Lateral stability analysis of frames for civilian
aircraft structures
Oscar Grossmann
Division of Solid Mechanics
Master Thesis
Department of Management and Engineering
LIU-IEI-TEK-A- -17/02959- -SE
Lateral stability analysis of frames for civilian
aircraft structures
Master Thesis in Structural Engineering
Department of Management and Engineering
Division of Solid Mechanics
Linköping University
by
Oscar Grossmann
LIU-IEI-TEK-A- -17/02959- -SESupervisors: Lars Johansson
IEI, Linköping University
Carolina Lara Silva
SAAB
Daniel Goodwin
SAAB
Examiner: Ulf Edlund
IEI, Linköping University
Upphovsrätt
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Abstract
The fuselage of a civilian airplane is a quite complex structure build from slender members and panels. Slender members are essential to minimize the weight of the airplane. These slender structures are at risk of buckling due to high compressive loads. The scope of this thesis is to evaluate the lateral stability of one of these structures, a curved beam later referred to as the frame. Three main factors that affect the lateral stability of the frame will be discussed: The effect of the surrounding geometry, the load conditions and the initial curvature of the frame. A literature study was carried out to find current information and methodolo-gies regarding such structures. The most promising techniques were later imple-mented to evaluate the lateral stability of the frame.
The results from the analysis indicate that the structures surrounding the frame are sufficiently stiff so they won’t deflect and rotate together with the frame when it buckles laterally. The whole cross section of the frame was assumed to be a rigid body in this thesis but the experience from the present work indicates that this might be wrong. The stress distribution in the frame was seen as a combination of two different load cases, plane bending and pure compression. By analyzing these load cases individually we found that for the pure compression case the critical buckling stress was lower than for the maximum stress level of the plane bending case. The effect of the initial curvature of the frame was evaluated for the plane bending load case. The initial curvature increased the resistance to the lateral buckling modes but had also local effects on the stability of the frame. The cross section of the frame is of the type of an open channel and if one were to take the initial curvature into consideration one would also have to make sure that the web doesn’t buckle locally due to increased radial compressive stress in the web.
It was found that the surrounding structures could be disregarded if one applies fixed boundary conditions to the fastener line of the frame where it is connected to the skin panel. Further evaluation of how the initial curvature affects the buckling mode is needed but the results look promising.
Acknowledgments
I would like to thank both of my supervisors at SAAB who have helped me along the progress of my work. I would also like to thank Ashok Kumar who have always been helpful regardless of a specific question or some open discussion.
Linköping, Month, Year Oscar Grossmann
Contents
1 Introduction 9
1.1 Lateral stability . . . 10
1.2 Aim of the thesis . . . 11
1.3 FE-model . . . 11
2 Theory 13 2.1 Coordinate system . . . 13
2.2 Degrees of freedom . . . 14
2.3 Center of gravity and centroid definition . . . 14
2.4 Shear center . . . 15 2.5 Torsion . . . 16 2.6 Warping . . . 18 2.7 Buckling . . . 19 2.8 Slenderness ratio . . . 20 2.9 Local buckling . . . 21 2.10 Global buckling . . . 21 2.10.1 Flexural buckling . . . 21 2.10.2 Torsional buckling . . . 22
2.10.3 Flexural torsional buckling . . . 23
2.10.4 Lateral torsional buckling . . . 24
2.10.5 Rotational and translational restraint of a beam with con-tinuous elastic supports . . . 26
2.11 Effective skin . . . 28
2.12 Stiffness calculations for the springs . . . 29
2.13 Radial stresses due to the initial curvature of the beam . . . 31
3 Method 33 3.0.1 Different load cases and constraints . . . 34
3.0.2 Determine the effect of the skin panel and the stringers on the buckling mode of the frame . . . 36
3.1 Effects of initial curvature of beam . . . 41
3.2 Full-scale linear model of the fuselage . . . 42 ix
4.1 Continuous elastic support . . . 46 4.2 Curved arc . . . 50
5 Conclusions 51
5.1 Future work . . . 52
Contents 1
List of Figures
1.1 Skin panel in grey, stringers in red and frames in green . . . 10 1.2 Example of how a beam may buckle laterally . . . 10 2.1 Location of the coordinate system . . . 13 2.2 Degrees of freedom for the beam, translation in the global directions
denotes as ui, rotations about the global axes denoted as ϕi and
warping axis denoted as ω. i=x,y,z . . . . 14 2.3 Geometrical properties of different cross sections, a) a doubly
sym-metrical I-beam and b) a mono-symsym-metrical open channel beam . 15 2.4 Pure torsion of a square beam taken from Lundh [5] page 324 . . . 16 2.5 Torsion of a beam which is free to warp. The warping angle denoted
as θw . . . 18
2.6 The axial compressive force, P, as a function of the displacement, w, of the beam. At the bifurcation point an increase in the displace-ment is possible without loading the beam further, also known as critical buckling load . . . 19 2.7 Local buckling occuring in the web and the flanges . . . 21 2.8 Axial loaded beam with a support placed at L/2 . . . . 22 2.9 Torsional buckling occuring for a beam with a double symmetrical
cruciform cross section [5] . . . 22 2.10 Rotation and translation of a beam subjected to uniform
compres-sion, due to a load P . . . 23 2.11 Rotation and translation of a beam subjected to a bending moment,
M . . . 25 2.12 Elastic rotation and translational support of an arbitrarily shaped
beam . . . 26 2.13 Substituting the whole panel with a width of sheet attached to each
stringer . . . 28 2.14 Cross section of the frame attached to a strip of skin . . . 29 2.15 Amount by which the stringer beam with effective skin is
com-pressed in the direction of the applied force . . . 29 2.16 Deflection of a stringer with effective skin . . . 30 2.17 Angle between the stringer with effective skin and the axis between
the supports when a unit moment was applied in the middle of the supports . . . 30 2.18 Radial stresses in the web due to tension of the outer flange and
compression in the inner flange . . . 31 3.1 Model of the frame from the global FE-model . . . 33 3.2 Stress distribution in the frame which had the highest peak stress . 34 3.3 Simply supported beam with fork supports . . . 35 3.4 Simply supported beam with fork supports . . . 38
3.5 Bending axis shown as the dashed line. For the beam in Figure a) the same formulas as for the critical moment of a double symmet-rical cross section may be applied. The calculations for the critical moment has to be modified for Figure b) . . . 39 3.6 Thin panel fastened to the frame . . . 40 3.7 Linear elastic model of a portion of the fuselage . . . 43 4.1 Critical buckling load for a continuous elastic supported beam with
only one spring that restrains the deflection in the x-direction. Unit of the critical buckling load in Newton and rigidity in the x-direction in N/mm . . . . 46 4.2 Critical buckling load as a function of the translational rigidity in
the y direction and the torsional modulus, kφ. Unit of the critical
buckling load in Newton, rigidity in the y-direction in N/mm and torsional modulus in N m/rad . . . . 47 4.3 Critical buckling load as a function of the rigidity of the support in
the y-direction for a continuous elastic supported beam with rigid supports kx and kφ. Unit of the critical buckling load in Newton
and rigidity in the y-direction in N/mm . . . . 48 4.4 Results from the full fuselage model . . . 49 4.5 Simplification of Figure 4.4b . . . 49 4.6 Different solutions to the same differential equations for a simply
supported curved arc compared with a straight beam. Unit of the critical buckling moment in N m . . . 50 4.7 Critical buckling load for a curved arc compared to a straight beam.
