v A Since the axial rigidity k ij is defined by P/v A, we obtain Pa 3

Transcription

1 The The rd rd Inernaional Conference on on Design Engineering and Science, ICDES 14 Pilsen, Czech Pilsen, Republic, Czech Augus Republic, 1 Sepember 1-, 14 In-plane and Ou-of-plane Deflecion of J-shaped Beam Tadashi HORIBE* 1 and Koaro MORI* *1 Deparmen of Mechanical Engineering, Ibaraki Universiy Nakanarusawa-cho, Hiachi-shi, Ibaraki, , JAPAN horibe@mx.ibaraki.ac.jp * Deparmen of Mechanical Engineering, Ibaraki Universiy Nakanarusawa-cho, Hiachi-shi, Ibaraki, , JAPAN mori-k@mx.ibaraki.ac.jp Absrac In his paper, an analyical soluion for a J-shaped beam deflecion is given when he beam, which is clamped a one end and free a he oher, is subjeced o a poin load (boh in-plane and ou-of-plane load). We assume ha he J-shaped beam is made up of wo pars, i.e., a sraigh beam and a ellipically curved beam and ha he beam is clamped a a sraigh end. The soluion which is based on Casigliano s heorem is deduced by using numerical inegraion of he modified ellipical inegral and he differeniaion of he beam s srain energy. The soluions are shown in exac expressions and he effecs of he curvaure on he J-shaped beam deflecion are clarified. Keywords: beam heory, ellipically curved beam, deflecion, Casigliano's heorem, numerical analysis 1 Inroducion Curved beams are widely used as pars of many kinds of machines and ools. Figure 1 is an example of a curved beam, which is generally called a J-bol or hook bol. The curvaure of he J-bol can be represened by he ellipical curve. For srucural design, sresses and deflecions of he curved beams are summarized in a exbook, e.g. [1]. For he analyical convenience, many researchers have been assumed ha he curved beams have a circular curvaure, e.g. []. The deformaion of curved elasic beams including he effec of large displacemen is also discussed by A. Shinohara e al. [] and C. Gonzalez e al. [4], assuming he curvaure as a circular arc. Only Dahlberg analyzed he ellipically curved beams [5] and considered boh saically deerminae and saically indeerminae problems for he beams. In spie of he pracical use of ellipically curved beams as pars of srucures, very lile daa on he beam wih ellipical curvaure is available. The purpose of he presen paper is o obain exac soluions for boh in-plane and ou-of-plane deflecions in a J-shaped beam due o a concenraed load based on he Casigliano s heorem and o demonsrae he accuracy and usefulness of he presen approach. The soluions obained from hese formulae are compared wih he alernaive soluions available, including he resuls of a hree dimensional finie elemen mehod (- D FEM). Fig. 1 Example of J-shaped beam, or hook-bol Mehod of Soluion.1 In-plane bending of J-shaped beam The J-shaped beam, made up of wo pars, a half ellipse curve and sraigh line, is invesigaed in his paper as shown in Fig.. The half-axes of he ellipse are denoed a and b. The beam is clamped a one end and loaded wih a force P a he free end. The force P acs parallel o he plane of he ellipse. The bending siffness of he curved beam is. The equaion of he ellipse is x y 1. (1) a b And we inroduce he following θ as a parameer. x acos, y bsin. () Therefore infiniesimal arc lengh ds on he ellipse is ds dx dy ( dx / d ) ( dy / d ) d a sin b cos d a 1e cos d, () where, e 1 ( b/ a) is he eccenriciy []. Here, he aspec raio = β has he relaionship wih e by b/ a 1 e. Copyrigh 14, The Organizing Commiee of he ICDES 14 5

