U.S. patent application number 15/632712 was filed with the patent office on 2018-02-15 for method for the structural analysis of panels consisting of an isotropic material and stiffened by triangular pockets.
This patent application is currently assigned to AIRBUS OPERATIONS (S.A.S). The applicant listed for this patent is AIRBUS OPERATIONS (S.A.S). Invention is credited to Gerard COUDOUENT, Paolo MESSINA.
Application Number | 20180046740 15/632712 |
Document ID | / |
Family ID | 42111874 |
Filed Date | 2018-02-15 |
United States Patent
Application |
20180046740 |
Kind Code |
A1 |
COUDOUENT; Gerard ; et
al. |
February 15, 2018 |
METHOD FOR THE STRUCTURAL ANALYSIS OF PANELS CONSISTING OF AN
ISOTROPIC MATERIAL AND STIFFENED BY TRIANGULAR POCKETS
Abstract
A method for dimensioning of panels stiffened by triangular
pockets that make it possible to take into account aeronautical
specifics and more particularly the stresses that are admissible
for the different types of buckling and the calculations of adapted
reserve factors is presented. The method relates to dimensioning of
a substantially plane panel of homogeneous and isotropic material
by an analytical procedure, wherein the panel is composed of a skin
reinforced by an assembly of three parallel bundles of stiffeners
integrated with the panel, and triangular pockets defined on the
skin, the stiffeners are strip-shaped and the panel must satisfy a
specification of mechanical resistance to predetermined external
loads, including steps organized in such a way that they can be
repeated iteratively for different values of input data until
reserve factors are obtained to determine the dimensions and
arrangement of the panel elements necessary to obtain the imposed
mechanical resistance.
Inventors: |
COUDOUENT; Gerard;
(Fontenilles, FR) ; MESSINA; Paolo; (Toulouse,
FR) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
AIRBUS OPERATIONS (S.A.S) |
Toulouse |
|
FR |
|
|
Assignee: |
AIRBUS OPERATIONS (S.A.S)
Toulouse
FR
|
Family ID: |
42111874 |
Appl. No.: |
15/632712 |
Filed: |
June 26, 2017 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
13395744 |
May 25, 2012 |
9690887 |
|
|
PCT/FR2010/051900 |
Sep 13, 2010 |
|
|
|
15632712 |
|
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06F 30/23 20200101 |
International
Class: |
G06F 17/50 20060101
G06F017/50 |
Foreign Application Data
Date |
Code |
Application Number |
Sep 14, 2009 |
FR |
0956286 |
Claims
1. (canceled)
2: A method for dimensioning of a substantially plane panel of
homogeneous and isotropic material by an analytical procedure, the
panel having a skin reinforced by an assembly of three parallel
bundles of stiffeners integrated with the panel, and triangular
pockets defined on the skin by the stiffeners, wherein the
stiffeners are strip-shaped and the panel must satisfy a
specification of mechanical resistance to predetermined external
loads, the method includes: acquiring values of input data
commensurate with a geometry of the panel, a material of the panel,
and loading applied to the panel; calculating stresses applied on
the skin of the panel and the stiffeners, based on the geometry of
the panel stiffened by the triangular pockets, and on applied
external loads; calculating internal loads induced on the panel;
performing a strength analysis including calculating reserve
factors of the material at a limit load and at an ultimate load;
calculating local stresses admissible by the panel, including
calculating the reserve factors at maximum stresses; and
calculating general instability of the panel, including calculating
a reserve factor for a stiffened flat panel under pure or combined
loading conditions.
3: The method of claim 2, wherein the input data includes the
mechanical parameters related to the material, dimensions of the
panel, cross sections of the stiffeners, dimensions of the core, a
constant thickness of the panel and limit loads of the panel.
4: The method of claim 2, wherein calculating the internal loads
induced on the panel further include performing a correction of the
applied loads that takes plasticity into account using an iterative
method of calculation of plastic stresses, carried out until five
parameters of the material (E.sub.0.degree..sup.st,
E.sub.+.quadrature..sup.st, E.sub.-.quadrature..sup.st, E.sub.skin,
.quadrature..sub.ep) acquired during the first step are
substantially equal to the same parameters obtained after
calculation of plastic stress.
5: The method of claim 2, wherein calculating the local stresses
further includes calculating admissible buckling flows and reserve
factors for isosceles triangular pockets, wherein the applied
stresses taken into account for calculating the reserve factor are
stresses acting exclusively in the skin and external flows used are
skin flows that do not correspond to a complete loading of the
panel.
6: The method of claim 5, wherein calculating admissible buckling
flows further includes calculating admissible values for plates
subjected to cases of pure loading (compression in two directions
in the plane, shear) by using a finite elements procedure, and
calculating curves of interaction between these cases of pure
loading.
7: The method of claim 6, wherein calculating admissible values
includes creating a parametric finite elements model of a
triangular plate, executing numerous different combinations to
obtain buckling results, obtaining parameters compatible with a
polynomial analytical formulation.
8: The method of claim 7, wherein, in the case of pure loading, the
interaction curves are defined by creating finite elements models
of a plurality of triangular plates having different isosceles
angles, wherein the isosceles angle .theta. is defined as a base
angle of the isosceles triangles, and for each isosceles angle the
following steps are performed: determining the admissible folding
flow (without plastic correction) for diverse plate thicknesses via
finite elements model, plotting a curve of admissible buckling flow
as a function of the ratio D h 2 ( D ##EQU00120## plate stiffness,
h height of the triangle), wherein this curve is determined for
small values of the ratio D h 2 ##EQU00121## by a second degree
equation that is a function of this ratio, in which the
coefficients K.sub.1 and K.sub.2 depend on the angle and on the
load case under consideration, and plotting the evolution of the
coefficients K.sub.1 and K.sub.2 of the polynomial equation as a
function of the base angle of the isosceles triangle, wherein these
coefficients are plotted as a function of the angle of the
triangular plates under consideration, then interpolating to
determine a polynomial equation with which these constants can be
calculated regardless of the isosceles angle.
9: The method of claim 7, wherein in the case of combined loading,
the following hypothesis is used: if certain components of the
combined load are in tension, then these components are not taken
into account for the calculation, and in that the interaction
curves are defined by creating finite elements models of a
plurality of triangular plates having different isosceles angles,
wherein the isosceles angle is defined as the base angle of the
isosceles triangle, the following step are performed determining
the inherent buckling value corresponding to different
distributions of external loads, plotting interaction curves for
each angle and each combination of loads, then approximating these
curves with a single equation covering all combinations: E eX A + R
cY B + R s C = 1 ( where R i = N i app N i crit , i = cX , cY or s
) , ##EQU00122## wherein A, B, C are empirical coefficients.
10: The method of one of claim 7, wherein in the case of isosceles
triangular plates in simple or nested bracing relationship, in case
of combined loading, one interaction curve:
R.sub.cX+R.sub.cY+R.sub.s.sup.3/2=1, is used for all loading
cases.
11: The method of claim 10, wherein calculating the reserve factor
is calculated on the basis of external flows of the stiffened panel
and of limit conditions in simple or nested bracing
relationship.
12: A non-transitory computer readable medium storing
computer-readable instructions therein which when executed by a
computer cause the computer to perform a method for dimensioning of
a substantially plane panel, the method comprising: acquiring
values of input data commensurate with a geometry of the panel, a
material of the panel, and loading applied to the panel;
calculating stresses applied on the skin of the panel and the
stiffeners, based on the geometry of the panel stiffened by the
triangular pockets, and on applied external loads; calculating
internal loads induced on the panel; performing a strength analysis
including calculating reserve factors of the material at a limit
load and at an ultimate load; calculating local stresses admissible
by the panel, including calculating the reserve factors at maximum
stresses; and calculating general instability of the panel,
including calculating a reserve factor for a stiffened flat panel
under pure or combined loading conditions.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a divisional of U.S. application Ser.
No. 13/395,744, filed May 25, 2012, which is the National Stage of
PCT/FR2010/051900, filed Sep. 13, 2010, which claims the benefit of
priority to French Application No. 0956286, filed Sep. 14, 2009,
the entire contents of each of which is incorporated herein by
reference.
FIELD OF THE INVENTION
[0002] The present invention is from the domain of structures. It
particularly concerns structures of a stiffened panel type, and
even more particularly such panels which are reinforced by
stiffeners. The invention is concerned with calculating the
resistance of such structures subjected to combined loads.
PRIOR ART AND THE PROBLEM POSED
[0003] Thin, stiffened structures represent the greater part of
primary commercial aircraft structures.
[0004] Panels are generally reinforced with stiffeners which are
perpendicular to each other and which define rectangular zones on
the skin of the panel, limited by stiffeners and referred to as
pockets.
[0005] The structure of an aeroplane is thus conceived with a
skeleton of stiffeners and a skin: [0006] longitudinal stiffeners
(generally referred to as longerons): they provide support for the
structure in the principal direction of loads [0007] transversal
stiffeners (generally called frame" or "rib"): their main function
is to provide support for the longerons [0008] a panel (generally
called skin): as a general rule, it takes up loading in the plane
(membrane)
[0009] The longerons and the stringers are set at 90.degree. to
each other and define rectangular pockets on the skin.
[0010] However, during the 1950's and 60's, for spacecraft
structures, NASA developed a new concept for stiffened structures
called "Isogrid" (see FIG. 1).
[0011] Such a stiffened structure thus composed of a reinforced
skin with a network of stiffeners set at .theta..degree.
(.theta.=60.degree., in the structures envisaged by NASA) between
them. The stiffeners are blade shaped and are built into the panel.
Because of its geometry, this configuration possesses orthotropic
qualities (isotropic when .theta.=60.degree.) and the pockets
formed on the skin are triangular.
[0012] In the following description, the terms structure stiffened
by triangular pockets or panel stiffened by triangular pockets are
used to define the structures or panels reinforced by crossed
stiffeners forming triangular pockets.
[0013] Limited data is available in literature for calculating the
resistance and the stability of such a structure stiffened by
triangular pockets.
[0014] State of the Art of Calculation Methods for Panels Stiffened
by Triangular Pockets
[0015] A method for the analytical calculation of panels stiffened
by equilateral triangular pockets is described in the NASA Contract
Report "Isogrid" design handbook" (NASA-CR-124075, 02/1973)
[0016] This method is well documented, but presents some serious
limitations: use of equilateral triangles only: angle=60.degree.,
calculation of applied stresses but no calculation of stress
capacity, Poisson coefficient of the material equal to only
1/3.
[0017] The prior method presents many limitations and does not take
into account all the problems which are presented on an aircraft
structure, in particular concerning boundary conditions and
plasticity. It cannot therefore be used reliably for the analytical
calculation of the structure of panels stiffened by triangular
pockets.
OBJECTIVES OF THE INVENTION
[0018] In order to carry out a structural analysis of panels
stiffened by triangular pockets, a method for structural analysis
was developed, based on a theory of composite plate and taking into
account its specific modes of failure. This method is applied to
flat panels made of a material with isotropic properties.
[0019] The method described herein envisages a modification of the
base angle between the stiffeners (which is 60.degree. in the
"Isogrid" structures). This signifies that the isotropic quality of
the panel is no longer guaranteed.
EXPLANATION OF THE INVENTION
[0020] The invention relates, to this effect, to a method of
dimensioning by an analytical method, an essentially flat panel
consisting of a homogenous and isotropic material, the panel being
composed of a skin reinforced by a set (known as "grid") of three
parallel bundles of stiffeners built into the panel, the pockets
determined on the skin by said groups of stiffeners are triangular,
the stiffeners are blade shaped and the stiffened panel must comply
with specifications for mechanical resistance to predetermined
external loads, the angles between bundles of stiffeners being such
that the triangular pockets have any kind of isosceles shape.
[0021] According to one advantageous implementation, the method
includes the steps:
[0022] Step 2 of calculating the stresses applied in the skin and
the stiffeners, as well as the flow in the skin and loads in the
stiffeners, based on the geometry of the stiffened panel, and the
external loads, assumed to be in the plane of the panel and applied
at the centre of gravity of the section (of the panel), the
stiffened panel being represented by an assembly of two orthotropic
plates, the grid of stiffeners being represented by an equivalent
panel.
[0023] Step 3--of calculating the internal loads of the stiffened
panel,
[0024] Step 4--of resistance analysis including a calculation of
reserve factors of the material at capacity and ultimate load,
[0025] Step 5--of calculating the local stress capacity,
[0026] In preference, the method takes into account the
redistribution of applied stresses between the panel and the grid
of stiffeners due:
[0027] to the post-buckling of stiffeners, by the definition of an
effective straight section for each type of stiffener (0.degree.,
+.theta. or -.theta.). A.sub.0.degree..sup.st,
A.sub.+.theta..sup.st and A.sub.-.theta..sup.st,
[0028] to the post-buckling of the pocket through the calculation
of an effective thickness of the panel: t.sub.s.sub._.sub.eff,
[0029] to the plasticity of applied external loads, through an
iterative process on the various properties of the material, in
particular Young's modulus and Poisson coefficients:
E.sub.0.degree..sup.st, E.sub.+.theta..sup.st,
E.sub.-.theta..sup.st for the stiffeners and E.sub.x.sup.s,
E.sub.y.sup.s and .nu..sub.ep.sup.st for the skin, using the
Ramberg-Osgood law.
[0030] According to a preferred mode of implementation of the
method according to the invention, this includes a step of
correcting the applied loads to take into account plasticity, using
an iterative method for calculating the plastic stresses, carried
out until the five parameters of the material
(E.sub.0.degree..sup.st, E.sub.+.theta..sup.st,
E.sub.-.theta..sup.st, E.sub.skin, .nu..sub.ep) entered at the
start of the process are noticeably equal to the same parameters
obtained after the calculation of plastic stress.
[0031] According to an advantageous implementation, the method
includes a step 4, of analysing resistance comprising a calculation
of the reserve factors of the material at a capacity and ultimate
load, carried out by comparing the applied loads calculated in the
stiffened panel components with the maximum stress capacity of the
material, the applied loads being corrected to take into account
the plasticity of the stiffened panel.
[0032] According to an advantageous implementation, the method
includes a step 5 for calculating the local stress capacity, which
includes a sub-step 5A of calculating the buckling flow capacity,
and the reserve factor for the isosceles triangular pockets, the
applied stresses to be taken into account for the calculation of
the reserve factor being only the stresses affecting the skin, the
external flows used being the flows of the skin do not correspond
to the stiffened panel being fully loaded.
[0033] In this case, step 5A of calculating the buckling flow
capacity and reserve factor for the isosceles triangular pockets
favourably includes two sub-steps: firstly of calculating the
capacity values for plates subjected to cases of pure loading
(compression according to two directions in the plane, shear load)
by using a finite element method, then calculating the interaction
curves between these cases of pure loading.
