<p>Cross-section profile along the span of a homogeneous, simply-supported, Euler-Bernoulli beam of constant depth is obtained through shape optimization, for minimum mean compliance, under uniformly distributed load. Scope is restricted to beam designs governed by deflection or stiffness criteria rather than strength criteria that requires imposition of stress constraints. It is analytically demonstrated that a generic shape optimization setup using calculus of variations framework results in a non-prismatic beam with the commonly known I-shaped cross-section. However, it is shown that a re-formulation in terms of the geometric parameters involving the I-shape leads to characteristic special cases of non-prismatic beams that further improve the optimality of the solutions. The optimal design solutions can be described through analytical equations as demonstrated through an example. The formulations proposed consider minimum and maximum dimensional constraints that are useful for structural design applications. With advancements in manufacturing and 3D printing, this work forms an important theoretical contribution to design beams with unconventional shapes but with desired optimal behavior.</p>