User:Eas4200c.f08.blue.a/Lecture 20

Problem Statement
Given the following:

1st wall at 1/4*c, 2nd wall at 3/4*c from leading edge.

You have 3 unknowns q1, q2, and q3

You can now use the following sets of equations and equalities:
 * 1) T=2Σ(qi*Ai)
 * 2) θ1=θ2
 * 3) θ2=θ3

Q: Find J, the Torsional Constant.

Note: You need t, the thickness, but G cancels out in the calculation.

Theory

T=GJθ

T=2qA

Engineering (ad hoc) Derivation

θ= 1/(2GA) Σ (q/t) * ds

This is the formula we shall use in order to find the amount of twist in a uniform bar with non-circular cross section subject to twist (torsion).



Displacement PP' due to α

PP'/OP=tanα =~ α (for small angles)

Project displacement PP' on the direction perpendicular to OP'

PP"=PP'cosα

PP"=OPtanα*cosα

=OP"tanα

Recall:


 * 1) OP=r
 * 2) OP"=ρ

PP"=rcosα*tanα

=ρα

which is the displacement of P in the direction tangent to the lateral surface of the bar.

Computing the strain:

STRAIN:    γ = PP"/dx = ρα/dx = ρθ

with θ=α/dx, which is also known as the rate of twist.

Note: This calculation is based on the assumption that alpha is a very small angle (also denoted as dα).