While there's technically nothing stopping you from using a bed without a frame, the frame serves many beneficial purposes, such as providing better airflow, which helps prevent mold and mildew. It's also much easier to get into and out of a bed that's not low to the ground. Mattress and box spring sets are also designed for use with a bed frame and will better stay in place with one.
Benefits of Airflow
A frame raises the mattress and box spring off the ground, allowing for airflow under and around the bed. Without that airflow, you're subject to drafts during cool weather and potential moisture issues during humid weather. In a humid environment, excessive moisture is more likely to cause mold or mildew growth on a bed sitting directly on the floor compared to one that's sitting atop a frame that allows for air circulation beneath the bed. Dust mites also prefer humid environments, so that lack of airflow could mean more mites near face level if your bed is on the ground.
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Though it's true that Japanese-style futon mattresses are designed for use without a bed frame, they're typically hung outside to air out on a regular basis, which helps prevent mold and mildew as well as moisture buildup. They also sit on a tatami mat rather than directly on the floor, which increases the comfort level of the thin futon mattress and helps keep ground-level moisture off the mattress.
Extra Storage Space
One great benefit of a bed frame is the storage space it creates underneath the bed. Lack of ample storage is a common issue in many homes and bedrooms in particular. Underbed storage bags and bins provide quite a bit of space for storing out-of-season clothing, shoes, excess bedding, or even craft and sewing supplies that aren't used often. While you could store things under the bed without storage bins, the bins help keep your belongings dust-free and better organized.
Dust still finds its way under the bed around the bins, so it's a good idea to run a dust mop beneath the bed regularly in a bedroom with a hardwood floor. Use a vacuum cleaner that allows for low angles to get under the bed in a carpeted room.
Benefits of Added Height
The average bed height is 25 inches measured from the floor to the top of the mattress. Common bed styles generally range from 18 inches tall for a platform bed to 36 inches tall for antique bed styles, such as some sleigh beds. A bed that is either too low or too high isn't quite as comfortable to use when getting into or out of bed or even when simply sitting on the side of the bed to put on socks.
A bed sitting directly on the floor may be a fun height for a young child, but for adults, it's not as fun to roll out of or get into a bed that sits too low to the ground. Lying in bed to watch TV or read is also awkward in a bed without a frame, as it makes no sense to have a headboard without a frame. It's also easy for crawling insects to get into the bed if the bed sits directly on the floor.
Box Spring Support
Mattress retailers typically sell the mattress and box spring as a set designed for use together. Simply putting the box spring and mattress on the floor without a frame means the box spring could move easily as you use the bed if you set it on a hardwood floor. The mattress/box spring combo is meant for use with a bed frame, as the frame stabilizes the box spring, which also helps the mattress stay in place.
Using a mattress without a box spring or frame isn't a good idea, as many standard mattress warranties require the use of the matching box spring or the warranty is void. The box spring offers support for a typical mattress, making the bed far more comfortable. As for platform beds, the slats that hold up a mattress serve a similar purpose as a box spring, providing just the right amount of support.