Dome Deflection Formula for Calculating Deflection from Point Load

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Discussion Overview

The discussion revolves around finding a formula to calculate the deflection of a dome under a point load applied at its center on the convex side. The focus is on deriving the appropriate formula given specific parameters of the dome, including material properties and geometric dimensions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant requests a specific formula for dome deflection due to a point load, indicating a lack of access to reference materials.
  • Another participant asks for detailed parameters such as Poisson's ratio, mean radius, shell thickness, subtended half angle, and edge support conditions to refine the inquiry.
  • A participant provides specific values for the parameters, including material type, Poisson's ratio, radius, thickness, half angle, and edge support conditions, clarifying that they are interested in deflection from a point load at the dome's center.
  • A subsequent post specifies that the point load is assumed to be evenly distributed over a small circular area, presenting a formula for deflection that includes variables for total applied load and modulus of elasticity, while noting the conditions under which this formula is applicable.

Areas of Agreement / Disagreement

Participants have not reached a consensus on a definitive formula for dome deflection, and there are varying assumptions about the load distribution and edge support conditions that may affect the calculations.

Contextual Notes

The discussion includes limitations related to the assumptions made about load distribution and edge support, as well as the specific conditions under which the provided formula is valid. There may be additional factors that influence deflection that have not been addressed.

Tadders
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I am looking for a formula to give me the deflection of a dome if all dome perameters are known from a point load at the center of the dome towards the dome on the convex side. I do not have the Roark book so I need the actual formula, not just a reference.
 
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Tadders: Can you provide numeric values for the following parameters, to narrow your question? Your question is currently slightly too generic to be easily answered.

nu = Poisson's ratio.
r = spherical dome mean radius.
t = spherical dome shell thickness.
phi = spherical dome subtended half angle, phi ≤ 90 deg, where phi = 90 deg is a hemispherical dome.
Also, type of edge support, if known (optional).
 
nvn,
Here is the data:
Material 1070 steel
nu = 0.29
r = 22.08 inches
t = 4.5mm thick
phi = 21.24 degrees
edge support = free to rotate, there will be some lateral restraint for my application but for calc purposes say no restraint, and complete vertical restraint (vertical meaning in the direction of the central axis i.e. if the dome was a roof on a building, the edge could not move vertically).
 
nvn,
PS to prior post. I am only interested in a point load deflection where the point load is at the center of the dome on the convex side towards the dome.
Thanks.
 
Tadders: I assumed your point load is evenly distributed over a small circular area having a diameter of 4.5 mm. Therefore, the deflection at the center of the load is y = -10.942*P/E, where y = deflection (mm), P = total applied load (N), and E = tensile modulus of elasticity (MPa).

This answer is applicable only if r = 560.83 mm, t = 4.5 mm, nu = 0.29, and the diameter of the circular area of the applied load is 4.5 mm.
 
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