Advanced Mechanics of Materials and Applied Elasticity

Equation of Advanced Mechanics of Materials and Applied Elasticity
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three-dimensional state of stress

 * $$[\tau_{i,j}]=

\begin{bmatrix} \sigma_x & \tau_{x,y} & \tau_{x,z} \\ \tau_{y,x} & \sigma_y & \tau_{y,z} \\ \tau_{z,x} & \tau_{z,y} & \sigma_z \end{bmatrix} $$

prismatic bars of linearly elastic material
\tau=\frac{T\rho}{J} $$ \sigma_x=-\frac{My}{I} $$ \tau=\frac{VQ}{Ib}$$
 * axial loading $$\sigma_x=\frac{P}{A}$$
 * torsion $$
 * bending $$
 * shear $$
 * $$T$$ torque.
 * $$V$$ vertical shear force from bending force.
 * $$I$$ moment of inertia about neutral axis(N.A.).
 * $$J$$ polar moment of inertia of circular cross section.
 * $$\rho$$ distance from the center of torque to the point.
 * $$Q$$ first moment about N.A. of the area beyond the point at which \tau_{x,y} is calculated.

thin-walled pressure vessels
\sigma_\theta=\frac{pr}{t}. $$
 * cylinder $$
 * $$\sigma_a=\frac{pr}{2t}.

$$ \sigma=\frac{pr}{2t}. $$
 * sphere $$
 * $$\sigma_\theta$$ tangential stress in cylinder wall.
 * $$\sigma_a$$ axial stress in cylinder wall.
 * $$\sigma$$ membrane stress in sphere wall.
 * $$p$$ internal pressure.
 * $$t$$ wall thickness.
 * $$r$$ mean radius.

axially loaded members
$$\sigma_{x'}=\sigma_x\cos ^2\theta.$$

$$\tau_{x'y'}=-\sigma_x\sin \theta\cos \theta$$

$$\sigma_{max}=\sigma_x$$

$$\tau_{max}=\pm0.5\sigma_x$$
 * $$\theta_{\max{\sigma}}=0^\circ,180^\circ. $$


 * $$\rho_{\max{\sigma}}=45^\circ,135^\circ. $$

differential equations of equilibrium
$$\frac{\partial \tau_{i,j}}{\partial x_j}+F_i=0, i,j=x,y,z.$$

plane-stress transformation
(2-dimensional stress, neglect the stress in the z coordinate.)

$$\sigma_{x'}=\tfrac{1}{2}(\sigma_x+\sigma_y)+\tfrac{1}{2}(\sigma_x-\sigma_y)\cos 2\theta+\tau_{xy}\sin 2\theta$$

$$\tau_{x'y'}=-\tfrac{1}{2}(\sigma_x-\sigma_y)\sin 2\theta+\tau_{xy}\cos 2\theta$$

$$\sigma_{y'}=\tfrac{1}{2}(\sigma_x+\sigma_y)-\tfrac{1}{2}(\sigma_x-\sigma_y)\cos 2\theta-\tau_{xy}\sin 2\theta$$

Stress tensor

$$\sigma_{x'}+\sigma_{y'}=\sigma_x+\sigma_y=$$constant.

$$\theta_{\min}=31.7^\circ+90^\circ(+180^\circ).$$

$$\theta_{\max}=31.7^\circ(+180^\circ).$$

$$\tau_{\min}=31.7^\circ(+90^\circ).$$

$$\tau_{\max}=31.7^\circ+45^\circ(+90^\circ).$$

principal stresses in plane
$$\sigma_{\max,\min}=\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{(\frac{\sigma_x-\sigma_y}{2})^2+\tau^2_{xy}}$$

$$\tau_{\max}=\pm\tfrac{1}{2}(\sigma_1-\sigma_2)$$

$$\tau'=\tau_{ave}=\tfrac{1}{2}(\sigma_1-\sigma_2)$$