Skip to main content
SpringerLink
Log in

Large and Small Deflection Analysis of a Cantilever Beam

  • Original Contribution
  • Published:
Journal of The Institution of Engineers (India): Series A Aims and scope Submit manuscript

Abstract

This research focuses on the geometrically nonlinear large deflection analysis of a cantilever beam subjected to a concentrated tip load. Initially, a step-by-step development of the theoretical solution is provided and is compared with numerical analysis based on beam and shell elements. It is shown that the large deflections predicted by numerical analysis using beam elements accurately capture the theoretical results as compared to shell elements. Comparison of above deflections with theoretical and numerical approaches based on small deflection theory is also provided to show the extent of latter’s applicability. Finally, it is shown that for a linear elastic working range of common engineering metals, both small and large deflection approaches yield same results and one can adopt the simple small deflection approach for engineering design. It is highlighted that the theoretical approach of large deflection commonly available in design texts is valid only within the linear elastic strain limit and recommends a careful approach to designers. Further, the effect of parametric variation in geometry and stiffness of beam on large deflection, and resulting bending strains and tip reactions are analyzed and discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

b :

Width of plate (mm)

E :

Modulus of elasticity (GPa)

I :

Area moment of inertia (mm4)

k :

Non-dimensional modulus parameter

L :

Length of plate (mm)

M :

Bending moment (N-mm)

P :

Concentrated load or reaction to applied tip displacement (N)

s :

Arc length (mm)

t :

Thickness of plate (mm)

u x :

Horizontal displacement (mm)

u y :

Vertical displacement (mm)

x :

Arbitrary distance from fixed end (mm)

α :

Non-dimensional load parameter

ϕ :

Slope (rad)

ϕ 0 :

Maximum slope (rad)

σ :

Normal stress (N/mm2)

ε :

Normal strain

θ :

Deflection angle (rad)

References

  1. S. Gross, E. Lehr, Die Federn (VDI-Verlag, Berlin, 1938)

    Google Scholar 

  2. K.E. Bisshopp, D.C. Drucker, Large deflection of cantilever beams. Q. Appl. Math. 3, 272–275 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  3. T.M. Wang, Non-linear bending of beams with concentrated loads. Int. J. Nonlinear Mech. 285, 386–390 (1968)

    Google Scholar 

  4. T.M. Wang, Non-linear bending of beams with uniformly distributed loads. Int. J. Nonlinear Mech. 4, 389–395 (1969)

    Article  MATH  Google Scholar 

  5. A. Love, The Mathematical Theory of Elasticity (Dover, New York, 1944)

    MATH  Google Scholar 

  6. R. Frisch-Fay, Flexible Bars (Butterworths, London, 1962)

    MATH  Google Scholar 

  7. J.M. Gere, S.P. Timoshenko, Mechanics of Materials (McGraw Hill, New York, 1972)

    Google Scholar 

  8. L.L. Howell, Compliant Mechanisms (Wiley, Hoboken, 2001)

    Google Scholar 

  9. K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals. Int. J. Numer. Methods Eng. 17(1), 145–153 (1981)

    Article  MATH  Google Scholar 

  10. F. De Bona, S. Zelenika, A generalized elastica-type approach to the analysis of large displacements of spring-strips. Proc. Instit. Mech. Eng. Part C: J. Mech. Eng. Sci. 211(7), 509–517 (1997)

    Article  Google Scholar 

  11. H.J. Su, A pseudo-rigid-body 3R model for determining large deflection of cantilever beams subject to tip loads. J. Mech. Robot. 1(2), 021008 (2009)

    Article  Google Scholar 

  12. H. Tari, On the parametric large deflection study of Euler-Bernoulli cantilever beams subjected to combined tip point loading. Int. J. Nonlinear Mech. 49, 90–99 (2013)

    Article  Google Scholar 

  13. H. Tari, G.L. Kinzel, D.A. Mendelsohn, Cartesian and piecewise parametric large deflection solutions of tip point loaded Euler-Bernoulli cantilever beams. Int. J. Mech. Sci. 100, 216–225 (2015)

    Article  Google Scholar 

  14. T. Beléndez, C. Neipp, A. Beléndez, Large and small deflections of a cantilever beam. Eur. J. Phys. 23, 371–379 (2002)

    Article  Google Scholar 

  15. T. Beléndez, C. Neipp, A. Beléndez, Numerical and experimental analysis of a cantilever beam: a laboratory project to introduce geometric nonlinearity in mechanics of materials. Int. J. Eng. Educ. 19, 885–892 (2003)

    Google Scholar 

  16. Y.V. Zakharov, Nonlinear bending of thin elastic rods. J. Appl. Mech. Tech. Phys 43, 739–744 (2002)

    Article  Google Scholar 

  17. M. Batista, Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions. Int. J. Solids and Struct. 51(13), 2308–2326 (2014)

