工字梁有限元报告.docx

1、工字梁有限元报告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

2、 theory : 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:

3、 Thus we 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 th

4、e left section: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 s

5、tress of 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

6、is around 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 s

7、tress (Mpa)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 analyt

8、ical results. 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 preceden

9、t model.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 membra

10、ne, and 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 r

11、eache the 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 di

12、rection 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

13、:3D FullSpan 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 s

14、ee that 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

15、the model. 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 results

16、We 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. The error with the analytical solution reaches 0.7%. The normal stress: reaches the maximum at the terminal vertexes. The max is 1.31282E+02 at these terminal vertexes,

17、 C/D/E/F. The error reaches 0.2%.5. The modeling of this beam with membrane and rod elements:We construct the model with membrane and rod elements. The section of the two rods is 960mm2. The model is shown as follow.We can see that the displacement of the beam in 1D is also changed linearly. At the

18、end of the beam the displacement is 1.0735427E+01mm. The error with the analytical solution reaches 4.3%.We can see that 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 is 1.190676E+02mm

19、 at these terminal vertexes. The error reaches 6.1%.We can see that the shear stress of the beam with membrane and rods 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 quadr

20、atic function of y along the web. The max is -9.084330E+00 mm, the error is 3.4%.3. Conclusion: In the following table, we list the comparison of different models with the analytical solution of beam theory.Difference withAnalytical solution of Beam theoryError of the displacement (%)Error of the no

21、rmal stress(%)Error of the shear stress(%)CalculationTimeSimplifiedtheory0.412.9-3D2.6/9.1/1.8/2D4.6/32/2.3/1D0.7/0.2/-Membrane and beams12.5/16.5/33.3/Membrane and rods4.3/6.1/3.4/ From the table we can see the 3D model is good enough for the accuracy, but the calculation time of the model is long.

22、 The model with membrane and beam has big error in the normal stress. The 2D model is less accurate in the normal stress, but accurate in displacement and shear stress. The 1D model is the most accurate, but the model is not suitable for every situation. The model with membrane and rods which is usually used for an aeronautical structure is accurate for all the parameter.