Analysis and Simulation of Slender Curved Beams

Analysis and Simulation of Slender Curved Beams

Amitkumar Premdas Chavan, Hong Zhou Department of Mechanical Engineering Texas A&M University-Kingsville Kingsville, Texas, USA

AbstractCurved beams have a wide variety of applications that include switches, clamps, suspensions, tools and other

I in Equation (1) is the moment of inertia of the cross

bt 3

devices. However, the existing formulas of stress and deflection calculations on beams are commonly for straight beams that

section, which is

I for rectangular cross section. M is

12

undergo small linear deflections. They are not applicable to curved beams. The depth of the cross section of a slender curved beam is small compared with its radius of curvature. It usually has large deformation. The load-deflection relationship of a slender curved beam is often nonlinear. It is much more challenging to analyze the deformation of a nonlinear slender curved beam than a linear straight beam. In this paper, the stress calculation formula is presented for slender curved beams. The nonlinear deformations of slender curved beams are analyzed. The deformations and stresses of slender curved beams are simulated. The results of this paper provide a useful roadmap for analyzing and designing slender curved beams.

Keywords Curved Beam; Slender Beam; Large Deformation; Stress Analysis; Simulation.

1. INTRODUCTION

   Curved beams have a wide variety of applications that include switches, clamps, suspensions, tools and many others [1-2]. However, the stress and deflection calculation formulas on beams in many textbooks of material mechanics are commonly for straight beams that undergo small linear deflections [3].

   Figure 1 shows a straight uniform beam with rectangular cross section that is under pure bending. The in-plane depth and out-of-plane width of the cantilever beam are t and b, respectively. The x axis is along the centroidal axis of the straight beam. The neutral axis of the straight beam coincides with its centroidal axis. The normal stress on the cross section from bending is along x direction and can be calculated from the following equation [4-5].

   the bending moment. y is the distance of the stress element

   away from the neutral axis. This equation is often referred to as flexure formula.

   Figure 2 shows an initially curved beam with rectangular cross section that is under bending. Plane assumption holds for curved beams, i.e., planar cross sections before bending remain planar after bending. The dash-dot line is the neutral axis of the curved beam. The normal stress on the cross section from bending can be analysed as follows [4-5].

   Fig. 2 An initially curved beam under bending.

   As shown in Figure 2, element AB is at a distance of y from the neutral axis. The angle formed by AB and its curvature center O of the curved beam element is 1 before bending. The angle changes to 2 after bending. The element on the neutral axis corresponding to AB is element CD. Because of plane assumption, element CD has the same initial and final angles as element AB. Element CD has initial and final radii of curvature of R1 and R2, respectively, on the neutral axis.

   The strain of element AB can be derived [6].

   AB

   AB

   (R2 y)2 (R1 y)1

   (2)

   (R1 y)1

   Fig. 1 An initially straight beam under bending.

   M y

   The arc length of element CD is R11 and R22, respectively, before and after bending. Since CD is on the neutral axis, it does not change its length during bending. We

   x

   (1)

   have

   R11 R22 . Equation (2) can be simplified based on

   I this relationship.

   AB

   y(2 1)

   (R y)

   (3)

   In Equation (10),

   rdA AR1 . Here

   A

   R1 references the

   1 1 location of the centroidal axis from its curvature center.

   Substituting

   2 R11 R2

   into Equation (3) yields the

   Substituting Equation (8) and

   R1 into Equation (10) leads to

   following equation.

   the following M equation.

   y(R1 R2 )

   (4)

   M EA(R1 R2 ) R

   • R

     (11)

     AB R (R

     • y)

       R 1 1

       2 1 2

       The bending stress of element AB is normal to the cross section and can be calculated from AB .

       Substituting Equation (11) into Equation (5) yields the calculation formula on the bending stress.

       E y(R1 R2 )

       R2 (R1 y)

       (5)

       M (r R1)

       Ar(R1 R1)

       (12)

       E in Equation (5) is Youngs modulus of the beam material. The stress distribution described in Equation (5) that is for a cross section of a curved beam is nonlinear, which makes the neutral and centroidal axes no longer coincide.

