Columns and Struts

Contents

**Define the term Column, Strut, and Tie** 1

**Explain the Failure Modes of Column** 2

**Differentiate between the Buckling and Crushing Load of Failure** 3

**Explain Euler’s Theory of Column** 4

**Describe and calculate the Euler’s formula for Crippling Load when both ends Hinged** 5

**Describe and calculate the Euler’s Formula for Crippling Load when both ends Fixed** 7

**Explain the Effective Length of a Column** 9

**Define Slenderness Ratio of a Column** 10

**Explain the assumptions and limitations of Euler’s Theory** 12

**Explain the Rankine formula for Crippling Load** 13

**Define the term Column, Strut, and Tie**

In structural engineering, columns, struts, and ties are structural elements used in construction to support vertical loads, prevent buckling or deformation, and maintain the stability of a structure.

A column is a vertical structural member that primarily supports compressive loads, which can include both dead and live loads, and transfers them to the foundation or lower floors of a building. Columns are commonly made of concrete, steel, or timber and can have various cross-sectional shapes such as circular, square, rectangular, or hexagonal.

A strut is a structural element that resists compressive forces in the direction of its axis, which can be either tensile or compressive. Struts are commonly used to support structures, such as bridges or towers, where long spans require intermediate support.

A tie is a structural member that resists tensile forces, such as those generated by wind loads or seismic forces, and transfers them to the foundations or other structural elements. Ties are typically used to keep structural elements from pulling apart.

Understanding the behaviour and strength of columns, struts, and ties is crucial in designing and analyzing complex structures to ensure they can withstand loads and forces and maintain their stability and safety.

**Explain the Failure Modes of Column**

Columns are the structural elements that are designed to resist axial compression loads. The failure of a column occurs when it is unable to resist the applied load, resulting in structural collapse. There are different failure modes of columns, which are:

- Buckling: Buckling is a type of failure that occurs when the compressive stress exceeds the critical buckling stress. Buckling occurs when the column is slender, and the critical buckling stress is lower than the compressive stress. When a column buckles, it loses its ability to resist the applied load, resulting in structural collapse.
- Crushing: Crushing is a type of failure that occurs when the compressive stress exceeds the ultimate compressive strength of the material. Crushing failure is more likely to occur in short columns, where the column is not slender, and the ultimate compressive strength is higher.
- Yielding: Yielding is a type of failure that occurs when the compressive stress exceeds the yield strength of the material. Yielding occurs in ductile materials like steel. When the column yields, it undergoes permanent deformation, and the load-carrying capacity is significantly reduced.
- Combined Failure: In some cases, the column may fail due to a combination of the above failure modes. For example, a column may buckle and then crush due to the applied load.

The design of columns involves consideration of these failure modes to ensure that the column can withstand the applied load safely. Proper design involves selection of an appropriate column cross-sectional area, the material of the column, the column’s length, and end conditions to ensure that the column does not fail.

**Differentiate between the Buckling and Crushing Load of Failure**

The terms buckling and crushing are often used in the context of structural engineering, particularly when describing the failure of columns. The key difference between these two modes of failure is the way in which the load is distributed and the direction of deformation.

Buckling is a failure mode that occurs when a column is subjected to a compressive load that exceeds its critical buckling load. In this case, the column will deflect laterally and may buckle or fold. Buckling can occur suddenly and catastrophically, with little warning. This type of failure is typically associated with slender columns or those that are relatively long and thin.

On the other hand, crushing occurs when the load on the column is such that the material cannot withstand the stress and begins to deform plastically. This can occur in materials such as concrete or masonry, which are brittle and do not have a well-defined yield point. In the case of crushing, the column will deform in a way that is more uniform and will not show significant lateral deflection.

It is important to note that both buckling and crushing can occur simultaneously in a column that is subjected to a large compressive load. In order to design structures that are safe and reliable, engineers must carefully consider the various modes of failure that can occur and design for the appropriate load conditions.

**Explain Euler’s Theory of Column**

Euler’s theory of column is used to predict the buckling load of a long, slender column subjected to an axial compressive load. The theory is based on the assumption that the column is perfectly straight, homogeneous, and elastic, and that the load is applied in the longitudinal direction.

According to Euler’s theory, the critical buckling load is given by the formula Pcr = π²EI/L², where Pcr is the critical load, E is the modulus of elasticity of the material, I is the moment of inertia of the cross-sectional area, and L is the length of the column. This formula assumes that the column is pinned at both ends.

The formula indicates that the buckling load is directly proportional to the modulus of elasticity of the material and the moment of inertia of the cross-sectional area and inversely proportional to the square of the length of the column. Therefore, to increase the buckling load, the modulus of elasticity and the moment of inertia should be increased, and the length of the column should be decreased.