[0034] Even more precisely, calculating the capacity values
includes the following sub-steps of: [0035] Creating an FEM
parametric model of a triangular plate [0036] Testing various
combinations to obtain buckling results, [0037] Obtaining
parameters that are compatible with an analytical polynomial
formula
[0038] In a particular mode of implementation, in the case of pure
loading, the interaction curves are defined by the following
sub-steps: [0039] of creating finite element models of several
triangular plates with different isosceles angles, the isosceles
angle (0) being defined as the base angle of the isosceles
triangle, [0040] for each isosceles angle: [0041] 1/ of calculating
by Finite Element Model to determine the flow capacity of wrinkling
(without plastic correction) for various plate thicknesses. [0042]
2/ of tracing a flow curve of buckling capacity according to
the
[0042] D h 2 ##EQU00001## ratio (D plate stiffness, h height of the
triangle), this curve being determined for the small values of
D h 2 , ##EQU00002## by a second degree equation according to this
ratio, of which the coefficients K.sub.1 and K.sub.2 depend on the
angle and the load case being considered, [0043] 3/ of tracing the
evolution of coefficients of the polynomial equation K.sub.1 and
K.sub.2 according to the base angle of the isosceles triangle,
these coefficients being traced according to the angle of the
triangular plates being considered, and interpolation to determine
a polynomial equation which makes it possible to calculate the
constants whatever the isosceles angle.
[0044] Again, in the case of calculation of the buckling flow
capacity and reserve factor of isosceles triangular pockets,
according to an advantageous implementation, in the case of
combined loading, the following hypothesis is used: if some
components of the combined load are under pressure, these
components are not taken into account for the calculation, and the
interaction curves are defined by the following sub-steps: [0045]
of creating finite element models of several triangular plates with
different isosceles angles, the isosceles angle (0) being defined
as the base angle of the isosceles triangle, [0046] for each angle,
[0047] 1/ of calculating by Finite Element Model (FEM) to determine
the eigenvalue of buckling that corresponds to the various
distributions of external loads. [0048] 2/ of tracing the
interaction curves, for each angle and each combination of loads
and approximating these curves with a unique equation covering all
these combinations:
[0048] R cX A + R cy B + R s C = 1 ( or R i = N i app N i crit ,
##EQU00003##
equations in which R.sub.i represents the load rate and
N.sub.i.sup.app and N.sub.i.sup.crit the applied flows and critical
flows for i=cX, cY or s, corresponding to cases of compression
according to axes X and Y, and to a case of shear load), A, B, C
being empirical coefficients.
[0049] Advantageously, the method also comprises a sub-step of
calculating reserve factors, by solving the following equation:
( R cY R ) A + ( R cX R ) B + ( R s R ) C = 1 ##EQU00004## with R =
N cY app N cYcomb crit = N cX app N cXcomb crit = N s app N scomb
crit = 1 RF ##EQU00004.2##
[0050] According to an advantageous implementation, the method
uses, for the calculation of plasticity corrected stress capacity,
a plasticity correction factor .eta., defined by: [0051] for all
cases of loading (pure and combined) with the exception of shear
load,
[0051] .eta. 5 = E tan E c ##EQU00005## [0052] for cases of pure
shear loading:
[0052] .eta. 6 = ( 1 + v e ) ( 1 + v ) E sec E c ##EQU00006##
[0053] the plasticity correction being calculated by using the
equivalent elastic stress of Von Mises.
[0054] According to an advantageous implementation, in the case of
simply supported or clamped isosceles triangular plates, in a case
of combined loading, an interaction curve is used:
R.sub.cX+R.sub.cY+R.sub.s.sup.3/2=1, for all cases of loading.
[0055] According to an advantageous implementation, the method
includes a step 5 for calculating the local stress capacity, which
includes a sub-step 5B of calculating the buckling stress capacity,
and reserve factor for the stiffener web, considered as a
rectangular panel, the stresses applied for calculations of reserve
factor being only the stresses in the stiffener webs.
[0056] According to an advantageous implementation of the method,
this includes a step 6, of calculating general instability,
providing data on buckling flow capacity, and reserve factors, for
a flat stiffened panel, under pure or combined loading conditions,
the flows applied, to be taken into account for the calculation of
reserve factor being the external flows of the stiffened panel.
[0057] In this case, more specifically, the method advantageously
includes the following sub-steps: [0058] of using a general
behaviour law (equation 6-8), defining the flows and moments
relations between flow and moments, on one hand, and strains, on
the other, a state of plane stresses being considered, [0059] using
general balance equations (equations 6-9 and 6-10) of an element of
the stiffened panel, linking the flows, moments and the density of
surface strengths, [0060] of solving a general differential
equation (equation 6-17) between the stress flows, the surface
strength density, strains and bending stiffeners.
[0061] According to a favourable implementation, the method
includes an iteration step, making it possible to modify the values
of applied stresses, or the dimensional values of panels, according
to the results of at least one of steps 3 to 6.
[0062] In another respect, the invention relates to a computer
programme product including a series of instructions adapted to
implement a method such as explained, where this set of
instructions is executed on a computer.
BRIEF DESCRIPTION OF FIGURES
[0063] The description which follows, given purely as an example of
an embodiment of the invention, is given in reference to the
annexed figures which represent:
[0064] FIG. 1--An example of a flat panel stiffened by triangular
pockets,
[0065] FIG. 2--A definition of loading and of the system of
coordinates,
[0066] FIG. 3--A geometric definition of a panel
[0067] FIG. 4--A junction in a structure stiffened by triangular
pockets,
[0068] FIG. 5--An example of general instability of a panel
stiffened by triangular pockets,
[0069] FIG. 6--The theory of effective width,
[0070] FIG. 7--A general organogram of the method according to the
invention,
[0071] FIG. 8--A decomposition of the grid in elementary
triangles,
[0072] FIG. 9--An elementary isosceles triangle used in the
calculation of the panel mass,
[0073] FIG. 10--An elementary rectangular triangle used in the
calculation of the panel mass,
[0074] FIG. 11--An elementary shape of a stiffener grid in a panel
stiffened by triangular pockets,
[0075] FIG. 12--A case of pure loadings of the stiffened plate,
[0076] FIG. 13--A diagram of loads on a stiffener,
[0077] FIG. 14--An expression of the Kc coefficients according to
cases of boundary conditions,
[0078] FIG. 15--A panel of stiffeners considered as an assembly of
two orthotropic plates,
[0079] FIG. 16--The loads on an elementary shape of a stiffener
grid for a panel stiffened by triangular pockets,
[0080] FIG. 17--A method for calculating plasticity corrected
applied loads,
[0081] FIG. 18--The notation conventions of the elementary
isosceles triangle,
[0082] FIG. 19--A linear or quadratic interpolation of the K
coefficient,
[0083] FIG. 20--A case of combined loading,
[0084] FIG. 21--Conventions of flow and moments,
[0085] FIG. 22--The value of h(.alpha.) according to the various
boundary conditions, for a case of compression,
[0086] FIG. 23--The shear buckling coefficient for a four-sided
simply supported configuration,
[0087] FIG. 24--A table of shear buckling coefficient values,
[0088] FIG. 25--The shear buckling coefficient for a clamped
four-sided configuration,
[0089] FIG. 26--The evolution of the K1 constant according to the
isosceles angle for a simply supported triangular plate,
[0090] FIG. 27--The evolution of the K2 constant according to the
isosceles angle for a simply supported triangular plate,
[0091] FIG. 28--The evolution of the K1 constant according to the
isosceles angle for a clamped triangular plate,
[0092] FIG. 29--The evolution of the K2 constant according to the
isosceles angle for a clamped triangular plate,
DETAILED DESCRIPTION OF A MODE OF EMBODIMENT OF THE INVENTION
[0093] The method for resistance analysis of a metal panel
stiffened by triangular pockets, principally plane, described is
intended to be implemented in the form of a programme on a computer
of a known type.
[0094] The method is intended to be used for a structure which is
principally plane (stiffeners and skin). The method here-described
applies exclusively to the calculation of typical structural
settings with the following boundaries: [0095] The edges of the
studied zone do not border an opening. [0096] None of the
stiffeners extend outside of the zone studied. [0097] Each cross
section must be bordered by stiffeners. [0098] All the triangular
pockets in the skin are assumed to have the same thickness. [0099]
All the stiffeners are assumed to have the same dimensions.
[0100] This method is used for calculating panels built from a
homogenous and isotropic material (for example--but not limited
to--metal) for which the describing monotonically increasing curves
(.sigma., .epsilon.) can be idealised by the means of formulas such
as R&O (see further on).
[0101] The simplified organogram of the method according to the
invention is illustrated by FIG. 7.
[0102] Two types of failure (the occurrence of which is evaluated
in steps 4 and 6 of the method) can occur on a structure stiffened
by triangular pockets: A fault in material (which is the object of
step 4): the applied stresses have reached the maximum stress
capacity of the material (F.sub.tu or F.sub.su), global failure:
generalised buckling (including the grid of stiffeners) occurs on
the whole panel (this verification is the object of step 6).
[0103] In addition, two types of instability (object of step 5)
weaken the global rigidity of the structure stiffened by triangular
pockets but do not cause the global failure of the complete
structure: [0104] Instability of the panel: buckling of the
triangular pockets [0105] Instability of the stiffeners: buckling
of stiffener webs
[0106] The buckled sections can only support a part of the load
which they could support before they were buckled. Because of this,
the applied loads are redistributed in the structure.
[0107] It is noted that in the present invention, post-buckling
calculations are not processed. Because of this, the two types of
buckling referred to above are considered as modes of failure.
[0108] Notation and Units
[0109] The conventions of notations and axis systems are explained
in FIG. 2
[0110] A local system of coordinates is defined for each stiffener.
An X axis is defined in the plane of the straight section of the
stiffener, this is the exit axis, in the direction of the principal
dimension of the stiffener. A Z axis is defined as the normal axis
in the plane of the skin, in the direction of the stiffener.
Finally, a Y axis is the third axis in a system of straight
coordinates.
[0111] For forces and loads, a negative sign on a force according
to the X axis signifies a compression of the stiffener, a positive
sign signifies a tension.
[0112] A positive bending moment causes a compression in the skin
and a tension in the stiffeners.
[0113] The general notations used are defined in the following
table.
TABLE-US-00001 Symbol Unit Description A mm.sup.2 Surface I
mm.sup.4 Inertia J mm.sup.4 Torsion constant K N/mm Normal rigidity
(tension/compression) of a plate D N mm Bending rigidity of a plate
.sigma. N/mm.sup.2 Stress .epsilon. -- Strain .kappa. -- Strain
outside of the plane .eta. -- Plastic correction factor z mm
Coordinates on the z axis k -- Buckling coefficient
[0114] Suffixes:
TABLE-US-00002 Symbol Unit Description g -- grid st -- stiffener s
-- skin
[0115] Geometric Characteristics
[0116] The geometric characteristics of a panel, considered here as
a non-limitative example, are given in FIG. 3.
[0117] For the rest of the description, several hypotheses are
used. It is supposed that the Z axis is a plane of symmetry for the
straight section of the stiffener. Also, the dimensions a and h are
defined according to the neutral fibre of a stiffener. In addition,
the envisaged panel stiffened by triangular pockets does not have
stiffeners on the two sides defined by: X=0 and X=Lx
TABLE-US-00003 Symbol Unit Description L.sub.X mm length L.sub.Y mm
width a mm Length of the side of a triangle .theta. .degree. Angle
of the triangle h mm Height of a triangle ( = a 2 tan .theta. )
##EQU00007## t mm Thickness of the skin d mm Height of the
stiffener web b mm Thickness of the stiffener web R.sub.n mm Fillet
node radius R.sub.f mm Pocket radius A.sub.x.degree..sup.st
mm.sup.2 Straight section of the stiffener according to axis x
v.sub.p mm Panel offset between its centre of gravity and the
origin point of the local coordinates system v.sub.w mm Stiffener
network offset between its centre of gravity and the origin point
of the local coordinates system
[0118] Materials
TABLE-US-00004 Symbol Unit Description F.sub.cy MPa Elastical
capacity of the material under compression F.sub.tu MPa Ultimate
tension resistance of the material F.sub.su MPa Ultimate shear
resistance of the material .sigma..sub.n MPa Stress reference
.epsilon..sub.ult -- Ultimate plastic strain (=e %) .nu..sub.e --
Elastic Poisson coefficient .nu..sub.p -- Plastic Poisson
coefficient (=0.5) .nu. -- Elasto-plastic Poisson coefficient
E.sub.c MPa Young's elastic modulus in compression E MPa Young's
elastic modulus under tension E.sub.sec MPa Secant modulus
E.sub.tan MPa Tangent modulus n.sub.ec -- Ramberg and Osgood
(R&O) coefficient in compression G MPa Shear modulus G.sub.sec
MPa Secant shear modulus .rho. kg/mm.sup.3 Material density
[0119] Stresses
TABLE-US-00005 Symbol Unit Description P.sub.i N Normal load with i
equal to: 0.degree. for the load applied on a stiffener at
0.degree. x.degree. for the load applied on a stiffener at
x.degree. N N/mm Flow M N mm Bending moment .tau. MPa Shear stress
.sigma. MPa Normal stress .sigma..sub.crit.sup.i MPa Buckling
stress of a panel i in compression .tau..sub.crit.sup.i MPa
Buckling stress of a panel i in shear load .sigma..sub.app.sup.i
MPa Applied stresses on the element i RF -- Reserve factor R.sub.c
-- Compression load rate R.sub.s -- Shear load rate R.sub.p -- Load
rate due to pressure LL Load limit UL Ultimate load
Definitions
[0120] For the rest of the description the following terms are
defined.
[0121] In a structure stiffened by triangular pockets, grid refers
to the complete network of single stiffeners.
[0122] The term node is used to describe an intersection of several
stiffeners in a structure stiffened by triangular pockets (see FIG.
4). In practice, it is an element of the complex design including
bending radii in both directions.
[0123] When a structure (subject to loads only in its plane)
suffers significant, visible transversal displacements of loads in
the plane, it is said to buckle. FIG. 5a illustrates such a case of
local instability of a panel stiffened by triangular pockets.
[0124] The buckling phenomenon can be demonstrated by pressing the
opposite sides of a flat cardboard sheet towards each other. For
small loads, the buckle is elastic (reversible) because the buckle
disappears when the load is removed.
[0125] Local buckling (or local instability) of plates or shells is
indicated by the appearance of lumps, undulations or waves and is
common in plates which compose thin structures. When considering
stiffened panels, local buckling, as opposed to general buckling,
describes an instability in which the panel between the longerons
(stiffeners) buckles, but the stiffeners continue to support the
panels and do not show any significant strains outside of the
plane.
[0126] The structure can therefore present two states of balance:
[0127] Stable: in this case, displacements increase in a controlled
manner when the loads increase, that is to say that the capacity of
the structure to support additional loads is maintained, or [0128]
Unstable: in this case, strains instantly increase and the capacity
to support loads rapidly declines
[0129] A neutral balance is also possible in theory during
buckling, this state is characterised by an increase in strain
without modifying the load
[0130] If buckling strains become too great, the structure fails.
