NUMERICAL VERIFICATION OF THE RELATIONSHIP BETWEEN THE IN-PLANE GEOMETRIC CONSTRAINTS USED IN FRACTURE MECHANICS PROBLEMS.
|
|
- Candace Mitchell
- 8 years ago
- Views:
Transcription
1 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems NUMERICAL VERIFICAION OF HE RELAIONSHIP BEWEEN HE IN-PLANE GEOMERIC CONSRAINS USED IN FRACURE MECHANICS PROBLEMS Marci GRABA * * ielce Uiversity of echology, Faculty of Mechatroics ad Machie Desig, Al. 1-lecia PP 7, ielce, Polad mgraba@tu.kielce.pl Abstract: I the paper, umerical verificatio ad catalogue of the umerical solutios based o Modify Boudary Layer Approach to determie the relatioship betwee -stress ad -stress are preseted. Based o the method proposed by Larsso ad Carlsso, the -stress value are calculated for some elastic-plastic materials for differet value of -stress ad exteral load expressed by J-itegral. he ifluece of the exteral load, -stress value ad material properties o -stress value were tested. Such catalogue may be useful durig solvig the egieerig problems, especially while is eeded to determie real fracture toughess with icludig the geometric costraits, what was proposed i FINE procedures. ey words: Fracture, Crack, Stress Fields, HRR, MBLA, i-plae Costraits, -Stress, -Stress. 1. DESCRIPION OF HE SRESS FIELD NEAR CRAC IP HEOREICAL BACGROUNDS For mode I of loadig, stress field ahead of a crack tip i elastic liear isotropic material ca be give by (Williams, 1957): θ θ 3θ 11 = = I xx cos 1 si si 2πr 2 2 2, θ θ 3θ 22 = = I yy cos 1 + si si 2πr 2 2 2, θ θ 3θ 12 = τ = I xy cos si cos 2πr 2 2 2, 33 = zz = for plae stress, ( ) + = zz= ν for plae strai, 33 xx yy where ij is the stress tesor, r ad θ are as defied i Fig. 1, ν is Poisso s ratio, I is the Stress Itesity Factor (SIF). crack y Fig. 1. Defiitio of the coordiate axis ahead of a crack tip he z directio is ormal to the page (ow Fig. based o (Williams, 1957) r τ xy θ yy τ yx xx (1) x he SIF defies the amplitude of the crack tip sigularity. hat is, stresses ear the crack tip icrease i proportio to. Moreover, the stress itesity factor completely defies the crack tip coditios; if is kow, it is possible to solve for all compoets of stress, strai, ad displacemet as a fuctio of r ad θ. his sigle-parameter descriptio of crack tip coditios turs out to be oe of the most importat cocepts i fracture mechaics (Williams, 1957). I 1968, Hutchiso (1968) ad Rice ad Rosegre (1968) derived the sigular stress-strai fields at a crack tip i a powerlaw hardeig material (which called the HRR-field). he hardeig law used by Hutchiso ad Rice ad Rosegre is give by: ε = α, (2) ε where is a referece stress value that is usually equal to the yield stregth, ε= /E, α is a dimesioless costat, ad is the strai hardeig expoet. Assumig the Ramberg-Osgood material, the results obtaied by Hutchiso ad Rice ad Rosegre for plae strai ad for plae stress may be expressed i the followig form: 1 J 1+ ij = ~ ij(, ) α ε Ir θ, (3) J 1+ εij = αε ~ εij(, ) α ε Ir θ, (4) where J is the J-itegral, I is a itegratio costat that depeds o, (, ) ad (, ) are dimesioless fuctios of ad. hese parameters also deped o the stress state (i.e. plae stress or plae strai). All this fuctio may be determied usig the algorithm ad computer program preseted by Gałkiewicz ad Graba (26). 38
2 acta mechaica et automatica, vol.6 o.2 (212) he J-itegral defies the amplitude of the HRR sigularity, just as the stress itesity factor characterizes the amplitude of the liear elastic sigularity. hus, J-itegral completely describes the coditios withi the plastic zoe. A structure i smallscale yieldig has two sigularity-domiated zoes: oe i the elastic regio, where stress varies as 1/ ad oe i the plastic zoe where stress varies as (1/ ) /( ). he latter ofte persists log after the liear elastic sigularity zoe has bee destroyed by crack tip plasticity. he HRR sigularity cotais the same apparet aomaly as the liear elastic fracture mechaics sigularity; amely, both predict ifiite stresses as r. he sigular field does ot persist all the way to the crack tip, however. he large strais at the crack tip cause the crack to blut, which reduces the stress triaxiality locally. he bluted crack tip is a free surface; thus xx must vaish at r =. he aalysis that leads to the HRR sigularity does t cosider the effect of the bluted crack tip o the stress fields, or does it take accout of the large strais that are preset ear the crack tip. his aalysis is based o small strai theory, which is the multi-axial equivalet of egieerig strai i a tesile test. Small strai theory breaks dow whe strais are greater tha about 1%. directio parallel to the crack advace directio. he mode I stress field becomes: = ij I 2πr f ij ( θ) +, (5) ν here is a uiform stress i the x directio (which iduces a stress ν i the z directio i plae strai). 