Homework2.docx

The Finite Element Method HOMEWORK 2

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The Finite Element Method The second exercise will emphasize the influence of the mesh refinement on the results. The difference between the exact solution and the approximate one, obtained by the F.E.M., depends on the mesh refinement and/or on the degree of the approximation functions. Usually, to improve the results (minimizing the error), two methods are available: - to refine the mesh of the model (especially in those areas where high gradients of the unknown parameters are expected); - to use elements with higher order of the polynomial functions. In both cases, the computing time increases. In the first case, the computing time increases by augmenting the element calculations (more element matrices, more equations in the global equation system, etc). In the second case, the integration process per element is more complicated. For the second exercise, the analyzed structure is a cantilever wall-beam (the name is due to the unusual h/L ratio, larger then for ordinary beams), subjected to a vertical concentrated force, acting on its right-hand side end. The parameter of interest is the vertical displacement of the same point (the maximum vertical deflection occurring at the end of the beam).

Working hypotheses: - The problem is considered a 2D one, the beam working in plane stress state. - The beam’s own weight is neglected. The geometry of the wall-beam is defined by its length, height and thickness (L., h, b). They are prescribed according to the student’s number. - 𝐿 = 1000 + 25𝑁 𝑐𝑚 = 1000 + 25 × 1 = 1025 𝑐𝑚 - ℎ = 200 + 5𝑁 𝑐𝑚 = 200 + 5 × 1 = 205 𝑐𝑚 - 𝑏 = 40 + 𝑁 𝑐𝑚 = 40 + 1 = 41 𝑐𝑚 The material properties: - Young modulus E = 25000 daN/cm2; - Poisson’s ratio μ = 0.25 The applied force: -𝑃 = 5000 + 100𝑁 𝑑𝑎𝑁 = 5000 + 100 × 1 = 5100 𝑑𝑎𝑁 Four finite element models will be created, increasing the elements’ number (10, 20, 40 and 80 elements).

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Preprocessing text files Model 1: /prep7 ! ET,1,42,,,3 R,1,41 ! MP,EX,1,2.5E4 MP,NUXY,1,0.25 ! N,1,0,0 N,6,1025,0 FILL,1,6 ! NGEN,3,6,1,6,1,,205/2 ! E,1,2,8,7 EGEN,5,1,1 EGEN,2,6,1,5,1 ! NSEL,S,LOC,X,0 D,ALL,ALL,0 NALL ! NSEL,S,LOC,X,1025 F,ALL,FY,-5100/3 NALL ! SAVE FINISH

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Model 2: /prep7 ! ET,1,42,,,3 R,1,41 ! MP,EX,1,2.5E4 MP,NUXY,1,0.25 ! N,1,0,0 N,11,1025,0 FILL,1,11 ! NGEN,3,11,1,11,1,,205/2 ! E,1,2,13,12 EGEN,10,1,1 EGEN,2,11,1,10,1 ! NSEL,S,LOC,X,0 D,ALL,ALL,0 NALL ! NSEL,S,LOC,X,1025 F,ALL,FY,-5100/3 NALL ! SAVE FINISH

Model 3: /prep7 ! ET,1,42,,,3 R,1,41 ! MP,EX,1,2.5E4 MP,NUXY,1,0.25 4

! N,1,0,0 N,11,1025,0 FILL,1,11 ! NGEN,5,11,1,11,1,,205/4 ! E,1,2,13,12 EGEN,10,1,1 EGEN,4,11,1,10,1 ! NSEL,S,LOC,X,0 D,ALL,ALL,0 NALL ! NSEL,S,LOC,X,1025 F,ALL,FY,-5100/5 NALL ! SAVE FINISH

Model 4: /prep7 ! ET,1,42,,,3 R,1,41 ! MP,EX,1,2.5E4 MP,NUXY,1,0.25 ! N,1,0,0 N,21,1025,0 FILL,1,21 ! 5

NGEN,5,21,1,21,1,,205/4 ! E,1,2,23,22 EGEN,20,1,1 EGEN,4,21,1,20,1 ! NSEL,S,LOC,X,0 D,ALL,ALL,0 NALL ! NSEL,S,LOC,X,1025 F,ALL,FY,-5100/5 NALL ! SAVE FINISH

The vertical displacement (v, or UY in ANSYS terms) of the nodes placed on the right-hand side of the beam will be determined in each node. The values will be represented on a “convergence” graph, together with the analytical (“exact”) solution of the problem. We get the analytical solution: Vmax =

I=

𝑏ℎ3 12

= 29435010,4 cm4

- I is the moment of inertia - E is Young modulus

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PL3 3EI

+

Ph2 L 24EI

= 2,553109792 cm

Postprocessing:

Model 1:

MAXIMUM ABSOLUTE VALUES NODE 18 (6) VALUE -2.5067

Model 2: MAXIMUM ABSOLUTE VALUES NODE 11 (33) VALUE -2.5348

Model 3:

MAXIMUM ABSOLUTE VALUES NODE 11 (55) VALUE -2.5396 Model 4:

MAXIMUM ABSOLUTE VALUES NODE 21 VALUE -2.5498

Model 1: Vmax = 2,5067 cm Model 2: Vmax = 2,5348 cm Model 3: Vmax = 2,5396 cm Model 4: Vmax = 2,5498 cm

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The convergence graph

In order to emphasize the convergence of the solution toward the analytical result by increasing the refinement of the mesh, a convergence graph should be represented. On the abscissa, we place the number of elements in the mesh. On the ordinate, we place the displacement values calculated with the analytical formula and with the 4 models. 2.56 2.5531

Maximum deflection (cm)

2.55

2.5531

2.5531

2.54

2.5531 2.5498

2.5396 2.5348

2.53

numerical solution analytical solution

2.52 2.51 2.5067 2.5 0

10

20

30 40 Number of elements in the mesh

8

50