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1、Finite Element Raport1. Analytical solution Displacement created by the bending moment: Displacement created by the shear force: Simplified theory : Beam theory : So, total displacement is The normal stress is： Simplified theory : Beam theory : , The shear stress is： Simplified theory : Beam theory
2、: 2. Finite Element solution1The modeling of this beam with a 3D model: The 3D model with a half of dimension: We add the limit condition by using the symmetry. We obtain for the symmetry plan XOY Tz=0. The left extremity is fixed, so the Tx,Ty,Tz=0.We can get the model with limit condition: Thus we
3、 can get the result. The displacement is shown as follow.We can see that the displacement of the beam in 3D is changed linearly. At the end of the beam the displacement is 1.092503E+01 mm. It is close to the analytical solution, and the error is 2.6%. The normal stress of component X :For the left s
4、ection:We can see that the stress of component X of the beam in 3D is changed linearly in the left section. It reaches the maximum at the terminal vertexes. The max is1.435663E+02 at these terminal vertexes, and the error is 9.1%. The shear stress of component XY :We can see that the shear stress of
5、 the beam in 3D is nearly constant in the middle part of the beam in the direction of X. We chose several nodes in the direction Y in the middle part. The precedent figure shows a range as a quadratic function of y along the web, which verifies the beam theory. The range of the shear stress is aroun
6、d 8-9Mpa, so we can make the simplified model which has constant shear stress along the web. The maximum shear is 9.570804E+00, the error with beam theory is 1.8%.Comparing with the analytical solution, we make a table to critique the results. DisplacementMax(mm)Normal stress max(Mpa)Shear stress (M
7、pa)Finite Element in 3D1.092503E+011.435663E+029.570804E+00(max)Analytical solution11.221319.4(max)The results of 3D finite element model are close to the analytical solution of beam theory no matter in the displacement and the stress. So the 3D model makes a good approximation to the analytical res
8、ults. We augment twice the size of the element with the same 3D model. We can get the result of the normal stress.The max normal stress is 183Mpa, which is far more than the precedent model compared with the analytical solution, but the delay of the calculation is much less than the precedent model.
9、So it is not a wise choice to use 3D model, because we should try to find a balance of the accuracy and the delay.2The modeling of this beam with a 2D model: We construct the model with membrane and beam elements by 5 surfaces. Among these five surface, the web surface is modeled by a membrane, and
10、the others are modeled by shells. From the model we can get the resultsWe can see that the displacement of the beam in 3D is changed linearly. At the end of the beam the displacement is 1.0702785E+01mm. It is close to the analytical solution, and the error is 4.6%. The normal stress doesnt reache th
11、e maximum at the terminal vertexes, but at the left section. The max is7.5603534, and the error compared with beam theory is 32.6%We can see that the shear stress of the beam in 2D model is nearly constant in the middle part of the beam in the direction of X. We chose several nodes in the direction
12、Y in the middle part. The precedent figure shows a range as a quadratic function of y along the web. The max is -9.624660E+00 mm, and the error is 2.3%.3. The modeling of this beam with membrane and beam elements: We construct the model with membrane and beam elements Display:1D line Display:3D Full
13、Span From the model we can get the resultsWe can see that the displacement of the beam in model of membrane and beam is changed linearly. At the end of the beam the displacement is 9.814795E+00mm. The error with the analytical solution reaches 12.5%.The normal stress of component X :We can see that
14、the normal stress of component X of the beam in this model is also changed linearly in the left section. It reaches the maximum at the terminal vertexes. The max is1.093545E+02at these terminal vertexes. The error reaches 16.5%.We chose several nodes in the direction Y in the middle part of the mode
15、l. The precedent figure shows a range as a quadratic function of y along the web. The max is -6.271123E+00. The error reaches 33.3%. 4. The modeling of this beam with a 1D model: We construct the model in 1D model The section of beam: Display:3D FullScan From the model we can get the resultsWe can see that the displacement of the beam in 1D is changed linearly. At the end of the beam the displacement is 1.1136911E+01mm.
