For a column of length (L) and flexural rigidity (EI) which has one end fixed and other end free, the expression for critical load is given as -

Question

Prepp Team · Accepted Answer

Understanding Column Buckling and Critical Load

Columns are structural elements that are primarily subjected to axial compressive loads. When a column is subjected to an increasing compressive load, it can fail in one of two ways: by yielding of the material or by buckling. Buckling is a sudden lateral instability that causes the column to bend significantly, even if the material stress is below the yield strength. The load at which buckling occurs is called the critical load or buckling load.

Euler's theory provides a method to calculate the critical load for ideal columns under various end conditions. An ideal column is perfectly straight, made of homogeneous material, and the load is applied perfectly axially.

Euler's Critical Load Formula

The general formula for Euler's critical load ($P_e$) is given by:

$P_e = \frac{\pi^2 EI}{L_e^2}$

Where:

$P_e$ is the Euler's critical load.
$E$ is the modulus of elasticity of the column material.
$I$ is the minimum moment of inertia of the column's cross-section.
$L_e$ is the effective length of the column.

The effective length ($L_e$) is a crucial concept in Euler's theory. It represents the length of an equivalent pin-ended column that would buckle under the same critical load. The effective length depends on the boundary conditions at the ends of the column. It is typically expressed as $L_e = kL$, where $L$ is the actual length of the column and $k$ is the effective length factor.

Analyzing the Fixed-Free Column End Condition

The question specifies a column with one end fixed and the other end free. Let's consider this specific boundary condition to determine the effective length factor ($k$).

For a column with one end fixed and the other end free:

The fixed end prevents both translation and rotation.
The free end allows both translation and rotation.

Under buckling, the fixed end remains vertical, while the free end displaces laterally and rotates. The shape of the buckled column for this end condition resembles half of a sine wave from a pin-pin column. This means the effective length is twice the actual length.

The effective length factor ($k$) for a fixed-free column is 2.

Therefore, the effective length ($L_e$) for a fixed-free column is:

$L_e = 2L$

Calculating the Critical Load for Fixed-Free Column

Now, we substitute the effective length $L_e = 2L$ into Euler's critical load formula:

$P_e = \frac{\pi^2 EI}{L_e^2}$

$P_e = \frac{\pi^2 EI}{(2L)^2}$

$P_e = \frac{\pi^2 EI}{4L^2}$

So, the expression for the critical load for a column of length $L$ and flexural rigidity $EI$ with one end fixed and the other end free is $\frac{\pi^2 EI}{4L^2}$.

Comparing with Given Options

Let's compare our derived formula with the given options:

Option 1: $P = \frac{\pi^2 EI}{L^2}$ (This corresponds to a pin-ended column, $k=1$).
Option 2: $P = \frac{4\pi^2 EI}{L^2}$ (This corresponds to a fixed-fixed column, $k=0.5$, $L_e = 0.5L$, $P_e = \frac{\pi^2 EI}{(0.5L)^2} = \frac{\pi^2 EI}{0.25L^2} = \frac{4\pi^2 EI}{L^2}$).
Option 3: $P = \frac{2\pi^2 EI}{L^2}$ (This does not correspond to a standard end condition).
Option 4: $P = \frac{\pi^2 EI}{4L^2}$ (This matches our derived formula for a fixed-free column, $k=2$).

Based on our derivation, the correct expression for the critical load for a fixed-free column is $\frac{\pi^2 EI}{4L^2}$.

Effective Length Factors for Different End Conditions
End Condition	Effective Length Factor (k)	Effective Length (L_e)	Critical Load (P_e)
Pin-Pinned (Hinged-Hinged)	1.0	L	$\frac{\pi^2 EI}{L^2}$
Fixed-Fixed	0.5 (theoretically), 0.65 (recommended)	0.5L or 0.65L	$\frac{\pi^2 EI}{(0.5L)^2} = \frac{4\pi^2 EI}{L^2}$ (theor.)
Fixed-Pinned (Fixed-Hinged)	0.7 (theoretically), 0.80 (recommended)	0.7L or 0.80L	$\frac{\pi^2 EI}{(0.7L)^2} \approx \frac{2\pi^2 EI}{L^2}$ (theor.)
Fixed-Free	2.0	2L	$\frac{\pi^2 EI}{(2L)^2} = \frac{\pi^2 EI}{4L^2}$

Revision Table: Column Buckling Concepts

Here's a quick summary of key concepts related to column buckling and critical load:

Buckling: Structural instability of a column under compression.
Critical Load: The minimum compressive load causing a column to buckle.
Euler's Theory: Predicts critical load for slender, ideal columns.
Flexural Rigidity (EI): Measure of a beam's or column's resistance to bending. E is Modulus of Elasticity, I is Moment of Inertia.
Effective Length (L_e): Length of an equivalent pin-ended column; $L_e = kL$.
Effective Length Factor (k): Depends on column end boundary conditions.

Additional Information: Beyond Euler's Formula

While Euler's formula is fundamental for understanding column buckling, it has limitations:

Applicability: It is primarily valid for long (slender) columns where buckling occurs at stresses below the material's proportional limit.
Short/Intermediate Columns: For shorter columns, material yielding or crushing may occur before or simultaneously with buckling. Formulas like the Rankine-Gordon formula or Johnson's parabolic formula are used for intermediate columns.
Ideal Conditions: Euler's theory assumes ideal conditions (perfectly straight column, perfectly axial load, homogeneous material). Real-world columns have imperfections that can reduce the actual buckling load.
Slenderness Ratio: The ratio of the effective length ($L_e$) to the minimum radius of gyration ($r$) of the cross-section ($\lambda = L_e/r$). This ratio is critical in determining whether a column is short, intermediate, or long, and thus which formula (Euler or other) is appropriate. $I = Ar^2$, where A is the cross-sectional area.

Understanding the end conditions and the concept of effective length is vital for correctly applying Euler's formula and analyzing column stability.

Prepp Team · Answer

P = &pi;&sup2;EI/L&sup2;

Prepp Team · Answer

P = 4&pi;&sup2;EI/L&sup2;

Prepp Team · Answer

P = 2&pi;&sup2;EI/L&sup2;