3. Modeling A#

3.1. Characteristics of modeling#

The model is composed of 10 Euler straight beam elements.

Section S1: variable circular section

embedded,

\(\mathrm{R1}=0.1m\) (solid section)

at the free end,

\(\mathit{R2}\mathrm{=}0.05m\) (solid section)

Section S2: variable rectangular section

embedded,

\({H}_{\mathit{y1}}\mathrm{=}0.05m\)

\({H}_{\mathit{z1}}\mathrm{=}0.10m\)

at the free end,

\({H}_{\mathit{y2}}\mathrm{=}0.05m\)

\({H}_{\mathit{z2}}\mathrm{=}0.05m\)

Section S3: general variable section

embedded,

\({A}_{1}\mathrm{=}{10}^{2}{m}^{4}\)

\({I}_{\mathit{y1}}\mathrm{=}8.3333{10}^{6}{m}^{4}\)

at the free end,

\({A}_{2}\mathrm{=}2.5{10}^{3}{m}^{2}\)

\({I}_{\mathit{y2}}\mathrm{=}5.20833{10}^{7}{m}^{4}\)

3.2. Characteristics of the mesh#

3 sections \(\times\) 10 items POU_D_E

3.3. Tested sizes and results#

Load Case

Section

Identification

Reference

Tolerance %

1

\(\mathit{S1}\)

\(u(l)\)

3.1831E—08

1.00E-05

\(n(0)\)

1.0000E+02

1.00E-05

\(n(l)\)

1.0000E+02

1.00E-05

\({\sigma }_{\mathit{xx}}(0)\)

3.1831E+03

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

1.2732E+04

1.00E-05

2

\(\mathit{S1}\)

\(v(l)\)

4.2441E—06

1.00E-05

\({\theta }_{z}(l)\)

8.4882E—06

1.00E-05

\({v}_{y}(0)\)

1.0000E+02

1.00E-05

\({v}_{y}(l)\)

1.0000E+02

1.00E-05

\({\mathit{mf}}_{z}(0)\)

1.0000E+02

1.00E-05

\({\mathit{mf}}_{z}(l)\)

0.0000E+00

1.00E-05

\({\sigma }_{\mathit{xx}}(0)\)

1.2732E+05

1.00E-05

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

0.0000E+00

1.00E-05

3

\(\mathit{S1}\)

\({\theta }_{x}(l)\)

3.8621E—05

1.00E-05

1.00E-05

\({m}_{x}(0)\)

1.0000E+02

1.00E-05

\({m}_{x}(l)\)

1.0000E+02

1.00E-05

4

\(\mathit{S1}\)

\(w(l)\)

—8.4882E—06

1.00E-05

\({\theta }_{y}(l)\)

2.9708E—05

1.00E-05

2.9708E-05

\({v}_{z}(0)\)

0.0000E+00

1.00E-05

\({v}_{z}(l)\)

0.0000E+00

1.00E-05

\({\mathit{mf}}_{y}(0)\)

1.0000E+02

1.00E-05

\({\mathit{mf}}_{y}(l)\)

1.0000E+02

1.00E-05

\({\sigma }_{\mathit{xx}}(0)\)

1.2732E+05

1.00E-05

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

1.0185E+06

1.00E-05

5

\(\mathit{S1}\)

\(u(l)\)

1.2296E—08

1.00E-02

\(n(0)\)

1.0000E+02

1.00E-05

\(n(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\sigma }_{\mathit{xx}}(0)\)

3.1831E+03

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

0.0000E+00

1.00E-03 (absolute)

6

\(\mathit{S1}\)

\(v(l)\)

1.3486E—06

1.00E-02

\({\theta }_{z}(l)\)

2.1220E—06

1.00E-02 (absolute)

\({v}_{y}(0)\)

1.0000E+02

1.00E-05

\({v}_{y}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\mathit{mf}}_{z}(0)\)

5.0000E+01

1.00E-02

\({\mathit{mf}}_{z}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\sigma }_{\mathit{xx}}(0)\)

6.3662E+04

1.00E-05

\({\sigma }_{\mathit{xy}}(0)\)

