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Large deflection of a non-linear, elastic, asymmetric Ludwick cantilever beam subjected to horizontal force, vertical force and bending torque at the free end

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Abstract

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated constant vertical load, to a concentrated constant horizontal load and to a concentrated constant bending torque at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler–Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper.

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Abbreviations

L :

Initial length of the beam and length of the beam neutral curve, (m)

b :

Width of rectangular cross-section, (m)

h :

Height of rectangular cross-section, (m)

F V :

Constant vertical force at the free end of the beam, (N)

F H :

Constant horizontal force at the free end of the beam, (N)

T :

Constant bending torque at the free end of the beam, (Nm)

Oxyz :

Coordinate system of reference configuration

O′x′y′z′ :

Coordinate system defined for each cross-section

h 1 , h 2 :

Quotes individuating the neutral axis of cross-section, (m)

E t :

Tensile Young modulus, (GPa)

E c :

Compressive Young modulus, (GPa)

n :

Tensile non-linearity coefficient or tensile strain exponential coefficient

m :

Compressive non-linearity coefficient or compressive strain exponential coefficient

σ t :

Tensile stress, (GPa)

σ c :

Compression stress, (GPa)

ε t :

Positive strain

ε c :

Negative strain

δ h :

Theoretical horizontal displacement of the free end, (m)

δ v :

Theoretical vertical displacement of the free end, (m)

δ h1 :

Horizontal displacement of the free end calculated by numerical algorithm given in this paper, (m)

δ v1 :

Vertical displacement of the free end calculated by numerical algorithm given in this paper, (m)

δ h2 :

Horizontal displacement of the free end calculated in Lewis and Monasa [5], (m)

δ v2 :

Vertical displacement of the free end calculated in Lewis and Monasa [5], (m)

δ h3 :

Horizontal displacement of the free end calculated in Bayakara et al. [8], (m)

δ v3 :

Vertical displacement of the free end calculated in Bayakara et al. [8], (m)

δ h4 :

Horizontal displacement of the free end calculated by ABACUS/CAE®, (m)

δ v4 :

Vertical displacement of the free end calculated by ABACUS/CAE®, (m)

M :

Bending moment, (Nm)

S 0 , S 1 :

Cross-sections delimiting an infinitesimal portion of the beam

ds :

Initial length of a horizontal fibre of an infinitesimal portion of the beam

dθ :

Infinitesimal angle

t :

Position of a horizontal fibre of an infinitesimal portion of the beam, (m)

L 1 :

Final length of a deformed horizontal fibre of an infinitesimal portion of the beam, (m)

ρ :

Radius of curvature of the neutral curve, (m)

σ s :

Stress along s axis, (GPa)

ε s :

Strain along s axis

f(x) :

Analytical expression of the neutral curve of the beam, (m)

References

  1. Borboni A, De Santis D, Faglia R (2010) Large deflection of a non-linear, elastic, asymmetric Ludwick cantilever beam. ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis, ESDA2010, 2: 99–106. DOI: 10.1115/ESDA2010-24257

  2. Borboni A, De Santis D, Faglia R (2010) Large deflection of a non-linear, elastic, asymmetric Ludwick cantilever beam. 16th Quadrennial ASME Conference of Theoretical and Applied Mechanics

  3. Bisshopp KE, Drucker DC (1945) Large deflections of cantilever beams. Q Appl Math 3:272–275

    MATH  MathSciNet  Google Scholar 

  4. Lo CC, Gupta D (1978) Bending of a nonlinear rectangular beam in large deflection. J Appl Mech ASME 45:213–215

    Article  Google Scholar 

  5. Lewis G, Monasa F (1981) Large deflections of cantilever beams of nonlinear materials. Comput Struct 14:357–360

    Article  Google Scholar 

  6. Lewis G, Monasa F (1982) Large deflections of cantilever beams of non-linear materials of the Ludwick type subjected to an end moment. Int J Non-Linear Mech 17:1–6

