Cross-Sectional Area of Column According to Johnson's Parabolic Formula Calculator

Johnson's Parabolic Formula:

\[ A = \frac{P}{\sigma_c - (r \times (\frac{L_{eff}}{r_{least}}))} \]

Critical Load On Column:

Compressive Yield Stress:

Johnson's Formula Constant:

Effective Column Length:

Least Radius of Gyration:

Column Cross-Sectional Area (m²):

1. What is Johnson's Parabolic Formula for Columns?

Definition: Johnson's parabolic formula is used to calculate the critical load for intermediate columns that fail due to a combination of crushing and buckling.

Purpose: It helps engineers determine the cross-sectional area required for a column to safely support a given load while considering material properties and column geometry.

2. How Does the Calculator Work?

The calculator uses Johnson's formula:

\[ A = \frac{P}{\sigma_c - (r \times (\frac{L_{eff}}{r_{least}}))} \]

Where:

\( A \) — Cross-sectional area of column (m²)
\( P \) — Critical load on column (N)
\( \sigma_c \) — Compressive yield stress (Pa)
\( r \) — Johnson's formula constant (±5%)
\( L_{eff} \) — Effective column length (m)
\( r_{least} \) — Least radius of gyration (m)

Explanation: The formula calculates the required cross-sectional area by considering both material strength and column slenderness effects.

3. Importance of Column Cross-Sectional Area Calculation

Details: Proper calculation ensures structural stability, prevents buckling failures, and optimizes material usage in construction projects.

4. Using the Calculator

Tips: Enter the critical load, material properties (compressive yield stress and Johnson's constant), and column geometry (effective length and least radius of gyration). All values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is the typical range for Johnson's constant?
A: Johnson's constant typically ranges from 1 to 10, with common values around 6 for many materials (±5%).

Q2: How is effective column length determined?
A: Effective length depends on end conditions - pinned-pinned (1.0L), fixed-fixed (0.5L), fixed-pinned (0.7L), fixed-free (2.0L).

Q3: What's a typical compressive yield stress for steel?
A: Structural steel typically has compressive yield stress between 250-400 MPa (250,000,000-400,000,000 Pa).

Q4: How do I find the least radius of gyration?
A: For standard sections, refer to engineering tables. For custom sections, calculate as \( r = \sqrt{I/A} \), where I is the smallest moment of inertia.

Q5: When is Johnson's formula applicable?
A: For intermediate columns where slenderness ratio (Leff/r) is between the limits for pure crushing and pure buckling failures.