Sign conventions: axis system and displacements

The sectional reference frame

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.

• Unit vectors b2 and b3 define the plane of the cross-section and
• Unit vector b1 is normal to the plane of the section and extends along the axis of the beam.

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.

The warping field

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.

• The words “axial” or “out-of-plane” refer to components along the axis of the beam, i.e., along unit vector b1. For instance, component w1 of the warping field is referred to as the “axial warping component” or “out-of-plane warping component.”
• The words “transverse” or “in-plane” refer to components within the plane of the cross-section, i.e., within the plane defined by unit vectors b2 and b3. For instance, components w2 and w3 of the warping field are referred to as the “transverse warping components” or “in-plane warping components.”

Figure 2. Sign convection for the externally applied loads

Figure 2 shows the loads externally applied to a section of the beam. These loads fall into two categories.

1. Three externally applied forces: the axial force, F1, and the two transverse shear forces, F2 and F3.
2. There externally applied moments: the torque, M1, and the two bending moments, M2 and M3.

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.

Sign conventions: stress field

The sign convention for the stress field follows the conventions used in mechanics [1]. The stress field comprises six components.

1. The axial stress component, σ11(x2, x3). This direct stress component acts on the plane of the cross-section along unit vector b1.
2. The two transverse shear stress components, τ12 (x2, x3) and τ13 (x2, x3). These shear stress components act in the plane of the cross-section along unit vectors b2 and b3, respectively.
3. The two in-plane stress components, σ22 (x2, x3) and σ33 (x2, x3). These direct stress component act on the planes normal to unit vectors b2 and b3, respectively, and along unit vectors b2 and b3, respectively.
4. The in-plane shear stress component, τ23 (x2, x3). This shear stress component act on the plane normal to unit vector b2 along unit vector b3.

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.

Sign conventions: strain field

The sign convention for the strain field follows the conventions used in mechanics [1]. The strain field comprises six components.

1. The axial strain component, ε11(x2, x3).
2. The two transverse shear strain components, γ12 (x2, x3) and γ13 (x2, x3).
3. The two in-plane strain components, ε22 (x2, x3) and ε33 (x2, x3).
4. The in-plane shear strain component, γ23 (x2, x3).

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.

[1] Bauchau, O.A. and Craig, J.I., Structural Analysis with Application to Aerospace Structures, Springer, Dordrecht, Heidelberg, London, New-York, 2009