Weld design calculation, as discussed in the previous weld sizing article, requires determining the moment of inertia equations of the whole fillet weld.

We will discuss the steps required for calculating the area moment of inertia as well as the polar moment of inertia for the welding joint of the previous article which has a rectangular shaped joint like below:

Where,

*a = 150 mm*

*b = 100 mm*

**Little theory: **

- The area or the second moment of inertia of a planer cross section of a beam define the beam’s ability to withstand the bending and torsional shear stress.
- The area moment of inertia of a composite section can be calculated by adding/subtracting the sub-areas.
- The area moment of inertia about Z axis of a planer area laid in XY plane is called polar moment of inertia (Jz).
- The polar moment of inertia (Jz) can be found by adding the area moment of inertia about the X axis (Ix) and that about the Y axis (Iy) or
**Jz=Ix+Iy**.

**Steps for calculating the area and polar moment of inertia equations of the above fillet weld joint:**

The area moment of inertia about the X and the Y axis are calculated by subtracting the second moment of inertia values of the inner rectangular area from that of the outer rectangular area.

Finally, the polar or torsional moment of inertia (Jz ) is calculated by summing up the Ix and Iy.

**Step-1: Calculating Ix:**

**Ix = (Ix of the outer rectangle) – (Ix of the inner rectangle) **

** = [a*b ^{3}/12] – [(a-2)*(b-2)^{3}/12]**

* Putting, a=150 and b=100*

** Ix = 12500000 – 11608034.66 = 891965.34 mm ^{3}**

**Step-2: Calculating Iy: **

** Iy = (Iy of the outer rectangle) – (Iy of the inner rectangle) **

** = [b*a ^{3}/12] – [(b-2)*(a-2)^{3}/12]**

*Putting, a=150 and b=100*

** Ix = 28125000 – 26474634.66 =1650365.34 mm ^{3}**

**Step-3: Calculating Jz: **

**Jz = Ix + Iy = 891965.34 + 1650365.34 = 2542330.68 mm ^{3 }**

Shouldn’t the units be in mm^4??

(mm)^3 * mm = mm^4

You are right, the units should be mm^4 instead of mm^3. Thanks for pointing it out, i will correct it soon.

is it allowed to add different section of moment of inertia of fillet weld.