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001895

2024-09-09

Formula for Ideal Buckling Load of Individual Members Subjected to Compression

Every day, thousands of structural engineers design structural components using design check formulas that include the critical buckling load. But where do these old formulas come from, that Leonard Euler established over 200 years ago and that form the basis of all three structural steel design concepts?

The origin lies in the differential equation that describes the equilibrium of the internal moment of a structural component and the moment from the axial force multiplied by the eccentricity at each location x of the structural component.

Flexural Buckling of Imperfect Column

Therefore, an imperfection must always be present for the stability failure. This is applied as a function in the literature, but can be infinitesimally small and can be determined incoherently to the critical buckling load itself.

- M (x) + N \cdot w (x) = 0

w'' (x) \cdot E I + N \cdot w (x) = 0 | \div (E I)

w'' (x) + \frac{N}{E I} \cdot w (x) = 0

M(x)	Inner moment at each x-location
N	Acting compressive axial force
w(x)	Deflection/displacement at any location x
E	Modulus of elasticity
I	Second moment of area

First Differential Equation for Determining Critical Buckling Load

To reduce the number of constants, a new constant is introduced, which is used to substitute for N, E, and I.

α = \sqrt{\frac{N}{EI}}

Constant α

The differential equation of the equilibrium of internal and external forces can thus be rewritten as:

w'' (x) + α^{2} \cdot w (x) = 0

Euler's Differential Equation with α

Now, Leonhard Euler created the first approximation function for the deflection.

w (x) = e^{λ x}

Trial Function for Deflection

After creating the second derivation and subsequent insertion into the equation of the equilibrium of forces, it results in the following equation:

λ^{2} e^{λx} + α^{2} e^{λx} = 0

Inserting Differential Equation for Deflection in Differential Equation of Equilibrium of Forces

Since exponential functions can never become 0, the entire term can be divided by them.

λ^{2} + α^{2} = 0 | - α^{2}

λ^{2} = - α^{2} | \sqrt{}

λ_{1} = - i α

λ_{2} = i α

Setting up Correct Differential Equation for Deflection

As there are two independent complex solutions here, the differential equation of the deflection can be set up using a real linear combination of both complex equations:

w (x) = e^{iαx} + e^{- iαx}

Real Linear Combination of Differential Equation for Deflection

In this case, the functions can be rewritten as complex combinations, which corresponds to the principle of the unit circle. This becomes clear if you visualize the e-function using the Taylor series expansion:

f (x) = \sum_{n = 0}^{n = \infty} \frac{f^{(' n)} (0)}{n!} \cdot x^{n}

e^{i x} = \frac{1}{1} + \frac{i x}{1} + \frac{- x^{2}}{2} + \frac{i x^{3}}{6} + \frac{x^{4}}{24} + \frac{i x^{5}}{120} + . . .

('n)	n'th derivative of the initial function
i	(-1)^0.5

Taylor Series Expansion

If you consider the Maclaurin series of the sine and cosine function, the principle of the unit circle and thus the following adjustments of the differential equation can be explained:

i \cdot \sin (x) = \frac{1}{1} + \frac{0}{1} + \frac{- x^{2}}{2} + \frac{0}{6} + \frac{i x^{4}}{24} + \frac{0}{120} + . . .

\cos (x) = \frac{0}{1} + \frac{i x}{1} + \frac{0}{2} + \frac{i x^{3}}{6} + \frac{0}{24} + \frac{i x^{5}}{120} + . . .

Maclaurin Series

When considering these three series, the Euler's function is immediately available:

e^{i x} = i \sin (x) + \cos (x)

e^{i α x} = i \sin (α x) + \cos (α x)

Euler's Function

Since i is a complex number and the function can thus be divided into a real and an imaginary part, it is possible first to introduce the integration constants A and B, which are composed of the imaginary and the real parts:

w (x) = A e^{- i α x} + B e^{i α x}

A = (a_{1} + a_{2})

B = (b_{1} + b_{2})

A	Complex integration constant A = c₁
B	Complex integration constant B = c₂
a₁	Real part of the integration constant controlling the real part of the function
a₂	Imaginary part of the integration constant controlling the imaginary part of the function
b₁	Real part of the integration constant B controlling the real part of the function
b₂	Imaginary part of the integration constant B controlling the imaginary part of the function

Differential Equation of Deflection with Integration Constants

w (x) = A \cdot [i \sin (α x) + \cos (α x)] + B \cdot [- i \sin (α x) + \cos (α x)]

Converting Differential Equation to Sine-Cosine Function

Since the function reflects the deflection, which only contains real values, the function need to be reduced to its real part so that the equation includes the new integration constants A and B of purely real parts:

w (x) = A \sin (α x) + B \cos (α x)

Differential Equation of Deflection After Reduction to Real Component

This can be derived again for the function of the component curvature:

w^{'} (x) = A \cdot α \sin (α x) - B \cdot α \sin (α x)

w^{''} (x) = - A \cdot α^{2} \sin (α x) - B \cdot α^{2} \cos (α x)

Deriving Function of Component Curvature Using Euler's Differential Equation

Since a column with fully hinged supports on both sides will be considered in this example, the constants can be solved using the following boundary conditions here:

w (0) = 0 = A \sin (α \cdot 0) + B \cos (α \cdot 0)

0 = A \cdot 0 + B

0 = B

w (L) = 0 = A \sin (α \cdot L) = 0

Compliance with Boundary Conditions

Due to the sine function, this results in several solutions as a multiples of a natural number of π. These solutions can be defined as several eigenvalues of only one imperfection. It is important to understand that not all eigenvalues are calculated, but only several eigenvalues that arise from the same imperfection approach.

α \cdot L = n \cdot π

Smaller eigenvalue for n=1 corresponds to the smallesr and thus critical buckling load.

\sqrt{\frac{N}{EI}} \cdot L = π | {(. .)}^{2}

\frac{N}{EI} \cdot L^{2} = π^{2} | \cdot \frac{EI}{L^{2}}

N = \frac{π^{2} \cdot EI}{L^{2}}

Solving Differential Equation Using Trivial Solutions of Boundary Conditions

The great thing about this derivation is the principle of how independent the critical buckling load is of a selected imperfection. Although an imperfection must be applied, its amplitude is completely negligible; the calculation of the imperfection is incoherent with the calculation of the stability failure.

This illustrates how similar the various steel design check methods are: While the methods that use the equivalent member design calculate the resulting bending moment using this ideal buckling load with further reduction and imperfection factors, which leads to the ultimate limit state design, the other design methods apply an imperfection independent of the design check formulas whose equivalent loads produce the same bending moment as that resulting from the stability analysis. The stability analysis is thus included in all ultimate limit state design checks.

For the application of the amplitude of the imperfection according to the design method based on the cross-section check, with the combination of which the design load resulting from the ideal buckling load is obtained, the following article provides more information:

For the application of the amplitude according to the equivalent member method, with the combination of which the design load resulting from the ideal buckling load is obtained, this article provides more information: KB 1897 | Imperfections in Design Check Formulas for Flexural Buckling Analysis

In the case of lateral-torsional buckling, which is not covered in this article, there is space for discussion. The equations are more complex due to shear forces along the beam length. In the course of the geometric imperfections, there are not only bending moments occurring.

Links

KB 1897 | Imperfections in Design Check Formulas for Flexural Buckling Analysis

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