3. Modeling A#

3.1. Characteristics of modeling#

We use a POU_D_EM model.

the beam is modelled with 1 finite element and the section is discretized with 8 fibers. The reference axis is deliberately chosen different from the barycenter, offset by h/2 downwards, with the keyword COOR_AXE_POUTRE from DEFI_GEOM_FIBRE.

The section is « cut » into 8 fibers (Figure 2). The coordinates of the fiber centers and their areas are given in table 1.

_images/1000020100000066000000987E9CF55EEACE648C.png

Figure 2: Dividing the section into 8 fibers

Fibers

\({y}_{i}\)

\({z}_{i}\)

\({S}_{i}\)

Fibers

\({y}_{i}\)

\({z}_{i}\)

\({S}_{i}\)

1

0.1

0.875

0.75

0.05

5

-0.1

0.875

0.05

2

0.1

0.625

0.05

0.05

6

-0.1

0.625

0.05

3

0.1

0.375

0.05

0.05

7

-0.1

0.375

0.05

4

0.1

0.125

0.05

0.05

8

-0.1

0.125

0.05

Table 1: Characteristics of fibers

3.2. Characteristics of the mesh#

The mesh contains 1 elements of type SEG2.

3.3. Tested sizes and results#

Numerical calculation of the quantities of the straight section gives:

\(S=0.4{m}^{2}\) \({A}_{G}=0.2{m}^{3}\) \({I}_{G}=0.13125{m}^{4}\) and \({I}_{{G}_{0}}=0.03125{m}^{4}\)

The arrow at the loaded end of the beam is: \(f=0.00035555m\)

At the level of embedding \(x=0\) we have the following values:

\(M=-{10}^{-6}\mathit{Nm}\) \({\chi }_{s}=0.00106666{m}^{-1}\) \({ϵ}_{s}=-0.00053333{m}^{-1}\)

At the level of the first Gauss point \(x=(1-\frac{1}{\sqrt{3}})/2=0.21132486540518503m\) we have the following values: \(M=-788675.13\mathit{Nm}\) \({\chi }_{s}=0.00084125{m}^{-1}\) \({ϵ}_{s}=-0.0004206{m}^{-1}\)

for fiber No. 1 (\(z=0.875m\)): \(ϵ=0.00031547\) and \(\sigma =9464101.6\mathit{Pa}\) and for fiber No. 4 to \(z=0.125m\) \(ϵ=-0.00031547\) and \(\sigma =-9464101.6\mathit{Pa}\)

Boom at the end of the beam (DEPL):

Point

Component

Reference Value

Tolerance

APPUI

DZ

\(-3.5555555555555E-4\)

\(1.E-6\)

Generalized deformations during embedding (DEGE_ELNO):

Mesh

Knot

Component

Reference Value

Tolerance

M1

N1

KY

\(1.066666666666667E-3\)

\(1.E-6\)

M1

N1

EPXX

\(-5.333333333333333E-3\)

\(1.E-6\)

Deformations and stresses in fibers (EPSI_ELGA and SIEF_ELGA):

Mesh

Point

Sub-Point

Sub-Point

Component

Reference Value

Tolerance

M1

1

1

EPXX

\(3.15470053837926E-4\)

\(1.E-6\)

M1

1

1

SIXX

\(9.46410161513778E6\)

\(1.E-6\)

M1

1

4

EPXX

\(-3.15470053837926E-4\)

\(1.E-6\)

M1

1

4

SIXX

\(-9.46410161513778E6\)

\(1.E-6\)