UNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL

Transcription

1 8th International DAAAM Baltic Conference "INDUSTRIAL ENGINEERING April 01, Tallinn, Estonia UNCERTAINTY FOCUSED STRENGTH ANALYSIS MODEL Põdra, P. & Laaneots, R. Abstract: Strength analysis is a critical phase when developing engineering soltions and determining respective components safe dimensions. Uncertainty based method for strength calclations will be investigated in this stdy. This concept reqires that a calclation model is developed for each particlar strength calclation case, where the initial data estimates along with the estimates of their ncertainties are the inpts. This approach allows to better optimise the vale of design factor and facilitate a new solid method for engineering analysis, where the correctness of calclation reslts is assred at the certain predefined confidence level. Key words: machine design, strength analysis, ncertainty, component safety 1. INTRODUCTION The objective of the strength analysis is often to calclate the vale of some component dimension in order to assre the component safe and reliable operation dring the whole predefined service life. Each strength analysis, based on statics or dynamics strength condition, is performed by the se of a nmber of inflence- or inpt qantities. And the vale of the selected otpt qantity is calclated sing analytical or experimental relationships and inpt qantities vales. The estimates of the vales needed for solving the case are determined based on known data, given in standards, handbooks or other sorces. The formla or analysis model, describing the relationship of the otpt qantity with other qantities, mst be developed according to the calclation task in qestion. The problem, however, is that the vales of inpt qantities are known only approximately, i.e. with some ncertainty. In order to assre the component performance and safety even at the most critical combination of inpt qantities vales, certain level of safety mst be assmed in the analysis. The needed level of safety is characterised by the reqired vale of the factor of safety, that also known as the design factor. The vale of design factor is sally determined based on engineering practice and following statistical gidelines presented in textbooks, handbooks, standards, etc. (Table 1) [1]. They are to be refined, ths, according to the experience of each particlar design engineer. Type of loading Type of material Dctile Brittle Static 6 Repeated 8 10 Shock 1 15 Table 1. Recommended vales of design factor Too large vale of design factor is followed by the excessive material se. Too small vale, however, decreases the strength and endrance of component below the critical level. The well-reasoned and optimm vale of design factor provides the prodct a competitive advantage. Development of the method, based on the analysis of ncertainties of parameters inflencing the component strength that allows to design safe prodcts with better optimised material se is therefore highly motivated.

2 . THEORY The calclation model sed in strength analysis cold in general terms be expressed as G(Y, X 1, X, X i, X n ) 0, (1) where G is the model basic fnctional relationship, Y is the otpt qantity of the model, i.e. the parameter to be calclated and X 1, X, X i, X n are the inpt qantities the otpt qantity is dependent on. The inpt qantities of the calclation model (1) cold also be dependent on other qantities, inclding constants. The model (1) cold therefore appear to be rather complicated, sometimes even to sch extent, that it is not possible to formlate it exactly. The fnctional relationship of the calclation model may be identifiable experimentally or may exist as an algorithm to be evalated nmerically. Ths, the interpretation of the fnction G mst be larger than that of a prely mathematical fnction. It is important, that the fnction G incldes all inpt qantities that inflence the vale of the otpt qantity Y. If the practical observations indicate, that the present fnction G does not model the calclation procedre with needed accracy, it mst be complemented with additional inpt qantities. Since the vales of the inpt qantities in calclation model (1) X 1, X, X i, X n are not known exactly, the estimates of the vales of these qantities mst be sed in practice instead. By replacing the qantities in calclation model (1) with respective estimates, the fnction general form appears y g(x 1, x, x i, x n ) () where y is the estimate (approximation) of the otpt qantity Y, g is the relationship sed in particlar calclation formla (mathematical, experimental or other relationship for obtaining the otpt qantity) and x 1, x, x i, x n are the estimates of inpt qantities vales. In order to calclate the reslt with the formla (), that fnctional relationship g mst be known. Since the calclation reslt y is the estimate (approximation) of Y, it mst be presented together with the data abot its ncertainty. In the case, that all inpt qantities in fnctional relationship () are ncorrelated, the combined ncertainty of the calclation reslt y, following the law of propagation of ncertainty and based on the ncertainties of inpt qantities estimates, [] can be expressed as n ( y) ( x ) i 1 g xi i () where (y) is the combined ncertainty of the calclation reslt and (x i ) is the standard ncertainty of the inpt qantity estimate x i. Depending on the predefined confidence level, that combined ncertainty () mst be mltiplied with a coefficient and the expanded ncertainty is expressed as U p k (y) (4) where U p is the expanded ncertainty of the calclation reslt y at the confidence level p, that is associated with coverage probability and k is the coverage factor, the vale of which depends on p. U p characterises the possible limits of the calclation reslt y vale and the choice of the vale of k depends on the reqirements of the coverage probability set forth for that calclation. Coverage probability 100 % is never possible. Determination of the sitable confidence level is a bsiness decision it shows the percentage of the components or prodcts in qestion that shold not fail. Assming the normal distribtion of the qantity y, the vales of selected coverage factors at respective confidence levels [] are given in Table.

