Module 5. Cables and Arches. Version 2 CE IIT, Kharagpur

Transcription

1 odule 5 Cable and Arche Veron CE IIT, Kharagpur

2 Leon 33 Two-nged Arch Veron CE IIT, Kharagpur

3 Intructonal Objectve: After readng th chapter the tudent wll be able to 1. Compute horzontal reacton n two-hnged arch b the method of leat work.. Wrte tran energ tored n two-hnged arch durng deformaton. 3. Anale two-hnged arch for external loadng. 4. Compute reacton developed n two hnged arch due to temperature loadng Introducton anl three tpe of arche are ued n practce: three-hnged, two-hnged and hngele arche. In the earl part of the nneteenth centur, three-hnged arche were commonl ued for the long pan tructure a the anal of uch arche could be done wth confdence. owever, wth the development n tructural anal, for long pan tructure tartng from late nneteenth centur engneer adopted two-hnged and hngele arche. Two-hnged arch the tatcall ndetermnate tructure to degree one. Uuall, the horzontal reacton treated a the redundant and evaluated b the method of leat work. In th leon, the anal of two-hnged arche dcued and few problem are olved to llutrate the procedure for calculatng the nternal force. 33. Anal of two-hnged arch A tpcal two-hnged arch hown n Fg. 33.1a. In the cae of two-hnged arch, we have four unknown reacton, but there are onl three equaton of equlbrum avalable. ence, the degree of tatcal ndetermnac one for twohnged arch. Veron CE IIT, Kharagpur

4 The fourth equaton wrtten conderng deformaton of the arch. The unknown redundant reacton b calculated b notng that the horzontal dplacement of hnge B zero. In general the horzontal reacton n the two hnged arch evaluated b traghtforward applcaton of the theorem of leat work (ee module 1, leon 4), whch tate that the partal dervatve of the tran energ of a tatcall ndetermnate tructure wth repect to tatcall ndetermnate acton hould vanh. ence to obtan, horzontal reacton, one mut develop an expreon for tran energ. Tpcall, an ecton of the arch (vde Fg 33.1b) ubjected to hear forcev, bendng moment and the axal compreon N. The tran energ due to bendng calculated from the followng expreon. U b U b = d (33.1) The above expreon mlar to the one ued n the cae of traght beam. owever, n th cae, the ntegraton need to be evaluated along the curved arch length. In the above equaton, the length of the centerlne of the arch, I the moment of nerta of the arch cro ecton, E the Young modulu of the arch materal. The tran energ due to hear mall a compared to the tran energ due to bendng and uuall neglected n the anal. In the cae of flat arche, the tran energ due to axal compreon can be apprecable and gven b, U a = N d AE (33.) Veron CE IIT, Kharagpur

5 The total tran energ of the arch gven b, U d = + N d AE (33.3) Now, accordng to the prncple of leat work U =, where choen a the redundant reacton. U = d + N AE N d = (33.4) Solvng equaton 33.4, the horzontal reacton evaluated Smmetrcal two hnged arch Conder a mmetrcal two-hnged arch a hown n Fg 33.a. Let C at crown be the orgn of co-ordnate axe. Now, replace hnge at B wth a roller upport. Then we get a mpl upported curved beam a hown n Fg 33.b. Snce the curved beam free to move horzontall, t wll do o a hown b dotted lne n Fg 33.b. Let and N be the bendng moment and axal force at an cro ecton of the mpl upported curved beam. Snce, n the orgnal arch tructure, there no horzontal dplacement, now appl a horzontal force a hown n Fg. 33.c. The horzontal force hould be of uch magntude, that the dplacement at B mut vanh. Veron CE IIT, Kharagpur

6 Veron CE IIT, Kharagpur

7 From Fg. 33.b and Fg 33.c, the bendng moment at an cro ecton of the arch ( D ), m be wrtten a = ( h ) (33.5) The axal compreve force at an cro ecton ( D ) m be wrtten a N = N + coθ (33.6) Where θ the angle made b the tangent at D wth horzontal (vde Fg 33.d). Subttutng the value of and N n the equaton (33.4), U = = ( h ) ( h ) d + N + coθ coθ d EA (33.7a) Veron CE IIT, Kharagpur

