Passive gust load alleviation systems have the potential to significantly reduce airframe mass without reliance on complex systems of sensors and actuators. Recent experimental work by the authors has shown that a passive, strain-actuated spoiler can rapidly reduce the lift coefficient of an aerofoil. In this work, we numerically investigate the efficacy of a strain-actuated spoiler in alleviating loads within the wider airframe. The airframe is represented by a beam model which is exposed to a series of One-Minus-Cosine gusts. The effect of the spoiler on the wing is captured by locally reducing lift when wingbox strains meet a triggering condition. The model spoiler is shown to be capable of reducing the sizing wing root bending moment by up to $17$% for the airframe and spoiler parameters considered. In addition, the sensitivity of this load alleviation to key spoiler design parameters is investigated. It is found that deploying the spoiler as early as possible in the gust provides the best load alleviation performance. In a few cases, the spoiler is found to induce a limit cycle oscillation in the wing by repeatedly deploying and stowing. This may be an artefact caused by the low fidelity structural model employed in this work. Nonetheless, two ways of preventing this behaviour are demonstrated. Our work demonstrates for the first time that a strain-actuated spoiler is capable of alleviating loads at the scale of a full aircraft.