This is the explanation for a bodyweight template I came up with to accompany the Jim Wendler 5/3/1 program. As I personally do run marathons the boring but big accessory work does not really appeal to me, because every kilo/pound that I put on, I have to carry around for 42km. That does not necessarily help my time. So here is the attempt at squaring the circle for increased strength at a minimum increase of body weight. Always consult a doctor before doing exercises.
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This is to be done after you have completed all your work on the barbell for the given day. You will do 4x cycles with 90 seconds rest between the cycles. One cycle is from Chin-ups to the Ring dips with no rest.
You start to aim for
Take a wider than shoulder double overhand grip on the bar and bring your arms to full extension so that you reach the lowest point. Pull yourself in one even moved up to the bar, so that your chin touches the bar, Lower yourself in a controlled manner at the same speed you went up. This completes one repetition.
I highly recommend grip variations on different sets to attack your muscles from all various angles. If you are not able to perform a full pull-up yet, start from a box or do negatives. Doing negatives means that you will jump from a box into the highest position of the chin up and then lower yourself slowly back to the box, which completes one repetition.
Do regular push-ups and emphasize proper form during the whole movement. Bring your arms to full lockout and keep your core tight. If look like a seal trying to get on land, you are properly not paying enough attention to form.
If you are not able to push-ups with proper form substitute for box push-ups, where the knees stay on the ground.
Hang from the pull-up bar and bring your feet to full extension pointing to the ground. Bring your feet up until they are parallel to the ground while still hanging from the pull-up bar and move slowly back to full extension. This completes one repetition.
Perform a regular air squat. Bring your feet roughly shoulder with apart and keep your back straight. Then lower yourself to the lowest possible point which is comfortable. Ideally, you bring your bottom as close to the ground as possible. If you have flexibility issues, stretch accordingly.
If you can not do this exercise, I probably think you should not do 5/3/1 at all.
You can also do bar dips or bench dips instead. I found ring dips to be the most challenging and rewarding version of dips. Use a dip bar or a bench and put your arms on top of it. Then lower yourself to the lowest point possible in a controlled manner and come up again. This completes one repetition.
Get to the highest position of the chin up and hold for 30 seconds then lower yourself slowly.