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

$[{\tau }_{i,j}]=\left[\begin{array}{ccc}{\sigma }_x& {\tau }_{x,y}& {\tau }_{x,z}\\ {\tau }_{y,x}& {\sigma }_y& {\tau }_{y,z}\\ {\tau }_{z,x}& {\tau }_{z,y}& {\sigma }_z\end{array}\right]$

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

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

${\tau }_{x^{\prime }{y}^{\prime }}=-{\sigma }_x\sin \theta \cos \theta$

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

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

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

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

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.)

${\sigma }_{x^{\prime }}=\frac{1}{2}({\sigma }_x+{\sigma }_y)+\frac{1}{2}({\sigma }_x-{\sigma }_y)\cos 2\theta +{\tau }_{xy}\sin 2\theta$

${\tau }_{x^{\prime }{y}^{\prime }}=-\frac{1}{2}({\sigma }_x-{\sigma }_y)\sin 2\theta +{\tau }_{xy}\cos 2\theta$

${\sigma }_{y^{\prime }}=\frac{1}{2}({\sigma }_x+{\sigma }_y)-\frac{1}{2}({\sigma }_x-{\sigma }_y)\cos 2\theta -{\tau }_{xy}\sin 2\theta$

Stress tensor

${\sigma }_{x^{\prime }}+{\sigma }_{y^{\prime }}={\sigma }_x+{\sigma }_y=$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

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

${\tau }_{max}=\pm \frac{1}{2}({\sigma }_1-{\sigma }_2)$

${\tau }^{\prime }={\tau }_{ave}=\frac{1}{2}({\sigma }_1-{\sigma }_2)$