8 Let, ~ = h ~ ~ N + coθ d + coθ d = EA (33.7b) Solvng for, eld ~ d + ~ d + N coθ d + EA co EA θ d = = ~ d ~ d + N coθ d EA co θ d EA (33.8) Ung the above equaton, the horzontal reacton for an two-hnged mmetrcal arch m be calculated. The above equaton vald for an general tpe of loadng. Uuall the above equaton further mplfed. The econd term n the numerator mall compared wth the frt term and neglected n the anal. Onl n cae of ver accurate anal econd term condered. Alo for flat arched, coθ 1a θ mall. The equaton (33.8) now wrtten a, ~ d ~ d d + EA = (33.9) A axal rgdt ver hgh, the econd term n the denomnator m alo be neglected. Fnall the horzontal reacton calculated b the equaton = ~ d ~ d (33.1) For an arch wth unform cro ecton contant and hence, Veron CE IIT, Kharagpur

9 = ~ ~ d d (33.11) In the above equaton, the bendng moment at an cro ecton of the arch when one of the hnge replaced b a roller upport. ~ the heght of the arch a hown n the fgure. If the moment of nerta of the arch rb not contant, then equaton (33.1) mut be ued to calculate the horzontal reacton Temperature effect Conder an unloaded two-hnged arch of pan L. When the arch undergoe a unform temperature change of T C, then t pan would ncreae b α LT f t were allowed to expand freel (vde Fg 33.3a). α the co-effcent of thermal expanon of the arch materal. Snce the arch retraned from the horzontal movement, a horzontal force nduced at the upport a the temperature ncreaed. Veron CE IIT, Kharagpur

10 Now applng the Catglano frt theorem, U ~ co θ = α LT = d + d (33.1) EA Solvng for, = α LT ~ co θ d d EA + (33.13) The econd term n the denomnator m be neglected, a the axal rgdt qute hgh. Neglectng the axal rgdt, the above equaton can be wrtten a α LT = ~ d (33.14) Veron CE IIT, Kharagpur

11 Example 33.1 A emcrcular two hnged arch of contant cro ecton ubjected to a concentrated load a hown n Fg 33.4a. Calculate reacton of the arch and draw bendng moment dagram. Soluton: Takng moment of all force about hnge B lead to, R 4 = = kn ( ) 3 F = R = 1.67 kn ( ) (1) b Veron CE IIT, Kharagpur

12 From Fg. 33.4b, ~ Rnθ = x = R( 1 coθ ) d = R dθ () tanθ c = θ c = 6.18 = π rad Now, the horzontal reacton m be calculated b the followng expreon, = ~ ~ d d (3) Now the bendng moment at an cro ecton of the arch when one of the hnge replaced b a roller upport gven b, Veron CE IIT, Kharagpur

13 = R x = R R(1 coθ ) θ θ c and, = R = R R(1 coθ ) 4( x 8) R(1 coθ ) 4{ R(1 coθ ) 8} Integratng the numerator n equaton (3), θ θ π c (4) ~ d = θ = R = R c R R R 3 3 R (1 coθ )nθ dθ + π / π θ [ R c R(1 coθ ) 4{ R(1 coθ ) 8}] Rnθ Rdθ π ( 1 coθ )nθ dθ + R [ R R(1 coθ )nθ 4{ R(1 coθ )nθ 8nθ}] dθ π /.895 π /.895 π π π [ coθ ] + R [ R R( coθ ) ] [ 4 R( coθ ) ] + [ 4 8( coθ ) ] π /.895 π /.895 π /.895 [ R R] [ 4 (1.4667)] + [ 4 8(1.4667) ] 3 =.533R R + R R = ( ) = (5) The value of denomnator n equaton (3), after ntegraton, ~ d = π = R ( Rnθ ) π 3 Rdθ 1 coθ 3 π dθ = R = (6) ence, the horzontal thrut at the upport, Bendng moment dagram = = 19.9 kn (7) Bendng moment at an cro ecton of the arch gven b, Veron CE IIT, Kharagpur

