Euler's Theory of Buckling MCQ Quiz - Objective Question with Answer for Euler's Theory of Buckling - Download Free PDF

Concept:

The crippling load for a strut with one end fixed and the other hinged is calculated using Euler's formula: \( P = \frac{\pi^2 E I}{L_e^2} \)

Where, \( L_e = \frac{L}{\sqrt{2}} \) for this end condition.

Given:

L = 4 m = 4000 mm, d = 80 mm, E = 2 × 10⁵ N/mm², π³ = 31

Calculation:

Effective length, \( L_e = \frac{4000}{\sqrt{2}} = 2828.4 \, \text{mm} \)

Moment of Inertia, \( I = \frac{\pi d^4}{64} = \frac{3.14 \times 80^4}{64} = 2010613 \, \text{mm}^4 \)

Crippling Load, \( P = \frac{\pi^2 \times 2 \times 10^5 \times 2010613}{(2828.4)^2} = 495142.4 \, \text{N} \approx 496 \, \text{kN} \)

Explanation:

Euler's Equation for Buckling

• Euler's equation for buckling describes the critical load at which a slender, straight column under axial compression will buckle.
• This equation is applicable to idealized columns that are long and slender, with ends that are pinned (hinged) or fixed, and it assumes that the column is perfectly straight and homogeneous.
• Buckling is a failure mode characterized by a sudden lateral deflection of a column under axial load. Euler's formula predicts the critical load (P_cr) at which a column will buckle. The critical load is given by:

Formula:

\(P_{cr} = \frac{{{n^2}{\pi ^2}EI}}{{{L^2}}}\)

Where:

• P_cr = Critical load at which buckling occurs
• E = Modulus of elasticity of the material
• I = Moment of inertia of the cross-section about the axis of buckling
• K = Column effective length factor (depends on end conditions, e.g., 1 for pinned-pinned, 0.5 for fixed-fixed)
• L = Actual length of the column

Applicability:

• Euler's equation is most applicable to long, slender columns where buckling occurs before the material yields. These columns are characterized by having a high slenderness ratio (length to radius of gyration). The formula is derived based on the assumption that the column will fail due to buckling rather than crushing or yielding of the material.

Advantages:

• Provides a simple and reliable method to predict the buckling load for slender columns.
• Helps in the design of structures by ensuring that columns are prevented from buckling under expected loads.

Disadvantages:

• Not applicable for short, stocky columns where yielding may occur before buckling.
• Assumes ideal conditions (perfectly straight, homogeneous material), which may not be true in real-world applications.

Applications:

•  Euler's buckling theory is widely used in the design of structural elements in buildings, bridges, towers, and other structures where long, slender columns are used. It helps in determining the safe load limits to avoid buckling failure.

Concept:

Euler's Crippling Load for a strut with both ends hinged is given by:

\( P = \frac{\pi^2 \cdot E \cdot I}{L^2} \)

• \( L = 4 \, \text{m} = 4000 \, \text{mm} \)
• \( d = 64 \, \text{mm} \)
• \( E = 2 \times 10^5 \, \text{N/mm}^2 \)
• \( I = \frac{\pi d^4}{64} \)

Substituting values and solving gives:

\( P \approx 101601.6 \, \text{N} \)

Concept:

Effective length is defined as the distance between two adjacent points of zero bending moment or contra flexure.

Clamped free column means one end fixed and other ends free.

The value of the effective length varies according to the support condition.

Explanation:

According to Euler's column theory, the critical load (P) on the column for different types of end conditions is as follows:

\({{\rm{P}}_{\rm{e}}} = \frac{{{\rm{\pi E}}{{\rm{I}}_{{\rm{min}}}}}}{{{\rm{L}}_{\rm{e}}^2}} = \frac{{{\rm{n}}{{\rm{\pi }}^2}{\rm{E}}{{\rm{I}}_{{\rm{min}}}}}}{{{{\rm{L}}^2}}}\)

Where Pe = Buckling load, Imin = Minimum of [Ixx & Iyy], L = Actual length of the column, α = Length fixity coefficient, n = End fixity coefficient 

Le = αL and \({\bf{n}} = \frac{1}{{{\alpha ^2}}}\)

Euler's formula holds good only for long columns.

The Length fixity coefficient and End fixity coefficient for the given end conditions are given in the following table:

Explanation:

The buckling load of a given material depends on Slenderness ratio, Area of a cross-section, and Modulus of elasticity.

