Calculate Section Modulus for an Irregular Section - (using X, Y Cordinates) 7

[DimzK]

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

 

[IDS]

dik - with inch units and principal axis parallel to the X axis, and origin at the centre of the circle I get:

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/

 

[BAretired]

@dik,

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

 

[dik]

(ERROR: Iy HAS TO BE TRANSFORMED) They should be identical if the angle of rotation is 0 deg/rad. It was my only quick check... I didn't check the initial formulae.

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

-Dik

 

[dik]

(ERROR: Iy HAS TO BE TRANSFORMED) Updated program:

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

-Dik

 

[IDS]

dik - could you post a summary of the results for the non-SMath users.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dik]

(ERROR: Iy HAS TO BE TRANSFORMED) Not much to it... attached. 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.



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

-Dik

 

[IDS]

Thanks dik. I probably do have SMath installed somewhere, but I prefer the spreadsheet style interface, so I'll stick wit Excel.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dik]

I use both... for AISC steel sections database and stuff related to it, Excel cannot be beat. I like SMath because of the formulas and the units.

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

-Dik

 

[IDS]

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/

 

[dik]

With phi primed equal to zero, there is absolutely no change to the original x and y axis configuration so all aspects of Ix', Iy' and Ixy' should be identical to the Ix, Iy and Ixy values. That's correct... rotation if any would be about the centroid. The only thing I checked was that with the x value calculated... the y value is consistent.

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

-Dik

 

[IDS]

But the prime coordinates have their origin at the centroid rather than the centre of the circle, so Iy' should be different even if the X axis is unchanged.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dik]

My error... I just realised the coordinates in the original drawing were at the centre of the circle... and not at the centroid. The Iy has to be corrected for the (area * x^2). Thanks, My apologies...


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

-Dik

 

[dik]

Corrected version:


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

-Dik

 

[dik]

and...



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

-Dik

 

[IDS]

How are X and Y-Axis Centroidal Distance defined?

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/

 

[dik]

I accidentally transformed the coordinates from the centre of the 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...


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

-Dik

 

[Celt83]

I get the same properties as IDS, for the 20" radius and 40 degree sweep. I also have the stress worked out with general bending formula or transformation of bending moment to the principal axis

:
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

 

[dik]

Yup...

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

-Dik

 

[Celt83]

nice.

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

 

[dik]

Correct... will fix designation and thanks.

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

-Dik