Numerical and Analytical Study on Curved Beams

Numerical and Analytical Study on Curved Beams

Shiv Pratap Singh Yadav, Sandeep G M, Avinash L Assistant Professors

Nitte Meenakshi Institute of Technology Bengaluru, India

Kishan Reddy R, Akshay Bhat, Aaron George Biju, Akash Doijode, Jayanth J.

UG Students

Nitte Meenakshi Institute of Technology Bengaluru, India

Dr. Sudheer Reddy Professor

Nitte Meenakshi Institute of Technology Benguluru, India

AbstractA Curved Beam is an element in which the neutral axis in the unloaded condition is curved as an alternative of straight. Curved beams eliminate the use of vertical columns in buildings. The horizontally Curved beam has a variety of uses such as in the design of buildings for good aesthetic looks, in the mechanical elements like crane hooks to lift huge loads. For the required load to withstand, it is tedious to design the beam for stresses at inner and outer layers manually. The parameters which are involved in the curved beams were analyzed. C Program for these parameters is designed and these results are compared with the outputs of FEA based software. Curved beam is modelled in the design software for the required dimensions.

KeywordsCurved Beams, Ring, Link.

1. INTRODUCTION

   Curved beams finds lot of applications in the fields of automotive and structural mechanics as the use of them reduces many linkages which are attached to the structures, even capable of withstanding diametral loads as in chain links, crane hooks etc. It is necessary to obtain quick solutions to the problems as they are of different cross sections like circular, rectangular and even trapezoidal, analysis of which is most necessary to know their failure conditions and the maximum stress conditions.

   In this project we chose circular cross section as it is the most used, we designed chain link and circular rings using CATIA V5 which accepts the dimensions, these dimensions, where also given as inputs to C program coded in CODE BLOCKS (16.01) and the results obtained were checked theoretically and also with the outputs of ANSYS 14.5.

   ANSYS being FEA approach approximate results were obtained but the regions of stress conditions was identified clearly. The outputs of C code were exact and quick. The stress conditions for different materials can be determined easily instead of solving theoretically which is tiring. The CATIA model displayed the exact model lookalike.

2. METHODOLOGY

   1. Theoretical method

      Steps and equations

      Step 1: Consider the force acting on the link, length of the link, centroidal diameter and diameter of the cross section.

      Step 2: Obtain inner radius, centroidal radius and outer radius.

      • Inner radius Ri = (1)

      • Outer radius Ro = (2)

        Step 3: Obtain values of Rn, e, Ci, Co.

      • Radius of Neutral fiber RN = (3)

      • Eccentricity e = Rc – RN (4)

      • Distance of N.A. to inner fiber Ci = RN – Ri (5)

      • Distance of N.A. to outer fiber CO = RO -RN (6)

      • Area of cross section A = (7)

        Step 4: calculate the axial and shear stresses across the cross section.

      • Bending moment at section A MbA = (8)

      • Inner fiber stress biA = (9)

      • Outer fiber stress boA = (10)

      • Bending moment at section B MBb = (11)

   • Inner fiber stress biB = +

     1. C output

3. RESULTS

   • Outer fiber stress boB –

   • Maximum shear stress =0.5×max

   1. C algorithm

      Step 1: Enter the type of curved beam, i.e S-Link, or Chain Link, or Ring.

      Step 2: Based on the type of Curved Beam, Run respective function.

      Step 3: Create a Function named Circular(). In circular(), Enter the diameter of circular cross section, and save it to variable d. Calculate x=d/2. Select the type of radius being entered, i.e. inner radius, outer radius, or centroidal radius. Based on the value entered, calculate the other radii, the neutral axis radius, the distance of inner fiber from neutral axis, the distance of the outer fiber from neutral axis.

      Step 4: Create functions slink(), chain-link(), ring(). In each of these functions, the moment is calculated by multiplying force and centroidal axis radius. The stresses at inner and outer fiber, at both cross sections, are calculated using formulae. The stresses and then converted to positive values, and the maximum value is then saved. The maximum shear stress is the half of the stress obtained by calculations

   2. ANSYS Method

      • Create a new file under the structural menu.

      • Import the .CAT file into the ANSYS.

      • Create element type of solid tetrahedron 4 node 285.

      • Assume model material to be mild steel (E=2.1e11) and possions ratio as 0.3.

      • Free mesh the link/ring in YZ plane and refine mesh to the accuracy of 3.

      • Constrain the bottom part of the link/ring in all degrees of freedom and apply the given load to the upper part of the link/ring in -Z direction.

      • Solve the problem.

      • Obtain the contour plot of VONMISES stresses for deformed and un-deformed shape.

      Fig. 1: Result obtained when F=12000N, d=30mm, Ro=90mm

      Fig. 2: Result obtained when F=9000N, d=30mm, Rc=45mm

      1. Theoretical method

         Chain Link is made up of 30mm diameter steel rod has a mean diameter of 90mm.the straight sides are each 60mm long, if the link carries a load of 9KN. Determine the Maximum fiber stress induced in the link.[1]

         Fig. 3: cross sectional area

         d = 30mm (cross section diameter)

         Dc = 90mm, Rc = 45mm (radius to centroidal axis) l = length of the straight sides=60mm

         F = 9000N

         Ri = = = 30mm

         Ro = = 60mm

         Step 1: RN = RN =

         RN = 43.713203 mm

         Step 2: e = Rc – RN = 45-43.713203 = 1.286797 mm Ci = RN – Ri = 43.713203-30 = 13.713203 mm CO = RO -RN = 60-43.713203 = 16.286797 mm A == 706.5 mm2

         Step 3: MbA =

         MbA

         MbA = 150894.1878 Nmm

         1. Bending stress at inner fiber biA =

            =

            biA = -75.869628 N/mm2 (compression) (2)Bending stress at outer fiber

            b0 (A) =

            boA = 45.054432 N/mm2 (tensile) MBb =

            =

            MBb = – 51605.81222 N-mm

            1. Combined stress at inner fiber

               biB = +

               = +

               biB = -19.580378 N/mm2 (compression)

            2. Combined stress at outer fiber

         boB = –

         = –

         boB = 21.7970 N/mm2 (tensile) max = 0.5×75.86

         max = 37.93 N/mm2

      2. ANSYS Output

         Results obtained from ANSYS for Ring and Chain Link

         Fig.4: Ring Analysis

         Fig.5: Chain link Analysis

      3. Comparison of Results

   TABLE I. COMPARISON OF RESULTS

   LINK
   Method Parameters	Analytical Method	C	FEA
   Max. stress (MPa)	75.869628	75.869656	67.8683
   Min. stress (MPa)	19.580378	19.577996	23.007
   RING
   Max. stress (MPa)	126.851493	127.10455	122.509
   Min. stress (MPa)	64.112296	63.931113	62.825

4. ACKNOWLEDGMENT

   The authors acknowledge, to Dr. HC Nagaraj, Principal. Nitte Meenakshi Institute of Technology for providing the support and infrastructure to carry out our research. We would also like to acknowledge Dr. Kiran Aithal S., HoD, Department of Mechanical Engineering of Nitte Meenakshi Institute of Technology for their valuable suggestions and support.

5. REFERENCES

1. V. B. Bhandari Design of Machine Elements. pp. 130-145

2. Dr. K. Lingaiah – Design data Hand Book Volume-II