22 / J = 116kN/m SEN(B) plae strai a/w =.5 W = 4mm = 315MPa = 5 E = 26MPa ν =.3 ε = /E =.153 large strai (FEM) small strai (FEM) HRR solutio ψ = x 22 /J Fig. 2. he stress distributio ear a crack tip (curves were obtaied usig the FEM for small ad fiite strai ad HRR formula) McMeekig ad Parks (1979) performed crack tip fiite elemet aalyses that icorporated large strai theory ad fiite geometry chages. Some of their results are show i Fig. 2, which is a plot of stress ormal to the crack plae versus distace. he HRR sigularity is also show o this plot. Note that both axes are dimesioless i such a way that both curves are ivariat, as log as the plastic zoe is small compared to specime dimesios. he solid curve i Fig. 2 reaches a peak whe the ratio x /J is approximately uity, ad decreases as x. his distace correspods approximately to twice the Crack ip Opeig Displacemet (COD). he HRR sigularity is ivalid withi this regio, where the stresses are iflueced by large strais ad crack blutig. Preseted above solutios for stress (Eq. (1) for liear fracture mechaics ad Eq. (3) for oliear fracture mechaics) oly describe the ear tip field ad cosider oly the first term of aylor expasio. I the liear case, the secod term i the aylor expasio correspods to the so called -stress which acts i the Fig. 3. Modified boudary layer aalysis. he first two terms of the Williams series are applied as boudary coditios (ow Fig. based o Aderso, (1995)) he more commo form of Eq. (9) is the followig etry: = ( θ) δ1 iδ, (6) 1j 2πr I ij fij + where δij is the roecker delta. We ca assess the ifluece of the stress by costructig a circular model that cotais a crack, as illustrated i Fig. 3. O the boudary of this model, let us apply i-plae tractios that correspod to Eq. (5). A plastic zoe develops at the crack tip, but its size must be small relative to the size of the model i order to esure the validity of the boudary coditios, which are iferred from a elastic solutio. his cofiguratio, ofte referred to as a modified boudary layer aalysis, simulates the ear tip coditios i a arbitrary geometry, provided the plasticity is well cotaied withi the body. Fig. 4 is a plot of fiite elemet results from a modified boudary layer aalysis (Neimitz et al., 27) that show the effect of the stress o stresses deep iside the plastic zoe, obtaied for large strai assumptio. he special case of = correspods to the small-scale yieldig limit, where the plastic zoe is a egligible fractio of the crack legth ad size of the body, ad the sigular term uiquely defies the ear-tip fields. he sigleparameter descriptio is rigorously correct oly for =. Note that egative values cause a sigificat dowward shift i the stress fields. Positive values shift the stresses to above the small-scale yieldig limit, but the effect is much less proouced tha it is for egative stress. Note that the HRR solutio does ot match the = case. he stresses deep iside the plastic zoe ca be represeted by a power series, where the HRR solutio is the leadig term. Fig. 4 idicates that the higher order plastic terms are ot egligible whe =. 39
3 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems Modified Boudry Layer Approach = 315 MPa = 5 ν =.3 E = 26 MPa 22 / Fig. 4. he stress distributio i frot of the crack, computed usig modified boudary layer approach for costat SIF, I, ad chagig -stress. Properties of material like i the Sumpter ad Forbes paper (Sumpter ad Forbes, 1992) I a cracked body subject to Mode I loadig, the stress, like I, scales with the applied load. he biaxiality ratio relates to stress itesity: πa β =, (7) I where a is the crack legth. For a through-thickess crack i a ifiite plate subject to a remote ormal stress β = -1. hus a remote stress iduces a stress of - i the x directio. I the oliear fracture mechaics, the factor was itroduced by O Dowd ad Shih (1991, 1992) to accout for differece betwee the HRR field ad fiite elemet results. he factor (also called -stress) correspods to a additioal hydrostatic pressure. he modified by O Dowd ad Shih stress field is obtaied as: ( ij) + δij =, (8) ij HRR where (ij)hrr is give by Eq. (3). For small scale yieldig, Eq. (8) ca be writte i the followig form: ( ij) δij = = + ij. (9) he parameter ca be iferred by subtractig the stress field for the = referece state from the stress field of iterest. O'Dowd ad Shih ad most subsequet researchers defied as follows: ( ) ( yy) yy = = for θ= ad r = 2. (1) J J = 3 kn/m / = -1,16 / = -1, / = -,75 / = -,5 / = -,25 / = / =,25 / =,5 / = 1, ψ = x 22 /J Referrig to Fig. 4, we see that is egative whe is egative. For the modified boudary layer solutio, ad are uiquely related. Fig. 