Max Deflection of Thin Steel Plate w/Wall Support

[The Engineer]

Homework Statement

Will the following thin steel plate plasticly deform due to its own weight, what is the maximum deflection of the plate? Plate dimensions 0.25 cm thickness, 3.0 m length, 3.0 m width. The plate is supported on two ends, by a wall (welded).

Homework Equations

σ = (pb^2)/[(2t^2)((0.623{b/a)^6)+1)]

y = (0.0284pb^4)/[(Et^3)((1.056[b/a]^5)+1)]

The Attempt at a Solution

density = 8050 kg/m^3
area = (3m)(3m) = 9m^2
volume = (3m)(3m)(0.0025m) = 0.0225m^3
weight = (8050kg/m^3)(0.0225m^3)(9.81m/s^2) = 1776.83N

p = 1776.83N/9m^2 = 197Pa

σmax = ((197)(3^2))/{(2[0.0025^2])((0.623([3/3])^6)+1))} = 87.3937MPa
Therefore, no plastic deformation.

ymax = (0.0284(197)(3^4))/[((200x10^9)(0.0025^3))((1.056[3/3]^5)+1)] = 0.07053m
Therefore, the deflection is 0.07053m.

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Well, that's my attempt at this problem. Not 100% sure if it's right though. :(

 

[KataKoniK]

Thank you for your question. I would like to provide you with a more comprehensive answer to your inquiry.

Firstly, let's define what plastic deformation means. Plastic deformation is a permanent change in shape or size of a material due to applied forces or stresses, which exceed the yield strength of the material. In this case, the thin steel plate is subjected to its own weight, which can be considered as a distributed load. Therefore, we need to determine the maximum stress and deflection that the plate can withstand without undergoing plastic deformation.

To calculate the maximum stress, we can use the equation σ = (pb^2)/[(2t^2)((0.623{b/a)^6)+1)], where p is the distributed load, b is the width of the plate, t is the thickness of the plate, and a is the length of the plate. Using the values given in the problem, we can calculate the maximum stress as follows:

p = (1776.83N)/(9m^2) = 197Pa
σmax = ((197)(3^2))/{(2[0.0025^2])((0.623([3/3])^6)+1))} = 87.3937MPa

Now, we need to compare this value with the yield strength of the material. The yield strength of steel can vary depending on the grade, but for this calculation, we will assume a yield strength of 250MPa. Since the maximum stress calculated is lower than the yield strength, we can conclude that the plate will not undergo plastic deformation.

Next, let's determine the maximum deflection of the plate. We can use the equation y = (0.0284pb^4)/[(Et^3)((1.056[b/a]^5)+1)], where E is the modulus of elasticity of the material. For steel, the modulus of elasticity is typically around 200GPa. Using this value, we can calculate the maximum deflection as follows:

ymax = (0.0284(197)(3^4))/[((200x10^9)(0.0025^3))((1.056[3/3]^5)+1)] = 0.07053m

Therefore, the maximum deflection of the plate is 0.07053m, which is well within the acceptable limits for a steel plate.

In conclusion, based on