3. Kinematics of a beam in finite rotations

Figure 3-a: Evolution of a beam stretch

Let’s follow the evolution of a section of a beam from its initial position - or of reference - [fig3‑a] (a) to its deformed position at the time \(t\) [fig 3-a] (b).

The right section of the center \(P\) of the beam in the reference position is indicated by the curvilinear abscissa \(s\) of \(P\) on the center line (or neutral fiber). To this section is attached the orthonormal trihedron \({E}_{1}\text{}{E}_{2}\text{}{E}_{3}\text{}:\text{}{E}_{1}\) is the unit tangent of the line of centers in \(P;{E}_{2}\text{et}{E}_{3}\) are directed along the main axes of inertia of the section.

As in [bib1] to [bib5], it is assumed that during the movement the initially straight sections remain flat and do not change shape.

From moment 0 to moment \(t\):

• \(P\) comes in \(P\text{'}\) and the position of \(P\text{'}\) is defined by the vector \({x}_{o}(s,t)\);

• the orthonormal trihedron \({E}_{1}{E}_{2}{E}_{3}\) becomes the orthonormal trihedron \({t}_{1}{t}_{2}{t}_{3}\text{.}{t}_{2}\) and \({t}_{3}\) are always directed along the main inertia axes of the section and \({t}_{1}\) is always normal unit to this section. But \({t}_{1}\) is not necessarily tangent to the center line in \(P\text{'}\): in other words, there may be, in a deformed position, a slip due to shear, as in the Timoshenko model.

The state of the section at time \(t\) is therefore defined by:

• the vector \({x}_{o}(s,t)\), which gives the position of the center of gravity;

• the rotation vector that goes from trihedron \({E}_{1}{E}_{2}{E}_{3}\) to trihedron \({t}_{1}{t}_{2}{t}_{3}\), and which is defined in [§4.1].

The combination of these two vectors makes up vector \(\phi (s,t)\).