Section 7

Stress ConcentrationStress concentrations produced by discontinuities in structures such as holes, notches, and fillets will be introduced in this section. The stress concentration factor will be defined. The concept of fracture toughness will also be introduced.

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Contents• Stress Concentration• Stress Concentration – Definition• Stress Concentration chart – Central hole• Stress Concentration Factor Formulae• Basic Design Rule – Yield Limited Design• Fatigue• Fracture Toughness• Example: Fracture Toughness• Credits & Notices

Stress Concentration

Engineering stress = P/A or the far-field stress is invalid.– In immediately vicinity where the external load is applied (St Venant)– Around discontinuities such as: holes, notches, or fillets

These irregularities (stress concentrations or stress raisers) cause a disruption to stress flow and stresses concentrations in localised regions. The change of section concentrates stress most strongly where the curvature of the surface is greatest. The far-field stresses are less useful as they underestimate the actual local stress.

Hole Notch Fillet

Stress Concentration

Abrupt changeStress “flow lines” crowd together causing high stress concentration in transition zone

Smooth change“Flow lines” more evenly distributed causing lower stress concentration in transition zone

Fillet

Stress Concentration

2 4

max 2 4

max

Mean stress over the irregular regions is given by:

= =-

The local stress can be calculated roughly by

31

2 - 2 -

It can be seen that when 2 then =3

On the contrary,

m

m

m

P P

A b d h

d d

b d b d

b d

max

when 0 (vanishing hole) then the stress

concentration becomes non-existent as .m

d

(Thickness h)P Pbd

Stress Concentration - Definition

max

Stress concentration factor defined as:

=conm

K

K

Stress concentration factors are:

• Dependent on irregularity, dimensions of irregularity, overall dimensions, loading

• Obtained experimentally, analytically, etc

• Published in charts (e.g. Roark’s Formulas)

• Very important in brittle materials

• In ductile materials:

– Important in fatigue calculation.

– Important if safety is critical.

– Localized yielding hardens material (strain hardening).

– Redistributes stress concentration.

Stress Concentration Chart – Central hole

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5

Stre

ss co

ncen

trati

on f

acto

r K

Diameter to width ratio d/b

P

P

d

b

Stress Concentration Factor Formulae

2 3

2

Flat bar of width with central circular hole diamter

3 - 3.13 3.66 - 1.53

Flat bar of width with non-central circular hole diamter

3 - 3.13 3.66 -2 2

con

con

b d

d d dK

b b b

b d

d dK

b b

3

2 3

1.532

Flat bar of width with single semicircular notch of depth

2.99 - 7.30 9.74 - 4.43

Flat bar of width with two semicircular notches of depth / 2

3.07 - 3.

con

con

d

b

b h

h h hK

b b b

b d

K

2 3

2 3

37 0.65 - 0.66

Round bar of diameter with mid section notch of depth / 2

3.04 - 5.42 6.27 - 2.89con

d d d

b b b

D d

d d dK

D D D

Basic Design Rule – Yield Limited Design

max max

max limit

Estimation of maximum or upper limit load using yield stress

Load in in irregular region:

Using =

So when =

If 3 then maximum allowab

y

m

conm con

y yy m

con con

con

P

P P A

AK P

K

AP OR

K K

K

le mean stress is 3

y

If limit load is reached we will get permanent deformation. Note that the holes, notches and fillets are being discussed in a structural sense. In reality we can get holes and notches through surface defects and manufacturing defects. These can become an issue in fatigue related issues. There are several examples of structural failure occurring relating to fatigue.

P Pbd

Example 1: Stress Concentration

b1 db2 PP b3

A stepped flat bar of 6 mm thick has a hole of 18 mm diameter. It has three widths of b1=40 mm, b2=50 mm and b3=36 mm. Stress concentration factors for the left fillet, hole and the right fillet are 1.24, 2.28 and 1.31, respectively. The allowable stress is 41 MPa.

What is the permissible load Pmax?

Example 1: Stress Concentration

max 11 m1 1

1

max 22 m2 2

2

max 33 m3 3

3

We need to do this calculation in three parts as follows:

41 40 6Left hand fillet: = = 7.94

1.24

41 32 6Hole: = = 3.45

2.28

41 36 6Right hand fillet: = = 6

1.31

AP A kN

K

AP A kN

K

AP A

K

max

1 2 3

.76

The hole therefore governs the maximum load and 3.45

(Actual mean stress values are 14.3MPa, 17.8MPa, 15.8MPa

kN

P kN

b1 db2 PP b3

Fatigue

Fatigue is concerned with materials getting ‘tired’ due to repeated load cycles. The number of cycles is generally quite high. For instance the vibration of a aircraft wing during a long flight can result in tens of thousands of load cycles. During the lifetime of the aircraft the wing will see millions of load cycles. A con rod in a F1 car will see well over 1 million load cycles. If designed properly these structures will not fail if the stress is below the endurance stress limit. An increase in the required number of load cycles reduces the endurance limit.

