Strain gauges burned out in half bridge
The power dissipated in strain gauge elements should always be far less than the power level at which you can do them any harm. Even connecting the "excitation voltage" across a single bridge element should do it no harm, eve though this would not be a useful thing to do. The implications are that you have done something which is fundamentally incorrect. The voltage used must be substantially greater than the rated voltage for the damaged part.

One possible reason for one only gauge being damaged is that the output is grounded or connected to v+ or to some other voltage so that theundamaged gauge "sees" little or no voltage.
 What is your half-bridge midpoint connected to?

Do you have a data sheet
 What is (or was) the gauge resistance for each part?
 What is the applied voltage?
 What power dissipation do you expect and what is the rated dissipation?