"for the sake of" means something is done to benefit someone or something else. "We go to war for the sake of our country!" = "we go to war for the benefit of our country!" "as a result of" means one event caused another "My injury occurred as a result of the car accident" = "I am injured because of the car accident." "on the strength of" is either very positive or very negative depending on the context. But it means that someone's success is due almost entirely to one factor, and all other things made little or no contribution. "He just got into the University on the strength of his recommendations" could be a negative way of saying his friends or family got him in to the University, but he didn't deserve it. "The proposal was accepted on the strength of it's research." would be a positive way of saying a proposal was very well researched and so it was accepted. "by virtue of" is used similarly to "on the strength of"

For the sake of - for someone, for someone's benefit (I work hard for the sake of my children) As a result of - because of some action (As a result of higher oil prices, the cost of air travel is up) On the strength of - relying on something (He got the job on the strength of his English) By virtue of - because of, formal (by virtue of my authority as the president…)