Curvature tensor of a 3D slender beam. Euler–Bernoulli beam theory

The figure below depicts a deformed 3D slender beam. The local reference frame $(\bar x,\bar y,\bar z)$ is defined as follows: the $\bar x$-axis is tangent to the deformed beam axis, while the $\bar y$- and $\bar z$-axes coincide with the principal centroidal axes of the beam cross-section. The reference frame $(x,y,z)$ is the global reference frame.

The orientation of the local reference frame is defined by the rotation matrix $\htmlClass{fea-math-bold-italic}{\boldsymbol{R}}$. In this section, $\htmlClass{fea-math-bold-italic}{\boldsymbol{R}}$ is used as a passive transformation matrix from the global reference frame to the local reference frame. Thus, for any vector,

Here $\bar x$ denotes the local beam-axis coordinate, defined along the locus of the cross-section centroids. For a general curved beam, an arc-length coordinate $s$ could also be used; here $\bar x$ is preferred because the formulation will later be specialized to straight beams with small deformations.

Let us consider two very close cross-sections located at a distance $d\bar x$. The first one is characterized by the rotation matrix $\htmlClass{fea-math-bold-italic}{\boldsymbol{R}}$. The second cross-section is rotated with respect to the first one by the infinitesimal angles $d\varphi_{\bar x}$, $d\varphi_{\bar y}$, and $d\varphi_{\bar z}$ about the $\bar x$-, $\bar y$-, and $\bar z$-axes of the local reference frame, respectively.

The corresponding infinitesimal relative rotation matrix, expressed in the local reference frame of the first cross-section, is

Consequently, the rotation matrix of the second cross-section is

Therefore,

Dividing by $d\bar x$,

We define the curvature components:

The curvature tensor is

and

The curvature tensor is skew-symmetric:

The curvature components can also be organized as a vector:

The components $\kappa_{\bar x},\kappa_{\bar y},\kappa_{\bar z}$ are not the curvatures of the beam axis in the usual differential-geometric sense. They are defined from the relative infinitesimal rotation of two neighbouring cross-sections per unit length about the local axes. Thus, $\kappa_{\bar x}\,d\bar x$ is the relative infinitesimal rotation about the local $\bar x$-axis, $\kappa_{\bar y}\,d\bar x$ is the relative infinitesimal rotation about the local $\bar y$-axis, and $\kappa_{\bar z}\,d\bar x$ is the relative infinitesimal rotation about the local $\bar z$-axis.

According to the sign convention adopted in the Nomenclature, the corresponding constitutive relations are

see Section 22.2.

It is also easy to prove directly that $\widetilde{\htmlClass{fea-math-bold-italic}{\boldsymbol{\kappa}}}$ is skew-symmetric [1]. Since $\htmlClass{fea-math-bold-italic}{\boldsymbol{R}}$ is an orthogonal matrix,

Differentiating with respect to $\bar x$,

Hence,

and therefore

References

1. Da Lozzo, E. C., Geometrically exact three-dimensional beam theory: modeling and FEM implementation for statics and dynamics analysis, Università degli Studi di Pavia, 2010.