Section Properties of an Irregular Shape with Holes

In some previous articles we discussed other section properties related topics, click the links below to learn more:

• Section Modulus of a Composite Beam System
• Area Moment of Inertia Calculation

It is relatively common in aerospace stress analysis to come across odd shaped fittings that need to be analyzed. But in order to analyze such fittings, you must pick one or more critical sections, and then calculate the properties of that section.

Some online resources let you use spread sheet tools, such as this article: Voided Irregular Sections. They may use manual entry excel tools, or classical hand calculations methods to find the properties.

In this post, I would like to show you a simpler but more powerful way to calculate the same properties. You can use this technique with any shaped section, with our without fastener holes or other cutout gaps.

As seen in Figure 1, this section’s shape may be more regular than irregular. But, let us assume that the center slot, the gussets, flanges, and fastener holes make it a bit more irregular.

Challenges:

This section (and others more irregular than this one) poses certain challenges:

1. It is not a single surface to easily measure the properties using the surface tool in FEMAP
2. We can use traditional hand calculations. But it will quickly get complicated if more separated and complicated shape segments exist

Solution:

So, what do we do now? FEMAP comes to the rescue again. OK so let us dive right into it.

• First, as shown in Figure 1, split the body at the critical section, hide the other solid part
• Then, as shown in Figure 2, create a “plot only” mesh (no need for properties) on all the section surfaces
• Following that, create a local Cartesian reference CSYS at a convenient corner
• And then, from the main menu bar at the top, we use the FEMAP command: Tools -> Section Properties -> Mesh Properties
  • Note that for a single closed loop surface, we can use the “Tools – Section Properties – Surface Properties” command
    • This command internally does the same thing for a single surface. It creates an internal surface mesh and then calculates the properties
• FEMAP will then calculate the following properties:
  • Area A
  • Centroid (from the specified origin), Cy and Cz
  • Moment of inertia, Iyy and Izz, Iyz
  • Principal moment of inertia, I1 and I2 (about axes with Iyz = zero)
  • Radius of gyration, Ry and Rz
  • Angle to principal axes, Ang
  • Polar moment of inertia, Ip

Benefits:

• The biggest benefit is that there is no need to link all the surfaces with slender surfaces to make it work
• No need for classical hand calculations or excel calculations
• The section can be irregular, unsymmetric, random, and with or without holes

Downsides:

• The downside to the Mesh Properties command:
  • If the section has any gaps in the mesh, then FEMAP cannot calculate the shear torsion and warping properties listed below (possible for a closed loop surface):
    • Scy and Scz: Shear Center from origin
    • Scy and Scz: Shear Center from centroid
    • Shear Areas, Asy and Asz
      • Effective shear areas along section y and z axes
      • Used for shear deformation stiffness matrix
      • These are basically effective areas: The total area times form factors
      • These areas are lower than total area
    • Torsional constant, J
    • Warping constant, W

Of course, we do get the commonly used properties for calculating section stresses. But there maybe situations where the shear load is not applied at the shear center. This results in twisting of the section. The twisting produces additional torsional shear. This needs to be accounted for separately, but we cannot easily do this using the above method as the shear center is not known yet, more work is required to get that.

Results:

After following the above steps in FEMAP, in other words creating the mesh, invoking the command, selecting all the surfaces, selecting the section Y axis in the new local coordinate system etc., FEMAP spits out the results:

Computing Properties…
Orientation of Section Properties:
Origin: X= 366.449 Y= -88.8868 Z= 192.095 (this is the origin of our local CSYS in the global CSYS)
Y Axis: X= 1. Y= 0. Z= 0.
Z Axis: X= 0. Y= 1. Z= 0.

Section Properties:
Area A= 0.92878
Centroid (from Origin): Cy= 1. Cz= 0.6774
Moment of Inertia: Iyy= 0.14851 Izz= 0.92077 Iyz= 0.
Principal Moment of Inertia: I1= 0.92077 I2= 0.14851
Radius of Gyration: Ry= 0.39988 Rz= 0.99568
Angle to Principal Axes: Ang= -2.4261E-9
Polar Moment of Inertia: Ip= 1.06929

Conclusions:

The top red point in Figure 3, at the CG of the section, was created using the results. The section properties we just determined are then used to calculate the critical section stresses listed below, based on the applied loading (for example, say the bottom red point shown in Figure 3, or directly using freebody loads and moments at this section if the whole fitting is modeled in FEM).

• Bending Stress
• Direct Shear Stress
• Axial Stress
• Torsional Shear Stress

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