Arched Or Flat Crossbeams ?

Your observation Ad Hoc is correct; not any load, but a similar load perpendicular to the primary axis of the beam. So it is true in a particular case only.

You made me rethink my statement, trimaran or catamaran beams are not loaded in pure bending. A load in the same direction as the axis of the beam, such as a column in compression, will yield VERY different results. A curved column will deflect much more than a straight one. But that is column loading not beam loading.

In the example above there is not just a bending load imposed by the dagger board but also a compression load into the beam. this would result in-fact in much higher deflections on the curved beam than the straight beam. So in this application, the straight beam beam resist deflections better than the curved beam.

the same cross sectional beam would have the same torsional resistance as well, except the curved beam is longer. so the curved beam again will have higher torsional deflections than a straight beam.