The target of the paper focuses in the required properties for successful behavior of the CuAlBe Shape Memory Alloy (SMA) in damping of steel structures under the action of earthquakes. The appropriate fracture – life, the long time of aging, the minor creep effects on cycling, the Clausius-Clapeyron equation and the self-heating effects are also, evaluated. Analysis via simulation using a proprietary model of the SMA behavior furnishes satisfactory results. Our main interest is focused in the experimental results produced by tests in one loaded steel portico (beam: HEB-120 of 4.10 m, pillars: HEB-100 of 2.47 m) free and with SMA dampers in the diagonals. The dampers are built with two trained wires of CuAlBe or two wires of NiTi. The experimental results without and with the dampers are, mainly analyzed using one excitation induced by a hydraulic device at 1 Hz. The portico is mounted over a chariot with wheels protected by supplementary reinforced walls to avoid “out-of-plane” oscillations.