2 Since he axial rigidiy k ij is defined by P/v A, we obain k ij P v A a a 4l E. (9). Ou-of-plane bending of J-shaped beam In his secion ou-of-plane deformaion of J-shaped beam will be invesigaed. The beam is clamped a one end and loaded wih a force P a he free end, see Fig.. The force P acs perpendicular o he plane of he ellipse. The bending siffness of he curved beam is and he orsional rigidiy. Differeniaion of Eq. (1) gives: dy b x x. (1) dx a y y Fig. Configuraion of J-shaped beam The srain energy of he beam as shown in Fig. is wrien as [7], [8] Pa l 1 U Pa 1 cos a 1 e cos d Pal Pa 1 cos 1e cos d, (4) where, l is he lengh of sraigh porion of he beam. The conribuion of he axial force and he shear force o he srain energy U has been negleced and we assume ha cross-secion of he beam is consan. Defining he inegral erm in Eq. (4) as and applying he Casigliano's heorem, we obain he deflecion v a load P as A 1 cos 1 e cos d 1 cos 1 1 cos d E1 E1, E, (5) / Pa a va 4 l E. () Inroducing he deflecion 8Pa /(), which is he maximum deflecion of he canilever beam of span a, we have a normalized deflecion of Eq. () as va l v A E( ). (7) (8 Pa /( )) a 1 The inegral erm E (β) is funcion of he parameer β only and can be evaluaed by he numerical inegraion. The case β=1 or β= gives ha he ellipse becomes a circle or a sraigh line, respecively. In hose cases, he definie inegral E (β) can be analyically inegraed as E 1 1 1, E.559. (8) 9 Fig. Ou-of-plane loading of J-shaped beam The infiniesimal lengh of he beam ds is also given: dy ds dx dy dx 1. (11) dx 54

3 Inroducing he angle as shown in Fig., one obains dx 1 sin, ds dy 1 dx dy dy / dx cos. (1) ds dy 1 dx Nex, we sudy he bending momen on he beam cross-secion siuaed a angle. The bending momen M b and he orsional momen M are acing a his cross-secion. Assuming is influence on he beam deflecion is negligible, he shear force has been omied. A he cross-secion a angle, he momen equilibriums hold as follows: M b Py cos P( a x)sin, M Py sin P( a x)cos. (1) The elasic srain energy sored in he beam can now be calculaed. We have where L is he oal arc lengh of he beam. Following he Casigliano's heorem, he deflecion w A of he beam end a he loaded poin can be calculaed. Then we obain, Pl 4Pa l wa P L y cos a x sin ds L P y sin a x cos ds. (15) Pa l 1 l U P 1 y dy 1 1 M ds, (14) L L M bds Subsiuion Eq. (1) ino Eq. (15) yields Pl 4Pa l Pa 4Pa w + + I ( ) + I ( ) A 1 Pa l l I1 + +I +4 + a a ( ) ( ), (1) where I 1(β), I (β) are he inegral erms in Eq. (15) and are rewrien I I 1/ / / d, d. 1/ Afer insering he values of β, which deermines he configuraion of he ellipse, he values of he definie inegrals in Eq. (17) are obained hrough he numerical (17) inegraion. For he limiing case β= (he arc reduce o he sraigh line) and β=1 (circular curvaure), we have I I 8.7, (18) 1 1 Dividing Eq. (1) by (8Pa )/(), one obains he following non-dimensional deflecion. wa w A 8 Pa / l l I1( ) I 4. (19) a a Axial rigidiy of he beam is given as k oj Numerical Calculaion Based on he above derivaions, he deflecions for boh he in-plane and he ou-of-plane behavior of he J-shaped beam are calculaed by a closed form soluion. The deflecions are evaluaed as a funcion of aspec raio or a funcion of beam lengh l/a and numerical resuls are compared wih he D-FEM soluions. Definie inegral E () Deflecion v A /(8Pa ) /() 1. () a 1 l l I1 I 4 a a Fig. 4 Definie inegral E(β) 1 l/a = Fig. 5 In-plane non-dimensional deflecion as funcion of 55

