Depending on the bridge, you might be trespassing in some way. Railway bridges are the property of the railway company, they don't allow to go there unless specifically allowed and even the absence of a sign is not an ok in those cases. Similarly, highway bridges usually are communal property and they don't need to be marked as off limits if there are laws that prohibit being there. The absence of signs and fencing is not a carte balance to go there: If your starting point for the climbing is not easily accessible, they don't have to fence it.
But besides trespassing, you might make a huge dookey:
Depending on the attorney general, you might be, in Austria, in violation of § 46 StVO for being a pedestrian on the Highway, as the bridge in whole is part of the highway structure. Even if you are under the lanes, you are technically on or inside the highway structure. If the attorney really wants you in, they could pull out § 89 StGB "endangerment" as well as the § 176 / § 177 "endangerment of the commonality" (by reckless or negligence). After all, you might or might not be climbing the bridge to perform something to endanger all the people using it.
In Germany, you will get fined for violating traffic laws if you go onto railway tracks or climb onto a railway bridge as you violate the railway code and the minimum charge is 25 €. You could get also sued for Reckless endangerment of the train traffic (§ 315 StGB) and locked up for between 6 months and 10 years. If you do delay trains in any way (and even if it is to allow a crew to get to you), you are also liable damages - this is very easy in the 5 to 6 digits. Similarly, if it was a highway, you'd get billed 10 € for being on foot on a motorway without special privilege under $ 25 StVO and again, you could get jailed in extreme cases under § 315b StGB.