5 is a plot of versus for a two work hardeig expoets. A relatio betwee ad stress, based o umerical calculatios, usig large strai theory ad icremetal strai plasticity is give by O Dowd ad Shih (1991) as: 2 ( ) + a ( ) a ( ) 3 a + a +, (11) where coefficiets a, a1, a2 ad a3 were give oly for two work hardeig expoets: for =5: a=-.1, a1=.76, a2=-.32, a3=-.1 ad for =1: a=-.1; a1=.76, a2=-.52, a3=. Proposed by Eq. (11) relatioship is the result of the matchig mathematical formula to umerical results. I 1995 O Dowd (O Dowd, 1995) was proposed, to describe the relatioship betwee ad stress by liear formula, as = for for <. (12) > Equatio (11) for =5 Equatio (11) for =1 Equatio (12) / Fig. 5. Relatioship betwee ad stress, based o Equatios (11) ad (12) (based o O Dowd ad Shih (1991)) Fig. 5 shows graphically the mutual relatioship betwee the ad stress, based o formulas (11) ad (12). he use of both depedecies is relatively simple. I the case of formula (11) must kow the values of the coefficiets a, a1, a2 ad a3. Seems to be easier to use formula (12), because it is required oly kowledge level of stress, which for various geometries ca be foud i the literature (Sherry et al., 1991), (Leevers ad Rado, 1983). Whe Equatio (12) will be used i aalysis of the real structural elemet, as ca be observed, this formula did ot take ito accout the geometry characterizatio, for example material properties, exteral load, kid of specime. Formula (12), takes ito accout oly crack legth, because, stress deped o relative crack legth (Leevers ad Rado, 1983). However, thaks to its simplicity, Equatio (12) has foud use i the solvig egieerig problems ad it was recommeded by FINE procedures (FINE, 26). It should be oted that both Equatios - (11) ad (12) were based o aalysis of small scale yieldig, based o Modify Boudary Layer Approach (MBLA). It should be oted also that both Equatios geerally do ot iclude exteral load, which i the case of plae strai strogly affects the stress level. 4
4 acta mechaica et automatica, vol.6 o.2 (212) 2. UILIZAION OF HE CONSRAIN PARAMEERS IN HE EVALUAION OF FRACURE OUGHNESS Both costrait parameters, the ad stress were foud applicatio i Europea Egieerig Programs, like SINAP (SINAP, 1999) ad FINE (FINE, 26). he -stress or -stress is applied uder costructio of the fracture criterio ad to assessmet the fracture toughess of the structural compoet. Real fracture toughess, may be evaluated usig the formula proposed by Aisworth ad O Dowd (1994). Aisworth ad O Dowd have show that the icrease i fracture i both the brittle ad ductile regimes may be represeted by a expressio of the form: C mat = mat mat k [ 1+ α( βl ) ] r β Lr > (13) βl < where mat is the fracture toughess for plae strai coditio obtaied usig FINE procedures, ad β is the parameter calculated usig followig formula: ( L ) r β =, Lr for elastic materials, for elastic-plastic materials, r (14) where Lr is the ratio of the actual exteral load P ad the limit load P (or the referece stress), which may be calculated usig FINE procedures (FINE, 26). he costats α ad k which are occurrig i Eq. (13), are material ad temperature depedet (ab. 1). Sherry ad et al., (25a, b) proposed the desigatio procedures to calculate the costats α ad k. hus J- ad - theories have practical applicatio i egieerig issues. ab 1. Some values of the α ad k parameters, which are occurrig i Eq. (13) (SINAP, 1999; FINE, 26) material temperature fracture mode α k A533B (steel) -75 C cleavage A533B (steel) -9 C cleavage A533B (steel) -45 C cleavage Low Carbo Steel -5 C cleavage A515 (steel) +2 C cleavage ASM 71 Grade A +2 C ductile he reciprocal relatioship betwee i-plae costrait parameters, for what ca be cosidered the -stress ad -stress may be very useful i practical egieerig problems, to determiatio of the real fracture toughess or i failure assessmet diagrams (FAD) aalysis whe correctio of the FAD curve usig costrait parameter is doe (SINAP, 1999), (FI- NE, 26). hus, i this paper, catalogue of the - trajectories obtaied usig MBLA aalysis will be preseted. he ifluece of the material properties will be tested. 