The effects of the ring depend on three factors:
- The height of the ring above earth ($h_{surface}$)
- The width of the ring ($w$)
- The direction of the ring's rotational axis
The first two work together to give you the apparent width of the ring in the sky ($w_<$):
$$w_< = w/h_{surface}$$
They also work together with the earth's radius ($R$) to give the total surface area of the ring ($A$):
$$A = w\cdot 2\pi(R + h_{surface})$$
Thus, if we assume that the ring has the same surface area as the earth (510 million square kilometers), we get:
$$A = 510\cdot 10^6 km^2$$
$$\Rightarrow w = \frac{510\cdot 10^6 km^2}{2\pi(R + h_{surface})}$$
If we put the ring in low earth orbit, just $1000km$ above the equator, the equation above yields a width of
$$w = \frac{510\cdot10^6km^2}{2\pi(6400km + 1000km)} = 11000km$$
The "ring" would be more like a can that's almost as high as the ball that's inside. I think, we don't need to talk about the climatic effects, the ring itself won't get much sunlight on its inner surface due to its own shadow.
Ok, so we move the ring up a bit, say to $10000km$ above the equator:
$$w = \frac{510\cdot10^6km^2}{2\pi(6400km + 10000km)} = 5000km$$
Now the earth is about 2.5 times as wide as the ring, and the ring diameter is still just about 6.5 times its width. The ring basically takes all the sunlight away from the equator, and the earth takes the better part of the sunlight away from the ring. Not a workable setup.
So we move the ring even higher to a level, where it can easily be assembled using a space elevator: geostationary orbit. Now we get:
$$w = \frac{510\cdot10^6km^2}{2\pi(6400km + 36000km)} = 2000km$$
$$w_< = \frac{2000km}{36000km} = 0.0556 rad$$
For comparison, the apparent width of the sun is $0.0093 rad$, so the ring is about six times as wide in the sky as the sun.
If the ring's axis were parallel to the axis of earth's orbit, the ring would be fully in its own shadow. Thus, I will assume that the ring's axis is parallel to earth's rotational axis.
During summer and winter, the ring's shadow will entirely miss the earth by a large margin, and will wander over the globe over the course of less than two months in fall and spring.
This will severely disrupt climate, because we have a stripe of the earth that is in total darkness for about a week. Within this stripe, temperatures will drop brutally. As the air cools down, it gets heavier, so we get strong, cold winds out of the shadow region. I guess, icy storms would be a better word. This will have quite a bit of destructive effects.
You can move the ring further up, another interesting point is the hight where the apparent width of the ring matches that of the sun:
$$w_< = \frac{w}{h_{surface}} = 0.0093 rad$$
$$\Rightarrow h_{surface} = 107\cdot w$$
$$w = \frac{510\cdot 10^6 km^2}{2\pi(R + h_{surface})} = \frac{510\cdot 10^6 km^2}{2\pi(R + 107w)}$$
$$\Leftrightarrow ...$$
$$\Rightarrow w = 842km$$
$$\Rightarrow h_{surface} = 90000km$$
The radius of the ring has increase by more than a factor of two, so the speed at which the shadow will swipe over the earth in spring/fall has more than doubled. This is assuming that the ring's axis is still aligned with the earth's rotational axis. Also, we now only experience total darkness for a few hours at most, with the sun being partially occluded for a few days. The climate effects should be benign now, and they will be restricted to the time around spring/fall equinox.
If you align the axis of the ring with the axis of earth's orbit around the sun, you get a thin, unlit stripe right where the sun is in zenith.
However, the earth still rotates under this shadow: At the equinoxes, points on the equator move right across the shadow region during the day, and at the solstices the tropic circles will experience the maximum shadowing, but only around noon. I.e. shadowing follows a dayly cycle, and affects a large range of latitudes.
As such, the effects are not local enough to trigger catastrophic small scale phenomena, but rather significantly reduce the sunshine across the tropics. This will definitely reduce the strengths of a) the tropic rains, including monsoon, b) the trade winds, c) cyclones (hurricanes/taifoon/whatever they are called), and d) the aridity of the deserts.
The reduction of the strength of the tropical air cycle will have other effects on the global system of winds, but I'm not enough of a meteorologist to make a decent guess at those.
