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.
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"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