Columns and Struts free study notes

1. Introduction

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 Column and Strut

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

Slenderness ratio determines the failure mode of a column.$$\text{Slenderness Ratio} = \frac{L_e}{k}$$Slenderness Ratio=kLe​​

Where:

• $L_e$Le​ = Effective length
• $k$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 column is subjected to compressive load beyond a critical value, it becomes unstable and bends sideways. This phenomenon is called buckling.

5. Euler’s Theory for Long Columns

Applicable for long, slender columns.

The critical load at which buckling occurs is given by:$$P_{cr} = \frac{\pi^2 E I}{L_e^2}$$Pcr​=Le2​π2EI​

$P_{cr} = \frac{\pi^2 E I}{L_e^2}$Pcr​=Le2​π2EI​

Where:

• $P_{cr}$Pcr​ = Critical (buckling) load
• $E$E = Young’s modulus
• $I$I = Moment of inertia
• $L_e$Le​ = 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:$$P = \frac{\sigma_c A}{1 + a\left(\frac{L_e}{k}\right)^2}$$P=1+a(kLe​​)2σc​A​

$P = \frac{\sigma_c A}{1 + a\left(\frac{L_e}{k}\right)^2}$P=1+a(kLe​​)2σc​A​

Where:

• $\sigma_c$σc​ = Crushing stress
• $A$A = Cross-sectional area
• $a$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

It represents how the area is distributed about an axis:$$k = \sqrt{\frac{I}{A}}$$

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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