I have recently added side planks to my regular workout in order to get a better workout for my obliques. However, once I've been in the plank for about a minute, I start to feel considerable joint strain on my shoulder and elbow. This is quite painful, and is a significant barrier to increasing my time. I've tried every modification of position I can, with no effect. Softer surfaces and cushions don't help either. Does anyone have any suggestions on how to fix this issue?
I cannot tell you how to avoid the pain, only how to reduce it.
Generally, I have a much better understanding of such kind of issues after I began doing yoga. I can not repeat what yogis say here. This would just make you laugh: Strange statements like "feel the energy running up from the earth and through all your body". However, when I do what I think they want me to do, I feel much better.
First, press your hips upwards and keep a high tension in the deep abs. This is not yoga, but how you perform this exercise in a correct manner. Then press your lower shoulder backwards and down and keep a high tension like you do in many other exercises. Never sack in the shoulder.
Now, if you know the warrior exercises from yoga and how to press your legs together, it is an advantage. While you keep the arm you rest on in a vertical position, then think about pressing your forearm down and towards the legs, and even it seems impossible, the legs down and towards the forearm.
If you do things right, you will feel it like the weight of your body is suddenly reduced. So are the pains.
As a part of writing this answer, I did this exercise for 2½ minute and the only reason to I went down was that my abs was shaking. I felt only insignificant pain in the lower elbow. Before yoga, I always had to stop because of pains in elbow and shoulder.