One of the basic exercises - the pushup. Aka the press-up in the King's English. It is a pretty simple exercise. You lay face-down on the floor, put your hands under your shoulders, keep your elbows tight to your sides, and push your rigid body away from the floor to arm's length, lower back down until your chest touches and come back up. It's a good test of your relative strength (strength compared to bodyweight) and a good endurance builder, too.
What happens when you get strong enough to do 20+ pushups in a row without a problem? Well, you can just doing more and more reps. But after a certain point it's not giving you much return on your effort.
Here are some ways you can make them harder.
Load them with external resistance. While you can have a partner place a 45# plate on your back, this is sub-optimal. The weight presses down on your shoulder blades as they try to move, the plate can shift (backwards or to the side is bad, forwards is potentially disasterous), and it's hard to load it by yourself. Here are some better options:
Resistance bands. Resistance tubing works as well, but it's easier to get quality bands (such as Jump Stretch or Iron Woody). Simply take a long band (41" bands work great) and loop it around your hands and over your back. Now, carefully get down on the floor into a pushup position. When you push up, the band tightens and makes the reps harder; at the bottom it gives less resistance. The longer your arms the more resistance you'll get out of any given band. Here is a band-resisted pushup.
Chains. Have a partner drape lengths of chain across your back. At the bottom, only the links on your back provide resistance. As you go up, the extra links come off the floor and the weight you're pushing goes up. Here is a nice video of someone doing chain pushups.
Weighted Vest. Strap on a weighted vest - a special vest sewn with pockets for small weights - and start pushing. You can combine these with chains or bands, and even wear multiple vests. These provide linear resistance - the weight you push is the same at the bottom as at the top.
A partner. You can have a partner sit on you. I wouldn't advise this unless you are very strong and your partner is pretty small though. Your partner can get into pushup position on your back, if that's easier. You can also have your partner stand over you and push down a little, giving you some extra resistance. But this resistance is hard to measure. That makes increasing the effort the next workout that much harder.
Change them up. You can also make the pushup harder by varying up the exercise itself. These can be combined with the loading ideas above.
Rings or Blast Straps. If you do a pushup with your hands in a pair of suspended gynmastic rings or blast straps, they become that much harder. The exercise is no longer a closed chain exercise, with your muscles pushing your body away from the floor. You now have to contend with keeping the straps lined up under you. This added demand for stabilization challenges your body to keep itself rigid in an unstable situation. This isn't the same as putting your feet up on a swiss ball or bosu ball to do pushups - that makes your feet unstable, which some research has show makes for a less-effective exercise. If you are going to make pushups unstable, it has to be around the hands and arms. Rings and blast straps do that.
Medicine Balls. You can push up off one or two medicine balls. You can use one (holding both hands on top), two (one hand per ball), or alternate one (one hand on the ball, the other on the floor). These, like the rings and blast straps, force you to stabilize yourself.
Raise your feet. You can raise your feet up on a box or bench. The higher the box, the harder the pushups, as you both increase the relative weight and increase the angle you push at. Eventually, you may incline your feet so far it becomes a mirror of an incline bench press, or even become a handstand pushup - the mirror of a standing press.
Clapping Pushups. Push up explosively, clap your hands in front of you, and then land and go down into the next rep. Very good for developing power. Try them the first time on a soft surface, like a wrestling mat. Your wrists will thank you!
Handles or Hex Dumbbells. Pushup handles (which range from inexpensive to very expensive) or hex-shaped dumbbells can be used as well. Put them parallel to each other and push up off of them. You will get a larger range-of-motion because the higher handles allow you to go much deeper. These may also be easier on your wrists if you've got delicate or sore wrists. A nice hex dumbbell pushup set.
Extra depth. You can do pushups in the shallow between two raises surfaces to get a greater range of motion.
Barbell Pushups. Place a barbell in a power rack, or in a pinch across two hex dumbbells. Get into position like a reversed bench press, and push up. Touch your chest to the bar on each rep, and keep the barbell pushed hard against the rack. This works with standard 1" thick barbells and olympic barbells, and with thicker bars (such as 2", 2.5", or 3" fat bars).
Getting creative is fun with pushups. Various options can be combined, such as chain pushups off of hex dumbbells, med ball pushups with feet raised, clapping pushups with a vest, band resisted barbell pushups, raised feet weighted vest pushups off of blast straps, etc. Each little tweak can add a lot of additional difficulty, so ease into new variations. If you just nailed 25 perfect pushups on the floor, suddenly going into raised-feet blast strap pushups with that new weighted vest you bought is going to be very hard indeed. Better to just add a little at a time - perhaps just feet up or the straps, then add again once your reach your target reps.
There have been some good articles on the pushup, here are a few:
Ross Enamait shows knuckle and fingertip pushups.
A Crossfit article on pushups, showing some additional progressions using parallel bars.
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