Much depends on the height of your building.
In addition to Zalka's books, this one is an excellent old standby for lateral calcs done by hand:
Link
I agree with others that the relative stiffness between the shear walls and the moment frames will probably limit the effectiveness of the moment frames. I don't think that the distribution will be quite so lopsided however. Here's why:
1) The concrete core that you've drawn as a box is really a three sided channel. It's got an elevator door opening on at least one side that takes up most of that side and makes coupling across the opening ineffective. A channel's torsional stiffness is drastically less than that of a closed section.
2) The center of rigidity of the system is likely pretty close to the core wall shaft. If the elevator door is on the north side, as is likely, the center of rigidity may be as far south as the south wall of the shaft. The effectiveness of a lateral resisting element varies with the square of its distance from the center of rigidity. That squaring is going to amplify the efficacy of that north wall moment frame.
If this is a low-rise building, doing a hand calculation for the distribution shouldn't be too bad. Estimating the stiffness of the moment frames is a cinch. Here are some pointers regarding the shear walls:
1) For a low-rise building, shear flexibility will govern the behavior of the shaft walls. Flexure and composite section torsion will fade to background noise. This is a big help as it will allow you to treat the shaft as three unconnected walls rather than as a complicated composite assembly. And you can only need to consider shear stiffness. Of course, you should do a quick check to verify shear dominant behavior for your building height.
2) Concrete wall stiffnesses need to be modified for cracking. Steel moment frames don't. This can be troublesome in mixed systems because the concrete cracking factors or pretty gross approximations and can skew the relative force distribution between lateral resisting elements quite a bit. You should play around with your numbers to assess sensitivity.
3) Torsion in the shaft eventually needs to become torsion in your foundations. Keep an eye on that.
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