If a component or a part of a component is likely to suffer
buckling, then its conception must comply with the stresses of both
resistance and buckling.
[0131] General instability refers to the phenomenon which appears
when the stiffeners are no longer able to counteract the
displacements of the panel outside of the plane during
buckling.
[0132] FIG. 5b shows an example of global buckling in compression
of a structure stiffened by triangular pockets, when the panel
reaches its first mode of general buckling.
[0133] Because of this, it is necessary to find out if the
stiffeners act as simple supports of the panel (in compression,
shear load and combination load). If this condition is not
fulfilled, it must be supposed that the panel assembly and the
stiffeners buckle in a global manner in a mode of instability,
something which must be avoided in a structure designed for
aeronautical use.
[0134] A general failure (or global) happens when the structure is
no longer capable of supporting additional loads. It can be said
therefore that the structure has reached failure loading or loading
capacity.
[0135] General failure covers all types of failure: [0136] Failure
due to an instability (general instability, post-buckling . . . )
[0137] Failure caused by exceeding the maximum load supported by
the material (for example after local buckling)
[0138] Effective width (or working width) of the skin of a panel is
defined as the portion of the skin which is supported by a longeron
in a stiffened panel structure which does not buckle when it is
subject to an axial compression load.
[0139] Buckling of the skin alone does not constitute a panel
failure; the panel will in fact support additional loads up to the
stress at which the column formed by the stiffener and the
effective panel starts to fail. When the stress in the stiffener
goes above the buckling stress of the skin, the skin adjacent to
the stiffener tolerates an additional stress because of the support
provided by the stiffeners. However, the stress at the centre of
the panel will not go above the initial buckling stress, whatever
the stress reached at the level of the stiffener.
[0140] The skin is more effective around the position of the
stiffeners because there is a local support against buckling. At a
given level of stress, lower than that of local buckling of the
skin, the effective width is equal to the width of the panel. The
theory of effective width is illustrated by FIG. 6
[0141] Idealisation of Material
[0142] It is herewith noted that up to the yield stress (F.sub.cy),
the stress-strain curve of material is idealised by the known law
of Ramberg and Osgood (referred to as the R&O formula in the
rest of the description):
= .sigma. E c + 0.002 ( .sigma. Fcy ) n c Equation 0 - 1
##EQU00008##
[0143] We can deduce the following expressions:
[0144] Secant Modulus:
E sec = .sigma. R & O law E sec = 1 1 E c + 0.002 F cy (
.sigma. F cy ) ( n c - 1 ) Equation 0 - 2 ##EQU00009##
[0145] Tangent Modulus
1 E tan = .differential. ( ) .differential. ( .sigma. ) = n c E sec
+ 1 - n c E c R & O law E tan = 1 1 E c + 0.002 F cy n c (
.sigma. F cy ) ( n c - 1 ) Equation 0 - 3 ##EQU00010##
[0146] Poisson Coefficient:
v = E sec E c v e + ( 1 - E sec E c ) v p With v p = 0.5 Equation 0
- 4 ##EQU00011##
[0147] It is noted that, with the R&O ratio (parameter n or n
corrected), known by those skilled in the art, these equations are
only correct in the zone [0; F.sub.cy]. For the following part of
this study, this zone must be extended from F.sub.cy to F.sub.tu.
Over F.sub.cy, different curves can be used up to the ultimate
stress, in particular: R&O formula using a modified n
coefficient, or elliptical method.
[0148] In the following, the R&O formula uses a modified
coefficient. Continuity between the two curves is maintained. The
modified n coefficient of the R&O formula is calculated by:
n = ln ( 2 p 1 p ) ln ( .sigma. 2 .sigma. 1 ) Equation 0 - 5
##EQU00012##
[0149] With: [0150] e.sub.2p=.epsilon..sub.ult [0151]
e.sub.1p=0.002 [0152] .sigma..sub.2=F [0153]
.sigma..sub.1=F.sub.cy
[0154] It is noted that to use this formula, the following
criterion must be respected: F.sub.tu>F.sub.cy and
.epsilon..sub.ult>0.002
[0155] In the elliptical method, above F.sub.cy, another curve is
used up to the ultimate stress: the elliptical extension curve.
Naturally, the continuity between the R&O curve and the
elliptical extension curve is ensured.
[0156] The stress-strain ratios of the elliptical extension
are:
E ( .sigma. ) = 3 - a 1 - ( .sigma. - .sigma. E_ref + b ) 2 b 2 ;
.sigma. E ( ) = .sigma. E_ref - b + b 1 - ( - 3 ) 2 a 2 Equation 0
- 6 ##EQU00013##
[0157] with:
a = - b 2 x 1 m ( b - D ) ; b = D ( m x 1 + D ) m x 1 + 2 D -
ellipse parameters ##EQU00014## x 1 = 2 - 3 ; D = .sigma. E_ref -
.sigma. RO_ref ; ##EQU00014.2## m = E 1 + RO_ref nE .sigma. RO_ref
##EQU00014.3##
[0158] Plasticity
[0159] Again it is noted that, it is known that plasticity
correction factors depend on the type of load and boundary
conditions.
[0160] The plasticity correction factors for flat rectangular
panels are presented in table 1 below.
TABLE-US-00006 Boundary Loading conditions Equation Compression and
bending Flange with a hinged edge unloaded (free- .eta. 1 = E sec E
c 1 - v e 2 1 - v 2 ##EQU00015## supported edges) Flange with a
fixed unloaded edge (clamped-free .eta. 2 = .eta. 1 ( 0.33 + 0.335
1 + 3 E tan E sec ) ##EQU00016## edges) Plate with unloaded hinged
edges (supported edges) .eta. 3 = .eta. 1 ( 0.5 + 0.25 1 + 3 E tan
E sec ) ##EQU00017## Plate with unloaded fixed edges (clamped
edges) .eta. 4 = .eta. 1 ( 0.352 + 0.324 1 + 3 E tan E sec )
##EQU00018## Compression Column .eta. 5 = E tan E c ##EQU00019##
Shear load all conditions .sigma..sub.eq = .tau. {square root over
(3)} .eta. 6 = G sec G ##EQU00020## Shear load re-tightened edges
.sigma..sub.eq = .tau. 2 .eta. 8 = .eta. 1 ( 0.83 + 0.17 E tan E
sec ) ##EQU00021##
[0161] In the specific case of shear load, the compression
stress-strain curve of the material is equally used as follows:
[0162] Calculation of the equivalent normal stress:
.sigma..sub.eq=.tau. {square root over (3)} [0163] Calculation of
the corresponding E.sub.s and .nu. values based on this stress:
.sigma..sub.eq
[0163] .eta. 6 = G s G = ( 1 + v e 1 + v ) E sec E c = ( 1 + v 1 +
v e ) .eta. 1 ##EQU00022##
[0164] Step 1--Data Entry Module: Geometry, Material, Loading
[0165] The method includes a first phase of entering data relating
to the panel stiffened by triangular pockets being considered and
to the loading applied to this panel. These data are entered using
known means and memorised in a data base which is also of a known
type.
[0166] The entry parameters for the analytical calculation of
panels stiffened by triangular pockets particularly include: [0167]
General dimensions: rectangular panel (dimensions: L.sub.x,
L.sub.y) [0168] Straight section of stiffeners: dimensions of the
web: b, d [0169] Constant thickness of the panel (t) [0170] Load
boundaries of the panel N.sub.x, N.sub.y, N.sub.xy
[0171] Calculation of Mass
[0172] This part is designed for the complete calculation of the
mass of the panel stiffened by triangular pockets, including taking
into account the radii of the fillet and the node. This step of
calculating mass is independent from the rest of the method
described herein. The mass is calculated in a known manner using
the geometrical definition of the panel.
[0173] Data entered for this process are the geometry of the panel
including the radii of pockets and nodes (R.sub.n and R.sub.f).
Exit data is the panel's mass.
[0174] Mass is calculated by adding up the mass of the skin and the
longerons. The radii of filets between two longerons and between
the skin and the longerons are also taken into account. Calculation
of mass is based on two elementary triangles: an isosceles triangle
and a rectangular triangle (see FIGS. 8, 9 and 10).
[0175] Step 2--Calculation of Applied Loads
[0176] This step makes it possible to calculate the stresses
applied in the skin and the stiffeners based on the geometry of the
panel stiffened by triangular pockets and the external loads. The
method takes into account a plasticity correction of applied loads,
done using an iterative process. It makes it possible to take into
account the post-buckling of stiffeners and pockets.
[0177] This represents substantial progress in relation to the NASA
"Isogrid" design handbook" (NASA-CR-124075, 02/1973) in that it
particularly takes into account the following points: grid of
stiffeners with .theta..noteq.60, panel stiffened by triangular
pockets considered as an assembly of two orthotropic plates.
[0178] The entry data for this step are: [0179] Geometric data:
[0180] .theta.: angle of the base of the triangle, [0181] a: base
of the triangle, [0182] A.sub.i.sup.st: straight section of the
stiffener, i=0.degree., .theta. or -.theta.. [0183] t.sub.s:
thickness of the skin, [0184] t.sub.g: thickness of the panel
equivalent to the grid [0185] Data on the material: [0186]
E.sub.x.sup.s, E.sub.y.sup.s: Young's modulus of the skin, [0187]
G.sub.xy.sup.s: shear modulus of the skin, [0188]
.nu..sub.xy.sup.s, .nu..sub.yx.sup.s: Poisson coefficient of the
skin, [0189] E.sup.st: Young's modulus of the stiffeners, [0190]
.nu..sup.st: Poisson coefficient of the stiffeners [0191] Material
data (n: Ramberg & Osgood coefficient, Fcy, Ftu,
.nu..sub.plast=0.5) [0192] Loads applied on the structure
(N.sub.x.sup.0, N.sub.y.sup.0, N.sub.xy.sup.0)
[0193] The data obtained at the end of this step are: [0194]
N.sub.x.sup.s, N.sub.y.sup.s, N.sub.xy.sup.s: flow in the skin,
[0195] .sigma..sub.x.sup.s, .sigma..sub.y.sup.s,
.tau..sub.xy.sup.s: stresses in the skin, [0196]
.sigma..sub.0.degree., .sigma..sub..theta., .sigma..sub.-.theta.:
stresses in the stiffeners, [0197] F.sub.0.degree., F.sub..theta.,
F.sub.-.theta.: loads in the stiffeners.
[0198] In the following part of the description, the skin is
assumed to be of an isotropic material.
[0199] The method provides entries for: [0200] The analysis of
resistance (step 4): stresses in the skin and in the stiffeners
[0201] the analysis of pocket buckling (step 5.1): stresses in the
skin [0202] the analysis of stiffener buckling (step 5.2): stresses
in the stiffeners [0203] the analysis of general instability (step
6): stresses in the skin and in the stiffeners to calculate the
bending rigidity of the panel stiffened by triangular pockets.
[0204] The calculation method requires entry data on the
post-buckling of stiffeners: A.sub.0.degree..sup.st,
A.sub.+.theta..sup.st and A.sub.-.theta..sup.st and on
post-buckling of pockets: t.sub.s.sub._.sub.eff
[0205] The method takes into account the redistribution of applied
stresses between the panel and the grid of stiffeners due in the
first instance, to the post-buckling of the stiffeners, by the
definition of an effective straight section for each type of
stiffener (0.degree., +.theta. or -.theta.):
A.sub.0.degree..sup.st, A.sub.+.theta..sup.st and
A.sub.-.theta..sup.st, and in the second instance, to the
post-buckling of the pocket through an effective thickness of the
panel: t.sub.s.sub._.sub.eff, finally, to the plasticity of applied
external loads, using an iterative process on the different
properties of the material: E.sub.0.degree..sup.st,
E.sub.+.theta..sup.st, E.sub.-.theta..sup.st for the stiffeners and
E.sub.x.sup.s, E.sub.y.sup.s and .nu..sub.ep.sup.st for the
skin.
[0206] The external load is assumed to be in the plane of the panel
and applied at the centre of gravity of the section:
{ N M } = [ A B B C ] { .kappa. } ##EQU00023## therefore : .noteq.
0 and .kappa. = 0 { N } = [ A ] { } with { N } = { Nx Ny Nxy }
##EQU00023.2##
[0207] In consequence, the stresses in the skin do not depend on
the thickness of said skin and the position in the plane. Also,
stresses in the stiffeners do not depend on the position on the
section of the stiffener, but only on the angle of the
stiffener.
[0208] The geometric definition of the grid of stiffeners used for
carrying out calculations is defined in FIG. 11:
[0209] To obtain a panel stiffened by triangular pockets, this
elementary shape is associated with the skin and is repeated as
many times as is required. Because of this, this method does not
take into account the concept of the geometry of the edges.
[0210] For each stiffener, the real section (A.sub.i.sup.st with i:
0.degree., +.theta. or -.theta.) is given by the ratio:
A.sub.i.sup.st=% A.sub.i.times.A.sup.st (in the present
non-limitative example, only the case of % A.sub.i=1 is
envisaged).
[0211] The straight section of the stiffeners includes the section
of the radius of the pocket
( 2 R f 2 ( 1 - .pi. 4 ) ) . ##EQU00024##
[0212] Whatever their position on the grid, the stresses and
strains are identical for each type of stiffener (0.degree.,
+.theta., -.theta.).
[0213] To take into account the plasticity which can occur in each
stiffener, the Young's modulus is specific to each type of
stiffener (0.degree., +.theta., -.theta.): E.sub.0.degree..sup.st,
E.sub.+.theta..sup.st, E.sub.-.theta..sup.st.
[0214] The "material" matrix E is defined by:
E g _ _ = ( E 0 .degree. st 0 0 0 E .theta. st 0 0 0 E - .theta. st
) . ##EQU00025##
[0215] Step 3--Calculation of Internal Loads
[0216] 3.1 Plate Equivalent to the Stiffeners
[0217] 3.1.1 Relation Between Global Strains and Stiffener
Strains
[0218] The geometric notations and conventions are illustrated by
FIG. 16. We are looking for the relation between (.epsilon..sub.x,
.epsilon..sub.y, E.sub.xy) and (.epsilon..sub.0.degree.,
.epsilon..sub..theta., .epsilon..sub.-.theta.). General strains are
defined by the following formulas:
.epsilon..sub.nn={right arrow over (n)}.epsilon.{right arrow over
(n)}
[0219] Therefore our strains are:
0 .degree. = i 0 .degree. .fwdarw. _ _ i 0 .degree. .fwdarw. = ( 1
0 0 ) ( xx xy xz yx yy yz zx zy zz ) ( 1 0 0 ) ##EQU00026## .theta.
