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# Moment of Inertia of Circular Tube Cross Section at an Angle13

## Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Does anyone have the formula for a circular tube cut at an angle to the axis? I have one for an ellipse, but if I make the major diameter equal to the minor diameter then I get a different number than for a straight circular cross section. They should be similar/exact.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Does that get weird because the apparent wall thickness will be different across the cut?

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Hi dik

I haven’t got one for an ellipse but I found this site, I am thinking you could do a graphical integration on a section if it’s drawn out.

https://studylib.net/doc/8800517/calculating-momen...

“Do not worry about your problems with mathematics, I assure you mine are far greater.” Albert Einstein

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

2
If you tilt a tube or rod so you have an elliptical area in a cross section, the moment of inertia in one direction increases by 1/cos(theta) and in the other direction increases by (1/cos(theta))^3.
If you look at the integral defining I, then every Y dimension becomes Y/cos(theta), that cos(theta) is a constant that can be pulled out of the integral, so you get the integral for a circular area with one of those two factors.
If you have a composite cross section that includes a tilted tube, I think that violates the assumptions made to derive beam bending in the first place, so I don't know that numbers you get using those I's are that significant. IE, the stresses won't all be tilted normal to your plan, they'll be axial in the tube.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Is this for a weld or the member itself? I'm not sure that elliptical treatment would even be valid for the member.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Thanks... I knew it wasn't easy... else I'd have posted the question on facebook or something of that ilk... I'll take a gander at the fox's post... it looks promising. If I have a solid elipse? and take away the equivalent inner ellipse, it should accommodate the difference in wall thickness due to the cut...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Koot... It's for a weld... and the difference between a tube with no angle and a circle is wildly different... it's to accommodate guardrails welded on a slope.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

2
If you find a formula for solid ellipse then outside minus inside with different minor/major axis dimensions.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

my I for a solid ellipse is pi*a^3*b/4 (like pi*r^4/4 for a circle), a is major semi-axis, b is minor semi axis, I about minor axis
and for a tube, subtract the hole.

you "could" derive it from the ellipse equation ... (x/a)^2 + (y/b)^2 = 1 ... but who's got time for that !

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)

#### Quote (If you find a formula for solid ellipse then outside minus inside with different minor/major axis dimensions.)

Maybe have to find a better ellipse... the formula I used initially was way out of whack... by a factor of nearly 2. I figured a weld on the perimeter would be similar to a sliced tube accommodating the weld size with different major and minor axis. Thanks...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)

thanks Agent... that formula is different than the one I used... will take a gander to see how it compares to a normal circle... I used agent's welding pattern program to do one in SMath and didn't want to use it for a mathematical ellipse... in a pinch, I would have, but didn't want to.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Does Bruhn have a formula about the weak axis?

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

The properties for a half ellipse are in Part 17 of the AISC Manual. Seems like one could use those to derive the properties for an entire ellipse.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

#### Quote (dik)

Does Bruhn have a formula about the weak axis?

Swap a and b...

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Can you draw the ellipse in AutoCAD and then use MassProp command to calculate the moment of inertia.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

"Does Bruhn have a formula about the weak axis?"

no, but as agent said, as his link says, swap the "a" and "b".

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Actually, I don't think this is valid even for the welds. Using the ellipse shape implicitly assumes that you're delivering load perpendicular to your tube walls which is surely inconsistent with your tube design. You could do the ellipse thing but, then, you'd just have to turn around and take the parallel to tube component of that anyhow. And that would just get you back to the circle.

As far as the formula for an ellipse goes, I'm on board with JStephen's method. When in doubt, trust the math.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
koot... it's not exact and I'm happy with it for welds... it's close enough... pi*a*b^3/4 works... thanks, gentlemen... problem solved...

#### Quote (If you tilt a tube or rod so you have an elliptical area in a cross section, the moment of inertia in one direction increases by 1/cos(theta) and in the other direction increases by (1/cos(theta))^3. If you look at the integral defining I, then every Y dimension becomes Y/cos(theta), that cos(theta) is a constant that can be pulled out of the integral, so you get the integral for a circular area with one of those two factors.)