    Article  Google Scholar 

  18. R. Kumar, L.S. Ramachandra, D. Roy, Techniques based on genetic algorithms for large deflection analysis of beams. Sadhana 29, 589–604 (2004)

    Article  MATH  Google Scholar 

  19. M. Dado, S. Al-Sadder, A new technique for large deflection analysis of non-prismatic cantilever beams. Mech. Res. Commun. 32, 692–703 (2005)

    Article  MATH  Google Scholar 

  20. B.S. Shvartsman, Large deflections of a cantilever beam subjected to a follower force. J. Sound Vib. 304, 969–973 (2007)

    Article  Google Scholar 

  21. M. Mutyalarao, D. Bharathi, B.N. Rao, On the uniqueness of large deflections of a uniform cantilever beam under a tip-concentrated rotational load. Int. J. Nonlinear Mech. 45, 433–441 (2010)

    Article  MATH  Google Scholar 

  22. M.A. Rahman, M.T. Siddiqui, M.A. Kowser, Design and non-linear analysis of a parabolic leaf spring. J. Mech. Eng. 37, 47–51 (2007)

    Article  Google Scholar 

  23. A. Banerjee, B. Bhattacharya, A.K. Mallik, Large deflection of cantilever beams with geometric non-linearity: analytical and numerical approaches. Int. J. Nonlinear Mech. 43, 366–376 (2008)

    Article  MATH  Google Scholar 

  24. L. Chen, An integral approach for large deflection cantilever beams. Int. J. Nonlinear Mech. 45, 301–305 (2010)

    Article  Google Scholar 

  25. D.K. Roy, K.N. Saha, Nonlinear analysis of leaf springs of functionally graded materials. Procedia Eng. 51, 538–543 (2013)

    Article  Google Scholar 

  26. C.A. Almeida, J.C.R. Albino, I.F.M. Menezes, G.H. Paulino, Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation. Mech. Res. Commun. 38, 553–559 (2011)

    Article  MATH  Google Scholar 

  27. M. Sitar, F. Kosel, M. Brojan, Large deflections of nonlinearly elastic functionally graded composite beams. Arch. Civ. Mech. Eng. 14, 700–709 (2014)

    Article  Google Scholar 

  28. N.D. Kien, Large displacement behaviour of tapered cantilever Euler-Bernoulli beams made of functionally graded material. Appl. Math. Comput. 237, 340–355 (2014)

    MathSciNet  MATH  Google Scholar 

  29. X.T. He, L. Cao, Z.Y. Li, X.J. Hua, J.Y. Sun, Nonlinear large deflection problems of beams with gradient: a bi-parametric perturbation method. Appl. Math. Comput. 219, 7493–7513 (2013)

    MathSciNet  MATH  Google Scholar 

  30. A.K. Nallathambi, C.L. Rao, S.M. Srinivasan, Large deflection of constant curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445 (2010)

    Article  Google Scholar 

  31. B.S. Shvartsman, Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force. Int. J. Nonlinear Mech. 50, 75–80 (2013)

    Article  Google Scholar 

  32. S. Ghuku, K.N. Saha, A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams. Eng. Sci. Tech. Int. J. 19, 135–146 (2016)

    Article  Google Scholar 

  33. Wahl AM. Mechanical Springs. 2nd edn. (Mc-Graw Hill Book Co., New York, 1964); Reprint, Spring Manufacturers Institute, USA, 1991

  34. J. Shigley, Machine Design (Tata Mc-Graw Hill Publishing Pvt Ltd, New York, 2015)

    Google Scholar 

  35. E. Jahnke, F. Emde, Tables of Functions with Formulae and Curves, 4th edn. (Dover, New York, 1945)

    MATH  Google Scholar 

  36. B.O. Pierce, A Short Table of Integrals (Ginn & Company, New York, 1899)

    Google Scholar 

  37. ANSYS-User’s Manual. Release 15, USA, 2015

Download references

Acknowledgements

Authors are thankful to Technology Director, Systems Integration (Mechanical), and Director, Research Centre Imarat, Hyderabad, for offering the Junior Research Fellowship to the first author.

Author information

Authors and Affiliations

  1. Research Centre Imarat, Vignyanakancha P.O., Hyderabad, 500 069, India

    D. Singhal & V. Narayanamurthy

Authors
  1. D. Singhal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Narayanamurthy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Narayanamurthy.

Appendices

Appendix 1

See Tables 1, 2 and 3

Table 1 Values of F(k) and E(k) for certain values of k
Full size table
Table 2 Values of F(k, θ1) for certain values of k and θ1
Full size table
Table 3 Values of E(k, θ1) for certain values of k and θ1
Full size table

Appendix 2

See Table 4

Table 4 Parameters used in computing theoretical solution of large deflection
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singhal, D., Narayanamurthy, V. Large and Small Deflection Analysis of a Cantilever Beam. J. Inst. Eng. India Ser. A 100, 83–96 (2019). https://doi.org/10.1007/s40030-018-0342-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40030-018-0342-3

Keywords

Search

Navigation