User:Eas4200c.f08.vqcrew.f/hw3

HW 3

 Torsion of a Uniform Non-Circular Bar 

A uniform non-circular bar leads to warping of the cross-section. Warping is defined as the axial displacement along the x-axis (i.e. along bar length) of a point on the deformed rotated cross-section.







$$U_y =$$ y component of displacement vector $$\vec{PP}'$$

$$U_z =$$ z component of displacement vector $$\vec{PP}'$$

Important Definition : $$\displaystyle \theta = \frac{\displaystyle \alpha}{x}$$       =   Rate of Twist

Referring to Figure-4, assuming that the cross-section at x=0 remains stationary, and if the rotation angle $$\displaystyle \alpha$$ is small, then the displacement components at P are given by the following:

$$u = -r \displaystyle \alpha sin\displaystyle \beta = -\alpha y$$, which can be simplified to Equation (3.11):

(1) $$U_y = - \displaystyle \theta x z$$

Now the we have the displacement in the y direction, we can solve for the displacement $$U_z$$ which yields:

$$U_z = + PP' cos\displaystyle \beta = +\displaystyle \alpha y_p$$, this can be simplified to Equation (3.12):

(2) $$U_z = \displaystyle \theta x y$$

Now that we have the vertical and horizontal displacements the warping displacement in the x-axis can be found as a function of the y and z directions.

(3) $$U_x = \displaystyle \theta \displaystyle \psi (y, z)$$

Based on these three equations come the Kinematic Assumptions.

Small Angle Approximation



Road Map for Torsional Analysis of Aircraft Wing:

A) Kinematic Assumption

B) Strain Displacement

C) Equilibrium Equation (stresses)

D) Prandtl Stress Function $$\displaystyle \phi$$

E) Strain Compatibility Equation

F) Equation for $$\displaystyle \phi$$

G) Boundary Conditions for $$\displaystyle \phi$$

H) $$T = 2 \iint_A \displaystyle \phi \,dA$$

$$T = G J \displaystyle \theta$$

$$ J =\frac{-4}{\nabla^2 \phi}\iint_A \displaystyle \phi \,dA$$

I) Thin-Walled Cross-Section, $$T = 2 q A $$ (3.48) This equation was also derived in Problem 1.1.

J) Twist Angle $\displaystyle \theta$ Method 1.

$$\displaystyle \theta = \frac{1}{2 G A} \oint \frac{q}{t}\,ds$$