Testing and Results Validation¶

sectionproperties has a (slowly) growing suite of tests. The testing suite serves to verify functionality and find exceptions from edge cases, but also validate the results against known cases. Since this library performs engineering calculations, it should have some manner of proving its accuracy. Each analyst who uses it is responsible for their own projects and calculations, but having a high level of confidence that the software can produce correct results, given the correct inputs, is a boon to all users. Some test results and explanations from the latter category will be outlined on this page, since the former really serves no use to the end user.

Textbook Examples¶

An obvious starting location is replicating examples from academic texts. “Aircraft Structures” by David J. Peery is a highly regarded text in the field of aerospace structures 1.

Peery - Symmetric Sections¶

The simplest example of a realistic section problem is the symmetric I-Beam, with a free-fixed boundary condition and a transverse tip load. The free-body-diagram and shear and moment diagrams are shown in the problem statement, referenced below. This problem is Example 1 in Section 6.2.

Check #1:¶

In sectionproperties, there are multiple ways to set this problem up. We could different shapely geometries and merge together, or a set of custom points, or a built-in constructor. For the sake of simplicity, this simpler I-section is identical to the Nastran I-section definition, so it makes sense to utilize the built-in constructor from pre.library.nastran_sections.nastran_i.

Using an arbitrarily coarse mesh, the properties can then be directly calculated from the class method Section.calculate_geomtric_properties().

Peery lists the second moment of area about the primary bending axis as a value of \(43.3 [in^4]\). For the automated tests in this library, we check against this hardcoded value, with a tolerance of \(\pm 0.1%\).

Check #2:¶

As a final check against this example, we can calculate the maximum bending stress on the I-beam. From simple statics, the maximum moment from the FBD will be \(800,000 [in-lbs]\) at the fixed end. Applying this moment to our section from before will allow computation of stress over the FEM.

Peery quotes the peak value at \(55.5 [ksi]\), which is rounded to the nearest decimal place. From the equatio listed in the text, the theoretical value is actually \(55,427.3 [psi]\).

\[f = \frac{My}{I} = 55,427.3 = 55,400\]

Again, the automated test against this checks the hardcoded value with a tolerance of \(\pm 0.1%\). For accuracy, 55,427.3 is used instead of the rounded value.

For full details and the most updated code of this exampe, see the examples page in the documentation gallery. For the exact test code execution, check the source.

Peery - Unsymmetric Sections¶

For a more complex example, we can turn to Example 1 in Section 7.2 of Peery. Here, we have a still-simplified Z-section, but bending about two axes. Note axes definitions in the problem statement. Beam axial direction in sectionproperties is always referenced as the z-axis, and loads must be applied in this coordinate system.

The construction of this geometry takes a similar approach to Ex 6.2.1, and utilizes a built-in factory: pre.library.nastran_sections.nastran_zed. The only difference you may notice in the test code is usage of a custom class for ease of initialization. This is not necessary.

Using an arbitrarily coarse mesh, the properties can then be directly calculated from the class method Section.calculate_geomtric_properties(). Each property listed directly by Peery is taken as a hardcoded value and checked against, within the testing suite.

Property	Peery Value
I_x	693.3 [in⁴]
I_y	173.3 [in⁴]
I_xy	-240 [in⁴]
I_p	787.1 [in⁴]
I_q	79.5 [in⁴]
theta	21.35 [deg]

For stress results, the theoretical values follow the biaxial bending equation. These values are checked against automatically in the testing suite. Note that again Peery rounds the values quoted directly, for simplicity. The testing suite also verifies that the theoretical value as per the equation matches the theoretical value quoted in the text, which also matches the computed value from the sectionproperties FEM.

\[f_b = \frac{M_x I_{xy} - M_y I_x}{I_x I_y - I_{xy}^2}x + \frac{M_y I_{xy} - M_x I_y}{I_x I_y - I_{xy}^2}y\]

Point	x	y	-494x	-315y	\(f_b\), [psi]
A	-5	4	2470	-1260	1210.0 = 1210
B	-5	6	2470	-1890	580.0 = 580
C	1	6	-494	-1890	-2384.0 = -2380

For full details and the most updated code of this example, see the examples page in the documentation gallery. For the exact test code execution, check the source.

1: D. J. Peery, Aircraft Structures. New York: Dover Publications, 2011. ISBN-10: 0-486-48580-3