Why are some water storage dams ellipsoidal in shape

It would firstly be a matter of cost. A spherical shape would require more material to achieve the same result. however multi arch dams are More spherical in shape.

A second will be the forces on the flanks will be closer to perpendicular on a ellipsoid than for a spherical design.

As you take the central segment of the dam, the water pushes it back. But, due its curvature, it translates into a push almost outward. Next segment carries it, and adds its own. When you finish this train of thinking, every segment have to carry the compression load from the previous one, and add its own to it. So you are ending up with one gradually thickening toward the side walls and gradually and cumulatively curving away from the water to transmit the backward push to the valley walls. That is an ellipsoid.

The beauty of it is, that it is all compression in the dam. After all, you might not be happy with a dam with tension in it, would you!?!

Hendrik pretty much nailed it in very few words. Not much explanation WHY there is less material in an ellipsoidal dam rather than a spherical dam, so, in case anyone is wondering about that, it is really basic geometry.

One could compare the formulae for the circumference of an ellipse to that of a circle, but the formula for the ellipse is quite complex:

where is the eccentricity of the ellipse (=0 for a circle) and a is its semi-major axis. So perhaps it would be better to compare the two graphically.

First of all, just as a circle is a special case of an ellipse (where the two foci are co-incident), so is a sphere a special case of the ellipsoid. If you can, imagine an ellipse within a circle such that the major axis of the ellipse falls exactly on the diameter of the circle, they have the same length, and the two figures intersect at the two points which are at the ends of the major axis and diameter.

Since the shortest distance between any two points is a straight line, any curve which deviates from the line will be longer. The more it deviates the longer is the line. A very shallow ellipse will deviate only slightly from the line. Its two foci are very near the end points of its axis. These end points would be the anchor points for the dam. As you "stretch" the ellipse so that the foci get closer and closer the perimeter of the ellipse deviates more and more from the straight line, getting longer and longer, until the shape becomes a circle when the foci meet at the center. Thus the semi-circular arc is longer than the semi-elliptical arc. Extending this to three dimensions the spherical section has more area than an ellipsoidal section having the same anchor points. It is a little like comparing a basketball (about 10 inches in diameter) to a slightly smaller than normal American football (actually a prolate spheroid, about 11 inches long) so that the basketball diameter was the same as the football length. The surface area of the football would be significantly smaller.

Since many dams have a minimum wall thickness, primarily due to construction techniques employed, a shorter, albeit weaker, ellipsoidal dam may have more than enough strength to retain the water pressure as compared to the alternate spherical dam. Therefore the choice would be made to use the smaller shape that conserves materials (and building time).