I should have clicked you were capturing when you mentioned .ts, my bad. It may vary here for you if you are capturing actual TV, home video, game footage and more besides.
There is software out there that will encode in 60fps, in fact it is nothing spectacular at all. However I sense we are going to have to discuss video concepts first, to that end *warning, boring technical video discussion incoming*
First it may only be that your capture card can capture in 30fps -- not all that much originally went above 30 fps and technically not a lot there even got to actual 30 fps (more on that later). If the video encoder is sensible it will try to only give you that option, likewise if the encoder is set to mimic a given format (say DVD, blu ray.....) it may stop that from happening.
Is the source interlaced? If so the f in fps in this case stands for "fields per second" if it is telling you it is at 60 fps. Interlacing is a horrible thing that stuck around because of legacy reasons and you may be facing it here (
http://www.theregister.co.uk/2013/06/25/the_future_of_moving_images_the_eyes_have_it/ goes into a bit of depth here). There are ways of increasing interlaced framerates from 30 to 60 or 25 to 50 that are not nasty and time consuming frame rate conversion type things, personally I do not think it does anything and so do most video editors hence most deinterlacing turning X fields per second into X/2 frames per second.
Pulldowns.... interlaced I can deal with, pulldowns are just horrible and it actually gets worse from there. Film cameras were traditionally 24fps (less for some really old stuff -- what bad video editors cause old war footage to look like everybody is on the Benny Hill show), PAL is 25fps so it is sped up but you can not speed up to NTSC (30fps). Few things were captured at 30fps so frames/fields were played multiple times (the most common being the so called 3:2 pulldown but various people did all sorts of things). Modern cameras and game consoles do output at full frame rates, also CGI footage tended to be rendered at 30fps which is fun for those ripping DVDs properly. To finish it off you also have "pulldown flags", various video decoders could see these flags and do their own pulldown at decode time. Naturally not everything supports this so how it plays out varies a bit, this is also where some people started getting into variable frame rate video.
If it turns out you have true 30fps footage and want it to be 60fps it is possible, personally I do not suggest it and it will see your encode times probably increase by a factor of 10 (no exaggeration there) unless you do just do the basic replay frames thing which is just pointless really.