    Article  MATH  Google Scholar 

  7. Lee K (2002) Large deflections of cantilever beams of non-linear elastic material under a combined loading. J Non-Linear Mech 37:439–443

    Article  MATH  Google Scholar 

  8. Bayakara C, Güven U, Bayer I (2005) Large deflections of a cantilever beam of nonlinear bimodulus material subjected to an end moment. J Reinf Plast Comp 24:1321–1326

    Article  Google Scholar 

  9. Solano-Carrillo E (2009) Semi-exact solutions for large deflections of cantilever beams of non-linear elastic behaviour. Int J Non-Linear Mech 44:253–256

    Article  Google Scholar 

  10. Brojan M, Videnic T, Kosel F (2009) Large deflections of nonlinearly elastic non-prismatic cantilever beams made from materials obeying the generalized Ludwick constitutive law. Meccanica 44:733–739

    Article  MATH  MathSciNet  Google Scholar 

  11. Holden JT (1972) On the finite deflections of thin beams. Int J Solids Struct 8:133–135

    Article  Google Scholar 

  12. Byoung KL, Wilson JF, Sang JO (1993) Elastica of cantilevered beams with variable cross sections. Int J Non-Linear Mech 28:579–589

    Article  Google Scholar 

  13. Baker G (1993) On the large deflections of non-prismatic cantilevers with a finite depth. Comput Struct 46:365–370

    Article  Google Scholar 

  14. Prathap G, Varadan TK (1976) The inelastic large deformation of beams. J Appl Mech 43:689–690

    Article  Google Scholar 

  15. Varadan TL, Joseph D (1987) Inelastic finite deflections of cantilever beams. J Aeronaut Soc India 39:39–41

    Google Scholar 

  16. Kang YA, Li XF (2009) Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force. Int J Non-Linear Mech 44:696–703

    Article  Google Scholar 

  17. Chainarong A, Boonchai P, Gissanachai J, and Somchai C (2012) Effect of material nonlinearity on large deflection of variable-arc-length beams subjected to uniform self-weight. Mathematical Problems in Engineering 2012. Article ID 345461

  18. Brojan M, Cebron M, Kosel F (2012) Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end. Acta Mech Sin 28(3):863–869

    Article  ADS  Google Scholar 

  19. Eren I (2013) Analyses of large deflections of simply supported nonlinear beams, for various arc length functions. Arabian J Sci Eng 38(4):947–952

    Article  Google Scholar 

  20. Amici C, Borboni A, Faglia R (2010) A compliant PKM mesomanipulator: kinematic and dynamic analyses advances in mechanical engineering 2010, art. no. 706023. doi: 10.1155/2010/706023

  21. Amici C, Borboni A, Faglia R, Fausti D, Magnani PL (2008) A parallel compliant meso-manipulator for finger rehabilitation treatments: Kinematic and dynamic analysis. 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS, art. no. 4651029, pp. 735–740. doi: 10.1109/IROS.2008.4651029

  22. Borboni A, De Santis D, Faglia R (2006) Stochastic analysis of free vibrations in piezoelectric bimorphs. In : Proceedings of 8th Biennial ASME Conference on Engineering Systems Design and Analysis, ESDA2006

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Authors and Affiliations

  1. Mechanical and Industrial Engineering Department, Università degli Studi di Brescia, Via Branze, 38, 25123, Brescia, Italy

    Alberto Borboni & Diego De Santis

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  1. Alberto Borboni
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  2. Diego De Santis
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Correspondence to Alberto Borboni.

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Borboni, A., De Santis, D. Large deflection of a non-linear, elastic, asymmetric Ludwick cantilever beam subjected to horizontal force, vertical force and bending torque at the free end. Meccanica 49, 1327–1336 (2014). https://doi.org/10.1007/s11012-014-9895-z

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  • DOI: https://doi.org/10.1007/s11012-014-9895-z

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