Stiffened panel buckling

Introduction

The following figure presents a stiffed panel subject to buckling on a military aircraft.

This use-case implements a simplified model of buckling for a stiffened panel (see [ko1994]).

buckling illustration

Figure 1. Buckling of a stiffened panel.

buckling simulation

Figure 2. 3D simulation of buckling.

stiffened panel geometry

Figure 3. Parameterization of the stiffened panel.

This test case is composed of nine random variables:

  • E\sim\mathcal{TN}(\num{110.0e9}, \num{55.0e9}, \num{99.0e9}, \num{121.0e9}) : Young modulus (\unit{\Pa})

  • nu\sim\mathcal{U}(0.3675, 0.3825) : Poisson coefficient (-)

  • h_c\sim\mathcal{U}(0.0285, 0.0315) : Distance between the mean surface of the hat and the foot of the Stiffener (\unit{m})

  • \ell\sim\mathcal{U}(0.04655, 0.05145) : Length of the stiffener side (\unit{m})

  • f_1\sim\mathcal{U}(0.0266, 0.0294) : Width of the stiffener foot (\unit{m})

  • f_2\sim\mathcal{U}(0.00627, 0.00693) : Width of the stiffener hat (\unit{m})

  • t\sim\mathcal{U}(\num{8.02e-5}, \num{8.181e-5}) : Thickness of the panel and the stiffener (\unit{m})

  • a\sim\mathcal{U}(0.6039, 0.6161) : Width of the panel (\unit{m})

  • b_0\sim\mathcal{U}(0.04455, 0.04545) : Distance between two stiffeners (\unit{m})

  • p\sim\mathcal{U}(0.03762, 0.03838) : Half-width of the stiffener (\unit{m})

The output of interest is:

  • (N_{xy})_{cr}: the critical shear force (\unit{N})

We assume that the input variables are independent except the f_1 and f_2 for which we measure a Spearman correlation of \rho^S_{12}=-0.8, modelled using a NormalCopula.

The critical load (\tau_{xy})_{cr} of a stiffened panel subject to shear load is:

(\tau_{xy})_{cr}=k_{xy}\frac{\pi^2D}{b_0^2t_s}

where:

  • a is the width of the panel;

  • b_0 is the width between too consecutive stiffener feet;

  • t_s is the thickness of the panel main surface;

  • E_s is the Young modulus of the panel main surface;

  • \nu_s is the Poisson coefficient of the panel main surface;

  • D is the bending coefficient of the panel main surface:

  • k_{xy} is the load factor associated to shear buckling. It is given as a function of \frac{b_0}{a} through the empirical equation:

k_{xy} = 5.35 + 4 \left(\frac{b_0}{a}\right)^2.

It is more convenient to use the shear force N_{xy} instead of the shear stress component \tau_{xy}. It leads to the equation:

N_{xy}=q_1+q_c

where q_1 abd q_c are the shear fluxes in the panel main surface and its stiffener. They are given by:

q_1=\tau_{xy}t_s=2G_sh_0t_s\frac{\partial^2w}{\partial x\partial y}

and:

q_c=\frac{G_ct_cp}{\ell} \left(h - 2h_0 + \frac{h_c}{2p}(f_1-f_2)\right) \frac{\partial^2w}{\partial x\partial y}

where:

  • G_s is the shear modulus of the panel main surface:

G_s = \frac{E_s}{2(1 + \nu_s)};

  • \frac{\partial^2w}{\partial x\partial y} is the torsion strain of the panel main surface;

  • G_c is the shear coefficient of the stiffener:

G_c = \frac{E_c}{2(1 + \nu_c)};

  • t_c is the thickness of the stiffener;

  • h_c is the distance between the mean surfaces of the stiffener hat and foot;

  • h is the distance between the mean surfaces of the stiffener hat and the panel main surface:

h = h_c+\frac{t_c + t_s}{2};

  • f_1 is the width of the foot of the stiffener;

  • f_2 is the width of the hat of the stiffener;

  • p is the half-widht of the stiffener;

  • R is the radius of the circular part of the stiffener;

  • \theta is the angle of the circular part of the stiffener;

  • \ell is the length of the stiffener flank;

  • d=\frac{\ell-f_2}{2}-R\theta is the half-lenght of the straight part of the side of the stiffener;

  • A=\ell t_c is the area of the section of an half-ondulation;

  • \bar{A} is the area of the section of the full panel (main surface and stiffener) bounded by p:

\bar{A} = A + pt_s + \frac{1}{2}(f_1 - f_2)t_c

  • h_0 is the distance between the mean surface of the panel main surface and the global geometric center of the panel:

h_0 = \frac{1}{2\bar{A}} \left(A(h_c+t_c+t_s)+\frac{1}{2}t_c(f_1-f_2)(t_c+t_s)\right).