3. DEAILS OF HE NUMERICAL ANALYSIS I the umerical aalysis, ADINA System (ADINA, 28a, b) was used. Computatios were performed for plae strai usig small strai optio ad the Modify Boudary Layer Approach (MBLA) model. he MBLA model cosists of big circle, which radius aroud the crack tip where the boudary coditios are modeled is 1 meters log. Due to the symmetry, oly a half of the circle was modeled (see Fig. 6). he boudary coditios are modeled usig the followig relatioship: u1 = + u2 = ν E ( 1+ ν) 2 ( 1 ν ) r cos( θ) E E ( 1+ ν) E ( 1+ ν) r si( θ) κ 1+ r θ cos + 2 θ 2π si 2, (15) κ + 1+ r θ si + 2 θ 2π 2 2 cos 2 where is the stress itesity factor (SIF) calculated form J-itegral value usig formula = /(1 ) i preseted i the paper umerical program the followig values of the J-itegral were tested: J={1, 25, 5, 1, 25, 5}kN/m; r ad θ are polar coordiates; ν is the Poisso s ratio; E is Youg s modulus; is the stress expressed i stress uit ( ) i preseted i the paper umerical program, the followig values of the parameter were tested: ={.5,.25,, -.25, -.5, -1}; κ=3-4ν for plae strai; κ=(3-ν)/(1+ν) for plae stress. he radius of the crack frot was equal to rw=5 1-6 m. he crack tip was modeled as half of arc. he crack tip regio about to m was divide ito 5 semicircles. he first of them, was at least 2 times smaller the the last oe. he fiite elemet mesh was filled with the 9-ode plae strai elemets. he size of the fiite elemets i the radial directio was decreasig towards the crack tip, while i the agular directio the size of each elemet was kept costat. It varied from θ=π/19 to θ=π/3 for various cases tested. he whole MBLA model was modeled usig 2584 fiite elemets ad 1647 odes. he example fiite elemet model for MBLA aalysis is preseted i Fig. 6. I the FEM simulatio, the deformatio theory of plasticity ad the vo Misses yield criterio were adopted. I the model the stress strai curve was approximated by the relatio: ε for =, where α=1. (16) ε α( ) for > he tesile properties for the materials which were used i the umerical aalysis are preseted below i ab. 2. I the FEM aalysis, calculatios were doe for sixtee material cofiguratios, which were differed by yield stress ad the work hardeig expoet. 41
5 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems a) a) b) b) c) Fig. 6. he MBLA model which was used i the umerical program a); Crack tip regio usig i the MBLA model b) he J-itegral were calculated usig two methods. he first method, called the virtual shift method, uses cocept of the virtual crack growth to compute the virtual eergy chage. he secod method is based o the J itegral defiitio: J = [wdx 2 t ( u x1 )ds ], (17) C where w is the strai eergy desity, t is the stress vector actig o the cotour C draw aroud the crack tip, u deotes displacemet vector ad ds is the ifiitesimal segmet of cotour C. ab. 2. he mechaical properties of the materials used i umerical aalysis ad the HRR parameters for plae strai [MPa] E [MPa] 26 ν ε=/e α ~θθ (θ = ) I RESULS OF HE NUMERICAL ANALYSIS Fig. 7 presets the ifluece of the stress parameter value o the shape ad size (deoted i Fig. as rp) of the plastic zoe ear crack tip. For smaller value of the stress parameter, the greater plastic zoe is observed, if the same level of the J-itegral was used to calculate the boudary coditios ad the same material characteristic was established. he bigger plastic zoe is observed for MBLA model characterized by smaller yield stress (see Fig. 8). 42 Fig. 7. Ifluece of the stress parameter o shape ad size of the plastic zoe for MBLA model characterized by =315MPa, =1, J=1kN/m: a) =.5, rp=.29m; b) =, rp=.33m; c) =-.5, rp=.9m (brighter regio is the plastic zoe) For smaller values of the stress parameter the, smaller value of the stress are observed. he ifluece of the work hardeig expoet o =() trajectories should be cosidered for a umber of cofiguratios. For the cases characterized by yield stress 5MPa ad for J-itegral values betwee 1 ad 25kN/m (which are used to determie the boudary coditios), the lower values of the stress are observed for weakly stregthe materials (see Fig. 9). For theses cases almost parallel arragemet of the =() trajectories is observed. he highest o the chart are situated curves for strogly stregthe material (=3). I other cases, whe the J-itegral value is equal to or greater tha 5kN/m, it ca be see the itersectig of the =() curves for differet values of the work hardeig expoet (see Fig. 1). For small values of the stress parameter (which meas that we are dealig with a case of high-level of the flat geometric costraits), the lower values of parameter are observed for weakly hardeig materials. Icrease value of the stress parameter makes cuttig curves ad the reversal of the tred o the chart - the a smaller values of the parameter are observed for the case of the strogly hardeig materials. Numerical aalysis show, that the ifluece of the yield stress o =() trajectories is quite complex. For small values of the J-itegral (J=1kN/m or J=25kN/m), it ca be cocluded, that =() curves characterized by regularity of arragemet, especially for strog hardeig materials (=3 ad =5). he lowest o the charts are arraged the =() curves, described by small value of the yield stress. his meas that for the materials characterized by higher yield stress, the higher values of the parameter are obtaied (see Fig. 11).
