Modeling A ============== Characteristics of modeling ----------------------------------- The model is composed of 10 Euler straight beam elements. **Section S1:** variable circular section .. csv-table:: "embedded,", ":math:`\mathrm{R1}=0.1m` (solid section)" "at the free end,", ":math:`\mathit{R2}\mathrm{=}0.05m` (solid section)" **Section S2:** variable rectangular section .. csv-table:: "embedded,", ":math:`{H}_{\mathit{y1}}\mathrm{=}0.05m` "," :math:`{H}_{\mathit{z1}}\mathrm{=}0.10m`" "at the free end,", ":math:`{H}_{\mathit{y2}}\mathrm{=}0.05m` "," :math:`{H}_{\mathit{z2}}\mathrm{=}0.05m`" **Section S3:** general variable section .. csv-table:: "embedded,", ":math:`{A}_{1}\mathrm{=}{10}^{2}{m}^{4}` "," :math:`{I}_{\mathit{y1}}\mathrm{=}8.3333{10}^{6}{m}^{4}`" "at the free end,", ":math:`{A}_{2}\mathrm{=}2.5{10}^{3}{m}^{2}` "," :math:`{I}_{\mathit{y2}}\mathrm{=}5.20833{10}^{7}{m}^{4}`" Characteristics of the mesh ---------------------------- 3 sections :math:`\times` 10 items POU_D_E Tested sizes and results ------------------------------ .. csv-table:: "**Load Case**", "**Section**", "**Identification**", "**Reference**", "**Tolerance %**" "1"," :math:`\mathit{S1}` "," :math:`u(l)` ", "3.1831E—08", "1.00E-05" "", "", ":math:`n(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`n(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "3.1831E+03", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "1.2732E+04", "1.00E-05" "2"," :math:`\mathit{S1}` "," :math:`v(l)` ", "4.2441E—06", "1.00E-05" "", "", ":math:`{\theta }_{z}(l)` ", "8.4882E—06", "1.00E-05" "", "", ":math:`{v}_{y}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{v}_{y}(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{z}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{z}(l)` ", "0.0000E+00", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "1.2732E+05", "1.00E-05", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "0.0000E+00", "1.00E-05" "3"," :math:`\mathit{S1}` "," :math:`{\theta }_{x}(l)` ", "3.8621E—05", "1.00E-05", "1.00E-05" "", "", ":math:`{m}_{x}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{m}_{x}(l)` ", "1.0000E+02", "1.00E-05" "4"," :math:`\mathit{S1}` "," :math:`w(l)` ", "—8.4882E—06", "1.00E-05" "", "", ":math:`{\theta }_{y}(l)` ", "2.9708E—05", "1.00E-05", "2.9708E-05" "", "", ":math:`{v}_{z}(0)` ", "0.0000E+00", "1.00E-05" "", "", ":math:`{v}_{z}(l)` ", "0.0000E+00", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{y}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{y}(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "1.2732E+05", "1.00E-05", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "1.0185E+06", "1.00E-05" "5"," :math:`\mathit{S1}` "," :math:`u(l)` ", "1.2296E—08", "1.00E-02" "", "", ":math:`n(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`n(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "3.1831E+03", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "6"," :math:`\mathit{S1}` "," :math:`v(l)` ", "1.3486E—06", "1.00E-02" "", "", ":math:`{\theta }_{z}(l)` ", "2.1220E—06", "1.00E-02 (absolute)" "", "", ":math:`{v}_{y}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{v}_{y}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\mathit{mf}}_{z}(0)` ", "5.0000E+01", "1.00E-02" "", "", ":math:`{\mathit{mf}}_{z}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "6.3662E+04", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xy}}(0)` ", "3.1831E+03", "1.00E-05" .. csv-table:: "**Load case**", "**Section**", "**Identification**", "**Reference**", "**Deviation %**" "1"," :math:`\mathit{S2}` "," :math:`u(l)` ", "1.3862E—07", "1.00E-05", "1.00E-05" "", "", ":math:`n(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`n(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "2.0000E+04", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "4.0000E+04", "1.00E-05" "2"," :math:`\mathit{S2}` "," :math:`v(l)` ", "1.8969E—04", "2.30E-02" "", "", ":math:`{\theta }_{z}(l)` ", "3.0238E—04", "2.70E-02" "", "", ":math:`{v}_{y}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{v}_{y}(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{z}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{z}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "2.4000E+06", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\sigma }_{\mathit{xy}}(0)` ", "2.0000E+04", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xy}}(l)` ", "4.0000E+04", "1.00E-05" "3"," :math:`\mathit{S2}` "," :math:`{\theta }_{x}(l)` ", "8.3506E—04", "5.70E-02" "", "", ":math:`{m}_{x}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{m}_{x}(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xy}}(0)` ", "1.5600E+06", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xy}}(l)` ", "4.0371E+06", "5.00E-02" "", "", ":math:`{\sigma }_{\mathit{xz}}(0)` ", "1.5600E+06", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xz}}(l)` ", "4.0371E+06", "5.00E-02" "4"," :math:`\mathrm{S2}` "," :math:`w(l)` ", "—1.2000E—04", "3.00E-03" "", "", ":math:`{\theta }_{y}(l)` ", "3.600E—04", "4.00E-03" "", "", ":math:`{v}_{z}(0)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{v}_{z}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\mathit{mf}}_{y}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\mathit{mf}}_{y}(l)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(0)` ", "1.2000E+06", "1.00E-05" "", "", ":math:`{\sigma }_{\mathit{xx}}(l)` ", "4.8000E+06", "1.00E-05" "5"," :math:`\mathit{S2}` "," :math:`u(l)` ", "6.1370E—08", "1.00E-05" "", "", ":math:`n(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`n(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\sigma }_{\mathrm{xx}}(0)` ", "2.0000E+04", "1.00E-05" "", "", ":math:`{\sigma }_{\mathrm{xx}}(l/2)` ", "1.3333E+04", "1.00E-05" "", "", ":math:`{\sigma }_{\mathrm{xx}}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "6"," :math:`\mathrm{S2}` "," :math:`v(l)` ", "6.8626E—05", "2.00E-02" "", "", ":math:`{\theta }_{z}(l)` ", "9.4847E—05", "2.40E-02", "9.4847E-02" "", "", ":math:`{v}_{y}(0)` ", "1.0000E+02", "1.00E-05" "", "", ":math:`{v}_{y}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\mathrm{mf}}_{z}(0)` ", "5.0000E+01", "1.00E-02" "", "", ":math:`{\mathrm{mf}}_{z}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "", "", ":math:`{\sigma }_{\mathrm{xx}}(0)` ", "1.2000E+06", "1.00E-05" "", "", ":math:`{\sigma }_{\mathrm{xx}}(l)` ", "0.0000E+00", "1.00E-03 (absolute)" "7"," :math:`\mathrm{S3}` "," :math:`w(l)` ", "—3.8259E—05", "1.00E-02" "", "", ":math:`{\theta }_{y}(l)` ", "5.7388E—05", "1.00E-02" "", "", ":math:`{v}_{z}(0)` ", "—4.4633E+02", "1.00E-03" "", "", ":math:`{\mathrm{mf}}_{y}(0)` ", "1.7535E+02", "1.00E-02" notes --------- Since the modeling is done in Euler beams, the shear coefficients are :math:`\mathit{ky}\mathrm{=}\mathit{kz}\mathrm{=}1`.