# 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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- Received:
- Accepted:

## 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.

### Keywords

Large deflections Asymmetric Ludwick constitutive law Material non-linearity Geometrical non-linearity Cantilever beam### List of symbols

*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)

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