       The force resultant from the normal stress of pure bending

       Let e R1 R1 . Here e is the distance between the centroidal and neutral axes. Substituting e, r R1 y and y r R1 into Equation (12) yields the following stress calculation equation.

       on a cross section of a curved beam is zero, which leads to the following equation.

       M y

       Ae (R1 y)

       (13)

       dA E y(R1 R2 )dA E(R1 R2 ) y dA 0 (6)

       A A R2 (R1 y) R2 A (R1 y)

       Let r R1 y . Here r references the location of element AB from its curvature center. Substituting r equation into Equation (6) yields the following equation.

       Equations (12) and (13) represent the two forms of the stress calculation formula for curved beams [4-5]. The stress distribution on the cross section is hyperbolic.

       From Equation (11), the change in curvature of the neutral axis before and after bending can be determined.

       y dA r R1 dA dA R dA 0

       1

       1

       (7)

       1 1 M

       (14)

       A (R1

     • y)

       A r A A r

       R2 R1

       EAeR1

       From Equation (7), the location of the neutral axis can be solved.

       In order to solve the bending stress by using equation (12) or (13), the bending moment (M) on a cross section has to be known. For curved beams that are mainly used for loading

       R1

       A

       dA

       A r

       (8)

       bearing such as crane hooks [6], M can be calculated based on initially curved shape or undeformed shape since deformation is small. However, slender curved beams often experience large deformation. It is inaccurate to use undeformed shape to

       r

       r

       2 1 2 1

       2 1 2 1

       A is area of the cross section. For a rectangular cross section, we have A b(r r ) bt , dA b ln(r r ) and

       A

       1

       1

       R (r2 r1) . Here r1 and r2 are the minimum and

       ln(r2 r1)

       maximum radii from the curvature center of the rectangular cross section.

       The bending moment M on the cross section can be

       calculate bending moment on a cross section. To have accurate stress and deformation analysis results, deformed shape has to be used for bending moment calculation. This makes it challenging to analyse slender curved beams.

       The in-plane depth (t) is usually much smaller than the radius of curvature (R1 and R2) of the neutral axis in slender curved beams. In Equation (4), the y part in R1 y can be reasonably neglected [6], which leads to the following strain equation.

       derived from the bending stress.

       y (R R ) 1 1

       E(R R ) y2

       1 2 y

       (15)

       M ydA 1 2 dA

       (9)

       R2 R1

       R2

       R1

       A R2

       A (R1 y)

       The bending stress of slender curved beams can then be

       Sustituting

       r R1 y

       and

       y r R1 into Equation (9)

       derived from Equation (15) as follows.

       yields the following equation.

       y (R

   • R )

   1 1

   E E 1 2 E y

   (16)

   1 1

   1 1

   M E(R1 R2 ) rdA 2R dA R 2 dA

   (10)

   R2 R1

   R2

   R1

   R2 A A

   A r

   The stress distribution on the cross section of slender curved beams is considered as linear as shown in Equation (16). The force resultant from the normal stress of pure

   bending on a cross section of a slender curved beam is zero, which leads to the following equation.

   fixed. We assume the displacements of the right free end are given together with the parameters of t and b of the

   y(R R ) E(R R )

   rectangular cross section of the curved beam and Youngs

   dA E 1 2 dA 1 2 ydA 0

   (17)

   modus (E) of the beam material. The required input forces of

   A A R2 R1

   R2 R1 A

   Fx and Fy at the right end are to be determined in order to generate the desired displacements.

   Equation (17) shows that the neutral and centroidal axes in

   slender curved beams coincide. The bending moment M on the cross section of slender curved beams can be derived from the bending stress.

   Let the coordinates of an arbitrary point on the solid curve be (x, y). The arc length at the point is s. s meets the condition of 0 s L . L here is the total arc length of the curved beam that is considered as inextensible during its deformation.

   M ydA E(R1 R2 ) y2dA EI (R1 R2 ) (18)

   s 0 at the fixed end while s L at the free end.

   A R2 R1 A

   R2 R1

   The bending moment at an arbitrary point on the solid

   Rearranging Equation (18) yields the following equation.

   curve can be derived.