6 acta mechaica et automatica, vol.6 o.2 (212) a) -.4 b) c) -.8 =315MPa J=5kN/m E=26GPa ν=.3 =3 =5 =1 = Fig. 1. he ifluece of the work hardeig expoet o the =() trajectories for MBLA characterized by =315MPa, J=5kN/m.5 Fig. 8. Ifluece of the yield stress o shape ad size of the plastic zoe for MBLA model characterized by =, =1, J=1kN/m: a) =5MPa, rp=.13m; b) =1MPa, rp=.3m; c) =15MPa, rp=.1m (brighter regio is the plastic zoe) =5MPa J=1kN/m E=26GPa ν=.3 =3 =5 =1 = Fig. 9. he ifluece of the work hardeig expoet o the =() trajectories for MBLA characterized by =5MPa, J=1kN/m he ifluece of the yield stress o =() trajectories is sigificat for the low level of the J-itegral which was used to determiatio of the boudary coditios. Sometimes the ifluece of the yield stress o stress value is egligible, whe material characterized by big work hardeig expoet ad the level of the J-itegral is quite large (J 5kN/m). For weakly hardeig materials, =() trajectories are parallel, whe J- itegral characterized by very high level (for example J=25kN/m or J=5kN/m) =3 J=25kN/m E=26GPa ν=.3 =315MPa =5MPa =1MPa =15MPa Fig. 11. he ifluece of the yield stress o the =() trajectories for MBLA characterized by =3, J=25kN/m he most importat coclusio cocers the ifluece of the J-itegral (which is used to determiatio of the boudary coditios for MBLA model) o stress value. As show by umerical calculatios, the value of the parameter as a fuctio of the stress parameter i a very small extet depeds o the J-itegral value adopted to determie the boudary coditios at MBLA issue. A very little impact (hardly isigificat), or the lack of impact is characterized for MBLA models, for which the J-itegral level used to determiatio of the boudary coditios was equal to or greater tha 5kN/m (see Fig. 12). Fig. 13 presets the ifluece of the work hardeig expoet o stress value for differet level of the stress parameter, which may be cosidered as a measure of the i-plae costrait parameter. For the case of low costraits (low value of the stress, equal to -.5 or -1.), stress value decreases whe the value of the work hardeig expoet icreases. Whe the value of the stress parameter is greater tha -.25, it ca be see that stress value is costat or slightly icreases if value of the work hardeig expoet icreases. 43
7 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems a) a).4 J=5kN/m =315MPa E=26GPa ν= b) J [kn/m] =3 =315MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. b) J=1kN/m =5MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. c) J [kn/m] =5 =315MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. c) J=25kN/m =1MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 = Fig. 12. he ifluece of the J-itegral value which was used to determiatio of the boudary coditios, o stress value for differet stress parameter: a) =3, =315MPa; b) =5, =315MPa; b) =1, =5MPa 5. CONCLUSIONS J [kn/m] =1 =5MPa E=26GPa ν=.3 =.5 =.25 = =-.25 =-.5 =-1. I the paper, catalogue of the umerical solutios based o Modify Boudary Layer Approach to determie the relatioship betwee -stress ad -stress were preseted. Based o method proposed by Larsso ad Carlsso, the -stress value were calculated for sixtee elastic-plastic materials for differet value of -stress ad exteral load expressed by J-itegral both parameter were used to determie the boudary coditios, which are ecessary to carry out the MBLA aalysis. he ifluece of the -stress parameter ad material properties o -stress value were tested =.5 =.25 = =-.25 =-.5 =-1. Fig. 13. he ifluece of the work hardeig expoet o stress value for differet level of the stress parameter: a) J=5kN/m, =315MPa; b) J=1kN/m, =5MPa; c) J=25kN/m, =1MPa Obtaied umerical results lead to followig coclusios: for smaller value of the stress parameter, the smaller value of the stress are observed; the ifluece of yield stress o the stress value is sigificat for the case characterized by low level of the J-itegral, adopted to determie the boudary coditios at MBLA issue; i case of a higher level of the J-itegral, it ca be observed a little impact of the yield stress o the stress value, sometimes this effect is egligible; for differet stress value, stress value depeds o work hardeig expoet ; this ifluece should be discussed for each MBLA models separately; the stress parameter very weak depeds (or i geeral does ot deped) o the level of the J-itegral adopted 44
