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Reliability approach to the random imperfection sensitivity of columns

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Summary

Buckling of stochastically imperfect finite columns on a mixed quadratic-cubic elastic foundation is studied. The imperfection function is assumed to be a normally distributed random function of the space coordinate with given mean function and autocorrelation function. The problem is solved by the Monte-Carlo-Method. The Fourier coefficients of the initial imperfection function, expanded in terms of the buckling modes of the associated perfect column, are simulated. For each trial the buckling load is obtained numerically, and the results are used in constructing the reliability function at a specified load (defined as the probability of the buckling load exceeding this specified load). The study is a sequel to an earlier work of the author concerning the reliability of stochastically imperfect columns on a purely cubic nonlinear elastic foundation.

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Abbreviations

A :

positive constant, Eq. (67)

A pqrm :

defined in Eq. (22)

a :

arbitrary constant, Eq. (3)

B :

positive constant, Eq. (67)

B pqm :

defined in Eq. (21)

B(p+q, m) :

defined in Eq. (21)

c mk :

elements ofC

c ijmp :

defined in Eq. (44)

d :

end-shortening of column

d k :

standard independent normal variables

E :

Young's modulus

e imp :

defined in Eq. (44)

erf (...):

error function

E[v(η)]:

mean function

\(F(u,\bar u)\) :

functional of nondimensional displacements

I :

section moment of inertia

I m :

defined in Eq. (20)

I m (N) :

defined in Eq. (38)

J m :

defined in Eq. (20)

J m (N) :

defined in Eq. (38)

\(K_{\bar u} (\eta _1 ,\eta _2 )\) :

nondimensional autocovariance function

\(K_{\bar v} (\eta _1 ,\eta _2 )\) :

nondimensional autocovariance function

k 1,k 2,k 3 :

elastic foundation constants

l :

length of column

M :

number of trials

m :

number of half-waves

m * :

number of half-waves at buckling

N :

number of terms taken into account

n :

subscript

P :

axial load

P E :

buckling load of column without foundation

P cl :

buckling load of column on linear foundation

R :

required reliability

R(α′):

reliability at load level α′

\(\bar U\) :

mean imperfection function

u :

nondimensional additional displacement

\(\bar u\) :

nondimensional initial imperfection

\(\bar v\) :

initial imperfection before cutting

w :

additional deflection

\(\bar w\) :

initial imperfection

x :

axial coordinate

α:

nondimensional load

αm :

defined in Eq. (23)

α* :

buckling load, load-carrying capacity

γ(ϰ1):

defined in Eq. (13)

δij :

Kronecker delta

Δ:

radius of gyration

ϰ1, ϰ2, ϰ3 :

elastic foundation constants

η:

nondimensional axial coordinate

ξk,\(\bar \xi _k\) :

Fourier coefficients

\(\langle \bar \xi _k \rangle\) :

mean value of\(\bar \xi _k\)

\(\bar \xi _{m(1,2)}\) :

defined in Eq. (57)

μM :

number of α* values exceeding α′

σmn :

elements of variance-covariance matrix

〈...〉:

mathematical expectation

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Authors and Affiliations

  1. Department of Aeronautical Engineering, Technion-Israel Institute of Technology, Haifa, Israel

    I. Elishakoff

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Dedicated to Prof. Dr. Ir. Warner Tjardus Koiter on the occasion of his 70th birthday.

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Elishakoff, I. Reliability approach to the random imperfection sensitivity of columns. Acta Mechanica 55, 151–170 (1985). https://doi.org/10.1007/BF01267987

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