   1 1 M

   (19)

   M (s) Fy (xB x) Fx ( yB y)

   (20)

   R2 R1 EI

   The deformation or deformed shape of slender curved beams can be analysed based on Equation (19) in which the bending moment (M) has to be calculated from the deformed shape.

   The material of the analysed slender curved beams in the paper is considered as homogeneous and isotropic. Although slender curved beams undergo large deformations, their strains ( ) remain small and are within the range of elastic deformation. Slender curved beams are analysed in the paper as Euler-Bernoulli curved beams for which plane hypothesis holds. Because of the thin depth and beam flexibility, slender curved beams are considered to be inextensible in the paper during their deformation.

   Although it is difficult to solve the large deformation of a slender curved beam from Equation (19) analytically, many different numerical approaches have been proposed and published [7-12]. When R1 in Equation (19) approaches

   Fig. 3 An analyzed slender curved beam.

   Let (s) be the angle that is from the positive x direction

   to the tangent line at a point on the solid curve. Then we have the following equation.

   infinity, a slender curved beam degenerates to a slender straight beam. There are more publications on solving large deformations of slender straight beams [13-19].

   d(s)

   ds

   1

   R2 (s)

   (21)

   Among the existing published papers on slender curved beams, most are focused on solving large deformations under

   Substituting Equations (20) and (21) into Equation (19) leads to the following equation.

   given loadings that include concentrated, distributed or combined. In this paper, the analysed slender curved beams are under given input displacements. The corresponding large

   EI d(s)

   ds

   • EI R1

     Fy (xB x) Fx ( yB y)

     (22)

     deformations and required input forces of analysed slender curved beams are to be solved. The authors of this paper are motived by the challenges facing slender curved beams. The

     Differentiating both sides of Equation (22) with respect to

     s leads to the following equation.

     research objective of this paper is to establish an approach and

     d 2

     dx dy

     provide a guideline for analysing slender curved beams.

     The remainder of the paper is organized as follows. The

     EI

     ds2

     Fy ds Fx ds

     (23)

     deformation analysis of slender curved beams is presented in

     For the solid curve, we have

     dx ds cos

     and

     section II. The simulations of deformations and stresses on slender curved beams are provided in section III. Section IV is on the shape design of slender curved beams. Conclusions are drawn in section V.

2. DEFORMATION ANALYSIS OF SLNDER CURVED BEAMS

   dyds sin . Substituting the expressions of ds and dy ds into Equation (23) and moving the terms on the right hand side to the left yields the following equation.

   d 2

   For the slender curved beam shown in Figure 3, the dashed

   EI

   ds2

   • Fy cos Fx sin 0

     (24)

     curve is the undeformed shape of the beam that has a constant radius of curvature of R1. The solid curve is the deformed shape of the beam that has radius of curvature of R2 that may change along the curve. The left end of the curved beam is

     From Equation (24), we have

     d 1 d 2

     dy

     EI sin d

     EI

     ds 2 ds

     • Fy sin Fx cos 0

       (25)

       2 F (sin sin) F (cos cos)

       (34)

       y m x m

       Equation (25) results in the following equation.

       Integrating Equation (34) from the left end to the right end of the solid curve yields the following equation.

       1 d 2

       EI

       2 ds

     • Fy sin Fx cos C

   (26)

   EI m sin d

   yB

   2

   2

   0 Fy (sinm

   • sin) Fx (cos m

     (35)

   • cos)

     C in Equation (26) is an arbitrary constant. It can be decided by one boundary condition of the deformed curved beam. At the right end of the solid curve, we have s L ,

     m , Fy, and Fx are the three unknowns here. They can be solved numerically by Equations (31), (33) and (35).

     M (L) 0 , and

     d

     ds s L

     0 . Assume the tangent angle of the

3. DEFORMATION AND STRESS SIMULATIONS

   solid curve at its right end is m . Substituting the expressions of d ds and at the right end into Equation (26) yields the

   The deformation and stress of a slender curved beam can be directly analyzed and simulated by finite element analysis

   following equation.