However, Euler’s theory assumes that the column is perfectly straight, which is rarely the case in practice. In addition, it does not take into account the effect of residual stresses, imperfections, and material properties on the column’s behaviour. Therefore, other more advanced theories have been developed to account for these factors and provide a more accurate prediction of the buckling load.

**Describe and calculate the Euler’s formula for Crippling Load when both ends Hinged**

Euler’s formula for crippling load is a theoretical equation used to determine the maximum compressive load that a column or strut can sustain without undergoing buckling failure. The formula was first developed by Leonhard Euler, a Swiss mathematician, and physicist in the 18th century.

Assuming a column with both ends hinged, Euler’s formula for crippling load can be derived as:

P_{c} = (π^{2} * E * I)/(l_{e})^{2}

where,

P_{c} is the critical buckling load

E is the modulus of elasticity of the column material

I is the moment of inertia of the column’s cross-sectional area

l_{e} is the effective length of the column, which depends on the boundary conditions of the column and its end connections.

The formula assumes that the column is long and slender, i.e., the length is much greater than its cross-sectional dimension. It also assumes that the column is subjected to an idealised loading condition, with a perfectly straight axis of the column, and a uniform load along its length.

In practice, the Euler’s formula provides only an idealised estimate of the critical load, since the actual behaviour of columns is influenced by several factors, such as the material properties, the cross-sectional shape, the imperfections in the geometry, the initial curvature, and the boundary conditions. Therefore, the design of columns and struts requires careful consideration of these factors and the use of appropriate safety factors to account for uncertainties in the loading and material properties.

**Describe and calculate the Euler’s formula for Crippling Load when one end is fixed and the other Hinged**

Euler’s formula is used to calculate the critical buckling load of a long, slender column under an axial compressive load. The formula is applicable for ideal columns that are homogeneous, isotropic, and have a uniform cross-section. Euler’s formula is derived using the following assumptions:

- The column is initially straight and has a constant cross-sectional area along its length.
- The material is homogeneous and isotropic, meaning that it has the same properties in all directions.
- The column is loaded with a purely axial compressive force.
- The deformation is elastic, meaning that the column returns to its original shape when the load is removed.
- The column is considered as a pin-ended column or fixed-ended column.

For a pin-ended column, the Euler’s formula for crippling load can be expressed as:

P_{cr} = (π^{2} * E * I)/(L_{e})^{2}

where P_{cr} is the critical buckling load, E is the modulus of elasticity of the material, I is the second moment of area of the column cross-section, and L_{e} is the effective length of the column. The effective length L_{e} is the distance between the points of contraflexure, which are the points in the column where the bending moment is zero. For a pin-ended column, the effective length is equal to the actual length of the column.

For a fixed-ended column, the Euler’s formula for crippling load can be expressed as:

P_{cr} = (π^{2} * E * I)/(4 * L_{e})^{2}

where L_{e} is the effective length of the column. For a fixed-ended column, the effective length L_{e} is equal to half of the actual length of the column.

The difference between the crippling load and buckling load is that crippling load refers to the maximum load that a column can withstand before it is crushed or fails in a material sense, while the buckling load refers to the critical load at which the column starts to buckle, resulting in a large deflection of the column.

**Describe and calculate the Euler’s Formula for Crippling Load when both ends Fixed**

The Euler’s formula for crippling load is a mathematical expression that calculates the critical load at which a slender column or beam under an axial compressive load will buckle or fail due to instability. This formula is a fundamental concept in structural engineering and is essential in designing stable structures that can withstand compressive loads without failing.

When both ends of a slender column or beam are fixed, the Euler’s formula for crippling load is expressed as:

P_{c} = ((pi^{2} x E x I)/(L^{2}))

Where:

- P
_{c}is the critical or crippling load - E is the modulus of elasticity of the material
- I is the moment of inertia of the cross-sectional area of the column or beam
- L is the length of the column or beam

This formula is derived from Euler’s buckling equation, which takes into account the geometric and material properties of the column or beam. The fixed ends condition in the formula assumes that the ends of the column or beam are restrained from moving in any direction, and therefore, the column or beam will buckle in a single direction.

It is important to note that the Euler’s formula for crippling load assumes that the material of the column or beam is homogenous, isotropic, and free from any initial imperfections or stresses. In reality, the material may have imperfections and may not be perfectly straight, which can affect the actual critical load. Therefore, the formula should only be used as a guide, and structural engineers should consider the actual behaviour of the material and other external factors in their design calculations.

