While not strictly about umbrellas, some arts have trained to use the cane as weapon. Check this:http://ejmas.com/jnc/jncart_barton-wright_0200.htm
Most normal umbrellas are not really suitable for SD, as they are too flimsy. There are some makers of heavy-duty umbrellas for SD purposes. Google "umbrella for self defense", you should find something.
"In case you ever wondered what it's like to be knocked out, it's like waking up from a nightmare only to discover it wasn't a dream." -Forrest Griffin