thus, the method according to the present invention allows iterative looping taken into account the design loops of a structural product, said design loops leading to iterative changes of the finite element model.

Figures 7a and 7b illustrate such a case in which the finite element model changes while the fasteners pattern remains the same. Figure 7a shows an initial finite element model which has been first considered. Said initial finite element model is defined at the second input data scale with which fasteners loads had potentially been computed. Figure 6b shows the finite element model corresponding to the reading 91 finite element model file, with said finite element model being defined at the third input data scale. This can be the result of design loops of the load-bearing members, the design loops consisting in departing from a roughly load-bearing members design (represented here by the finite element model defined at the second input data scale) to a refined load-bearing members design (represented here by the finite element model defined at the third input data scale). Owing to the present method which desynchronizes the numerical solution of the finite element model and the fasteners loads computation, the need to rerun the complete computation of fasteners loads included in the finite element model is avoided when the finite element model is modified.

More generally, during a design process, the refinement of the total finite element model from which are issued the loads progressively increases, notably due to more and more fine analyses: general efforts in first, static strength in second, then fatigue verification, and eventually damage tolerance. The here proposed method has the advantage to be continuous and to be able to use the same data describing the fasteners between two evolutions of the finite element model.

In order to provide a complete specification of the present method, the following mathematical algorithm is included which describes in a mathematical manner the above considerations, and more particularly distributing 95 and computing 96 steps of the present method, applied to the case illustrated on figure 8 and to the parameters values of the tables.

Nomenclature

CSO Cartesian coordinate system in which the coordinate triplets defining the vectors and points are expressed

sph. i Radius of fastener i interest sphere

head i Initial interest distance of fastener i

Φ fast . i Diameter of fastener i

fast, i

fast, i Surface of fastener i in load-bearing section A_{fast Λ} = π

Point locating node j in CSO

Qi Point locating fastener i centre in CSO

Qn Point locating fastener i fastener head in CSO

Point locating fastener i fastener foot in CSO

^GJ Point locating the centroid of fasteners interesting node j in CSO

δ_ϋ Binary term (0 or 1 ) of interest matrix (nodes j interested by fasteners i ) y Binary term (0 or 1 ) of moment transfer matrix from ^{G j}J t ton a

P G

β_;. Binary term (0 or 1 ) of moment transfer matrix from ^J to ^J

coord(M) Coordinate triplet of any point M in CSO

a b cross product such as (a1 , a2, a3) (b1 , b2, b3)

ab Scalar product such as (a1 , a2, a3) . (b1 , b2, b3) = a1 b1 + a2b2+ a3b3 a Norm of any vector (a1 , a2, a3) equal to a1² + a2² + a3²

F_Gj Node j nodal force transferred at centroid of fasteners interesting node j

^M°^j Node j nodal moment transferred at centroid of fasteners interesting node j ¹ CSOj Node j nodal force expressed in CSO

Mcsoj Node j nodal moment expressed in CSO

X Unit vector of (1 ,0,0) coordinates in CSO

y Unit vector of (0,1 ,0) coordinates in CSO

z Unit vector of (0,0,1 ) coordinates in CSO

¹ CSOi Calculated forces applied to fastener i expressed in CSO

sqrt(x) Square root of any x positive scalar value

Fx, Fy, Fz Scalar force components of a torsor expressed in CSO

Mx, My, Mz Scalar moment components of a torsor expressed in CSO

X, Y, Z Scalar components of a coordinate vector expressed in CSO

Context

The context of the hereafter description is the critical airworthiness assessment of two non-conform flight structures. As illustrated on the middle of figure 8, the first aircraft has three missing fasteners; as illustrated on the bottom of figure 8, the second aircraft has only two missing fasteners.

Aim

The structural engineer has fifteen minutes to provide relevant engineering data in order to substantiate the flight clearance decision. The only available FEM is those of conform flight structure, which appears to be a FEM defined at the second input scale.

Methodology

The nodal loads are allocated to the different fasteners with respect to the hereafter non-limitative procedure.

Identify for the considered load path the nodes relevant for finite element cut, so as to associate with load path fasteners i only these identified nodes j by iterative adjustment of the interest spheres.

Choose a relevant value of incremental length for interest sphere adjustment. For instance, for a rivet, the rivet head diameter is relevant, for a nail, the nail head diameter is relevant. For each fastener i, store r_{sph i} = §_{head i}

Store lo ^" norms, terms of "fastener i - node j" distance matrix

1 if

4. Calculate terms of initial interest matrix δ,, = Q_tP, < r s.ph. i

otherwise

5. While 0do r , . = r , . +<j>_{Aeed i} then re-assess δ.

Conserve the first 1 as

the final interest matrix

Store G . centroids locations of the different fasteners interesting each node j

^≠fa_St_.fioord Q^

coord (G )

Store the norms , terms of "node j - centroid of fasteners interesting node j" distance matrix. Store the β .. terms such as β.. =

0 otherwise Store the nodal loads transferred at centroids

Store the norms G.Q, , terms of "centroid j - fastener i" distance matrix

> -φ

Store the γ.. terms such as 1 if fjQ, fast, i

2

0 otherwise Store x A G .Q_I , y A G .Q_I , and z A G .Q_T vectors Store x A GJQ_J , \\y A GJQ_J , and z Λ G O vectors norms Store the allocated fasteners loads in CSO so as:

A fast, i δ ij

First user responsibility is to identify for the considered load path the relevant nodes for the finite element cut, so as to associate with load path fasteners only these identified nodes (by iterative adjustment of the interest spheres).

