The immediate roof behind a long wall face often considered to be a cantilever beam as shown

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The immediate roof behind a long wall face often considered to be a cantilever beam as shown in section in the sketch. Beginning with a safety factor design criterion, derive an expression for the distance X the face can advance before R1 failure. Assume bed separation between R1 and R2 occurs, flat strata, no gas pressure and that currently L = Lo. Show shear and moment diagrams and location of expectedfailure.

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