# 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

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