Let \( x \) be the total number of plates shipped. According to the problem, 13% of the plates arrive broken, which can be expressed mathematically as:
\[ 0.13x = 117 \]
To find the total number of plates shipped, we can solve for \( x \):
\[ x = \frac{117}{0.13} \]
Calculating that:
\[ x = 117 \div 0.13 \approx 900 \]
Thus, the total number of plates shipped is \( \boxed{900} \).