User:Eas4200c.f08.wiki.b/HW3

The rate of twist angle, $$\displaystyle \theta $$, is defined as


 * $$\displaystyle \theta = {\alpha \over x} $$

Where $$\displaystyle \alpha $$ is the angle of rotation. The horizontal displacement (displacement in the y-direction) is


 * (1) $$\displaystyle u_y = -\theta x z $$

The vertical displacement of a point of the cross section (displacement in the z-direction) can be determined from


 * $$\displaystyle u_z = +(PP') \cos \beta = +\alpha \zeta_p $$

Where $$\displaystyle (OP)\alpha $$ can be substituted in place of $$\displaystyle PP' $$ and $$\displaystyle \theta_x $$ can be substituted in place of $$\displaystyle \alpha $$, thus resulting in the following


 * (2) $$\displaystyle u_z = +\theta x y $$

The displacement due to warping along the x-axis is defined as


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

Equations 1, 2, and 3 make up the kinematic assumptions.

=Roadmap for Torsional Analysis=

The following list lays out a general step by step roadmap of equations to apply for the torsional analysis of an aircraft wing:


 * A. Kinematic assumptions (equations 1, 2, and 3 above)
 * B. Strain-displacement relation given by the two equations:
 * $$\displaystyle \gamma_{yz} = {\partial u_x \over \partial y} + {\partial u_y \over \partial x}$$
 * $$\displaystyle \gamma_{zx} = {\partial u_x \over \partial z} + {\partial u_z \over \partial x}$$
 * C. Equilibrium equations for stresses:
 * $$\displaystyle {\partial \sigma_{yy} \over \partial y} + {\partial \tau_{zy} \over \partial z} + {\partial \tau_{xy} \over \partial x} = 0 $$
 * $$\displaystyle {\partial \tau_{yz} \over \partial y} + {\partial \sigma_{zz} \over \partial z} + {\partial \tau_{xz} \over \partial x} = 0 $$
 * $$\displaystyle {\partial \tau_{yx} \over \partial y} + {\partial \tau_{zx} \over \partial z} + {\partial \sigma_{xx} \over \partial x} = 0 $$
 * But, since $$\displaystyle \tau_{yx} $$ and $$\displaystyle \tau_{zx} $$ are independent of x in the displacement field, the equations reduce to the following:
 * $$\displaystyle {\partial \tau_{yx} \over \partial y} + {\partial \tau_{zx} \over \partial z} = 0 $$
 * D. Prandt stress funtion $$\displaystyle \phi (y,z) $$:
 * $$\displaystyle \tau_{yx} = {\partial \phi \over \partial z} $$
 * $$\displaystyle \tau_{zx} = -{\partial \phi \over \partial y} $$
 * E. Strain compatibility equation:
 * $$\displaystyle {\partial \gamma_{zx} \over \partial y} - {\partial \gamma_{yx} \over \partial z} = 2 \theta $$
 * F. Equation for $$\displaystyle \phi $$
 * $$\displaystyle {\partial^2 \phi \over \partial y^2} - {\partial^2 \phi \over \partial z^2} = -2G \theta $$
 * G. Boundary conditions for $$\displaystyle \phi $$:
 * The traction free boundary condition $$\displaystyle t_x = 0 $$ on a lateral surface yields the following:
 * $$\displaystyle {d \phi \over ds} = 0 $$
 * thus, $$\displaystyle \phi $$ is a constant. But, this constant becomes arbitrary since it is a solid section with only one contour boundary, thus
 * $$\displaystyle \phi = 0 $$
 * H. Resultant torque:
 * $$\displaystyle T = 2 \iint_A \phi dA $$
 * I. $$\displaystyle T = G J \theta $$ where
 * $$\displaystyle J = -{4 \over \triangledown^2 \phi} \iint_A \phi dA $$
 * $$\displaystyle T = 2q \bar A $$
 * J. Twist angle
 * $$\displaystyle \theta = {1 \over 2G \bar A} \oint {q \over t}ds $$
 * where $$\displaystyle s $$ is the curvilinear coordinate along a thin wall