= i .theta. .fwdarw. _ _ i .theta. .fwdarw. = ( cos .theta. sin
.theta. 0 ) ( xx xy xz yx yy yz zx zy zz ) ( cos .theta. sin
.theta. 0 ) ##EQU00026.2## - .theta. = i - .theta. .fwdarw. _ _ i -
.theta. .fwdarw. = ( cos .theta. - sin .theta. 0 ) ( xx xy xz yx yy
yz zx zy zz ) ( cos .theta. - sin .theta. 0 ) ##EQU00026.3##
[0220] And finally:
( 0 .degree. .theta. - .theta. ) = ( 1 0 0 cos 2 .theta. sin 2
.theta. 2 sin .theta. cos .theta. 2 cos 2 .theta. 2 sin 2 .theta. -
2 sin .theta. cos .theta. ) ( x y xy ) Equation 3 - 1
##EQU00027##
[0221] The above matrix is denoted Z:
Z _ _ = ( 1 0 0 cos 2 .theta. sin 2 .theta. 2 sin .theta. cos
.theta. 2 cos 2 .theta. 2 sin 2 .theta. - 2 sin .theta. cos .theta.
) ##EQU00028##
[0222] 3.1.2 Relation Between Stresses and Strains
[0223] As stated, the geometric notations and conventions are
illustrated by FIG. 16. The loads in stiffeners are given by the
following expressions:
{right arrow over
(P)}.sub.i=.epsilon..sub.iE.sup.stA.sup.st.sub.i{right arrow over
(i)}.sub.i,(i=0.degree.,.theta.,-.theta.) Equation 3-2
[0224] Therefore the base element below is subjected to: {right
arrow over (P.sub.0.degree.)}, {right arrow over (P.sub..theta.)},
{right arrow over (P.sub.-.theta.)}({right arrow over
(P.sub.0.degree.)} counted twice because the dimension of the base
element according axis Y is 2 h, therefore the stiffener
corresponding to 0.degree. should also be taken into account).
[0225] According to Axis x:
P x g = i P .fwdarw. i x .fwdarw. = ( 2 P 0 .degree. .fwdarw. + P
.theta. .fwdarw. + P - .theta. .fwdarw. ) x .fwdarw. = 2 E 0
.degree. st A 0 .degree. st 0 .degree. + E .theta. st A .theta. st
.theta. cos .theta. + E _.theta. st A - .theta. st - .theta. cos
.theta. ##EQU00029##
[0226] According to Axis y:
P.sub.y.sup.g=E.sub..theta..sup.stA.sub..theta..sup.st.epsilon..sub..the-
ta. sin
.theta.+E.sub.-.theta..sup.stA.sub.-.theta..sup.st.epsilon..sub.-.-
theta. sin .theta.
[0227] Shear Load in the Plane ({right arrow over (x)},{right arrow
over (y)}):
P.sub.xy.sup.g=E.sub..theta..sup.stA.sub..theta..sup.st.epsilon..sub..th-
eta. sin
.theta.-E.sub.-.theta..sup.stA.sub.-.theta..sup.st.epsilon..sub.--
.theta. sin .theta. Equation 3-3
[0228] To obtain the stresses, the load is divided by the surface
of a base element. The section of the base element on a normal
surface according to axis X is 2ht.sub.g=a tan .theta.t.sub.g The
section of the base element on a normal surface according to axis Y
is at.sub.g.
[0229] In terms of stresses we have:
.sigma. x g = P x g 2 ht g = 1 2 ht g ( 2 E 0 .degree. st A 0
.degree. st E .theta. st A .theta. st cos .theta. E - .theta. st A
- .theta. st cos .theta. ) ( 0 .degree. .theta. - .theta. )
##EQU00030## .sigma. x g = 1 at g ( 2 E 0 .degree. st A 0 .degree.
st tan .theta. E .theta. st A .theta. st cos 2 .theta. sin .theta.
E - .theta. st A - .theta. st cos 2 .theta. sin .theta. ) ( 0
.degree. .theta. - .theta. ) ##EQU00030.2##
[0230] For .sigma..sub.y and .sigma..sub.xy, we obtain with the
same method:
.sigma. y g = 1 at g ( 0 E .theta. st A .theta. st sin .theta. E -
.theta. st A - .theta. st sin .theta. ) ( 0 .degree. .theta. -
.theta. ) ##EQU00031## .tau. xy g = 1 at g ( 0 E .theta. st A
.theta. st sin .theta. - E - .theta. st A - .theta. st sin .theta.
) ( 0 .degree. .theta. - .theta. ) ##EQU00031.2##
[0231] The same results can be presented in matrix form:
( .sigma. x g .sigma. y g .tau. xy g ) = ( 2 E 0 .degree. st A 0
.degree. st tan .theta. E .theta. st A .theta. st cos 2 .theta. sin
.theta. E - .theta. st A - .theta. st cos 2 .theta. sin .theta. 0 E
.theta. st A .theta. st sin .theta. E - .theta. st A .theta. st sin
.theta. 0 E .theta. st A .theta. st sin .theta. - E - .theta. st A
.theta. st sin .theta. ) ( 0 .degree. .theta. - .theta. ) Equation
3 - 4 ##EQU00032##
[0232] The above matrix is denoted T:
T _ _ = 1 at g ( 2 E 0 .degree. st A 0 .degree. st tan .theta. E
.theta. st A .theta. st cos 2 .theta. sin .theta. E - .theta. st A
- .theta. st cos 2 .theta. sin .theta. 0 E .theta. st A .theta. st
sin .theta. E - .theta. st A .theta. st sin .theta. 0 E .theta. st
A .theta. st sin .theta. - E - .theta. st A .theta. st sin .theta.
) ##EQU00033##
[0233] Thus by using Equation 3-1 and the Z matrix notation:
( .sigma. x g .sigma. y g .tau. xy g ) = T _ _ Z _ _ ( x y xy )
Equation 3 - 5 ##EQU00034##
[0234] The previous matrix is denoted W: W=TZ
[0235] This relation (Equation 3-5) signifies that the behaviour of
the panel equivalent to the stiffeners is similar to an anisotropic
material (the W matrix can be completed: all these cells have
non-null values).
[0236] 3.2 Stiffened Panels
[0237] We use the Kirchhoff hypothesis: plane sections remain plane
after straining. The network of stiffeners is modelled by an
equivalent panel with a W matrix behaviour (see Equation 3-5). This
disposition of modelling a panel stiffened by triangular pockets,
by two orthotropic plates is illustrated by FIG. 15.
[0238] For the calculation of .nu..sub.yx, we have:
v xy s E x s = v yx s E y s ##EQU00035##
[0239] 3.2.1. Stresses and Loads of Stiffeners
[0240] Flow in the Panel Equivalent to the Stiffeners
[0241] The general expression of flows is:
N .alpha. .beta. = .intg. h .sigma. .alpha..beta. dz Equation 3 - 6
##EQU00036##
[0242] The flow according to axis X is expressed as follows:
N xx = .intg. h .sigma. xx dz = .intg. - h 2 - h 2 + t s .sigma. xx
s dz + .intg. - h 2 + t s h 2 .sigma. xx g dz ##EQU00037## N xx = E
s x t s 1 - v xy s v yx s xx + v xy s E y s t s 1 - v xy s v yx s
yy + .sigma. xx g t g ##EQU00037.2##
[0243] by using Equation 3-5:
N x = [ E x s t s 1 - v xy s v yx s + t g W _ _ 1 , 1 ] x + [ V xy
s E y s t s 1 - v xy s v yx s + t g W _ _ 1 , 2 ] y + t g W _ _ 1 ,
3 xy ##EQU00038##
[0244] And, by using the same method for the N.sub.y and N.sub.xy
flows.
N y = [ v yx s E x s t s 1 - v xy s v yx s + t g W _ _ 2 , 1 ] x +
[ E y s t s 1 - v xy s v yx s + t g W _ _ 2 , 2 ] y + t g W _ _ 2 ,
3 xy N xy = t g W _ _ 3 , 2 x + t g W _ _ 3 , 3 y + ( 2 G xy t s +
t g W _ _ 3 , 1 ) xy Equation 3 - 7 ##EQU00039##
[0245] These expressions clearly show the distribution of flow
between the skin and the panel equivalent to the stiffeners. In the
skin, the relation between the flows and strains is:
( N x s N y s N xy s ) = ( N x N y N xy ) - ( N x g N y g N xy g )
= ( E x s t s 1 - v xy s v yx s v yx s E y s t s 1 - v xy s v yx s
0 v yx s E x s t s 1 - v xy s v yx s E y s t s 1 - v xy s v yx s 0
0 0 2 G xy s t s ) ( x y xy ) Equation 3 - 8 ##EQU00040##
[0246] We note X the previous matrix:
X _ _ = ( E x s t s 1 - v xy s v yx s v yx s E y s t s 1 - v xy s v
yx s 0 v yx s E x s t s 1 - v xy s v yx s E y s t s 1 - v xy s v yx
s 0 0 0 2 G xy s t s ) Thus : ( N x N y N xy ) = ( N x g N y g N xy
g ) + 1 t g X _ _ W _ _ - 1 ( N x g N y g N xy g ) Equation 3 - 9
##EQU00041##
[0247] By inversing this relation, the flows in the grid are
expressed according to the globally applied flows:
( N x g N y g N xy g ) = V ( N x N y N xy ) With : V _ _ = ( I d _
_ + 1 t g X _ _ W _ _ - 1 ) - 1 ( I d _ _ is the identity matrix )
Equation 3 - 10 ##EQU00042##
[0248] Stresses and Loads in Stiffeners
[0249] The flow in the panel equivalent to the stiffeners can be
expressed by:
( N x g N y g N xy g ) = t g T _ _ ( 0 .degree. .theta. - .theta. )
Equation 3 - 11 ( N x g N y g N xy g ) = t g T _ _ E g _ _ - 1 (
.sigma. 0 .degree. .sigma. .theta. .sigma. - .theta. ) Equation 3 -
12 ##EQU00043##
[0250] By using the following notation: U=t.sub.gTE.sub.g.sup.-1 We
have:
( .sigma. 0 .degree. .sigma. .theta. .sigma. - .theta. ) = U - 1 (
N x g N y g N xy g ) Equation 3 - 13 ##EQU00044##
[0251] Finally, the loads and stresses in the stiffeners are
expressed according to the flows of external loads:
( .sigma. 0 .degree. .sigma. .theta. .sigma. - .theta. ) = U - 1 V
( N x N y N xy ) and ( F 0 .degree. F .theta. F - .theta. ) = ( A 0
.degree. st A .theta. st A - .theta. st ) ( .sigma. 0 .degree.
.sigma. .theta. .sigma. - .theta. ) Equation 3 - 14
##EQU00045##
[0252] 3.2.2 Flows and Stresses in the Skin
[0253] As shown by Equation 3-8, flows in the skin are expressed as
below:
( N x s N y s N xy s ) = ( N x N y N xy ) - ( N x g N y g N xy g )
Equation 3 - 15 ##EQU00046##
[0254] thus, the stresses in the skin are expressed by:
( .sigma. x s .sigma. y s .tau. xy s ) = 1 t s ( N x s N y s N xy s
) Equation 3 - 16 ##EQU00047##
[0255] 3.3 The method of calculating plasticity corrected applied
loads is presented here with references to the matrices introduced
in the description (FIG. 17).
[0256] We note that the theory used for calculating plasticity
supposes the isotropic quality of the skin which forms the skin.
The solution for plasticity corrected applied loads is provided by
an iterative method.
[0257] A process of convergence must be carried out until the five
material parameters (E.sub.0.degree..sup.st, E.sub.+.theta..sup.st,
E.sub.-.theta..sup.st, E.sub.skin, .nu..sub.ep) entered at the
start of the iterative process are equal to the same parameters
calculated at the exit (after calculation of plastic stress). In
FIG. 17, already cited, the convergence parameters are indicated by
a grey background.
[0258] More precisely, in this iterative process, the initial
entries are the loads applied in the grid of stiffeners and in the
skin.
[0259] For the grid of stiffeners, the data of the n.sup.th
iteration of Young's modulus of stiffeners E.sub.0.degree..sup.st,
E.sub.+.theta..sup.st, E.sub.-.theta..sup.st in 3 directions:
0.degree., +.theta., -.theta., make it possible, in association
with the value of the angle .theta. and the geometry, to calculate
the matrix [T] (equation 3-4). The values of the angle .theta. and
geometry supplying the matrix [Z] (equation 3-1). The matrices [T]
and [Z] giving the matrix [W](equation 3-5).
[0260] For the skin, the material data of the isotropic skin
E.sub.skin, .nu..sub.ep) make it possible to calculate the matrix
[X] (equation 3-8).
[0261] The [W] and [X] matrices make it possible to calculate the
[U](equation 3-13) and [V] matrices (equation 3-10).
[0262] The results drawn from these matrices include: the flows,
elastic stresses, plasticity corrections on stresses, the values of
the n.sup.th+1 iteration of the Young's modulus of corrected
stiffeners and skin, and of the Poisson coefficient of the
corrected skin, and the loads in the stiffeners.
[0263] We understand that the calculation is iterated until the
Young's modulus and Poisson coefficients vary during an iteration
of a value which is lower than a predetermined threshold.
[0264] Impact of Plasticity Correction on the Calculation of
General Buckling:
[0265] Naturally, plasticity correction changes the behaviour law
matrix calculated in the general instability modulus throughout the
5 material parameters (see the section on general instability).
##STR00001##
[0266] Because of this, plasticity correction also alters the
coefficients .OMEGA..sub.i (i=1 . . . 3) used for calculating
general buckling. Plasticity correction in the analysis of general
buckling is provided by these modified coefficients
.OMEGA..sub.i.
[0267] 3.4 Example: Distribution of Loads Under Bi-Compression and
Shear Load
[0268] In the step of calculating the applied stresses the radius
of the pocket fillet is taken into account for the calculation of
the stiffener section. In addition, there is no post-buckling, we
can therefore write:
% A.sub.0.degree..sup.st=% A.sub.+e.sup.st=%
A.sub.-e.sup.st=100%
t.sub.s.sub._.sub.eff=t.sub.s
[0269] In the present non-limitative example described herein, the
geometry of the panel stiffened by triangular pockets is defined
by:
TABLE-US-00007 L.sub.x = 1400.45 mm a = 198 mm node radius: R.sub.n
= 9 mm L.sub.y = 685.8 mm t = 3.64 mm pocket radius: R.sub.f = 4 mm
.theta. = 58.degree. b = 2.5 mm d = 37.36 mm
[0270] We consider an isotropic material. The elasto-plastic law
used is that of Ramberg & Osgood.
