Calculating Strain Energy of Circular Clamped Plate

[harpreet singh]

I need to calculate the strain energy of a symmetrical circular clamped plate using the equillibrium equation in polar coordinates.. Can somebody help me with the method??

 

[Jupiter6]

I need to calculate the strain energy of a symmetrical circular clamped plate using the equillibrium equation in polar coordinates.. Can somebody help me with the method??

Ack..lol. I would recommend getting a copy of "Theory of Plates and Shells" by Timoshenko. Any egr. library should have it.

 

[oswaler]

Sure, I can help you with the method for calculating the strain energy of a symmetrical circular clamped plate using the equilibrium equation in polar coordinates.

First, let's define some variables for clarity. Let R be the radius of the circular plate, t be the thickness of the plate, and E be the Young's modulus of the material.

To calculate the strain energy of the plate, we will use the strain energy density equation, which is given by:

u = 1/2 * σ * ε

Where u is the strain energy density, σ is the stress, and ε is the strain.

In polar coordinates, the equilibrium equation for a circular clamped plate is given by:

∂^2u/∂r^2 + 1/r * ∂u/∂r + 1/r^2 * ∂^2u/∂θ^2 = 0

We can rewrite this equation in terms of stress and strain using the following relationships:

σ_r = E * ε_r

σ_θ = E * ε_θ

Where σ_r and σ_θ are the radial and tangential stresses, and ε_r and ε_θ are the radial and tangential strains, respectively.

Using these relationships, we can rewrite the equilibrium equation as:

∂^2u/∂r^2 + (1/r) * E * (ε_r + ε_θ) + (1/r^2) * E * (ε_r + ε_θ) = 0

Since the plate is clamped, we know that the radial and tangential displacements must be zero at the edges. Therefore, we can simplify the equation to:

∂^2u/∂r^2 + (2/r) * E * ε_r = 0

Integrating this equation twice with respect to r, we get:

u = (1/4) * E * ε_r * r^2 + c1 * r + c2

Applying the boundary conditions of zero displacement at the edges, we get:

c1 = 0 and c2 = 0

Therefore, the strain energy density equation becomes:

u = (1/4) * E * ε_r * r^2

To calculate the total strain energy of the plate, we need to integrate this equation over the entire plate area. Since the plate is symmetrical, we can integrate from r = 0 to