Strength of material . lesson 7

– 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 ra

Section 7

Stress Concentration

Stress 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.

© Loughborough University 2010 This work is licensed under a Creative Commons Attribution 2.0 Licence

• 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  = PP/A Por 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 change

Stress “flow lines” crowd

together causing high stress

Fillet

-3 1

Stress Concentration - Definition

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

b

Stress Concentration Factor Formulae

Basic Design Rule – Yield Limited Design

Estimation of maximum or upper limit load using yield stress

Load in in irregular region:

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.

Example 1: Stress Concentration

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

K A

The hole therefore governs the maximum load and 3.45

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

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 P

P1

Example 2: Stress Concentration

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, K c

– Critical strain energy release rate, G c

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

• 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.

P P P P P P P

(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 following

three basic fast fracture modes.

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

 c

a 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 G c 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

Gc: Toughness – strain energy release rate

Kc: Fracture toughness – critical stress intensity release factor

If K<Kc then crack is stable

If 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 or

a is half length of central crack)

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?

C c

This resource was created by Loughborough University and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.

© 2010 Loughborough University.

Except where otherwise noted this work is licensed under a Creative Commons Attribution 2.0 Licence

The name of Loughborough University, and the Loughborough University logo are the name and registered marks of Loughborough University To the fullest extent permitted by law Loughborough University reserves all its rights in its name and marks, which may not

be used except with its written permission.

The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England

& Wales Licence All reproductions must comply with the terms of that licence.

The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher.

Credits