TABLE-US-00008 E 78000 Nu 0.3 Ftu 490 Fty 460 e % 20% n 40
[0271] N.sub.x=-524.65 N/mm
[0272] N.sub.y=-253.87 N/mm
[0273] N.sub.z=327.44 N/mm
[0274] The method of calculating internal loads and applied loads
including plasticity correction is written in the form of a
matrix:
##STR00002##
[0275] The obtained results are therefore the flows, stresses and
loads in the stiffeners:
TABLE-US-00009 N.sub.xg = -81.1 N/mm .sigma..sub.0.degree. =
-101.14 MPa F.sub.0.degree. = -104417N N.sub.yg = -39 N/mm
.sigma..sub.+.theta. = 44.25 MPa F.sub.+.theta. = 4437N N.sub.xyg =
48.1 N/mm .sigma..sub.-.theta. = -135.07 MPa F.sub.-.theta. =
-13537N
[0276] As well as the flows and stresses in the skin:
[0277] N.sub.xs=-443.5 N/mm .sigma..sub.xs=-121.85 MPa
[0278] N.sub.ys=-214.87 N/mm .sigma..sub.ys=-59.03 MPa
[0279] N.sub.xys=279.3 N/mm .tau..sub.xys=76.74 MPa
[0280] In this example, the applied stresses reside in the elastic
domain.
[0281] The flow distribution between the skin and the grid of
stiffeners is summarised by the following table:
TABLE-US-00010 Distribution of loads - Distribution of loads - Flow
External flows percentages (N/mm) flows Grid Skin Grid Skin Nx
-524.65 -81.13 -443.52 15.46% 84.54% Ny -253.87 -39.00 -214.87
15.36% 84.64% Nxy +327.44 +48.12 +279.32 14.7% 85.3%
[0282] Step 4--Resistance Analysis Module:
[0283] This phase is aimed at calculating the reserve factors (RF)
by comparison with the applied loads calculated in the components
of the panel stiffened by triangular pockets, and the maximum
stress capacity of the material.
[0284] The stress capacities in the ultimately loaded material are
determined by: F.sub.tu (the material's ultimate tension
resistance) compared to the stresses applied in the stiffener webs,
F.sub.tu compared to the principal stresses applied in the skin,
F.sub.su (the material's ultimate shear resistance) compared to the
maximum shear capacity in the skin.
[0285] Analysis of resistance consists of calculating the reserve
factors of the limit loaded and ultimate loaded material. The
applied stresses come from loads in the plane (compression, shear
load) or outside of the plane (pressure).
[0286] The entry data for this calculation are: [0287] Capacity
values for the material: F.sub.ty, F.sub.cy, F.sub.sy, F.sub.tu,
F.sub.su [0288] Stresses applied to the structure: [0289] Stresses
on the skin (.sigma..sub.xs, .sigma..sub.ys et .tau..sub.xys)
[0290] Normal stresses in the stiffeners
[0291] Note: applied stresses are corrected for plasticity in the
applied stress calculation method, as was stated above.
[0292] The exit data are the reserve factors.
[0293] The analysis of resistance in the plane of the panel is
based on the following hypotheses. The stresses in the skin do not
depend on the thickness of the skin and the position in the plane.
The stresses in a stiffener do not depend on the position on the
section of the stiffener, but only on the angle of the
stiffener.
[0294] These hypotheses are not valid when post-buckling and
behaviour outside of the plane are equally taken into account. In
these cases, the max/min functions, of known types, must be
implemented to take into account these phenomena.
[0295] Calculation of Principal Stresses:
[0296] To calculate the skin's reserve factor, the principal
stresses (.sigma..sub.max, .sigma..sub.min and .tau..sub.max) are
used:
.sigma. max _ s = .sigma. xs + .sigma. ys 2 + ( .sigma. xs -
.sigma. ys ) 2 2 + .tau. xys 2 .sigma. min _ s = .sigma. xs +
.sigma. ys 2 - ( .sigma. xs - .sigma. ys ) 2 2 + .tau. xys 2 .tau.
max = .sigma. max _ s - .sigma. min _ s 2 Equation 4 - 1
##EQU00048##
[0297] The value .sigma..sub.max used in the calculation of the
reserve factor is defined as the absolute maximum between
.sigma..sub.max.sub._.sub.s and .sigma..sub.min.sub._.sub.s
calculated in Error! Reference source not found.1.
[0298] Reserve Factor at Load Limit (LL):
[0299] Reserve Factor on the Stiffener Webs:
RF in _ plan e _ LL blade _ 0 = F y .sigma. blade _ 0 _ LL
##EQU00049## RF in _ plan e _ LL blade _ + .theta. = F y .sigma.
blade _ + .theta._ LL ##EQU00049.2## RF in _ plan e _ LL blade _ -
.theta. = F y .sigma. blade _ - .theta._ LL ##EQU00049.3##
[0300] Reserve Factor on the Skin:
[0301] Shear capacity:
RF in _ plan e _ LL skin _ shear = F sy .tau. max _ LL
##EQU00050##
[0302] In this formula, if F.sub.sy is unknown, F.sub.su/ 3 can be
used
[0303] Principal Stress:
RF in _ plan e _ LL skin = F y .sigma. max _ LL ##EQU00051## [0304]
An envelope reserve factor is calculated at load limit:
[0304] RF material _ LL = min { RF in _ plan e _ LL blade _ 0 ; RF
in _ plan e _ LL blade _ + .theta. ; RF in _ plan e _ LL blade _ -
.theta. ; RF in _ plan e _ LL skin _ shear ; RF in _ plan e _ LL
skin } Equation 4 - 2 ##EQU00052##
[0305] Reserve Factor at Ultimate Load UL:
[0306] Reserve Factor on the Stiffener Webs:
RF in _ plan e _ UL blade _ 0 = F u .sigma. blade _ 0 _ UL (* ) RF
in _ plane _ UL blade _ + .theta. = F u .sigma. blade _ + .theta._
UL (* ) RF in _ plane _ UL blade _ - .theta. = F u .sigma. blade _
- .theta._ UL ##EQU00053##
[0307] Note: if F.sub.cu is unknown, F.sub.cy or F.sub.tu can be
used
[0308] Reserve Factor on the Skin:
[0309] Maximum Shear Capacity:
RF in _ plan e _ UL skin _ shear = F su .tau. max _ UL
##EQU00054##
[0310] Principal Stress:
RF in _ plan e _ UL skin = F u .sigma. max _ UL ##EQU00055## [0311]
An envelope reserve factor is calculated at ultimate load:
[0311] RF material _ UL = min { RF in _ plan e _ UL blade _ 0 ; RF
in _ plan e _ UL blade _ + .theta. ; RF in _ plan e _ UL blade _ -
.theta. ; RF in _ plan e _ UL skin _ shear ; RF in _ plan e _ UL
skin } ##EQU00056##
[0312] The analysis of resistance outside of the plane of the panel
lies outside of the scope of the present invention.
[0313] In an example of an implementation of this part of the
method for analysing resistance in the plane, the same geometry and
the same case of loads as in the preceding sections is studied. In
the first instance, the ratio of the loads factor is assumed to be
equal to 1.
[0314] Reserve factor of the stiffener webs: [0315]
RF.sub.in.sub._.sub.plane.sub._.sub.UL.sup.blade.sup._.sup.0=4.71
[0316]
RF.sub.in.sub._.sub.plane.sub._.sub.UL.sup.blade.sup._.sup.+.theta.=11.14
[0317]
RF.sub.in.sub._.sub.plane.sub._.sub.UL.sup.blade.sup._.sup.-.theta-
.=3.63
[0318] Reserve Factor of the Skin:
[0319] Calculation of Principal Stresses: [0320]
.sigma..sub.max.sub._.sub.s=-7.52 MPa [0321]
.sigma..sub.min.sub._.sub.s=-173.35 MPa [0322] .tau..sub.max=82.91
MPa [0323] .sigma..sub.max.sub._.sub.skin.sub._.sub.UL=173.35
MPa
[0324] Shear capacity:
RF.sub.in.sub._.sub.plane.sub._.sub.UL.sup.skin.sup._.sup.shear=3.41
[0325] Principal stress:
RF.sub.in.sub._.sub.plane.sub._.sub.UL.sup.skin=2.82
[0326] Reserve factor envelope RF at ultimate load (UL) [0327]
RF.sub.materal.sub._.sub.UL=2.82
[0328] In the case of plastic calculation, an iterative calculation
is carried out until the applied loads reach the resistance failure
load (in order to carry out plasticity correction on these). At
each iteration loop, the same calculations as previously described
are carried out.
[0329] Step 5--Calculating Local Stress Capacity,
[0330] Two types of instability weaken the global rigidity of the
structure stiffened by triangular pockets, but do not cause the
global failure of the complete structure [0331] Instability of the
panel: buckling of triangular pockets [0332] Instability of
stiffeners: buckling of stiffener webs
[0333] The buckled sections can only support a part of the load
which they could support before they were buckled. Because of this,
the applied loads are redistributed in the structure.
[0334] It is noted that in the present invention, post-buckling
calculations are not processed. Because of this, the two types of
buckling referred to above are considered as modes of failure.
[0335] 5A--Calculation of the Panel's Local Buckling:
[0336] In panels stiffened by triangular pockets, the pockets are
triangular plates subjected to combined loads in the plane. In
order to calculate the buckling flow capacity and reserve factors
under pure loading of the complete stiffened panel, a method based
on a finite element model (FEM) is used.
[0337] This section provides a calculation of pocket buckling flow
capacity for isosceles triangular pockets: the base of the triangle
can vary between all values, and, in the present non-limitative
example, between 45.degree. and 70.degree.. The flow calculation is
carried out with two types of boundary conditions: simply supported
and clamped
[0338] The applied stresses to be taken into account for the
calculation of the reserve factor are solely the stresses affecting
the skin, and determined in the section, described above, of the
applied stress calculation.
[0339] The entry data for this section are: [0340] geometric data
(base of the triangle, isosceles angle, thickness of the skin)
[0341] material data (linear (E, .nu.) and non-linear (F.sub.cy,
F.sub.tu, e %, n.sub.c)) [0342] uniquely isotropic material [0343]
the plastic buckling flow capacity values are only pertinent up to
F.sub.cy [0344] Boundary conditions: simply supported or clamped
[0345] Loads applied to the skin
[0346] It should be noted that all the external flows used in this
section are skin flows and do not correspond with a complete
loading of the stiffened panel.
[0347] In addition, the height of the triangle (h) used as
reference length for the buckling calculation is reduced to the
semi-thickness of stiffener webs.
[0348] In this section, the formula used for the height of the
triangle is: h=h_red
h_red = a_red 2 tan .theta. ##EQU00057## a_red = a - b [ 1 tan
.theta. + 1 sin .theta. ] ##EQU00057.2##
[0349] The exit data are the following: [0350] Pocket buckling
capacity. [0351] Reserve factor
[0352] This section is aimed at calculating the flow capacities for
isosceles triangular plates.
[0353] It includes two parts: 1/A calculation of capacity values
for simply supported triangular plates (part 5A.5), 2/A calculation
of capacity values for clamped triangular plates (part 5A.6).
[0354] The two following parts follow the same approach: firstly a
calculation of capacity values for triangular plates subject to
cases of pure loading (compression according to X, compression
according to Y and shear load), then a calculation of interaction
curves between the three cases of pure loading.
[0355] 5A.1 Calculation Principle
[0356] The cases of pure loading envisaged are presented in FIG.
12.
[0357] Several analytical formula methods for local buckling of
triangles exist in written documentation. The comparison of these
methods shows that there is a large difference between the
previously cited buckling stresses. Moreover, some of the
parameters used in these methods are derived from calculations by
finite elements, tests are often empirical and certain methods
provide absolutely no data for angles other than those of
60.degree..
[0358] The development of a complete theory of this problem being
somewhat lengthy, the method such as is herein described, which is
non-limitative, implements a method based entirely on a Finite
Element Model (FEM): [0359] Creating an FEM parametric model of a
triangular plate (parameters: base angle, thickness, height of
triangle, boundary conditions), [0360] Testing various combinations
to obtain linear results of buckling, [0361] Obtaining the
parameters which will be used in an analytical formula
(coefficients K).
[0362] The induced plasticity effects must also be taken into
account in the calculation of capacity values. The applied loads
are either simple loads or combinations of these simple loads.
[0363] 5A.2 Case of Pure Loading
[0364] Interaction curves are defined as follows. Six finite
element models of triangular plates were created with angles of
between 45.degree. and 70.degree. in the present non-limitative
example which concerns angles of the triangle. In this section, the
isosceles angle (.theta.) is defined as the base angle of the
isosceles triangle (see FIG. 18). For each isosceles angle and for
each case of pure loading, the study is organised in three
points:
[0365] 1/Calculation by Finite Element Model (FEM)
[0366] Linear calculations of local buckling of triangles by finite
element model (FEM) of a known type were carried out to determine
the wrinkle flow capacity (without plastic correction) for various
thicknesses, and therefore various stiffnesses of a plate. We note
that the first mode observed always presents a single buckle (a
single lump).
[0367] 2/Tracing the Curve of Buckling Flow Capacity According to
D/h.sup.2
[0368] In general, in written documentation, the buckling flow
capacity is expressed as follows:
N crit = K D h 2 ##EQU00058##
[0369] K is a constant,
[0370] D the stiffness of the plate:
D = E t 3 12 ( 1 - v 2 ) , ##EQU00059##
[0371] h is the height of the triangle:
h = a 2 tan .theta. . ##EQU00060##
[0372] But this study demonstrates that, in the case of triangular
plates, and for the small values of the ratio
D h 2 , ##EQU00061##
an expression of the buckling flow capacity using a first degree
equation in
D h 2 ##EQU00062##
is not pertinent. Better results are obtained with a second degree
equation in the following form:
N crit = K 1 ( D h 2 ) 2 + K 2 D h 2 ( elastic value ) Equation 5 -
1 ##EQU00063##
[0373] Constants K.sub.1 and K.sub.2 depend on the angle and the
load case being considered. Therefore, for each case and each
angle, a value for the K.sub.1 and K.sub.2 constants is
obtained.
[0374] 3/ Tracing the Evolution of K.sub.1 and K.sub.2 According to
the Base Angle of the Isosceles Triangle
[0375] K.sub.1 and K.sub.2 are traced according to the angle and an
interpolation is carried out to determine a polynomial equation
which makes it possible to calculate these constants of any angle
between 45.degree. and 70.degree.. FIG. 19 illustrates a linear or
quadratic interpolation for the K coefficients. It is clear that
this function also makes it possible to extrapolate values outside
of, but close to the domain going from 45 to 70.degree.. Thus, by
knowing the isosceles angle and the boundary conditions, it is
possible to directly calculate the buckling flow capacity of the
triangular plate being studied.
[0376] 5A.3 Case of Combined Loading
[0377] In this case the following hypothesis is used: if some
components of the combined load are under tension, these components
are reduced to zero (are not taken into account for the
calculation). It is, in effect, conservative to consider that the
components under tension have no affect with regards to the
buckling flow being studied, and do not improve the buckling stress
on the plate. For example, if N.sub.x.sup.app=+200 N/mm (which
shows a tension) and N.sub.s.sup.app=300 N/mm, the combined load
capacity is reduced to the pure shear load capacity.
