Combined Bending and Direct Stresses, Columns, and Struts

Definition

A column or strut is a structural member subjected to axial compressive loads. When the load is not perfectly axial or the member is subjected to lateral forces, it experiences "combined stresses," which is the superposition of direct (compressive) stress and bending (flexural) stress. Euler's theory provides a mathematical approach to determine the critical buckling load of long columns, while empirical formulas bridge the gap for intermediate columns where theoretical assumptions fail.

Main Content

1. Combined Bending and Direct Stresses

When an axial load $P$ acts at an eccentricity $e$ from the centroid, it creates a direct stress ($\sigma_d = P/A$) and a bending moment ($M = P \times e$).
The total stress at any point in the cross-section is the algebraic sum of the direct stress and the bending stress ($\sigma = \sigma_d \pm \sigma_b$), where $\sigma_b = My/I$.

2. Euler’s Theory for Columns

Euler's theory assumes the column is perfectly straight, homogeneous, and isotropic, with the load applied exactly along the axis.
It defines the "Crippling Load" ($P_e$) as the load at which the column becomes unstable and begins to buckle (deflect laterally).

3. Empirical Formulae for Loads

Euler's theory overestimates the strength of short and medium columns because they fail by crushing rather than buckling.
Empirical formulas like the Rankine-Gordon formula or the Johnson’s Parabolic formula are used to calculate safe loads by accounting for both crushing and buckling.

Visualizing Eccentric Loading:
    |   ^ P (Load)
    |   |
    |  ( )  <-- Centroid
    |   | e (Eccentricity)
    |   |
    |   |
    +-------+
    |       |
    |       |  Column cross-section
    +-------+

Working / Process

1. Calculation of Combined Stress

Calculate Direct Stress: $\sigma_d = \frac{P}{A}$
Calculate Bending Stress: $\sigma_b = \frac{M}{Z}$, where $Z$ is the section modulus ($I/y$).
Combine them: $\sigma_{max} = \frac{P}{A} + \frac{P \cdot e}{Z}$ and $\sigma_{min} = \frac{P}{A} - \frac{P \cdot e}{Z}$.

2. Application of Euler’s Formula

Determine the effective length ($L_e$) based on end conditions (e.g., both ends hinged, one end fixed, etc.).
Apply the formula: $P_e = \frac{\pi^2 EI}{L_e^2}$.
Ensure the slenderness ratio ($\lambda = L_e/k$) is high enough to satisfy the validity of Euler's assumptions.

3. Using Rankine’s Formula

Calculate the crushing load ($P_c = \sigma_c \cdot A$) and the Euler buckling load ($P_e$).
Use the Rankine formula: $\frac{1}{P_R} = \frac{1}{P_c} + \frac{1}{P_e}$.
This provides a combined result that accounts for both material strength and buckling geometry.

Advantages / Applications

Structural Integrity: Ensures that bridge piers and building columns do not fail under off-center loads.
Optimization: Empirical formulas allow engineers to design columns that are safe for both short-column crushing and long-column buckling.
Safety: Combined stress analysis prevents localized tensile failure in brittle materials (like cast iron) used in compression.

Summary

This unit covers the mechanics of vertical structural members subjected to axial and eccentric forces. It emphasizes that while Euler’s theory accurately predicts the failure of long, slender columns through elastic buckling, empirical formulas are essential for real-world intermediate columns where material crushing limits the load capacity.

Important terms to remember: - Slenderness Ratio: The ratio of effective length to the least radius of gyration. - Eccentricity: The distance between the applied load and the centroidal axis. - Crippling Load: The maximum axial load a column can sustain before buckling. - Effective Length: The length of an equivalent column hinged at both ends.