In summary, the Euler’s formula for crippling load is an essential tool in structural engineering to design stable structures that can withstand compressive loads without failing. By calculating the critical load, engineers can ensure that the columns or beams are designed to meet the required safety standards and avoid catastrophic failure.

**Describe and calculate the Euler’s Formula for Crippling Load when one end is fixed and the other is Free**

The Euler’s formula for crippling load is a mathematical expression that calculates the critical load at which a slender column or beam under an axial compressive load will buckle or fail due to instability. When one end of a slender column or beam is fixed and the other end is free, the Euler’s formula for crippling load is expressed as:

P_{c} = ((pi^{2} * E * I)/(L^{2})) * (n^{2})

Where:

- P
_{c}is the critical or crippling load - E is the modulus of elasticity of the material
- I is the moment of inertia of the cross-sectional area of the column or beam
- L is the length of the column or beam
- n is the number of half-waves in the buckled shape

This formula is derived from Euler’s buckling equation, which takes into account the geometric and material properties of the column or beam. The fixed end condition in the formula assumes that the fixed end of the column or beam is restrained from moving in any direction, while the free end can move in any direction. The number of half-waves, n, is the number of regions where the column or beam buckles, and it depends on the boundary conditions and the shape of the column or beam.

It is important to note that the Euler’s formula for crippling load assumes that the material of the column or beam is homogenous, isotropic, and free from any initial imperfections or stresses. In reality, the material may have imperfections and may not be perfectly straight, which can affect the actual critical load. Therefore, the formula should only be used as a guide, and structural engineers should consider the actual behaviour of the material and other external factors in their design calculations.

In summary, the Euler’s formula for crippling load when one end is fixed and the other end is free is a fundamental concept in structural engineering. By calculating the critical load, engineers can ensure that the columns or beams are designed to meet the required safety standards and avoid catastrophic failure. It is important to consider the number of half-waves in the buckled shape and the actual behaviour of the material in design calculations.

**Explain the Effective Length of a Column**

The effective length of a column is a critical concept in structural engineering that determines the buckling behaviour of a slender column under an axial compressive load. In general, a column is a long and slender structural member that supports the weight of the structure above it. When a compressive load is applied to a column, it tends to buckle or fail due to instability, which can cause catastrophic damage to the structure.

The effective length of a column is the length of the column that is free to buckle or deflect under an axial compressive load. It depends on the boundary conditions at the ends of the column and the geometry of the column. The boundary conditions can be fixed, pinned, or free, and they determine the degree of restraint at the ends of the column. The geometry of the column can be slender or stocky, and it determines the resistance of the column to buckling.

For example, in a column with both ends fixed, the effective length is equal to the distance between the two fixed ends. In a column with one end fixed and the other end free, the effective length is equal to 2L, where L is the length of the column. In a column with both ends pinned, the effective length is equal to 2L, where L is the length of the column.

The effective length of a column is essential in determining the critical load at which the column will buckle or fail due to instability. It is used in various design equations and formulas, such as the Euler’s formula for crippling load, to calculate the buckling load of a slender column. Structural engineers use the effective length to design columns that can withstand the required load without failing and to ensure the safety of the structure.

In summary, the effective length of a column is the length of the column that is free to buckle or deflect under an axial compressive load. It depends on the boundary conditions at the ends of the column and the geometry of the column. The effective length is essential in determining the buckling behaviour of a slender column and is used in various design equations and formulas in structural engineering.

**Define Slenderness Ratio of a Column**

The slenderness ratio of a column is a dimensionless parameter that describes the ability of a column to resist buckling under an axial compressive load. It is the ratio of the effective length of the column to the radius of gyration of the cross-sectional area of the column. The slenderness ratio is a critical concept in structural engineering, as it determines the buckling behaviour of a slender column and is used in the design of structural systems.

The radius of gyration of the cross-sectional area of the column is a measure of the distribution of the area around the centroid of the cross-section. It is the square root of the moment of inertia of the cross-sectional area divided by the area of the cross-section. The moment of inertia of the cross-sectional area is a measure of the resistance of the cross-section to bending, and it depends on the shape and size of the cross-section.

The slenderness ratio is calculated as follows:

λ = (L_{e} / r)

Where:

- λ is the slenderness ratio
- L
_{e}is the effective length of the column - r is the radius of gyration of the cross-sectional area of the column

In general, a column with a low slenderness ratio is considered stocky and can resist buckling under a compressive load without significant deflection. Conversely, a column with a high slenderness ratio is considered slender and is more susceptible to buckling under a compressive load, which can cause significant deflection and even failure.

The slenderness ratio is used to determine the design requirements for columns in various structural systems. For example, building codes and standards specify the maximum allowable slenderness ratio for different types of columns and materials to ensure that the columns can resist the required load without buckling or failing.

