Analysis and Simulation of Slender Curved Beams

Transcription

1 nalsis and Siulation of Slender Curved eas itkuar Predas Chavan, Hong Zhou Departent of Mechanical Engineering Teas &M Universit-Kingsville Kingsville, Teas, US bstract Curved beas have a wide variet of applications that include switches, claps, suspensions, tools and other devices. However, the eisting forulas of stress and deflection calculations on beas are coonl for straight beas that undergo sall linear deflections. The are not applicable to curved beas. The depth of the cross section of a slender curved bea is sall copared with its radius of curvature. It usuall has large deforation. The load-deflection relationship of a slender curved bea is often nonlinear. It is uch ore challenging to analze the deforation of a nonlinear slender curved bea than a linear straight bea. In this paper, the stress calculation forula is presented for slender curved beas. The nonlinear deforations of slender curved beas are analzed. The deforations and stresses of slender curved beas are siulated. The results of this paper provide a useful roadap for analzing and designing slender curved beas. I in Equation ( is the oent of inertia of the cross 3 bt section, which is I for rectangular cross section. M is the bending oent. is the distance of the stress eleent awa fro the neutral ais. This equation is often referred to as fleure forula. Figure shows an initiall curved bea with rectangular cross section that is under bending. Plane assuption hol for curved beas, i.e., planar cross sections before bending reain planar after bending. The dash-dot line is the neutral ais of the curved bea. The noral stress on the cross section fro bending can be analsed as follows [4-5]. Kewor Curved ea; Slender ea; Large Deforation; Stress nalsis; Siulation. I. INTRODUCTION Curved beas have a wide variet of applications that include switches, claps, suspensions, tools and an others [-]. However, the stress and deflection calculation forulas on beas in an tetbooks of aterial echanics are coonl for straight beas that undergo sall linear deflections [3]. Figure shows a straight unifor bea with rectangular cross section that is under pure bending. The in-plane depth and out-of-plane width of the cantilever bea are t and b, respectivel. The ais is along the centroidal ais of the straight bea. The neutral ais of the straight bea coincides with its centroidal ais. The noral stress on the cross section fro bending is along direction and can be calculated fro the following equation [4-5]. Fig. n initiall straight bea under bending. M ( I Fig. n initiall curved bea under bending. s shown in Figure, eleent is at a distance of fro the neutral ais. The angle fored b and its curvature center O of the curved bea eleent is θ before bending. The angle changes to θ after bending. The eleent on the neutral ais corresponding to is eleent CD. ecause of plane assuption, eleent CD has the sae initial and final angles as eleent. Eleent CD has initial and final radii of curvature of R and R, respectivel, on the neutral ais. The strain of eleent can be derived [6]. ( R ( R ( ( R The arc length of eleent CD is R θ and R θ, respectivel, before and after bending. Since CD is on the neutral ais, it does not change its length during bending. We have R R. Equation ( can be siplified based on this relationship. IJERTV5IS

