PRACTICE EXERCISES

$1-6.$ The stress states in Problems $1-6$ were all determined for various loaded components from planes, trains, and automobiles by an accomplished Stress Guesser who suceessfully completed this ccurse.

a$.)$ Determine the principal stresses and sketch them on an element at this orientation relative to theoriginal $x,y,$ and $z$ ccordinate systems.

b$.)$ In addition, determine the complete state of stress at the

maximum shear stress orientation and show them on an element relative to the priacipal axes.

$1.){}_{ij}=|[0,15,0],$

$[15,0,0],$

$[0,0,0]|ksi$

$2.){}_{ij}=|[25,15,0],$

$[15,-5,0],$

$[0,0,5]|MPa$

$**$Note$**$ For other answeres, they use the stress state equation for alpha$\_$p to find the angle, but mathmatically, the steps don't make sense to me$.$

Let a $=$ alpha$\_$p

tan$(2$a$)=(2*$ tau$\_$xy$)/($sigma$\_$x $-$ sigma$\_$y$)$

tan$(2$a$)=(2*15)/(0-0)$

tan$(2$a$)=30/0$

they then state that $2$a $=90,$ and get $45$ degrees as an answer $.$ Isn't $30/0$ illegal math? Any help would be appreciated.