   Fy sinm Fx cosm C

   Substituting Equation (27) into Equation (26), we have

   (27)

   (FEA) software ANSYS [20-22]. With the specified input

   displacements, the required input forces can also be directly obtained from ANSYS through ANSYS simulation.

   Figure 4 shows a slender curved beam with the shape of half a circle. The material of the beam is structural steel with

   d 2

   2

   Youngs modules (E) of 2000 GPa, Poissons ratio () of 0.3,

   Fy (sinm sin) Fx (cosm cos) (28)

   ds EI

   Taking square root on both sides of Equation (28) yields the following equation.

   yield strength (y) of 350 MPa. The diameter of the beam is 150 mm. The depth (t) and width (b) of the rectangular cross section of the beam are 0.25 mm and 15 mm, respectively. The left end of the beam is fixed at the origin O of the coordinate system. The right free end A of the beam is on the x axis when the curved beam is undeformed. When A is

   d 2

   ds EI

   Fy (sinm

   • sin) Fx (cos m

   • cos)

     (29)

     displaced to A1, A2, A3 and A4, the deformed beam shape, the stress distribution and the input forces at A are to be

     Rearranging Equation (29) yields the following equation.

     EI d

     determined by simulation in ANSYS.As shown in Figure 4, A1A2A3A4 forms a square with its center at the un-displaced point A and its length of 60 mm.

     ds (30)

     2 Fy (sinm sin) Fx (cos m cos)

     Integrating Equation (30) from the left end to the right end of the solid curve yields the following equation.

     EI m d

     2

     2

     L

     0

     Fy (sinm

   • sin) Fx (cos m

   • cos)

     (31)

     Substituting Equation (30) into following equation.

     dx cos ds

     yields the

     dx

     EI cos d

     (32)

     2 Fy (sinm sin) Fx (cos m cos)

     Integrating Equation (32) from the left end to the right end of the solid curve yields the following equation.

     Fig. 4 A slender curved beam with half a circle.

     To analyse the curved beam, the solid model of the curved beam is first created in the Design Modeler [23] of ANSYS. ANSYS Design Modeler is an ANSYS Workbench

     EI m

     cos d

     application that provides modeling tool. The solid model

     xB

     2

     2

     0 Fy (sinm

   • sin) Fx (cosm

     (33)

   • cos)

   created in ANSYS Design Modeler is then meshed and analysed in ANSYS Mechanical [24] that is also an application of ANSYS Workbench.

   Substituting Equation (30) into

   following equation.

   dy sin ds

   yields the

   Figure 5 shows the meshing model of the analysed curved beam. When A is displaced to A1, the deformed shape of the beam is shown in Figure 6. Because of the thin depth of the curved beam, the deformed shape line is very light in the

   figure. The colour map and numbers in the figure represent the directional deformation along the x direction. The maximum stress in the deformed beam is 114.61 MPa that is shown in Figure 7. To generate the displacements at A, the required input forces (Fx and Fy) are 0.42261 N and -0.01739 N, respectively. These input forces are the reaction forces in ANSYS.

   When A is displaced to A2, the deformed shape of the beam is shown in Figure 8. The maximum stress in the deformed beam is 246.69 MPa that is shown in Figure 9. The input forces (Fx and Fy) are -0.45321 N and 0.20784 N, respectively. The maximum stress in the deformed beam at A2 is more than doubled than that at A1.

   When A is displaced to A3, the deformed shape of the beam is shown in Figure 10. The maximum stress in the deformed beam is 71.098 MPa that is shown in Figure 11. The input forces (Fx and Fy) are -0.11983 N and 0.00818 N, respectively.

   Fig. 5 The meshing model of the curved beam.

   Fig. 6 The deformation of the curved beam at A1.

   Fig. 7 The stress of the curved beam at A1.

   When A is displaced to A4, the deformed shape of the beam is shown in Figure 12. The maximum stress in the deformed beam is 363.06 MPa that is shown in Figure 13. The input forces (Fx and Fy) are 0.85707 N and -0.48320 N, respectively. The maximum stress in the deformed beam at A4 is the highest among all four deformed shapes of the curved beam, which is beyond that of the yield strength of the beam material.

