8. Shear projection

In this chapter we describe the projection process on the initial axis, or first axis, and the second axis. We recall that the projection on these two axes is orthogonal.

8.1. Case where axis 1 is the initial axis

This case is represented on [Figure 8.1-a]. Let’s move on to frame \((O,\overrightarrow{u},\overrightarrow{v},\overrightarrow{n})\). The definitions of \(\overrightarrow{u}\), \(\overrightarrow{v}\), and \(\overrightarrow{n}\) are given in reference [bib6]. In the \((\overrightarrow{u},\overrightarrow{v})\) plane of normal \(\overrightarrow{n}\), the points \(A\), \(B\), \(C\),,, \(D\) and \(O\) have the coordinates of \(({U}_{\text{min}},{V}_{\text{max}})\), \(({U}_{\text{max}},{V}_{\text{max}})\), \(({U}_{\text{max}},{V}_{\text{min}})\),,,,,,, and, respectively. \(({U}_{\text{min}},{V}_{\text{min}})\) \(({U}_{O},{V}_{O})\)

Figure 8.1-a: Projection in the case where axis 1 is the initial axis

8.1.1. Determining the second axis

Here to determine the second axis we solve the equation:

\(\overrightarrow{\text{DB}}\text{.}\overrightarrow{\text{OM}}=0\) eq 8.1.1-1

where the coordinates \({U}_{M},{V}_{M}\) of the point \(M\) are the unknowns.

Equation [éq 8.1.1-1] is also written in the following form:

\(({U}_{\text{max}}-{U}_{\text{min}})({U}_{M}-{U}_{O})+({V}_{\text{max}}-{V}_{\text{min}})({V}_{M}-{V}_{O})=0\)

which leads to:

\({V}_{M}={V}_{O}-\frac{({U}_{\text{max}}-{U}_{\text{min}})}{({V}_{\text{max}}-{V}_{\text{min}})}({U}_{M}-{U}_{O})\)

By giving ourselves a value of \({U}_{M}\) that is different from \({U}_{O}\) we immediately get \({V}_{M}\).

8.1.2. Projection of any point on the initial axis

From any known point \(P\), the first step consists in calculating the coordinates of a point \(P\text{'}\) such as:

\(\overrightarrow{\mathrm{DB}}\mathrm{.}\overrightarrow{{\mathrm{PP}}^{\text{'}}}=0\)

Proceeding as before, we get the relationship:

\({V}_{{P}^{\text{'}}}\mathrm{=}{V}_{P}\frac{({U}_{\text{max}}\mathrm{-}{U}_{\text{min}})}{({V}_{\text{max}}\mathrm{-}{V}_{\text{min}})}({U}_{{P}^{\text{'}}}\mathrm{-}{U}_{P})\)

where \({V}_{{P}^{\text{'}}}\) results from a value of \({U}_{{P}^{\text{'}}}\) that is different from

.

In the plan

the initial axis and the segment \(\stackrel{ˉ}{\mathrm{PP}\text{'}}\) are linear lines respectively described by \(\nu ={a}_{i}u+{b}_{i}\) and \(\nu ={a}_{P}u+{b}_{P}\), so to know the coordinates of the point projected on the initial axis

we solve the equation:

\({a}_{i}u+{b}_{i}={a}_{P}u+{b}_{P}\)

where

\({a}_{i}=\frac{({V}_{\text{max}}-{V}_{\text{min}})}{({U}_{\text{max}}-{U}_{\text{min}})}\),	\({b}_{i}=\frac{({U}_{\text{max}}{V}_{\text{min}}-{U}_{\text{min}}{V}_{\text{max}})}{({U}_{\text{max}}-{U}_{\text{min}})}\),
\({a}_{P}=\frac{({V}_{{P}^{\text{'}}}-{V}_{P})}{({U}_{{P}^{\text{'}}}-{U}_{P})}\),	\({b}_{P}=\frac{({U}_{P\text{'}}{V}_{P}-{U}_{P}{V}_{P\text{'}})}{({U}_{P\text{'}}-{U}_{P})}\).

We get:

\({U}_{{P}_{i}}=\frac{{b}_{p}-{b}_{i}}{{a}_{i}-{a}_{p}}\)

\({V}_{{P}_{i}}=\frac{{a}_{i}{b}_{P}-{a}_{P}{b}_{i}}{{a}_{i}-{a}_{P}}\)

The projection of any point on the second axis is described in Appendix 2.

