SectionBuilder uses a number of sign conventions for the various quantities used in the analysis. Figure 1 shows the reference system used by SectionBuilder. The origin of the axis system is at point O.
The coordinates of a point of the cross-section are denoted x2 and x3 along unit vectors b2 and b3, respectively, as indicated in fig. 1. Because the cross-section is in plane (b2, b3), coordinate x1 = 0.
When the beam is loaded, the cross-section will deform both in-plane and out-of-plane. Typically, the displacement field is called the “warping displacement field”, or simply, the “warping field”. The components of the warping displacement vector along unit vectors b1, b2, and b3 are denoted w1, w2, and w3, respectively, as indicated in fig. 1.
The following vocabulary is used in beam theory.
Figure 2 shows the loads externally applied to a section of the beam. These loads fall into two categories.
Forces F1, F2, and F3 are positive when acting along unit vectors b1, b2, and b3, respectively. Moments M1, M2, and M3 are positive when acting about unit vectors b1, b2, and b3, respectively.
Unless otherwise specified, the lines of action of forces F1, F2, and F3 pass through point O, as illustrated in fig. 2.
The sign convention for the stress field follows the conventions used in mechanics . The stress field comprises six components.
Note that in Euler-Bernoulli or Timoshenko beam theory, the in-plane stress components are assumed to vanish, i.e., σ22 (x2, x3) ≈ 0, σ33 (x2, x3) ≈ 0, and τ23 (x2, x3) ≈ 0.
The sign convention for the strain field follows the conventions used in mechanics . The strain field comprises six components.
Note that in Euler-Bernoulli beam theory, all strain components are assumed to vanish, except for the axial strain component that is assumed to vary linearly over the cross-section. Timoshenko beam theory relaxes these assumptions by allowing non-vanishing values for the two transverse shear strain components that are uniformly distributed over the cross-section.
 Bauchau, O.A. and Craig, J.I., Structural Analysis with Application to Aerospace Structures, Springer, Dordrecht, Heidelberg, London, New-York, 2009