26.6:
Design of Columns under a Centric Load
Euler's formula is used to determine the critical load of a column, and Secant's formula is used to calculate the deformations and stresses of a column under eccentric loadings.
Here, it was assumed that the column is initially a straight homogenous prism and all the stresses are within the proportional limit.
In real life, the materials of columns are not ideal, and the design of a column is based on an empirical formula derived from numerous laboratory experiments.
For example, the data of many steel columns are recorded for an applied centric load and increasing it until failure.
For large columns, Euler's formula can be used to predict the failure, and the critical stress depends on the modulus of elasticity.
On the other hand, for short columns, failure occurs majorly because of yield strength.
For the columns of intermediate lengths, failure is a complex process and it depends on both yield and the modulus of elasticity.
For each of the above cases, the empirical formula is modified slightly to design steel columns.
26.6:
Design of Columns under a Centric Load
The design of columns under centric load is a fundamental aspect of structural engineering and is critical for ensuring the stability and integrity of structures. Euler's and Secant's formulas are central to understanding and calculating the critical load and deformation behaviors of columns, providing a basis for safe and effective structural design.
Euler's formula is applicable under the assumption that the column is a perfect, straight, homogenous prism, and it is operating within the elastic limit of the material. The critical load, according to Euler's formula, is directly dependent on the column's modulus of elasticity and its geometric properties. However, it is essential to note that Euler's formula is most accurate for long, slender columns where buckling is the predominant mode of failure. In practical applications, the materials used for columns exhibit imperfections, and their behavior under load does not always align with ideal, elastic assumptions. Real-life columns might have initial slight bends, variations in cross-sectional area, or material inconsistencies, all of which can significantly influence their load-bearing capacity and failure modes. Therefore, empirical formulas derived from extensive laboratory experiments are used to design columns that can withstand real-world conditions. These empirical formulas take into account the material properties, such as yield strength and modulus of elasticity, as well as the column's length, cross-sectional dimensions, and boundary conditions.
For columns that are long enough for Euler's formula to predict failure accurately, the critical stress depends primarily on the material's modulus of elasticity. These columns fail by buckling before the yield strength of the material is exceeded. Failure in short columns is predominantly due to the material reaching its yield strength, leading to a crushing failure rather than buckling. In these cases, the design focuses on the material's yield strength rather than its elasticity. Columns of intermediate length present a complex scenario where both the yield strength and modulus of elasticity influence failure. The empirical formulas for these columns are adjusted to consider the intricate interaction between material yielding and elastic buckling.
These considerations ensure that the design of columns, regardless of their length and the material used, is robust, reliable, and capable of supporting the intended loads without failure.