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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
  • 347 Downloads

Abstract

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.

Keywords

Nonlocal theory Timoshenko beam Bernstein polynomials Collocation Free vibration Buckling 

Abbreviations

A

Area of the cross section

a

Characteristic length

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

Undetermined parameters for w

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

Undetermined parameters forψ

Bk,n

Bernstein kth polynomial of degree n

Cijkl

Fourth-order elasticity tensor

cb

Taper ratio for the width

cd

Taper ratio for diameter

ch

Taper ratio for depth

[D]

Differential matrix

E

Young’s modulus

e0

Material constant

f

Distributed axial force

G

Rigidity modulus

I

Moment of inertia of the section about y-axis

\({k_{d0}},{k_{dL}}\)

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

\({k_{r0}},{k_{rL}}\)

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

K

Flexural stiffness

KG

Geometric stiffness

M

Mass matrix

M

Moment

N

Axial force

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

Buckling load parameter (P cr—critical buckling load)

\({p_{\text{f}}}=\frac{P}{{{P_{{\text{cr}}}}}}\)

Axial load parameter

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

Axial load

Q

Shear force

q

Distributed transverse load

\([{R_0}],[{R_1}],[{R_2}]\)

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)

\(s={e_0}a\)

Scale coefficient, nonlocal parameter

t

Time

T

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

w

Transverse displacement in z direction

Wext

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

δ

Variation

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

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

\(\nu\)

Poisson’s ratio

ρ

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

\(\omega\)

Circular natural frequency

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

Non-dimensional parameter for length

\(f(\xi )\)

Function value at ξ

Notes

Acknowledgements

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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