Contents 3
List of Tables
3.1 Solution matrix . . . 37 4.1 Results from of all the different tests, the numbers in the first
col-umn represents the different load cases. *Exceeds the compressive yield stress . . . 45 4.2 Calculated stiffness values of the springs for the model with
Nomenclature
The following list describes the most important symbols to be used within the body of the text
δx Deflection of the stringer in the direction of an applied compressive force
δy Deflection of the stringer in the direction of an applied force normal to the
longitudinal axis
γ Used for simplicity when calculating the critical bending moment for a curved simply supported beam
λ Slenderness ratio
φ Angle of twist around the longitudinal axis
σsk Varying stress distribution for the skin panel loaded in compression
σst Stress distribution in the stringers loaded in compression
θ Angle where a unit moment was applied to a stringer
θw Warping angle
A Area of cross section of the column
b Width of the flange
C Product of the shear modulus and the torsion constant
C1 Product of the Young’s modulus and the warping constant
Cw Warping constant
CG Center of gravity
E Young’s modulus
e Distance from the SC to the center of the web
G Shear modulus
h Length between the center lines of the flanges 5
hx Distance between the centroid and the point where the continuous elastic
supports are connected in the x-direction
hy Distance between the centroid and the point where the continuous elastic
supports are connected in the y-direction
I0 Polar moment of inertia about the centroid
Ix Area moment of inertia about the x-axis
Iy Area moment of inertia about the y-axis
J Torsion constant
kφ Rotational regidity
kx Stiffness of the spring that restrains the translational movement in the
x-direction
ky Stiffness of the spring that restrains the translational movement in the
y-direction
L Length of the column
M Applied bending moment
Mt Applied moment
Mz Distributed moment along the beam
Mcr Critical bending moment if loaded more the column will buckle
N Point where the continuous elastic supports are connected
P Applied compression force
Pi Stress in the inner flange for a curved beam in bending
Po Stress in the outer flange for a curved beam in bending
Pcr Critical compressive force if loaded more the column will buckle
R Radius of the column
Rg Radius of gyration
SC Shear center
tf Thickness of the flange
tw Thickness of the web
u Deflection of the shear center of a column in the x-direction loaded in compression or bending
Contents 7
v Deflection of the shear center of a column in the y-direction loaded in compression or bending
x0 Distance between the centroid and the shear center in the x-direction
Chapter 1
Introduction
The work described in the present report was done at the Swedish company Saab, which among other things builds military aircraft. There are many challenges when it comes to constructing an aircraft. The weight of an airplane is one of the biggest concerns and it is determined by the structure and components of the aircraft. Weight minimization is crucial, so structure has become more and more efficient, leading to very thin structures with complex geometry. These slender structures are susceptible to buckling. The present thesis is concerned with problems of elastic buckling of such structures. Elastic buckling is an instability condition which could lead to total failure of the airplane. In order to evaluate Saab’s present methodology, alternative ways of calculating these types of global buckling modes were investigated. Concerning the body of the airplane, also known as the fuselage, the objective is to absorb and redistribute loads efficiently. The structure can, in its simplest form be modelled as beam that is exposed mainly to torsion, bending and axial forces. The type of structure investigated here can be divided into three different parts that together supports the loads. The parts are:
• Frame • Skin panel • Stringers
Figure 1.1 shows a segment of the fuselage of an airplane which will be studied as a model problem in this thesis.
Accurate and efficient structural analysis methods to obtain slender structures are not only crucial in order to make an aircraft fly but will also lower its fuel con-sumption. The present work is mainly concerned with civilian aircraft structures, but SAAB as a company is active in the defence market. All activities in this regard is in accordance with Swedish law for weapon export. Since they work with military equipment they make sure their employers understand issues that arises from this. Their philosophy when it comes to weapon export is mainly from a defensive perspective and they work a lot with civilian protection.
(a) Section of the fuselage (b) Close-up of the geometry
Figure 1.1: Skin panel in grey, stringers in red and frames in green
1.1
Lateral stability
The lateral stability of the frame is the main concern of this thesis, see figure 1.2. The overall geometry of the frame can be seen as a circular beam with an open channel cross section. The geometric properties are discussed in more detail in the second chapter. The web and inner flange of the frame, as shown in figure 1.2a, are at risk of torsion and lateral deflection due to buckling. This type of buckling, that distorts the frame out of plane, is going to be evaluated in this thesis.
Skin Outer flange
Web
Inner flange
(a) Cross section of the frame mounted to the skin panel
(b) Example of how a straight beam (green wireframe) buckles laterally, the color is re-lated to the magnitude of the displacement
1.2 Aim of the thesis 11
Four aspects that each has a large impact on the buckling load has been identified for further study.
• How much the skin panel and stringers restrain the rotation and translation of the outer flange
• Geometrical properties of the cross section of the frame • Load distribution on the frame
• Initial curvature of the frame
1.2
Aim of the thesis
The work for this thesis has been divided into two parts. The first part is to investigate how different load distributions on the frame will affect the critical buckling load. The effect of the skin panel and stringers will also be taken into consideration in the first part. The theory of continuous elastic supports will be implemented. The second part is to evaluate whether or not an initial curvature of the frame affects the critical buckling mode when compared to a straight beam.
1.3
FE-model
Since the available computational power has increased over the last decade, more complex FE-models can be set up and solved within reasonable time. Local models will be set up to correlate the results of the analytic calculations. A linear buckling analysis model will also be set up for a small portion of the fuselage structure. The program HyperMesh, created by Altair, will be used. Optistruct is the solver for the FEM calculation, also made by Altair. The different load cases will be explained later in detail. The investigation will contain a number of steps:
• Understand the instability behaviour of the fuselage structure and the buck-ling theory
• Model the fuselage and set up the FEM model
Chapter 2
Theory
2.1
Coordinate system
A right handed Cartesian coordinate system will be used to define the orientation of the beam. The z axis is taken to lie in the direction along the beam, the y and
x axes are chosen along the principal axes of the cross section, see figure 2.1. The
origin of the coordinate system is denoted OP and lies along the centroid axis, see section 2.3 regarding the definition of the centroid.
