User:Eas4200c.f08.blue.a/Lecture 15

$$\theta = \frac{\alpha}{x} $$ rate of twist    (1)

$$u_{y}=-\theta \times z $$ is the horizontal portion

$$u_{z}=+(PP' )\cos \beta$$

$$=\alpha y_{p}$$

$$u_{z}=+\theta \simeq y$$    )(2)

z is the displacement of a point on the cross section

For the warping displacement along the x-axis is

$$ u_{x} = \theta \psi(y, z)$$    (3)

The kinematic assumptions are equations (1), (2), and (3)

After the following road map, this derivation will continue.

Road map for torsional analysis of an aircraft wing

A. Kinematic Assumption

B. Strain Displacement Relationship

C. Equilibrium Equations

D. Prandth Stress Function $$/phi$$

E. Strain Compatibility Equations

F. Equation for $$\phi$$

G. Boundary Condition for $$\phi$$

H. $$T = 2\int \int \phi dA $$

T = GJ$$\theta$$

J = moment of inertia for a circular cross section

Moment of inertia for a non circular cross section is

$$J = \frac{-4}{\bigtriangledown ^{2}\phi}\int \int \phi dA$$

I. Thin walled cross section

Ad-Hoc Assumption

Formal Derivation

$$T = 2q\bar{A}$$

J. Twist angle $$\theta$$: Method 1

$$\theta = \frac{1}{2G\bar{A}} \oint_{}^{}{}\frac{q}{t}ds$$

s = curvilinear coordinate along this wall

K. Sec. 3.6 on multi cell thin walled cross-section