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Design and Calculations

Better Understanding Section Modulus, Design Moment and Moment of Inertia

Both the Moment of Inertia and the section modulus are measurements of the relative stiffness of a cross section of steel piling.

Generally speaking, I (Moment of Inertia) is a geometrical value, used for stiffness determination and is therefore important to determine deflections in the vertical cross section and is used for more general calculations when compared to section modulus which is usually used to determine the resistance in the horizontal cross section against bending moments.

When calculating the stress in a steel pile, the formula using I is:

stress = M*y / I

where M is the bending moment at a point on the steel pile (called Design Moment) and y is the vertical distance from the bending axis at the middle (centroid) of the cross section. This is a general formula, because you can determine what the stress is at any point in the cross section by plugging in a value for y.

However, for most civil engineering work using steel, the engineer is not as concerned about what the stress is at a given distance from the centroid of the steel pile as they are concerned about when it will yield. Therefore, section modulus is a more important and useful comparison and design criteria. To determine the section modulus, Z, you divide the Moment of Inertia by y.

Therefore,

Z = I/y

Why is this more useful for engineers? Because if you switch this around, it also means that

I = Z*y

Substitute this into the stress formula, and you get:

stress = M*y / Z*y

The y’s cancel out and you now have:

stress = M/Z

This is the stress at the extreme fiber of the beam, which is the worst case scenario. And obviously the worst case scenario is what civil engineers usually design for, in terms of designing a steel sheet pile for maximum strength.

Note: On most steel piling projects that are to be bid for construction, it is best to have a Design Moment specified (e.g., 100 k-in/ft.) that engineers can work from, rather than a specified steel section, as this does not tell engineers the exact stresses that they need to work from.

Formulas for calculating neutral axis, moment of inertia and section modulus

e = (Ab x hb/2) + ((hb-fb/2) x Ac1Ac2)) / AbAc1+Ac2

Then the total Moment of Inertia is calculated by:

Ik = ((e-hb/2)2 x Ab) + ((e – (hb-fb/2))2 x (Ac1 + Ac2))

and:

IT = n (Ib + Ik + Is) + 2Is

Therefore the section modulus can then be calculated by:

(IT/e) / l

where:

Ib = beam’s Moment of Inertia
Is = sheet’s Moment of Inertia
Ic1 = connector 1’s Moment of Inertia
Ic2 = connector 2’s Moment of Inertia
l = panel width
n = number of beams
hb = beam height
fb = beam flange thickness
Ab = beam area
Ac1 = connector 1’s area
Ac2 = connector 2’s area

Calculating Section Modulus From a Given Design Moment

Calculating Section Modulus from a Given Design Moment

S = M / Fa

S = section modulus
M = design moment
Fa = allowable bending stress

Here is a specific example of how to determine the required section modulus given a design moment of 650 k-ft/ft for different grades of steel:

Using ASTM 572 Grade 50:
Given that the US Army Corps of Engineers has an allowable bending stress of 25 ksi for A572 Grade 50 steel (see below), the section modulus = 650 k-ft/ft x 12 in/ft / 25 k/in2
Therefore, the required section modulus is 312 in3/ft
Using S430 GP:
Section modulus = 650 k-ft/ft x 12 in/ft / 31.2 k/in2
Therefore, the required section modulus is 250 in3/ft
The US Army Corps of Engineers Design of Sheet Pile Walls Engineer Manual from 1994 recommends accounting for a safety factor for the allowable bending stress of 50% (.50). Hence (Fa = .50 x _____ ksi of the given steel grade)
Therefore:
A572 Grade 50 (50 ksi) has an allowable bending stress Fa = 25 ksi
S355 GP (52 ksi): Fa = 26 ksi
A572 Grade 60 (60 ksi): Fa = 30 ksi
S430 GP (62.4 ksi): Fa = 31.2 ksi

1 kip (k) = 1,000 lb
1 ksi = 1,000 lb/sq in

ASTM 572 Grade 60 vs. S430 GP

The two are very similar. In essence, S430 GP should be considered a stronger alternative to a ASTM A572 Grade 60 steel due to the fact that it carries a minimum KSI of 62 versus a KSI of 60 in ASTM A572 Grade 60.

S430 GP is a non-alloy steel grade used in sheet piling applications per EN 10248. The chemical requirements and mechanical properties of S430 GP are comparable with ASTM A572-60 as shown below. (YS and UTS for S430 GP are converted to ksi from MPa.)

Chemistry

All values are maximums.

Element EN 10248 S 430 GP ASTM A572-60
C 0.24 0.26
Mn 1.60 1.35
P 0.040 -
S 0.040 0.050
Si 0.55 -
N 0.009 -

Mechanical Properties

EN 10248 S 430 GP ASTM A572-60
YS MIN 430 Mpa (62.4 ksi) 60 ksi
UTS MIN 510 Mpa (74.0 ksi) 75 ksi
E* min 19% 16% (8"GL), 18% (2" GL)

Elongation for S430 GP is calculated using a gauge length Lo proportional to the CSA of the test specimen Lo = 5.65√So. Elongation for A572-60 is calculated using a gauge length of 8" or 2" per ASTM A370.

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