You will need to tighly grip the pairs of dumbbells so that your hands do not slip down the handles. Additionally, you will need to press the dumbbells hard into the floor so that they do not fall over. This will work the muscles of your upper body with an emphasis on your triceps. Tighten your abs as if you are bracing to take a punch. Tense your glutes to help protect your back. Concentrate on generating as much tension as possible. Gradually increase the amount of time that you hold this position. You can also do Crush Grip Push-ups, but it is very hard to lock out your elbows at the top of the push-up because your wrists are placed in an awkward position.
The percentage of your bodyweight that you are supporting with this exercise can be manipulated by varying the incline. If you are not currently able to perform the exercise as described above, place the dumbbells on a stair step, chair, or picnic table, for example, with your feet on the floor. The greater the incline, the smaller the percentage of your bodyweight that you will be supporting. Gradually reduce the incline by resting the dumbbells on progressively lower objects until you can hold the position on the floor. If holding the crush grip plank on the floor becomes too easy, you can gradually shift more weight to one of your arms. If you do this, make sure that you do another set with more weight shifted to the opposite arm. You could also add more resistance by elevating your feet on a step, bench, or chair with the dumbbells on the floor. I generally move on to a more difficult progression when I can hold the current position for 30 seconds.
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You'll find more bodyweight training articles by Brad Johnson in MILO: A Journal for Serious Strength Athletes.