Stability

Stability of Columns

Introduction

Fig.1 Buckling

The stability of a structure is its ability to withstand a load without undergoing an unexpected deformation. If the equilibrium of a column subjected to a load P is disturbed, then it will return to its original position. This will not occur if the load applied is greater than the critical load of the column, P_cr. In this case the column is said to be unstable. The accompanying sudden change in shape and size is known as buckling (Fig. 1). Since buckling is one of the major causes of failures structures, in structure design the possibility of buckling should always be considered.

Euler's Formula (Ideal Column)

Fig. 2 Pin-supported ideal column

A column that is pinned at both ends is one of the more common column structure and is referred to as an ideal column (Fig.2). The column will buckle when the load P reaches a critical level, called the critical load P_cr. In order to find the critical load, the load must be increased up to a point where a slight disturbance will cause the column to assume a curved shape as a result of buckling. This critical load is determined from Euler's formula:

where E is the modulus of elasticity of the material,
I is the least moment of inertia, and L is the unsupported length of the column. It should be noted that critical load is not dependent on the strength of the material. It is only dependent on the dimensions and modulus of elasticity of the column.

Columns With Various Supports

Fig. 3 Various support conditions and their efffective lengths

Fig. 4 Various support conditions and their efffective lengths

The ideal column equation above is only valid for a column with both ends pinned. Other equations must be used if the end conditions are other than pinned. The end conditions will effect the P_cr loads.

For columns with different types of support, Euler's formula may still be used if the distance L is now used to indicate the distance between the zero moment points. This length is also called the effective length L_e (Fig.3 and Fig. 4). Thus the critical load and the critical stress would now be:

Critical Stress

The critical stress is the stress corresponding to the critical load in the column. It is given by the equation:

where r is the smallest radius of gyration and (L / r) is known as the slenderness ratio and is a measure of the column's flexibility.

Slenderness Ratio

Fig. 5 Slenderness ratio

The slenderness ratio is used to help determine when the Euler's column buckling equation is valid. If the column is short (low slenderness ratio) then the column will be crushed before it buckles (Fig. 5). This there is an lower limit of when the Euler equation can be used. The graphic on the left shows how the buckling load P decreases as the the member becomes slender. The upper limit is based on the yield strength of the material. The graphic on the left shows a plot of the critical stress vs. the slenderness ratio for a structural steel with an assumed modulus of elasticity, E of 200 GPa.

Stability of Columns

		Introduction
Fig.1 Buckling		The stability of a structure is its ability to withstand a load without undergoing an unexpected deformation. If the equilibrium of a column subjected to a load P is disturbed, then it will return to its original position. This will not occur if the load applied is greater than the critical load of the column, P_cr. In this case the column is said to be unstable. The accompanying sudden change in shape and size is known as buckling (Fig. 1). Since buckling is one of the major causes of failures structures, in structure design the possibility of buckling should always be considered.

		Euler's Formula (Ideal Column)
Fig. 2 Pin-supported ideal column		A column that is pinned at both ends is one of the more common column structure and is referred to as an ideal column (Fig.2). The column will buckle when the load P reaches a critical level, called the critical load P_cr. In order to find the critical load, the load must be increased up to a point where a slight disturbance will cause the column to assume a curved shape as a result of buckling. This critical load is determined from Euler's formula: where E is the modulus of elasticity of the material, I is the least moment of inertia, and L is the unsupported length of the column. It should be noted that critical load is not dependent on the strength of the material. It is only dependent on the dimensions and modulus of elasticity of the column.

		Columns With Various Supports
Fig. 3 Various support conditions and their efffective lengths Fig. 4 Various support conditions and their efffective lengths		The ideal column equation above is only valid for a column with both ends pinned. Other equations must be used if the end conditions are other than pinned. The end conditions will effect the P_cr loads. For columns with different types of support, Euler's formula may still be used if the distance L is now used to indicate the distance between the zero moment points. This length is also called the effective length L_e (Fig.3 and Fig. 4). Thus the critical load and the critical stress would now be:

		Critical Stress
		The critical stress is the stress corresponding to the critical load in the column. It is given by the equation: where r is the smallest radius of gyration and (L / r) is known as the slenderness ratio and is a measure of the column's flexibility.

		Slenderness Ratio
Fig. 5 Slenderness ratio		The slenderness ratio is used to help determine when the Euler's column buckling equation is valid. If the column is short (low slenderness ratio) then the column will be crushed before it buckles (Fig. 5). This there is an lower limit of when the Euler equation can be used. The graphic on the left shows how the buckling load P decreases as the the member becomes slender. The upper limit is based on the yield strength of the material. The graphic on the left shows a plot of the critical stress vs. the slenderness ratio for a structural steel with an assumed modulus of elasticity, E of 200 GPa.