OP
X Y
Z
Figure 2.1: Location of the coordinate system
The area moment of inertia is larger about the major axis x, than the minor axis
y. Major and minor axes are cross section properties and denotes the axes that
show the largest and also smallest resistance to bending, respectively. 13
2.2
Degrees of freedom
A section of the open channel beam is shown in figure 2.2. A three-dimensional beam element in a finite element code usually has two nodes that have six degrees of freedom each. These are translations in the direction of the x-, y- and z-axes and rotations about each axis. A beam element where warping of the cross section occurs has an additional degree of freedom denoted ω in figure 2.2. When warping,
ω, occurs the upper and lower flanges rotate in opposite directions, while φy is a
rotation about the y-axis of the end of the beam as a whole. Warping occurs due to a torsion moment around the z-axis. See section 2.6 for more information regarding warping. uz ux uy ϕz ϕy ϕx ω uz ux uy ϕz ϕy ϕx ω
Figure 2.2: Degrees of freedom for the beam, translation in the global directions denotes as ui, rotations about the global axes denoted as ϕi and warping axis
denoted as ω. i=x,y,z
2.3
Center of gravity and centroid definition
For a linear analysis the density of the material is considered constant so that the centroid coincides with the center of gravity. The definition of the center of gravity is the point where all the static moments of an arbitrary shaped body adds up to zero, [5]. The location of the center of gravity, denoted CG, for two different types of cross sections are shown in figure 2.3. The first cross section is a doubly symmetrical I-beam and the second cross section corresponds to an open channel
2.4 Shear center 15
beam. The thickness of the flanges are equal in both cases.
y CG SC x tw tf b h a) y x tw tf SC CG e h b b)
Figure 2.3: Geometrical properties of different cross sections, a) a doubly symmet-rical I-beam and b) a mono-symmetsymmet-rical open channel beam
Where e = Distance from the SC to the center of the web
h = Length between the center lines of the flanges
b = Width of the flange
tf = Thickness of the flange
tw = Thickness of the web
2.4
Shear center
The shear center of a cross section is the point where the line of action of the shear force must pass to get plane bending of the beam. The position of the shear center, denoted as SC, can be seen in figure 2.3 for a double symmetrical I-beam and a mono-symmetrical open channel beam. For a double symmetrical I-beam the shear center coincides with the centroid. For a mono-symmetrical cross section the shear center lies on the axis of symmetry, [5]. The distance from the shear center to the center of the web, denoted e in figure 2.3, can be calculated using equation 2.1,[9]. e = 3b 2t f 6btf+ htw (2.1)
In the case where the beam is loaded at a point that is not the shear center, torsion of the beam will occur in addition to the bending.
2.5
Torsion
For pure torsion conditions no bending of a beam occurs only torsion around an axis which lies still in its plane. The torsion axis lies on the torsion center of the cross section. For double symmetrical cross sections the torsion center coincides with the CG and SC. In the case of mono-symmetrical cross sections such as the open channel section the torsion center coincides with the SC and both lie on the axis of symmetry [5]. Figure 2.4 shows pure torsion of a rectangular beam.
Figure 2.4: Pure torsion of a square beam taken from Lundh [5] page 324 Pure torsion occurs when a moment is applied about the normal axis of the cross section passing by the torsion center and the beam is free to warp. The angle of twist, ϕ, per unit length due to a torsion moment is given by equation 2.2 according to the de Saint Venant torsion theory.
dϕ dz =
Mt
C (2.2)
Here, Mtis the applied moment and C is the torsional rigidity. C is the product
of the shear modulus, G, and the torsion constant J . For a thin walled open cross section consisting of different rectangular shapes the torsion constant is calculated by equation 2.3 [9].
J = 1
3 X
mit3i (2.3)
Where J = Torsion constant
m = Length of the center line of the cross section
t = Thickness of the cross section
2.5 Torsion 17
This method of obtaining the torsion constant from the length of each rectangular cross section is a conservative estimate according to the Canadian Institute of Steel Construction [7]. Another way of calculating the torsion constant is to take the length from the intersection point between the center line of the web and the flange to the end of the flange instead. Fillets of the cross section are not taken into consideration.
The de Saint Venant theory of torsion is also known as uniform torsion and is only valid for pure torsion conditions. In reality, this is rarely the case. Warping can, for example, be restrained by supports and braces of the beam, by the end attachments of the beam and how the torsion moment is applied. For open channel cross sections the stiffness is increased due to warping constraints.
In 1940 the Russian mathematician V.Z Vlasov developed a torsion theory that also covers warping restraints.The theory is also known as non-uniform torsion because the angle of twist of the cross section, ϕ, is not constant along the z-axis. The angle of twist along the z-axis can be obtained from the following differential equation [3]. ECw d4ϕ dz4 − GJ d2ϕ dz2 = Mz (2.4)
Where Cw = Warping constant
GJ = Torsion constant
Mz = Distributed moment along the beam
Where warping is restrained, a bi-moment is needed to keep the flange that is restrained from rotating. Let MF lange, from equation 2.5, be a moment that
coun-teracts the stress distribution in the flanges due to the torsion of the beam. The bi-moment, B, is obtained by multiplying the MF lange and the distance between
the flanges, equation 2.5. The second equality was given by [3].
B = Mf langea = −ECw
d2ϕ
dz2 (2.5)
Where MF lange= Moment due to axial stresses in each of the
flanges
a = Distance between the flanges
The moment, Mt, that is applied to the beam can be written as equation 2.6, [9].
Mt= GJ dϕ dz + dB dz = GJ dϕ dz − ECw d3ϕ dz3 (2.6)
In [3] it is shown that Saint Venant’s torsion theory is conservative in the sense that a structure that actually have restraints to warping is stiffer than what is assumed by Saint Venant. The next section, 2.6, explains warping and how to calculate the warping constant, Cw.
2.6
Warping
As mentioned before, when a torsion moment is applied about the z-axis of the beam, the cross section is twisted around its torsion center TC which coincides with the SC for mono-symmetrical cross sections. In addition to the twisting of the cross section around TC it may also rotate around the y- and/or the x-axis, this type of rotation is called warping see figure 2.5.
y x z Mt Mt θw
Figure 2.5: Torsion of a beam which is free to warp. The warping angle denoted as θw
Torsion of the beam leads to a stress distribution in the flanges which is the reason for warping. The warping constant, Cw, describes the flanges resistance to
warping. Like the torsion constant the warping constant can also be calculated in different ways depending on the geometrical properties of the cross section. Equation 2.7 was derived by Timoshenko and Gere [9] for mono-symmetrical open channel cross sections having equally thick flanges.