3 Allowed nmber of failed components ot of Confidence level p/% Coverage factor k 17 68, , , Table. Vales of the coverage factor k In classical strength calclations the component is considered safe, if the strength condition is flfilled S lim max [ S], (5) where S and [S] are the factor of safety and design factor respectively, lim is material limit stress (yield stress R e or ltimate stress R m ) and max is the maximm vale of stress in that component. In order to determine the optimm design factor [S] vale, several risks and aspects cold be considered, sch as: severity of risks related to the failre of that component and the possibilities of failre processes early detection; ncertainties of the vales of loads, geometrical dimensions and material properties; simplifications of the calclation model and formlae, etc. All relevant isses, inflencing the component safety, mst be taken into accont sing respective inpt qantities X i with their estimates x i (these cold be regarded as partial factors of safety) in the calclation models (1) and (). In the best case the concept of design factor cold be abandoned altogether (taking [S] 1) and the safe estimate of the qantity Y is the respective limit vale y lim (y + U p ) or y lim (y U p ). Now the strength condition (5) takes a new shape S max lim ( y ) lim 1, (6) where max (y lim ) is the maximm vale of the component stress in the case that the vale of the qantity, calclated sing () is safe y lim. The strength analysis example of the cantilever beam, together with respective ncertainty analysis is presented below.. RESULTS AND DISCUSSION The safe vale of the diameter of a niform circlar cross-section cantilever beam (Fig. 1) mst be calclated. The confidence level (coverage probability) of the calclation reslt mst be 99,7 %, giving the vale of the coverage factor k (Table ). In that simple case the initial qantities are as follows: load of the beam F 000 N is acting perpendiclar to the beam axis; beam is to be manfactred of steel E95 with the limit stress eqal to the yield stress, according to EVS-EN [] lim R e 85 MPa, if the thickness of the material is in the limits ( ) mm; distance of the load acting point from the hosing wall l 50 mm. Figre 1. Cantilever beam Under the inflence of F the beam deforms elastically with the maximm vale of the bending moment M M max Cantilever beam Hosing F 000 N Fl 000 0,5 750 N m located at the beam attachment area to the hosing (Fig. ).

4 Figre. Calclation scheme and bending moment M diagram of the beam Mathematical fnction for the beam diameter calclation is [1] d Fl [ S] Fl, (7) πre πr e when taking [S] 1 and the beam diameter, i.e. the otpt qantity, vale estimate is obtained 000 0,5 d 0,099 m 0 mm. 6 π This calclation is followed by the ncertainty analysis and the inpt qantities standard ncertainties mst be determined first, sing methods otlined above [, 4] and formlae () and (4), becase the inpt qantities are ncorrelated. Giving the formla (7) another form d F l Re π, (8) the combined ncertainty of the beam diameter d can be calclated according to (), sing the relationship ( d ) M / N m F 000 N l 0,5 m ( F ) + ( l) + F l, (9) + ( Re ) R e where the partial derivatives of (8) appear d l d F d R l Re F π F Re l π e l F Re π, (10) and (11). (1) There is no data given abot the load F in addition to its vale. Since all the vales in the deviation range may characterise the load F with eqal probability, its vales have a rectanglar distribtion in the limits of that deviation range. And the standard ncertainty of the load F estimate is [] ( F ) { F} (1) where {F} is the parameter of the deviation range, expressed as (F ± {F}). The distance of the load to the hosing wall l may change in the limits of its tolerance T l. Also in this case, the vales of the dimension l follow the rectanglar distribtion and the standard ncertainty of the dimension l 50 mm estimate is [] ( l ) T l. (14) The vale of the yield stress R e is taken from the standard EVS-EN and assming the rectanglar distribtion of the R e in the deviation range limits, its standard ncertainty is calclated as [] ( R ) { R } e e, (15) where {R e } is the parameter of the deviation range, given as (R e ± {R e }). The parameter {F} for the load F 000 N may have a rogh vale of {F} 500 N. The reasonable tolerance for the distance l 0,5 m is considered T l 10 mm. The vale of {R e } for R e 85 MPa is determined here by the ronding interval

