## Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

## Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

(OP)

Hi all,

I have a simple mechanics of solids type question which I have been googling, but I have not been able to find an answer.

I have an irregular shaped section with bending moments imposed about its X axis as shown below.

The centroid of the shape coincide with the x=0 aand y = 0 of the co-ordinate system.

Now for me to check my bending stresses at point 1 , 2 and 3, I need calculate my section modulus about Section X for each of these points.

If this was a simple rectangle, we could simply use B * H^2/6.

Is there any formular which would let me calculate the corresponding section modulus for these three points if I provide co-ordinates and Ix, Iy and Ixy values which I cam get of Autocad.

For context, I am trying model a deteriorated timber pile to check load capacity.

Any help would be appreciated.

Cheers,

Dimuth

I have a simple mechanics of solids type question which I have been googling, but I have not been able to find an answer.

I have an irregular shaped section with bending moments imposed about its X axis as shown below.

The centroid of the shape coincide with the x=0 aand y = 0 of the co-ordinate system.

Now for me to check my bending stresses at point 1 , 2 and 3, I need calculate my section modulus about Section X for each of these points.

If this was a simple rectangle, we could simply use B * H^2/6.

Is there any formular which would let me calculate the corresponding section modulus for these three points if I provide co-ordinates and Ix, Iy and Ixy values which I cam get of Autocad.

For context, I am trying model a deteriorated timber pile to check load capacity.

Any help would be appreciated.

Cheers,

Dimuth

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

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

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

and maybe the second...I think that will do it... just a matter of putting the stuff together, I think. That goes back so far, it makes my head hurt... over 50 years. Having the new moments of inertia, it's a matter of determining the Section Modulii. Let me know if it works.Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

https://newtonexcelbach.com/2018/06/24/using-secti...

will find section properties for a variety of defined shapes.

The standard shape for a circular segment is symmetrical about the x-axis, but the group feature allows you to rotate any shape about the origin, which should give the results you need. The link above gives an example of rotating a circular segment through 90 degrees, but you can use any angle.

Alternatively (and the way I'd probably go) you can define the shape with coordinates and get the section properties for any shape.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

I may have to go for the old method of picking Ixx / distance from centroid to extreme comrpession/tension fiber, <- and guess the extreme points

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

On the DefShapes sheet, select "Circular Sector" and enter the radius and theta:

Scroll down to the Group properties input, enter -Xcentroid as "X offset" and the rotation angle, then click "Create New Group":

Go to the Coords_Group sheet and click Plot Group. You can then find the coordinates of the three corner points, with the origin at the centroid, which you can use to calculate the 3 section modulus values:

I have attached a copy of the spreadsheet. Please ask if any problems.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

you can always go back to the basic definition of moments of area ... intergral of xdA and integral of x^2dA.

or you can divide your shape into smaller standard pieces (squares and triangles) and sum.

another day in paradise, or is paradise one day closer ?

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

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

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

another day in paradise, or is paradise one day closer ?

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

(Inverted commas because that's an impressive amount of deterioration you seem to be showing.)## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

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

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

That formula is only correct for symmetric/principle axis bending, when Ixy=0.

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

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

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

https://res.cloudinary.com/engineering-com/raw/upload/v1637463102/tips/Moment_of_Inertia-Segment_prsdxc.sm

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

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

_{xy}= 0 is surprising!Edit: See below!

BA

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

I guess the formula's correct... I didn't check the derivation... just plugged numbers into a formula... and if you set the angle of rotation equal to 0, you get the same values for no rotation... something seems to be OK.

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

BA

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Rotating the axis by 20 degrees and moving the origin to the CoG I get:

Note that your Iy seems to be with the Y axis through the circle centre, and your Iy' has the same value, but should be much smaller.

Your Ix' agrees with my Iv.

You have zero for Ixy' but it should be non-zero for non-principal axes.

Note that my second screenshot uses point coordinates with a large number of segments for the curve; hence the small non-zero values for the centroid and first moments of area.

Also I think we are over-complicating this. Why not find the section properties and section modulus values with the principal axes parallel to horizontal and vertical, then rotate the applied moments, rather than rotating the cross section?

Note:

I posted that before reading your most recent post.

So we agree that we don't need to rotate the section, but the Iy does need to be with the Y axis through the centroid (i.e. the IYc value in my first screenshot).

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Iy/Ix = 24.3 seems wrong.

Ixy = 0 is okay when X axis is an axis of symmetry.

The primed values, Ix', Iy' and Ixy' can't be equal to the non-primed values, Ix, Iy and Ixy.

BA

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

It was my only quick check... I didn't check the initial formulae.-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

If you're not running SMath... you should. Other than a couple of minor hiccups, it's a great free program and really useful.Only downside is political and that it was written by a Russian.