Cw= tfb3h2 12 3btf+ 2htw 6btf+ htw (2.7)
Where Cw = Warping constant
h = Length between the center lines of the flanges
b = Width of the flanges
tf = Thickness of the flanges
tw = Thickness of the web
Technically the whole cross section will warp but the amount of warp for the web may be neglected compared to that of the flanges in most cases.
2.7 Buckling 19
2.7
Buckling
The instability phenomenon called buckling is a bifurcation of equilibrium states. There are three conditions for the validity of the theory used in the present work.
• Elastic material
• Small deflections due to the applied load • Perfect structures, ideal structural elements
Slender beams are at risk of failing when subjected to a compressive stress less than the ultimate compressive stress that the material can withstand because of sudden onset of deformation in the direction transverse to the load. The deformation of the structure leads to changes to the equilibrium. This type of failing mode is referred to as buckling.
P
w
Bifurca+on point
Figure 2.6: The axial compressive force, P, as a function of the displacement, w, of the beam. At the bifurcation point an increase in the displacement is possible without loading the beam further, also known as critical buckling load
The FEM-solver calculates the critical buckling load by solving an eigenvalue prob-lem. In order to set up the eigenvalue problem, a linear static analysis is needed first to obtain the stiffness matrix K. In the linear static analysis a unit load is applied to the beam to obtain the stresses needed to determine the stiffness matrix. The shortening of the beam is neglected.
KD ref =F ref (2.8)
Where K = Stiffness matrix D ref = Displacement vector F ref = Applied unit load vector
The finite element equilibrium equation including effects of a displaced geometry is written:
K − λKgeo
where λ is a multiplier relating the membrane stresses to the reference state at whichKgeo was calculated. Simply put, the deformation will change the moment
arms of membrane forces. In a linearized approach this effect will appear as stiffness contributions depending on the state of deformation and therefore on the state of stress. To avoid having to deal with such a contribution depending on all the nodal forces in the entire FEM-model, a matrixKgeo is calculated such that
λKgeo is the apparent change in stiffness from deformation when the loads are
changed by a multiplier λ from a reference state. The geometric stiffness reduces the structural stiffness when the displacement increases. Buckling occurs when multiple equilibrium deformations exist at the same load so that:
K − λKgeo D ref + δD = λF ref (2.10) Subtracting equation 2.9 and 2.10:
K − λKgeo δD =0 (2.11)
Buckling values of λ are thus found from a generalized eigenvalue problem:
det K − λKgeo = 0 (2.12)
A necessary and sufficient condition for non-trivial solutions is that the determi-nant of the matrix in Equation 2.12 is zero, and that led to a cubic equation. The smallest root to the problem is referred to the bifurcation point and multiplying the reference load with the root yields the critical buckling load.
Similar behavior is found for a beam that is subjected to plane bending condition. When the applied moment reaches a critical value, Mcr, the beam will be on the
border between a stable and an unstable equilibrium state. If the moment is not increased beyond this critical value the beam wont buckle but the slightest increase will make it buckle and the beam will then be in an unstable equilibrium state.
2.8
Slenderness ratio
The slenderness ratio, λ, can be visualized as the tendency of a column to buckle. It is defined by the following equation 2.14.
λ = l Rg
(2.13) Where l = Length of the column
Rg = Radius of gyration
The radius of gyration, Rg, is calculated as follows.
Rg=
r
I
2.9 Local buckling 21
Where I = Area moment of inertia of the column
A = Area of the column
2.9
Local buckling
In columns built from slender parts subjected to a compression load the web and the flanges might buckle locally. Local buckling affects the load carrying ability of the column due to the loss of stiffness in the locally buckled elements. The flanges tends to buckle before the web since they have a free edge [8]. This type of buckling won’t affect the longitudinal axis.
Figure 2.7: Local buckling occuring in the web and the flanges
2.10
Global buckling
Global buckling affects the beam globally. If a beam buckles globally the whole structure will fail. Depending on the loading and the restraints of the beam, the buckling mode can be subdivided into parts that are failing in the first mode of buckling. Take for example the beam in Figure 2.8, an axial loaded beam with a support placed at L/2. The critical buckling load can be determined as the second buckling mode of the whole beam. Another way to calculate the critical buckling mode would be to divide the beam into two parts at x = L/2 and to calculate the first buckling mode of each part separately.
2.10.1
Flexural buckling
Flexural buckling, also know as Euler buckling, occurs when a column subjected to a compression load buckles without twisting. If the beam is unbraced it will buckle about the weakest axis. Bracing of the beam will affect the buckling length
P P
L
Figure 2.8: Axial loaded beam with a support placed at L/2
of the beam and also the shape of the buckling mode. See the example explained in Section 2.10.
2.10.2
Torsional buckling
A thin-walled beam with low torsional stiffness may twist around its longitudinal axis without bending under a compressional load. This type of instability is called torsional buckling and the longitudinal axis remains straight as no translational motion occurs, see figure 2.9.
Figure 2.9: Torsional buckling occuring for a beam with a double symmetrical cruciform cross section [5]
2.10 Global buckling 23
2.10.3
Flexural torsional buckling
An open channel beam subjected to uniform compression, normally fails due to flexural torsional buckling, a combination of bending and torsion. The cross section undergoes translation and rotation.
L SC CG x0 u v x y P P φ
Figure 2.10: Rotation and translation of a beam subjected to uniform compression, due to a load P
The translation is defined as the deflection of the shear center denoted u and v in the x and y directions, respectively. The rotation about the shear center, φ, rotates the centroid around the longitudinal axis of the beam, see figure 2.10. For small φ the final deflection of the centroid is:
x = u + y0φ
y = v − x0φ
y0 is zero for a beam with a open channel cross section, as shown in figure 2.10,
since there is no distance between the shear center and the centroid in the y direction of the undeformed beam.
A system of three differential equations may be set up in order to determine the critical buckling load of the structure. For the case where the centroid coincides with the shear center, x0 = y0 = 0, all equations 2.15, 2.16 and 2.17 below can
be treated separately. In this case the first two differential equations concern the deflection of the centroid in the x and y direction due to the bending moment on a small element dz. The third differential equation concerns the rotation of the centroid due to the uniform torsion of the beam. For more complex geometries the critical buckling load is dependent on both the bending and the twisting of the beam, see equations 2.15, 2.16 and 2.17, taken from ref [9] pages 230-232.
