Sign conventions: axis system and displacements
Figure 1. Reference system for SectionBuilder
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 b_{2} and b_{3} define the plane of the cross-section and
- Unit vector b_{1} 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 x_{2} and x_{3} along unit vectors b_{2} and b_{3}, respectively, as indicated in fig. 1. Because the cross-section is in plane (b_{2}, b_{3}), coordinate x_{1} = 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 b_{1}, b_{2}, and b_{3} are denoted w_{1}, w_{2}, and w_{3}, 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 b_{1}. For instance, component w_{1} 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 b_{2} and b_{3}. For instance, components w_{2} and w_{3} of the warping field are referred to as the “transverse warping components” or “in-plane warping components.”
Sign conventions: externally applied loads
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.
- Three externally applied forces: the axial force, F_{1}, and the two transverse shear forces, F_{2} and F_{3}.
- There externally applied moments: the torque, M_{1}, and the two bending moments, M_{2} and M_{3}.
Forces F_{1}, F_{2}, and F_{3} are positive when acting along unit vectors b_{1}, b_{2}, and b_{3}, respectively. Moments M_{1}, M_{2}, and M_{3} are positive when acting about unit vectors b_{1}, b_{2}, and b_{3}, respectively.
Unless otherwise specified, the lines of action of forces F_{1}, F_{2}, and F_{3} 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.
- The axial stress component, σ_{11}(x_{2}, x_{3}). This direct stress component acts on the plane of the cross-section along unit vector b_{1}.
- The two transverse shear stress components, τ_{12} (x_{2}, x_{3}) and τ_{13} (x_{2}, x_{3}). These shear stress components act in the plane of the cross-section along unit vectors b_{2} and b_{3}, respectively.
- The two in-plane stress components, σ_{22} (x_{2}, x_{3}) and σ_{33} (x_{2}, x_{3}). These direct stress component act on the planes normal to unit vectors b_{2} and b_{3}, respectively, and along unit vectors b_{2} and b_{3}, respectively.
- The in-plane shear stress component, τ_{23} (x_{2}, x_{3}). This shear stress component act on the plane normal to unit vector b_{2} along unit vector b_{3}.
Note that in Euler-Bernoulli or Timoshenko beam theory, the in-plane stress components are assumed to vanish, i.e., σ_{22} (x_{2}, x_{3}) ≈ 0, σ_{33} (x_{2}, x_{3}) ≈ 0, and τ_{23} (x_{2}, x_{3}) ≈ 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.
- The axial strain component, ε_{11}(x_{2}, x_{3}).
- The two transverse shear strain components, γ_{12} (x_{2}, x_{3}) and γ_{13} (x_{2}, x_{3}).
- The two in-plane strain components, ε_{22} (x_{2}, x_{3}) and ε_{33} (x_{2}, x_{3}).
- The in-plane shear strain component, γ_{23} (x_{2}, x_{3}).
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