TU erlin, Fakultät V Institut ür Mechanik (LKM) Pro. W. rocks Dipl.-Ing. R. Falkenberg Pro. K. Weinberg Dr. R. Wille..9 Plastizität und ruchmechanik. Projektaugabe (D) Zu olgendem Thema ist ein 5- minütiger Vortrag zu halten. Die wesentlichen Unterlagen sind der Gruppe in elektronischer Form zur Verügung zu stellen. Problem ruchmechanische uslegung eines kreiszylindrischen Druckbehälters mit axialem innerem Oberlächenriss unter Innendruck (siehe Figure 5. in der nlage). erechnen Sie die Spannungintensitätsaktoren an der tiesten Stelle des Risses und an der inneren ehälteroberläche unter nnahme einer konstanten Umangsspannung (Membranspannung) nach der "Kesselormel", mit der Spannungsverteilung σ ϕϕ (r) nach der LMÉschen Formel ür dickwandige Druckbehälter (siehe Projekt ). Gegeben: Innenradius R i = 4 mm, ußenradius R a = 5 mm, Risstiee a = mm, Risslänge c = 5 mm, Innendruck p = 5 MPa. Termin ür den Vortrag ist der..9 nlage: uszug aus Zerbst et al.: Fitness-or Service Fracture ssessment o Structures Containing Deects, Elsevier 7. Die Unterlagen (als *.ppt, *.doc, *.tex, *.pd, auch eingescannte handschritliche Notizen) werden au der Seite http://mechanik.tu-berlin.de/weinberg/lehre/bruch/pbruch.html öentlich gemacht. Daher geben Sie bitte darau wahlweise Ihren Namen oder Ihre Matrikelnummer oder ein Pseudonym an.
96 U. Zerbst et al.: Fitness-or-Service Fracture ssessment o Structures Containing Deects, Elsevier 7 5. THE MODEL PRMETERS 5. The Stress Intensity Factor (K Factor) 5... Sources or analytical K actor solutions 5... Types o analytical K actor solutions 5... K actor solutions or speciic geometries and speciic loading Speciic geometries are lat and curved plates, pipes, pressure vessels, nozzles, etc. Speciic loading may be tensile orces, bending moments, internal pressure and others. n example or a K solution or both a speciic geometry and a speciic loading is provided in Eq. (5.). It reers to a hollow cylinder with an axial semi-elliptical crack at the inner side such as shown in Figure 5.. Figure 5.: Geometry used in Examples 5., 5. and 5.5: Hollow cylinder with an axial semielliptical crack at the inner surace. stress intensity actor solution or internal pressure is provided in [5.]. For the deepest point o the crack K is obtained as: t Ro+ Ri a KI =.97σ πa Y.5 R + i Ro Ri t (5.) with the hoop stress σ being σ= pr t. (5.) i The geometry unction Y is: a a Y= M M M Q + + t t 4 (5.) 96
97 U. Zerbst et al.: Fitness-or-Service Fracture ssessment o Structures Containing Deects, Elsevier 7 with M..9( a c) =, (5.4).89 M =.54+. + a c ( ) (5.5) and M =.5 + 4 a c.65+ a c ( ) ( ) 4. (5.6) The elliptical crack shape actor is given by a Q = +.464 c.65. (5.7) For the surace points o the crack, K is: KI = KI..5 a t + a c ( ) ( ). (5.8) Example 5.: tube with dimensions R i = mm, R o = mm and t = R o R i = mm contains an internal crack with a = mm and c = mm. I the applied load is an internal pressure o MPa the K actor at the deepest point o the crack is determined as. MPa. m /. 5... K actor solutions or speciic geometries but arbitrary loading in the thickness direction Frequently the loading is provided as a three-dimensional stress ield, e.g., by inite element calculations in the un-cracked section. However, the K solutions introduced in this chapter are based on variable stress proiles in the thickness direction only, i.e. the stress proile is assumed to be constant parallel to the surace. This assumption is reasonable in many cases, e.g. or pressurized tubes. n example or the latter is the hollow cylinder o Figure 5. above. In that case the stress distribution in the axial direction can be assumed to be uniorm. For a stress proile in the thickness direction given by a polynomial j j (5.9) j= ( xa) ( xa) ( xa) ( xa) ( xa) σ=σ = σ =σ +σ +σ +σ the K actor can be determined by: or x a ( ) (5.) K = πa σ a t; a c; R t =σ +σ +σ +σ I j j i j= [5., 5.]. The solution is a shortened version o solutions in [5.] and [5.]. Its geometry unction j is given in Tables 5. and 5.. 97