EIy d2u dz2 + P (u + y0φ) = 0 (2.15) EIx d2v dz2 + P (v − x0φ) = 0 (2.16) ECw d4φ dz4 − GJ − PI0 A d2φ dz2 − P x0 d2v dz2 + P y0 d2u dz2 = 0 (2.17)
Where E = Young’s modulus
G = Shear modulus
Ix = Area moment of inertia about the x axis
Iy = Area moment of inertia about the y axis
P = Compression force on the structure
Cw = Warping constant
J = Torsion constant
I0 = Polar moment of inertia about the centroid
A = Area of the cross section
These equations are derived for an arbitrary shaped cross section. Each equation contains the angle φ which indicates that in a general case, bending and torsion of the beam occurs simultaneously. With a channel cross sectioned beam y0 = 0,
which means that Equation 2.15 may be treated separately. In order to solve this system of differential equation an ansatz that corresponds to the boundary conditions is needed. Further information regarding the ansatz and how to solve the system will be discussed in chapter 3.
2.10.4
Lateral torsional buckling
A beam that is exposed to plane bending can under certain loads and constraints fail due to lateral torsional buckling. Traditional Euler buckling fails due to axial loading and not bending loading. When an open channel beam or an I-beam is exposed to bending loading, one flange will be in compression and the other flange in tension. The flange that is in compression will buckle laterally and try to push the beam sideways. The flange in tension tries to pull the beam back into its original position. This results in a combination of rotation and bending of the beam, see figure 2.11.
In the treatment of Timoshenko and Gere, ref [9], the initial curved-shape deflec-tion of the axis of the beam due to the applied bending moment was considered to be very small. When setting up equilibrium equations, the axis was assumed to be straight as for the case of flexural torsional buckling. Therefore the expression for the deflection of the centroid was written as before:
x = u + (y0)φ
2.10 Global buckling 25 L SC CG x0 u v x y M M φ
Figure 2.11: Rotation and translation of a beam subjected to a bending moment, M
When the applied bending moment acts along one of the two principal axes, two differential equations may be set up in order to find the critical buckling moment. In our case, the moment was applied in the x-direction yielding the equations 3.3 and 3.4. Had the moment been applied about the y-axis instead, the equations would look slightly different.
EIy d4u dz4 − M d2φ dz2 = 0 (2.18) ECw d4φ dz4 − (GJ − M β1) d2φ dz2 − M d2u dz2 = 0 (2.19)
where β1 is a function of x and y:
β1= 1 Ix Z A y3dA + Z A x2ydA − 2y0 (2.20)
If the moment is applied about the axis of symmetry of the cross section, β1equals
zero. Equations 2.18 and 2.19 can then be written as:
EIy d4u dz4 − M d2φ dz2 = 0 (2.21) ECw d4φ dz4 − GJ d2φ dz2 − M d2u dz2 = 0 (2.22)
A general form of the differential equation derived for an arbitrary shaped cross section and regardless of the direction in which the moment is applied can be found in [9] page 244-247.
2.10.5
Rotational and translational restraint of a beam with
continuous elastic supports
In the present work, the stringer are modelled as elastic supports of the outer flanges of the frame, providing torsional restraint about the longitudinal axis of the beam and translational stiffness in the x and y direction. Figure 2.12 shows an arbitrarily shaped beam with elastic supports fastened at a point N. N is, in our case, the point of the fastener between the skin and the outer flange.
x0 y0 hy N kφ kx ky hx SC CG x y
Figure 2.12: Elastic rotation and translational support of an arbitrarily shaped beam
Where kx = Stiffness of the elastic support in the x direction
ky = Stiffness of the elastic support in the y direction
kz = Torsional modulus of the elastic support
N = Point on the beam where it is supported
CG = Centroid
SC = Shear center
x0 = Distance from the SC to the CG in the x direction
y0 = Distance from the SC to the CG in the y direction
hx = Distance from the CG to the point of the beam
where it is supported in x direction
hy = Distance from the CG to the point of the beam
2.10 Global buckling 27
Vlasov introduced three differential equations that described the buckling for a beam with continious elastic supports. By exploring the intensities of the dis-tributed loads in a beam he could implement the effects of the reaction forces of the springs, Equations 2.23, 2.24 and 2.25. [9]
EIy d4u dz4 + P d2u dz2 + y0 d2φ dz2 + kx[u + (y0− hy)φ] = 0 (2.23) EIx d4v dz4+ P d2v dz2 − x0 d2φ dz2 + ky[v − (x0− hx)φ] = 0 (2.24) ECw d4φ dz4− GJ − PI0 A d2φ dz2− P x0 d2v dz2− y0 d2u dz2 + kx[u + (y0− hy)φ](y0− hy) − ky[v − (x0− hx)φ](x0− hx) + kφφ = 0 (2.25)
When the stiffness values of the springs are equal to zero, these equations are reduced to the same equations as for flexural torsional buckling, in Section 2.10.3. The stiffnesses of the springs depend on the skin and the stringers. The contribu-tion of the skin will be explained in the following Seccontribu-tion 2.11.
2.11
Effective skin
When a section of the skin together with its attached stringers is loaded with a compressive force, the skin buckles at quite low stress levels. The skin can’t buckle where it is connected to the stringers and therefore deforms together with the stringers. This results in a varying stress distribution in the panel see Figure 2.13a. It is difficult to handle a variation, so a simplification was introduced where the skin panels are substituted by a strip of skin attached to each stringer. The uniform compression in these skin strips equals the total varying stress distribution of the whole skin panel. [1] and [6]. There are different ways of calculating the effective width of the skin stripes, one approximation is using 30 times the thickness of the skin.
𝜎𝑠𝑡
𝜎𝑠𝑘
(a) Varying stress distribution in the skin shown in red, uniform compression in the stringers shown in black
𝜎𝑠𝑡
(b) Effective skin and stringers carrying a uniform stress equal to the total stress distribution of the panel
Figure 2.13: Substituting the whole panel with a width of sheet attached to each stringer
One could therefore substitute the skin panel and attach an effective skin width to both the stringers and the frames. This will affect the geometrical properties of both components but also the buckling equations for a beam loaded for plane bending. The shear center is now displaced perpendicularly to the bending axis relative to the centroid. The bending axis is still in the x-direction, the horizontal direction in Figure 2.14. The buckling equations have to be modified. The torsional buckling equation has to be modified to include extra terms. These terms affect the torsional buckling mode and occur when calculating the intensities of the distributed torque along the shear center axis of the beam due to the applied torque. In the book Theory of elastic stability by Timoshenko and Gere the terms are denoted β1and β2 on page 245 [9].
