The sheet flexure is commonly used to provide support stiffness in flexure mechanisms for precision applications. While the sheet flexure is often analyzed in a simplified form, e.g. by assuming planar deformation or linearized stiffness, the deformation in practice is spatial and sufficiently large that nonlinear effects due to the geometric stiffness are significant.

This paper presents a compact analytical model for the nonlinear stiffness characteristics of spatially deforming sheet flexures under general 3-D load conditions at moderate deformations. This model provides closed-form expressions in a mixed stiffness and compliance matrix format that is tailored to flexure mechanism analysis. The effects of bending, shear, elongation, torsion and warping deformation are taken into account, so that the stiffness in all directions, including the in-plane lateral support direction, is modeled accurately. The model is verified numerically against beam and shell-based finite elements. The approach for deriving closed-form solutions in a nonlinear context is detailed in this paper. The Hellinger–Reissner variational principle with a specific physically motivated set of low-order interpolation functions is shown to be well-suited to the geometrically nonlinear analysis of flexures.

An extension of the derivation approach to the nonlinear closed-form analysis of general flexure mechanisms consisting of multiple sheet flexures connected in parallel is presented. This is demonstrated with the case of a spatially deforming parallelogram flexure mechanism and a cross-hinge flexure mechanism.

薄板挠曲通常用于为精密应用提供挠曲机构中的支撑刚度。尽管通常以简化形式来分析薄板挠曲，例如通过假设平面变形或线性刚度来进行分析，但实际上变形是空间上的，并且足够大，以至于几何刚度引起的非线性影响非常明显。

本文提出了一个紧凑的分析模型，用于在中等变形条件下一般3-D载荷条件下空间变形的片状挠曲的非线性刚度特性。该模型以混合刚度和柔度矩阵格式提供闭合形式的表达式，该格式专门针对挠曲机制分析而设计。考虑到弯曲，剪切，伸长，扭转和翘曲变形的影响，因此可以精确建模包括平面内横向支撑方向在内的所有方向的刚度。该模型针对梁和基于壳的有限元进行了数值验证。本文详细介绍了在非线性情况下得出封闭形式解的方法。

提出了将推导方法扩展到对一般挠曲机构进行非线性闭合形式分析的方法，该挠曲机构由并联连接的多个板挠曲组成。这在空间变形平行四边形挠曲机构和交叉铰链挠曲机构的情况下得到证明。