[0378] The presentation of the envisaged loading cases is
illustrated in FIG. 21. In this section, three finite element
models were used: three isosceles triangular plates with angles
equal to 45.degree., 60.degree. and 70.degree.. For each angle, the
study is organised in two points:
[0379] 1/Calculation by Finite Element Model (FEM)
[0380] For all the combinations presented below, linear
calculations by finite element method were carried out to determine
the eigenvalue of buckling corresponding to different distributions
of external loads.
[0381] We can see that the first mode observed always presents a
single blister.
[0382] It appears that the interaction curve depends little on the
value of
D h 2 . ##EQU00064##
[0383] 2/Tracing the Interaction Curves
[0384] The interaction curve is traced for each angle and each
combination of loads. Next, the various curves are approximated
with classical curves for which the calculation takes the following
form:
R.sub.1.sup.A+R.sub.2.sup.B=1
[0385] with
R i = N i app N i crit , i = cX , cY or s . ##EQU00065##
[0386] The results and the choices made have shown that the
equations of interaction curves do not depend on the base angle of
the isosceles triangle and are therefore compatible and can be
unified by a single equation covering all the combinations in the
following form:
R.sub.cX.sup.A+R.sub.cY.sup.B+R.sub.s.sup.C=1
[0387] Based on this equation, to determine the reserve factors, we
can solve the following equation:
( R cy R ) A + ( R cX R ) B + ( R s R ) C = 1 ##EQU00066##
[0388] with
R = N cY app N cYcomb crit = N cX app N cXcomb crit = N s app N
scomb crit = 1 RF ##EQU00067##
[0389] 5A.4 Plasticity Correction Factor
[0390] Obtaining a plasticity correction for cases of pure loading,
according to the isosceles angle and the boundary conditions is
very complex. In fact, for triangular plates, deflection functions
are complex and give rise to numerous digital integration
problems.
[0391] As a result, it was decided to use a conservative .eta.
factor, based on the NACA Report 898 ("A Unified Theory of Plastic
Buckling of Columns and Plates", July 1947).
[0392] This factor is defined for all cases of loading (pure and
combined) with the exception of shear load, by:
.eta. 5 = E ta n E c ##EQU00068##
[0393] And for cases of pure shear load, by:
.eta. 6 = ( 1 + v e ) ( 1 + v ) E se c E c ##EQU00069##
[0394] The correction is calculated by using the equivalent elastic
stress of Von Mises:
.sigma..sub.VM= {square root over
(.sigma..sub.x.sub._.sub.comb.sup.crit
2+.sigma..sub.y.sub._.sub.comb.sup.crit
2-.sigma..sub.x.sub._.sub.comb.sup.crit.sigma..sub.y.sub._.sub.comb.sup.c-
rit+3.tau..sub.xy.sub._.sub.comb.sup.crit 2)}
[0395] Therefore, the corrected stress capacities can be
calculated:
.sigma..sub.x.sub._.sub.comb.sup.plast=.eta..sigma..sub.x.sub._.sub.comb-
.sup.crit
.sigma..sub.y.sub._.sub.comb.sup.plast=.eta..sigma..sub.y.sub._.sub.comb-
.sup.crit
.sigma..sub.xy.sub._.sub.comb.sup.plast=.eta..tau..sub.xy.sub._.sub.comb-
.sup.crit
[0396] For cases of pure loading (compression according to X,
compression according to Y or shear load), the plasticity
correction is also applied to the Von Mises stress, therefore for
cases of pure shear loading, the corrected stress is: {square root
over (3)}.tau..sub.xy.
[0397] 5A.5 Isosceles Triangular Simply Supported Plates
[0398] Case of Pure Loading
[0399] FIGS. 26 and 27 show the evolution of the K.sub.1 and
K.sub.2 constants according to the isosceles triangle. The
equations of these curves are (with .theta. in degrees):
K.sub.1cX=-0.0000002417.theta..sup.4+0.0000504863.theta..sup.3-0.0039782-
194.theta..sup.2+0.1393226958.theta.-1.8379213492
K.sub.1cYc=-0.0000007200.theta..sup.4+0.0001511407.theta..sup.3-0.011924-
7778.theta..sup.2+0.4177844180.theta.-5.4796530159
K.sub.1s=-0.0000018083.theta..sup.4+0.0003804181.theta..sup.3-0.03007439-
72.theta..sup.2+1.0554840265.theta.-13.8695053175
K.sub.2cX=0.0029565.theta..sup.3-0.4291321.theta..sup.2+21.1697836.theta-
.-291.6730902
K.sub.2cY=0.0068664.theta..sup.3-1.0113413.theta..sup.2+51.3462358.theta-
.-852.1945224
K.sub.2s=0.013637.theta..sup.3-2.017207.theta..sup.2+102.120039.theta.-1-
674.287384
[0400] Whether this is for K.sub.1 or K.sub.2, their values under
pure compression X and pure Y are equal for an isosceles angle of
60.degree.. The intersection point at 60.degree. is proof of the
isotropic behaviour of the structure stiffened by triangular
pockets at 60.degree., in terms of local buckling in the skin.
[0401] Case of Combined Loading
[0402] In the case of combined loading, an analysis is carried out
by finite element model of the linear calculation of buckling for
simply supported triangular plates. We therefore choose a
conservative interaction curve, close to the calculated interaction
curves, but in a simple formula, which becomes the curve used in
the method herein described. Its equation is:
R.sub.1.sup.A+R.sub.2.sup.B=1. With
R i = N i app N i crit , ##EQU00070##
i=cX, cY or s.
[0403] Interaction Compression X+Compression Y (Case 1)
[0404] In this case of loading, for angles between 45.degree. and
70.degree., we chose a conservative interaction curve. That is to
say a curve which declares an interaction to a value lower than the
sum of compressions R.sub.cX on X and R.sub.cY on Y, in respect of
all the interaction curves calculated for the angle values between
45.degree. and 70.degree.. This curve is defined by the following
equation: R.sub.cX+R.sub.cY=1
[0405] Interaction Compression X+Shear Load (Case 2)
[0406] In this case of loading, for angles between 45.degree. and
70.degree., we chose a conservative interaction curve, in respect
of the different interaction curves according to angles between
45.degree. and 70.degree., defined by the following equation:
R.sub.cX+R.sub.s.sup.3/2=1
[0407] Interaction Compression Y+Shear Load (Case 3)
[0408] For angles between 45.degree. and 70.degree., in order to
determine the reserve factor in the case of a combined compression
loading according to Y and of shear load, we chose a conservative
equation of the following formula: R.sub.cY+R.sub.s.sup.2=1. In
order to arrive at a single equation covering all cases of loading,
we chose to use another, even more conservative curve
R.sub.cY+R.sub.s.sup.3/2=1
[0409] Interaction Compression X+Compression Y+Shear Load (Case
4)
[0410] The equation chosen for these cases of loading is:
R.sub.cX+R.sub.cY+R.sub.s.sup.3/2=1. This unique equation is used
for all cases of combined loading
[0411] 5A.6 Clamped Isosceles Triangular Plates
[0412] Case of Pure Loading
[0413] FIGS. 28 and 29 show the evolution of the K.sub.1 and
K.sub.2 constants according to the isosceles triangle. The
equations of these curves are (with .theta. in degrees):
K.sub.1cX=-0.0000018547.theta..sup.4+0.0003940252.theta..sup.3-0.0314632-
778.theta..sup.2+1.1143937831.theta.-14.8040153968
K.sub.1cY=-0.0000027267.theta..sup.4+0.0005734489.theta..sup.3-0.0453489-
667.theta..sup.2+1.5921323016.theta.-20.9299676191
K.sub.1s=-0.0000069990.theta..sup.4+0.0014822211.theta..sup.3-0.11790804-
17.theta..sup.2+4.1617623127.theta.-54.9899559524
K.sub.2cX=0.0110488.theta..sup.3-1.6258419.theta..sup.2+81.82784200-1254-
.6580819
K.sub.2cY=0.0158563.theta..sup.3-2.3439723.theta..sup.2+119.5038876-1970-
.9532998
K.sub.2s=0.0252562.theta..sup.3-3.7563673.theta..sup.2+191.0642156.theta-
.-3113.4527806
[0414] Case of Combined Loading
[0415] In the case of combined loading, an analysis is carried out
by finite element model of the linear calculation of buckling for
clamped triangular plates. We therefore chose a conservative
interaction curve, close to the calculated interaction curves, but
in a simple formula, which becomes the curve used in the method
herein described. Its equation is: R.sub.1.sup.A+R.sub.2.sup.B=1
with
R i = N i app N i crit , ##EQU00071##
i=cX, cY or s.
[0416] Interaction Compression X+Compression Y (Case 1)
[0417] In this case of loading, for angles between 45.degree. and
70.degree., we chose a conservative interaction curve, in respect
of the different interaction curves according to angles between
45.degree. and 70.degree., defined by the following equation:
R.sub.cX+R.sub.cY=1
[0418] Interaction Compression X+Shear Load (Case 2)
[0419] In this case of loading, for angles between 45.degree. and
70.degree., we chose a conservative interaction curve, in respect
of the different interaction curves according to angles between
45.degree. and 70.degree., defined by the following equation:
R.sub.cX+R.sub.s.sup.3/2=1
[0420] Interaction Compression Y+Shear Load (Case 3)
[0421] For angles of between 45.degree. and 70.degree., to
determine the reserve factor in the case of a combined compression
load according to Y and of shear load, we chose a conservative
equation of the following formula: R.sub.cY+R.sub.s.sup.2=1. In
order to arrive at a single equation covering all cases of loading,
we chose to use another, even more conservative curve
R.sub.cY+R.sub.s.sup.3/2=1
[0422] Interaction: Compression X+Compression Y+Shear Load (Case
4)
[0423] The equation, used for this case of loading, is
R.sub.cX+R.sub.cY+R.sub.s.sup.3/2=1. This unique equation is used
for all cases of combined loading.
[0424] 5B Calculation of Local Buckling of the Stiffener:
[0425] This modulus calculates the buckling stress and the reserve
factor of the stiffener web, considered as a rectangular panel with
diverse boundary conditions to be defined by the user.
[0426] The applied stresses to take into account for the
calculations of reserve factor are uniquely the stresses in the
stiffener webs, derived from the modulus of calculation of applied
stresses.
[0427] On the stiffener grid, one or several types of stiffener
webs are compression loaded. Because of this the compression stress
capacity must be calculated.
[0428] The entry data for this module are: [0429] Geometric data:
dimensions of stiffener webs (length, height, thickness), [0430]
Material data (linear (E,.nu.) and non-linear (F.sub.cy, F.sub.tu,
.epsilon..sub.ult, n.sub.c)).
[0431] In the present example we are only considering an isotropic
material, [0432] boundary conditions (four are available), [0433]
Loads applied to the stiffener webs.
[0434] The exit data are the buckling capacity of the stiffener web
and a reserve factor. The buckling stress capacity of the stiffener
web is (see FIG. 13 for the notation conventions):
.sigma. blade crit = k c .eta. E c .pi. 2 12 ( 1 - v e 2 ) ( b d )
2 Equation 5 - 2 ##EQU00072## [0435] With: [0436] b: thickness of
the stiffener web [0437] d: height of the stiffener web [0438]
L.sub.b: length of the stiffener web [0439] E.sub.c: Young's
modulus in compression [0440] .nu..sub.e: Poisson coefficient in
the elastic domain [0441] k.sub.c: Local buckling factor (dependent
on boundary conditions and geometry) [0442] .eta.: Plasticity
correction factor
[0443] Note: the length of a stiffener web is given by [0444]
(L.sub.b)=a (for stiffener webs in the X direction)
[0444] ( L b ) = a 2 1 + tan 2 .theta. ##EQU00073##
(for stiffener webs in transversal directions)
[0445] Numerous boundary conditions can be applied on the stiffener
web according to the surrounding structure. (see FIG. 14). We note
that if Lb/d is greater than the value of Lim, then k.sub.c is
worth k.sub.c infinitively. The recommended conservative buckling
factor for calculations based on numerous finite element analyses
case 2 (2 clamped edges--one simply supported edge--one free
edge)
[0446] According to the boundary conditions cited above and
according to table 1 of plastic correction factors for rectangular
plates, the plasticity correction factor used in this case is:
.eta. = .eta. 1 = E s ec E c 1 - v e 2 1 - v 2 ##EQU00074##
[0447] The formula for reserve factor calculation for buckling of
the stiffener web is valid for all types of stiffener webs used in
the present panel stiffened by triangular pockets (0.degree.,
+.theta., -.theta.):
RF buck blade = .sigma. blade crit .sigma. blade app
##EQU00075##
[0448] The following example is based on the same geometry as was
used in the previous sections. The geometry of stiffeners is: b=2.5
mm, and d=37.36 mm. The length of stiffener webs is (L.sub.b):
L.sub.b=a=198 mm for stiffener webs in the X direction and
L b = a 2 1 + tan 2 .theta. = 186.82 mm ##EQU00076##
for transversal stiffener webs. The boundary conditions used are: 2
sides clamped--1 side simply supported--1 side free.
[0449] Thus:
k c _ 0 .degree. = 4.143 ( d L b ) 2 + 0.384 = 0.5315
##EQU00077##
for stiffener webs in the X direction and
k c _ .theta..degree. = 4.143 ( d L b ) 2 + 0.384 = 0.5497
##EQU00078##
for transversal stiffener webs
[0450] And the buckling load for each stiffener is:
[0451] The plasticity correction factor at stiffener web buckling
is:
.eta. = .eta. 1 = E sec E c 1 - v e 2 1 - v 2 = 1 ( elastic )
##EQU00079##
[0452] The loads applied in the stiffeners are: [0453]
.sigma..sub.0.degree.=-101.14 MPa [0454] .sigma..sub.+.theta.=44.25
MPa [0455] .sigma..sub.-.theta.=-135.07 MPa
[0456] The results of reserve fracture calculation are the
following:
[0457] Step 6--Calculation of General Instability:
[0458] This step provides data on buckling flow capacity for a flat
panel stiffened by triangular pockets, in the conditions of pure or
combined loading.
[0459] The formulae are based on the buckling of orthotropic
plates. Two or four boundary conditions are possible according to
the loading case (4 simply-supported edges, 4 clamped edges, 2
loaded simply supported edges and 2 lateral clamped edges, 2 loaded
clamped edges and 2 simply supported lateral edges). The applied
flows, to take into account for reserve factor calculation, are the
external flows of the panel stiffened by triangular pockets which
are the entry data.
