Ex20Stress Concentration

7/28/2019 Ex20Stress Concentration

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Exercise 20 Stress Concentration 1

Exercise 20

Stress Concentration

In Exercise 15, forces are uniformly applied on a block of

regular shape, and the resulting stresses are uniformly

distributed among the block (15-6[17-21]). In many other

cases, stresses are not uniformly distributed: some locations

have higher stress than others. As an example, in the C-Bar

example, the stress is higher at an inner corner [1]. This

phenomenon is called stress concentration. After performing a

stress analysis, you should pay attention on locations of stress

concentration, and make sure the material doesn't fail,

according to the failure criteria discussed in 19-1.

Besides an inner corner, stress concentration may occurs

in many other situations. For example, when a force is applied

on a very small area, it would produce a high stress on the

area. In this exercise, however, we'll focus on the stress

concentration on an inner corner.

In this exercise, we'll show that, given an external loading,the magnitude of concentrated stress depends on the radius of

the fillet of an inner corner. The smaller the radius, the larger

the stress. Fillets should be designed in a structure so that the

stresses are well below the safety margin.

In the next exercise (Exercise 21), we'll show that, as theradius of a fillet approaches zero, the stress concentration willapproach a theoretical value of infinity. A stress of infinite value

is called a singular stress. In computer simulations, to reduce

computing time, we often simplify the geometry by neglecting

detailed features such as fillets, and model the fillets as sharp

inner corners. The results usually remain quite satisfactory in a

global sense. Locally, however, singular stresses occur at these

sharp inner corners. Singular stresses may not exist in the

real-world, since zero-radius fillets are not common in the real-

world, but they exist everywhere in a simulation model.

What we want to emphasize is that when you see a largestress value, check again to see if it is caused by a unrealistic

simplification of geometry. If it is, then you should neglect it.

[1] The stress isconcentrated here.(This picture is a

duplicate of 19-4[5].)

20-1 Introduction


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2 Copyright by Huei-Huang Lee

100 100

50

10 kN 10 kN

Unit: mm.

100

Problem DescriptionStress concentration occurs most commonly on concave corners. In this exercise, we'll use a filleted plate [2-4] to

show that the degree of stress concentration depends on the radius of the fillets. The smaller the radius, the larger the

degree of stress concentration. The stress concentration factor K, which will be defined later, is commonly used to

describe the degree of stress concentration. We'll complete a chart that shows how the stress concentration factor

increases as the radius of the fillets decrease.

20-2 Start a New Project

Launch Workbench. Create a

system. Save the project as "Fillet." Start up

DesignModeler [1]. Select as length

unit.

[3] The platehas a thickness

of 2 mm.[2] The plate ismade of steel.

[1] Double-click to

start upDesignModeler.

[4] The radiusof the fillets

ranges from 5to 15 mm.


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20-3 Create Geometry in DesignModeler

[1] Create thissketch on XYPlane.

The sketch issymmetric about the

X-axis.

[2] Click.

[5] Click.

[4] Type 2 (mm)for .

[1] Don't close DesignModel.We'll be back again. Double-click

to start up.

20-4 Start Up

[3] Click.


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20-5 Apply Forces

[1, 4] Highlight andselect .

[2] Apply a forceon this face. Fordetails, see next

step.

[3] Details of.

[5] Apply a force onthis face. For details,

see next step.[6] Details of.


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20-6 Set Up Mesh Controls

[2] With highlighted, select .

20-7 Set Up a Result Object and Solve

[1] With

highlighted, select

to insert a. Leave allsettings in the as

their defaults.

[2] Solve themodel.

[3] The maximum

stress is max

=168

MPa.

[4] The average

stress is

ave=100 MPa.

[1] With

highlighted, select for and type 100 for

. This is thefinest mesh we can obtainusing these two settings.

[3] Meshcount.


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Stress Concentration FactorThe stress concentration factor K, used to describe the degree of stress concentration, is defined by the ratio between

the maximum stress and the average stress; in this case,

K=

max

ave

=

168

100=1.68

20-8 Modify the Radius of the Fillets in DesignModeler

[1] In DesignModeler,

change the radius to10 (mm).

