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I am a very weak and puny man with legs like twigs. I write enjoy 'strength' work in the gym for the legs an core conditioning for variety and to add a little power and robustness.
I do sometime question why i'm doing it though...
It can occasionally feel detrimental. The DOMS if i've not done a session in a while or if i've upped the weights can fuck me the next day, but if you get a regular routine its ok. If i had less training time i would ditch the gym though.
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My sessions were timed around speedwork and I usually managed to recover ok. I'd get DOMS the day after my turbo session. Scherrit might have a contact for me but my riding is likely ramping up to the point where I won't be recovering for days so I need to manage it. Like skinny said, it's the first thing I'll drop if I feel I can't manage it along with the cycling.
You might. I might not.
I do enough (too much) time on the bike. Gym was refreshing and fun for me. It didn't seem to slow me down last year. My coach is injured though so I'm looking for a new one.
Have a read:
https://www.trainingpeaks.com/blog/year-round-strength-training-for-cyclists/
https://www.theguardian.com/lifeandstyle/2009/jan/11/cycling-strength-training
http://www.cyclingweekly.co.uk/fitness/strength-training-for-cyclists-is-it-worth-it-125222