An 18 -kip centric load is applied to a rectangular sawn...

An 18 -kip centric load is applied to a rectangular sawn lumber column of 22 -ft effective length. Using lumber for which the adjusted allowable stress for compression parallel to the grain is $\sigma_{C}=1050$ psi and the adjusted modulus is $E=440 \times 10^{3} \mathrm{psi}$ determine the smallest cross section that can be used. Use $b=2 d$.

Key Concepts

Allowable Compressive Stress

This is the maximum stress that the material is designed to withstand under compression along the grain. It is an adjusted value that takes into account various factors such as moisture, imperfections, and load duration, ensuring the safety and performance of the column under the applied load.

Modulus of Elasticity

The modulus of elasticity (E) represents the stiffness of the material. It is a measure of its ability to deform elastically under load. In column design, a higher modulus indicates a stiffer member, which is critical for analyzing and predicting buckling behavior through formulas like Euler's buckling equation.

Euler Buckling

Euler buckling refers to the phenomenon where a slender column under axial compressive load becomes unstable and deflects laterally. The critical buckling load is calculated using Euler’s formula, which incorporates the modulus of elasticity, the moment of inertia, and the effective length of the column, highlighting the interplay between material properties and geometry in preventing failure.

Cross-Sectional Properties

The moment of inertia of a cross section is a key property that reflects how the area is distributed about a bending axis. In the case of a rectangular cross-section with a specified width-to-depth ratio (b = 2d), the moment of inertia is fundamentally linked to the dimensions of the section (specifically d in this configuration), influencing the column's resistance to buckling under load.

Effective Length

The effective length of a column is a theoretical length that represents the actual buckling behavior considering the boundary conditions or end restraints. It is used in buckling calculations to account for how the supports affect the column's stability, directly influencing the critical buckling load and overall design.

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