ME 5657 FINITE ELEMENT METHOD

Submitted by: ALPAY BURAK DEMIRYUREK

FEM ANSYS PROJECT Evaluation of different element types in bending Effect of a Circular Hole on Stress Distributions in a Plate

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Contents Evaluation of different element types in bending ........................................................................................ 2

Exact Solution (and Single Beam Element) ................................................................................................ 2

FE Solution using Q4 Elements – 1x10 Elements ...................................................................................... 4

FE Solution using Q4 Elements – 2x20 Elements ...................................................................................... 5

FE Solution using Q4 Elements – 4x40 Elements ...................................................................................... 5

FE Solution using Q4 Elements – 8x80 Elements ...................................................................................... 6

FE Solution using LST Elements – 1x20 Elements ...................................................................................... 6

FE Solution using LST Elements – 2x40 Elements ...................................................................................... 7

FE Solution using LST Elements – 4x80 Elements ...................................................................................... 7

FE Solution using LST Elements – 8x160 Elements .................................................................................... 8

Exact Normal Stress and FEM Stresses at Top of the Beam ...................................................................... 8

Effect of a Circular Hole on Stress Distributions in a Plate .......................................................................... 13

Stress concentration factor ..................................................................................................................... 13

Contour plot of σ x the converged solution ............................................................................................. 14

Contour plot of σ y the converged solution ............................................................................................. 15

Contour plot of τxy the converged solution ............................................................................................. 16

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Evaluation of different element types in bending

Goal of this project is to compare different element types in pure bending.

Often in analysis one tends to think that using solid elements (such as Q4,Q8,CST and LST) will provide

accurate results when modeling thin slender sections. In this project, a cantilever beam subjected to pure

bending is modeled by using different elements and different mesh sizes. The results of the analysis are

compared to the exact solution given by the single beam element.

The dimensions and material constants of the beam are given below.

h = 8 mm, L=80 mm, b=1000 mm, E= 10000 N/mm2, v=0.3, M=8 N.mm

Exact Solution (and Single Beam Element)

𝐸𝐼𝑑2𝑣

𝑑𝑥2= 𝑀𝑜

Integrating with respect to x, two times

𝐸𝐼𝑑𝑣

𝑑𝑥= 𝑀𝑜 𝑥 + 𝐶1

𝐸𝐼 𝑣 = 𝑀𝑜

𝑥2

2+ 𝐶1𝑥 + 𝐶2

Constants C1 and C2 should be defined in order to calculate vertical deflection v.

Applying boundary conditions,

@ 𝑥 = 0 → 𝑣 = 0 𝑎𝑛𝑑 𝑑𝑣

𝑑𝑥= 0

𝐶1 𝑎𝑛𝑑 𝐶2 𝑐𝑜𝑚𝑒𝑠 𝑜𝑢𝑡 𝑡𝑜 𝑏𝑒 𝐶1 = 𝐶2 = 0

∴ 𝑣 =𝑀𝑜𝑥2

2𝐸𝐼 → @ 𝑥 = 80 𝑚𝑚 , 𝑀𝑜 = 8 𝑁. 𝑚𝑚, 𝐸 = 10000, 𝐼 =

1

12 𝑥 1000 𝑥 83 𝑚𝑚4

𝑣 = 0.00006 𝑚𝑚 𝑎𝑛𝑑 𝜃 = 15x10−7 rad

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In ANSYS Mechanical APDL 15.0

ANSYS Mechanical APDL 15.0

New AnalysisMain Menu → Preferences→

Structural (check)

