Module 3 Design for Strength. Version 2 ME, IIT Kharagpur


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1 Module 3 Design for Srengh
2 Lesson 2 Sress Concenraion
3 Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress concenraion facor; experimenal and heoreical mehods. Faigue srengh reducion facor and noch sensiiviy facor. Mehods of reducing sress concenraion Inroducion In developing a machine i is impossible o avoid changes in crosssecion, holes, noches, shoulders ec. Some examples are shown in figure KEY COLLAR BEARING GEAR GRUB SCREW F Some ypical illusraions leading o sress concenraion. Any such disconinuiy in a member affecs he sress disribuion in he neighbourhood and he disconinuiy acs as a sress raiser. Consider a plae wih a cenrally locaed hole and he plae is subjeced o uniform ensile load a he ends. Sress disribuion a a secion AA passing hrough he hole and anoher
4 secion BB away from he hole are shown in figure Sress disribuion away from he hole is uniform bu a AA here is a sharp rise in sress in he viciniy of he hole. Sress concenraion facor k is defined as k 3 = av, where av a secion AA is simply Pw ( 2 b) and P 1 =. This is he heoreical or w geomeric sress concenraion facor and he facor is no affeced by he maerial properies. P B A 2b B A w P F Sress concenraion due o a cenral hole in a plae subjeced o an uniaxial loading. I is possible o predic he sress concenraion facors for cerain geomeric shapes using heory of elasiciy approach. For example, for an ellipical hole in an infinie plae, subjeced o a uniform ensile sress 1 (figure ), sress disribuion around he disconinuiy is disurbed and a poins remoe from he disconinuiy he effec is insignifican. According o such an analysis 2b 3 = 1 1+ a If a=b he hole reduces o a circular one and herefore 3 =3 1 which gives k =3. If, however b is large compared o a hen he sress a he edge of ransverse
5 crack is very large and consequenly k is also very large. If b is small compared o a hen he sress a he edge of a longiudinal crack does no rise and k = a 2b F Sress concenraion due o a cenral ellipical hole in a plae subjeced o a uniaxial loading. Sress concenraion facors may also be obained using any one of he following experimenal echniques: 1. Srain gage mehod 2. Phooelasiciy mehod 3. Brile coaing echnique 4. Grid mehod For more accurae esimaion numerical mehods like Finie elemen analysis may be employed. Theoreical sress concenraion facors for differen configuraions are available in handbooks. Some ypical plos of heoreical sress concenraion facors and r raio for a sepped shaf are shown in figure d
6 F Variaion of heoreical sress concenraion facor wih r/d of a sepped shaf for differen values of D/d subjeced o uniaxial loading (Ref.[2]). In design under faigue loading, sress concenraion facor is used in modifying he values of endurance limi while in design under saic loading i simply acs as sress modifier. This means Acual sress= k calculaed sress. For ducile maerials under saic loading effec of sress concenraion is no very serious bu for brile maerials even for saic loading i is imporan. I is found ha some maerials are no very sensiive o he exisence of noches or disconinuiy. In such cases i is no necessary o use he full value of k and
7 insead a reduced value is needed. This is given by a facor known as faigue srengh reducion facor k f and his is defined as k f Endurance lim i of noch free specimens = Endurance lim i of noched specimens Anoher erm called Noch sensiiviy facor, q is ofen used in design and his is defined as k 1 = f q k 1 The value of q usually lies beween 0 and 1. If q=0, k f =1 and his indicaes no noch sensiiviy. If however q=1, hen k f = k and his indicaes full noch sensiiviy. Design chars for q can be found in design handbooks and knowing k, k f may be obained. A ypical se of noch sensiiviy curves for seel is shown in figure F Variaion of noch sensiiviy wih noch radius for seels of differen ulimae ensile srengh (Ref.[2]).
8 3.2.2 Mehods of reducing sress concenraion A number of mehods are available o reduce sress concenraion in machine pars. Some of hem are as follows: 1. Provide a fille radius so ha he crosssecion may change gradually. 2. Someimes an ellipical fille is also used. 3. If a noch is unavoidable i is beer o provide a number of small noches raher han a long one. This reduces he sress concenraion o a large exen. 4. If a projecion is unavoidable from design consideraions i is preferable o provide a narrow noch han a wide noch. 5. Sress relieving groove are someimes provided. These are demonsraed in figure (a) Force flow around a sharp corner Force flow around a corner wih fille: Low sress concenraion. (b) Force flow around a large noch Force flow around a number of small noches: Low sress concenraion.
9 (c) Force flow around a wide projecion Force flow around a narrow projecion: Low sress concenraion. (d) Force flow around a sudden change in diameer in a shaf Force flow around a sress relieving groove F Illusraions of differen mehods o reduce sress concenraion (Ref.[1]) Theoreical basis of sress concenraion Consider a plae wih a hole aced upon by a sress. S. Veran s principle saes ha if a sysem of forces is replaced by anoher saically equivalen sysem of forces hen he sresses and displacemens a poins remoe from he region concerned are unaffeced. In figure a is he radius of he hole and a r=b, b>>a he sresses are no affeced by he presence of he hole.
