Тест 1.15 Vertical cantilever rod of square cross-section, loaded with longitudinal and transverse concentrated loads at the free end
Verification of the LIRA-FEM software / Тест 1.15 Vertical cantilever rod of square cross-section, loaded with longitudinal and transverse concentrated loads at the free end / Pisarenko G.S., Yakovlev A.P., Matveev V.V. Reference Book on Strength of Materials. - Kiev: Scientific Opinion, 1988.
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<b>Аналитическое решение</b>: Pisarenko G.S., Yakovlev A.P., Matveev V.V. Reference Book on Strength of Materials. - Kiev: Scientific Opinion, 1988.
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<b>Task formulation</b>: A vertical cantilever rod of square cross-section, loaded with longitudinal and transverse concentrated loads at the free end. Determine the displacements of the free end x, y, z.
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<b>Source data</b>:
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E = 3.0·10<sup>7</sup>Па
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- flexural modulus,
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μ = 0.2
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- Poisson's ratio,
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l = 10 м
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- cantilever rod height;
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b = h = 0.5 м
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- cross-sectional dimensions;
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Px = 10 кН
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- value of the concentrated force acting along the X-axis (loading 1);
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Py = 10 кН
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- value of the concentrated force acting along the Y axis (loading 2);
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N = 10000 кН
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- value of the concentrated force acting along axis Z (loading 3).
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<b>Calculation results</b>:
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At the analytical solution according to the undeformed displacement schemes of the free end x, y, z:
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<span type="#_x0000_t75" style="width: 341.25pt; height: 47.85pt;"> <span src="file:///C:/Users/DA62~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png"> <span type="tight"> </span></span></span>
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<img width="614" alt="Формула.png" src="/upload/medialibrary/e02/Formula.png" height="125" title="Формула.png" class="lazy img-responsive img-bordered ">
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<b>Примечание</b>: Three computational models are considered: rod model FE 10 - universal spatial rod FE; shell model FE 48 - universal quadrangular shell FE with intermediate nodes on the sides; volume model FE 35 - universal spatial eight-node isoparametric FE with intermediate nodes on the sides.
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<b>CALCULATION RESULTS</b>:
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<table class="table table-bordered">
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Model
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Loading 1
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Loading 2
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Loading 3
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Movements<br>
y (мм)
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Uncertainty, %
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Movements<br>
x (мм)
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Uncertainty, %
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Movements<br>
z (мм)
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Uncertainty, %
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Rod
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21.3
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0.0
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21.3
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0.0
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13.3
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0.0
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Shell
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21.3
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0.0
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21.3
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0.0
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13.4
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0.008
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Volumetric
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21.2
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0.005
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21.3
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0.005
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13.8
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0.038
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Analytical solution
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21.3
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-
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21.3
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-
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13.3
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-
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<img width="1920" alt="Fig.1 The displacements of the free end x y z for the rod shell and volume models.png" src="/upload/medialibrary/31a/Ris.1-Peremeshcheniya-svobodnogo-kontsa-x_-y_-z-dlya-sterzhnevoy_-obolochechnoy-i-obemnoy-modeley.png" height="1080" title="Рис.1 Перемещения свободного конца x y z для стержневой оболочечной и объемной моделей.png" class="lazy img-responsive img-bordered ">
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<span type="#_x0000_t75" style="width: 757.5pt; height: 426pt;"> <span src="file:///C:/Users/DA62~1/AppData/Local/Temp/msohtmlclip1/01/clip_image003.png"> </span></span>
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Fig.1 Movements of free end x, y, z for rod, shell and volume models
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<img width="1920" alt="Рис.2 Эпюры изгибающих моментов и поперечных усилий в стержневой модели и в целевых стержнях СА.png" src="/upload/medialibrary/881/Ris.2-Epyury-izgibayushchikh-momentov-i-poperechnykh-usiliy-v-sterzhnevoy-modeli-i-v-tselevykh-sterzhnyakh-SA.png" height="1080" title="Рис.2 Эпюры изгибающих моментов и поперечных усилий в стержневой модели и в целевых стержнях СА.png" class="lazy img-responsive img-bordered ">
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Fig.2 Bending moment and shear force diagrams in the rod model and in the CA target rods
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<b>Note</b>: rod analogs were modeled for planar and volumetric elements. The forces in their target rods fully coincide with the forces in the rod version of the model.
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</p> Preview image: Array
Note:
Three computational models are considered: rod model FE 10 - universal spatial rod FE; shell model FE 48 - universal quadrangular shell FE with intermediate nodes on the sides; volumetric model FE 35 - universal spatial eight-node isoparametric FE with intermediate nodes on the sides.
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