# Advanced Mechanics of Materials and Applied Elasticity

 Subject classification: this is an engineering resource.
 Subject classification: this is a physics resource.

## Equation of Advanced Mechanics of Materials and Applied Elasticity

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### Chapter 1:Analysis of Stress

#### three-dimensional state of stress

${\displaystyle [\tau _{i,j}]={\begin{bmatrix}\sigma _{x}&\tau _{x,y}&\tau _{x,z}\\\tau _{y,x}&\sigma _{y}&\tau _{y,z}\\\tau _{z,x}&\tau _{z,y}&\sigma _{z}\end{bmatrix}}}$

#### prismatic bars of linearly elastic material

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

#### thin-walled pressure vessels

• cylinder ${\displaystyle \sigma _{\theta }={\frac {pr}{t}}.}$
${\displaystyle \sigma _{a}={\frac {pr}{2t}}.}$
• sphere ${\displaystyle \sigma ={\frac {pr}{2t}}.}$
${\displaystyle \sigma _{\theta }}$ tangential stress in cylinder wall.
${\displaystyle \sigma _{a}}$ axial stress in cylinder wall.
${\displaystyle \sigma }$ membrane stress in sphere wall.
${\displaystyle p}$ internal pressure.
${\displaystyle t}$ wall thickness.
${\displaystyle r}$ mean radius.

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

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

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

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

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

#### differential equations of equilibrium

${\displaystyle {\frac {\partial \tau _{i,j}}{\partial x_{j}}}+F_{i}=0,i,j=x,y,z.}$

#### plane-stress transformation

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

${\displaystyle \sigma _{x'}={\tfrac {1}{2}}(\sigma _{x}+\sigma _{y})+{\tfrac {1}{2}}(\sigma _{x}-\sigma _{y})\cos 2\theta +\tau _{xy}\sin 2\theta }$

${\displaystyle \tau _{x'y'}=-{\tfrac {1}{2}}(\sigma _{x}-\sigma _{y})\sin 2\theta +\tau _{xy}\cos 2\theta }$

${\displaystyle \sigma _{y'}={\tfrac {1}{2}}(\sigma _{x}+\sigma _{y})-{\tfrac {1}{2}}(\sigma _{x}-\sigma _{y})\cos 2\theta -\tau _{xy}\sin 2\theta }$

Stress tensor

${\displaystyle \sigma _{x'}+\sigma _{y'}=\sigma _{x}+\sigma _{y}=}$constant.

${\displaystyle \theta _{\min }=31.7^{\circ }+90^{\circ }(+180^{\circ }).}$

${\displaystyle \theta _{\max }=31.7^{\circ }(+180^{\circ }).}$

${\displaystyle \tau _{\min }=31.7^{\circ }(+90^{\circ }).}$

${\displaystyle \tau _{\max }=31.7^{\circ }+45^{\circ }(+90^{\circ }).}$

#### principal stresses in plane

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

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

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