On the Job Stress Analysis

Hand calc's are best in my opinion. SW is a round about the right answer or close enough to some companies. You should always back it up with your own calculations. I would ask senior engineer for help if your in doubt of anything.

I do a lot of hand calcs for back of the envelope type stuff. I also do hand calcs for situations that FEA handles poorly.

In general, hand calcs involve canned solutions to highly simplified problems (see Roarke's Formulas for Stress and Strain). There are not closed form solutions to general geometries, so what you wind up doing is making heroic assumptions to get a linear, solvable equation. Heroic assumptions are not usually good assumptions.

So, it is easy to get the hand-calc  von Mises stress for a prismatic, cantilever beam with almost arbitrary loading conditions, but your reality will involve a series of approximations and simplifications that eventually tend to invalidate the solution. In my experience, engineers very rarely carry errors through their calcs, and I think most would be surprised at what they find if they did. The advantage, though, is that the equations are easy to understand, and intuitive--they tend to give you an feel for the situation that is frequently good enough for design work.

The advantage of FEA is that the equations are entirely general. You can solve an arbitrary problem using exactly one equation for stress. You can also model arbitrarily complicated situations--if you know what ypou are doing. The disadvantage is that most users don't understand the equations, and, as a result, have little feel for the problem, or whether the solution they've achieved is correct, or even meaningfull.

It depends on what you are doing. Shaun d puts it well, if life is at risk, you should do everything you can do to validate your design before implementing.

Hand calcs, fea and physical testing.

Hand calcs and fea are similar in that they are approximations. In either  case if you want to use them as absolute you need to correlate them to physical. But up to that point, they can be used to compare design under know approximations and assumptions.