The barbell bench press is a popular horizontal push exercise that trains your chest, triceps, and front deltoids. But when some people perform the bench press, they tend to feel shoulder discomfort. In this case, I suggest you check your form. A common cause for shoulder pain during the bench press is flaring your elbows excessively.
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To keep your shoulder in a neutral position and press from a more stable base, you want your elbows to be tucked in slightly. An easy way you can accomplish this is by aiming to lower the bar towards your lower chest. Basically, lowering the bar toward "nipple height." You can swipe left to see a video example.
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Also, I'd like to emphasize that flaring your elbows more than shown in this post is not by definition "wrong". The point of this post is that if you experience discomfort while benching, keeping your elbows tucked in (~75 degrees) is worth a try.
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If for some reason the barbell bench keeps feeling uncomfortable, you can also consider switching to dumbbell bench press instead. The dumbbell bench allows you more freedom in terms of hand placement and shoulder movement + it targets the push muscles of your upper body similarly well [1].
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References:
1. A comparison of muscle activity and 1-RM strength of three chest-press exercises with different stability requirements. - PubMed - NCBI