Euler's Theory of Buckling MCQ Quiz in मराठी - Objective Question with Answer for Euler's Theory of Buckling - मोफत PDF डाउनलोड करा

Concepts:

Buckling load for the column,

Where,

P_e = Buckling load, I_min = Moment of inertia about centroids axis, L_e = Effective length.

Effective length (L_e),

When both ends of the column hinged (L_e) = L

When both ends of the column fixed (L_e) = 

Calculation:

Given:

Buckling load for both ends hinged = 20kN and L_e = L

  = 20 kN.

Then, buckling load for the column when both ends fixed and L_e = 

P_e= 4 × 20 = 80 kN.

Concept:

Euler’s crippling load:

Assuming the guided end to be fixed and the other end is given as hinged. Euler’s crippling load formula is used to find the buckling load of long columns.

It is given by

Where E = Modulus of elasticity, I = moment of inertia, leff = Effective length of the column

The Effective length of the column depending on ends conditions is as follows:

Calculations:

Given : Diameter (d) = 20 mm

Length (L) = 700 mm

Force (P) = 10 kN (compressive)

Assuming guided end to be fixed and other end given as hinged. The crippling load according to Euler’s equation.

Factor of safety 

Explanation:

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

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:

Columns and Struts:

A structural member, subjected to an axial compressive force, is called a strut. A strut may be horizontal, inclined or even vertical. But a vertical strut, used in buildings or frames, is called a column.

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

Where P_e = Buckling load, I_min = Minimum of [I_xx & I_yy], L = Actual length of the column, α = Length fixity coefficient, n = End fixity coefficient 

L_e = αL and 

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:

Additional Information

(1) The vertical column will have two moments of inertia (i.e. I_xx and I_yy). Since the column will tend to buckle in the direction of the least moment of inertia, therefore the least value of the two moments of inertia is to be used in Euler's formula.

(2) Euler's formula is given by

∵  I = Ak²,

Where A is the area and k is the least radius of gyration of the section

(3) The ratio (L/k) is known as the slenderness ratio.

(4) Sometimes, the columns whose slenderness ratio is more than 80, are known as long columns, and those whose slenderness ratio is less than 80 are known as short columns.

(5) Euler's formula holds good only for long columns.

(6) For short or long columns Rankine’s Formula is used.

Concept:

Elastic critical load in columns is given by Euler’s Formula which is,  

Where, is the critical load of the column, E is the modulus of elasticity of the column, I is the moment of inertia of the column cross section,  is the effective length of the column.

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

Now, moment of inertia can be expressed as,  where A is the cross sectional area of the column and r is the radius of gyration for the column cross section.

 where λ is slenderness ratio.

∴ Elastic critical stress, 

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 analysed with the Euler’s column formulas can be given as

Calculation:

P₁ = both ends hinged = 100 kN, P₂ = one end fixed and other end free

P ∝ 1/L_eq²

∴ P₂ = 25 kN

Concept:

Euler’s critical buckling load 

Where, Imin = minimum area moment of Inertia, E = young’s modulus of elasticity, Le = Equivalent length = L ⋅ α

α for different loading conditions are different

Calculation:

Given:

Two columns,

both ends are fixed, L_e = L × 0.5 

both ends are hinged L_e = L × 1

Concept:

Calculation:

A steel column is simply supported at both ends means it is a case of both ends hinged:

Given: b = 15 mm, d = 10 mm, L = 1.5 m = 1500 mm, E = 200 GPa = 200 × 1000 MPa

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 analysed with the Euler’s column formulas can be given as

• 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:

Application:

n_Fixed-Fixed > n_Fixed-Hinged > n_Hing-Hing > n_Fixed-Free

PFixed-Fixed > PFixed-Hinged > PHing-Hing > PFixed-Free

So highest buckling load will be when both end fixed.

The following assumptions are made in the Euler’s column theory about long column:

• The column is initially perfectly straight, and the 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