Buckling load:

Analysis of long column is done using Euler’s formula:

Elastic Critical stress (fcr)

\({{\rm{f}}_{{\rm{cr}}}}{\rm{\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{E}}}}{{{{\rm{\lambda }}^2}}}\)

where E = Modulus of elasticity of the material, and λ = slenderness Ratio

\({\rm{\lambda \;}} = {\rm{\;}}\frac{{{\rm{l}}{{\rm{e}}_{{\rm{ff}}}}}}{{{{\rm{r}}_{{\rm{min}}}}}}\)  & Imin = A × rmin2

where rmin = Minimum radius of gyration of the section, Imin = Minor principle Moment of Inertia, A = Cross-sectional Area, and Leff­ = It depends on the end conditions of the column section.

Eulers Load is given as

\({\rm{P\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{E}}}}{{{\rm{l}}_{{\rm{eff}}}^2}}{\rm{r}}_{{\rm{min}}}^2 \times {\rm{A\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{E\;Imin}}}}{{{\rm{l}}_{{\rm{eff}}}^2}}\)

Thus by Euler’s load formula,

\({\rm{Load\;}} \propto {\rm{\;}}{{\rm{I}}_{{\rm{min}}}}{\rm{\;and\;Load}} \propto \frac{1}{{{{\left( {{{\rm{l}}_{{\rm{eff}}}}} \right)}^2}}}\)

The load at which column buckle is termed as buckling load. Buckling load is given by:

\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{L_e^2}}\)

where E = Young’s modulus of elasticity, Imin = Minimum moment of inertia, and Le = Effective length

We know that MOI for a circular section is \(I = \frac{\pi d^4}{64}\)

Explanation:

Euler buckling load for a column is given by:

\({{\rm{P}}_{\rm{E}}}{\rm{\;}} = {\rm{\;}}\frac{{{{\rm{\pi }}^2}{\rm{EI}}}}{{{\rm{L}}_{\rm{e}}^2}}\)

where, Le = is the effective length of the column that depends on the end support conditions and EI is the flexural rigidity of the column.

Now, effective length (Le) for different end conditions in terms of actual length (L) are listed in the following table:

Putting the values of Le we will get:

\({{\rm{P}}_{\rm{E}}}{\rm{}} = {\rm{}}\frac{{{{\rm{\pi }}^2}{\rm{EI}}}}{{{\rm{L}}_{\rm{}}^2}}\)   [Both ends hinged]

\({{\rm{P}}_{\rm{E}}}{\rm{}} = {\rm{}}\frac{{{{\rm{2\pi }}^2}{\rm{EI}}}}{{{\rm{L}}_{\rm{}}^2}}\)  [One end hinged other end fixed]

\({{\rm{P}}_{\rm{E}}}{\rm{}} = {\rm{}}\frac{{{{\rm{4\pi }}^2}{\rm{EI}}}}{{{\rm{L}}_{\rm{}}^2}}\)  [Both ends fixed]

\({{\rm{P}}_{\rm{E}}}{\rm{}} = {\rm{}}\frac{{{{\rm{\pi }}^2}{\rm{EI}}}}{{{\rm{4L}}_{\rm{}}^2}}\)  [One end is fixed and another end free]

Thus the maximum value we get is for both ends fixed.

Additional Information

Compression member:

A compression member is a structural member which is straight and subjected to two equal and opposite compressive forces applied at its ends. They are of two types:

1. Column
2. Strut

Column:

• The structural member carries the axial load and it is vertical & both ends are fixed.

Strut:

• The structural member which carries axial load & it can be or cannot be vertical, one end can be free & other fixed, one end can be hinged & other fixed.

Explanation:

According to Euler's column theory, the crippling load for a column of length L,

\({P_{cr}} = \frac{{{\pi ^2}EA}}{{{{\left( {\frac{{{L_{eq}}}}{r}} \right)}^2}}}\)

Where Leq is the effective length of the column.

The following assumptions are made in Euler's column theory:

• The column is initially straight and load is applied axially
• The cross-section of the column is uniform throughout its length
• The column material is perfectly elastic, homogeneous and isotropic and obeys Hooke’s law
• The length of the column is very large as compared to its lateral dimensions
• The direct stress is very small as compared to the bending stress
• The column will fail by buckling alone
• The self-weight of the column is negligible

Explanation:

Column:

• If a beam element is under a compressive load and its length is an order of magnitude larger than either of its other dimensions such a beam is called a column.
• Due to its size, its axial displacement is going to be very small compared to its lateral deflection called buckling.

