5. Columns, CE 3003 notes in English, Mechanics of Materials notes in English

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5. Columns

5.1 Concept of Compression Member, Short and Long Columns, Effective Length, Radius of Gyration, Slenderness Ratio, Types of End Conditions for Columns, Buckling of Axially Loaded Columns

Compression Member:

A compression member is a structural element subjected primarily to compressive forces. Columns are a common example of compression members. They are designed to carry loads that compress them along their axis, as opposed to beams, which are designed to resist bending.

Short and Long Columns:

• Short Column: A column is considered short when its length is small compared to its cross-sectional dimensions. For short columns, the material strength governs, and they fail by direct crushing under the compressive load.
• Long Column: A column is considered long if its length is large relative to its cross-sectional dimensions. For long columns, the failure is typically due to buckling (lateral deformation) rather than material crushing.

Effective Length (Le):

The effective length of a column is the length of the column that is used in buckling calculations, considering the type of end conditions (restraining effects at both ends). The effective length is not always the actual physical length of the column.

Radius of Gyration (r):

The radius of gyration of a column is defined as:

$$r = \sqrt{\frac{I}{A}}$$

where:

• $I$ is the moment of inertia of the column's cross-sectional area about the axis of bending.
• $A$ is the cross-sectional area of the column.

The radius of gyration is a measure of how the cross-sectional area is distributed about the axis and impacts the column's susceptibility to buckling.

Slenderness Ratio ($\lambda$):

The slenderness ratio is a dimensionless measure of a column's susceptibility to buckling and is defined as the ratio of the effective length to the radius of gyration:

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

where:

• $L_e$ = Effective length of the column.
• $r$ = Radius of gyration.

Columns with a higher slenderness ratio are more likely to buckle under a given load.

Types of End Conditions for Columns:

The behavior of a column during buckling depends on the boundary conditions at its ends. The most common types of end conditions are:

1. Both Ends Fixed: The column is fixed at both ends, preventing both translation and rotation. This provides the greatest resistance to buckling.
2. One End Fixed, Other End Free: One end of the column is fixed, while the other end is free to move. This results in a high degree of buckling and lower buckling strength.
3. Both Ends Pinned: The column is pinned at both ends, allowing it to rotate but preventing translation. This condition typically results in an intermediate level of buckling resistance.
4. One End Pinned, Other End Fixed: One end is pinned, and the other end is fixed, resulting in moderate buckling strength.

Buckling of Axially Loaded Columns:

When a column is subjected to an axial compressive load, it may undergo buckling if the load exceeds a critical value. Buckling is the sudden lateral deflection of the column, caused by the instability of the column under axial load. The tendency to buckle increases with column length and decreases with the column's cross-sectional area.

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5.2 Euler’s Theory, Assumptions Made in Euler’s Theory, and Its Limitations, Application of Euler’s Equation to Calculate Buckling Load

Euler’s Theory:

Euler’s theory of buckling describes the buckling behavior of long columns. According to this theory, the critical buckling load (also called the Euler's buckling load) is given by the equation:

$$P_{cr} = \frac{\pi^2 E I}{(L_e)^2}$$

where:

• $P_{cr}$ = Critical buckling load (N)
• $E$ = Modulus of elasticity of the material (N/m² or Pa)
• $I$ = Moment of inertia of the column's cross-section (m⁴)
• $L_e$ = Effective length of the column (m)

Assumptions Made in Euler’s Theory:

1. The column is straight and has a uniform cross-section.
2. The column is subjected to a centrally applied axial load.
3. The column material is homogeneous and behaves according to Hooke's law (elastic material).
4. The column is initially straight, and any lateral displacement occurs due to the applied load.
5. The column is perfectly pinned or fixed at the ends, as per the boundary conditions assumed.

Limitations of Euler’s Theory:

1. Euler’s theory applies primarily to long columns (slenderness ratio $\lambda > 100$).
2. It assumes the column deforms elastically (i.e., it does not take into account any inelastic or plastic deformations).
3. The theory assumes no imperfections or initial deflection in the column.
4. It assumes that the material’s modulus of elasticity (E) is constant and does not change under load.

Euler’s theory provides an idealized and theoretical prediction for buckling, but real-world columns may fail at loads lower than the Euler load due to factors such as imperfections, material nonlinearity, and lateral forces.

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5.3 Rankine’s Formula and Its Application to Calculate Crippling Load

Rankine’s Formula:

Rankine’s formula provides a more practical approach for calculating the crippling load for both short and long columns. It combines the effects of both material strength (for short columns) and buckling (for long columns). The formula is:

$$P_{cr} = \frac{A}{\left(\frac{1}{\sigma_{cr}} + \frac{1}{\sigma_{b}}\right)}$$

where:

• $P_{cr}$ = Crippling load (N)
• $A$ = Cross-sectional area of the column (m²)
• $\sigma_{cr}$ = Compressive strength of the material (N/m² or Pa)
• $\sigma_b$ = Theoretical buckling stress based on Euler’s theory (N/m² or Pa)

Application of Rankine’s Formula:

Rankine’s formula is widely used to determine the critical load for columns where both buckling and material strength factors are significant. It is typically applied when:

• The column is not purely short or long, but somewhere in between.
• It accounts for the material's yield stress and the column's geometrical properties.

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5.4 Concept of Working Load/Safe Load, Design Load, and Factor of Safety

Working Load/Safe Load:

The working load (or safe load) is the maximum load that a column or structural member is designed to carry during its normal service. It is always less than or equal to the crippling load to ensure safety and avoid failure. The working load is determined based on the strength of the material and the design codes.

Design Load:

The design load is the load considered in the design process, which may include safety factors. The design load is the load for which the column or structural member is designed, considering all potential load combinations (dead load, live load, etc.) and the load-bearing capacity.

Factor of Safety (FoS):

The factor of safety is a measure of the safety margin built into the design. It is the ratio of the load at which the structure fails (or the maximum allowable load) to the working load or design load. The factor of safety accounts for uncertainties in material properties, loading conditions, manufacturing defects, and other factors. It is typically defined as:

$$\text{Factor of Safety (FoS)} = \frac{\text{Ultimate Load (or Crippling Load)}}{\text{Working Load (or Design Load)}}$$

A factor of safety greater than 1 ensures that the column will not fail under the working load.