Mechanical Engineering Question

I had hoped to find a calculator online for this...  The wooden sheet on which my big reflector sits is beginning to deform at one of the corners where a caster goes.  I want to replace this with something more durable before I have a catastrophic failure.  So what thickness of steel do I need to support 150 pounds without too muh deformation?  that chatgpt gave me this formula during a lucid, hopefully nonhallucinatory phase:

To calculate the maximum deflection of a rectangular plate supported at the four corners, you can use the classical plate theory (Kirchhoff-Love theory). For a rectangular plate under uniform load, the maximum deflection $w_{\text{max}}$ can be estimated using the following formula e:

$$w_{\text{max}} = \frac{q \cdot a^4}{D}$$

where:

• $q$ is the uniform load per unit area (N/m²),
• $a$ is the shorter side length of the rectangular plate (m),
• $D$ is the flexural rigidity of the plate, calculated as:

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

In this formula:

• $E$ is the modulus of elasticity of the material (Pa),
• $h$ is the thickness of the plate (m),
• $\nu$ is the Poisson's ratio of the material.

Steps to Calculate Maximum Deflection:

1. Determine Material Properties: Find the values for $E$ and $\nu$ for the plate material.
2. Calculate Flexural Rigidity $D$: Use the thickness $h$ of the plate.
3. Define Loading Conditions: Determine the uniform load $q$ and the dimensions of the plate.
4. Apply the Formula: Substitute the values into the formula to find $w_{\text{max}}$.

The load is not centrally located but this seems a worst case assumption.  The casters at the four corners of the current 28" plywood square are not sitting at the true corners of course but close.

load = 150 pounds / 2.2 * 9.8 giving 668N/m^2 (okay the load is not the same eveywhere but this is an approximation)

a = 28 inches/39.37

thickness is .25 inches /39.37

E is 200 GPa for steel

v is Poisson's ratio, which all examples seem to regard as .3

D calculates as 4385.9

maximum deformation is .04 m or about 1.5 inches.

This seems absurdly high for 1/4" steel plate.  The plywood sheet there right now is not eforming that much.  Do you see what I am doing wrong?