126x

2024-09-10

Imperfections in Design Check Formulas for Flexural Buckling Analysis

Flexural buckling analysis is a combination of stability analysis and ultimate limit state design that has been used in steel structures for hundreds of years. The critical buckling load is the starting point here for considering the stability problem, but no design has yet been performed without considering imperfections. How exactly are these imperfections determined?

Info

For a better understanding of this article, we recommend first reading the article of the Euler's buckling load: KB 1895 | Origin of Formula for Ideal Buckling Load of Individual Members Subjected to Compression . The background and similar equation transformations of the differential equations are described in detail there.

The imperfection of a structural component consists of the following components:

Geometric imperfections
Material imperfections
Eccentric Load Introduction

Since these imperfections are distributed along the length of the structural component, a function is first applied to them:

e (x) = e_{0} \cdot \sin (\frac{π \cdot x}{L})

e₀	Initial imperfection

Imperfection Function

Imperfection Function Visualized Using Mode Shapes

Furthermore, this design includes additional effects from the second-order analysis. However, they are strictly to be considered as additional and only virtually present in the derivation of the correct bending function from the imperfection. It is explicitly not possible to dispense with the calculation of internal forces according to the second-order analysis because of this individual design that is based on the effects of the second-order analysis. The additional effects of the second-order analysis included in this design concept are very small and should be applied to the already modified internal forces of the entire structure. As in the case of the determination of the ideal buckling load, the initial function is the equilibrium of the internal moments and the axial force with its eccentricity.

The following image shows how the Euler's buckling load is determined, and compares it using the Euler's formulas and the determination based on numerical methods. In this case, the deviation is only 0.47%. This illustrates how the basis of a modern stability analysis was laid several hundred years ago.

Comparison of Critical Buckling Loads According to Modern Stability Analysis and Formulas Resulting from Euler's Differential Equations

However, this eccentricity is now composed of a part that is purely the imperfection function, and a part of the second-order deformation analysis. This gives us the following formula:

EI \cdot w^{''} (x) = - M (x) = - N \cdot [w (x) + e (x)]

w^{''} (x) + \frac{N}{EI} \cdot w (x) = - \frac{N}{EI} \cdot e (x)

Application of Imperfection

Again, the constant α is introduced for substitution:

α = \sqrt{\frac{N}{EI}}

Constant α

Inserting the constant and the imperfection function results in the following differential equation:

w^{''} (x) + α^{2} \cdot w (x) = - α^{2} \cdot e_{0} \cdot \sin (\frac{π \cdot x}{L})

Differential Equation with Imperfections and Second-Order Analysis

This corresponds to an inhomogeneous differential equation:

- α^{2} \cdot e_{0} \cdot \sin (\frac{π \cdot x}{L})

Inhomogeneous Part of Differential Equation of Imperfection

It represents the imperfection function, and thus cannot be 0.

First, the solution of the homogeneous equation is sought:

w'' (x) + α^{2} \cdot w (x) = 0

Euler's Differential Equation with α

w (x) = A \sin (α x) + B \cos (α x)

Differential Equation of Deflection After Reduction to Real Component

Then, it is necessary to solve the particular part of the equation:

w_{p} (x) = C \cdot \sin (\frac{π \cdot x}{L})

{w_{p}}^{''} (x) = - {(\frac{π}{L})}^{2} \cdot C \cdot \sin (\frac{π \cdot x}{L})

Particulate Part of Differential Equation for Imperfections

If we substitute this particulate solution into the homogeneous equation, the result is:

- {(\frac{π}{L})}^{2} \cdot C \cdot \sin (\frac{π \cdot x}{L}) + α^{2} \cdot C \cdot \sin (\frac{π \cdot x}{L}) = - α^{2} \cdot e_{0} \cdot \sin (\frac{π \cdot x}{L})

- {(\frac{π}{L})}^{2} \cdot C + α^{2} \cdot C = - α^{2} \cdot e_{0}

C \cdot (α^{2} - {(\frac{π}{L})}^{2}) = - α^{2} \cdot e_{0} | / (α^{2} - {(\frac{π}{L})}^{2})

C = \frac{α^{2} \cdot e_{0}}{{(\frac{π}{2})}^{2} - α^{2}}

Inserting Particulate Solution of Imperfection Differential Equation

It can be divided by sin(π·x/L), because the components and thus x have never a length of 0 m.