Generally fatigue problems are divided into high cycle and low cycle fatigue. (High and low refers to the number of load cycles). In the former the stress is not allowed to exceed yield in the latter the stress can exceed yield.

Example 2: Stress Concentration

A round straight bar with a diameter of d1 = 20 mm is being compared with a bar of the same diameter, which has an enlarged portion with a diameter of d2 = 25 mm. The radius of the fillets is 2.5 mm and the associated stress concentration factor is 1.74.

Does enlarging the bar in this manner make it stronger?

Justify the answer by determining the maximum permissible load P1 for the straight bar and the maximum permissible load P2 for the enlarged bar if an allowable tensile stress of the material is 80 MPa.

d1d2 P2P2

d1P1P1d1

Example 2: Stress Concentration

2

1 1 1 max 1

2max 1 max 1

2 m1 11

For left hand uniform bar

20= 80. 25100 kN (3 s.f.)

4

For enlarged bar (at left or right hand fillet):

80. 20= = 14400 kN (3 s.f.)

1.74 1.74 4

Surprisingly (?) enlarged

P A A

A AP A

K

bar is weaker not stronger.

P1P1d1 d1

d2 P2P2d1

Fracture Toughness

• Structures which were properly designed to avoid large elastic deflections and plastic fail in a catastrophic way by fast fracture. Common to all such structures is the presence of cracks. Catastrophic failure is caused by the crack growing at the speed of sound in the material. This mechanism is called fast fracture.

• Two parameters are used to represent fracture toughness– Critical stress intensity factor, Kc

– Critical strain energy release rate, Gc.

• Gc is a measure of the material’s ability to yield and absorb strain energy released by crack propagation.

Fracture Toughness

• Strength is defined as the resistance to plastic flow– Yield strength– Strength increases in plastic zone due to work hardening reaching maximum

tensile strength• Toughness is resistance of material to propogation of a crack

– Glasses and ceramics have low toughness– Ductile metals have high toughness

The crack in the tough material, shown in (b), does not propagate when the sample is

loaded; that in the brittle material propagates without general plasticity, and thus at a stress

less than the yield strength.

(a) Cracked (b)Tough (c) Brittle sample behaviour behaviour

Fracture Toughness

Mode I is by far the most common in all the engineering materials.Mode II is dominant only in fibre-reinforced composites.

All the cracks can be represented by one or a combination of the followingthree basic fast fracture modes.

Mode I Mode II Mode III

It can be shown that the onset of fast fracture is governed by the following condition:

ca G E

The LHS says fast fracture will occur when in a material subject to a stress , a crack reaches some critical size a; or when a material containing cracks of size a is subjected to a critical stress . The RHS depends on material properties only. E is material constant and Gc energy required to propagate crack which depends on material. HENCE the critical combination of stress and crack length is a material constant.

Fracture Toughness

The term crops up quite frequently in fast fracture mechanics and is usually given the symbol K. The units of K are MN m-3/2. It is called the stress intensity factory (!)

Fast fracture occurs when

where

c

c c

K K

K G E

a

Gc: Toughness – strain energy release rateKc: Fracture toughness – critical stress intensity release factorIf K<Kc then crack is stableIf K=Kc then crack will propagate (at speed of sound in material 1 mile s-1 !!)Kc is material property

To measure Kc, (Gc): experimental set up for mode I:(Caution: a is length of edge crack ora is half length of central crack)

a

2a

A sheet of aluminium alloy has a 4mm central crack through the thickness. If a stress of 200 MPa is reached on the point of fast fracture, determine the fracture toughness of the sheet. If the Young’s Modulus is 70 GPa what is its toughness GC?

Example: Fracture Toughness

A sheet of aluminium alloy has a 4mm central crack through the thickness. If a stress of 200 MPa is reached on the point of fast fracture, determine the fracture toughness of the sheet. If the Youngs Modulus is 70 GPa what is its toughness GC?

Solution:

Example: Fracture Toughness

3

2 2

4 10200 15.9

2

15.90.00359

70000

c c C

Cc

K a

MPa m

K K and K G E

KHence G MPa m

E

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Credits