Solid Mechanics

Formulae: Stress Transformation

The necessary formulae for the plane stress case are as follows(ref. Solid Mechanics, Popov).Stresses on the $\theta$-plane.\begin{displaymath} \sigma_{x'}=\frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2} cos(2\theta) + \tau_{xy}sin(2\theta) \end{displaymath}\begin{displaymath} \tau_{x'y'}=-\frac{\sigma_x-\sigma_y}{2} sin(2\theta) + \tau_{xy}cos(2\theta) \end{displaymath}Principal planes\begin{displaymath} tan(2\theta_{1})=\frac{2\tau_{xy}}{\sigma_x-\sigma_y} \end{displaymath}This gives principal planes at $\theta_{11}$ and $\theta_{12}(=\theta_{11}+\pi/2)$. Principal stresses $(\sigma_{1}$ and $\sigma_2)$ corresponding to these planes are found from the formula for $(\sigma_{x'})$ above.Maximum shear stress $(\sigma_1$ ~$ \sigma_2)/2$ and its planes

Planes at 

and,

 

are the planes on which maximum shear stress occurs. Sense of maximum shear stresses corresponding to these planes are found from the formula for $(\tau_{x'y'})$ above.


Principal stresses and maximum shear stress can also be found directly as,

Computational Mechanics of Laminted plates

My research

PhD Thesis  produced following publications:
1. Stability and Vibration of Mindlin Sector Plates: An Analytical Approach, AIAA JOURNAL, Ashish Sharma, H.B. Sharda and Y. Nath, Vol. 43, No. 5, May 2005, pp. 1109-1116 
2. Stability and vibration of thick laminated composite sector plates, Ashish Sharma, H.B. Sharda and Y. Nath, Journal of Sound and Vibration, Vol. 287, 2005, pp. 1-23
3. Non-linear analysis of moderately thick sector plates, Y. Nath , H.B. Sharda and Ashish Sharma, Communications in Nonlinear Science and Numerical Simulation 10 (2005) 765-778
4. Nonlinear transient analysis of moderately thick laminated composite sector plates, Ashish Sharma, Y. Nath and H.B. Sharda, Communications in Nonlinear Science and Numerical Simulation

• The thesis basically involved obtaining different solutions of five simultaneous partial differential equations
• Two-dimensional Chebyshev polynomials were used for spatial discretization
• Houbolt time-marching was used for temporal discretization for simulating the non-linear dynamic model
• The code was written in C++ on LINUX platform
• For linear algebra, the C++ library named TNT/JAMA were used
• For word-processing,  pdflatex was used
• For Figures & Plots, gnuplot and xfig were used