The stiffness maximization of elastic straight Euler-Bernoulli beams under the action of linearly distributed loads is addressed. The goal is achieved by minimizing the average compliance, which is given by the value of internal elastic energy distributed over the length of the beam. Studies in the literature suggest considering this approach since it provides, unlike the minimization of the maximum deflection, a constant bending stress behavior along the beam axis. An isoperimetric constraint on the material volume is also considered and optimal solutions are derived by means of calculus of variations. Three types of boundary conditions are discussed, namely cantilever, simply supported and guided-simply supported beams. Introducing a well known relation between the cross sectional area and moment of inertia, closed-form solutions for several cross sections commonly used in engineering are derived. Finally, a sensitivity analysis with respect to the load parameters is addressed within a numerical example.

提出了在线性分布载荷作用下弹性直欧拉-伯努利梁的刚度最大化。该目标是通过使平均柔度最小化来实现的，该平均柔度由在梁的整个长度上分布的内部弹性能的值给出。文献中的研究建议考虑使用这种方法，因为与最大挠度的最小化不同，它可以沿光束轴提供恒定的弯曲应力行为。还考虑了对物料体积的等操作约束，并通过微积分计算得出了最佳解决方案。讨论了三种边界条件，即悬臂梁，简支梁和导引简支梁。在横截面积和惯性矩之间引入一个众所周知的关系，推导了工程中常用的几个横截面的封闭式解决方案。最后，在一个数值示例中解决了关于负载参数的敏感性分析。