   Fig. 8 The deformation of the curved beam at A2.

   Fig. 9 The stress of the curved beam at A2.

   Fig. 10 The deformation of the curved beam at A3.

   Fig. 11 The stress of the curved beam at A3.

   Fig. 12 The deformation of the curved beam at A4.

   Fig. 13 The stress of the curved beam at A4.

4. SHAPE DESIGN OF SLENDER CURVED BEAMS

   As shown in Figure 13, the maximum stress within the deformed slender curved beam at A4 is above the yield strength of the beam material. This maximum stress can be reduced by either increasing the arc length or decreasing the cross section depth of the slender curved beam. Both ways belong to shape design of a slender curved beam that is to improve its performance and better meet its needs and requirements by changing its geometric parameters.

   The central angle of the slender circular beam analyzed last section is 180°. The angle is increased to 270° in this section while other parameters of the beam remain unchanged. Figure 14 shows the slender curved beam with the increased arc length. The meshing model of the increased curved beam is shown in Figure 15.

   When A of the increased curved beam is displaced to A1, the deformed shape of the beam is shown in Figure 16. The maximum stress in the deformed beam is 76.54 MPa that is shown in Figure 17, which is smaller than that of 114.61 MPa for the half a circle case shown in Figure 7. To generate the displacements at A, the required input forces (Fx and Fy) are 0.01223 N and 0.10798 N, respectively.

   When A of the increased curved beam is displaced to A2, the deformed shape of the beam is shown in Figure 18. The maximum stress in the deformed beam is 65.154 MPa that is shown in Figure 19, which is much lower than that of 246.69 MPa for the half a circle case in Figure 9. The input forces (Fx and Fy) are -0.01156 N and 0.09961 N, respectively.

   Fig. 14 The slender curved beam with increased length.

   Fig. 15 The meshing model of the increased curved beam.

   Fig. 16 The deformation of the increased curved beam at A1.

   When A of the increased curved beam is displaced to A3, the deformed shape of the beam is shown in Figure 20. The maximum stress in the deformed beam is 86.245 MPa that is shown in Figure 21, which is above that of 71.098 MPa for the half a circle case in Figure 10. The input forces (Fx and Fy) are

   -0.06167 N and -0.1382 N, respectively.

   Fig. 17 The stress of the increased curved beam at A1.

   Fig. 18 The deformation of the increased curved beam at A2.

   Fig. 19 The stress of the increased curved beam at A2.

   When A of the increased curved beam is displaced to A4, the deformed shape of the beam is shown in Figure 22. The maximum stress in the deformed beam is 68.005 MPa that is shown in Figure 23, which is far below that of 363.06 MPa for the half a circle case in Figure 13. The input forces (Fx and Fy) are 0.039885 N and -0.12613 N, respectively.

   Among all four displacements, the maximum stress within the deformed curved beam with increased arc length is

   86.245 MPa, which is well below the yield strength of 350 MPa for the beam material.

   Fig. 20 The deformation of the increased curved beam at A3.

   Fig. 21 The stress of the increased curved beam at A3.

   Fig. 22 The deformation of the increased curved beam at A4.

   Fig. 23 The stress of the increased curved beam at A4.

5. CONCLUSIONS

When slender curved beams have much smaller depth of cross section than radius of curvature, their neutral and centroidal axes can be reasonably considered to coincide for stress and deformation analyses. Slender curved beams usually undergo large deformations and have nonlinear load- deflection relationships. Because of the large deformation, the bending moment on a cross section has to be calculated from the deformed shape in order to have a decent accuracy. Large deformation might cause high bending stress within a deformed slender curved beam. The high bending stress can be significantly reduced by either increasing the beam length or decreasing the cross section depth. The stress and deformation analyses presented in the paper provide guidelines for analysing and designing slender curved beams.

ACKNOWLEDGMENT

The authors of this paper gratefully acknowledge the research instrument support of the US National Science Foundation under Grant No. 1337620. Any opinions, findings, recommendations or conclusions expressed in this paper are those of the authors and do not necessarily reflect the views of the US National Science Foundation.

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