I built a tabletop gluing up seven 2x4s together, and it warped a lot. Two opposing corners are up and the others down.
I know cheap construction 2x4s can warp, but can this be fixed?
The only way I can think of is to plane it a lot with my hand plane.
This is not really an answer but a large comment that could point out several pitfalls that could have brought you to this point. I am not sure of the best advice for your situation as the warping looks significant. Like discussed in comments I think you might be better off starting over and paying attention to points below.
Knots and Pith
Depending on where you sourced these it is likely that you could have large amount of knots and pith. When you got to purchase lumber you need to check for these things as they will largely affect the drying process.
Why should I avoid using the pith section of quarter sawn lumber?
Normalizing Wood
The wood you purchased should sit in the environment you plan on keeping it in for several weeks or months depending. While it might not looked warped in the store you sometimes don't know how long its been there and more importantly I doubt your build location has the exact same humidity. While gluing can help discourage some movement the wood is still going to lose / gain moisture.
Wood Selection
While I would still try this, and look forward to this summer, many would caution against using softwoods like pine, spruce and fir as they can be prone to warping. Also, depending on the use for the table softwood would damage more easily. This point might not be for you but you are likely using pine.
How can I deal with wood from DIY stores that is crooked and may or may not be dry?
The Glue up Process
It is of immense importance that the gluing surfaces are perfectly parallel and flat. A failing in this area would give the wood more room to move as well. Your choice of clamping tools and process would also impact this.
What is a good method for laminating pallet wood/reclaimed lumber?
Alternate Grain During Lamination
When you are gluing up it is important to try and have the grain directions of the boards compliment each other so that when they do try to warp that the other boards will help mitigate the movement. A decent example is on the end of this table:
Image from bobreuterstl.com
Special considerations for a very large, engraved food serving board
There are impressive answers using laminating that might be of interest to you that cover these topics in more detail. I added some question links above to some of those questions.
If you do choose to start over and have access to a table saw (With a good outfeed table) you could rip down the table on the glue lines. That could be hard on the blade and hard to do depending on the size of table.
That won't do anything for the warp but what you can do to address that but save the wood is make some cross cuts on the warped boards. Putting them back together will reduce the overall warp of the boards.
Again keeping in mind all of the other point I made above this would be overkill for big box store lumber. It is a solution nonetheless.
I know cheap construction 2x4s can warp
There are multiple possible causes for distortion in a glued-up panel like this. Often more than one of them are to blame, although there may be one major culprit. I suspect that's the case here.
Since your tabletop ended up 'in wind' (twisted) and didn't cup or bow I think the wood itself is likely to be mostly or solely to blame, with a possible contribution from the jointing of the board edges or stresses introduced during clamping (one edge tight and another looser, due to irregular clamp pressure).
If the wood when you bought it was not well dried, sadly very common in 2x material, it might have had an equilibrium moisture content (EMC) well above the norm, e.g 18% or higher. The further drying of the wood down to 10-14% could account for this, in concert with the grain pattern in individual boards which can lead to more or less of a tendency to change shape during shrinkage, see diagram below.
The only way I cab think of is to plane it a lot with my hand plane.
You can in theory plane warped glue-ups down to flat just as one does with an individual board that's in wind, but you can lose a lot of thickness this way. In this case the warping appears to be so severe that it's not a viable fix as the high corners will end up much too thin.
I suppose the best fix for this is to start again from scratch with new material (ideally selected for even, straight grain and minimal knots) but it may be possible to rectify the problem using the existing wood.
First step would be to rip it back into individual boards, sawing by hand if necessary. Then you flatten each one as much as needed to allow you to proceed and when that's done you re-joint the edges and glue the tabletop back together again. Once the glue has fully dried you'd complete your flattening of the top and then affix it to the leg/apron assembly.
There are a few techniques that can be employed to help a tabletop stay flat over time that you may wish to incorporate into the table. Firstly some general info on allowing for expansion:
What general considerations do I need to take into account for wood movement?
Hardwood buttons and expansion plates:
Will these table legs support this table?
Battens to help hold a tabletop flat:
Stabilizer for softwood tabletop that bows
In addition to the concerns in other answers addressing the selection, preparation, and arranging of the wood used in the top I would also suggest two design changes. First, the overall thickness of the top shown in your picture is only 1 1/2". A thicker top (3-4") would be much better resisting the internal forces that curve/curl the profile. Second, consider adding "breadboard" ends to the top. Breadboard ends consist of running a board perpendicular to the top boards, side to side rather than end to end. Since this end board will be resisting the warping force of the top, turn the 2x4 vertical for maximum strength. Join the breadboard to the end of the top using a continuous tongue and groove connection at the center depth. This groove connection will insure that the top stays aligned with the end boards. Although the tongue and groove joint should be a snug fit, it should not be glued in place, except at the center of the joint. This is because the table top width will expand and contract with the weather, while the end board will not grow and shrink in the direction of its length. Secure the breadboard ends to the top using lag bolts with an over-sized hole (1/4" bigger than the lag bolt should do it) through the breadboard to allow for the differential expansion/contraction.
It may be possible to apply a breadboard to the end of the top in your picture, but you will need a lot of clamping power to twist it into a flat shape and the top will have a great deal of internal stress. It may be better to make a new top.
