# Advanced Mechanics of Materials and Applied Elasticity

## Equation of Advanced Mechanics of Materials and Applied Elasticity

REDIRECT User:Oh_Isaac

### Chapter 1:Analysis of Stress

#### 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

• axial loading $\sigma _{x}={\frac {P}{A}}$
• torsion $\tau ={\frac {T\rho }{J}}$
• bending $\sigma _{x}=-{\frac {My}{I}}$
• shear $\tau ={\frac {VQ}{Ib}}$
$T$  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

• cylinder $\sigma _{\theta }={\frac {pr}{t}}.$
$\sigma _{a}={\frac {pr}{2t}}.$
• sphere $\sigma ={\frac {pr}{2t}}.$
$\sigma _{\theta }$  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.

$\sigma _{x'}=\sigma _{x}\cos ^{2}\theta .$

$\tau _{x'y'}=-\sigma _{x}\sin \theta \cos \theta$

$\sigma _{max}=\sigma _{x}$

$\tau _{max}=\pm 0.5\sigma _{x}$

$\theta _{\max {\sigma }}=0^{\circ },180^{\circ }.$
$\rho _{\max {\sigma }}=45^{\circ },135^{\circ }.$

#### 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

$\sigma _{\max ,\min }=\sigma _{1,2}={\frac {\sigma _{x}+\sigma _{y}}{2}}\pm {\sqrt {({\frac {\sigma _{x}-\sigma _{y}}{2}})^{2}+\tau _{xy}^{2}}}$

$\tau _{\max }=\pm {\tfrac {1}{2}}(\sigma _{1}-\sigma _{2})$

$\tau '=\tau _{ave}={\tfrac {1}{2}}(\sigma _{1}-\sigma _{2})$