Section 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 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 σ = 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 Hole Notch Fillet 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 Stress Concentration Fillet Abrupt change Stress “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 Stress Concentration P d b P (Thickness h) Mean stress over the irregular regions is given by: σm= P P = A (b-d) h The local stress can be calculated roughly by  d2 3d  ÷ σ max = σ m  + +  2( b - d ) 2( b - d ) ÷   It can be seen that when b = 2d then σ max =3 σ m On the contrary, when d → (vanishing hole) then the stress concentration becomes non-existent as σ max ≈ σ m Stress Concentration - Definition Stress concentration factor K defined as: σ K = max σm 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 P d b P Stress Concentration Factor Formulae Flat bar of width b with central circular hole diamter d d  d  d  K = - 3.13  ÷ + 3.66  ÷ - 1.53  ÷ b b b Flat bar of width b with non-central circular hole diamter d  d   d   d  K = - 3.13  + 3.66  - 1.53  ÷ ÷ ÷  2b   2b   2b  Flat bar of width b with single semicircular notch of depth h 3 h h h K = 2.99 - 7.30  ÷+ 9.74  ÷ - 4.43  ÷ b b b Flat bar of width b with two semicircular notches of depth d / d  d  d  K = 3.07 - 3.37  ÷ + 0.65  ÷ - 0.66  ÷ b b b Round bar of diameter D with mid section notch of depth d / 2 K d  d  d  = 3.04 - 5.42  ÷+ 6.27  ÷ - 2.89  ÷ D D D Basic Design Rule – Yield Limited Design Estimation of maximum or upper limit load P using yield stress σ y Load in P in irregular region: Using K = σ max σm So when σ max =σ y P = σmA P= P σ max A K Plimit = σyA K If K = then maximum allowable mean stress is OR d σm = b P σy K σy 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 Example 1: Stress Concentration P b1 b2 d b3 P A stepped flat bar of 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 P b1 b2 d b3 P We need to this calculation in three parts as follows: Left hand fillet: Hole: Right hand fillet: σ max A1 41× 40 × = = 7.94kN K1 1.24 σ A 41× 32 × P2 =σ m A2 = max = = 3.45kN K2 2.28 P1=σ m1A1= P3 =σ m A3 = σ max A3 41× 36 × = = 6.76kN K3 1.31 The hole therefore governs the maximum load and Pmax = 3.45kN (Actual mean stress values are σ = 14.3MPa, σ = 17.8MPa, σ = 15.8MPa 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 rod in a F1 car will see well over 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 P1 d1 P1 P2 d1 d2 d1 P2 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 P 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 Example 2: Stress Concentration P1 d1 P1 P2 d1 For left hand uniform bar π 202 P1=σ 1A1 = σ max A1 = 80 = 25100 kN (3 s.f.) For enlarged bar (at left or right hand fillet): σ max A1 σ max A1 80.π 202 P2 =σ m1A1= = = = 14400 kN (3 s.f.) K1 1.74 1.74 × Surprisingly (?) enlarged bar is weaker not stronger d2 d1 P2 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 (a) Cracked sample (b)Tough behaviour (c) Brittle behaviour 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 Fracture Toughness All the cracks can be represented by one or a combination of the following three basic fast fracture modes Mode I Mode II Mode III Mode I is by far the most common in all the engineering materials Mode II is dominant only in fibre-reinforced composites It can be shown that the onset of fast fracture is governed by the following condition: σ π a = Gc E The LHS says fast fracture will occur when in a material subject to a stress σ, a crack reaches some critical size a; or when a material containing cracks of size a is subjected to a critical stress σ 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 σ π a 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 K = Kc where K c = Gc E Gc: Toughness – strain energy release rate Kc: Fracture toughness – critical stress intensity release factor If K