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This paper presents a plastic design method of steel frames with a vibration control system with U-shaped steel dampers based on a quasi-linear motion mechanism. Firstly, a plastic collapse load of the damper system is evaluated using the upper bound theorem where the U-shaped dampers are assumed to have three plastic hinges. The accuracy of the evaluation is examined by comparing the cyclic loading test results. Secondly, the plastic design is discussed for steel frames considering somedoi:10.3130/aijs.76.379 fatcat:zo7gnbtcsjff7bqfjukmsx6ema