Polyurethane is known among chemists as a "reaction polymer." The most common reaction polymers people use in everyday life are plastics. Polyurethane finishes are paint-on liquid coatings used for floors, decks and furniture. Polyurethane finishes dry into a tough, clear, lustrous finish. A polyurethane finish that is microtextured to be slightly "softer" than the glassier "gloss" finish called "stain finish polyurethane."
Short Answer
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The short answer is that stain polyurethane finishes can be applied over any paint or stain that is already properly applied to the surface and that has completely dried. That includes the various semigloss paints and stains, whether they are latex, acrylic or oil-based. If the paint has been applied without proper priming or on surfaces that contain moisture, then the paint itself can blister free of the surface, damaging the satin polyurethane finish.
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Is Polyurethane What You Need?
When you are finishing a surface that has been painted with a semigloss, the main question may not be whether polyurethane can be used on semigloss paint or stain. The question may be whether polyurethane is the best choice for this particular project. Shellacs, lacquers, varnishes, oil finishes and water-based finishes are options you need to consider; because each of them has slightly different characteristics from each other and from polyurethane.
Stir Well and Often
Some woodworkers and household do-it-yourselfers have noticed that satin polyurethane finishes can be inconsistent. A common complaint is that the satin finish they expected became a finish that had highly glossy patches alongside much flatter finishes. Polyurethane is made satiny by mixing a paste into the liquid finish. The finish formula has very low viscosity (it is thin) and this paste can settle. To prevent this problem, mix often and mix well.
Put Sand in Your Hand
Your satin polyurethane finish needs to adhere tightly with your semigloss to get the best result. The best way to ensure that adhesion is with elbow grease and some 280-grit or finer sandpaper. Lightly sand the entire semigloss surface with sandpaper. Then wipe the surface clean with a camp cloth to remove all the dust. When it is dry, apply one coat of the polyurethane finish. Let the first coat of polyurethane dry and sand it the same way, wiping it clean, letting it dry and applying the second coat.