2 (3 ( R Substituting R R into Equation (3 iel the R R (4 R ( R The bending stress of eleent is noral to the cross section and can be calculated fro. R R E (5 R ( R E in Equation (5 is Young s odulus of the bea aterial. The stress distribution described in Equation (5 that is for a cross section of a curved bea is nonlinear, which akes the neutral and centroidal aes no longer coincide. The force resultant fro the noral stress of pure bending on a cross section of a curved bea is zero, which lea to the R R E( R R d E d d 0 (6 R ( R R ( R Let r R. Here r references the location of eleent fro its curvature center. Substituting r equation into Equation (6 iel the d ( R r R d r d R d 0 r Fro Equation (7, the location of the neutral ais can be solved. R (8 d r is area of the cross section. For a rectangular cross d section, we have b( r r bt, b ln( r r and r ( r r R. Here r and r are the iniu and ln( r r aiu radii fro the curvature center of the rectangular cross section. The bending oent M on the cross section can be derived fro the bending stress. M (7 E( R R d d (9 R ( R Substituting r R and r R into Equation (9 iel the E( R R M rd R d R d R r (0 In Equation (0, rd R. Here R references the location of the centroidal ais fro its curvature center. Substituting Equation (8 and R into Equation (0 lea to the following M equation. E( R R M R R ( R Substituting Equation ( into Equation (5 iel the calculation forula on the bending stress. M ( r R ( r( R R Let e R R. Here e is the distance between the centroidal and neutral aes. Substituting e, r R and r R into Equation ( iel the following stress calculation equation. M (3 e ( R Equations ( and (3 represent the two fors of the stress calculation forula for curved beas [4-5]. The stress distribution on the cross section is hperbolic. Fro Equation (, the change in curvature of the neutral ais before and after bending can be deterined. M (4 R R EeR In order to solve the bending stress b using equation ( or (3, the bending oent (M on a cross section has to be known. For curved beas that are ainl used for loading bearing such as crane hooks [6], M can be calculated based on initiall curved shape or undefored shape since deforation is sall. However, slender curved beas often eperience large deforation. It is inaccurate to use undefored shape to calculate bending oent on a cross section. To have accurate stress and deforation analsis results, defored shape has to be used for bending oent calculation. This akes it challenging to analse slender curved beas. The in-plane depth (t is usuall uch saller than the radius of curvature (R and R of the neutral ais in slender curved beas. In Equation (4, the part in R can be reasonabl neglected [6], which lea to the following strain equation. R R (5 RR R R The bending stress of slender curved beas can then be derived fro Equation (5 as follows. R R E E E (6 RR R R The stress distribution on the cross section of slender curved beas is considered as linear as shown in Equation (6. The force resultant fro the noral stress of pure IJERTV5IS

3 bending on a cross section of a slender curved bea is zero, which lea to the R R E( R R d E d d 0 (7 R R R R Equation (7 shows that the neutral and centroidal aes in slender curved beas coincide. The bending oent M on the cross section of slender curved beas can be derived fro the bending stress. E( R R ( R R M d d (8 R R R R Rearranging Equation (8 iel the R R M (9 The deforation or defored shape of slender curved beas can be analsed based on Equation (9 in which the bending oent (M has to be calculated fro the defored shape. The aterial of the analsed slender curved beas in the paper is considered as hoogeneous and isotropic. lthough slender curved beas undergo large deforations, their strains ( reain sall and are within the range of elastic deforation. Slender curved beas are analsed in the paper as Euler-ernoulli curved beas for which plane hpothesis hol. ecause of the thin depth and bea fleibilit, slender curved beas are considered to be inetensible in the paper during their deforation. lthough it is difficult to solve the large deforation of a slender curved bea fro Equation (9 analticall, an different nuerical approaches have been proposed and published [7-]. When R in Equation (9 approaches infinit, a slender curved bea degenerates to a slender straight bea. There are ore publications on solving large deforations of slender straight beas [3-9]. ong the eisting published papers on slender curved beas, ost are focused on solving large deforations under given loadings that include concentrated, distributed or cobined. In this paper, the analsed slender curved beas are under given input displaceents. The corresponding large deforations and required input forces of analsed slender curved beas are to be solved. The authors of this paper are otived b the challenges facing slender curved beas. The research objective of this paper is to establish an approach and provide a guideline for analsing slender curved beas. The reainder of the paper is organized as follows. The deforation analsis of slender curved beas is presented in section II. The siulations of deforations and stresses on slender curved beas are provided in section III. Section IV is on the shape design of slender curved beas. Conclusions are drawn in section V. II. DEFORMTION NLYSIS OF SLNDER CURVED EMS For the slender curved bea shown in Figure 3, the dashed curve is the undefored shape of the bea that has a constant radius of curvature of R. The solid curve is the defored shape of the bea that has radius of curvature of R that a change along the curve. The left end of the curved bea is fied. We assue the displaceents of the right free end are given together with the paraeters of t and b of the rectangular cross section of the curved bea and Young s odus (E of the bea aterial. The required input forces of F and F at the right end are to be deterined in order to generate the desired displaceents. Let the coordinates of an arbitrar point on the solid curve be (,. The arc length at the point is s. s eets the condition of 0 s L. L here is the total arc length of the curved bea that is considered as inetensible during its deforation. s 0 at the fied end while s L at the free end. The bending oent at an arbitrar point on the solid curve can be derived. M( s F ( F ( (0 Fig. 3 n analzed slender curved bea. Let (s be the angle that is fro the positive direction to the tangent line at a point on the solid curve. Then we have the d( s R ( s ( Substituting Equations (0 and ( into Equation (9 lea to the d( s R F ( F ( ( Differentiating both sides of Equation ( with respect to s lea to the d F d F d For the solid curve, we have (3 d cos and d sin. Substituting the epressions of d and d into Equation (3 and oving the ters on the right hand side to the left iel the d F cos sin 0 F (4 Fro Equation (4, we have IJERTV5IS