[0460] The entry data are the following: [0461] Geometric data:
[0462] L.sub.x: length of the panel equivalent to the grid [0463]
L.sub.y: width of the panel equivalent to the grid, [0464] t.sub.s:
thickness of the skin, [0465] t.sub.g: thickness of the panel
equivalent to the grid, [0466] Data on the material: [0467]
E.sub.x.sup.s, E.sub.y.sup.s: Young's modulus of the skin, [0468]
G.sub.xy.sup.s: shear modulus of the skin, [0469]
.nu..sub.xy.sup.s, .nu..sub.yx.sup.s: Poisson coefficient of the
skin, [0470] E.sub.x.sup.g, E.sub.y.sup.g: Young's modulus of the
grid, [0471] G.sub.xy.sup.g: shear modulus of the grid, [0472]
.nu..sub.xy.sup.g, .nu..sub.yx.sup.g: Poisson coefficient of the
grid, [0473] loads applied to the structure (N.sub.x.sup.0,
N.sub.y.sup.0, N.sub.xy.sup.0, p.sub.z) [0474] boundary conditions
(2 or 4 are possible according to the type of loading)
[0475] The exit data are: [0476] N.sub.x.sup.c, N.sub.y.sup.c,
N.sub.xy.sup.c: buckling flow capacity, [0477]
N.sub.x.sup.c.sub.comb, N.sub.y.sup.c.sub.comb,
N.sub.xy.sup.c.sub.comb: combined buckling flow capacity, [0478]
Reserve factors
[0479] We use the Kirchhoff hypothesis: plane sections remain
principally plane following strain. The grid (of stiffeners) is
modelled here by an equivalent panel. The skin and the panel
equivalent to the grid are considered as plates of an orthotropic
nature.
[0480] The material parameters verify the following relation:
v xy i E x i = v yx i E y i ##EQU00080##
[0481] The conventions of flow and moments are illustrated by FIG.
21.
[0482] 6.1.1 Displacements
[0483] The vector {right arrow over (U)} represents the
displacement of a point M(x,y) of the median surface: {right arrow
over (U)}=[u,v,w]=u(x,y){right arrow over (x)}+v(x,y){right arrow
over (y)}+w(x,y){right arrow over (z)}
[0484] The different variables do not depend on z because a plane
state of stresses has been envisaged (.sigma..sub.zz=0).
[0485] 6.1.2 Strains
[0486] The general expression of strains in a section of the plate
situated at a z distance from the median axis is:
{ xx = xx 0 + z . .kappa. xx yy = yy 0 + z . .kappa. yy xy = xy 0 +
z . .kappa. xy Equation 6 - 1 ##EQU00081##
[0487] With:
{ xx 0 = u , x + 1 2 ( w , x ) 2 yy 0 = v , y + w R + 1 2 ( w , y )
2 xy 0 = 1 2 ( u , y + v , x ) + 1 2 w , x . w , y Equation 6 - 2
##EQU00082##
[0488] The terms .epsilon..sub.xx.sup.0, .epsilon..sub.yy.sup.0,
and .epsilon..sub.xy.sup.0 represent the contribution in strain in
the plane of the plate. The terms 1/2(w.sub.,x).sup.2,
1/2(w.sub.,y).sup.2, and 1/2w.sub.,xw.sub.,y represent the
non-linear contribution in strain in the plane of the plate. The
term R represents the radius of the shell, but here we are
considering a plane plate, therefore
1 R = 0. ##EQU00083##
{ .kappa. xx = - w , x 2 .kappa. yy = - w , y 2 .kappa. xy = - w ,
xy Equation 6 - 3 ##EQU00084##
[0489] The terms z.kappa..sub.xx, z.kappa..sub.yy, and
z.kappa..sub.xy represent the contribution in strain due to the
change of the plate curve (z is the distance from the median axis
of the plate).
[0490] 6.1.3 Behaviour Laws
[0491] The sink and the panel equivalent to the stiffeners are
considered as orthotropic plates. Because of this the relations
between stresses and strains are:
( .sigma. xx i .sigma. yy i .sigma. xy i ) = ( E x i 1 - v xy i v
yx i v xy i E y i 1 - v xy i v yx i 0 v yx i E x i 1 - v xy i v yx
i E y i 1 - v xy i v yx i 0 0 0 2 G xy i ) ( xx yy xy ) Equation 6
- 4 ##EQU00085##
[0492] with i=(s, g) (indices s for the values relative to the skin
and indices g for the values relative to the stiffener grid).
[0493] 6.1.4 Flow and Moments
[0494] The expressions of flow and moments by unit of length
are:
N .alpha..beta. = .intg. h .sigma. .alpha..beta. dz M .alpha..beta.
= .intg. h .sigma. .alpha..beta. . zdz Equation 6 - 5
##EQU00086##
[0495] with (.alpha.,.beta.)=(x,y).
[0496] Flow:
N .alpha..beta. = .intg. h .sigma. .alpha..beta. dz = .intg. - h 2
- h 2 + t s .sigma. .alpha..beta. s dz + .intg. - h 2 + t s h 2
.sigma. .alpha..beta. g dz ##EQU00087##
[0497] By using Equation 6-5 and the relation h=t.sub.s+t.sub.g we
find:
N xx = ( E x s t s 1 - v xy s v yx s + E x g t g 1 - v xy g v yx g
) xx 0 + ( v xy s E y s t s 1 - v xy g v yx g + v xy g E y g t g 1
- v xy g v yx g ) yy 0 + [ 1 2 t s t g ( E x g 1 - v xy g v yx g -
E x s 1 - v xy s v yx s ) ] .kappa. xx + [ 1 2 t s t g ( v xy g E y
g 1 - v xy g v yx g - v xy s E y s 1 - v xy g v yx g ) ] .kappa. yy
Equation 6 - 6 N yy = ( v yx s E x s t s 1 - v xy g v yx g + v yx g
E x g t g 1 - v xy g v yx g ) xx 0 + ( E y s t s 1 - v xy s v yx s
+ E y g t g 1 - v xy g v yx g ) yy 0 + [ 1 2 t s t g ( v yx g E x g
1 - v xy g v yx g - v yx s E x s 1 - v xy g v yx g ) ] .kappa. xx +
[ 1 2 t s t g ( 1 2 t s t g E y g 1 - v xy g v yx g - E y s 1 - v
xy s v yx s ) ] .kappa. yy N xy = 2 ( G xy s t s + G xy g t g ) xy
0 + [ t s t g ( G xy g - G xy s ) ] .kappa. xy ##EQU00088##
[0498] Moments by Unit of Length:
M .alpha..beta. = .intg. h .sigma. .alpha..beta. zdz = .intg. - h 2
- h 2 + t s .sigma. .alpha..beta. s zdz + .intg. - h 2 + t s h 2
.sigma. .alpha..beta. g zdz ##EQU00089##
[0499] By using Equation 6-5 and the relation h=t.sub.s+t.sub.g we
find:
- M xx = [ 1 2 t s t g ( E x g 1 - v xy g v yx g - E x s 1 - v xy s
v yx s ) ] xx 0 + [ 1 2 t s t g ( v xy g E y g 1 - v xy g v yx g -
v xy s E y s 1 - v xy g v yx g ) ] yy 0 + [ 1 12 ( E x s t s ( t s
2 + 3 t g 2 ) 1 - v xy s v yx s + E x g t g ( t g 2 + 3 t s 2 ) 1 -
v xy g v yx g ) ] .kappa. xx + [ 1 12 ( v xy s E y s t s ( t s 2 +
3 t g 2 ) 1 - v xy g v yx g + v xy g E y g t g ( t g 2 + 3 t s 2 )
1 - v xy g v yx g ) ] .kappa. yy Equation 6 - 7 M yy = [ 1 2 t s t
g ( v yx g E x g 1 - v xy g v yx g - v yx s E x s 1 - v xy g v yx g
) ] xx 0 + [ 1 2 t s t g ( E y g 1 - v xy g v yx g - E y s 1 - v xy
s v yx s ) ] yy 0 + [ 1 12 ( v yx s E x s t s ( t s 2 + 3 t g 2 ) 1
- v xy g v yx g + v yx g E x g t g ( t g 2 + 3 t s 2 ) 1 - v xy g v
yx g ) ] .kappa. xx + [ 1 12 ( E y s t s ( t s 2 + 3 t g 2 ) 1 - v
xy s v yx s + E y g t g ( t g 2 + 3 t s 2 ) 1 - v xy g v yx g ) ]
.kappa. yy - M xy = [ t s t g ( G xy g - G xy s ) ] xy 0 + [ 1 6 (
G xy s t s ( t s 2 + 3 t g 2 ) + G xy g t g ( t g 2 + 3 t s 2 ) ) ]
.kappa. xy ##EQU00090##
[0500] Once the general behaviour law (between flow and moments on
one hand and strains on the other), is obtained:
##STR00003##
[0501] Matrices A, B and C are symmetrical.
[0502] 6.2 Balance Equations
[0503] General balance equations of an element of the panel (or
shell) are given by the following expressions, linking flows,
moments and surface strength density:
{ N xx , x + N xy , y + p x = 0 N yy , y + N yx , x + p y = 0 ( N
xx w , x ) , x + ( N yy w , y ) , y + ( N xy w , y ) , x + ( N yx w
, x ) , y - N yy R - M xx , x 2 + M yy , y 2 - 2 M xy , xy + p z =
0 Equation 6 - 9 ##EQU00091##
[0504] where {right arrow over (f)}=p.sub.x{right arrow over
(x)}+p.sub.y{right arrow over (y)}+p.sub.z{right arrow over (z)} is
the surface strength density acting on the shell element. The
surface strength density only acts along the Z radial direction,
and not in the other directions. Therefore, p.sub.x=p.sub.y=0.
Furthermore, in this instance we are considering the case of a flat
plate, therefore
1 R = 0. ##EQU00092##
Because of this we obtain simplified balance equations:
{ N xx , x + N xy , y = 0 N yy , y + N yx , x = 0 N xx w , x 2 + N
yy w , y 2 + 2 N xy w , xy + p z = M xx , x 2 - M yy , y 2 + 2 M xy
, xy Equation 6 - 1 ##EQU00093##
[0505] 6.3 For the general solution of these equations, we define
the following vectors
[ ] = [ xx yy xy ] [ N ] = [ N xx N yy N xy ] [ K ] = [ .kappa. xx
.kappa. yy .kappa. xy ] [ W ] = [ w , xx w , yy w , xy ] [ M ] = [
- M xx M yy - M xy ] Equation 6 - 11 ##EQU00094##
[0506] The usual loads applied on the plate are: [0507] Uniform
compression flow along the x axis: -N.sub.x.sup.0 [0508] Uniform
compression flow along the y axis: -N.sub.y.sup.0 [0509] Uniform
shear flow in the x-y plane: -N.sub.xy.sup.0 [0510] Uniform
Pressure along the z axis: p.sub.z{right arrow over (z)}
[0511] Because of this, since the applied loads defined below are
uniform, it can be deduced that the two first equations of Equation
6-10 are verified.
N.sub.xx,x=N.sub.yy,y=N.sub.xy,y=N.sub.yx,x=0.
[0512] Expression of Moments:
[0513] By using Equation 6-8 and Equation 6-11, the following
relations are found:
[N]=A[.epsilon.]+B[K]
[M]=B[.epsilon.]+C[K] Equation 6-12
[0514] As the applied flows are uniform, the following relations
are obtained:
[N].sub.,x.sub.2=0A[.epsilon.].sub.,x.sub.2+B[.epsilon.].sub.,x.sub.2=0[-
.epsilon.].sub.,x.sub.2=A.sup.-1B[W].sub.,x.sub.2
[N].sub.,y.sub.2=0A[.epsilon.].sub.,y.sub.2+B[.epsilon.].sub.,y.sub.2=0[-
.epsilon.].sub.,y.sub.2=A.sup.-1B[W].sub.,y.sub.2
[N].sub.,xy=0A[.epsilon.].sub.,xy+B[.epsilon.].sub.,xy=0[.epsilon.].sub.-
,xy=A.sup.-1B[W].sub.,xy Equation 6-13
[0515] With the fact that: [W]=-[K].
[0516] We therefore have for moments:
[M].sub.,x.sub.2=B[.epsilon.].sub.,x.sub.2+C[K].sub.,x.sub.2=(BA.sup.-1B-
-C)[W].sub.,x.sub.2=-D[W].sub.,x.sub.2
[M].sub.,y.sub.2=B[.epsilon.].sub.,y.sub.2+C[K].sub.,y.sub.2=(BA.sup.-1B-
-C)[W].sub.,y.sub.2=-D[W].sub.,y.sub.2
[M].sub.,xy=B[.epsilon.].sub.,xy+C[K].sub.,xy=(BA.sup.-1B-C)[W].sub.,xy=-
-D[W].sub.,xy Equation 6-14
with
D=C-BA.sup.-1B Equation 6-15
[0517] D is the global stiffness matrix, and is symmetrical.
D _ _ = ( D 11 D 12 0 D 12 D 22 0 0 0 D 33 ) ##EQU00095##
[0518] Derivations of the moments of the Equation 6-10 are
therefore obtained:
M.sub.xx,x.sub.2=D.sub.11w.sub.,x.sub.4+D.sub.12w.sub.,x.sub.2.sub.y.sub-
.2
M.sub.yy,y.sub.2=D.sub.12w.sub.,x.sub.2.sub.y.sub.2+D.sub.22w.sub.,y.sub-
.4
M.sub.xy,xy=D.sub.33w.sub.,x.sub.2.sub.y.sub.2 Equation 6-16
[0519] The expression of Equation 6-10 in terms of displacements
therefore gives the general differential equation:
-N.sub.x.sup.0w.sub.,x.sub.2-N.sub.y.sup.0w.sub.,y.sub.2-2N.sub.xy.sup.0-
w.sub.,xy+p.sub.z=.OMEGA..sub.1w.sub.,y.sub.4+.OMEGA..sub.2w.sub.,y.sub.4+-
.OMEGA..sub.3w.sub.,x.sub.2.sub.y.sub.2 Equation 6-17
with:
.OMEGA..sub.1=D.sub.11
.OMEGA..sub.2=D.sub.22
.OMEGA..sub.3=2(D.sub.12+D.sub.33) Equation 6-18
[0520] In the following section, the panel stiffened by triangular
pockets is modelled with its three bending stiffeners (.OMEGA.1,
.OMEGA.2 and .OMEGA.3) in order to calculate the buckling flows in
an orthotropic plate.
[0521] Here again, the following hypothesis is used in this case:
if some components of the combined load are in tension, these
components are not taken into account for the calculation. It is,
in effect, conservative to consider that the components in tension
have no affect with regards to the buckling flow being studied, and
do not improve the buckling stress on the plate.
