工字梁有限元报告.docxVIP

Finite Element Raport 1. 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. Finite Element solution 1．The 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 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 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 HYPERLINK app:ds:terminal terminal HYPERLINK app:ds:vertex vertexes. The max is1.435663E+02 at these HYPERLINK app:ds:terminal terminal HYPERLINK app:ds:vertex vertexes, and the error is 9.1%. The shear stress of component XY : We can see that the shear stress 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 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. Displacement Max(mm) Normal stress