Main Menu → Preprocessor→Element Type→Add/Edit/Delete

Add→Structural Mass→

Beam→2 Node 188

Main Menu → Preprocessor→

Material Props→Material Models

Structural →Linear→Elastic→Isotropic

EX = 10000 →PRXY = 0.3 →

Main Menu → Preprocessor→

Sections→Beam→

Common Sections

Name = BEAM →B=8→

H=1000

Main Menu → Preprocessor→

Modeling→Create→

Nodes→In Active CS

Node Number = 1X Y Z = 0 0 0 →

Node Number =2X Y Z = 80 0 0

Node Number = 1X Y Z = 0 0 0 →

Node Number =2X Y Z = 80 0 0

Main Menu → Preprocessor→

Modeling→Create→

Elements→

Auto Numbered→Thru Nodes→

Select Node 1&2

Main Menu → Preprocessor→

Loads→Apply→

Structural→

Displacement →On Nodes→

Select Node 1→ All DOF = 0

Force/Moment →On Nodes→

Select Node 2→ Mz=8

Force/Moment →On Nodes→

Select Node 2→ Mz=8

Main Menu → Preprocessor→

Solution→Solve→

Current LS

Main Menu → Preprocessor→

General PostProc→Plot Results→

Deformed Shape

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FE Solution using Q4 Elements – 1x10 Elements

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FE Solution using Q4 Elements – 2x20 Elements

FE Solution using Q4 Elements – 4x40 Elements

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FE Solution using Q4 Elements – 8x80 Elements

FE Solution using LST Elements – 1x20 Elements

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FE Solution using LST Elements – 2x40 Elements

FE Solution using LST Elements – 4x80 Elements

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FE Solution using LST Elements – 8x160 Elements

Exact Normal Stress and FEM Stresses at Top of the Beam

𝜎𝑒𝑥𝑎𝑐𝑡 =𝑀 𝑐

𝐼=

8 𝑥 4

112

1000 𝑥 83 = 0.00075

𝑁

𝑚𝑚2= 0.75 𝑘𝑃𝑎 = 750 𝑃𝑎 (𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛)

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Q4 Element - 1x10 Mesh

LST Element - 1x20 Mesh

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Q4 Element - 2x20 Mesh

LST Element - 2x40 Mesh

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Q4 Element - 4x40 Mesh

LST Element - 4x80 Mesh

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Q4 Element - 8x80 Mesh

LST Element - 8x160 Mesh

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Element Type Number of Elements Through Beam Thickness

1 2 3 4

νFEA/νexact Q4 0.406/0.600=0.677 0.537/0.600=0.895 0.585/0.600=0.975 0.600/0.600 =1 LST 0.600/0.600 = 1 0.603/0.600 = 1.005 0.605/0.600 = 1.0083 0.606/0.600 = 1.01

Beam 0.600/0.600 = 1

σFEA/σexact Q4 0.556/0.75 = 0.741 0.702/0.75 = 0.936 1.185/0.75 = 1.58 2.281/0.75 = 3.041 LST 0.75/0.75 = 1 1.184/0.75 = 1.579 2.368/0.75 = 3.157 4.763/0.75= 6.351

In Ansys, PLANE182(Q4 Element) with full integration exhibits shear locking(parasitic/spurious shear) in

bending. The elements behave in a more stiff manner when the mesh is coarse. However, for increasing

number of elements used in meshing (refined mesh) and for use of high order elements (LST) in

modeling the results of analysis are reasonably accurate.

Effect of a Circular Hole on Stress Distributions in a Plate The goal of the project is to find the stress concentration factor by using

different element types (Q4 and LST-elements)

different mesh sizes (50,40,30,20 and 10 mm)

Selected Material Properties, Loads and

Dimensions

E = 200 kN/mm2

v = 0.3

a = 100 mm

S = 1 kN/mm

Stress concentration factor

Element Type Stress Concentration Factor

50 40 30 20 10

LST 3.8235 3.89313 4.04125 4.05871 4.12216

Q4 3.59528 3.89652 3.90934 4.12722 4.19236

Stress concentration factors are summarized above table. For LST element, solution converged for mesh

size 20 mm, and for Q4 element solution converged for mesh size 30 mm.

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Contour plot of σ x the converged solution

LST Element

Q4 Element

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Contour plot of σ y the converged solution

LST Element

Q4 Element

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Contour plot of τxy the converged solution

LST Element

Q4 Element