4 .1 In-plane bending of J-shaped beam Firs, we evaluae he inegral E (), expressed by Eq. (5). Figure 4 shows he definie inegral values as funcion of. When =1, one obains E (1)=1. This agrees wih wha can be found in some handbooks. The non-dimensional displacemen curves provided by Eq. (7) for J-shaped beams are ploed in Fig. 5 and Fig.. I can been seen ha he deflecion is proporional o he parameer and have maximum values when =1, i.e., circular arc. 1.. (1) Deflecion v A /(8Pa ) /() =1 1 4 l/a Fig. In-plane non-dimensional deflecion as funcion of l/a In order o validae he proposed mehod, he hree dimensional finie elemen resuls are compared wih hose obained by he presen mehod under he following dimensions and physical properies: Young's Modulus: E=GPa, he Poisson's raio: ν=., In-plane load: P=1N, Beam lengh of sraigh porion: l=75, 15mm, Major axis of he ellipse: a=5mm (l/a=., ), Beam diameer: d=1mm, Minor axis of he ellipse:b=5, 1.5,.mm. The J-beam is divided ino abou 1, quadraic erahedra elemens as shown in Fig. 7. The resuls obained by he wo mehods are compared in Table 1 and Fig. 8. Good agreemen beween he wo mehods can be observed. The difference beween he wo heory is mainly caused by he shear force acing on he cross-secion.. Ou-of-plane bending of J-shaped beam Nex, we consider ou-of-plane deflecions of J-shaped beam. Figure 9 shows he definie inegral values of I 1(), I (), expressed by Eq. (17). The inegrals I 1() and I () are relaed o he bending and he orsional deformaion, respecively. Hence, i is found ha he orsion conribues o he deflecion when 1. For a beam wih a circular cross-secion, we have, using ν=.: Fig. 7 D-FEM mesh examples of J-shaped beam (Quadraic erahedral elemen is employed) Table 1 Comparison of in-plane deflecion a loading poin l/a Presen [mm] D-FEM [mm] Deflecion v A /(8Pa ) /() Presen D FEM l/a = l/a = Fig.8 Comparison beween presen heory and D-FEM resuls in in-plane deflecion of J-shaped beam The normalized ou-of-plane displacemen provided by Eq.(19) for J-shaped beams are ploed in Fig. 1 and Fig. 11. When l/a= and =, he J-shaped beam coincides wih he sraigh one of he lengh a. I can be seen ha he deflecions increase as he parameers and l/a increase. Comparison of D-FEM soluions wih presen soluions are abulaed in Table and ploed in Fig. 1. The J-shaped beam has he same geomerical and physical properies as hose given in previous secion oher han ou-of-plane load P=1N. In spie of concise 5

5 calculaion of he presen approach, good agreemens are again observed beween he presen resuls and he D-FEM ones. Defnie inegral I 1 (), I () 4 I Fig. 9 Definie inegrals I1() and I() v A /(8Pa ) /() 1 9 / =1. l/a = Fig. 1 Ou-of-plane non-dimensional deflecion of he J-shaped beam as funcion of v A / (8Pa ) /() =1. 1 l/a I / =1. Fig. 11 Ou-of-plane non-dimensional deflecion of he J-shaped beam as funcion of l/a Table Comparison of ou-of-plane deflecion a loading poin l/a Presen [mm] D-FEM [mm] v A / ( 8Pa )/() D FEM Presen l/a = Fig. 1 Comparison of presen heory and D-FEM resuls in ou-of-plane deflecion 7 Conclusions In his paper, analyical expressions are obained for boh in-plane and ou-of-plane displacemen of a J-shaped beam subjeced o a concenraed load a free end. In he analysis, he Casigliano s heorem is employed and he deflecions of he beam are given including definie inegrals and are evaluaed hrough a numerical inegraion algorihm. Comparing wih he hree dimensional finie elemen mehod, i is demonsraed ha he presen mehod is useful and accurae, while requiring only a limied calculaion. If in-plane and ou-of-plane load ac on a J-shaped beam simulaneously, we can esimae he deflecions or sresses of he beam by superposing each resuls. References [1] W. C. Young, and R. G. Budynas, Roark s Formulas for Sress and Srain (7h ed.), (), McGraw-Hill. [] A Shinohara and M. Hara, Large deflecion of a circular C-shaped spring, In. J. Mech. Sci.. Vol.1, (1978), pp [] C. Gonzalez and J. LLorca, Siffness of a curved beam subjeced o axial load and large displacemens, In. J. Sol. and Sruc., Vol.4, (5), pp [4] K, Kang and J. Han, Analysis of a curved beam using classical and shear deformable beam heories, KSME In. J., Vol. 1, No., (1998), pp [5] T. Dahlberg, Procedure o calculae deflecions of curved beams, In. J. Engng. Ed.,Vol., No. 57

6 (4), pp [] M. Toda, Inroducion of ellipic funcion (in Japanese) (), Nippon-hyouron-sha. [7] S. Timoshenko and D. H. Young, Elemens of srengh of maerials (5h ed.), (1974), van Nosrand Co. [8] A. Ausin and J. H. Swannell, Sresses in a pipe bend of oval cross-secion and varying wall hickness loaded by inernal pressure, In. J. Pres. Ves. and Piping, Vol.7 (1979), pp Received on Sepember, 1 Acceped on January 1, 14 58