Thus, the total solution of the inhomogeneous differential equation results in:

w (x) = w_{h} (x) + w_{p} (x)

w (x) = Asin (αx) + Bcos (αx) + \frac{α^{2} \cdot e_{0}}{{(\frac{π}{2})}^{2} - α^{2}} \cdot \sin (\frac{π \cdot x}{L})

w_h	Homogeneous solution of the differential equation
w_p	Particulate solution of the differential equation

Total Solution of Inhomogeneous Differential Equation

Thus, the boundary conditions W(x=0) result in B and w(x=L) = 0. This results in B = 0 and the differential equation is reduced to:

w (x) = A \cdot \sin (α \cdot x) + \frac{{(α \cdot L)}^{2}}{{(π^{2} - αL)}^{2}} \cdot e_{0} \cdot \sin (\frac{π x}{L})

Reduced Differential Equation for Deflection with Imperfections

Since the Euler's buckling load must result for A ≢ 0 and the imperfection must become infinitesimal again, it is necessary to consider Case A = 0!

Thus, the equation is:

w (x) = \frac{{(α \cdot L)}^{2}}{π^{2} - {(αL)}^{2}} \cdot e_{0} \cdot \sin (\frac{πx}{L})

Differential Equation of Bending Line for Applied Imperfections

Now, the equilibrium of the internal and external forces can be calculated:

M (x) = N \cdot [w (x) + e (x)]

M (x) = N \cdot (\frac{{(α \cdot L)}^{2}}{π^{2} - {(α \cdot L)}^{2}} + 1) \cdot e_{0} \cdot \sin (\frac{πx}{L})

= N \cdot (\frac{{(α \cdot L)}^{2}}{π^{2} - {(α \cdot L)}^{2}} + \frac{π^{2} - {(α \cdot L)}^{2}}{π^{2} - {(α \cdot L)}^{2}}) \cdot e_{0} \cdot \sin (\frac{πx}{L})

= N \cdot \frac{(α \cdot L)^{2} + π^{2} - {(α \cdot L)}^{2}}{π^{2} - {(α \cdot L)}^{2}} \cdot e_{0} \cdot \sin (\frac{πx}{L})

= N \cdot \frac{π^{2}}{π^{2} - {(α \cdot L)}^{2}} \cdot e_{0} \cdot \sin (\frac{πx}{L})

= N \cdot \frac{1}{1 - \frac{{(α \cdot L)}^{2}}{π^{2}}} \cdot e_{0} \cdot \sin (\frac{πx}{L})

α = \sqrt{\frac{N}{EI}} = > N \cdot \frac{1}{1 - \frac{N \cdot L^{2}}{{EIπ}^{2}}} \cdot e_{0} \cdot \sin (\frac{πx}{L})

Determination of Maximum Bending Moment as Function of Axial Force

When using the Euler's buckling load and calculating extreme values, you obtain the maximum bending moment.

M (x) = \frac{1}{1 - \frac{N}{N_{cr}}} \cdot N \cdot e_{0} \cdot \sin (\frac{π \cdot x}{L})

M^{'} (x) = \frac{π}{L} \cdot \frac{1}{1 - \frac{N}{N_{cr}}} \cdot N \cdot e_{0} \cdot \cos (\frac{π \cdot x}{L}) = 0

\cos (\frac{π \cdot x}{L}) = 0 = > x = 0

M (x = 0) = \frac{1}{1 - \frac{N}{N_{cr}}} \cdot N \cdot e_{0} = M_{max}

Determination of Extreme Value of Bending Moment

The magnification factor (1-N/N_cr)^-1 is called the Dischinger factor.

In order to ensure the most accurate results, a surface moment of inertia is applied that neglects the web. Furthermore, other imperfection factors, depending on the steel grade and ratios of the cross-section dimensions (buckling curves), are used to design the buckling failure on the basis of plasticizing flanges. Therefore, you can consider the selection of the buckling curve as a simplified type of cross-section classification, which reduces the application of plasticizing cross-sections in the case of buckling-prone sections. For simplification, a linear interaction is assumed.