In summary, the slenderness ratio of a column is a dimensionless parameter that describes the ability of a column to resist buckling under an axial compressive load. It is the ratio of the effective length of the column to the radius of gyration of the cross-sectional area of the column. The slenderness ratio is used in the design of structural systems to ensure the safety and stability of the columns.

**Explain the assumptions and limitations of Euler’s Theory**

Euler’s theory, also known as the Euler-Bernoulli beam theory, is a mathematical model used to describe the behaviour of beams under load. This theory makes a number of assumptions and has some limitations that should be considered when using it to analyze beams.

The following are the main assumptions of Euler’s theory:

- The beam is initially straight and any deformations are small: Euler’s theory assumes that the beam is initially straight, and that any deformation caused by loading is small. This is known as the small deflection assumption. If the deformation is large, then the assumptions of Euler’s theory may not be valid.
- The beam is made of a homogeneous and isotropic material: Euler’s theory assumes that the beam is made of a material that is homogeneous (i.e., has uniform properties throughout) and isotropic (i.e., has the same properties in all directions). This assumption may not be valid for composite or non-homogeneous materials.
- The cross-section of the beam is constant: Euler’s theory assumes that the cross-section of the beam remains constant along its length. This is known as the constant cross-section assumption. If the cross-section varies along the length of the beam, then the assumptions of Euler’s theory may not be valid.
- The beam is subjected to axial loads only: Euler’s theory assumes that the beam is subjected to axial loads only, meaning that the load is applied along the axis of the beam. If the load is applied at an angle or is not axial, then the assumptions of Euler’s theory may not be valid.
- The deformation of the beam is caused by bending only: Euler’s theory assumes that the deformation of the beam is caused by bending only, and not by shear or torsion. This is known as the bending-only assumption. If shear or torsion is significant, then the assumptions of Euler’s theory may not be valid.

The following are some limitations of Euler’s theory:

- Large deformations: As mentioned earlier, Euler’s theory assumes that the deformations are small. If the deformations are large, then the assumptions of Euler’s theory may not be valid. In such cases, a more advanced theory, such as the Timoshenko beam theory or finite element analysis, may be needed.
- Non-homogeneous or anisotropic materials: Euler’s theory assumes that the beam is made of a homogeneous and isotropic material. If the material is non-homogeneous or anisotropic, then the assumptions of Euler’s theory may not be valid. In such cases, a more advanced theory that considers material properties, such as the shear modulus and Poisson’s ratio, may be needed.
- Complex loading conditions: Euler’s theory assumes that the beam is subjected to axial loads only. If the loading conditions are more complex, such as combined axial and transverse loads, then the assumptions of Euler’s theory may not be valid. In such cases, a more advanced theory, such as the theory of elasticity, may be needed.
- Non-uniform cross-section: Euler’s theory assumes that the cross-section of the beam is constant along its length. If the cross-section varies along the length of the beam, then the assumptions of Euler’s theory may not be valid. In such cases, a more advanced theory that considers the effects of varying cross-sections, such as the Timoshenko beam theory or finite element analysis, may be needed.
- Failure due to material yielding: Euler’s theory assumes that the material does not yield under load. If the material yields, then the assumptions of Euler’s theory

**Explain the Rankine formula for Crippling Load**

The Rankine formula is a mathematical equation used to determine the maximum load, known as the crippling load, that a long, slender column can withstand before buckling or collapsing under the applied load. This formula was first developed by William Rankine, a Scottish engineer and physicist, in the mid-19th century.

The Rankine formula is expressed as:

Pcr = (π² x E x I) / L²

where Pcr is the crippling load, E is the modulus of elasticity of the material, I is the moment of inertia of the cross-section of the column, and L is the unsupported length of the column.

The formula assumes that the column is long and slender, with a constant cross-section throughout its length, and that it is subjected to an axial compressive load. It also assumes that the material of the column is isotropic, meaning it has the same properties in all directions.

The moment of inertia, I, is a measure of the resistance of the column to bending. A column with a larger moment of inertia is more resistant to bending, and thus more able to resist buckling. The unsupported length, L, is the length of the column that is not supported by any other structure.

The formula can be used to determine the maximum load that a column can withstand before buckling occurs. If the applied load is less than the crippling load, the column will not buckle and will remain stable. However, if the applied load exceeds the crippling load, the column will buckle and may collapse.

It is important to note that the Rankine formula is based on several assumptions, and may not be applicable to all situations. For example, the formula does not take into account the effects of eccentric loading, material yielding, or initial imperfections in the column. In some cases, a more advanced theory, such as the Euler buckling theory or finite element analysis, may be needed to accurately predict the buckling behaviour of a column.