The shear flow in each wall segment there are two cells and

The shear flow in each wall segment (there are two cells, and three segments here) The rate of twist (d theta/dz) The torsion constant \"J\" for the beam (defined such that T = GJ (d theta/dz))

Solution

Let, the reference line from where the centroid is to be measured is tangent to the dome of the semi-ellipse.

Calculation of the centroid is given in the following table:

Section

Area (a) in inch²

Distance of individual section from the reference line (x) in inch

a*x

a*x²

I_self

Elliptical

1

1.67

1.67

2.79

Rectangular

1.48

5.72

8.47

48.42

Triangular

21.2

19.28

408.74

7880.43

23.68

418.88

7931.64

The centroid = ( a*x)/ ( a) = 418.88/23.68 = 17.69 inch

Polar moment of inertia = Sum of the Polar moment of inertis of the individual sections about the centroid

= I_self + a*x²

This will give the value of the torsion constant (J)

Section	Area (a) in inch2	Distance of individual section from the reference line (x) in inch	a*x	a*x2	Iself
Elliptical	1	1.67	1.67	2.79
Rectangular	1.48	5.72	8.47	48.42
Triangular	21.2	19.28	408.74	7880.43
	23.68		418.88	7931.64