[2] Click.

20-9 Update the Model and Solve It in

[2] Solve thenew model.

[1] Don't closeDesignModeler. In

, select .


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[3] The maximumstress is

max

=189 MPa.

[4] The averagestress is

ave

=100 MPa.

The stress concentration factor in this case is

K=max

ave

=189

100=1.89

20-10 Modify the Radius of the Fillets in DesignModeler

[1] InDesignModeler,

change the radiusto 5 (mm).

[2] Click.

20-11 Update the Model and Solve It in

[2] Solve the

new model.

[1] Don't closeDesignModeler. In

, select .


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[3] The maximumstress is

max

= 219 MPa.

[4] The averagestress is

ave

=100 MPa.

The stress concentration factor in this case is

K=max

ave

=219

100= 2.19

20-12 Discussion

1.6

1.7

1.8

1.9

2.0

2.1

2.2

5 10 15

StressConcentrationFactor(K)

Radius of Fillets (mm)

Radius(mm)

Stressconcentration

factor (K)

15 1.68

10 1.89

5 2.19

According to foregoing stress analyses, we may plot a curve to show a relationship between the radius of the fillets and

the stress concentration factor [1]. We conclude that, as the radius becomes smaller, the degree of stress

concentration becomes larger.

[1] The curveshows, as the radius

becomes smaller,the degree of stress

concentrationbecomes larger.

Are the calculated stresses accurate enough?The stresses calculated are actually not very accurate. In an area where the stress changes abruptly (i.e., high stress

gradient), it usually requires afi

ner mesh [2]; thefi

ner the mesh, the more accurate the stress. If the mesh is notfi

neenough, the stress is usually underestimated. On the other hand, in an area where the stress distribution is smooth

(i.e., low stress gradient), the mesh can be coarser [3]. In the following exercises, we'll refine the mesh near the areas

of stress concentration, to obtain more accurate results.


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[2] In an areawhere the stress

changes abruptly, it

usually requires afiner mesh.

[3] In an areawhere the stress

distribution issmooth, the mesh

can be coarser.

20-13 Refine Mesh Around the Fillets

[1] With highlighted, select

for andtype 0 for .

[2] Select to

insert a meshcontrol method.

[3] Select thebody and click

.


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[4] Select to

insert a sizing control.

[6] Select this point.

[8] Select toinsert another sizing

control.

[5] Turn on .

[7] Click .

[9] Select this point.

[10] Click .


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[11] With highlighted, select .

[12] Solve themodel.

[13] Themaximum stress

is max

= 234

MPa.

The stress concentration factor is

K=

max

ave

=

234

100= 2.34


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20-14 A More Accurate Results

Modify the radius offillets to 10 mm and 15 mm respectively (20-8[1, 2]), refresh the geometry (20-9[1]), and solve the

model repeatedly (20-9[2]). We'll come up with stresses of 190 MPa and 168 MPa respectively. A new curve may be

plotted [1]. Note that the difference is significant.

Radius(mm)

Stressconcentration

factor (K)

15 1.68

10 1.90

5 2.34

Numerical simulations are mesh dependent!?A lesson we've learned in this section, besides the behavior of stress concentration, is that solution accuracy dependson mesh adequacy. As a general rule, the finer the mesh, the more accurate the solutions. This is an important nature

of numerical simulations. Meshing techniques are inseparable part of computer simulations. As a first course of

computer simulation, we will not discuss further on meshing techniques. For now, however, following questions must

be answered. In 20-13[13], the calculated maximum stress is 234 MPa. How accurate is this value? Is this value the

most accurate one we can obtain? If we refine the mesh further, can we obtain an even more accurate stress? The

same questions may be applied to the stresses calculated in 17-4[18]. We will answer these question in next exercise

(Exercise 21).

Wrap UpClose both DesignModeler and , save the project, and exit Workbench.

1.6

1.8

2.0

2.1

2.3

2.5

5 10 15

StressConcentrationFactor(K)

Radius of Fillets (mm)

[1] This curve is moreaccurate

[2] The curveduplicated from

20-12[1].

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