Engineering with Computers

, Volume 34, Issue 3, pp 543–563 | Cite as

Analysis of axially functionally graded nano-tapered Timoshenko beams by element-based Bernstein pseudospectral collocation (EBBPC)

  • Sundaramoorthy Rajasekaran
Original Article


In this paper, static bending, buckling and free vibration of axially functionally graded (AFG) nano-tapered Timoshenko (NTTB) or Bernoulli Euler (NTEB) beams are examined based on the nonlocal Timoshenko beam theory (NTBT). This theory incorporates the length scale parameter (nonlocal parameter) to capture the small-scale effect. The material properties and geometry properties of the nanobeam are assumed to vary along the length direction. The governing equations and the associated boundary conditions are derived using Hamilton’s principle. The model is then applied on the studies of static, buckling and free vibration analysis of NTTB or NTEB using element-based Bernstein pseudo-spectral collocation approach (EBBPC). After the Bernstein pseudo-spectral collocation method is validated, detailed numerical analyses about the effect of boundary conditions, load types are carried out. Non-local parameter and axial load effects on the static and dynamic response of AFG-NTTB and AFG-NTEB are discussed. The approach is tested on benchmark problems of static, buckling and free vibration analyses, showing high accuracy.


Nonlocal theory Timoshenko beam Bernstein polynomials Collocation Free vibration Buckling 



Area of the cross section


Characteristic length

\(\{ a\} ,\,\,\{ \underline {a} \}\)

Undetermined parameters for w

\(\{ b\} ,\,\,\{ \underline {b} \}\)

Undetermined parameters forψ


Bernstein kth polynomial of degree n


Fourth-order elasticity tensor


Taper ratio for the width


Taper ratio for diameter


Taper ratio for depth


Differential matrix


Young’s modulus


Material constant


Distributed axial force


Rigidity modulus


Moment of inertia of the section about y-axis


Spring constants of translational springs at x = 0 and x = L


Spring constants of rotational springs at x = 0 and x = L


Flexural stiffness


Geometric stiffness


Mass matrix




Axial force

\(p=\frac{{{P_{{\text{cr}}}}{L^2}}}{{{\pi ^2}EI}}\)

Buckling load parameter (P cr—critical buckling load)


Axial load parameter

\({P^{\prime \prime \prime }}\)

Axial load


Shear force


Distributed transverse load


Matrices to define \(w,\,\frac{{{\text{d}}w}}{{{\text{d}}\xi }};\;\frac{{{{\text{d}}^2}w}}{{{\text{d}}{\xi ^2}}}\)

\(r=\frac{{{P_{{\text{cr}}}}{L^2}}}{{{\pi ^2}EI}}\)

Buckling load parameter (P cr—critical buckling load)


Scale coefficient, nonlocal parameter




Kinetic energy

\({\hat {u}_{,x}}\)

Displacement at any point in x direction

\({\hat {u}_{,y}}\)

Displacement at any point in y direction

\({\hat {u}_{,z}}\)

Displacement at any point in z direction

u = u(x,t)

x displacement of the mid-plane


Transverse displacement in z direction


External work

\(\hat {w}=100\,w\,\frac{{EI}}{{q\,{L^4}}}\)

Non-dimensional factor for deflection

x, y and z

Coordinate axes

\(\alpha =\frac{s}{L}\)

Nonlocal parameter



\({\varepsilon _{xx}}\)

Normal strain

\({\gamma _{xz}}\)

Shear strain

\({\kappa _x}=\frac{{{\text{d}}\psi }}{{{\text{d}}x}}\)

Bending curvature

\(\lambda =\sqrt \Omega\)

Frequency parameter

\(\mu ={s^2}\)

Nonlocal parameter


Poisson’s ratio


Mass density of the material

\({\sigma _{ij}}\)

Stress tensor

\({\sigma _{xx}}\)

Normal stress

\({\sigma _{xz}}\)

Shear stress

\(\Omega =\omega \,{L^2}\,\sqrt {\frac{{\rho \,A}}{{E\,I}}}\)

Frequency parameter

ψ = ψ(x,t)

Bending rotation of the beam


Circular natural frequency

\(\xi =\frac{x}{L}\)

Non-dimensional parameter for length

\(f(\xi )\)

Function value at ξ



The author thanks the management and Principal Dr. R. Rudramoorthy for giving necessary facilities to carry out the work reported in this paper. The author also thanks Dr. Hossein Bakshi Khaniki of Iran University of Science and Technology for his suggestions and advice in improving the standard of the manuscript.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Civil EngineeringPSG College of TechnologyCoimbatoreIndia

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