4 d d F sin F cos 0 Equation (5 results in the d F sin F cos C (5 (6 sin d d (34 F (sin sin F (cos cos Integrating Equation (34 fro the left end to the right end of the solid curve iel the sin d (35 0 F (sin sin F (cos cos C in Equation (6 is an arbitrar constant. It can be decided b one boundar condition of the defored curved bea. t the right end of the solid curve, we have s L, d M ( L 0, and 0. ssue the tangent angle of the sl solid curve at its right end is. Substituting the epressions of d and at the right end into Equation (6 iel the F sin F cos C (7 Substituting Equation (7 into Equation (6, we have d F (sin sin F (cos cos (8 Taking square root on both sides of Equation (8 iel the d F (sin sin F (cos cos Rearranging Equation (9 iel the (9 d (30 F (sin sin F (cos cos, F, and F are the three unknowns here. The can be solved nuericall b Equations (3, (33 and (35. III. DEFORMTION ND STRESS SIMULTIONS The deforation and stress of a slender curved bea can be directl analzed and siulated b finite eleent analsis (FE software NSYS [0-]. With the specified input displaceents, the required input forces can also be directl obtained fro NSYS through NSYS siulation. Figure 4 shows a slender curved bea with the shape of half a circle. The aterial of the bea is structural steel with Young s odules (E of 000 GPa, Poisson s ratio (ν of 0.3, ield strength (σ of 350 MPa. The diaeter of the bea is 50. The depth (t and width (b of the rectangular cross section of the bea are 0.5 and 5, respectivel. The left end of the bea is fied at the origin O of the coordinate sste. The right free end of the bea is on the ais when the curved bea is undefored. When is displaced to,, 3 and 4, the defored bea shape, the stress distribution and the input forces at are to be deterined b siulation in NSYS. s shown in Figure 4, 3 4 fors a square with its center at the un-displaced point and its length of 60. Integrating Equation (30 fro the left end to the right end of the solid curve iel the L 0 F (sin sin F (cos cos d (3 Substituting Equation (30 into d cos iel the cos d d (3 F (sin sin F (cos cos Integrating Equation (3 fro the left end to the right end of the solid curve iel the cos d (33 0 F (sin sin F (cos cos Substituting Equation (30 into d sin iel the Fig. 4 slender curved bea with half a circle. To analse the curved bea, the solid odel of the curved bea is first created in the Design Modeler [3] of NSYS. NSYS Design Modeler is an NSYS Workbench application that provides odeling tool. The solid odel created in NSYS Design Modeler is then eshed and analsed in NSYS Mechanical [4] that is also an application of NSYS Workbench. Figure 5 shows the eshing odel of the analsed curved bea. When is displaced to, the defored shape of the bea is shown in Figure 6. ecause of the thin depth of the curved bea, the defored shape line is ver light in the IJERTV5IS

