The stiffness of a proportionally loaded structure may continuously increase or decrease. As a special exception, it may be constant. On the other hand, an initially stiffening (softening) structure may turn into a softening (stiffening) structure. At the load level of such a change the stiffness of the structure attains an extreme value. The task of this work is to present mathematical conditions for these load levels. Lack of them represents a void in the pertinent literature. The practical significance of the aforementioned changes is the one of indicators of the mechanical behavior to be expected after their occurrence. The second and the last author have recently presented a condition for the load level at which the stiffness of a proportionally loaded structure becomes a minimum value. It is given as , representing the condition for a point of inflection of the real part of a complex eigenvalue function , where denotes a dimensionless load parameter. The underlying linear eigenvalue problem has two indefinite coefficient matrices, which is a necessary condition for complex regions of eigenvalue functions. These matrices are established with hybrid elements, available in a commercial finite element program. In the present work, is shown to be the condition for the load level at which the stiffness of a proportionally loaded structure attains a maximum value. The eigenvalue function concerned has no complex region. It is also shown that the displacement elements, which are the basis for their extension to the employed hybrid elements, are unable to indicate the load level at an extreme value of the stiffness.

中文翻译：

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特殊特征值函数的拐点作为比例加载结构的刚度最大值/最小值的指标


按比例加载的结构的刚度可以连续增加或减少。作为特殊例外，它可能是恒定的。另一方面，最初的硬化（软化）结构可能会变成软化（硬化）结构。在这种变化的载荷水平下，结构的刚度达到极值。这项工作的任务是提出这些负载水平的数学条件。缺乏它们代表着相关文献的空白。上述变化的实际意义是其发生后预期的机械行为的指标之一。第二位也是最后一位作者最近提出了负载水平的条件，在该条件下，比例负载结构的刚度变为最小值。它给出为 ，表示复特征值函数 实部拐点的条件，其中 表示无量纲载荷参数。潜在的线性特征值问题具有两个不定系数矩阵，这是特征值函数的复杂区域的必要条件。这些矩阵是使用混合元素建立的，可在商业有限元程序中使用。在目前的工作中，表明了负载水平的条件，在该条件下，比例负载结构的刚度达到最大值。相关特征值函数没有复区域。还表明，作为其扩展到所采用的混合元件的基础的位移元件无法指示刚度极值处的负载水平。