14 = = R ~ R(1 coθ ) R nθ θ θ c (8) = (1 coθ ) 98.5nθ (1 coθ ) 98.5nθ 4(15(1 coθ ) 8) θ θ π (9) = c Ung equaton (8) and (9), bendng moment at an angle θ can be computed. The bendng moment dagram hown n Fg. 33.4c. Veron CE IIT, Kharagpur

15 Example 33. A two hnged parabolc arch of contant cro ecton ha a pan of 6m and a re of 1m. It ubjected to loadng a hown n Fg.33.5a. Calculate reacton of the arch f the temperature of the arch raed b 4 C. Aume co-effcent 6 of thermal expanon a α = 1 1 / C. Takng A a the orgn, the equaton of two hnged parabolc arch m be wrtten a, = x x (1) The gven problem olved n two tep. In the frt tep calculate the horzontal reacton due to 4 kn load appled at C. In the next tep calculate the horzontal reacton due to re n temperature. Addng both, one get the horzontal reacton at the hnge due to combned external loadng and temperature change. The horzontal reacton due to 4 kn load m be calculated b the followng equaton, Veron CE IIT, Kharagpur

16 1 = ~ d d (a) For temperature loadng, horzontal reacton gven b, α LT = d (b) Where L the pan of the arch. For 4 kn load, 1 6 d = R x dx + [ R x 4 ( x 1) ] dx (3) 1 Pleae note that n the above equaton, the ntegraton are carred out along the x-ax ntead of the curved arch ax. The error ntroduced b th change n the varable n the cae of flat arche neglgble. Ung equaton (1), the above equaton (3) can be eal evaluated. The vertcal reacton A calculated b takng moment of all force about B. ence, 1 R = [ 4 5 ] = kn 6 R = 6.67 kn. Now conder the equaton (3), l 1 1 dx = (33.33) x( x x ) dx b [(33.33) x 4( x 1) ]( x x ) dx 1 = = (4) Veron CE IIT, Kharagpur

17 l 6 1 dx = x 3 3 = 3 x dx (5) ence, the horzontal reacton due to appled mechancal load alone gven b, 1 l dx = = = 3.71 kn l 3 dx (6) The horzontal reacton due to re n temperature calculated b equaton (b), = = Takng E = kn/mm and I =.333m 4 = kn. (7) ence the total horzontal thrut = 1+ = kn. When the arch hape more complcated, the ntegraton d and d are accomplhed numercall. For th purpoe, dvde the arch pan n to n equal dvon. Length of each dvon repreented b (Δ ) (vde Fg.33.5b). At the mdpont of each dvon calculate the ordnate b ung the 1 equaton = x x. The above ntegral are approxmated a, 3 3 n 1 d = ( ) ( Δ) = 1 (8) n 1 d = = 1 ( ) ( Δ) The complete computaton for the above problem for the cae of external loadng hown n the followng table. (9) Veron CE IIT, Kharagpur

18 Table 1. Numercal ntegraton of equaton (8) and (9) Segme orzontal Correpond oment at ( ) ( Δ) ( ) ( Δ) nt dtance x ng that No eaured (m) Pont ( ) from A (m) (knm) ( ) ( Δ) = = = 3.73 kn ( ) ( Δ) (1) Th compare well wth the horzontal reacton computed from the exact ntegraton. Summar Two-hnged arch the tatcall ndetermnate tructure to degree one. Uuall, the horzontal reacton treated a the redundant and evaluated b the Veron CE IIT, Kharagpur

19 method of leat work. Toward th end, the tran energ tored n the twohnged arch durng deformaton gven. The reacton developed due to thermal loadng are dcued. Fnall, a few numercal example are olved to llutrate the procedure. Veron CE IIT, Kharagpur