8 acta mechaica et automatica, vol.6 o.2 (212) to determie the boudary coditios i MBLA issue, especially for the case whe the level of the J-itegral is equal to or greater tha 5kN/m. Preseted i the paper such catalogue may be useful durig solvig the egieerig problems, especially while is eeded to determie real fracture toughess with icludig the geometric costraits, what was proposed i FINE procedures (FINE, 26). REFERENCES 1. ADINA (28a), ADINA 8.5.4: ADINA: heory ad Modelig Guide - Volume I: ADINA, Report ARD 8-7, ADINA R&D, Ic., ADINA (28b), ADINA 8.5.4: ADINA: User Iterface Commad Referece Maual - Volume I: ADINA Solids & Structures Model Defiitio, Report ARD 8-6, ADINA R&D, Ic., Aisworth R.A., O'Dowd N.P. (1994), A Framework of Icludig Costrait Effects i the Failure Assessmet Diagram Approach for Fracture Assessmet, ASME Pressure Vessels ad Pipig Coferece, PVP-Vol 287/MD-Vol47, ASME. 4. Betego, C. ad Hacock, J.W. (1991), wo Parameter Characterizatio of Elastic-Plastic Crack ip Fields, Joural of Applied Mechaics, Vol. 58, Bilby, B.A., Cardew, G.E., Goldthorpe, M.R., Howard, I.C. (1986), A Fiite Elemet Ivestigatio of the Effects of Specime Geometry o the Fields of Stress ad Strai at the ips of Statioary Cracks, Size Effects i Fracture, Istitute of Mechaical Egieers, Lodo, FINE Fitess for Serwice Procedure Fial Draft (26), Edited by M. oçak, S. Webster, JJ. Jaosh, RA. Aisworth, R. oers. 7. Gałkiewicz J., Graba M. (26), Algorithm for Determiatio of (, ), (, ), (, ), ( ) ad ( ) Fuctios i Hutchiso-Rice-Rosegre Solutio ad its 3d Geeralizatio, Joural of heoretical ad Applied Mechaics, Vol. 44, No. 1, Hutchiso J. W. (1968), Sigular Behaviour at the Ed of a esile Crack i a Hardeig Material, Joural of the Mechaics ad Physics of Solids, 16, Irwi, G.R. (1957), Aalysis of Stresses ad Strais ear the Ed of a Crack raversig a Plate, Joural of Applied Mechaics, Vol. 24, Leevers P.S., Rado J.C. (1983), Iheret Stress Biaxiality i Various Fracture Specime Geometries, Iteratioal Joural of Fracture, 19, McMeekig, R.M. ad Parks, D.M. (1979), O Criteria for J-Domiace of Crack ip Fields i Large-Scale Yieldig, ASM SP 668, America Society for estig ad Materials, Philadelphia, Neimitz A., Graba M., Gałkiewicz J. (27), A Alterative Formulatio of the Ritchie-ott-Rice Local Fracture Criterio, Egieerig Fracture Mechaics, Vol. 74, O Dowd N. P. (1995), Applicatios of two parameter approaches i elastic-plastic fracture mechaics, Egieerig Fracture Mechaics, Vol. 52, No. 3, O Dowd N. P., Shih C. F. (1992), Family of Crack-ip Fields Characterized by a riaxiality Parameter II. Fracture Applicatios, J. Mech. Phys. Solids, Vol. 4, No. 5, O Dowd N. P., Shih C.F. (1991), Family of Crack-ip Fields Characterized by a riaxiality Parameter I. Structure of Fields, J. Mech. Phys. Solids, Vol. 39, No. 8, Rice J. R., Rosegre G. F. (1968), Plae Strai Deformatio Near a Crack ip i a Power-law Hardeig Material, Joural of the Mechaics ad Physics of Solids, 16, Rice, J.R. (1968), A Path Idepedet Itegral ad the Approximate Aalysis of Strai Cocetratio by Notches ad Cracks, Joural of Applied Mechaics, Vol. 35, pp Sherry A.H., Hooto D.G., Beardsmore D.W., Lidbury D.P.G. (25), Material costrait parameters for the assessmet of shallow defects i structural compoets Part II: costrait based assessmet of shallow cracks, Egieerig Fracture Mechaics, 72, Sherry A.H.,Frace C.C., Goldthorpe M.R. (1995), Compedium of -stress solutios for two ad three dimesioal cracked geometries, Fatigue & Fracture of Egieerig Materials & Structures, Vol. 18, No. 1, Sherry, A.H., Wilkes M.A., Beardsmore D.W., Lidbury D.P.G. (25), Material costrait parameters for the assessmet of shallow defects i structural compoets Part I: Parameter solutios, Egieerig Fracture Mechaics, 72, SINAP (1999), SINAP: Structural Itegrity Assessmet Procedures for Europea Idustry. Fial Procedure, Brite-Euram Project No BE Rotherham: British Steel. 22. Seddo, I.N. (1946), he Distributio of Stress i the Neighbourhood of a Crack i a Elastic Solid, Proceedigs, Royal Society of Lodo, Vol. A-187, Sumpter J. G. D., Forbes A.. (1992), Costrait based aalysis of shallow cracks i mild steels, Proceedigs of WI/EWI/IS It. Cof o Shallow Crack Fracture Mechaics, oughess ests ad Applicatios, Paper 7, Cambridge U Westergaard, H.M. (1939), Bearig Pressures ad Cracks, Joural of Applied Mechaics, Vol. 6, Williams, M.L. (1957), O the Stress Distributio at the Base of a Statioary Crack, Joural of Applied Mechaics, Vol. 24, Ackowledgmets: he support of the ielce Uiversity of echology Faculty of Mechatroics ad Machie Desig through grat No 1.22/7.14 is ackowledged by the author of the paper. ANNEX - NUMERICAL RESULS OBAINED FOR ALL MBLA MODELS ab. A.1. Results for MBLA models, characterized by J=1kN/m J=1kN/m =315MPa J=1kN/m =5MPa
