Thermal post-buckling of slender composite and FGM columns through a simple and novel FE formulation

Abstract

A simple and novel finite element (FE) formulation is proposed to study the thermal post-buckling of composite and FGM columns with axially immovable ends and operating in severe thermal environment. A linear eigenvalue analysis gives the critical buckling temperature but practically the buckled columns can withstand additional thermal load beyond critical temperature, which can be obtained using von-Karman geometric nonlinearity, applicable for moderately large deflections. In the present study, the solution of the non-linear post-buckling problem is obtained by treating it as a linear eigenvalue problem using the concept of effective stiffness. Here, the total degrees of freedom (dof) of the discretized column are reduced and the post-buckling load is obtained without the need for iterative analysis. Comparison of the numerical results obtained from this FE formulation is in very good agreement with those obtained from the earlier FE formulations.

Appendices

Nomenclature

a :

central transverse deflection

A :

cross sectional area

A ₁₁ :

extensional stiffness

B ₁₁ :

bending-extension coupling stiffness

D ₁₁ :

bending stiffness

E ₁₁ :

elastic modulus in fiber direction

E ₂₂ :

elastic modulus in transverse direction

E :

effective modulus of the column (= $\frac{{A_{11} }}{t}$)

ν ₁₂, ν ₂₁ :

in-plane major and minor Poisson ratio

G ₁₂ :

in-plane shear modulus

[G]:

geometric stiffness matrix

h :

thickness of column

H _i :

Hermite shape functions

$I$ :

area moment of inertia of the column cross-section

[K]:

linear elastic stiffness matrix

$\left[ {K_{E} } \right]$ :

effective stiffness matrix

L :

length of column

$M_{xx}$ :

moment resultant

N :

total number of layers

$N_{xx}$ :

stress resultant

NE :

number of elements

P :

mechanical equivalent of the thermal load

$Q_{11}$ :

reduced stiffness

$\bar{Q}_{11}$ :

transformed reduced stiffness

n :

volume fraction exponent

r :

radius of gyration of the column cross-section

t :

temperature increment

T :

tensile load induced in column due to large transverse deflections

u :

axial displacement

u′ :

du/dx

V :

volume fraction

w :

transverse deflection of column

w′ :

dw/dx

x :

coordinate along axis of column

z :

coordinate along thickness of column

$z_{k} , z_{k - 1}$ :

depth coordinates of a particular layer

$\varepsilon_{xx}$ :

axial strain

$\varepsilon_{xx}^{0}$ :

axial strain at Neutral Axis

$_{xx}$ :

curvature

$\theta$ :

angle made by a layer with respect to the axial direction x

$\lambda$ :

eigenvalue

$\lambda_{L} , \lambda_{T}$ :

linear buckling load parameter, tension parameter

{$\delta$}:

eigenvector (buckled mode shape)

α:

coefficient of linear thermal expansion

Subscripts

1, 2:

node numbers

L :

linear

NL :

nonlinear

m, c :

metal, ceramic

0:

mid-thickness

11, 22:

fiber and transverse directions