1. Reference problem#

1.1. Geometry#

1.1.1. Straight beam with variable circular cross section#

_images/1000000000000165000000485356E243AEEE10BE.png

Figure 1.1.1-a : Beam with variable circular cross section.

Length

: \(1m\)

Built-in radius

: \(\mathrm{0,1}m\)

Free End Radius

: \(\mathrm{0,05}m\)

1.1.2. Straight beam with variable rectangular cross section#

_images/10000000000001F30000008EC41478BE35E47508.png

Figure 1.1.2-a : Beam with variable rectangular section

Length

: \(1m\)

embedded

: \(\mathit{Hy}\mathrm{=}\mathrm{0,05}m\)

\(\mathit{Hz}\mathrm{=}\mathrm{0,10}m\)

at the free end

: \(\mathit{Hy}\mathrm{=}\mathrm{0,05}m\)

\(\mathit{Hz}\mathrm{=}\mathrm{0,05}m\)

1.1.3. Straight beam with variable general cross section#

_images/10000000000001E70000006AA8DFB717622898B0.png

Figure 1.1.3-a : Beam with variable general section

Length

: \(1m\)

embedded

: \(A\mathrm{=}{10}^{\mathrm{-}2}{m}^{2}\)

\(\mathit{Iy}\mathrm{=}\mathrm{8,3333}{10}^{\mathrm{-}6}{m}^{4}\)

at the free end

: \(A\mathrm{=}{\mathrm{2,510}}^{\mathrm{-}3}{m}^{2}\)

\(\mathit{Iy}\mathrm{=}\mathrm{5,20833}{10}^{\mathrm{-}7}{m}^{4}\)

1.2. Material properties#

Young’s module:

\(E\mathrm{=}2.{10}^{11}\mathit{Pa}\)

Poisson’s ratio:

\(\nu \mathrm{=}\mathrm{0,3}\)

Density:

\(\rho \mathrm{=}7800{\mathit{kg.m}}^{\mathrm{-}3}\)

1.3. Boundary conditions and loading#

Boundary condition:

Recessed end: \(\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}\mathit{DRX}\mathrm{=}\mathit{DRY}\mathrm{=}\mathit{DRZ}\mathrm{=}0\)

Charging:

On the right beam of variable circular cross section and on the right beam of variable rectangular cross section, the following are applied successively:

Load case

Nature

1

a one-time effort following \(X\) at the free end, \(\mathit{Fx}\mathrm{=}100N\)

2

a one-time effort following \(Y\) at the free end, \(\mathit{Fy}\mathrm{=}100N\)

3

a point moment around the \(X\) axis at the free end, \(\mathit{Mx}\mathrm{=}100\mathit{m.N}\)

4

a point moment around the \(Z\) axis at the free end, \(\mathit{Mz}\mathrm{=}100\mathit{m.N}\)

5

a load distributed over the entire beam, \(\mathit{fx}\mathrm{=}100{\mathit{N.m}}^{\mathrm{-}1}\)

6

a load distributed over the entire beam, \(\mathrm{fy}=100{\mathrm{N.m}}^{-1}\)

On the right beam of variable general cross section, the following are applied:

Load case

Nature

7

a heavy-handed effort following \(z\) with \(g=\mathrm{9,81}{\mathrm{m.s}}^{-2}\)