Section Modulus Formula

no, they aren't necessarily non-standard shapes but I would like to have my spreadsheet figure the section of a particular shape whenever I plug in the outer dimensions and the wall thickness.

An example of my inquiry is; TS 8x8x.375 steel tube... in the manual, it's section modulus is listed as 26.4 but if I use the formula for figuring the section of a hollow rectangle I get 27.77 which is obviously more section because it is not compensating for the radiused corners. Where you referenced I_xx =I_yy =0.0549r⁴, e_x =e_y = 0.424r, is the 'r' = radius of gyration or radius? if it is radius of gyration, then I would figure it for a rectangle with a shape of 6x.375 and one with a shape of .375x6" and then the moment of inertia with =0.0549r⁴ ?

Have a look here: http://www.engineeringtoolbox.com/

thanks, i already checked there and that site only has the formula for hollow rectangle. same as in the AISC manual.

This might be of some use. I use it as a reference table for spreadsheets I write when needing the properties of members.

This is the spreadsheet by Alex Tomanovich showing the AISC 13.0 section properties of all the standard shapes.

I simply use it as a lookup reference for my spreadsheets, and build from there. Don't know if that helps, but the other suggestions of figuring the Ixx, Iyy and the polar moment of the individual sides and the 4 corner arcs is probably the best way to work out a non-standard section. That is how I have done it in the past.

now thats a nice little spreadsheet. thanks. I do still want to figure out this formula though, lol...

Consider a steel tube of width b, depth d and thickness t. Corner outer radius is R and inner radius r.

The shape consists of 2 sidewalls, 2 horizontal walls and 4 - quarter rings. You must first compute Ix and Iy, then find Sx and Sy by the relationship Sx = 2*Ix/d and Sy = 2*Iy/b.

The two top quarter rings can be represented by a single half circle of radius R and a negative half circle of radius r. The area of the larger half circle is A = πR²/2 or aaproximately 1.57R². The centroid measured from the diameter is c = R*4/3π = 0.424R. The moment of inertia of the large half circle about its centroid is I = R⁴(π/8-8/9π) or 1.098R⁴. Similar expressions apply to the small half circle substituting r for R.

The centroid of the large half circle is y_R = (d/2+0.424R) above the neutral axis of the member. The half circle representing the bottom corners is the same distance below the N.A.

The contribution of all four corners to overall Moment of Inertia Ix is 2*1.098(R⁴-r⁴) + 2(A*y_R² - a*y_r²) where a is the area of the smaller half circle.

The sidewalls taken together are 2t x (d-2R) and have Ix of 2t(d-2R)³/12.

The top and bottom walls are each (b-2R) x t. Together, they contribute Ix = (b-2R)*(d³-(d-2t)³)/12.

Finally, Ix = sum of individual Ix above and Sx = 2*Ix/d. The procedure for calculating Iy and Sy is similar.

In standard hollow sections, r = t and R = 2t, but I thought it better to keep it general so you could vary these values. In any case, to retain constant thickness, R = r + t.

Hope I didn't make too many errors in typing and I hope it helps.

Re-reading my earlier post, I noted an error in the expressions for y_R and y_r. They should read:

y_R = (d/2 -R + 0.424R) or y_R = (d/2 - 0.576R)

y_r = (d/2 -R + 0.424r)

Sorry I wasn't logged in when I made my last post... Thank you guys! let hear some feedback. Also I found it obviously more accurate to use the actual formula instead of replacing them with rounded constants...

This is from the

Steel Tube Institute

HSS DESIGN GUIDE: Section Properties & Design Information, V1 pg#28-29

HSS Properties

The tables in this section provide the dimensions and properties for hollow structural sections(HSS). Effort was made to include all shapes available in the United States; however, inclusion in these tables does not imply that a section is commonly produced or stocked. Please refer to the Section Selection Guide in this manual for more information.

It is highly important that special care be taken in selecting the appropriate section properties for the material being specified.

The following are assumptions made in the generation of section properties given in the tables:

• Design Thickness

• For ASTM A500, design thickness = 0.93 * nominal thickness.
• For ASTM A1085 and ASTM A1065, design thickness = nominal thickness.

• Order of shapes

• For square and rectangular, thicknesses divisible by 16 are listed first, then gage thicknesses, then miscellaneous thicknesses.
• For circular, shapes are listed in order of decreasing thickness.

• The nominal weight is based on the nominal thickness.

• ID, A, I, S, Z, r, J, C all are based on design thickness.

• Corner radii

• 2*tdes for A500 and A1085o For A1065, corner radii are based on 3*tdes for large shapes and 2*tdes for smaller shapes (shapes based on 2*tdes are shaded in the tables).

• b/t and h/t are based on R = 1.5*tdes.

• Workable flats are based on the nominal thickness accurate to the nearest 1/16th inch and are based on 2.25*tnom for A500 and A1085. For A1065, workable flats are based on 3*tnom for large shapes and 2.25*tnom for smaller shapes (shapes based on 2.25*tnom are shaded in the tables). Values of two inches or smaller are not shown.

• Properties o Moment of inertia, I Square and rectangular:
I = (tdes/6)(H-2*R)^3+(tdes^3/6)(B-2*R)+(tdes/2)(B-2*R)(H- tdes)^2+Ir+Ar[(H-2*R)(1/2)+yr]^2

• Elastic section modulus, S

Square and rectangular: S = 2*I/H

• Radius of gyration, r

Square and rectangular: r = Sqrt(I/A)

• Plastic section modulus, Z

Square and rectangular: Z =(tdes/2)(H − 2R)^2 + tdes(B-2*R) ∗ (H − tdes) + Ar[1/2(H − 2*R) +ye]

• Torsional stiffness constant, J

Square and rectangular: J = 4*tdes*(Ao)^2/(Pc)

• Torsional shear constant, C

Square, rectangular, and circular: C = 2*tdes*Ao

• Radius properties for square and rectangular shapes:

Moment of inertia of corner radii, total:

Ir = 4[R^4 − (R − tdes)^4] ∗ (pi/16 − (4/9*pi))

Area of corner radii, total: Ar = pi[R^2 − (R − tdes)^2]

Center of gravity of corner radius, measured from end of radius:

Xr, Yr = [4/(3*pi)]*[R^3 − (R − tdes)^3]/[(R^2 − (R − tdes)^2]

• Torsional properties for square and rectangular shapes, measured at tdes/2:

Gross area enclosed by shear flow path: Ao = (B − tdes) ∗ (H − tdes) − Rc^2(4 − pi)

Perimeter of shear flow path: Pc = 2*[(B + H − 4*R)] + 2*pi*Rc

{Note: The original formula contained an error that is corrected by multiplying the function in the [square brackets] by two}

Radius of shear flow path: Rc = R − tdes/2