The second user responsibility is to choose a relevant incremental length for interest sphere adjustment. The initial sphere size has to be sufficiently small, for instance as rivet's head diameter, to ensure that, generally, not all the nodes of the junction are associated with all the rivets at the first search iteration. The initial size of the sphere equal to the rivet's head diameter is a preferred initial value.

In the context of the present example, these actions were already performed during certification phase in order to calculate certification loads from FEM defined at the second input data scale. The entry data are summarized in Table 1 Fasteners locations in CSO), Table 2 Fasteners sizes), Table 3 Nodes locations in CSO), Table 4 Nodal torsors in CSO). The initial result was an acceptable critical shear load of 1395N in rivet n°3 of the aircraft illustrated on the top of figure 8, as reported in Table 5 Obtained fasteners loads in CSO).

Table 1 Fasteners locations in CSO i Ai [mm²]

1 8.0424772

2 8.0424772

3 8.0424772

4 8.0424772

5 8.0424772

6 8.0424772

7 8.0424772

8 8.0424772

9 8.0424772

10 8.0424772

1 1 8.0424772

12 8.0424772

13 8.0424772

14 8.0424772

Table 2 Fasteners sizes

Table 3 Nodes locations in CSO

j 1 2 3 4 5 6 7 8

Fx [N] 1337.2 1330.6 1219.4 1 1 16.7 172.91 22.639 464.31 988.33

FCSOj Fy [N] 1469.6 709.78 -157.85 -458.24 1371 .1 517.43 -183.03 -513.83

Fz [N] -7.9088 -47.145 19.093 -140.69 145.24 53.933 66.122 -88.642

Mx [N.mm] 2621 .7 121 1 .8 -366.68 -966.73 1948.4 391.08 -509.84 -945.58

MCSOj My [N.mm] -2283.3 -2367.9 -2108.1 -1364.9 -91.835 83.139 -633.44 -1043.0

Mz [N.mm] -239.16 -522.56 -634.06 -580.08 362.39 1 17.93 -106.21 -1 1 1.59

Table 4 Modal torsors in CSO

a e a ne as eners oa s n

Results

Considering the same input data except three missing rivets as on the aircraft illustrated on the middle of figure 8, the result is a non acceptable critical shear load of 1609N in rivet n°1 1 illustrated on the middle of figure 8, as reported in Table 6 Obtained fasteners loads in CSO with three missing rivets). The corresponding aircraft gets no flight clearance.

Table 6 Obtained fasteners loads in CSO with three missing rivets

Considering the same input data except two missing fasteners as on the aircraft illustrated on the bottom of figure 8, the result is an acceptable critical shear load of 1395 N in rivet n°3 illustrated on the bottom of figure 8, as reported in Table 7 Obtained fasteners loads in CSO with two missing rivets). The corresponding aircraft gets flight clearance.

Table 7 Obtained fasteners loads in CSO with two missing rivets

The method may be implemented by a computer program product that is able to implement or help to implement the method steps as described above when loaded and run on computer means of an image resizing apparatus. The computer program may be stored/distributed on a suitable medium supplied together with or as a part of other hardware, but may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems.

An integrated circuit may be arranged to perform or help to perform the method steps in accordance with the disclosed embodiments.

Other variations to the disclosed embodiments can be understood and effected by those skilled on the art in practicing the claimed invention, from a study of the drawings, the disclosure and the appended claims. In the claims, the word "comprising" does not exclude other elements and the indefinite article "a" or "an" does not exclude a plurality.

Claims

1 . Method of post-processing a structural system (100) comprising at least two load-bearing members (101 , 102) and at least one fastener (1 , 2, ... , 14), said load-bearing members (101 , 102) being at least partially superimposed on each other and said fastener (1 , 2, 14) joining according to a fasteners pattern and under constraint said load-bearing members by their superimposed parts between which a junction is defined which carries a junction load, the method comprising:

- reading (91 ) a finite element model file comprising at least two associated sets of data:

a first set of data comprising at least spatial position of each node (201 , 202, 209) of a set of nodes distributed on a finite element mesh (300), with this latter modeling said at least two load-bearing members (101 , 102), and a second set of data comprising a node force balance consisting in a list of loads applied to each node (201 , 202, 209) of the finite element mesh (300),