2.12 Stiffness calculations for the springs 29
Centroid Shear center x
y
Figure 2.14: Cross section of the frame attached to a strip of skin
2.12
Stiffness calculations for the springs
As mentioned in the previous section 2.10.5, a small portion of the skin with a given width may be attached to the stringers to represent the skin in a post buckled state. The cross section of a stringer is an asymmetrical T-shaped beam, which means that the web is not taken to lie in the middle of the flange. The size and shape of the stringer is used to calculate the total area and the area moments of inertia. When calculating the stiffness of the springs one can use Hooke’s law and elementary beam theory. The translational restraint in the x-direction can be obtained by applying a unit compression force to a simply supported stringer see Figure 2.15. The supports represents two frames with a simply supported stringer.
𝛿𝑥=
𝑃𝐿 𝐸𝐴 𝑃
Figure 2.15: Amount by which the stringer beam with effective skin is compressed in the direction of the applied force
Where δx = Deflection of the stringer in the direction of the applied force
P = Unit compressive force
L = Length of the column
E = Young’s modulus
ky was calculated in the same way but instead of a unit compression force a unit
force was applied perpendicular to the beam see Figure 2.16.
𝑃
Figure 2.16: Deflection of a stringer with effective skin
Where δy = Deflection of the stringer in the direction of the applied force
P = Unit force
L = Length of the column
E = Young’s modulus
I = Area moment of inertia of the stringer plus the effective skin For the rotational stiffness a unit moment was applied in the middle of the beam and the angle, θ, was measured, see Figure 2.17. If the moment is applied in the middle between the support the angle at the support is the same as θ.
𝜃 =𝑀 6 𝐿 4𝐸𝐼− 3𝑥2 𝐿𝐸𝐼 𝑀 𝜃 x
Figure 2.17: Angle between the stringer with effective skin and the axis between the supports when a unit moment was applied in the middle of the supports
Where M = Unit couple
x = Point where the moment is applied, here x = L/2
θ = Angle where the moment is applied
L = Length of the column
E = Young’s modulus
I = Area moment of inertia of the stringer plus the effective skin The stiffness was then obtained by Hooke’s law, the translational stiffness in the x-and y-direction was calculated by F = kδ x-and the rotational stiffness by M = kθ. The stiffness of each spring was then distributed over a frame section, therefore the stiffness was divided by the width between two frames.
2.13 Radial stresses due to the initial curvature of the beam 31
2.13
Radial stresses due to the initial curvature
of the beam
When a curved beam is loaded with a bending moment radial stresses in the web will occur. Depending on the direction of the moment there will either be tension or compression in the web. If the inner flange is in compression and the the outer flange is in tension the flanges want to compress the web and it is destabilized see figure 2.18. A curved beam is more resilient to lateral torsional buckling but caution is required regarding the local effect of the radial stresses. This thesis will not investigate this local instability but it is worth mentioning.
Outer Flange Inner Flange Web 𝑃0 𝑃𝑖 Outer Flange Inner Flange 𝑃0 𝑃𝑖 Web
Figure 2.18: Radial stresses in the web due to tension of the outer flange and compression in the inner flange
Chapter 3
Method
An FE-model of the whole airplane was developed by Saab to obtain the stress distribution in the aircraft structure. The stress level in the frames varies depend-ing on where the frame is located in the fuselage. For a specific frame the stress distribution varies along circumference of the fuselage. The frames were modeled using 2D shell elements, each connected to two 1D rod elements, see figure 3.1. The shell elements represent the web of the frame and the rod elements the inner and outer flange. The necessary computational power was reduced by simulating the frame using shell and rod elements elements and not modeling them using 3D elements. This reduced the number of finite elements and consequently the number of equations to solve. Shell elements are useful to use for structures where one dimension is much smaller than the other two, for example slabs and walls. Each element had a material property assigned to it. In addition to the material property the shell elements were also assigned a specific thickness. The rod ele-ments had the cross section properties of the flanges assigned to them, i.e. area and moments of inertia.
Figure 3.1: Model of the frame from the global FE-model
The stress distributions for all frames were obtained from the full-scale FE-model of the airplane provided by SAAB. The frame with the highest compression-stress level was found and the stress distribution of this cross section was determined. As mentioned before, the stress distribution varies throughout a certain frame. At the point where the stress reached its peak, both the inner and outer flanges were found to be in compression. The inner flange had approximately three times higher compression-stress than the outer flange, facing the skin. The stress distribution of the cross section at this point can be seen in figure 3.2, where the initial curvature of the frame is not included.
Compression
Outer flange
Inner flange
Figure 3.2: Stress distribution in the frame which had the highest peak stress
The stress distribution in the frame could be seen as a combination of two different load conditions, uniform compression and plane bending.
3.0.1
Different load cases and constraints
Since the frame was subjected to both pure compression and plane bending, we will now extend the original work in this thesis by separately evaluate these load conditions further, in order to see how each load case affected the buckling in-stability. The model shown in figure 1.1a needed to be broken down into simple models in order to understand which parameters affect the lateral stability of the frame the most and which assumptions could be used in the analytic model in order to still obtain an accurate result. Local models were set up with increasing complexity to gain an understanding of how the skin panel and the stringers affect the instability of the frame. In these models the beam was assumed to be a straight beam. The effects of the initial curvature were taken into consideration later on. In order for the model to be in equilibrium the end points of the beam had to be restrained somehow. For any single span beam with end supports the translation in the x- and y-direction is constrained regardless of support. The translation in the z-direction, along the beam axis, had to be restrained at at least one of the end supports. If the z-direction is locked at one of the end supports the beam is seen as simply supported and if it is restrained at both end supports the beam is fixed supported. The rotation about the longitudinal axis, z-axis, had to be restrained
35
to prevent the beam from rotating about it’s own axis. The remaining degrees of freedom of the end supports are:
1. Warping
2. Rotation about the x-axis 3. Rotation about the y-axis
The frame is a continuous beam with a varying stress distribution where only a part of the frame buckles under a certain load. In order to break down the system to local models, only the part of the continuous frame that actual buckles was of interest. The lengths of the local models were set to the same length as the section of the frame that buckles. Which boundary condition for a section of the frame most accurately corresponds to a continuous beam was uncertain. That was the reason that all models were taken to be simply supported since that is the most conservative way of calculating the buckling phenomenon. The supports used to restrain the beam are called fork supports and are shown in figure 3.3. The degrees of freedom of the end supports for a simply supported beam were rotation about the x- and y-axis, flanges were free to warp and for one of the end supports translation along the z-axis was possible.
y
x
z
Figure 3.3: Simply supported beam with fork supports
The approach for evaluating the system was divided into three major parts. • Determine the effect of the skin panel and the stringers on the buckling mode
of the frame
• Determine the effect of the initial curvature of the beam on the critical buckling load
3.0.2
Determine the effect of the skin panel and the stringers
on the buckling mode of the frame
The fuselage is a complex structure, so it was hard to know how the stringers and skin panel would effect the buckling mode of the frame. Therefore, the structure was broken down into smaller models to be able to correlate the analytical result with the FE calculations. Three different models were set up with increasing complexity.
