I am thinking of implementing this by daisy-chaining multiple splitters but I am not sure if it will work.
Does anyone know of a cost-effective solution to this?
You can't use a passive splitter for this as the signal level would drop far too low, however it should be trivial to accomplish with some simple active distribution amplifiers.
There are two real solutions for this and the best bet really depends on the details of where the projectors are. More than just splitting the signal, the amount of distance to the projectors may also be an issue as most video signals degrade over long distance cable runs.
If all the runs are fairly short or you are able to use either fiber or signal boosters, you can use a large distribution amplifier or matrix switch to feed one signal to all 40 outputs at one location.
Alternately, you could use a series of smaller distribution amplifiers to spider out from the middle. As each DA will also act as a repeater, it should help you cover a longer distance and you only need a chain of about 3 to cover all 40 projectors if you are using 4 way DAs. You'd start with the signal in the middle, run it in to a distribution amplifier and run each of the 4 outputs in to another distribution amplifier heading off towards your other projectors. Those 4 DAs would give you 16 total outputs. Each one of those would run in to another 4 way DA and give you a total of up to 64 outputs, which easily covers your 40 projectors.
Another option if exact timing doesn't matter and you already have a solid Ethernet network in place would be to stream the video over the network and have each projector play back the stream independently.
A final option would be to setup a short range broadcast using wireless gear, however this is probably the most costly and potentially most error prone of the options though it can also be the easiest to setup if the environment supports it as it doesn't require multiple splitters or any complicated wiring, however transmitters tend to not be cheap as the range gets longer and beyond a certain range, you start needing licenses.