That's what the a b^3 part works on... again thanks so much.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

In my opinion it's not just inexact dik. Rather, it's wholly incorrect to use the ellipse without adjusting for the component of weld resistance that would be parallel to the tube walls.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

You should be using the moment of inertia of a circular ring. That is the section you are considering, even though it is cut at an angle to its axis.

BA

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

As an example, consider if you were doing this to an square tube that you were diagonalizing to a rectangle. I would say that:

1) there would be some "stretch" improvement on the long / web sides but;

2) no improvement on the short / flange sides because pushing on them at an angle would bow them out.

Trying to think about how that pans out for for an ellipse hurts me brain.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

I'm with BA. The stresses will be distributed in the post as if it's a circle, and that's the stress the weld will see. Because it's cut on angle, you could argue you have more weld per unit length for an equivalent throat size, but I would just ignore that and size it as if it were a flat circle.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Blodgett weld as line formula for ellipse.

Sorry for the fuzzy.

I believe it's from "Solutions to Design of Weldments"

https://www.jflf.org/ProductDetails.asp?ProductCod...

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

A little less fuzzy.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)

#### Quote (You should be using the moment of inertia of a circular ring.)

That's how the post is designed (I hope no one thought I was using the elliptical perimeter for the tube design) and the weld checked based on the circular perimeter as well as the elliptical perimeter. I'm aware the elliptical perimeter is a bit off... but not likely significant... the circular perimeter greatly underestimates the weld strengthstress. I just wanted a bit of a handle on where it stood. I didn't mean to imply that I was using the increased section modulus based on the sloped cut. The tube stands by itself, as if it were normal and the weld is similar to that... I just wanted to get an idea of how much greater weld capacity there was if the ellipse were considered instead of the circle.

Thanks gentlemen.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

See ROARK`S Formulas for Stress & Strain.

Regards

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

In my note above, I'm assuming you section a tube at an angle, so the thickness normal to the section actually varies as you go around. That's different from laying out a uniform-size weld around the perimter of an ellipse.
That said, my solution for the problem is to just treat it as a circle and figure the elliptical weld is a little stronger, which is quick and easy.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
... JStephen... as a rough idea I used the Ix using inverse Cos() to get the major diameter with the tube OD as the minor diameter and subtracted this from the Ix using the major and minor diameters +D (I could have used 2xD, but chose the centreline of the weld).

Not exact, but a good ballpark. Thanks... and for small angles, it becomes a better guess.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

While my post does not provide the formulas to obtain the inertia of an inclined section of a circular tube, it's an alternative way to obtain the same result. I often use Inventor to calculate inertias of the sections of structural members.

Here's how you could obtain the inertia of an inclined section of a circular tube:

1) Create a tubular part:

2) Cut the tube using an inclined plane:

3) Obtain the inertia using the region properties:

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Thanks... I needed the formulae for an SMath program... done, and I'm happy with it. As the wise ol' owl says, "It's been a hoot." Thanks very much gentlemen. My original formula was in error, and I don't remember the source... and didn't record the source. With 2xD, before modified... and I didn't record the source for this one, either...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

My section properties spreadsheet will do a hollow ellipse:
https://newtonexcelbach.com/2020/08/06/section-pro...

The example has the long axis set to root 2 x the short axis, with the inner ellipse with a radius 2 units shorter on both axes.

The observant will note that the results suggest that Iy = Ix = Ixc, which is obviously wrong. Since the origin is at the centroid of both ellipses Ix = Ixc and Iy = Iyc. I will fix that and upload to the same address.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

While it is true that the thickness of a circular tube varies when cut at an angle, the weld material should be of constant size. So the formula could be based on an elliptical line element corresponding to the outer dimension of the section.

BA

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Yup that's how the formula treats it... I've modified that by using D rather than 2xD... to reduce the capacity by a tad... the added perimeter over the straight circular section with the full D, works, too. Thanks for everyone's help...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Some caution is warranted if the angle θ between the cut plane and the member axis is small. A fillet weld will not engage much metal at the toe and may be hard to access at the heel.

BA

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

BAretired: That's why fillet welds are not pre-qualified with a dihedral angle less than 60 degrees. See W59 figure 4.8 for example or equivalent in AWS. Once you have less than 60 degrees (or more than 135) the weld is no longer considered a fillet and must be specified as a partial penetration groove weld or like with effective throat clearly specified.

