Step 9: Validate the results
We checked in the previous step that the BC's have been applied correctly and the deflection looks plausible.
Comparison With Plate Theory
An FEA analyst displays his mettle by going headtohead with theory. The theoretical solution for a clamped plate subjected to pressure is presented by Timoshenko and WoinowskyKrieger, page 202. This solution can be modified to account for the presence of stiffeners using the parallel axis theorem. This calculation and the resulting stress and displacement values have been generously provided by Prof. Alan Zehnder and are sumarized in this pdf document. Do take the time to review this document.
The headtohead comparison of the analytical values with the FEA values from Step 8 at the center of the structure is presented in the following table.
Entity 
Location 
FEA 
Theory 

w 

3.8 mm 
3.6 mm 
σ_{xx} 
Bottom 
56 MPa 
50 MPa 
σ_{xx} 
Top 
312 MPa 
350 MPa 
σ_{yy} 
Bottom 
28 MPa 
20 MPa 
σ_{yy} 
Top 
38 MPa 
105 MPa 
The deflection at the center is about 8% larger than the theoretical value due to shear deformation in the FEA model that is not accounted for in the analytical result. Correlation of stresses in the center is reasonably good. It might improve if solution for anisotropic plate were used. Also, the stiffeners are not at the centerline of the model, while maximum stresses in theory are calculated at the center.
The headtohead comparison of the edge stresses is given in the following table.
Entity 
Location 
FEA 
Theory 

σ_{xx} 
Bottom 
116 MPa 
100 MPa 
σ_{xx} 
Top 
653 MPa 
704 MPa 
σ_{yy} 
Bottom 
77 MPa 
78 MPa 
σ_{yy} 
Top 
300 MPa 
403 MPa 
There is good correlation of the edge stress on the bottom of facesheet but the edge stress at the top of the stiffeners is overestimated by the theory. This is possibly due to the stiffeners not being at the centerline of the model while the maximum stresses in theory are calculated at the center.
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