3.1831E+03

1.00E-05

Load case

Section

Identification

Reference

Deviation %

1

\(\mathit{S2}\)

\(u(l)\)

1.3862E—07

1.00E-05

1.00E-05

\(n(0)\)

1.0000E+02

1.00E-05

\(n(l)\)

1.0000E+02

1.00E-05

\({\sigma }_{\mathit{xx}}(0)\)

2.0000E+04

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

4.0000E+04

1.00E-05

2

\(\mathit{S2}\)

\(v(l)\)

1.8969E—04

2.30E-02

\({\theta }_{z}(l)\)

3.0238E—04

2.70E-02

\({v}_{y}(0)\)

1.0000E+02

1.00E-05

\({v}_{y}(l)\)

1.0000E+02

1.00E-05

\({\mathit{mf}}_{z}(0)\)

1.0000E+02

1.00E-05

\({\mathit{mf}}_{z}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\sigma }_{\mathit{xx}}(0)\)

2.4000E+06

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\sigma }_{\mathit{xy}}(0)\)

2.0000E+04

1.00E-05

\({\sigma }_{\mathit{xy}}(l)\)

4.0000E+04

1.00E-05

3

\(\mathit{S2}\)

\({\theta }_{x}(l)\)

8.3506E—04

5.70E-02

\({m}_{x}(0)\)

1.0000E+02

1.00E-05

\({m}_{x}(l)\)

1.0000E+02

1.00E-05

\({\sigma }_{\mathit{xy}}(0)\)

1.5600E+06

1.00E-05

\({\sigma }_{\mathit{xy}}(l)\)

4.0371E+06

5.00E-02

\({\sigma }_{\mathit{xz}}(0)\)

1.5600E+06

1.00E-05

\({\sigma }_{\mathit{xz}}(l)\)

4.0371E+06

5.00E-02

4

\(\mathrm{S2}\)

\(w(l)\)

—1.2000E—04

3.00E-03

\({\theta }_{y}(l)\)

3.600E—04

4.00E-03

\({v}_{z}(0)\)

0.0000E+00

1.00E-03 (absolute)

\({v}_{z}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\mathit{mf}}_{y}(0)\)

1.0000E+02

1.00E-05

\({\mathit{mf}}_{y}(l)\)

1.0000E+02

1.00E-05

\({\sigma }_{\mathit{xx}}(0)\)

1.2000E+06

1.00E-05

\({\sigma }_{\mathit{xx}}(l)\)

4.8000E+06

1.00E-05

5

\(\mathit{S2}\)

\(u(l)\)

6.1370E—08

1.00E-05

\(n(0)\)

1.0000E+02

1.00E-05

\(n(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\sigma }_{\mathrm{xx}}(0)\)

2.0000E+04

1.00E-05

\({\sigma }_{\mathrm{xx}}(l/2)\)

1.3333E+04

1.00E-05

\({\sigma }_{\mathrm{xx}}(l)\)

0.0000E+00

1.00E-03 (absolute)

6

\(\mathrm{S2}\)

\(v(l)\)

6.8626E—05

2.00E-02

\({\theta }_{z}(l)\)

9.4847E—05

2.40E-02

9.4847E-02

\({v}_{y}(0)\)

1.0000E+02

1.00E-05

\({v}_{y}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\mathrm{mf}}_{z}(0)\)

5.0000E+01

1.00E-02

\({\mathrm{mf}}_{z}(l)\)

0.0000E+00

1.00E-03 (absolute)

\({\sigma }_{\mathrm{xx}}(0)\)

1.2000E+06

1.00E-05

\({\sigma }_{\mathrm{xx}}(l)\)

0.0000E+00

1.00E-03 (absolute)

7

\(\mathrm{S3}\)

\(w(l)\)

—3.8259E—05

1.00E-02

\({\theta }_{y}(l)\)

5.7388E—05

1.00E-02

\({v}_{z}(0)\)

—4.4633E+02

1.00E-03

\({\mathrm{mf}}_{y}(0)\)

1.7535E+02

1.00E-02

3.4. notes#

Since the modeling is done in Euler beams, the shear coefficients are \(\mathit{ky}\mathrm{=}\mathit{kz}\mathrm{=}1\).