Warning to non-enginerds: this question gets a little technical, feel free to skip it if you want. The takeaway is YES, it usually does, so make your anchor extra stout. If you want to know why, read on.
Ahh yes, this is a very interesting question, and one subject to much debate on the inter-webs. It's a classic example of the difference between MA theory on paper, and MA in the real world, where we add the complex variable of friction. And, the correct answer is actually kind of tricky, because it depends on a lot of different things.
There's also the issue of what's happening to the load. If you have a progress capture device on your anchor, there are three different things that can be happening to the load at any time.
You can be actively pulling
You can let the load rest with the progress capture device on the anchor
You can be holding the load with your body but not pulling.
In each of three cases, the load on the anchor changes!
Lots of people have the misguided idea that an MA system always magnifies the load on the anchor. For example, many folks think if you have a 2 :1 system lifting a 100 pound load, then the anchor is holding 200 pounds. A 3 to 1 system, the anchor is holding 300 pounds, etc.
This is NOT correct!
An MA system, per se, does NOT put more force on the anchor. In some cases, it actually puts LESS force on the anchor! Let’s have another look at one of our previous diagrams.
Here, we have a 2:1 MA, and Sticky is doing a direct arm haul of the rope. Note that the anchor is only holding a 50 pound load, and the other 50 pound is being held by Sticky. So, in this case, the 2:1 MA system DECREASES the load on the anchor by 50%.
Now, with the same 2:1 MA system in place, say Sticky redirects her haul rope through a pulley high on the tree, and attaches a load capturing prusik knot, shown in orange. Here, you can see she is not pulling it all, resting and admiring her anchor handiwork. But, now with the load not moving, both strands of the rope are hanging from the anchor with 100% of the load, so the anchor is holding 100 pounds.
Again, with the same 2:1 system in place, Sticky starts hauling. To move her half of the rope, she needs to pull down with a force of 50 pounds. This extra force gets applied onto the anchor, so when she is hauling, the anchor is now holding 150 pounds.
So, when actually hauling with this set up, the load on the anchor DOES increase.
In the real world, friction magnifies force on the anchor even more.
Let's revisit Sticky, who is pulling a simple 1:1 pull, redirected through the pulley high up in the tree. Only this time, instead of the tree branch conveniently hanging out over the edge of the cliff, the tree is set back, so the rope is running over a rock ledge. For this discussion, let's say that Sticky needs to pull with an extra 50 pounds of effort to overcome the friction of the rope running over the ledge. For her to lift the 100 pound load, she needs to generate 150 pounds of effort.
If she pulls with 150 pounds of effort to raise the load, that means there is now 150 pounds on the strand coming out the other side of the pulley. Which means the anchor is holding 300 pounds rather than 200 pounds. Remember, when you redirect your pull, that redirect point will receive twice the force that you apply.
Is this a problem? Maybe, maybe not. Do you have an anchor on a stout tree limb or three well equalized points of rock protection? Probably not.
Or, is your anchor a a vertical snow picket, and you're about to set up a 3 to 1 haul with two strong people trying to pull somebody out of a crevasse? Then, that lone anchor might be a significant problem. Take a few more minutes and make a deadman anchor from that picket, or better yet, two that share the load.
With this in mind, let's consider a real world crevasse rescue scenario.
One person from your 3 person rope team fell into a crevasse. The rope going to them is cut deeply into the crevasse lip, adding a lot of friction.
You only have 50% efficient carabiners instead of 90% efficient pulleys.
On your alpine climb, you’re using small diameter, stretchy dynamic ropes.
You set up at 3:1 Z drag, and you and your partner both start pulling with your entire body weight. To move the load, your pulling force has to overcome all the extra friction from the biners, the rope against the snow, (and the inefficiency of the stretchy rope) and that pulling force has to be transmitted to the anchor. How much force? Hard to exactly say, but two people pulling as hard as they can on a Z drag with a lot of real world friction can generate a BUNCH! That extra force has to be absorbed by something - most of it’s going onto the anchor. (Better bury another picket as a deadman!)
You may be thinking: “Here I am with my buddy, and we’re both pulling as hard as we can on this 3:1 Z drag, trying to lift our 150 pound partner out of the crevasse. Me and my partner weigh a combined 300 pounds, so in theory, if we’re pulling with a 3 to 1 we should be creating 900 pounds of pulling force, but we’re still only barely lifting my 150 pound friend. This is way harder than it should be . . .”
To use a more extreme example, let’s say you get your car stuck in a ditch, and you rig a 9:1 with a big tree as anchor to try to pull it out. Now you have basically a tug of war between the tree and your car. When you pull onto 9:1, the anchor (in theory) gets your pulling force multiplied by 8.
Now, say someone else steps up to help you pull. How much force on the anchor components can two strong people apply? That’s now the force of two people pulling multiplied by eight. Now you’re probably getting pretty close to the safe working limits of some of your equipment. There’s a chance the weakest links on the system could start to fail, like prusiks sliding or breaking or even hardware failing. Hopefully the car moves before anything breaks or slips, but the point is, your anchor and the components of your pulley system need to be stout enough to handle these magnified forces.
So, here is the final answer - In the real world, mechanical advantage systems often result in extra force on the anchor, because of the extra effort needed to overcome friction. The greater the MA of your system, and the heavier the load you’re trying to lift, and the more friction is involved, the stronger your anchor needs to be.
This is discussed in the excellent book “The Mountain Guide Manual”, by Marc Chauvin and Rob Coppolillo, pg 276.