5 figure. The colour ap and nubers in the figure represent the directional deforation along the direction. The aiu stress in the defored bea is 4.6 MPa that is shown in Figure 7. To generate the displaceents at, the required input forces (F and F are 0.46 N and N, respectivel. These input forces are the reaction forces in NSYS. When is displaced to, the defored shape of the bea is shown in Figure 8. The aiu stress in the defored bea is MPa that is shown in Figure 9. The input forces (F and F are N and N, respectivel. The aiu stress in the defored bea at is ore than doubled than that at. When is displaced to 3, the defored shape of the bea is shown in Figure 0. The aiu stress in the defored bea is MPa that is shown in Figure. The input forces (F and F are N and N, respectivel. When is displaced to 4, the defored shape of the bea is shown in Figure. The aiu stress in the defored bea is MPa that is shown in Figure 3. The input forces (F and F are N and N, respectivel. The aiu stress in the defored bea at 4 is the highest aong all four defored shapes of the curved bea, which is beond that of the ield strength of the bea aterial. Fig. 8 The deforation of the curved bea at. Fig. 5 The eshing odel of the curved bea. Fig. 9 The stress of the curved bea at. Fig. 6 The deforation of the curved bea at. Fig. 0 The deforation of the curved bea at 3. Fig. 7 The stress of the curved bea at. Fig. The stress of the curved bea at 3. IJERTV5IS

6 Fig. The deforation of the curved bea at 4. Fig. 4 The slender curved bea with increased length. Fig. 3 The stress of the curved bea at 4. IV. SHPE DESIGN OF SLENDER CURVED EMS s shown in Figure 3, the aiu stress within the defored slender curved bea at 4 is above the ield strength of the bea aterial. This aiu stress can be reduced b either increasing the arc length or decreasing the cross section depth of the slender curved bea. oth was belong to shape design of a slender curved bea that is to iprove its perforance and better eet its nee and requireents b changing its geoetric paraeters. The central angle of the slender circular bea analzed last section is 80. The angle is increased to 70 in this section while other paraeters of the bea reain unchanged. Figure 4 shows the slender curved bea with the increased arc length. The eshing odel of the increased curved bea is shown in Figure 5. When of the increased curved bea is displaced to, the defored shape of the bea is shown in Figure 6. The aiu stress in the defored bea is MPa that is shown in Figure 7, which is saller than that of 4.6 MPa for the half a circle case shown in Figure 7. To generate the displaceents at, the required input forces (F and F are 0.03 N and N, respectivel. When of the increased curved bea is displaced to, the defored shape of the bea is shown in Figure 8. The aiu stress in the defored bea is MPa that is shown in Figure 9, which is uch lower than that of MPa for the half a circle case in Figure 9. The input forces (F and F are N and N, respectivel. Fig. 5 The eshing odel of the increased curved bea. Fig. 6 The deforation of the increased curved bea at. When of the increased curved bea is displaced to 3, the defored shape of the bea is shown in Figure 0. The aiu stress in the defored bea is MPa that is shown in Figure, which is above that of MPa for the half a circle case in Figure 0. The input forces (F and F are N and N, respectivel. IJERTV5IS

7 Fig. 7 The stress of the increased curved bea at. Fig. 0 The deforation of the increased curved bea at 3. Fig. 8 The deforation of the increased curved bea at. Fig. The stress of the increased curved bea at 3. Fig. 9 The stress of the increased curved bea at. When of the increased curved bea is displaced to 4, the defored shape of the bea is shown in Figure. The aiu stress in the defored bea is MPa that is shown in Figure 3, which is far below that of MPa for the half a circle case in Figure 3. The input forces (F and F are N and N, respectivel. ong all four displaceents, the aiu stress within the defored curved bea with increased arc length is MPa, which is well below the ield strength of 350 MPa for the bea aterial. Fig. The deforation of the increased curved bea at 4. Fig. 3 The stress of the increased curved bea at 4. IJERTV5IS