Linear M/N Interaction

The simplified radius of gyration is determined as follows:

i = \sqrt{\frac{I}{A}} = \sqrt{\frac{2 \cdot ((b \cdot tf) \cdot {(\frac{h}{2})}^{2}}{2 \cdot (b \cdot tf)}} = \frac{h}{2}

Radius of Gyration Without Web

In this case, only the Steiner components of the flanges about the major axis are considered.

Now, the equation for the maximum moment as a function of the ideal buckling load can be used in the equation of the linear interaction:

\frac{\frac{1}{1 - \frac{N}{N_{cr}}} \cdot N \cdot e_{0}}{N_{pl} \cdot \frac{h}{2}} + \frac{N}{N_{pl}} = \frac{1}{1 - \frac{N}{N_{cr}}} \cdot \frac{N}{N_{pl}} \cdot \frac{e_{0}}{\frac{h}{2}} + \frac{N}{N_{pl}} = 1

N	Maximum axial force before buckling
e₀	Selected pre-deformation
N_cr	Critical buckling load
N_pl	Plastic resistance to axial forces
h	Cross-section depth

Inserting Moment Equation Depending on Elastic Buckling Load in Interaction Formula

Furthermore, the following constants are introduced:

χ = \frac{N}{N_{pl}}; λ = \frac{N_{pl}}{N_{cr}}; \frac{e_{0}}{\frac{h}{2}}; e_{0} = \frac{L}{m}; α = \frac{λ_{1}}{m}

= > \frac{e_{0}}{\frac{h}{2}} = \frac{(\frac{L}{m})}{i} = \frac{(\frac{L}{i})}{m} = \frac{λ}{m} = \frac{\bar{λ} \cdot λ_{1}}{m} = \bar{λ} \cdot α

e₀	Pre-deformation
L	Structural length of the component
λ₁	Limit slenderness
α	Imperfection factor from various buckling curves
λ	Slenderness
$\overline{λ}$	Related slenderness
m	Divisor of the pre-deformation is adjusted to the buckling curves

Factors

In this case, a value determined from tests is applied for α:

Imperfection Factors of Buckling Curves

When substituting the constants in the equation of the linear interaction, the following term results:

\frac{1}{1 - χ \cdot {\bar{λ}}^{2}} \cdot χ \cdot α + χ = 1

χ	Reduction factor from the ratio of the axial force at which the column fails due to the flexural buckling and the plastic resistance
α	The imperfection factor results from the ratio of the limit slenderness to the imperfection amplitude.
λ^-	Limit slenderness

Equation for Reduction Factor Based on Linear M/N Interaction

\frac{1}{1 - χ \cdot {\bar{λ}}^{2}} \cdot χ \cdot α + χ = 1 | - 1

\frac{1}{1 - χ \cdot {\bar{λ}}^{2}} \cdot χ \cdot α + χ - 1 = 0 | \cdot 1 - χ \cdot {\bar{λ}}^{2}

= > χ^{2} - \frac{2 ϕ}{λ^{2}} \cdot χ + \frac{1}{λ^{2}} = 0

χ_{1, 2} = + \frac{ϕ}{λ^{2}} \pm \sqrt{{(\frac{ϕ}{λ^{2}})}^{2} - \frac{1}{λ^{2}}} = \frac{ϕ}{λ^{2}} \pm \frac{1}{λ^{2}} \cdot \sqrt{ϕ^{2} - λ^{2}} = \frac{1}{λ^{2}} (ϕ \pm \sqrt{ϕ^{2} - λ^{2}})

= \frac{1}{λ^{2}} \cdot \frac{(ϕ \pm \sqrt{ϕ^{2} - λ^{2}}) \cdot (ϕ \pm \sqrt{ϕ^{2} - λ^{2}})}{\cdot (ϕ \pm \sqrt{ϕ^{2} - λ^{2}})} = \frac{1}{λ^{2}} \cdot \frac{ϕ^{2} - (ϕ^{2} - λ^{2})}{(ϕ \pm \sqrt{ϕ^{2} - λ^{2}})}

χ = \frac{1}{ϕ \pm \sqrt{ϕ^{2} - λ^{2}}}

Solving Quadratic Equation for Flexural Buckling

Author

Mr. Böttcher provides technical support for customers of Dlubal Software and takes care of their requests.

References

European Committee for Standardization. (2005). Eurocode 3: Design of Steel Structures, Part 1-1: General Rules and Rules for Buildings, EN 1993-1-1:2005. Brussels: CEN.

Do you have any questions?