User:Eas4200c.f08.gator.edwards/Plate Buckling



First, we start by creating a 3-D perspective plot of the buckling shape of a simply supported rectangular plate.In the formula for u_z used to create the plot, c = 1, a = 1.5 m, b = 1.0 m, and therefore $$\frac{a}{b}=1.5$$

For Case 1, we set m = 1 and n = 2

For Case 2, we set m = 2 and n = 1

Now we found the period T of the function using:

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

therefore,

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

canceling out the x terms, we have

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

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

Looking at this, we can determine that m is indeed the number of half-wavelengths.

Reproducing the k_c vs. a/b graph found on the MIT lecture notes page:



CRITICAL STRESS

Now we will create another perspective plot similar to before, except now the plate will be under clamped conditions instead of the previous simply supported conditions.