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1 TNSILVNI UNIVSITY OF BSOV CHNICL NGINING FCULTY DPTNT OF CHNICL NGINING Couse 9 Cuved as 9.. Intoduction The eams with plane o spatial cuved longitudinal axes ae called cuved as. Thee ae consideed two classes of polems: a) initiall cuved eams whee the depth of coss-section can e consideed small in elation to the initial adius of cuvatue. ) those eams whee the depth of coss-section and initial adius of cuvatue ae appoximatel of the same ode, i.e. deep eams with high cuvatue. c) The high cuved as ending theo was developed mile Winkle. 9.. Initiall cuved slende eams In this case the ation of , whee is the cuvatue o the a and h is the height h of the coss section. Let consideed the cuved a fom Figue 9.. Unde the complex load consisting of foces and moments geneic denoted with P i ( i,,..., n,..., q,...), in the coss sections ae P n developed tensile and shea foces N, and T and ending moments. G s is known the tensile foce N geneate N T a nomal stess unifoml distiuted on the coss section suface: P k O Figue 7. P q N, (9.) whee is the value of the coss section aea. The shea foce T develop a shea stess that is calculated with Juavski elationship: T S I. (9.) STNGTH OF TILS - PT II Pof.d.ing. Ioan Calin OSC
2 TNSILVNI UNIVSITY OF BSOV CHNICL NGINING FCULTY DPTNT OF CHNICL NGINING The nomal stess geneated the ending moment is found out with the classical Navie s elationship:. (9.3) I 9.3. Deep eams with high initial cuvatue In case of a ation of , whee is the cuvatue o the a and h is the height of h the coss section, the as ae consideed to have high initial cuvatue (i.e. small adius of cuvatue). In this case the ending stess has to e calculate with the theo developed mile Winkle. The theo is ased on the following assumptions: The longitudinal axes ae situated in a single plane; This plane is, in the same time, a smmet plane of the a; ll the loads ae applied in the same plane that is the smmet plane; The coss section is consideed to e constant along the a; The mateial satisf the Hooke s law; It is espected the Benoulli s assumption which states that the plane coss sections ae nomal to the longitudinal axes efoe and afte defomation (the shape of the coss section is changed, unde loads, in a neglected atio and so one can conside that emains the same); It is neglected the compession developed on adial diection the ending moment etween the fies. It is consideed a pat of a plane cuved a defined angle d (Figue 9.). Thee ae made the following Figue 9. notations: - inne fie adius; - extenal fie adius; - distance fom the cuvatue cente C to the cente of gavit G ; - is the distance to the neutal axis O O ; - is the cuvatue adius of a fie. STNGTH OF TILS - PT II Pof.d.ing. Ioan Calin OSC
3 TNSILVNI UNIVSITY OF BSOV CHNICL NGINING FCULTY DPTNT OF CHNICL NGINING Unde the action of the ending moment the end sections (end edges) of the consideed element otate one to the othe one with an angle equal with d. To simplif the calculation one can conside that onl one end edge is otated aound the neutal axe. Thee ae made the following specifications: The fie that is situated on the neutal axe dose not changes its length. The neutal axe divides the coss section in two pats: one whee the nomal stess is positive (tensile), and the othe one whee the value of stess is negative one (compession); In the case of cuved as the neutal axe is not the same with the axe of the coss sections centes of gavit (longitudinal axe) and esults that it is necessa to find out the position of neutal axe. It is consideed a fie that has the length equal with ds that is situated at a distance fom the neutal axe. The length of the fie can e calculate, ased on Figue 9., as: ds d. (9.4) Unde the ending load, the fie length gowth with a quantit Benoulli s assumption, is: ds that, accoding with ds d. (9.5) s it can e seen in Figue (9.), etween the two quantities and exists the elationship:. (9.6) s was mentioned, it is consideed that the mateial satisfies the Hooke s law. This assumption leads to the possiilit to wite, ased on elationships (9.4) (9.6) the stain mateial as: ds ds d d d, (9.7) d and the nomal stess as: ds d. (9.8) d d Osevations: Fom elationship (9.8) esults that the vaiation of the nomal stess, on the coss sections, is epesented a hpeolic function; The highest values ae developed in the fies that ae situated at the exteme edges of the coss section; In the neutal axe o 0 the nomal stess is eo 0. 3 STNGTH OF TILS - PT II Pof.d.ing. Ioan Calin OSC
4 TNSILVNI UNIVSITY OF BSOV CHNICL NGINING FCULTY DPTNT OF CHNICL NGINING s it is known thee ae two elationships of equivalenc witten as: d 0 ; d, (9.9) that, ased on (9.8) ecome: dx dx dx dx d 0 ; d. (9.0) Fom the fist equation of (9.0) one can otain the geometic position of neutal axe: d d 0, (9.) that leads to: d, (9.) and the integal fom denominato has diffeent values accoding with the coss section shape. Fom the second elationship of (9.0) is otained the elationship of stain: d d. (9.3) d The value of integal fom elationship (9.3) is: d d d d Intoducing (9.4) in (9.3) one can otain: e. (9.4) d, (9.5) d e that comined with (9.8) leads to the nomal stess fomula:, e 4 STNGTH OF TILS - PT II Pof.d.ing. Ioan Calin OSC
5 TNSILVNI UNIVSITY OF BSOV CHNICL NGINING FCULTY DPTNT OF CHNICL NGINING o,. (9.6) e Consideing the geometical notations thee ae otained the following elationships fo the stesses developed in extemel edges: inne edge: oute edge: ; (9.7) e. (9.8) e : In the case of a tensile load that is supeposed on the ending load, the total stess is given N. (9.9) e 5 STNGTH OF TILS - PT II Pof.d.ing. Ioan Calin OSC
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