[0522] 6.4 Buckling Flow Capacity
[0523] 6.4.1 Longitudinal Compression Flow (Compression According
to X)
[0524] The plate is subjected to a uniform longitudinal compression
flow (according to the X axis): -N.sub.x.sup.0. Because of this:
N.sub.y.sup.0=N.sub.xy.sup.0=p.sub.z=0. The general differential
equation (Equation 6-17) therefore expresses:
-N.sub.x.sup.0w.sub.,x.sub.2=.OMEGA..sub.1w.sub.,x.sub.4+.OMEGA..sub.2w.-
sub.,y.sub.4+.OMEGA..sub.3w.sub.,x.sub.2.sub.y.sub.2 Equation
6-19
[0525] Firstly, we consider a simply supported plate (the boundary
conditions are generalised further on):
[0526] For x=0 and x=L.sub.x: w=0 and -M.sub.xx=0
[0527] For y=0 and y=L.sub.y: w=0 and M.sub.yy=0
[0528] The following expression for displacement w satisfies all
the boundary conditions detailed above:
w ( x , y ) = C mn sin ( m .pi. x L x ) sin ( n .pi. y L y ) ( m ,
n ) .di-elect cons. N 2 Equation 6 - 20 ##EQU00096##
[0529] The previous expression for w must satisfy the general
differential equation (Equation 6-19), therefore we obtain:
N x 0 ( m .pi. L x ) 2 = .OMEGA. 1 ( m .pi. L x ) 4 + .OMEGA. 2 ( n
.pi. L y ) 4 + .OMEGA. 3 ( m .pi. L x ) 2 ( n .pi. L y ) 2 ( m , n
) .di-elect cons. N 2 ##EQU00097##
[0530] The minimum value of N.sup.0 corresponds to the value of
flow capacity of general buckling N.sub.x.sup.c. We demonstrate
that this value is:
N x c = 2 ( .pi. L y ) 2 ( .OMEGA. 1 .OMEGA. 2 + .OMEGA. 3 2 )
Equation 6 - 21 ##EQU00098##
[0531] This formula can be generalised for different boundary
conditions (loaded edges and simply supported or clamped lateral
edges):
N x c = k c ( .pi. L y ) 2 .OMEGA. 1 .OMEGA. 2 Equation 6 - 22
##EQU00099##
[0532] with
k c = h ( .alpha. _ ) + q .beta. ##EQU00100## .alpha. _ = L x L y
.OMEGA. 2 .OMEGA. 1 4 ##EQU00100.2## .beta. = .OMEGA. 3 2 .OMEGA. 1
.OMEGA. 2 ##EQU00100.3## h ( .alpha. _ ) = { ( 1 .alpha. _ ) 2 +
.alpha. _ 2 , si .alpha. _ .ltoreq. 1 2 , sin on q = 2
##EQU00100.4##
[0533] FIG. 22 shows the value of h(a) according to different
boundary conditions (the case of four simply supported edges is the
base curve).
[0534] 6.4.2 Transversal Compression Flow (Compression According to
Y)
[0535] The plate is subjected to a uniform transversal compression
flow (according to the Y axis): -N.sub.y.sup.0. Because of this:
N.sub.x.sup.0=N.sub.xy.sup.0=p.sub.z=0. The solution is the same as
the one described in the previous section.
[0536] The buckling capacity flow N is expressed by:
N y c = k c ( .pi. L x ) 2 .OMEGA. 2 .OMEGA. 1 Equation 6 - 23
##EQU00101##
[0537] with
k c = h ( .alpha. _ ) + q .beta. ##EQU00102## .alpha. _ = L y L x
.OMEGA. 1 .OMEGA. 2 4 ##EQU00102.2## .beta. = .OMEGA. 3 2 .OMEGA. 2
.OMEGA. 1 ##EQU00102.3## h ( .alpha. _ ) = { ( 1 .alpha. _ ) 2 +
.alpha. _ 2 , si .alpha. _ .ltoreq. 1 2 , sin on q = 2
##EQU00102.4##
[0538] 6.4.3 Shear Flow
[0539] The plate is subjected to a uniform shear flow:
-N.sub.xy.sup.0. Because of this:
N.sub.x.sup.0=N.sub.y.sup.0=p.sub.z=0. We note that in this
paragraph, the following formulae are only validated for Ly<Lx.
In the opposite case, some terms must be exchanged: L.sub.xL.sub.y,
and .OMEGA..sub.1.OMEGA..sub.2. The buckling flow capacity N.sup.0
is expressed by:
N xy c = k s ( .pi. L y ) 2 .OMEGA. 1 .OMEGA. 2 3 4 Equation 6 - 24
##EQU00103##
[0540] With k.sub.s obtained from the graph in FIG. 23 and the
table in FIG. 24 (for the case of simply supported edges), and in
FIG. 25 (case of four clamped edges) (source: S. G. Lekhnitskii:
Anisotropic Plates. Gordon and Breach).
[0541] The entry data of the table and the graph are defined
with:
.alpha. _ = L y L x .OMEGA. 1 .OMEGA. 2 4 ##EQU00104## .beta. =
.OMEGA. 3 2 .OMEGA. 2 .OMEGA. 1 ##EQU00104.2##
[0542] 6.4.4 Biaxial Compression Flow
[0543] The plate is subjected to combined loading: a uniform
longitudinal compression flow (according to the x axis) and a
uniform transversal compression flow (according to the y axis):
-N.sub.x.sub.comb.sup.0 and -N.sub.y.sub.comb.sup.0 Because of
this: N.sub.xy.sup.0=p.sub.z=0. We define .lamda. by:
N.sub.y.sub.comb.sup.0=.lamda.N.sub.x.sub.comb.sup.0. The general
differential equation is expressed
-N.sub.x.sub.comb.sup.0w.sub.,x.sub.2-N.sub.y.sub.comb.sup.0w.sub.,y.sub-
.2=.OMEGA..sub.1w.sub.,x.sub.4+.OMEGA..sub.2w.sub.,y.sub.4+.OMEGA..sub.3w.-
sub.,x.sub.2.sub.y.sub.2 Equation 6-25
[0544] Boundary Conditions Four Simply Supported Edges:
[0545] For x=0 and x=L.sub.x: w=0 and -M.sub.xx=0
[0546] For y=0 and y=L.sub.y: w=0 and M.sub.yy=0
[0547] Expression of Displacement:
[0548] The following expression of displacement w satisfies all the
boundary conditions detailed above:
w ( x , y ) = C mn sin ( m .pi. x L x ) sin ( n .pi. y L y ) ( m ,
n ) .di-elect cons. N 2 Equation 6 - 26 ##EQU00105##
[0549] Buckling flow capacity N.sub.x.sub.comb.sup.c,
N.sub.y.sub.comb.sup.c):
[0550] The previous expression for w must satisfy the general
differential equation (Equation 6-25), and we therefore obtain:
N x comb 0 ( m .pi. L x ) 2 + N y comb 0 ( n .pi. L y ) 2 = .OMEGA.
1 ( m .pi. L x ) 4 + .OMEGA. 2 ( n .pi. L y ) 4 + .OMEGA. 3 ( m
.pi. L x ) 2 ( n .pi. L y ) 2 ( m , n ) .di-elect cons. N 2
##EQU00106##
[0551] The expression of N.sub.x.sub.comb.sup.0 according to A must
satisfy:
N x comb 0 = .pi. 2 L y 2 m 2 + .lamda. L x 2 n 2 [ .OMEGA. 1 ( L y
L x ) 2 m 4 + .OMEGA. 2 ( L x L y ) 2 n 4 + .OMEGA. 3 m 2 n 2 ] ( m
, n ) .di-elect cons. N 2 ##EQU00107##
[0552] thus, the buckling flow capacities are obtained:
N x comb c = Min { .pi. 2 L y 2 m 2 + .lamda. L x 2 n 2 [ .OMEGA. 1
( L y L x ) 2 m 4 + .OMEGA. 2 ( L x L y ) 2 n 4 + .OMEGA. 3 m 2 n 2
] , ( m , n ) .di-elect cons. N 2 } N y comb c = .lamda. N x comb c
Equation 6 - 1 ##EQU00108##
[0553] Boundary Conditions: Four Clamped Edges:
[0554] For x=0 and x=L.sub.x:
w = 0 and .differential. w .differential. y = 0 ##EQU00109##
[0555] For y=0 and y=L.sub.y:
w = 0 and .differential. w .differential. x = 0 ##EQU00110##
[0556] Expression of Displacement:
[0557] The following expression of displacement w satisfies all the
boundary conditions detailed above:
w ( x , y ) = C mn [ 1 - cos ( m 2 .pi. x L x ) ] [ 1 - cos ( n 2
.pi. y L y ) ] ( m , n ) .di-elect cons. N 2 Equation 6 - 28
##EQU00111##
[0558] Buckling flow capacity (N.sub.x.sub.comb.sup.c,
N.sub.y.sub.comb.sup.c):
[0559] The expression of N.sub.x.sub.comb.sup.0 according to
.lamda. must satisfy:
N x comb 0 = 4 .pi. 2 L y 2 m 2 + .lamda. L x 2 n 2 [ .OMEGA. 1 ( L
y L x ) 2 m 4 + .OMEGA. 2 ( L x L y ) 2 n 4 + 1 3 .OMEGA. 3 m 2 n 2
] ( m , n ) .di-elect cons. N 2 ##EQU00112##
[0560] thus, the buckling flow capacities are obtained:
N x comb c = Min { 4 .pi. 2 L y 2 m 2 + .lamda. L x 2 n 2 [ .OMEGA.
1 ( L y L x ) 2 m 4 + .OMEGA. 2 ( L x L y ) 2 n 4 + 1 3 .OMEGA. 3 m
2 n 2 ] , ( m , n ) .di-elect cons. N 2 } N y comb c = .lamda. N x
comb c Equation 6 - 2 ##EQU00113##
[0561] 6.4.5 Longitudinal and Shear Compression Flow
[0562] The plate is subjected to combined loading: uniform
longitudinal (according to the axis X) and shear compression flow:
-N.sub.x.sub.comb.sup.0 and -N.sub.y.sub.comb.sup.0
[0563] Because of this: N.sub.y.sup.0=p.sub.z=0.
[0564] Interaction Equation:
[0565] The interaction equation for the combined flows of
longitudinal and shear compression is:
R.sub.x+R.sub.xy.sup.1.75=1 Equation 6-30
[0566] With
R x = N x comb c N x c ##EQU00114##
ratio of longitudinal compression flow
R xy = N xy comb c N xy c ##EQU00115##
ratio of shear flow
[0567] where N.sub.x.sup.c and N.sub.xy.sup.c are the buckling flow
capacities calculated above for a uniaxial loading.
[0568] 6.4.6 Transversal and Shear Compression Flow
[0569] The plate is subjected to a uniform transversal compression
flow (according to the y axis) and a shear flow:
-N.sub.y.sub.comb.sup.0 and -N.sub.xy.sub.comb.sup.0
[0570] Because of this: N.sub.x.sup.0=p.sub.z=0.
[0571] The interaction equation for the combined flows of
transversal and shear compression is:
R.sub.y+R.sub.xy.sup.1.75=1 Equation 6-31
[0572] With:
R y = N y comb c N y c ##EQU00116##
ratio of transversal compression flow
R xy = N xy comb c N xy c ##EQU00117##
ratio of shear flow
[0573] where N.sub.y.sup.c and N.sub.xy.sup.c are the buckling
capacity flows calculated above for a uniaxial loading.
[0574] 6.4.7 Biaxial and Shear Compression Flow
[0575] The plate is subjected to combined loadings: a uniform
longitudinal compression flow (according to the x axis) and a
uniform transversal compression flow (according to the y axis) as
well as a buckling flow: -N.sub.x.sub.comb.sup.0;
-N.sub.y.sub.comb.sup.0 and -N.sub.xy.sub.comb.sup.0.
[0576] Because of this: p.sub.z=0.
[0577] The interaction equation is obtained in two steps. Firstly,
we determine a reserve factor RF.sub.bi corresponding to the
biaxial compression flow:
RF bi = N x comb c N x comb 0 = N y comb c N y comb 0 Equation 6 -
32 ##EQU00118##
[0578] Then this value is used in an interaction equation for the
combined flows of biaxial and shear compression:
R.sub.bi+R.sub.xy.sup.1.75=1 Equation 6-33
[0579] With: [0580] R.sub.bi=RF.sub.bi.sup.-1 ratio of biaxial
compression flow
[0580] R xy = N xy comb 0 N xy c ##EQU00119##
ratio of shear flow
[0581] where N.sub.xy.sup.c is the buckling flow capacity
calculated in pure shear load.
[0582] The method according to the invention also includes an
iteration loop (see FIG. 7). This loop makes it possible to modify
the value of applied loads, or the dimensional values of panels
stiffened by triangular pockets in consideration, according to the
results of at least one of steps 3 to 6.
[0583] The method, such as has been described, can be implemented
at least partially in the form of a macro on a spreadsheet type
programme.
[0584] Such a programme used thus, for example, for entering
material and geometry data stored in a dedicated zone, as well as
various cases of considered loads and boundary conditions, and
supplying exit values for panel mass, reserve factor at ultimate
load concerning in particular triangular pockets, stiffeners and
general failure. These exit data thus highlight the cases of loads
or dimensioning which are incompatible with the desired reserve
factors.
ADVANTAGES OF THE INVENTION
[0585] We understand that the NASA process previously known, has
been substantially extended in the frame of the present invention
to take into account the particularities of the aeronautical
domain: [0586] Local capacity values for stiffeners (destruction,
lateral instability etc.) for compression according to direction X
or Y and shear load, [0587] Local capacity values of the triangular
skin for compression according to direction X or Y and shear load,
[0588] Plasticity correction, [0589] Preliminary mass calculation,
[0590] Calculation of general buckling for a compression according
to direction X or Y and a shear load.
[0591] The principal improvements are the calculation of stress
capacities for the different types of buckling and the calculations
of adapted reserve factors.
[0592] Extensions: Case of Loading [0593] double-compression (for
local and global buckling) [0594] Combined loading: compression and
shear load
[0595] Extensions: Improvements of the Parameters of the Method
[0596] No limitation on the material's Poisson coefficient [0597]
Variation of the grid angle (different by 60.degree.) [0598]
Plasticity where the structure stiffened by the triangular pockets
is considered as an equivalent stiffened panel [0599] Boundary
conditions (clamping or intermediate boundary conditions) on the
local or global buckling One of the most significant advantages of
the method of dimensioning according to the invention is the
possibility of installing panels stiffened by triangular pockets,
instead of and in place of the panels previously created with two
perpendicular families of stiffeners (longerons and ribs)
underneath, resulting, for equal mechanical resistance, in a mass
gain reaching 30% on some pieces.
VARIATIONS OF THE INVENTION
[0600] The scope of the present invention is not limited to details
of the types of embodiment considered above as an example, but
extends on the contrary to modifications to the scope of those
skilled in the art.
[0601] In the present description we have referred to isosceles
triangle base angles of between 45.degree. and 70.degree., which
correspond to current requirements for aeronautical structures. It
is however clear that a similar method can be implemented for all
isosceles angle values in panels stiffened by triangular
pockets.
* * * * *