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The Fourier transform of $y\left(x,t\right)$ from the time to the frequency domain is given by $Y\left(x,\omega\right)=\int_{-\infty}^{\infty}y\left(x,t\right)e^{i\omega t}dt$ and satisfies the differential equation: $$ EI\frac{\partial^{4}Y\left(x,\omega\right)}{\partial ...


Strength of materials is affected by defects. A perfect crystal of iron would be extremely strong. Once a crack starts, it is not so hard to make it advance one atom deeper. Think of tearing open a plastic bag. Much easier once the tear starts. Brittle materials can be easier to break because they stretch less. It is easier to tear a sheet of paper than a ...


So, buckling is the bifurcation of static equilibrium. And thus: More technically, consider the continuous dynamical system described by the ODE $\dot x=f(x,\lambda)\quad > f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.$ A local bifurcation occurs at $(x0,λ0)$ if the Jacobian matrix $\textrm{d}f_{x_0,\lambda_0}$ has an ...


Actually the data presented by You show that iron/steel is more brittle than diamond. Precise tensile strength of diamond is unknown, however values of up to 60000 MPa have been observed. Typical values of tensile strength of iron/steel varies from 100 to 11000 MPa. Therefore diamond can withstand more than iron/steel.


"different values of n represent different modes of buckling" (http://emweb.unl.edu/NEGAHBAN/Em325/21-Buckling%20of%20columns/Buckling%20of%20columns.htm )


Generalities The problem has spherical symmetry, so it makes sense to use spherical coordinates ($r$, $\theta$, $\phi$). We can divide the vessel into differential elements like the one shown in this post. Deformation and strain Only radial deformations are allowed by the spherical symmetry, so let's parametrize the deformation by $$r \rightarrow r + ...


The bucking formula comes from a stability analysis of the restoring moment inside the beam. Actually as compressive forces are applied to a beam, its natural frequency drops, and when you reach the Euler's limit the natural frequency of the beam becomes zero. By definition this is the point the beam will not behave at all in a static fashion and will move ...


Buckling is not limited to thin columns, it is also important, e.g., for thin shells under compression; for example, if pressure in a poorly designed tank is below atmospheric pressure, the tank can buckle under atmospheric pressure (it happens to large oil tanks, railway car tanks - you name it; you can easily find a lot of impressive images on the net).

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