1. Model 1
• Simply supported beam with fork supports • No additional restraints of the beam 2. Model 2
• Simply supported beam with fork supports
• One flange restrained in the x-direction, could be seen as an imaginary thin panel fastened to that flange
• The panel would not add any stiffness in the normal direction of the flange nor any rotational stiffness
• An initial attempt to see how the skin would affect the buckling mode 3. Model 3
• Simply supported beam with fork supports
• One flange restrained by adding continuous elastic supports
• Translational motion in the x- and y-direction was restrained and also the rotational motion around the z-axis
• The stiffnesses of the elastic springs, depend on the material and geo-metrical properties of the skin panel and stringers
The first model was a simply supported beam that did not contain any type of restrains of the beam other than the boundary conditions. This way one could see how the cross section properties affect the buckling mode of the frame.
The second model restrains the translational motion of one of the flanges of the beam in the x-direction in addition to the fork supports. This type of constraint could be visualized as an imaginary thin panel fixed to the frame. The panel would not add any stiffness in the normal direction of the flange nor any rotational stiffness. Adding this restraint was an initial attempt to see how the skin would affect the buckling mode.
In the third model the skin and stringers were simulated by three elastic supports that restrain the rotation along the beam and the translation in two directions in addition to the fork supports. The stiffnesses of the elastic springs, depending on
37
the material and geometrical properties of the panel and stringers were calculated, see Section 2.12. These springs would restrain the beam more than in the second model, the frame would no longer be able to rotate freely and translate in the y-direction. Further details regarding the models is found in the following sections. The beams in these sections were all straight i.e had no initial curvatures. For each model both uniform compression and plane bending load cases were evaluated. The determination of the buckling length was based on the stress distribution in the frame obtained from the global FE model of the whole airplane.
Solution matrix
A solution matrix was setup to plan the progress of the work and also to keep track of the different results that were obtained, see Table 3.1.
Compression Plane bending
Analytic result FEA result Analytic result FEA result
Straight beam Straight beam Straight beam Curved beam Straight beam Curved beam
1 x x x x x x
2 x x x x
3 x x x x
Table 3.1: Solution matrix
Model 1 = Simply supported beam
2 = Thin panel fastened to the frame 3 = Continuous elastic support
For the simply supported beam, the critical buckling load was calculated ana-lytically and the result was correlated with a local FE-model of the structure. This was to see that the loads and boundary conditions were set up properly in the FE-model. For the more complex models, 2 and 3, the FE-results were only correlated to the analytic result from the pure compression case. In this way the additional constraints in the FE-model could be verified. The results for the plane bending could then be obtained by relying on the FE-model. Four cells in Table 3.1 were colored red since no analytic results for these models were obtained. The following sections will discuss each of these steps in more detail. The equations were solved by programming the differential equations in PTC Mathcad.
Uniform compression for a simply supported straight beam
A simple model of a straight beam was set up in order to see how the cross sectional properties affect the buckling mode see figure 3.4. The beam was loaded with a uniform compression force at each end, see section 2.10.3 for the theory behind flexural torsional buckling.
y
x
z
Figure 3.4: Simply supported beam with fork supports
The boundary conditions for a simply supported beam were satisfied with the following solutions to the three different differential equations 2.15, 2.16 and 2.17.
u = A1sin πz n l v = A2sin πz n l φ = A3sin πz n l (3.1) Where Ai = Constant
n = Modal shape factor
l = Length of the beam
Inserting the trial solutions from equation 3.1 into equations 2.15, 2.16 and 2.17, the following system of equations was found.
P − EIyπ 2 l2 0 P y0 0 P − EIxπ 2 l2 −P x0 P y0 −P x0 PIA0 − ECwπ 2 l2 − GJ A1 A2 A3 = 0 0 0 (3.2)
A necessary and sufficient condition for non-trivial solutions is that the determi-nant of the 3x3 matrix in equation 3.2 is zero, and that led to a cubic equation for the buckling load.
P − EIyπ 2 l2 0 P y0 0 P − EIxπ 2 l2 −P x0 P y0 −P x0 PIA0 − ECwπ 2 l2 − GJ = 0
As mentioned in Section 2.10.3, the differential equation for the bending of the beam in the x-direction could be treated separately, since y0= 0. The first buckling
mode that occurred was a pure flexural buckling mode in the x-direction, with no rotation of the beam at all.
39
Plane bending for a simply supported straight beam
Whether or not the equations that describe the lateral torsional buckling phenom-ena of a double-symmetric cross section also can be applied to a mono-symmetric cross section depends on which direction the bending moment or bending force is applied. If the shear center is not displaced perpendicularly to the bending axis relative to the centroid, the same formulas for double-symmetrical cross sections can be applied see Figure 3.5, [4] and [9]. For this assumption to be valid the bending force has to be applied in the shear center so that no extra torque is ap-plied. The local models did not take the effective skin into consideration because this effect was found close to the end of the project. If one were to model the frame together with the effective skin the analytic equation would have to be modified.
Centroid
Shear center Centroid Shear center
a) Both the shear center and centroid lies on the bending axis
b) Both the shear center and centroid displaced above the bending axis
Figure 3.5: Bending axis shown as the dashed line. For the beam in Figure a) the same formulas as for the critical moment of a double symmetrical cross section may be applied. The calculations for the critical moment has to be modified for Figure b)
The governing equations for the buckling state as discussed in Section 2.10.4 are.
EIy d4u dz4 − M d2φ dz2 = 0 (3.3) ECw d4φ dz4 − GJ d2φ dz2 − M d2u dz2 = 0 (3.4)
The solutions to the two differential equations above, which satisfied the boundary conditions, were the same as before:
u = A1sin
πz n
l φ = A2sin πz n
l (3.5)
Inserting the trial solutions into the differential equations yielded:
EIyπ 2 l2 M M ECwπ 2 l2 + GJ ! A1 A2 =0 0 (3.6)
A quadratic equation for the critical bending moment was obtained by setting the determinant of the 2 × 2 matrix from equation 3.6 equal to zero resulting in
Mcr= π L r (EIyGJ 1 +π 2EC w GJ L2 (3.7)
Equation 3.7 has two solutions since there are two roots to the characteristic polynomial of the differential equations. The solutions depend on the direction of the applied moment. The magnitude of the critical bending moment for a straight beam was the same regardless of the sign of the applied moment.