Euler's Buckling Load:

\( {P_{cr}} = \frac{{{{\rm{\pi }}^2}E{I_{\min }}}}{{l_e^2}}\)

where, P_cr = crtical load for buckling; E = Young's Modulus (GPa); I_min = Area moment of inertia, l_e = effective length; k_min  = minimum raduis of gyration

Slenderness Ratio (S): 

The ratio between the length and least radius of gyration.

\(S = \frac{{{l_e}}}{{{k_{\min }}}}\)

Hence, buckling load does not depend on the area of the cross-section.

Additional Information

Buckling load for various end conditions is given in the table below.

Concept:

Buckling load:

The load at which column buckle is termed as buckling load. Buckling load is given by:

\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{L_e^2}}\)

where E = Young’s modulus of elasticity, I_min = Minimum moment of inertia, and L_e = Effective length

Calculation:

Given:

Critical Buckling load for column fixed at both ends (Pcr₁) \(= \frac{{4{\pi ^2}EI}}{{\begin{array}{*{20}{c}} {{L^2}} \end{array}}}\)

Critical Buckling load for a column hinged at both ends (Pcr₂) = \(\frac{{{\pi ^2}EI}}{{{L^2}}}\)

\(\therefore\frac{(P_{cr})_1}{(P_{cr})_2}=4\)

Concept:

The maximum load at which the column tends to have lateral displacement or tends to buckle is known as buckling or crippling load.

Load columns can be analyzed with the Euler’s column formulas, can be given as:

\(P = \frac{{{n^2}{\pi ^2}EI}}{{{L^2}}}\)

• For both end hinged, n = 1
• For one end fixed and other free, n = 1/2
• For both end fixed, n = 2
• For one end fixed and other hinged, n = √2

Effective Length:

\(L_{eq}=\frac{L}{n}\)

Calculation:

\({P_{fixed - fixed}} > {P_{fixed - hinged}} > {P_{hinged - hinged}} > {P_{fixed - free}}\)

p₂ > p₄ > p₁ > p₃

Explanation:

Buckling load:
The load at which column buckle is termed as buckling load. Buckling load is given by:

\({P_b} = \frac{{{\pi ^2}EI}}{{L_e^2}}\)
where E = Young's modulus of elasticity, Imin = Minimum moment of inertia, and Le = Effective length

\({P_{b1}} = \frac{{{\pi ^2}EI}}{{L_e^2}}\)------------(1)

\({P_{b2}} = \frac{{{\pi ^2}EI}}{{4L_e^2}}\)------------(2)

Dividing equation (2) by (1), we get

\(\frac{{{P_{b2}}}}{{{P_{b1}}}} = \frac{1}{4}\)

Hence the critical load will be One-fourth of the original value.

Concept:

Struts:

These are long and slender structural members in an assembly that are subjected only to an axial compressive force.

They predominantly fail by buckling but sometimes the yielding failure occurs in compression before buckling.

These members are structurally analogous to columns. So, the buckling strength of struts is similar to that of the column and can be taken as,

\({{\rm{P}}_{{\rm{buck}}}} = {\rm{K}}\frac{{{{\rm{\pi }}^2}{\rm{EI}}}}{{{\rm{L}}_{\rm{}}^2}},{\rm{\;K\;}}\)is a factor depending upon the end conditions.

\(\therefore {\rm{\;}}{{\rm{P}}_{{\rm{buck}}}} \propto \frac{{{\rm{EI}}}}{{{\rm{L}}_{\rm{e}}^2}} \propto \frac{{{\rm{EA}}{{\rm{r}}^2}}}{{{\rm{L}}_{\rm{e}}^2}} \propto \frac{{{\rm{EA}}}}{{{{\left( {\frac{{{{\rm{L}}_{\rm{e}}}}}{{\rm{r}}}} \right)}^2}}} \propto \frac{{{\rm{EA}}}}{{{{\left( {\rm{\lambda }} \right)}^2}}}{\rm{\;\;}}\;\;\)

\(\therefore {{\rm{P}}_{{\rm{buck}}}} \propto \frac{1}{{{{\left( {\rm{\lambda }} \right)}^2}}}\;\) where λ is the slenderness ratio.

where,

P_buck is the buckling strength of the strut,

E is the modulus of elasticity of the strut,

I is the moment of inertia, 

A is the cross-section and

L_e effective length of the strut respectively.

Note: The net area of the strut is mainly the deciding factor in case for yielding failure but not for the buckling strength failure.