- reading (92) a fasteners pattern file containing the description of at least the fasteners pattern according to which said two load-bearing members (101 , 102) are joined and comprising at least spatial position of at least one modeled fastener (401 , 402, 414), this latter modeling said at least one fastener (1 , 2, 14),

spatial positions of said nodes (201 , 202, 209) and at least one modeled fastener (401 , 402, ... , 414) being defined in a same coordinate system (CS0), - in the coordinate system (CS0), representing (93) nodes (201 , 202, 209) and said at least one modeled fastener (401 , 402, 414) in order to define a finite element cut (310) consisting in a subset of selected nodes (201 , 202, 208) which is fully representative of a load path carrying the junction load,

- associating (94) each modeled fastener (401 , 402, ... , 414) with at least one selected node of said subset in accordance with an interest zone sizing process which consists in defining for each modeled fastener a zone of interest (501 , 502, 514) to which the modeled fastener (401 , 402, ... , 414) belongs, with each zone of interest (501 , 502, 514) including at least one selected node and each selected node belonging to at least one zone of interest (501 , 502, 514), in such a way to ensure that each selected node

(201 , 202, ... , 208) is associated with at least one modeled fastener (401 , 402, 414) and that each modeled fastener (401 , 402, 414) is associated with at least one selected node (201 , 202, ... , 208), and

- in accordance with the interest zone sizing process, distributing (95) the loads of each selected node (201 , 202, 208) among its at least one associated modeled fastener (401 , 402, 414) by using a distribution method which respects the loads conservation rules like force balance and moment balance, and computing (96) the loads applied on each modeled fastener (401 , 402, ... , 414),

this post-processing method allowing to assess the junction strength of the structural system (100) with said fasteners pattern.

2. Method according to claim 1 , wherein the interest zone sizing process is an interest sphere sizing process, with each zone of interest (501 , 502, 514) being a sphere of interest, with each sphere of interest being centered on its associated modeled fastener and with the radius of each sphere defining an interest distance for each modeled fastener (401 , 402,

414).

3. Method according to claim 2, wherein the interest sphere sizing process is an iterative process which consists in adjusting the interest sphere size of each modeled fastener (401 , 402, 414) by iteratively enlarging it from an initial sphere size having an initial interest radius as interest distance.

4. Method according to claim 3, wherein the iterative enlargement of the interest sphere size consists in incrementing, iteration by iteration, the interest radius by an amount equal to the initial interest radius.

5. Method according to any of claims 3 and 4, wherein the initial interest radius of the interest sphere of each modeled fastener (401 , 402, 414) is equal to a head diameter of each corresponding fastener (1 , 2, 14).

6. Method according to any of claims 1 to 5, wherein, with the loads of each node (201 , 202, 209) consisting in a torsor, the appropriate distribution method consists in that the torsor of each selected node (201 , 202,

208) contributes to loads applied on each of its at least one associated modeled fastener in a determined amount, in such a way that, per selected node (201 , 202, 208), the sum of the determined amounts in which the torsor of said selected node contributes to loads applied on all of its at least one associated modeled fastener is equal to or lower than the torsor of said selected node.

7. Method according to claim 6, wherein said determined amounts in which the torsor of a determined selected node contributes to loads applied on all of its at least one associated modeled fastener depend on the distances between the selected node (201 , 202, 208) and each of its at least one associated modeled fastener (401 , 402, ... , 414).

8. Method according to any of claims 6 and 7, wherein said appropriate distribution method consists in a bolt group method.

9. Method according to claim 8, wherein, when a lever arm length, that is the distance between a centroid of modeled fasteners associated with a determined selected node and one of its associated modeled fasteners, is less than a predetermined threshold value, the corresponding moment constituting a part of the torsor of the selected node is not transferred to said one of its associated modeled fasteners.

10. Method according to any of claims 6 to 9, wherein, when more than one selected node (201 , 202, 208) is in the interest zone (501 , 502, 514) of a single modeled fastener, computing the loads applied on the modeled fastener (401 , 402, 414) consists in calculating the sum of determined amounts of loads in which the torsors of the selected nodes belonging to the interest zone of the modeled fastener contribute to the loads applied on this latter.

1 1 . Method according to any of claims 1 to 10, wherein said finite element model file corresponds to the result of a given finite element model modeling the structure (100) comprising said at least two load-bearing members (101 , 102) with an initial fasteners pattern, and wherein said fasteners pattern file is representative of a modified fasteners pattern in comparison with the initial fasteners pattern.

12. Method according to claim 1 1 , wherein modification of said initial fasteners pattern consists in subtracting, adding, displacing and/or changing the kind of one or several fasteners (1 , 2, ... , 14).

13. Method according to any of claims 1 to 10, wherein the part of the finite element mesh (300) which surrounds the finite element cut (310) changes between two implementations of the method according to any of claims 1 to 10.

14. Computer program product comprising portions of program code for processing and/or aiding to process steps of the method according to any one of claims 1 to 13, when said program is executed on a computer.