9 Marci Graba Numerical Verificatio of the Relatioship Betwee he I-Plae Geometric Co-Straits used i Fracture Mechaics Problems J=1kN/m =1MPa J=1kN/m =15MPa ab. A.2. Results for MBLA models, characterized by J=25kN/m J=25kN/m =315MPa J=25kN/m =5MPa J=25kN/m =1MPa J=25kN/m =15MPa ab. A.3. Results for MBLA models, characterized by J=5kN/m J=5kN/m =315MPa J=5kN/m =5MPa J=5kN/m =1MPa J=5kN/m =15MPa ab. A.4. Results for MBLA models, characterized by J=1kN/m J=1kN/m =315MPa
10 acta mechaica et automatica, vol.6 o.2 (212) J=1kN/m =5MPa J=1kN/m =1MPa J=1kN/m =15MPa ab. A.5. Results for MBLA models, characterized by J=25kN/m J=25kN/m =315MPa J=25kN/m =5MPa J=25kN/m =1MPa J=25kN/m =15MPa ab. A.6. Results for MBLA models, characterized by J=5kN/m J=5kN/m =315MPa J=5kN/m =5MPa J=5kN/m =1MPa J=5kN/m =15MPa
Research Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationNon-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationConvention Paper 6764
Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationUsing Four Types Of Notches For Comparison Between Chezy s Constant(C) And Manning s Constant (N)
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH OLUME, ISSUE, OCTOBER ISSN - Usig Four Types Of Notches For Compariso Betwee Chezy s Costat(C) Ad Maig s Costat (N) Joyce Edwi Bategeleza, Deepak
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationHow To Solve The Phemean Problem Of Polar And Polar Coordiates
ISSN 1 746-733, Eglad, UK World Joural of Modellig ad Simulatio Vol. 8 (1) No. 3, pp. 163-171 Alterate treatmets of jacobia sigularities i polar coordiates withi fiite-differece schemes Alexys Bruo-Alfoso
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More information19. LINEAR VISCOUS DAMPING. Linear Viscous Damping Is a Property of the Computational Model And is not a Property of a Real Structure
19. LINEAR VISCOUS DAMPING Liear Viscous Dampig Is a Property of the Computatioal Model Ad is ot a Property of a Real Structure 19.1 INRODUCION { XE "Viscous Dampig" }I structural egieerig, viscous velocity-depedet
More informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationCantilever Beam Experiment
Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Catilever Beam Experimet Backgroud A disk drive maufacturer is redesigig several disk drive armature mechaisms. This is the result of evaluatio
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationHow To Solve The Homewor Problem Beautifully
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationData Analysis and Statistical Behaviors of Stock Market Fluctuations
44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationBENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationLog-Logistic Software Reliability Growth Model
Log-Logistic Software Reliability Growth Model Swapa S. Gokhale ad Kishor S. Trivedi 2y Bours College of Egg. CACC, Dept. of ECE Uiversity of Califoria Duke Uiversity Riverside, CA 9252 Durham, NC 2778-29
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More information, a Wishart distribution with n -1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationCOMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationFinite Element Analysis of Rubber Bumper Used in Air-springs
Available olie at www.sciecedirect.com Procedia Egieerig 48 (0 ) 388 395 MMaMS 0 Fiite Elemet Aalysis of Rubber Bumper Used i Air-sprigs amas Makovits a *, amas Szabó b a Departmet of Mechaical Egieerig,
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationProblem Solving with Mathematical Software Packages 1
C H A P T E R 1 Problem Solvig with Mathematical Software Packages 1 1.1 EFFICIENT PROBLEM SOLVING THE OBJECTIVE OF THIS BOOK As a egieerig studet or professioal, you are almost always ivolved i umerical
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationProbabilistic Engineering Mechanics. Do Rosenblatt and Nataf isoprobabilistic transformations really differ?
