I have a tube with a circular cross-section, this tube protrudes from the ground vertically for a section, then arcs over to one side. At the end of the tube, a flat plate is mounted to the end of it. The arc is of a constant radius but is not necessarily a 90 degree bend. (i.e. the angle of the flat plate to the ground plane is not necessarily 90 degrees). How would you go about calculating, by hand, how much load the flat plate could hold before the tube bends beyond it's yield point.

Assuming all geometries are known and the material properties are known and constant.

I've been working on a solution using a combination of formulas from Roarks book. But, I'd like to see how others would approach this problem. "Solve using FEA" is not an answer i'm looking for, but rather a reasonably accurate hand calculation approach.

Edit: -dimensions and loading direction added

Approximate tube dimensions:

Vertical straight section: 1m

'Bent' section arc radius: 125mm

Flat end, 10 degrees to floor

Tube OD - 25mm

Tube ID - 23mm

Loading direction will be in the negative z-axis i.e. gravitational loading only.

Tube Sketch

  • $\begingroup$ Including a sketch would help to clarify your problem and to specify which variables are available or of interest $\endgroup$ Mar 24, 2017 at 17:46
  • $\begingroup$ Knowing at least the relative size of the arch to the pipe diameter is absolutely essencial. Without that, the question becomes unanswerable: a long arch can be analyzed as a linear element; a tight arch can't. $\endgroup$
    – Wasabi
    Mar 24, 2017 at 17:55
  • $\begingroup$ Also, how will the loads be applied? Always vertical (gravity loads) or perpendicular/parallel to the plate? $\endgroup$
    – Wasabi
    Mar 24, 2017 at 18:10
  • $\begingroup$ Under what conditions would it be allowable to treat the curved section as linear? I'm trying to remember but is there a rule of thumb that suggests conditions for that simplification? (edit is underway with an image) $\endgroup$
    – Diesel
    Mar 24, 2017 at 19:06

1 Answer 1


I'd check for a maximum overturning moment which will cause yield stress at the base of the member where it is attached to a base plate or enters the foundation as a flag pole, of course with any required safety. Then It is a simple arithmetic $M=\sigma\cdot s$


$$ s=\pi\frac{d_o^4-d_i^4}{32\cdot d_o^2},$$

in which $d_i$ is the inner diameter and $d_o$ is the outer diameter.

A more detailed diagram as to the attachment of the post to the base plate or the configuration of the plate at the top would allow more detailed observation and checking for local buckling, shear, normal loading.

I am adding this clarification as to mode of failure.

The pipe will start to rotate counter clockwise and undergo elastic defleection which will add to the moment arm, but as we add the vertical loading it will yield from the base which already has the greatest bending moment of approximately m = P.16 cm.

  • $\begingroup$ While I know the question specifically asks about yielding, I wouldn't be surprised if the controlling condition here is buckling instead. $\endgroup$
    – Wasabi
    Mar 25, 2017 at 13:07
  • $\begingroup$ @Kamran I edited your answer please check if it is still what you wanted to express. $\endgroup$
    – MrYouMath
    Apr 24, 2017 at 10:27
  • $\begingroup$ @Wasabi, buckling is usually critical in cases where the column top is constrained laterally. In flag poles and free end loadings like this case, it is not. However local buckling under concentrated loads should be checked. $\endgroup$
    – kamran
    Apr 26, 2017 at 19:51
  • $\begingroup$ I would say that buckling is more likely to occur with a fixed-free condition due to the increased effective length. Buckling does not occur in flag-poles due to the lack of vertical load. However in the OP's case we have a vertical load at the top. The stiffness of the base fixity is therefore critical in assessing the stability of this structure. $\endgroup$ Nov 21, 2017 at 0:29

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