Columns and Struts free study notes

1. What is Columns and Struts?

Columns and struts are structural members subjected primarily to axial compressive loads. They are widely used in civil, mechanical, and structural engineering applications.

• A column is a vertical member carrying compressive load (e.g., building pillars).
• A strut is a member in any orientation (vertical, horizontal, or inclined) subjected to compressive force (e.g., truss members, machine components).

2. Difference Between Columns and Struts

Feature	Column	Strut
Orientation	Vertical	Any direction
Usage	Buildings, structures	Trusses, frames, machines
Example	Pillar in a building	Member in a bridge truss

3. Classification of Columns

(a) Based on Slenderness Ratio (λ)

Slenderness ratio determines the failure mode of a column. Slenderness ratio is the ratio of the effective length of a column to its least radius of gyration. It indicates the tendency of a column to buckle under compressive load.$$\text{Slenderness Ratio} = \frac{L_e}{k}$$

Where:

• $\lambda$ = Slenderness ratio
• $L_e$​ = Effective length
• $k$ = Radius of gyration

$\lambda = \frac{L_e}{k}$

Types:

1. Short Columns
   • Low slenderness ratio
   • Fail by crushing
2. Long Columns
   • High slenderness ratio
   • Fail by buckling
3. Intermediate Columns
   • Combination of crushing and buckling

(b) Based on End Conditions

Different end supports affect the effective length:

End Condition	Effective Length $L_e$Le​
Both ends hinged	$L$L
One end fixed, other free	$2L$2L
Both ends fixed	$L/2$L/2
One end fixed, other hinged	$L/\sqrt{2}$L/2​

4. Buckling of Columns

When a compressive load is applied to a perfectly straight column, it initially shortens slightly. As the load reaches a certain value called the critical load, the column suddenly bends sideways and becomes unstable. Long and slender columns are more likely to fail by buckling before the material fails in compression.

This minimum load at which buckling begins is known as the critical buckling load or Euler load.

Causes of Buckling

• High compressive load
• Large slenderness ratio
• Long unsupported length
• Initial imperfections in the column
• Eccentric loading
• Residual stresses
• Inadequate lateral support

Factors Affecting Buckling

The buckling strength depends on:

• Length of the column
• Cross-sectional shape
• Material properties (Young’s Modulus, E)
• End support conditions
• Moment of inertia (I)
• Radius of gyration (r)

5. Euler’s Theory for Long Columns

Applicable for long, slender columns.

The critical load at which buckling occurs is given by:$$P_{cr} = \frac{\pi^2 E I}{L_e^2}$$

Where:

• $P_{cr}$ = Critical (buckling) load
• $E$ = Young’s modulus
• $I$ = Moment of inertia
• $L_e$​ = Effective length

Assumptions:

• Column is perfectly straight
• Load is axial
• Material is homogeneous and elastic
• No initial imperfections

6. Rankine’s Formula (For All Columns)

Used for intermediate columns, combining crushing and buckling effects:$$P = \frac{\sigma_c A}{1 + a\left(\frac{L_e}{k}\right)^2}$$

Where:

• $\sigma_c$ = Crushing stress
• $A$ = Cross-sectional area
• $a$ = Rankine constant

7. Failure of Columns

Columns may fail due to:

1. Crushing (Short Columns)
   • Direct compressive stress exceeds material strength
2. Buckling (Long Columns)
   • Lateral deflection causes instability
3. Combined Failure (Intermediate Columns)

8. Radius of Gyration

Radius of gyration is the distance from the centroid of a cross-section at which the entire area of the section can be assumed to be concentrated without changing its moment of inertia.

It is a measure of how the cross-sectional area is distributed about an axis. A larger radius of gyration means the material is distributed farther from the centroid, making the column more resistant to buckling. It represents how the area is distributed about an axis:$$k = \sqrt{\frac{I}{A}}$$

Where:

• k = Radius of gyration (mm or m)
• I = Moment of inertia about the axis (mm⁴ or m⁴)
• A = Cross-sectional area (mm² or m²)

9. Factors Affecting Strength of Columns

• Length of column
• Cross-sectional shape
• End conditions
• Material properties
• Slenderness ratio
• Eccentricity of load

10. Practical Applications

• Building columns and pillars
• Bridge supports
• Transmission towers
• Machine components (connecting rods, frames)
• Structural frameworks

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