8 V. CONCLUSIONS When slender curved beas have uch saller depth of cross section than radius of curvature, their neutral and centroidal aes can be reasonabl considered to coincide for stress and deforation analses. Slender curved beas usuall undergo large deforations and have nonlinear loaddeflection relationships. ecause of the large deforation, the bending oent on a cross section has to be calculated fro the defored shape in order to have a decent accurac. Large deforation ight cause high bending stress within a defored slender curved bea. The high bending stress can be significantl reduced b either increasing the bea length or decreasing the cross section depth. The stress and deforation analses presented in the paper provide guidelines for analsing and designing slender curved beas. CKNOWLEDGMENT The authors of this paper gratefull acknowledge the research instruent support of the US National Science Foundation under Grant No n opinions, findings, recoendations or conclusions epressed in this paper are those of the authors and do not necessaril reflect the views of the US National Science Foundation. REFERENCES []. Shinohara, M. Hara, Large Deflection of a Circular C-Shaped Spring International Journal of Mechanical Sciences, 979, (3: [] C.Y. Wang, L.T. Wanson, On the Large Deforations of C-Shaped Springs International Journal of Mechanical Sciences, 980, (7: [3] J.M. Gere,.J. Goodno, Mechanics of Materials, Eighth Edition Toebben Drive, Independence, KY: Cengage Learning, 0. [4] R.C. Hibbeler, Mechanics of Materials, Tenth Edition. Upper Saddle River, New Jerse: Pearson, 06. [5] F.P. eer, E.R. Johnston Jr., J.T. DeWolf, D.F. Mazurek, Mechanics of Materials, Seventh Edition. Penn Plaza, New York, NY: McGraw- Hill Education, 04. [6] P.P. enha, R.J. Crowford, C.G. rstrong, Mechanics of Engineering Materials, Second Edition. urnt Mill, Harlow, England: Longan Group Liited, 996. [7] D.G. Fertis, Nonlinear Structural Engineering. Springer-Verlag erlin Heidelberg, 006. [8] H.D. Conwa, N.Y. Ithaca, The Nonlinear ending of Thin Circular Ro Journal of pplied Mechanical, 956, 3: 7-0. [9].E. Seaes, H.D. Conwa, N.Y. Ithaca, Nuerical Procedure for Calculating the Large Deflections of Straight and Curved eas Journal of pplied Mechanical, 957, 4: [0] F.D. ona, S. Zelenika, generalized elastica-tpe approach to the analsis of large displaceents of spring-strips, Journal of Mechanical Engineering Science, 997, (7: [] T. Dahlberg, Procedure to Calculate Deflections of Curved eas, International Journal of Engineering Education, 004, 0(3: [] S. Ghuku, K.N. Saha, Theoretical and Eperiental Stud on Geoetric Nonlinearit of Initiall Curved Cantilever eas, Engineering Science and Technolog, 06, 9(: [3] K.E. isshopp, D.C. Drucker, Large Deflection of Cantilever eas, Quarterl of pplied Matheatics, 945, 3(3: [4] J.H. Lau, Large deflection of cantilever bea, SCE Journal of Engineering Mechanics, 98, 07(: [5].N. Rao,.G. Rao, Large Deflection of a Unifor Cantilever ea with End Rotational Load, Forschung i Ingenieurwesen, 988, 54(: 4-6. [6] K.M. Hsiao, F.Y. Hou, Nonlinear Finite Eleent nalsis of Elastic Fraes, Coputers & Structures, 988, 6(4: [7] D.G. Fertis, Vibration of eas and Fraes b Using Dnaicall Equivalent Sstes, Structural Engineering Review, 995, 7(: 3-7. [8] L. Chen, n Integral pproach for Large Deflection Cantilever eas, International Journal of Non-Linear Mechanics, 00, 45(3: [9] G.K. Suresh, H. Zhou, Shape Design of Cantilever Springs, International Journal of Engineering Research & Technolog, 06, 5(8: [0] E.H. Dill, The Finite Eleent Method for Mechanics of Soli with NSYS pplications roken Sound Parkwa, NY: CRC Press, 0. [] S. Moaveni, Finite Eleent nalsis Theor and pplication with NSYS, Fourth Edition. Upper Saddle River, NJ: Pearson, 05. [] H.H. Lee, Finite Eleent Siulations with NSYS Workbench Martwa Drive, Mission, KS: SDC Publications, 05. [3] NSYS, Design Modeler User's Guide, Canonsburg, P: NSYS, 05 [4] NSYS, NSYS Mechanical User's Guide, Canonsburg, P: NSYS, 05. IJERTV5IS080