8.2. Case where axis 2 is the initial axis

This case is represented on [Figure 8.2-a]. As before, in plane \((\overrightarrow{u},\overrightarrow{v})\), the points \(A\), \(B\),, \(C\),, \(D\) and \(O\) have, respectively, as coordinates

,

,

, \(({U}_{\mathit{min}},{V}_{\mathit{min}})\) and \(({U}_{\mathrm{0,}}{V}_{0})\).

Figure 8.2-a: Projection in the case where axis 2 is the initial axis

8.2.1. Determining the second axis

Here to determine the second axis we solve the equation:

\(\overrightarrow{\text{AC}}\text{.}\overrightarrow{\text{OM}}=0\) eq 8.2.1-1

where the coordinates \(({U}_{M},{V}_{M})\) of the point \(M\) are the unknowns.

Equation [éq 8.2.1-1] is also written in the following form:

\(({U}_{\text{max}}-{U}_{\text{min}})({U}_{\text{M}}-{V}_{\text{O}})({V}_{\text{max}}-{V}_{\text{min}})({V}_{\text{M}}-{V}_{\text{O}})=0\)

which leads to:

\({V}_{M}={V}_{O}+\frac{({U}_{\text{max}}-{U}_{\text{min}})}{({v}_{\text{max}}-{V}_{\text{min}})}({U}_{M}-{U}_{O})\)

By giving yourself a value of

different from

we immediately get

.

8.2.2. Projection of any point on the initial axis

From a point

Whatever is known, the first step consists in calculating the coordinates of a point \(P\text{'}\) such as:

\(\overrightarrow{\mathrm{AC}}\mathrm{.}\overrightarrow{\mathrm{PP}\text{'}}=0\)

Proceeding as before, we get the relationship:

\({V}_{P\text{'}}={V}_{P}\frac{({U}_{\text{max}}-{U}_{\text{min}})}{({V}_{\text{max}}-{V}_{\text{min}})}({U}_{P\text{'}}-{U}_{P})\)

where for a value of \({U}_{P\text{'}}\) different from

we calculate \({V}_{P\text{'}}\).

In the plan

the initial axis and the segment \(\stackrel{ˉ}{\mathrm{PP}\text{'}}\) are linear lines respectively described by \(\nu ={a}_{i}u+{b}_{i}\) and \(\nu ={a}_{p}u+{b}_{p}\), so to know the coordinates of the point projected on the initial axis

we solve the equation:

\({a}_{i}u+{b}_{i}={a}_{p}u+{b}_{p}\)

where

\({a}_{i}=-\frac{({V}_{\text{max}}-{V}_{\text{min}})}{({U}_{\text{max}}-{U}_{\text{min}})}\),	\({b}_{i}=\frac{({U}_{\text{max}}{V}_{\text{max}}-{U}_{\text{min}}{V}_{\text{min}})}{({U}_{\text{max}}-{U}_{\text{min}})}\),
\({a}_{p}=\frac{({V}_{{P}^{\text{'}}}-{V}_{P})}{({U}_{{P}^{\text{'}}}-{U}_{P})}\),	\({b}_{p}=\frac{({U}_{P\text{'}}{V}_{p}-{U}_{P}{V}_{P\text{'}})}{({U}_{P\text{'}}-{U}_{P})}\).

We get:

\({U}_{{P}_{i}}=\frac{{b}_{P}-{b}_{i}}{{a}_{i}-{a}_{P}}\),

\({V}_{{P}_{i}}=\frac{{a}_{i}{b}_{P}-{a}_{P}{b}_{i}}{{a}_{i}-{a}_{P}}\).

The projection of any point on the second axis is described in Appendix 2.

8.3. Definition of the module and orientation of the projection axis

We propose to define the sign of the modulus of the projected point in relation to the initial axis. This is the \((O,\overrightarrow{u},\overrightarrow{v},\overrightarrow{n})\) frame in which the split evolves. In this coordinate system, if the \({U}_{{P}_{i}}\) component of the projected point is greater than or equal to zero, the sign of the module is positive, otherwise it is negative. In summary, the module and the sign of the modulus of the projected point are defined as follows:

\(\begin{array}{cc}{P}_{\text{mod}}=\sqrt{{\overline{{\text{OP}}_{i}}}^{2}+{\overline{{\text{OP}}_{s}}}^{2}}& \text{si}{U}_{{P}_{i}}\ge 0,\\ {P}_{\text{mod}}=-\sqrt{{\overline{{\text{OP}}_{i}}}^{2}+{\overline{{\text{OP}}_{s}}}^{2}}& \text{si}{U}_{{P}_{i}}<0\text{.}\end{array}\)

The module definition differentiates between affine loads and circular loads. In accordance with experience, a circular loading will be considered to be more damaging than a fine loading [bib1].