The weight limit of a bike varies considerably, based on construction, quality of materials and other factors. Mass-marketed adult bikes usually have weight limits between 200 and 300 lbs., with the exception of racing bikes, which may have lower limits. Obtain the manufacturer's recommended weight limit before riding a bike if you're concerned about its strength.
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If two bikes are built identically, one with aluminum and one with steel, the steel bike will be three times as strong. However, as the late bicycle mechanic and bicycling writer Sheldon Brown noted, aluminum tubing tends to be thicker and larger, and often less round, to compensate for its weakness. In addition, different alloys of steel -- aluminum and titanium -- have different strengths. In short, you can't gauge a bike's strength by its material. However, carbon fiber bikes are still generally unsuitable for high weights, as they're built for speed and to be lightweight, not to be strong under consistently heavy loads.
A bike rack rated to hold 20 lbs. can collapse, bend or twist under the weight of full touring gear. An ultralight wheel is designed to hold up the weight of a thin racing cyclist, so it is inappropriate for any bike with heavy loads and/or heavy riders. The Big Folks Sports and Activities FAQ notes that some heavier cyclists prefer to use wheels and other parts made for tandem bicycles, as they are designed to withstand greater weights. If you're a heavy person and you stand up on the pedals a lot, the FAQ recommends metal, rather than plastic, pedals for durability.
Sheldon Brown notes that a strong aluminum frame doesn't have the same stiffness as a strong steel frame, so when the bike is under a heavy load, it will feel less stable, even if it can hold you up just fine. This is just one of the reasons it's important to try to find the manufacturer's recommended weight limit; a bike that feels incapable of holding you up because of the material's flexibility may actually be strong enough if built right, whereas a stiff frame can also be too weak.
Many regular bicycles have recommended weight limits of 250 to 300 lbs, and some manufacturers make bikes especially for people over 300 lbs., even as high as 550 lbs., though you should expect to pay much more for a bike with a high weight limit. Those providing advice for heavy bicyclists, such as cyclist Charlotte Cooper and the Big Folks Sports and Activities FAQ, recommend steel frames for their durability and strength.