98 U. Zerbst et al.: Fitness-or-Service Fracture ssessment o Structures Containing Deects, Elsevier 7 Table 5.: Geometry unctions o Eq. (5.) or the deepest point o the crack. Component geometry according to Figure 5.. a/t a/c = ; R i /t = 4 a/c = ; R i /t =..659.47.87.7.659.47.87.7..64.454.75.6.647.456.75.6.5.66.46.78.8.669.464.8.8.8.74.489.97.4.694.484.94.9 a/t a/c =.4; R i /t = 4 a/c =.4; R i /t =..99.58.44.55.99.58.44.5..99.579.45.8.9.584.455.8.5.7.6.474.95.58.69.477.97.8.55.7.54.44..7.5.49 a/t a/c =.; R i /t = 4 a/c =.; R i /t =..5.66.44.57.5.66.44.57..45.64.487.46.6.64.489.47.5.8.79.54.48.59.746.544.44.8.865.948.659.56.78.94.69.54 Eq. (5.) is obtained by the weight unction method. Note the stresses acting on the un-cracked section at the position o the imaginary crack are o particular importance or the analysis. Thus, in order to improve the quality o the polynomial approximation it can be o advantage to restrict the included stresses to a region close to, but larger than the crack area. 98
99 U. Zerbst et al.: Fitness-or-Service Fracture ssessment o Structures Containing Deects, Elsevier 7 Table 5.: Geometry unctions o Eq. (5.) or the surace points o the crack. Component geometry according to Figure 5.. a/t a/c = ; R i /t = 4 a/c = ; R i /t =..76.8.4..76.8.4...79.4.46.4.76.6.47.4.5.759.6.5.7.777.4.54.8.8.867.58.6..859.6.6. a/t a/c =.4; R i /t = 4 a/c =.4; R i /t =..67.4..6.67.4..5..67.7.7.8.676.9.7.8.5.8.5.59..84.5.6..8.6.9.95.5.6.5.9.49 a/t a/c =.; R i /t = 4 a/c =.; R i /t =..56.69.7.9.56.69.7.9..577.75...578.75...5.759.4.5.7.75..5.6.8.44.5..56..4.99.5 Example 5.: From a inite element analysis o a stiened tubular structure a stress proile across the wall without a crack is given in Table 5.. Table 5.: Stress proile across the un-cracked wall o a stiened tubular structure. The coordinate x is oriented in the crack depth direction and measured rom the ree surace The component dimensions (without the stiener) reer to Example 5. (R i = mm, R o = mm, t = mm). The crack dimensions are a = mm and c = 4 mm. x in mm x/a σ in MPa x in mm x/a σ in MPa 4 5.5..5..5 6 9 6 6 7 8 9..5 4. 4.5 5. 97 94 9 88 89 This stress proile is approximated by the polynomial o Eq. (5.9) giving the coeicients σ o = 5.8 MPa, σ = -.554 MPa, σ =.49 MPa and σ = -5.88. - MPa. σ ( x a) = 5.8.554( x a).49( x a) 5.88 ( x a) + MPa For a crack depth-to-wall thickness ratio o a/t =., a crack shape a/c = and a tube radius to thickness ratio R i /t = (Figure 5.) the geometry unctions are taken rom Tables 5. and 5. as (a) Deepest point o the crack: 99
U. Zerbst et al.: Fitness-or-Service Fracture ssessment o Structures Containing Deects, Elsevier 7 =.647, =.456, (b) Surace points o the crack: =.76, =.6, =.75, =.47, =.6, =.4. Extrapolation o the tabulated values is perormed as described in Chapter 5..4. s a result the stress intensity actors are determined as (a) Deepest point o the crack: K I = 6.7 MPa. m /, (b) Surace points o the crack: K I = 8.4 MPa. m /. Weight unction solutions such as Eq. (5.) can be given or various kinds o stress distributions across the wall such as uniorm, square root, linear, power law, quadratic, cubic or even polynomial.