Advanced Mechanics of Materials and Applied Elasticity

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Equation of Advanced Mechanics of Materials and Applied Elasticity

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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.

axially loaded members

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

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

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

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

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

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

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

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

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

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