2 Bar Subject to Extension 17 Inspired by the structure of wood, the member is considered to consist

of a very large number of parallel fibres in the longitudinal direction.Later we will look at the limiting case in which the number of fibres isso large that the area A of a single fibre approaches zero.

The fibres are kept together by a very large number of absolutely rigidplanes perpendicular to the direction of the fibres. These rigid planesare known as cross-sections. Later we will look at the limiting casein which the number of cross-sections is so large that the distance xbetween two consecutive cross-sections approaches zero.

The plane cross-sections remain plane and normal to the longitudinalfibres of the beam, even after deformation. This assumption is knownas Bernoullis hypothesis.1

To describe the behaviour of the model, we use an xyz coordinate systemwith the x axis parallel to the fibres and the yz plane parallel to the cross-sections, perpendicular to the direction of the fibres.

The location of a cross-section is defined by its x coordinate; the locationof a fibre is defined by its y and z coordinates.

Later we will see that the behaviour of the bar is most easily described whenthe x axis is selected along a particular preferred fibre through the normalcentre NC. This fibre is known as the bar axis. As long as the location ofthe normal centre and bar axis are not yet known, the x axis is defined alongan arbitrary fibre that may even lie outside the cross-section.

The following assumptions are made with respect to the material behav-iour:

1 Named after the Swiss Jacob Bernoulli (16541705), from a famous family ofmathematicians and physicists.