5 {R e } 5 MPa. The vales of the inpt qantities standard ncertainties, calclated sing the vales above and formlae (1), (14) and (15), are given in Table. (F), N (l), m (R e ), Pa 90,9 10 -, Table. Standard ncertainties of inpt qantities Writing the formla (9) in the form ( d ) A + B C +, (16) its parameters based on (9) will be A ( F ), F (17) B ( l) and l (18) C ( Re ). (19) Re The vales of A, B and C, according to relationships (10), (11) and (1) together with the data in Table, are shown in Table 4. A, m B, m C, m 9, , , Table 4. Components of the combined ncertainty of the otpt qantity The vales of the combined ncertainty and expanded ncertainty of the beam diameter d can be calclated next sing formlae (16) and (4) respectively ( d ) 8 9, , ,0 10 0, m 0,98 mm U 0,98,9 mm. 99,7 % 8 Now the calclation reslt of the beam diameter is d 0 mm ± mm. The safe vale of the beam is d mm. For comparison, another calclation is made, taking the relative ncertainty of all inpt qantities eqal to 15 %. The inpt qantities standard ncertainties, following the method otlined above, are shown in Table 5 and the respective components of otpt qantity combined ncertainty are given in Table 6. (F), N (l), m (R e ), Pa 60, Table 5. Standard ncertainties of inpt qantities with eqal relative ncertainty A, m B, m C, m 7, , , Table 6. Components of combined ncertainty of the otpt qantity with the eqal relative ncertainty of inpt qantities. Now the combined and expanded ncertainty of the beam diameter were calclated ( d ) 7, , ,7 10 1,51 10 m 1,5 mm U 1,5 4,5 5 mm. 99,7 % The calclation reslt in this particlar case appears d 0 mm ± 5 mm with the beam safe diameter eqal to d 5 mm. The comparison of the combined ncertainties components according to Table 4 and Table 6 is presented on Fig.. It can be seen in the present cases, that if the relative ncertainties of the inpt qantities are: eqal, their inflence on the vale of the otpt qantity combined ncertainty is also approximately eqal; different, their impact on the otpt qantity combined ncertainty may be considerably different. In this example the ncertainties of yield stress R e and distance dimension l may safely be left ot of consideration.

6 Combined ncertainty (d) / mm C: inflence of R e B: inflence of l A: inflence of F Varying relative inpt ncertainties Figre. Components of the combined ncertainties For verification of the ncertainty based calclation reslts, the beam diameter was also calclated sing the classical approach, with the vale of design factor [S] (Table 1) and formla (7) 000 0,5 d 6 π ,077 m 8 mm. It can be conclded, that the ncertainty based strength analysis model provides well-reasoned reslts and in this case allows to jstify the se of smaller beam diameter than that calclated by classical method. 4. CONCLUSIONS Same relative inpt ncertainties 15 % The analysis of the calclation model above shows that the approach with partial derivatives may be complicated in terms of mathematics, bt this problem cold be solved sing appropriate compter software and programs. When sing the concept of ncertainty, there is possible to, following the predefined level of confidence, give a better reasoning for the choice of machine components dimensions, loads and material properties. This method also allows to analyse the inflence of different inpt qantities and respective ncertainties on the vales of the calclation reslt and its ncertainty, especially in the cases of complex engineering calclations involving many parameters, coefficients and physical constants. The se of respective compter programs allows, for instance, to efficiently analyse the need for inpt qantities standard ncertainties in order to obtain the reasonable vale of otpt qantity expanded ncertainty, etc. More sophisticated cases will be analysed in frther research. 5. REFERENCES [1] Mott, R. L. Applied Strength of Materials 4th ed. Prentice-Hall, 00. [] Laaneots, R and Mathiesen, O. An Introdction to Metrology The second, thoroghly completed edition. Tallinn: TUT Press, 011. [] EVS-EN 1005-:005. Hot rolled prodcts of strctral steels Part : Technical delivery conditions for non alloy strctral steels. Tallinn: EVS, 005. [4] Raba, K. Uncertainty Focsed Prodct Improvement Models Thesis on mechanical and instrmental engineering E46. Tallinn: TUT Press, 009. Acknowledgements. This work was spported by Estonian Ministry of Edcation Research Project No. SF s ADDITIONAL DATA Priit Põdra and Rein Laaneots, Tallinn University of Technology, Dept. of Mechatronics, Ehitajate tee 5, Tallinn, Estonia. priit.podra@tt.ee; rein.laaneots@tt.ee