User:Eas4200c.f08.wiki.d/HW6

= Airfoil =



Above is a graph of $$\sigma_{xx}$$ at all y positions in the middle wing section on the top skin. Due to the graph being concave up, it can be seen that the load is compressive on the top skin of the wing section.



Above is a graph of $$\sigma_{xx}$$ at all y positions in the middle wing section on the bottom skin. Due to the graph being concave down, it can be seen that the load is tensile on the top skin of the wing section.

The average compressive stress was found to be 144.2 MPa.

= Plate Buckling =

Simply-Supported Boundary Conditions
A simply-supported rectangular plate under an in-plane distributed compressive load is to be evaluated:

Buckling mode shapes can be determined by the following equation: $$\displaystyle \psi (x,y) = c_{mn}\sin\left(\frac{m \pi x}{a}\right)\sin\left(\frac{n \pi y}{b}\right)\, \ {\rm for} \ m,n = 1, 2, 3, \ldots$$ 1

Perspective views of the buckling shape of a simply-supported rectangular plate for various cases of $$\displaystyle\ m$$ and $$\displaystyle\ n$$



Now considering the factor with $$\displaystyle\ x$$ as the variable in the expression for $$\displaystyle\ \psi_{mn}(x,y)$$, the period, $$\displaystyle\ T$$, is calculated as follows:

$$ \sin ( \frac{m \pi (x + T)}{a} ) = \sin ( \frac{m \pi x}{a} )$$

$$\frac{m\pi (x+T)}{a}=\frac{m\pi (x)}{a}+2\pi n$$ where $$\displaystyle\ n=1,2,3,...$$

Evaluating at $$\displaystyle\ n=1$$

$$\frac{m\pi (x+T)}{a}=\frac{m\pi (x)}{a}+2\pi$$

$$ x+T = x + \frac{2\pi a}{m\pi}$$

$$ T = \frac{2a}{m}$$

The values of $$\displaystyle\ T$$ along with plots of the function are calculated and displayed below for values of $$\displaystyle\ m=1,...,5$$ to show that $$\displaystyle\ m$$ is indeed the number of half wave-lengths generated.

The critical buckling load $$\displaystyle\ (P_x)_{cr}$$ is calculated as follows:

$$\displaystyle (P_x)_{cr}=k_c\frac{\pi ^2 D}{b}$$ 1

where $$\displaystyle k_c(m, a/b):=\left(\frac{m b}{a}+\frac{a}{m b}\right)^2$$

and $$\displaystyle D:=\frac{E h^3}{12 (1 - \nu^2)}$$

A graph of $$\displaystyle\ k_c$$ versus aspect ratio, $$\displaystyle\ a/b$$, for various values of $$\displaystyle\ m$$ is shown below

Going one step further, the critical buckling stress can be calculated by

$$\displaystyle (\sigma_{xx})_{cr}=\frac{(P_x)_{cr}}{bh}=k_c\frac{\pi ^2 D}{b^2 h}$$ 1

Clamped Boundary Conditions
Next a rectangular plate with clamped boundary conditions is to be examined

The critcal buckling stress under an in-plane compressive load can be calculated as follows:

$$\displaystyle\ (\sigma_{xx})_{cr}=K\frac{E}{1-\nu ^{2}}(\frac{t}{b})^2$$

where $$\displaystyle K$$ is a function different from $$\displaystyle k_c$$, and $$\displaystyle t \equiv h$$ the plate thickness.

The buckling shape is now calculated using

$$\displaystyle\psi (x,y)=\frac{c}{4}\left(1 - \cos \frac{2 \pi x}{a}\right)\left(1 - \cos \frac{2 \pi y}{b}\right)$$



= Conclusions =

The figure below shows the critical buckling stress as a function of aspect ratio for the top panel (under compressive load from prescribed conditions) of the NACA 2415 Airfoil under both simply-supported and clamped boundary conditions. Also shown in green is the average compressive stress in the panel. Since stringers support the panel, it is neither exactly simply-supported or clamped; therefore, the actual critical stress must fall within the envelope prescribed by the simply-supported and clamped cases. It can be seen that the average compressive stress of the panel does encounter the critical stress for the clamped case but not the simply-supported one; consequently, depending on how close the panel relates to the two cases, buckling is a possibility if the panel behaves closely to the clamped boundary condition.