10 y a Q P b x F A plae wih a cenral hole subjeced o a uniaxial sress Here, x =, y = 0, τ xy = 0 For plane sress condiions: = cos θ+ sin θ+ 2τ cosθsinθ 2 2 r x y xy = sin θ+ cos θ 2τ cosθsinθ θ 2 2 x y xy ( ) sin cos ( cos 2 sin 2 ) τ = θ θ+τ θ θ rθ x y xy This reduces o 2 r =cos θ= ( cos2θ+ 1) = + cos2θ θ = sin θ= ( 1 cos2θ ) = cos2θ τ rθ = sin2θ 2 such ha 1 s componen in r and θ is consan and he second componen varies wih θ. Similar argumen holds for τ rθ if we wrie τ rθ = sin 2θ. The 2 sress disribuion wihin he ring wih inner radius r i = a and ouer radius r o due o 1 s componen can be analyzed using he soluions of hick cylinders and = b
11 he effec due o he 2 nd componen can be analyzed following he Sressfuncion approach. Using a sress funcion of he form φ = ( ) Rrcos2θ he sress disribuion due o he 2 nd componen can be found and i was noed ha he dominan sress is he Hoop Sress, given by 2 4 a 3a θ = 1+ 1 cos θ 2 r 2 r This is maximum a θ=±π 2 and he maximum value of 2 4 a 3a θ = r r Therefore a poins P and Q where r = a θ is maximum and is given by θ = 3 i.e. sress concenraion facor is Problems wih Answers Q.1: The fla bar shown in figure is 10 mm hick and is pulled by a force P producing a oal change in lengh of 0.2 mm. Deermine he maximum sress developed in he bar. Take E= 200 GPa. Fille wih sress concenraion facor 2.5 Hole wih sress concenraion facor 2 50 mm 25 mm 25 mm P Fille wih sress concenraion facor mm 300 mm 250 mm F A.1: Toal change in lengh of he bar is made up of hree componens and his is given by 0.2x P = x x x x10 3 This gives P= KN. 9
12 Sress a he shoulder s =k ( )x0.01 where k=2. This gives h = MPa. Q.2: Find he maximum sress developed in a sepped shaf subjeced o a wising momen of 100 Nm as shown in figure Wha would be he maximum sress developed if a bending momen of 150 Nm is applied F r = 6 mm d = 30 mm D = 40 mm. A.2: Referring o he sress concenraion plos in figure for sepped shafs subjeced o orsion for r/d = 0.2 and D/d = 1.33, K T Torsional shear sress is given by τ=. Considering he smaller diameer and π 3 d he sress concenraion effec a he sep, we have he maximum shear sress as τ = K 16x100 π ( 0.03) max 3 This gives τ max = MPa. Similarly referring o sressconcenraion plos in figure for sepped shaf subjeced o bending, for r/d = 0.2 and D/d = 1.33, K M Bending sress is given by = πd 3 Considering he smaller diameer and he effec of sress concenraion a he sep, we have he maximum bending sress as = K 32x150 π max 3 ( 0.03) This gives max = MPa.
13 F Variaion of heoreical sress concenraion facor wih r/d for a sepped shaf subjeced o orsion (Ref.[5]) F Variaion of heoreical sress concenraion facor wih r/d for a sepped shaf subjeced o a bending momen (Ref.[5]). Q.3: In he plae shown in figure i is required ha he sress concenraion a Hole does no exceed ha a he fille. Deermine he hole diameer.
14 5 mm 100 mm d' 50 mm P F A.3: Referring o sressconcenraion plos for plaes wih filles under axial loading (figure ) for r/d = 0.1 and D/d = 2, sress concenraion facor, K 2.3. From sress concenraion plos for plaes wih a hole of diameer d under axial loading ( figure ) we have for K = 2.3, d /D = This gives he hole diameer d = 35 mm F Variaion of heoreical sress concenraion facor wih r/d for a plae wih filles subjeced o a uniaxial loading (Ref.[5]).
15 F Variaion of heoreical sress concenraion facor wih d/w for a plae wih a ransverse hole subjeced o a uniaxial loading (Ref.[5]) Summary of his Lesson Sress concenraion for differen geomeric configuraions and is relaion o faigue srengh reducion facor and noch sensiiviy have been discussed. Mehods of reducing sress concenraion have been demonsraed and a heoreical basis for sress concenraion was considered.
A = p 4 (0.05)2 = 1.9635(103 ) m 2. J = p 2 (0.025)4 = 0.61359(104 ) m 4. s = P A = 2(10 3 ) 1.9635(103 = 1.019 MPa. t = Tc J = 500(0.
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