Probabilistic Egieerig Mechaics 4 (009) 577 584 Cotets lists available at ScieceDirect Probabilistic Egieerig Mechaics joural homepage: wwwelseviercom/locate/probegmech Do Roseblatt ad Nataf isoprobabilistic
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More information3 Energy. 3.3. Non-Flow Energy Equation (NFEE) Internal Energy. MECH 225 Engineering Science 2
MECH 5 Egieerig Sciece 3 Eergy 3.3. No-Flow Eergy Equatio (NFEE) You may have oticed that the term system kees croig u. It is ecessary, therefore, that before we start ay aalysis we defie the system that
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationLecture 7: Stationary Perturbation Theory
Lecture 7: Statioary Perturbatio Theory I most practical applicatios the time idepedet Schrödiger equatio Hψ = Eψ (1) caot be solved exactly ad oe has to resort to some scheme of fidig approximate solutios,
More informationDecomposition of Gini and the generalized entropy inequality measures. Abstract
Decompositio of Gii ad the geeralized etropy iequality measures Stéphae Mussard LAMETA Uiversity of Motpellier I Fraçoise Seyte LAMETA Uiversity of Motpellier I Michel Terraza LAMETA Uiversity of Motpellier
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationA Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationODBC. Getting Started With Sage Timberline Office ODBC
ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.
More informationTHE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY
- THE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY BY: FAYE ENSERMU CHEMEDA Ethio-Italia Cooperatio Arsi-Bale Rural developmet Project Paper Prepared for the Coferece o Aual Meetig
More informationVolatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina
Overcomig the Crisis: Ecoomic ad Fiacial Developmets i Asia ad Europe Edited by Štefa Bojec, Josef C. Brada, ad Masaaki Kuboiwa http://www.hippocampus.si/isbn/978-961-6832-32-8/cotets.pdf Volatility of
More informationExample: Probability ($1 million in S&P 500 Index will decline by more than 20% within a
Value at Risk For a give portfolio, Value-at-Risk (VAR) is defied as the umber VAR such that: Pr( Portfolio loses more tha VAR withi time period t)
More informationDAME - Microsoft Excel add-in for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2
Itroductio DAME - Microsoft Excel add-i for solvig multicriteria decisio problems with scearios Radomir Perzia, Jaroslav Ramik 2 Abstract. The mai goal of every ecoomic aget is to make a good decisio,
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More information3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationTradigms of Astundithi and Toyota
Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationZ-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module The Sciece of Surface ad Groud Water Versio CE IIT, Kharagpur Lesso 8 Flow Dyamics i Ope Chaels ad Rivers Versio CE IIT, Kharagpur Istructioal Objectives O completio of this lesso, the studet shall
More informationModeling of Ship Propulsion Performance
odelig of Ship Propulsio Performace Bejami Pjedsted Pederse (FORCE Techology, Techical Uiversity of Demark) Ja Larse (Departmet of Iformatics ad athematical odelig, Techical Uiversity of Demark) Full scale
More informationSemiconductor Devices
emicoductor evices Prof. Zbigiew Lisik epartmet of emicoductor ad Optoelectroics evices room: 116 e-mail: zbigiew.lisik@p.lodz.pl Uipolar devices IFE T&C JFET Trasistor Uipolar evices - Trasistors asic
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More informationInstallment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate
Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi
More informationChatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com
SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTI-TRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial
More informationLECTURE 13: Cross-validation
LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More information