Let \( x \) be the total number of glass plates shipped. According to the problem, 13% of the plates arrive broken, which can be expressed as:
\[ 0.13x = 117 \]
To find \( x \), we can solve for it by dividing both sides by 0.13:
\[ x = \frac{117}{0.13} \]
Calculating this gives:
\[ x = 900 \]
Therefore, the total number of plates shipped is \( \boxed{900} \).