Thin panel fastened to the frame
The first attempt to model the skin was to fasten an imaginary thin panel with infinite stiffness in its plane to the frame, see Figure 3.6. The added panel did not affect the cross section properties of the frame and did not add any stiffness in the y direction nor rotational stiffness about the longitudinal axis of the beam.
y
x
z
Figure 3.6: Thin panel fastened to the frame
The fastener line is located in the middle of the outer flange of the frame where the rivets are fastened in the real fuselage. In Hypermesh the panel was modeled with multiple single point constraints, SPCs, along the fastener line and the degree of freedom in the x direction was quenched. The same equations as for the continuous elastic supports were used to determine the critical buckling load analytically see section 2.12. EIy d4u dz4 + P d2u dz2 + y0 d2φ dz2 + kx[u + (y0− hy)φ] = 0 (3.8) EIx d4v dz4 + P d2v dz2 − x0 d2φ dz2 + ky[v − (x0− hx)φ] = 0 (3.9)
3.1 Effects of initial curvature of beam 41 ECw d4φ dz4− GJ − PI0 A d2φ dz2− P x0 d2v dz2− y0 d2u dz2 + kx[u + (y0− hy)φ](y0− hy) − ky[v − (x0− hx)φ](x0− hx) + kφφ = 0 (3.10)
Since the model only restrained the translational motion in the x-direction for one of the flanges, the stiffnesses ky and kφ were set to zero. Instead of setting the
stiffness of kx to infinity the displacement of the nodes along the line where the
spring was applied was set to zero, uN = u + (y0− hy)φ = 0. Equations 3.8, 3.9
and 3.10 were then reduced to Equation 3.11 and 3.12, [9] page 241-244.
EIx d4v dz4 + P d2v dz2 − EIxy(y0− hy) d4φ dz4 − P x0 d2φ dz2 = 0 (3.11) ECw+ EIy(y0− hy)2 d4φ dz4 − GJ −I0 AP + P y 2 0− P h 2 y d2φ dz2 − EIxy(y0− hy) d4v dz4− P x0 d2v dz2 = 0 (3.12)
Ixy is the product of inertia and it is zero for the open channel cross section. The
differential equations can be solved by using an ansatz that satisfies the boundary conditions: v = A1sin πz n l φ = A2sin πz n l (3.13)
Inserting these trial solutions into Equation 3.11 and 3.12 yields.
EIx(π nl )4− P (π nl )2 P x0(π nl )2 P x0(π nl ) 2 (EC w+ EIyh2y)( π n l ) 4+ (GJ −I0 AP − P h 2 y)( π n l ) 2 A1 A2 =0 0 (3.14)
3.1
Effects of initial curvature of beam
An FE model of a curved beam was set up that had the same radius as the fuselage. The length of the centroidal axis was the same for this model as for the straight beam.
Plane bending of a curved beam
The differential equations, which take the curvature of the beam into account, were derived by P. Vacharajittiphan and N. S. Trahair [10]. When the radius of the curved beam goes to infinity, the results should converge to the same value as for a straight beam. When a large radius was inserted into their solution, the critical buckling moment was almost half of that for the straight beam. Having
solved the differential equations, our solution differed a little from their result. The solution that was obtained for a large radius converged to the result for a straight beam, as was expected. The only difference between the two solutions, our solution and that of article [10], was that their solution contained a term π/L2where as we
had found π2/L2. Since that term comes from the boundary conditions π and L
should have the same exponent. Our solution converged to the same value as for a straight beam. This was probably just a typo in the article. The critical buckling moment for a curved member can be calculated by using equation 3.15.
Mcr= EIy+ γGJ 2R r EIy+ γGJ 2R 2 +π 2 L2 − 1 R2 EIyγGJ (3.15) γ =1 + π 2EC w GJ L2
Where E = Young’s modulus
Iy = Area moment of inertia about the y-axis
Cw = Warping constant
G = Shear modulus
J = Torsion constant
R = Radius of the initial curvature
L = Length of the centroidal axis
Equation 3.15 has two solutions since there are two roots to the characteristic equations of the differential equations. The solutions depend on the direction of the applied moment. If the outer flange was in tension and the inner flange in compression, the second term in equation 3.15 was positive. A curved beam, where the inner flange is in compression, is more stable than a straight beam in terms of lateral torsional stability. On the other hand, there are induced radial compression forces in the web that might jeopardize the integrity of the structure and may lead to local buckling modes in the web, see section 2.13. Therefore it is important to calculate the allowable load for the web in order to see if local buckling will occur before the global buckling mode.
3.2
Full-scale linear model of the fuselage
A small portion of the fuselage was set up in a linear elastic model in HyperMesh. The geometry was based on the CAD-files provided by SAAB and is a typical aircraft geometry. The section modeled consisted of four frames together with the 46 stringers and the skin panel, as shown in Figure 3.7. Bending forces, shear forces and also loads from the radar legs, which was a from a radar that was installed on top of the airplane, were applied to the model.
3.2 Full-scale linear model of the fuselage 43
Figure 3.7: Linear elastic model of a portion of the fuselage
Chapter 4
Results and discussion
Table 4.1 shows the results from the different models of the beam when loaded in compression and plane bending. The results shows the critical buckling stress of each model.
Compression [MPa] Plane bending [MPa]
Analytical result FEA result Analytical result FEA result
Straight beam Straight beam Straight beam Curved beam Straight beam Curved beam
1 9,18 9,16 22,23 91,66 22,05 80,75
2 15,86 15,75 x x 22,71 80,88
3 470,68* x x x x x
Table 4.1: Results from of all the different tests, the numbers in the first column represents the different load cases. *Exceeds the compressive yield stress
Load case 1 = Simply supported beam
2 = Thin panel fastened to the frame 3 = Continuous elastic support
No result for the beam with continuous elastic supports, loaded in bending, was obtained since no correlation between the analytic value and the FE-model could be made for the beam loaded with a central thrust.
The buckling shape of the simply supported beam in compression was a pure flexural mode, the beam was bent in the x-direction. The buckling shape of the other models was either a lateral torsional or flexural torsional buckling mode. That means that the deflection of the beam after buckling was a combination of bending and rotation.
