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Analytical effective width equations for limit state design of thin plates under non-homogeneous in-plane loading

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Abstract

The effective width concept has been widely used in engineering practice for the computation of ultimate strength of slender members. Many design codes employ this concept in order to compensate for the stiffness reduction in the post-buckling state. Extensive work was done to develop effective width equations for plates under uniform compression, while little attention has been given for plates under non-homogeneous in-plane loading. North American, British and European design codes provide only expressions for the computation of the elastic buckling loads for plates under this load combination, while the effective width calculation is based on the uniformly compressed plates. It will be shown that due to the non-uniformity of the applied load, the stress characteristics in the post-buckling state are different from the uniform compression case, thus requiring special treatments. The paper presents analytical closed form expressions for the computation of effective width of thin plates under non-homogeneous in-plane loading. The longitudinal edges are assumed to be straight and free to translate in the plane of the plate. The proposed expressions are very useful for limit state design of slender I-sections of beam columns or channel sections under this general type of loading. They enable the designers to compute the effective width of the section with the aid of simple formulas that, for design purposes, are suitable for hand-calculation and avoid the cost and effort that any numerical non-linear analysis may require.

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Abbreviations

a :

Length of the plate

b :

Width of the plate

b e , be1, be2:

Effective widths

D :

Plate bending rigidity per unit width, Et 3/12(1−v 2)

E :

Elastic modulus

f, fo:

Amplitudes of the out-of plane deflection and initial imperfection, respectively

m :

Number of half waves in the longitudinal direction

N 1 :

Compressive force at the heavily loaded edge

N x , N y :

Compressive forces per unit length in the x and y directions, respectively

t :

Thickness of the plate

u, v:

In-plane displacements in the x, y direction, respectively

w, wo:

Out of plane deflection and initial imperfection, respectively

β :

Plate aspect ratio

η :

Non-dimensional width, y/b

v :

Poisson ratio

ξ :

Non-dimensional length, x/a

Φ:

Stress function

\({\varphi}\) :

Limit state resistance factor of safety

σ 1 :

Stress intensity at the heavily loaded edge

σ x , σ y :

Membrane stresses acting in the x, y direction, respectively

σ xy :

Membrane shear stress acting in the x, y plane

σ o :

Yield stress of the plate

σ cr :

Critical stress of the plate

σmax1, σmax2:

Maximum compressive edge stresses

σ i :

Induced stresses due to applied loads

ε x , ε y :

Middle surface membrane strains in the x, y directions, respectively

γ xy :

Middle surface shear strain

κ :

A parameter equals m/β

Ψ:

Stress gradient coefficient

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  1. AMEC Americas Ltd., 116 Mount Aberdeen Manor SE, Calgary, AB, T2Z 3N8, Canada

    Osama Bedair

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Bedair, O. Analytical effective width equations for limit state design of thin plates under non-homogeneous in-plane loading. Arch Appl Mech 79, 1173–1189 (2009). https://doi.org/10.1007/s00419-009-0296-z

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