Short Answer:

The crippling load is the maximum compressive load at which a column or strut fails by buckling or bending sideways. It is also known as the buckling load or critical load. When a column is subjected to an axial compressive force, it remains straight up to a certain limit, but beyond this load, it suddenly bends and loses stability — this load is called the crippling load.

In simple terms, the crippling load is the highest safe compressive load that a column can carry without bending or collapsing. It depends on the material properties, length, cross-section, and end conditions of the column.

Detailed Explanation :

Crippling Load

The crippling load is defined as the axial compressive load at which a long and slender column becomes unstable and bends or buckles. It is the limiting load that the column can safely bear before it loses its ability to carry the load in a straight position. This phenomenon occurs due to buckling, which is a form of elastic instability rather than material failure.

When a column is gradually loaded, it remains straight up to a certain value of load. As the load increases beyond a particular limit, the column suddenly bends sideways even though the stress in the material is below the crushing strength. This sudden bending is called buckling, and the corresponding load is known as the crippling load or critical buckling load.

Behavior of Column under Axial Load

1. When the column is short and thick, it usually fails due to crushing, as the compressive stress reaches the material’s yield strength before any bending occurs.
2. When the column is long and slender, it fails by buckling under a load much smaller than the crushing load.
3. When the column is intermediate, the failure may occur due to a combination of crushing and buckling.

Thus, the crippling load mainly applies to long slender columns, where buckling dominates over crushing failure.

Expression for Crippling Load (Euler’s Formula)

For long slender columns, the Euler’s theory gives the mathematical expression for crippling load as:

Where,

•  = Crippling or critical load (N)
•  = Modulus of elasticity of the material (N/m²)
•  = Moment of inertia of the least cross-sectional axis (m⁴)
•  = Effective length of the column (m)

The effective length depends on the end conditions of the column:

• Both ends hinged →
• Both ends fixed →
• One end fixed, other free →
• One end fixed, other hinged →

The crippling load varies inversely with the square of the effective length, meaning that longer columns buckle under smaller loads.

Derivation (Conceptual)

Consider a long column of length , fixed or hinged at the ends, subjected to an axial compressive load .
As per bending theory:

But for a column under compression:

Substituting, we get:

Solving this differential equation with appropriate boundary conditions, we obtain:

This is the Euler’s buckling or crippling load equation, representing the maximum load a column can carry before buckling.

Factors Affecting Crippling Load

1. Length of Column:
   Longer columns have lower crippling load since buckling tendency increases with length.
2. End Conditions:
   Columns with fixed ends can carry more load than those with hinged or free ends because of shorter effective length.
3. Moment of Inertia (I):
   Cross-sections with higher moment of inertia resist bending better and hence have higher crippling load.
4. Material Property (E):
   Materials with higher modulus of elasticity can withstand greater loads before buckling.
5. Slenderness Ratio (L/r):
   Columns with higher slenderness ratio are more prone to buckling, reducing crippling load.

Modes of Failure in Columns

1. Crushing Failure:
   Occurs in short columns when the compressive load exceeds the material’s yield strength.

where  = crushing stress and  = cross-sectional area.

1. Buckling (Crippling) Failure:
   Occurs in long columns due to instability before the material reaches its yield point.

1. Combined Failure:
   In intermediate columns, failure occurs due to both crushing and buckling effects.

Rankine’s Formula for Crippling Load

For intermediate columns, neither pure Euler’s formula (for long columns) nor pure crushing load (for short columns) gives accurate results.
Hence, Rankine’s formula is used to combine both effects:

or

where,

•  = Crushing strength of the material
•  = Cross-sectional area
•  = Constant depending on material type

This formula gives a more realistic value of the crippling load, applicable to all column lengths.

Importance of Crippling Load

1. Ensures Structural Safety:
   It helps engineers design columns that can carry the load without sudden buckling.
2. Prevents Instability:
   Determines the load beyond which the column becomes unstable.
3. Used in Design Codes:
   Building codes and standards use crippling load calculations to ensure safe design.
4. Applies to Various Structures:
   Used in designing bridges, towers, machine frames, and building supports.

Example

A steel column with , , and .
The crippling load is:

 

Hence, the column will start to buckle if the load exceeds 24.67 MN.

Conclusion

The crippling load is the maximum compressive load a column can bear before it becomes unstable and buckles. It depends on the material’s modulus of elasticity, moment of inertia, and effective length. For long columns, Euler’s formula accurately determines the crippling load, while for intermediate columns, Rankine’s formula gives a better approximation. Understanding the crippling load helps engineers design safe and stable structures that can resist compressive stresses without bending or collapsing.