It leads to:

N_{xy}=q_1(1+q_c/q_1) = \tau_{xy}t_s \left(1 + \frac{1}{4}\frac{G_ct_c}{G_st_s} \frac{2p(h-2h_0) - h_c(f_1-f_2)}{h_0\ell}\right)

and finally, (N_{xy})_{cr} is given by:

(N_{xy})_{cr}=\left(5.35 + 4\left(\frac{b_0}{a}\right)^2\right)\left(\frac{\pi^2}{b_0^2}\frac{E_st_s^3}{12(1-\nu_s^2)}\right)\left(1+\frac{1}{4}\frac{G_ct_c}{G_st_s}\frac{2p(h-2h_0)-h_c(f_1-f_2)}{h_0\ell}\right)

For industrial constraints, the stiffener and the main surface are cut in the same metal sheet, so E_c=E_s=E, \nu_c=\nu_s=\nu, t_c=t_s=t. The final expression of the critical shear force is then:

(N_{xy})_{cr}=\left(5.35 + 4\left(\frac{b_0}{a}\right)^2\right)\left(\frac{\pi^2}{b_0^2}\frac{Et^3}{12(1-\nu^2)}\right)\left(1+\frac{1}{4}\frac{2p(h-2h_0)-h_c(f_1-f_2)}{h_0\ell}\right)

with:

  • A=\ell t;

  • \bar{A}=A+t\left(p+\frac{f_1-f_2}{2}\right);

  • h_0=\frac{A(h_c+2t)+t^2(f_1-f_2)}{2\bar{A}};

  • h=h_c+t.

References

Load the use case

We can load this model from the use cases module as follows :

>>> from openturns.usecases import stiffened_panel
>>> sp = stiffened_panel.StiffenedPanel()
>>> # Load the stiffened panel use case
>>> model = sp.model()

API documentation

class StiffenedPanel

Data class for the stiffened panel model.

Examples

>>> from openturns.usecases import stiffened_panel
>>> # Load the stiffened panel model
>>> panel = stiffened_panel.StiffenedPanel()
>>> print("Inputs:", panel.model.getInputDescription())
Inputs: [F,L,a,De,di,E]
>>> print("Outputs:", panel.model.getOutputDescription())
[Deflection,Left angle,Right angle]
Attributes:
dimConstant, the dimension of the problem.

dim=10

modelSymbolicFunction

Model of the critical shearing load. The model has input dimension 10 and output dimension 1. More precisely, we have \vect{X} = (E, \nu, h_c, \ell, f_1, f_2, t, a, b_0, p) and Y = (N_{xy})_{cr}.

EYoung modulus (Pa), TruncatedNormal distribution

ot.TruncatedNormal(110.0e9, 55.0e9, 99.0e9, 121.0e9)

nuPoisson coefficient (-), Uniform distribution

ot.Uniform(0.3675, 0.3825)

h_cDistance between the mean surface of the hat and the foot of the Stiffener (m), Uniform distribution

ot.Uniform(0.0285, 0.0315)

ellLength of the stiffener flank (m), Uniform distribution

ot.Uniform(0.04655, 0.05145)

f_1Width of the stiffener foot (m), Uniform distribution

ot.Uniform(0.0266, 0.0294)

f_2Width of the stiffener hat (m), Uniform distribution

ot.Uniform(0.00627, 0.00693)

tThickness of the panel and the stiffener (m), Uniform distribution

ot.Uniform(8.02e-5, 8.181e-5)

aWidth of the panel (m), Uniform distribution

ot.Uniform(0.6039, 0.6161)

b_0Distance between two stiffeners (m), Uniform distribution

ot.Uniform(0.04455, 0.04545)

pHalf-width of the stiffener (m), Uniform distribution

ot.Uniform(0.03762, 0.03838)

distributionJointDistribution

The joint distribution of the input parameters.

Examples based on this use case

Estimate a buckling probability

Estimate a buckling probability