For raker design (HSS tubes connected to flat plates top and bottom on a 45 degree angle) we show a cut section on our drawings through the joint so it can be clearly seen where a fillet stops and a partial penetration weld begins. Typically we indicate the effective throat required on the small angle side, and the desired fillet weld on the larger angle side. On plan we indicate this via different line thicknesses.

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

....“it's to accommodate guardrails welded on a slope”.

@ dik. Please, can you tell us the tube diameter and the design loads?

Regards

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
Very often the 1.66x0.125 is the guard post of choice and sometimes the 0.140 wall thickness (properties for the 0.191 wall thickness are shown).

and design loads (the data columns line up... just from different parts of the program):

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

#### Quote (Enable)

BAretired: That's why fillet welds are not pre-qualified with a dihedral angle less than 60 degrees. See W59 figure 4.8 for example or equivalent in AWS. Once you have less than 60 degrees (or more than 135) the weld is no longer considered a fillet and must be specified as a partial penetration groove weld or like with effective throat clearly specified.

That's good information. So in this case, the minimum angle between member axis and cut plane would be 45 degrees if a fillet weld is to be used.

BA

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Fillet weld size at base (inclined or not) of post is 0,7 x thickness of pipe post

For design loads See OSHA 1926.502 “Fall protection systems criteria and practices”.

Regards

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)

#### Quote (with a dihedral angle less than 60 degrees. See W59 figure 4.8 for example or equivalent in AWS. Once you have less than 60 degrees (or more than 135))

Just to clear up a bit of confusion on my part. I don't have access to W59. Is the minimum dihedral angle, for a prequalified fillet weld, 45 degrees or 60 degrees?

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

Figure 4.8 equivalency of fillets relative to 90 degree joint (F = 1.0)

#### Quote (Clause 4.5.6)

The relation between dihedral angle and the effective throat shall be as follows:
(a) 60° to 135°: the effective throat (E) shall be considered equal to the theoretical throat (T) (see Figure 4.8);
(b) 45° to 60°: the effective throat shall equal the theoretical throat reduced by 3 mm (1/8 in), unless a larger effective throat is established through procedure qualification (see Figure 4.8); and
(c) Theta < 45°: the effective throat shall be established through procedure qualification (see Figure 4.8)

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
They are for the sngles on the opposite sides... thanks for the clarification, and the weld capacity would be modified by the F factor based on the fillet weld size. For example a fillet weld on a 60 deg angle on the RHS would have 0.71 times the equivalent 90 deg fillet weld. Is that correct?

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

It's the other way around. These are equivalency factors for leg length.

For example lets say you need a 6mm weld at 90 degrees. The equivalent leg length on a 60 degree joint would be S = 6mm*0.71 = 4.26mm. If you put a 6mm leg length then your capacity would be 1 / 0.71 (1.41x) as much as for a 6mm weld at a 90 degree joint. For the same leg length the effective throat is greater in a joint with a smaller angle and less in a joint with a larger angle.

https://app.aws.org/mwf/attachments//73/286473/SMA...

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)

thanks... I would have thought it would be reversed.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

#### Quote (dik)

thanks... I would have thought it would be reversed.

I believe the strength of the weld is proportional to the length of the green line shown below. Doesn't seem to be precisely equal to Enable's reference, but his is conservative.

BA

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
I realised that looking at the cross section shown that the acute weld was 'stockier'... and more robust... thanks... My inital thought was that the space was limited and the weld capacity would be smaller...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

I cannot understand how something so simple is treated here in something so complicated.
Is this railing for a nuclear power plant?
Even in a nuclear power plant they don't make it that complicated.

Sorry

Regards

### RE: Moment of Inertia of Circular Tube Cross Section at an Angle

(OP)
It's all part of a learning process... I often connect HSS tubes on an angle and have just use a circular cross section of the attachment moment; since I write a lot of programs to do my work... I thought I would check to see what sort of overkill there was. I've now got a method shown in the moment of inertia and section modulus program above that gives me a reasonable more exact solution. I do maybe 20 or 30 guardrails a month... In 15 minutes, I can check half a dozen conditions from various spans and cantilevers on various slopes... just making life easier actually. If I have one with a different base, I modify the template to cover it. I've attached a print of the various bases I have so far. The guardrail program is up to about 15 pages of calcs, now...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

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