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

I like SMath because of the formulas and the units.-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

With the origin at the centre of the circle and X axis passing through the centroid I agree with your Ix, Iy and Ixy values, and X axis centroidal distance, but:

Yc should be 0

For Theta' = 0 the Ix' and Ixy' are unchanged, but Iy' should be much smaller.

I get: Iy' = 2987.p in^4

Also the diagram is confusing, but I am assuming for Theta' = 0 the transformed axes should be parallel to the original axes.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

If the origin of the original section is at the centroid, shouldn't they both be zero?

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

~~circle~~segment to the centroid and should have modified Iy. Segment is symmetric about the x-axis so change in Iy is simply the original Iy less the (area times the x distance squared).Ix doesn't change. I've noted the error in the messages above.A log of changes to the program show up in the SMath program, but not in the *.pdf file.I think I'll go sit in the corner now and put on my pointed hat...

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

A: 139.6263 in2

Cx: 12.2763 in

Cy: 4.4682 in

Centroidal Axis Properties Parallel to global x and y axis

Ixx: 1326.9146 in4

Iyy: 2767.8334 in4

Ixxyy: 604.5372 in4

Jzz: 4094.7480 in4

rxx: 3.0827 in

ryy: 4.4523 in

rzz: 5.4154 in

Principle Axis Properties

2Θ -2.4435 rad

A 2047.3740

B -720.4594

I1 2987.8669 in4

I2 1106.8811 in4

Iuu 2987.8669 in4

Ivv 1106.8811 in4

Iuuvv 0.0000 in4

Θu -70.0000 degrees

Θv 20.0000 degrees

Jkk 4094.7480 in4

ruu 4.6259 in

rvv 2.8156 in

rkk 5.4154 in

Elastic Section Moduli

Sx+ 158.2009 in3

Sx- 296.9671 in3

Sy+ 358.3576 in3

Sy- 225.4608 in3

At User Point

x 19 in Dist. To U 6.8421 in

y 6 in Dist. To V -0.8602 in

Sx 866.2574 in3 Su 436.6901 in3

Sy 411.6556 in3 Sv -1286.7314 in3

Elastic Stresses at user point from loads applied at shape centroid

P = 0.0000 kips 0.000 lbs positive = tension

Mx = 10.0000 ft-kips 120000.000 in-lbs positive creates Tension on top surface

My = 0.0000 ft-kips 0.000 in-lbs positive creates Tension on left surface

σ,axial = P/A = 0.0000 ksi 0.0000 psi

Mx:

y (in) dy = (y-cy) (in) σy =( (Mx Iyy - My Ixxyy) / ( Ixx Iyy - Ixxyy^2) ) * dy [ General Elastic Bending Stress Formula which captures non principle axis bending, if Ixxyy = 0 reduces to the common stress formula = Mx dy /Ixx ]

6.0000 1.5318 153.8349 psi

My:

x (in) dx = (x-cx) (in) σx =( (My Ixx - Mx Ixxyy) / ( Ixx Iyy - Ixxyy^2) ) * dx [ General Elastic Bending Stress Formula which captures non principle axis bending, if Ixxyy = 0 reduces to the common stress formula = My dx /Iyy ]

19.0000 6.7237 -147.4850 psi

σ,user point = 0.0063 ksi 6.3499 psi [From General Bending Stress Formulas using Geometric Bending]

σ,user point = 0.0063 ksi 6.3499 psi [From Transformation of Bending Moments to Principal Axis]

Principal Axis Moments

Mv = 9.397 ft-kips 112763.114 in-lbs σ,v = -87.6353 psi

Mu = 3.420 ft-kips 41042.417 in-lbs σ,u = 93.9852 psi

σ,bending = 6.3499 psi

Spreadsheet can be downloaded from here: Link

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Small correction to your Smath sheet Ixy and Ixy' are the products of inertia not the polar moments of inertia.

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

Github Link

Original Eng-Tips thread where they announce it

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

A good link, Bookmarked, thanks.

-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

I have never used Smath before.

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

The text in blue are the units and the program takes care of them, too. Only downside to the program is that it was written by a Russian dude and that my have political fallout... I'm not concerned about such stuff. Larger programs can be written by cutting and pasting parts of other programs. Most of my programs are over 10 pages.-Dik

## RE: Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates)

First find the offsets from the points where stresses are required to the principal axes (points 1-3 are the corners, point 4 is the point used in Celt's calcs:

Then calculate stresses using the principal axes then the global XY axes:

So the results are consistent and also agrees with Celts results.

Having worked through the process I now think it is simplest to use the original axes with the full bending stress formulas:

Sigy =( (Mx Iy - My Ixy) / ( Ix Iy - Ixy^2) ) * dy

Sigx =( (My Ix - Mx Ixy) / ( Ix Iy - Ixy^2) ) * dx

The spreadsheet is attached below.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/